Modelling and Simulation of GPS Multipath Propagationmultipath have proven much more difficult to...

Modelling and Simulation

of

GPS Multipath Propagation

Bruce M Hannah

Modelling and Simulation

of

GPS Multipath Propagation

Bruce M. Hannah

B.Eng. (Hons)

The Cooperative Research Centre for Satellite Systems

Queensland University of Technology

THIS DISSERTATION IS SUBMITTED IN PARTIAL FULFILMENT

OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE

DOCTOR OF PHILOSOPHY

March 2001

v

Statement of Authorship

The work contained in this thesis has not been previously submitted for a degree or

diploma at any other higher education institution. To the best of my knowledge and

belief, the thesis contains no material previously published or written by another

person except where due reference is made.

Signed:...................................................

Date: ......................................................

vi

Key Words

Global Positioning System, GPS, Multipath, Radio Frequency, RF, Propagation,

Parabolic Equation, PE, Modelling, Simulation, Correlation, DLL, Discrimination,

Range Error, Mitigation, Reflection, Diffraction, Fresnel Zone.

vii

Acknowledgements

This work was carried out in the Cooperative Research Centre for Satellite Systems

with financial support from the Commonwealth of Australia’s Cooperative Research

Centres Program. Other work presented in this dissertation was funded in part by the

British Council under a postgraduate bursary scheme. Additional funding was

provided by the QUT grant-in-aid scheme. The author thanks all for their financial

assistance.

Special thanks to Professor Kurt Kubik and Professor Miles Moody for providing

insightful respective supervision of this research work within the Space Centre for

Satellite Navigation and the Cooperative Research Centre for Satellite Systems.

The Radio Communications Research Unit at Rutherford Appleton Laboratory

Oxfordshire, UK, specifically Dr Mireille Levy for inspiring and maintaining the use

of the Parabolic Equation as an effective GPS propagation modelling tool.

Dr Rodney Walker of the Queensland University of Technology for his interest,

insight, inspiration and most all his friendship.

To my parents Mervyn and Maureen, my wife Tanya and her parents, Barry and Rae,

all of whom I owe a great personal debt for the investment they have made in me.

This dissertation is dedicated to them.

ix

Abstract

Multipath remains a dominant error source in Global Positioning System (GPS)

applications that require high accuracy. With the use of differential techniques it is

possible to remove many of the common-mode error sources, but the error effects of

multipath have proven much more difficult to mitigate. The research aim of this work

is to enhance the understanding of multipath propagation and its effects in GPS

terrestrial applications, through the modelling of signal propagation behaviour and the

resultant error effects.

Multipath propagation occurs when environmental features cause combinations of

reflected and/or diffracted replica signals to arrive at the receiving antenna. These

signals, in combination with the original line-of-sight (LOS) signal, can cause

distortion of the receiver correlation function and ultimately the discrimination

function and hence errors in range estimation.

To date, a completely satisfactory mitigation strategy has yet to be developed. In the

search for such a mitigation strategy, it is imperative that a comprehensive

understanding of the multipath propagation environment and the resultant error effects

exists. The work presented here, provides a comprehensive understanding through the

use of new modelling and simulation techniques specific to GPS multipath.

This dissertation unites the existing theory of radio frequency propagation for the GPS

L1 signal into a coherent treatment of GPS propagation in the terrestrial environment.

To further enhance the understanding of the multipath propagation environment and

the resultant error effects, this dissertation also describes the design and development

of a new parabolic equation (PE) based propagation model for analysis of GPS

multipath propagation behaviour.

The propagation model improves on previous PE-based models by incorporating

terrain features, including boundary impedance properties, backscatter and time-

domain decomposition of the field into a multipath impulse response. The results

x

provide visualisation as well as the defining parameters necessary to fully describe the

multipath propagation behaviour.

These resultant parameters provide the input for a correlation and discrimination

model for visualisation and the generation of resultant receiver error measurements.

Results for a variety of propagation environments are presented and the technique is

shown to provide a deterministic methodology against real GPS data.

The unique and novel combined modelling of multipath propagation and reception,

presented in this dissertation, provides an effective set of tools that have enhanced the

understanding of the behaviour and effect of multipath in GPS applications, and

ultimately should aid in providing a solution to the GPS multipath mitigation

problem.

xi

Table of Contents

Chapter 1 Introduction and Overview.................................................................... 1

1.1 Introduction...................................................................................................... 1

1.2 Overview of Research Presented ..................................................................... 2

1.3 Research Contribution ..................................................................................... 5

1.4 References........................................................................................................ 6

Chapter 2 Nature of GPS Multipath Propagation ................................................. 9

2.1 Multipath Environment ..................................................................................... 9

2.2 Specular Reflection ......................................................................................... 10

2.2.1 Linear Reflection Coefficient Representation......................................... 10

2.2.2 Circular Reflection Coefficient Representation ...................................... 16

2.2.3 Ray-based Reflection Geometry: Relative Time Delay and Phase ......... 27

2.2.3.1 Summary of Amplitudes, Relative Delays and Phases ....................... 31

2.2.4 Multipath Modes and Coupled Reflection Coefficients ......................... 32

2.3 Rough Surface Scatter..................................................................................... 45

2.4 Fresnel Zones .................................................................................................. 49

2.5 Diffraction....................................................................................................... 54

2.5.1 Knife-Edge Diffraction ........................................................................... 55

2.6 GPS Fading Signal Characteristics ................................................................. 59

2.6.1 Signal Fade Envelopes ............................................................................ 59

2.6.2 Signal Fading Characteristics for Specular Reflection ........................... 72

2.7 GPS Receiver Context .................................................................................... 80

2.7.1 Aspects of Physical Antenna Location.................................................... 80

2.8 Path Loss in the Terrestrial Domain................................................................ 86

2.9 Summary ......................................................................................................... 87

2.10 References ................................................................................................... 90

Chapter 3 Overview of Propagation Modelling ................................................... 93

3.1 Review of Maxwell’s Equations.................................................................... 93

3.2 Overview of Computational Electromagnetics Techniques .......................... 96

3.2.1 Finite-Elements Technique ..................................................................... 96

xii

3.2.2 Finite-Difference Time-Domain Technique............................................ 98

3.2.3 Finite-Difference Frequency-Domain Technique ................................. 102

3.2.4 Method of Moments .............................................................................. 102

3.2.5 Geometrical and Uniform Theory of Diffraction .................................. 104

3.2.6 Generalised Multipole Technique ......................................................... 105

3.2.7 Parabolic Equation Method................................................................... 106

3.3 Comparison of Modelling Techniques......................................................... 109

3.3.1 Requirements for Modelling GPS Signal Propagation ......................... 110

3.3.2 Comparison of EM Techniques............................................................. 110

3.4 References.................................................................................................... 112

Chapter 4 GPS Parabolic Equation Model ......................................................... 115

4.1 Development of the PE for Electromagnetic Propagation ............................ 115

4.2 The Free-Space Parabolic Equation .............................................................. 116

4.3 Limitations of Refractive Index Terms in PE Forms .................................... 119

4.3.1 Fourier Split-Step Solution of the SPE ................................................. 123

4.3.2 Phase Errors in Rational-Linear Approximation Forms of the PE........ 126

4.4 Numerical Implementation for GPS Satellite Propagation ........................... 128

4.4.1 Domain Sampling.................................................................................. 129

4.4.2 Incident Boundary Condition ................................................................ 130

4.4.3 Upper Boundary Condition ................................................................... 131

4.4.4 Lower Boundary Condition................................................................... 132

4.4.5 Implementation Algorithm.................................................................... 134

4.5 Implementing Arbitrary Terrain in the PE Model ......................................... 135

4.5.1 Boundary-Shift Technique for Arbitrary Terrain .................................. 136

4.6 Implementing Backscatter for a Two-Way PE Model .................................. 138

4.7 Summary ....................................................................................................... 141

4.8 References ..................................................................................................... 143

Chapter 5 Time Series Analysis with the FSPE.................................................. 149

5.1 Introduction.................................................................................................. 149

5.2 Implementation ............................................................................................ 150

5.3 Domain Considerations................................................................................ 155

xiii

5.4 Deriving the MCIR from the PETA Model ................................................. 157

5.5 Multipath Phase Information from the PETA Model .................................. 159

5.6 Antenna Gain Pattern from Angle of Arrival Information .......................... 161

5.7 PETA Implementation Issues ...................................................................... 164

5.8 PETA Domain Representation and Performance ........................................ 166

5.9 Summary...................................................................................................... 168

5.10 References.................................................................................................... 169

Chapter 6 Model Validation................................................................................. 171

6.1 Validation of FSPE with an Exact Solution.................................................. 171

6.1.1 Phase Error ............................................................................................ 171

6.2 Forward Multipath Propagation .................................................................... 174

6.2.1 Static Test.............................................................................................. 174

6.2.2 Dynamic Tests....................................................................................... 177

6.3 Forward Diffraction ...................................................................................... 181

6.3.1 Static Test.............................................................................................. 181

6.3.2 Dynamic Test ........................................................................................ 185

6.4 BA/BB-Mode: Backscatter ........................................................................... 187

6.4.1 Static Test.............................................................................................. 188

6.4.2 Dynamic Test ........................................................................................ 192

6.5 Summary ....................................................................................................... 199

6.6 References ..................................................................................................... 200

Chapter 7 Simulation of GPS Propagation......................................................... 203

7.1 Introduction ................................................................................................... 203

7.2 Candidate Simulation Cases.......................................................................... 203

7.2.1 Pine Dam............................................................................................... 204

7.2.2 Caboolture Soccer Field........................................................................ 214

7.3 Summary ....................................................................................................... 220

7.4 References ..................................................................................................... 221

Chapter 8 SNR for Deriving a Height Observable............................................. 223

8.1 Introduction ................................................................................................... 223

xiv

8.2 Estimating Antenna Height from the GPS S/N Ratio ................................... 225

8.3 Results ........................................................................................................... 228

8.4 North Pine Dam............................................................................................. 229

8.5 Bribie Island .................................................................................................. 230

8.6 System Aspects ............................................................................................. 230

8.7 Summary ....................................................................................................... 236

8.8 References ..................................................................................................... 237

Chapter 9 Receiver Correlation and Discrimination ......................................... 239

9.1 Introduction ................................................................................................... 239

9.2 Fundamentals of a GPS Receiver Model ...................................................... 240

9.2.1 Generic GPS Receiver Functions .......................................................... 241

9.2.1.1 Antenna and RF Section.................................................................... 241

9.2.1.2 Reference Oscillator and Frequency Synthesis ................................. 242

9.2.1.3 Down-conversion and IF ................................................................... 242

9.2.1.4 Signal Processing .............................................................................. 242

9.3 The Key GPS Receiver Elements.................................................................. 243

9.3.1 PN Code Generation.............................................................................. 243

9.3.2 Delay Lock Loop................................................................................... 245

9.4 A Receiver Model for use with the FSPE/PETA Model............................... 252

9.5 The Modelled Code Correlation Function and Multipath............................. 254

9.5.1 Variation of Relative Multipath Delay Time ........................................ 254

9.5.1.1 In-Phase Case .................................................................................... 254

9.5.1.2 Anti-phase Case................................................................................. 258

9.5.2 Variation of Relative Phase................................................................... 261

9.6 Error Envelopes for a Single multipath Signal.............................................. 265

9.6.1 Variation of Relative Time Delay ......................................................... 265

9.6.2 Variation of Relative Amplitude ........................................................... 268

9.6.3 Variation of Phase ................................................................................. 269

9.7 Error Envelopes for Two Multipath Signals ................................................. 270

9.8 Summary ....................................................................................................... 275

9.9 References ..................................................................................................... 276

xv

Chapter 10 Conclusions.......................................................................................... 279

Chapter 11 Recommendations ............................................................................... 281

11.1 References ................................................................................................. 283

List of Appendices

Appendix A – Research Publications.................................................................... A-1

Appendix B – Raw GPS Multipath Data.............................................................. B-1

B.1 Data Results Fresh Water - North Pine Dam 2 December 1999....................B-1

(a) SV 5............................................................................................................B-1

(b) SV 6............................................................................................................B-2

(c) SV 8............................................................................................................B-2

(d) SV 9............................................................................................................B-3

(e) SV 10..........................................................................................................B-3

(f) SV 17..........................................................................................................B-4

(g) SV 24..........................................................................................................B-4

(h) SV 25..........................................................................................................B-5

(i) SV 26..........................................................................................................B-5

(j) SV 30..........................................................................................................B-6

B.2 Data Results Soil - Caboolture 30 November 1999.......................................B-7

(a) SV 5............................................................................................................B-7

(b) SV 6............................................................................................................B-8

(c) SV 8............................................................................................................B-8

(d) SV 17..........................................................................................................B-9

(e) SV 21..........................................................................................................B-9

(f) SV 23........................................................................................................B-10

(g) SV 26........................................................................................................B-10

(h) SV 30........................................................................................................B-11

B.3 Data Results Sea Water - Bribie Island 11 November 1999 ........................B-12

(a) SV 3..........................................................................................................B-12

(b) SV 6..........................................................................................................B-13

xvi

(c) SV 10........................................................................................................B-13

(d) SV 17........................................................................................................B-14

(e) SV 21........................................................................................................B-14

(f) SV 22........................................................................................................B-15

(g) SV 23........................................................................................................B-15

(h) SV 30........................................................................................................B-16

Appendix C – MATLAB Code............................................................................... C-1

C.1 Propagation Modelling Code .........................................................................C-1

(a) GOPE.M.....................................................................................................C-1

(b) MPE.M.......................................................................................................C-7

(c) FIELDCALC.M........................................................................................C-22

(d) PETASETUP.M.......................................................................................C-26

(e) LOADPROFILE.M ..................................................................................C-27

C.2 GPS Receiver Modelling..............................................................................C-29

(a) RUNCORR.M..........................................................................................C-29

(b) RXCORR.M.............................................................................................C-32

(c) PNGEN.M................................................................................................C-37

C.3 GPS Multipath Data Acquisition .................................................................C-39

(a) ASHTEQC.M...........................................................................................C-39

(b) RIN2QC.M...............................................................................................C-41

(c) QC2MAT.M.............................................................................................C-43

(d) PLOTQC.M..............................................................................................C-46

xvii

List of Figures

Figure 2.1 — Multipath environment ........................................................................... 9

Figure 2.2 — Linear reflection coefficients for Concrete ........................................... 12

Figure 2.3 — Linear reflection coefficients for Dry Ground ...................................... 13

Figure 2.4 — Linear reflection coefficients for Medium Dry Ground........................ 13

Figure 2.5 — Linear reflection coefficients for Wet Ground...................................... 14

Figure 2.6 — Linear reflection coefficients for Fresh Water...................................... 14

Figure 2.7 — Linear reflection coefficients for Sea Water......................................... 15

Figure 2.8 — Circular reflection coefficients for Concrete ........................................ 17

Figure 2.9 — Circular reflection coefficients for Dry Ground ................................... 17

Figure 2.10 — Circular reflection coefficients for Medium Dry Ground................... 18

Figure 2.11 — Circular reflection coefficients for Wet Ground................................. 18

Figure 2.12 — Circular reflection coefficients for Fresh Water ................................. 19

Figure 2.13 — Circular reflection coefficients for Sea Water .................................... 19

Figure 2.14 — Incident RHCP-RC for Concrete ........................................................ 21

Figure 2.15 — Incident RHCP-RC for Dry Ground ................................................... 22

Figure 2.16 — Incident RHCP-RC for Medium Dry Ground..................................... 22

Figure 2.17 — Incident RHCP-RC for Wet Ground................................................... 23

Figure 2.18 — Incident RHCP-RC for Fresh Water................................................... 23

Figure 2.19 — Incident RHCP-RC for Sea Water ...................................................... 24

Figure 2.20 — Incident LHCP-RC for Concrete ........................................................ 25

Figure 2.21 — Incident LHCP-RC for Dry Ground.................................................... 25

Figure 2.22 — Incident LHCP-RC for Medium Dry Ground..................................... 26

Figure 2.23 — Incident LHCP-RC for Wet Ground................................................... 26

Figure 2.24 — Incident LHCP-RC for Fresh Water ................................................... 27

Figure 2.25 — Incident LHCP-RC for Sea Water ...................................................... 27

Figure 2.26 — Forward scatter geometry ................................................................... 28

Figure 2.27 — Backscatter geometry 1....................................................................... 29

Figure 2.28 — Backscatter geometry 2....................................................................... 31

Figure 2.29 — Decoupled Polarisation Zone 1 BB-Mode.......................................... 33

Figure 2.30 — Coupled Polarisation Zone 1 BB-Mode.............................................. 34

Figure 2.31 — Modified Coupled RC Zone 1 Concrete............................................. 37

xviii

Figure 2.32 — Modified Coupled RC Zone 1 Dry Ground ........................................ 37

Figure 2.33 — Modified Coupled RC Zone 1 Medium Dry Ground.......................... 38

Figure 2.34 — Modified Coupled RC Zone 1 Wet Ground........................................ 38

Figure 2.35 — Modified Coupled RC Zone 1 Fresh Water........................................ 39

Figure 2.36 — Modified Coupled RC Zone 1 Sea Water ........................................... 39

Figure 2.37 — Modified Coupled RC Zone 2 Concrete ............................................. 40

Figure 2.38 — Modified Coupled RC Zone 2 Dry Ground ........................................ 40

Figure 2.39 — Modified Coupled RC Zone 2 Medium Dry Ground.......................... 41

Figure 2.40 — Modified Coupled RC Zone 2 Wet Ground........................................ 41

Figure 2.41 — Modified Coupled RC Zone 2 Sea Water ........................................... 42

Figure 2.42 — Modified Coupled RC Zone 2 Fresh Water........................................ 42

Figure 2.43 — Resultant RC BB-Mode Concrete....................................................... 43

Figure 2.44 — Resultant RC BB-Mode Dry Ground.................................................. 43

Figure 2.45 — Resultant RC BB-Mode Medium Dry Ground ................................... 44

Figure 2.46 — Resultant RC BB-Mode Wet Ground ................................................. 44

Figure 2.47 — Resultant RC BB-Mode Fresh Water ................................................. 45

Figure 2.48 — Resultant RC BB-Mode Sea Water..................................................... 45

Figure 2.49 — Surface roughness geometry ............................................................... 46

Figure 2.50 — Rayleigh Roughness Criterion ............................................................ 47

Figure 2.51 — Rough Surface Reduction Factor ........................................................ 48

Figure 2.52 — Fresnel Zones for Reflection............................................................... 50

Figure 2.53 — First Fresnel Zone Dimensions (1-5 degs).......................................... 51

Figure 2.54 — First Fresnel Zone Dimensions (5-10 degs)........................................ 51

Figure 2.55 — First Fresnel Zone Dimensions (10-90 degs)..................................... 52

Figure 2.56 — Fresnel Zones for LOS........................................................................ 53

Figure 2.57 — Diffraction at Obstruction................................................................... 54

Figure 2.58 — Diffraction at Obstacle........................................................................ 55

Figure 2.59 — Knife-edge Diffraction Gain vs v........................................................ 57

Figure 2.60 — Diffraction Parameter and Gain .......................................................... 57

Figure 2.61 — Time of Arrival Error.......................................................................... 58

Figure 2.62 — Concrete F-mode................................................................................. 60

Figure 2.63 — Dry Ground F-mode............................................................................ 60

Figure 2.64 — Medium Dry Ground F-mode ............................................................. 61

xix

Figure 2.65 — Wet Ground F-mode........................................................................... 61

Figure 2.66 — Fresh Water F-mode ........................................................................... 62

Figure 2.67 — Sea Water F-mode .............................................................................. 62

Figure 2.68 — Concrete BA-mode ............................................................................. 63

Figure 2.69 — Concrete2 BB-mode............................................................................ 64

Figure 2.70 — Dry Ground2 BB-mode ....................................................................... 64

Figure 2.71 — Medium Dry Ground2 BB-mode......................................................... 65

Figure 2.72 — Wet Ground2 BB-mode....................................................................... 65

Figure 2.73 — Concrete to Sea Water ........................................................................ 66

Figure 2.74 — Dry Ground to Sea Water ................................................................... 66

Figure 2.75 — Medium Dry Ground to Sea Water..................................................... 67

Figure 2.76 — Concrete to Fresh Water ..................................................................... 67

Figure 2.77 — Dry Ground to Fresh Water ................................................................ 68

Figure 2.78 — Medium Dry Ground to Sea Water..................................................... 68

Figure 2.79 — Wet Ground to Concrete..................................................................... 69

Figure 2.80 — Wet Ground to Dry Ground................................................................ 69

Figure 2.81 — Wet Ground to Medium Dry Ground ................................................. 70

Figure 2.82 — Medium DryGround to Concrete........................................................ 70

Figure 2.83 — Medium Dry Ground to Dry Ground .................................................. 71

Figure 2.84 — Dry Ground to Concrete ..................................................................... 71

Figure 2.85 — Linear Variation of Propagation Angle (Fwd).................................... 73

Figure 2.86 — Linear Variation (Reduced gradient) .................................................. 73

Figure 2.87 — Variation of Antenna Height............................................................... 74

Figure 2.88 — Backscatter from above ...................................................................... 75

Figure 2.89 — Equal Antenna Distance and Height................................................... 76

Figure 2.90 — Antenna Height > Distance................................................................. 77

Figure 2.91 — Antenna Height < Distance................................................................. 77

Figure 2.92 — Addition of Multipath Modes ............................................................. 78

Figure 2.93 — Variation of Relative Multipath Amplitude........................................ 79

Figure 2.94 — BA-mode Decorrelation Distance/Height Bound ............................... 82

Figure 2.95 — BA-mode Existence-Boundary........................................................... 83

Figure 2.96 — BA-mode Existence Region Metrics .................................................. 83

Figure 2.97 — BB-mode Existence Region................................................................ 84

xx

Figure 2.98 — BB-Mode Decorrelation Distance....................................................... 85

Figure 2.99 — BB-mode Antenna Height Bounds for Decorrelation......................... 86

Figure 2.100 — Additional Path Loss......................................................................... 87

Figure 3.1 — 3-D FD-TD grid .................................................................................. 102

Figure 3.2 — PE solution domain............................................................................. 109

Figure 4.1 — Q-functions ......................................................................................... 127

Figure 4.2 — Phase error with common approximates............................................. 128

Figure 4.3 — FSPE Solution Domain....................................................................... 129

Figure 4.4 — Initial Field.......................................................................................... 131

Figure 4.5 — Upper Absorption Region................................................................... 132

Figure 4.6 — Reflection Coefficient in P-Space....................................................... 133

Figure 4.7 — Implementation Domain ..................................................................... 134

Figure 4.8 — Implementation Algorithm.................................................................. 135

Figure 4.9 — Solution Domain Representation ........................................................ 137

Figure 4.10 — Boundary Shift Technique ................................................................ 137

Figure 4.11 — Boundary-Shift Algorithm ................................................................ 138

Figure 4.12 — Backscatter implementation.............................................................. 140

Figure 4.13 — Backscatter Implementation Algorithm............................................ 141

Figure 5.1 — SINC pulse.......................................................................................... 150

Figure 5.2 — Input spectrum .................................................................................... 152

Figure 5.3 — Propagation domain ............................................................................ 155

Figure 5.4 — Corrections for spatial time reference................................................. 157

Figure 5.5 — Time delay estimation error ................................................................ 159

Figure 5.6 — LOS Phase Estimation Error............................................................... 160

Figure 5.7 — Forward Multipath Phase Estimation Error ........................................ 161

Figure 5.8 — LOS AOA Estimation Error................................................................ 163

Figure 5.9 — Forward Multipath AOA Estimation Error......................................... 163

Figure 5.10 — Non-aliased time response ................................................................ 165

Figure 5.11 — Aliased time response ....................................................................... 165

Figure 5.12 — Domain representation...................................................................... 166

xxi

Figure 6.1 — Phase error in Narrow-Angle SPE...................................................... 172

Figure 6.2 — Phase error in FSPE............................................................................ 173

Figure 6.3 — FSPE Field vs Exact Solution............................................................. 174

Figure 6.4 — Field over flat perfect conductor......................................................... 175

Figure 6.5 — User Received Minimum L1 C/A Signal Level.................................. 176

Figure 6.6 — Time series for forward propagation .................................................. 176

Figure 6.7 — Dynamic situation............................................................................... 177

Figure 6.8 — FSPE field at 8 degrees....................................................................... 178

Figure 6.9 — PETA result at 8 degrees .................................................................... 178

Figure 6.10 — FSPE field and reconstructed PETA field comparison..................... 179

Figure 6.11 — Comparison of RC Magnitudes ........................................................ 180

Figure 6.12 — Fade Pattern Comparison.................................................................. 180

Figure 6.13 — Diffraction over terrain element ....................................................... 181

Figure 6.14 — Diffraction geometry......................................................................... 182

Figure 6.15 — Time series no terrain ....................................................................... 183

Figure 6.16 — Time series with terrain .................................................................... 184

Figure 6.17 — Dynamic diffraction situation ........................................................... 185

Figure 6.18 — FSPE field at 10 degrees.................................................................. 185

Figure 6.19 — FSPE field 5 degrees to 45 degrees .................................................. 186

Figure 6.20 — LOS and diffracted propagation time comparison............................ 187

Figure 6.21 — Forward propagation over terrain ..................................................... 188

Figure 6.22 — Back propagation from reflected interfaces...................................... 189

Figure 6.23 — Total propagated field....................................................................... 189

Figure 6.24 — Two-way field with 20 m high vertical reflector at 20m.................. 190

Figure 6.25 — Time series back-propagation........................................................... 190

Figure 6.26 — Model domain................................................................................... 191

Figure 6.27 — Stepped backscatter geometry........................................................... 192

Figure 6.28 — Stepped backscatter PETA results for 5 degrees .............................. 193

Figure 6.29 — Stepped backscatter PETA results for 12.5 degrees ......................... 194

Figure 6.30 — Stepped backscatter PETA results for 15 degrees ............................ 194

Figure 6.31 — Stepped backscatter PETA results for 5 to 15 degrees ..................... 195

Figure 6.32 — Stepped backscatter knife-edge geometry......................................... 195

xxii

Figure 6.33 — Diffraction loss over B1 interface ..................................................... 196

Figure 6.34 — Normalised diffraction loss PETA vs CCIR for B1 interface........... 197

Figure 6.35 — PETA path delays ............................................................................. 198

Figure 6.36 — Reconstructed PETA field and FSPE field comparison ................... 198

Figure 6.37 — Reconstructed GO field and FSPE field comparison........................ 199

Figure 7.1 — Pine Dam Data Collection Site ........................................................... 204

Figure 7.2 — Data Collection Basis.......................................................................... 205

Figure 7.3 — Location Orientation ........................................................................... 205

Figure 7.4 — SV17 Results....................................................................................... 207

Figure 7.5 — SV17 AZ-EL Data .............................................................................. 207

Figure 7.6 — SV17 Fresnel Data .............................................................................. 208

Figure 7.7 — SV6 Results......................................................................................... 209

Figure 7.8 — SV6 AZ-EL Data ................................................................................ 210

Figure 7.9 — SV6 Fresnel Data ................................................................................ 210







Figure 7.16 — Caboolture Data Collection Site ....................................................... 215

Figure 7.17 — SV21 Results..................................................................................... 215

Figure 7.18 — SV21 AZ-EL Data ............................................................................ 216

Figure 7.19 — SV21 Fresnel Data ............................................................................ 216

Figure 7.20 — SV23 Results..................................................................................... 217

Figure 7.21 — SV23 AZ-EL Data ............................................................................ 217

Figure 7.22 — SV23 Fresnel Data ............................................................................ 218




Figure 8.1 — SV21 Bribie Island Tidal Variation .................................................... 223

xxiii

Figure 8.2 — Relative Height Measurement ............................................................ 225

Figure 8.3 — Sampled received S/N......................................................................... 226

Figure 8.4 — Sampled elevation angle data ............................................................. 226

Figure 8.5 — Interference pattern ............................................................................. 228

Figure 8.6 — Height estimation above fresh water .................................................. 229

Figure 8.7 — Height estimation above sea surface................................................... 230

Figure 8.8 — System representation ......................................................................... 231

Figure 8.9 — Raw S/N sea water.............................................................................. 232

Figure 8.10 — Raw S/N fresh water......................................................................... 232

Figure 8.11 — Height Error Bound 0.1 Deg Error ................................................... 234

Figure 8.12 — Height Error Bound 0.01 Deg Error ................................................. 234

Figure 9.1 — Multipath Modelling Environment ..................................................... 239

Figure 9.2 — General form of GPS user equipment................................................. 240

Figure 9.3 — Generic GPS receiver.......................................................................... 241

Figure 9.4 — C/A Code Generation ......................................................................... 244

Figure 9.5 — C/A autocorrelation function .............................................................. 246

Figure 9.6 — Receiver tracking loops ...................................................................... 247

Figure 9.7 — Correlation process ............................................................................. 248

Figure 9.8 — DLL discriminator curves................................................................... 250

Figure 9.9 — Receiver Correlation Model ............................................................... 253

Figure 9.10 — DLL discriminator curves................................................................. 253

Figure 9.11 — Multipath-free Correlation and Discrimination ................................ 255

Figure 9.12 — 0.05 Chip delay In-phase Multipath.................................................. 256

Figure 9.13 — 0.5 chip delay In-phase Multipath .................................................... 257

Figure 9.14 — Correlation Distortion (0°)................................................................ 257

Figure 9.15 — Short-delay Multipath (180°)............................................................ 259

Figure 9.16 — Long-delay Multipath (180°) ............................................................ 260

Figure 9.17 — Correlation Distortion (180°)............................................................ 260

Figure 9.18 — 200ns relative delay , 0.5 Multipath Power Ratio (0°) ..................... 262

Figure 9.19 — 200ns relative delay, 0.5 Multipath Power Ratio (60°) .................... 262

Figure 9.20 — 200ns relative delay, 0.5 Multipath Power Ratio (90°) .................... 263

xxiv

Figure 9.21 — 200ns relative delay, 0.5 Multipath Power Ratio (125°) .................. 264

Figure 9.22 — 200ns relative delay, 0.5 Multipath Power Ratio (180°) .................. 264

Figure 9.23 — ½-Chip 0.5 Multipath Power Ratio................................................... 265

Figure 9.24 — 1/20 Chip 0.5 Multipath Power Ratio............................................... 266

Figure 9.25 — ½-Chip 0.5 Multipath Power Ratio (2MHz)..................................... 267

Figure 9.26 — 1/20-Chip 0.5 Multipath Power Ratio (8MHz)................................. 267

Figure 9.27 — Variation of Multipath Power Ratio ½-Chip (0°) ............................. 268

Figure 9.28 — Variation of Multipath Power Ratio ½-Chip (180°) ......................... 269

Figure 9.29 — Range Error 0.5 Multipath Power Ratio (200 ns 0°-180°) ............... 270

Figure 9.30 — Dual Multipath Error (0° and 0°)...................................................... 271

Figure 9.31 — Dual Multipath Error (0° and 180°).................................................. 272

Figure 9.32 — Dual Multipath Error (180° and 0°).................................................. 272

Figure 9.33 — Dual Multipath Error (180° and 180°).............................................. 273

Figure 9.34 — S/N Fade Pattern ............................................................................... 274

List of Tables

Table 2.1 — Electrical Properties ............................................................................... 12

Table 2.2 — Geometric Delays and Phases ................................................................ 32

Table 2.3 — Coupled Reflection Coefficient Magnitudes.......................................... 36

Table 3.1 — Model Comparison............................................................................... 110

Table 4.1 — PE Coefficients..................................................................................... 122

Table 6.1 — Modelling Errors Forward.................................................................... 177

Table 6.2 — Modelling Errors Forward and Back.................................................... 191

Table 9.1 — C/A-Code Selection ............................................................................. 245

Table 9.2 — DLL discriminator algorithms.............................................................. 249

Table 9.3 — PLL discriminator algorithms .............................................................. 251

Table 9.4 — FLL discriminator algorithms .............................................................. 252

1

Chapter 1 Introduction and Overview

1.1 Introduction

The Global Positioning System (GPS) is currently revolutionising the disciplines of

navigation and positioning [1, 2]. Since the first navigation satellite was

commissioned in June 1977, the research community has devoted a significant

amount of time and energy into reducing the errors associated with satellite based

positioning systems. Initially the best accuracy obtainable with the system was 30m

(in three dimensions and 95% of the time). With the use of differential techniques it is

possible to remove many of the common-mode error sources, but there is one error

source that has proven much more difficult to mitigate — multipath. Multipath is the

dominant error source in applications that require high accuracy [3, 4].

Examples of applications where high accuracy is required are numerous. There is a

concerted effort in the open-cut mining community to reduce human exposure to

hazardous tasks. The automation of mining machines such as haul trucks is a prime

example. The use of GPS to navigate these huge machines is mandated by the

accuracy and availability requirements. However, the environment in which these

machines operate is harsh for the operation of GPS. Indeed the principle error source

in this hostile propagation environment is multipath propagation.

Surveying also requires the highest accuracy and in an attempt to achieve the highest

precision possible all attempts are made at reducing potential error sources. The

development and expansion of the use of differential GPS (DGPS) is an example of

the user community desiring higher accuracy.

Further to these obvious requirements any reduction in error sources opens the way to

new applications of GPS. Although this work is concerned with terrestrial

applications, it is also noted that space-based applications of GPS are likewise limited

by localised multipath effects [5].

2

There has been substantial research into the topic of multipath and it remains an active

research area. Work by Braasch, van Nee [3, 4] and others [6-11] has provided an

understanding of multipath effects for defined and controlled multipath parameters.

The derivation of multipath propagation parameters, amplitude, phase, phase rate-of-

change, and delay, from actual propagation models has, however, received little

attention.

This research describes the use of modelling techniques, to gain an insight into

multipath propagation behaviour, and to extract the relevant multipath parameters for

simulated environments. In conjunction with receiver modelling, the use of a

propagation model — that more realistically represents actual propagation conditions

— provides the potential to further the understanding of multipath and its related

effects on GPS receivers. A comparative analysis of this model system with results

from actual receiver measurements (observables), obtained in field trials, provides

verification of an accurate understanding of the underlying principles used in the

simulation of the multipath environment.

This research aims to provide an improved understanding of the behaviour and effects

of Global Positioning System (GPS) signal propagation in precision application

environments, through the use of novel modelling and simulation techniques.

1.2 Overview of Research Presented

The research work and specific contributions presented in this dissertation are as

follows:

The fundamental issues of multipath propagation, and the concepts necessary for

development of modelling techniques are presented. Previous work has not offered

complete definition or explanation of the specific multipath environment for GPS.

The concept of reflection and diffraction are not new, but in this work a more

3

thorough analysis of reflection and diffraction for the GPS L1 (1.575 GHz) signal is

given.

Specifically the scattering of the right-hand-circularly-polarised (RHCP) GPS signal is

represented in linear and circular reflection coefficients. A novel representation, more

suitable for single boundary propagation modelling, is presented. This coupled

polarisation reflection coefficient makes use of the fact that both co- and cross-

polarised representations have approximately similar phase shifts. The axial-ratio, or

effective left-hand-circular-polarisation (LHCP) rejection ratio of the GPS antenna is

then incorporated, with the cross-polarised component, into a single reflection

coefficient that may be used in propagation models designed for modelling of linear

polarisation propagation.

Aspects of rough surface effects, Fresnel zones, and diffraction are examined in detail.

As are the physical dimensions, in which GPS multipath propagation has an effect on

the receiver, for defined propagation states.

Providing an insight into multipath propagation behaviour; an enhanced Parabolic

Equation (PE)-based GPS propagation model is developed specifically for multipath

analysis of the GPS L1 signal. This work is based on a newly developed Free-Space

PE propagation technique. The model presented is an extension of an initial PE model

developed by Walker. This improved model includes the effects of backscatter and the

more realistic coupled GPS reflection coefficient for modelling of terrain boundary

effects.

In addition to providing relevant propagation data, the new model provides a

visualisation of the complete propagation of the GPS signal and its interaction with

the localised terrain. The visualisation aspects of the model are a valuable aid in the

understanding of GPS multipath propagation.

To extract the necessary multipath parameters, amplitude, phase, time delay, a Fourier

time synthesis technique has been developed. The complete PE-based Time Analysis

4

(PETA) model provides a comprehensive description of the multipath RF

environment.

In addition, as for the FSPE on which it is based, the PETA provides a visualisation of

the multipath channel impulse response. This visualisation allows an interpretation of

the propagation mechanisms and their interaction in the modelled domain. The

resultant data output also provides the necessary input for a generic GPS correlation

model that provides range error estimation for the given propagation environment.

The FSPE and PETA models have been validated against exact solutions and accepted

standards, and multipath data was collected from a variety of sites and a comparison

made with the simulations provide by the FSPE model. In all cases the FSPE and

PETA models showed excellent agreement with accepted theoretical aspects of

propagation and the real-world data collected in field sessions.

The undesirable effect of signal fading on the GPS L1 signal was investigated and

utilised to provide a novel observable for relative antenna height above a reflecting

surface.

Receiver concepts are introduced and an open loop GPS code-correlation receiver

model is implemented. The results of the FSPE/PETA models form the input to the

receiver model and provide an insight into the effects of multipath propagation on the

code measurement process.

In this work a novel modelling environment has been researched, developed and

implemented. The Free-Space Parabolic Equation model, the PE-based Time Analysis

model and the GPS correlation/discrimination model provide a comprehensive suite

of modelling and simulation tools for the investigation of GPS multipath propagation

and its resultant error effects. It is recommended that further investigation be made of

any benefit in pursuing 3D modelling. In addition a thorough comparative analysis

should be made — using the modelling tools developed in this work — of multipath

data collected from a large variety of propagation environments.

5

1.3 Research Contribution

The program of work carried out and presented in this dissertation has resulted in the

contribution of the following:

A thorough analysis of the terrestrial multipath propagation environment for the GPS

L1 signal. The nature of the propagation mechanisms and the resultant behaviour

including reflection, diffraction, rough surface effects and the spatial context of the

environment are examined for the first time, in such a way as to bring coherence to

our understanding of GPS multipath propagation.

The development and implementation of a new electromagnetic propagation model

specifically for GPS multipath investigation. The model builds on the accepted

parabolic equation propagation model with the inclusion of boundary conditions that

deal with the right-hand circularly polarised nature of the GPS signal. In addition

backscatter has been incorporated to provide a more realistic representation of real-

world propagation environments.

A novel time-analysis model that uses the GPS parabolic-equation propagation

method to provide a complete description and visualisation of the multipath

propagation behaviour.

The development of an open-loop correlation/discrimination model that allows the

variation of correlation/discrimination design parameters. Combined with the

propagation models, this novel comprehensive modelling system permits the testing

of performance for particular implementations in defined and repeatable multipath

environments.

6

1.4 References

[1] National Academy of Public Administration and National Research Council,

"The Global Positioning System: Charting the Future," May 1995 1995.

[2] National Research Council, "The Global Positioning System - A Shared

National Asset." Washington DC: National Academy Press, 1995.

[3] M. S. Braasch, "Multipath Effects," in Global Positioning System: Theory and

Applications, vol. 1, B. W. Parkinson and J. R. Spilker Jr., Eds. Washington:

American Institute of Aeronautics and Astronautics, 1996, pp. 547-568.

[4] R. D. J. van Nee, "Multipath and Multi-Transmitter Interference in Spread-

Spectrum Communication and Navigation Systems," in Faculty of Electrical

Engineering, Telecommuncation and Traffic Control Systems Group. Delft:

Delft University of Technology, 1995, pp. 205.

[5] P. Axelrad and L. M. Ward, "Spacecraft Attitude Estimation Using the Global

Positioning System: Methodology and results for RADCAL," Journal of

Guidance, Control, and Dynamics, vol. 19, pp. 1201-1209, 1996.

[6] E. Breeuwer, "Modelling and Measuring GPS Multipath Effects," in Faculty

of Electrical Engineering. Delft: Delft University of Technology, 1992.

[7] B. Eissfeller and J. O. Winkel, "GPS Dynamic Multipath Analysis in Urban

Areas," presented at The 9th International Technical Meeting of The Satellite

Division of The Institute of Navigation., Kansas City, Missouri, 1996.

[8] A. El-Rabbany, "Temporal Characteristics of Multipath Errors," presented at

8th International Technical Meeting of The Satellite Division of The Institute

of Navigation., Palm Springs, California, 1995.

[9] T. Lo and J. Litva, "Use of a Highly Deterministic Multipath Signal Model in

Low-Angle Tracking," presented at IEE Proceedings-F [Radar and Signal

Processing], 1991.

[10] H. Leung and T. Lo, "A Spatial Temporal Dynamical Model for Multipath

Scattering From the Sea," IEEE Transactions on Geoscience & Remote

Sensing, vol. 33, pp. 441-448, 1995.

7

[11] S. U. Hwu, B. P. Lu, R. J. Panneton, and B. A. Bourgeois, "Space Station GPS

Antennas Multipath Analysis," presented at IEEE Antennas and Propagation

Society International Symposium, Newport Beach, California, 1995.

8

THIS PAGE IS INTENTIONALLY BLANK

9

Chapter 2 Nature of GPS Multipath Propagation

To further the understanding of GPS multipath propagation and to develop and

implement an accurate GPS propagation model for simulation studies, it is necessary

that the environmental aspects of radio frequency propagation at the GPS L1

frequency be well understood. In this chapter the fundamental issues of environmental

factors, reflection, diffraction and rough surface effects on the GPS L1 signal are

examined in detail. The relevant concepts are examined within the context of being

ultimately incorporated into a GPS multipath propagation model.

2.1 Multipath Environment

Multipath is the unwanted distortion, of the direct line-of-sight satellite signal, by

localised reflected and/or diffracted signals. An example of these multipath signals is

shown in Figure 2.1.

Reflected path

Edge-diffracted path

Reflected pathDirect path

GPS antenna

Figure 2.1 — Multipath environment

The nature of the localised terrain determines the composition of the radio frequency

environment. To understand this multipath environment, it is necessary to consider

not only the physical relationship of the GPS receiver to the surrounding terrain

10

elements (geometry), but also the propagation characteristics (electromagnetic

properties ) of the terrain.

The term multipath obviously describes the separate propagation paths taken by the

reflected/diffracted signals. Since the multipath signals travel additional distances

they are delayed relative to the line-of-sight (LOS) signal. This relative time delay, is

one of the defining parameters for describing the characteristics of multipath [1, 2].

In addition to the relative time delay, multipath is characterised by its amplitude,

phase, and phase rate-of-change, all relative to the LOS signal. The relative phase is a

function of the additional path length and the electrical properties of the

reflecting/diffracting medium, whilst the phase rate-of-change accounts for the

changing multipath propagation environment—due to the relative satellite-user

dynamics. Finally, the relative amplitude of the multipath signals is determined by the

nature of the reflecting surface structure. Critical aspects of the scattering of the GPS

L1 signal are presented in the following sections.

2.2 Specular Reflection

The theory of specular reflection is well understood [3-5], and aspects of the theory

are of importance for the understanding of multipath propagation of the Right-Hand

Circularly Polarised (RHCP) GPS L1 signal. In the following sections the concept of

the reflection coefficient is examined.

2.2.1 Linear Reflection Coefficient Representation

As was outlined previously, the behaviour of multipath is determined by the geometry

and electrical properties of the propagation environment.

The reflection coefficients, derived from the Fresnel equations for a smooth earth

surface provide information on the nature of reflected signals. The GPS signal is

11

RHCP, and since circular polarisation is the vector sum of the linearly polarised

waves (horizontal and vertical), it is appropriate to initially consider the reflection

coefficients for the linear cases. The reflection coefficients for horizontal and vertical

polarisation are given respectively as,

ΓH =− −

+ −

sin cos

sin cos

θ ε θ

θ ε θ

2

2(2.1)

and

ΓV =− −

+ −

ε θ ε θ

ε θ ε θ

sin cos

sin cos

2

2(2.2)

where

ε εσ

ωε= −r j0

(2.3)

is the complex dielectric constant with assumed time dependence in tje ω− .

Substituting for ω and ε0, in equation (2.3) gives

ε ε λσ= −r j60 (2.4)

The calculation of each linear reflection coefficient is now straightforward, for a given

frequency, grazing angle (θ), dielectric constant and conduction value for the

reflecting surface medium. The resultant complex reflection coefficient then defines

the relative amplitude and phase of the specular reflection, and is of the general form,

Γ = ≤ ≤−ρ ρφe j , 0 1 (2.5)

12

By now using equations (2.2) and (2.3), the linear reflection coefficients for

representative materials at 1 GHz (man-made and natural) given in Table 2.1 [6] can

be plotted.

Material Conductivity Relative Permittivity

Concrete 2 x 10-5 3

Dry Ground 1 x 10-5 4

Medium Dry Ground 4 x 10-2 7

Wet Ground 2 x 10-1 30

Fresh Water (fresh) 2 x 10-1 80

Sea Water (sea) 4 20

Table 2.1 — Electrical Properties

Figure 2.2 through to Figure 2.7, show the magnitude and phase of the linear

reflection coefficients as a function of propagation angle, for the given materials, at

the GPS L1 frequency, 1.575 GHz.

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)

Propagation angle (degs)

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient - Concrete

Mag

nitu

de

HV

Figure 2.2 — Linear reflection coefficients for Concrete

13

0 10 20 30 40 50 60 70 80 900

50

100

150

200P

hase

(de

gs)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient - Dry Ground

Mag

nitu

de

HV

Figure 2.3 — Linear reflection coefficients for Dry Ground

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient - Medium Dry Ground

Mag

nitu

de

HV

Figure 2.4 — Linear reflection coefficients for Medium Dry Ground

14

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient - Wet Ground

Mag

nitu

de

HV

Figure 2.5 — Linear reflection coefficients for Wet Ground

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient - Fresh Water

Mag

nitu

de

HV

Figure 2.6 — Linear reflection coefficients for Fresh Water

15

0 10 20 30 40 50 60 70 80 900

50

100

150

200P

hase

(de

gs)


0 10 20 30 40 50 60 70 80 900.2

0.4

0.6

0.8

1Reflection Coefficient - Sea Water

Mag

nitu

de

HV

Figure 2.7 — Linear reflection coefficients for Sea Water

As can be seen, the variation of the reflection coefficient magnitude is different for

each linear polarisation. The horizontal component for all media decreases smoothly

for increasing propagation angles, with a near constant 180 degree phase shift for all

propagation angles. The vertical component magnitude, however, decreases quickly

and has a near constant 180 degree phase shift for angles less than the Brewster angle.

For propagation angles greater than the Brewster angle the vertical reflection

coefficient magnitude increases with a near constant zero degree phase shift.

On considering the vector addition of the two linear reflection coefficients, it is

obvious that the resultant polarisation of the GPS RHCP will be elliptical when the

coefficients are different, circular when they are equal, and linearly polarised when the

vertical component goes to zero. The nature of the final polarisation is determined by

the relative phase relationship of each linear component upon reflection.

For angles less than the Brewster angle the reflected signal is right-hand-elliptically-

polarised (RHEP), becoming increasingly more elliptic as the Brewster angle is

approached. Generally at the Brewster angle the polarisation approaches that of pure

horizontal polarisation. For angles greater than the Brewster angle the reflected signal

is left-hand-elliptically-polarised (LHEP) with the ellipticity decreasing as

propagation angles increase.

16

The implication for modelling the GPS RHCP signal using the linear reflection

coefficients, is that two separate (horizontal and vertical) boundary models would

need to be implemented. An alternative representation of specular reflection

coefficients is given in the following section.

2.2.2 Circular Reflection Coefficient Representation

As an alternative representation of the reflection of the GPS RHCP L1 signal, the

resultant reflection can be considered as the sum of two circularly polarised (CP)

signals; one that maintains the co-polarisation (original) and a cross-polarisation

(opposite) component. The copolar ( oΓ ) and crosspolar ( xΓ ) reflection coefficients, as

a function of the horizontal and vertical reflection coefficients, are respectively given

as [7]:

2vh

o

Γ+Γ=Γ (2.6)

and

2vh

x

Γ−Γ=Γ (2.7)

From the previous analysis of the linear reflection coefficients for RHCP, it is an

intuitive observation that for propagation angles less than the Brewster angle the

copolar component predominates and for angles greater than the Brewster angle the

crosspolar component becomes dominant. This understanding is verified in the

following plots of copolar and crosspolar reflection coefficients for the same media as

those used for the linear reflection coefficient case.

17

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient: circ (original & cross) - Concrete

Mag

nitu

de

orig cross

0 10 20 30 40 50 60 70 80 900

50

100

150

200P

hase

(de

gs)


Figure 2.8 — Circular reflection coefficients for Concrete

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient: circ (original & cross) - Dry Ground

Mag

nitu

de

orig cross

Figure 2.9 — Circular reflection coefficients for Dry Ground

18

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient: circ (original & cross) - Medium Dry Ground

Mag

nitu

de

orig cross

Figure 2.10 — Circular reflection coefficients for Medium Dry Ground

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient: circ (original & cross) - Wet Ground

Mag

nitu

de

orig cross

Figure 2.11 — Circular reflection coefficients for Wet Ground

19

0 10 20 30 40 50 60 70 80 900

50

100

150

200P

hase

(de

gs)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient: circ (original & cross) - Fresh Water

Mag

nitu

de

orig cross

Figure 2.12 — Circular reflection coefficients for Fresh Water

0 10 20 30 40 50 60 70 80 900

50

100

150

200

Pha

se (

degs

)


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1Reflection Coefficient: circ (original & cross) - Sea Water

Mag

nitu

de

orig cross

Figure 2.13 — Circular reflection coefficients for Sea Water

The behaviour of the reflected signal is more easily recognised in the CP

representation, than in the linear case. For circular polarisation, copolar and crosspolar

reflection coefficients are of nearly equal magnitude at the Brewster angle. When the

magnitudes of the CP components are different the resultant polarisation will be

elliptic. This confirms that for propagation angles less than the Brewster angle the

resultant reflection is RHEP, becoming increasingly elliptic, until near linear

20

polarisation occurs at the Brewster angle. For angles greater than the Brewster angle

the signal is highly LHEP becoming less elliptic for increasing propagation angles,

and ultimately, fully LHCP at 90 degrees.

On further examination of the circular reflection coefficients it is apparent that the

phase of both the copolar and crosspolar components tend towards 180 degrees. From

this observation an assumption can be made that both circular components, that

represent the original signal reflection, experience the same phase shift. However, to

model the complete reflection of the RHCP GPS signal it would still be necessary to

model two separate boundary conditions to account for the two separate polarisation

cases and then couple the resultant fields at the antenna location.

In considering the actual signal reception at the GPS antenna of a single reflected

signal, a further assumption can be made to simplify the boundary condition

requirements to a single effective CP reflection coefficient. This can be achieved by

making use of the fact that the GPS antenna is designed in an optimal sense, to

receive RHCP signals. The antenna thus rejects the LHCP or cross-polarised

component, at some level. By incorporating the LHCP rejection ratio into the cross-

polarised component, a simplified coupled single boundary representation is given,

one which can be readily implemented in a GPS propagation model. This effective

reflection coefficient for an incident RHCP signal can be written as:

πρρ jx

K

cR e−−

+=Γ 2010 (2.8)

where

cρ is the co-polarisation circular reflection coefficient magnitude (RHCP)

xρ is the cross-polarised circular reflection coefficient magnitude (LHCP)

and

21

K is the GPS antenna LHCP rejection ratio in dB—which may be a constant or

modelled as a function of elevation angle (and azimuth angle). In general the gain

characteristics of high quality GPS antennas are essentially omnidirectional for both

polarisations. With this assumption the rejection ratio is essentially constant and the

use of K as a constant is justified. Typical values for commercial GPS antennas are

axial ratios of 3dB which equate to LHCP rejection ratios of about 10-11dB.

The effective coupled reflection coefficient magnitude for an incident L1 GPS RHCP

signal is given in the following figures, for constant rejection ratios ranging from 0 to

30 dB, in 3 dB increments.

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled RC Mag: Concrete

Mag

nitu

de

Propagation Angle (degs)

0 3 6 912151821242730

Figure 2.14 — Incident RHCP-RC for Concrete

30 dB

0 dB

22

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled RC Mag: Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.15 — Incident RHCP-RC for Dry Ground

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled RC Mag: Medium Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.16 — Incident RHCP-RC for Medium Dry Ground

30 dB

0 dB

30 dB

0 dB

23

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled RC Mag: Wet Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.17 — Incident RHCP-RC for Wet Ground

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled RC Mag: Fresh Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.18 — Incident RHCP-RC for Fresh Water

30 dB

0 dB

30 dB

0 dB

24

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled RC Mag: Sea Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.19 — Incident RHCP-RC for Sea Water

In addition to the coupled reflection coefficients for incident RHCP, it is necessary to

consider the resultant coupled reflection coefficient for an incident LHCP signal. A

component of LHCP will be present after a single reflection of RHCP, if this LHCP

signal component is incident upon another reflection boundary, then the resultant

coupled LHCP reflection coefficient is given by:

πρρ jxc

K

L e−−

+=Γ 2010 (2.9)

here

cρ is the co-polarisation circular reflection coefficient magnitude (LHCP)

xρ is the cross-polarised circular reflection coefficient magnitude (RHCP)

and again

K is the GPS antenna LHCP rejection ratio in dB.

30 dB

0 dB

25

The effective coupled reflection coefficient magnitude for an incident L1 LHCP signal

is given in the following figures, for rejection ratios ranging from 0 to 30 dB, in 3 dB

increments.

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled LHCP Incident RC Mag: Concrete

Mag

nitu

de


0 3 6 912151821242730

Figure 2.20 — Incident LHCP-RC for Concrete

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled LHCP Incident RC Mag: Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.21 — Incident LHCP-RC for Dry Ground

26

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled LHCP Incident RC Mag: Medium Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.22 — Incident LHCP-RC for Medium Dry Ground

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled LHCP Incident RC Mag: Wet Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.23 — Incident LHCP-RC for Wet Ground

27

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled LHCP Incident RC Mag: Fresh Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.24 — Incident LHCP-RC for Fresh Water

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coupled LHCP Incident RC Mag: Sea Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.25 — Incident LHCP-RC for Sea Water

2.2.3 Ray-based Reflection Geometry: Relative Time Delay and Phase

Before considering reflection coefficients for specific GPS multipath propagation

cases we first need to develop an understanding of propagation situations that describe

the majority of GPS multipath propagation scenarios. In this section we consider

geometric optics to investigate simple ray-based geometry and formulate equations for

28

the defining multipath parameters of relative amplitude, delay and phase, for three

primary modes of multipath propagation. These equations form a canonical set of

reflection situations that can be used in the verification of multipath parameters

derived from a GPS L1 propagation model. They also form the basis for an

understanding of the nature of multipath propagation in terrestrial environments.

Consider the forward-scatter problem confined to an arbitrary two-dimensional

domain with a flat reflecting lower boundary, Figure 2.26. The GPS antenna is located

at point P, at a distance d from the left-hand boundary, height h above the reflecting

surface, and with a Line-of-sight (LOS) signal propagating into the domain at angle θ.

h

-h

θ θ

d

θ

P

Pi

θ

∆R

LOS

Figure 2.26 — Forward scatter geometry

Using image theory, the reflected signal must travel an additional distance ∆R to the

image point Pi. This additional path length is given by:

θsin2hR =∆ (2.10)

Also the distance of propagation of the LOS signal into the domain—from the

arbitrary left-hand-side reference boundary—is given by

Dd

p =cosθ

(2.11)

29

Now consider the backscatter problem with the addition of a vertical reflecting

surface forming the right-side boundary, Figure 2.27.

h

-h

θ

P

Pi

2θ

x

l1

l2

l3

l5

l4

LOS

Figure 2.27 — Backscatter geometry 1

For this given geometry the following inequality must be satisfied:

xh

>tanθ

The region in which this occurs will be known as Zone 1. The individual path lengths

are then given by:

lx

1

2=

cos

cos

θθ

, l2 =x

cosθ, l3 = 2hsinθ +

x cos2θcosθ

, l4 =x

cosθ−

h

sinθ, l5 =

h

sinθ,

and the total path-length differences, relative to the line-of sight, for the two multipath

propagation paths are given by,

∆R xa = 2 cosθ (2.12)

30

and

∆R h xb = +2 2sin cosθ θ (2.13)

Here the subscripts a and b represent the multipath, either arriving from above or

below the horizontal plane containing the antenna.

For the special case of a geometry with xh

=tanθ

(corner reflection), the path lengths

are:

lx

1

2=

cos

cos

θθ

, l2 = l3 = l5 =h

sinθ=

x

cosθ, l4 = 0

and the total path difference is,

∆Rx h

= =2 2

cos sinθ θ(2.14)

Finally, if xh

<tanθ

, the case is such that the reflected signal arriving from below the

horizontal is produced by a different geometric arrangement, as shown in Figure 2.28.

31

h

θP

2θ

x

l1

l2

l3

l5

l4

LOS

Figure 2.28 — Backscatter geometry 2

For this case, the region of occurrence will be known as Zone 2, and the relevant paths

are:

lx

1

2=

cos

cos

θθ

, l2 = l5 =x

cosθ, l3 = 2x cosθ −

h cos2θsinθ

, l4 =h

sinθ−

x

cosθ

However, the total path differences are the same as those given by equations (2.10)

and (2.11).

2.2.3.1 Summary of Amplitudes, Relative Delays and Phases

In summary, the amplitudes, relative time delays, and phases, for each of these simple

multipath propagation environments, can be given in terms of the effective reflection

coefficient and the geometric relationship of the antenna and the terrain.

32

Path

(Mode)

Amplitude Relative TimeDelay

Total Phase Retardation

Line-of-sight

(LOS)

E0 1

c

d

cos

*

θ

2πλ θ

d

cos

*

Forward

scatter (F)

ρg E0 ( )12

ch sinθ φ

πλ

θg

h+

4sin

Backscattera

(BA)

ρr E0 ( )12

cx cosθ φ

πλ

θr

x+

4cos

Backscatterb

(BB)

0Egr ρρ ( )12 2

ch xsin cosθ θ+ ( )φ φ

πλ

θ θr g h x+ + +4

sin cos

*These values are relative to the arbitrary incident boundary of Figure 2.26

Table 2.2 — Geometric Delays and Phases

Here d is the horizontal distance from the left-side arbitrary boundary to the antenna

location, h is the vertical height of the antenna and x is the horizontal distance from

the vertical reflector to the antenna location. The speed of propagation is given as the

reference speed of light C. The reflection coefficients, ρg , φg , ρr , and φr , refer to the

ground and back-reflector respectively. For our derived effective coupled CP

reflection coefficients, the phase terms are all assumed as being π.

2.2.4 Multipath Modes and Coupled Reflection Coefficients

The three types of multipath propagation characterised in the previous section can be

considered as primary modes of GPS multipath propagation. F-Mode represents the

forward reflection of the signal from a ground-bounce source, BA-Mode is the back-

reflection that arrives from above the antenna. F-mode multipath propagation is a case

for direct application of the coupled RHCP reflection coefficient at the lower incident

boundary. For BA-Mode, the reflection coefficient magnitude terms are derived from

the same coupled RHCP reflection coefficient but using the transpose of the

33

propagation angle (π/2-θ). That is for a propagation angle of 10 degrees the mode BA-

mode reflection coefficient is given by the coupled incident RHCP reflection

coefficient for an incident angle of 80 degrees at the backscatter vertical interface.

BB-Mode specifies the back-reflection arriving from below. In addition the BB-Mode

has two zones (Zone 1 and Zone 2) that specify the nature of the production of this

mode of multipath. In Zone 1 the first reflection is a BA-mode from the backscatter

interface incident upon the lower boundary and reflected to the antenna spatial

location. Consider Figure 2.29 where the circular reflection coefficients are decoupled

in polarisation and treated separately.

incident RHCP (Eo)

x and ρρ ′′c

(LHCP)

(RHCP)

xρρ

′′c

x and ρρc

(LHCP) and

(RHCP) and

cxx

xx

ρρρρρρρρ

′′′′

c

cc

Figure 2.29 — Decoupled Polarisation Zone 1 BB-Mode

Coupling the final resultant magnitudes at the antenna location with a LHCP rejection

factor of k gives:

( ) 0Ekk cxxcxxcc ρρρρρρρρ ′+′+′+′ (2.15)

34

Now for BA-mode the back-reflector has the coupled RHCP incident reflection

coefficient magnitude applied. Therefore all reflected signals from the backscatter

interface in BA-mode are coupled correctly and the field travelling down towards the

lower boundary is fundamentally correct. However at the lower boundary the

reflection coefficient magnitude must be such that the resultant magnitude is the same

as that given in Equation (2.15) for the decoupled polarisation.

incident RHCP (Eo)

xρρ ′+′ kc

( ) 0Ek xc ′+′

σ

( ) 0Ekk cxxcxxcc ′+′+′+′

Figure 2.30 — Coupled Polarisation Zone 1 BB-Mode

If an arbitrary reflection coefficient is applied at the lower boundary then

( ) cxxcxxccxc kkk ρρρρρρρρσρρ ′+′+′+′=′+′ (2.16)

and

xc ηρρσ += (2.17)

where

35

The required lower boundary reflection coefficient magnitude for BB-mode in Zone 1

is seen to be a coupled incident RHCP reflection coefficient with the cross-

polarisation component modified by the ratio of the incident LHCP and incident

RHCP coupled reflection coefficient for the back-scatter interface. This ratio increases

the cross-polarised influence on the final field magnitude compensating for the initial

coupling of polarisation states at the back-scatter interface.

Following the same procedure for Zone 2 gives:

xc ρµρβ ′+′= (2.18)

where

Likewise the BB-mode Zone 2 back-scatter reflection coefficient is a coupled

coefficient modified by the ratio of the incident LHCP to incident RHCP coupled F-

mode reflection coefficients. The associated modes and reflection coefficient

magnitudes are given in Table 2.3.

′+′′+′

=xc

xc

k

k

ρρρρη

++=

xc

xc

k

k

ρρρρµ

36

Multipath Mode Coupled RC Magnitude Boundary

F-Mode xc kρρ + Lower in forward direction

BA-Mode xc kρρ ′+′ Back Reflector at positive

angles of incidence

BB-Mode Zone 1 xxc

xcc k

k ρρρρρρ

′+′′+′

+ Lower in back direction

BB-Mode Zone 2 xxc

xcc k

k ρρρρρρ ′

+++′ Back Reflector at negative

angles of incidence

Table 2.3 — Coupled Reflection Coefficient Magnitudes

The nature of the modified coupled reflection coefficients for BB-Mode in Zone 1 are

illustrated in Figure 2.31 to Figure 2.36 given below. It should be noted that these

reflection coefficients are contrived in the sense that they are necessitated by the

initial coupling of polarisation through the use of a coupled reflection coefficient at

the initial reflection boundary.

We recall that the coupling of the polarisation is done such that a single boundary

condition is satisfied and in terms of modelling there is no longer any need to treat the

polarisation boundary conditions separately, thus simplifying the boundary modelling.

37

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4Coupled BB-Mode Zone 1 RC Mag: Concrete

Mag

nitu

de


0 3 6 912151821242730

Figure 2.31 — Modified Coupled RC Zone 1 Concrete

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Coupled BB-Mode Zone 1 RC Mag: Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.32 — Modified Coupled RC Zone 1 Dry Ground

38

0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

6

7Coupled BB-Mode Zone 1 RC Mag: Medium Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.33 — Modified Coupled RC Zone 1 Medium Dry Ground

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14Coupled BB-Mode Zone 1 RC Mag: Wet Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.34 — Modified Coupled RC Zone 1 Wet Ground

39

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18Coupled BB-Mode Zone 1 RC Mag: Fresh Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.35 — Modified Coupled RC Zone 1 Fresh Water

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18Coupled BB-Mode Zone 1 RC Mag: Sea Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.36 — Modified Coupled RC Zone 1 Sea Water

The coupled RC for BB-Mode in Zone 2 is again the propagation angle transpose of

the coupled reflection coefficient magnitudes and are included for completeness.

40

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4Coupled BB-Mode Zone 2 RC Mag: Concrete

Mag

nitu

de


0 3 6 912151821242730

Figure 2.37 — Modified Coupled RC Zone 2 Concrete

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Coupled BB-Mode Zone 2 RC Mag: Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.38 — Modified Coupled RC Zone 2 Dry Ground

41

0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

6

7Coupled BB-Mode Zone 2 RC Mag: Medium Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.39 — Modified Coupled RC Zone 2 Medium Dry Ground

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14Coupled BB-Mode Zone 2 RC Mag: Wet Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.40 — Modified Coupled RC Zone 2 Wet Ground

42

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18Coupled BB-Mode Zone 2 RC Mag: Sea Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.41 — Modified Coupled RC Zone 2 Sea Water

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

18Coupled BB-Mode Zone 2 RC Mag: Fresh Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.42 — Modified Coupled RC Zone 2 Fresh Water

The resultant reflection coefficient for the double reflection mode specified by BB-

Mode multipath is identical for Zone 1 and Zone 2, and is simply the product of the

two incident boundary reflection coefficient magnitudes. Indeed the modified

coupling at the second incident boundary ensures that the resultant product represents

the decoupled polarisation case. The resultant reflection coefficients for common

43

material (that is both boundaries have the same characteristics) BB-Mode are shown

in the figures below for LHCP rejections ranging from 0dB to 30dB.

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35Resultant Coupled BB-Mode RC Mag: Concrete

Mag

nitu

de


0 3 6 912151821242730

Figure 2.43 — Resultant RC BB-Mode Concrete

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35Resultant Coupled BB-Mode RC Mag: Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.44 — Resultant RC BB-Mode Dry Ground

44

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Resultant Coupled BB-Mode RC Mag: Medium Dry Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.45 — Resultant RC BB-Mode Medium Dry Ground

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7Resultant Coupled BB-Mode RC Mag: Wet Ground

Mag

nitu

de


0 3 6 912151821242730

Figure 2.46 — Resultant RC BB-Mode Wet Ground

45

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Resultant Coupled BB-Mode RC Mag: Fresh Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.47 — Resultant RC BB-Mode Fresh Water

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Resultant Coupled BB-Mode RC Mag: Sea Water

Mag

nitu

de


0 3 6 912151821242730

Figure 2.48 — Resultant RC BB-Mode Sea Water

2.3 Rough Surface Scatter

The coupled reflection coefficients account only for specular reflection. For many

practical cases of GPS L1 propagation the surface cannot be considered as smooth. In

these cases the signal is scattered in a non-coherent, non-directional sense. If the

46

surface contains features that lead to diffuse scattering of the incident wave the

coupled reflection coefficients for the GPS RHCP L1 signal require modification.

In explaining diffuse scatter from rough surfaces we consider the reflection of rays

incident on an irregular surface, Figure 2.49.

∆H

ψ

ψ ψ

Figure 2.49 — Surface roughness geometry

The phase difference, between the two rays reflecting from the different levels, is

given by [8],

ψλ

πφ sin4 H

H

∆=∆ (2.19)

Setting the phase difference to π, full cancellation of the reflected components in the

forward direction occurs. If there is no energy transferred in the forward direction then

diffuse scattering of the signal in other directions must have occurred. The Rayleigh

criterion (where the phase difference is considered π/2) for the roughness of a surface

can be written as [9],

ψλ

sin8≥∆H (2.20)

47

In the case where ∆H is less than this criterion, the surface roughness effects are

considered negligible—specular reflection occurs—and the coupled reflection

coefficient applies without modification. In general ∆H is considered as the standard

deviation of the surface height about the local mean value within the first Fresnel

zone (which is examined in the next section).

For GPS L1 signal propagation a plot of ∆H for all propagation angles provides only a

qualitative indication of the likelihood of the resultant scatter being diffuse or

specular, Figure 2.50.

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Surface Roughness Criterion >delta-H

delta

-H:

Sur

face

Irre

gula

rity

(m)


Figure 2.50 — Rayleigh Roughness Criterion

For cases where the surface roughness needs to be accounted for, the effective

specular reflection coefficient, is modified by applying a magnitude reduction factor

to the coefficient. The modified reflection coefficient is then given as [10]:

Γ=Γ ss ρ (2.21)

where;

region of increaseddiffuse scattering

48

−= 2

02

2

1

2

1exp HHs I φφρ (2.22)

is the rough surface specular reduction factor, and 0I the modified Bessel function of

zero order.

A plot of sρ at the GPS L1 frequency for RMS surface roughness heights of 0-10cm

at 2.5cm steps, is given below.

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Rough Surface Scatter

Diff

use

Sca

tter

Red

uctio

n F

acto

r


Figure 2.51 — Rough Surface Reduction Factor

This treatment of rough surface scattering accounts only for the reduction of the

coherent scatter in the specular direction. Generally for a rough surface, the reflected

signal has an additional component that accounts for the nature of the diffuse scatter.

Experimental results indicate that the diffuse component is statistically random with a

Rayleigh distribution. In this work, only the rough surface specular reduction factor

will be considered for implementation of rough surface effects in the developed

propagation model through the use of equation (2.22).

2.5cm

5.0cm7.5cm10cm

0cm

49

2.4 Fresnel Zones

In geometrical optics representations of propagation, specular reflection is considered

to occur from some single geometric point. However, the source illuminates a large

portion of the Earth, and there is contribution from an area of the surface, to the total

reflected signal. In understanding the nature of GPS L1 propagation it is worthwhile

asking the question, ”which regions of the scattering surface most contribute to the

total field at the GPS receiving antenna”.

In Figure 2.52, we consider a smooth plane illuminated by the GPS source at A and

received at the GPS antenna located at B. The interleaving space can be sub-divided

by a family of Fresnel ellipsoids. These ellipsoids have focal points located at A and

B, such that at any point M on any one ellipsoid, the following relation holds [11]:

2

λnABMBAM +=+ (2.23)

Where n is an integer with n=1, defining the first Fresnel ellipsoid and so forth. The

Fresnel zone concept for the case of reflection is such that the path of reflection is

between the source located at A, and an image of the antenna located at B’. This is

illustrated in Figure 2.52.

50

A

B’

B

M

h

y

x

n=1 2 3

Figure 2.52 — Fresnel Zones for Reflection

The ellipses generated in the xy-plane, by the intersection of the Fresnel ellipsoids,

determine the Fresnel zones. The nth zone is defined as the area between the ellipses

obtained from ellipsoids n and n-1 respectively. It is generally accepted that the first

Fresnel zone contributes most to the reflected incident energy, when the reflecting

surface is much larger than the first Fresnel zone.

The defining measurements, radius and semi-major axis, of the first Fresnel zones for

the GPS L1 frequency, as a function of antenna height and propagation angle are

respectively [11]:

θλ

sin

hRF = (2.24)

51

and θsin

FSMA

RL = (2.25)

The following plots give an indication of the dimension of the first Fresnel zone for

antenna heights ranging from 1 to 5 metres.

1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

1st

Zone

Rad

ius

(m)

GPS First Fresnel Zone

1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

300

400

500

Sem

i-maj

or a

xis

(m)

Propagation Angle

1m2m3m4m5m

Figure 2.53 — First Fresnel Zone Dimensions (1-5 degs)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 101

1.5

2

2.5

3

3.5

1st

Zone

Rad

ius

(m)


5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

10

20

30

40

Sem

i-maj

or a

xis

(m)

Propagation Angle


52

10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

1st

Zone

Rad

ius

(m)


10 20 30 40 50 60 70 80 900

5

10

15

Sem

i-maj

or a

xis

(m)

Propagation Angle


The size and location of the first Fresnel zone (for reflection) with relation to the GPS

antenna allows interpretation of the level of multipath that could be expected to result.

When there is a surface that is dimensionally larger than the first zone, we can expect

specular reflection to occur. The nature of the surface determines the level of the

reflection.

53

GPS SV

Fresnel Zones

Obstruction in firstFresnel zone

GPS antenna

Figure 2.56 — Fresnel Zones for LOS

Considering now Fresnel zones for line-of-sight transmission, Figure 2.56, the nature

of the propagation is determined by obstructions within the Fresnel zones. Signal

blocking is apparent if there is an obstructing object larger than the first Fresnel zone.

If there is an obstruction lying within the first Fresnel zone that is not larger than the

zone (but larger in relation to the wavelength) then the expected multipath

propagation behaviour is that of diffraction. Diffraction of the GPS L1 signal is

considered in the next section.

54

2.5 Diffraction

As was seen in the previous sections, specular reflection is the primary multipath

producer. There is however, one other propagation characteristic that can lead to

secondary multipath effects—that of signal diffraction. Whenever a signal encounters

an obstructing object, some of the energy is diffracted at the edges of the object—

effectively bending the signal around the edge. This results in the area behind the

obstructing object not being completely shadowed from the direct signal (as would be

expected from geometric optics), and some residual energy will exist within the

expected shadow region.

Constant Phase Fronts

Diffracted Signal

LOS

Obstruction

Shadow Boundary

Figure 2.57 — Diffraction at Obstruction

The most obvious effect of diffraction on the GPS L1 signal is the shadowing effect of

the obstacle. The expectation is that the signal level will be reduced and the satellite

being tracked will become unusable because of lack of signal strength. From a

multipath perspective however the main effect of diffraction of the GPS L1 signal is

the very presence of a signal in a shadow region. The nature of the signal is such that

55

it cannot have travelled by the direct LOS path, since the GPS antenna LOS is blocked

by the obstacle. The signal must therefore have travelled some additional distance

around the obstruction.

2.5.1 Knife-Edge Diffraction

For diffraction of the GPS L1 frequency we consider the case of knife-edge

diffraction. The geometric parameters that define the knife-edge diffraction problem

are incorporated in a single dimensionless Fresnel diffraction parameter denoted by v.

There are many representations of v depending on the given geometric relationships.

For GPS it is most appropriate to utilise the following geometric description.

LOS

θβ

h

ha

d2

d1

l

da

Constant phase fronts

Figure 2.58 — Diffraction at Obstacle

The angle between the line-of-sight vector (propagation angle θ ) and the diffracted

ray is given by,

θβ −

= −

a

a

d

h1tan (2.26)

56

with

222 aa dhd += (2.27)

and

d d1 2= cosβ (2.28)

In the GPS far field the Fresnel diffraction parameter can then be written as:

v hd

=2

2λ(2.29)

where

h d= 2 sin β (2.30)

For the knife-edge case the signal distance between the LOS and the diffracted path is

seen to be d2-d1. If the dynamic range of the GPS receiver allows tracking of the

diffracted signal then it will have an error in time-of-travel estimation of:

( )it ddc

−= 2

1ε (2.31)

To determine if the receiver can track the signal we need to quantify the expected loss

in the diffraction region. For values of v greater than –1 the loss can be approximated

by [11];

57

( ) ( )vvvJ +++= 1log204.6 2 (2.32)

Which is plotted below in Figure 2.59 for 31 ≥≥ v- .

-1 -0.5 0 0.5 1 1.5 2 2.5 3-25

-20

-15

-10

-5

0

5

J(v)

(db

)

v

Figure 2.59 — Knife-edge Diffraction Gain vs v

As a descriptive example we place a GPS receiver 5 metres from and below a knife-

edge diffractor. The resultant plots of parameter v and the diffraction gain are given

below, Figure 2.60.

10 15 20 25 30 35 40 45 50-2

0

2

4

6

Diff

ract

ion

Par

amet

er (

v)

10 15 20 25 30 35 40 45 50-30

-20

-10

0

Diff

ract

ion

Gai

n (d

b)


Figure 2.60 — Diffraction Parameter and Gain

58

The critical angle at which the GPS signal is expected to be in the clear is 45 degrees.

From the plot the loss at this critical angle is approximately 6dB. In addition the

signal is not in the clear (that is at 0dB or line-of-sight) until approximately 50

degrees.

If the receiver’s dynamic range is such that it can track signals down 30 dB then the

diffracted signal could be tracked (for a satellite that is setting). The LOS time-of-

travel error for tracking this example diffracted signal is given below in Figure 2.61.

10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Err

or (

ns)

Propagation Angle (Degs)

Figure 2.61 — Time of Arrival Error

This time error is low but for this example the estimate of range is in error by a

maximum of 1.2 metres. In a stand-alone application of GPS this range error would

not have a influential effect on the total position solution. It would however be a

substantial range error for DGPS applications.

59

2.6 GPS Fading Signal Characteristics

2.6.1 Signal Fade Envelopes

Having established a set of coupled reflection coefficients for the GPS L1 signal for

the three primary modes (F, BA, BB) of specular multipath propagation, it is now

possible to consider the effect of these selected modes on the received GPS signal

strength.

The total electric field above the earth, due to the interference of the direct LOS signal

with a single reflected signal component, is termed the interference region [12]. The

interference region field can be written as,

( ) ( ) ( )E = + − +E f E f ei ij k R

0 0θ ρ θ φ∆ (2.33)

where f(θi) and f(θr) represent the field distribution, as a function of the angle of

incidence, and angle of reflection respectively. In terrestrial propagation these factors

would account for the transmission antenna’s radiation pattern. For the GPS satellite

propagation problem, these pattern shape terms are not required—as at the Earth’s

surface the far-field is essentially uniform plane-wave only.

In equation (2.33), the total phase retardation of the reflected component is the sum of

the phase shift upon reflection (φ), and the phase shift due to the additional path

length (∆R) taken by the multipath signal—which is simply k∆R.

The concept of the interference region provides a visualisation of the two-ray

problem. It is obvious that the fading pattern of the signal will be bounded by the in-

phase and anti-phase limits. The normalised received signal strength envelopes of the

GPS L1 signal with one dominant reflection can then be written as:

60

( ) ( )Γ+→Γ− 1log201log20 (2.34)

Considering first F-mode—a single forward reflection. Plots (Figure 2.62 to Figure

2.67)of the signal strength fade envelopes reveal the bounds of the fading received

signal strength for this mode of GPS L1 multipath affected signal.

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)


6dB LHCP Rejection 12dB LHCP Rejection18dB LHCP Rejection

Figure 2.62 — Concrete F-mode

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.63 — Dry Ground F-mode

61

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Medium Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.64 — Medium Dry Ground F-mode

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Wet Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.65 — Wet Ground F-mode

62

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Fresh Water

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.66 — Fresh Water F-mode

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Sea Water

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.67 — Sea Water F-mode

From the above plots it is apparent that for the F-mode, single forward-scatter

multipath signal, the effect on the received signal strength is more conspicuous at

lower angles of propagation.

For the BA-mode case, the fade envelopes are transposed in propagation angle and it

is obvious that the effect on the signal strength envelope is opposite. That is, deeper

63

fades of the signal strength occur at higher propagation angles. This is evident in

Figure 2.68 given as an example of the mode BA fade envelope for a concrete back

reflector.

0 10 20 30 40 50 60 70 80 90-50

-40

-30

-20

-10

0

10Fade Envelopes: Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.68 — Concrete BA-mode

The fade envelopes for all other media in BA-mode can be read directly as the

propagation angle transpose of the F-mode plots, and are not repeated here.

For mode BB the total reflection coefficient of the multipath signal is the product of

the ground and back reflector individual modified coupled reflection coefficients. The

four plots shown below indicate the expected signal envelopes for the case of the

ground and back-reflector being of the same medium (hence the symmetry).

64

0 10 20 30 40 50 60 70 80 90-1.5

-1

-0.5

0

0.5

1

1.5BB-Mode Fade Envelope: Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)


6 LHCP Rejection12 LHCP rejection18 LHCP rejection

Figure 2.69 — Concrete2 BB-mode

0 10 20 30 40 50 60 70 80 90-2

-1.5

-1

-0.5

0

0.5

1

1.5BB-Mode Fade Envelope: Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.70 — Dry Ground2 BB-mode

65

0 10 20 30 40 50 60 70 80 90-3

-2

-1

0

1

2

3BB-Mode Fade Envelope: Medium Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.71 — Medium Dry Ground 2 BB-mode

0 10 20 30 40 50 60 70 80 90-8

-6

-4

-2

0

2

4BB-Mode Fade Envelope: Wet Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.72 — Wet Ground2 BB-mode

From these plots the fading of the signal strength due to the double reflection of the

BB-mode multipath is greatest at 45 degrees. For the representative media given here,

the fades for concrete are unlikely to exceed 1 dB, for dry ground 1.5 dB, for medium

dry ground 2 dB, and finally for wet ground 6 dB. In general the fading expected due

to BB-mode multipath is not as large as the fades for F and BA mode multipath.

However the fading remains fairly constant across a large range of propagation angles,

66

whereas the fading for the other two modes is dominant at the extremes of the

propagation incident angle range. Additional BB-mode fade envelopes are included

here for completeness. They indicate the variation of the two propagation media.

0 10 20 30 40 50 60 70 80 90-5

-4

-3

-2

-1

0

1

2

3Fade Envelopes: Sea Water-Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.73 — Concrete to Sea Water

0 10 20 30 40 50 60 70 80 90-5

-4

-3

-2

-1

0

1

2

3Fade Envelopes: Sea Water-Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.74 — Dry Ground to Sea Water

67

0 10 20 30 40 50 60 70 80 90-5

-4

-3

-2

-1

0

1

2

3Fade Envelopes: Sea Water-Medium Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.75 — Medium Dry Ground to Sea Water

0 10 20 30 40 50 60 70 80 90-5

-4

-3

-2

-1

0

1

2

3Fade Envelopes: Fresh Water-Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.76 — Concrete to Fresh Water

68

0 10 20 30 40 50 60 70 80 90-5

-4

-3

-2

-1

0

1

2

3Fade Envelopes: Fresh Water-Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.77 — Dry Ground to Fresh Water

0 10 20 30 40 50 60 70 80 90-5

-4

-3

-2

-1

0

1

2

3Fade Envelopes: Fresh Water-Medium Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.78 — Medium Dry Ground to Sea Water

69

0 10 20 30 40 50 60 70 80 90-4

-3

-2

-1

0

1

2

3Fade Envelopes: Wet Ground-Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.79 — Wet Ground to Concrete

0 10 20 30 40 50 60 70 80 90-4

-3

-2

-1

0

1

2

3Fade Envelopes: Wet Ground-Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.80 — Wet Ground to Dry Ground

70

0 10 20 30 40 50 60 70 80 90-4

-3

-2

-1

0

1

2

3Fade Envelopes: Wet Ground-Medium Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.81 — Wet Ground to Medium Dry Ground

0 10 20 30 40 50 60 70 80 90-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Fade Envelopes: Medium Dry Ground-Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.82 — Medium DryGround to Concrete

71

0 10 20 30 40 50 60 70 80 90-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Fade Envelopes: Medium Dry Ground-Dry Ground

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.83 — Medium Dry Ground to Dry Ground

0 10 20 30 40 50 60 70 80 90-2

-1.5

-1

-0.5

0

0.5

1

1.5Fade Envelopes: Dry Ground-Concrete

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)



Figure 2.84 — Dry Ground to Concrete

In this section the bounds of the GPS signal fade pattern have been examined for the

three primary modes of multipath propagation. In the next section the nature of the

fading pattern within the envelope bounds is examined.

72

2.6.2 Signal Fading Characteristics for Specular Reflection

For the case of m specular multipath signals (ignoring receiver antenna gain pattern),

the received signal strength at the GPS receiver can be written as ;

( )

+= ∑ +

m

i

ji

j iLOSLOS eeAS ψφφ ρ0 (2.35)

where for the ith multipath signal, ρi is the magnitude of the reflection coefficient, ψi

is the total phase retardation due to the reflection and subsequent additional path

distance travelled, and φLOS is the phase of the LOS signal.

We now consider the normalised case of a single forward multipath signal. From

Table 2.2 the nature of the total received signal strength is a function of the reflection

coefficient amplitude, the height of the GPS antenna, and the propagation angle from

the satellite.

We consider first a linear variation of the propagation angle from 1-90 degrees in 900

measurement cycles (epochs). The height of the receiving antenna is set at 0.5 meters

with Multipath to LOS Ratio (MPR) of 1 (ie the multipath signal is of equal amplitude

to the LOS signal). The reflection coefficient amplitude has not been considered for

this descriptive example.

73

0 100 200 300 400 500 600 700 800 900-40

-30

-20

-10

0

10H=0.5: MPR=1.0

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100A

ngle

(de

gs)

Measurement Epochs

Figure 2.85 — Linear Variation of Propagation Angle (Fwd)

The gradient of the linear variation in the propagation angle is now reduced by half

such that in the same 900 epochs the propagation angle varies from 0-45 degrees.

0 100 200 300 400 500 600 700 800 900-40

-30

-20

-10

0

10H=0.5: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

Ang

le (

degs

)

Measurement Epochs

Figure 2.86 — Linear Variation (Reduced gradient)

Note that the peaks of the fading pattern peak at 6 dB—since the direct plus multipath

add up to twice the strength of the direct signal only. From these plots we also note

that the periodicity of the fading is a function of the rate-of-change of propagation

74

angle. This is expected, since the rate-of-change in phase of the single multipath

signal, as a function of the rate-of-change in propagation angle, is given by:

dt

dh

dt

d θθλπψ

= cos

4(2.36)

This equation explains one aspect of the fading pattern—in that the phase-rate varies

as the cosine of the propagation angle. So for a given positive rate-of-change of

propagation angle the fade separation increases as the rate-of-change of the multipath

phase decreases.

It is also apparent from equation (2.36) that the phase rate is also directly related to

the height of the antenna above the reflecting surface. This can be seen in Figure 2.87

where for a given rate-of-change in propagation angle the fading periodicity increases

directly as a function of antenna height.

0 10 20 30 40 50 60 70 80 90-40

-35

-30

-25

-20

-15

-10

-5

0

5

10H=0.2: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

Propagation Angle (degs)0 10 20 30 40 50 60 70 80 90

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10H=0.5: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)


0 10 20 30 40 50 60 70 80 90-40

-35

-30

-25

-20

-15

-10

-5

0

5

10H=1: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

Propagation Angle (degs)0 10 20 30 40 50 60 70 80 90

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10H=2: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)


Figure 2.87 — Variation of Antenna Height

75

For the case of BA-mode, backscatter from above, the phase-rate is given by

dt

dx

dt

d θθλπψ

−= sin

4(2.37)

Here the phase rate-of-change varies as the negative sine of the propagation angle. In

a similar fashion as for the forward scatter case we can infer that for a given positive

rate-of-change of propagation angle the fade separation now decreases as the rate-of-

change of the multipath phase increases. This is illustrated in Figure 2.88.

0 100 200 300 400 500 600 700 800 900-40

-30

-20

-10

0

10X=0.5: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

Measurement Epochs

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

Ang

le (

degs

)

Measurement Epochs

Figure 2.88 — Backscatter from above

Comparing Figure 2.85 with Figure 2.88 for the same variation of propagation angle it

is possible (for these two cases) to infer where the multipath signal has originated

from.

Similarly for BB-mode, the backscatter from below, the phase rate-of-change is given

as:

76

( )dt

dxh

dt

d θθθλπψ

sincos4 −= (2.38)

The resultant fading pattern for an antenna located 3m from the ground and 3m from

the backscatter interface is shown in Figure 2.89.

0 100 200 300 400 500 600 700 800 900-40

-30

-20

-10

0

10X=3 H=3: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

Measurement Epochs

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

Ang

le (

degs

)

Measurement Epochs

Figure 2.89 — Equal Antenna Distance and Height

For equal height and distance, and with linear variation of propagation angle, we note

that the first 450 epochs are dominated by the variation due to the antenna height. The

cosine variation being apparent. The final 450 epochs are dominated by the distance

from the backscatter interface with its sine variation. This dominant variation depends

solely on the relevant distances. Decreasing the distance to the backscatter interface

will make the antenna height the predominant factor in the fading pattern, Figure 2.90.

77

0 100 200 300 400 500 600 700 800 900-40

-30

-20

-10

0

10X=1 H=3: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)Measurement Epochs

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100A

ngle

(de

gs)

Measurement Epochs

Figure 2.90 — Antenna Height > Distance

Likewise, if the antenna height is reduced relative to the distance to the backscatter

interface this distance factor has increased influence on the resultant fade pattern,

Figure 2.91.

0 100 200 300 400 500 600 700 800 900-40

-30

-20

-10

0

10X=3 H=1: MPR=1

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

Measurement Epochs

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

Ang

le (

degs

)

Measurement Epochs

Figure 2.91 — Antenna Height < Distance

Having examined these three cases, it is apparent that individually it would be

possible to infer the nature of the reflection from the resultant fade pattern. However

78

the true nature of the multipath environment may be such that all three of these basic

multipath propagation modes exist at the same time and are received by the GPS

antenna. The increasing complexity of the resultant signal strength fading pattern is

obvious as more multipath modes are introduced, Figure 2.92.

0 100 200 300 400 500 600 700 800 900-40

-20

0

X=3 H=3: MPR=1

dB

0 100 200 300 400 500 600 700 800 900-40

-20

0

dB

0 100 200 300 400 500 600 700 800 900-40

-20

0

dB

Measurement Epochs

Figure 2.92 — Addition of Multipath Modes

In the first plot the resultant signal fading pattern is due only to mode F—a single

forward multipath signal. The next plot is the addition of modes F and BA—the

backscatter from above multipath with the forward multipath. Lastly the backscatter

from below (mode BB) is also included. As can be seen the variations that were

apparent for the individual cases are no longer recognisable.

Having examined the periodicity of the fading, we now consider the effect on the

received signal strength of variation in the MPR. For the forward scatter case, the

resultant variation of signal strength fade depth is shown in Figure 2.93, for an

antenna height of 0.5 metres.

79

0 100 200 300 400 500 600 700 800 900-40

-35

-30

-25

-20

-15

-10

-5

0

5

10Variation of Relative MP Amplitude (H=0.5)

Nor

mal

ised

Sig

nal S

tren

gth

(dB

)

Measurement Epochs

00.20.40.60.8 1

Figure 2.93 — Variation of Relative Multipath Amplitude

From a generalised viewpoint of signal reception the critical point of note is the effect

of the signal fades on the receiver’s ability to search for or track the received GPS

signal. The signal-to-noise ratio at deep fades may be so low that in the search mode

the receiver cannot lock onto the desired signal, and if it is tracking a signal it may

lose lock for a period of time. All of these effects are undesirable.

The exact nature of the received signal strength for the three primary multipath modes

is the product of the appropriate fade envelope as given in Section 2.6.1 and the

fading patterns examined in this section.

80

2.7 GPS Receiver Context

Having examined the nature of multipath propagation of the GPS L1 signal it is

necessary to consider these findings in the context of the location and limits of the

GPS receiver.

2.7.1 Aspects of Physical Antenna Location

In the previous sections the nature of GPS multipath has been examined. It is apparent

from the three primary modes of specular reflection that the relative delay of the

multipath signal depends solely on the physical relationship of the reflector to the

antenna. For the GPS SPS with a standard ½-chip correlator spacing, signals delayed

by more than 1.5 chips (1466 nanoseconds) are effectively decorrelated and as such

have no impact on the accuracy of the range measurement [1, 13, 14]. For the three

primary modes of specular reflection this decorrelation limit provides a basis for

bounding the physical relationship of the antenna to the terrain within the multipath

environment.

Considering F-mode multipath, the height bound for decorrelation of the multipath

signal can be derived from Table 2.2 and is written approximately as:

θsin

220=Fh (2.39)

This height represents the height of the antenna above a region of terrain that contains

a relatively flat area larger than the first Fresnel zone. The intervening terrain

introduces only blockage and diffractive effects and the multipath is not otherwise

constrained by the physical distance to the area of reflection. Equation (2.39) shows

that the minimum height for decorrelation of the multipath signal occurs at 90 degrees

(zenith) where the required height for decorrelation is 220 metres or higher. For most

land-based applications of GPS, the antenna height would be well below this level. At

81

lower elevation angles the decorrelation height increases markedly and the likelihood

of decorrelation decreases dramatically. It is apparent then that F-mode multipath is

likely to introduce range errors at practical antenna heights, ignoring receiver antenna

gain pattern effects.

For a narrow-correlator with 1/10-chip spacing, signals delayed by more than 1.1

chips (1075 nanoseconds) are effectively decorrelated [15, 16]. The height bound for

decorrelation of the multipath signal in this case is given approximately as:

θsin

160=Fh (2.40)

In this case the minimum decorrelating antenna height is 160 metres.

Similarly for the BA-mode the relative delay is dependant upon the distance of the

antenna from the back-reflector. For a standard correlator receiver, the bound of

decorrelation distance is given approximately by:

θcos

220=BAx (2.41)

For a back-reflector of height Hb, the distance bound must satisfy the inequality

θtan0 BAb xHh −≤≤ (2.42)

In most land-based applications the height of the back-reflector is unlikely to exceed

30 metres. This height represents the height of a typical tall urban structure but is

arbitrary in choice and is selected from a purely practical viewpoint. This results in

values of decorrelation distance from the back-reflector and corresponding height

given in Figure 2.94.

82

220 220.2 220.4 220.6 220.8 221 221.2 221.4 221.6 221.8 2220

5

10

15

20

25

30BA-Mode Decorrelation Bound

Ant

enna

Hei

ght

(m)

Distance (m)

0 degs

1 degs

2 degs

3 degs

4 degs

5 degs

6 degs

7 degs

0 degs

1 degs

2 degs

3 degs

4 degs

5 degs

6 degs

7 degs

Figure 2.94 — BA-mode Decorrelation Distance/Height Bound

The implication of these results—for an arbitrary 30 metre high back reflector—is that

if the antenna is located more than the required bounded distance from the back-

reflector, then the multipath is either decorrelated or non-existent (as far as the GPS

receiver is concerned).

For the signal to exist the antenna must be located below the bound shown, for the

corresponding decorrelation distance. If the antenna is above this height then it is not

located within the back-reflected signal region and is not received. At propagation

angles greater than 8 degrees decorrelation will not occur, since the maximum

distance for the multipath—relative to the existence (or shadow) boundary, Figure

2.95— can never exceed the decorrelation distance. For the narrow-correlator receiver

the equivalent decorrelation distance limits are from 160 metres at 0 degrees to 163

metres at 10.5 degrees.

The actual spatial region of existence for the BA-mode multipath signal is shown

below in Figure 2.95.

83

Hb

ExistenceBoundary

θ

Figure 2.95 — BA-mode Existence-Boundary

At all propagation angles the BA-mode multipath signal exists only within the lower

region (ignoring diffractive effects) defined by the existence-boundary. The relevant

distances related to the limit of the back-reflector height (30 metres) and the

propagation angle is illustrated in Figure 2.96 below.

0 20 40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30BA-Mode Existence Boundary

Bac

k R

efle

ctor

(m

)

Distance (m)

5 degs20 degs35 degs60 degs75 degs80 degs

Figure 2.96 — BA-mode Existence Region Metrics

It is obvious, that to avoid BA-mode multipath, the antenna needs to as far as

practicable from any vertical interface, and that the multipath mode is more evident at

low propagation angles.

84

For the BB-mode multipath the existence region is shown in Figure 2.97.

Hb

UpperExistenceBoundary

θ

LowerExistenceBoundary

Zone 2

Zone 1

Figure 2.97 — BB-mode Existence Region

This region is defined by an upper and a lower boundary, which bounds the antenna

height for a given distance from the back-reflector. In addition the region is defined by

two distinct zones related to the process of the generation of the BB-mode multipath

signal (see Section 2.2.3). For the lower height bound the decorrelation distance is

given by

θcos220=BBLx (2.43)

Likewise, the upper height bound limits the decorrelation distance as:

( ) θθ cossin220 bBBU Hx −= (2.44)

85

The height of the antenna as a function of distance from the back-reflector must

satisfy

bBBUBBL Hxhx +≤≤ θθ tantan (2.45)

for the BB-mode multipath signal to be received at the antenna. Applying the bounds

given in equation (2.45) the resultant decorrelation distances are shown in Figure

2.98.

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250BB-Mode Decorrelation Distance

Dis

tanc

e (m

)


At Lower Height BoundAt Upper Height Bound

Figure 2.98 — BB-Mode Decorrelation Distance

The corresponding height bounds are shown below in Figure 2.99.

86

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250BB-Mode Decorrelation Height

Hei

ght

(m)


At Lower BoundAt Upper Bound

Figure 2.99 — BB-mode Antenna Height Bounds for Decorrelation

2.8 Path Loss in the Terrestrial Domain

In general the free space path loss in a radio link is specified by the Friss transmission

formula given by [12],

=

2log10

rP

P

t

r α(2.46)

where α is a function of the frequency and the effective apertures of the transmit and

receive antennas, and r is the separation distance.

In modelling the propagation of GPS in a terrestrial solution domain we need to know

if free space transmission loss is required in the implementation. The GPS satellite is

located approximately 20000 km from the receiving antenna. In addition the

minimum received signal strength specified for the GPS L1 signal is –160 dBw. By

equating these values an estimate of additional path loss can be made, Figure 2.100.

87

0 100 200 300 400 500 600-160.3

-160.25

-160.2

-160.15

-160.1

-160.05

-160

-159.95

Min

imum

Sig

nal L

evel

(dB

w)

Additional Path Distance (km)

Path Loss - Over 20,000km

Figure 2.100 — Additional Path Loss

We see that for additional path distances of 100 km the additional path loss is less

than 0.05 dB. In terms of the physical dimension of a model solution domain the

exclusion of path loss in the model would result in very small errors in the estimate of

the received signal strength. In general the solution domain would not exceed a few

kilometres and the path loss variation over these distances would be overwhelmed by

actual variation of the incident received field strength being modelled.

From a multipath perspective the additional path loss due to the additional distance

travelled by the multipath signal, relative to the LOS path is inconsequential.

2.9 Summary

In this chapter the nature of multipath propagation of the GPS L1 signal has been

examined. The theory of specular reflection, rough surface effects, Fresnel zones, and

diffraction have been applied directly to aid in describing the expected propagation

behaviour of the GPS L1 signal.

Considering specular reflection, an effective coupled CP reflection coefficient was

developed that combines the nature of the reflected RHCP GPS signal with the

88

rejection capability of the GPS antenna. By integrated these parameters, a single

effective circular polarisation reflection coefficient, can be used to describe the nature

of the specular reflection. From the viewpoint of modelling GPS signal propagation,

the necessary constraint of modelling two separate polarisation modes of propagation

has been removed. The adaptation of a single-mode propagation model is thus more

readily achievable.

Three primary propagation modes for specular multipath were introduced. The F-

mode, which describes the specular reflection of the GPS signal from the forward

direction over the ground. The BA-mode, which describes specular reflection from a

vertical interface located behind the GPS antenna, resulting in a back-reflected signal

arriving from above the horizontal plane containing the antenna. Lastly the BB-mode

that describes a double reflected signal that includes a forward and a backward

component of specular reflection, arriving from below the horizontal plane containing

the antenna.

The individual nature of these three primary multipath modes were examined in

relation to the limit of signal fading, as shown in the fade envelopes for the received

signal strength. The fading nature of the modes were also examined from the

perspective of phase-rate of change of the received GPS signal.

The effect of rough surfaces on the reflection of the GPS L1 signal was examined. In

general, rough surfaces result in a reduction of the specular reflection. The rough

surface reduction factor was derived for the GPS L1 signal. In considering GPS

multipath propagation, rough surface effects are considered desirable as they limit the

effect of specular reflection.

Fresnel zone descriptions were derived for the GPS L1 signal. These descriptions

provide a geometric interpretation of the spatial metrics of the physical terrain in

which the GPS L1 signal is propagating. The requirements for specular reflection,

signal blockage and diffraction were given physical meaning. Diffraction of the GPS

L1 signal was investigated and the nature of this multipath examined.

89

Finally the nature of the three modes of multipath propagation were examined within

the context of the physical relationship of the GPS antenna with the terrain, and the

effect of decorrelation of the multipath signals. It was found that in general F-mode

multipath is unlikely to be decorrelated. That BA-mode multipath has a very confined

region of influence, spatially and in terms of propagation angle range. For this mode it

was found that as long as the antenna was located more than approximately 220

metres from the reflecting interface then the multipath mode either did not exist or

was effectively decorrelated. For BB-mode the region of influence is much greater

due to the large region of existence of this multipath mode. However the double-

reflected nature of the BB-mode multipath limits it’s ultimate effect on the GPS

receiver.

These propagation conditions form the theoretical platform for the development of a

model tailored to the propagation of the GPS L1 signal.

90

2.10 References

[1] M. S. Braasch, "On the Characteristics of Multipath Errors in Satellite-Based

Precision Approach and Landing Systems," in Department of Electrical and

Computer Engineering. Athens: Ohio University, 1992, pp. 203.

[2] R. D. J. van Nee, "Multipath Effects on GPS Code Phase Measurements,"

Navigation: Journal of The Institute of Navigation, vol. 39, pp. 177-190, 1992.

[3] H. R. Reed and C. M. Russell, Ultra High Frequency Propagation, Second ed.

New York: Chapman and Hall, 1966.

[4] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in

Communication Electronics. New York: John Wiley & Sons, 1984.

[5] H. Bremmer, Terrestrial Radio Waves: Theory of Propagation. London:

Elsevier Publishing Company, 1949.

[6] ITU-R, "Electrical Characteristics of the Surface of the Earth,"

Recommendation Rec. 527-3, 1992.

[7] W. L. Flock, "Propagation Effects on Satellite Systems at Frequencies Below

10 GHz: A Handbook for Satellite Systems Design," NASA December 1987.

[8] P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves

From Rough Surfaces. Norwood: Artech House, 1987.

[9] D. E. Kerr, "Propagation of Short Radio Waves." Boston: Boston Technical

Publishers, 1964.

[10] CCIR International Radio Consultive Commitee, "Report 1008 Reflection

from the Surface of the Earth," International Telecommunications Union,

Dubrovnik 1986.

[11] CCIR International Radio Consultive Commitee, "Report 715-2 Propagation

by Diffraction," International Telecommunications Union, Dubrovnik 1986.

[12] J. Doble, Introduction to Radio Propagation for Fixed and Mobile

Communications. Boston: Artech House, 1996.

[13] M. S. Braasch, "Isolation of GPS Multipath and Receiver Tracking Errors,"

Navigation: Journal of The Institute of Navigation, vol. 41, pp. 415-434, 1994.

[14] M. S. Braasch, "GPS and DGPS Multipath Effects and Modeling," in ION

GPS-95 Tutorial: Navtech Seminars, 1995.

91

[15] A. J. van Dierendonck, P. Fenton, and T. Ford, "Theory and Performance of

Narrow Correlator Spacing in a GPS Receiver," presented at The Institute of

Navigation National Technical Meeting, San Diego, CA, 1992.

[16] A. Montalvo and A. Brown, "A Comparison of Three Multipath Mitigation

Approaches for GPS Receivers," presented at 8th International Technical

Meeting of The Satellite Division of The Institute of Navigation., Palm

Springs, California, 1995.

92


93

Chapter 3 Overview of Propagation Modelling

This Chapter introduces some of the more popular techniques that can be applied to

solving electromagnetic problems. The requirements for modelling GPS signal

propagation introduces some restrictions on which models are appropriate. In

particular the size of the domain in which the modelling takes place is of primary

importance for modelling GPS signal propagation. The size of the domain will be of

the order of several tens of metres up to hundreds of metres, in both height and range.

In characterising multipath propagation the relative delay of signals needs to be

accurately modelled. With time-domain based models the time information is

inherent, but for frequency-domain models a Fourier-synthesis technique, for deriving

time-domain information, needs to be considered. In this Chapter the applicability of

various methods is compared, for the modelling of GPS signal propagation.

3.1 Review of Maxwell’s Equations

This section provides a brief review of Maxwell’s equations of electromagnetics and

the development of the Helmholtz wave equation [1, 2]. As electromagnetic problems

require the solution or approximate (numerical) solution of these equations it is

appropriate that they be reviewed here. The development is started by defining the

following parameters for a linear, isotropic medium.

B = µ H

D = ε E

J = σ E

Here µ is the permeability, ε is the permittivity, and σ is the conductivity of the

medium. The vector components are:

D — Electric Flux Density ( Displacement current)

94

B — Magnetic Flux Density

E — Electric Field Intensity

H — Magnetic Field Intensity

J — Current Density

Maxwell’s equations in differential form are given by:

∇⋅ D = ρ (3.1)

∇ ⋅ B = ∇ ⋅H = 0 (3.2)

∇ × = − = −EB H∂

∂ µ∂∂t t

(3.3)

∇ × = + = +H JD

EE∂

∂ σ ε∂∂t t

(3.4)

and in integral form by:

D dS⋅ =∫ ∫S VdVρ (3.5)

B dS⋅ =∫S0 (3.6)

E dl B dS⋅ = − ⋅∫ ∫∂

∂ t S(3.7)

H dl J dS D dS⋅ = ⋅ + ⋅∫∫ ∫S St

∂∂

(3.8)

The development of the Helmholtz wave equation is started by using the vector

identity, ( )∇ × ∇ × = ∇ ∇⋅ − ∇A A A2 . The curl equation in H can then be rewritten as:

95

( )∇ × ∇ × = ∇ ∇⋅ − ∇H H H2

now substituting (3.2) and (3.4) gives

( )− ∇ = ∇ × +

= ∇ × +

2H E

EEσ ε

∂∂

σ ε∂∂t t

and substituting (3.3) results in

∇ = +22

2HH H

µσ∂∂

µε∂∂t t

(3.9)

Similarly for the case of E in a source-free region (∇ ⋅ E = 0 )

∇ = +22

2EE E

µσ∂∂

µε∂∂t t

(3.10)

The differential forms for the time-periodic case ( e j tω time dependence) are reduced

by replacing ∂∂ t

with jω and ∂∂

2

2t with −ω 2 giving the vector wave equations,

∇ =2 2H Hγ and ∇ =2 2E Eγ (3.11)

whereγ µσ ω µεω α β= − = +j j2

This is known as the propagation constant with real part α, the attenuation factor, and

imaginary part β, the phase shift constant. Now withγ = jk the three-dimensional

Helmholtz’s wave equations for phasor fields are given by:

∇2E + k2E = 0 and ∇2H + k 2H = 0 (3.12)

96

3.2 Overview of Computational Electromagnetics Techniques

Computational techniques have revolutionised the way in which electromagnetic

problems are analysed. Electrical engineers rely on efficient and accurate computer

models to analyse and evaluate electromagnetic behaviour of antennas, propagation,

scattering and component designs.

Although most electromagnetic problems ultimately involve solving only one or two

partial differential equations subject to boundary constraints, very few practical

problems can be solved without computer-based methods [3]. Computational

electromagnetics involves the development of numerical algorithms for the solution of

Maxwell’s equations, and their subsequent use in analysing electromagnetic problems.

Whereby, analytical techniques make simplifying assumptions about the geometry of

the problem in order to apply a closed-form solution, numerical techniques attempt to

solve the fundamental field equations directly, subject to the boundary constraints

posed by the geometry.

Numerical techniques generally require more computation than analytical techniques

but they are very powerful EM analysis tools. Without making a priori assumptions

about which field interactions are most significant, numerical techniques analyse the

entire geometry, and calculate the solution based on a full-wave analysis [4]. A

number of different numerical techniques for solving electromagnetic problems are

available. The following sections provide an introduction to several of the most

widely used techniques in computational electromagnetics.

3.2.1 Finite-Elements Technique

An increasing availability of computer resources coupled with a desire to model more

complex electromagnetic problems has resulted in a wave of renewed interest in

Finite Element (FE) methods for solving EM radiation problems. FE techniques [3-8]

require the entire domain to be divided into several sub-domains or elements. Certain

97

boundary conditions are satisfied at the boundaries and each element may have

completely different material properties from those of neighbouring elements.

The first step in FE analysis is to divide the configuration into a number of small

elements, the selection of the element shape is based upon the geometry, material

constants, excitations and boundary constraints of the problem. In each finite element,

a simple variation of the field quantity is assumed. The corners of the elements are

known as nodes. The basis of the FE analysis is to determine the field quantities at the

nodes.

Most FE methods use variational techniques, minimising or maximising an

expression that is known to be stationary about the true solution. Generally, finite-

element analysis techniques solve for the unknown field quantities by minimising an

energy functional. The energy functional is an expression describing all the energy

associated with the configuration being analysed. For 2-D, time-harmonic problems

this functional is a surface integral while for 3-D problems, it can be represented as a

volume integral,

FH E J E

= + −⋅

∫µ ε

ω

2 2

2 2 2v jdv (3.13)

The first two terms in the integrand represent the energy stored in the magnetic and

electric fields and the third term is the energy dissipated (or supplied) by conduction

currents. Expressing H in terms of E and setting the derivative of this functional with

respect to E equal to zero, an equation of the form f(J, E) = 0 is obtained. An

approximation(k-th order) of this function is then applied at each of the N nodes and

boundary conditions are satisfied, resulting in a system of equations,

J

J

J

y y y

y y y

y y y

E

E

En

n

n

n n nn n

1

2

11 12 1

21 22 2

1 2

1

2

M

K

K

M M OM

K

M

=

(3.14)

98

The values of J in Equation (3.14) are referred to as the source terms. They represent

the known excitations. The elements of the Y-matrix are functions of the problem

geometry and boundary constraints. Since each element only interacts with elements

in its own neighbourhood, the Y-matrix is generally sparse. The terms of the vector E

represent the unknown electric field at each node. These values are obtained by

solving the system of equations.

In general, finite element techniques are appropriate for modelling complex

inhomogeneous configurations. However, they do not model unbounded radiation

problems as effectively as other techniques. The major advantage that finite element

methods have over other EM modelling techniques stems from the fact that the

electrical and geometric properties of each element can be defined independently.

This permits the problem to be set up with a large number of small elements in

regions of complex geometry and fewer, larger elements in relatively open regions.

Thus it is possible to model configurations that have complicated geometry’s and

many arbitrarily shaped dielectric regions in a relatively efficient manner

3.2.2 Finite-Difference Time-Domain Technique

Finite difference time domain (FD-TD) techniques [3-6, 8] require the entire domain

volume to be meshed. Normally, this mesh must be uniform, so that the mesh density

is determined by the smallest detail of the configuration. The FD-TD is, as the name

suggests, a time-domain technique well-suited to transient analysis problems. The

technique is best explained by the numerical FD-TD solution of the one-dimensional

scalar wave equation which is given by:

∂∂

∂∂

2

22

2

2

u

tc

u

x= (3.15)

where ( )u u x t= , .

Now defining a function of the form u x t F x ct G x ct( , ) ( ) ( )= + + − , where F and G

are arbitrary, and differentiating twice with respect to t and x provides

99

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

∂∂

∂∂

∂∂

∂∂

u

t

dF x ct

d x ct

x ct

t

dG x ct

d x ct

x ct

tcF x ct cG x ct

u

tc F x ct c c G x ct c F x ct c G x ct

= ++

⋅+

+ −−

⋅−

= ′ + − ′ −

= ′′ + − − ′′ − = ′′ + + ′′ −

( )

( )

( )

( )2

22 2 2

and

( ) ( ) ( ) ( )

( ) ( )

∂∂

∂∂

∂∂

∂∂

u

x

dF x ct

d x ct

x ct

x

dG x ct

d x ct

x ct

xF x ct G x ct

u

xF x ct G x ct

= ++

⋅+

+ −−

⋅−

= ′ + + ′ −

= ′′ + + ′′ −

( )

( )

( )

( )2

2

Substituting these equations into the equation (3.15) gives

c2 ′ ′ F x + ct( ) + c2 ′ ′ G x − ct( ) = c2 ′ ′ F x + ct( )+ ′ ′ G x − ct( )[ ]

F and G are known as propagating wave solutions and the identity holds regardless of

their choice. After some time, ∆t, the wave solution F must have moved to the left (-x

direction) since the argument of F has increased by c∆t, the spatial part of the

argument has to correspondingly decrease by c∆t. The converse is true for G and

therefore F(x+ct) and G(x-ct) are leftward and rightward travelling waves

respectively. The factor c represents the wave velocity in the ±x direction. A Taylor’s

series expansion of u x tn( , ) for a fixed tn , about the space point xk to ( )x xk + ∆ gives:

u x + ∆x( )tn

= u x k ,t n+ ∆x ⋅

∂u

∂x x k ,tn

+∆x 2

2⋅∂ 2u

∂x2x k ,tn

+∆x 3

6⋅∂3u

∂x3xk ,t n

+∆x4

24⋅

∂4u

∂x4ξ1,t n

The last term is an error term where ξ1 is a space point located somewhere in the

interval ( )x x xk k, + ∆ . Now for the Taylor’s series expansion of u x tn( , ) for a fixed tn ,

about the space point xk to ( )x xk − ∆ .

100

( )u x x u xu

x

x u

x

x u

x

x u

xt x tx t x t x t t

nk n

k n k n k n n

+ = − ⋅ + ⋅ − ⋅ + ⋅∆ ∆ ∆ ∆ ∆,

, , , ,

∂∂

∂∂

∂∂

∂∂ ξ

2 2

2

3 3

3

4 4

42 6 242

Here ξ 2 in the error term represents a space point in the interval ( )x x xk k, − ∆ .

Adding these two expressions results in

( ) ( )u x x u x x u xu

x

x u

xt x tx t t

nk n

k n n

+ + − = + ⋅ + ⋅∆ ∆ ∆ ∆2

122

2

2

4 4

4

3

,, ,

∂∂

∂∂ ξ

With ξ 3 a space point in the new interval ( )x x x xk k− +∆ ∆, . Rearranging to give an

expression for the second derivative

( ) ( ) ( )( )

( )[ ]∂∂

β2

2 2

22u

x

u x x u x u x x

xx

x t

k

tk nn

,

=+ − + −

+∆ ∆

∆∆ (3.16)

Equation (3.16) is a second-order, central-difference approximation to the second

partial space derivative of u, with β representing the error term. Using subscript k for

spatial position and superscript n for observation epoch gives

( )( )[ ]∂

∂β

2

21 1

2

22u

x

u u u

xx

x t

kn

kn

kn

k n,

=− +

++ −

∆∆ (3.17)

Similarly the second-order, central difference approximation to the second partial time

derivative of u is given by

( )( )[ ]∂

∂β

2

2

1 1

2

22u

t

u u u

tt

x t

kn

kn

kn

k n,

=− +

++ −

∆∆ (3.18)

For these ukn is a wave or field quantity calculated at the spatial point x k xk = ∆ and

epoch t n tn = ∆ .

101

Substituting into the scalar wave equation (3.15),

( )( )[ ] ( )

( )[ ]u u u

tt c

u u u

xxk

nkn

kn

kn

kn

kn+ −

+ −− ++ =

− ++

1 1

2

2 2 1 12

22 2

∆∆

∆∆β β (3.19)

and solving for the latest value of u at the spatial point k gives

( )( )

( )[ ] ( )[ ]u c tu u u

xu u t xk

n kn

kn

kn

kn

kn+ + − −=

− +

+ − + +1 2 1 1

21 2 22

2∆∆

∆ ∆β β (3.20)

This is an explicit second-order expression for ukn+1 in which all wave quantities given

in the RHS of the equation are known from previous values at epochs tn and tn−1 . The

FD-TD solution is thus obtained by solving ukn+1 for all space points and iterating in

time.

Similarly the FD-TD method can be applied to solve Maxwell’s time-dependent curl

equations (2.3, 2.4), repeated here

t∂∂µ H

E −=×∇ and t∂

∂εσ EEH +=×∇ (3.21)

The domain is represented as two interleaved grids. Each grid contains respective

points of electric and magnetic field values. With the spatial domain as represented in

Figure 3.1, a first-order central-difference approximation of curl equation in E (2.21)

is expressed as:

[ ] [ ]1

4 1 2 3 40

01

01

∆ ∆ ∆x yE E E E

tH Hz

nx

nz

ny

ny

ny

n+ − − = − −+ −µ(3.22)

This equation (3.22) is solved for Hy0n+1 . The same technique is applied to the

alternate curl equation of the magnetic field, and in this way, at each time step, the

102

magnetic and electric field components are alternately solved on their respective grids.

Thus the field is time-step propagated throughout the grid until either a steady-state

solution or the desired response is obtained.

Ex1

Ey1

Ez1

Ex2

Ey2

Ey3Ex4

Hx0

Ez4

Ez3

Hz0

Hy0

z

y

x

scatterer

Figure 3.1 — 3-D FD-TD grid

3.2.3 Finite-Difference Frequency-Domain Technique

Although conceptually the Finite Difference Frequency Domain (FD-FD) method [3-

6, 8] is similar to the Finite Difference Time Domain (FD-TD) method, from a

practical standpoint it is more closely related to the finite element method. Like FD-

TD, this technique results from a finite difference approximation of Maxwell’s curl

equations. However, the time-harmonic versions of these equations are employed, and

since there is no time stepping it is not necessary to keep the mesh spacing uniform.

Therefore optimal FD-FD meshes generally resemble optimal finite element meshes.

The FD-FD technique generates a system of linear equations, where the corresponding

matrix is sparse like that of the finite element method. Although it is conceptually

much simpler than the finite element method, very little attention has been devoted to

this technique in the literature.

3.2.4 Method of Moments

Like finite-element analysis, the moment method [3-6, 8, 9]] is a technique for solving

complex integral equations (2.7, 2.8) by reducing them to a system of simpler linear

103

equations. In contrast to the approach of the finite element method however, moment

methods employ a technique known as the method of weighted residuals. Harrington

[10]] first demonstrated the power and flexibility of this numerical technique for

solving problems in electromagnetics.

All weighted residual techniques begin by establishing a set of trial solution functions

with one or more variable parameters. The residuals are a measure of the difference

between a trial solution and the true solution. The variable parameters are determined

in a manner that guarantees a best fit of the trial functions based on a minimisation of

the residuals. The equation solved by moment method techniques is generally a form

of the Electric Field Integral Equation (EFIE) or the Magnetic Field Integral Equation

(MFIE). Both of these equations can be derived from Maxwell’s equations by

considering the problem of a field scattered by a perfect conductor (or a lossless

dielectric). As an example, the EFIE is written in the form,

( )E J= f e (3.23)

where E is the incident field and J is the induced current. The form of the integral

equation used determines which types of problems a moment-method technique is

best suited to solve. For example one form of the EFIE may be particularly well suited

for modelling thin-wire structures, while another form is better suited for analysing

metal plates. Usually these equations are expressed in the frequency domain, however

the method of moments can also be applied in the time domain. The first step in the

moment-method solution process is to expand J as a finite sum of basis (or

expansion) functions,

J ==∑ J bi ii

M

1

(3.24)

where bi is the i-th basis function and Ji is an unknown coefficient. Next, a set of M

linearly independent weighting functions, wj, are defined. An inner product of each

weighting function is formed with both sides of the equation being solved. This

104

results in a set of M independent equations, and by expanding J using equation (3.24)

the result is a set of M equations in M unknowns.

[ ] [ ][ ]E J= Z (3.25)

Where the vector E contains the known incident field quantities and the terms of the

Z-matrix are functions of the geometry. The unknown coefficients of the induced

current are the terms of the J vector. These values are obtained by solving the system

of equations. Other parameters such as the scattered electric and magnetic fields can

be calculated directly from the induced currents.

Depending on the form of the field integral equation used, moment methods can be

applied to configurations of conductors only, homogeneous dielectrics only, or very

specific conductor-dielectric geometry’s. Moment method techniques applied to

integral equations are not very effective when applied to arbitrary configurations with

complex geometry’s or inhomogeneous dielectrics. Moment method techniques do an

excellent job of analysing a wide variety of important three-dimensional

electromagnetic radiation problems, particularly the modelling of wire antennas or

wires attached to large conductive surfaces. They are also widely used for antenna and

electromagnetic scattering analysis.

3.2.5 Geometrical and Uniform Theory of Diffraction

The Geometrical Theory of Diffraction was proposed by Keller [11], as an extension

and improvement to the classic ray-based high-frequency approximation, Geometric

Optics (GO), by introducing additional rays (diffraction coefficients) to account for

diffraction [12]. Diffraction is a local phenomena at high frequencies and the

behaviour of the diffracted wave at edges, corners, and surfaces can be determined

from an asymptotic form of the exact solution for simpler canonical problems. For

example, the diffraction around a sharp edge is found by considering the asymptotic

form of the solution for an infinite wedge. Both GO and GTD are only accurate when

the dimensions of objects being analysed are large relative to the wavelength of the

105

field. In general, as the wavelengths of an electromagnetic field approach zero, the

fields can be determined using geometric optics.

In GTD the high-frequency approximate solutions for general scatterers are derived

from a set of exact solutions which in turn are derived for a selection simple

geometric shapes. The common canonical solutions are for infinite wedges,

conducting half sheets, circular disks, cylinders, spheres, and others. Therefore the

problem geometry needs to constructed around these various shapes before a solution

can be found.

The Uniform Theory of Diffraction (UTD) is an extension of the GTD made by

implementing improvements to the diffraction coefficients [13]. The use of uniform

coefficients solves one limitation of the GTD, the infinities produced by the

asymptotic evaluation of the Fresnel integral used in deriving the diffraction

coefficients.

GTD and UTD are approximation methods, but are still analytical; their physical

meaning being quite clear, in contrast to other computational electromagnetics

techniques. No special techniques are required for the implementation of GTD/UTD

codes, but implementations that require detailed terrain are very complex. and have

high computational loads.

3.2.6 Generalised Multipole Technique

The Generalised Multipole Technique (GMT) [4, 5] is a relatively new method for

analysing EM problems. It is a frequency domain technique that (like the method of

moments) is based on the method of weighted residuals. However, this method is

unique in that the expansion functions are analytic solutions of the fields generated by

sources located some distance away from the surface where the boundary condition is

being enforced.

106

Moment methods generally employ expansion functions representing quantities such

as charge or current that exist on a boundary surface. The expansion functions of the

GMT are spherical wave field solutions corresponding to multipole sources. By

locating these sources away from the boundary, the field solutions form a smooth set

of expansion functions on the boundary and singularities on the boundary are avoided.

Like the method of moments, a system of linear equations is developed and then

solved to determine the coefficients of the expansion functions that yield the best

solution. Since the expansion functions are already field solutions, it is not necessary

to do any further computation to determine the fields. Conventional moment methods

determine the currents and/or charges on the surface first and then must integrate

these quantities over the entire surface to determine the fields. This integration is not

necessary at any stage of the GMT solution.

There is little difference in the way dielectric and conducting boundaries are treated

by the GMT. The same multipole expansion functions are used. For this reason, a

general purpose implementation of the GMT models configurations with multiple

dielectrics and conductors much more readily than a general purpose moment-method

technique. On the other hand, moment method techniques, which employ expansion

functions that are optimised for a particular type of configuration (e.g. thin wires), are

generally much more efficient at modelling that specific type of problem.

3.2.7 Parabolic Equation Method

The parabolic equation was first proposed for use in radiowave propagation by

Leontovich and Fock in 1946 [14, 15]. However at that time there were no suitable

numerical techniques for solving the equation and hence the method was not pursued.

Past research has seen the parabolic equation being applied to seismology and

underwater acoustics problems [16]. From this work and with advances in numerical

techniques there has been increased interest in applying the method to electromagnetic

propagation problems. An excellent reference may found in [17].

107

The parabolic equation is a reduction of the wave equation making use of the fact that

horizontal variations in refractive index occur at a slower rate than vertical variations.

The starting point for the development of an electromagnetic parabolic equation is

with the Helmholtz wave equation (3.12) for a field component, ( )ψ x z, with

assumed time dependence e− jωt .

∇ + =20

2 2 0ψ ψk n (3.26)

where k0

2=

πλ

is the free space wavenumber, and nk

k=

0

is the refractive index.

Since the propagation is above a spherical earth, a spherical co-ordinate system would

be appropriate. However, the problem can be greatly simplified if some assumptions

are made. The first assumption is that propagation takes place over a flat earth; thus

allowing the use of a cylindrical co-ordinate system. The field is then assumed

invariant in azimuth (azimuthal symmetry). Equation (3.26) can now be expressed in

cylindrical co-ordinates (r,φ, z)—with the φ - co-ordinate terms removed. For the

problem of radiowave propagation the co-ordinate r represents range distance from

the source, and z represents the height above the earth.

∂ ψ∂

∂ψ∂

∂ ψ∂

ψ2

2

2

2 02 21

0r r r z

k n+ + + = (3.27)

It is now assumed that the solution of Equation (3.27) is in terms of a Hankel function

that satisfies the Bessel differential equation

( ) ( ) ( ) ( ) ( ) ( )∂∂

∂∂

2 10

2

10

02 1

0

10

H k r

r r

H k r

rk H k ro o

o+ + = (3.28)

108

and takes the form

( ) ( ) ( ) ( )ψ r z u r z H k r, ,= 01

0 (3.29)

which is an outgoing cylindrical wave solution. The envelope function u(r, z) is

assumed to be slowly varying in range. Substituting the trial solution (3.29) into

equation (3.27) and using the Hankel function property given by equation (3.28)

gives:

( ) ( )( ) ( ) ( )∂

∂∂

∂∂∂

∂∂

2

2 10

10

2

2 02 22 1

1 0u

r H k r

H k r

r r

u

r

u

zk n u

o

o+ +

+ + − = (3.30)

Using the far-field assumption,k0r ⟩⟩ 1, the Hankel function is then given by the phase

factor

( ) ( )H k r eojk r1

00≈ (3.31)

Substitution of this approximation into equation (3.30) yields the simplified elliptic

equation,

( )∂∂

∂∂

∂∂

2

2 0

2

2 02 22 1 0

u

rjk

u

r

u

zk n u+ + + − = (3.32)

There are several methods for deriving various parabolic forms from equation (3.32).

One method is to assume that,

2 0

2

2jku

r

u

r

∂∂

∂∂

⟩⟩ (3.33)

which is valid since the main radial dependence of the field is contained in the e jkr

term of the Hankel function, while the envelope function u is slowly varying with

range [18]. With this approximation the following wave equation is obtained.

109

( )2 1 00

2

2 02 2jk

u

r

u

zk n u

∂∂

∂∂

+ + − = (3.34)

This is the Standard Parabolic Equation (SPE) and is limited in propagation angle

(≈15°) by the paraxial approximation given by Equation (3.33). Equation (3.34) can

be solved by marching the solution out in range, see Figure 3.2, since it is

fundamentally an open boundary problem.

∆x

∆z

0,0

Figure 3.2 — PE solution domain

3.3 Comparison of Modelling Techniques

The use of any particular computational electromagnetic technique is application

dependent. Although it is theoretically possible to base all computational

electromagnetics codes on one numerical method, it is obvious that this is not possible

from a practical viewpoint. The selection of an appropriate numerical technique for

modelling GPS signal propagation is the subject of this section.

110

3.3.1 Requirements for Modelling GPS Signal Propagation

The study, to be undertaken, is concerned with modelling the propagation of GPS

signals in the local vicinity. The effects of secondary path signals also need to be

modelled. Therefore the main criterion is the size of the domain over which the

modelling is to take place. The wavelength of the GPS L1 frequency is of the order of

19 cm. For representative domains of say, 100 metres, the total size of the domain,

expressed in wavelengths, will be of a size of about 500.

3.3.2 Comparison of EM Techniques

Table 3.1 [3] gives an indication of the applicability of the EM modelling techniques

discussed in the previous sections, as a function of performance for domains ranging

in size from less than a wavelength to domains greater than 100 wavelengths.

Table 3.1 — Model Comparison

As can be seen, the methods that are applicable to the requirements of modelling GPS

signal propagation, solely based on domain size, are the high-frequency techniques

and the parabolic equation method. The high-frequency techniques include the ray-

based methods of GTD and UTD.

Modelling Approach <λ/10 Åλ Å10λ Å100λ >100λParabolic Equation Method á á á á áHigh-Frequency Techniques à à á á áMethod of moments á á à à àFD-TD and FE methods à á á á àHybrid Methods á á á à à

111

The GTD/UTD implementation, requires that complex arbitrary terrain shapes be

represented as combinations of canonical problems, and therefore, to include the

appropriate diffraction coefficients requires a complex representation of the terrain

elements. To more fully characterise GPS multipath propagation accurate

representations of terrain are required. It is the intention in this work to use accurate

Digital Terrain Models (DTM) as input to the EM modelling code. The use of

GTD/UTD models preclude the use of DTMs as input for the terrain.

In addition to terrain implementation problems, there are many different propagation

modes, representing families of rays, that need to be considered and calculated in ray-

based code. Thus the resultant GTD/UTD code is complex and has long development

times. Finally these high-frequency methods are only suitable when the structures are

larger than a wavelength.

From these considerations it is determined that the most applicable technique is the

Parabolic Equation method. Not only does it allow for the large electrical domain

sizes required for the analysis but it has no apparent limitations for the analysis of

smaller-scale structures. In addition, it will allow the use of DTMs for terrain input, is

simple to implement, and is not nearly as computationally intensive as GTD/UTD

implementations

112

3.4 References

[1] S. Y. Liao, Microwave Devices and Circuit Theory, 2nd ed. Englewood Cliffs

N.J.: Prentice Hall, 1985.

[2] D. K. Cheng, Field and Wave Electromagnetics. Reading, MA: Addison-

Wesley, 1989.

[3] K. Umashankar and A. Taflove, Computational Electromagnetics. Boston,

MA: Artech House, 1993.

[4] T. H. Hubling, “Survey of Numerical Electromagnetic Modeling Techniques,”

University of Missouri-Rolla TR91-1-001.3, 1991.

[5] C. Hafner, The Generalized Multipole Technique for Computational

Electromagnetics. Boston: Artech House, 1990.

[6] N. Ida, Numerical Modeling for Electromagnetic Non-Destructive Evaluation.

London: Chapman & Hall, 1995.

[7] P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers,

Second ed. Cambridge: Cambridge University Press, 1990.

[8] E. Yamashita, “Analysis Methods for Electromagnetic Problems,” . Boston:

Artech House, 1990.

[9] J. T. Johnson, R. T. Shin, J. C. Eidson, L. Tsang, and J. A. Kong, “Method of

Moments Model for VHF Propagation,” IEEE Transactions on Antennas &

Propagation, vol. 45, pp. 115-125, 1997.

[10] R. F. Harrington, Field Computation by Moment Methods. New York:

Macmillan, 1968.

[11] J. B. Keller, “Geometrical Theory of Diffraction,” Journal of the Optical

Society of America, vol. 52, pp. 116-130, 1962.

[12] M. Ando, “The Geometrical Theory of Diffraction,” in Analysis Methods for

Electromagnetic Problems , E. Yamashita, Ed. Boston: Artech House, 1990,

pp. 213-242.

[13] D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to

the Uniform Geometrical Theory of Diffraction. Boston: Artech House, 1990.

113

[14] M. A. Leontovich and V. A. Fock, “Solution of the Problem of Propagation of

Electromagnetic Waves along the Earth's Surface by the Method of Parabolic

Equations,” Journal of Physics of the USSR, vol. 10, pp. 13-24, 1946.

[15] V. A. Fock, Electromagnetic Diffraction and Propagation Problems. Oxford:

Pergamon, 1965.

[16] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational

Ocean Acoustics. New York: AIP Press, 1994.

[17] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, “Parabolic

Equations,” in Computational Ocean Acoustics , R. T. Beyer, Ed. New York:

AIP Press, 1994.

[18] B. W. Parkinson and J. R. Spilker Jr., “Global Positioning System: Theory and

Applications Volume I,” in Progress in Astronautics and Aeronautics Series,

vol. 163, P. Zarchen, Ed. Washington: American Institute of Aeronautics and

Astronautics, 1996.

[19] C. D. McGillem and G. R. Cooper, Continuous and Discrete Signal and

System Analysis, Third ed. Philadelphia: HRW Saunders, 1991.

[20] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, “Broadband

Modelling,” in Computational Ocean Acoustics , R. T. Beyer, Ed. New York:

AIP Press, 1994.


Wesley, 1989.

[2] S. Y. Liao, Microwave Devices and Circuit Theory, 2nd ed. Englewood Cliffs

N.J.: Prentice Hall, 1985.

[3] K. Umashankar and A. Taflove, Computational Electromagnetics. Boston,

MA: Artech House, 1993.

[4] T. H. Hubling, "Survey of Numerical Electromagnetic Modeling Techniques,"

University of Missouri-Rolla TR91-1-001.3, September 1 1991.

[5] C. Hafner, The Generalized Multipole Technique for Computational

Electromagnetics. Boston: Artech House, 1990.

[6] N. Ida, Numerical Modeling for Electromagnetic Non-Destructive Evaluation.

London: Chapman & Hall, 1995.

[7] P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers,

Second ed. Cambridge: Cambridge University Press, 1990.

114

[8] E. Yamashita, "Analysis Methods for Electromagnetic Problems." Boston:

Artech House, 1990.

[9] J. T. Johnson, R. T. Shin, J. C. Eidson, L. Tsang, and J. A. Kong, "Method of

Moments Model for VHF Propagation," IEEE Transactions on Antennas and

Propagation, vol. 45, pp. 115-125, 1997.

[10] R. F. Harrington, Field Computation by Moment Methods. New York:

Macmillan, 1968.

[11] J. B. Keller, "Geometrical Theory of Diffraction," Journal of the Optical


[12] M. Ando, "The Geometrical Theory of Diffraction," in Analysis Methods for

Electromagnetic Problems, E. Yamashita, Ed. Boston: Artech House, 1990,

pp. 213-242.

[13] D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to

the Uniform Geometrical Theory of Diffraction. Boston: Artech House, 1990.

[14] M. A. Leontovich and V. A. Fock, "Solution of the Problem of Propagation of

Electromagnetic Waves along the Earth’s Surface by the Method of Parabolic

Equations," Journal of Physics of the USSR, vol. 10, pp. 13-24, 1946.


Pergamon, 1965.



[17] M. F. Levy, Parabolic Equation Methods for Electromagnetic Wave

Propagation, 1 ed. London: IEE, 2000.

[18] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, "Parabolic

Equations," in Computational Ocean Acoustics, R. T. Beyer, Ed. New York:

AIP Press, 1994.

[19] B. W. Parkinson and J. R. Spilker Jr., "Global Positioning System: Theory and

Applications Volume I," in Progress in Astronautics and Aeronautics, vol.

163, P. Zarchen, Ed. Washington: American Institute of Aeronautics and

Astronautics, 1996.

115

Chapter 4 GPS Parabolic Equation Model

4.1 Development of the PE for Electromagnetic Propagation

The parabolic equation method was first proposed, as a solution method for

propagation of electromagnetic waves, by Leontovich and Fock [1, 2]. Since that time

the PE method has been used for physics, seismic, atmospheric and underwater

acoustics, and electromagnetics applications [3].

The potential of the PE method was not fully realised until an efficient numerical

technique known as the Fourier split-step was introduced, to the ocean acoustics

community, by Hardin and Tappert [4]. The PE method was then proposed as a useful

modelling technique for seismic wave propagation [5] as well as receiving increased

attention in ocean acoustics [6].

The acoustic PE method was modified for electromagnetics by Ko et al [7]. Both

Craig and Dockery [8, 9] continued research on the PE—using the method to model

electromagnetic wave propagation in the troposphere. Development of the PE method,

for tropospheric propagation modelling, continued in the late 1980’s, early 1990’s,

with Craig [10-14], Levy [15-18], Dockery[19], and others [20-25], refining the

standard PE for communications and radar predictions.

The PE research for ocean acoustics was, in the meantime, developing methods for

dealing with wide-angle propagation [26-36], a requirement for particular ocean

environment problems.

Further significant contributions to the development of the PE method have been

made. In particular the implementation of backscatter [37-41] and the inclusion of

terrain interactions [42-50] have extended the usefulness of the PE method, for

various applications. An excellent reference on the PE method can be found in [51].

116

The use of the PE method for modelling GPS signal propagation is one such

application to be enabled by this ongoing research and significantly, was first

introduced by Walker [52, 53] for modelling the operation of GPS in harsh

environments. In this chapter the PE modelling method for GPS is introduced.

Limitations of the standard PE forms are investigated. Improvements of the original

PE model are made including finite impedance conditions at the boundary and two-

way propagation through the inclusion of backscatter.

4.2 The Free-Space Parabolic Equation

We start with the simplified elliptic equation introduced in Chapter 3 as equation

(3.32) and repeated here for clarity.

( ) 012 2202

2

02

2

=−+++ unkz

u

r

ujk

r

u

∂∂

∂∂

∂∂

(4.1)

By defining the operators

rP

∂∂= and

2

2

20

2 1

zknQ

∂∂+= (4.2)

the simplified elliptic equation given by equation (4.1) can now be written in the

form,

( )[ ] 012 2200

2 =−++ uQkPjkP (4.3)

The factorisation of this equation permits it to be formulated into incoming and

outgoing wave components.

117

( )( ) [ ] 0,00000 =−++−+ uQPjkuQjkjkPQjkjkP

(4.4)

where

[ ] QPPQQP −=, (4.5)

is the commutator of the P and Q operators. If the refractive index n is independent of

range then the operators commute and the final term in equation (4.4) is zero. This is

true for free-space propagation but is also assumed valid for weak range dependence

and thus the commutator term can be neglected. The outgoing component of equation

(4.4) is now selected, giving

( ) 010 =−+ uQjkPu (4.6)

By replacing the operator notation, the result is a one-way wave equation, exact for

range-independent environments, within the limits imposed by the far-field

approximation, and is given by

01

12

2

20

20 =

+−+ u

zknjk

r

u

∂∂

∂∂

(4.7)

This equation is evolutionary in range but is not easily solved, due to the inclusion of

the refractive index term, n. In this work we are concerned only with propagation

effects from localised terrain interaction, and not atmospheric effects. If the refractive

index is set to 1 (free-space propagation) then the result is the Free-Space Parabolic

Equation (FSPE) which is given by

118

01

112

2

20

0 =

+−+ u

zkjk

r

u

∂∂

∂∂

(4.8)

Replacing r with x, to emphasise that this is a two-dimensional problem, and by

taking a Taylor series expansion of the square-root operator we can make use of the

Fourier transform property

( ) ( ) ( )pXjpzxdz

d nF

n

n

⇔ (4.9)

This allows an efficient Fourier transform based stepping technique to be used, where

the solution at a range-step ( x∆ ) is given by

( ) ( )[ ]

=∆+

−−∆

− zxuFeFzxxuk

pxjk

,,11

12

2

(4.10)

Here p is the vertical wave-number and is related to k by θsinkp = , with θ , the

propagation angle relative to the horizontal. The p-domain defines the angular

spectrum (Chapter 3) of the field, and together with the z-domain (spatial domain),

they form a Fourier transform pair. The exponential term in equation (4.10) is often

referred to as the PE propagator—which may take different forms for different PE

implementations. Specifically for this free-space implementation it will be known as

the FSPE propagator.

As can be seen from the form of equation (4.10) an initial field condition is defined at

x=0, and the solution marched out in range in discrete steps using Fourier transforms.

The solution of the FSPE is an exact solution and is not limited in either propagation

angle or range. This is not true for PE forms that include refractive index terms. In the

119

following section alternate, traditional PE models are examined for their limitations in

modelling GPS signal propagation.

4.3 Limitations of Refractive Index Terms in PE Forms

For propagation in media, various forms of parabolic equation can be derived from

equation (4.7) by using approximations for the pseudo-differential operator Q. The

resultant non-free-space parabolic equations may then be solved by a variety of

numerical techniques, but none are as efficient as the Fourier-step technique used to

solve the FSPE.

The development of these alternate PE forms is initiated by writing the square-root

operator Q given in equation (4.2) as

qQ += 1 (4.11)

where

µε +=q (4.12)

with

12 −= nε and 2

2

20

1

zk ∂∂µ = (4.13)

These abbreviations allow the operator to be cast in a form that implies something

about the nature of the medium and the angle of propagation. This is shown in the

following development.

Consider a trial plane-wave solution, for the reduced function, of the form

120

( )θθ sincos zrjkeu ±= (4.14)

The medium wave-number can be related to the horizontal (kr) and vertical (kz or p )

wave-numbers by the dispersion relation

222zr kkk += (4.15)

giving

θsinkkz = (4.16)

where θ is the angle of propagation with respect to the horizontal.

Using the trial solution given by equation (4.14) in equation (4.7) allows evaluation of

the differential operator µ.

20

2

k

kz−=µ (4.17)

by making use of equation (4.16) and substituting for the refractive index, n, gives

θµ 22 sinn−= (4.18)

Thus the operator µ is seen to be a function of the propagation angle and refractive

index, while ε is a function only of the refractive index. A further implication

becomes apparent when q is reformed.

121

( ) θµε 222 sin1 nnq −−=+= (4.19)

Using Snell’s law, the refractive index can be related to the propagation angles

through

θθ

cos

cos 0=n (4.20)

giving

( ) 022

20

2

sin1sin1cos

cos θθθθ −=−−=q (4.21)

The implication of this result is that the operator q is only a function of propagation

angle from the source.

Approximations of the Q-operator can be written in a general rational-linear form as

qba

qbaqQ

11

001++≈+= (4.22)

This equation and equation (4.7) provide the basis for the development of a general

parabolic wave equation given by

02

2

2

3

=+++z

uDCu

rz

uB

r

uA

∂∂

∂∂∂

∂∂ (4.23)

where

122

( )

( ) ( )( )( )( )01

0

201010

20

1

211

1

1

bbk

jD

nbbaajkC

k

bB

nbaA

−=

−−+−=

=

−+=

This equation can be solved by a variety of methods including finite-differences,

finite-elements, and in certain cases, the Fourier split-step technique [4]. Various

coefficients have been used in parabolic equation implementations of equation (4.23),

in acoustics, seismology, and electromagnetic propagation studies. The following

table summarises the coefficients for three of the most common.

Coefficients Tappert Claerbout (Padé 1) Greene

a0 1.0 1.00 0.99987

b0 0.5 0.75 0.79624

a1 1.0 1.00 1.00000

b1 0.0 0.25 0.30102

Table 4.1 — PE Coefficients

The first set of coefficients are attributed to Tappert and are the first two terms of a

Taylor series expansion of Q given by

K+−+=+82

112qq

q

The requirement for convergence of this series is that 1<q . Substitution of these

coefficients into the general form given by equation (4.23) gives

( ) 012 2202

2

0 =−++ unkz

u

r

ujk

∂∂

∂∂

(4.24)

123

Which is known as the Standard (or narrow-angle—due to the convergence

requirement) Parabolic Equation (SPE).

Greene derived the coefficients used for his implementation by optimisation of the

rational-linear approximation for defined angle intervals [54]. The coefficients were

chosen to minimise the maximum error of the square-root approximation.

The coefficients given by Claerbout [5] are those of the first term of a Padé series

expansion of Q in the form of

( )∑=

+++

+=m

i

m

mi

mi qOb

qaQ

1

12

,

,

11 (4.25)

All of these forms provide approximate solutions to the full-wave equation and as

such are limited in their usefulness for modelling GPS signal propagation. These

limitations are examined in the following sections.

4.3.1 Fourier Split-Step Solution of the SPE

By making use of operator formalism a solution scheme for the standard parabolic

equation is developed [55].

Equation (4.24) can rewritten, with the range variable r replaced by x (highlighting the

fact that it is a two-dimensional problem), in the following compact form

[ ]uBAjx

u +=∂∂

(4.26)

where

124

( )12

2 −= nk

A and 2

2

2

1

zkB

∂∂=

The solution of equation (4.26) at a range step, x + ∆x, is

( ) ( ) ( )zxuezxxu xBAj ,, ∆+=∆+ (4.27)

A split-form of the exponential operator, ( ) xBAje ∆+

, is now introduced.

( ) xjBxjAxBAj eee ∆∆∆+ ≅ (4.28)

giving

( ) ( )zxueezxxu xjBxjA ,, ∆∆=∆+ (4.29)

The split form of this equation is exact only when the operators A and B commute.

This is the case if the refractive index, n, is assumed constant. The form of this

splitting now allows a relatively straight forward numerical solution. Since A is a

multiplication operator, a Fourier transform solution of the B-term is sought.

Let

( ) ( )zxxgezxxu xjA ,, ∆+=∆+ ∆ (4.30)

where g is given by

( ) ( )zxuezxxg zkxj

,,2

2

2

1

∂∂

∆

=∆+ (4.31)

125

Making use of the Fourier transform property

( ) ( ) ( )pXjpzxdz

d nF

n

n

⇔ (4.32)

the Fourier transform of equation (4.31) can be written directly as

( ) ( )pxUepxxGp

k

xj

,,2

2

∆−

=∆+ (4.33)

This allows the solution to g to written in terms of two Fourier transforms

( ) ( )[ ]g x x z F e F u x zj x

kp

+ =

−−

∆∆

, ,1 22

(4.34)

The complete solution of the reduced function, u, also includes the refractive effects

of the medium—through the refractive index term, n—and is given by

( ) [ ] ( )[ ]

=∆+

∆−−∆−

zxuFeFezxxup

k

xjxn

jk

,,22

211

2 (4.35)

It is now evident why Tappert coined the terms Fourier Split-Step for this algorithm,

since the solution involves Fourier techniques and accounts for diffraction and

refraction in two distinct (split) steps.

The solution of g given by equation (4.34) is the solution of the SPE for propagation

in free-space (n = 1) and only accounts for diffractive effects. Note that the PE

Propagator, for this solution, varies from that given for the FSPE solution, equation

(4.10). This variation of PE-propagator is crucial and it will become evident in the

next section how this effects the field solution.

126

4.3.2 Phase Errors in Rational-Linear Approximation Forms of the PE

Now equations (3.18 and 3.28) show that the exact form (Helmholtz) of the square-

root operator Q, and hence the phase of the solution [3], is given by

θ2sin1−=Q (4.36)

Which is valid for a plane wave propagating at angle θ in free space.

Likewise, substituting for q, equation (4.22), into the general rational-linear form

given by equation (4.23) allows the phases for each case to be written directly [32] as

QTappert = −1 05 2. sin θ (4.37)

QClaerbout Pade,

. sin

. sin1

2

2

1 0 75

1 0 25=

−−

θθ

(4.38)

QGreene =−

−0 99987 0 79624

1 0 30102

2

2

. . sin

. sin

θθ

(4.39)

A plot of these various Q-approximates over the angle interval {-90 degrees, +90

degrees} gives an insight into the asymptotic behaviour of each approximation. This

is shown in Figure 4.1.

127

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Q Approximations

Angle (deg)

Q F

unct

ion

Exact Tappert Greene Claerbout

Figure 4.1 — Q-functions

As can be seen the approximations for Q are only valid for a narrow range of angles,

centred about zero degrees (horizontal propagation). The divergence of the

approximations from the exact Q function results in phase errors within the stepped

solution of each PE approximate form.

These associated phase errors can be defined as,

aa QQ −=ξ (4.40)

where Qa represents each of the approximations given above, and Q the exact function

given by equation (4.36). These phase errors, plotted as a function of positive

propagation angle, see Figure 4.2, give a numerical interpretation of the angular

limitations associated with each of these approximations.

128

Tappert

..............

Greene

Claerbout

Figure 4.2 — Phase error with common approximates

For an arbitrary phase error limit of 0.002 the Tappert, Claerbout (Padé1), and Greene

approximations have acceptable propagation angle limits of approximately 20°, 36°,

and 47° respectively.

From this it is seen that the requirements for modelling of GPS signal propagation (up

to 90 degrees) is not easily met by these non-free-space PE forms. This provides

justification for using the FSPE, which has no phase error limitations, related to

propagation angles (achieved by not including refractive atmospheric effects). In

addition the Fourier-step solution, given by equation (4.10), is highly efficient, indeed

optimal, when there are no refractive index variations and terrain can be modelled

with non-complex boundary conditions. There are alternate implementations, such as

a split-step Padé method [29], that allows wider-angle limits, and is highly efficient

for more complex propagation problems.

4.4 Numerical Implementation for GPS Satellite Propagation

Having established that the FSPE solution, given by equation (4.10), will form the

basis for a GPS propagation model, we can now consider the implementation

requirements of the model. The Fourier step solution is an open boundary problem

129

that is solved by modifying the previous solution and marching it to the next array

position, using equation (4.10).

4.4.1 Domain Sampling

The representative solution domain for the FSPE model is shown in Figure 4.3.

∆x

∆z

0,0

zmax

range

Figure 4.3 — FSPE Solution Domain

The first step in the implementation is to select the sampling rates for the spatial

domain parameters x and z. For the vertical spatial sampling the sampling rate must

satisfy the Nyquist sampling criterion [56] where an analogy between the

frequency⇔time domains of Fourier analysis in signal processing, and the spatial-

frequency⇔spatial-distance domains for this modelling, is made. Therefore the

sampling must satisfy,

max2

2

pz

π≤∆ (4.41)

130

where for a selected maximum propagation angle

maxmax sinθkp = (4.42)

In the corresponding, angular spectrum domain (p-domain) the sample spacing must

satisfy

zNp

∆=∆ π2

(4.43)

where N is the total sample number in the Fast Fourier Transform (FFT), and should

be a power of two for efficiency. If the maximum propagation angle is taken to be 90

degrees (zenith) then ∆z is simply λ/2.

The selection of the spatial sampling in range (∆x) is made such that the solution of

the FSPE is within acceptable error bounds. The error can be made as small as desired

by making ∆x sufficiently small. Jensen et al [57] suggest that the only safe way to

ensure numerically accurate PE results is through a convergence test , where ∆x and

∆z are systematically reduced until a stable answer is obtained.

4.4.2 Incident Boundary Condition

The left-hand boundary represents the incident, or initial, field condition and as such

must represent the field values entering the domain from the selected GPS satellite.

The most effective way to implement the incident boundary condition is to use an

initial plane-wave field representation at the first array position (x = 0)—following

the implementation given by Walker [53]—this is satisfied by

( ) ( )θθ sin0

sin0

zjkzjkinc eEeEE −= − (4.44)

131

This plane-wave incident field represents a direct signal from the satellite in addition

to a ground reflected signal modelled as occurring from terrain in front of the initial

boundary. It is an arbitrary choice to include the reflected field components in the

initial boundary condition, and if they are considered of no consequence in the

subsequent analysis they may be omitted. The nature of this initial field is represented

in Figure 4.4

GPS Satellitepropagation

path

θ

Direct SignalConstant Phase Fronts

Reflected SignalConstant Phase Fronts

Z

X

X=0

- Initial Field Value

Figure 4.4 — Initial Field

There are other incident field conditions that may be utilised, Levy [58] suggested that

the field can be represented as incoming energy, thus allowing the vertical domain

size to be reduced and improving computation times. In this work the plane-wave

field condition is adequate (in a computational sense) for the modelling of GPS signal

propagation.

4.4.3 Upper Boundary Condition

The upper boundary is required to absorb all of the energy incident upon it. If the field

were simply set to zero the upper boundary would act as an ideal conductor and all the

132

incident energy would be reflected back into the domain and corrupt the results. The

solution proposed by Craig [14], and implemented by Walker [53] for modelling GPS

propagation, was to set a domain height, 2Zmax, as being twice the maximum height of

interest and to apply a Hamming or Hanning window in this extended region,

effectively attenuating the signal with filtering techniques.

Figure 4.5 shows the implementation of the upper boundary condition for the FSPE.

Here the incident plane-wave initial field is attenuated by a Hanning window in the

upper region (Zmax to 2Zmax) of the implementation-domain. We define the solution-

domain as the positive first half of the implementation domain.

0 500 1000 1500 2000 2500-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Initial Field - 10 deg

Fie

ld L

evel

Z-Samples

Hanning WindowInitial Field

Figure 4.5 — Upper Absorption Region

4.4.4 Lower Boundary Condition

For the lower boundary the effective reflection coefficient is implemented through the

use of FFT’s in the angular spectrum domain. The incident field is used to form the

source image where the 180 degree phase shift is then implemented. The magnitude

of the reflection coefficient is calculated for the complete propagation angular

spectrum (0 degrees - 90 degrees). This is then applied directly, to the p-space

(angular spectrum space) image representation of the field, to form the product. The

133

resultant modified image field is then propagated with the source to form the FSPE

solution at the next range step. The p-space application of the effective reflection

coefficient is illustrated in Figure 4.6 for a plane wave at 10 degrees propagation

angle incident upon a fresh water boundary.

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1N

orm

alis

ed V

alue

Propagation Angle (degrees)

Reflection Coefficient in P-Space

Angular Spectrum (0-90)Coefficient Magnitude

Figure 4.6 — Reflection Coefficient in P-Space

For this case the resultant angular spectrum completely includes the coupled reflection

coefficient derived in Chapter 2 for GPS signal propagation. As was shown in Chapter

2, the effective reflection coefficient can also be modified to include any rough

surface effects.

It should be noted that the image method proposed here is an approximation. A more

correct method would involve a mixed Fourier transform as proposed by Kuttler and

Dockery [59, 60].

134

4.4.5 Implementation Algorithm

The complete implementation domain is shown below in Figure 4.7.

FieldSource Solution Domain

UpperAbsorption Region

2Zmax

LowerAbsorption Region

ImageSourceincludingEffective

Reflection Coefficient

-2Zmax

Zmax

-Zmax

0 X

Z

HanningWindow

HanningWindow

Image Domain

Figure 4.7 — Implementation Domain

The solution-domain specifies the region of interest in the modelling being

undertaken. The implementation methodology is outlined in the algorithm flow-chart

shown below in Figure 4.8.

135

FFTReflectionCoefficient

InverseFFT

SurfaceRoughness

ModifiedImage Field

Total Field

Image Field

Source Field

Combine Fields

FFTFSPE

PropagatorInverse

FFT

PropagationAngle

Initial Field

ApplyHanning Window

Figure 4.8 — Implementation Algorithm

At this point the FSPE modelling is in its basis form and does not include the effect of

terrain interactions. The following section introduces the implementation of terrain

interaction into the basic FSPE model.

4.5 Implementing Arbitrary Terrain in the PE Model

The Fourier split-step method has been extensively used for modelling tropospheric

propagation over smooth earth and sea profiles. There is now however increased

interest in modelling propagation over arbitrary terrain profiles. The modelling of

136

propagation over arbitrary terrain is a mandatory requirement for the GPS propagation

model being developed as part of this research.

The following section introduces the Boundary-Shift (BS) technique for inclusion of

terrain interaction.

4.5.1 Boundary-Shift Technique for Arbitrary Terrain

The boundary-shift technique, for handling arbitrary terrain within the PE code,

involves the shifting of the field array (aperture) either up or down to account for the

shift in the boundary position, and thus satisfy the terrain boundary conditions. The

field aperture immediately to the left of any obstructing terrain is stored then shifted

down according to the height of the terrain element. The lower elements — those that

would propagate into the terrain — are discarded and zeros inserted at the top of the

array to maintain the correct number of elements. This modified field array is then

propagated to the next array, with the FSPE Fourier-step technique. At negative

terrain transitions, the reverse procedure is applied. The array is shifted up by the

corresponding height, with the top elements discarded, and zeros inserted at the

element positions where the field is obscured by the terrain. The result of the

boundary shifting technique is simply a restructuring of the domain representation to

that of a field propagating over a plane earth while accounting for diffractive effects

over terrain.

The development of the boundary-shift technique is based on intuitive concepts and

approximations, more so than any sound mathematical or physical formulations [61].

The method has however shown excellent agreement with other more mathematically

correct methods.

The implementation is straight forward and allows use of the Fourier step routine, for

propagation over a smooth earth, to be used (without modification) to propagate the

field, with aperture shifts made at the appropriate range steps. The technique is

presented graphically in Figure 4.9 and Figure 4.10. Figure 4.9 is the representation of

137

the problem within the solution-domain space. The implementation of the boundary

shift method is shown in Figure 4.10.

The boundary shift method, for this simple example, is explained as follows:

The field aperture immediately to the left of the first terrain block (array 1) is stored

then shifted down four elements. The four lower elements—those that would

propagate into the terrain—are discarded and zeros inserted at the top of the array to

maintain the correct number of elements. This modified field array is then propagated

to the second array, with the Fourier split-step. At the negative terrain transition, the

reverse technique is applied. The array is shifted up, with the four top elements

discarded, and zeros inserted at the element positions where the field is obscured by

the terrain. Similarly the technique is applied for all terrain transitions. The result of

the boundary shifting technique is seen, in Figure 4.10, as a restructuring of the

domain representation to that of a field propagating over a plane earth.

∆x

∆z

0,0Range

Height

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

N∆z

Figure 4.9 Figure 4.10

138

The inclusion of the boundary-shift technique into the basic FSPE model involves

modification of the source field element as given in Figure 4.8. This modification is

implemented as follows.

Field Array

TerrainTransition

ShiftField Array

Up

ShiftField Array

DownField Array

Source Field

-ve +ve

No

Figure 4.11 — Boundary-Shift Algorithm

4.6 Implementing Backscatter for a Two-Way PE Model

In the development of the FSPE, it was necessary to assume that the field was

outgoing only. For the one-way FSPE model the incident boundary condition is set to

represent the incoming plane-wave propagation from a representative GPS satellite.

The FSPE method propagates the field components in the +x direction with the

Fourier-step technique. The terrain profile is accounted for by the boundary shift

method (aperture shift) as outlined in the previous section. The one-way restriction

139

can be lifted by using a store and forward method of back-propagating field

components.

In this implementation a method is proposed whereby the field components that will

propagate into terrain are identified, stored and utilised as initial boundary conditions.

The PE is then propagated, with these initial boundary conditions, in the reverse

direction with the terrain mirrored and accounted for with the same boundary-shift

techniques, as used in the one-way implementation. This method has been utilised by

Levy [41] for electromagnetic propagation over terrain, and by Collins [38, 39] for the

analogous problem in underwater acoustics.

The steps for implementation of a two-way PE model derived from a one-way model

are as follows;

1. The field is propagated with the one-way PE model in the forward (+x) direction

2. The field components that will propagate into terrain (potential back-scatterers),

are identified.

3. These field values and indexes to their positions within the domain are stored for

later use.

4. The terrain profile and domain are mirrored vertically such that the one-way

implementation can again be used without modifying the existing PE model code.

5. The one-way PE is then used to propagate the stored field values—with

appropriate application of reflection coefficients and surface roughness—which

are added into the model as initial field conditions of the back-propagation.

6. The field components of the forward and back implementations are then added to

provide the resultant full field.

The backscatter implementation in the solution-domain is shown graphically in Figure

4.12.

140

a b

Figure 4.12 — Backscatter implementation

The forward field condition is shown as (a) with the mirrored reverse case indicated

as (b).

The use of this technique is justified by image theory, where the components at a

vertical interface would travel to an image of the domain mirrored vertically about the

vertical reflector. In addition, the method is complementary to the boundary-shift

technique, where the down-shifted components normally discarded, are stored for use

as the initial field values for a two-way PE implementation. It should be noted that

reflection coefficients can be directly applied to the backscattered field components.

The implementation algorithm for the two-way FSPE model is given in Figure 4.13.

+x -x

141

MoreArrays

Insert Array

Reverse FSPE

StoredReverse

Field Arrays

Total ReverseField

StoredForward

Field

Sum Fields

Total Field

YES

No

Figure 4.13 — Backscatter Implementation Algorithm

4.7 Summary

In this chapter the Free-Space Parabolic Equation propagation model for GPS was

introduced. The two-dimensional FSPE model accounts for reflection coefficients,

terrain interaction, diffractive effects and backscatter. Its efficiency is in the nature of

the solution using Fourier stepping techniques. The solution-domain is unbounded in

range and as such the total field can be calculated at any desired range position.

In addition the FSPE model, being a full field solution, provides information on the

field at any discrete spatial position within the solution domain. Although this is a

positive aspect in a model for analysis of GPS multipath effects there are drawbacks.

The most apparent limitation for multipath analysis is that the FSPE, as introduced in

this chapter, provides only field information, and as such does not describe the GPS

142

multipath environment. It does however, provide the ability to visualise the field at

any spatial position within the solution domain.

The key multipath parameters of relative amplitude, time delay, phase and phase rate

are lost with the assumed time dependence removed. This time dependence can be

reinstated and the issue of deriving the Multipath Channel Impulse Response (MCIR

from the FSPE is covered in the next chapter.

143

4.8 References


Pergamon, 1965.

[2] M. A. Leontovich and V. A. Fock, "Solution of the Problem of Propagation of

Electromagnetic Waves along the Earth’s Surface by the Method of Parabolic

Equations," Journal of Physics of the USSR, vol. 10, pp. 13-24, 1946.



[4] R. H. Hardin and F. D. Tappert, "Applications of the Split-Step Fourier

Method to the Numerical Solution of Nonlinear and Variable Coefficient

Wave Equations," SIAM Review, vol. 15, pp. 423, 1973.

[5] J. F. Claerbout, Fundamentals of Geophysical Data Processing. New York:

McGraw Hill, 1976.

[6] F. R. DiNapoli and R. L. Deavenport, "Numerical Methods of Underwater

Acoustic Propagation," in Ocean Acoustics, J. A. DeSanto, Ed. Berlin:

Springer-Verlag, 1979.

[7] H. W. Ko, J. W. Sari, M. E. Thomas, P. J. Herchenroeder, and P. J. Martone,

"Anomalous Propagation and Radar Coverage Through Inhomogeneous

Atmospheres," presented at AGARD CP-346, 1984.

[8] K. H. Craig, "Propagation Modelling in the Troposphere: Parabolic Equation

Method," Electronics Letters, vol. 24, pp. 1136-1139, 1988.

[9] G. D. Dockery, "Modeling Electromagnetic Wave Propagation in the

Troposphere using the Parabolic Equation," IEEE Transactions on Antennas

and Propagation, vol. 36, 1988.

[10] K. H. Craig and M. F. Levy, "Field Strength Forecasting with the Parabolic

Equation: Wideband Applications," presented at Sixth International

Conference on Antennas and Propagation ICAP 89, Coventry, UK, 1989.

[11] K. H. Craig and M. F. Levy, "Recent Developments in Propagation

Forecasting: Channel Characterisation for Radar Systems," presented at IEE

Colloquium on ’Radar Clutter and Multipath Propagation’ (Digest No.62),

London, UK, 1989.

144

[12] K. H. Craig and M. F. Levy, "A Forecasting System using the Parabolic

Equation- Application to Surface-to-Air Propagation in the Presence of

Elevated Layers," presented at Operational Decision Aids for Exploiting or

Mitigating Electromagnetic Propagation Effects (AGARD-CP-453), San

Diego, CA, USA, 1989.

[13] K. H. Craig and M. F. Levy, "A PC-based Microwave Propagation Forecasting

Model," presented at Seventh International Conference on Antennas and

Propagation ICAP 91, York, UK, 1991.

[14] K. H. Craig and M. F. Levy, "Parabolic Equation Modelling of the Effects of

Multipath and Ducting on Radar Systems," presented at IEE Proceedings-F

[Radar and Signal Processing], 1991.

[15] M. F. Levy and K. H. Craig, "Case Studies of Transhorizon Propagation:

Reliability of Predictions using Radiosonde Data," presented at Sixth

International Conference on Antennas and Propagation ICAP 89, Coventry,

UK, 1989.

[16] M. F. Levy and K. H. Craig, "Assessment of Anomalous Propagation

Predictions using Minisonde Refractivity Data and the Parabolic Equation

Method," presented at Operational Decision Aids for Exploiting or Mitigating

Electromagnetic Propagation Effects (AGARD-CP-453), San Diego, CA,

USA, 1989.

[17] M. F. Levy and K. H. Craig, "Millimeter-Wave Propagation in the

Evaporation Duct," presented at Atmospheric Propagation in the UV, Visible,

IR and MM-Wave Region and Related Systems Aspects (AGARD-CP-454),

Copenhagen, Denmark, 1990.

[18] M. F. Levy, "PE Modelling of Radiowave Propagation over the Sea,"

presented at IEE Colloquium on ’The Interaction of Radiowaves with the Sea

Surface’ (Digest No.037). London, UK, 1990.

[19] G. D. Dockery, "Method for Modelling Sea Surface Clutter in Complicated

Propagation Environments," IEE Proceedings-F [Radar and Signal

Processing], vol. 137, pp. 73-79, 1990.

[20] J. P. Reilly and G. D. Dockery, "Influence of Evaporative Ducts on Radar Sea

Return," IEE Proceedings-F, vol. 137, pp. 80-88, 1990.

145

[21] P. L. Slingsby, "Modelling Tropospheric Ducting Effects on VHF/UHF

Propagation," IEEE Transactions on Broadcasting, vol. 37, pp. 25-34, 1991.

[22] A. E. Barrios, "Parabolic Equation Modeling in Horizontally Inhomogeneous

Environments," IEEE Transactions on Antennas and Propagation, vol. 40, pp.

791-797, 1992.

[23] D. Rouseff, "Simulated Microwave Propagation Through Tropospheric

Turbulance," IEEE Transactions on Antennas and Propagation, vol. 40, pp.

1076-1083, 1992.

[24] H. V. Hitney, "Hybrid Ray Optic and Parabolic Equation Methods for Radar

Propagation Modeling," presented at International Conference Radar 92,

Brighton, UK, 1992.

[25] W. P. M. N. Keizer and R. B. Boekema, "Within the Horizon Propagation

Measurements over sea at 10.5 GHz," presented at IEEE Antennas and

Propagation Society International Symposium, Chicago, IL, USA, 1992.

[26] M. D. Collins, "Applications and Time-Domain Solution of Higher-Order

Parabolic Equations in Underwater Acoustics," Journal of the Acoustic Society

of America, vol. 86, pp. 1097-1102, 1989.

[27] M. D. Collins, "Benchmark Calculations for Higher-Order Parabolic

Equations," Journal of the Acoustic Society of America, vol. 87, pp. 1535-

1538, 1990.

[28] M. D. Collins and E. K. Westwood, "A Higher-Order Energy-Conserving

Parabolic Equation for Range-Dependent Ocean Depth, Sound Speed, and

Density," Journal of the Acoustic Society of America, vol. 89, pp. 1068-1075,

1991.

[29] M. D. Collins, "A Split-Step Padé Solution for the Parabolic Equation

Method," Journal of the Acoustic Society of America, vol. 93, pp. 1736-1742,

1993.

[30] R. A. Dalrymple, L. C. Munasinghe, D. H. Wood, and J. T. Kirby, "A Very-

Wide-Angle Acoustic Model for Underwater Sound Propagation," Journal of

the Acoustic Society of America, vol. 88, pp. 1863-1876, 1990.

[31] N. Dodd, "Efficient Higher-Order Finite-Difference Schemes for Parabolic

Models," Coastal Engineering, vol. 28, pp. 57-92, 1996.

146

[32] R. J. Hill, "Wider-Angle Parabolic Wave Equation," Journal of the Acoustic


[33] G. H. Knightly, D. Lee, and D. F. St. Mary, "A Higher-Order Parabolic Wave

Equation," Journal of the Acoustic Society of America, vol. 82, pp. 580-587,

1987.

[34] D. J. Thomson and N. R. Chapman, "A Wide-Angle Split-Step Algorithm for

the Parabolic Equation," Journal of the Acoustic Society of America, vol. 74,

pp. 1848-1854, 1983.

[35] D. J. Thomson, "Wide-Angle Parabolic Equation Solutions to Two Range-

Dependent Benchmark Problems," Journal of the Acoustic Society of America,

vol. 87, pp. 1514-1520, 1990.

[36] D. Yevick and D. J. Thomson, "Split-Step/Finite-Difference and Split-

Step/Lanczos Algorithms for Solving Alternative Higher-Order Parabolic

Equations," Journal of the Acoustic Society of America, vol. 96, pp. 396-405,

1997.

[37] P. P. Borsboom and M. F. Levy, "Scattering with Parabolic Equation Methods:

Application to RCS Computation," presented at IEE Colloquium on Common

Modelling Techniques for Electromagnetic Wave and Acoustic Wave

Propagation, London, UK, 1996.

[38] M. D. Collins and R. B. Evans, "A Two-way Parabolic Equation for Acoustic

Backscattering in the Ocean," Journal of the Acoustic Society of America, vol.

91, pp. 1357-1368, 1992.

[39] M. D. Collins, "A Two-Way Parabolic Equation for Elastic Media," Journal of

the Acoustic Society of America, vol. 93, pp. 1815-1825, 1993.

[40] M. F. Levy, "Parabolic Equation Modelling of Backscatter from the Rough

Sea Surface," presented at Target and Clutter Scattering and their Effects on

Military Radar Performance (AGARD-CP-501), Ottawa, Ont., Canada, 1991.

[41] M. F. Levy and P. P. Borsboom, "Radar Cross-section Computations using the

Parabolic Equation Method," Electronics Letters, vol. 32, pp. 1234-1236,

1996.

[42] A. E. Barrios, "Terrain Modelling using the Split-step Parabolic equation

Method," presented at International Conference Radar 92, Brighton, UK,

1992.

147

[43] A. E. Barrios, "A Terrain Parabolic Equation Model for Propagation in the

Troposphere," IEEE Transactions on Antennas and Propagation, vol. 42, pp.

90-98, 1994.

[44] M. F. Levy, "Parabolic Equation Modelling of Propagation over Irregular

Terrain," presented at Seventh International Conference on Antennas and

Propagation ICAP 91, York, UK, 1991.

[45] M. F. Levy, "Horizontal Parabolic Equation Solution of Radiowave

Propagation Problems on Large Domains," IEEE Transactions on Antennas

and Propagation, vol. 43, pp. 137-144, 1995.

[46] S. W. Marcus, "A Parabolic Approximation Method for Propagation

Prediction in an Inhomogeneous Atmosphere over Irregular Terrain,"

presented at 17th Convention of Electrical and Electronics Engineers in Israel,

Tel Aviv, Israel, 1991.

[47] C. Mattiello, "Use of Parabolic Equation for Computing Diffraction by Terrain

Undulations," Cselt Tech Rep, vol. 21, pp. 947-963, 1993.

[48] R. J. McArthur, "Propagation Modelling over Irregular Terrain using the Split-

step Parabolic Equation Method," presented at International Conference Radar

92, Brighton, UK, 1992.

[49] D. J. Donohue and J. R. Kuttler, "Modeling Radar Performance over Terrain,"

John Hopkins APL Technical Digest, vol. 18, pp. 279-287, 1997.

[50] C. C. Lin and J. P. Reilly, "A Site-Specific Model of Radar Terrain

Backscatter and Shadowing," John Hopkins APL Technical Digest, vol. 18,

pp. 432-447, 1997.

[51] M. F. Levy, Parabolic Equation Methods for Electromagnetic Wave

Propagation, 1 ed. London: IEE, 2000.

[52] R. A. Walker, "Numerical Modelling of GPS Signal Propagation:

Development of a Numerical Electromagnetic Wave Propagation Model for

the Modelling of GPS Positioning," Space Centre for Satellite Navigation,

SIDC Technical Report 002-95, 1996.

[53] R. A. Walker, "Operation and Modelling of GPS Sensors in Harsh

Environments," in School of Electrical and Electronic Systems Engineering:

Queensland University of Technology, 1999.

148

[54] R. R. Greene, "The Rational Approximation to the Acoustic Wave with

Bottom Interaction," Journal of the Acoustic Society of America, vol. 76, pp.

1764-1773, 1984.

[55] M. Cayer, B. Philibert, M. Lecours, and D. Dion, "Analysis of the Fourier

Split-Step Method for Resolution of Radio Propagation Over the Sea,"

presented at 1994 Canadian Conference on Electrical and Computer

Engineering, Halifax, Canada, 1994.

[56] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing.

Englewood Cliffs: Prentice Hall, 1989.

[57] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, "Parabolic

Equations," in Computational Ocean Acoustics, R. T. Beyer, Ed. New York:

AIP Press, 1994.

[58] M. F. Levy, "Transparent Boundary Conditions for Parabolic Equation

Solutions of Radiowave Propagation Problems," IEEE Transactions on

Antennas and Propagation, vol. 45, pp. 66-72, 1997.

[59] J. R. Kuttler and G. D. Dockery, "Theoretical Description of the Parabolic

Approximation/Fourier Split-Step Method of Representing Electromagnetic

Propagation in the Troposphere," Radio Science, vol. 26, pp. 381-393, 1991.

[60] G. D. Dockery and J. R. Kuttler, "An Improved Impedance-Boundary

Algorithm for Fourier Split-Step Solutions of the Parabolic Wave Equation,"

IEEE Transactions on Antennas and Propagation, vol. 44, pp. 1592-1599,

1996.

[61] A. E. Barrios, "Terrain and Refractivity Effects on Non-Optical Paths,"

presented at AGARD Electromagnetic Wave Propagation Panel Symposium,

Rotterdam, The Netherlands, 1993.

149

Chapter 5 Time Series Analysis with the FSPE

In the previous chapter the FSPE method for modelling GPS signal propagation was

introduced. The major limitation of the full-field FSPE solution is the lack of defining

multipath information. The FSPE provides only the solved field value within the

solution domain. The defining parameters of relative delay, amplitude, and phase are

fully incorporated within the solution. This defining multipath information can,

however, be extracted from the FSPE field results. This chapter introduces the PE-

based Time Analysis (PETA) model, for extraction of the multipath parameters from

the FSPE full-field solution.

5.1 Introduction

The determination of relative propagation path delays is important in the

understanding of GPS multipath errors. The superposition of delayed replicas of the

direct ranging signal leads to distortion of the signal at the GPS receiver antenna [1].

The receiver requires an undistorted signal to provide an accurate estimate of the

pseudorange to the satellite. In trying to understand the impact these multipath signals

have on the receiver it is necessary to characterise the multipath signal. Two of the

most important multipath parameters are; the relative time delay between the direct

and multipath signals, and the relative amplitude of the delayed signals[2, 3].

Together these parameters form the Multipath Channel Impulse Response (MCIR)

[4].

Determining the MCIR is an example of a pulse propagation problem which can be

solved, via the frequency domain, by the use of Fourier synthesis of the modelled

FSPE results [5]. This method is attractive since the FSPE propagation model that has

been developed for analysis of GPS signals lends itself readily to processing by this

technique. The technique involves Fourier synthesis based on a number of FSPE

calculations over a band of frequencies and the solution of the time-dependent field

150

equation can be obtained by the Fourier transformation of the PE field solution,

namely:

( ) ( ) ( ) dfefzxufStzxu ftj π2,,,, ∫∞

∞−

= (5.1)

where S(f) is the spectrum of a source pulse and u(x,z,f) is the spatial transfer function

derived from the FSPE modelling process. This integral is evaluated using Fast

Fourier Transform (FFT) techniques at the spatial point of interest in the model

solution domain, that is, the antenna location. For this work we have chosen as our

source, a sinc pulse—of a duration to be determined by modelling requirements—

modulated at the GPS L1 frequency. The MCIR is the output of the PE-based Time

Analysis (PETA), and is given as a time series of time delayed, and attenuated source

pulses. The factors for determining the computational load are made clear as the

technique is further explained.

5.2 Implementation

Consider a time-domain sinc pulse as depicted in Figure 5.1.

Figure 5.1 — SINC pulse

151

This sinc pulse is defined as:

( ) ( )Wt

WtWth

ππ

2

2sin2= (5.2)

here the function ( )usinc is defined as ( )u

u

ππsin

.

Now n/2W (-∞<n<∞) represents the zero crossings of the pulse. The Fourier transform

[6] of this pulse is given by

( ) ( ){ }

=

W

fWtWFfH

2rect=2sinc2 (5.3)

here rect is defined as rectangular function of frequency such that:

( )elsewhere0

2

11rect

=

<= xx

and W is defined as the half bandwidth of the signal.

The selection of a sinc pulse as the input signal guarantees that the spectrum of input

frequencies require no modification to their respective amplitudes, thus simplifying

the implementation. Since GPS signal propagation is to be modelled, the function is

shifted such that it is centred at the GPS L1 frequency—1.575.42 GHz (f1). This

frequency shift is achieved by convolving the rectangular function with a delta

function at f1. The inverse Fourier transform of the shifted rectangular function is the

sinc pulse modulated by 12 fje π . The frequency spectrum is now given by

( ) ( ){ }

−=

W

ffeWtWFfS fj

2rect=2sinc2 12 1π (5.4)

152

Figure 5.2 shows the complete source frequency spectrum.

f1 f1+Wf1-W

|S(f)|

0f

Figure 5.2 — Input spectrum

Assuming now, that the time response is required at some spatial point p(x, z), in a

two-dimensional, frequency-domain, electromagnetic propagation model, and that the

time response will be modelled within a time window of length T; the time and

frequency axes can be discretised as

( )11,0,min −=∆+= Nktkttk K

and

( )( ) ( )12/1,0,1112/, −−−−−=∆= NNlflf l KK

The first zero crossing points of the sinc pulse are used to define its pulse width. The

relationship between the time domain pulse width (τ) and the bandwidth (2W) of the

rectangular frequency function is given by

W =1

τ(5.5)

Since the half-bandwidth cannot exceed the central frequency, the time-domain pulse

width satisfies:

153

nse

63.0957542.1

1 =≥τ

Therefore for the modelling of GPS L1 pulse propagation a pulse cannot be smaller

than 0.63 nanoseconds in width at the first zero crossing points. In this work one

nanosecond is selected as an appropriate pulse width in which to resolve separately

propagated pulses.

Now consider the spacing of samples in the frequency domain:

∆ fT

=1

(5.6)

With the frequency spacing now known, the total number of frequencies ( M )

required, the sample number (L) of the mid-spectrum frequency (f1), and the required

value of each discrete frequency can be found. The total number of frequencies

required is given by,

∆

=f

WM

2round (5.7)

and the sample number of the mid-spectrum frequency,

∆

=f

fL 1round (5.8)

The minimum frequency can now be defined by,

fM

Lf ∆

−=

2min (5.9)

154

giving the required discrete frequency spectrum,

Mkfkff k K1,0,min =∆+= (5.10)

To satisfy the Nyquist sampling criterion the sampling frequency in the time domain

must be at least twice the maximum frequency of the source spectrum.

max2

1

ft =∆ (5.11)

where

fM

Lf ∆

+=

2max (5.12)

The time window length is given by

tNttT ∆=−= minmax (5.13)

In order to use efficient FFT algorithms it is necessary that N is a power of 2. For a

specified window length T the minimum number of time samples, N, is found.

t

TN

∆=min (5.14)

The next highest power of two is then selected as the required sample number.

min2 2 NN x ≥= (5.15)

155

Finally it is now necessary to calculate the time-domain sample spacing.

2N

Tt =∆ (5.16)

5.3 Domain Considerations

Now consider a solution domain (in range and height) that contains a portion of the

spatial plane x, z, referenced to the origin and bounded by zmax and xmax, Figure 5.3.

This domain is an arbitrary selection but is chosen to represent, as closely as possible,

a two-dimensional electromagnetic domain in which the propagation occurs. The

domain can include elements of terrain [7].

θi

ζζ’

θr

z

x

E+ E-

H+ H-

domainzmax

xmax(0,0)

Figure 5.3 — Propagation domain

In a simplified plane-wave model representation, the lower boundary, represented by

the x-axis, is considered a perfect conductor, and as such all EM energy that impinges

is reflected according to the governing laws. Two directions are specified, such that

the incident and reflected waves have directions of travel, ζ and negative ζ’,

respectively. Therefore, for the case of a Transverse Electric (TE) plane wave, the

incident and reflected fields at any point x, z (z > 0) are given by [8]:

156

( )ii xzjkinc eEE θθ cossin +−

+= (5.17)

( )rr xzjkrefl eEE θθ cossin −

−= (5.18)

The condition imposed on the lower boundary (z = 0) is such that the field must be

zero for all x. Therefore the two amplitudes and angles must be equal giving the total

field anywhere in the domain as:

( ) ( )θθθθ cossin0

cossin0

xzjkxzjk eEeEE −+− −= (5.19)

Figure 5.4 illustrates the concept of the time reference spatial corrections necessary in

the implementation. Since the reference plane-wave source point in the domain is at

the origin (point A), an adjustment is required so that all of the calculated field values

are referenced, at t = 0, to the arbitrary left boundary of the domain (x = 0), for all

values of z. This is achieved by shifting the plane-wave reference from point A to

point C. The first step is to correct for the height of the antenna (A to B), then to

correct for the distance, from the arbitrary boundary, to the antenna (B to C).

157

planes of constant phase

arbitrary boundary

θ

∆z

∆x

∆z sinθ

∆x(sinθtanθ)

transmission distance referenced to (0,0) at t = 0

A

B

C

antenna location

Domain

Figure 5.4 — Corrections for spatial time reference

The required correction to the total field calculations is then of the form;

( )θθθ tansinsin xjkzjkc eeuu ∆∆= (5.20)

Here the calculated field is corrected for a zero time reference relative to the spatial

point where the line-of-sight signal enters the domain.

5.4 Deriving the MCIR from the PETA Model

These equations now form the basis for time series analysis where the FSPE

propagation model is executed for each of the frequencies and the spectrum recreated

from the resultant field values at the point of interest. The inverse FFT of the

constructed spectrum will result in the complex MCIR which is given by

( )∑=

−=M

iii tMCIR

1

sinc τ (5.21)

158

In estimating the relative amplitude and time delay from the MCIR a gradient ascent

technique is used to locate the peaks. The sample numbers of these peaks are then

used to find the value of amplitude and time delay, of the signal pulses, within the

selected time window. These respective parameters are given by

[ ]iki MCIRabs=α (5.22)

tkii ∆=τ (5.23)

Here ki is the sample number of the ith peak and ∆t is the time sample spacing.

The complex field in terms of the MCIR parameters, at a spatial point (x,z), is then

given by the addition of the decomposed plane waves.

( ) ( )∑=

++=M

i

ftji

ftjPETA

iieezx1

22 0, φππ αψ (5.24)

Here the first term represents the line-of-sight signal with a propagation time of t0,

from an arbitrary domain incident boundary at, x=0. The summation term represents,

the M multipath signals, where αi and ti represent respectively, the ith multipath

amplitude and time of arrival. The phase term, φi, is the resultant phase shift due to the

boundary reflection(s) for the ith multipath signal. This equation can be normalised by

assuming zero reference phase for the LOS signal. This normalisation is simply a

change from absolute time delay, as presented by the PETA, to relative time delay,

and is given by,

( ) ( )∑=

++=M

i

fjiPETA

iiezx1

21, φτπαψ (5.25)

159

where τi is the time delay relative to the LOS signal [9].

An example of time delay estimation error from the PETA based on an exact plane-

wave solution, and for a canonical geometric optics forward propagation problem, is


5 10 15 20 25 30-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Relative Delay Error

Del

ay e

rror

(ns

)


Figure 5.5 — Time delay estimation error

The mean relative time delay estimation error is 0.00 ns with a standard deviation of

0.02 ns.

5.5 Multipath Phase Information from the PETA Model

The MCIR directly provides estimates of αI and τi. The phase term, φi, is not so readily

extracted as it remains embedded in the complex MCIR information. The total MCIR

phase is a combination of the transformed frequency-domain phase of each

component making up the total PE field. This composite phase is dependant on the

distance travelled (time delay) and the phase upon reflection (if reflection has

occurred) of each plane wave present in the model. The determination of the phase

term, φi, can be achieved by taking the difference between the MCIR phase results

160

from the PETA and a reference phase produced by a plane-wave MCIR at the

estimated amplitude and time delay of the signal path under consideration. The

residual phase, for the given time delay, is then the phase variation resulting from

terrain interaction. This procedure can be written as

( )[ ] ( )[ ]ii kiiREFkiiiPETAi ,MCIRphase,,MCIRphase −= φφ (5.26)

Here the MCIR derived from the PETA is a combined function of amplitude, time

delay, and the phase term. The reference result differs in that it does not include the

unknown phase term and the differencing of the time-domain phases, at each pulse

peak, gives the desired phase term.

Estimation of the phase of the LOS signal, using the technique outlined, for a simple

forward propagation problem, is shown in Figure 5.6.

5 10 15 20 25 30-25

-20

-15

-10

-5

0

5

10

15LOS Phase

Pha

se (

degr

ees)


Figure 5.6 — LOS Phase Estimation Error

The mean is 0.39 degrees with a standard deviation of 5.41 degrees, where the true

phase for the LOS is zero degrees. For the single specular reflection the error in phase

estimation is shown in Figure 5.7.

161

5 10 15 20 25 30165

170

175

180

185

190

195

200

205Reflection Phase

Pha

se (

degr

ees)


Figure 5.7 — Forward Multipath Phase Estimation Error

In this case the mean phase is 180.3 degrees with a standard deviation of 5.56 degrees.

The expected phase of the reflected signal is 180 degrees for the effective reflection

coefficient.

5.6 Antenna Gain Pattern from Angle of Arrival Information

The multipath parameters derived from the PETA model provide relative amplitude,

time delay, and phase of each multipath component. To more fully model a realistic

situation it is necessary to include the effects of the GPS antenna gain pattern [10]. No

effective way of including the antenna gain pattern within the FSPE has yet been

found, and the gain pattern is presently implemented post-modelling.

The gain pattern specifies the relative gain of the antenna in a particular orientation,

namely horizontal or vertical. The horizontal gain pattern specifies a field distribution

factor in azimuth, while the vertical gain pattern specifies a field distribution factor in

elevation. For the two-dimensional PETA model it is only the latter that needs to be

considered. The incorporation of the antenna pattern gives the field as [11]:

162

( ) ( ) ( ) ( )∑=

++=M

i

ftji

ftjPETA

iiefefzx1

22 0, φππ αθθψ (5.27)

where the term f(θ) represents the receiving antenna’s vertical gain factor. To

incorporate this factor it is necessary to have an estimation of the Angle of Arrival

(AOA) for each signal component. In practice this is achieved with an array of

antennas, such as the Multiple Emitter Location and Signal Parameter Estimation

technique introduced by Schmidt [12]. However, with the PETA model the results are

time-based and it is possible to determine the AOA from the relative time

information. This is achieved by selected a field cell spatially diverse from the

antenna location, and comparing relative time shifts, of each plane-wave component,

as provided by the PETA model. For a point vertically displaced from the antenna

location the AOA is given by

( )

−

∆= ∆+− zzx

izx

iAOA z

C ,,1sin ττϕ (5.28)

where ∆z is the vertical displacement, with its sign determining relative displacement

(positive for above, negative for below). The AOA thus derived is then used to

incorporate the antenna gain characteristics relative to each signal component in post

processing.

Examples of errors in AOA estimates, using this technique, are given respectively in

Figure 5.8 and Figure 5.9. For the LOS estimate the mean error is 0 degrees with a

standard deviation of 2.91 degrees.

163

5 10 15 20 25 30-8

-6

-4

-2

0

2

4

6

8LOS Angle of Arrival Error

AO

A e

rror

(de

gs)


Figure 5.8 — LOS AOA Estimation Error

5 10 15 20 25 30-8

-6

-4

-2

0

2

4

6Reflection Angle of Arrival Error

AO

A e

rror

(de

gs)


Figure 5.9 — Forward Multipath AOA Estimation Error

For a single forward reflection the error in AOA estimate has a mean of 0.01 degrees

with a standard deviation of 2.68 degrees.

164

5.7 PETA Implementation Issues

Some observations on resolution and computational load of this Fourier-based time

analysis technique are now made. Firstly the required resolution of the time domain

solution should be considered as necessary for the analysis being made. If the

resolution is not considered adequate then other parameters, of the technique, require

adjustment to achieve the required resolution. The most obvious way to increase the

time resolution is to increase the sample number. This can be achieved by

oversampling in the time-domain.

Secondly the effect of aliasing (periodicity of the time window T), introduced by the

discretisation in frequency, needs to be considered. The actual time response (in terms

of the inverse FFT), in the selected time window {tmin, tmin + T} is given by [5],

( ) ( )[ ]( )( )∑∑

≠

−

=

−− +−

∆=0

12/

0

2

2 ,,,,Re2,,2

2min

nk

N

l

N

tj

ftjllk nTtzxueefzxuftzxu

k

l

ππαπ (5.29)

The last term, of equation (5.29), represents the aliasing from the periodic time

windows. Although the time window needs to be made as small as possible, to reduce

the computational load of the technique, aliasing can only be avoided by making

certain the time window selected will contain all of the time response information.

These conflicting requirements can be met by selected tmin from the expected

propagation time requirements of the problem being analysed. The total length of the

time window is then selected, likewise, by considering the range and geometry of the

propagation environment. An example of a non-aliased time-window is shown in

Figure 5.10.

165

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

Absolute time delay (ns)

peak 1 delay: 037.0955 nspeak 2 delay: 045.5206 nstime delay: 8.4251 ns

Figure 5.10 — Non-aliased time response

We note the absolute propagation times are 37.0955ns for the LOS and 45.5206ns for

a single forward reflection. This is for the simulation of a spatial point 10 metres into

the modelling domain and 5 metres above the reflecting surface.

If the time-window is now halved in length aliasing will occur. This aliasing effect is

shown in Figure 5.11

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Absolute time delay (ns)


Figure 5.11 — Aliased time response

166

Here the LOS time of arrival is 12.3326ns aliased in the 25ns window such that the

true LOS time is the addition of the modelled delay and the aliasing window length,

giving 37.3326ns. Likewise the true reflection time is 45.7659ns, being the addition of

the modelled time and aliasing window length.

5.8 PETA Domain Representation and Performance

The propagation domain is represented by a two-dimensional plane that is specified

by the azimuthal direction to the satellite, the maximum height, and the maximum

range to be modelled. The antenna can be located at any point on the plane, above the

terrain. Terrain information can be derived from Digital Terrain Models (DTMs) [13].

The model domain is depicted in Figure 5.12.

Forward propagation

Back- propagation

Antenna

Satelliteazimuth

(0,zmax)

(0,0)(xmax,zmax)

(xmax,0)

Terrain

Incidentboundary

Satellite signal

Figure 5.12 — Domain representation

The definitions of forward and back-propagation are relative to the directions

specified in Figure 5.12.

167

Since the time information is essentially contained within the phase variation at each

frequency, it is necessary that the rapid phase variation be reinstated. This is achieved

by modifying the reduced PE field result as follows

( ) ( ) jkxezxuzx −= ,,ψ (5.30)

The phase must be reinstated not only in the forward direction but also for each

individual backscattering interface. This requirement can only be met if each back-

scatterer is treated as an individual FSPE propagation problem. The total field is then

simply the addition of the forward and backward fields.

The model simulation time for single frequency field values, with forward

propagation only, is given by the proportionality

AkT wayPE θ∝−1 (5.31)

where, k is the wave-number, θ is the propagation angle, and A is the area of the

domain plane. With inclusion of back-scatter this increases to

( ) wayPE

wayPE TLT −− += 12 1 (5.32)

for L back-scatterers. For the PETA the simulation time is

wayPE

pulse

winPETA TT −= 22

ττ

(5.33)

where τwin is the width of the time analysis window, and τpulse is the source pulse width.

168

5.9 Summary

In this chapter the PETA was introduced as an effective method for deriving the

MCIR and all relevant multipath parameters from the FSPE. It was shown that the

method is capable of accurately estimating the relative amplitude, time delay and

phase of the multipath signals.

The method as an adjunct to the FSPE provides a comprehensive modelling technique

for the analysis of GPS multipath propagation.

169

5.10 References

[1] L. R. Weill, "GPS Multipath Mitigation by Means of Correlator Reference

Waveform Design," presented at The National Technical Meeting of The

Institute of Navigation., Santa Monica, CA, 1997.







[4] H. Hashemi, "Impulse Response Modeling of Indoor Radio Propagation

Channels," IEEE Journal on Selected Areas in Communications, vol. 11, pp.

967-978, 1993.

[5] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, "Broadband

Modelling," in Computational Ocean Acoustics, R. T. Beyer, Ed. New York:

AIP Press, 1994.

[6] C. D. McGillem and G. R. Cooper, Continuous and Discrete Signal and

System Analysis, Third ed. Philadelphia: HRW Saunders, 1991.

[7] J. T. Hviid, J. B. Andersen, J. Toftgard, and J. Bojer, "Terrain-based

Propagation Model for Rural Area - An Integral Equation Approach," IEEE

Transactions on Antennas and Propagation, vol. 43, pp. 41-46, 1995.


Wesley, 1989.

[9] R. J. C. Bultitude, P. Melancon, H. Zaghloul, G. Morrison, and M. Prokki,

"The Dependence of Indoor Radio Channel Multipath Characteristics on

Transmit Receive Ranges," IEEE Journal on Selected Areas in

Communications, vol. 11, pp. 979-990, 1993.




170

[11] H. Bremmer, Terrestrial Radio Waves: Theory of Propagation. London:

Elsevier Publishing Company, 1949.

[12] R. O. Schmidt, "Multiple Emitter Location and Signal Parameter Estimation,"

IEEE Transactions on Antennas and Propagation, vol. 34, pp. 276-280, 1986.

[13] R. A. Walker, "Operation and Modelling of GPS Sensors in Harsh

Environments," in School of Electrical and Electronic Systems Engineering:


171

Chapter 6 Model Validation

In the previous chapters, the methodology for modelling GPS multipath propagation

was introduced, namely the Free Space Parabolic Equation (FSPE) model and the PE-

based Time Analysis (PETA) model. The combination of the FSPE and PETA

provides a comprehensive tool for the analysis and visualisation of the behaviour of

multipath propagation for terrestrial applications of GPS. In this chapter an analyse of

the results of the FSPE/PETA implementation is made. The FSPE-PETA modelling

system is implemented in MATLAB.

6.1 Validation of FSPE with an Exact Solution

With the MATLAB implementation of the FSPE completed, a validation of the results

is made by comparison with results from an exact field solution. By validating against

an exact plane-wave field solution any implementation errors are easily determined [1,

2]. The theory of the PE model is well tested [3, 4] and no attempt is made within this

work to retest the underlying validity of the PE as a EM modelling tool [5]. The

testing performed here is to show that the particular implementation of the FSPE and

the PETA, for modelling and simulation of GPS multipath propagation, is valid, and

that the models are suitable for their intended purpose within reasonable bounds of

accuracy [6].

6.1.1 Phase Error

Before completing the validation of the free-space parabolic equation propagation

model, it is worthwhile considering the problem of phase error in the standard PE

forms [7]. In chapter 3 the limitation of including the refractive index term in the

standard PE was examined. To highlight this limitation on angle of propagation, the

SPE (the standard parabolic equation) was implemented and compared to an exact

solution.

172

Figure 6.1 represents plots of the phase variation over range, in a narrow-angle

Tappert PE implementation [2] (dashed) and an exact plane-wave solution [8] (solid)

for propagation at 40 degrees over an ideal conductor.

0 10 20 30 40 50 60 70 80 90 100-4

-3

-2

-1

0

1

2

3

4SPE vs Exact Phase Variation over Range

range (m)

phas

e (r

ads)

exactSPE

Figure 6.1 — Phase error in Narrow-Angle SPE

The results of an error analysis shows a mean error -0.04 rads (-2.3 degs), and an error

standard deviation of 2.59 radians (148 degs). The obvious nature of the error is that it

is cumulative and increases as a direct function of range, as the SPE solution is

marched out in range. As mentioned previously the narrow-angle Tappert PE, or SPE,

is therefore unacceptable if accurate phase information is required at high propagation

angles.

The FSPE is now compared for the same propagation problem, see Figure 6.2.

173

0 10 20 30 40 50 60 70 80 90 100-4

-3

-2

-1

0

1

2

3

4

range (m)

phas

e (r

ads)

FSPE vs Exact Phase Variation over Range

exactFSPE

Figure 6.2 — Phase error in FSPE

There is no error in the FSPE solution with a mean error of 0.00, error standard

deviation 0.00 using the same error analysis as for the SPE. The FSPE as given by

equation (4.8 Chapter 4) is shown to be an exact solution for the GPS propagation

problem and is not limited by angle of propagation but achieves this by neglecting

refractive index effects.

For the GPS propagation problem — local terrestrial multipath, where we are within

less than 300 metres of the receiver — refractive index effects are negligible and can

be ignored. Results of comparison of the FSPE field and an exact solution is shown in

Figure 6.3.

174

5 10 15 20 25 30-70

-60

-50

-40

-30

-20

-10

0

10

Propagtion angle (degs)

Fie

ld L

evel

(dB

)

Signal Strength (x=25,z=3)

Exact FSPE (-30dB)

Figure 6.3 — FSPE Field vs Exact Solution

In this test example the signal strength has been simulated using both the FSPE and

exact plane-wave solution. The simulation is of a GPS satellite rising in elevation

(linearly) from 5 degrees to 30 degrees. The simulated antenna position is located 25

metres from the incident boundary and at a height 3 metres above the local terrain.

The FSPE solution has been offset by 30 dB for clarity of comparison.

Again, the FSPE solution agrees extremely well with the exact solution of the wave

equation [9].

6.2 Forward Multipath Propagation

6.2.1 Static Test

Having established that the FSPE is valid against an exact plane-wave solution we set

about testing the PETA against an equivalent exact solution in the time-domain — the

results derived from geometric interpretation of propagation problems [10].

175

The FSPE model was tested for the forward propagation case over a perfect

conductor. Here the domain size is 40 metres in range and height.

Figure 6.4 — Field over flat perfect conductor

The standing wave pattern represents the constructive and destructive interference of

the incident field and the ground reflections and is defined, in the interference region

[11], as

( ) ( ) ( )φθρθ +∆−+= Rkjri efEfE 00E (6.1)

It is interesting to note that for this geometry the distance between minima is 1.147

metres. Here f(θi) and f(θr) represent the field distribution, as a function of the angle of

incidence, and angle of reflection respectively. In terrestrial propagation these factors

would account for the transmission antenna’s radiation pattern. For the GPS satellite

propagation problem, these elevation angle-dependent pattern shape terms account for

the variation of received power due to the transmitting antenna array design as shown

in Figure 6.5.

176

Figure 6.5 — User Received Minimum L1 C/A Signal Level

The Fourier synthesis technique (PETA) was applied to the FSPE model and the

results for a spatial position, x = 20 m, z = 10 m, are shown in Figure 6.6.

Figure 6.6 — Time series for forward propagation

A comparison of results for the modelled and calculated delays is given in Table 6.1.

177

Delay Calculated (ns) Modelled (ns) Error (ns)

Forward 66.9213 66.9189 0.0024 (0.004%)

Forward Scatter 5.8104 5.8350 0.0246 (0.4%)

Table 6.1 — Modelling Errors Forward

It should be noted that the results are for propagation over a perfect conductor and

hence the multipath signals are of equal amplitude to the line-of-sight (LOS) incident

GPS signal. The results show excellent agreement between the PETA results and

those derived from geometric interpretation of the problem.

6.2.2 Dynamic Tests

We now consider the case of simulated GPS satellite motion and forward specular

reflection, as depicted in Figure 6.7,. Again in this multipath situation we have the

direct LOS signal and a single multipath signal arriving at the antenna.

5 m

Figure 6.7 — Dynamic situation

The modelling is for a GPS satellite rising in elevation from 5 degrees to 10 degrees.

Figure 6.8 and Figure 6.9 show, respectively, the calculated FSPE field, and the PETA

result for a propagation angle of 8 degrees.

178

Figure 6.8 — FSPE field at 8 degrees

This plot of the field strength again shows the classical interference region pattern,

with constructive and destructive interference clearly evident in height. The distance

between minima is calculated at 0.684 metres.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Relative time delay (ns)


Figure 6.9 — PETA result at 8 degrees

The time-domain analysis clearly shows the LOS and the multipath signals. Each of

the multipath parameters is extracted from the PETA results at 0.1 degree increments,

179

and by using equation (5.24 Chapter 5), we can reconstruct the total field. Figure 6.10

shows a comparison of the PETA estimated field compared to the full field solution as

given by the FSPE propagation model.

5 6 7 8 9 10-40

-35

-30

-25

-20

-15

-10

-5

0

5

10


Fie

ld a

mpl

itude

(dB

)

Figure 6.10 — FSPE field and reconstructed PETA field comparison

This figure again shows the classical fading pattern for a single multipath reflection.

The results from the PETA reconstruction are in good agreement with the full field

result given by the FSPE. having compared the PETA and FSPE field results we now

make a comparison of the PETA multipath relative amplitude with the implemented

reflection coefficient of concrete for an antenna height of 2 metres, F-mode

propagation from 1 to 25 degrees. The results are shown below in Figure 6.11.

180

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Reflection Coefficient Magnitude: circ(combined) - Concrete

Mag

nitu

de


PETA Calculated

Figure 6.11 — Comparison of RC Magnitudes

The reflection coefficient, as implemented in the FSPE and ultimately in the PETA, is

in strong agreement with the calculated reflection coefficient for this problem—LHCP

rejection of 6dB. The fade pattern for this problem is also tested against that of an

exact interpretation of the refection coefficient, Figure 6.12.

0 5 10 15 20 25-30

-25

-20

-15

-10

-5

0

5

10FSPE Fade Pattern

Fie

ld L

evel

(dB

)


FSPE Calculated Envelope

Figure 6.12 — Fade Pattern Comparison

181

Again the FSPE-PETA implementation of the boundary refection coefficients is in

excellent agreement with the expected results.

6.3 Forward Diffraction

6.3.1 Static Test

The FSPE propagation model was tested in the diffracted signal shadow area behind a

terrain block with a propagation angle of 2 degrees. The terrain block may represent a

building for the purpose of this study. The height of the terrain block is 20 metres and

is situated in the domain 20 metres from the left side boundary. The total domain

range is 60 metres and the domain height is 40 metres. The spatial point is selected as

being 19 metres above the baseline terrain and 50 metres in range giving a diffracted

ray angle of approximately 5 degrees. This places the spatial point in the geometric

optics shadow zone, a zone that is treated correctly for diffraction by this PE model.

The field plot for this situation is shown in Figure 6.13.

Figure 6.13 — Diffraction over terrain element

182

The diffractive effects that are expected can be quantified by considering the

geometrical arrangement represented here.

LOS

αβ

h

y1

y2

d2

d1y3

l

x1 x2

x3

Figure 6.14 — Diffraction geometry

The implementation of the model used for this problem calculates time delay relative

to the line-of-sight. The total distance modelled is the line-of-sight ray-path to the

intersection of the perpendicular marked h, (this distance is represented on the

diagram as l ), plus the distance from this point to the spatial point of interest in the

domain (distance d1) . The total diffracted ray path distance is l+d2, giving the path

difference as d2-d1. The relevant calculations are as follows.

The angle between the line-of-sight vector (propagation angle α) and the diffracted

ray is given by,

β α=

−−tan 1 2

2

y

x(6.2)

with

d x y2 22

22= + (6.3)

183

and

d d1 2= cosβ (6.4)

The relative time delay between the line-of-sight and the diffracted signal is then

given by

( )path delayd

c= −2 1 cosβ (6.5)

The path delay for this particular problem is calculated as 0.070226 ns. Time spacing

resolution for this modelling was 0.00244 ns.

Figure 6.15 shows the time series of the line-of-sight without terrain obstruction. Note

that the ground reflection is present in this plot.

Figure 6.15 — Time series no terrain

The amplitude for this plot has been normalised, with a maximum recorded value of

25.76e-3. The line-of-sight delay to the spatial point of interest is modelled at

184

166.7725 ns with forward-scatter delay of 4.4189 ns. Actual time delay calculations

for this case are, 166.7683 ns with forward-scatter delay of 4.4206 ns.

Modelling the time series with the inclusion of the terrain block resulted in the plot


Figure 6.16 — Time series with terrain

Again this plot has been normalised against the maximum amplitude, for this case

3.23e-3. The forward delay is seen to be 166.8457 ns. Note there is no dominant

ground reflection, as would be expected due to the terrain obstruction. However it is

interesting to note the reflected diffracted signal to the right of the diffracted solely

signal.

The modelled diffraction delay is simply the difference between the line-of-sight

propagation time and the propagation time of the diffracted signal. Therefore, the time

delay for this case is 0.0732 ns, this compares reasonably, with the calculated delay of

0.0702 ns. It must be remembered however that the time resolution for this particular

case is ±0.00244 ns.

185

6.3.2 Dynamic Test

We now include satellite dynamics and present the case of a GPS satellite rising over

a terrain obstruction, from an initial elevation angle of 5 degrees to a final angle of 15

degrees. In this case we can expect diffraction effects to dominate. The situation is

depicted in Figure 6.17.

5 m

3 m

2 m

Figure 6.17 — Dynamic diffraction situation

A plot of the instantaneous PE field, for a propagation angle of10 degrees, is given in

Figure 6.18.

Figure 6.18 — FSPE field at 10 degrees

186

Diffractive effects are evident behind the terrain obstacle. These diffractive effects are

clearly presented in Figure 6.19.

5 10 15 20 25 30 35 40 45-20

-15

-10

-5

0

5Diffracted Field Value

Fie

ld a

mpl

itude

(dB

)


Figure 6.19 — FSPE field 5 degrees to 45 degrees

The field result for an elevation change of 5 degrees to 15 degrees, exhibits the usual

features of diffraction by an obstacle, with field strength rising, overshooting and

oscillating about the 0dB LOS level. At about 35 degrees we see the diffractive effects

reducing and ground reflection interference starting to dominate.

187

5 10 1523.4

23.5

23.6

23.7

23.8

23.9

24

24.1

24.2

24.3Diffraction Time Delay

Tim

e de

lay

(ns)


Figure 6.20 — LOS and diffracted propagation time comparison

Figure 6.20 presents a comparison of the delay of the diffracted signal (upper plot) to

that of the unobstructed line-of-sight (lower plot) as provided from the PETA results.

This clearly indicates the additional path length caused by the diffraction of the signal

around the terrain edge. If a receiver has a dynamic range of better than 20 dB then it

is able to acquire and maintain track of the diffracted signal. At 5 degrees the

diffracted path delay is 0.2 ns representing approximately a 6 cm range error. The

convergence of the curves corresponds to the fact that the path becomes line-of-sight

as the satellite rises.

6.4 BA/BB-Mode: Backscatter

This section considers a two-way FSPE model developed from the one-way model

with the inclusion of backscatter. Results are presented for the static case and for

satellite motion.


188

6.4.1 Static Test

Figure 6.21 shows the field for a 5 degree forward (one-way) GPS signal, propagating

over simple terrain elements. The interference region (standing wave) pattern,

generated by the incident and reflected fields, is clearly evident in the unobstructed

areas of the domain. The shadowing/diffraction effects of the propagation over the

terrain blocks is also shown.

Figure 6.21 — Forward propagation over terrain

The back propagation for this example is shown in Figure 6.22. Here the field values

are for the back reflected field only. The interference pattern in front of the block at

the 40 metre range is as expected. There is no field after the 100 metre range since

there are no reflected interfaces after this point. The amplitude of the reflected field

components has been arbitrarily set at 80% of the incident components at the

reflecting boundary and with a 180 degree phase shift.

189

Figure 6.22 — Back propagation from reflected interfaces

The total field (two-way) propagation for this example is shown in Figure 6.23. This

is resultant of the addition of the forward and back fields.

Figure 6.23 — Total propagated field

The time delay analysis using the Fourier synthesis technique is now tested with a

domain consisting of a single 20 metre high vertical reflector located at a 20 metre

range point. The total resultant two-way field is shown in Figure 6.24.

190

Figure 6.24 — Two-way field with 20 m high vertical reflector at 20m

The time series for the spatial point, x = 10 m, z = 10 m, is shown below—Figure

6.25.

Figure 6.25 — Time series back-propagation

The stored and reflected field components were arbitrarily set to 80% of the incident

field level. This was done to provide clarity in the normalised plot of Figure 6.25, and

also to demonstrate that the backscatter algorithm can implement reflection

coefficient conditions on the reflecting boundary. This will allow the reflections, as

191

implemented in the backscatter algorithm, to have the correct amplitude and phase

variation, as determined by the electrical characteristics of the modelled terrain.

A comparison between calculated and modelled results is given in Table 6.2. Here

Backscattera refers to backscatter arriving from above the horizontal, and Backscatterb

refers to backscatter arriving from below.

Delay Calculated (ns) Modelled (ns) Error (ns)

Forward 33.4607 33.4961 0.0354 (0.1%)

Forward Scatter 5.8104 5.7617 0.0487 (0.8%)

Backscattera 66.4130 66.4062 0.0068 (0.01%)

Backscatterb 72.2234 72.2656 0.0422 (0.06%)

Table 6.2 — Modelling Errors Forward and Back

The modelled domain is shown below in Figure 6.26.

10 m 10 m

20 m

10 m

reflector

forward Bb

Ba

LOS

Figure 6.26 — Model domain

192

6.4.2 Dynamic Test

To demonstrate satellite dynamics we present a more complicated arrangement that

may occur in urban environments. The situation is depicted in Figure 6.27.

5 m 5 m

6 m

2 m

2 m

5 m

B1

B2

Figure 6.27 — Stepped backscatter geometry

We call this a stepped backscatter, where in addition to the forward scatter, we have

signals scattered in the reverse propagation direction, from two distinct interfaces.

Figure 6.28 shows the time-domain results for a 5 degree propagation angle. This

modelling case presents all elements of the previous propagation examples, namely;

forward reflection, backscatter and diffractive effects.

193

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ised

am

plitu

de

Propagation angle - 5 °

peak 1 delay: 016.7383 nspeak 2 delay: 019.6484 nspeak 3 delay: 049.9609 nspeak 4 delay: 052.8613 ns

Figure 6.28 — Stepped backscatter PETA results for 5 degrees

Here we see the LOS, forward scatter, and two additional multipath signals reflected

from interface B1. The first of these multipath signals is identified as backscatter from

above, that is, the LOS is reflected from the B1 interface and arrives at the antenna

location from a positive elevation angle. The next signal is backscatter from below,

and is reflection of the LOS from a combination of ground and interface. At 5 degrees

refection the B2 interface is obstructed by the B1 step. Close examination of Figure

6.28 shows some low level signal from the B2 interface, but diffractive effects have

reduced its influence.

The propagation mechanisms in this situation become evident at higher propagation

angles. In Figure 6.29 we show the PETA results for 12.5 degrees.

194

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ised

am

plitu

de

Propagation angle - 12.5 °

peak 1 delay: 017.1680 nspeak 2 delay: 024.3457 nspeak 3 delay: 049.7168 nspeak 4 delay: 056.8945 nspeak 5 delay: 062.7148 ns

Figure 6.29 — Stepped backscatter PETA results for 12.5 degrees

Here we see the expected variation in influence of the reflecting interfaces. The

multipath signal from B1 (above) is now affected by diffraction effects, and the

multipath signal from B2 (above) is beginning to dominate. At 15 degrees (Figure

6.30) the effect is more pronounced and the reflection from B2 is essentially line-of-

sight. We note the reflection from B1 (below) is unaffected, and that there is reflection

from B2 (below).

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ised

am

plitu

de

Propagation angle - 15 °

peak 1 delay: 017.3828 nspeak 2 delay: 025.9863 nspeak 3 delay: 049.5996 nspeak 4 delay: 058.1836 nspeak 5 delay: 062.5000 ns

Figure 6.30 — Stepped backscatter PETA results for 15 degrees

195

The total multipath propagation situation is presented in Figure 6.31. Here we see the

full influence of the diffractive effects for this situation.

3040

5060 70

80

2010

05

15

10

Time Delay (ns)

PropagationAngle (degs)

Figure 6.31 — Stepped backscatter PETA results for 5 to 15 degrees

We now compare these diffraction results to the theory of knife-edge diffraction. The

edge represented is the top of the B1 interface with geometry as shown below in

Figure 6.32.

5 m 2 m

5 m

T

R

d1

d2

h

Figure 6.32 — Stepped backscatter knife-edge geometry

In this representation the energy reflected from the B2 interface is assumed to act as

an electromagnetic radiation source, hence it appears as a transmitter (T). The receiver

196

(R) is located at the indicated spatial position, with d1 and d2 respectively

representing the distance of transmitter and receiver from the diffracting knife-edge. It

is now possible to apply the CCIR recommended formula [13] for knife-edge

obstacles, which is given by

+=

21

112

ddhv

λ(6.6)

which, for large values of d1 reduces to

v hd

=2

2λ(6.7)

For v greater than –1 the approximate value of loss in decibels is given by

( ) ( )

−++−+= 1.011.0log209.6 2 vvvJ (6.8)

A plot of the diffraction loss for the given geometry is shown in Figure 6.33.

11 11.5 12 12.5 13 13.5 14 14.5 150.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Knife Edge Diffraction Loss

Rel

ativ

e am

plitu

de


Figure 6.33 — Diffraction loss over B1 interface


197

We note from Figure 6.33 that the propagation is not line-of-sight until about 14.4

degrees. A plot of the normalised CCIR diffraction loss is made in comparison with

the loss as given by the PETA results, Figure 6.34.

5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Back-Scatter Amplitudes

Rel

ativ

e am

plitu

de


Figure 6.34 — Normalised diffraction loss PETA vs CCIR for B1 interface

Here we see that for the B1 interface diffraction, the CCIR (circles) and PETA (line)

results are in good agreement. The additional plot given in Figure 6.34 is the PETA

diffraction results for the B2 interface.

The total plot of PETA derived time delays, for the stepped backscatter situation, is



198

0 10 20 30 40 50 60 70 805

6

7

8

9

10

11

12

13

14

15Multipath Time Profile

Pro

paga

tion

angl

e (d

egs)

Time dealy (ns)

Figure 6.35 — PETA path delays

Starting from the left we have the LOS, forward scatter, B1 backscatter (above), B1

backscatter (below) and B2 backscatter (above) delay profiles. Results are now

presented for the reconstruction of the total field from the PETA results. In Figure

6.36 the reconstructed field (diamonds) shows excellent agreement with the FSPE

field result.

5 10 15-20

-15

-10

-5

0

5

10

15


Fie

ld a

mpl

itude

(dB

)

Figure 6.36 — Reconstructed PETA field and FSPE field comparison

Time delay (ns)

199

The RMS error is 1.953 dB. A comparison of the FSPE results to those expected by

geometric optics (GO) is now shown in Figure 6.37. The geometric optics solution is

simply the field calculated from the trigonometric ray paths and their relative delays,

which do not account for any diffraction. Here the RMS error is 4.8391 dB and it is

obvious that by neglecting diffraction effects large errors are produced in the

estimation of the field for this situation.

5 10 15-25

-20

-15

-10

-5

0

5

10

15


Fie

ld a

mpl

itude

(dB

)

Figure 6.37 — Reconstructed GO field and FSPE field comparison

6.5 Summary

Results were presented that show the validity of the FSPE and PETA in a variety of

static and dynamic GPS modelling scenarios. The visualisation of the full-field and

the time-domain results given by the FSPE-PETA model were shown to give an

insight into the behaviour of GPS multipath in a variety of static and dynamic GPS

simulations.

200

6.6 References

[1] G. D. Dockery, “Development and Use of Electromagnetic Parabolic Equation

Propagation Models For Us Navy Applications,” Johns Hopkins APL

Technical Digest, vol. 19, pp. 283-292, 1998.

[2] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, “Parabolic

Equations,” in Computational Ocean Acoustics, R. T. Beyer, Ed. New York:

AIP Press, 1994.

[3] G. D. Akrivis, V. A. Dougalis, and N. A. Kampanis, “Error Estimates for

Finite Element Methods for a Wide-Angle Parabolic Equation,” Applied

Numerical Mathematics, vol. 16, pp. 81-100, 1994.

[4] G. D. Akrivis, V. A. Dougalis, and G. E. Zouraris, “Error Estimates for Finite-

Difference Methods for a Wide-Angle Parabolic Equation,” SIAM Journal of

Numerical Analysis, vol. 33, pp. 2488-2509, 1996.

[5] M. A. Leontovich and V. A. Fock, “Solution of the Problem of Propagation of

Electromagnetic Waves along the Earth's Surface by the Method of Parabolic

Equations,” Journal of Physics of the USSR, vol. 10, pp. 13-24, 1946.

[6] M. D. Collins, “Comparison of algorithms for Solving Parabolic Wave

Equations,” Journal of the Acoustic Society of America, vol. 100, pp. 178-182,

1996.

[7] M. D. Collins, “A Split-Step Padé Solution for the Parabolic Equation

Method,” Journal of the Acoustic Society of America, vol. 93, pp. 1736-1742,

1993.

[8] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in

Communication Electronics. New York: John Wiley & Sons, 1984.

[9] R. A. Walker, “Operation and Modelling of GPS Sensors in Harsh

Environments,” in School of Electrical and Electronic Systems Engineering:


[10] M. Ando, “The Geometrical Theory of Diffraction,” in Analysis Methods for

Electromagnetic Problems, E. Yamashita, Ed. Boston: Artech House, 1990,

pp. 213-242.

201

[11] D. E. Kerr, “Propagation of Short Radio Waves,” . Boston: Boston Technical

Publishers, 1964.

[12] J. Doble, Introduction to Radio Propagation for Fixed and Mobile

Communications. Boston: Artech House, 1996.

[13] CCIR International Radio Consultive Commitee, “Report 715-2 Propagation

by Diffraction,” International Telecommunications Union, Dubrovnik 1986.

202


203

Chapter 7 Simulation of GPS Propagation

In the previous chapters the FSPE and PETA were introduced as novel methods for

the modelling and simulation of GPS multipath propagation. In this chapter we

examine collected GPS data and FSPE simulations. The complete data results can be

found in Appendix B.

7.1 Introduction

The previous chapter validated the FSPE against an exact solution; validated the

PETA to geometric optics; and validated the PETA to the FSPE field results. This

combinational validation allows us now to compare the FSPE to measured GPS signal

data, with confidence in the underlying modelling techniques used for the following

simulation cases.

7.2 Candidate Simulation Cases

A data collection exercise was initiated for testing of the FSPE model. Several

candidate sites were examined and it was determined the most appropriate multipath

data could be collected in a classic F-mode situation. Water features were considered

the most appropriate as the surface would be uniform and only affected by wind

turbulence. This turbulence would generate rough surface effects [1] and for pure

specular reflection a calm situation was required [2].

With this in mind — and with a desire to test different reflection coefficients — two

sites were selected; one a fresh-water feature and the other a flat grass playing field .

The comparison of the results of the GPS signal data and the FSPE simulations are

given in the following sections. The full data results can be found in Appendix B, as

well as data for a third site at Bribie Island for the investigation of sea water multipath

and tidal variation effects.

204

7.2.1 Pine Dam

The first data collection exercise was made at Pine Dam, north of Brisbane in

Queensland, Australia. The large fresh-water surface provided an excellent

environment for collection of specular multipath data. The data was collected on a day

with very little wind, thus the surface was considered specular, and surface roughness

effects could be neglected. In Figure 7.1 the set-up for the data collection at Pine Dam

is shown.

Figure 7.1 — Pine Dam Data Collection Site

The antenna was mounted on a tripod with the antenna oriented vertically, such that

the full antenna gain pattern, see Figure 7.2, incorporated the expected specular

multipath reflection [3]. The height of the antenna above the water level was

measured at approximately 1.26 metres.

205

Water Feature

AntennaGain Pattern

LOS

MP

1.26m

Figure 7.2 — Data Collection Basis

The orientation of the antenna was selected such that as much of the water feature as

possible was within the radiation pattern of the GPS receiver antenna. The bore-sight

orientation was also selected for the greatest horizontal distance across the water

feature before terrain was encountered in the opposite shoreline. In this way the

maximum number of satellites would be propagating within the known modelling

domain. The orientation of the GPS receiver antenna is illustrated in Figure 7.3 below.

N

Shoreline

Antenna Boresight

Water Surface

NW

Figure 7.3 — Location Orientation

206

The limit in azimuth that is permitted for reflection off the water surface is

approximately SSW to approximately NNE. This represents from approximately –165

degrees to approximately +25 degrees azimuth angle range.

Data was collected using an Ashtech Z-Surveyor GPS receiver and the Ashtech

700700 L1/L2 antenna. The data collection interval was set at 1 second epochs. The

data was analysed using a combination of TEQC from UNAVCO [4] and MATLAB

scripts. After approximately 2 hours of data collection four satellites were seen to

have maintained full specular reflection with the desired surface. The three satellites

were SV’s 17, 6, 8, and 9.

Before going further it is worthwhile noting that it is usual and correct to term the

signal strength reported by a GPS receiver as carrier-to-noise C/No, which is typically

in units of dB-Hz, thus inferring some knowledge of the tracking bandwidth. This

refers to the fact that the GPS signal is direct sequence spread spectrum and the

receiver estimates the C/No since it is a stochastic process. .In this work we are

seeking to validate the FSPE and hence the exact nature of the signal is not important,

and we revert back to the more generic term of signal-to-noise.

In addition, the trend of the power received at the GPS receiver is a function of many

variables and as such is difficult to estimate. In this validation no attempt has been

made to trend the simulated incident power other than the power variation specified in

the Standard Positioning Service specification. The simulated values (which are

inherently normalised to a SNR of 0 dB) are shifted in relative terms by an estimated

received incident signal strength.

The simulation results for SV 17 are shown below in Figure 7.4.

207

0 500 1000 1500 2000 2500 3000 3500 4000 450042

44

46

48

50

52

54

Epoch Samples

S/N

(dB

)

S/N Comparison SV17-Fresh Water

MeasuredModelled

Figure 7.4 — SV17 Results

The total data simulated in this case was for a total period of 4500 epochs or 75

minutes. For the first 3000 epochs or 50 minutes the simulation is in excellent

agreement with the data. The LHCP rejection ratio was selected to be 11 dB (typical

range for a 3dB axial ratio specification). SV 17 is seen to be rising from

approximately 10 degrees to approximately 32 degrees in elevation for the simulation

period, Figure 7.5. In azimuth the SV is moving from North-West to the West,

starting approximately in-line with the bore-sight at North-West.

0 500 1000 1500 2000 2500 3000 3500 4000 45000

10

20

30

40Az-El SV17

Ele

vatio

n (d

eg)

0 500 1000 1500 2000 2500 3000 3500 4000 4500-80

-70

-60

-50

Azi

mut

h (d

eg)

Figure 7.5 — SV17 AZ-EL Data

208

The geometric point of reflection; which represents the centre of the Fresnel zone is

very close to the antenna, ranging in distance from approximately 7 metres to 2 metres

horizontally distant from the antenna for the observation period, Figure 7.6.

0 500 1000 1500 2000 2500 3000 3500 4000 45000

5

10Specular Reflection Point SV17 :antenna height = 1.26m

Ran

ge D

ista

nce

(m)

0 500 1000 1500 2000 2500 3000 3500 4000 45000.5

1

1.5

1st

Zone

Rad

ius

(m)

0 500 1000 1500 2000 2500 3000 3500 4000 45000

5

10

sem

i-maj

or a

xis

(m)

Measurement Epochs

Figure 7.6 — SV17 Fresnel Data

We note the Fresnel zone dimensions are quite small. The differing value of the

simulation and data from epoch 3000 appears to be a variation in the received signal

strength of SV17, with an apparent downward trend. The result could also be a factor

of the proximity of the shore line impinging within the Fresnel zone of the higher

elevation angles. Any variation in the surface condition; height, reflection coefficient,

or surface roughness will affect the simulation results.

The second set of simulation data is that for SV6, Figure 7.7.

209

0 500 1000 1500 2000 2500 3000 3500 400020

25

30

35

40

45

50

Epoch Samples

S/N

(dB

)


MeasuredModelled


We note that the fades in the signal strength data recorded, are of such depth that the

tracking loop has in fact lost lock on two occasions. The first is approximately

between epochs 510 and 800, with the second occurring between epochs 2800 and

3200. We note that the loss of tracking exists for approximately 5-6 minutes on each

occasion.

The simulation again shows excellent agreement with the data. We note that this SV

is also rising in elevation (approximately 25 degrees to 42 degrees), and moving

approximately from South-West to approximately South-South-West, Figure 7.8.

210

0 500 1000 1500 2000 2500 3000 3500 400020

25

30

35

40

45Az-El SV6

Ele

vatio

n (d

eg)

0 500 1000 1500 2000 2500 3000 3500 4000-150

-140

-130

-120

Azi

mut

h (d

eg)


As for the previous SV the Fresnel zone becomes progressively smaller and closer to

the antenna location as the SV rises in elevation angle, Figure 7.9. The slight variation

in the simulation results near the last peak in the fading pattern would appear to be the

local variable terrain being included in the reflective properties of the water surface

and affecting the multipath signal slightly.

0 500 1000 1500 2000 2500 3000 3500 40001

2


Ran

ge D

ista

nce

(m)

0 500 1000 1500 2000 2500 3000 3500 40000.4

0.6

0.8

1st

Zone

Rad

ius

(m)

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5

2

sem

i-maj

or a

xis

(m)

Measurement Epochs


211

Our third simulation is for SV8, Figure 7.10.

0 500 1000 1500 2000 2500 3000 3500 4000 450044

46

48

50

52

54

56

Epoch Samples

S/N

(dB

)


MeasuredModelled


The simulation provides excellent agreement with the results from epoch 1700. We

again note that at the higher propagation angles that there is some difference between

the simulation and the actual received signal strength results.

0 500 1000 1500 2000 2500 3000 3500 4000 450010

15

20

25

30

35Az-El SV8

Ele

vatio

n (d

eg)

0 500 1000 1500 2000 2500 3000 3500 4000 4500-70

-60

-50

-40

-30

Azi

mut

h (d

eg)


212

This would appear to indicate that the closeness of the reflection zone — at the higher

elevation angles — to the antenna is again the underlying cause. This is apparent since

the relative peak-to-fade of the simulation and the data does not vary significantly. If

there is a variation between the simulated and the actual reflection coefficient then

this is indicated within the comparison as variation in the depth of the fades. The only

other probable cause for the variation in the simulation, is a variation in the received

signal power at this elevation angle range. This is possible, but would be extremely

difficult to include in a propagation simulation unless the complete power profile is

available.

0 500 1000 1500 2000 2500 3000 3500 4000 45002

4

6


Ran

ge D

ista

nce

(m)

0 500 1000 1500 2000 2500 3000 3500 4000 45000.5

1

1.5

1st

Zone

Rad

ius

(m)

0 500 1000 1500 2000 2500 3000 3500 4000 45000

2

4

6

sem

i-maj

or a

xis

(m)

Measurement Epochs


The final simulation for Pine Dam is shown below in Figure 7.13, for SV9.

213

0 500 1000 1500 2000 2500 3000 3500 400038

40

42

44

46

48

50

52

54

56

Epoch Samples

S/N

(dB

)


MeasuredModelled


Again the simulation using the FSPE is in excellent agreement with the recorded data.

The SV is setting, starting at approximately 20 degrees before disappearing below the

horizon. The SV is moving across from approximately the bore-sight of the antenna at

NW around to the North, Figure 7.14.

0 500 1000 1500 2000 2500 3000 3500 40000

5

10

15

20

25Az-El SV9

Ele

vatio

n (d

eg)

0 500 1000 1500 2000 2500 3000 3500 4000-25

-20

-15

-10

-5

0

Azi

mut

h (d

eg)


214

We also note that the Fresnel zone dimensions are increasing and the distance to the

zone is increasing, Figure 7.15.

0 500 1000 1500 2000 2500 3000 3500 40000

50


Ran

ge D

ista

nce

(m)

0 500 1000 1500 2000 2500 3000 3500 40000

2

4

1st

Zone

Rad

ius

(m)

0 500 1000 1500 2000 2500 3000 3500 40000

100

200

sem

i-maj

or a

xis

(m)

Measurement Epochs


7.2.2 Caboolture Soccer Field

The second data collection and simulation exercise was undertaken at a grass field

located in the township of Cabooture. The experimental set-up and the nature of the

environment is shown in Figure 7.16.

215

Figure 7.16 — Caboolture Data Collection Site

The data collected permitted the simulation of SV’s 21, 23, and 5. The simulated

fading patterns from the FSPE and the elevation, azimuth and Fresnel zone data is

given in the following figures.

4000 4500 5000 5500 6000 6500 7000 750025

30

35

40

45

50

55

60

Epoch Samples

S/N

(dB

)

S/N Comparison SV21-Wet Ground

MeasuredModelled


216

4000 4500 5000 5500 6000 6500 7000 75000

5

10

15

20

25Az-El SV21

Ele

vatio

n (d

eg)

4000 4500 5000 5500 6000 6500 7000 7500-55

-50

-45

-40

-35

Azi

mut

h (d

eg)


4000 4500 5000 5500 6000 6500 7000 75000

50


Ran

ge D

ista

nce

(m)

4000 4500 5000 5500 6000 6500 7000 75000

2

4

6

1st

Zone

Rad

ius

(m)

4000 4500 5000 5500 6000 6500 7000 75000

100

200

300

sem

i-maj

or a

xis

(m)

Measurement Epochs


217

0 500 1000 1500 2000 2500 3000 350036

38

40

42

44

46

48

50

52

54

56

Epoch Samples

S/N

(dB

)


MeasuredModelled


0 500 1000 1500 2000 2500 3000 35005

10

15

20

25

30Az-El SV23

Ele

vatio

n (d

eg)

0 500 1000 1500 2000 2500 3000 3500-32

-30

-28

-26

-24

Azi

mut

h (d

eg)


218

0 500 1000 1500 2000 2500 3000 35000

5

10


Ran

ge D

ista

nce

(m)

0 500 1000 1500 2000 2500 3000 35000.5

1

1.5

21s

t Zo

ne R

adiu

s (m

)

0 500 1000 1500 2000 2500 3000 35000

10

20

sem

i-maj

or a

xis

(m)

Measurement Epochs


1500 2000 2500 3000 3500 4000 4500 5000 5500

25

30

35

40

45

50

Epoch Samples

S/N

(dB

)


MeasuredModelled


219

1500 2000 2500 3000 3500 4000 4500 5000 55000

10

20

30

40Az-El SV5

Ele

vatio

n (d

eg)

1500 2000 2500 3000 3500 4000 4500 5000 55006

8

10

12

14A

zim

uth

(deg

)


1500 2000 2500 3000 3500 4000 4500 5000 55000

10


Ran

ge D

ista

nce

(m)

1500 2000 2500 3000 3500 4000 4500 5000 55000.5

1

1.5

2

1st

Zone

Rad

ius

(m)

1500 2000 2500 3000 3500 4000 4500 5000 55000

10

20

30

sem

i-maj

or a

xis

(m)

Measurement Epochs


In all three cases for the grass field at Caboolture the FSPE simulation provided

excellent agreement to the received signal strength.

220

7.3 Summary

In this chapter results of comparisons of simulated propagation with real collected

data has been shown. In all cases the FSPE model has provided simulations that are in

excellent agreement with actual received GPS signals.

Two representative environments were chosen, one representing fresh water and the

other wet ground. Results from the FSPE and the PETA models were in excellent

agreement with GPS multipath propagation measurements in these simple

environments. This provides the basis for simulation of more complex environments.

221

7.4 References



[2] CCIR International Radio Consultive Commitee, “Report 1008 Reflection

from the Surface of the Earth,” International Telecommunications Union,

Dubrovnik 1986.

[3] A. Kavak, G. Xu, and W. Vogel, “GPS Multipath Fade Measurements to

Determine L-Band Ground Reflectivity Properties,” presented at 20th NASA

Propagation Experimenters Meeting, Pasadena, 1996.

[4] UNAVCO/UCAR, “TEQC,” , 2000 Feb 29 ed. P.O. Box 3000 Boulder,

Colorado 80307-3000: UNAVCO, 2000.

222


223

Chapter 8 SNR for Deriving a Height Observable

In this chapter an analysis is made of the fading effect on the received GPS signal

caused by multipath over defined surfaces. This analysis suggest that it is possible to

use this effect, of multipath on the signal-to-noise ratio (SNR) or more correctly the

carrier-to-noise C/No, to derive information about the reflecting surface.

8.1 Introduction

In the process of collecting and analysing GPS multipath data it became apparent that

variation in the signal strength fading pattern contained extractable information that

described aspects of the reflecting surface [1, 2]. Consider the comparison of GPS

SNR data and the modelled simulation for propagation over tidal sea water, Figure

8.1.

0 1000 2000 3000 4000 5000 6000 700025

30

35

40

45

50

55

60

Figure 8.1 — SV21 Bribie Island Tidal Variation

To accurately model the fading pattern for this situation, accurate tidal information

was required such that the relative height between the reflecting surface and the GPS

224

antenna could be used in the model. Without this accurate information the modelling

results were very inaccurate. This highlighted the fact that the fading pattern is a direct

function of relative antenna height, as was shown by equation (2.36) given in Chapter

2.

With this in mind, a technique was investigated that makes use of the often

undesirable fading effect of GPS multipath on the SNR, to determine the relative

change in height between a reflecting surface and the phase centre of a GPS antenna.

Others [3, 4] have used multipath fading characteristics to infer antenna height but

these techniques have relied on expensive or multiple receivers . This relative height

change technique does not rely on special receiver techniques and can be applied to a

number of practical applications from observations of the local tide heights, or

observation of the change in water level in a dam.

Land subsidence observations [5] could also be made by deliberately placing ground

reflectors to produce multipath. As the antenna moves with respect the located ground

reflector a change in height would be recorded. The advantage that this technique has

over standard GPS positioning, is that an expensive carrier phase receiver is not

required [6]. So for slope monitoring applications, networks of cheap OEM GPS

receivers could feasibly be used to determine subsidence.

This work shows that processing of the SNR from a standard GPS receiver yields cm

level accuracy for relative height change. We assume that a single reflection is caused

by the reflecting surface. The GPS antenna is orientated such that the gain pattern

includes both the line-of-sight and the reflection from the surface. The signal received

is a combination of these two signals and the fade pattern in the received S/N depends

on the antenna height. Analysis of these fade patterns over a period of time allows an

estimation of the height variation. Results are presented verifying the technique.

225

Reflected path

Direct path

GPS antenna

Relative Height

Figure 8.2 — Relative Height Measurement

Figure 8.2 shows the geometry and the relative height measurement that is estimated

with this technique.

8.2 Estimating Antenna Height from the GPS S/N Ratio

The model chosen for this work is a simple two-ray plane-wave representation that

assumes that there is [7-9]:

• A single specular reflection from the reflecting surface.

• No diffuse scattering.

• No variation in the reflecting surface height at the various geometric reflection

points.

Under these assumptions the complex received signal is given by:

( )RjkSVSV eAS ∆Γ+= 1 (8.1)

The amplitude of the LOS svA , and the reflection coefficient magnitude ρ , only

affect the size of the variation of the peaks and fades in the received signal. Therefore

only the phase upon reflection, φ , affects the signal shape at any given epoch

(elevation angle sample).

226

The received signal-to-noise data is now sampled at the peaks and fades and the

corresponding angle of propagation determined from the elevation angle to the

satellite in use. The fade pattern for SV3 over sea water is shown in Figure 8.3. Here

the signal-to-noise data has been averaged with a sliding 101-point averaging window.

The corresponding samples for the peaks and fades are then determined.

0.85 0.9 0.95 1 1.05 1.1 1.15

x 104

25

30

35

40

45

50

55

60Averaged S/N SV3

S/N

(dB

)

epochs

Raw AveragedSample

Figure 8.3 — Sampled received S/N

The elevation angle and corresponding sampling ( iθ ) is shown in Figure 8.4.

0.85 0.9 0.95 1 1.05 1.1 1.15

x 104

0

2

4

6

8

10

12

14

16

18

20Elevation Angle SV3

Ele

vatio

n A

ngle

(de

gs)

epochs

SV ElevationSample

Figure 8.4 — Sampled elevation angle data

227

We can now solve, at each peak/fade sample, for the local antenna height ih , above

the reflecting surface using

samples ...3,2,1sin2

mik

nh

i

iii =−=

θφπ

(8.2)

where

( )111 −− −+= iiii signnn θθ (8.3)

and 0n , the starting count, is some initial integer value that is even for a peak or odd

for a fade. The sign of the change in propagation angle determines whether n is

incremented or decremented by 1 from the previous value.

In this way as each peak and fade passes through the antenna’s vertical phase-centre

position an integer count is maintained. The initial value for 0n is determined from a

priori knowledge of the antenna height at the first corresponding peak or fade. Once

this initialisation is set the procedure is independently capable of estimating antenna

height from the signal-to-noise ratio.

To clarify the technique, a visual representation of a FSPE derived forward scatter

interference pattern is shown in Figure 8.5.

228

Figure 8.5 — Interference pattern

This is for a low elevation satellite, and represents a domain 25 metres in height and

range. The lower boundary is the plane reflecting surface. An antenna is represented at

3.9 metres in height and 15 metres into the domain. The peaks in the interference

pattern are seen as the dark portions, with the fades lightest. For this particular

representation the antenna is seen to be located at a peak for this propagation angle.

Note that the distance between minima for this case is 2.63 metres. If the height

estimation technique was initialised at this point, the starting value of 0n would be

two, as it is the second peak in the pattern above the reflecting boundary. Peaks in the

interference pattern occur at n = 0, 2, 4…, and correspondingly the first fade is at n =

1, with further fades at n = 3, 5, 7 …etc.

8.3 Results

The technique was validated in a number of experiments. Results are presented for

both still water (where there is no variation of the surface height) and marine

environments.

229

8.4 North Pine Dam

The first experimental arrangement was the determination of the antenna height above

a water feature with no tidal variation. Data was collected at a fresh water dam with

the antenna oriented vertically and facing the main water surface.

The local height from the antenna to the water surface was measured at approximately

1.3 metres. The estimation of antenna height from three SV’s (6, 8 and 9) is shown in

Figure 8.6.

0 500 1000 1500 2000 2500 3000 3500 4000

1.255

1.26

1.265

1.27

Antenna Height Estimation

Ant

enna

Hei

ght

(met

res)

epochs

SV6SV8SV9

Figure 8.6 — Height estimation above fresh water

As the reflection surface height is essentially static it is appropriate to take some

statistics of the resultant height estimation data. The mean of the resultant data is an

estimated antenna height of 1.26 metres, and the standard deviation is 0.0044 metres.

230

8.5 Bribie Island

Data was collected at a tidal coastal site, with a relatively smooth sea surface. Three

SV’s, 10, 21 and 22 were observed with direct reflection from the sea surface. The

estimation of antenna height is shown in Figure 8.7.

3500 4000 4500 5000 5500 6000 6500 7000 7500 80003.3

3.4

3.5

3.6

3.7

3.8

3.9

4

Ant

enna

Hei

ght

(m)

epochs

Antenna Height Estimation

SV10SV21SV22

Figure 8.7 — Height estimation above sea surface

The total tidal variation for the estimation period was a rise of 53 cm [10]. The

estimates of antenna height agree with this variation showing a total decrease in

antenna height of approximately 57 cm for SV22. The other two SV’s observed did

not cover the full observation period of SV22 but do provide confirmation of a

decrease in relative antenna height, corresponding to an increase in the sea-surface

height.

8.6 System Aspects

The presented technique could be implemented as a surface monitor for a variety of

applications. A system-level representation is shown below in Figure 8.8.

231

monitored surface

Suitable SV’s

GPS Rx

surface heightestimation

reflector

S/N

ephemeris data

initialheight

estimate

Figure 8.8 — System representation

Here a GPS receiver is located at a monitoring site and an estimate of antenna height

(surface height) is provided for a selected surface area. However, there are a number

of system level aspects that are required in order to make this technique work. These

requirements are:

• An initial height estimate must be provided.

• The location of the antenna must be precisely known.

• Current ephemeris is required.

• The monitored surface must have a known, preferably invariable reflection

coefficient.

The initial antenna height is required for the initialisation of the estimation technique

as mentioned previously. The antenna location and ephemeris is required so accurate

SV elevation angle information can be generated. Finally, the amount of multipath

must be controlled. It is not desirable to have a very strong multipath signal that

causes the GPS receiver to lose lock, as this would not allow any measurements to be

made. Loss of lock is evident in Figure 8.9 (reflection from salt water), where large

decreases in received signal strength have exceeded the dynamic range of the

receiver’s tracking loops. The resulting data gaps make it difficult to accurately

predict the true position of the fades.

232

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 21

S/N

(dB

)

Figure 8.9 — Raw S/N sea water

Figure 8.10 shows a corresponding fade pattern for SV8 over fresh water. The

reduced reflection coefficient allows the receiver to maintain lock for the full

observation period.

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 8

S/N

(dB

)

Figure 8.10 — Raw S/N fresh water

233

The technique also lends itself to the intentional placing of a ground reflector onto to

surface to be monitored. In this case, careful consideration of the shape and properties

of that ground reflector must be made.

From Figure 8.6 it is seen that multiple observations, from separate satellites in

differing elevation ranges, are combined to produce the best estimate of the GPS

antenna height. To provide a large number of height estimations it is desirable that the

variation of satellite elevation is high, for any given SV.

The accuracy of the height estimation is also a function of the satellite elevation angle.

Equation (8.2) can be recast in the form

FBh = (8.4)

where

θsin2

1

kF = and φπ −= nB (8.5)

If there is an error in the estimate of θ, then ε±=′ FF and the new estimate of height

is given by Bhh ε±=′ . For a given dθ, and h, the error in height is then

approximately bounded by

θε

d

dF

F

hh ±= (8.6)

From equation (8.6) we see that the error in height is a function of the actual height h,

the SV elevation angle θ, and the size of the angle error, dθ. A plot of the error

bounds for a height of 1m, an error in θ of 0.1 degree, for an elevation range of 1

degree to 20 degrees is given in Figure 8.11.

234

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

SV Elevation (degs)

Hei

ght

Err

or (

m)

1m Height Error Bounds: 0.1 deg error

Figure 8.11 — Height Error Bound 0.1 Deg Error

Reducing the error in θ to 0.01 degree, results in the height error bounds given below

in Figure 8.12.

0 2 4 6 8 10 12 14 16 18 20-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

SV Elevation (degs)

Hei

ght

Err

or (

m)

1m Height Error Bounds: 0.01 deg error

Figure 8.12 — Height Error Bound 0.01 Deg Error

As expected there is a ten-fold reduction in the error bounds. From this error analysis

the following observations can be made:

235

• Larger relative heights have proportionally larger errors for a given error in the

estimation of the SV elevation angle.

• The error in height estimation decreases inversely to the SV elevation angle.

• The error in height estimation decreases proportionally to the error in the

estimation of the SV elevation angle.

From Figure 8.3 and Figure 8.4, it is obvious that the error in estimation of the SV

elevation angle, is dependent upon accurate location of the peaks and fades in the

received S/N. In addition, the rate of change, and accuracy of the elevation data

ultimately sets the resolution of the elevation angle error.

236

8.7 Summary

In this chapter use was made of the fading effect on the received GPS signal caused

by multipath over defined surfaces. We saw that it is possible — through

undemanding techniques — to make use of GPS multipath effects on the received

signal-to-noise ratio (SNR), to derive height information about the reflecting surface.

237

8.8 References

[1] A. Kavak, G. Xu, and W. Vogel, "GPS Multipath Fade Measurements to

Determine L-Band Ground Reflectivity Properties," presented at 20th NASA

Propagation Experimenters Meeting, Pasadena, 1996.

[2] K. D. Anderson, "Determination of Water Level and Tides Using

Interferometric Observations of GPS Signals," Journal of Atmospheric and

Oceanic Technology, vol. 17, pp. 1118–1127, 2000.

[3] C. E. Cohen, "Attitude Determination Using GPS," in The Department of

Aeronautics and Astronautics. Stanford: Stanford University, 1993, pp. 184.

[4] S. Wu, T. Meehan, and L. Young, "The Potential Use of GPS Signals as

Ocean Altimetry Observables," presented at ION National Technical Meeting,

Santa Monica, 1997.

[5] R. A. Walker, "Multipath Issues in GPS Monitoring," presented at

International Workshop on Advances in GPS Deformation Monitoring, Perth,

Australia, 1998.

[6] E. D. Kaplan, "Understanding GPS: Principles and Applications," in Mobile

Communications Series, J. Walker, Ed. Boston: Artech House, 1996.



[8] CCIR International Radio Consultive Commitee, "Report 715-2 Propagation

by Diffraction," International Telecommunications Union, Dubrovnik 1986.

[9] CCIR International Radio Consultive Commitee, "Report 1008 Reflection

from the Surface of the Earth," International Telecommunications Union,

Dubrovnik 1986.

[10] G. J. Broadbent, "Tidal Readings - Bongaree Jetty," Maritime Services

Branch, Queensland Transport Maritime Division 1999.

238


239

Chapter 9 Receiver Correlation and Discrimination

In the previous chapters the FSPE and PETA models have been shown to provide a

novel and worthwhile simulation environment, that provides not only a unique

environment for visualisation but also all of the necessary defining parameters for

describing multipath propagation within the simulation domain. In this chapter the

value of incorporating a GPS receiver model with the propagation modelling

environment is investigated.

9.1 Introduction

The advantage of the FSPE/PETA modelling method, as shown in previous chapters,

is that the exact multipath nature of a complicated environment can be understood and

decomposed. By combining a receiver model with the propagation models a complete

software-based satellite to user modelling system is developed. A block diagram of

this system is shown in Figure 9.1

Satellite

Signal

FSPE/PETAGPS Propagation

Model

RXModel

RXParameters

MultipathParameters

MultipathAnalysis

Environment

MultipathParameters

ErrorParametersMitigation

Figure 9.1 — Multipath Modelling Environment

240

No assumptions will be made regarding the nature of the input signal to the receiver,

the only inputs to the system will be the receiver parameters (discrimination type, chip

spacing, bandwidth, sampling), satellite parameters (elevation, azimuth, code), signal

parameters (frequency, power) and the environmental parameters (terrain

characteristics).

9.2 Fundamentals of a GPS Receiver Model

A typical GPS user equipment configuration is shown in Figure 9.2. The separation of

the functional blocks, the GPS Receiver, and the Application Processing is based on

the premise that not all GPS receivers perform navigation processing. The application

may be that of time transfer, data collection or differential surveying. For this reason it

is appropriate to consider that the user equipment performs two separate functions.

Firstly, tracking of received signals, utilising some form of Phase or Frequency Lock

Loop (PLL/FLL) for carrier recovery and a Delay Lock Loop (DLL) for code

recovery, and secondly the processing of the observables provided by the receiver, to

provide application-specific requirements.

GPS Receiver

Application Processing

pseudoranges carrier phase nav data

position velocity

User Clock

GPSSatellite Signals

Observables

Figure 9.2 — General form of GPS user equipment

The following sections will look more closely at the operation of the GPS receiver,

the technological aspects and their implementations.

241

9.2.1 Generic GPS Receiver Functions

The system level functions of a GPS receiver vary, in only a few aspects, from the

fundamental functions performed by many modern RF-based communications

systems. A generic GPS receiver would consist of the following functional elements

as shown in Figure 9.3: antenna, Radio Frequency (RF) amplification, reference

oscillator, frequency synthesis, down-conversion, Intermediate Frequency (IF) section,

and signal processing.

RFSection

DownConverter

FrequencySynthesiser

ReferenceOscillator

IFSection

SignalProcessing

ClocksLO

pseudorangesdelta pseudorangeintegrated Doppler

Figure 9.3 — Generic GPS receiver

These functions are described in various excellent GPS-specific sources including [1,

2], and since the first six functions are also common to other RF systems the reader

can find information regarding these topics in this literature. All functions are briefly

described here for completeness, but it is the signal processing function that forms the

core of a GPS receiver and it is this function that is the focus of later sections.

9.2.1.1 Antenna and RF Section

The GPS signals, received from all GPS satellites, are right-hand circularly polarised

(RHCP), and of very low power (-160 dBw) [3]. The antenna and RF sections

therefore need to maximise the signal reception. The antenna needs to have a near

hemispherical gain pattern, so as to maximise the number of satellites for tracking.

242

The antenna may be designed as an active device with some amplification occurring

before the signal is sent to the RF section.

The RF section usually consists of filtering and amplification functions. The filtering

is used to reduce the effects of out-of-band noise and interference. The amplification

is achieved with a low-noise-amplifier (LNA), with the gain selected so as to establish

the designed receiver noise figure.

9.2.1.2 Reference Oscillator and Frequency Synthesis

The reference oscillator provides the time and frequency reference for the GPS

receiver. The parameters of the reference oscillator (size, stability, and phase noise)

are trade-offs between cost and performance. The higher the stability, the more costly

the oscillator becomes. In GPS receivers the reference oscillator is used by the

frequency synthesiser to generate all Local Oscillators (LO) and clocks.

9.2.1.3 Down-conversion and IF

The local oscillators are used in the downconverter to convert the RF signal down to a

lower Intermediate Frequency (IF). The IF section provides additional amplification,

filtering and provides a conditioned signal that can be used by the signal processing

section.

9.2.1.4 Signal Processing

The signal processing section is really the core function in a GPS receiver. It performs

multi-channel acquisition and carrier/code tracking of satellites, navigation data

demodulation, code-phase (pseudorange), carrier-phase (delta pseudorange) and

carrier frequency (integrated Doppler) measurements, in addition to extracting the

Signal-to-Noise Ratio (SNR) of the received signals.

243

9.3 The Key GPS Receiver Elements

The key elements of a GPS receiver are located in the signal processing section. These

elements provide the ability for the receiver to acquire and track the GPS ranging-

code. There are four fundamental sub-sections necessary.

1. Pseudo-Random Noise (PRN) code generator — all of the possible satellite PRN

codes need to be replicated within the receiver, for in the acquisition and tracking

of the GPS ranging signal. The generation of these codes is achieved with code

generators similar in design to those used on the GPS satellites.

2. Signal Acquisition Process — acquires the satellite signal so that tracking may be

performed.

3. Delay Lock Loop (DLL) — tracks the code-phase of the received signal.

4. Phase or Frequency lock Loop (PLL/FLL) — tracks the carrier-phase of the

received signal.

These elements will be examined in the following sections.

9.3.1 PN Code Generation

The GPS C/A-code is a 1023 bit (chip) Gold code, with chipping rate of 1.023 MHz.

This gives a code with a period of 1 millisecond and a chip width of 977.5

nanoseconds. The receiver PN C/A coder implementation is specified in ICD-GPS-

200 and the GPS SPS Signal Specification. The implementation of a two-tap selection

C/A-code generator is given in Figure 9.4.

244

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

PHASE SELECTOR

G1 GENERATOR

G2 GENERATOR

S1

S2

TAP SELECTIONG2i

G1

G2

XGi

÷ 1024

÷ 20

÷ 10

RESETALL

ONES

RESET

RESET

CLOCK

CLOCK

10.23 Mbps

X1 EPOCH

C/A CODE

G EPOCH

DATA CLOCK

50 bps

1 kbps

SYNCSYNC

+

+

+

+

Figure 9.4 — C/A Code Generation

Following this implementation it is seen that the C/A-code is the Modulo-2 sum of

two 1023 chip linear sequences, G1 and G2i. Here G2i is a delayed G2 sequence, with the

effective delay selected by the Modulo-2 sum of two phases from the G2 shift register.

The G1 and G2 sequences are generated by 10-stage shift registers initialised with the

vector 1111111111, and specified by the following polynomials

13101 ++= XXG and

123689102 ++++++= XXXXXXG

The specified code phase assignments for the selection of the G2i sequence, provide a

total of thirty-six codes for GPS use. The code phase assignments, as a function of the

GPS PRN code number, are summarised in Table 9.1.

245

PRN Signal Number (N) Code Phase Selection

1, 2, 3, 4 ( ) ( )51 +⊕+ NN

5, 6 ( ) ( )44 +⊕− NN

7, 8, 9 ( ) ( )16 +⊕− NN

10, 11, 12, 13, 14, 15, 16 ( ) ( )78 −⊕− NN

17, 18, 19, 20, 21, 22 ( ) ( )1316 −⊕− NN

23 ( ) ( )2022 −⊕− NN

24, 25, 26, 27, 28 ( ) ( )1820 −⊕− NN

29, 30, 31, 32, 33 ( ) ( )2328 −⊕− NN

34 ( ) ( )2430 −⊕− NN

35, 36 ( ) ( )2834 −⊕− NN

Table 9.1 — C/A-Code Selection

9.3.2 Delay Lock Loop

GPS signal acquisition and tracking is a two-dimensional process. The receiver must

correlate the received code with a shifted internally generated code replica as well as

maintain lock with the carrier of the signal.

In the code-phase dimension, when the code-phases are matched there is maximum

correlation. Minimum correlation occurs when the replica code is offset by more than

one chip. The C/A-code autocorrelation function is given in Equation (9.1),

246

( ) ( ) ( )RG

CAG

itt

t Gi

t dtτ τ= ==∫ +

1

1023 01023

Τ(9.1)

and the resultant plot of the autocorrelation function is shown in Figure 9.5.

0 1977.5 x 10 6

-1/1023

A2 RG(t)

t (ms)

Figure 9.5 — C/A autocorrelation function

The code-phase tracking loop is of a similar form to that used in other Direct

Sequence Spread Spectrum (DSSS) systems—a Delay Lock Loop (DLL). The DLL

used can be either coherent or noncoherent. The coherent DLL requires parallel

tracking of the carrier-phase whilst the noncoherent DLL requires no carrier tracking.

The form of the DLL is determined by the type of discriminator used.

In addition to detecting the signal in the code-phase dimension, the receiver must also

detect the signal in the carrier-phase dimension. The tracking of the carrier-phase can

be achieved by a variety of tracking loops that are of either PLL, Costas PLL, or FLL

form. In each of these forms, there is a variety of discriminator functions that can be

utilised. A generic form of the carrier and code tracking loops is shown in Figure 9.6.

247

PN CodeGenerator

Accumulator

Accumulator

Accumulator

Accumulator

Accumulator

Accumulator

Early Prompt LateCarrierNCO

SINMap

COSMap

Carrier LoopDiscriminator

Carrier LoopFilter

CodeNCO

Code LoopFilter

Code LoopDiscriminator

carrier aiding

I

Q

E

P

L

QE

E

P

L

QL

QP

IE

IP

IL

IPQP

Figure 9.6 — Receiver tracking loops

The input is digital IF, and the carrier is stripped or wiped-off by mixing with a

replica carrier (plus carrier Doppler, to account for relative receiver dynamics). The

outputs are in-phase (I) and quadrature-phase (Q) samples. This signal is then

collapsed back to baseband by code stripping using early/late correlators. The

baseband signal is then processed by some form of discriminator. The code and carrier

loops are controlled by Numerically Controlled Oscillators (NCO).

The early/late correlators provide the input to the discriminator, which is implemented

by one of several different early minus late forms. The code correlation process is

illustrated in Figure 9.7.

248

-1/2-1 0 +1E

P

L

Chip offset

Normalised correlator output

-1/2-1 0 +1

E

P

L

Chip offset

Received code

Rx generated codes

Early

Prompt

Late

Figure 9.7 — Correlation process

This shows how the early, prompt, and late correlation envelopes vary, for one chip of

the C/A-code, as the phase of the replica code is varied. The early minus late

discrimination of these envelopes produces a discrimination characteristic (tracking

error) that controls the closed loop operation of the DLL. Some common DLL

discrimination algorithms [4, 5] are given in Table 9.2.

249

Discriminator Algorithm Description

( ) ( )I I I Q Q QE L P E L P− + −∑ ∑ Dot Product Power — noncoherent

( ) ( )∑∑ +−+ 2222LLEE QIQI Early - Late Power — noncoherent

( ) ( )∑∑ +−+ 2222LLEE QIQI Early - Late Envelope — noncoherent

( ) ( )( ) ( )∑∑

∑∑+++

+−+2222

2222

LLEE

LLEE

QIQI

QIQI Normalised Early - Late Envelope —

noncoherent

sign IP( ) IE − IL( )∑ Early - Late — coherent

Table 9.2 — DLL discriminator algorithms

Each consecutive noncoherent algorithm increases in computational load, with the

dot-product having the lowest, and the normalised envelope the highest. The coherent

algorithm is included here but its performance is marginal at low SNR and is

obviously unusable when carrier tracking is not possible. The noncoherent DLL

implementations are the most robust. The discriminator characteristic (also known as

the s-curve) for the envelope and normalised envelope discriminators are shown in

Figure 9.8.

250

-0.5-1.5

+1.5+0.5ε/T - chips

D(ε) - chips

-1.0

+1.0

+1.0

-1.0

envelope

normalised envelope

Figure 9.8 — DLL discriminator curves

The discriminator output is the relative tracking error of the DLL and is filtered and

applied to the code NCO, which in turn provides the necessary adjustment to the code

generator phase for correct code-phase alignment of the replica and received codes.

For the carrier tracking loop there are, again, several implementations that may be

used. If the receiver does not need to demodulate the navigation message (pure phase

tracking) then a pure PLL implementation can be used. Since however, most receivers

will need to demodulate this data, a different PLL (Costas PLL) is often used. Other

implementations may make use of Frequency Lock Loops (FLL). The FLL achieves

carrier wipe-off by replicating the approximate frequency of the carrier.

The common algorithms [4, 5], for the true PLL and Costas PLL implementations, are

presented in Table 9.3.

251

PLL Discriminator

Algorithm

Phase Error Description

( )sign I QP P⋅ sinφ Decision-Directed PLL/Costas

I QP P⋅ sin2φ Generic PLL/Costas

Q

IP

P

tanφ Tangent PLL/Costas

TANQ

IP

P

−

1

φ Arctangent — two-quadrant (Costas)

four-quadrant (PLL)

Table 9.3 — PLL discriminator algorithms

The performance of these discriminator algorithms is different under variable SNR

conditions. The decision-directed, generic, and tangent discriminator algorithms are

identical for the pure PLL and Costas PLL implementations. The decision-directed

algorithm is near optimal for high SNR and has the least computational burden. The

Generic algorithm is near optimal at low SNR with moderate computational burden,

while the tangent algorithm is suboptimal, but performs reasonable well at high and

low SNR, with high computational burden. In addition, the tangent algorithm must

check for divide by zero errors near ± 90 degrees. The final algorithm—arctangent—

is an optimal Maximum Likelihood (ML) estimator at high and low SNR. In the pure

PLL implementation it is a four-quadrant arctangent, while for the Costas PLL, it is

two-quadrant.

The common FLL discriminator algorithms [5] are given in Table 9.4.

252

FLL Discriminator Algorithm Frequency

Error

Description

( ) ( )sign I I Q Q I Q I Q

t tP P P P P P P P1 2 1 2 1 2 2 1

2 1

⋅ + ⋅ ⋅ ⋅ − ⋅−

( )[ ]sin 2 2 1

2 1

φ φ−−t t

Sign(dot)

cross

I Q I Q

t tP P P P1 2 2 1

2 1

⋅ − ⋅−

( )sin φ φ2 1

2 1

−−t t

Cross

( ) ( )[ ]( )

tan I I Q Q I Q I Q

t t

P P P P P P P P− ⋅ + ⋅ ⋅ − ⋅

−

11 2 1 2 1 2 2 1

2 1 360

,( )

φ φ2 1

2 1 360

−−t t

Arctangent

cross, dot

Table 9.4 — FLL discriminator algorithms

9.4 A Receiver Model for use with the FSPE/PETA Model

The aspects of the GPS receiver, covered in the preceding sections, are sufficient for

the development and implementation of a GPS receiver model. Models that have been

developed for the characterisation of the effects of multipath propagation, include

those of Braasch [6], van Nee [7], and others [8-21]. These models were developed

using the well defined fundamentals of the GPS receiver; signal down-conversion via

mixing, correlation, and discrimination. In this work, the primary focus is on the

investigation and characterisation of multipath effects on the code-phase tracking of

the ranging signal. In this context it is sufficient that the receiver model is developed

around a variable, generic DLL implementation, shown in Figure 9.9. The MATLAB

implementation can be found in Appendix C.

253

CHANNELC/A-CODE

GEN

N Multipath signal parameters[relative amplitudes, delays, phases]

+

RECEIVERC/A-CODE

GEN

EARLYCORRELATOR

PROMPTCORRELATOR

LATECORRELATOR

PE L

[Correlator spacing]

LOW-PASS

FILTER

[Cut-off frequency]

DISCRIMINATOR

[Discriminator select]

Range error

N

1

LOS

Figure 9.9 — Receiver Correlation Model

The resultant correlation and discrimination allows the visualisation of these functions

for any multipath scenarios produced by FSPE/PETA model-based simulations. The

various discrimination functions as outlined above were implemented in the receiver

model and are shown below, Figure 9.10.

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5DLL Discrimination Functions

Nor

mal

ised

Dis

crim

inat

ion

Val

ue

Chip Offset

E-L Norm E-L Env E-L Power Dot Product

Figure 9.10 — DLL discriminator curves

254

These discrimination functions are as expected for the various implementation types,

and it can be assumed that the correlation and discrimination functions are correctly

implemented within the model.

9.5 The Modelled Code Correlation Function and Multipath

Having established a DLL model that provides visualisation and calculation of

correlation and discrimination functions (and hence the range error), we can consider

the actual effect that multipath propagation has on the correlation function shape and

the consequential tracking error generated by the receiver discrimination function. In

general the correlation function is distorted, but with use of these modelling

techniques a unique visualisation of the corruption of the signal can be made. The

visualisation is a valuable aid in increasing our understanding of multipath effects and

how they may be mitigated.

9.5.1 Variation of Relative Multipath Delay Time

The first modelling situation presented is the case of variation in relative time delay of

a fixed amplitude single multipath signal. Firstly we consider the two limiting cases of

multipath error effect on the correlation function shape. That is; when the multipath

signal is completely in phase (zero degrees phase difference) with the LOS signal and

when it is 180 degrees out of phase with the LOS signal. These two cases; the in-

phase and anti-phase, are presented below.

9.5.1.1 In-Phase Case

In this first series of correlation and discrimination plots for the in-phase case, the

relative multipath-to-LOS amplitude is one. This implies that the multipath signal is

of equal amplitude to the LOS signal. The correlator modelled is a standard ½-chip

correlator and infinite bandwidth is selected (no pre-correlation filtering), this is done

255

to provide clarity in the plots. The first plot, Figure 9.11, is the multipath-free

correlation of the LOS signal and the resultant discrimination function for this

condition.

Figure 9.11 — Multipath-free Correlation and Discrimination

As we see the discrimination or tracking error is zero for this case as is expected. In

the next plot, Figure 9.12, a single multipath signal is present as well as the LOS

signal.

256

Figure 9.12 — 0.05 Chip delay In-phase Multipath

The total correlation function of the combined multipath and LOS signal is shown to

have increased from the previous normalised value of one. Although the

discrimination function shows very little distortion there is in fact a fractional offset

of the zero-crossing point from the zero chip offset. In other words there is a tracking

error whenever the discrimination function has a non-zero-offset for a prompt arrival

of the LOS signal.

We now increase the in-phase multipath relative delay and observe that the distortion

of the discrimination function has increased dramatically, Figure 9.13.

257

Figure 9.13 — 0.5 chip delay In-phase Multipath

The range offset that the DLL tracks is the zero-crossing offset which in this case is a

positive offset. This infers that the range estimate for this case is larger than the true

range. We now consider the correlation function and describe the distortion points

caused by in-phase multipath.

-2000 -1500 -1000 -500 0 500 1000 1500 2000-0.5

0

0.5

1

1.5

2

2.5x 10

4 Correlation Function

Time Offset (ns)

Cod

e C

orre

latio

n

Delayed Prompt Combined

Figure 9.14 — Correlation Distortion (0°)

C

A

B

D

E

F

258

Consider the figure of a multipath corrupted correlation function shown above, Figure

9.14. The following points are observed and correspond to those indicated on the

figure:

A. The undistorted leading portion of the combined correlation function has the same

gradient as the LOS correlation function.

B. The gradient of the combined correlation function above the multipath breakpoint

(the breakpoint occurs at the point where the multipath signal arrives after the

LOS) is the linear combination of the LOS and multipath correlation functions.

C. The actual peak of the correlation is not displaced or distorted (for the infinite

bandwidth case).

D. The second breakpoint occurs here. Note that the gradient immediately following

this point is substantial different to that just prior to the peak of the correlation

function.

E. The absolute value of the gradient on the rear side of the combined correlation

function is the same value as the absolute of the leading edge gradient (B). If these

gradients are extended to intercept then the displacement of the tracked peak

becomes obvious.

F. The last position of the combined correlation function is undistorted after the third

and final breakpoint.

9.5.1.2 Anti-phase Case

In the plot of correlation and discrimination given below in Figure 9.15, the combined

signal is made up of a prompt LOS and a very short delayed multipath signal that

arrives out-of-phase with a relative amplitude of 0.5. That is the multipath has power

half that of the LOS signal.

259

Figure 9.15 — Short-delay Multipath (180°)

As for the in-phase case we note that the short-delay multipath has very little apparent

effect on the discrimination and hence the tracking or zero-crossing point.

The true distortion becomes apparent as the relative multipath delay is increased,

Figure 9.16.

260

Figure 9.16 — Long-delay Multipath (180°)

As for the in-phase case there is an offset of the zero-crossing point. For the anti-

phase multipath the offset is negative and the range estimate from the tracking loop

will be shorter than the true range. Again we consider this correlation distortion.

-2000 -1500 -1000 -500 0 500 1000 1500 2000-1

-0.5

0

0.5

1

1.5

2x 10

4 Correlation Function

Time Offset (ns)

Cod

e C

orre

latio

n

Delayed Prompt Combined

Figure 9.17 — Correlation Distortion (180°)

A

C

B

D

E

F

261

Consider Figure 9.17, which represents the anti-phase case for a single multipath

delayed relative to the line-of-sight by approximately 100 nanoseconds, and at a

relative amplitude of 0.5. The observations are restricted to the shape of the

correlation function and do not include any mathematical interpretation of the

function, other than geometric. Firstly the relative delay of the multipath signal is

represented at the breakpoints on the leading and trailing slopes, this is also true of the

in-phase case. Indeed if it is possible to track the breakpoints (A and B, or C and D, or

E and F) then the relative delay could be inferred.

These breakpoints occur as singular transitions in the slope of the function. The

function includes six breakpoints (A-F) within the 4-delta spacing used in the figure.

Obviously point C represents the peak correlation point, and we note again that the

multipath does not alter the location of this peak.

9.5.2 Variation of Relative Phase

We now consider the distortion of the correlation function and the resultant

discrimination function for variation in the relative phase of the multipath signal. In

in-phase and anti-phase limiting cases were examined above. Consider the case of a

multipath signal arriving 200 nanoseconds after the line-of-sight signal from the GPS

satellite and at ½ of the LOS signal power.

262

-3 -2 -1 0 1 2 3-2

-1

0

1

2Correlation Function

Nor

mal

ised

Cor

rela

tion

LOSMP P

-3 -2 -1 0 1 2 3-2

-1

0

1

2E-L Envelope Discriminator

Offset (chips)

Nor

mal

ised

Dis

crim

inat

ion

Figure 9.18 — 200ns relative delay , 0.5 Relative Multipath Ratio (0°)

In Figure 9.18, shown above, the multipath arrives in-phase relative to the LOS signal.

There is a small positive offset in the discrimination function as is expected. We now

retard the relative phase to 60 degrees relative to the LOS, Figure 9.19.

-3 -2 -1 0 1 2 3-2

-1

0

1


Nor

mal

ised

Cor

rela

tion

LOSMP P

-3 -2 -1 0 1 2 3-2

-1

0

1


Offset (chips)

Nor

mal

ised

Dis

crim

inat

ion

Figure 9.19 — 200ns relative delay, 0.5 Relative Multipath Ratio (60°)

263

Note the effect on the multipath correlation function is a reduction in the correlation

value. This is the same effect expected if the relative amplitude of the multipath is

also reduced without altering the relative phase relationship between the multipath

and the LOS signal. The reduction in the multipath correlation reduces the distortion

of the combined correlation and hence the tracking error of the discrimination

function is also reduced. The chosen delay of 200 ns represents a delay that would

occur in forward scatter for an antenna height above a reflector of 30 m, with the

satellite at zenith. It is also representative of an antenna distance of 30 m from a back

reflector at a satellite angle of zero degrees.

Further change in the relative phase relationships highlights the reduction of

correlation relative to the phase relationship of the multipath and LOS signal, Figure

9.20.

-3 -2 -1 0 1 2 30

0.5

1

1.5


Nor

mal

ised

Cor

rela

tion

LOSMP P

-3 -2 -1 0 1 2 3-2

-1

0

1


Offset (chips)

Nor

mal

ised

Dis

crim

inat

ion


Here the multipath signal is in quadrature phase with the LOS signal and there is zero

correlation of the multipath signal. Further retardation of the relative phase leads to a

relative change of the correlation function, which is equivalent to a anti-phase

multipath signal arriving at reduced amplitude, Figure 9.21.

264

-3 -2 -1 0 1 2 3-2

-1

0

1


Nor

mal

ised

Cor

rela

tion

LOSMP P

-3 -2 -1 0 1 2 3-2

-1

0

1


Offset (chips)

Nor

mal

ised

Dis

crim

inat

ion


Finally we arrive at the 180 degree relative phase retardation or the anti-phase case,

where our multipath signal is in full negative correlation, Figure 9.22.

-3 -2 -1 0 1 2 3-2

-1

0

1


Nor

mal

ised

Cor

rela

tion

LOSMP P

-3 -2 -1 0 1 2 3-2

-1

0

1


Offset (chips)

Nor

mal

ised

Dis

crim

inat

ion


265

9.6 Error Envelopes for a Single multipath Signal

We now consider the examples given above for a single multipath signal, over a range

of variables for the limiting cases (i.e. in-phase and anti-phase). This includes the

variation over the complete range of correlated delays, relative multipath signal

strength, and phase for a fixed delay.

9.6.1 Variation of Relative Time Delay

In this section we validate the receiver model by generating known results from

previous work, notably the error curves for GPS receivers developed by Braasch [22,

23].

0 200 400 600 800 1000 1200 1400 1600-80

-60

-40

-20

0

20

40

60

80C/A-code; 0.5 MP/LOS; 1/2-chip Dot Product

Relative MP Delay (ns)

Pse

udor

ange

Err

or (

m)

Figure 9.23 — ½-Chip 0.5 Relative Multipath Ratio

We have not generated every possible combination for the relative multipath ratio

value, but have selected 0.5 relative multipath ratio as a representative case. In Figure

9.23 we have the range error envelope (0° and 180° limiting cases) for an infinite pre-

266

correlation bandwidth (remembering that the DLL model developed allows any pre-

correlation bandwidth to be selected) for a standard ½ chip correlator spaced DLL.

0 200 400 600 800 1000 1200 1400 1600-8

-6

-4

-2

0

2

4

6

8C/A-code; 0.5 MP/LOS; 1/20-chip Dot Product


Pse

udor

ange

Err

or (

m)

Figure 9.24 — 1/20 Chip 0.5 Relative Multipath Ratio

The equivalent range error envelope for a narrow correlation DLL is given above in

Figure 9.24. Again we note that the model developed allows any correlation sampling

spacing (relative to the sampling resolution) to be selected. We now incorporate pre-

correlation filtering into the model.

267

0 200 400 600 800 1000 1200 1400 1600-80

-60

-40

-20

0

20

40

60

80C/A-code; 0.5 MP/LOS; 1/2-chip Dot Product (2Mhz)


Pse

udor

ange

Err

or (

m)

Figure 9.25 — ½-Chip 0.5 Relative Multipath Ratio (2MHz)

The pre-correlation bandwidth for the standard ½-chip correlation spacing is usually 2

Megahertz for the most commercially available receivers [1]. In Figure 9.25 above we

have selected this bandwidth and we see the resultant variation of the range error

envelope. Figure 9.26 shows the equivalent case for the accepted narrow correlation

design, where the pre-correlation bandwidth is 8 Megahertz [24].

0 200 400 600 800 1000 1200 1400 1600-8

-6

-4

-2

0

2

4

6

8C/A-code; 0.5 MP/LOS; 1/20-chip Dot Product (8Mhz)


Pse

udor

ange

Err

or (

m)

Figure 9.26 — 1/20-Chip 0.5 Relative Multipath Ratio (8MHz)

268

In all of the above examples the results of the implemented correlation/discrimination

model are similar to the results presented in a variety of published sources see [22, 25-

28] for examples.

9.6.2 Variation of Relative Amplitude

We now consider the variation of relative amplitude of a single multipath signal over

a range of relative delays.

Figure 9.27 — Variation of Relative Multipath Ratio ½-Chip (0°)

269

Figure 9.28 — Variation of Relative Multipath Ratio ½-Chip (180°)

Figure 9.27 and Figure 9.28 show the level of complex information that can be

derived from the developed DLL model. Here the X-axis is relative time delay of the

multipath signal with respect to the line-of-sight signal, and the Y-axis is the relative

amplitude. We see that for fixed delays the range error varies in a non-linear manner,

although the range errors at fixed amplitudes are piece-wise linear in form.

9.6.3 Variation of Phase

In the cases given above we have considered the range error for the limiting cases of

in-phase and anti-phase multipath signals. The variation of relative phase was shown

to distort the discrimination function in a non-linear manner.

270

0 20 40 60 80 100 120 140 160 180-60

-50

-40

-30

-20

-10

0

10

20C/A-Code 1:180 degs,0.5MPR, 200ns

Relative Code Phase (degs)

Ran

ge E

rror

(m

)

Figure 9.29 — Range Error 0.5 Relative Multipath Ratio (200 ns 0°-180°)

As an example of the nature of this variation we present the case of phase variation,

from zero to 180 degrees phase retardation, for a fixed 200 nanosecond relative delay

half-power multipath signal, Figure 9.29. This example shows the wideband nature of

the range error (note the non-sinusoidal shape) found by Braasch in his research [22].

9.7 Error Envelopes for Two Multipath Signals

In the above section we considered the effect of the relative parameters that define

multipath on the receiver and the corresponding range error for a single multipath

signal. In real-world situations the effects of multiple multipath signals on the receiver

need to be considered [29].

The model developed in this work is not limited in the number of multipath signals

that may be present with the LOS signal. As an example of this multiple multipath

capability we present the limiting cases (range error envelopes) for two multipath

signals, where both multipath signals are half-power relative to the LOS signal.

271

For all of the following figures the X-axis is relative delay in nanoseconds of first

multipath (MP1) with respect to the LOS signal and the Y-axis is the relative delay in

nanoseconds of the second multipath (MP2) with respect to the first multipath signal

(MP1). Both multipath signals are 0.5 relative amplitude with respect to the LOS

signal.

Figure 9.30 — Dual Multipath Error (0 ° and 0°)

272


Figure 9.32 — Dual Multipath Error (180° and 0°)

273


These are limiting cases for a standard ½-chip correlator of infinite bandwidth for

C/A-code PRN1, and infer the envelope of the range error for stated phase

relationships. Occurrences of these limiting phase relationships are as follows

(assuming zero reference phase for LOS and 180 degree phase shift upon reflection of

each multipath signal):

1. MP1 and MP2 in-phase with LOS: when relative delay of MP1 with-respect-to

LOS is 0.3174ns (1/2f) or some multiple of 0.6348ns (1/f) from this initial delay,

and relative delay of MP2 writ MP1 is some multiple of 0.6348ns (1/f).

2. MP1 in-phase and MP2 anti-phase with LOS: when relative delay of MP1 wrt

LOS is 0.3174ns (1/2f) or some multiple of 0.6348ns (1/f) from this initial delay,

and relative delay of MP2 with-respect-to MP1 is some odd multiple of 0.3174ns

(1/2f).

3. MP1 anti-phase and MP2 in-phase with LOS: when relative delay of MP1 with-

respect-to LOS is 0.6348ns (1/f) or some multiple of 0.6348ns (1/f) from this

initial delay, and relative delay of MP2 with-respect-to MP1 is some odd multiple

of 0.3174ns (1/2f).

274

4. MP1 anti-phase and MP2 anti-phase with LOS: when relative delay of MP1 with-

respect-to LOS is 0.6348ns (1/f) or some multiple of 0.6348ns (1/f) from this

initial delay, and relative delay of MP2 with-respect-to MP1 is some multiple of

0.6348ns (1/f).

These relationships are evident when a plot of signal strength for the single multipath

case is made, Figure 9.34.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8

-6

-4

-2

0

2

4in-phase

anti-phase

S/N Single 0.5 MPR Multipath

S/N

(db

)

Relative delay (ns)

Figure 9.34 — S/N Fade Pattern

In this figure the signal is the addition of the normalised LOS with a single 0.5

(relative multipath ratio) multipath, with 180 degree phase shift upon reflection. The

peaks represent the case when both signals are in-phase and the fades indicate anti-

phase relationships.

For the three-dimensional plots of range error, the true behaviour of the error within

the envelopes is such that the plots would be of such complexity as to render them

impossible to interpret. The complete plot of range error would involve the transition

of the error from one state to any other state of the three remaining limiting cases. In

addition there are two other intermediate states; where the relative phase of either or

both multipath results in zero contribution to the range error. This occurs when the

multipath signal is in quadrature-phase with the LOS. For this case the multipath

275

correlation is zero and the range error bounds to a single multipath occurrence. This

bound is also met when the second multipath is effectively decorrelated at delays

exceeding 1500ns from the LOS (relative delay of MP1+ relative delay of MP2 =

1500ns). For the limiting cases these zones of decorrelation of the second multipath

signal are clearly evident in the figures given above (Figure 9.30-Figure 9.33).

9.8 Summary

In this chapter the developed correlation/discrimination model has been shown to

provide complete control of multipath variables. The resultant multipath errors are

given directly and any combination of multipath signals is possible.

The implementation of the model is not limited by the need to solve iterative

equations and is very closely modelled on the physical implementation that would be

seen in a modern GPS receiver.

The model provides the ability to provide complete visualisation of the correlation,

discrimination functions and the resultant range error results.

The model presented here can utilise directly the resultant output of the PETA

propagation model and as such provides the ability to generate realistic interpretations

of resultant effects of multipath propagation on a GPS receiver with selected

particular parameters.

Indeed it is the logical next step, in future research work, that the modelling is

combined fully, and that a comparison of measured and modelled data be made for a

large variety of propagation environments.

276

9.9 References

[1] B. W. Parkinson and J. R. Spilker Jr., "Global Positioning System: Theory and

Applications Volume I," in Progress in Astronautics and Aeronautics, vol.

163, P. Zarchen, Ed. Washington: American Institute of Aeronautics and

Astronautics, 1996.

[2] E. D. Kaplan, "Understanding GPS: Principles and Applications," in Mobile

Communications Series, J. Walker, Ed. Boston: Artech House, 1996.

[3] NAVSTAR Joint Program Office, "Technical Characteristics of the

NAVSTAR GPS," 1991.

[4] A. J. van Dierendonck, "GPS Receivers," in Global Positioning System:

Theory and Applications Volume I, vol. 163, Progress in Astronautics and

Aeronautics, B. W. Parkinson and J. R. Spilker Jr., Eds. Washington:


[5] P. W. Ward, "Satellite Signal Aquisition and Tracking," in Understanding

GPS: Principles and Applications, Mobile Communications Series, E. D.

Kaplan, Ed. Boston: Artech House, 1996.

[6] M. S. Braasch, "GPS Multipath Model Validation," presented at IEEE 1996

Position and Location Symposium, Atlanta, Georgia, 1996.





[8] G. D. Akrivis, V. A. Dougalis, and N. A. Kampanis, "Error Estimates for

Finite Element Methods for a Wide-Angle Parabolic Equation," Applied

Numerical Mathematics, vol. 16, pp. 81-100, 1994.

[9] J. R. Auton and J. Cruz, "Simulating GPS Receiver Measurement Errors,"

presented at The 9th International Technical Meeting of The Satellite Division

of The Institute of Navigation., Kansas City, Missouri, 1996.

[10] P. Axelrad, C. J. Comp, and P. F. MacDoran, "SNR-Based Multipath Error

Correction for GPS Differential Phase," IEEE Transactions on Aerospace and

Electronic Systems, vol. 32, pp. 650-660, 1996.

277

[11] M. E. Cannon, G. Lachapelle, and G. Lu, "Kinematic Ambiguity Resolution

with High-Precision C/A Code Receiver," Journal of Surveying Engineering

ASCE, vol. 119, pp. 147-155, 1993.

[12] A. J. R. M. Coenen and A. J. de Vos, "FFT-Based Interpolation for Multipath

Detection in GPS/GLONASS Receivers," Electronics Letters, vol. 28, pp.

1787-1788, 1992.

[13] D. Doris and A. Benhallam, "On Correlation Processes Reducing Multipath

Errors in the L1 GPS Receiver," presented at The 9th International Technical

Meeting of The Satellite Division of The Institute of Navigation., Kansas City,

Missouri, 1996.

[14] G. Lachapelle, M. E. Cannon, and G. Lu, "A Comparison of P-Code and High

Performance C/A Code GPS Receivers for on the Fly Ambiguity Resolution,"

Bulletin Geodesique, vol. 67, pp. 185-192, 1993.

[15] G. Lachapelle, M. E. Cannon, G. Lu, and B. Loncarevic, "Shipborne GPS

Attitude Determination During MMST-93," IEEE Journal of Oceanic

Engineering, vol. 21, pp. 100-105, 1996.

[16] W. Lippencott, T. Milligan, and D. Igli, "Method for Calculating Multipath

Environment and Impact on GPS Receiver Solution Accuracy," presented at

ION National Technical Meeting, Santa Monica, California, 1996.

[17] A. Montalvo and A. Brown, "A Comparison of Three Multipath Mitigation

Approaches for GPS Receivers," presented at 8th International Technical

Meeting of The Satellite Division of The Institute of Navigation., Palm

Springs, California, 1995.

[18] G. D. Morley and W. D. Grover, "Improved Location Estimation with Pulse-

Ranging in Presence of Shadowing and Multipath Excess-Delay Effects,"

Electronics Letters, vol. 31, pp. 1609-1610, 1995.

[19] J. Shi and M. E. Cannon, "Critical Error Effects and Analysis in Carrier Phase-

Based Airborne GPS Positioning over Large Areas," Bulletin Geodesique, vol.

69, pp. 261-273, 1995.

[20] B. J. H. van den Brekel and R. D. J. van Nee, "GPS Multipath Mitigation by

Antenna Movements," Electronics Letters, vol. 28, pp. 2286-2288, 1992.

278

[21] W. Zhuang and J. M. Tranquilla, "Effects of Multipath and Antenna on GPS

Observables," IEE Proceedings - Radar, Sonar and Navigation, vol. 142, pp.

267-275, 1995.

[22] M. S. Braasch, "On the Characteristics of Multipath Errors in Satellite-Based

Precision Approach and Landing Systems," in Department of Electrical and

Computer Engineering. Athens: Ohio University, 1992, pp. 203.



[24] A. J. van Dierendonck, P. Fenton, and T. Ford, "Theory and Performance of

Narrow Correlator Spacing in a GPS Receiver," presented at The Institute of

Navigation National Technical Meeting, San Diego, CA, 1992.

[25] M. S. Braasch, "Multipath Effects," in Global Positioning System: Theory and

Applications, vol. 1, B. W. Parkinson and J. R. Spilker Jr., Eds. Washington:


[26] L. R. Weill, "GPS Multipath Mitigation by Means of Correlator Reference

Waveform Design," presented at The National Technical Meeting of The

Institute of Navigation., Santa Monica, CA, 1997.

[27] L. R. Weill, "Conquering Multipath: The GPS Accuracy Battle," GPS World,

vol. 8, pp. 59-66, 1997.

[28] L. R. Weill, "Achieving Theoretical Accuracy Limits for Pseudoranging in the

Presence of Multipath," presented at The 8th International Technical Meeting

of The Satellite Division of The Institute of Navigation., Palm Springs, CA,

1995.

[29] C. Macabiau, B. Roturier, E. Chatre, and R. Yazid, "N-Multipath Performance

of GPS Receivers," presented at Position Location and Navigation

Symposium, San Diego, 2000.

279

Chapter 10 Conclusions

Multipath remains a dominant error source in Global Positioning System (GPS)

applications that require high accuracy. Multipath propagation occurs when

environmental features cause combinations of reflected and/or diffracted replica

signals to arrive at the GPS receiving antenna. These signals, in combination with the

original line-of-sight (LOS) signal, can cause distortion of the receiver correlation

function and hence errors in range estimation.

With the use of differential techniques it is possible to remove many of the common-

mode error sources, but the error effects of multipath have proven much more difficult

to mitigate.

The research aim of this work was to enhance the understanding of multipath in GPS

terrestrial applications. This was achieved through the use of novel models of signal

propagation behaviour and its effects. To this end, the work presented in this

dissertation describes the research, development, implementation and validation of:

• a Free-Space Parabolic Equation (FSPE) based propagation model for analysis of

multipath propagation field behaviour, and;

• a PE-based Time Analysis (PETA) model that accurately provides defining

multipath propagation information namely; relative delay , amplitude and phase of

multipath signals in realistic terrestrial propagation environments, and;

• a correlation/discrimination model that processes the results of the PETA (or

derived input) into resultant error effects.

In addition the existing theory of radio frequency propagation, for the GPS L1 signal,

has been united into a coherent treatment of GPS propagation in the terrestrial

environment.

280

The propagation models developed and implemented herein improve on previous PE-

based models by incorporating terrain features, including boundary impedance

properties with a novel coupled circular reflection coefficient, backscattering and

time-domain decomposition of the terrestrial field into a unique multipath impulse

response.

The results provide visualisation as well as the parameters necessary to fully describe

the multipath propagation behaviour. These resultant parameters may be used as input

to the newly developed correlation/discrimination model, for the visualisation and the

generation of resultant error parameters.

Results for a variety of propagation environments were presented and the technique

was shown to provide a deterministic methodology against real GPS data and

accepted solutions. In addition a novel method was introduced for the use of the

undesirable effects of multipath propagation on the received GPS signal for the

determination of relative antenna height.

The comprehensive modelling system developed allows the testing of various

mitigation scenarios such that the effectiveness of each strategy can be ascertained.

The various parameters of the correlation and discrimination model can be adjusted

and tested using real-world propagation input. This flexible approach can also be

extended into the propagation domain where multiple antenna location scenarios can

also be investigated.

The unique and novel combined modelling of multipath propagation and reception,

presented in this dissertation, provides an effective set of tools that can be used to

further the understanding of the behaviour and effect of multipath in GPS

applications, and ultimately should aid in providing a solution to this problem.

281

Chapter 11 Recommendations

The author’s recommendations for further research in the topic area are presented

below.

In the work presented, a novel and unique modelling and simulation system for

terrestrial GPS multipath propagation has been introduced. The concept of a coupled

circular reflection coefficient allows the use of a single boundary condition in the

Free-Space PE and PE-based time analysis models.

The inclusion of an accurate antenna gain model in the FSPE would ultimately

provide improved simulation accuracy for the models and the resultant simulations[1].

Although the author has utilised time-domain information [2] to derive an angle of

arrival for the signals incident upon the antenna — as a means to implement the

antenna gain pattern — it is felt that a more inclusive methodology [3] could be

adopted for future implementations of the FSPE. The gain pattern could, for instance,

be implemented as part of the field image which is propagated in the FSPE. In

addition the LHCP rejection ratio of the antenna should be more correctly modelled as

a function of elevation angle.

An obvious extension to the models presented in this work is the development of

three-dimensional model implementations. The additional information and improved

accuracy brought by three-dimensional modelling may not be considered worthwhile,

in view of the much greater computational load, but nonetheless the question needs to

be asked. The benefit of a three-dimensional implementations [4, 5] over that

provided by the present two-dimensional FSPE/PETA implementation therefore

requires further investigation.

The FSPE and the PETA implementations used in this work were written in

MATLAB. This environment is excellent as a research and development tool but both

models would benefit greatly in terms of speed and performance from a software

implementation such as in C or C++.

282

It is also evident that there is lead-on work in the area of using GPS multipath as an

additional observable. Although not completely novel [6], but considered at similar

times, we saw in Chapter 7 how it is possible to make use of GPS multipath by

manipulation of the received Signal-to-Noise Ratio [7]. A more rigorous and thorough

analysis (which is outside of the scope of this work) is required to extend this concept.

The correlation model developed in Chapter 8 provides the basis for comparison of

modelled receiver errors with measured results. Further research should be made to

provide complete verification and validation of the correlation models presented.

As previously introduced, the concept of temporal variation of correlation samples

and subsequent biasing of the discrimination function requires further investigation.

Finally the complete simulation system developed, which incorporates the

FSPE/PETA GPS multipath propagation models and the DLL receiver model should

be utilised in an extensive investigation of GPS multipath using recorded multipath

measurements from a large number of propagation domains.

283

11.1 References

[1] F. Amoroso and W. W. Jones, "Geometric Model for DSPN Satellite

Reception in the Dense Scatterer Mobile Environment," IEEE Transactions on

Communications, vol. 41, pp. 450-453, 1993.

[2] B. M. Hannah, R. A. Walker, and K. Kubik, "Parabolic Equation-Based Time

Analysis of GPS Multipath Propagation," presented at International

Conference of Spatial Information Science and Technology, Wuhan, China,

1998.




[4] W. M. O’Brien, E. M. Kenny, and P. J. Cullen, "An Efficient Implementation

of a Three-Dimensional Microcell Propagation Tool for Indoor and Outdoor

Urban Environments," IEEE Transactions on Vehicular Technology, vol. 49,

pp. 622-630, 2000.

[5] C. A. Zelley and C. C. Constantinou, "A Three-Dimensional Parabolic

Equation Applied to VHF/UHF Propagation over Irregular Terrain," IEEE

Transactions on Antennas and Propagation, vol. 47, pp. 1586-1596, 1999.

[6] K. D. Anderson, "Determination of Water Level and Tides Using

Interferometric Observations of GPS Signals," Journal of Atmospheric and

Oceanic Technology, vol. 17, pp. 1118–1127, 2000.

[7] B. M. Hannah and R. A. Walker, "Determination of Tide Height Variation

using GPS Multipath," presented at 4th International Symposium on Satellite

Navigation Technology & Applications, Brisbane, Australia, 1999.

284


A-1

Appendix A Research Publications

Kubik, K, Hannah, B., Sang, J., Hahn, M., “Attitude Sensing System Using Stereo

Imagery Tracking”, Griffith University Contract Report for Electro Optic Systems

Pty. Ltd. May 1997.

Hannah, B., Walker, R. and Kubik, K., “Towards a Complete Virtual Multipath

Analysis Tool”, Proceedings of The 11th International Technical Meeting of The

Satellite Division of The Institute of Navigation, Nashville TN, USA, September

1998.

Hannah, B., Walker, R. and Kubik, K., “Parabolic Equation-Based Time Analysis of

GPS Multipath Propagation”, International Conference of Spatial Information Science

and Technology, Wuhan, China, December 13-16 1998.

Hannah, B., “Parabolic Equation Research at RAL”, Cooperative Research Report,

British Council, December 1998.

Hannah, B., Walker, R. and Kubik, K., “Electromagnetic Propagation Modelling for

GPS”, SS04, Cooperative Research Centre for Satellite Systems Conference, Paradise

Wirrina Cove Resort, South Australia, CRCSS Technical Memoranda 99/1, 16-19

February, 1999.

Hannah, B. and Walker, R., “Determination of Tide Height Variation using GPS

Signal-to-Noise Ratio”, The 4th International Symposium on Satellite Navigation

Technology and Applications, Brisbane, Australia, 20-23 July 1999.

Hannah, B., “PE-Based Fourier Synthesis Time Analysis of GPS Multipath

Propagation”, Interim PhD report, July 1999.

A-2

Hannah, B., Kubik, K., and Walker, R., “Propagation Modelling of GPS Signals”,

Quo Vadis Geodesia Symposium, Technical Report Nr. 1999.6, Department of

Geodesy and Geoinformatics, Universität Stuttgart, October 1999.

Hannah, B. and Walker, R. “Dual Multipath Error Envelopes for SPS GPS”, CRCSS

Centre Conference 2000, Adelaide, South Australia, CRCSS Technical Memoranda

00/1, 15-17 February, 2000.

Walker, R. and Hannah, B. and Kubik, K., “Deterministic GPS Multipath Mitigation

for Spacecraft Precise Orbit Determination Applications”, SS02, Cooperative

Research Centre for Satellite Systems Conference, Paradise Wirrina Cove Resort,

South Australia, CRCSS Technical Memoranda 99/1, 16-19 February, 1999.

Walker, R., Hannah, B. and Kubik, K. “Multipath Issues in GPS Monitoring”,

International Workshop Proceedings for Advances in Deformation Monitoring, Curtin

University of Technology, Perth Western Australia, 24-25 September 1998. Invited

paper.

Walker, R. and Hannah, B., “Deterministic GPS Multipath Mitigation for Spacecraft

Precise Orbit Determination Applications”, The 4th International Symposium on

Satellite Navigation Technology and Applications, Brisbane, Australia, 20-23 July

1999.

Walker, R. and Hannah, B. “Deterministic GPS Multipath Mitigation for Spacecraft

Applications”, CRCSS Centre Conference 2000, Adelaide, South Australia, CRCSS

Technical Memoranda 00/1, 15-17 February, 2000.

B-1

Appendix B Raw GPS Multipath Data

B.1 Data Results Fresh Water - North Pine Dam 2 December 1999

Antenna Height 1.26 metres.

(a) SV 5

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 5

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 5

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs

Satellite Azimuth SV 5

Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs

Satellite Elevation SV 5

Ele

vatio

n (d

egs)

B-2

(b) SV 6

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 6

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 6

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(c) SV 8

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 8

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 8

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-3

(d) SV 9

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 9

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 8

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(e) SV 10

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 10

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

L1 MULTIPATH N/A

B-4

(f) SV 17

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 17

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 17

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(g) SV 24

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 24

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

L1 MULTIPATH N/A

B-5

(h) SV 25

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 25

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 25

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(i) SV 26

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 26

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 26

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-6

(j) SV 30

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

epochs

L1 S/N SV 30

S/N

(dB

)

0 500 1000 1500 2000 2500 3000 3500 4000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 30

Ran

ge E

rror

(m

)

0 500 1000 1500 2000 2500 3000 3500 4000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 500 1000 1500 2000 2500 3000 3500 40000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-7

B.2 Data Results Soil - Caboolture 30 November 1999

Antenna height 1.3 metres.

(a) SV 5

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 5

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 5

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-8

(b) SV 6

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 6

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 6

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(c) SV 8

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 8

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 8

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-9

(d) SV 17

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 17

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 17

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(e) SV 21

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 21

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 21

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-10

(f) SV 23

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 23

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 23

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(g) SV 26

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 26

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 26

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-11

(h) SV 30

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

epochs

L1 S/N SV 30

S/N

(dB

)

0 1000 2000 3000 4000 5000 6000 7000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 30

Ran

ge E

rror

(m

)

0 1000 2000 3000 4000 5000 6000 7000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-12

B.3 Data Results Sea Water - Bribie Island 11 November 1999

Antenna Height 3.4 metres (approx – tidal variation)

(a) SV 3

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 3

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 3

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-13

(b) SV 6

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 6

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 6

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(c) SV 10

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 10

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 10

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-14

(d) SV 17

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 17

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 17

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(e) SV 21

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 21

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 21

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-15

(f) SV 22

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 22

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 22

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

(g) SV 23

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 23

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 23

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

B-16

(h) SV 30

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

epochs

L1 S/N SV 30

S/N

(dB

)

0 2000 4000 6000 8000 10000 12000-5

-4

-3

-2

-1

0

1

2

3

4

5

epochs

L1 Multipath SV 30

Ran

ge E

rror

(m

)

0 2000 4000 6000 8000 10000 12000

-300

-200

-100

0

100

200

300

epochs


Azi

mut

h (d

egs)

0 2000 4000 6000 8000 10000 120000

10

20

30

40

50

60

70

80

90

epochs


Ele

vatio

n (d

egs)

C-1

Appendix C MATLAB Code

C.1 Propagation Modelling Code

(a) GOPE.M

%***************************************************************************% GOPE - control file for FSPE and PETA modelling%%% Bruce Hannah% Version date 14/8/00%***************************************************************************

clear % clear the workspace to start

%***************************************************************************% The following code is an example of using the code with data files% created from data collection exercises modified to MATLAB format by% teqc routines.% When not required code is commented out%***************************************************************************

%sv_num=17; % set SV number%start_epoch=1; % start of data epoch%end_epoch=5000; % end of data epoch%epoch_step=50; % data resolution%epoch_range=start_epoch:end_epoch; % set epoch range%resampled_epoch_range=...% start_epoch:epoch_step:end_epoch; % resample data%cd g:\BMH_data\30_11_98\mat_files; % set data path%load cab_ele % load elevation data%ele_data=sv_data(sv_num,:); % save elevation data%load cab_sn1 % load SNR data%sn_data=sv_data(sv_num,:); % save SNR data%clear sv_data % clear data

%***************************************************************************% start timing of simulation and zero floating point operation counter%***************************************************************************ticflops(0)

%***************************************************************************% global variable declarations%***************************************************************************

global TERRAIN_PROFILE % stores dtm terrain profileglobal TRUEglobal FALSEglobal MHZglobal GHZglobal Cglobal RADIANSglobal DEGREES

C-2

%***************************************************************************% constants%***************************************************************************

TRUE=1; % logical true keywordFALSE=0; % logical false keywordMHZ=1e6; % megahertz constantGHZ=1e9; % gigahertz constantC=3e8; % speed of light in a vacuum (m/s)RADIANS=pi/180; % convert degs to rads (ex: 10*RADIANS)DEGREES=180/pi; % convert rads to degs (ex: pi*DEGREES)

%***************************************************************************% control flags%*************************************************************************** % if setfield_plot=1; % PE field is plottedtime_plot=1; % time series is plottedterrain=0; % terrain is loadedbackscatter=0; % backscatter algorithm is implemented in mpe.mtime_analysis=0; % time analysis routine is implemented in mpe.mini_reflection=1; % initial reflection component includedsv_power=0; % sv power level profile usedfilter_time=0; % a hanning window is applied for time filteringsave_spectrum=0; % angular spectrum saved for later usesave_results=0; % results are saved in MAT filesave_images=0; % images are saved in format specifieddo_fwd_rays=0; % forward rays are plotted on the field plotdo_rev_rays=0; % reverse rays are plotteddo_interp=0; % interpolates dtm data if necessarydo_pie_progress=0; % user feedback of calculation timeuser_ant=0; % user selectable antenna positioning

%***************************************************************************% variable selections%***************************************************************************

domain_height=10; % default domain heightx_max=20; % default domain range distancedbmax=10; % maximum scale value for field displaydbmin=-40; % minimum scale value for field displayt_amp=1.2; % maximum amplitude scale for time displayf=1.575*GHZ; % centre frequency of analysislb_roughness=0; % surface roughness height for lower boundarybs_roughness=0; % surface roughness height for backscatterersrange_step=1; % selected range step size for PE calculationsrange_dist=6; % default location of antenna in x-dimensionant_height=1; % default location of antenna in z-dimensionangle_range=1; % user selects single angle or% angle_range=2:0.05:10; % or range of arbitrary linear values or% angle_range=... % or load angle info from data to simulate% ele_data(resampled_epoch_range);LHCP_reject=6; % antenna rejection of LHCP signals in dBrc_type=’concrete’;% reflection coefficient material type flag % if empty or 0 then not invoked % use only the following strings: % sea_water, fresh_water, ground_dry, % ground_med, ground_wet, concrete

if time_plot % catch flags for missed settings time_analysis=1; % user must want time analysisendif backscatter terrain=1; % user must have terrain selected to use bsend

C-3

%***************************************************************************% setup for image saving%***************************************************************************

if save_images % save image option ON prntype=’-dbitmap’; % set saved image format dirpath= ’c:\bruce\PEfiles\movfiles\’; % set path dirname=[’F_mode2’];%,num2str(sv_num)]; % directory name for images if ~exist([dirpath,dirname],’dir’) % check if directory exists eval([’!mkdir ’,dirpath,dirname]); % if not make directory end eval([’cd ’,dirpath,dirname]); % change to new directoryend

%***************************************************************************% terrain and backscatter setup routines%***************************************************************************

if terrain % use terrain option ON [TERRAIN_PROFILE,domain_height,x_max]=... % load the terrain information loadprofile(range_step,do_interp); fwd_delta_h=get_heights(TERRAIN_PROFILE); % load delta heights ofterrain if backscatter % backscatter option ON rev_delta_h=-fliplr(fwd_delta_h); % reversed delta heights scatterer=find(fwd_delta_h>0); % index to back scatterers endend

%***************************************************************************% get antenna height and distance from user input if option selected%***************************************************************************

if user_ant&terrain % available only with terrain figure % open figure hold on terrain_x=0:range_step:(x_max-2*range_step); % set up terrain display dk_green=[ 0.3 .5 0.3]; % set terrain block colour col=dk_green; for index=1:length(terrain_x) % fill in terrain profile h=fill([terrain_x(index) terrain_x(index)... terrain_x(index)+range_step... terrain_x(index)+range_step],... [0 TERRAIN_PROFILE(index) TERRAIN_PROFILE(index) 0],col); set(h,’EdgeColor’,col); end; set(gca,’YLim’,[0 domain_height],’XLim’,[0 x_max-2*range_step]); title(’Use the mouse to position the Antenna’); xlabel(’Range (m)’); ylabel(’Domain height (m)’);

ans=ginput(1); % get user selection range_dist=round(ans(1)/range_step)*range_step; % set range toantenna ant_height=ans(2); % set height of antenna

%bring the antenna height back down to height above terrain

ant_height=ant_height-TERRAIN_PROFILE(round(range_dist/range_step)); close;end

C-4

%***************************************************************************% time analysis setup routines if option selected% calls petasetup.m%***************************************************************************

if time_analysis % time analysis option ON: load time analysisparameters [f_max,f_array,lower_pad,upper_pad,ts,tmin]=... petasetup(f,x_max,range_dist,range_step,... ant_height,angle_range,backscatter,terrain);else % time analysis option OFF: limit to centre frequency f_array=f; % frequency array is centre frequency f_max=f; % maximum frequency is centre frequencyend

%***************************************************************************% sampling in the z (height) domain% at 90 degrees dz is half the wavelength of the maximum frequency% other selections are allowable within nyquist bound or better%***************************************************************************

dz=0.01; % C/(4*f_max) is the alternative

%***************************************************************************% adjust terrain profile for dz sample resolution%***************************************************************************

if terrain TERRAIN_PROFILE=round(TERRAIN_PROFILE/dz)*dz;end

%***************************************************************************% FSPE setup routines%***************************************************************************

ant_height=round(ant_height/dz)*dz; % antenna height at dz stepcell_X=round(range_dist/range_step); % antenna range cellcell_Z=round(ant_height/dz); % antenna height cellif time_analysis % arrays for time analysis delay_time=zeros(length(angle_range),8); % initialise time delay delay_amp=zeros(length(angle_range),8); % initialise amplitude delay_phase=zeros(length(angle_range),8); % initialise phase delay_aoa=zeros(length(angle_range),8); % initialise angle of arrivalend

angle_index=1; % initialise loop index (angle)

%***************************************************************************% Run Simulation for all angles through calls to mpe.m (Multiple PE)%***************************************************************************

for theta=angle_range*RADIANS %======START OF SIMULATION====== disp([’Running FSPE for ’,... num2str(angle_range(angle_index)),... ’ degs :’, rc_type]); mpe;

C-5

%***************************************************************************% Store PETA results for each multipath signal% delay_time is delay% delay_amp is normalised amplitude {0 1}% delay_phase is phase% delay_aoa is angle of arrival% t_out is MCIR data%***************************************************************************

if time_analysis delay_time(angle_index,1:length(delay))=delay(1,:); delay_amp(angle_index,1:length(mp_mag))=mp_mag(1,:); delay_phase(angle_index,1:length(phase_diff))=phase_diff(1,:); delay_aoa(angle_index,1:length(aoa))=aoa(1,:); t_out(:,:,angle_index)=time_out; end

%***************************************************************************% Store FSPE results for each propagation angle% field_result is total FSPE field returned by mpe.m% field_pe is absolute field value at antenna location (f)% field_db is field value in dB%***************************************************************************

field_pe(angle_index)=result_field(cell_Z,cell_X); field_db(angle_index)=20*log10(abs(field_pe(angle_index)));

%***************************************************************************% set-up routines to save images of simulation%***************************************************************************

if (save_images)&(field_plot|time_plot) if angle_index<10 filename=[’ image0’,num2str(angle_index)]; else filename=[’ image’,num2str(angle_index)]; end eval([’print ’,prntype,filename]); close end angle_index=angle_index+1;end %=======END OF SIMULATION=======

%***************************************************************************% Simulation timing and operations report%***************************************************************************

runtime=toc; % time of simulationdisp([’simulation time: ’,num2str(runtime),’s. ’,’FLOPS: ’,num2str(flops)]);

%***************************************************************************% Save results to MAT file if required%***************************************************************************

if save_results fname=input(’save results filename?’,’s’); if time_analysis t_out=squeeze(t_out); % remove single dimension in t_out array eval([’save ’,fname,’_td’,... ’ delay_time delay_amp ... delay_phase delay_aoa ... angle_range time_axis ... t_out field_db field_pe’]); else eval([’save ’,fname,’_fd’,’ field_db field_pe angle_range’]); endend

C-6

%***************************************************************************% Memory cleanup%***************************************************************************

clear globalclear do_* db*clear field_plot time_plot terrain backscatter time_analysis... filter_time save_spectrum save_results save_images user_ant angle_index... aoa delay domain_height f_max lower_pad upper_pad mp_mag phase_diff ... stored_field full_field t_amp time_out theta

C-7

(b) MPE.M

%***************************************************************************% MPE - called by gope.m for range of propagation angles%% Free-space parabolic equation Fourier step method% Options:% Backscatter% Time series analysis option%% This file is called recursively by gope so that a range of propagation% angles can be used to provide a simulation of GPS satellite motion.% Depending on options selected it will calculate field results% with and without terrain, backscattering, variable sv power profiles,% or reflection coefficients. Time domain analysis is also carried out% if selected.%% Results are saved and/or displayed as required.%% Bruce M. Hannah% Cooperative Research Centre for Satellite Systems% Queensland University of Technology% Version date 22/8/00%***************************************************************************

%***************************************************************************% global variables%***************************************************************************

global TRUE; % Logical true keywordglobal FALSE; % Logical false keywordglobal MHZ; % Megahertzglobal GHZ; % Gigahertzglobal C; % Speed of light in a vacuum (m/s)global RADIANS; % Convert degs to rads (ex: 10*RADIANS)global DEGREES; % Convert rads to degglobal TERRAIN_PROFILE; % Terrain profile data

%***************************************************************************% Linearised user minimum received signal level profile% as per ICD-GPS-200 6.3.1 if option selected%***************************************************************************

if sv_power if theta*DEGREES<=4 sv_power_ratio=1; elseif theta*DEGREES<=20 sv_power_ratio=10^(((1.1/16)*(theta*DEGREES-4))/20); elseif theta*DEGREES<=32 sv_power_ratio=10^((1.1+(0.75/12)*(theta*DEGREES-20))/20); elseif theta*DEGREES<=40 sv_power_ratio=10^((1.85+(0.16/8)*(theta*DEGREES-32))/20); elseif theta*DEGREES<=48 sv_power_ratio=10^((2.01-(0.14/8)*(theta*DEGREES-40))/20); elseif theta*DEGREES<=60 sv_power_ratio=10^((1.87-(0.66/12)*(theta*DEGREES-48))/20); elseif theta*DEGREES<=90 sv_power_ratio=10^((1.21-(1.21/30)*(theta*DEGREES-60))/20); endelse sv_power_ratio=1;end

bck_call=FALSE; % flag for use with backscatter routine

C-8

%***************************************************************************% z_max: maximum height of domain allowing for entry of line-of-sight signal% into the area of interest. This is achieved by taking into account the% angle of propagation and then doubling the domain height.%***************************************************************************

z_max=2*(domain_height+((x_max*1.1)*tan(theta)));

%***************************************************************************% setup sampling and initilise arrays, dz is calculated in gope.m%***************************************************************************

N_minus_1=nextpow2(z_max/dz); % number of z samplesN=N_minus_1+1;if save_spectrum U=zeros(2^N,x_max/range_step); % initialise spectrum arrayendif backscatter % with backscatter initialise % an initial fwd field array fwd_initial=zeros(2^N_minus_1,x_max/range_step);endu=zeros(2^N_minus_1,x_max/range_step); % initialise generic field arrayfwd_field=zeros(2^N_minus_1,x_max/range_step); % initialise forward fieldarrayu_len=length(u(:,1)); % length of z domain in samplesz=dz:dz:((2^N_minus_1)*dz); % generate array of z elementsz=reshape(z,u_len,1);

%***************************************************************************% generate Hanning window and modify the shape of the window so that it% tapers off the aperture field for values between Zmax and 2*Zmax.%***************************************************************************

hn=hanning(u_len);hn(1:u_len/2)=ones(size(hn(1:u_len/2)));

%***************************************************************************% setup the sampling in the angular-spectrum or p-domain%***************************************************************************

dp=2*pi/((2^N)*dz);p=0:dp:(((2^N_minus_1)-1)*dp);

%***************************************************************************% fft setup%***************************************************************************

fft_size=2^N;

clear N u_len % memory clean up

%***************************************************************************% Do the calculations for the frequencies specified in f_array. These %frequencies are specified for the Fourier-based time analysis. If time% analysis is not being done then the f_array is simply the GPS frequency% specified in gope.m%***************************************************************************

for array_index=1:length(f_array) ========Start field calculations========== % if selected option if do_pie_progress % provide visual feedback l=length(f_array)-array_index+1; % of calculation progress pie([array_index l]); title(’Percent Complete’); pause(0.01); set(gca,’Selected’,’on’); end

frequency=f_array(array_index); % get current frequency k=2*pi*frequency/C; % calculate wavenumber

C-9

%**************************************************************************% setup reflection coefficients for modifying the angular-spectrum% or p-space%**************************************************************************

% Set the 0 to 90 degrees theta limit from p-space

rc_theta=asin((0:2*dp:k)/k);

% Electrical properties of mediums if rc_type if strcmp(rc_type,’sea_water’) cond=4; ep_r=20; elseif strcmp(rc_type,’fresh_water’) cond=1e-3; ep_r=80; elseif strcmp(rc_type,’ground_dry’) cond=1e-5; ep_r=4; elseif strcmp(rc_type,’ground_med’) cond=1e-3; ep_r=7; elseif strcmp(rc_type,’ground_wet’) cond=1e-2; ep_r=30; elseif strcmp(rc_type,’concrete’) cond=2e-5; ep_r=3; end

ep=ep_r-(j*60*(C/frequency)*cond);

sin_term=sin(rc_theta); cos_term=sqrt(ep-(cos(rc_theta).^2)); refl_h=(sin_term-cos_term)./(sin_term+cos_term); refl_v=(ep*sin_term-cos_term)./(ep*sin_term+cos_term); po=(refl_h+refl_v)/2; px=(refl_h-refl_v)/2;

% Co-polarised rc magnitude: forward propagation on lower boundary co_mag=abs(po);

% Cross-polarised rc magnitude: forward propagation on lower boundary cross_mag=abs(px);

% Coupled forward reflection coefficient for RHCP incidence fmode_rc_R=(co_mag+(10^(-LHCP_reject/20))*cross_mag);

% Coupled forward reflection coefficient for LHCP incidence fmode_rc_L=((10^(-LHCP_reject/20))*co_mag+cross_mag);

%***************************************************************************% Surface roughness reduction factor for lower boundary%***************************************************************************

g_term=(4*pi*lb_roughness*sin(rc_theta))/(C/frequency); bessel_factor=besseli(0,0.5*(g_term.^2));

% Rough surface reduction factor for lower boundary fwd_sr_factor=exp(-0.5*(g_term.^2)).*bessel_factor;

% Total combined rc mag for fwd prop over lower boundary fwd_rc_lb=fmode_rc_R.*fwd_sr_factor;

clear cond sin_term cos_term refl_h refl_v po px g_term... bessel_factor ep_r

C-10

%***************************************************************************% backscatter reflection coefficients%***************************************************************************

if backscatter

% Theta at the vertical backscatter interface % for the calculation of reflection coefficients % is 90 degrees - propagation angle bs_theta=pi/2-rc_theta;

% Calculate bs reflection coefficient magnitudes sin_term=sin(bs_theta); cos_term=sqrt(ep-(cos(bs_theta).^2)); refl_h=(sin_term-cos_term)./(sin_term+cos_term); refl_v=(ep*sin_term-cos_term)./(ep*sin_term+cos_term); po=(refl_h+refl_v)/2; px=(refl_h-refl_v)/2;

% Co-polarised component at bs interface bs_co_mag=abs(po);

% Cross-polarised component at bs interface bs_cross_mag=abs(px);

% BA-mode rc for RHCP incidence on bs interface bamode_rc_R=(bs_co_mag+(10^(-LHCP_reject/20))*bs_cross_mag);

% BA-mode rc for LHCP incidence on bs interface bamode_rc_L=((10^(-LHCP_reject/20))*bs_co_mag+bs_cross_mag);

% Coupling factor for BB-mode zone 1 z1_factor=bamode_rc_L./bamode_rc_R;

% Coupling factor for BB-mode zone 2 z2_factor=fmode_rc_L./fmode_rc_R;

%***********************************************************************% surface roughness reduction factor on backscatter interface%***********************************************************************

g_term=(4*pi*bs_roughness*sin(bs_theta))/(C/frequency); bessel_factor=besseli(0,0.5*(g_term.^2)); bs_sr_factor=exp(-0.5*(g_term.^2)).*bessel_factor;

%***********************************************************************% Calculation of coupled and modified% coupled reflection coefficient magnitudes%***********************************************************************

% Lower boundary modified coupled rc in back direction bck_rc_lb=(co_mag+z1_factor.*cross_mag).*fwd_sr_factor;

% Backscatter interface coupled rc for BA-mode propagation bamode_rc_bs=bamode_rc_R.*bs_sr_factor;

% Backscatter interface modified couplied rc BB-mode propagation bbmode_rc_bs=(bs_co_mag+z2_factor.*bs_cross_mag).*bs_sr_factor;

clear sin_term cos_term refl_h refl_v po px bs_co_mag... bs_cross_mag bamode_rc_R bamode_rc_L z1_factor z2_factor... g_term bessel_factor bs_sr_factor fmode_rc_L fmode_rc_R... bs_theta

end

C-11

%***************************************************************************% Calculation of coupled reflection coefficient for initial reflected% incident signal%***************************************************************************

if ini_reflection

% Calculate initial incident rc from initial propagation angle sin_term=sin(theta); cos_term=sqrt(ep-(cos(theta).^2)); refl_h=(sin_term-cos_term)./(sin_term+cos_term); refl_v=(ep*sin_term-cos_term)./(ep*sin_term+cos_term); po=(refl_h+refl_v)/2; px=(refl_h-refl_v)/2; orig_mag=abs(po); cross_mag=abs(px); inc_rc_mag=(orig_mag+(10^(-LHCP_reject/20))*cross_mag);

g_term=(4*pi*lb_roughness*sin(theta))/(C/frequency); bessel_factor=besseli(0,0.5*(g_term.^2));

% Rough surface reduction factor mod_factor=exp(-0.5*(g_term.^2)).*bessel_factor;

% initial value for incident reflection inc_rc_mag=inc_rc_mag*mod_factor;

clear sin_term cos_term refl_h refl_v po px... orig_mag cross_mag g_term bessel_factor mod_factor end

% If no medium defined do total reflection % This does not couple polarisations else fwd_rc_lb=ones(1,length(rc_theta)); bck_rc_lb=ones(1,length(rc_theta)); bamode_rc_bs=ones(1,length(rc_theta)); bbmode_rc_bs=ones(1,length(rc_theta)); inc_rc_mag=1; end

rc_mag=fwd_rc_lb;

clear ep co_mag cross_mag fmode_rc_L fmode_rc_R fwd_sr_factor... fwd_rc_lb rc_theta

%***************************************************************************% setup initial field and pe propagator%***************************************************************************

incident=exp(-j*k*z*sin(theta)); % incident plane-wave reflected=-exp(j*k*z*sin(theta)); % reflected plane-wave

if ini_reflection u(:,1)=(sv_power_ratio*incident)+... (sv_power_ratio*inc_rc_mag*reflected); % initial reduced field hasreflection clear inc_rc_mag else u(:,1)=(sv_power_ratio*incident); % just initial incident field end

% FSPE propagator propagator=exp(i*k*range_step*(sqrt(1-p.^2/k^2)-1));

% Image of FSPE propagator mirror_prop=fliplr(propagator);

% Combined propagator for FSPE solution combined_prop=[propagator,mirror_prop];

C-12

% Initial propagation is in forward direction % Therefore terrain jumps are for forward terrain if terrain delta_h=fwd_delta_h; end

%***************************************************************************% The forward propagated reduced field is calculated by the fieldcalc% routine%***************************************************************************

fieldcalc;

%***************************************************************************% The FSPE calculates the reduced field which has the rapid phase variation% in x removed. For time-domain analysis this phase variation needs% to be reinstated.%***************************************************************************

index=1;

for x=range_step:range_step:x_max fwd_field(:,index)=u(:,index)*exp(j*k*x); index=index+1; end

if terrain stored_count=bin_count; end

%***************************************************************************% If the backscatter option is selected we need to find the initial% backscatter field components and propagate them using the FSPE% field calculation (fieldcalc) routine.%***************************************************************************

if backscatter

% Identify backscatter interface locations and element sizes for back_index=1:length(scatterer) ref_x_index=scatterer(back_index); num_cells=round((1/dz)*delta_h(ref_x_index)); fwd_initial(1:num_cells,ref_x_index)=... fwd_field(1:num_cells,ref_x_index); end

% Mirror initial backscatter fields so fieldcalc can be used bck_initial=fliplr(fwd_initial); clear back_index num_cells ref_x_index [row_pos,col_pos,field_value]=find(bck_initial); row_index=find(row_pos==1); col_pos=col_pos(row_index); delta_h=rev_delta_h;

% Insert initial bs field values and calculate field % for each backscattering interface for index=1:length(col_pos)

u(:,1)=bck_initial(:,1); bck_call=col_pos(index); row_start=row_index(index);

if index==length(col_pos) row_end=length(row_pos); else row_end=row_index(index+1)-1; end

num_rows=row_end-row_start+1;

% The lower boundary reflection coefficient is now % the modified coupled rc for backscatter

C-13

rc_mag=bck_rc_lb;

% Use fieldcalc to solve field fieldcalc;

x=0; col_index=col_pos(index);

% Reinstate phase variation for backscatter for phase_index=col_pos(index)*range_step:range_step:x_max u(:,col_index)=u(:,col_index)*exp(j*k*x); x=x+range_step; col_index=col_index+1; end

clear phase_index

% For all caculated fields in the reverse direction % the total reverse field is the summation of all solved fields if index==1 rev_field=u; else rev_field=rev_field+u; end

end

% Back propagation completed bck_call=FALSE;

% The total back propagated field is the mirrored reverse FSPE result bck_field=fliplr(rev_field);

%**************************************************************************% Back propagation is being done therefore the total field is addition of% both forward and back propagated fields.%**************************************************************************

total_field=bck_field+fwd_field;

clear num_rows rev_field bck_initial bck_field field_value row_pos... row_start col_pos col_index bamode_rc_bs bbmode_rc_bs bck_rc_lb... row_end row_index

else

**************************************************************************% Back propagation is not being done therefore the total field is the% forward field only. %**************************************************************************

total_field=fwd_field;

end

clear delta_bins bin_count incident reflected fwd_field... combined_prop propagator mirror_prop

C-14

%***************************************************************************% To correct the time domain results to the incident boundary% we must first correct for the relative height of the antenna location% to the incident field origin point.If there is terrain the relative height% is the height difference between the height of the terrain% at the incident boundary and the terrain height at the antenna,% plus the actual antenna height above the terrain.% If there is no terrain the relative height is simply the antenna height.% This is the first correction term applied to the field result.%% To correct to the line-of-sight entry point on the incident boundary% a correction is made for the antenna distance.% This is the second correction term to the field result.

% In addition a correction is applied to shift the time domain results% relative to the minimum time set for the time window. This correction is% the last correction term applied to the field result.%% The resultant field value is stored for each frequency in the array% field_cell.%% An angle of arrival estimation uses the variation of time arrival of the% signals by looking at the time results one cell above the antenna.% This result is stored in aoa_cell.%***************************************************************************

if time_analysis

if terrain del_h=ant_height+(round(TERRAIN_PROFILE(cell_X)/dz)*dz)-... (round(TERRAIN_PROFILE(1)/dz)*dz); else del_h=ant_height; end

% Phase correction field_cell(array_index)=total_field(cell_Z,cell_X)... *exp(j*k*del_h*sin(theta))... *exp(j*k*range_dist*(1/cos(theta)-cos(theta)))... *exp(-j*2*pi*frequency*tmin); aoa_cell(array_index)=total_field(cell_Z+1,cell_X)... *exp(j*k*del_h*sin(theta))... *exp(j*k*range_dist*(1/cos(theta)-cos(theta)))... *exp(-j*2*pi*frequency*tmin);

clear del_h

end

%***************************************************************************% When the frequency is the centre frequency store the field result% at antenna location%***************************************************************************

if frequency==f stored_field=total_field; result_field=stored_field(cell_Z,cell_X); end

clear total_field

end

C-15

%===========================================================================%===========Field calculations at all frequencies completed=================%===========================================================================

clear sv_power_ratio k frequency dp fft_size array_index ... fwd_initial hn N_minus_1 p z bck_call u

% Visual progress indication stoppedif do_pie_progress closeend

%***************************************************************************% Time analysis is done here. The frequency-domain spectrum is created from% the stored field results of each frequency. The inverse fft then gives the% required time-domain results. Residual phase and angle of arrival (aoa)for% each peak are also estimated.%***************************************************************************

if time_analysis

% Time results filtered if option selected % Filtering results in widening of time pulses % Resolution is therefore reduced with filtering if filter_time win_fun=transpose(hanning(length(field_cell))); field_cell=win_fun.*field_cell; aoa_cell=win_fun.*aoa_cell; threshold=0.1; % threshold for filtered results clear win_fun else threshold=0.4; % threshold value for peak search end

pos_spec=[lower_pad,field_cell,upper_pad]; % create frequency spectrum td=fft(pos_spec); % do fft to get time-domain result time_out=abs((td)); % get the amplitude time spectrum t_step=2*ts; % the sampling is twice since using half the samples max_value=max(time_out); % find the maximum value peak time_out=time_out./max_value; % prenormalise time_out for peak search samples=find(time_out>threshold); % get all the samples above thethreshold max_index=1; % initialise the index of peaks sample_num=0; % initilise sample storage array store=FALSE; % recursive detection flag

%***************************************************************************% Use an gradient ascent technique to find the peak of each pulse% above the threshold.%***************************************************************************

for index=1:length(samples)-1 % get gradient grad=time_out(samples(index+1))-time_out(samples(index));

if grad>0 % if a positive gradient find_max=samples(index+1); % store the sample number store=TRUE; % and set store flag else % if the gradient is negative

if store % and there is a stored sample, this a peak sample_num(max_index)=find_max; % the sample number of the peakis stored store=FALSE; % reset flag to search for more peaks max_index=max_index+1; % increment the peak index for the nextpeak end

end end

num_peaks=length(sample_num); % number of peaks above the threshold

C-16

if num_peaks>0 % if there is at least one peak norm_amp1=time_out(sample_num(1)); % get the prenormalised peak valueof peak 1 time_out=time_out./norm_amp1; % normalise the time output to thefirst peak

%**************************************************************************% The time delays, amplitudes and phases of the peaks are now determined.%**************************************************************************

delay=(tmin/1e-9)+(t_step/1e-9*(sample_num-1)); % array of time delays mp_mag=time_out(sample_num); % array of corresponding magnitudes td_phase=angle(td(sample_num)); % array of phase at each peak

for phase_index=1:length(td_phase) % for each delay

field_ref=exp(j*2*pi*f_array*(delay(phase_index)*1e-9-tmin));

pos_spec_ref=[lower_pad,field_ref,upper_pad]; td_ref=fft((pos_spec_ref)); % reference phase of nth delay td_ref_phase=angle(td_ref(sample_num(phase_index)));

%***********************************************************************% The resultant phase from reflection is the difference between the% actual phase and the reference phase created solely using the% absolute time delay.%***********************************************************************

phase_diff(phase_index)=td_ref_phase-td_phase(phase_index); end

for index=1:num_peaks delay_txt=num2str(delay(index),’%9.4f’); delay_len=length(delay_txt);

if delay_len==7 delay_txt=[’0’,delay_txt]; elseif delay_len==6 delay_txt=[’0’,’0’,delay_txt]; elseif delay_len==5 delay_txt=[’0’,’0’,’0’,delay_txt]; end

text_str(index,:)=[delay_txt,’ ns’]; disp([’peak ’,num2str(index),’: ’,text_str(index,:)]); end

clear index delay_txt delay_len text_str

end

C-17

%***************************************************************************% This is the routine that looks at the cell above the antenna location% and calculates angle of arrival of the peaks in the time spectrum%***************************************************************************

aoa_spec=[lower_pad,aoa_cell,upper_pad]; % create frequency spectrum aoa_td=fft(aoa_spec); % do fft to get time-domain result aoa_time_out=abs((aoa_td)); % get the amplitude time spectrum aoa_max_value=max(aoa_time_out); % find the maximum value peak % prenormalise time_out for peak search aoa_time_out=aoa_time_out./max_value; % get all the samples above the threshold aoa_samples=find(aoa_time_out>threshold); max_index=1; % initialise the index of peaks aoa_sample_num=0; % initilise sample storage array store=FALSE; % recursive detection flag

%***************************************************************************% Use an gradient ascent technique to find the peak of each pulse% above the threshold.%***************************************************************************

for index=1:length(aoa_samples)-1 grad=aoa_time_out(aoa_samples(index+1))-... aoa_time_out(aoa_samples(index)); % store the gradient

if grad>0 % if a positive gradient find_max=aoa_samples(index+1); % store the sample number store=TRUE; % and set store flag else % if the gradient is negative

if store % and there is a stored sample, this a peak % sample number of the peak is stored

aoa_sample_num(max_index)=find_max; store=FALSE; % reset flag to search for more peaks % increment the peak index for the next peak max_index=max_index+1; end

end end

% number of peaks above the threshold num_aoa_peaks=length(aoa_sample_num);

if num_aoa_peaks>0 % if there is at least one peak norm_amp1=time_out(aoa_sample_num(1)); % normalise the time output aoa_time_out=aoa_time_out./norm_amp1; % to the first peak

%**************************************************************************% The time delay for aoa analysis is now determined.%**************************************************************************

% array of time delays aoa_delay=... (tmin/1e-9)+(t_step/1e-9*(aoa_sample_num-1)); end

%**************************************************************************% Angle of arrival (aoa) estimates are found for each pulse in the time% domain by finding the difference between the arrival times in the two% cells. The inverse sin of the relative distance travelled by the pulse% gives the aoa estimate.%***************************************************************************

aoa=asin(C*(delay-aoa_delay)*1e-9/dz)*DEGREES;

time_axis=(0:t_step/1e-9:length(time_out)*t_step/1e-9)+tmin/1e-9;

C-18

%***************************************************************************% Plotting Routines - Time plot only%***************************************************************************

if time_plot&~field_plot plot(time_axis,[time_out,0]); axis([round(min(time_axis)) round(max(time_axis)) 0 t_amp]); xlabel(’Absolute time delay (ns)’); x_text_pos=time_axis(round(length(time_out)/1.64));

for index=1:num_peaks text(x_text_pos,1-index*0.05,... [’peak ’,num2str(index),’ delay: ’,text_str(index,:)]); end

if num_peaks>1

for index=2:num_peaks delay_diff=delay(index)-delay(1); text(x_text_pos,1-(index+num_peaks-1)*.05,... [’time delay: ’,num2str(delay_diff),’ ns’]); end

end clear index text_str x_text_pos delay_diff end clear max_index find_max grad aoa_* max_value field_cell field_ref... norm_amp1 num_peaks num_aoa_peaks pos_spec pos_spec_ref samples... sample_num store t_step td td_phase td_ref td_ref_phase threshold... tmin tsend

%**************************************************************************% Setup rountine for field alone or field and time plots together.%***************************************************************************

if field_plot|(field_plot&time_plot)

%***************************************************************************% Shift the field to account for the boundary shift technique. Zeros are% inserted at the terrain positions.%***************************************************************************

if terrain

x_index=1;

num_zeros=round((dz+TERRAIN_PROFILE(x_index))/dz);

for x=0:range_step:x_max-2*range_step

if x_index==1 stored_bins=0; else stored_bins=stored_count(x_index-1); end

num_zeros=num_zeros+stored_bins;

clear stored_bins

if ~(num_zeros==0) clear temp temp(abs(num_zeros))=0; field=stored_field((1:(length(stored_field(:,x_index))-... length(temp))),x_index); us=size(field); field=reshape(field,1,us(1)); temp_field=[temp,field]; stored_field(:,x_index)=temp_field(:); clear field temp temp_field us end

C-19

x_index=x_index+1; end

clear stored_count x_index

end

num_z_elem=round(domain_height/dz); % number of Z elements X=0:range_step:x_max-2*range_step; % X array of range elements Z=0:dz:(num_z_elem)*dz; % Z array of height elements warning off % suppress log of zero warnings log_field=20*log10(abs(stored_field(1:length(Z),1:length(X)))); warning on clear stored_field num_z_elem num_zerosend

%***************************************************************************% Field plot only%***************************************************************************

if field_plot&~time_plot pcolor(X,Z,log_field); colormap(jet); caxis([dbmin dbmax]); shading interp; colorbar;

clear X Z log_field

if terrain % overlay the terrain on the field plot hold on terrain_x=0:range_step:(x_max-2*range_step); dk_green=[ 0.3 .5 0.3]; col=dk_green;

for step=1:length(terrain_x); h=fill([terrain_x(step) terrain_x(step) ... terrain_x(step)+range_step terrain_x(step)+range_step],... [0 TERRAIN_PROFILE(step) TERRAIN_PROFILE(step) 0],col); set(h,’EdgeColor’,col); end

hold off y_ant=ant_height+TERRAIN_PROFILE(cell_X);

clear step col dk_green h terrain_x

else y_ant=ant_height; end

if do_fwd_rays fr_height=plot_fwd_rays(range_dist,range_step,... ant_height,domain_height,theta,terrain); end

if do_rev_rays&backscatter [bra_dist,brb_dist,brb_height]=plot_rev_rays... (range_dist,ant_height,domain_height,theta,... x_max,scatterer,range_step,cell_X); end

hold on plot(range_dist,y_ant,’kv’); % plot antenna location markers plot(range_dist,y_ant,’k*’); hold off

xlabel(’Range (m)’); ylabel(’Height (m)’); title([’theta = ’,num2str(theta*DEGREES),' °']);end

C-20

%***************************************************************************% Field and time plotted together.%***************************************************************************

if field_plot&time_plot subplot(2,1,1),pcolor(X,Z,log_field); colormap(jet); caxis([dbmin dbmax]); shading interp; colorbar;

clear log_field X Z

if terrain hold on terrain_x=0:range_step:(x_max-2*range_step); dk_green=[ 0.3 .5 0.3]; col=dk_green; for step=1:length(terrain_x);

h=fill([terrain_x(step) terrain_x(step) ... terrain_x(step)+range_step terrain_x(step)+range_step],... [0 TERRAIN_PROFILE(step) TERRAIN_PROFILE(step) 0],col); set(h,’EdgeColor’,col); end

hold off y_ant=ant_height+TERRAIN_PROFILE(cell_X);

clear step col dk_green h terrain_x

else y_ant=ant_height; end

if do_fwd_rays fr_height=plot_fwd_rays(range_dist,range_step,ant_height,... domain_height,theta,terrain); end

if do_rev_rays&backscatter [bra_dist,brb_dist,brb_height]=plot_rev_rays(range_dist,... ant_height,domain_height,theta,x_max,scatterer,range_step,cell_X); end

hold on plot(range_dist,y_ant,’kv’); plot(range_dist,y_ant,’k*’); hold off

title([’Theta = ’,num2str(theta*DEGREES),' °']); xlabel('Range (m)'); ylabel('Height (m)');

subplot(2,1,2),plot(time_axis,[time_out,0]);

axis([round(min(time_axis)) round(max(time_axis)) 0 t_amp]); % setamplitude limit

if do_fwd_rays t_los=(range_dist/(C*cos(theta)))*1e9; t_fr=t_los+(2*fr_height*sin(theta)/C)*1e9; hold on plot(t_los,0,'gd') plot(t_fr,0,'kd') hold off clear t_los t_fr end

if exist('bra_dist')&bra_dist

for index=1:length(bra_dist)

C-21

t_bra=(range_dist*1e9/(C*cos(theta)))+... (2*bra_dist(index)*cos(theta)*1e9/C); hold on plot(t_bra,0,’bd’) hold off clear t_bra end

end

if exist(’brb_dist’)&brb_dist

for index=1:length(brb_dist) t_brb=(range_dist*1e9/(C*cos(theta)))+... (2*brb_dist(index)*cos(theta)*1e9/C)+... (2*brb_height(index)*sin(theta)*1e9/C); hold on plot(t_brb,0,’cd’) hold off clear t_brb end

end

xlabel(’Absolute time delay (ns)’); ylabel(’Relative amplitude’);end

clear rev_delta_h y_ant x delta_h fwd_delta_h fr_height... brb_dist brb_height bra_dist

C-22

(c) FIELDCALC.M

%***************************************************************************% FIELDCALC - Free-space parabolic field solution% Called by mpe.m for each angle and frequency if doing time analysis%% Bruce Hannah% version date 22/8/00%***************************************************************************

clear bin_count delta_bins;

%***************************************************************************% March FSPE solution out in range%***************************************************************************

x_index=1;

for x=0:range_step:(x_max-(2*range_step))

%***************************************************************************% Filter upper limit of solution domain with Hanning Window%***************************************************************************

u(:,x_index)=u(:,x_index).*hn;

%***************************************************************************% Previous field from field array%***************************************************************************

prev_field=u(:,x_index);

ps=size(prev_field); % Reshape field prev_field=reshape(prev_field,1,ps(1));

%***************************************************************************% Set up parameters for boundary shifting%***************************************************************************

if terrain if delta_h(x_index)~=0 delta_bins=round(delta_h(x_index)/dz); if (delta_bins>0) prev_field=prev_field((abs(delta_bins)+1):length(prev_field)); prev_field(length(prev_field)+abs(delta_bins))=0; end if (delta_bins<0) clear temp; temp(abs(delta_bins))=0; prev_field=[temp,prev_field(1:... (length(prev_field)-length(temp)))]; end else delta_bins=0; end bin_count(x_index)=delta_bins; end

%***************************************************************************% Create image field and apply 180 deg phase shift%***************************************************************************

odd_part=fliplr(-prev_field);

C-23

%***************************************************************************% Create the image angular spectrum%***************************************************************************

odd_ang_spec=fft(odd_part);

%***************************************************************************% This is the 0:90 degree region for the image spectrum%***************************************************************************

pos_ang_spec=odd_ang_spec(1:length(rc_mag));

%***************************************************************************% Apply reflection coefficient for the reflection% In forward direction this is F-mode reflection coefficient% In back direction (when there is backscatter) this is the% modified coupled reflection coefficient to compensate for% polarisation coupling at the backscatter interface%***************************************************************************

mod_pos_spec=(rc_mag.*abs(pos_ang_spec)).*exp(j*angle(pos_ang_spec));

%***************************************************************************% Insert modified spectrum into total spectrum%***************************************************************************

odd_ang_spec(1:length(rc_mag))=mod_pos_spec;

%***************************************************************************% Inverse fft to give total image field with correct rc applied%***************************************************************************

odd_part=ifft(odd_ang_spec);

clear odd_ang_spec pos_ang_spec mod_pos_spec

%***************************************************************************% Combine incident and image fields to create total field%***************************************************************************

combined=[prev_field,odd_part];

%***************************************************************************% Then take the FFT of it for total angular spectrum%***************************************************************************

U_x=fft(combined,fft_size);

%*************************************************************************** % Save incident spectrum if required%***************************************************************************

if save_spectrum if (x_index==1) U(:,1)=U_x(:); end end

%*************************************************************************** % U_x is the angular spectrum for the previous step. % We now multiply by the Propagator%***************************************************************************

U_x=U_x.*combined_prop;

if save_spectrum U(:,x_index+1)=U_x(:); end

C-24

%***************************************************************************% Take the IFFT to create FSPE propagated field%***************************************************************************

u_x=ifft(U_x,fft_size);

clear U_x combined

%***************************************************************************% Only want the upper half of the domain (solution domain)%***************************************************************************

u_x=u_x(1:(2^(N_minus_1)));

%***************************************************************************% This is now the next field array%***************************************************************************

u(:,x_index+1)=u_x(:);

%***************************************************************************% Insert the backscattered field components with reflection coefficients% The postive angles of the angular spectrum at the bs interface are bb-mode% The negative angles are ba-mode% Each region of angular spectrum is seperated into -90-0 and 0-90 degree% The appropriate reflection coefficient is then applied in p-space%***************************************************************************

if bck_call==x_index+1

%*************************************************************************% Apply 180 deg phase shift to field components%*************************************************************************

u(1:num_rows,x_index+1)=-field_value(row_start:row_end);

%*************************************************************************% Full angular spectrum (p-space) of backscattered field%*************************************************************************

bs_p_space=fft(u(:,x_index+1));

%*************************************************************************% Angles of propagation at 0:90 degs are bb-mode%*************************************************************************

pos_ang_spec=bs_p_space(1:length(bbmode_rc_bs));

%*************************************************************************% Angles of propagation at -90:0 degs are ba-mode%*************************************************************************

neg_ang_spec=bs_p_space(length(bs_p_space)-... length(bamode_rc_bs)+1:length(bs_p_space));

%*************************************************************************% Apply bb-mode reflection coeff magnitude (includes surface roughness)%*************************************************************************

mod_pos_spec=(bbmode_rc_bs’.*... abs(pos_ang_spec)).*exp(j*angle(pos_ang_spec));

%*************************************************************************% Apply ba-mode reflection coeff magnitude%*************************************************************************

mod_neg_spec=(fliplr(bamode_rc_bs)’.*... abs(neg_ang_spec)).*exp(j*angle(neg_ang_spec));

C-25

%*************************************************************************% Insert modified positive angular spectrum components%*************************************************************************

bs_p_space(1:length(bbmode_rc_bs))=mod_pos_spec;

%*************************************************************************% Insert modified negative angular spectrum components%*************************************************************************

bs_p_space(length(bs_p_space)-length(bbmode_rc_bs)+... 1:length(bs_p_space))=mod_neg_spec;

%*************************************************************************% IFFT gives modified elements of the backscattered field%*************************************************************************

u(:,x_index+1)=ifft(bs_p_space);

clear bs_p_space pos_ang_spec neg_ang_spec mod_pos_spec mod_neg_spec

end

x_index=x_index+1;

end

clear x_index rc_mag prev_field odd_part u_x ps

C-26

(d) PETASETUP.M

%***************************************************************************% PETASETUP – PE-based time analysis set-up function% Called by gope.m if doing time analysis%% Bruce Hannah% version date 22/8/00%***************************************************************************

function [f_max,f_array,lower_pad,upper_pad,ts,tmin]=... petasetup(f,x_max,range_dist,range_step,... ant_height,angle_range,backscatter,terrain)

global TERRAIN_PROFILEglobal C;

theta_max=max(angle_range)*pi/180;theta_min=min(angle_range)*pi/180;pulse_width=1e-9; % required width to resolve multipath delays

% round in 10 units below LOS timing

tmin=floor(1e9*range_dist/(10*C*cos(theta_min)))*10e-9;if backscatter % allow time to end of domain and back to ant location tmax=ceil(1e9*(x_max+(x_max-range_dist))/(10*C))*10e-9;else % allow only for forward scatter tmax=ceil((1e9*range_dist/(10*C*cos(theta_max)))... +(2e9*ant_height*sin(theta_max)/(10*C)))*10e-9; if terrain tmax=ceil((1e9*range_dist/(10*C*cos(theta_max)))... +(2e9*(ant_height+TERRAIN_PROFILE(round(range_dist/range_step)))... *sin(theta_max)/(10*C)))*10e-9;% allow additional time for terrain endendtime_window=tmax-tmin;M=round(f*time_window);time_window=M/f;f_step=1/time_window;

if time_window<200e-9 os_factor=128;elseif time_window<500e-9 os_factor=64;elseif time_window<1000e-9 os_factor=32;end

b=f_step*round((1/pulse_width)/f_step); % 2*b is the bandwidth of RECTf_max=f+b; % maximum frequencyf_min=f-b; % minimum frequencymin_fs=(os_factor*f_max); % Nyquist samplingmax_ts=1/min_fs; % maiximum time stepN_min=time_window/max_ts; % minimum sample sizex=nextpow2(N_min); % set at power of twoN_ir=2^x;ts=time_window/N_ir; % time domain sample spacingf_array=f_min:f_step:f_max; % create full array of frequencieslower_pad=zeros(1,round((f_min/f_step)-1)); % pad outupper_pad=zeros(1,(N_ir/2)-length(lower_pad)-length(f_array));

C-27

(e) LOADPROFILE.M

%***************************************************************************% LOADPROFILE – function for loading of DTM terrain data%% Bruce Hannah% Version date 14/8/00%***************************************************************************

function[TERRAIN_PROFILE,domain_height,x_max]=loadprofile(range_step,do_interp)

% function loadprofile.m loads the dtm terrain profile for FSPE-PETA

% dtm range resolution as provided in the dtmdtm_step=1; % dtm range resolution as provided in the dtm

% additional height for the domain above terrain with terrain option ONmedium_height=10;

%***************************************************************************% The following code is an example of created DTM profiles for testing%***************************************************************************

flat(1:10)=0;back_reflector=20;block1(1:3)=3;block2(1:5)=6;block3(1:15)=3;block4(1:15)=1;wedge=0:0.2:5;profile1=[0,0.05,0,0.05,0,0.05,0,0.05,0,0.05,0,0.05,0,0.05,0,0.05,0];profile2=[flat,block1,flat];profile3=[flat,flat,flat,block,flat,flat,block,flat];profile4=[flat,4,flat,flat];profile5=[[4 4 4 3 2 1],flat,[1 2 3 4 4 5 6 6 6 ]];

%***************************************************************************% The DTM profile is loaded at this point%***************************************************************************

dtm=profile3;

%***************************************************************************%sampling of dtm for use by PE%***************************************************************************

dtm=dtm-min(dtm)+1;domain_height=(max(dtm)-1)+medium_height;terrain_scale=range_step/dtm_step;

%***************************************************************************%undersample dtm, if dtm has higher resolution than is used by PE%***************************************************************************

if terrain_scale>1 terrain_index=1; for scale_index=1:terrain_scale:length(dtm) TERRAIN_PROFILE(terrain_index)=dtm(scale_index); terrain_index=terrain_index+1; end

%***************************************************************************% oversample dtm by interpolating between points if dtm resolution is not as% high as required by PE%***************************************************************************

elseif terrain_scale<1 terrain_index=1;

C-28

for dtm_index=2:length(dtm) %counts all the elements in the dtm %measure the size and sign of terrain excursions h=dtm(dtm_index)-dtm(dtm_index-1); %for terrain transitions, interpolate if h~=0&do_interp for h_step=0:h*terrain_scale:h TERRAIN_PROFILE(terrain_index)=dtm(dtm_index-1)+h_step; terrain_index=terrain_index+1; end %when no transitions just oversample at set terrain height else; for count=1:1/terrain_scale TERRAIN_PROFILE(terrain_index)=dtm(dtm_index-1); terrain_index=terrain_index+1; end end end%when resolutions are the same just load the dtm into the terrain profileelse TERRAIN_PROFILE=dtm;endx_max=length(TERRAIN_PROFILE)*range_step;

C-29

C.2 GPS Receiver Modelling

(a) RUNCORR.M

%***************************************************************************% RUNCORR – control file for rxcorr%% Bruce Hannah% Version date 14/8/00%***************************************************************************

%clear;tic

%***************************************************************************% option selections for correlation and display%***************************************************************************

save_images=0;filter_on=0;corr_select=0; % 0=standard 1 chip spacing % 1=narrow 0.1 chip spacingplot_select=11; % 0=no plotting % 1=plot prompt correlation % 2=plot early correlation % 3=plot late correlation % 4=plot LOS correlation % 5=plot MP correlation % 6=plot early & late % 7=plot prompt, early & late % 8=plot prompt, LOS & MP % 9=plot discriminator function % 10=plot prompt & discriminator % 11=plot prompt, LOS, MP & discriminatordiscrim_type=3; % 1=dot product (E-L)P [coherent & noncoherent] % 2=early minus late power (E^2-L^2) [noncoherent] % 3=early minus late envelope (E-L) [noncoherent] % 4=early minus late envelope normalised (E-L)/(E+L) [noncoherent]

%***************************************************************************% defaults%***************************************************************************

C=3e8; % speed of propagation of the C/A-codeprn_num=3; % prn number of the c/a codeca_code_period=1e-3; % c/a code is 1 ms long

filter_order=2; % filter order; for filtfilt order is twice this

cutoff_freq=1e6; % 2 MHz precorrealtion filter for standard correlatorcorr_spacing=0.5; % standard 1 chip (1/2 chip early and 1/2 chip late) % correlator spacingchip_samples=40; % this is the resample rate for the C/A-code with % the standard correlatorsample_mult=1;lag_chips=2; % the lag in chips for the correlation function display

if corr_select % if narrow correlator sample_mult=2; cutoff_freq=8e6; % 8 MHz pre-correlation filter for narrow correlator corr_spacing=0.05; % 1/10th chip narrow correlator spacing % (1/20 chip early and 1/20 chip late) % resample rate of C/A-code for narrow is 10x that for standard chip_samples=chip_samples*sample_mult;

C-30

lag_chips=0.1; % the lag in chips for the correlation functiondisplayend

%***************************************************************************% the rounded lag in integer samples for the correlation function%***************************************************************************

lag=round(lag_chips*chip_samples);

sample_time=1e-3/(1023*chip_samples);sample_freq=1/sample_time;

if ~exist(’prompt_code’)

%***************************************************************************% generate the (1023 chip x chip_samples) sampled c/a code from pngen.m%***************************************************************************

prompt_code=pngen(prn_num,chip_samples);end

%***************************************************************************% set up filtering%***************************************************************************

if filter_on filt_cutoff=cutoff_freq/(sample_freq/2);else filt_cutoff=1; % no filtering of codeend

[B,A]=butter(filter_order,filt_cutoff);

%***************************************************************************% lowpass filter the prompt C/A-code to replicate the receiver% precorrelation filtering.%***************************************************************************

filtered_prompt_code=filtfilt(B,A,prompt_code);

clear B A filt_cutoff sample_freq % free up memory

%***************************************************************************% set up for saving images%***************************************************************************

if save_images % save image option ON prntype=’-dbitmap’; % set saved image format dirpath= ’c:\bruce\PEfiles\movfiles\’; % set path dirname=[’corr\run3’]; % directory name for storing images if ~exist([dirpath,dirname],’dir’) % check if directory already exists eval([’!mkdir ’,dirpath,dirname]); % if not make directory end eval([’cd ’,dirpath,dirname]); % change to new directory to save imagesend

%***************************************************************************% default delay parameters% note: rxcorr accepts unlimited array of multipath signals%***************************************************************************

% the relative amplitude of the multipath code to the LOS codeMP_delay_amp=[0.5];

% the delay time of the multipath code to the LOS codeMP_delay_time=[5*sample_time];

% phase of multipath code in degreesMP_delay_phase=[0];

C-31

index=1;

%***************************************************************************% run rxcorr for set parameters%***************************************************************************

for MP_delay_time=0:10*sample_time*sample_mult:1600e-9 [discrim,tau_error,e_p_l]=... rxcorr(MP_delay_time,MP_delay_amp,MP_delay_phase,LOS_delay_time,... LOS_delay_amp,prompt_code,filtered_prompt_code,chip_samples,... sample_time,corr_spacing,discrim_type,lag,plot_select);

range_error(diff_index,time_index)=C*tau_error; e_p_l_sample(time_index,:)=e_p_l;

%***************************************************************************% range error and early,late and prompt sample data%***************************************************************************

range_error(index)=C*tau_error; e_p_l_sample(index,:)=e_p_l;

if save_images filename=[’ mp_out_time2_’,num2str(index)]; eval([’print ’,prntype,filename]); close end index=index+1;end

toc

C-32

(b) RXCORR.M

%***************************************************************************% RXCORR – receiver correlation/discrimnation function% Called by runcorr.m% delay_time,MP_delay_amp,MP_delay_phase are the relative multipath% parameters. Prompt_code is the generated C/A-code for a given PRN number.%% Bruce Hannah% version date 22/8/00%***************************************************************************

function [discrim,tau_error,e_p_l]=... rxcorr2(MP_delay_time,MP_delay_amp,MP_delay_phase,... LOS_delay_time,LOS_delay_amp,prompt_code,... filtered_prompt_code,chip_samples,sample_time,... corr_spacing,discrim_type,lag,plot_select)

corr_spacing_samples=corr_spacing*chip_samples;

zero_lag_sample=lag+1; % the sample number of the zero lag correlation

%***************************************************************************% early and late sample numbers%***************************************************************************

early_sample=zero_lag_sample-(corr_spacing*chip_samples);late_sample=zero_lag_sample+(corr_spacing*chip_samples);

%***************************************************************************% generate the early and late local reference codes%***************************************************************************

early_code=[prompt_code(corr_spacing_samples+1:length(prompt_code)),... prompt_code(1:corr_spacing_samples)];

late_code=[prompt_code(length(prompt_code)-... corr_spacing_samples+1:length(prompt_code)), prompt_code(1:length(prompt_code)-corr_spacing_samples)];

%***************************************************************************% the LOS, multipath delayed, and attenuated replicas are generated% the delayed code is addition of delayed replicas of the prompt (LOS) code%***************************************************************************

LOS_code=filtered_prompt_code; %initialise LOS as the unshifted prompt

if LOS_delay_time<0

LOS_delay_chips=-LOS_delay_time/(1e-3/1023); LOS_delay_samples=round(LOS_delay_chips*chip_samples); LOS_code=LOS_delay_amp*[filtered_prompt_code(LOS_delay_samples+... 1:length(filtered_prompt_code)),filtered_prompt_code(... 1:LOS_delay_samples)];% the LOS code arrives earlier

elseif LOS_delay_time>0

LOS_delay_chips=LOS_delay_time/(1e-3/1023); LOS_delay_samples=round(LOS_delay_chips*chip_samples); LOS_code=LOS_delay_amp*[filtered_prompt_code(length(filtered_prompt_... code)-LOS_delay_samples+1:length(filtered_prompt_code)),... filtered_prompt_code(1:length(filtered_prompt_code)-... LOS_delay_samples)]; % the LOS code arrives laterend

C-33

%***************************************************************************% initialise the Multipath code as all zero%***************************************************************************

MP_code=zeros(size(LOS_code));

%***************************************************************************% generate the Multipath code as the linear combination of all multipath% signals%***************************************************************************

for index=1:length(MP_delay_time)

MP_delay_chips=MP_delay_time(index)/(1e-3/1023);

% the delay in samples for the multipath code MP_delay_samples=round(MP_delay_chips*chip_samples); MP_code=MP_code+(cos(MP_delay_phase(index)*pi/180)*... MP_delay_amp(index)*[LOS_code(length(LOS_code)-... MP_delay_samples+1:length(LOS_code)),... LOS_code(1:length(LOS_code)-MP_delay_samples)]);end

%***************************************************************************% generate the autocorrelation function% and normalise for display%***************************************************************************

%***************************************************************************% this is the cross-correlation of the prompt filtered c/a code with the% unfiltered reference%***************************************************************************

corr_auto=xcov(prompt_code,filtered_prompt_code,lag);

%***************************************************************************% the reference autocorrelation coeff at the zeroth lag%***************************************************************************

norm_corr_coeff=corr_auto(zero_lag_sample);

clear corr_auto

%***************************************************************************% the true input code to the receiver is now the addition of the LOS% and the delayed MP codes%***************************************************************************

input_code=LOS_code+MP_code;

%***************************************************************************% ontime correlation of the combined input code%***************************************************************************

corr_prompt=xcov(prompt_code,input_code,lag)/norm_corr_coeff;

%***************************************************************************% late correlation of the combined input code%***************************************************************************

corr_late=(xcov(late_code,input_code,lag)/norm_corr_coeff);

%***************************************************************************% early correlation of the combined input code%***************************************************************************

corr_early=(xcov(early_code,input_code,lag)/norm_corr_coeff);

C-34

%***************************************************************************% LOS correlation%***************************************************************************

corr_LOS=xcov(prompt_code,LOS_code,lag)/norm_corr_coeff;

%***************************************************************************% correlation of the delayed multipath code%***************************************************************************

corr_MP=xcov(prompt_code,MP_code,lag)/norm_corr_coeff;

e_p_l=[corr_prompt(early_sample) corr_prompt(zero_lag_sample) ... corr_prompt(late_sample)];

%***************************************************************************% selected receiver discrimination%***************************************************************************

if discrim_type==1 discrim=(corr_early-corr_late).*corr_prompt; discrim_text=’Dot Product’;elseif discrim_type==2 discrim=(corr_early.^2-corr_late.^2); discrim_text=’E-L Power’;elseif discrim_type==3 discrim=(corr_early-corr_late); discrim_text=’E-L Envelope’;elseif discrim_type==4 discrim=(corr_early-corr_late)./(corr_early+corr_late); discrim_text=’E-L Normalised Envelope’;elseif discrim_type==5 discrim=(corr_early+corr_late)-corr_prompt; discrim_text=’NEW BIASED’;end

%***************************************************************************% plot selection and set up%***************************************************************************

if plot_select if plot_select==4 min_corr=round(min(corr_LOS)*11)/10; max_corr=round(max(corr_LOS)*11)/10; elseif (plot_select==5) min_corr=round(min(corr_MP)*11)/10; max_corr=round(max(corr_MP)*11)/10; elseif (plot_select==8)|(plot_select==11)|(plot_select==12) min_corr=round(min([min(corr_LOS) min(corr_MP)min(corr_prompt)])*11)/10; max_corr=round(max([max(corr_LOS) max(corr_MP)max(corr_prompt)])*11)/10; else min_corr=round(min(corr_prompt)*11)/10; max_corr=round(max(corr_prompt)*11)/10; endend

max_discrim=round(max(discrim)*11)/10;

min_corr=-2;max_corr=2;max_discrim=2;

C-35

%***************************************************************************% track zero crossing of discrimination function%***************************************************************************

pos_error_sample=min(find(discrim(early_sample:late_sample)>0))+... early_sample-1; % the first sample that is positiveneg_error_sample=max(find(discrim(early_sample:late_sample)<0))+... early_sample-1; % the last sample that is negative

%***************************************************************************% timing error measurement%***************************************************************************

if (discrim(zero_lag_sample)<1e-6)&(discrim(zero_lag_sample)>-1e-6) tau_error=0;elseif neg_error_sample>=zero_lag_sample tau_error=sample_time*((neg_error_sample-zero_lag_sample)... +(discrim(neg_error_sample)/(discrim(neg_error_sample)-discrim(pos_error_sample))));elseif pos_error_sample<=zero_lag_sample tau_error=-sample_time*((zero_lag_sample-pos_error_sample)... +(discrim(pos_error_sample)/(discrim(pos_error_sample)-discrim(neg_error_sample))));end

%***************************************************************************% plot results if required%***************************************************************************

if plot_select

plot_prompt=’plot((-lag:lag)/chip_samples,corr_prompt,’’b’’);’; plot_early=’plot((-lag:lag)/chip_samples,corr_early,’’r’’);’; plot_late=’plot((-lag:lag)/chip_samples,corr_late,’’g’’);’; plot_discrim=’plot((-lag:lag)/chip_samples,discrim,’’b’’);’; plot_LOS=’plot((-lag:lag)/chip_samples,corr_LOS,’’g’’);’; plot_MP=’plot((-lag:lag)/chip_samples,corr_MP,’’r’’);’; corr_axes=’axis([-lag/chip_samples lag/chip_samples min_corr ... max_corr]);’; discrim_axes=’axis([-lag/chip_samples lag/chip_samples ... -max_discrim max_discrim]);’; corr_title=’title(’’Correlation Function’’);’; discrim_title=’title([discrim_text,’’ Discriminator’’]);’; x_text=’xlabel(’’Offset (chips)’’);’; y_corr=’ylabel(’’Normalised Correlation’’);’; y_discrim=’ylabel(’’Normalised Discrimination’’);’;

if plot_select==1 % prompt plot_control=[plot_prompt,corr_axes,corr_title,x_text,y_corr,... ’grid on;’];

elseif plot_select==2 % early plot_control=[plot_early,corr_axes,corr_title,x_text,y_corr,... ’grid on;’];

elseif plot_select==3 % late plot_control=[plot_late,corr_axes,corr_title,x_text,y_corr,... ’grid on;’];

elseif plot_select==4 % LOS plot_control=[plot_LOS,corr_axes,corr_title,x_text,y_corr,’grid on;’];

elseif plot_select==5 % MP plot_control=[plot_MP,corr_axes,corr_title,x_text,y_corr,’grid on;’];

elseif plot_select==6 % early & late plot_control=[plot_early,’hold on;’,plot_late,corr_axes,corr_title,... x_text,y_corr,’grid on;legend(’’E’’,’’L’’);’];

elseif plot_select==7 % early, late & prompt plot_control=[plot_early,’hold on;’,plot_late,plot_prompt,...

C-36

corr_axes,corr_title,x_text,y_corr,’grid on;legend... (’’E’’,’’L’’,’’P’’);’];

elseif plot_select==8 % LOS, MP & prompt plot_control=[plot_LOS,’hold on;’,plot_MP,plot_prompt,corr_axes,... r_title,x_text,y_corr,’grid on;legend(’’LOS’’,’’MP’’,’’P’’);’];

elseif plot_select==9 % discriminator plot_control=plot_discrim;

elseif plot_select==10 % prompt & discriminator plot_control=[’subplot(2,1,1);’,plot_prompt,corr_axes,corr_title,... y_corr,’grid on;’,’subplot(2,1,2);’,plot_discrim,discrim_axes,... discrim_title,x_text,y_discrim,’grid on;’];

elseif plot_select==11 % LOS, MP, prompt & discriminator plot_control=[’subplot(2,1,1);’,plot_LOS,’hold on;’,plot_MP,... plot_prompt,corr_axes,corr_title,y_corr,’grid on;legend... (’’LOS’’,’’MP’’,’’P’’);’,’subplot(2,1,2);’,’hold on;’... plot_discrim,discrim_axes,discrim_title,x_text,y_discrim,... ’grid on;’];

elseif plot_select==12 % early, late & discriminator plot_control=[’subplot(2,1,1);’,plot_early,’hold on;’,plot_late,... corr_axes,corr_title,y_corr,’grid on;legend... (’’E’’,’’L’’);’,’subplot(2,1,2);’,’hold on;’... plot_discrim,discrim_axes,discrim_title,x_text,y_discrim,... ’grid on;’];

end

eval(plot_control);

end

C-37

(c) PNGEN.M

%***************************************************************************% PNGEN – C/A-code generationo function% Called by runcorr.m

% 1023 bit C/A code generator function% input:% prn(SV) number via prn_num% chip_sampling via chip_samples% output:% sampled C/A code via ca_code% generates the 1023 bit C/A code available in prn_code% saves prn_code to MAT file if save_code flag set%% Bruce Hannah% version date 22/8/00%***************************************************************************

function ca_code=pngen(prn_num,chip_samples)

save_code=0;

N=1023; % number of bitsM=10; % number of shift registersca_code=[];

if prn_num<5 % prn phase tap lookup table S=[prn_num+1,prn_num+5];elseif prn_num<7 S=[prn_num-4,prn_num+4];elseif prn_num<10 S=[prn_num-6,prn_num+1];elseif prn_num<17 S=[prn_num-8,prn_num-7];elseif prn_num<23 S=[prn_num-16,prn_num-13];elseif prn_num<24 S=[prn_num-22,prn_num-20];elseif prn_num<29 S=[prn_num-20,prn_num-18];elseif prn_num<34 S=[prn_num-28,prn_num-23];end

% initialise registers

for k=1:M reg_g1(k)=1; % initialise G1 generator registers reg_g2(k)=1; % initialise G2 generator registersend

%***************************************************************************% see p14-17 GPS SPS signal specification for details%***************************************************************************

for clock=1:N input_g1=xor(reg_g1(3),reg_g1(10)); input_g2=xor(xor(xor(reg_g2(2),reg_g2(3)), xor(reg_g2(6),reg_g2(8))),xor(reg_g2(9),reg_g2(10))); G1(clock)=reg_g1(10); % output of G1 stored G2(clock)=reg_g2(10); % output of G2 stored G2i(clock)=xor(reg_g2(S(1)),reg_g2(S(2))); % phase selector output stored prev_reg_g1=reg_g1; % store current G1 register states prev_reg_g2=reg_g2; % store current G2 register states for k=2:M reg_g1(k)=prev_reg_g1(k-1); % shift registers 2-10 of G1 reg_g2(k)=prev_reg_g2(k-1); % shift registers 2-10 of G2 end

C-38

reg_g1(1)=input_g1; % register 1 bit set from given input reg_g2(1)=input_g2;end

prn_code=xor(G1,G2i); % prn code generated from G1 and G2i

for k=1:N if prn_code(k)==0 prn_code(k)=1; % map code from [0,1] to [1,-1] else prn_code(k)=-1; end for n=1:chip_samples chip_code(k,n)=prn_code(k); end ca_code=[ca_code,chip_code(k,:)];end

%***************************************************************************% code saved to MAT file%***************************************************************************

if save_code~=0 var_name=’prn_code’; filename=[’prn’,num2str(prn_num),’code’]; save(filename,var_name);end

C-39

C.3 GPS Multipath Data Acquisition

(a) ASHTEQC.M

%***************************************************************************% ASHTEQC Converts ashtech download files (B,E) to rinex using teqc% The location of teqc.exe must be specified in ’teqc_dir’% The location of the data directory must be specified in ’data_dir’% Ashtech B, E files must exist in a directory named ’ash_files’% If not already existing a directory is created called ’rinex_files’% The resultant obs and/or nav rinex files are placed here%% User specifies observables and file creation options (nav or obs or both)%% Bruce Hannah% version date 3/2/00%***************************************************************************

clear

teqc_dir=’g:\teqc\’; % location of teqc.exedata_dir=’g:\BMH_data\1_04_00\’; % set the path where data is locatedash_dir=[data_dir,’ash_files’];rinex_dir=[data_dir,’rinex_files\’];

eval([’cd ’,ash_dir]); % change to existing ashtech file directory

disp(’convert Ashtech files to RINEX format’)disp(’’)

while 1

if ~exist(rinex_dir,’dir’) % check if directory already exists eval([’!mkdir ’,rinex_dir]); % if not make rinex directory end

l1=’’; % initilise strings l2=’’; c1=’’; p1=’’; p2=’’; d1=’’; d2=’’; s1=’’; s2=’’; nav=’’; n_file=’’; obs=’’; obs_str=’’;

dir

disp(’option 1: obs + nav’) disp(’option 2: obs only’) disp(’option 3: nav only’) opt=input(’enter option number (q to quit): ’,’s’);

if strcmp(opt,’q’) % quit break else b_file=input(’enter B-file name: ’,’s’); out_file=input(’enter output file name (no extension): ’,’s’); opt=str2num(opt); if opt==1|opt==2 % if obs required disp(’enter observables (1 include 0 exclude)’) % user includes or excludes the observations from % the command line string

C-40

if input(’L1: ’) l1=’L1+’; end if input(’L2: ’); l2=’L2+’; end if input(’C1: ’); c1=’C1+’; end if input(’P1: ’); p1=’P1+’; end if input(’P2: ’); p2=’P2+’; end if input(’D1: ’); d1=’D1+’; end if input(’D2: ’); d2=’D2+’; end if input(’S1: ’); s1=’S1+’; end if input(’S2: ’); s2=’S2+’; end % command line string for observations obs_str=[l1,l2,c1,p1,p2,d1,d2,s1,s2]; obs_str(length(obs_str))=’ ’; % replace last ’+’ with a space obs=’ -O.obs ’; % obs option file_ext=’.98o’; % obs file extension if opt==1 % do both nav and obs nav=’ +nav ’; % include nav option n_file=[out_file,’.98n’]; % nav file name end

else % nav file conversion only nav=’n ’; file_ext=’.98n ’; end eval([’!’,teqc_dir,’teqc -ash d’,nav,rinex_dir,n_file,obs,obs_str,... b_file,’ > ’,rinex_dir,out_file,file_ext])

endend

C-41

(b) RIN2QC.M

%***************************************************************************% RIN2QC Teqc Quality Check that creates UNAVCO compact files from rinex% data files. The location of teqc.exe must be specified in ’teqc_dir’% The location of the data directory must be specified in ’data_dir’% Rinex files must exist in a directory named ’rinex_files’% If not already existing a directory is created called ’qc_files’% where the resultant qc files are saved%% Bruce Hannah% version date 3/2/00%***************************************************************************

clear

data_dir=’g:\BMH_data\1_04_00\’; % specify data location pathteqc_dir=’g:\teqc\’; % specify location of teqc.exe

rinex_dir=[data_dir,’rinex_files\’]; % rinex directory pathqc_dir=[data_dir,’qc_files\’]; % qc directory path

if ~exist(qc_dir,’dir’) % check if qc directory already exists eval([’!mkdir ’,qc_dir]); % if not make qc directoryend

eval([’cd ’,rinex_dir]); % change to rinex directory

while 1 dir % display rinex files f_name=input(’enter RINEX obs filename (q to quit): ’,’s’);

if strcmp(f_name,’q’) break else eval([’!g:\teqc\teqc +qc ’,f_name]);

% move the qc files to the qc directory

qc_name=f_name(1:length(f_name)-4);

if exist([qc_name,’.azi’],’file’) eval([’!copy ’,qc_name,’.azi ’,qc_dir]); eval([’!del ’,qc_name,’.azi’]); end

if exist([qc_name,’.ele’],’file’) eval([’!copy ’,qc_name,’.ele ’,qc_dir]); eval([’!del ’,qc_name,’.ele’]); end

if exist([qc_name,’.mp1’],’file’) eval([’!copy ’,qc_name,’.mp1 ’,qc_dir]); eval([’!del ’,qc_name,’.mp1’]); end

if exist([qc_name,’.mp2’],’file’) eval([’!copy ’,qc_name,’.mp2 ’,qc_dir]); eval([’!del ’,qc_name,’.mp2’]); end

if exist([qc_name,’.sn1’],’file’) eval([’!copy ’,qc_name,’.sn1 ’,qc_dir]); eval([’!del ’,qc_name,’.sn1’]); end

if exist([qc_name,’.sn2’],’file’) eval([’!copy ’,qc_name,’.sn2 ’,qc_dir]); eval([’!del ’,qc_name,’.sn2’]);

C-42

end

if exist([qc_name,’.ion’],’file’) eval([’!copy ’,qc_name,’.ion ’,qc_dir]); eval([’!del ’,qc_name,’.ion’]); end

if exist([qc_name,’.iod’],’file’) eval([’!copy ’,qc_name,’.iod ’,qc_dir]); eval([’!del ’,qc_name,’.iod’]); end

endend

C-43

(c) QC2MAT.M

%***************************************************************************% QC2MAT Convert UNAVCO compact plot files to matlab data format% User will be prompted for a teqc UNAVCO format file% Data is then loaded from UNAVCO compact format% and saved in matlab .mat format for later plotting using plotqc.mat%% The location of the data directory must be specified in ’data_dir’% QC files must exist in a directory named ’qc_files’% If not already existing a directory is created called ’mat_files’% where the resultant files are saved%% The variables ’xxx_data’, ’total_sats’, ’epoch’, ’t_samp’, ’start-time’% are stored in the .mat file. xxx indicates the data type stored% for example, azi file data will be stored in variable ’azi_data’%% Bruce Hannah% version date 7/2/00%***************************************************************************

clear

dirpath=’g:\BMH_data\1_04_00\’; % set path

qc_dir=[dirpath,’qc_files\’]; % qc directory pathmat_dir=[dirpath,’mat_files\’]; % mat directory path

if ~exist(mat_dir,’dir’) % check if directory already exists eval([’!mkdir ’,mat_dir]); % if not make mat directoryend

while 1 fid=-1; eval([’cd ’,qc_dir]); % change to qc directory dir % display qc files f_name=input(’enter qc file name (q to quit): ’,’s’);

if (strcmp(f_name,’q’)) break else f_front=f_name(1:length(f_name)-4); f_ext=f_name(length(f_name)-2:length(f_name)); [fid,message]=fopen(f_name,’rt’); disp(message); end

if fid~=-1 line=fgetl(fid); % read in the first header line s1=sscanf(line,’%s’);

if strcmp(s1,’COMPACT’) % check for UNAVCO file compatibility compact_format=1; else disp(’this file is not UNAVCO compact format’) fclose(fid); compact_format=0; end

if compact_format % if a compatible file

line=fgetl(fid); % read in the second header line (not used) % read in the third header line (time sample rate) line=fgetl(fid); t_samp=sscanf(line,’%*s %g’); % get the sample time line=fgetl(fid); % read the fourth header line start_time=sscanf(line,’%*s %g’); % get the start time % all other lines are sat or observation data

C-44

epoch=1; % initialise epoch count counter=1; % initialise decade counter sat_line=1; % initialise sat line flag true obs_line=0; % initialise obs line flag false visible_sats(1:32)=0; % initialise array of visible_sats line=fgetl(fid); % go to first observation line

while 1 % while not the end of file if sat_line % look for sats % read in the number of sats and sv numbers sat_info=sscanf(line,’%d’);

if sat_info(1)==0 % if there are no sats % set data for all sats as no data (non number) sv_data(1:32,epoch)=nan; line=fgetl(fid); % move to the next line if ~isstr(line) % if end of the file, stop reading file break end sat_line=1; % set flag to check for sat data again epoch=epoch+1; % increment epoch counter else % otherwise there are sats sat_line=0; % sat line flag is reset obs_line=1; % observation line flag is set num_sats=sat_info(1); % number of sats is read for y=2:num_sats+1 % satellite order is read sat_order(y-1)=sat_info(y); % and stored in sat_order % for each sat line read flag each visible sat visible_sats(sat_order(y-1))=1; end sat_info=0; % reset sat_info for next sat line read end

end

if obs_line % if there is a line of observation data % initialise data for all sats as no data (non number) sv_data(1:32,epoch)=nan; line=fgetl(fid); % go to the observation line obs=sscanf(line,’%f’); % read the observations for y=1:num_sats % for each sat % store data in variable sv sv_data(sat_order(y),epoch)=obs(y); end obs_line=0; % reset observation line flag line=fgetl(fid); % go to the next line

if ~isstr(line) % if end of the file, stop reading file break else % else the line is a repeat obs flag ornew sat info flag=sscanf(line,’%d’); % read the line if flag==-1 % if it’s a data flag obs_line=1; % there are more obs for these sats % otherwise there’s a new satellite configuration else % set the sat line flag to read the new sat info sat_line=1; end end

epoch=epoch+1; % increment the epoch counter end

if epoch/100==counter disp([num2str(100*counter),’ epochs of ’,... f_name,’ processed’]); counter=counter+1; end

end disp([num2str(epoch),’ total epochs processed’]);

C-45

fclose(fid); total_sats=find(visible_sats); if strcmp(f_ext,’azi’) data_type=’azi_data’; azi_data=sv_data; elseif strcmp(f_ext,’ele’) data_type=’ele_data’; ele_data=sv_data; elseif strcmp(f_ext,’ion’) data_type=’ion_data’; ion_data=sv_data; elseif strcmp(f_ext,’iod’) data_type=’iod_data’; iod_data=sv_data; elseif strcmp(f_ext,’mp1’) data_type=’mp1_data’; mp1_data=sv_data; elseif strcmp(f_ext,’mp2’) data_type=’mp2_data’; mp2_data=sv_data; elseif strcmp(f_ext,’sn1’) data_type=’sn1_data’; sn1_data=sv_data; elseif strcmp(f_ext,’sn2’) data_type=’sn2_data’; sn2_data=sv_data; end % save info in MAT file eval([’save ’,mat_dir,f_front,’_’,f_ext,’ ’,data_type,... ’ total_sats epoch t_samp start_time’]) end endend

C-46

(d) PLOTQC.M

%***************************************************************************

% PLOTQC Plot azi,ele,sn1,sn2,mp1,mp2,ion,iod qc’d data% PLOTQC Loads and plots data from MATLAB .mat file format% The location of the data directory must be specified in ’data_dir’% The mat files must exist in a directory named ’mat_files’%% Bruce Hannah% version date 18/1/00%***************************************************************************

data_dir=’g:\BMH_data\30_11_00\’; % set data directory patheval([’cd ’,data_dir,’mat_files’]); % change to mat file directory

while 1 clear % clear all variables dir

f_name=input(’enter mat filename ("q" to quit): ’,’s’); if strcmp(f_name,’q’) break else f_ext=f_name(length(f_name)-2:length(f_name)); eval([’load ’,f_name],[’disp(lasterr)’]) end

data_type=[f_ext,’_data’]; % thsi is for early data

if exist(’sv_data’)==1 data_type=’sv_data’; end

if exist(data_type)==1 disp([’data for SV: ’,num2str(total_sats)]) disp(’type "r" to return to top, "a" to plot all, "sv" for sat data’)

while 1 % plot routine until user quits using ’q’ key_in=input(’: ’,’s’); % keyboard prompt line for display

if ~(strcmp(key_in,’r’)|strcmp(key_in,’a’)|strcmp(key_in,’sv’)) key_in=str2num(key_in); if (key_in>=1)&(key_in<=32) eval([’plot(’,data_type,’(key_in,:))’]); sv_text=num2str(key_in); plot_flag=1; else disp(’not a correct SV number, or command’) plot_flag=0; end elseif strcmp(key_in,’a’) line_hdl=plot(1:epoch,eval(data_type)); legend(line_hdl(total_sats),num2str(transpose(total_sats))); sv_text=num2str(total_sats); plot_flag=1; elseif strcmp(key_in,’sv’) disp([’SV: ’,num2str(total_sats)]) elseif strcmp(key_in,’r’) break end

if plot_flag xlabel(’epochs’) if strcmp(f_ext,’azi’) title([’Satellite Azimuth SV ’,sv_text]) ylabel(’Azimuth (degs)’)

C-47

axis([0 epoch -360 360]); elseif strcmp(f_ext,’ele’) title([’Satellite Elevation SV ’,sv_text]) ylabel(’Elevation (degs)’) axis([0 epoch 0 90]); elseif strcmp(f_ext,’ion’) title([’L2 Ionospheric Observable SV ’,sv_text]) ylabel(’Range Error (m)’) elseif strcmp(f_ext,’iod’) title([’Derivative of L2 Ionospheric ... Observable SV ’,sv_text]) ylabel(’Range Error (m)’) elseif strcmp(f_ext,’mp1’) title([’L1 Multipath SV ’,sv_text]) ylabel(’Range Error (m)’) axis([0 epoch -5 5]); elseif strcmp(f_ext,’mp2’) title([’L2 Multipath SV ’,sv_text]) ylabel(’Range Error (m)’) axis([0 epoch -5 5]); elseif strcmp(f_ext,’sn1’) title([’L1 S/N SV ’,sv_text]) ylabel(’S/N (dB)’) axis([0 epoch 0 60]); elseif strcmp(f_ext,’sn2’) title([’L2 S/N SV ’,sv_text]) ylabel(’S/N (dB)’) axis([0 epoch 0 60]); end zoom on end end endend

Modelling and Simulation of GPS Multipath Propagationmultipath have proven much more difficult to...

Documents

Transcript of Modelling and Simulation of GPS Multipath Propagationmultipath have proven much more difficult to...