DESIGN OF LINEAR ARRAY GEOMETRY FOR HIGH RESOLUTION

ARRAY PROCESSING

By

XINPING HUANG, B.Sc.(Eng.), University of Science and Technology'of China

M.Eng., Nanjing A"ronautical Institute

A Thesis

Submitted to the School of Graduate Studies

in Partial Fulfilment of the Requirements

for the Degree

Doctor of Philosophy

McMaster University

September 9, 1993

TO ALL MEMBERS IN MY LOVELY FAMILY

DESIGN OF LINEAR ARRAY GEOMETRY

FOR HIGH RESOLUTION ARRAY PROCESSING

DOCTOR OF PHILOSOPHY (September 9, 1993)

(Electrical and Computer Engineering)

II

MCMASTER UNIVERSITY

Hamilton, Ontario

TITLE:

AUTHOR:

SUPERVISOR(S):

Design of Linear Array Geometry for High Resolution

Array Processing

Xinping Huang

B.Sc.(Eng.), University of Science and Technology of China

M.Eng., Nanjing Aeronautical Institute

Dr. Kon Max Wong

Professor, Chairman of Department of Electrical and Com­

puter Engineering

B.Sc.(Eng.), Ph.D., (University of London)

D.I.C. (University of London)

Fellow, I.E.E.

Fellow, Royal Statistical Society

Fellow, Institute of Physics

Dr. James P. Reilly

Professor, Department of Electrical and Computer Engineer­

ing

B.A.Sc. (University of Waterloo)

M.Eng., Ph.D. (McMaster University)

NUMBER OF PAGES: xvii, 141

Abstract

The linear array is one of the most important types of multi-element sensor arrays, being

extensively used in radar, sonar, telecommunications, radio astronomy and medical imaging

systems. Traditionally, the array assumes uniform geometry with an inter-sensor spacing

of %' which limits resolvability because of the axed aperture. Since the 1950's, much work

has been done on designing nonuniform arrays with focus on conventional beamforming

techniques. One of the typical results is the Minimum Redundancy (MR) arrays which

provide improved performance over the uniform array.

In this thesis, this issue is re-investigated from the viewpoint of high resolution

array processing. A new criterion (called DOBC), based on D-Optimality, is developed,

which yields a new array geometry by minimizing a measure of joint estimation error with

respect to the array geometry parameters. The sensor gain and phase calibration errors

and their effects on high resolution array signal processing are also examined, and formulae

are developed to evaluate such effects.

In addition, the Modified Forward-Backward Linear Prediction (MFBLP) method

is modified to substantially improve the low SNR performance without increase in compu­

tationalload. A form of Cramer-Rao lower bound (CRLB) is derived for a reduced model

which facilitates performance comparisons in directions of arrival (DOA) estimation.

Compliter experiments are conducted to verify our analysis. We conclude that (1).

the DOBC design outperforms the conventional uniform array and the MR array; (2). the

iii

IY

formulae developed predict very well the behaviour of high resolution algorithms in the

presence or absence of calibration errors. The design criterion and formulae can be used by

the system designer to design a new array geometry given the performance requirement and

hardware specifications, to evaluate the expected performance of an array, given information

about hardware specifications, or to develop hardware specifications given the performance

requirement.

Acknowledgement

The author wishes to express his deep appreciation to Professors K. M Wong and J. P. Reilly

for their encouragement, continued assistance, expert guidance and supervision throughout

the course ofthis work. He also thanks Professors. J. F. McGregor and Z. Q. Luo, members

of his Supervisory Committee, for their continuing interest and useful suggestions.

The author is very grateful to Drs. Q. WU, Q. Jin, and many other researchers

and fellow graduate students in the Communications Research Laboratory for providing

stimulating discussions, helpful suggestions and generous assistance.

He is also d.leply indebted to his wife, Ping He, and his son, Andy, for their under­

standing, constant encouragement and support.

v

Contents

Abstract

Acknowledgement

List of Tables

List of Figures

Glossary

1 INTRODUCTION

1.1 What Is A Sensor Array?

1.2 Array Signal Processing . .

1.3 Geometry of Sensor Arrays

1.4 Errors in Sensor Arrays

1.5 Organization of thesis .

vi

iii

v

xi

xiv

xv

1

1

2

3

6

8

CO~TE~7S

2 SENSOR ARRAY SIGNAL PROCESSING

2.1 Introduction .

2.2 Model of Array Observations

2.3 Detection of Signals

!1

10

\~

2.3.1 Akaike's Information Theoretic Criterion \3

2.3.2 Minimum Description Length Criterion 1:1

2.3.3 Detection of Number of Signals Using Information Theoretic Criteria 14

2.3.4 Example......

2.4 Estimation of Parameters

2.4.1

2.4.2

Maximum Likelihood Estimator

MUSIC Estimator . . . . . .

16

18

18

21

2.4.3 The FBLP Based Algorithms

2.4.4 Examples . . . . . .

2.5 Geometry of Sensor Array . . . .

24

29

31

2.5.1

2.5.2

Co-array of Linear Arrays . . .

Minimum Redundancy Arrays

33

35

2.6 Conclusions.................................... 36

:I OPTIMAL DESIGN OF LINEAR SENSOR ARRAYS

viii

37

3.1 Introduction . 37

:1.3 Design Criterion ..

Formu!ation of Problem3.2

3.3.1

3.3.2

D-optimality ...

Covariance Matrix

40

41

42

45

3.3.3 The DOBC Criterion Applied to Array Signal Processing

3.3.4 Some Design Examples .....

3.4 Aperture Selection and Threshold Effect

3.5 Sensitivity of Design .

3.6 Relationship of New Criterion to CRLB

49

52

56

63

65

C-l Maximum Likelihod Estimator in Colored Gaussian Noise

3.7 Conclusions • . . . . . . . . . . . . . . . . . . . . . . . . . 67

69

C-2 Proof of Equation 3.30 ..

C·3 Proof of Equation 3.31 ...

C·4 Proof of Equation 3.32 . . .

C·5 Proof of Equation 3.37 . . .

C·6 Proof of Invari~nce Property ..

71

72

73

74

77

CO~TESH

e·6.1 Inyariance property of E {yyT }

e·6.2 Inyariance property of T .

C· i Cramer·Rao Lower Bound . . . .

C· i.l Probability density function: model reduction.

·IX

, ,

iH

C-7.2 CRLB for Reduced ~lodel . . . . . . . . . . . . . . . . . . . . . . .. sa

4 ANALYSIS OF CALIBRATION ERROR 85

4.1 Introduction . 85

4.2 Calibration Error Model . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Covariance Matrix of DOA Estimates Given Calibration Error .... 90

4.4 The Second Order Approximation To Conditional Covariance Matrix. 92

4.4.1 Expanding E ...

4.4.2 Approximating F ....

4.4.3 Evaluating the Product

92

93

94

4.5 A Matrix Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95

4.6 The Averaged Covariance Matrix In the Presence of Gain and Phase Error 96

4.7 Examples . . . . . 97

4.8 Two Special Cases . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102

4.8.1

oJ.8.2

4.8.3

Gain Error ~Iodel .

Averaged Covariance Matrix in the Presence of Gain Error

Phase Error Model .

x

102

103

104

4.8.4 Averaged Covariance Matrix in Presence of Phase Error . . . . . .. 105

4.8.5 Examples

4.9 Conclusions ...

D-1 Proof of Matrix Lemma 1 . . . . . . . . . . . . . . .

. . . . . . . . . . . .. 106

. . . . . . . . . . . .. 106

110

D-2 Five Ensemble Averages In the Presence of Gain and Phase Error.

D-3 Analysis of Calibration Error: Amplitude Error Only.

114

118

D-3.1 Matrix Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119

D-3.2 Ensemble Averaged Covariance Matrix of DOA estimates . . . . .. 123

D-4 Analysis of Calibration Error: Phase Error Only 127

D-4.1 Matrb: Lemma 3 . . . . . . . . . . • . . . . . . . . . . . . . . . . .. 128

D-4.2 Ensemble Averaged Covariance Matrix of DOA estimates . . . . .. 130

5 CONCLUSIONS

BIBLIOGRAPHY

134

136

List of Tables

2.1

2.2

A uniform array of 4 elements and its coarray . . . .

A non-uniform array of 4 element and its coarray . 3·1

2.3 List of non-redundancy arrays .

2.4 Some Minimum-Redundancy Array Configurations .

34

36

3.2 Optimal arrays of 6 elements with different apertures.

3.1 List of various array designs . . . . . . . . . . . . . . . 53

61

4.1 Comparison between designs of Chapter 3 and Chapter 4, Colored noise .. 101

xi

List of Figures

1.1 Functional relationship between the environment, the sensor array, and the

processor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 illustration of a linear array and signal environment. . . . . . . . . . . . 10

2.2 Valuer. of AIC(k) and MDL;,,) versus k. . . . . . . . . . . . . . . . . . . . . 17

2.3 Probability of detection error using AIC and MDL.. .......... 17

2.4 Mean-squared error of the MLE and MUSIC estimators. . . . . . . . . . . . 31

2.5 Mean-squared error of the FBLP-based techniques. . . . . . . . . . . . . .. 32

3.1 illustration of a linear array .

3.2 Grating lobes of various array configurations. . ......•.

3.3 Spatial autocorrelation of noise, the spatial lag is in unit of ~..

41

51

53

3.4 Normalized spatial spectum of noise. . . . • . . . • . . . . • . . . . . 54

3.5 Mean-squared error of various arrays, M = 6, N = 13, Colored noise. 56

3.6 Mean-squared error of various arrays, M =8, N =23, Colored noise. 57

xii

LIST OF FIGt'RES xiii

3.7 Mean-squared error of various arrays. M = 6. S = 13. White noise. 57

3.S slean-squared error of various arrays. M = 8. ;\' = 23. White noise..

3.9 :Vlean-squared error of various arrays. M = 6. N = 30, Colored noise.

3.10 Mean-squared error of various arrays, AI = 6, N = 30, White noise.

3.11 Effect of the aperture on heights of spurious peaks.

3.12 Mean-squared errors versus the array aperture. . .

3.13 Probability of erroneous estimation versus the array aperture.

3.14 Mean-squared error of various arrays for different DOAs...

58

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59

60

62

63

65

3.15 Comparison between the covariance matrix and the CRLB. .. . . . . . .. 67

4.1 Decomposition of actual gain and phase .

4.2 MSE of MLE in the presence of calibration error (-20dB), M=6.

4.3 MSE of MLE in the presence of calibration error (-20dB). M=8. .

4.4 Composition of MSE in the presence of Calibration error (-30dB).

89

98

98

99

4.5 Theoretical value of MSE for different level of calibration errors. 100

4.6 Performance comparison between designs of Chapters 3 and 4. 101

4.7 illustration of the ideal and actual values of gain. . . . . . . . . . . . . . .. 103

4.8 illustration of Ideal and actual values of phase. . . . . . . . . . . . . . . .. 105