Co-registered Vibrometry & Imaging: A Combined Synthetic-Aperture Radar & Fractional-Fourier Transform Approach

University of New MexicoFY2008

University Project

May 2009NCMR Technology Review

PI & Presenter: Majeed Hayat

Project Information

• Title of project: Co-registered Vibrometry and Imaging: A Combined Synthetic-Aperture Radar and Fractional-Fourier Transform Approach

• Lead organization: University of New Mexico, Electrical & Computer Engineering Department

• Project lead: Professor Majeed M. Hayat• Personnel:

UNM Faculty:

Prof. Majeed Hayat (ECE, 15%)

Prof. Balu Santhanam (ECE,15%)

Prof. Walter Gerstle (CIVIL Engr,15%)

Sandia collaborators: Tom Atwood and Toby Townsend (10%)

Graduate students: Qi Wang (50%) Srikanth Narravula (50%) Tong Xia (50)% Tom Baltis (25)%Post Doc: Matt Pepin (DOE funded)

Program Details

• Date of award ($190,959 for FY08): Aug. 1, 2008• Date of receipt of funds: Aug. 1, 2008• Date work actually started: May 15, 2008 (via Pre-

award)• Percent of FY-08 funds spent to date: ~80%• Percent of total work completed (over three year

period) to date: ~33%

Project Narrative:Objectives

•To exploit a powerful signal-processing tool, called the fractional Fourier transform, which is suitable for representing non-stationary signals, to design a novel synthetic-aperture radar imaging strategy that yields simultaneous imaging and vibrometry.

•To test the new approach using both simulated and real SAR data; the latter may be provided by our collaborators at Sandia National Laboratories.

•Tasks were revised in May 2008 to insure there is no duplication with newly awarded DoE award.

Background: 2-D SAR process

The SAR signal is chirped in two dimensions: 1)in the u-dimension by the chirp pulse and 2)in the v-dimension by the change in range to the scatterer.

The returned signal after this step is:

1

1

0 0

0 04

2 2( ) ( ) exp{ 2 ( )}

2( )exp ( 2 )

u

u

at

c

u ur t A g u j j a t du

c c

ug u j a

c

F

(| | / 2)ct

Resolution is limited by the bandwidth of the sent chirped microwave pulse and the size of the synthetic aperture

Step 3: Inverse Fourier transform in each dimension creates an image

A/D 1{ } F | | | ( , ) |g u v

Step 1: Deramp quadrature demodulation removes the u-chirp

Step 2: Aperture compression and range compensation remove v-chirp

Azimuth Deramped Data

Range (meters)

Syn

thet

ic A

pert

ure

(ste

p)

-300 -200 -100 0 100 200 300

100

200

300

400

500

600

700

800

Range Deramped Data

Range (meters)

Syn

thet

ic A

pert

ure

(ste

p)

-300 -200 -100 0 100 200 300

100

200

300

400

500

600

700

800

Previous Work: Non-stationary case

• When the ground is vibrating the reflectance becomes time varying: • • The return signal after steps 1 & 2 becomes

• Different processing is required to extract g(u,t)

• To proceed, we need to specialize g(u,t) to practical forms

1

1

0 0 0

2 2( ) ( , / 2)exp{ 2 ( )}

u

u

u ur t A g u t j j a t du

c c

( )g u ( , )g u t

(| | / 2)ct

By using the existing quadratic demodulation process and low-pass filtering, the return signal of each sent pulse becomes a superposition of chirp signals

• Modulates the magnitude of each chirp

•Linear dependence between and the pair [central frequency, chirp rate]

•Need a method to measure the central frequency and chirp rate of each chirp signal simultaneously (FRFT)

• We use the Fractional Fourier Transform and its discretization

0 0 0 0 0 0 2

0 0 0 0

chirp ratephase shift central frequency

2( ) exp [( 2 ) (2 ) 2 ]i i i i i i i

i

jr t A K u av au v v t av t

c

(| | / 2)ct

( , )n ni iu v

Analysis of Discrete Vibrating Points

| |iK

Previous Work: Discrete FRFT

The discrete fractional Fourier transform (DFRFT) has the capability to concentrate linear chirps in few coefficients

MA-CDFRFT

1

0

2

][][N

p

prN

j

kk epzrX

1

0

][][N

nnpkkpk vnxvpz

2 2T

a G G

A W V Λ V

1

0

}{N

p

jpnpkpkn evv

A

• Each “peak” relates to each target point

• Position of each peak is related to position & velocity of point target

DFRFT Estimates

0 (0)0.48 (0.5)37.5/37.5 (37.5)Target3

472 (500)0.42 (0.5)1.27/6.5 (0)Target2

945 (1000)0.78 (0.9)-39.4/-28.5 (-37.5)Target1

Est. velocity (actual)Est. reflectivity (actual)Est. position: FRFT/FT (actual)Vibrating Targets

Previous Work: Vibration Identification Methodology

Compute the central frequencies, and chirp rates

Compute positions, and velocities

1

1

20 0 0

2 2( ) Re ( , ) exp ( ) ( )

u

c

u

u ur t A g u t j t a t du

c c

Return echo

quadratic demodulation& low-pass filtering (A/D)

0 0 0 0 0 0 20 0 0 0

2( ) exp ( 2 ) (2 ) 2i i i i i i i

i

jr t A R u av au v v t av t

c

MA-CDFRFT

Read out the positions of peaks 3

tan( / 2) / 22 1.41

0.85( / 2)

p pr

c p p

cN N

0 0

2(2 )

8

c

r

au v vc

av cc

Co-registration with

traditional SAR imagery

New Work: 2-D Non-stationary case

• When the ground is vibrating the reflectance becomes time varying: • The return signal becomes

• Different processing is required to extract g(u,v,t)

• Practical forms:

Instantaneous velocity and sum of sinusoidal modes

1

1

0 0 0

2 ( ) 2 ( ) 2 ( )( , ) ( / 2 )exp{ 2 ( )}

R

R

R m R m R mr t m A g t j j a t dR

c c c

( , )g u v ( , , )g u v t

(| | / 2)ct

,( ) ( )u v

mR m R V t

PRF

,ˆ( ) sin( )u v i i i i

i

mR m R R D f t f

PRF

How can we estimate the motion of each discrete target?•Piece-wise linear approximation:

•Send successive pulses to estimate

Model for Discrete Vibrating Points

•Pulse duration must be much shorter than vibrating period (at Nyquist rate)•Low frequency vibration measurement limited by maximum collection time•High frequency vibrations proportional to Doppler of single measurement instantaneous velocity

and m mi iR V

( , ) ( / ) for ( 1) ,0m mi i i c cR t m R V t m PRF m t m m M

( ) iR t

2 4 6 8

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Measured Vibration Spectrum

Frequency (Hz)

Am

plit

ud

e

Single Look: Vibration Frequency and Direction

Multi-Look

Vibrating Target

Changing aperture splits vibration into two sin waves

Complex amplitudes estimate vibration direction ᶿ

( ) 2 cos( ) cos(2 )j j j j jj

V t f D f t (cos(2 ) cos(2 ))j j j j j j j j

j

f D f t t f t t

Fit of V(t) cos envelope also estimates direction ᶿ

0 500 1000 1500-3

-2

-1

0

1

2

3x 10

-5 Simulated Phase Shift Change

Pulse Number

Rat

e o

f C

han

ge

of

Ph

ase

Vibration Angle-85

Vibration Angle-65

Vibration Angle-45

Vibration Angle-25

Vibration Angle 0

Vibration Angle 25

Vibration Angle 45

Vibration Angle 65

Vibration Angle 85

-100 -50 0 50 100-100

-50

0

50

100Fit of cosine envelope

Vibration Angle (Degrees)F

itte

d A

ng

le (

Deg

rees

)

True AngleFitted Angle

Single Look Approach Envelope Fit

• Fitting the phase change envelope uses the slight change in amplitude of the vibration over the synthetic aperture

• This method is least accurate around zero degrees when the vibration is directly aligned with the electromagnetic direction of propagation

-100 -80 -60 -40 -20 0-100

-80

-60

-40

-20

0Fit of cosine envelope

Vibration Angle (Degrees)

Fit

ted

An

gle

(D

egre

es)

True AngleFitted Angle

( ) 2 cos( ) cos(2 )j j j j jj

V t f D f t

0 500 1000 1500-3

-2

-1

0

1

2

3x 10

-5 Simulated Phase Shift Change

Pulse Number

Rat

e o

f C

han

ge

of

Ph

ase

Vibration Angle-85

Vibration Angle-65

Vibration Angle-45

Vibration Angle-25

Vibration Angle 0

Multilook Approach :Frequency and Direction Estimates

, , , ,( ) cos( ) 2 cos(2 )i i j i j i j i j

j

V t f D f t How to calculate at multiple look angles

By taking two looks with different squint angles, the average energy ratio these two looks is

The vibration direction can be resolved this way using multiple look angles and fitting the expected change in energy over the different squint angles to resolve the vibration direction

Amplitude Modulation

2 21 1

222 2

[ ( )] cos ( ) 1 cos(2 2 )ˆ cos ( ) 1 cos(2 2 )[ ( )]i

i

E V t

E V t

Actual Θ -60° -45° -30° 0.0° 30° 45° 60°

Estimation -59.7° -45.2° -29.7° -0.05° 30.02° 45.4° 59.95°

Results:

The patch of ground

The vibrating point target

Θ

Amplitude Modulation

Amplitude Modulation

Animated Demonstration

Summary: 2-D Methodology

Compute the frequencies, chirp rates, positions, and velocities

Estimate vibration frequencies and directions

Form SAR image and overlay vibration information

1

1

20 0 0

2 2( ) Re ( , ) exp ( ) ( )

u

c

u

u ur t A g u t j t a t du

c c

Return echo

quadratic demodulation& low-pass filtering (A/D)

0 0 0 0 0 0 20 0 0 0

2( ) exp ( 2 ) (2 ) 2i i i i i i i

i

jr t A R u av au v v t av t

c

MA-CDFRFT

Read out the positions of peaks

3

tan( / 2) / 22 1.41

0.85( / 2)

p pr

c p p

cN N

0 0

2(2 )

8

c

r

au v vc

av cc

Multiple looks to measure and refine vibration direction

( ) 2 cos( ) cos(2 )j j j j jj

V t f D f t

Actual Θ -60° -45° -30° 0.0° 30° 45° 60°

Estimation -59.7° -45.2° -29.7° -0.05° 30.02° 45.4° 59.95°

Enhancing Resolution via Non-uniform Frequency Sampling

• DFRFT: DFT of the sequence zk[p]:

• Non-uniform DFT:

• Evaluates Z-transform at locations of interest in the set zk

Nonuniform Sampling: NDFT

• Provides better peak resolution for larger in-band/out-band ratios

(¼ 0.8-1). • Frequency domain

samples can be concentrated around DFRFT peaks.

• Sharper peak locations translate to better center-frequency & chirp-rate estimates.

Subspace Approach

• DFRFT peak detection & chirp parameter estimation akin to DFT -- based sinusoidal frequency estimation: location of peak gives frequency estimate

• Periodogram approach is statistically inconsistent. Subspace approaches yield asymptotically consistent estimates.

• Covariance matrix of zk[p] is full-rank & eigenvalue spectrum not separable into S+N and N subspaces.

• Subspace approach rank reduction needed.

Modeling Electromagnetic Wave Interactions with Vibrating Structures

• Goals:– Construct full-Maxwell’s equations models of the interaction

of specific synthetic aperture radar pulses with vibrating objects– Produce simulated Doppler shift information for single / multi-mode

vibrating buildings encompassing a variety of geometrical and material features.

• Methodology:– Employ the finite-difference time-domain (FDTD) method, a

grid-based, wide-band computational technique of great robustness

(~ 2,000 FDTD-related publications/year as of 2006, 27 commercial/proprietary FDTD software vendors)

Monica Madrid (Ph.D. student) and Jamesina Simpson (Assistant Professor)Electrical and Computer Engineering Department, University of New Mexico

Leveraging DOE Funding

FDTD Modeling Details

• Model the structures using an advanced algorithm that accommodates both the surface perturbations1, as well as their internal density modulations2.

• Perform a near-to-far-field (NTFF) transformation to obtain the unique signatures of vibrating objects as would be recorded by a remote antenna system.

• Complete the model with the advanced convolutional perfectly matched layer (CPML) to terminate the grid and a total-field/scattered-field formulation (TFSF) to generate the plane wave illumination of objects.

[1] A. Buerkle, K. Sarabandi, “Analysis of acousto-electromagnetic wave interaction using sheet boundary conditions and the finite-difference time-domain method,” IEEE TAP, 55(7), 2007.[2] A. Buerkle, K. Sarabandi, “Analysis of acousto-electromagnetic wave interaction using the finite-difference time-domain method,” IEEE TAP, 56(8), 2008.

Ongoing and Future FDTD Work

• Current status and ongoing work:– We have implemented a 2-D FDTD model incorporating the CPML

boundary conditions, NTFF transformation, TFSF formulation and surface vibrating perturbations.

– Next steps will be to use the validated code to model a variety of structural geometries (rough surfaces, edges, corners) and materials (concrete, etc.), vibrating at specific modes as specified by the civil engineers on our team.

• Future Work:– Extend the 2-D model to a fully 3-D simulation of synthetic aperture

radar signals interacting with vibrating structures.

Modeling Vibrations and Physical Structures

vibrating mass

f(t) = F0 sin(Ωt) frictionless tube (A, L)

m2

k2

gas (B, ρ, A)

Lx

m1

k1

- Tests simulate theoretical model

- A speaker simulates the vibrating mass m1

- An aluminum disk and two steel beams simulate the spring- mass system response

- Matlab code controls the vibration frequency generating a sinusoidal excitation with well-controlled frequencies

Forcing Frequencies (Hz)

The speaker (inside the box) generates harmonic forces causing the box to vibrate. The transducer will measure the pressure of the sound, an accelerometer attached to the box will measure the acceleration of the walls

Structural Acoustics ExperimentPressure transducermeasures the pressure of a sound excitation.A steel box will simulate a room

SAR Vibrometry Laboratory Planning

• Simple laboratory for the experimental demonstration SAR-based vibrometry

• Initial equipment concept complete• UNM Space allocated

Summary of Effort Against Objectives

Original Objectives Work Completed

DSP strategy for multi-pulse SAR data acquisition (Q1-Q3)

• 1D and 2D analytical model for return signals

• FRFT-based deramp process

• Investigate practical multi-pulse implementations

• 1D and 2D practical model for vibrating objects

• Simulation tools for SAR signal generation

• Multi-pulse generalizations are in progress

Microwave pulse design and DFRFT processing (Q2-Q5)

• Tradeoff analysis between pulse width, chirp rate and detectable vibration frequency and speed

• 2D extensions in progress

Summary of Effort against Objectives

• Side-by-side summary of the effort

Original Objectives Work Completed

Understanding and Modeling Physical Characteristics of Ground Vibrations (Q1-Q3)

- Analytical models of physical objects developed.

- Models validated via experiments

(Revised) Develop subspace-based estimation algorithms to increase robustness to noise (Q4-Q6)

- In progress

Summary of Effort against Objectives

• Side-by-side summary of the effort

Original Objectives Work Completed

(Revised) A simple laboratory platform to demonstrate the proposed sensing concept (Q7-Q8)

- Microwave testing platform designed and equipment identified

(Revised) Solutions to inverse problem of identifying structures based upon signatures generated by the proposed approach (Q7-Q12)

-

Project Self-Assessment

• Several 1D and 2D vibration estimation algorithms have been developed

• A wide variety of vibrations may be estimated with range and cross-range methods• Two methods for estimating multiple vibration frequencies and angles completed • Signal processing method to improve vibration frequency resolution completed• Subspace methods to improve robustness to noise underway

• Initial physical modeling of vibrating structures completed; Extension to more complex structures underway• Experimental testbed underway

Patents, Publications, and Experiments Associated with Project

• Q. Wang, M. M. Hayat, B. Santhanam, and T. Atwood, “SAR Vibrometry using fractional-Fourier-transform processing,” SPIE Defense & Security Symposium: Radar Sensor Technology XIII (Conference DS304), Orlando, FL, April 2009.

• B. Santhanam, S. L. Reddy, and M. M. Hayat, “Co-channel FM Demodulation Via the Multi Angle-Centered Discrete Fractional Fourier Transform,” 2009 IEEE Digital Signal Processing Workshop," Marcos Islands, Jan. 2009, FL, 2009.

• M. Madrid, J. J. Simpson, B. Santhanam, W. Gerstle, T. Atwood, and M. M. Hayat, "Modeling electromagnetic wave interactions with vibrating structures," IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, Charleston, SC, June 2009, accepted.

Summary

• Phase history information in SAR data can be exploited via DFRFT-based signal processing to estimate vibrations while performing usual imaging• Vibration-axis ambiguities can resolved using a multiple-look approach combined with 2D analysis.• We have developed an understanding of the capabilities and limitations of the DFRFT based approach for SAR vibrometry• Additional validations are needed using simulations and experiments