Co-registered Vibrometry & Imaging:
A Combined Synthetic-Aperture Radar &
Fractional-Fourier Transform Approach
University of New Mexico
FY2008
University Project
May 2009
NCMR 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
Graduate students:
• Personnel:
Qi Wang (50%)
UNM Faculty:
Srikanth Narravula (50%)
Prof. Majeed Hayat (ECE, 15%)
Tong Xia (50)%
Prof. Balu Santhanam (ECE,15%)
Tom Baltis (25)%
Prof. Walter Gerstle (CIVIL Engr,15%) Post Doc: Matt Pepin (DOE funded)
Sandia collaborators: Tom Atwood and Toby Townsend (10%)
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 Preaward)
• 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:
u1
r (t )  A  g (u )exp{ j0
Step 1: Deramp quadrature
demodulation removes the u-chirp
 u1

 2u

 F  g (u )exp   j (0  2a 0 )  
 c
   4 at

(| t |  c / 2)
Range Deramped Data
100
Azimuth Deramped Data
200
100
300
200
400
Synthetic Aperture (step)
Synthetic Aperture (step)
Step 2: Aperture compression and
range compensation remove v-chirp
500
600
Step 3: Inverse Fourier transform
in each dimension creates an image
A/D
1
F {}
2u
2u
 j 2a (t   0 )} du
c
c
300
400
500
700
600
800
-300
-200
-100
0
Range (meters)
100
200
300
700
800
-300
||
-200
-100
0
Range (meters)
100
200
300
| g (u, v) |
Resolution is limited by the bandwidth of the sent chirped microwave pulse
and the size of the synthetic aperture
c
Previous Work: Non-stationary case
• When the ground is vibrating the reflectance
becomes time varying:
• g (u )  g (u, t )
• The return signal after steps 1 & 2 becomes
u1
2u
2u
r (t )  A  g (u, t   0 / 2) exp{ j0
 j 2a (t   0 )} du
c
c
 u1
(| t |  c / 2)
• Different processing is required to extract g(u,t)
• To proceed, we need to specialize g(u,t) to practical
forms
Analysis of Discrete Vibrating Points
By using the existing quadratic demodulation process and lowpass filtering, the return signal of each sent pulse becomes a
superposition of chirp signals


 2 j
0
0
0
0
0
0
2 
r (t )  A Ki exp 
[(0ui  2avi  0 )  (2aui  vi  0  0vi ) t  2avi t ]
c
i


chirp rate
phase shift
central frequency


(| t |  c / 2)
• | Ki | Modulates the magnitude of each chirp
n
n
•Linear dependence between (ui , vi ) 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
Previous Work: Discrete FRFT
The discrete fractional Fourier transform (DFRFT) has the capability to
concentrate linear chirps in few coefficients
MA-CDFRFT
Aa  W
2

N 1
2

 VG Λ VGT
N 1
{A }kn   vkp vnp e
p 0
jp
X k [r ]   z k [ p ]e
j
2
pr
N
N 1
• Each “peak” relates to each target
point
n 0
• Position of each peak is related to
position & velocity of point target
p 0
z k [ p]  v kp  x k [n]v np
DFRFT Estimates
Previous Work: Vibration
Identification Methodology
u

2u
2u 2  
 1


r
(
t
)

A
Re
g
(
u
,
t
)
exp
j

(
t



)

a
(
t



)   du

Return echo c
0
0

 0
c
c 
  u1


quadratic demodulation
 2 j

(0ui0  2avi0 0 )  (2aui0  vi0 0  0vi0 )t  2avi0t 2  
r (t )  A Ri exp 
& low-pass filtering (A/D)
 c

i
tan( p   / 2)
p  / 2
cr  2
 1.41
N
N
c   p  0.85( p   / 2)3
MA-CDFRFT
Read out the positions of peaks
Compute the central frequencies,
and chirp rates
Compute positions, and velocities
Co-registration with
traditional SAR imagery
2
 (2au  v 0  0 v)  c
c
8
av  cr
c
Vibrating Targets Est. position: FRFT/FT (actual) Est. reflectivity (actual) Est. velocity (actual)
Target1
-39.4/-28.5 (-37.5)
0.78 (0.9)
945 (1000)
Target2
1.27/6.5 (0)
0.42 (0.5)
472 (500)
Target3
37.5/37.5 (37.5)
0.48 (0.5)
0 (0)
New Work: 2-D Non-stationary case
• When the ground is vibrating the reflectance
g (u , v)  g (u , v, t )
becomes time varying:
• The return signal becomes
R1
2 R(m)
2 R(m)
2 R(m)
r (t , m)  A  g (t   0 / 2 
)exp{ j0
 j 2a
(t   0 )} dR
c
c
c
 R1
(| t |  c / 2)
• Different processing is required to extract g(u,v,t)
• Practical forms:
Instantaneous velocity and sum of sinusoidal modes
m
R(m)  Ru ,v  V (t 
)
PRF
m
ˆ
R(m)  Ru ,v   R Di sin( fit  fi
 i )
PRF
i
Model for Discrete Vibrating Points
How can we estimate the motion Ri (t ) of each discrete target?
•Piece-wise linear approximation:
Ri (t , m)  Rim  Vi m (t  m / PRF ) for m c  t  (m  1) c , 0  m  M
•Send successive pulses to estimate Rim and Vi m
•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
Single Look:
Vibration Frequency
and Direction




V (t )   2 f j D j cos( j  ) cos(2 f j t 
j
)
j
   f j D j (cos(2 f j t   t   j   j )  cos(2 f j t   t   j   j ))
j
Vibrating
Target
ϴ
Range
Measured Vibration Spectrum
0.045
Changing aperture
splits vibration into
two sin waves
0.04
Amplitude
0.035
Complex amplitudes estimate 0.03
0.025
vibration direction
0.02
0.015
Fit of V(t) cos envelope also
0.01
estimates direction 0.005
Multi-Look
2
4
6
Frequency (Hz)
8
Single Look Approach
Envelope Fit
V (t )   2 f D cos(   ) cos(2 f t   )
j
j
j
x 10
Vibration
Simulated Phase Shift
Change
Rate of Change of Phase
-5
3
2
1
0
-1
-2
x 10
Angle-85
Vibration Angle-65
Vibration Angle-85
2


Vibration
Angle-45
Vibration
Angle-65


Vibration Angle-45
1
Vibration Angle-25
Vibration Angle-25
Vibration Angle 0
Vibration Angle 0
Vibration Angle 25
0
Vibration Angle 45
Vibration Angle 65
-1
Vibration Angle 85
-2
-3
0
500
1000
Pulse Number
-3
0
1500
500
1000
Pulse Number
Fit of cosine envelope
0
1500
Fit of cosine envelope
100
True Angle
True Angle
Fitted AngleFitted Angle
Fitted Angle (Degrees)
• 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
3
j
Simulated Phase Shift Change
-5
-20
-40
-60
-80
-100
-100
Fitted Angle (Degrees)
j
Rate of Change of Phase
j
50
0
-50
-100
-100
-80
-50
0
50
Vibration Angle (Degrees)
-60
-40
-20
Vibration Angle (Degrees)
100
0
Multilook Approach :
Frequency and Direction Estimates
How to calculate
 at multiple look angles
Amplitude Modulation
Vi (t )  cos(   ) 2 fi , j Di , j cos(2 fi , j t  i , j )
j
By taking two looks with different squint angles,
the average energy ratio these two looks is
E[Vi 2 (t )] cos 2 (  1 ) 1  cos(2  21 )


2
2
ˆ
E[Vi (t )] cos (   2 ) 1  cos(2  2 2 )
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
Results:
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°
Animated Demonstration
Amplitude Modulation
The vibrating point target
Amplitude Modulation
Θ
The patch of ground
Summary: 2-D Methodology
u

2u
2u 2  
 1


r
(
t
)

A
Re
g
(
u
,
t
)
exp
j

(
t



)

a
(
t



)   du

Return echo c
0
0

 0
c
c 
  u1


quadratic demodulation
 2 j

(0ui0  2avi0 0 )  (2aui0  vi0 0  0vi0 )t  2avi0t 2  
r (t )  A Ri exp 
& low-pass filtering (A/D)
 c

i
cr  2
tan( p   / 2)
 1.41
N
c   p  0.85( p   / 2)3
MA-CDFRFT
Read out the positions of peaks
Compute the frequencies, chirp
rates, positions, and velocities
Estimate vibration frequencies
and directions
p  / 2
N
2
 (2au  v 0  0 v)  c
c
8
av  cr
c
V (t )   2 f j D j cos( j   ) cos(2 f j t   j )
j
Form SAR image and overlay
vibration information
Multiple looks to measure
and refine vibration direction
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 inband/out-band ratios
(¼ 0.8-1).
• Frequency domain
samples can be
concentrated around
DFRFT peaks.
• Sharper peak locations
translate to better centerfrequency & 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 fullrank & eigenvalue spectrum not
separable into S+N and N
subspaces.
• Subspace approach  rank
reduction needed.
Modeling Electromagnetic Wave
Interactions with Vibrating Structures
• Goals:
Monica Madrid (Ph.D. student) and Jamesina Simpson (Assistant Professor)
Electrical and Computer Engineering Department, University of New Mexico
Leveraging DOE Funding
– 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)
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 finitedifference 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
f(t) = F0 sin(Ωt)
frictionless tube (A, L)
k1
k2
m1
m2
gas (B, ρ, A)
vibrating mass
x
- Tests simulate theoretical
model
- A speaker simulates the
vibrating mass m1
- An aluminum disk and two steel
beams simulate the springmass system response
- Matlab code controls the
vibration frequency generating
a sinusoidal excitation with
well-controlled frequencies
L
Forcing Frequencies (Hz)
Structural Acoustics Experiment
Pressure
transducer
measures the
pressure of a
sound excitation.
A steel box will
simulate a room
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
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• 1D and 2D analytical model for return
pulse SAR data acquisition signals
(Q1-Q3)
• 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