Demonstration of Target Vibration Estimation
in Synthetic Aperture Radar Imagery
Qi Wang1,2, Matthew Pepin1,2, Ryan J. Beach2, Ralf Dunkel3,
Tom Atwood2,4, Armin W. Doerry4, Balu Santhanam2, Walter
Gerstle5, and Majeed M. Hayat1,2
1Center
for High Technology Materials, University of New Mexico, Albuquerque, NM,
87131 USA
2Depart. of Electrical and Computer Science, University of New Mexico, Albuquerque,
NM, 87131 USA
3General Atomics Aeronautics Systems, Inc., San Diego, CA 92064 USA
4Sandia National Laboratories, Albuquerque, NM 87185 USA
5Depart. of Civil Engineering, University of New Mexico, Albuquerque, NM, 87131 USA
IGARSS 2011, July 24 -29, Vancouver, Canada
Content
• Motivation
• Signal model
• Algorithm development
– The discrete fractional Fourier transform
• Experimental results
• Performance analysis
• Conclusions
IGARSS 2011, July 24 -29, Vancouver, Canada
Motivation
 Vibration signatures associated with objects such as active
structures (e.g. bridges and buildings) and vehicles can bear
vital information about the type and integrity of these objects
 The ability of remote sensing target vibrations is important:
 Avoids the cost of acquiring and installing accelerometers
on remote structures
 Enables sensing vibrations of structures that not easily
accessible
 Synthetic aperture radar has the potential due to its active
illumination feature
IGARSS 2011, July 24 -29, Vancouver, Canada
Synthetic Aperture Radar
Data collection
positions
Platform
height
A single static target
A single vibrating target
The image is generated by the Lynx system built by
General Atomics Aeronautical Systems. Inc
IGARSS 2011, July 24 -29, Vancouver, Canada
Micro-Doppler effect
RF pulse:
p(t )  cos(2f ct ), | t | tc /2
X0
Frequency (Hz)
y (azimuth)
Doppler frequency
caused by the plane’s
motion
t (sec)
Doppler frequency caused by
both the plane’s motion and
target vibration
IGARSS 2011, July 24 -29, Vancouver, Canada
Signal model
• The SAR phase history is de-ramped and reformatted; autofocus is
performed if necessary. The signal from a vibrating scatterer after
range compression is:
  4f Vy

4f c
c

s[n]   exp j
n
rd [n]   , 0  n  N I .


c
  c R0 f prf

• σTaylor
: reflectivity
the vibrating
scattererto r [n] in a short time window
• The
seriesofexpansion
is applied
d
• fc: carrier frequency
starting at m, the signal in the time window is approximately :
• c: speed of the microwave
• V: speed of the antenna
  4f

 2f c ad [m] 2
4the
f c  Vy
c
•
y:
azimuth
position
of
vibrating
scatterer

  vd [m] n 
s[n]   exp j
rd [m] 
n   , m  n  m  N w.


c repetition
cf prf frequency
cf prf
• fprf: the
 pulse
 R0


• rd[n]: time-varying range of the vibrating scatterer
• φ: other (constant) phase terms
•NI : total number of pulses
IGARSS 2011, July 24 -29, Vancouver, Canada
The discrete fractional Fourier transform
 Uses a new parameter α to exploit the linear time-frequency
relation of the signal (equals to DFT for α = π/2)
 Concentrates a linear chirp into a few coefficients we obtain an
impulse-like transform analogous to what the discrete Fourier
transform produces for a sinusoid.
Intersection for α = π/2
The DFRFT of a complex-valued signal containing two component: a pure 150 Hz
sinusoid and a chirp with a center frequency of 200 Hz and chirp rate of 400 Hz/s.
IGARSS 2011, July 24 -29, Vancouver, Canada
The DFRFT (cont’d)
Let W denote the transformation matrix of the centered-DFT, the
fractional power of W is defined as
2
W  VG   VGT .
The DFRFT of a signal x[n] is the DFT of an intermediate signal xˆ[ p ]
for each index k (k = 0,1, …,N-1), that is
2 

X k [r ]   xˆ[ p] exp  j
pr ,
N


p 0
N 1
where r = 0,1,…,N-1 is the newly introduced angular index and
α=2πr/N. The intermediate signal xˆ[ p ] is calculated by
xˆ[ p]  v
N 1
(k )
p
( n)
x
[
n
]
v
 p,
p 0
where vp is the pth column vector of VG.
J. G. Vargas-Rubio and B. Santhanam, “On the multiangle centered discrete fractional Fourier
transform,” IEEE signal Processing Letters, vol. 12, pp. 273-276, 2005
IGARSS 2011, July 24 -29, Vancouver, Canada
Incorporating the chirp z-transform
Left: regular DFRFT; right: CZT-incorporated DFRFT with a zoom-in factor of 2
• The CZT can be easily incorporated into the DFRFT to provide a fine
resolution in estimating the angular position of the peak.
• The relation between the peak location and the chirp rate is
2


cr 
tan  p  
N

2
IGARSS 2011, July 24 -29, Vancouver, Canada
Vibration estimation procedure
1.
2.
3.
4.
5.
Demodulate and re-format the SAR phase history, perform
autofocus;
Apply range compression to the SAR phase history, identify the
signal from the vibrating scatterer;
For all m = 0 to m = N – Nw + 1 do
Apply the DFRFT to the signal of interest in each time window
and calculate the vibration acceleration via
cf prf2


a(m)  
tan (pm)  
2Nw fc
2

End for
Reconstruct the history of the vibration acceleration and calculate
its DFT spectrum
IGARSS 2011, July 24 -29, Vancouver, Canada
Experiment
static targets
Vibrating target
•Aluminum triangular trihedral with
lateral length of 15 inches
• The vibration is caused by the
rotation of an unbalanced mass.
SAR image of the test ground near Julian, CA. • Vibration magnitude: 1.5 mm;
It was generated by the Lynx system on 2010. vibration frequency: 5 Hz
IGARSS 2011, July 24 -29, Vancouver, Canada
Experiment: the DFRFT spectra
Angle (rad)
The changing of position of the peak in the DFRFT plain indicates a timevarying vibration acceleration of the target.
Frequency (Hz)
IGARSS 2011, July 24 -29, Vancouver, Canada
Estimation results
Estimated vibration acceleration
(x: time (s); y: acceleration ( m/s2)
Estimated DFT spectrum of the vibration
(x: frequency (Hz); y: magnitude (AU)
Actual vibration
frequency: 5 Hz
IGARSS 2011, July 24 -29, Vancouver, Canada
Performance analysis
• The vibration frequency resolution is limited by the SAR
observation time of the target
• For the Lynx system, it ranges from 0.3 Hz to 1.0 Hz
depending on the data collection geometry and the radar
cross section of the target.
• The maximum measurable vibration frequency is upperbounded by fprf /2 theoretically
• In practice, vibration frequencies up to 25 Hz can be easily
estimated when fprf = 1000 Hz;
IGARSS 2011, July 24 -29, Vancouver, Canada
Conclusions
• A DFRFT-based method is proposed for SAR vibration
estimation
• In the experiment, the proposed method successfully
estimated a 1.5 mm, 5-Hz vibration from a corner reflector
• Performance analysis of the proposed method is carried out in
terms of vibration frequency resolution and maximum
measurable vibration frequency
IGARSS 2011, July 24 -29, Vancouver, Canada
Acknowledgement
This work was supported by the United States Department of
Energy (Award No. DE-FG52-08NA28782), the National
Science Foundation (Award No. IIS-0813747), National
Consortium for MASINT Research, and Sandia National
Laboratories. The authors also thank GA-ASI for making the
Lynx system available to this project.
IGARSS 2011, July 24 -29, Vancouver, Canada