A Study on The Resolution Limit
of Polarimetric Radar
and The Performance
of Polarimetric Bandwidth Extrapolation
Technique
Kei Suwa, Toshio Wakayama, Masafumi Iwamoto
Mitsubishi Electric Co.
Information Technology R&D Center
Outline
•
•
•
•
•
•
•
Background -- SAR resolution, PRF, and Polarimetry -Polarimetric Bandwidth Extrapolation (PBWE)
The Signal Model
Statistical Resolution Limit (SRL)
The Cramér-Rao Bound (CRB)
SRL and The Performance of PBWE
Conclusion
Background -- SAR resolution, PRF, and Polarimetry -Range resolution : signal bandwidth / Azimuth resolution : synthetic aperture length
 Range resolution is determined by the signal bandwidth.
c
c
r =
2B
: Signal Bandwidth
r
B
FT
B : Speed of light
range
freq.
 Azimuth resolution is determined by the synthetic aperture length.
a 

2

 R

2 L

: wave length
L : Synthetic Aperture length
R : Range

 
Strip map mode
 

D
L
orbit

R
Spotlight mode

 D D
a 
  
2 2  2
Higher resolution with
smaller aperture size D .
Background -- SAR resolution, PRF, and Polarimetry
Due to PRF limitation, Polarimetric SAR often give up the resolution.
orbit
 Conditions of the PRF
 PRF must be higher than the signal
Doppler bandwidth, which is determined by
the azimuth beam width.
 PRF must be low enough so that the range
ambiguity is sufficiently low.
 The problem with Polarimetry
 Polarimetry requires sending H-polarization
pulse and V-polarization pulse.
 PRF gets lower for each polarization
channel, so polarimetric SAR often need to
give up the resolution.
It would be nice
if polarization information helps enhancing
the resolution
Beam
Illumination
area
Doppler Freq.
Time
PRF
Platform trajectory
Background -- SAR resolution, PRF, and Polarimetry
Dr. Mihai Datcu “Semantic Content Extraction from High Resolution Earth Observation Images”
Background -- SAR resolution, PRF, and Polarimetry
What is “resolution”?
… resolution refers to separate two or more nearby targets,
i.e., to determine that there are two instead of one.
~Radar Handbook (1970) 4-2~
Background -- SAR resolution, PRF, and Polarimetry
It seems to be possible that polarization information helps improving the “resolution”.
1. Flat plate and dihedral corner reflector case
HH
Polarization
vector
range
Two vectors are
orthogonal to each other.
VV
2. Two flat plates case
HH
range
Polarization information
would not help
VV
polarization vectors
Flat plate
Shh
Shv
Svv
=
1
0
1
Dihedral
corner
reflector
1
0
-1
Polarimetric Bandwidth Extrapolation (PBWE)
• A polarimetric linear prediction model is fitted to the spectral data.
• Then the HH,HV,VV spectral data is extrapolated up to B’.
i
c
2B
B
ii
FFT
VV
HV
HH
v
VV
HV
HH
iii
range
freq.
Polarimetric Linear Prediction
Model Estimation
c
2B’
B’
iv
IFFT
VV
HV
HH
range
VV
HV
HH
freq.
Polarimetric Bandwidth Extrapolation (PBWE)
We have empirically shown that Polarization information does contribute to
the resolution enhancement
azimuth
range
Storebaelt : the great belt
EMISAR : dual frequency (L- and C-band) polarimetric Synthetic Aperture Radar (SAR)
system developed at Technical University of Denmark.
http://www.emi.dtu.dk/research/DCRS/Emisar/emisar.html
Resolution
Quantitative analysis on the influence of polarization information on
the resolution is provided in this presentation.
“Nominal” Resolution does not reflect polarization properties
range
r =
c
2B
azimuth
a 

2

 R

2 L
 Fourier analysis would give the nominal resolution.
 Parametric spectrum analysis would achieve higher resolution.
--- e.g. PBWE, MUSIC, ESPRIT, MVM, MEM, etc
We are interested in the resolution achievable
by “polarimetric” parametric spectrum analysis method such as PBWE.
Statistical Resolution Limit (SRL)
SRL is a simple metric that provides the highest resolution achievable
by any unbiased parametric spectral estimator.
•
“Statistical Resolution” : Δd
d  d

s.t.  dˆ  d   d , where  dˆ  d   E dˆ  d
true target separation = d
d
•

2
The estimate of target separation dˆ
contains error due to noise.
If the expected error is sufficiently small
compared to the real separation, we can
claim that these targets are resolvable.
dˆ
“Statistical Resolution Limit” : min(Δd)
min d   d


s.t. min  dˆ  d 

 d , where  dˆ  d   E dˆ  d

2
[1] S.T.Smith, “Statistical Resolution Limits and the Complexified Cramér-Rao Bound,”
IEEE Trans. Signal Process., vol. 53, no. 5, pp1597—1609, May 2005
The Signal Model
The signal at each resolution cell is represented by a polarization vector.
target
clutter
scene
range
Polarization
property
power
VV
Polarimetric
channels
HV
HH
The Signal Model
Two closely located point targets signal model is considered.
B
target #1
VV
HH
a11
VV
HH
freq.
d
z  Va  : n
 N 1 

 j
  
 2 
1 e


V  

 N 1 


j
 
 2 
1 e

 1
a  a  j
e



e j    
1

E nn H  R  I
The probability density function of the data z is:
f  z a,   
1
N R
a12
a21

target #2
a22
range
n: Gaussian noise
V: matrix of steering vectors
a: complex amplitudes of
the targets
N: the number of samples

exp   z  Va  : R 1  z  Va  :
H
"X : " denotes the long column vector formed by concatenating the columns of X.
The Cramér-Rao Bound (CRB)
The CRB for the parameters are derived.
•
Likelihood function of the parameters
f  z a,   
•
1
N R


exp   z  Va  : R 1  z  Va  :
H
Fisher Information Matrix


G a;    E 2a;  log f  z | a,  
•
Cramér-Rao Bound on the covariance of the estimator of the parameters
Var  | a  CRB  | a  G1|a
•
Convert the CRB to the lower bound of the target separation d
 dˆ  d  
N
2
CRB  | a
The Cramér-Rao Bound (CRB)
The CRB for 2 polarimetric channels case is derived.
•
The lower bound for the target separation
(Polarimetric / 2 channels / White Gaussian noise)


N
Std d   Tr DG D 
2 a
d0
[ FRC ]
2
2
2
d 0  d1 cos   / 2 
s (0) s '(  ) 2
d 0   s "(0) 
s (0) 2  s( ) 2
s ( ) s '( ) 2
d1   s "( ) 
s (0) 2  s( ) 2
•
cf) when ∆φ →large (two targets are far away)
d0  s "(0), d1  0
Std d  
N
1

2 a
s "(0)
N

sin   
2

s ( ) 
1

sin   
2

SRL and the Performance of PBWE
The PBWE achieves CRL and SRL fairly well.
Δd=0.4
(a) ∆Ψ = 0
•
•
•
Δd=0.1
5
(b) ∆Ψ = π/6
Monte Carlo iteration 1,000 times.
N=50 / SNR=40dB / order of the linear prediction filter = 6
If the polarization properties of the two targets are the same
polarization information does not help (a).
a
a
ae j
ae j  
VV
HH
d
range
SRL and the Performance of PBWE
The SRL is minimum when the polarization properties of the two targets are orthogonal
•
•
(c) ∆Ψ = π/2
(d) ∆Ψ = π
j
ae
a
The SRL is improved to ¼ by using polarization information,
a ae j  
when the polarization properties of the two targets are
VV
orthogonal (d).
HH
(d) corresponds to a case where two targets are trihedral
range
d
and dihedral corner reflectors.
Conclusion
• The SRL of polarimetric radar is derived and compared with that of
single polarization radar
• It has been shown analytically that the polarization information helps
improving the resolution when the polarization properties of the two
closely located targets are different, and thus on the average, the
polarimetric radar outperforms single polarization radar.
– e.g. If the polarization properties of the two closely located
targets are orthogonal to each other, the SRL is improved to ¼
by using polarization information.
• It has been shown that the resolution of the PBWE previously
proposed by the authors almost achieves the SRL.