Transcript Document
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.
r = c 2B c B
: Signal Bandwidth : Speed of light
B
FT freq.
Azimuth resolution is determined by the synthetic aperture length.
2 2
R L
L R
: wave length : Synthetic Aperture length : Range
L
r
range orbit Strip map mode
D
Spotlight mode
a
2 2
D
D
2 Higher resolution with
R
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.
Beam Illumination area 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.
PRF Platform trajectory Doppler Freq.
Time It would be nice if polarization information helps enhancing the resolution
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 VV Two vectors are orthogonal to each other.
2. Two flat plates case HH range Polarization information would not help VV polarization vectors Flat plate
S hh S hv S vv
= 1 0 1 Dihedral corner reflector 1 0 -1
Polarimetric Bandwidth Extrapolation (PBWE)
i • A polarimetric linear prediction model is fitted to the spectral data. • Then the HH,HV,VV spectral data is extrapolated up to
B’
.
c 2B
ii
B
FFT VV HV HH v
c 2B’ range
VV HV HH iii Polarimetric Linear Prediction Model Estimation
freq.
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 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 2 2
R 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
s.t.
d
ˆ
d
, where
d
ˆ
E
d
ˆ
d
2 • d true target separation = d The estimate of target separation
d
ˆ contains error due to noise. If the expected error is sufficiently small compared to the real separation, we can ˆd claim that these targets are resolvable.
“Statistical Resolution Limit” : min(Δd) min
d
s.t. min
d
ˆ
d
, where
d
ˆ
E
d
ˆ
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
VV HH Two closely located point targets signal model is considered.
B
target #1
a
11
a
12
a
21 target #2
a
22
freq.
VV HH d
range z
:
n V
1 1
e
j e j N
1 2
N
2 1
a
1
e j
e j
1
I
n: Gaussian noise V: matrix of steering vectors a: complex amplitudes of the targets N: the number of samples The probability density function of the data z is: ,
N
1
R
exp :
H R
1
z
: "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 ,
N
1
R
exp Fisher Information Matrix
G
a
; 2
a
; log :
H R
1
z
: • 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
d
ˆ
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)
G
D
N
2
a d
0
s
"(0)
s s
(0) 2
s d
1
s
"(
s
(
s
(0) 2 ) 2 ) 2
s
) 2 ) 2
d
0 2
d
1 2
d
0 cos 2 / 2 [
FRC
]
s
( sin sin
N
2 1 2 • cf) when ∆φ →large (two targets are far away)
d
0
s
"(0),
N
2
d
1 0
a
1
s
"(0)
SRL and the Performance of PBWE
The PBWE achieves CRL and SRL fairly well.
Δd=0.4
Δd=0.1
5 • • • (a) ∆Ψ = 0 (b) ∆Ψ = π/6 a 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).
VV HH a ae j ae j 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 The SRL is improved to ¼ by using polarization information, when the polarization properties of the two targets are orthogonal (d).
(d) corresponds to a case where two targets are trihedral and dihedral corner reflectors.
VV HH a a ae j ae j d range
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.