Joint MIMO Radar Waveform and Receiving Filter Optimization
description
Transcript of Joint MIMO Radar Waveform and Receiving Filter Optimization
Joint MIMO Radar Waveform and Receiving Filter Optimization
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of TechnologyElectrical Engineering/DSP Lab
ICASSP 2009
Outline
Problem Formulation– Extended target and clutter– Detection– MIMO radar
Proposed Algorithm– Iterative algorithm– Receiver– Waveforms
Numerical Examples Conclusions
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1Problem Formulation
3
Extended Target vs. Point Target
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)(tfPoint Target
Extended Target vs. Point Target
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)(tf )( trfPoint Target
r : radar cross section : delay
Extended Target vs. Point Target
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)(tf
)(tf )( ii tfr
)( trfPoint Target
Extended Target vs. Point Target
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dtfr )()(
)(tf
)(tf
)(tf
)( ii tfr
)( trfPoint Target
Extended Target
Extended Target and Clutter
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)(tf
Extended Target Extended Clutter
Extended Target and Clutter
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dtfc
dtfr
)()(
)()(
)(tf
Extended Target Extended Clutter
Extended Target and Clutter
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dtfc
dtfr
)()(
)()(
)(tf
Extended Target
R(s)
C(s)
v(t)f(t)
Extended Clutter
Baseband Equivalent Model
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Modulation R(s)
C(s)
Demodulation
v (t)f(n) D/A A/D r(n)
Baseband Equivalent Model
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Modulation R(s)
C(s)
Demodulation
v (t)f(n) D/A A/D r(n)
R(z)
C(z)
v (n)
f(n)
)()(
)()(
mnfmc
mnfmr
Detection Problem
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H0
H1 Target
Clutter
Detection Problem
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H0
H1 Target
Clutter
R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
[Delong & Hofstetter 67] [Pillai et al. 03]
Transmittedwaveform
Detection Problem
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H0
H1 Target
Clutter
R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
[Delong & Hofstetter 67] [Pillai et al. 03]
Transmittedwaveform
SINR Maximization
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
[Delong & Hofstetter 67] [Pillai et al. 03]
u
)( vCfRfh Hu vhCfhRfh HHH
vhCfhRfh HHH
SINR Maximization
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
[Delong & Hofstetter 67] [Pillai et al. 03]
u
)( vCfRfh Hu
1 subject to
max22
2
,
f
vhCfh
Rfhfh
HH
H
EE
vhCfhRfh HHH
SINR Maximization
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
[Delong & Hofstetter 67] [Pillai et al. 03]
u
)( vCfRfh HuSignal
1 subject to
max22
2
,
f
vhCfh
Rfhfh
HH
H
EE
vhCfhRfh HHH
SINR Maximization
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
[Delong & Hofstetter 67] [Pillai et al. 03]
u
)( vCfRfh HuClutter
1 subject to
max22
2
,
f
vhCfh
Rfhfh
HH
H
EE
vhCfhRfh HHH
SINR Maximization
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
[Delong & Hofstetter 67] [Pillai et al. 03]
u
)( vCfRfh HuNoise
1 subject to
max22
2
,
f
vhCfh
Rfhfh
HH
H
EE
vhCfhRfh HHH
SINR Maximization
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
[Delong & Hofstetter 67] [Pillai et al. 03]
u
)( vCfRfh Hu
Power constraint1 subject to
max22
2
,
f
vhCfh
Rfhfh
HH
H
EE
The MIMO Case
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[Friedlander 07]
The MIMO Case
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
)( vCfRfh Hu vhCfhRfh HHH
[Friedlander 07]
Prior Information
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1 subject to
max22
2
f
vhCfh
Rfhfh
HH
H
,EE
Assumptions:
RTarget impulse response is known
Prior Information
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Assumptions:
RTarget impulse response is known
''*jiijCCE2nd order statistics of clutter is known
1 subject to
max22
2
f
vhCfh
Rfhfh
HH
H
,EE
2 Proposed Algorithm
26
Iterative Algorithm
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
1. Fixed f, solve for h
22
2
maxvhCfh
Rfhh
HH
H
EE
Iterative Algorithm
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
1. Fixed f, solve for h2. Fixed h, solve for f
1 subject to
max22
2
f
vhCfh
Rfhf
HH
H
EE
Iterative Algorithm
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
1. Fixed f, solve for h2. Fixed h, solve for f3. Fixed f, solve for h
22
2
maxvhCfh
Rfhh
HH
H
EE
Iterative Algorithm
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R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
1. Fixed f, solve for h2. Fixed h, solve for f3. Fixed f, solve for h
22
2
maxvhCfh
Rfhh
HH
H
EE
SINR is guaranteed to be non-decreasing in each iterative step.
Solving for the Receiver
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22
2
maxvhCfh
Rfhh
HH
H
EE
R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
Solving for the Receiver
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22
2
maxvhCfh
Rfhh
HH
H
EE
hvvhhCCffh
Rfhh
max
2
HHHHH
H
EE
Solving for the Receiver
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hvvhhCCffh
Rfhh
max
2
HHHHH
H
EE
Solving for the Receiver
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hvvhhCCffh
Rfhh
max
2
HHHHH
H
EE
1 subject to
min
Rfh
hvvCCffhhH
HHHH EE MVDR (Minimum Variance Distortionless)
Solving for the Receiver
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hvvhhCCffh
Rfhh
max
2
HHHHH
H
EE
1 subject to
min
Rfh
hvvCCffhhH
HHHH EE MVDR (Minimum Variance Distortionless)
RfvvCCffh-1 HHH EE
Solving for the Waveforms
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1 subject to
max22
2
f
vhCfh
Rfhf
HH
H
EE
R(z)
C(z)
v (n)f(n) H(z) LRT
Receiving filter
H0 or H1
Likelihood ratio test
Transmittedwaveform
u
Solving for the Waveforms
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1 subject to
max22
2
f
vhCfh
Rfhf
HH
H
EE
1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
Solving for the Waveforms
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1 subject to
max22
2
f
vhCfh
Rfhf
HH
H
EE
1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
Cannot be solved using MVDR
Solving for the Waveforms
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1 subject to
max22
2
f
vhCfh
Rfhf
HH
H
EE 1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
Try Lagrange Method:
Solving for the Waveforms
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1 subject to
max22
2
f
vhCfh
Rfhf
HH
H
EE 1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
0
22
22
fvhfChhCf
fChhCRfhvhfChhCfRfhhRλ
EE
EEE
HHHH
HHHHHHHHH
cannot be solved easily
Try Lagrange Method:
Recasting the Waveform Optimization Problem
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1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
Recasting the Waveform Optimization Problem
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1 f
Recasting the Waveform Optimization Problem
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1 f
1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
1 subject to
max
22
2
f
fvhfChhCf
Rfhf
HHHH
H
EE
Recasting the Waveform Optimization Problem
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1 subject to
max
2
2
f
vhfChhCf
Rfhf
HHHH
H
EE
1 f
1 subject to
max
22
2
f
fvhfChhCf
Rfhf
HHHH
H
EE
Recasting the Waveform Optimization Problem
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22
2
max
fvhfChhCf
Rfhf
HHHH
H
EE
Recasting the Waveform Optimization Problem
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22
2
max
fvhfChhCf
Rfhf
HHHH
H
EE
MVDR (Minimum Variance Distortionless)
1 subject to
min2
Rfh
fIvhChhCffH
HHHH EE
Recasting the Waveform Optimization Problem
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22
2
max
fvhfChhCf
Rfhf
HHHH
H
EE
MVDR (Minimum Variance Distortionless)
1 subject to
min2
Rfh
fIvhChhCffH
HHHH EE
hRIvhChhCf H-12
HHH EE
Proposed Algorithm
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R(z)
C(z)
v (n)f(n) H(z)
Receiving filterTransmittedwaveform
HHE CCffR f Compute .1
RfvvRh 1f ])[( .2 HE
Initialize: Choose a start point for f
Proposed Algorithm
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R(z)
C(z)
v (n)f(n) H(z)
Receiving filterTransmittedwaveform
HHE CCffR f Compute .1
RfvvRh 1f ])[( .2 HE
ChhCR HHEh Compute .3
hRIhvvhRf HHHE 1h )][( .4
Initialize: Choose a start point for f
Proposed Algorithm
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HHE CCffR f Compute .1
RfvvRh 1f ])[( .2 HE
ChhCR HHEh Compute .3
hRIhvvhRf HHHE 1h )][( .4
fff .5
RepeatR(z)
C(z)
v (n)f(n) H(z)
Receiving filterTransmittedwaveform
Initialize: Choose a start point for f
Numerical Examples
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0 5 10 15 20 25 30 35 40 45 50
20
22
24
26
28
30
32
34
36
38
40
SIN
R (d
B)
# of iterations
Proposed
Method in [Pillai et al. 03]
LFM (Linear Frequency Modulation)
Matched Filter Bound
Parameters# of transmitters: 2# of receivers: 2Randomly generated impulse response
Numerical Examples
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-10 -5 0 5 10 15 20 25 30 35 40-50
-40
-30
-20
-10
0
10
20
30
CNR (dB)
SN
R (d
B)
Proposed
Method in [Pillai et al. 03]
Matched Filter Bound
Parameters# of transmitters: 2# of receivers: 2Averaging 1000randomly generated examples
LFM (Linear Frequency Modulation)
Conclusions Detection of Extended Target in Clutter
– Prior information• Target impulse response• Clutter statistics
Iterative Algorithm– Recast the problem– MVDR solution
More General Target Impulse Response are considered in the Journal Version– Uncertainty Set (Worst case optimization)– Random
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[Chen & Vaidyanathan, TSP under review]
Q&AThank You!
Any questions?
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