compressed sensing in mimo radar chun-yang chen and p. p. vaidyanathan california institute of...
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Compressed Sensing in MIMO Radar
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
Asilomar 2008
Outline
Review of the background– Compressed sensing [Donoho 06, Candes&Tao 06…]
• Compressed sensing in radar [Herman & Strohmer 08]– MIMO radar [Bliss & Forsythe 03, Robey et al. 04, Fishler et al. 04….]
Compressed sensing in MIMO radar– Compressed sensing receiver– Waveform optimization– Examples
Conclusion
2Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Brief Review of Compressed Sensing
4Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
)dim()dim( sy
y Φ s Goal: Reconstruct s from y.
Brief Review of Compressed Sensing
5Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
)dim()dim( sy
y Φ s Goal: Reconstruct s from y.
jiji
φφ ,max
Incoherence:
is small.
Brief Review of Compressed Sensing
6Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
)dim()dim( sy
y Φ s Goal: Reconstruct s from y.
jiji
φφ ,max
Incoherence:
is small. 0| isiSparsity:
is small.
0| isiSparsity:
is small.
Brief Review of Compressed Sensing
7Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
)dim()dim( sy
y Φ s Goal: Reconstruct s from y.
jiji
φφ ,max
Incoherence:
is small.
Given y and F, s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).
Given y and F, s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).
0| isiSparsity:
is small.
Brief Review of Compressed Sensing
8Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
)dim()dim( sy
y Φ s Goal: Reconstruct s from y.
jiji
φφ ,max
Incoherence:
is small.
This concept can be applied to sampling and compression.This concept can be applied to sampling and compression.
Given y and F, s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).
Given y and F, s can be perfectly recovered by sparse approximation methods even when dim(y)<dim(s).
Review: Compressed Sensing in Radar
9Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
[Herman & Strohmer08]
u
ytargets
Range
Doppler
Review: Compressed Sensing in Radar
10Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
[Herman & Strohmer08]
u
ytargets
Range
Doppler
si: target RCS in the i-th Range-Doppler cell.
*
sy Φ
**
*
Review: Compressed Sensing in Radar
11Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
[Herman & Strohmer08]
u
ytargets
Range
Doppler
si: target RCS in the i-th Range-Doppler cell.
F is a function of the transmitted waveform u.
*
sy Φ
**
*
*
sy Φ
**
*
Review: Compressed Sensing in Radar
12Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
[Herman & Strohmer08]
u
ytargets
Range
Doppler
si: target RCS in the i-th Range-Doppler cell.
Assumption: s is sparse.
Transmitted waveform u can be chosen such that F is incoherent.
F is a function of the transmitted waveform u.
Review: Compressed Sensing in Radar
13Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
si: target RCS in the i-th Range-Doppler cell.
Assumption: s is sparse.
Transmitted waveform u can be chosen such that F is incoherent.
Target scene s can be reconstructed by compressed sensing method. High resolution can be achieved. [Herman & Strohmer08]
Target scene s can be reconstructed by compressed sensing method. High resolution can be achieved. [Herman & Strohmer08]
F is a function of the transmitted waveform u.
*
sy Φ
**
*
Brief Review of MIMO Radar
u2( )tu1( )t
u0( )t
w2u( )tw1u( )t
w0u( )t
Advantages– Better spatial resolution [Bliss & Forsythe 03]– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]– Improved parameter identifiability [Li et al. 07]
Phased array radar (Traditional)Each element transmits a scaled version of a single waveform.
MIMO RadarEach element can transmit an arbitrary waveform.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
MIMO Radar Signal Model
16Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
MIMO Radar Signal Model
17Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)
…
y0(t) y1(t) yN-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
1
0
2)(2
)()(M
m
tfjj
mnD
nmT
eetuty
yxp
MIMO Radar Signal Model
18Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)
…
y0(t) y1(t) yN-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
Received signals
MIMO Radar Signal Model
19Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)
…
y0(t) y1(t) yN-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
1
0
2)(2
)()(M
m
tfjj
mnD
nmT
eetuty
yxp
Range
1
0
2)(2
)()(M
m
tfjj
mnD
nmT
eetuty
yxp
MIMO Radar Signal Model
20Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)
…
y0(t) y1(t) yN-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
xm: location of the m-th transmitteryn: location of the n-th transmitter
Cross range
1
0
2)(2
)()(M
m
tfjj
mnD
nmT
eetuty
yxp
MIMO Radar Signal Model
21Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)
…
y0(t) y1(t) yN-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
xm: location of the m-th transmitteryn: location of the n-th transmitter
)(sin2
nm yxje
for linear array
1
0
2)(2
)()(M
m
tfjj
mnD
nmT
eetuty
yxp
MIMO Radar Signal Model
22Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
u0(t) u1(t) uM-1(t)
(p,t, fD)
…
y0(t) y1(t) yN-1(t)
(p,t, fD)t:delayfD :Dopplerp: direction
xm: location of the m-th transmitteryn: location of the n-th transmitter
Doppler
1
0
)(2
)1(2
2
)'(
1
M
m
yxNM
j
m
LL
j
Lj
LLL
L
L
n
nm
D
D
e
e
e
u
0
I
0
y
1
0
2)(sin2
)()(M
m
tfjyxj
mnD
nm
eetuty
MIMO Radar Signal Model
23Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Discrete Model:
MIMO Radar Signal Model
24Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
0
2)(sin2
)()(M
m
tfjyxj
mnD
nm
eetuty
1
0
)(2
)1(2
2
)'(
1
M
m
yxNM
j
m
LL
j
Lj
LLL
L
L
n
nm
D
D
e
e
e
u
0
I
0
y
Discrete Model:Range
12,1,0 LRange Cell: L: Length of um
MIMO Radar Signal Model
25Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
0
)(2
)1(2
2
)'(
1
M
m
yxNM
j
m
LL
j
Lj
LLL
L
L
n
nm
D
D
e
e
e
u
0
I
0
y
1
0
2)(sin2
)()(M
m
tfjyxj
mnD
nm
eetuty
Discrete Model:
Doppler
12,1,0 LRange Cell: L: Length of um
12,1,0 LD Doppler Cell:
MIMO Radar Signal Model
26Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
0
)(2
)1(2
2
)'(
1
M
m
yxNM
j
m
LL
j
Lj
LLL
L
L
n
nm
D
D
e
e
e
u
0
I
0
y
12,1,0 LRange Cell: L: Length of um
M: # of transmitting antennasN: # of receiving antennas
12,1,0 LD 12,1,0 NM
Doppler Cell:
Angle Cell:
1
0
2)(sin2
)()(M
m
tfjyxj
mnD
nm
eetuty
Discrete Model:
Angle
MIMO Radar Signal Model
27Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
0
)(2
)1(2
2
)'(
1
M
m
yxNM
j
m
LL
j
Lj
LLL
L
L
n
nm
D
D
e
e
e
u
0
I
0
y
H
DH
nm
H
MIMO Radar Signal Model
28Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
0
)(2
)1(2
2
)'(
1
M
m
yxNM
j
m
LL
j
Lj
LLL
L
L
n
nm
D
D
e
e
e
u
0
I
0
y
H
DH
nm
H
uHHH
y
y
y
y
D
N
1
1
0
1
1
0
Nu
u
u
u
OverallInput-outputrelation:
uHHH
y
y
y
y
D
N
1
1
0
MIMO Radar Signal Model
29Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
1
0
Nu
u
u
u
OverallInput-outputrelation:
αH),,( D
α
uHHH
y
y
y
y
D
N
1
1
0
MIMO Radar Signal Model
30Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
1
0
Nu
u
u
u
OverallInput-outputrelation:
αH),,( D
α
D
12,1,0 LRange Cell:12,1,0 LD
12,1,0 NMDoppler Cell:
Angle Cell:
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
31Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
D
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
32Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
y Received waveforms
D
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
33Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
y Received waveforms
u Transmitted waveforms
D
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
34Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
y Received waveforms
αHu Transmitted waveforms
Transfer function for the target in the a cell D
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
35Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
D
y Received waveforms
αHu
αs
Transmitted waveforms
Transfer function for the target in the a cell
RCS of the target in a cell
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
36Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
D
y Received waveforms
αHu
αs
Transmitted waveforms
RCS of the target in a cell
sΦφα
αα s
αφ
Transfer function for the target in the a cell
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
37Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
D
sΦφα
αα s
s is sparse if the target scene is sparse.
αφ
α
ααuHy s
Compressed Sensing MIMO Radar Receiver
38Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,( Dα
D
sΦφα
αα s
s is sparse if the target scene is sparse.
Compressed sensing algorithm can effectively recover s if F is incoherent.
αφ
sΦφα
αα s α
ααuHy s
Waveform Optimization
39Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
D
Goal: Design u such that
is small.
uHuH αααα
''
,max
αφ
Waveform Optimization
40Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Goal: Design u such that
is small.
uHuH αααα
''
,max
…
α'α
u
…
α'α
uHuH αααα ''ss
TX RX
Waveform Optimization
41Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Goal: Design u such that
is small.
uHuH αααα
''
,max
…
α'α
u
…
α'α
uHuH αααα ''ss
Small Correlation
TX RX
Waveform Optimization: Dimension Reduction
42Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
uHuH αα ',
uHHHHHHu )()( ''' DD αααH
αααH
Waveform Optimization: Dimension Reduction
43Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
uHuH αα ',uHHHHHHu )()( ''' DD ααα
Hααα
H
uHHHHHHu )( ''' DD ααH
αH
ααH
αH
Waveform Optimization: Dimension Reduction
44Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
uHuH αα ',uHHHHHHu )()( ''' DD ααα
Hααα
H
uHHHHHHu )( ''' DD ααH
αH
ααH
αH
uHCHHu )( ''' DD αααααH
αH
1
1
1
KC
k
Waveform Optimization: Dimension Reduction
45Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
uHuH αα ',uHHHHHHu )()( ''' DD ααα
Hααα
H
uHHHHHHu )( ''' DD ααH
αH
ααH
αH
uHCHHu )( ''' DD αααααH
αH
),,',( Dαααα
1
1
1
KC
k
Goal: Design u such that
is small.
),,',(max)0,0,'(
),,(D
αααα
ααααD
Waveform Optimization: Dimension Reduction
46Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
uHuH αα ',uHHHHHHu )()( ''' DD ααα
Hααα
H
uHHHHHHu )( ''' DD ααH
αH
ααH
αH
uHCHHu )( ''' DD αααααH
αH
),,',( Dαααα
1
1
1
KC
k
Waveform Optimization: Beamforming
47Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Bα
αα
)0,0,,( B: the set consisting of angles of interest.
To concentrate the transmit energy on the angles of interest, we want the following term to be small
Waveform Optimization: Beamforming
48Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Bα
αα
)0,0,,(
Bα Bα
ααB
αα
2
)0,0,,(1
)0,0,,(
To uniformly illuminate the angles of interest, we want the following term to be small
To concentrate the transmit energy on the angles of interest, we want the following term to be small
B: the set consisting of angles of interest.
Waveform Optimization: Cost function
49Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,',(max)0,0,'(
),,(D
αααα
ααααD
Bα
αα
)0,0,,(
Incoherent
Stopband
Passband
Bα Bα
ααB
αα
2
)0,0,,(1
)0,0,,(
Waveform Optimization: Cost function
50Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
),,',(max)0,0,'(
),,(D
αααα
ααααD
)1(
+
+
Bα Bα
ααB
αα
2
)0,0,,(1
)0,0,,(
Bα
αα
)0,0,,(
Waveform Optimization: Cost function
51Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Bα Bα
BαD
αααα
ααB
αα
ααααααD
2
)0,0,'(),,(
)0,0,,(1
)0,0,,()1(
)0,0,,(),,',(max
minu
Incoherent Stopband
Passband
Phase Hopping Waveform
52Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Consider constant-modulus signal:
mljm el 2)( u
Phase Hopping Waveform
53Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Consider constant-modulus signal:
mljm el 2)( u
Consider phase on a lattice:
1,2,1,0 , KCK
Cml
mlml
Phase Hopping Waveform
54Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Consider constant-modulus signal:
mljm el 2)( u
Consider phase on a lattice:
1,2,1,0 , KCK
Cml
mlml
Bα Bα
BαD
αααα
ααB
αα
ααααααD
2
22
2
)0,0,'(),,(
)0,0,,(1
)0,0,,()1(
)0,0,,(),,',(max
mlCmin
Simulated Annealing Algorithm
Simulated annealing– Create a Markov chain on the set A with the equilibrium distribution
– Run the Markov chain Monte Carlo (MCMC)– Decrease the temperature T from time to time
55
Csubject to
…
CC’
…
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
)(min CCf
C
C
CC
T
fZ
T
f
Z
T
TT
)(exp
)(exp
1)(
Example: Histogram of correlations
56Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Parameters:Uniform linear array# of RX elements N=10# of TX elements M =4Signal length L=31# of phase K=15Angle of interest ALL
0 2 4 6 8 10 12 14 16 18 200
100
200
300
0 2 4 6 8 10 12 14 16 18 200
100
200
300
# of
(a,a
’) pa
irs
Alltop Sequence
Proposed Method',, ' uHuH αα
Example: Histogram of correlations
57Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
',, ' uHuH αα
0 2 4 6 8 10 12 14 16 18 200
100
200
300
0 2 4 6 8 10 12 14 16 18 200
100
200
300
# of
(a,a
’) pa
irs
Alltop Sequence
Proposed Method
Parameters:Uniform linear array# of RX elements N=10# of TX elements M =4Signal length L=31# of phase K=15Angle of interest ALL
Example: Histogram of correlations
58Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
',, ' uHuH αα
0 2 4 6 8 10 12 14 16 18 200
100
200
300
0 2 4 6 8 10 12 14 16 18 200
100
200
300
# of
(a,a
’) pa
irs
Alltop Sequence
Proposed Method
Parameters:Uniform linear array# of RX elements N=10# of TX elements M =4Signal length L=31# of phase K=15Angle of interest ALL
200 400 600 800 1000 12000
1
2
3
200 400 600 800 1000 12000
20
40
60
200 400 600 800 1000 12000
10
20
30
Cross Range
Ran
ge
10 20 30 40
10
20
30
10 20 30 40
10
20
30
Ran
ge
Cross Range10 20 30 40
10
20
30
Example: Recovering Target Scene
59Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Target Scene
CompressedSensing
Matched Filter
SNR=10dB
200 400 600 800 1000 12000
1
2
3
200 400 600 800 1000 12000
20
40
60
200 400 600 800 1000 12000
10
20
30
Cross Range
Ran
ge
10 20 30 40
10
20
30
10 20 30 40
10
20
30
Ran
ge
Cross Range10 20 30 40
10
20
30
Example: Recovering Target Scene
60Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Target Scene
CompressedSensing
Matched Filter
SNR=10dB
200 400 600 800 1000 12000
1
2
3
200 400 600 800 1000 12000
20
40
60
200 400 600 800 1000 12000
10
20
30
Cross Range
Ran
ge
10 20 30 40
10
20
30
10 20 30 40
10
20
30
Ran
ge
Cross Range10 20 30 40
10
20
30
Example: Recovering Target Scene
61Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Target Scene
CompressedSensing
Matched Filter
SNR=10dB
200 400 600 800 1000 12000
1
2
3
200 400 600 800 1000 12000
20
40
60
200 400 600 800 1000 12000
10
20
30
Cross Range
Ran
ge
10 20 30 40
10
20
30
10 20 30 40
10
20
30
Ran
ge
Cross Range10 20 30 40
10
20
30
Example: Recovering Target Scene
62Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Target Scene
CompressedSensing
Matched Filter
SNR=10dB
Conclusion
Compressed sensing based receiver– Applicable when the target scene is sparse– Better resolution than the matched filter receiver
Waveform design– Incoherent– Beamforming– Simulated annealing
63Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Simulated Annealing Algorithm
Simulated annealing– Create a Markov chain on the set A with the equilibrium distribution
65
)(min CCf Csubject to
C
C
CC
T
fZ
T
f
Z
T
TT
)(exp
)(exp
1)(
…
CC’
…
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008