Transcript Compressed sensing in MIMO radar

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
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
2
Review of the keywords: Compressed sensing,
MIMO Radar
3
Brief Review of Compressed Sensing
  
y  

  
  
Φ
 
 
  
  
 s

 
  
 
 
dim( y )  dim( s )
Goal: Reconstruct s from y.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
4
Brief Review of Compressed Sensing
  
y  

  
  
Φ
 
 
  
  
 s

 
  
 
 
dim( y )  dim( s )
Goal: Reconstruct s from y.
Incoherence:
max
i j
φi,φ
j
is small.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
5
Brief Review of Compressed Sensing
  
y  

  
  
Φ
 
 
  
  
 s

 
  
 
 
Incoherence:
max
i j
φi,φ
j
dim( y )  dim( s )
Goal: Reconstruct s from y.
Sparsity:
is small.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
i | s i
 0  is small.
6
Brief Review of Compressed Sensing
  
y  

  
  
Φ
Incoherence:
max
i j
φi,φ
j
is small.
 
 
  
  
 s

 
  
 
 
dim( y )  dim( s )
Goal: Reconstruct s from y.
Sparsity:
i | s i
 0  is small.
Given y and F, s can be perfectly recovered by
sparse approximation methods even when dim(y)<dim(s).
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
7
Brief Review of Compressed Sensing
  
y  

  
  
Φ
Incoherence:
max
i j
φi,φ
j
is small.
 
 
  
  
 s

 
  
 
 
dim( y )  dim( s )
Goal: Reconstruct s from y.
Sparsity:
i | s i
 0  is small.
Given y and F, s can be perfectly recovered by
sparse approximation methods even when dim(y)<dim(s).
This concept can be applied to sampling and compression.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
8
Review: Compressed Sensing in Radar
[Herman & Strohmer08]
Range
u
y
Doppler
targets
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
9
Review: Compressed Sensing in Radar
[Herman & Strohmer08]
Range
u
y
  
  

y  
  
Φ
Doppler
targets
 *
 
  *
  
 s

 
  *
 *
 
si: target RCS in the i-th
Range-Doppler cell.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
10
Review: Compressed Sensing in Radar
[Herman & Strohmer08]
Range
u
Doppler
y
  
  

y  
  
Φ
targets
 *
 
  *
  
 s

 
  *
 *
 
si: target RCS in the i-th
Range-Doppler cell.
F is a function of the
transmitted waveform u.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
11
Review: Compressed Sensing in Radar
[Herman & Strohmer08]
Range
u
Doppler
y
  
  

y  
  
Φ
targets
 *
 
  *
  
 s

 
  *
 *
 
F is a function of the
transmitted waveform u.
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.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
12
Review: Compressed Sensing in Radar
Target scene s can be reconstructed by compressed
sensing method. High resolution can be achieved.
[Herman & Strohmer08]
  
  

y  
  
Φ
 *
 
  *
  
 s

 
  *
 *
 
F is a function of the
transmitted waveform u.
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.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
13
Brief Review of MIMO Radar
MIMO Radar
Each element can transmit an
arbitrary waveform.
u2(t)
u1(t)
u0(t)
Phased array radar (Traditional)
Each element transmits a scaled
version of a single waveform.
w2u(t)
w1u(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]
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
Compressed Sensing in MIMO Radar
15
MIMO Radar Signal Model
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
16
MIMO Radar Signal Model
(p,t, fD)
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
…
uM-1(t)
y0(t) y1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
yN-1(t)
17
MIMO Radar Signal Model
(p,t, fD)
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
…
uM-1(t)
M 1
y n (t ) 
u
m
(t  t )  e
y0(t) y1(t)
j
2

yN-1(t)
T
p (xm y n )
e
j 2f D t
m 0
Received signals
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
18
MIMO Radar Signal Model
(p,t, fD)
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
…
uM-1(t)
M 1
y n (t ) 
u
m 0
m
(t  t )  e
y0(t) y1(t)
j
2

yN-1(t)
T
p (xm y n )
e
j 2f D t
Range
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
19
MIMO Radar Signal Model
(p,t, fD)
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
…
uM-1(t)
M 1
y n (t ) 
u
m
(t  t )  e
m 0
xm: location of the m-th transmitter
yn: location of the n-th transmitter
y0(t) y1(t)
j
2

yN-1(t)
T
p (xm y n )
e
j 2f D t
Cross range
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
20
MIMO Radar Signal Model
(p,t, fD)
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
…
uM-1(t)
y0(t) y1(t)
M 1
y n (t ) 
u
m
(t  t )  e
j
m 0
xm: location of the m-th transmitter
yn: location of the n-th transmitter
2

j
e
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
2

yN-1(t)
T
p (xm y n )
e
sin  ( x m  y n )
j 2f D t
for linear array
21
MIMO Radar Signal Model
(p,t, fD)
(p,t, fD)
t:delay
fD :Doppler
p: direction
…
u0(t) u1(t)
…
uM-1(t)
M 1
y n (t ) 
u
m
(t  t )  e
y0(t) y1(t)
j
2

m 0
xm: location of the m-th transmitter
yn: location of the n-th transmitter
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
yN-1(t)
T
p (xm y n )
e
j 2f D t
Doppler
22
MIMO Radar Signal Model
M 1
y n (t ) 

u m (t  t )  e
j
2

sin  ( x m  y n )
e
j 2f D t
m 0
Discrete Model:
 0 L
t
M 1 
yn   
IL
m 0 
 0 ( L '  L   t ) L
1


 
 
 
 


j 2
e
D
L

Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008


2  

j
( xm  yn )

NM
 um e


j 2  D ( L 1 ) 

L
e

23
MIMO Radar Signal Model
M 1
y n (t ) 

m 0
u m (t  t )  e
j
2

sin  ( x m  y n )
e
j 2f D t
Range
Discrete Model:
 0 L
t
M 1 
yn   
IL
m 0 
 0 ( L '  L   t ) L
Range Cell:
1


 
 
 
 


j 2
e
D
L

 t  0 ,1, 2  L  1
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008


2  

j
( xm  yn )

NM
 um e


j 2  D ( L 1 ) 

L
e

L: Length of um
24
MIMO Radar Signal Model
M 1
y n (t ) 

u m (t  t )  e
j
2

sin  ( x m  y n )
m 0
e
j 2f D t
Doppler
Discrete Model:
 0 L
t
M 1 
yn   
IL
m 0 
 0 ( L '  L   t ) L
1


 
 
 
 


j 2
e
D
L

Range Cell:  t  0 ,1, 2  L  1
Doppler Cell: D  0 ,1, 2  L  1
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008


2  

j
( xm  yn )

NM
 um e


j 2  D ( L 1 ) 

L
e

L: Length of um
25
MIMO Radar Signal Model
M 1
y n (t ) 

u m (t  t )  e
m 0
j
2

sin  ( x m  y n )
e
j 2f D t
Angle
Discrete Model:
 0 L
t
M 1 
yn   
IL
m 0 
 0 ( L '  L   t ) L
1


 
 
 
 


j 2
e
D
L

Range Cell:  t  0 ,1, 2  L  1
Doppler Cell: D  0 ,1, 2  L  1
Angle Cell:    0 ,1, 2  NM  1
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008


2  

j
( xm  yn )

NM
 um e


j 2  D ( L 1 ) 

L
e

L: Length of um
M: # of transmitting antennas
N: # of receiving antennas
26
MIMO Radar Signal Model
 0 L
t
M 1 
yn   
IL
m 0 
 0 ( L '  L   t ) L
1


 
 
 
 


j 2
e
D
L



2  

j
( xm  yn )

NM
 um e


j 2  D ( L 1 ) 

L
H 
e

nm
(
H t
)
HD
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
27
MIMO Radar Signal Model
 0 L
t
M 1 
yn   
IL
m 0 
 0 ( L '  L   t ) L
1


 
 
 
 


j 2
e
D
L

(
H t
Overall
Input-output
relation:


2  

j
( xm  yn )

NM
 um e


j 2  D ( L 1 ) 

L
H 
e

nm
)
HD
y0 


 y
 1 
y 
 H   H   H  D  u

t



y

 N 1 
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
u0 


 u
 1 
u 




u

 N 1 
28
MIMO Radar Signal Model
Overall
Input-output
relation:
y0 


 y
 1 
y 
 H   H   H  D  u

t



y

 N 1 
Hα
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
u0 


 u
 1 
u 




u

 N 1 

α  ( t ,  D ,   )
29
MIMO Radar Signal Model
Overall
Input-output
relation:
y0 


 y
 1 
y 
 H   H   H  D  u

t



y

 N 1 
Hα
t
D

u0 


 u
 1 
u 




u

 N 1 

α  ( t ,  D ,   )
Range Cell:  t  0 ,1, 2  L  1
Doppler Cell: D  0 ,1, 2  L  1
Angle Cell:    0 ,1, 2  NM  1
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
30
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα
α
α  ( t ,  D ,   )
t
D
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

31
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα
α
α  ( t ,  D ,   )
y
Received waveforms
t
D
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008

32
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα
α
α  ( t ,  D ,   )
y
u
Received waveforms
Transmitted waveforms
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
t
D

33
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα
α
α  ( t ,  D ,   )
t
y Received waveforms
u Transmitted waveforms
H α Transfer function for the target in the  cell
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
D

34
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα
α
α  ( t ,  D ,   )
t
y Received waveforms
u Transmitted waveforms
H α Transfer function for the target in the  cell
s α RCS of the target in  cell
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
D

35
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα 
φα
α  ( t ,  D ,   )
α
φ
α
 sα  Φ  s
α
t
y Received waveforms
u Transmitted waveforms
H α Transfer function for the target in the  cell
s α RCS of the target in  cell
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
D

36
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα 
φ
φα
α  ( t ,  D ,   )
α
α
 sα  Φ  s
α
t
s is sparse if the target
scene is sparse.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
D

37
Compressed Sensing MIMO Radar Receiver
y 
H
α
u  sα 
φ
φα
α  ( t ,  D ,   )
α
α
 sα  Φ  s
α
t
s is sparse if the target
scene is sparse.
D

Compressed sensing algorithm can
effectively recover s if F is incoherent.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
38
Waveform Optimization
y 
H
α
α
u  sα 
φα
φ
α
 sα  Φ  s
α
Goal: Design u such that
max
α α '
H α u , H α 'u
t
D

is small.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
39
Waveform Optimization
α
TX
α'
α
RX
…
u
α'
…
s α H α u  s α ' H α 'u
Goal: Design u such that
max
α α '
H α u , H α 'u
is small.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
40
Waveform Optimization
α
TX
α'
α
RX
…
u
α'
…
s α H α u  s α ' H α 'u
Goal: Design u such that
max
α α '
H α u , H α 'u
Small Correlation
is small.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
41
Waveform Optimization: Dimension Reduction
H α u , H α 'u
 u ( H α '  H αt '  H α D ' ) ( H α  H αt  H α D ) u
H
H
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
42
Waveform Optimization: Dimension Reduction
H α u , H α 'u
 u ( H α '  H αt '  H α D ' ) ( H α  H αt  H α D ) u
H
H
H
H
H
 u ( H α  ' H α   H α D ' H αt ' H αt H α D ) u
H
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
43
Waveform Optimization: Dimension Reduction
H α u , H α 'u
 u ( H α '  H αt '  H α D ' ) ( H α  H αt  H α D ) u
H
H
H
H
H
 u ( H α  ' H α   H α D ' H αt ' H αt H α D ) u
H
H
 u ( H α  ' H α   C αt  αt ' H α D  α D ' ) u
H
CK
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
k









1
1






1


44
Waveform Optimization: Dimension Reduction
H α u , H α 'u
 u ( H α '  H αt '  H α D ' ) ( H α  H αt  H α D ) u
H
H
H
H
H
 u ( H α  ' H α   H α D ' H αt ' H αt H α D ) u
H
H
 u ( H α  ' H α   C αt  αt ' H α D  α D ' ) u
H
k
  ( α  , α  ' ,  αt ,  α D )
CK
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008









1
1






1


45
Waveform Optimization: Dimension Reduction
H α u , H α 'u
 u ( H α '  H αt '  H α D ' ) ( H α  H αt  H α D ) u
H
H
H
H
H
 u ( H α  ' H α   H α D ' H αt ' H αt H α D ) u
H
H
 u ( H α  ' H α   C αt  αt ' H α D  α D ' ) u
H
  ( α  , α  ' ,  αt ,  α D )
k
Goal: Design u such that
max
( α  ,  αt ,  α D )
 ( α  ', 0 , 0 )
 ( α , α ' ,  αt ,  α D )
is small.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
CK









1
1






1


46
Waveform Optimization: Beamforming
 To concentrate the transmit energy on the angles of
interest, we want the following term to be small

α  B
 ( α , α ,0 ,0 )
B: the set consisting of
angles of interest.
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
47
Waveform Optimization: Beamforming
 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.
 ( α , α ,0 ,0 )
α  B
 To uniformly illuminate the angles of interest, we want the
following term to be small
2

α  B
 ( α , α ,0 ,0 ) 
1
B
  ( α , α ,0 ,0 )
α  B
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
48
Waveform Optimization: Cost function
Incoherent
max
( α  ,  αt ,  α D )
 ( α  ', 0 , 0 )
Stopband
 ( α , α ' ,  αt ,  α D )
  ( α , α ,0 ,0 )
α  B
2
Passband

 ( α , α ,0 ,0 ) 
α  B
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
B
  ( α , α ,0 ,0 )
α  B
49
Waveform Optimization: Cost function

max
( α  ,  αt ,  α D )
 ( α  ', 0 , 0 )
 ( α , α ' ,  αt ,  α D )
+
 
  ( α , α ,0 ,0 )
α  B
+
2
(1     ) 

 ( α , α ,0 ,0 ) 
α  B
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
1
B
  ( α , α ,0 ,0 )
α  B
50
Waveform Optimization: Cost function
Incoherent
Stopband


  max
 ( α , α ' ,  αt ,  α D )     ( α , α ,0 ,0 )  
 (α(α , ',α0t, 0, ) α D )

α  B


min 
2
u


1
 ( α , α ,0 ,0 )
 (1     )   ( α  , α  , 0 , 0 ) 




B α  B
α  B


Passband
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
51
Phase Hopping Waveform
Consider constant-modulus signal:
u m (l )  e
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
j 2  ml
52
Phase Hopping Waveform
Consider constant-modulus signal:
u m (l )  e
j 2  ml
Consider phase on a lattice:
 ml 
C ml
K
, C ml  0 ,1, 2 ,  K  1
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
53
Phase Hopping Waveform
Consider constant-modulus signal:
u m (l )  e
j 2  ml
Consider phase on a lattice:
 ml 
min
C ml 
C ml
K
, C ml  0 ,1, 2 ,  K  1


2
  max
 ( α , α ' ,  αt ,  α D )     ( α , α ,0 ,0 )  
 (α(α , ',α0t, 0, ) α D )

α  B



2


1
2
2
 ( α , α ,0 ,0 )
 (1     )   ( α  , α  , 0 , 0 ) 




B
α  B
α  B


Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
54
Simulated Annealing Algorithm
min f ( C )
subject to
C
C
 Simulated annealing
– Create a Markov chain on the set A with the equilibrium distribution
 T (C ) 
ZT 
 f (C ) 
exp  

ZT
T


1
C’
C
…
…
 f (C ) 
exp
  T 


C 
– Run the Markov chain Monte Carlo (MCMC)
– Decrease the temperature T from time to time
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
55
Example: Histogram of correlations
Alltop Sequence
# of (,’) pairs
300
200
100
0
0
2
4
6
8
10
12
14
16
18
20
16
18
20
Parameters:
Uniform linear array
# of RX elements N=10
# of TX elements M =4
Signal length
L=31
# of phase
K=15
Angle of interest ALL
HProposed
u , H α 'u Method
,   '
α
300
200
100
0
0
2
4
6
8
10
12
14
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
56
Example: Histogram of correlations
Alltop Sequence
# of (,’) pairs
300
200
100
0
0
2
4
6
8
10
12
14
16
18
20
16
18
20
Parameters:
Uniform linear array
# of RX elements N=10
# of TX elements M =4
Signal length
L=31
# of phase
K=15
Angle of interest ALL
Proposed Method
300
200
100
0
0
2
4
6
8
10
12
14
H α u , H α 'u ,    '
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
57
Example: Histogram of correlations
Alltop Sequence
# of (,’) pairs
300
200
100
0
0
2
4
6
8
10
12
14
16
18
20
16
18
20
Parameters:
Uniform linear array
# of RX elements N=10
# of TX elements M =4
Signal length
L=31
# of phase
K=15
Angle of interest ALL
Proposed Method
300
200
100
0
0
2
4
6
8
10
12
14
H α u , H α 'u ,    '
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
58
Example: Recovering Target Scene
3
Range
Target Scene
2
1
0
10
20
30
200
400
600
800
1000
1200
10
20
Cross Range
30
40
10
20
30
40
10
20
Cross Range
30
40
60
Matched Filter
40
10
20
20
0
30
200
400
600
800
1000
1200
30
20
Range
Compressed
Sensing
10
0
10
20
30
200
400
600
800
1000
1200
SNR=10dB
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
59
Example: Recovering Target Scene
3
Range
Target Scene
2
1
0
10
20
30
200
400
600
800
1000
1200
10
20
Cross Range
30
40
10
20
30
40
10
20
Cross Range
30
40
60
Matched Filter
40
10
20
20
0
30
200
400
600
800
1000
1200
30
20
Range
Compressed
Sensing
10
0
10
20
30
200
400
600
800
1000
1200
SNR=10dB
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
60
Example: Recovering Target Scene
3
Range
Target Scene
2
1
0
10
20
30
200
400
600
800
1000
1200
10
20
Cross Range
30
40
10
20
30
40
10
20
Cross Range
30
40
60
Matched Filter
40
10
20
20
0
30
200
400
600
800
1000
1200
30
20
Range
Compressed
Sensing
10
0
10
20
30
200
400
600
800
1000
1200
SNR=10dB
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
61
Example: Recovering Target Scene
3
Range
Target Scene
2
1
0
10
20
30
200
400
600
800
1000
1200
10
20
Cross Range
30
40
10
20
30
40
10
20
Cross Range
30
40
60
Matched Filter
40
10
20
20
0
30
200
400
600
800
1000
1200
30
20
Range
Compressed
Sensing
10
0
10
20
30
200
400
600
800
1000
1200
SNR=10dB
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
62
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
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
63
Thank You!
Q&A
Any questions?
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
64
Simulated Annealing Algorithm
min f ( C )
subject to
C
C
 Simulated annealing
– Create a Markov chain on the set A with the equilibrium distribution
 T (C ) 
ZT 
 f (C ) 
exp  

ZT
T


1
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
…
…
 f (C ) 
exp
  T 


C 
C’
C
65
Simulated Annealing Algorithm
min f ( C )
subject to
C
C
 Simulated annealing
– Create a Markov chain on the set A with the equilibrium distribution
 T (C ) 
ZT 
 f (C ) 
exp  

ZT
T


1
…
…
 f (C ) 
exp
  T 


C 
C’
C
– Run the Markov chain Monte Carlo (MCMC)
Chun-Yang Chen, Caltech DSP Lab | Asilomar 2008
66