Properties of the MIMO Radar
Ambiguity Function
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
ICASSP 2008
Outline
 Review of the background
– Radar ambiguity function and its properties
– MIMO radar
– MIMO radar ambiguity function
 Properties of the MIMO ambiguity function
–
–
–
–
Signal component
Energy
Symmetry
Linear frequency modulation (LFM)
 Conclusion
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
Review: Ambiguity function and MIMO radar
3
Radar Ambiguity Function
u(t)
u(t-t)ej2pnt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t: delay
n: Doppler
4
Radar Ambiguity Function
u(t)
Matched filter
output

u(t-t)ej2pnt
t: delay
n: Doppler
(u (t  t )e j 2pnt )(u (t  t ' )e j 2pn 't )* dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
5
Radar Ambiguity Function
u(t)
Matched filter
output

u(t-t)ej2pnt
t: delay
n: Doppler
(u (t  t )e j 2pnt )(u (t  t ' )e j 2pn 't )* dt
  u (t )u * (t  (t  t ' ))e j 2p (n n ')t dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
6
Radar Ambiguity Function
u(t)
Matched filter
output

u(t-t)ej2pnt
t: delay
n: Doppler
(u (t  t )e j 2pnt )(u (t  t ' )e j 2pn 't )* dt
  u (t )u * (t  (t  t ' ))e j 2p (n n ')t dt
Radar ambiguity
function
 (t ,n )   u (t )u * (t  t )e j 2pnt dt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
7
Radar Ambiguity Function
u(t)
Matched filter
output

u(t-t)ej2pnt
t: delay
n: Doppler
(u (t  t )e j 2pnt )(u (t  t ' )e j 2pn 't )* dt
  u (t )u * (t  (t  t ' ))e j 2p (n n ')t dt
Radar ambiguity
function
 (t ,n )   u (t )u * (t  t )e j 2pnt dt
 Ambiguity function characterizes the Doppler and range
resolution.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
8
Radar Ambiguity Function
u (t )
Multiple targets
(tk,nk)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
9
Radar Ambiguity Function
u (t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
Multiple targets
(tk,nk)
10
Radar Ambiguity Function
u (t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Matched filter
output
K

k 1
k
Multiple targets
(tk,nk)
  (t  t k ,n n k )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
11
Radar Ambiguity Function
u (t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Matched filter
output
K

k 1
k
Multiple targets
(tk,nk)
  (t  t k ,n n k )
n
target 1 (t1,n1)
target 2 (t2,n2)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
12
Radar Ambiguity Function
u (t )
K
j 2pn k t
u
(
t

t
)
e

k
k 1
Matched filter
output
n
K

k 1
k
Multiple targets
(tk,nk)
  (t  t k ,n n k )
 (t t1,n n1 )
target 1 (t1,n1)
target 2 (t2,n2)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
13
Radar Ambiguity Function
 Ambiguity function characterizes the Doppler and range
resolution.
n
 (t t1,n n1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
 (t ,n )   u (t )u (t  t ) e

j 2pnt
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
14
Radar Ambiguity Function
 Ambiguity function characterizes the Doppler and range
resolution.
n
 (t t1,n n1 )
target 1 (t1,n1)
target 2 (t2,n2)
t
 (t ,n )   u (t )u (t  t ) e

j 2pnt
dt
Ambiguity function
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
15
Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
n
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
16
Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
 Energy

2
 (t ,n ) dtdn  1
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
n
t
17
Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
 Energy

2
 (t ,n ) dtdn  1
 Symmetry
n
t
 (t ,n )   (t ,n )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
18
Properties of Radar Ambiguity Function
 Signal component
 (0,0)  1   (t ,n )
 Energy

2
 (t ,n ) dtdn  1
n
 Symmetry
t
 (t ,n )   (t ,n )
 Linear frequency modulation (LFM)
u
LFM
(t )  u(t )e
jpkt 2
 LFM (t ,n )   (t ,n  kt )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
19
MIMO Radar
The radar systems which emits orthogonal (or noncoherent)
waveforms in each transmitting antennas are called MIMO radar.
MIMO radar
f2(t)
f1(t)
f0(t)
SIMO radar (Traditional)
w2f(t)
w1f(t)
w0f(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 | ICASSP 2007 student paper contest
Ambiguity Function in MIMO Radar
(t,n,f) t:delay
n:Doppler
f: Spatial freq.
TX
dT
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
21
Ambiguity Function in MIMO Radar
(t,n,f)
TX
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
RX
…
uM-1(t)
(t,n,f)
…
dR
MF
…
MF
…
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
MF
…
22
Ambiguity Function in MIMO Radar
(t,n,f)
TX
dT
u0(t) u1(t)
t:delay
n:Doppler
f: Spatial freq.
RX
…
uM-1(t)
(t,n,f)
…
dR
MF
…
MF
…
y
(t ,n , f )
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
MF
…
(t )
23
Ambiguity Function in MIMO Radar
(t,n,f)
TX
RX
…
dT
u0(t) u1(t)
uM-1(t)
Matched filter output
 (y
(t ',n ', f ')
t:delay
n:Doppler
f: Spatial freq.
(t,n,f)
…
dR
MF
…
MF
…
y
(t ,n , f )
MF
…
(t )
(t ) )  y (t ,n , f ) (t )dt
H
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
24
Ambiguity Function in MIMO Radar
Matched filter output
 (y

(t ',n ', f ')
(t ) )  y (t ,n , f ) (t )dt
H
  ( u
N 1
M 1 M 1
n 0
m  0 m ' 0
j 2p ( f  f ') n
e


t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
)
*
j 2p (n v ') t
j 2p ( fm f 'm ')
(
t

t
)
u
(
t

t
'
)
e
dt
e
m
m
Receiver beamforming
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
25
Ambiguity Function in MIMO Radar
Matched filter output
 (y

(t ',n ', f ')
(t ) )  y (t ,n , f ) (t )dt
H
  ( u
N 1
M 1 M 1
n 0
m  0 m ' 0
j 2p ( f  f ') n
e


t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
)
*
j 2p (n v ') t
j 2p ( fm f 'm ')
(
t

t
)
u
(
t

t
'
)
e
dt
e
m
m
Receiver beamforming
 m,m ' (t ,n )   um (t )um* ' (t  t )e j 2pn t dt
Cross ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
26
Ambiguity Function in MIMO Radar
Matched filter output
 (y

(t ',n ', f ')
(t ) )  y (t ,n , f ) (t )dt
H
  ( u
N 1
M 1 M 1
n 0
m  0 m ' 0
j 2p ( f  f ') n
e


t:delay
n:Doppler
f: Spatial freq.
um(t): m-th waveform
xm: m-th antenna location
n: receiving antenna index
)
*
j 2p (n v ') t
j 2p ( fm f 'm ')
(
t

t
)
u
(
t

t
'
)
e
dt
e
m
m
Receiver beamforming
 m,m ' (t ,n )   um (t )um* ' (t  t )e j 2pn t dt
[San Antonio et al. 07]
M 1 M 1
 (t ,n , f , f ' )     m,m ' (t ,n )e j 2p ( fm f 'm ')
m  0 m ' 0
MIMO ambiguity function
Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007
27
Properties of the MIMO ambiguity function
28
Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
 (0,0, f , f ' )
f ' f
f'
 (0,0, f , f )
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
29
Properties of the signal component
 Ambiguity function:
 Signal component:
 (0,0, f , f ' )
 (t ,n , f , f ' )
 (0,0, f , f )
For orthogonal waveforms,

um (t )um* ' (t )dt   mm'
f ' f
f'
 (0,0, f , f )
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
30
Properties of the signal component
 Ambiguity function:
 Signal component:
 (0,0, f , f ' )
 (t ,n , f , f ' )
 (0,0, f , f )
For orthogonal waveforms,

f ' f
f'
 (0,0, f , f )
um (t )um* ' (t )dt   mm'
  (0,0, f , f )  M , f
If the waveforms are orthogonal,
the signal component will be a
constant for all angle.
f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
31
Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
For general waveforms,

For orthogonal waveforms,
2
u m (t ) dt  1

um (t )um* ' (t )dt   mm'
  (0,0, f , f )  M , f
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
32
Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
For general waveforms,

 If
dT

For orthogonal waveforms,
2
u m (t ) dt  1
is integer,

um (t )um* ' (t )dt   mm'
  (0,0, f , f )  M , f
   (0,0, f , f ) df  M , f
The integration of the signal
component is a constant if dT is
a multiple of the wavelength.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
33
Properties of the signal component
 Ambiguity function:
 Signal component:
 (t ,n , f , f ' )
 (0,0, f , f )
For general waveforms,

 If
For orthogonal waveforms,

2
u m (t ) dt  1
dT

dT is the spacing
between the
transmitting antennas
is integer,
um (t )um* ' (t )dt   mm'
  (0,0, f , f )  M , f
   (0,0, f , f ) df  M , f
 For the general case,
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
In general, the
integration of the
signal component is
confined.
34
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 mm' (t ,n ) dt dn
2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
35
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 
 mm' (t ,n ) dt dn
2
u
m
(t )u (t  t )e
*
m'
j 2pnt
2
dt dndt
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
36
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 
 mm' (t ,n ) dt dn
2
u
m
(t )u (t  t )e
*
m'
j 2pnt
2
dt dndt
   um (t )u (t  t ) dtdt
*
m'

2
Parserval
relation
( u (t ) dt )  1
2
2
m
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
37
Energy of the cross ambiguity function
 Cross ambiguity function:
 mm' (t ,n )   um (t )um* ' (t  t )e j 2pnt dt
 Energy of the cross ambiguity function:

 mm' (t ,n ) dt dn  1
2
The energy of the cross
ambiguity function is a
constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
38
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
2
dtdndfdf '
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
39
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
 
M 1 M 1
 
m  0 m ' 0
2
dtdndfdf '
2
j 4pdT /  ( fm f 'm ')
(
t
,
n
)
e
dfdf ' dtdn
mm'
dT is the spacing
between the
transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
40
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
 
M 1 M 1
 
m  0 m ' 0
M 1 M 1
2
dT is the spacing
between the
transmitting antennas
dtdndfdf '
2
j 4pdT /  ( fm f 'm ')
(
t
,
n
)
e
dfdf ' dtdn
mm'
      mm' (t ,n ) dtdn
2
m  0 m ' 0
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
If dT is a multiple of
the wavelength, we
can apply Parserval
relation for 2D DFT.
41
Energy of the MIMO ambiguity function
 MIMO ambiguity function:
M 1 M 1
 (t ,n , f , f ' )     mm' (t ,n )e j 4pd
T
/  ( fm f 'm ')
m  0 m ' 0
 Energy of the ambiguity function
     (t ,n , f , f ' )
 
M 1 M 1
 
m  0 m ' 0
M 1 M 1
2
dtdndfdf '
2
j 4pdT /  ( fm f 'm ')
(
t
,
n
)
e
dfdf ' dtdn
mm'
      mm' (t ,n ) dtdn
2
m  0 m ' 0
M 1 M 1
  1  M
m  0 m ' 0
2
Cross ambiguity
function has
constant energy
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
42
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,
2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

2
dT is the spacing
between the
transmitting antennas
If dT is a multiple of the
wavelength, the energy of
the MIMO ambiguity function
is a constant.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
43
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,
2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

2
dT is the spacing
between the
transmitting antennas
 Recall that the signal component satisfies,
  (0,0, f , f ) df
 M , f
– Because energy and the signal component are both constants,
we can only spread the energy to minimize the peak.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
44
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,
2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

2
dT is the spacing
between the
transmitting antennas
 In general, the energy satisfies,
2dT /  
2
2dT /   M 2 

2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

(2dT /  )2
(2dT /  )2
2
2
In general, the energy of the
MIMO ambiguity function is
confined in a certain range.
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
45
Energy of the MIMO ambiguity function
 If dT is a multiple of the wavelength,
2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

2
dT is the spacing
between the
transmitting antennas
 In general, the energy satisfies,
2dT /  
2
2dT /   M 2 

2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'

M

(2dT /  )2
(2dT /  )2
2
2
 In general, the signal component satisfies,
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
46
Symmetry properties
 Symmetry of the cross ambiguity function
 mm' (t ,n )   m'm (t ,n )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
47
Symmetry properties
 Symmetry of the cross ambiguity function
 mm' (t ,n )   m'm (t ,n )
 Symmetry of the MIMO ambiguity function
 (t ,n , f , f ' )   (t ,n , f ' , f )
It suffices to show only half of
the ambiguity function (t>0).
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
48
Linear frequency modulation (LFM)
 Linear frequency modulation
u
LFM
m
(t )  um (t )e
jpkt 2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
49
Linear frequency modulation (LFM)
 Linear frequency modulation
u
LFM
m
(t )  um (t )e
jpkt 2
 Cross ambiguity function

LFM
mm '
(t ,n )   mm' (t ,n  kt )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
50
Linear frequency modulation (LFM)
 Linear frequency modulation
u
LFM
m
(t )  um (t )e
jpkt 2
 Cross ambiguity function

LFM
mm '
(t ,n )   mm' (t ,n  kt )
 MIMO ambiguity function

LFM
Shear off
(t ,n , f , f ' )   (t ,n  kt , f , f ' )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
51
Linear frequency modulation (LFM)
 (t ,n , f , f ' )
n
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
52
Linear frequency modulation (LFM)
 (t ,n  kt , f , f ' )
 (t ,n , f , f ' )
LFM
n
Shear off
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
n
t
53
Linear frequency modulation (LFM)
 (t ,n  kt , f , f ' )
 (t ,n , f , f ' )
LFM
Shear off
n
The range
resolution is
improved by
LFM.
t
n
n
t
n
t
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
t
54
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2d T / 
55
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
– Energy
2d T / 
2
2dT /   M 2 
2dT /   M 2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'


(2dT /  )2
(2dT /  )2
2
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
2
56
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
– Energy
– Symmetry
2d T / 
2
2dT /   M 2 
2dT /   M 2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'


(2dT /  )2
(2dT /  )2
2
2
 (t ,n , f , f ' )   (t ,n , f ' , f )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
57
Conclusion
 Properties of the MIMO ambiguity function
– Signal component
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
– Energy
– Symmetry
– LFM
2d T / 
2
2dT /   M 2 
2dT /   M 2

(
t
,
n
,
f
,
f
'
)
d
t
d
n
dfdf
'


(2dT /  )2
(2dT /  )2
2
2
 (t ,n , f , f ' )   (t ,n , f ' , f )
 LFM (t ,n , f , f ' )   (t ,n  kt , f , f ' )
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
58
Thank You!
Q&A
Any questions?
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
59
Properties of the signal component
If the waveforms are
orthogonal, the
signal component
will be a constant
for all angle.
For orthogonal waveforms,
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008

um (t )um* ' (t )dt   mm'
  (0,0, f , f )  M , f
60
Properties of the signal component
For general waveforms,

 If
dT

2
u m (t ) dt  1
is integer,
   (0,0, f , f ) df  M , f
The integration of
the signal
component is a
constant if dT is a
multiple of the
wavelength.
dT is the spacing
between the
transmitting antennas
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
61
Properties of the signal component
In general, the
integration of the signal
component is confined
in a certain range.
 For the general case,
2dT /   M   (0,0, f , f ) df  2dT /   M

2d T / 
2d T / 
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
dT is the spacing
between the
transmitting antennas
62
MIMO Radar
TX
RX
…
SIMO
Radar
…
MF
MF
MF
u (t)
RX
TX
MIMO
Radar
…
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MF
…
…
MF
…
…
MF
…
…
63
MIMO Radar
 Advantages
– Better spatial resolution [Bliss & Forsythe 03]
– Flexible transmit beampattern design [Fuhrmann & San Antonio 04]
– Improved parameter identifiability [Li et al. 07]
RX
TX
MIMO
Radar
…
u0(t) u1(t)
…
uM-1(t)
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2008
MF
…
…
MF
…
…
MF
…
…
64