Signal Processing Algorithms for MIMO Radar

Signal Processing Algorithms for
MIMO Radar
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Electrical Engineering/DSP Lab
Candidacy

Chun-Yang Chen, Caltech DSP Lab | Candidacy
Outline
 Review of the background
– MIMO radar
– Space-Time Adaptive Processing (STAP)
 The proposed MIMO-STAP method
– Formulation of the MIMO-STAP
– Prolate spheroidal representation of the clutter signals
– Deriving the proposed method
– Simulations
 Conclusion and future work.

MIMO Radar
f2(t)
f1(t)
f0(t)
The radar systems which emits orthogonal (or noncoherent)
waveforms in each transmitting antennas are called MIMO radar.
w2f(t)
w1f(t)
w0f(t)
MIMO radar SIMO radar (Traditional)

MIMO Radar
MIMO radar
f2(t)
f1(t)
f0(t)
SIMO radar (Traditional)
w2f(t)
w1f(t)
w0f(t)
The radar systems which emits orthogonal (or noncoherent)
waveforms in each transmitting antennas are called MIMO radar.
[D. J. Rabideau and P. Parker, 03]
[D. Bliss and K. Forsythe, 03]
[E. Fishler et al. 04]
[F. C. Robey, 04]
[D. R. Fuhrmann and G. S. Antonio, 05]

Radar Systems
t
Radar
target
R
Received Signal
Matched filter output
threshold
R=ct/2
Detection
Ranging
Time
Radar was an acronym for Radio Detection and Ranging.

Beampattern of Antennas
target
Beampattern is the antenna gain as a function of angle of arrival.

d/2
-d/2


2
/
2
/
sin
2
0
)
(
d
d
y
j
dy
e
A
E





sin
y
target
Plane
wave-front

)
(
E

)
sin
sinc(
sin
2
2
/
2
/
0






 d
dy
e
A y
d
d
j





d/2
-d/2


2
/
2
/
sin
2
0
)
(
d
d
y
j
dy
e
A
E





sin
y
target
Plane
wave-front

)
(
E

d/2
-d/2


2
/
2
/
sin
2
0
)
(
d
d
y
j
dy
e
A
E





sin
y
target
Fourier transform
Plane
wave-front

)
(
E
)
sin
sinc(
sin
2
2
/
2
/
0






 d
dy
e
A y
d
d
j






Antenna Array
N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
+
…
y
wH
By linearly combining the output
of a group of antennas, we can
control the beampattern digitally.

Antenna Array
)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
+
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 
y
wH









sin
2
1
0
*
1
0
sin
2
*
)
(
d
M
n
jn
n
M
n
n
d
j
n
e
w
e
w
E












Antenna Array
)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
+
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 
y
wH









sin
2
1
0
*
1
0
sin
2
*
)
(
d
M
n
jn
n
M
n
n
d
j
n
e
w
e
w
E











Discrete time
Fourier transform

Antenna Array (2)





1
0
*
)
(
M
n
jn
ne
w
E 

 Advantages of antenna array:
…
target
Beampattern can be steered digitally.

Antenna Array (2)





1
0
*
)
(
M
n
jn
ne
w
E 

…
target
interferences
Beampattern can be adapted to the interferences.

Antenna Array (2)





1
0
*
)
(
M
n
jn
ne
w
E 

…
target
interferences
Beampattern can be adapted to the interferences.
The signal processing techniques to control the beampattern
is called beamforming.

Phased Array Beamforming
)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
+
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 
y
wH
The response of a desired angle
of arrival q can be maximized
by adjust wi.

)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
+
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 
y
wH
T
N
d
j
d
j
e
e 








 )
1
(
sin
2
sin
2
1







s
1
subject to
max
2

w
s
w
w
H
by adjust wi.

)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
+
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 
y
wH
T
N
d
j
d
j
e
e 








 )
1
(
sin
2
sin
2
1







s
1
subject to
max
2

w
s
w
w
H
s
w 

by adjust wi.

Adaptive Beamforming








 2
2
v
w
s
w
v
s
y
H
H
E
SINR
The beamformer can be further designed to maximize the SINR
using second order statistics of received signals.









 2
2
v
w
s
w
v
s
y
H
H
E
SINR
 
H
H
H
E yy
R
s
w
Rw
w
w

1
subject to
min
The SINR can be maximized by minimizing the total variance
while maintaining unity signal response.









 2
2
v
w
s
w
v
s
y
H
H
E
SINR
 
H
H
H
E yy
R
s
w
Rw
w
w

1
subject to
min
s
1

 R
w  [Capon 1969]
MVDR beamformer
(Minimum Variance Distortionless Response)
The SINR can be maximized by minimizing the total variance
while maintaining unity signal response.

An Example of Adaptive Beamforming
0 10 20 30 40 50 60 70 80 90
-60
-50
-40
-30
-20
-10
0
10
20
Angle
Beam
pattern
(dB)
 Parameters
 Noise: 0dB
 Signal: 10dB, 43 degree
 Jammer1: 40dB, 30 degree
 SINR
 Phased array: -20.13dB
 Adaptive: 9.70dB
However, the MVDR beamformer is very sensitive to target
DoA (Direction of Arrival) mismatch.
Adaptive beamforming can be very effective when there exists
strong interferences.

Beamforming under Direction-of-Arrival Mismatch
[2] Chun-Yang Chen and P. P. Vaidyanathan, “Quadratically Constrained Beamforming Robust
Against Direction-of-Arrival Mismatch,” IEEE Trans. on Signal Processing, July 2007.
 SINR
 Matched DoA: 9.70dB
 Mismatched DoA: -8.80dB
 Parameters
 Noise: 0dB
 Signal: 10dB, 43 degree
0 10 20 30 40 50 60 70 80 90
-60
-50
-40
-30
-20
-10
0
10
20
Angle
Beam
pattern
(dB)

Transmit Beamforming
N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
…
By weighting the input of a group
of antennas, we can also control
the transmit beampattern
digitally.
transmitted waveform

)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 









sin
2
1
0
*
1
0
sin
2
*
)
(
d
M
n
jn
n
M
n
n
d
j
n
e
w
e
w
E











digitally.

)
2
( x
kT
ft
j
e 

N-1
I/Q Down-
Convert
and ADC
w*
N-1
1
I/Q Down-
Convert
and ADC
w*
1
0
I/Q Down-
Convert
and ADC
w*
0
…
Plane
wave-front 

sin
d

sin
)
1
( d
N 









sin
2
1
0
*
1
0
sin
2
*
)
(
d
M
n
jn
n
M
n
n
d
j
n
e
w
e
w
E











Discrete time
Fourier transform
digitally.

SIMO Radar (Traditional)
Transmitter: M antenna elements
dT
ej2(ft-x/)
w2f(t) w1f(t) w0f(t)
Transmitter emits
coherent waveforms.
(transmit beamforming)
Receiver: N antenna elements
dR
ej2(ft-x/)
Number of received signals:
N

MIMO Radar
dT
ej2(ft-x/)
f2(t) f1(t) f0(t)
Transmitter emits
orthogonal waveforms.
(No transmit beamforming)
dR
ej2(ft-x/)
MF MF
…
…
Matched filters extract
the M orthogonal waveforms.
Overall number of signals:
NM

MIMO Radar – Virtual Array
Virtual array: NM elements

dT=NdR
ej2(ft-x/)
f2(t) f1(t) f0(t)

dR
ej2(ft-x/)
MF MF
…
…


MIMO Radar – Virtual Array (2)
Receiver: N elements
Transmitter: M elements
+ =
[D. W. Bliss and K. W. Forsythe, 03]
The spatial resolution for clutter is the same as a receiving array
with NM physical array elements.
NM degrees of freedom can be created using only N+M physical
array elements.
However, a processing gain of M is lost because of the broad
transmitting beam.

MIMO Transmitter vs. SIMO Transmitter
dT
w2f(t) w1f(t) w0f(t)
dT=NdR
f2(t) f1(t) f0(t)
…
In the application of scanning or imaging, global illumination is required. In
this case the SIMO system needs to steer the transmit beam. This cancels
the processing gain obtained by the focused beam in SIMO system.

Space-Time Adaptive Processing

v
vsini
airborne
radar
jammer
target
i-th clutter
vt
i
The adaptive techniques for processing the data from airborne
antenna arrays are called space-time adaptive processing (STAP).
The goal in STAP is to detect
the moving target on the
ground and estimate its
position and velocity.

Doppler Processing
Radar
target
v
ft
j
e 
2

Doppler Processing
f
c
v
fd
2

 Doppler
effect:
Radar
target
v
ft
j
e 
2
t
f
f
j d
e )
(
2 

Radar
target
v
The phenomenon can be used to estimate
velocity.

Adaptive Temporal Processing
t
f
j d
e 
2
I/Q
Down-
Convert
and ADC
w*
0 w*
1 w*
L-1
T T
…
+
y
wH
The same idea in adaptive
beamforming can be applied
in Doppler processing.

Adaptive Temporal Processing
t
f
j d
e 
2
I/Q
Down-
Convert
and ADC
w*
0 w*
1 w*
L-1
T T
…
+
y
wH
s
1

 R
w 
The same idea in adaptive
beamforming can be applied
in Doppler processing.
 
H
H
H
E yy
R
s
w
Rw
w
w

1
subject to
min

Separable Space-Time Processing
N-1
I/Q
Down-
Convert
and ADC
w*
N-1
1
I/Q
Down-
Convert
and ADC
w*
1
0
I/Q
Down-
Convert
and ADC
+
…
w*
0 w*
1 w*
L-1
T T
…
+
w*
0
Filtered out
the unwanted angles
Filtered out
the unwanted frequencies
When the Doppler frequencies
and looking-directions are independent,
the spatial and temporal filtering
can be implemented separately.

Example of Separable Space-Time Processing
Normalized Spatial Frequency
Normalized
Doppler
Frequency
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-70
-60
-50
-40
-30
-20
-10
 Parameters
 Noise: 0dB
 Signal: 10dB, (0.11, 0.15)
 Jammer1: 40dB, (-0.22, x )
 Jammer2: 20dB, (0.33, x )
 Clutter: 40dB, (x , 0 )
However, the beampattern is not always separable.
Space-time beampattern is the antenna gain as a function of
angle of arrival and Doppler frequency.

v
vsini
airborne
radar
jammer
target
i-th clutter
vt
i
f
c
v
f i
Di

sin
2


v
vsini
airborne
radar
jammer
target
i-th clutter
vt
i
The clutter Doppler frequencies
depend on angles. So, the
problem is non-separable in
space-time.
f
c
v
f i
Di

sin
2


Example of a Non-Separable Beampattern
Normalized
Doppler
Frequency
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Normalized
Doppler
Frequency
-0.5 0 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-70
-60
-50
-40
-30
-20
-10
Non-Separable Separable
In a stationary radar,
clutter Doppler frequency is
zero for all angle of arrival.
In airborne radar, clutter
Doppler frequency is proportional to
the angle of arrival. Consequently,
the beampattern becomes non-separable.

Space-Time Adaptive Processing (2)
Separable: N+L taps
Non separable: NL taps
Jointly process
Doppler frequencies and angles
Independently process
Doppler frequencies and angles
Angle
processing
Doppler
processing
Space-time
processing
L: # of radar pulses N: # of antennas
L

NL signals
As in beamforming and Doppler
processing, the maximum SINR can be
obtained by minimizing the
total variance while maintaining
unity signal response.
Optimal Space-Time Adaptive Processing

NL signals
 
H
H
H
E yy
R
s
w
Rw
w
w

1
subject to
min

s
1

 R
w 
NL signals
 
H
H
H
E yy
R
s
w
Rw
w
w

1
subject to
min

An Efficient Space-Time Adaptive Processing Algorithm
for MIMO Radar

MIMO Radar STAP
STAP MIMO Radar
NL signals
MIMO
STAP
M waveforms
NML signals
N: # of receiving antennas
M: # of transmitting antennas
L: # of pulses
[D. Bliss and
K. Forsythe 03]
+
NM signals

MIMO Radar STAP (2)
1
)
,
(
subject to
min

D
H
H
f

s
w
Rw
w
w
NML signals
MVDR (Capon) beamformer:

MIMO Radar STAP (2)
1
)
,
(
subject to
min

D
H
H
f

s
w
Rw
w
w
NML signals
MVDR (Capon) beamformer:
)
,
(
1
D
f

s
R
w 


Very good spatial resolution
Pros Cons
High complexity
Slow convergence
NMLxNML

The Proposed Method
)
,
(
1
D
f

s
R
w 

NML
J
c I
R
R
R 2




We first observe each of the matrices Rc and RJ has
some special structures.
clutter jammer noise
We show how to exploit the structures of these
matrices to compute R
-1
more accurately and efficiently.
[Chun-Yang Chen and
P. P. Vaidyanathan,
ICASSP 07]

The MIMO STAP Signals
 Received signal: yn,m,l
 n: receiving antenna index
 m: transmitting antenna index
 l: pulse trains index
 The signals contain four
components:
 Target
 Noise
 Jammer
 Clutter
v
vsinqi
airborne
radar
jammer
vt
target
i
i-th clutter
Target Noise Jammer Clutter

Formulation of the Clutter Signals
Matched
filters
Pulse 2
Pulse 1
Pulse 0
Matched
filters
Matched
filters
c002 c012 c102
c001 c011 c101
c000 c010 c100
c112 c202 c212
c111 c201 c211
c110 c200 c210
cnml: clutter signals
…
Clutter
points
n-th antenna
m-th matched filter output
l-th radar pulse








c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,










Formulation of the Clutter Signals
Matched
filters
Pulse 2
Pulse 1
Pulse 0
Matched
filters
Matched
filters
…
Clutter
points
n-th antenna
m-th matched filter output
l-th radar pulse
 Nc: # of clutter points
 ri: i-th clutter signal amplitude
 Receiving antenna
 Transmitting antenna
 Doppler effect
c002 c012 c102
c001 c011 c101
c000 c010 c100
c112 c202 c212
c111 c201 c211
c110 c200 c210
cnml: clutter signals

Simplification of the Clutter Expression







c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,










R
T
d
d


R
d
vT
2


, sin
R
s i i
d
f 



5
.
0
5
.
0
)
1
(
)
1
(
1
, 








i
s
f
L
M
N
X 



 


otherwise
,
0
0
),
2
exp(
)
;
( ,
,
X
x
x
f
f
x
c i
s
i
s








c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,










R
T
d
d


R
d
vT
2


, sin
R
s i i
d
f 



5
.
0
5
.
0
)
1
(
)
1
(
1
, 








i
s
f
L
M
N
X 



 


otherwise
,
0
0
),
2
exp(
)
;
( ,
,
X
x
x
f
f
x
c i
s
i
s








c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,










R
T
d
d


R
d
vT
2


, sin
R
s i i
d
f 









c
N
i
l
m
n
i
s
i f
x
c
1
, )
;
( 



5
.
0
5
.
0
)
1
(
)
1
(
1
, 








i
s
f
L
M
N
X 



 


otherwise
,
0
0
),
2
exp(
)
;
( ,
,
X
x
x
f
f
x
c i
s
i
s








c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,










R
T
d
d


R
d
vT
2


, sin
R
s i i
d
f 









c
N
i
l
m
n
i
s
i f
x
c
1
, )
;
( 


Trick: We can view the three dimensional signal as
non-uniformly sampled one dimensional signal.

Simplification of the Clutter Expression (2)







c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,

















c
N
i
l
m
n
i
s
i f
x
c
1
, )
;
( 


-2 0 2 4 6 8 10 12
-1.5
-1
-0.5
0
0.5
1
1.5
x
Re{c(x;fs,i)}
Re{c(n+m+l;fs,i)}


 


otherwise
,
0
0
),
2
exp(
)
;
( ,
,
X
x
x
f
f
x
c i
s
i
s


-50 0 50 100 150
-1
0
1
-1 -0.5 0 0.5 1
0
20
40
60
80
100
“Time-and-Band” Limited Signals


 


otherwise
,
0
0
),
2
exp(
)
;
( ,
,
X
x
x
f
f
x
c i
s
i
s

5
.
0
5
.
0
)
1
(
)
1
(
1
, 








i
s
f
L
M
N
X 

[0 X]
[-0.5 0.5]
Time
domain
Freq.
domain
The signals are well-localized in
a time-frequency region.
To concisely represent these
signals, we can use a basis which
concentrates most of its energy
in this time-frequency region.

is called PSWF.
Prolate Spheroidal Wave Functions
(PSWF)





 d
x k
X
k
k )
(
))
-
sinc((x
)
(
0

 ( )
k x

in [0,X]
Frequency window
-0.5 0.5
Time window
X
0
( )
k x
 ( )
k x

k


is called PSWF.
Prolate Spheroidal Wave Functions
(PSWF)
, ,
0
( ; )
X
s i i k
k
c x f 

 





 d
x k
X
k
k )
(
))
-
sinc((x
)
(
0


[D. Slepian, 62]
( )
k x

in [0,X]
Only X+1 basis functions are required to well represent the
“time-and-band limited” signal
Frequency window
-0.5 0.5
Time window
X
0
( )
k x
 ( )
k x

k

( )
k x





X
k
k
k
i
i
s x
f
x
c
0
,
, )
(
)
;
( 








c
N
i
l
m
n
i
s
i f
x
c
1
,
l
m,
n, )
;
(
c 


[D. Slepian, 62]
Concise Representation of the Clutter Signals




X
k
k
k
i
i
s x
f
x
c
0
,
, )
(
)
;
( 








c
N
i
l
m
n
i
s
i f
x
c
1
,
l
m,
n, )
;
(
c 

 
 





X
k
k
k
i
N
i
i l
m
n
c
0
,
1
)
( 











X
k
k
k l
m
n
0
)
( 


 )
1
(
)
1
(
1 




 L
M
N
X 

[D. Slepian, 62]




X
k
k
k
i
i
s x
f
x
c
0
,
, )
(
)
;
( 








c
N
i
l
m
n
i
s
i f
x
c
1
,
l
m,
n, )
;
(
c 

 
 





X
k
k
k
i
N
i
i l
m
n
c
0
,
1
)
( 











X
k
k
k l
m
n
0
)
( 


 )
1
(
)
1
(
1 




 L
M
N
X 

H
c Ψ
ΨR
R 

Ψ )
( l
m
n
k 


 


consists of

c Ψξ
NML X+1
[D. Slepian, 62]

)
1
(
)
1
(
1
,
,
1
,
0 




 L
M
N
k 


Concise Representation of the Clutter Signals (2)
H
c Ψ
ΨR
R 
 Ψ )
( l
m
n
k 


 


consists of
NML
N+(M-1)+(L-1)

)
1
(
)
1
(
1
,
,
1
,
0 




 L
M
N
k 


H
c Ψ
ΨR
R 
 Ψ )
( l
m
n
k 


 


consists of
can be obtained by sampling from . The PSWF
can be computed off-line
Ψ k

NML
N+(M-1)+(L-1)
k

Concise Representation of the Clutter
Signals (2)

)
1
(
)
1
(
1
,
,
1
,
0 




 L
M
N
k 


H
c Ψ
ΨR
R 
 Ψ )
( l
m
n
k 


 


consists of
can be obtained by sampling from . The PSWF
can be computed off-line
Ψ k

NML
N+(M-1)+(L-1)
k

The NMLxNML clutter covariance matrix has
only N+(M-1)+(L-1) significant eigenvalues. This is
the MIMO extension of Brennan’s rule (1994).
c
R
Concise Representation of the Clutter Signals (2)
[Chun-Yang Chen and P. P. Vaidyanathan, IEEE Trans SP, to appear]

Jammer Covariance Matrix
Matched
filters
jammer
Pulse 2
Pulse 1
Pulse 0
Matched
filters
Matched
filters
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210
jnml: jammer signals

Matched
filters
jammer
Pulse 2
Pulse 1
Pulse 0
Jammer signals in different pulses are
independent.
Matched
filters
Matched
filters
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210

Matched
filters
jammer
Pulse 2
Pulse 1
Pulse 0
independent.
Jammer signals in different matched
filter outputs are independent.
Matched
filters
Matched
filters
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210

Matched
filters
jammer
Pulse 2
Pulse 1
Pulse 0
independent.
Jammer signals in different matched
filter outputs are independent.













Js
Js
Js
J
R
0
0
0
R
0
0
0
R
R







Matched
filters
Matched
filters
Block diagonal
j002 j012 j102
j001 j011 j101
j000 j010 j100
j112 j202 j212
j111 j201 j211
j110 j200 j210

The Proposed Method
low rank
block diagonal
NML
J
c I
R
R
R 2



 H
v

 
ΨR Ψ R

The Proposed Method
low rank
block diagonal
NML
J
c I
R
R
R 2



 H
v

 
ΨR Ψ R
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
By Matrix Inversion Lemma

The Proposed Method
low rank
block diagonal
NML
J
c I
R
R
R 2



 H
v

 
ΨR Ψ R
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
 The proposed method
– Compute Y by sampling the prolate spheroidal wave functions.

– Instead of estimating R, we estimate Rv and Rx. The matrix Rv can
be estimated using a small number of clutter free samples.k
The Proposed Method
low rank
block diagonal
NML
J
c I
R
R
R 2



 H
v

 
ΨR Ψ R
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 

The Proposed Method
low rank
block diagonal
NML
J
c I
R
R
R 2



 H
v

 
ΨR Ψ R
– Instead of estimating R, we estimate Rv and Rx. The matrix Rv can
be estimated using a small number of clutter free samples.
– Use the above equation to compute R-1
.
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 

The Proposed Method – Advantages
v
R

R
:block diagonal
:small size
Inversions are
easy to compute
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 

v
R

R
:block diagonal
:small size
Inversions are
easy to compute
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
Low
complexity

v
R

R
:block diagonal
:small size
Inversions are
easy to compute
Fewer parameters
need to be estimated
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
Low
complexity

v
R

R
:block diagonal
:small size
Inversions are
easy to compute
Fewer parameters
need to be estimated
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
Low
complexity
Fast
convergence

The Proposed Method – Complexity
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
Complexity:
1 3
: (( ( 1) ( 1)) )
O N M L
  

   
R
)
(
: 3
1
N
O
v

R

1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
Complexity:
1 3
: (( ( 1) ( 1)) )
O N M L
  

   
R
)
(
: 3
1
N
O
v

R
Direct method The proposed method
)
,
(
1
D
f

s
R
)
( 3
3
3
L
M
N
O
1

R )
( 3
3
3
L
M
N
O

1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
Complexity:
1 3
: (( ( 1) ( 1)) )
O N M L
  

   
R
)
(
: 3
1
N
O
v

R
Direct method The proposed method
)
,
(
1
D
f

s
R
)
( 3
3
3
L
M
N
O )
))
1
(
)
1
(
(( 3



 L
M
N
O 

1

R )
))
1
(
)
1
(
(( 2
2
2
L
M
N
L
M
N
O 


 

)
( 3
3
3
L
M
N
O

The Zero-Forcing Method
 Typically we can assume that the clutter is very
strong and all eigenvalues of Rx are very large.
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
1
0


 
R

The Zero-Forcing Method
 Typically we can assume that the clutter is very
strong and all eigenvalues of Rx are very large.
1 1 1 1 1 1
( )
H H
v v v v
     
 
R R R Ψ Ψ R Ψ Ψ R
 Zero-forcing method
– The entire clutter space is nulled out without estimation
1
1
1
1
1
1
1
)
( 








 v
H
v
H
v
v R
Ψ
Ψ
R
Ψ
R
Ψ
R
R
R 
1
0


 
R

Proposed method K=300,Kv=20
Simulations – SINR
MVDR known R (unrealizable)
Proposed ZF method Kv=20
Sample matrix inversion K=1000
Diagonal loading K=300
Principal component K=300
SINR of a target at angle zero and
Doppler frequencies [-0.5, 0.5]
Parameters:
N=10, M=5, L=16
CNR=50dB
2 jammers, JNR=40dB
K: number of samples
Kv: number of clutter free samples
collected in passive mode
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-16
-14
-12
-10
-8
-6
-4
-2
0
Normalized Doppler frequency
SINR
(dB)

Parameters:
N=10, M=5, L=16, CNR=50dB
2 jammers, JNR=40dB
Target: (0,0.25)
Simulations – Beampattern
Proposed ZF Method
Normalized
Doppler
Frequency
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Target
Jammer
Clutter
Jammer

Conclusion and Future Work
 Conclusion
– The clutter subspace is derived using the geometry of the problem.
(data independent)
– A new STAP method for MIMO radar is developed.
– The new method is both efficient and accurate.
 Future work
– This method is entirely based on the ideal model.
– Find algorithms which are robust against clutter subspace mismatch.
– Develop clutter subspace estimation methods using a combination of
both the geometry and the received data.

Research Topics

Research Topics
Robust Beamforming
Algorithm against
DoA Mismatch [2]
An Efficient STAP
Algorithm for
MIMO Radar [3]
Precoded V-BLAST
Transceiver for MIMO
Communication [1]
Beamforming techniques for Radar systems
An Efficient STAP
Algorithm for
MIMO Radar [3]

Publications
[1] Chun-Yang Chen and P. P. Vaidyanathan, “Precoded FIR and Redundant V-
BLAST Systems for Frequency-Selective MIMO Channels,” IEEE Trans. on
Signal Processing, July, 2007.
[2] Chun-Yang Chen and P. P. Vaidyanathan, “Quadratically Constrained
Beamforming Robust Against Direction-of-Arrival Mismatch,” IEEE Trans. on
Signal Processing, Aug., 2007.
[3] Chun-Yang Chen and P. P. Vaidyanathan, “MIMO Radar Space-Time
Adaptive Processing Using Prolate Spheroidal Wave Functions,” accepted to
IEEE Trans. on Signal Processing.
[4] Chun-Yang Chen and P. P. Vaidyanathan, “MIMO Radar Space-Time
Adaptive Processing and Signal Design,” invited chapter in MIMO Radar
Signal Processing, J. Li and P. Stoica, Wiley, to be published.
Journal Papers
Book Chapter

Publications
Chun-Yang Chen, Caltech DSP Lab | ICASSP 2007 student paper contest
[5] Chun-Yang Chen and P. P. Vaidyanathan, “A Subspace Method for MIMO Radar
Space-Time Processing,” IEEE International Conference on Acoustics, Speech, and
Signal Processing Honolulu, Hi, April 2007.
[6] Chun-Yang Chen and P. P. Vaidyanathan, “Beamforming issues in modern MIMO
Radars with Doppler,” Proc. 40th Asilomar Conference on Signals, Systems, and
Computers, Pacific Grove, CA, Nov. 2006.
[7] Chun-Yang Chen and P. P. Vaidyanathan, “A Novel Beamformer Robust to Steering
Vector Mismatch,” Proc. 40th Asilomar Conference on Signals, Systems, and Computers,
Pacific Grove, CA, Nov. 2006.
[8] Chun-Yang Chen and P. P. Vaidyanathan, “Precoded V-BLAST for ISI MIMO
channels,” IEEE International Symposium on Circuit and System Kos, Greece, May 2006,
[9] Chun-Yang Chen and P. P. Vaidyanathan, “IIR Ultra-Wideband Pulse Shaper Design,”
Proc. 39th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA,
Nov. 2005.
Conference Papers

Future Topic – Waveform Design in MIMO Radar
 In SIMO radar, chirp waveform is often used in the transmitter to
increase the range resolution. This technique is called pulse
compression.
Radar
target
R
Received Signal
Matched filter output
Time
Range
resolution

Future Topic – Waveform Design in MIMO Radar
 In MIMO radar, multiple orthogonal waveforms are
transmitted.
 These waveforms affects not only the range resolution but also
angle and Doppler resolution.
 It is not clear how to design multiple waveforms which provide
good range, angle and Doppler resolution.
f2(t)
f1(t)
f0(t)
Range resolution
Angle
resolution
Doppler

Q&A
Thank You!
Any questions?

Parameters:
N=10, M=5, L=16, CNR=50dB
2 jammers, JNR=40dB
Normalized
Doppler
Frequency
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Jammer 1
Clutter
Target
Jammer 2
Target: (0,0.25)
Simulations – Beampattern
Proposed ZF Method

Space-Time Beam Pattern
Normalized Spatial Freq.
Normalized
Doppler
Freq.

Normalized
Doppler
Freq.
Velocity mismatch

Normalized
Doppler
Freq.
Velocity misalignment

Normalized
Doppler
Freq.
Internal clutter motion (ICM)

MIMO vs. SIMO

Simulations

Clutter Power in PSWF Vector Basis
0 50 100 150 200
-200
-150
-100
-50
0
50
100
Basis element index
Clutter
power
(dB)
Proposed subspace method
Eigen decomposition
N+(M-1)+(L-1)

Proposed method K=300,Kv=20
Simulations
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-16
-14
-12
-10
-8
-6
-4
-2
0
Normalized Doppler frequency
SINR
(dB)
MVDR perfect R
Proposed ZF method Kv=20
Sample matrix inversion K=2000
Diagonal loading K=300
Principal component K=300
SINR of a target at angle zero and
Doppler frequencies [-0.5, 0.5]
Parameters:
N=10, M=5, L=16
CNR=50dB
2 jammers, JNR=40dB
K: number of samples
Kv: number of clutter free samples
collected in passive mode

MIMO Radar – Virtual Array (2)
Receiver: N elements
Transmitter: M elements
+ =
[D. W. Bliss and K. W. Forsythe, 03]
The spatial resolution for clutter is the same as a receiving array
with NM physical array elements.
NM degrees of freedom can be created using only N+M physical
array elements.
A processing gain of M is lost because of the broad transmitting
beam.

Efficient Representation for the Clutter



X
k
k
k
i
i
s x
f
x
c
0
,
, )
(
)
;
( 








c
N
i
l
m
n
i
s
i f
x
c
1
,
l
m,
n, )
;
(
c 

 
 





X
k
k
k
i
N
i
i l
m
n
c
0
,
1
)
( 











X
k
k
k l
m
n
0
)
( 


 )
1
(
)
1
(
1 




 L
M
N
X 

[D. Slepian, 62]

Efficient Representation for the Clutter



X
k
k
k
i
i
s x
f
x
c
0
,
, )
(
)
;
( 








c
N
i
l
m
n
i
s
i f
x
c
1
,
l
m,
n, )
;
(
c 

 
 





X
k
k
k
i
N
i
i l
m
n
c
0
,
1
)
( 











X
k
k
k l
m
n
0
)
( 


 )
1
(
)
1
(
1 




 L
M
N
X 

H
c Ψ
ΨR
R 

Ψ )
( l
m
n
k 


 


consists of

c Ψξ
NML X+1
[D. Slepian, 62]







c
N
i
i
s
i l
m
n
f
j
1
, ))
(
2
exp( 










c i
i
T
i
R
N
i
v
T
l
j
d
m
j
d
n
j
i
l
m
n e
e
e
c
1
sin
2
2
sin
2
sin
2
,
,










R
T
d
d


R
d
vT
2


, sin
R
s i i
d
f 









c
N
i
l
m
n
i
s
i f
x
c
1
, )
;
( 




 


otherwise
,
0
0
),
2
exp(
)
;
( ,
,
X
x
x
f
f
x
c i
s
i
s

5
.
0
5
.
0
)
1
(
)
1
(
1
, 








i
s
f
L
M
N
X 

-2 0 2 4 6 8 10 12
-1.5
-1
-0.5
0
0.5
1
1.5
x
Re{c(x;fs,i)}
Re{c(n+m+l;fs,i)}
Receiver Transmitter Doppler

T
T
…
T
T
T
…
T
T
T
…
T
…
…
T
T
T
…

Time window Frequency window
X -W W
0 in [0,X]
( )
k x
 ( )
k k x
 

Signal Processing Algorithms for MIMO Radar

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