Beamforming Issues in Modern MIMO Radars with Doppler

1
Beamforming Issues in Modern MIMO Radars with
Doppler

• Chun-Yang Chen and P. P. Vaidyanathan
• California Institute of Technology

2
Outline

• Review of the MIMO radar
• Spatial resolution. D. W. Bliss and K. W.
  Forsythe, 03
• MIMO space-time adaptive processing (STAP)
• Problem formulation
• Clutter rank in MIMO STAP
• Clutter subspace in MIMO STAP
• Numerical example

3
SIMO Radar
Transmitter M elements
Receiver N elements
ej2p(ft-x/l)
ej2p(ft-x/l)
dR
dT
w1f(t)
w0f(t)
w2f(t)
Number of received signals N

• Transmitter emits coherent waveforms.

4
MIMO Radar
Transmitter M elements
Receiver N elements
ej2p(ft-x/l)
ej2p(ft-x/l)
dR
dT

MF
MF
f0(t)
f2(t)
f1(t)

Matched filters extract the M orthogonal
waveforms. Overall number of signals NM

• Transmitter emits orthogonal waveforms.

5
MIMO Radar (2)
ej2p(ft-x/l)
ej2p(ft-x/l)
q
q
dR
dTNdR

MF
MF
f0(t)
f2(t)
f1(t)

Transmitter M elements
Receiver N elements
q
The spacing dT is chosen as NdR, such that the
virtual array is uniformly spaced.
Virtual array NM elements
6
MIMO Radar (3)
D. W. Bliss and K. W. Forsythe, 03


Virtual array NM elements
Transmitter M elements
Receiver N elements

• The clutter resolution is the same as a receiving
  array with NM physical array elements.
• A degree-of-freedom NM can be created using only
  NM physical array elements.

7
Space-Time Adaptive Processing (STAP)
The adaptive techniques for processing the data
from airborne antenna arrays are called
space-time adaptive processing (STAP).
airborne radar
v
vsinqi
qi

• The clutter Doppler frequencies depend on looking
  directions.
• The problem is non-separable in space-time.

jammer
target
vt
i-th clutter
8
Formulation of MIMO STAP
target
target
ej2p(ft-x/l)
ej2p(ft-x/l)
vt
vt
q
q
vsinq
vsinq
dR
dTNdR

MF
MF
f2(t)
f1(t)
f0(t)

Transmitter M elements
Receiver N elements
target
noise
NML
clutter
jammer
NML x NML
9
Clutter in MIMO Radar
size NML
size NMLxNML
10
Clutter Rank in MIMO STAP Integer Case
Integer case g and b are both integers.
The set ngmbl has at most Ng(M-1)b(L-1)
distinct elements.
Theorem If g and b are integers,
This result can be viewed as the MIMO extension
of Brennans rule.
11
Clutter Signals and Truncated Sinusoidal Functions
ci is NML vector which consists of
It can be viewed as a non-uniformly sampled
version of truncated sinusoidal signals.
X
2W
The time-and-band limited signals can be
approximated by linear combination of prolate
spheroidal wave functions.
12
Prolate Spheroidal Wave Function (PSWF)

• Prolate spheroidal wave functions (PSWF) are the
  solutions to the integral equation van tree,
  2001.

X
-W
W
0
in 0,X
Frequency window
Time window

• Only the first 2WX1 eigenvalues are significant
  D. Slepian, 1962.
• The time-and-band limited signals can be well
  approximated by the linear combination of the
  first 2WX1 basis elements.

13
PSWF Representation for Clutter Signals
The time-and-band limited signals can be
approximated by 2WX1 PSWF basis elements.
clutter rank in integer case
14
PSWF Representation for Clutter Signals (2)
non-uniformly sample
U NML x rc A rc x rc

• The PSWF yk(x) can be computed off-line
• The vector uk can be obtained by sampling the
  PSWF.

15
truncated sinusoidal
Linear combination
PSWF
Non-uniformly sample
Non-uniformly sample
i-th clutter signal
Linear combination
Sampled PSWF
Stack
Stack
Linear combination
Sampled PSWF
i-th clutter signal
Clutter covariance matrix
U NML x rc A rc x rc
16
Numerical Example
qkH Rcqk

• The figure shows the clutter power in the
  orthonormalized basis elements.
• The proposed method captures almost all the
  clutter power.
• Parameters

Proposed method
N10 M5 L16 gN10 b1.5 NML800 Ng(M-1)b(L-1)
72.5
Eigenvalues
k
17
Conclusion

• The clutter subspace in MIMO radar is explored.
• Clutter rank for integer/non-integer g and b.
• Data-independent estimation of the clutter
  subspace.
• Advantages of the proposed subspace estimation
  method.
• It is data-independent.
• It is accurate.
• It can be computed off-line.

18
Further and Future Work

• Further work
• The STAP method applying the subspace estimation
  is submitted to ICASSP 07.
• Future work
• In practice, some effects such as internal
  clutter motion (ICM) will change the clutter
  space.
• Estimating the clutter subspace by using a
  combination of both the geometry and the data
  will be explored in the future.

New method
19
References

• 1 D. W. Bliss and K. W. Forsythe,
  Multiple-input multiple-output (MIMO) radar and
  imaging degrees of freedom and resolution,
  Proc. 37th IEEE Asilomar Conf. on Signals,
  Systems, and Computers, pp. 54-59, Nov. 2003.
• 2 D. Slepian, and H. O. Pollak, "Prolate
  Spheroidal Wave Functions, Fourier Analysis and
  Uncertainty-III the dimension of the space of
  essentially time-and-band-limited signals," Bell
  Syst. Tech. J., pp. 1295-1336, July 1962.
• 3 D. J. Rabideau and P. Parker, "Ubiquitous
  MIMO Multifunction Digital Array Radar," Proc.
  37th IEEE Asilomar Conf. on Signals, Systems, and
  Computers, pp. 1057-1064, Nov. 2003.
• 4 N. A. Goodman and J.M. Stiles, "On Clutter
  Rank Observed by Arbitrary Arrays," accepted to
  IEEE Trans. on Signal Processing.

20
Thank you
21
Comparison of the Clutter Rank in MIMO and SIMO
Radar
MIMO SIMO
Clutter rank Ng(M-1)b(L-1) Nb(L-1)
Total dimension NML NL
Ratio (gN)
gt
gt
lt

• The clutter rank is a smaller portion of the
  total dimension.
• The MIMO radar receiver can null out the clutter
  subspace without affecting the SINR too much.

22
Formulation of MIMO STAP (2)
target
target
ej2p(ft-x/l)
ej2p(ft-x/l)
vt
vt
q
q
vsinq
vsinq
dR
dT

MF
MF
f2(t)
f1(t)
f0(t)

Transmitter M elements
Receiver N elements
T Radar pulse period
23
Fully Adaptive STAP for MIMO Radar
Solution

• Difficulty The size of Ry is NML which is often
  large.
• The convergence of the fully adaptive STAP is
  slow.
• The complexity is high.

24
Clutter Subspace in MIMO STAP Non-integer Case

• Non-integer case g and b not integers.
• Basis need for representation of clutter steering
  vector ci.
• Data independent basis is preferred.
• Less computation
• Faster convergence of STAP
• We study the use of prolate spheroidal wave
  function (PSWF) for this.

25
Extension to Arbitrary Array

• This result can be extended to arbitrary array.

XR,n is the location of the n-th receiving
antenna. XT,m is the location of the m-th
transmitting antenna. ui is the location of the
i-th clutter. v is the speed of the radar station.
26
Review of MIMO radar Diversity approach

• If the transmitting antennas are far enough, the
  received signals of each orthogonal waveforms
  becomes independent. E. Fishler et al. 04
• This diversity can be used to improve target
  detection.

Receiver
27
Prolate Spheroidal Wave Function (PSWF) (2)

• By the maximum principle, this basis concentrates
  most of its energy on the band -W, W while
  maintaining the orthogonality.

• Only the first 2WX1 eigenvalues are significant
  D. Slepian, 1962.
• The time-and-band limited signals can be well
  approximated by the linear combination of the
  first 2WX1 basis elements.

28
Review of MIMO Radar Degree-of-Freedom Approach
ej2p(ft-x/l)
q


dR

MF
MF

Receiver N elements

• The clutter resolution is the same as a receiving
  array with NM physical array elements.
• A degree-of-freedom NM can be created using only
  NM physical array elements.

q
Virtual array NM elements
D. W. Bliss and K. W. Forsythe, 03