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Beamforming Issues in Modern MIMO Radars with Doppler. Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology. Outline. Review of the MIMO radar Spatial resolution. [D. W. Bliss and K. W. Forsythe, 03] MIMO space-time adaptive processing (STAP) Problem formulation

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beamforming issues in modern mimo radars with doppler

Beamforming Issues in Modern MIMO Radars with Doppler

Chun-Yang Chen and P. P. Vaidyanathan

California Institute of Technology

DSP Group, EE, Caltech, Pasadena CA

outline
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

DSP Group, EE, Caltech, Pasadena CA

simo radar
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.

DSP Group, EE, Caltech, Pasadena CA

mimo radar
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 theMorthogonal waveforms.

Overall number of signals:

NM

Transmitter emits orthogonal waveforms.

DSP Group, EE, Caltech, Pasadena CA

mimo radar 2
MIMO Radar (2)

ej2p(ft-x/l)

ej2p(ft-x/l)

q

q

dR

dT=NdR

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

DSP Group, EE, Caltech, Pasadena CA

mimo radar 3
MIMO Radar (3)
  • The clutter resolution is the same as a receiving array with NM physical array elements.
  • A degree-of-freedom NMcan be created using only N+M physical array elements.

[D. W. Bliss and K. W. Forsythe, 03]

+

=

Virtual array: NM elements

Transmitter : M elements

Receiver: N elements

DSP Group, EE, Caltech, Pasadena CA

space time adaptive processing stap
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

DSP Group, EE, Caltech, Pasadena CA

formulation of mimo stap
Formulation of MIMO STAP

target

target

ej2p(ft-x/l)

ej2p(ft-x/l)

vt

vt

q

q

vsinq

vsinq

dR

dT=NdR

MF

MF

f0(t)

f2(t)

f1(t)

Transmitter : M elements

Receiver: N elements

target

noise

NML

clutter

jammer

NML x NML

DSP Group, EE, Caltech, Pasadena CA

clutter in mimo radar
Clutter in MIMO Radar

size: NML

size: NMLxNML

DSP Group, EE, Caltech, Pasadena CA

clutter rank in mimo stap integer case
Clutter Rank in MIMO STAP: Integer Case

Integer case:gandb are bothintegers.

The set {n+gm+bl} has at most N+g(M-1)+b(L-1) distinct elements.

Theorem: If g and b are integers,

This result can be viewed as the MIMO extension of Brennan’s rule.

DSP Group, EE, Caltech, Pasadena CA

clutter signals and truncated sinusoidal functions
Clutter Signals and Truncated Sinusoidal Functions

ci is NML vector which consists of

It can be viewed as a non-uniformly sampled version oftruncated sinusoidal signals.

X

2W

The “time-and-band limited” signals can be approximated by linear combination of prolate spheroidal wave functions.

DSP Group, EE, Caltech, Pasadena CA

prolate spheroidal wave function pswf
Prolate Spheroidal Wave Function (PSWF)
  • Prolate spheroidal wave functions (PSWF) are the solutions to the integral equation [van tree, 2001].

0

X

-W

W

in [0,X]

Frequency window

Time window

  • Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962].
  • The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.

DSP Group, EE, Caltech, Pasadena CA

pswf representation for clutter signals
PSWF Representation for Clutter Signals

The “time-and-band limited” signals can be approximated by 2WX+1 PSWF basis elements.

clutter rank in integer case

DSP Group, EE, Caltech, Pasadena CA

pswf representation for clutter signals 2
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.

DSP Group, EE, Caltech, Pasadena CA

slide15

truncated sinusoidal

Linear

combination

PSWF

Non-uniformly

sample

Non-uniformly

sample

Linear

combination

i-th clutter signal

Sampled PSWF

Stack

Stack

Linear

combination

Sampled PSWF

i-th clutter signal

Clutter covariance matrix

U:NML x rc

A:rc x rc

DSP Group, EE, Caltech, Pasadena CA

numerical example
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

N=10

M=5

L=16

g=N=10

b=1.5

NML=800

N+g(M-1)+b(L-1)=72.5

Eigenvalues

k

DSP Group, EE, Caltech, Pasadena CA

conclusion
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.

DSP Group, EE, Caltech, Pasadena CA

further and future work
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

DSP Group, EE, Caltech, Pasadena CA

references
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.

DSP Group, EE, Caltech, Pasadena CA

thank you

Thank you

DSP Group, EE, Caltech, Pasadena CA

comparison of the clutter rank in mimo and simo radar
Comparison of the Clutter Rank in MIMO and SIMO Radar

>

>

<

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

DSP Group, EE, Caltech, Pasadena CA

formulation of mimo stap 2
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

f0(t)

f2(t)

f1(t)

Transmitter : M elements

Receiver: N elements

T: Radar pulse period

DSP Group, EE, Caltech, Pasadena CA

fully adaptive stap for mimo radar
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.

DSP Group, EE, Caltech, Pasadena CA

clutter subspace in mimo stap non integer case
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.

DSP Group, EE, Caltech, Pasadena CA

extension to arbitrary array
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.

DSP Group, EE, Caltech, Pasadena CA

review of mimo radar diversity approach

ej2p(ft-x/l)

dR

MF

MF

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:

DSP Group, EE, Caltech, Pasadena CA

prolate spheroidal wave function pswf 2
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 2WX+1 eigenvalues are significant [D. Slepian, 1962].
  • The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.

DSP Group, EE, Caltech, Pasadena CA

review of mimo radar degree of freedom approach

ej2p(ft-x/l)

q

dT=NdR

f0(t)

f2(t)

f1(t)

Transmitter : M elements

Review of MIMO Radar: Degree-of-Freedom Approach
  • The clutter resolution is the same as a receiving array with NM physical array elements.
  • A degree-of-freedom NMcan be created using only N+M physical array elements.

ej2p(ft-x/l)

q

+

=

dR

MF

MF

Receiver: N elements

q

Virtual array: NM elements

[D. W. Bliss and K. W. Forsythe, 03]

DSP Group, EE, Caltech, Pasadena CA