signal processing algorithms for mimo radar chun-yang chen and p. p. vaidyanathan california...

Signal Processing Algorithms for MIMO Radar

Chun-Yang Chen and P. P. Vaidyanathan

California Institute of TechnologyElectrical 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.

1MIMO Radar and Beamforming

MIMO Radar

The radar systems which emits orthogonal (or noncoherent) waveforms in each transmitting antennas are called MIMO radar.

MIMO radar SIMO radar (Traditional)

MIMO Radar

MIMO radar

SIMO radar (Traditional)

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

Radartarget

Received Signal

Matched filter outputthreshold

R=ct/2

Detection

Ranging

Radar was an acronym for Radio Detection and Ranging.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.Beampattern is the antenna gain as a function of angle of arrival.

target

Plane wave-front

sinc(sin

target

Plane wave-front

target

Fourier transform

Plane wave-front

sinc(sin

Antenna Array

I/Q Down-Convert and ADC

By linearly combining the output of a group of antennas, we can

control the beampattern digitally.

Antenna Array

)2( xkTftje

Plane wave-front

sindsin)1( dN

Antenna Array

)2( xkTftje

Plane wave-front

sindsin)1( dN

Discrete timeFourier transform

Antenna Array (2)

jnnewE

Advantages of antenna array:

target

Beampattern can be steered digitally.Beampattern can be steered digitally.

Antenna Array (2)

jnnewE

target

interferences

Beampattern can be adapted to the interferences.Beampattern can be adapted to the interferences.

Antenna Array (2)

jnnewE

target

interferences

Beampattern can be adapted to the interferences.Beampattern can be adapted to the interferences.

The signal processing techniques to control the beampattern

is called beamforming.

The signal processing techniques to control the beampattern

is called beamforming.

Phased Array Beamforming

)2( xkTftje

Plane wave-front

sindsin)1( dN

The response of a desired angle of arrival q can be maximized

by adjust wi.

)2( xkTftje

Plane wave-front

sindsin)1( dN

TNdjdj

)1(sin2

1 subject to

by adjust wi.

)2( xkTftje

Plane wave-front

sindsin)1( dN

TNdjdj

)1(sin2

1 subject to

by adjust wi.

Adaptive Beamforming

The beamformer can be further designed to maximize the SINR

using second order statistics of received signals.

1 subject to

The SINR can be maximized by minimizing the total variance

while maintaining unity signal response.

1 subject to

mins1 Rw [Capon 1969]

MVDR beamformer

(Minimum Variance Distortionless Response)

An Example of Adaptive Beamforming

0 10 20 30 40 50 60 70 80 90-60

Parameters Noise: 0dB Signal: 10dB, 43 degree Jammer1: 40dB, 30 degree Jammer2: 20dB, 75 degree

SINR Phased array: -20.13dB Adaptive: 9.70dB

However, the MVDR beamformer is very sensitive to target DoA (Direction of Arrival) mismatch.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.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 Jammer1: 40dB, 30 degree Jammer2: 20dB, 75 degree

0 10 20 30 40 50 60 70 80 90-60

Transmit Beamforming

By weighting the input of a group of antennas, we can also control

the transmit beampattern digitally.

transmitted waveform

)2( xkTftje

Plane wave-front

sindsin)1( dN

)2( xkTftje

Plane wave-front

sindsin)1( dN

Discrete timeFourier transform

SIMO Radar (Traditional)

Transmitter: M antenna elements

ej2(ft-x/)

w2 w1 w0

Transmitter emits

coherent waveforms.

(transmit beamforming)

Transmitter emits

coherent waveforms.

(transmit beamforming)

Receiver: N antenna elements

ej2(ft-x/)

Number of received signals: N

MIMO Radar

ej2(ft-x/)

Transmitter emits

orthogonal waveforms.

(No transmit beamforming)

Transmitter emits

orthogonal waveforms.

(No transmit beamforming)

ej2(ft-x/)

MF MF…

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

MIMO Radar – Virtual Array

Virtual array: NM elements

dT=NdR

ej2(ft-x/)

MF MF…

MIMO Radar – Virtual Array (2)

Receiver: N elements

Virtual array: NM 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

w2 w1 w0 dT=NdR

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.

2Space-Time Adaptive Processing

Space-Time Adaptive Processing

vvsini

airborne radar

jammertarget

i-th clutter

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.

The goal in STAP is to detect the moving target on the ground and estimate its

position and velocity.

Doppler Processing

Radartarget

ftje 2

Doppler Processing

2 Doppler

effect:

Radartarget

ftje 2

tffj de )(2

Radartarget

The phenomenon can be used to estimate velocity.

Adaptive Temporal Processing

tfj de 2

w*0 w*

1 w*L-1

The same idea in adaptive beamforming can be applied

in Doppler processing.

Adaptive Temporal Processing

tfj de 2

w*0 w*

1 w*L-1

1 subject to

Separable Space-Time Processing

w*0 w*

1 w*L-1

Filtered outthe unwanted angles

Filtered outthe unwanted frequencies

When the Doppler frequencies

and looking-directions are independent,

the spatial and temporal filtering

can be implemented separately.

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

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

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.However, the beampattern is not always separable.

Space-time beampattern is the antenna gain as a function of angle of arrival and Doppler frequency.

vvsini

airborne radar

jammertarget

i-th clutter

vvsini

airborne radar

jammertarget

i-th clutter

iThe clutter Doppler frequencies

depend on angles. So, the problem is non-separable in

space-time.

The clutter Doppler frequencies depend on angles. So, the

problem is non-separable in space-time.

Example of a Non-Separable Beampattern

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.5 0 0.5

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.

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 tapsNon separable: NL taps

Jointly processDoppler frequencies and angles

Independently process Doppler frequencies and angles

Angle processing

Doppler processingSpace-time

processing

L: # of radar pulses N: # of antennas

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

1 subject to

NL signals

1 subject to

Optimal Space-Time Adaptive Processing

3An Efficient Space-Time Adaptive Processing Algorithm for MIMO Radar

MIMO Radar STAPSTAP MIMO Radar

NL signals

MIMOSTAP

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

NML signals

MVDR (Capon) beamformer:

MIMO Radar STAP (2)

1),( subject to

NML signals

MVDR (Capon) beamformer:

),(1DfsRw

Very good spatial resolutionVery good spatial resolution

Pros ConsCons

High complexityHigh complexity

Slow convergenceSlow convergence

NMLxNML

The Proposed Method

),(1DfsRw

NMLJc IRRR 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

vvsinqi

airborne radar

jammer

target

i-th clutter

Target Noise Jammer Clutter

Formulation of the Clutter Signals

Matchedfilters

Pulse 2

Pulse 1

Pulse 0

Matchedfilters

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 antennam-th matched filter outputl-th radar pulse

ilmn eeec1

sin22sin2sin2

Formulation of the Clutter Signals

Matchedfilters

Pulse 2

Pulse 1

Pulse 0

Matchedfilters

Clutter points

n-th antennam-th matched filter outputl-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

ilmn eeec1

sin22sin2sin2

, sinRs i i

5.05.0

)1()1(1

otherwise,0

0),2exp();( ,

Xxxffxc is

ilmn eeec1

sin22sin2sin2

, sinRs i i

5.05.0

)1()1(1

otherwise,0

0),2exp();( ,

Xxxffxc is

ilmn eeec1

sin22sin2sin2

, sinRs i i

ilmnisi fxc

1, );(

5.05.0

)1()1(1

otherwise,0

0),2exp();( ,

Xxxffxc is

ilmn eeec1

sin22sin2sin2

, sinRs i i

ilmnisi fxc

1, );(

Trick: We can view the three dimensional signal as non-uniformly sampled one

dimensional signal.

Simplification of the Clutter Expression (2)

ilmn eeec1

sin22sin2sin2

ilmnisi fxc

1, );(

-2 0 2 4 6 8 10 12-1.5

Re{c(x;fs,i)} Re{c(n+m+l;fs,i)}

otherwise,0

0),2exp();( ,

Xxxffxc is

-50 0 50 100 150-1

-1 -0.5 0 0.5 10

“Time-and-Band” Limited Signals

otherwise,0

0),2exp();( ,

Xxxffxc is

5.05.0

)1()1(1

[-0.5 0.5]

Timedomain

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. is called PSWF.

Prolate Spheroidal Wave Functions (PSWF)

kk )())-sinc((x)(0 ( )k x

in [0,X]

Frequency window

-0.5 0.5

Time window

X0( )k x ( )k xk

is called PSWF. is called PSWF.

Prolate Spheroidal Wave Functions (PSWF)

( ; )X

s i i kk

kk )())-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

X0( )k x ( )k xk

( )k x

kkkiis xfxc

0,, )();(

ilmnisi fxc

1,lm,n, );(c

[D. Slepian, 62]

Concise Representation of the Clutter Signals

kkkiis xfxc

0,, )();(

ilmnisi fxc

1,lm,n, );(c

ii lmn

kkk lmn

)( )1()1(1 LMNX

[D. Slepian, 62]

kkkiis xfxc

0,, )();(

ilmnisi fxc

1,lm,n, );(c

ii lmn

kkk lmn

)( )1()1(1 LMNX

Hc ΨΨRR

Ψ )( lmnk consists ofc Ψξ

NML X+1

[D. Slepian, 62]

)1()1(1,,1,0 LMNk

Concise Representation of the Clutter Signals (2)

Hc ΨΨRR Ψ )( lmnk consists of

NMLN+(M-1)+(L-1)

)1()1(1,,1,0 LMNk

can be obtained by sampling from . The PSWF can be computed off-line can be obtained by sampling from . The PSWF can be computed off-lineΨ k

NMLN+(M-1)+(L-1)

)1()1(1,,1,0 LMNk

can be obtained by sampling from . The PSWF can be computed off-line can be obtained by sampling from . The PSWF can be computed off-lineΨ k

NMLN+(M-1)+(L-1)

The NMLxNML clutter covariance matrix has only N+(M-1)+(L-1) significant eigenvalues. This is the MIMO extension of Brennan’s rule (1994).

[Chun-Yang Chen and P. P. Vaidyanathan, IEEE Trans SP, to appear]

Jammer Covariance Matrix

Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0

Matchedfilters

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210

jnml: jammer signals

Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0

Jammer signals in different pulses are independent.

Matchedfilters

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210

Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0

Jammer signals in different matched filter outputs are independent.

Jammer signals in different matched filter outputs are independent.Matched

filtersMatched

filters

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210

Matchedfilters

jammer

Pulse 2

Pulse 1

Pulse 0

Jammer signals in different matched filter outputs are independent.

Matchedfilters

Block diagonalBlock diagonal

j002 j012 j102

j001 j011 j101

j000 j010 j100

j112 j202 j212

j111 j201 j211

j110 j200 j210

The Proposed Methodlow rank

block diagonalNMLJc IRRR 2 H

v ΨR Ψ R

1111111 )( vH

vv RΨΨRΨRΨRRR

By Matrix Inversion Lemma

v ΨR Ψ R

1111111 )( vH

vv RΨΨRΨRΨRRR

The proposed method

– Compute by sampling the prolate spheroidal wave functions.

The proposed method

– Instead of estimating R, we estimate Rv and Rx. The matrix Rv can

be estimated using a small number of clutter free samples.k

v ΨR Ψ R

1111111 )( vH

vv RΨΨRΨRΨRRR

v ΨR Ψ R

The proposed method

– 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.

1111111 )( vH

vv RΨΨRΨRΨRRR

The Proposed Method – Advantages

:block diagonal

:small size

Inversions are easy to compute

1111111 )( vH

vv RΨΨRΨRΨRRR

:block diagonal

:small size

1111111 )( vH

vv RΨΨRΨRΨRRR

Low complexityLow complexity

:block diagonal

:small size

Fewer parameters need to be estimated

1111111 )( vH

vv RΨΨRΨRΨRRR

:block diagonal

:small size

Fewer parameters need to be estimated

1111111 )( vH

vv RΨΨRΨRΨRRR

FastconvergenceFastconvergence

The Proposed Method – Complexity

1111111 )( vH

vv RΨΨRΨRΨRRR

Complexity:1 3: (( ( 1) ( 1)) )O N M L R

)(: 31 NOvR

1111111 )( vH

vv RΨΨRΨRΨRRR

)(: 31 NOvR

Direct method

The proposed method

),(1DfsR )( 333 LMNO

1R )( 333 LMNO

1111111 )( vH

vv RΨΨRΨRΨRRR

)(: 31 NOvR

Direct method

The proposed method

),(1DfsR )( 333 LMNO )))1()1((( 3 LMNO

1R )))1()1((( 222 LMNLMNO )( 333 LMNO

The Zero-Forcing Method

Typically we can assume that the clutter is very strong and all eigenvalues of Rx are very large.

1111111 )( vH

vv RΨΨRΨRΨRRR

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 Hv v v v

R R R Ψ Ψ R Ψ Ψ R

Zero-forcing method

– The entire clutter space is nulled out without estimation

1111111 )( vH

vv RΨΨRΨRΨRRR

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=16CNR=50dB2 jammers, JNR=40dB

K: number of samplesKv: 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

Normalized Doppler frequency

Parameters:N=10, M=5, L=16, CNR=50dB2 jammers, JNR=40dB

Target: (0,0.25)

Simulations – Beampattern

Proposed ZF MethodProposed ZF Method

-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.

4Research Topics

Research Topics

Robust Beamforming Algorithm against DoA Mismatch [2]

An Efficient STAPAlgorithm for

MIMO Radar [3]

Precoded V-BLAST Transceiver for MIMO

Communication [1]

Precoded V-BLAST Transceiver for MIMO

Communication [1]

Beamforming techniques for Radar systems Beamforming techniques for Radar systems

An Efficient STAPAlgorithm 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.

Radartarget

Received Signal

Matched filter outputTime

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.

Range resolution

Angleresolution

Doppler

Q&AThank You!

Any questions?

Parameters:N=10, M=5, L=16, CNR=50dB2 jammers, JNR=40dB

-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 MethodProposed ZF Method

Space-Time Beam Pattern

Normalized Spatial Freq.

Normalized Doppler

Velocity mismatchVelocity mismatch

Normalized Doppler

Velocity misalignmentVelocity misalignment

Normalized Doppler

Internal clutter motion (ICM)Internal clutter motion (ICM)

MIMO vs. SIMO

Simulations

Clutter Power in PSWF Vector Basis

0 50 100 150 200

Basis element index

Proposed subspace methodEigen 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

Normalized Doppler frequency

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=16CNR=50dB2 jammers, JNR=40dB

K: number of samplesKv: number of clutter free samples collected in passive mode

MIMO Radar – Virtual Array (2)

Receiver: N elementsVirtual array: NM 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

kkkiis xfxc

0,, )();(

ilmnisi fxc

1,lm,n, );(c

ii lmn

kkk lmn

)( )1()1(1 LMNX

[D. Slepian, 62]

Efficient Representation for the Clutter

kkkiis xfxc

0,, )();(

ilmnisi fxc

1,lm,n, );(c

ii lmn

kkk lmn

)( )1()1(1 LMNX

Hc ΨΨRR

Ψ )( lmnk consists ofc Ψξ

NML X+1

[D. Slepian, 62]

iisi lmnfj

1, ))(2exp(

ilmn eeec1

sin22sin2sin2

, sinRs i i

ilmnisi fxc

1, );(

otherwise,0

0),2exp();( ,

Xxxffxc is

5.05.0

)1()1(1

-2 0 2 4 6 8 10 12-1.5

Re{c(x;fs,i)}Re{c(n+m+l;fs,i)}

Receiver Transmitter Doppler

Time window Frequency window

X -W W0 in [0,X]

( )k x ( )k k x

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