3rd international workshop on mathematical issues...

SPAC Lab, ECE

Signal Processing for MIMO and

Passive Radar

Hongbin Li Signal Processing and Communication (SPAC) Laboratory

Department of Electrical and Computer Engineering

Stevens Institute of Technology

Hoboken, NJ, USA

July 9, 2014

3rd International Workshop on Mathematical Issues in

Information Sciences (MIIS’2014)

SPAC Lab, ECE

Acknowledgement

• Collaborators

– Guolong Cui (UESTC)

– Braham Himed (AFRL)

– Jun Liu (Stevens)

– Muralidhar Rangaswamy (AFRL)

– Pu Wang (Schlumberger-Doll Research Center)

SPAC Lab, ECE

Outline

• Part I: Moving target detection with distributed MIMO radars

– Non-homogeneous clutter

– Subspace based approach

– Parametric approach

• Part II: Waveform design for MIMO radar with constant modulus

and similarity constraints

– Design with practical constraints

– Two sequential optimization algorithms

• Part III: Passive detection with noisy reference

– Effect of noise in the reference signal

– Four different detectors

SPAC Lab, ECE

Part I:

Moving Target Detection with Distributed

MIMO Radars

SPAC Lab, ECE

MIMO Radars

• MIMO radar vs. Phased Array

– high spatial resolution

– more degrees of freedom

– better parameter identifiability

– flexible transmit beampattern

– increased spatial diversity

– detection diversity gain

Distributed MIMO Radar

with widely separated antennas

SPAC Lab, ECE

Backscattering

• Radar target RCS is angle

selective

• Conventional radars experience

target fluctuation of 5-25 dB

• Distributed MIMO radar exploits

the angular spread of the target

backscatter in a variety of ways to

improve radar performance

Detection/estimation

performance improvement

through diversity gain

• Clutter response has similar

angular selectivity, causing non-

homogeneous clutter

Target Backscattering vs.

azimuth angle [Skolnik’03]

Angle-Selective Backscatterring

SPAC Lab, ECE

Signal Model

• Systems Setup

– M transmit antennas (Tx)

– N receive antennas (Rx)

– K pulses in one coherent processing interval (CPI)

– Orthogonal probing waveforms from Tx

– M matched filters at each Rx

SPAC Lab, ECE

Target Signal

• Target Signal:

– Doppler frequency (for a given target velocity)

– Doppler Steering Vector

– Amplitude

Complex-valued

Unknown but deterministic

Different for different Tx-Rx pairs

8

SPAC Lab, ECE

Noise and Clutter

• Noise wm,n2 CK × 1 : zero-mean, spatially/temporally white

• Clutter cm,n2 CK × 1

– No clutter [Fishler et al.’05]

– Homogeneous clutter model: shares the same covariance

matrix [He-Lehmann-Blum-Haimovich’10]

– Subspace-based clutter model [Wang-Li-Himed’11]

Clutter is spanned by l Fourier bases

Different m,n for different TX-RX ) non-homogenous clutter power

Cutter covariance matrix structure is still homogeneous

Clutters fall within the column space of H

Buildings Uninterested slow-moving

vehicles

Plants (wind effects)

angle-selective backscattering non-homogeneous clutter

SPAC Lab, ECE

The MTD Problem

• Moving target detection (MTD) is concerned with the following

composite hypothesis testing problem

• Target-free training signals drawn from neighboring resolution cells

may be available. Generally, they are non-homogeneous across

resolution cells

SPAC Lab, ECE

Covariance Matrix Based Solutions

• The sample covariance matrix (SCM) based detector was introduced for MTD in homogeneous clutter [He-Lehmann-Blum-Haimovich’10]

• SCM require Kt ¸ 2K homogeneous training signals for each Tx-Rx pair

• A robust detector based on a compound Gaussian model [Chong-Pascal-Ovarlez-Lesturgie’10]

• Covariance is obtained by solving a fixed point equation (FPE)

SPAC Lab, ECE

• GLRT based on the subspace clutter model [Wang-Li-Himed’11]

• Test variable has central/non-central beta distribution [Wang-Li-Himed’11]

• The SGLRT

– Achieves constant false alarm rate (CFAR)

– Needs no training signal

– Works if the clutter can be expressed using a few Fourier bases

Subspace Based GLRT

SPAC Lab, ECE

Non-Homogeneous Clutter Modeling

A Parametric Approach

• Autoregressive models are capable of capturing the correlation of

radar clutter with a variety of power spectrums [Wang-Li-Himed’13]

• Clutter speckle is characterized by AR coefficients

• Clutter texture is characterized by the driving noise variance

• Different AR processes to model the disturbance observed at different

TX-RX pair ) truly non-homogeneous

• Parameter estimates can be obtained from test signal ) no need for

training signals

SPAC Lab, ECE

Parametric MTD

• Recall the moving target detection (MTD) problem

SPAC Lab, ECE

• The MIMO-PGLRT can be obtained by

• The second equality is due to statistical independence among different

Tx-Rx pairs

with i = 0, 1 denoting H0 and H1, respectively

MIMO Parametric GLRT

SPAC Lab, ECE

• The simplified MIMO-PGLRT test statistic is [Wang-Li-Himed’13]

Parametric GLRT for MIMO Radar

• Local detector adaptively projects test signal into two different subspaces

– Orthogonal complement of regression matrix Ym,n

– Orthogonal complement of the target-suppressed Ym,n, using the

highlighted projection matrix

• Energy of projected signals are computed, compared, and integrated

over MN pairs

• PGLRT is an adaptive subspace detector, notably different from the

previous fixed subspace detector SGLRT

SPAC Lab, ECE

• Asymptotic distribution of the MIMO-PGLRT test statistic is

where the non-centrality parameter is given by

• denotes the temporally whitened steering

vector

• Test statistic under H0 is independent of the disturbance

parameters ) asymptotically achieves CFAR

Asymptotic Performance

SPAC Lab, ECE

CRB of Target Velocity

• CRB provides a lower bound on all unbiased estimate

• CRB is also useful for sensor placement/selection

• An general expression for the CRB is

– Geometry related terms

– Fisher information (FI) related term

Highlights:

Geoometry-related terms (cmn and smn ) are known in advance

Need to compute the Fisher information-related term ψmn

Both exact and asymptotic expressions for ψmn are obtained,

resulting in exact and asymptotic CRB

SPAC Lab, ECE


• The exact CRB is obtained by plugging the exact fisher

information-related term ψmn into the general CRB expression

– temporally whitened steering vector

– first derivatives of w.r.t.

– matrix consisting of first derivative of regressive steering vector

w.r.t.

– target amplitude

– driving noise variance for (m,n)-th AR model

– coefficient vector for (m,n)-th AR model

Observations:

Exact expression of the Fisher information-related term ψmn is a function of target amplitude, Doppler steering vector, AR coefficients, and AR driving noise variance

This expression is complicated and offers limited intuition

SPAC Lab, ECE


• The asymptotic CRB is obtained by plugging the asymptotic Fisher

information-related term ψmn into the general CRB expression

– is the power spectrum density of the (m,n)-th AR

interference at the (m,n)-th Doppler frequency

Observations:

Fisher information-related term ψmn is proportional to the SINR

|mn|2/mn(fmn), and inversely proportional to K3

The asymptotic CRB is simpler to compute

SPAC Lab, ECE

Simulation Results

• Scenario --- 2 X 2 configuration

– M = 2 Tx

– N = 2 Rx

– Normalized target velocity

• Signal-to-noise ratio

• Clutter-to-noise ratio (subspace model)

• Signal-to-interference-plus-noise ratio

SPAC Lab, ECE

Subspace GLRT

• Clutter is generated over

Fourier basis with non-

homogeneous power

• Two cases with

known/estimated target

velocity

• Results are averaged

over random target

velocity (direction) and

amplitude

• PA-AMF: phased-array

with adaptive matched

filter

• Two MIMO detectors:

GLRT and SCM

P. Wang, H. Li, and B. Himed, "Moving target detection using

distributed MIMO radar in clutter with non-homogeneous

power," IEEE-TSP, no.10, 2011

SPAC Lab, ECE

A General Clutter Model

• Clutter temporal correlation function

• Covariance matrix

• Clutter covariance matrix for (m,n)th TX-RX pair

Clutter power spectrum density

SPAC Lab, ECE

Parametric GLRT

• Clutter is from general

clutter model, non-

homogeneous across

different TX-RX pairs

• Two cases with

known/estimated target

velocity

• Two covariance matrix

based detectors are

included in comparison:

SCM and robust MIMO

• Results are averaged

over random target

velocity (direction) and

amplitude P. Wang, H. Li, and B. Himed, "A parametric moving target

detector for distributed MIMO radar in non-homogeneous

environment," IEEE-TSP, no.9, 2013

SPAC Lab, ECE

Target Velocity Estimation

SPAC Lab, ECE

Conclusions

• Examined the moving target detection (MTD) of distributed MIMO

radars in non-homogeneous clutter

• Proposed a subspace based GLRT

– Requires no training

– Can handle clutter with non-homogeneous power

– Works if the clutter can be expanded on a few Fourier bases (e.g.,

stationary platforms)

• Proposed a parametric GLRT

– No training needed

– Different AR models for different Tx-Rx transmit pairs

– Can handle fully non-homogeneous clutter

• Future directions

– Senor placement and optimization

– Non-orthogonal waveforms

– Moving platforms

SPAC Lab, ECE

Part 2:

Waveform Design for MIMO Radar with

Constant Modulus and Similarity Constraints

SPAC Lab, ECE

MIMO Waveform Design

• Transmitter only based designs: employ transmit beam pattern (BP) or

radar ambiguity function (AF) Fuhrmann and San Antonio, “Transmit beamforming for MIMO radar systems using signal cross-

correlation,” IEEE-AES, no.1, 2008

Stoica, Li, and Xie, “On probing signal design for MIMO radar,” IEEE-TSP, no.8, 2007

Wang, Wang, Liu, and Luo, “On the design of constant modulus probing signals for MIMO radar,”

IEEE-TSP, no.8, 2012

San Antonio, Fuhrmann, and Robey, “MIMO radar ambiguity functions,” IEEE-SP, no.1, 2007

Chen and Vaidyanathan, “MIMO radar ambiguity properties and optimization using

frequency–hopping waveforms,” IEEE-TSP, no.12, 2008

• Joint transmitter-receiver designs: based on mutual information or max

SINR criterion Yang and Blum, “MIMO radar waveform design based on mutual information and minimum mean –

square error estimation,” IEEE-AES, no.1, 2007

Leshem, Naparstek, and Nehorai, “Information theoretic adaptive radar waveform design for multiple

extended targets,” IEEE-TSP, no.1, 2007

Li, Xu, Stoica, Forsythe and Bliss, “Range compression and waveform optimization for MIMO radar: A

Cram´er–Rao bound based study,” IEEE-TSP, no.1, 2008

Friedlander, “Waveform design for MIMO radars,” IEEE-AES, no.3, 2007

Chen and Vaidyanathan, “MIMO radar waveform optimization with prior information of the extended

target and clutter,” IEEE TSP, no.9, 2009

SPAC Lab, ECE

This Work

• Two constraints are imposed for practical MIMO radar

waveform design

– Constant modulus (CM) constraint: power amplifiers

often work in saturated mode, prohibiting amplitude

modulation in radar waveforms

– Similarity constraint: allows the designed waveform to

share some good ambiguity properties of a known

waveform

• We present a framework for joint TX-RX based MIMO radar

waveform design

– In the presence of signal-dependent interferences (e.g.,

clutter)

– Taking into account CM and similarity constraints

SPAC Lab, ECE

• A co-located MIMO radar with NT TX antennas and NR RX antennas

• Let s(n) be the NT ×1 waveform vector and at(θ) the NT ×1 TX

steering vector. The signal seen at a location/angle θ is given by

• Let ar(θ) be the NR×1 RX steering vector. The received signal is given

by

• Stacking vectors x(n), s(n) and v(n) in time

• The problem of interest is to design the NR radar waveforms contained in the NRN £ 1 vector s

Signal Model

interference noise signal

SPAC Lab, ECE

Waveform Design Criterion

• Pass the received signal x through a linear FIR receive filter w

• Output signal-to-interference-and-noise ratio (SINR)

where SNR = E[|0|2]/(v)

2 and INRk = E[|k|2 ]/(v)

2

• Constant modulus (CM) constraint

SPAC Lab, ECE

Waveform Design Criterion

• Similarity Constraint: Let s0 be the reference waveform

where 𝜖 is a real parameter ruling the extent of the similarity

• The similarity constraint is equivalent to [De Maio et al.’09]:

• It is noted that 0 · · 2. For = 0, s is identical to s0 . For = 2, similarity

constraint vanishes and only the constant modulus constraint is in effect

• The constrained optimization problem (non-convex)

De Maio, Nicola, Huang, Luo, and Zhang, “Design of phase codes for radar performance

optimization with a similarity constraint,” IEEE-TSP, no. 2, 2009

SPAC Lab, ECE

Sequential Optimization

Algorithm #1

Proposed SOA1

Luo, Ma, So, Ye, and Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE-SPM, no. 3, 2010

can be solved iteratively by semidefinite

relaxation (SDR)

Optimize ρ(s,w)

with respect to

w in terms of s

Substitute w

back into ρ(s,w)

and simplify

Fix (s) from last iteration,

iteratively update s by SDR

MVDR problem

SPAC Lab, ECE


Algorithm #1

• Relaxation by dropping the similarity and rank-one constraints

• Randomization to impose the rank-one and similarity constraints

– Generate L random vectors

– Construct feasible solutions to original problem

– Select the best solution among the L randomizations

De Maio, Nicola, Huang, Luo, and Zhang, “Design of phase codes for radar performance

optimization with a similarity constraint,” IEEE-TSP, no. 2, 2009

SPAC Lab, ECE


Algorithm #2

Proposed SOA 2

Optimize w by

maximizing the

SINR for a given s

Optimize s by

maximizing the

SINR for a given w

MVDR problem

Repeat above 2 steps till

convergence Solvable by SDR

SPAC Lab, ECE


Algorithm #2

• Relaxation

• Let X = yZ. Via Charnes-Cooper transform, above fractional problem

reduces to SDP

• Suppose that (X*,y*) is a solution to the SDP. Then, Z* = X*/y* is a

solution to the fractional problem

• Randomization can be applied in a similar way as in SOA1 to generate

solutions with rank-one and similarity constraints

SPAC Lab, ECE

Simulation Results

Reference signal: orthogonal LFM

MIMO

antennas

NT 4

NR 8

Target 0

|0|2 20dB

Interferences

1 -50o

|1|2 30dB

2 -10o

|2|2 30dB

3 40o

|3|2 30dB

Noise v2 0dB

Parameters Set up

Beam pattern

optimal receive filter optimal waveform

SPAC Lab, ECE 0 50 10013

14

15

16

17

18

19

20

Iteration index

SIN

R (

dB

)

-50 0 50-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Angle ()

Magnitu

de (

dB

)

SOA1-EC

SOA2-EC

+ SOA1-CMC

SOA2-CMC

SOA1-EC

SOA2-EC

+ SOA1-CMC

SOA2-CMC

Simulation Results

• Consider waveforms obtained from the proposed algorithms with only

constant modulus constraint (i.e., SOA1-CMC and SOA2-CMC)

• SOA1-CMC and SOA1-EC increase with the iteration number, and both

are converge very fast (i.e., after 2-3 iterations). For SOA2-EC and

SOA2-CMC, the convergence speed is slower

• Optimal SINRs are nearly the same and, therefore, there is no

significant loss of SINR by imposing the constant modulus constraint

SPAC Lab, ECE

Simulation Results

• The similarity constraint incurs an SINR loss. For example, with = 1.5,

the loss for SOA1-CMSC and SOA2-CMSC is 1.3 dB and, respectively,

2.4 dB

• In general, the smaller the value of , the higher the SINR loss. The

beampatterns show that as the similarity constraint becomes stronger,

the interference null also becomes higher

0 50 10013

14

15

16

17

18

19

20

Iteration index

SIN

R (

dB

)

-50 0 50-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Angle ()

Magnitude (

dB

)

SOA1-CMSC, =2

SOA2-CMSC, =2

+ SOA1-CMSC, =1.5

SOA2-CMSC, =1.5

SOA1-CMSC, =2

SOA2-CMSC, =2

+ SOA1-CMSC, =1.5

SOA2-CMSC, =1.5

0 50 10013

14

15

16

17

18

19

20

Iteration index

SIN

R (

dB

)

SOA1-CMSC, =2

SOA2-CMSC, =2

+ SOA1-CMSC, =0.5

SOA2-CMSC, =0.5

-50 0 50-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Angle ()

Ma

gn

itu

de

(d

B)

SOA1-CMSC, =2

SOA2-CMSC, =2

+ SOA1-CMSC, =0.5

SOA2-CMSC, =0.5

SPAC Lab, ECE

Simulation Results

• As increases, the side lobe level becomes higher and higher. It is

important to recall from previous simulation results a larger generally

yields a higher output SINR. Hence, in practice, the choice of should

be made by an appropriate tradeoff between the range solution and

output SINR of the resulting waveform.

-500 -400 -300 -200 -100 0 100 200 300 400 500-90

-80

-70

-60

-50

-40

-30

-20

-10

0

IFFT bin index

Ma

gn

itu

de

(d

B)

LFM

SOA1-CMSC, =2

SOA1-CMSC, =1

SOA1-CMSC, =0.5

SOA1-CMSC, =0.1

-500 -400 -300 -200 -100 0 100 200 300 400 500-90

-80

-70

-60

-50

-40

-30

-20

-10

0

IFFT bin index

Magnitude (

dB

)

LFM

SOA2-CMSC, =2

SOA2-CMSC, =1

SOA2-CMSC, =0.5

SOA2-CMSC, =0.1

SPAC Lab, ECE

Conclusions

• Addressed the problem of MIMO radar waveform design in an

environment with signal-dependent interference plus noise

• Proposed two sequential optimization algorithms, named SOA1

and SOA2, by maximizing the receiver output SINR, accounting

for the constant modulus constraint as well as a similarity

constraint

• Numerical results indicate that

the constant envelope constraint leads to waveforms with little SINR

loss compared with those obtained without the constraint. This

clearly motivates the use of our constant modulus waveforms which

can be used with efficient nonlinear power amplifiers.

the larger the similarity parameter, the larger the output SINR, but

the poorer the pulse compression performance. This suggests a

suitable tradeoff between the target detection probability and the

range resolution should be considered in practice

SPAC Lab, ECE

Part 3:

Passive Detection with Noisy Reference

SPAC Lab, ECE

Passive Radar

Target

Reference

Channel

Surveillance

Channel

Passive Radar

Non-cooperative

illuminators

Passive Radar: A class of radar systems that detect and tract objects

by processing reflections from non-cooperative sources of illumination

• Advantages

– Smaller, lighter, and cheaper over

active radars

– Less prone to jamming

– Resilience to anti-radiation

missiles

– Stealth operations

• Disadvantages

– Rely on third-party illuminators

– Waveforms out of control poor

spatial/doppler resolution

SPAC Lab, ECE

• Cross-correlation: cross-correlates the received data from the

reference and surveillance channels – H. D. Griffiths and C. J. Baker, “Passive coherent location radar systems. Part 1: performance

prediction,” IEE RSN, 2005

– P. E. Howland, D. Maksimiuk, and R. Reitsma, “FM radio based bistatic radar,” IEE RSN, 2005

• Generalized canonical correlation: based on the largest eigenvalue of

the Gram matrix of the received data – K. S. Bialkowski, I. Vaughan L. Clarkson and S. D. Howard, “Generalized canonical correlation

for passive multistatic radar detection,” IEEE SSP, 2011

• Autocorrelation-based detection – K. Polonen and V. Koivunen, “Detection of DVB-T2 control symbols in passive radar system,”

IEEE 7th SAM, 2012

• Passive MIMO radar detection: employ multiple illuminators of

opportunity and multiple receivers – D. E. Hack, L. K. Patton, B. Himed and M. A. Saville, “Detection in passive MIMO radar

networks,” IEEE TSP, 2014

– D. E. Hack, L. K. Patton, B. Himed and M. A. Saville, “Centralized passive MIMO radar

detection without direct-path reference signals,” IEEE TSP, 2014

Related Work

SPAC Lab, ECE

• Reference channel:

• Surveillance channel:

– s is the unknown transmitted signal, nr and nt are time delays

– d is a Doppler shift, and are propagation parameters

– v and w are i.i.d. Gaussian noise

• After delay and Doppler compensation

Signal Model

SPAC Lab, ECE

• Cross-correlation is a widely used passive detector

– Simple to implement

– Need no prior knowledge about the transmitted signal

– Equivalent to the optimum MF used in active sensing when the

reference channel is noiseless

– Performance degrades significantly with noisy reference channel

• Noise always exists in RC ) need new passive detectors capable of

dealing with noise in RC

• We propose GLRT based detectors by taking into account the noise in

the RC for the following four cases: the signal model is deterministic or

stochastic, the noise power is known or unknown

Motivation of Proposed Solutions

SPAC Lab, ECE

• Consider the GLRT with unknown noise power

– The likelihood function under hypothesis H1 is

– ML estimates of and are:

– Using these estimates, L1 becomes

– The ML estimate of is

Detectors in Deterministic Model

SPAC Lab, ECE

– The likelihood function under hypothesis H0 is

– The ML estimate of is:

– Using the estimate, L0 becomes

– The ML estimate of is

• The GLRT detector with unknown noise power is


SPAC Lab, ECE

• The GLRT detector with known noise power can be obtained in a

similar way

– Equivalently, the test variable can be written in terms of

eigenvalues


SPAC Lab, ECE

• Stochastic model:

– transmitted signal s(n) are modeled as i.i.d. complex Gaussian with

zero-mean and unit variance

– Justified for sources with multiplexing techniques (e.g., DVB-T signal)

• With known noise power, the GLRT is

• With unknown noise power, the GLRT is

• The above two detectors are referred to as B-GLRT detectors

Detectors in Stochastic Model

SPAC Lab, ECE

• As an example, consider B-GLRT detector with known noise power

• stimates of a and b can be obtained by numerically solving the

following equations:

where

• Use the Newton-Raphson iterative method to solve the equations, and

obtain the estimates of a and b

Detectors in Stochastic Model

SPAC Lab, ECE

• For comparison, we consider two detectors

– cross-correlation (CC) detector:

– matched filter (MF) detector:

• Define the SNRs in the surveillance and reference channels as,

respectively,

Numerical Results

SPAC Lab, ECE

Numerical Results

-20 -15 -10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Dete

ctio

n P

roba

bili

tyN = 100, SNR

r = -10 dB and P

fa = 0.01

GLRT, known

GLRT, unknown

Tcc

B-GLRT, known

B-GLRT, unknown

TMF

SPAC Lab, ECE

Numerical Results

-20 -15 -10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Dete

ctio

n P

roba

bili

tyN = 100, SNR

r = 0 dB and P

fa = 0.01

GLRT, known

GLRT, unknown

TCC

B-GLRT, known

B-GLRT, unknown

TMF

SPAC Lab, ECE

Conclusions

• Investigated passive detection with a reference channel and

a surveillance channel

• Proposed four GLRT detectors:

– Deterministic signal model, known noise power

– Deterministic signal model, unknown noise power

– Stochastic signal model, known noise power

– Stochastic signal model, unknown noise power

• The proposed four GLRT except the one developed with

unknown noise power in the stochastic model outperform the

CC detector, especially at low SNRr

• Detection performance of the proposed four detectors highly

depends on the SNRr in the reference channel: the higher

the SNRr, the better the detection performance

SPAC Lab, ECE

Thank you!

3rd international workshop on mathematical issues...

Documents