Signal Processing for MIMO and Passive Radar

3rd International Workshop on Mathematical Issues in Information Sciences (MIIS 2014) 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

Acknowledgement Collaborators Guolong Cui (UESTC) Braham Himed (AFRL) Jun Liu (Stevens) Muralidhar Rangaswamy (AFRL) Pu Wang (Schlumberger-Doll Research Center)

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

Part I: Moving Target Detection with Distributed MIMO Radars

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

Backscattering Angle-Selective Backscatterring 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 nonhomogeneous clutter Target Backscattering vs. azimuth angle [Skolnik 03]

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

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

Noise and Clutter Noise w m,n 2 C K 1 : zero-mean, spatially/temporally white Clutter c m,n 2 C K 1 No clutter [Fishler et al. 05] Homogeneous clutter model: Uninterested shares the same covariance Buildings matrix [He-Lehmann-Blum-Haimovich 10] slow-moving vehicles angle-selective backscattering non-homogeneous clutter Subspace-based clutter model [Wang-Li-Himed 11] Clutter is spanned by l Fourier bases Plants (wind effects) Clutters fall within the column space of H Different m,n for different TX-RX ) non-homogenous clutter power Cutter covariance matrix structure is still homogeneous

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

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 K t 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)

Subspace Based GLRT 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

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

Parametric MTD Recall the moving target detection (MTD) problem

MIMO Parametric GLRT 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 H 0 and H 1, respectively

Parametric GLRT for MIMO Radar The simplified MIMO-PGLRT test statistic is [Wang-Li-Himed 13] Local detector adaptively projects test signal into two different subspaces Orthogonal complement of regression matrix Y m,n Orthogonal complement of the target-suppressed Y m,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

Asymptotic Performance 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 H 0 is independent of the disturbance parameters ) asymptotically achieves CFAR

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 Highlights: Geoometry-related terms (c mn and s mn ) are known in advance Need to compute the Fisher information-related term ψ mn Both Fisher exact information and asymptotic (FI) related expressions term for ψ mn are obtained, resulting in exact and asymptotic CRB

CRB of Target Velocity The exact CRB is obtained by plugging the exact fisher information-related term ψ mn 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 ψ mn is a function of target amplitude, Doppler steering vector, AR coefficients, and AR driving noise variance This expression is complicated and offers limited intuition

CRB of Target Velocity The asymptotic CRB is obtained by plugging the asymptotic Fisher information-related term ψ mn 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 ψ mn is proportional to the SINR mn 2 / mn (f mn ), and inversely proportional to K 3 The asymptotic CRB is simpler to compute

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

Subspace GLRT 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 Clutter is generated over Fourier basis with nonhomogeneous 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

A General Clutter Model Clutter temporal correlation function Covariance matrix Clutter power spectrum density Clutter covariance matrix for (m,n)th TX-RX pair

Parametric GLRT 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 Clutter is from general clutter model, nonhomogeneous 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

Target Velocity Estimation

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

Part 2: Waveform Design for MIMO Radar with Constant Modulus and Similarity Constraints

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 crosscorrelation, 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

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

Signal Model A co-located MIMO radar with N T TX antennas and N R RX antennas Let s(n) be the N T 1 waveform vector and a t (θ) the N T 1 TX steering vector. The signal seen at a location/angle θ is given by Let a r (θ) be the N R 1 RX steering vector. The received signal is given by signal Stacking vectors x(n), s(n) and v(n) in time interference noise The problem of interest is to design the N R radar waveforms contained in the N R N 1 vector s

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 INR k = E[ k 2 ]/( v ) 2 Constant modulus (CM) constraint

Waveform Design Criterion Similarity Constraint: Let s 0 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 s 0. 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

Proposed SOA1 Optimize ρ(s,w) with respect to w in terms of s Sequential Optimization Algorithm #1 MVDR problem Substitute w back into ρ(s,w) and simplify Fix (s) from last iteration, iteratively update s by SDR can be solved iteratively by semidefinite relaxation (SDR) Luo, Ma, So, Ye, and Zhang, Semidefinite relaxation of quadratic optimization problems, IEEE-SPM, no. 3, 2010

Sequential Optimization 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

Sequential Optimization Algorithm #2 Proposed SOA 2 Optimize w by maximizing the SINR for a given s MVDR problem Optimize s by maximizing the SINR for a given w Repeat above 2 steps till convergence Solvable by SDR

Relaxation Sequential Optimization Algorithm #2 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

Simulation Results MIMO antennas Target Parameters Set up Interferences N T 4 N R 8 0 0 2 20dB 1-50 o 1 2 30dB 2-10 o 2 2 30dB 3 40 o 3 2 30dB Noise v 2 0dB Reference signal: orthogonal LFM Beam pattern optimal receive filter optimal waveform

SINR (db) Magnitude (db) 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 20 0 19-10 18-20 -30 17-40 16-50 15 SOA1-EC 14 SOA2-EC + SOA1-CMC SOA2-CMC 13 0 50 100 Iteration index -60-70 -80-90 SOA1-EC SOA2-EC + SOA1-CMC SOA2-CMC -50 0 50 Angle ( )

SINR (db) Magnitude (db) SINR (db) Magnitude (db) 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 20 0 20 0 19-10 19-10 -20-20 18 17 16-30 -40-50 -60 18 17 16 SOA1-CMSC, =2 SOA2-CMSC, =2 + SOA1-CMSC, =0.5 SOA2-CMSC, =0.5-30 -40-50 -60 15 14 SOA1-CMSC, =2 SOA2-CMSC, =2 + SOA1-CMSC, =1.5 SOA2-CMSC, =1.5 13 0 50 100 Iteration index -70-80 -90-100 SOA1-CMSC, =2 SOA2-CMSC, =2 + SOA1-CMSC, =1.5 SOA2-CMSC, =1.5-50 0 50 Angle ( ) 15 14 13 0 50 100 Iteration index -70-80 -90-100 SOA1-CMSC, =2 SOA2-CMSC, =2 + SOA1-CMSC, =0.5 SOA2-CMSC, =0.5-50 0 50 Angle ( )

Magnitude (db) Magnitude (db) 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. 0-10 -20-30 -40-50 -60 LFM -70 SOA1-CMSC, =2 SOA1-CMSC, =1-80 SOA1-CMSC, =0.5 SOA1-CMSC, =0.1-90 -500-400 -300-200 -100 0 100 200 300 400 500 IFFT bin index 0-10 -20-30 -40-50 -60 LFM -70 SOA2-CMSC, =2 SOA2-CMSC, =1-80 SOA2-CMSC, =0.5 SOA2-CMSC, =0.1-90 -500-400 -300-200 -100 0 100 200 300 400 500 IFFT bin index

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

Part 3: Passive Detection with Noisy Reference

Passive Radar Passive Radar: A class of radar systems that detect and tract objects by processing reflections from non-cooperative sources of illumination Non-cooperative illuminators Passive Radar Reference Channel Target Surveillance Channel 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

Related Work 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 7 th 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

Signal Model Reference channel: Surveillance channel: s is the unknown transmitted signal, n r and n t 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

Motivation of Proposed Solutions 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

Detectors in Deterministic Model Consider the GLRT with unknown noise power The likelihood function under hypothesis H 1 is ML estimates of and are: Using these estimates, L 1 becomes The ML estimate of is

Detectors in Deterministic Model The likelihood function under hypothesis H 0 is The ML estimate of is: Using the estimate, L 0 becomes The ML estimate of is The GLRT detector with unknown noise power is

Detectors in Deterministic Model 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

Detectors in Stochastic Model 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 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

Numerical Results 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,

Detection Probability Numerical Results 1 N = 100, SNR r = -10 db and P fa = 0.01 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 GLRT, known GLRT, unknown T cc B-GLRT, known B-GLRT, unknown T MF 0-20 -15-10 -5 0 5 10 SNR (db)

Detection Probability Numerical Results 1 N = 100, SNR r = 0 db and P fa = 0.01 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 GLRT, known GLRT, unknown T CC B-GLRT, known B-GLRT, unknown T MF 0-20 -15-10 -5 0 5 10 SNR (db)

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 SNR r Detection performance of the proposed four detectors highly depends on the SNR r in the reference channel: the higher the SNR r, the better the detection performance

Thank you!