Adaptive Array Detection, Estimation and Beamforming

Transcription

1 Adaptive Array Detection, Estimation and Beamforming Christ D. Richmond Workshop on Stochastic Eigen-Analysis and its Applications 3:30pm, Monday, July 10th 2006 C. D. Richmond-1 *This work was sponsored by Defense Advanced Research Projects Agency under Air Force contract FA C Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government.

2 Outline Introduction Radar/Sonar problem Detection algorithms Estimation algorithms Open problems Summary C. D. Richmond-2

3 Airborne Surveillance Radars RAdio Detection And Ranging = RADAR Goals / Mission: Long range surveillance Airborne Moving Target Indication (AMTI) Ground Moving Target Indication (GMTI) Synthetic Aperture Radar (SAR) Imaging C. D. Richmond-3

4 Airborne Surveillance Radars: Signals and Interference Hostile Jamming Interferer Azimuth Target s TX TX/RX Waveform { } ( t)= Re p ( t) e j 2πf c t Ground Clutter s RX { } ()= t Re α p ( t τ) e j 2π ( f c + f d )t Transmit Power Pattern v Time Delay (Range) Doppler (Velocity) C. D. Richmond-4

5 Radar Data Model and Optimum Linear Filter Primary snapshot (target range gate) x T = Sv( T, f T )+ n C. D. Richmond-5 Ground Clutter ( ) E{ x T }= Sv T, f T cov( x T )= E{ nn H }= R *Brennan and Reed, IEEE T-AES 1973 First to propose this for Radar Sig. Proc Power (db) Clutter Null NOISE Sin (Azimuth) Filter Response of w R 1 v 0 CLUTTER TARGET Two-dimensional filtering required to cancel ground clutter Space-Time Adaptive Processing (STAP) 1 Jammer Null JAMMER 0 Doppler (Hz) 0.5

6 Outline Introduction Detection algorithms Estimation algorithms Open problems Summary C. D. Richmond-6

7 Adaptive Detection Problem Test Cell: Analogy to 1-D H 0 : x T = n cov(x T ) = R CFAR Statistic H 1 : x T = Sv T + n Two Unknowns R & S Use Noise Only Training Set [ x x ] X = 1 2 x L cov(x i ) = R Assumptions: All Data Complex Gaussian Training Samples Homogeneous with Test Cell cov(x T ) = cov(x i ) Perfect Look ( v = v T ) t = #Cells ˆ 2 2 η σ N < C. D. Richmond-7

8 Summary of Adaptive Detection Algorithms Adaptive Matched Filter (AMF) Robey, et. al. IEEE T-AES 1992 Reed & Chen 1992, Reed et. al Generalized Likelihood Ratio Test (GLRT) Kelly IEEE T-AES 1986, Khatri 1979 Adaptive Cosine Estimator (ACE) Conte et. al. IEEE T-AES 1995, Scharf et. al. Asilomar 1996 Adaptive Sidelobe Blanker (ASB) Kreithen, Baranoski, 1996 Richmond Asilomar 1997 More t AMF = vh ˆ R 1 x T 2 v H ˆ R 1 v t GLRT = t ACE = t AMF H ˆ 1 L + x T f (t AMF,t ACE ) R 1 x T t AMF x T H ˆ R 1 x T C. D. Richmond-8 Each Algorithm is a function of the Sample Covariance ˆ R = 1 L x 1x 1 H + x 2 x 2 H + + x L x L H ( )

9 1 0.8 R known PD Adaptive Detection Performance: An Example PD vs SINR Loss Due to Covariance Estimation GLRT ASB, Fixed Thr. ACE AMF Max ASB PD Optimal MF N=10, L=2N, PFA=1e R unknown Output Array SINR (db) Random matrix theory predicts performance loss due to covariance estimation C. D. Richmond-9

10 Outline Introduction Detection algorithms Estimation algorithms Open problems Summary C. D. Richmond-10

11 Mean-Squared Error Performance: No Mismatch vs Mismatch No Mismatch Array Element Positions ˆ ML = argmax Ambiguity Function t ML (,data) Noise Free Mean Squared Error (db) No Information Threshold T Cramr-Rao Bound Driven by Global Ambiguity/Sidelobe Errors Asymptotic Driven by Local Mainlobe Errors Large Errors ˆ ML Small Errors T T Scan Angle Low SNR High SNR SNR TH SNR (db) T ˆ ML C. D. Richmond-11

12 Mean-Squared Error Performance: No Mismatch vs Mismatch No Mismatch Array Element Positions Assumed True Signal Mismatch Array Element Positions T T Sidelobe Target Mean Squared Error (db) No Information Threshold Cramr-Rao Bound Mismatch affects threshold and asymptotic region leading to atypical performance curves Asymptotic Mean Squared Error (db) No Information CRB Threshold Asymptotic C. D. Richmond-12 SNR SNR (db) TH SNR (db) SNR TH SNR TH Mismatch

13 Data Model: ML Estimator*: Data Model: ML Estimator*: Maximum-Likelihood Signal Parameter Estimation π N R 1 exp x Sv [ ()] H R 1 x Sv() [ ] { } ML = argmaxt MF ( ) t MF ()= vh ()R 1 x 2 v H ()R 1 v() π N(L +1) R (L +1) exp x Sv ML = argmaxt AMF ( ) t AMF {[ ()] H R 1 [ x Sv() ] tr( R 1 XX H )} ()ˆ ()= vh R 1 x 2 R ˆ 1 v H ()ˆ R 1 v() L XX H Test Cell Training Data Complex Gaussian data model: All snapshots N x 1 Arbitrary N x N Colored Covariance Deterministic Signal ( Conditional ) Colored noise only training samples available S unknown Clairvoyant Matched Filter R unknown S unknown Adaptive Matched Filter *See Swindlehurst & Stoica Proc. IEEE 1998 C. D. Richmond-13

14 Approximating MSE Performance: Based on Interval Errors MSE given by E ˆ 1 ( ) 2 1 E ˆ ML 1 ( ) 2 ω 1 ( ) 2 p ˆ ()dω ω ( ω 1 ) 2 dω 1 K k= 2 p ˆ ML = k 1 IE Local Errors ( ) NIE IE Global Errors K 2 σ ML ( 1 )+ p( ˆ ML = k ) 1 k 1 k= 2 ( ) 2 Challenge is calculation of error probabilities p( ˆ ML = k ) =? 1 and asymptotic MSE: 2 σ ML ( )=? 1 Both are functions of the estimated covariance C. D. Richmond-14

15 Broadside Planewave Signal in White Noise: No Mismatch, R known, ULA ULA Element Positions z n Distance (in units of λ) RMSE in Beamwidths (db) From 4000 Monte Carlo Simulations Threshold SNR Var. Uniform CRB Asympt. MSE MSE Prediction Monte Carlo Element Level SNR (db) N=18 element uniform linear array (ULA), (λ/2.25) element spacing 3dB Beamwidth 7.2 degs, search space [60 120] degs 0dB white noise, True 90 degs (broadside) Asymptotic ML MSE agrees with CRB above threshold SNR MIE MSE predictions very accurate above and below threshold C. D. Richmond-15

16 ULA Element Positions 2 ( 0 I ) z n + N σ 3, 3 RMS σ RMS = 0. 1λ Signal in White Noise: Perturbed ULA, R unknown, L = 3N Distance (in units of λ) RMSE in Beamwidths (db) From 4000 Monte Carlo Simulations Element Level SNR (db) Asympt. MSE MSE Prediction CRB Threshold SNR N=18 element ULA positions perturbed by 3-D Gaussian noise Zero mean with stand. dev. 0.1λ; use single realization MC Known R MC Unknown R Adaptivity Loss Estimated colored noise covariance from L = 3N samples ~15dB SNR adaptivity loss limits beam split ratio to 16:1 as opposed to 22:1 when R is known C. D. Richmond-16

17 The Capon-MVDR Algorithm T Capon proposed filterbank approach to spectral estimation that designs linear filters optimally: Given N x 1 vector snapshots for l =1,2,,L ( ) x l with covariance choose filter weights w according to min w H Rw such that w H v( )=1 Minimum Variance { ( )} R = E x( l)x H l Distortionless Response Capon 1969 Solution well-known: R 1 v E w H v { ()x () l 2 } = 1 H w()= C. D. Richmond-17 () ()R 1 v() where ˆ R = 1 L L l=1 Average Output Power of Optimal Filter: x()x l H () l v H ()R 1 v() Capon s Spectrum: P Capon () is sample covariance matrix Parameter estimate ˆ given by location of maximum power 1 Ambiguity Estimation Function v H ()R 1 v () Error T Scan Angle ˆ T v H v H 1 ()ˆ R 1 v() 1 R 1 v ()ˆ () Scan Angle

18 Diagonally Loaded Capon Algorithm In practice it is common to diagonally load the sample covariance: ˆ R α = α I + 1 L L l=1 x()x l H () l I P Capon (,α )= v H 1 R 1 α v ()ˆ () * Robustify Processing Diagonal loading mitigates undesired finite sample effects* Slow convergence of small/noise eigenvalues (DL compresses) High sidelobes (DL provides sidelobe [white noise gain] control) Excessive loading can degrade performance Diagonal loading is necessary to invert matrix in snapshot deficient aacase, i.e. L N Eigenvectors of sample covariance remain unaffected by diagonal aaloading Featherstone et al. showed diagonally loaded Capon to be a robust aadirection finding algorithm C. D. Richmond-18 *Cox, IEEE T-SP 1987, Carlson, IEEE T-AES 1988

19 Single Signal Broadside to Array in Spatially White Noise, L = 0.5N RMSE in Beamwidths (db) Monte Carlo CRB MSE Prediction L = 0.5N α = -10dB RMSE in Beamwidths (db) Threshold SNRs L = 0.5N α = +10dB Output Array SNR (db) Output Array SNR (db) C. D. Richmond-19 N=18 element uniform linear array (ULA), (λ/2.25) element spacing 3dB Beamwidth 7.2 degs 0dB white noise, True 90 degs (broadside) 4000 Monte Carlo simulations VB MSE prediction not applicable for L < N

20 Mismatch Example: Perturbed Array Positions Assumed Nominal Array Position Actual Perturbed Array Position z n z n + e n RMSE in Beamwidths (db) Monte Carlo MSE Prediction VB MSE Prediction L = 1.5N α = +10dB 8dB Error in VB Prediction of 15:1 Beamsplit Ratio SNR 10dB Error for 17:1 Based on Single Realization of Gaussian Perturbation: 2 e n ~ N 3 ( 0,I 3 σ RMS ), C. D. Richmond-20 σ RMS = 0.04λ Output Array SNR (db) N=18 element ULA with perturbed positions but assumed straight VB MSE prediction can lead to large errors in required SNRs DL Capon is more robust DF approach: 18:1 vs 40dB ASNR

21 Outline Introduction Detection algorithms Estimation algorithms Summary Open problems C. D. Richmond-21

22 What About Robust Detection? Adaptive Matched Filter (AMF) Robey, et. al. IEEE T-AES 1992 Reed & Chen 1992, Reed et. al Generalized Likelihood Ratio Test (GLRT) Kelly IEEE T-AES 1986, Khatri 1979 Adaptive Cosine Estimator (ACE) Conte et. al. IEEE T-AES 1995, Scharf et. al. Asilomar 1996 Adaptive Sidelobe Blanker (ASB) Kreithen, Baranoski, 1996 Richmond Asilomar 1997 C. D. Richmond-22 Each Algorithm is a function of the Sample Covariance t AMF α t GLRT t ACE ˆ R α = α I + 1 L ()= vh ˆ ( α)= ( α)= R 1 α x T 2 v H R ˆ 1 α v t AMF 1 + x T H ˆ R α 1 x T t AMF x T H ˆ R α 1 x T [ ( )] f t AMF ( α ), t ACE α L l=1 x()x l H () l

23 Magneto-encephalography (MEG) z (meters) False Peaks Source True Location P LCMV ( ) (db) y (meters) Inflated Cortical Surface LCMV Cost Function - Based on 74 Channel Dual Sensor Magnes II Biomagnetometer - SNR = -23 db Dipolar source located in the center of the Somatosensory Region C. D. Richmond-23 x (meters)

24 Composite Localization Accuracy vs Signal-to-Noise Ratio (SNR) Localization MSE (db) No Information: No Signal Threshold: Weak Signal -log(snr) Large Errors Due To False Peaks of Cost Function Asymptotic: Strong Signal Cost Function: Output of LCMV spatial filter as signal location hypothesis is varied when using true R Residual Error Due to Jitter About True Source Location Cost Fnc Height High Outstanding Problem THRESHOLD SNR H Pr tr V ˆ 1 R 1 V 1 ( ) 1 SNR (db) >tr ( V H ˆ 2 R 1 V ) 1 2 =? Low C. D. Richmond-24

25 Outline Introduction Detection algorithms Estimation algorithms Open problems Summary C. D. Richmond-25

26 Summary Random matrix theory provides insight into the performance of adaptive arrays systems Finite random matrix theory has been most common approach Infinite random matrix theory quickly gaining momentum as tool for analyses and design of robust signal processing algorithms C. D. Richmond-26

27 Distributions of 1-D Detectors Homogeneous Case Recall that PD of ASB is PD ASB = Pr( t ACE > η ace, t AMF > η amf ) Requires knowledge of Dependence! ~ t Define the following GLRT t GLRT /(1 t GLRT ) ~ t ACE t ACE /(1 t ACE ) K = L N + 2 It can be shown that Found in this Summary! Distributions of Adaptive Detectors t GLRT t AMF = d F 1,K 1 ( δ β ) = d F 1,K 1 ( δ β )/β where δ β = β S v H R -1 v t ACE = d F 1,K 1 ( δ β )/(1 β ) Richmond Asilomar 1997 Richmond IEEE SP 2000 C. D. Richmond-27

28 The AMF Detector Form the optimal Neyman-Pearson test statistic, that is, the LRT. Assume complex Gaussian statistics H 0 : H 1 : g H 0 = π N R 1 exp[ x H T R 1 x T ] g H1 = π N R T 1exp x T vs [ ( ) H R 1 ( x T vs) ] Likelihood = RatioTest max S g H 0 g H1 = v H ˆ R 1 x T 2 v H ˆ R 1 v t MF Matched Filter Simply replace true data covariance with an estimate XX H = ˆ R R t AMF = v H v Rˆ H 1 Rˆ x 1 T v 2 Known as the Adaptive Matched Filter (AMF) detector C. D. Richmond-28 Return

29 The Generalized LRT (GLRT) Form the LRT based on the totality of data: Assume homogeneous complex gaussian statistics where [ Test Cell Interference Training Set ]= [ x T X] X 0 H 0 : H 1 : M = [ vs 0] g H 0 [ ] = π N(L +1) R (L +1) exp trr 1 X 0 X 0 H [ ( )( X 0 M) H ] g H 1 = π N ( L +1) R ( L +1) exp trr 1 X 0 M Maximize likelihood functions over all unknown parameters: t GLRT = max S,R max R g H1 g H 0 1 L +1 = H 1+ x ˆ T R 1 x T ˆ H 1+ x ˆ T R 1 x T vh R 1 2 x T v H R ˆ 1 v C. D. Richmond-29 Known as Kelly s / Khatri s GLRT Return

30 The Adaptive Cosine Estimator (ACE) ψ v Target array response Measured data vector x The ACE statistic provides a measure of correlation between the test data vector x T and the assumed target array response v Inner product space defined wrt inverse of data covariance in whitened space t ACE = v H ˆ R 1 x T 2 ( H x ˆ T R 1 x T )v H R ˆ 1 v ( ) = cosψ 2 C. D. Richmond-30 Return

31 Simplest: The AMF Detector Practical Issues: AMF Computationally Attractive: Linear Filter AMF is an Adaptive Beamformer Measures Power in Assumed Target Direction Interference Suppression Based on Covariance Estimate Inhomogeneities Frustrate Interference Suppression Covariance Estimate Uncharacteristic of Data Results in high False Alarm Rates C. D. Richmond-31

32 Classical Sidelobe Blanking Directional Channel Threshold < Power in Target Direction Gate Output Input Omni-directional Channel Threshold < Comparator Total Power from All Directions Channel Magnitude Response Typical Comparator Input Ch 2 Ch 1 Ch 1 Ch 2 Strong Signal Time Azimuth C. D. Richmond-32

33 2-D ASB Detection Algorithm Step 1: Beamforming t AMF Power in Target Direction > η amf Step 2 : Sidelobe Blanking t AMF > η ace x T H ˆ R -1 x T Power in Target Direction Total Power From All Directions Sidelobe Blanking t ACE 1 0 η ace 2-D ASB Detector Passes ACE Fails AMF & ACE η amf Region of Declared Detections Passes AMF Directional Beamformer t AMF Return C. D. Richmond-33

34 The Complex Wishart Random Matrix If the training data is complex Gaussian s.t. X ~ CN( 0,I L R) then the sample covariance matrix is the Maximum-Likelihood estimator of the covariance parameter R: L N L ˆ R XX H = L k=1 x k x k H If then its PDF exists and is given by LR ˆ L N R L / Γ N (L) [ ( )] where 0 < ˆ exp tr R 1ˆ R L and the differential volume element is given by ( dr ˆ )= dr ˆ 11 dr ˆ 22 dr ˆ NN d Re( R ˆ 12 )d Im( R ˆ 12 ) d Re( R ˆ 13 )d Im ˆ d Re( R ˆ N 1,N )d Im( R ˆ N 1,N ) R ( R ) 13 C. D. Richmond-34