The Probability Distribution of the MVDR Beamformer Outputs under Diagonal Loading. N. Raj Rao (Dept. of Electrical Engineering and Computer Science)

Transcription

1 The Probability Distribution of the MVDR Beamformer Outputs under Diagonal Loading N. Raj Rao (Dept. of Electrical Engineering and Computer Science) & Alan Edelman (Dept. of Mathematics) Work supported by NSF Grant DMS

2 Detection Estimation Classification Main Message Capon-MVDR Beamformer Diagonal Loading used for Robust Adaptive Beamforming since 1960 s Open Problem: For over 40 years no analytical results on Diagonal Loading Distribution Bias Variance Solved! Initial results: Single Source and Source + Interferer in White Noise Technique: Infinite Random Matrix Theory 2

3 Outline Capon-MVDR Beamformer Why is Diagonal Loading used? Related work Capon-Goodman result New results Infinite Random Matrix Theory Simulations Snapshot Deficient case Future work Rank detection algorithm 3

4 Capon-MVDR Beamformer X = [x(1)... x(l)] is an N L data matrix N = Number of sensors, L = Number of snapshots x(l) for l = 1, 2,..., L has covariance R = E[x(l)x(l) H ] Capon s Power Spectral Estimator: ˆP Capon (θ) = 1 v H (θ)ˆr 1 v(θ) where ˆR = 1 L XXH is the Sample Covariance Matrix Also known as Minimum Variance Distortionless Response (MVDR) Beamformer Minimum variance since weight vector w(θ) chosen to minimize w H Rw Distortionless since w H v(θ) = 1 4

5 Diagonally Loaded Capon-MVDR Beamformer ˆR δ = 1 L XXH + δ I N ˆP Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) 30 N = 18, L = 36 : δ = 0 30 N = 18, L = 36 : δ = Eigenvalues (db) Eigenvalues (db) Index Index 5

6 Diagonally Loaded Capon-MVDR Beamformer ˆR δ = 1 L XXH + δ I N ˆP Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) 30 N = 18, L = 4 : δ = 0 30 N = 18, L = 4 : δ = Eigenvalues (db) Eigenvalues (db) Index Index 6

7 Diagonally Loaded Capon-MVDR Beamformer ˆR δ = 1 L XXH + δ I N Type I Estimator: ˆP I Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) Type II Estimator: Capon (θ, δ) = vh (θ)ˆr 1 δ ˆRˆR 1 δ v(θ) (v H (θ)ˆr 1 δ v(θ))2 ˆP II Results focus on Type I Estimator; can be extended for Type II as well. 7

8 Outline Capon-MVDR Beamformer Why is Diagonal Loading used? Related work Capon-Goodman result New results Infinite Random Matrix Theory Simulations Snapshot Deficient case Future work Rank detection algorithm 8

9 Related work (partial list) Carlson Cox, Zeskind and Owen Baggeroer and Cox Van Trees Stoica and Li Hawkes and Nehorai Owsley Simulations Fertig Evans and Tse Tse and Zeitouni Mestre and Lagunas Special Cases 9

10 Capon-Goodman Result ˆR δ = 1 L XXH + δ I N ˆP Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) Statistic/Scenario δ = 0, L N δ 0, L N, L < N Distribution Bias Variance complex chi-squared L N + 1 P Capon (θ) L L N + 1 L 2 P Capon (θ) 2??? Capon-Goodman 1970 Open Problem 10

11 Perspectives on Diagonal Loading Detection Estimation Classification Capon-MVDR Beamformer Diagonal Loading used for Robust Adaptive Beamforming since 1960 s Known to robustify beamforming to steering vector errors and finite sample effects Canonical cases: Single Source and Source plus Interferer in white noise not understood analytically No analytical results Finite Sample Effects not fully understood Inadequacy of existing Large Array feasibility studies 11

12 Outline Capon-MVDR Beamformer Why is Diagonal Loading used? Related work Capon-Goodman result New results Infinite Random Matrix Theory Simulations Snapshot Deficient case Future work Rank detection algorithm 12

13 New Results for Canonical Cases ˆR δ = 1 L XXH + δ I N ˆP Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) Statistic/Scenario δ = 0, L N δ 0, L N, L < N Distribution Bias Variance complex chi-squared L N + 1 P Capon (θ) L L N + 1 L 2 P Capon (θ) 2 Capon-Goodman 1970 Solved! 13

14 Probability Distribution (No Signal in White Noise) 1 ˆP Capon (θ, δ) = vh (θ)ˆr 1 δ v(θ) d = G 1 as N, L with N/L c > 0 G 1 = Gaussian Mean of Gaussian depends on N/L and δ Variance of Gaussian depends on N/L, N and δ Variance 0 as N, L with N/L c > 0 c > 1: Snapshot Deficient Scenario c << 1: Snapshot Abundant Scenario 14

15 Probability Distribution (Single θ S in White Noise) 1 ˆP Capon (θ, δ) = vh (θ)ˆr 1 δ v(θ) d = as N, L with N/L c > 0 G 1, G 2, and G 3 are independent Gaussians G 1 const. + G 1 + Variance of Gaussians 0 as N, L with N/L c > 0 c > 1: Snapshot Deficient Scenario c << 1: Snapshot Abundant Scenario G 2 const. + G 1 + G 3 Unifies analysis for Snapshot Deficient and Snapshot Abundant scenarios 15

16 Techniques used ˆR δ = 1 L XXH + δ I N ˆP Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) Capon-Goodman Result (δ = 0): Key property: Invariance allows decoupling of deterministic functional from random matrix based functional v H (θ)ˆr 1 0 v(θ) = vh (θ)r 1 v(θ) v H (θ)w 1 v(θ) W is complex null Wishart! 16

17 New Results (δ 0): Techniques used ˆR δ = 1 L XXH + δ I N ˆP Capon (θ, δ) = 1 v H (θ)ˆr 1 δ v(θ) Observation: No decoupling of deterministic functional from random matrix based functional v H (θ)ˆr 1 δ v(θ) vh (θ)r 1 v(θ) v H (θ)g(w)v(θ)v(θ) g(x) = 1/(δ + x) is a matrix valued function, W is a complex null Wishart Key property: Distribution of v H (θ)g(w)v(θ) known as N, L Known + New results from infinite random matrix theory Matrix Inversion Lemma + Key Property Desired Result Rigorous for N, L with N/L c > 0 In practice N = 8 is good enough 17

18 Outline Capon-MVDR Beamformer Why is Diagonal Loading used? Related work Capon-Goodman result New results Infinite Random Matrix Theory Simulations Snapshot Deficient case Future work Rank detection algorithm 18

19 Single Source in White Noise (Probability Distribution) Probability x N = 100, L = 250, σ 2 S = 10, θ S = 90, δ =

20 Single Source in White Noise (Probability Distribution) Probability x N = 10, L = 50, σ 2 S = 10, θ S = 90, δ =

21 Single Source in White Noise (Probability Distribution) Probability x N = 4, L = 5, σ 2 S = 10, θ S = 90, δ =

22 Source + Interferer in White Noise (N = 18, L = 4,σ 2 S = σ2 I = 100, θ S = 90, θ I = 70,BW = 7.2) Monte Carlo DL Capon MVDR (δ = 10 db) Capon MVDR (Reference) E[1/P Capon (θ)] (db) f Am θ (Degrees) (a) δ = 10 db 30 Monte Carlo DL Capon MVDR (δ = 10 db) 40 Capon MVDR (Reference) θ (Degrees) (b) δ = 10 db 22

23 Source + Interferer in White Noise (N = 18, L = 4,σ 2 S = σ2 I = 100, θ S = 90, θ I = 70,BW = 7.2) Monte Carlo DL Capon MVDR (δ = 10 db) 40 Capon MVDR (Reference) θ (Degrees)

24 Source + Interferer in White Noise (N = 18, L = 4,σS 2 = σ2 I = 100, θ S = 90, θ I = 95,BW = 7.2) E[1/P Capon (θ)] (db) Monte Carlo 40 DL Capon MVDR (δ = 10 db) Capon MVDR (Reference) θ (Degrees) 24

25 Source + Interferer in White Noise (N = 18, L = 4, σ 2 S = σ2 I = 1,θ S = 90,θ I = 95,BW = 7.2) 14 E[1/P Capon (θ)] (db) Monte Carlo DL Capon MVDR (δ = 10 db) Capon MVDR (Reference) θ (Degrees) 25

26 Outline Capon-MVDR Beamformer Why is Diagonal Loading used? Related work Capon-Goodman result New results Infinite Random Matrix Theory Simulations Snapshot Deficient case Future work Rank detection algorithm 26

27 Work in progress Multiple plane wave signals Arbitrary number of Sources + Interferers non-gaussian noise models Capon-MVDR Type II estimator Characterization of the estimation/detection/classification performance of DL Capon- MVDR Rank detection algorithm Dramatically outperforms MDL/AIC in high sensor/low sample support regime 27

28 Rank Detection Algorithm (N = 200) 5 True Rank = 120 Eigenvalues (Magnitude) Index 28

29 Rank Detection Algorithm (N = 200, L = 250) Magnitude (db) True rank = 120, Estimate = , MDL/AIC Estimate = index Red is Actual Rank, Black is New Algorithm, Green is MDL/AIC index Magnitude 29

30 Conclusions The distribution of Capon-MVDR successfully characterized under Diagonal Loading Single Source and Source + Interferer Case in White Noise Infinite random matrix theory facilitates required computation Applies for Snapshot Deficient and Snapshot Abundant cases Results encompass array response mismatch scenario 30

31 Detection Estimation Classification Main Message Capon-MVDR Beamformer Diagonal Loading used for Robust Adaptive Beamforming since 1960 s Open///////////// Problem: For over 40 years no analytical results on Diagonal Loading Distribution Bias Variance Solved! Initial results: Single Source and Source + Interferer in White Noise Technique: Infinite Random Matrix Theory 31