Coprime Coarray Interpolation for DOA Estimation via Nuclear Norm Minimization

Transcription

1 Coprime Coarray Interpolation for DOA Estimation via Nuclear Norm Minimization Chun-Lin Liu 1 P. P. Vaidyanathan 2 Piya Pal 3 1,2 Dept. of Electrical Engineering, MC California Institute of Technology, cl.liu@caltech.edu 1, ppvnath@systems.caltech.edu 2 3 Dept. of Electrical and Computer Engineering University of Maryland, College Park ppal@umd.edu ISCAS 2016 Liu et al. Coprime Coarray Interpolation ISCAS / 19

2 Outline 1 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 2 Coarray Interpolation via Nuclear Norm Minimization 3 Numerical Examples 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS / 19

3 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) Outline 1 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 2 Coarray Interpolation via Nuclear Norm Minimization 3 Numerical Examples 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS / 19

4 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) Direction-of-arrival (DOA) estimation 1 θ i Monochromatic Uncorrelated Sources Sensor Arrays DOA Estimators Estimated DOA θ i 1 Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Liu et al. Coprime Coarray Interpolation ISCAS / 19

5 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) ULA and sparse arrays ULA (not sparse) Identify at most N 1 uncorrelated sources, given N sensors. 1 Can only find fewer sources than sensors. Sparse arrays 1 Minimum redundancy arrays 2 2 Nested arrays 3 3 Coprime arrays 4 4 Super nested arrays 5 Identify O(N 2 ) uncorrelated sources with O(N) physical sensors. More sources than sensors! 1 Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Moffet, IEEE Trans. Antennas Propag., Pal and Vaidyanathan, IEEE Trans. Signal Proc., Vaidyanathan and Pal, IEEE Trans. Signal Proc., Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu et al. Coprime Coarray Interpolation ISCAS / 19

6 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) Coprime arrays 1 The coprime array with (M, N) = 1 is the union of 1 an N-element ULA with spacing Mλ/2 and 2 a 2M-element ULA with spacing Nλ/2. ULA (1) ULA (2) Physical array S (M = 3, N = 4): Difference coarray D = {n 1 n 2 n 1, n 2 S} Holes 1 Vaidyanathan and Pal, IEEE Trans. Signal Proc., Liu et al. Coprime Coarray Interpolation ISCAS / 19

7 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) The spatial smoothing MUSIC Algorithm 1 1 Sample covariance matrix: R S = 1 K K k=1 x S(k) x H S (k). 2 Sample autocorrelation function on the difference coarray: x D. x D x U Hermitian Toeplitz matrix R (indefinite matrix). x U 0 x U 1... x U 14 x U 1 x U 0... x U 13 R = x U 14 x U x U 0 DU 4 MUSIC on R resolves ( U 1)/2 = O(N 2 ) uncorrelated sources. 1 Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2010; Liu and Vaidyanathan, IEEE Signal Proc. Letter, Liu et al. Coprime Coarray Interpolation ISCAS / 19

8 Outline Coarray Interpolation via Nuclear Norm Minimization 1 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 2 Coarray Interpolation via Nuclear Norm Minimization 3 Numerical Examples 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS / 19

9 Coarray Interpolation via Nuclear Norm Minimization Why coarray interpolation? x D D D = 3MN + M N Not all information is used. x U U U = 2MN + 2M 1 All information is used. x V V V = 4MN 2N + 1 Liu et al. Coprime Coarray Interpolation ISCAS / 19

10 Coarray Interpolation via Nuclear Norm Minimization Previous work 1 Spatial smoothing MUSIC 1 : No coarray interpolation. 2 Positive-definite Toeplitz matrix completion 2 : Not always feasible. 3 Coarray interpolation (ICA-AI) 3 : Non-convex optimization. 4 Sparse support recovery techniques 4 : Predefined dense grid and parameters. 5 Gridless DOA estimator via low-rank recovery 5 : Not used for interpolation, but for denoising. 1 Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2010; Liu and Vaidyanathan, IEEE Signal Proc. Letter, Abramovich, Spencer, and Gorokhov, IEEE Trans. Signal Proc., Friedlander and Weiss, IEEE Trans. Aero. Elec. Sys., 1992; Tuncer, Yasar, and Friedlander, Radio Science, Zhang, Amin, and Himed, IEEE ICASSP, 2013; Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2015; 5 Pal and Vaidyanathan, IEEE Signal Proc. Letter, Liu et al. Coprime Coarray Interpolation ISCAS / 19

11 Coarray Interpolation via Nuclear Norm Minimization The proposed method (via nuclear norm minimization) R V = arg min R V C V+ V + R V = R H V, R V s. t. R V n1,n 2 = x D n1 n 2, n 1, n 2 V + = {n n V, n 0}. x D D x V autocorrelation functions V R V covariance matrices Low rank, Hermitian Toeplitz R V has a low-rank structure for sufficient number of snapshots. The nuclear norm (sum of singular values) is a convex relaxation of the matrix rank. R V is Hermitian. R V is a Toeplitz matrix with some known entries. Liu et al. Coprime Coarray Interpolation ISCAS / 19

12 Coarray Interpolation via Nuclear Norm Minimization Advantages over the previous work 1 All the information is used. 2 Gridless. 3 Always feasible, even though R V can be indefinite. 4 Convex program. 5 It is possible to resolve beyond the limit of U. x V autocorrelation functions V R V covariance matrices Low rank, Hermitian Toeplitz Coarray interpolation R V = arg min R V subject to MUSIC R V C V+ V + R V = R H V, P MUSIC () = R V n1,n 2 = x D n1 n 2. R V = ŨΛŨH, ] Ũ = [Ũs Ũ n, 1 ŨH n v V +() 2 2 Liu et al. Coprime Coarray Interpolation ISCAS / 19

13 Numerical Examples Outline 1 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 2 Coarray Interpolation via Nuclear Norm Minimization 3 Numerical Examples 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS / 19

14 Numerical Examples Simulation parameters A coprime array with M = 3 and N = 5: (10 sensors) S = {0, 3, 5, 6, 9, 10, 12, 15, 20, 25}, S = 10, S 1 = 9, D = { 25, 22, 20, 19, 17,..., 17, 19, 20, 22, 25}, D = 43, ( D 1)/2 = 21, U = { 17,..., 17}, U = 35, ( U 1)/2 = 17, V = { 25,..., 25}, V = 51. S : D : U : V : Liu et al. Coprime Coarray Interpolation ISCAS / 19

15 Numerical Examples Simulation parameters (Cont.) 1 The maximum number of resolvable uncorrelated sources: using U: 17, using D: Equal-power uncorrelated sources. 3 0 db SNR and 500 snapshots. 4 Root-mean-squared error: E = 1 D D ( i ) 2, i i=1 where { i } D i=1 is the estimated normalized DOA, and { i } D i=1 is the true normalized DOA. Liu et al. Coprime Coarray Interpolation ISCAS / 19

16 Numerical Examples Example 1: Two closely spaced sources (10 sensors) SS-MUSIC 1 Co-LASSO 4 Spline 3 P () E ICA-AI 3 P.D. Toeplitz Completion 2 Nuclear norm (Proposed) P () E See page 10 for all the references Liu et al. Coprime Coarray Interpolation ISCAS / 19

17 Numerical Examples Example 2: D = 19 sources (10 sensors) Co-LASSO (E = ) Spline (E = ) ICA-AI (E = ) Nuclear norm (E = ) Liu et al. Coprime Coarray Interpolation ISCAS / 19

18 Concluding Remarks Outline 1 Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 2 Coarray Interpolation via Nuclear Norm Minimization 3 Numerical Examples 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS / 19

19 Concluding remarks Concluding Remarks Coarray interpolation via nuclear norm minimization: The correlation information on the difference coarray is fully utilized. The estimation error is reduced. More sources than ( U 1)/2 (the limit of SS MUSIC using x U ) can be resolved. In the future, it will be interesting to incorporate the matrix denoising idea into the interpolation approach. Thank you! Liu et al. Coprime Coarray Interpolation ISCAS / 19