Coprime Coarray Interpolation for DOA Estimation via Nuclear Norm Minimization Chun-Lin Liu 1 . Vaidyanathan 2 Piya Pal 3 P . P 1 , 2 Dept. of Electrical Engineering, MC 136-93 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 2016 1 / 19
Outline Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 1 Coarray Interpolation via Nuclear Norm Minimization 2 Numerical Examples 3 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS 2016 2 / 19
Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) Outline Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 1 Coarray Interpolation via Nuclear Norm Minimization 2 Numerical Examples 3 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS 2016 3 / 19
Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) Direction-of-arrival (DOA) estimation 1 Monochromatic Uncorrelated θ i Sources Sensor • • • • • • • • • Arrays Estimated DOA Estimators DOA � θ i 1Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory , 2002. Liu et al. Coprime Coarray Interpolation ISCAS 2016 4 / 19
Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) ULA and sparse arrays ULA (not sparse) Sparse arrays 1 Minimum redundancy arrays 2 Identify at most N − 1 uncorrelated sources, 2 Nested arrays 3 given N sensors. 1 3 Coprime arrays 4 Can only find fewer 4 Super nested arrays 5 sources than sensors. Identify O ( N 2 ) uncorrelated sources with O ( N ) physical sensors. More sources than sensors! 1Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory , 2002. 2Moffet, IEEE Trans. Antennas Propag. , 1968. 3Pal and Vaidyanathan, IEEE Trans. Signal Proc. , 2010. 4Vaidyanathan and Pal, IEEE Trans. Signal Proc. , 2011. 5Liu and Vaidyanathan, IEEE Trans. Signal Proc. , 2016. Liu et al. Coprime Coarray Interpolation ISCAS 2016 5 / 19
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 2 M -element ULA with spacing Nλ/ 2 . ULA (1) ULA (2) Physical array S ( M = 3 , N = 4 ): • 0 ×× • • × × • • • ×× ××× ××× • • • 3 4 6 8 9 12 16 20 Difference coarray D = { n 1 − n 2 | n 1 , n 2 ∈ S } • ×× •• ••••••••••••••••••••••••••••• •• × × ×× • − 20 − 14 0 14 20 Holes 1Vaidyanathan and Pal, IEEE Trans. Signal Proc. , 2011. Liu et al. Coprime Coarray Interpolation ISCAS 2016 6 / 19
Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) The spatial smoothing MUSIC Algorithm 1 � K 1 Sample covariance matrix: � R S = 1 x H k =1 � x S ( k ) � S ( k ) . K 2 Sample autocorrelation function on the difference coarray: � x D . � x D x U � • × × •• ••••••••••••••••••••••••••••• •• × × × × • − 20 − 14 0 14 20 U D 3 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 � 13 � � x U � 0 . . . 4 MUSIC on � R resolves ( | U | − 1) / 2 = O ( N 2 ) uncorrelated sources. 1Pal and Vaidyanathan, IEEE Trans. Signal Proc. , 2010; Liu and Vaidyanathan, IEEE Signal Proc. Letter , 2015. Liu et al. Coprime Coarray Interpolation ISCAS 2016 7 / 19
Coarray Interpolation via Nuclear Norm Minimization Outline Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 1 Coarray Interpolation via Nuclear Norm Minimization 2 Numerical Examples 3 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS 2016 8 / 19
Coarray Interpolation via Nuclear Norm Minimization Why coarray interpolation? x D � • •• ••••••••••••••••••••••••••••• •• • × × × × × × − 20 − 14 0 14 20 | D | = 3 MN + M − N D Not all information is used. x U � ••••••••••••••••••••••••••••• − 14 0 14 | U | = 2 MN + 2 M − 1 U All information is used. x V � • •• •• ••••••••••••••••••••••••••••• •• • • •• • − 20 − 14 0 14 20 | V | = 4 MN − 2 N + 1 V Liu et al. Coprime Coarray Interpolation ISCAS 2016 9 / 19
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. 1Pal and Vaidyanathan, IEEE Trans. Signal Proc. , 2010; Liu and Vaidyanathan, IEEE Signal Proc. Letter , 2015. 2Abramovich, Spencer, and Gorokhov, IEEE Trans. Signal Proc. , 1999. 3Friedlander and Weiss, IEEE Trans. Aero. Elec. Sys. , 1992; Tuncer, Yasar, and Friedlander, Radio Science , 2007. 4Zhang, Amin, and Himed, IEEE ICASSP , 2013; Pal and Vaidyanathan, IEEE Trans. Signal Proc. , 2015; 5Pal and Vaidyanathan, IEEE Signal Proc. Letter , 2014. Liu et al. Coprime Coarray Interpolation ISCAS 2016 10 / 19
Coarray Interpolation via Nuclear Norm Minimization The proposed method (via nuclear norm minimization) x D � • • × × × • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • × × • • × � � � R ⋆ V = arg min R V � ∗ s. t. D R V ∈ C | V + |×| V + | � R V = � � R H � x V ≈ autocorrelation functions V , � � • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • R V � n 1 ,n 2 = � � x D � n 1 − n 2 , V n 1 , n 2 ∈ V + = { n | n ∈ V , n ≥ 0 } . � 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 2016 11 / 19
Coarray Interpolation via Nuclear Norm Minimization Advantages over the previous work Coarray interpolation 1 All the information is used. � � � R ⋆ V = arg min R V � ∗ 2 Gridless. R V ∈ C | V + |×| V + | � 3 Always feasible, even though subject to � R ⋆ V can be indefinite. R V = � � R H V , 4 Convex program. � � R V � n 1 ,n 2 = � � x D � n 1 − n 2 . 5 It is possible to resolve beyond the limit of U . MUSIC R ⋆ � V = � UΛ � U H , � � � x V ≈ autocorrelation functions � � � U = U s U n , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 P MUSIC (¯ θ ) = V � � 2 � � � R V ≈ covariance matrices � � n v V + (¯ U H θ ) � Low rank, Hermitian Toeplitz 2 Liu et al. Coprime Coarray Interpolation ISCAS 2016 12 / 19
Numerical Examples Outline Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC) 1 Coarray Interpolation via Nuclear Norm Minimization 2 Numerical Examples 3 4 Concluding Remarks Liu et al. Coprime Coarray Interpolation ISCAS 2016 13 / 19
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 . • × × × • 3 • 5 • × × • 9 • 10 • × × × × • × × × × • × × × • S : 0 6 12 15 20 25 • • •• ••••••••••••••••••••••••••••••••••• •• • • × × × × × × × × D : − 25 − 17 17 25 ••••••••••••••••••••••••••••••••••• U : − 17 17 ••••••••••••••••••••••••••••••••••••••••••••••••••• V : − 25 25 Liu et al. Coprime Coarray Interpolation ISCAS 2016 14 / 19
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