Tensor MUSIC in Multidimensional Sparse Arrays Chun-Lin Liu 1 and P . Vaidyanathan 2 . P Dept. of Electrical Engineering, MC 136-93 California Institute of Technology, Pasadena, CA 91125, USA cl.liu@caltech.edu 1 , ppvnath@systems.caltech.edu 2 Asilomar Conference on Signals, Systems, and Computers, 2015 Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 1 / 24
Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 2 / 24
Introduction Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 3 / 24
Introduction Motivation Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 4 / 24
Introduction Motivation Harmonic Retrieval in Planar Array Processing 1 Planar arrays Incoming Received waveforms • • • • • • • • • • plane • • • • • • • • • • • • • • • • • • • • waves t • • • • • • • • • • • • • • • • • • • • Spatial information Temporal information Utimate Goal Estimate source profiles (azimuth, elevation, range, Doppler, etc.) from sensor measurements efficiently and accurately. 1 Harry L. Van Trees. Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory . Wiley Interscience, 2002. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 5 / 24
Introduction Motivation Sparse Array Processing 2 , 3 Uniform Linear Arrays (ULAs) Linear Sparse Arrays ULA with N sensors and Nested array with N 1 , N 2 and sensor separation λ/ 2 . min. separation λ/ 2 . N N 1 N 2 •••••••••• • • • • • • • • • • λ/ 2 λ/ 2 ( N 1 + 1) λ/ 2 Identify at most N − 1 Identify O ( N 2 ) uncorrelated sources using N sensors. ✗ sources using O ( N ) sensors. ✓ 2 Alan T Moffet. “Minimum-redundancy linear arrays”. In: IEEE Trans. Antennas Propag. 16.2 (1968), pp. 172–175. 3 Piya Pal and P . P . Vaidyanathan. “Nested Arrays: A Novel Approach to Array Processing With Enhanced Degrees of Freedom”. In: IEEE Trans. Signal Process. 58.8 (2010), pp. 4167–4181. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 6 / 24
Introduction Motivation Tensor Model 4,5, etc. Measurements Vector Model Tensor Model x = X = Spatial/temporal relations Spatial/temporal relations are mixed ✗ are separated ✓ 4 M. Haardt, F. Roemer, and G. Del Galdo. “Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems”. In: IEEE Trans. Signal Process. 56.7 (2008), pp. 3198–3213. 5 D. Nion and N.D. Sidiropoulos. “Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar”. In: IEEE Trans. Signal Process. 58.11 (2010), pp. 5693–5705. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 7 / 24
Introduction Motivation Main Goal of this Work Proposed Scheme Azimuth Input Sparse Tensor Tensor Elevation arrays Models MUSIC Doppler etc. . . Related work: ULA, tensors, and MUSIC ⇒ DOA and polarization 6,7 . Nested arrays, tensors, and MUSIC ⇒ azimuth, elevation, and polarization 8 . 6 Sebastian Miron, Nicolas Le Bihan, and Jerome I Mars. “Vector-Sensor MUSIC for Polarized Seismic Sources Localization”. In: EURASIP Journal on Advances in Signal Processing 2005.1 (2005), pp. 74–84. 7 M. Boizard et al. “Numerical performance of a tensor MUSIC algorithm based on HOSVD for a mixture of polarized sources”. In: Proc. European Signal Process. Conf. 2013, pp. 1–5. 8 Keyong Han and A. Nehorai. “Nested Vector-Sensor Array Processing via Tensor Modeling”. In: IEEE Trans. Signal Process. 62.10 (2014), pp. 2542–2553. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 8 / 24
Introduction Tensors Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 9 / 24
Introduction Tensors Notations 9 A Tensor A B = Outer product A ◦ B A � � , B = A Inner product � A , B � = U A n -mode product A × n U A × 1 U 9 Tamara G. Kolda and Brett W. Bader. “Tensor Decompositions and Applications”. In: SIAM Review 51.3 (2009), pp. 455–500. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 10 / 24
Introduction Tensors Tensor Decomposition 10 CANDECOMP/PARAFAC (CP) decomposition: X ≈ � R r =1 a r ◦ b r ◦ c r . High-order SVD (HOSVD): X ≈ G × 1 A × 2 B × 3 C . 10 Tamara G. Kolda and Brett W. Bader. “Tensor Decompositions and Applications”. In: SIAM Review 51.3 (2009), pp. 455–500. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 11 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 12 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 13 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Sparse Array Processing Vector Model : Estimate Auto- Hermitian � � x S ( k ) � � � θ i , R S x D R covariance correlation Toeplitz MUSIC . . . matrix 11 matrix vector Physical array S Difference coarray D Tensor Model (Proposed) : Estimate Auto- � � � � X S ( k ) � R S X D θ i , R Coarray Tensor covariance correlation . . . tensor MUSIC tensor tensor Existing Proposed 11 S.U. Pillai, et al. “A new approach to array geometry for improved spatial spectrum estimation”. Proc. IEEE 73.10 (1985); C.-L. Liu and P . P . Vaidyanathan. “Remarks on the Spatial Smoothing Step in Coarray MUSIC”. IEEE SPL 22.9 (2015). Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 14 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Some Discussions on the Coarray Tensor � R Tensor model Vector model 12 � � R � p 1 ,p 2 ,...,p R ,p ′ 1 ,p ′ 2 ,...,p ′ R � � = � � R � p 1 ,p ′ 1 = � � x D � m 1 , X D � m 1 ,m 2 ,...,m R , p 1 − p ′ p r − p ′ 1 = m 1 . r = m r , r = 1 , 2 , . . . , R. � R avoids implementing spatial smoothing in tensors. � R admits the (tensor) MUSIC algorithm. 12 S.U. Pillai, et al. “A new approach to array geometry for improved spatial spectrum estimation”. Proc. IEEE 73.10 (1985); C.-L. Liu and P . P . Vaidyanathan. “Remarks on the Spatial Smoothing Step in Coarray MUSIC”. IEEE SPL 22.9 (2015). Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 15 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum Outline Introduction 1 Motivation Tensors Contribution: Tensor MUSIC in Multidimensional Sparse Arrays 2 Coarray tensor Tensor MUSIC spectrum Numerical Examples 3 4 Concluding Remarks Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 16 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum Tensor MUSIC MUSIC Tensor MUSIC 13 1 Eigen- 1 HOSVD: decomposition: R = � � K × 1 � U 1 × 2 � U 2 · · · × R � U R R = � � U � Λ � U H . × R +1 � U ∗ 1 × R +2 � U ∗ 2 · · · × 2 R � U ∗ R . 2 Signal and noise 2 Signal and noise subspace: subspace: � � � � � � � � � � U r = is a unitary matrix. U = U r,s U r,n U s U n 3 Tensor MUSIC spectrum 3 MUSIC spectrum: 1 P HOSV D (¯ µ ) = P (¯ θ ) = � � n v (¯ U H θ ) � 2 1 v (¯ θ ) : steering µ ) × 1 � U 1 ,n � 1 ,n . . . × R � U R,n � U H U H R,n � 2 � V (¯ F vectors. V (¯ µ ) : steering tensors. 13 M. Boizard et al. “Numerical performance of a tensor MUSIC algorithm based on HOSVD for a mixture of polarized sources”. In: Proc. European Signal Process. Conf. 2013, pp. 1–5. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 17 / 24
Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum Problem with tensor MUSIC via HOSVD Our observation: P HOSV D (¯ µ ) is a separable MUSIC spectrum R � 1 µ ( r ) ) , µ ( r ) ) = P HOSV D (¯ µ ) = P r (¯ P r (¯ � � µ ( r ) ) � 2 U H r,n v U + r (¯ r =1 2 P HOSV D (¯ µ ) has cross-terms • • Actual µ (2) ¯ P HOSV D (¯ µ ) • µ (1) ¯ Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 18 / 24
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