Iterative Algorithms for Polynomial Eigenvalue Decomposition and Applications Stephan Weiss UDRC — Loughborough, Surrey, Strathclyde & Cardiff Consortium Department of Electonic & Electrical Engineering University of Strathclyde, Glasgow, Scotland, UK UDRC Summer School, Heriot-Watt University, 23-27 June 2014 With many thanks to: J.G. McWhirter, S. Redif, C.H. Ta, W. Al-Hanafy, S. Lambotharan, M. Alrmah, J. Corr, K. Thompson and I.K. Proulder
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Presentation Overview 1. Overview 2. Eigenvalue (EVD) and singular value decomposition (SVD) 3. Narrowband source separation 4. Broadband problem 5. Polynomial matrices and basic operations 6. Polynomial eigenvalue decomposition algorithms 6.1 sequential best rotation (SBR2) 6.2 sequential matrix diagonalisation (SMD) 7. Applications 7.1 broadband / polynomial subspace decomposition 7.2 polynomial MUSIC 8. Summary
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Model ◮ Scenario with sensor array and far-field sources: x 1 [ n ] s 1 [ n ] x 2 [ n ] x 3 [ n ] x M [ n ] ◮ for the narrowband case, the source signals arrive with delays, expressed by phase shifts in a steering vector ◮ data model: x [ n ] =
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Model ◮ Scenario with sensor array and far-field sources: x 1 [ n ] s 1 [ n ] x 2 [ n ] x 3 [ n ] x M [ n ] ◮ for the narrowband case, the source signals arrive with delays, expressed by phase shifts in a steering vector s 1 ◮ data model: x [ n ] = s 1 [ n ] · s 1
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Model ◮ Scenario with sensor array and far-field sources: x 1 [ n ] s 1 [ n ] x 2 [ n ] x 3 [ n ] s 2 [ n ] x M [ n ] ◮ for the narrowband case, the source signals arrive with delays, expressed by phase shifts in a steering vector s 1 ◮ data model: x [ n ] = s 1 [ n ] · s 1
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Model ◮ Scenario with sensor array and far-field sources: x 1 [ n ] s 1 [ n ] x 2 [ n ] x 3 [ n ] s 2 [ n ] x M [ n ] ◮ for the narrowband case, the source signals arrive with delays, expressed by phase shifts in a steering vector s 1 , s 2 ◮ data model: x [ n ] = s 1 [ n ] · s 1 + s 1 [ n ] · s 2
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Model ◮ Scenario with sensor array and far-field sources: x 1 [ n ] s 1 [ n ] x 2 [ n ] s R [ n ] x 3 [ n ] s 2 [ n ] x M [ n ] ◮ for the narrowband case, the source signals arrive with delays, expressed by phase shifts in a steering vector s 1 , s 2 ◮ data model: x [ n ] = s 1 [ n ] · s 1 + s 1 [ n ] · s 2
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Model ◮ Scenario with sensor array and far-field sources: x 1 [ n ] s 1 [ n ] x 2 [ n ] s R [ n ] x 3 [ n ] s 2 [ n ] x M [ n ] ◮ for the narrowband case, the source signals arrive with delays, expressed by phase shifts in a steering vector s 1 , s 2 , . . . s R ; ◮ data model: R � x [ n ] = s 1 [ n ] · s 1 + s 1 [ n ] · s 2 + · · · + s R [ n ] · s R = s r [ n ] · s r r =1
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Steering Vector ◮ A signal s [ n ] arriving at the array can be characterised by the delays of its wavefront (neglecting attenuation): x 0 [ n ] s [ n − τ 0 ] δ [ n − τ 0 ] x 1 [ n ] s [ n − τ 1 ] δ [ n − τ 1 ] = = ∗ s [ n ] ◦ — • a ϑ ( z ) S ( z ) . . . . . . . . . x M − 1 [ n ] s [ n − τ M − 1 ] δ [ n − τ M − 1 ] ◮ if evaluated at a narrowband normalised angular frequency Ω i , the time delays τ m in the broadband steering vector a ϑ ( z ) collapse to phase shifts in the narrowband steering vector a ϑ, Ω i , e − jτ 0 Ω i e − jτ 1 Ω i a ϑ, Ω i = a ϑ ( z ) | z = e j Ω i = . . . . e − jτ M − 1 Ω i
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Data and Covariance Matrices ◮ A data matrix X ∈ C M × L can be formed from L measurements: � � X = x [ n ] x [ n + 1] x [ n + L − 1] . . . ◮ assuming that all x m [ n ] , m = 1 , 2 , . . . M are zero mean, the (instantaneous) data covariance matrix is ≈ 1 � � x [ n ] x H [ n ] L XX H R = E where the approximation assumes ergodicity and a sufficiently large L ; ◮ Problem: can we tell from X or R (i) the number of sources and (ii) their orgin / time series? ◮ w.r.t. Jonathon Chamber’s introduction, we here only consider the underdetermined case of more sensors than sources, M ≥ K , and generally L ≫ M .
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion SVD of Data Matrix ◮ Singular value decomposition of X : X U Σ V H = ◮ unitary matrices U = [ u 1 . . . u M ] and V = [ v 1 . . . v L ] ; ◮ diagonal Σ contains the real, positive semidefinite singular values of X in descending order: σ 1 0 . . . 0 0 . . . 0 . . . ... . . . 0 . . . σ 2 Σ = . . . ... ... . . . . 0 . . 0 0 σ M 0 . . . 0 with σ 1 ≥ σ 2 ≥ · · · ≥ σ M ≥ 0 .
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Singular Values ◮ If the array is illuminated by R ≤ M linearly independent sources, the rank of the data matrix is rank { X } = R ◮ only the first R singular values of X will be non-zero; ◮ in practice, noise often will ensure that rank { X } = M , with M − R trailing singular values that define the noise floor: 1 0.8 0.6 σ m 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 ordered index m ◮ therefore, by thresholding singular values, it is possible to estimate the number of linearly independent sources R .
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Subspace Decomposition ◮ If rank { X } = R , the SVD can be split: � Σ s � � V H � 0 s X = [ U s U n ] V H 0 Σ n n ◮ with U s ∈ C M × R and V H s ∈ C R × L corresponding to the R largest singular values; ◮ U s and V H s define the signal-plus-noise subspace of X : M R � � σ m u m v H σ m u m v H X = m ≈ m m =1 m =1 ◮ the complements U n and V H n , U H V s V H s U n = 0 n = 0 , define the noise-only subspace of X .
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion SVD via Two EVDs ◮ Any Hermitian matrix A = A H allows an eigenvalue decomposition A = QΛQ H with Q unitary and the eigenvalues in Λ real valued and positive semi-definite; ◮ postulating X = UΣV H , therefore: XX H ( UΣV H )( VΣ H U H ) = UΛU H = (1) X H X ( VΣ H U H )( UΣV H ) = VΛV H = (2) ◮ (ordered) eigenvalues relate to the singular values: λ m = σ 2 m ; ◮ the covariance matrix R = 1 L XX has the same rank as the data matrix X , and with U provides access to the same spatial subspace decomposition.
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband MUSIC Algorithm ◮ EVD of the narrowband covariance matrix identifies signal-plus-noise and noise-only subspaces � Λ s � � U H � 0 s R = [ U s U n ] U H 0 Λ n n ◮ scanning the signal-plus-noise subspace could only help to retrieve sources with orthogonal steering vectors; ◮ therefore, the multiple signal classification (MUSIC) algorithm scans the noise-only subspace for minima, or maxima of its reciprocal 1 S MUSIC ( ϑ ) = � U n a ϑ, Ω i � 2 2 S MUSIC ( ϑ ) /[dB] 0 −20 −40 −80 −60 −40 −20 0 20 40 60 80 ϑ / [o]
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Narrowband Source Separation ◮ Via SVD of the data matrix X or EVD of the covariance matrix R , we can determine the number of linearly independent sources R ; ◮ using the subspace decompositions offered by EVD/SVD, the directions of arrival can be estimated using e.g. MUSIC; ◮ based on knowledge of the angle of arrival, beamforming could be applied to X to extract specific sources; ◮ overall: EVD (and SVD) can play a vital part in narrowband source separation; ◮ what about broadband source separation?
Overview E&SVD Narrowband Source Separation Broadband Poly PEVD Apps Simulations Conclusion Broadband Array Scenario s 1 [ n ] x 0 [ n ] x 1 [ n ] x 2 [ n ] x M − 1 [ n ] ◮ Compared to the narrowband case, time delays rather than phase shifts bear information on the direction of a source.
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