An Improved DOA Estimator Based On Partial Relaxation Approach Minh Trinh Hoang, Mats Viberg and Marius Pesavento Communication Systems Group Department of Electrical Engineering Darmstadt University of Technology Chalmers University of Technology Darmstadt, Germany Gothenburg, Sweden April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 1
Motivation ◮ Wide application of DOA estimation ◮ Multiple families of DOA estimators: ◮ Maximum likelihood estimators ◮ Subspace-based estimators ◮ ... ◮ Proposal of a DOA estimator under the Partial Relaxation approach ◮ Closely related to conventional DOA estimators ◮ Efficient implementation for updating eigenvalues April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 2
Table of Contents Motivation Signal Model Partial Relaxation Approach Computational Aspects Simulation Results Conclusions and Outlook April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 3
Signal Model Multiple Snapshots Model X = A ( θ ) S + N . . . Source 1 Source N ◮ T : Number of available snapshots ◮ X ∈ C M × T : Received signal matrix ◮ S ∈ C N × T : Source signal matrix θ N θ 1 ◮ N ∈ C M × T : Sensor noise matrix Arbitrary array with M sensors ◮ A ( θ ) = [ a ( θ 1 ), ..., a ( θ N )] ∈ C M × N : Steering matrix Array Manifold A ∈ C M × N | A = [ a ( ϑ 1 ), ... , a ( ϑ N )] with 0 ≤ ϑ 1 < ... < ϑ N < 180 ◦ � � A N = April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 4
Signal Model Covariance Matrix R = AR s A H + σ 2 n I M ◮ R = E � x ( t ) x ( t ) H � ∈ C M × M : Covariance matrix of the received signal ◮ R s = E � s ( t ) s ( t ) H � ∈ C N × N : Covariance matrix of the transmitted signal ◮ σ 2 n : Noise power at the sensors Sample Covariance Matrix R = 1 T XX H = ˆ H H ˆ U s ˆ Λ s ˆ s + ˆ U n ˆ Λ n ˆ U U n ◮ Signal subspace spanned by ˆ ◮ Noise subspace spanned by ˆ U s U n � ˆ λ 1 , ... , ˆ � ◮ ( M − N ) smallest eigenvalues ◮ N largest eigenvalues λ N � ˆ λ N +1 , ... , ˆ � of ˆ R are contained in ˆ of ˆ R are contained in ˆ λ M Λ n Λ s April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 5
Table of Contents Motivation Signal Model Partial Relaxation Approach Computational Aspects Simulation Results Conclusions and Outlook April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 6
Partial Relaxation Approach Revision of Conventional DOA Estimators General Formulation � ˆ � A = arg min f ( A ) A ∈A N Remarks ◮ A N : Highly structured and non-convex set ◮ f ( · ): Generally non-convex function with multiple local minima ◮ Highly computational cost to obtain the global minimum Example: Deterministic Maximum Likelihood (DML) Estimator � ˆ � − 1 A H � �� � � � A H A ˆ I M − A A DML = arg min tr R A ∈A N April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 7
Partial Relaxation Approach Revision of Conventional DOA Estimators General Formulation � ˆ � A = arg min f ( A ) A ∈A N Remarks ◮ A N : Highly structured and non-convex set ◮ f ( · ): Generally non-convex function with multiple local minima ◮ Highly computational cost to obtain the global minimum Example: Covariance Fitting (CF) Estimator � ˆ � 2 � � �� � ˆ R − AR s A H � �� A CF = arg min min F A ∈A N R s � 0 April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 7
Partial Relaxation Approach Revision of Conventional DOA Estimators General Formulation � ˆ � A = arg min f ( A ) A ∈A N Remarks ◮ A N : Highly structured and non-convex set ◮ f ( · ): Generally non-convex function with multiple local minima ◮ Highly computational cost to obtain the global minimum Objective: Find a suboptimal solution without substantial performance degradation April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 7
Partial Relaxation Approach Concept General Formulation of Conventional Estimators � ˆ � A = arg min f ( A ) A ∈A N Relaxed Array Manifold ¯ � A ∈ C M × N | A = [ a ( ϑ ), B ] , a ( ϑ ) ∈ A 1 , B ∈ C M × ( N − 1) � A N = Partial Relaxation A = [ a , B ] ∈ ¯ ¯ A ∈ A N A N April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 8
Partial Relaxation Approach Concept General Formulation of Conventional Estimators � ˆ � A = arg min f ( A ) A ∈A N Relaxed Array Manifold ¯ � A ∈ C M × N | A = [ a ( ϑ ), B ] , a ( ϑ ) ∈ A 1 , B ∈ C M × ( N − 1) � A N = Formulation of the Partial Relaxation (PR) Approach a PR } = N arg min { ˆ B ∈ C M × ( N − 1) f ([ a , B ]) min a ∈A 1 ◮ Relax the manifold structure of the signals from interfering directions ◮ Grid-search for N -deepest local minima to obtain the estimated DOAs April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 8
Partial Relaxation Approach Proposed Estimators Formulation of PR-Constrained Covariance Fitting (PR-CCF) s aa H − EE H � � 2 a PR-CCF } = N arg min � ˆ � �� R − σ 2 �� { ˆ min F σ 2 s ≥ 0, E a ∈A 1 s aa H − EE H � 0 subject to ˆ R − σ 2 subject to rank( E ) ≤ N − 1 PR-CCF Estimator M � � 1 a PR-CCF } = N arg min � λ 2 aa H ˆ { ˆ R − k − 1 a a H ˆ R a ∈A 1 k = N Remarks ◮ Not applicable if ˆ R is singular ◮ Eigenvalues are extensively required April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 9
Partial Relaxation Approach Proposed Estimators PR-Unconstrained Covariance Fitting (PR-UCF) s aa H − EE H � � 2 a PR-UCF } = N arg min � ˆ �� � R − σ 2 �� { ˆ min F σ 2 s ≥ 0, E a ∈A 1 subject to rank( E ) ≤ N − 1 Equivalent formulation of the inner optimization M � ˆ � λ 2 R − σ 2 s aa H � min k σ 2 s ≥ 0 k = N ◮ No closed-form solution for the minimizer ˆ σ 2 s ,U � ˆ ◮ λ 2 R − σ 2 s aa H � is continuously differentiable with respect to σ 2 k s Minimization by Bisection Search or Newton’s Method possible April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 10
Table of Contents Motivation Signal Model Partial Relaxation Approach Computational Aspects Simulation Results Conclusions and Outlook April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 11
Computational Aspects Core Numerical Problem Objective: Efficient computation of the eigenvalue decomposition D − ρ zz H = ¯ H U ¯ D ¯ U ◮ D = diag( d 1 , ... , d K ) ∈ R K × K with d 1 > ... > d K ◮ ρ > 0 ◮ z = [ z 1 , ... , z K ] T ∈ C K × 1 has no zero component d K ) ∈ R K × K with ¯ D = diag( ¯ d 1 , ... , ¯ d 1 > ... > ¯ ◮ ¯ d K u K ] ∈ C K × K contains the normalized eigenvectors ¯ ◮ ¯ U = [¯ u 1 , ... , ¯ u k associated with the eigenvalues ¯ d k April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 12
Computational Aspects Core Numerical Problem Interlacing Property a) The modified eigenvalues ¯ d k satisfy h ( ¯ d k ) = 0 where the secular function h ( x ) is defined as: h ( x ) = 1 − ρ z H ( D − x I ) − 1 z K | z k | 2 � = 1 − ρ d k − x k =1 b) The modified eigenvalues ¯ d k down-interlace with the initial eigenvalues d k d 1 > ¯ d 1 > d 2 > ¯ d 2 > ... > d K > ¯ d K Objective: Determine ¯ d k which satisfies h ( ¯ d k ) = 0 April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 13
Computational Aspects Iterative Algorithm Rewriting the secular function for determining ¯ d k k K | z i | 2 | z i | 2 � � 0 = 1 − d i − x − d i − x i =1 i = k +1 k | z i | 2 K | z i | 2 � � ⇐ ⇒ d i − x = 1 − d i − x i =1 i = k +1 ⇐ ⇒ − ψ k ( x ) = 1 + φ k ( x ) Idea: Approximate ψ k and φ k with a rational function of first degree q p + if 0 ≤ k ≤ K − 1 d k +1 − x R k ; p , q ( x ) = 0 if k = K , April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 14
Computational Aspects Iterative Algorithm Rational Approximation − ψ k ( x ) 1 + φ k ( x ) × ¯ d k x April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 15
Computational Aspects Iterative Algorithm Rational Approximation − ψ k ( x ) 1 + φ k ( x ) ◦ × ◦ ¯ x ( τ ) d k x April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 15
Computational Aspects Iterative Algorithm Rational Approximation − ψ k ( x ) 1 + φ k ( x ) − R k − 1; p , q ( x ) 1 + R k ; r , s ( x ) ◦ × × ◦ ¯ x ( τ ) x ( τ +1) d k x April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 15
Computational Aspects Iterative Algorithm Rational Approximation − ψ k ( x ) 1 + φ k ( x ) ◦ × ◦ ¯ x ( τ +1) d k x ◮ Closed-form update ◮ Quadratic rate of convergence April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 15
Table of Contents Motivation Signal Model Partial Relaxation Approach Computational Aspects Simulation Results Conclusions and Outlook April 18, 2018 | NTS TUD | Minh Trinh Hoang, Mats Viberg and Marius Pesavento | 16
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