Eigenspace estimation for source localization using large random - PowerPoint PPT Presentation

Eigenspace estimation for source localization using large random matrices Pascal Vallet (1) Joint work with Philippe Loubaton (1) and Xavier Mestre (2) (1) LabInfo IGM (CNRS-UMR 8049) / Université Paris-Est (2) Centre Tecnologic de Telecomunicacions de Catalunya (CTTC) / Barcelona

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations Table of Contents 1 Introduction 2 Random matrix theory results Consistent estimation of eigenspace 3 Numerical evaluations 4 2 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We will assume that K source signals are received by an antenna array of M elements, and K < M . At time n , we receive y n = As n + v n , with A = [ a ( θ 1 ),..., a ( θ K )] the M × K "steering vectors" matrix with a ( θ 1 ),..., a ( θ K ) linearly independent. s n = [ s 1, n ,..., s n , K ] the vector of non-observable transmitted signals, assumed deterministic, v n a gaussian white noise (zero mean, covariance σ 2 I M ). θ 1 ,..., θ K are the parameters of interest of the K sources, it can be either frequencies, direction of arrival (DoA)... 3 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We will assume that K source signals are received by an antenna array of M elements, and K < M . At time n , we receive y n = As n + v n , with A = [ a ( θ 1 ),..., a ( θ K )] the M × K "steering vectors" matrix with a ( θ 1 ),..., a ( θ K ) linearly independent. s n = [ s 1, n ,..., s n , K ] the vector of non-observable transmitted signals, assumed deterministic, v n a gaussian white noise (zero mean, covariance σ 2 I M ). θ 1 ,..., θ K are the parameters of interest of the K sources, it can be either frequencies, direction of arrival (DoA)... 4 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We will assume that K source signals are received by an antenna array of M elements, and K < M . At time n , we receive y n = As n + v n , with A = [ a ( θ 1 ),..., a ( θ K )] the M × K "steering vectors" matrix with a ( θ 1 ),..., a ( θ K ) linearly independent. s n = [ s 1, n ,..., s n , K ] the vector of non-observable transmitted signals, assumed deterministic, v n a gaussian white noise (zero mean, covariance σ 2 I M ). θ 1 ,..., θ K are the parameters of interest of the K sources, it can be either frequencies, direction of arrival (DoA)... 5 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We will assume that K source signals are received by an antenna array of M elements, and K < M . At time n , we receive y n = As n + v n , with A = [ a ( θ 1 ),..., a ( θ K )] the M × K "steering vectors" matrix with a ( θ 1 ),..., a ( θ K ) linearly independent. s n = [ s 1, n ,..., s n , K ] the vector of non-observable transmitted signals, assumed deterministic, v n a gaussian white noise (zero mean, covariance σ 2 I M ). θ 1 ,..., θ K are the parameters of interest of the K sources, it can be either frequencies, direction of arrival (DoA)... 6 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We will assume that K source signals are received by an antenna array of M elements, and K < M . At time n , we receive y n = As n + v n , with A = [ a ( θ 1 ),..., a ( θ K )] the M × K "steering vectors" matrix with a ( θ 1 ),..., a ( θ K ) linearly independent. s n = [ s 1, n ,..., s n , K ] the vector of non-observable transmitted signals, assumed deterministic, v n a gaussian white noise (zero mean, covariance σ 2 I M ). θ 1 ,..., θ K are the parameters of interest of the K sources, it can be either frequencies, direction of arrival (DoA)... 7 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We collect N observations of the previous model, stacked in Y N = [ y 1 ,..., y N ], and we can write Y N = AS N + V N with S N and V N built as Y N . The goal is to infer the angles θ 1 ,..., θ K from Y N . There are essentially two common methods: Maximum Likelihood (ML) estimation Subspace method. 8 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We collect N observations of the previous model, stacked in Y N = [ y 1 ,..., y N ], and we can write Y N = AS N + V N with S N and V N built as Y N . The goal is to infer the angles θ 1 ,..., θ K from Y N . There are essentially two common methods: Maximum Likelihood (ML) estimation Subspace method. 9 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations We collect N observations of the previous model, stacked in Y N = [ y 1 ,..., y N ], and we can write Y N = AS N + V N with S N and V N built as Y N . The goal is to infer the angles θ 1 ,..., θ K from Y N . There are essentially two common methods: Maximum Likelihood (ML) estimation Subspace method. 10 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations The ML estimator is given by 1 I M − A ( ω )( A ( ω ) ∗ A ( ω )) − 1 A ( ω ) ∗ � Y N Y ∗ � argmin N Tr N , ω where A ( ω ) is the matrix in which we have replaced [ θ 1 ,..., θ K ] by the variable ω = [ ω 1 ,..., ω K ]. This estimator is consistent when M , N → ∞ , however, it clearly requires a multidimensional optimization. An alternative, requiring a monodimensional search, has been found through the subspace method. 11 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations The ML estimator is given by 1 I M − A ( ω )( A ( ω ) ∗ A ( ω )) − 1 A ( ω ) ∗ � Y N Y ∗ � argmin N Tr N , ω where A ( ω ) is the matrix in which we have replaced [ θ 1 ,..., θ K ] by the variable ω = [ ω 1 ,..., ω K ]. This estimator is consistent when M , N → ∞ , however, it clearly requires a multidimensional optimization. An alternative, requiring a monodimensional search, has been found through the subspace method. 12 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations The ML estimator is given by 1 I M − A ( ω )( A ( ω ) ∗ A ( ω )) − 1 A ( ω ) ∗ � Y N Y ∗ � argmin N Tr N , ω where A ( ω ) is the matrix in which we have replaced [ θ 1 ,..., θ K ] by the variable ω = [ ω 1 ,..., ω K ]. This estimator is consistent when M , N → ∞ , however, it clearly requires a multidimensional optimization. An alternative, requiring a monodimensional search, has been found through the subspace method. 13 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations N A ∗ has K non null Assuming S N has full rank K , then 1 N AS N S ∗ eigenvalues 0 = λ 1, N = ... = λ M − K , N < λ M − K + 1, N < ... < λ M , N . We denote by Π N the projector onto the eigensubspace associated with eigenvalue 0. Since span{ a ( θ 1 ),..., a ( θ K )} is also the eigenspace associated with non null eigenvalues λ M − K + 1, N ,..., λ M , N , it is possible to determine the ( θ k ) k = 1,..., K . MUSIC algorithm The angles θ 1 ,..., θ K are the (unique) solutions of the equation η ( θ ) : = a ( θ ) ∗ Π N a ( θ ) = 0. 14 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations N A ∗ has K non null Assuming S N has full rank K , then 1 N AS N S ∗ eigenvalues 0 = λ 1, N = ... = λ M − K , N < λ M − K + 1, N < ... < λ M , N . We denote by Π N the projector onto the eigensubspace associated with eigenvalue 0. Since span{ a ( θ 1 ),..., a ( θ K )} is also the eigenspace associated with non null eigenvalues λ M − K + 1, N ,..., λ M , N , it is possible to determine the ( θ k ) k = 1,..., K . MUSIC algorithm The angles θ 1 ,..., θ K are the (unique) solutions of the equation η ( θ ) : = a ( θ ) ∗ Π N a ( θ ) = 0. 15 / 68

Introduction Random matrix theory results Consistent estimation of eigenspace Numerical evaluations N A ∗ has K non null Assuming S N has full rank K , then 1 N AS N S ∗ eigenvalues 0 = λ 1, N = ... = λ M − K , N < λ M − K + 1, N < ... < λ M , N . We denote by Π N the projector onto the eigensubspace associated with eigenvalue 0. Since span{ a ( θ 1 ),..., a ( θ K )} is also the eigenspace associated with non null eigenvalues λ M − K + 1, N ,..., λ M , N , it is possible to determine the ( θ k ) k = 1,..., K . MUSIC algorithm The angles θ 1 ,..., θ K are the (unique) solutions of the equation η ( θ ) : = a ( θ ) ∗ Π N a ( θ ) = 0. 16 / 68