Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvalues

Abstract

An analytic parahermitian matrix admits an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors except in the case of multiplexed data. In this paper, we propose an iterative algorithm for the estimation of the analytic eigenvalues. Since these are generally transcendental, we find a polynomial approximation with a defined error. Our approach operates in the discrete Fourier transform (DFT) domain and for every DFT length generates a maximally smooth association through EVDs evaluated in DFT bins; an outer loop iteratively grows the DFT order and is shown, in general, to converge to the analytic eigenvalues. In simulations, we compare our results to existing approaches.

Detection of Weak Transient Signals Using a Broadband Subspace Approach

“…The evaluation of the discrimination is thresholdindependent, and future work would have to focus on setting a suitable detection threshold for a hypothesis test on the absence or presence of a transient signal. Also, the computational burden of the processor rests on the order of the polynomial factorisation of the space-time covariance matrix, with shorter order decompositions emerging [20], [21].…”

“…Similar to narrowband subspace methods, a diagonalisation of this spacetime covariance is required to access a subspace decomposition akin to the narrowband case. For the broadband problem, we are looking towards polynomial EVD methods that can decouple the space-time covariance for every lag value [14] -such decompositions have been shown to exist in most case [15], [16] and a number of algorithms have been developed to solves this diagonalisation often with guaranteed convergence [14], [17]- [21].…”

Support Estimation of Analytic Eigenvectors of Parahermitian Matrices

“…While DFT-based methods are superior to time-domain approaches in terms of order of the extracted factors, the algorithms themselves do not scale well with the spatial dimension -in case of the eigenvalues [28], [32] -and the temporal dimension -in case of the eigenvectors [30], [33] of the input para-hermitian matrix. For the eigenvectors [30], [33], the computational bottleneck is a phase-smoothing operation to establish phase coherence between eigenvectors across frequency bins [28], [30], [31], which is non-convex in nature [30] and NP-hard to solve [33]. The DFT size, which is directly related to the time-domain support of the analytic eigenvectors, therefore determines the algorithm complexity.…”

Polynomial Eigenvalue Decomposition-Based Target Speaker Voice Activity Detection in the Presence of Competing Talkers

“…In [20], a broadband subspace approach is used to detect weak transient signals. The approach uses an iterative polynomial matrix eigenvalue decomposition (PEVD) algorithm such as the family of second-order sequential best rotation (SBR2) [21,22] and sequential matrix diagonalization (SMD) approaches [23,24] in the time-domain or [25,26] in the frequency-domain. PEVD algorithms have been found useful for many broadband signal processing applications such as speech enhancement [27,28], source separation [29,30], source localizaton [31,32] and beamforming [33].…”