Iterative Approximation of Analytic Eigenvalues of a Parahermitian Matrix EVD

Abstract

We present an algorithm that extracts analytic eigenvalues from a parahermitian matrix. Operating in the discrete Fourier transform domain, an inner iteration re-establishes the lost association between bins via a maximum likelihood sequence detection driven by a smoothness criterion. An outer iteration continues until a desired accuracy for the approximation of the extracted eigenvalues has been achieved. The approach is compared to existing algorithms.

Sample Space-time Covariance Matrix Estimation

“…A spectrally majorised, not necessarily analytic version of this factorisation is the McWhirter decomposition [3], which approximates the factorisation by polynomial paraunitary and diagonal parahermitian matrices. A number of algorithms for the latter have emerged [3][4][5][6][7][8][9][10] and in turn triggered various applications ranging from broadband multiple-input and multipleoutput (MIMO) systems [11,12], to coding [13], beamforming [14,15], source separation [16] and angle of arrival estimation [17,18], to name but a few.…”

Measuring Smoothness of Real-Valued Functions Defined by Sample Points on the Unit Circle

“…1. In comparison, discrete Fourier transform (DFT) domain algorithms [10]- [13] can permit a choice to extract approximations of both spectrally majorised and analytic solutions.…”

“…In [10]- [12] this association is based on the continuity of eigenvectors, which in principle is easier to detect than a non-differentiability of eigenvalues. The association decisions are most crucial near Q-fold algebraic multiplicities of eigenvalues, where eigenvectors can be arbitrarily selected as an orthogonal basis within a Q-dimensional subspace [2], thus creating challenges for an eigenvector-driven association [13].…”

“…Since analyticity implies infinite differentiability, in this paper we explore a suitable cost function that can distinguish between analytic and alternative (such as spectrally majorised) solutions: the power of derivatives of a function F (e jΩ ). In [20], we have explored this metric and successfully applied it to drive an analytic eigenvalue extraction in [13]. The extraction algorithm in [13] aims to create associations for maximally smooth functions from an initially small but iteratively increasing number of sample points, similar to the 'missing samples problem' [21].…”

Efficient Implementation of Iterative Polynomial Matrix EVD Algorithms Exploiting Structural Redundancy and Parallelisation

“…Existing PEVD algorithms include second-order sequential best rotation (SBR2) [5], sequential matrix diagonalisation (SMD) [30], and various evolutions of both algorithm families [31]- [33]. Different from fixed order time domain PEVD schemes in [34], [35] and DFT-based approaches in [36]- [38], the SBR2 and SMD algorithm families have proven convergence. Both SBR2 and SMD algorithms employ iterative time domain schemes to approximately diagonalise a parahermitian matrix, and encourage -or even guarantee [39] -spectral majorisation such that the power spectral densities (PSDs) of the resulting eigenvalues are ordered at all frequencies [7].…”