Enforcing Eigenvector Smoothness for a Compact DFT-Based Polynomial Eigenvalue Decomposition

Abstract

A variety of algorithms have been developed to compute an approximate polynomial matrix eigenvalue decomposition (PEVD). As an extension of the ordinary EVD to polynomial matrices, the PEVD will generate paraunitary matrices that diagonalise a parahermitian matrix. While iterative PEVD algorithms that compute a decomposition in the time domain have received a great deal of focus and algorithmic improvements in recent years, there has been less research in the field of frequency-based PEVD algorithms. Such algorithms have shown promise for the decomposition of problems of finite order, but the state-of-the-art requires a priori knowledge of the length of the polynomial matrices required in the decomposition. This paper presents a novel frequency-based PEVD algorithm which can compute an accurate decomposition without requiring this information. Also presented is a new metric for measuring a function's smoothness on the unit circle, which is utilised within the algorithm to maximise eigenvector smoothness for a compact decomposition, such that the polynomial eigenvectors have low order. We demonstrate through the use of simulations that the algorithm can achieve superior levels of decomposition accuracy to a state-of-the-art frequency-based method.