Smooth QR Decomposition of Polynomial Matrices

This paper presents a novel algorithm for determining a compact order QR decomposition of a polynomial matrix, where both the
$\mathrm{Q}$
and
$\mathrm{R}$
factors themselves are approximated by polynomial matrices. The QR factorisation is subject to an allpass ambiguity; existing time domain methods can lead to factorisations of high order. The proposed algorithm performs the conventional QR decomposition the discrete Fourier transform domain. Subsequently, it establishes phase coherence between adjacent bins through a phase smoothing procedure, aimed at obtaining compact-order factors. The method is validated through experiments over an ensemble of randomized polynomial matrices and shown to outperform state-of-the-art algorithms.