Convolution and Combination Matrices in Non-stationary Filtering

TL;DR

The paper addresses non-stationary filtering where time-dependent masks destroy the circulant structure. It introduces the cyclic convolution matrix conv(C) and the cyclic combination matrix comb(C), and connects their time-domain definitions to a full Fourier-domain representation via $V^*conv(C)V = comb(F)$ with $F=(1/N)\,DFT2(C^T)$. Key results include a precise link between conv and comb (Proposition 1.1), fast FFT-based computations for rank-one masks, and a detailed treatment of how real masks induce Fourier conjugate symmetry in the frequency response. The work provides practical, undergraduate-accessible techniques for non-stationary frequency filtering and demonstrates the theory with explicit constructions and examples, bridging theory and computation in a linear-algebraic framework.

Abstract

Time independent convolution yields circulant matrices whose eigenvectors are the Fourier exponentials with the eigenvalues being the Fourier transform of the mask. The case of time dependent convolution, the non-stationary case, no longer has this property and two matrices are then introduced, the cyclic convolution matrix and the cyclic combination matrix. In our paper, we prove results on the properties of these matrices. We give results in the context of the non-stationary frequency response in the Fourier domain, where the Fourier matrix is a full matrix in general. The techniques used here are attainable at the advanced undergraduate linear algebra settings and can be introduced into a relevant linear algebra undergraduate course.