Convolution and Combination Matrices in Non-stationary Filtering

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.