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Time invariant property of weighted circular convolution and its application to continuous wavelet transform

Yi Hua, Ru Yu-Le, Dai Yin-Yun

Bulletin of the Polish Academy of Sciences. Technical Sciences

2021

Vol. 69, nr 4

art. no. e137726

Time invariant linear operators are the building blocks of signal processing. Weighted circular convolution and signal processing framework in a generalized Fourier domain are introduced by Jorge Martinez. In this paper, we prove that under this new signal processing framework, weighted circular convolution also has a generalized time invariant property. We also give an application of this property to algorithm of continuous wavelet transform (CWT). Specifically, we have previously studied the algorithm of CWT based on generalized Fourier transform with parameter 1. In this paper, we prove that the parameter can take any complex number. Numerical experiments are presented to further demonstrate our analyses.

Two relations for generalized discrete Fourier transform coefficients

Spałek D.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2018

Vol. 66, nr 3

275--281

A new generalized discrete Fourier transform DFT that allows for sample shift δ 2 [0, T/N] in time-domain is defined. Two relations are proved for the sum of errors between generalized DFT coefficients and theirs theoretical values. The first is the equation for samples received for continuous and piecewise–smooth functions. The second relation is the inequality for samples generated by discontinuous functions. Moreover, the influence of samples shift on generalized DFT coefficients values, which leads to aliasing phenomenon, is presented.