Scaling the time and Fourier domains to align periodically and their convolution

TL;DR

This paper presents a method for aligning periodic signals with their Fourier transforms through time and frequency scaling, facilitating new algorithm development such as pitch estimation, and explores convolution in the combined frequency-time domain.

Contribution

It introduces a novel approach to align periodic signals with their Fourier transforms using scaling, and discusses convolution in the frequency-time domain.

Findings

Alignment of signals via scaling is feasible and effective.

The method enables potential improvements in pitch estimation algorithms.

Convolution in the frequency-time domain is characterized and utilized.

Abstract

This note shows how to align a periodic signal with its the Fourier transform by means of frequency or time scaling. This may be useful in developing new algorithms, e.g. for pitch estimation. This note also convolves the signals and the frequency time convolution is denoted fxt.