Studies on Generalized Fourier Representations and Phase Transforms

TL;DR

This paper introduces the generalized Fourier representation (GFR) and phase transform (PT), providing new methods for phase manipulation, fractional derivatives, and wavelet-based phase analysis with efficient FFT implementation.

Contribution

It presents the GFR framework, derives the PT kernel, and develops wavelet phase and quadrature transforms, advancing phase analysis and signal processing techniques.

Findings

PT enables arbitrary phase shifts and constant time-delays.

GFR allows fractional derivatives and integrals of signals.

Wavelet transforms facilitate phase manipulation in time-frequency analysis.

Abstract

Fourier representation (FR) is an indispensable mathematical formulation for modeling and analysis of physical phenomenon, engineering systems and signals in numerous applications. In this study, we present the generalized Fourier representation (GFR) that is completely based on the FR of a signal, and introduce the phase transform (PT) which is a special case of the GFR and a true generalization of the Hilbert transform. We derive the PT kernel to obtain any constant phase shift, discuss the various properties of the PT, and demonstrate that (i) a constant phase shift in a signal corresponds to variable time-delays in all harmonics, (ii) to obtain a constant time-delay in a signal, one need to provide variable phase shift in all harmonics, (iii) a constant phase shift is same as the constant time-delay only for single frequency sinusoid. The time derivative and time integral, including fractional order, of a signal can be obtained using the GFR. We propose to use discrete cosine transform (DCT) based implementation to avoid end artifacts due to discontinuities present in both end of the signal. We introduce fractional delay of a discrete time signal using the FR, and present the fast Fourier transform (FFT) implementation of all the above proposed representations. Using the analytic wavelet transform (AWT), we propose wavelet phase transform (WPT) to obtain a desired phase-shift in a signal under-analysis, and propose the two representations of wavelet quadrature transform (WQT) which is special case of the WPT where phase-shift is $\pi/2$ radians.