Convolution, Product and Correlation Theorems for Simplified Fractional Fourier Transform: A Mathematical Investigation

TL;DR

This paper develops simplified convolution, product, and correlation theorems for the fractional Fourier transform, making their mathematical properties more elegant and generalizable, akin to classical Fourier transform theorems.

Contribution

It introduces simplified and elegant theorems for SmFrFT that generalize classical Fourier transform properties, enhancing mathematical clarity and applicability.

Findings

Theorems are mathematically derived for SmFrFT.

Theorems generalize classical Fourier transform properties.

Enhanced mathematical elegance in fractional Fourier analysis.

Abstract

The notion of fractional Fourier transform (FrFT) has been used and investigated for many years by various research communities, which finds widespread applications in many diverse fields of research study. The potential applications includes ranging from quantum physics, harmonic analysis, optical information processing, pattern recognition to varied allied areas of signal processing. Many significant theorems and properties of the FrFT have been investigated and applied to many signal processing applications, most important among these are convolution, product and correlation theorems. Still many magnificent research works related to the conventional FrFT lacks the elegance and simplicity of the convolution, product and correlation theorems similar to the Euclidean Fourier transform (FT), which for convolution theorem states that the FT of the convolution of two functions is the product of their respective FTs. The purpose of this paper is to devise the equivalent elegancy of convolution, product and correlation theorems, as in the case of Euclidean FT. Building on the seminal work of Pei et al. and the potential of the simplified fractional Fourier transform (SmFrFT), a detailed mathematical investigation is established to present an elegant definition of convolution, product and correlation theorems in the SmFrFT domain, along with their associated important properties. It has been shown that the established theorems along with their associated properties very nicely generalizes to the classical Euclidean FT.