The Multidimensional Quadratic Phase Fourier Transform: Theoretical Analysis and Applications

TL;DR

This paper introduces the multidimensional quadratic phase Fourier transform (QPFT), explores its theoretical properties, and demonstrates its applications in filter design and solving integral equations.

Contribution

It extends the QPFT to multiple dimensions, establishing new theoretical results and convolution operations for advanced signal processing applications.

Findings

Established Parseval's identity and inversion theorems for multidimensional QPFT

Proposed generalized convolutions and correlation for multiple variables

Applied the multidimensional QPFT to filter design and integral equations

Abstract

The quadratic phase Fourier transform (QPFT) is a generalization of several well-known integral transforms, including the linear canonical transform (LCT), fractional Fourier transform (FrFT), and Fourier transform (FT). This paper introduces the multidimensional QPFT and investigates its theoretical properties, including Parseval's identity and inversion theorems. Generalized convolutions and correlation for multiple variables, extending the conventional convolution for single-variable functions, are proposed within the QPFT setting. Additionally, a Boas-type theorem for the multidimensional QPFT is established. As applications, multiplicative filter design and the solution of integral equations using the proposed convolution operation are explored.