The windowed quadratic phase Fourier transform: structure, convolution theorem and application

TL;DR

This paper introduces the windowed quadratic phase Fourier transform (WQPFT), explores its mathematical properties, and demonstrates its application in solving convolution equations, enhancing signal analysis techniques.

Contribution

It provides a comprehensive analysis of the WQPFT's properties, including kernel, reconstruction, and convolution theorems, which were not previously established.

Findings

Derived the reproducing kernel and reconstruction formula.

Established convolution theorems in spectral and spatial domains.

Applied the transform to solve convolution equations.

Abstract

The windowed quadratic phase Fourier transform (WQPFT) combines the localization capabilities of windowed transforms with the phase modulation structure of the quadratic phase Fourier transform (QPFT). This paper investigates fundamental properties of the WQPFT, including linearity, shifting, modulation, conjugation, and symmetry. In addition, we derive the reproducing kernel, establish a reconstruction formula, and characterize the range of the transform. Convolution theorems in both the spectral and spatial domains are developed, along with the existence results and norm estimates for the convolution operation associated with the WQPFT. Finally, as an application, the solution of a convolution equation is given using the convolution theorem of the WQPFT.