Convolution theorems associated with quaternion linear canonical transform and applications

TL;DR

This paper introduces new convolution theorems for the quaternion linear canonical transform (QLCT), enabling efficient computation and applications in solving integral equations, PDEs, and filter design within quaternionic signal processing.

Contribution

It proposes novel convolution operators for QLCT in spatial and spectral domains, expanding the theoretical framework and practical applications of quaternionic transforms.

Findings

Derived convolution formulas for QLCT

Applied convolution theorems to solve integral equations

Facilitated the design of multiplicative filters

Abstract

Novel types of convolution operators for quaternion linear canonical transform (QLCT) are proposed. Type one and two are defined in the spatial and QLCT spectral domains, respectively. They are distinct in the quaternion space and are consistent once in complex or real space. Various types of convolution formulas are discussed. Consequently, the QLCT of the convolution of two quaternionic functions can be implemented by the product of their QLCTs, or the summation of the products of their QLCTs. As applications, correlation operators and theorems of the QLCT are derived. The proposed convolution formulas are used to solve Fredholm integral equations with special kernels. Some systems of second-order partial differential equations, which can be transformed into the second-order quaternion partial differential equations, can be solved by the convolution formulas as well. As a final point, we demonstrate that the convolution theorem facilitates the design of multiplicative filters.