Hypercomplex Signal Energy Concentration in the Spatial and Quaternionic Linear Canonical Frequency Domains

TL;DR

This paper extends the energy concentration problem to 2D hypercomplex signals using quaternionic linear canonical transforms, introducing improved QPSWFs that demonstrate superior energy concentration properties.

Contribution

It develops improved definitions of quaternionic prolate spheroidal wave functions and analyzes their energy concentration in 2D spatial and frequency domains.

Findings

QPSWFs are more energy concentrated than Gaussian functions.

Enhanced properties of QPSWFs are established.

Energy concentration measurements are effectively characterized.

Abstract

Quaternionic Linear Canonical Transforms (QLCTs) are a family of integral transforms, which generalized the quaternionic Fourier transform and quaternionic fractional Fourier transform. In this paper, we extend the energy concentration problem for 2D hypercomplex signals (especially quaternionic signals). The most energy concentrated signals both in 2D spatial and quaternionic linear canonical frequency domains simultaneously are recently recognized to be the quaternionic prolate spheroidal wave functions (QPSWFs). The improved definitions of QPSWFs are studied which gave reasonable properties. The purpose of this paper is to understand the measurements of energy concentration in the 2D spatial and quaternionic linear canonical frequency domains. Examples of energy concentrated ratios between the truncated Gaussian function and QPSWFs intuitively illustrate that QPSWFs are more energy concentrated signals.