Generalized Sampling Expansions Associated with Quaternion Fourier Transform

TL;DR

This paper develops a generalized sampling theory for quaternion-valued signals using quaternion Fourier and linear canonical transforms, enabling reconstruction from linear system outputs.

Contribution

It introduces new sampling expansions for quaternion signals in QFT and QLCT domains, expanding the theoretical framework for vector-valued signal processing.

Findings

Reconstruction of {}-bandlimited quaternion signals from linear system samples

Derived sampling formulas for quaternion signals in QFT and QLCT domains

Illustrated results with practical examples

Abstract

Quaternion-valued signals along with quaternion Fourier transforms (QFT)provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a {\sigma}-bandlimited quaternion valued signal in QFT sense can be reconstructed from the samples of output signals of M linear systems based on QFT. Quaternion linear canonical transform (QLCT) is a generalization of QFT with six parameters. Using the relationship between QFT, we derive the sampling formula for {\sigma}-bandlimited quaternion-valued signal in QLCT sense. Examples are given to illustrate our results.