Towards Quaternion Quadratic Phase Fourier Transform

TL;DR

This paper introduces the quaternion quadratic phase Fourier transform (QQPFT), extending the quadratic phase Fourier transform to quaternion signals, and explores its properties, inverse, and uncertainty principles for potential applications.

Contribution

The paper generalizes the quadratic phase Fourier transform to quaternion-valued signals and derives key properties, inverse formulas, and uncertainty principles for the QQPFT.

Findings

Derived the inverse QQPFT and associated formulas.

Established properties like linearity, shift, and modulation.

Formulated various uncertainty principles for QQPFT.

Abstract

The quadratic phase Fourier transform QPFT is a neoteric addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic phase Fourier transform to quaternion valued signals, known as the quaternion QPFT QQPFT. We initiate our investigation by studying the QPFT of 2D quaternionic signals, then we introduce the QQPFT of 2D quaternionic signals. Using the fundamental relationship between the QQPFT and quaternion Fourier transform QFT, we derive the inverse transform and Parseval and Plancherel formulas associated with the QQPFT. Some other properties including linearity, shift and modulation of the QQPFT are also studied. Finally, we formulate several classes of uncertainty principles UPs for the QQPFT, which including Heisenberg type UP, logarithmic UP, Hardys UP, Beurlings UP and Donohon Starks UP. It can be regarded as the first step in the applications of the QQPFT in the real world.