Short time quaternion quadratic phase Fourier transform and its uncertainty principles

TL;DR

This paper introduces the quaternion quadratic phase Fourier transform (QQPFT) for quaternion-valued functions, explores its properties, and establishes uncertainty principles relating it to other quaternion transforms.

Contribution

It extends the quadratic phase Fourier transform to quaternion-valued functions, defines the short time version, and derives new uncertainty principles for these transforms.

Findings

Established the sharp Hausdorff-Young inequality for QQPFT.

Derived Lieb's and entropy uncertainty principles for QQPFT and related transforms.

Connected QQPFT with quaternion ambiguity function and Wigner-Ville distribution.

Abstract

In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff-Young inequality for QQPFT. We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner-Ville distribution associated with QQPFT and obtain the Lieb's uncertainty and entropy uncertainty principles for these three transforms.