Quadratic phase wave packet transform

TL;DR

The paper introduces the quadratic phase wave packet transform (QP WPT), enhancing localizing the quadratic phase spectrum in signal processing, with theoretical properties and uncertainty principles derived.

Contribution

It proposes the QP WPT based on wave packet transform and quadratic phase Fourier transform, providing new properties and uncertainty inequalities.

Findings

Defined the QP WPT and related it to WFT

Derived properties like boundedness and reconstruction formula

Formulated various uncertainty inequalities

Abstract

The quadratic phase Fourier transform has gained much popularity in recent years because of its applications in image and signal processing. However, the QPFT is inadequate for localizing the quadratic phase spectrum which is required in some applications. In this paper, the quadratic phase wave packet transform QP WPT is proposed to address this problem, based on the wave packet transform WPT and QPFT. Firstly, we propose the definition of the QP WPT and gave its relation with windowed Fourier transform WFT. Secondly, several notable inequalities and important properties of newly defined QP WPT, such as boundedness, reconstruction formula, Moyals formula, Reproducing kernel are derived. Finally, we formulate several classes of uncertainty inequalities such as Leibs uncertainty principle, logarithmic uncertainty inequality and the Heisenberg uncertainty inequality.