Quadratic-phase wave packet transform

Abstract

The quadratic-phase Fourier transform (QPFT) 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 give its relation with windowed Fourier transform (WFT). Secondly, several notable inequalities and important properties of newly defined QP-WPT, such as boundedness, reconstruction formula, Moyal’s formula, reproducing kernel are derived. Finally, we formulate several classes of uncertainty inequalities, such as Leib’s uncertainty principle, logarithmic uncertainty inequality and the Heisenberg uncertainty inequality.