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.
