Some properties of windowed linear canonical transform and its logarithmic uncertainty principle

Abstract

Based on the relationship between the Fourier transform (FT) and linear canonical transform (LCT), a logarithmic uncertainty principle and Hausdorff–Young inequality in the LCT domains are derived. In order to construct the windowed linear canonical transform (WLCT), Gabor filters associated with the LCT is introduced. Using the basic connection between the classical windowed Fourier transform (WFT) and the WLCT, a new proof of inversion formula for the WLCT is provided. This relation allows us to derive Lieb’s uncertainty principle associated with the WLCT. Some useful properties of the WLCT such as bounded, shift, modulation, switching, orthogonality relation, and characterization of range are also investigated in detail. By the Heisenberg uncertainty principle for the LCT and the orthogonality relation property for the WLCT, the Heisenberg uncertainty principle for the WLCT is established. This uncertainty principle gives information how a complex function and its WLCT relate. Lastly, the logarithmic uncertainty principle associated with the WLCT is obtained.

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

“…The QWLCT is not only a linear transform, but also has similar properties as the WLCT [2,9]. In this paper, Pitt's inequality and Lieb inequality for the QWLCT are investigated and different forms of uncertainty principles associated with the QWLCT are proposed.…”

“…The windowed linear canonical transform (WLCT) is a method devised to study signals whose spectral content changes in time. As shown in [2,9,42,43] some important properties of the WLCT such as the analogue of the Poisson summation formula, sampling formulas, Paley-Wiener theorem, and dual window solution are discussed. The discrete WLCT is discussed in [44].…”

Quaternion Windowed Linear Canonical Transform of Two-Dimensional Signals

“…As shown in Refs. [4,16] some important properties of the WLCT are discussed. Those include the analogue of the Poisson summation formula, sampling formulas, Paley-Wiener theorem, and uncertainly relations.…”

Convolution and correlation theorems for the windowed offset linear canonical transform

“…In [14], [15], [17], the window function based on linear canonical transform (WLCT) were presented. It is believed to be a new and important signal processing tool.…”