Quaternion Windowed Linear Canonical Transform of Two-Dimensional Signals

Abstract

We investigate the 2D quaternion windowed linear canonical transform (QWLCT) in this paper. Firstly, we propose the new definition of the QWLCT, and then several important properties of newly defined QWLCT, such as bounded, shift, modulation, orthogonality relation, are derived based on the spectral representation of the quaternionic linear canonical transform (QLCT). Secondly, by the Heisenberg uncertainty principle for the QLCT and the orthogonality relation property for the QWLCT, the Heisenberg uncertainty principle for the QWLCT is es-tablished. Finally, we give an example of the QWLCT.

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

“…Firstly, we present some important properties of the QWLCT. These properties have been proved in [22]. Secondly, based on the relationship between the QWLCT and the QFT, the Pitt's inequality and Lieb inequality associated with the QWLCT are demonstrated.…”

“…In this subsection, we present several basic properties of the QWLCT. These properties have been proved in [21,22] .…”

“…According to the QLCT, Xiong and Fu [21] have proposed the two-side quaternion linear canonical transform associated with the real window function-called the quaternion window linear canonical transform (QWLCT), and they studied several properties and Heisenberg's uncertainty principle for the QWLCT [21]. Different from the [21], based on the quaternion window function, we [22] have investigated some properties of the two-side QWLCT, and gave an example to illustrative the QWLCT is highly effective.…”

“…In signal processing, an uncertainty principle states that the product of the variances of the signal in the time and frequency domains has a lower bound. There are many different kinds of uncertainty principles, for instance, Heisenberg uncertainty principle [22,30,31,39], Logarithmic uncertainty principle [32], Hardy's uncertainty principle [39,40], Beurling's uncertainty principle [50], Lieb uncertainty principle [7,35], Donoho-Stark's uncertainty principle [36][37][38]. The Lieb uncertainty principle for the WLCT has been discussed in [7], which takes the LCT version as one of its special cases.…”

Quaternion Linear Canonical Wavelet Transform and The Corresponding Uncertainty Inequalities

“…Because of the non-commutative property of multiplication of quaternions, there are mainly three various types of 2D quaternion linear canonical transform (QLCTs): two-sided QLCTs, left-sided QLCTs and right-sided QLCTs (refer to [14]). Based on the (two-sided) QLCT [13], the quaternion windowed linear canonical transform of 2D quaternionic signals has been introduced by Wen-Biao Gao and Bing-Zhao Li in [30]. It can reveal the local QLCT-frequency contents and enjoys high concentrations and eliminates the cross term, but it has limitation of having fixed window localization.…”