A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

Abstract

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

The Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

“…The scalar part of a quaternion q ∈ H is q 0 denoted by Sc(q), the non scalar part(or pure quaternion) of q is iq 1 + jq 2 + kq 3 denoted by V ec(q). The quaternion conjugate of q ∈ H, given by 3 , is an anti-involution, namely, qp = p q, p + q = p + q, p = p. The norm or modulus of q ∈ H is defined by…”

“…In this part of this paper, we are going to give a version of Lieb's theorem for the WVD-QOLCT. In the following theorem [3], we state Lieb's theorem related to the QLCT. Theorem 4.11.…”

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

“…As in the case of the LCT [38,39], the uncertainty principle is an important result of the QLCT. In [18,19,21], the authors have established the uncertainty principle for the two-sided QLCT. Based on the definition of our proposed QLCT in this article, we investigate an uncertainty principle related to the QLCT.…”

“…Our attention in this article is to introduce a definition of the QLCT (compared to [21][22][23]). is definition is obtained by substituting the Fourier kernel with the type II QFT kernel (see [24]) in the LCT definition, which is essentially different from the kernel of the type III QFT.…”

A Convolution Theorem Related to Quaternion Linear Canonical Transform

“…The (right-sided) quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition. Some important properties of the quaternion linear canonical transform such as the Parseval's theorem, reconstruction formula, and uncertainty principles are also discussed (see [9][10][11][12][13][14] and the references mentioned therein). However, there is no literature for establishing the convolution theorem associated with the QLCT as far as we know.…”