A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

Abstract

We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied.

Quaternion Windowed Linear Canonical Transform of Two-Dimensional Signals

“…The QLCT was firstly studied in Ref. [23] including prolate spheroidal wave signals and uncertainty principles [27]. Some useful properties of the QLCT such as linearity, reconstruction formula, continuity, boundedness, positivity inversion formula and the uncertainty principle were established in Refs.…”

“…Some useful properties of the QLCT such as linearity, reconstruction formula, continuity, boundedness, positivity inversion formula and the uncertainty principle were established in Refs. [12,15,18,19,23,33]. An application of the QLCT to study of generalized swept-frequency filters was introduced in Ref.…”

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.…”

A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

“…The component-wise and directional uncertainty principles associated with the QFT were proposed in [11]. In [26,27], the authors established a component-wise uncertainty principle for the QLCT and proved that the equality is achieved for optimal quaternion Gaussian function. Recently, the authors [23] proposed the logarithmic uncertainty principle associated with the QLCT which is the generalization of the logarithmic uncertainty principle for the QFT.…”