Abstract
The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.
Highlights
The quaternion Fourier transform (QFT) is a nontrivial generalization of the real and complex classical Fourier transforms (FT) using quaternion algebra
We show that some fundamental properties of the quaternion Wigner-Ville distribution (QWVD)-linear canonical transform (LCT) such as inversion formula and Moyal formula can be obtained by combining this relation and the properties of the QFT
A useful property of the QFT is stated in the following lemma, which is needed to derive Moyal formula of the quaternion Wigner-Ville distribution associated with the linear canonical transform (QWVD-LCT)
Summary
The quaternion Fourier transform (QFT) is a nontrivial generalization of the real and complex classical Fourier transforms (FT) using quaternion algebra. Many useful properties of the QFT were obtained such as shift, modulation, convolution, correlation, differentiation, energy conservation, and uncertainty principle. It was first introduced in [1] for the analysis of 2D linear time-invariant partial differential systems and applied in color image processing [2, 3]. It is a natural question to extend the QFT to the linear canonical transform (LCT) domains and it is the so-called quaternionic linear transform (QLCT). In [8], the authors developed this idea to derive an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by using the fundamental relationship between the QLCT and the QFT [8].
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