Abstract
This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in L2(Rn). The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, we formulate several uncertainty inequalities, including the Heisenberg’s, logarithmic, and Nazorav’s inequalities for the proposed transform in the linear canonical domain. The obtained results are reinforced with illustrative examples.
Highlights
The origin of the multi-dimensional linear canonical transform (LCT) dates back to the early 1970s with the foundational work of Moshinsky and Quesne [1] in quantum mechanics to study the linear maps of phase space
Soon after its inception in quantum mechanics, the linear canonical transform has been exclusively studied both in theory and applications [2,3]
We present an example for the lucid illustration of the proposed non-separable linear canonical wavelet transform in Equation (14)
Summary
The origin of the multi-dimensional linear canonical transform (LCT) dates back to the early 1970s with the foundational work of Moshinsky and Quesne [1] in quantum mechanics to study the linear maps of phase space. The theory of multi-dimensional non-separable LCT involving a general 2n × 2n real, symplectic matrix M = (A, B : C, D) with n(2n + 1) independent parameters offers a canonical formalism for the representation of several physical systems in a lucid and insightful way. With major modifications to the existing multi-dimensional wavelet transform in Equation (2), we propose the non-separable linear canonical wavelet transform of any f ∈ L2(Rn) concerning the free symplectic matrix M = (A, B : C, D) as. We introduce the notion of the non-separable linear canonical wavelet transform in L2(Rn), followed by some fundamental properties of the proposed transform, including the orthogonality relation, energy preserving relation, range theorem, and the inversion formula. Given a free symplectic matrix M = (A, B : C, D), the non-separable linear canonical transform of any f ∈ L2(Rn) is denoted by F M f and is defined as.
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