A new fractional wavelet transform
type: Item
instance of: scholarly article
title: A new fractional wavelet transform
zbMATH Open document ID: 1472.42054
author: Hongzhe Dai
author: Zhibao Zheng
author: Wei Wang
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2020-10-07T00:00:00Z
review text: The fractional Fourier transform and the continuous wavelet transform are useful in many applications. In this paper, using the fractional Fourier transform, the authors generalized the continuous wavelet transform. For a given signal \(x\in L^2(\mathbb{R})\), its continuous wavelet transform \(\mbox{WT}_x\) is defined to be \[ \mbox{WT}_x(a,b)=\langle x(t), \varphi_{a,b}(t)\rangle= \int_{-\infty}^\infty x(t) \overline{\varphi_{a,b}(t)}dt \] with \[ \varphi_{a,b}(t)=\frac{1}{\sqrt{a}} \varphi\left(\frac{t-b}{a}\right),\qquad t\in \mathbb{R} \] for \(a\in \mathbb{R}^+\) and \(b\in \mathbb{R}\), where \(\varphi\) is an admissible mother wavelet satisfying \(\int_0^\infty |\omega|^{-1} |\Phi(\omega)|^2 d\omega<\infty\) and \(\Phi\) is the Fourier transform of \(\varphi\). For \(\alpha\in \mathbb{R}\), the proposed \(\alpha\)-order fractional wavelet transform of a given signal \(x\in L^2(\mathbb{R})\) is defined to be \[ W_x^\alpha(a,b)=\langle x(t), \varphi_{\alpha,a,b}(t)\rangle= \int_{-\infty}^\infty x(y) \overline{\varphi_{\alpha,a,b}(t)}dt =\mbox{WT}_{\tilde{x}}(a,b) \] for \(a\in \mathbb{R}^+\) and \(b\in \mathbb{R}\), where \(\tilde{x}(t)= x(t)e^{\frac{j}{2}(t^2-b^2-(\frac{t-b}{a})^2)\cot \alpha}\) and \[ \varphi_{\alpha,a,b}(t)=e^{-\frac{j}{2}(t^2-b^2-(\frac{t-b}{a})^2)\cot \alpha} \varphi_{a,b}(t) \] and here \(j\) is the imaginary unit. Because \(\cot(\pi/2)=0\), the \(\alpha\)-order fractional wavelet transform with \(\alpha=\pi/2\) becomes the conventional continuous wavelet transform. Then the authors generalized many properties of the standard continuous wavelet transform to the \(\alpha\)-order fractional wavelet transform using the \(\alpha\)-order fractional Fourier transform. By discretizing the \(\alpha\)-order fractional wavelet transform, similar results for discrete wavelets such as the multiresolution analysis and orthogonal wavelets have been discussed for discretizing \(\alpha\)-order fractional wavelet transform. In particular, a Haar-type ort…
reviewed by: Bin Han
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94A12
zbMATH DE Number: 7257312
zbMATH Keywords: fractional Fourier transform
zbMATH Keywords: fractional wavelet transform
zbMATH Keywords: multiresolution analysis
MaRDI profile type: Publication
full work available at URL: j.cnsns.2016.06.034
OpenAlex ID: W2487714310
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DOI: 10.1016/J.CNSNS.2016.06.034
description: scientific article; zbMATH DE number 7257312