A new fractional wavelet transform
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 orthogonal fractional wavelet is constructed. A few examples are provided to illustrate the applications of the \(\alpha\)-order fractional wavelet transform to signal processing.
scientific article; zbMATH DE number 7257312