The continuous wavelet derived by smoothing function and its application in cosmology

Abstract

The wavelet analysis technique is a powerful tool and is widely used in broad disciplines of engineering, technology, and sciences. In this work, we present a novel scheme of constructing continuous wavelet functions, in which the wavelet functions are obtained by taking the first derivative of smoothing functions with respect to the scale parameter. Due to this wavelet constructing scheme, the inverse transforms are only one-dimensional integrations with respect to the scale parameter, and hence the continuous wavelet transforms (CWTs) constructed in this way are more ready to use than the usual scheme. We then apply the Gaussian-derived wavelet constructed by our scheme to computations of the density power spectrum for dark matter, the velocity power spectrum and the kinetic energy spectrum for baryonic fluid. These computations exhibit the convenience and strength of the CWTs. The transforms are very easy to perform, and we believe that the simplicity of our wavelet scheme will make CWTs very useful in practice.

Continuous Wavelet Analysis of Matter Clustering Using the Gaussian-derived Wavelet

“…This is illustrated by substituting = w a 2 into Equation (1), where a is the scale parameter for the traditional CWT. 4 However, as pointed out by Wang & He (2021), the 3D GDW is an anisotropic separable wavelet function, which is not a 3D Mexican hat wavelet. At present, we are only concerned with the 1D case.…”

“…In principle, there is a diverse set of wavelets available in the CWT. For instance, the most common ones are the Poisson wavelet, the Mexican hat wavelet, the spline wavelet, the Meyer wavelet, and the Morlet wavelet (see, e.g., scale along each of the three axes (x, y, and z), as pointed out by Wang & He (2021). For simplicity, we consider the isotropic case, in which the same scale parameter w is used for all directions.…”

“…Although there is an alternative inverse transform formula in the form of a single integral for the complex-valued wavelets, the real wavelets have generally been considered to have no such simple inverse transformation (Delprat et al 1992;Aguiar-Conraria & Soares 2014). In our previous work (Wang & He 2021), we proposed a novel scheme of constructing continuous wavelets to overcome this problem, in which the wavelet functions are obtained by taking the first derivative of smoothing functions with regard to the positively defined scale parameter. With this scheme, the original signal is recovered easily by integrating the wavelet coefficients with respect to the scale parameter.…”

“…In Wang & He (2021), we simply define GDW as ψ(w, x) ≡ ∂G(w, x)/∂w. 4 A detailed comparison between the GDW and the Mexican hat wavelet is given inWang & He (2021).…”

Comparisons between fast algorithms for the continuous wavelet transform and applications in cosmology: the one-dimensional case

“…So the accuracy of their numerical CWTs can be verified by the corresponding analytical results. Finally, it should be noted that the CWT for 1D signals is not trivial in astrophysics and cosmology, as it is also applicable to a wide range of scenarios, such as analyzing the light curves of astronomical sources (e.g., Tarnopolski et al 2020;Ren et al 2022), subtracting the foreground emission from the 21 cm signal (e.g., Gu et al 2013;Li et al 2019), measuring the smallscale structure in the Lyman-α forest (e.g., Lidz et al 2010;Garzilli et al 2012;Wolfson et al 2021), investigating the time-frequency properties of the gravitational waves (e.g., Tary et al 2018), characterizing the 1D density fields (e.g., da Cunha et al 2018;Wang & He 2021;, and so on. We publicly release the Fortran 95 implementation of the fast CWT algorithms described in this manuscript 1 , in the hope that the community will use them to perform wavelet analysis of 1D signals.…”