New Fractional Wavelet with Compact Support and Its Application to Signal Denoising

Abstract

Since their appearance, the wavelets have been developed very rapidly and have attracted the attention of many researchers, which resulted in the birth of several wavelet families: real, complex and fractional. However, the choice of an adequate analyzing wavelet remains an important problem; there is no wavelet suitable for all cases, for some applications, it is possible that we do not find among the known wavelets the one that suits. Therefore, it is necessary to try to build new wavelets that can adapt and cover a wide panorama of problems. In this context, we propose in the present research work a new wavelet family based on fractional calculus. The construction generally begins with the choice of an orthogonal digital low-pass filter associated with a base of wavelet with compact support; the filter will be generalized through the fractional delay (FD) Z-D which is approximated by a RIF filter using the Lagrange interpolation method, while ensuring correct properties of orthogonality, compact support and regularity. Then, the high-pass fractional filter is deduced from the low-pass filter by a simple modulation. However, the scale and wavelet functions are built using the cascade Daubechies algorithm. In order to illustrate the potential and efficiency of the fractional wavelets designed within this paper compared to the different wavelets existing in the literature, an application example is presented; this is the denoising of signals by thresholding fractional wavelet coefficients. The experimental results obtained are satisfactory and promising; they show that the performance of fractional wavelets is superior to those of classical wavelets; this is due to the flexibility and high selectivity of fractional filters associated with these fractional bases.