Some applications of wavelet transform: continuous and fractional approach

Abstract

Non-stationary signals are quite common in everyday life. It is well known that the conventional Fourier analysis is not capable of describing the evolution of the spectral features of a signal as this evolves in time. Since wavelets can represent signals locally in time and frequency, their application in various fields of science and engineering. Wavelets are mathematical functions generated from one basic function by the dilatation (scale parameter) (W(x) → W(2x)) and a translation (shift parameter) (W(x) → W(x+1)). Also, the mathematical idea of fractional derivatives and integrals have represented the subject of interest for various branches of science. The fractional integro-differential operators (fractional calculus) present a generalization of integration and derivation to non-integer order (fractional) operators. As it is already known the splines play a significant role on the early development of the theory of the wavelet transform. In this presentation a new wavelet transform is introduced and considered. The generalization of the splines (fractional B-splines) constructions will be proposed, namely new wavelet bases with a continuous order parameter will be obtained. The main advantage of this construction is that we can build the wavelet bases parameterized by the continuously-varying regularity parameter α. Specially, in the spectral analysis, the continuous wavelet transform and fractional wavelet transform are powerful tools for the data reduction, de-noising, compressing and baseline correction of the analytical signals and resolution of multi-component overlapping signals.