Wavelets and Wavelet Transforms

Abstract

One collecting manner of experimental data by engineers and scientists is as sequences of values at regularly spaced intervals in time. These sequences are named time-series. The fundamental problem with the data in the form of time-series is how to process them for extracting meaningful and correct information, i.e., the possible signals embedded in them. If a time-series is stationary, one can think that it can have harmonic components that can be detected by applying the Fourier analysis, i.e., ${\text{Fourier Transform}}$ (${\text{FT}}$). Although, it is evident that many time-series are not stationary and their mean properties are variables over time. The waves of infinite support that form the harmonic components are not adequate in the latter case in which one needs waves ${\text{localized}}$ not only in frequency but in time too. They named wavelets and permit a time-scale decomposition of a signal. Considerable progress in understanding the wavelet processing of non-stationary signals was attained. Although, for getting the dynamics that provides a non-stationary signal, it is crucial that in the corresponding time-series a correct separation of the fluctuations from the average behavior, or trend, is performed. Therefore, people should invent new statistical techniques for detrending the data that have to be combined with the wavelet analysis. In this chapter, we deal with wavelets and wavelet transforms.