Introduction: Wavelet transforms in biomedical engineering

Wavelets analysis is finding a rapidly growing number of applications in fields ranging from communications to medicine. It has become a powerful alternative for the analysis of nonstationary signals whose spectral characteristics are changing over the course of time, since the traditional Fourier transform method gives the frequency contents of the signals without providing the time localization of the observed frequency components. This is very important in biological signal analysis since most of the statistical characteristics of these signals are nonstationary. In particular, the analysis of biological signals consisting of short-lived, high-frequency components closely located in time and long duration components closely spaced in frequency requires an appropriate method. This method should exhibit good frequency resolution to localize the closely located low-frequency components and time resolution to resolve the closely located high frequency components. Therefore, the thorough analysis of biological signals requires a method to represent the signals both in frequency and time domain at the same time. The first attempt to construct such an approach was proposed by Gabor, who satisfied the stationary condition for nonstationary signals by dividing the signal into blocks of short segments in which the signal segment can be assumed to be stationary (4). This method is known as the short-time Fourier transform (STFT). However, the problem with the STFT is that both time and frequency resolutions of the transform are fixed over the entire time-frequency plane. In addition, choosing a short analysis window may cause poor frequency resolution. On the other hand, while a long analysis window may improve the frequency resolution, it compromises the assumption of stationarity within the window. Alternative time-frequency analysis methods, including the Gabor representation (4), Wigner-Ville Distribution (2, 7, 8), Binomial transform (9), Choi-Williams (1) Distribution methods, etc . , have been proposed. All the time-frequency methods have been the subject of active research areas.