Wavelet Analysis In Signal Processing And Application

Signal is the carrier of information and the physical manifestations of information. The variety of signals with different information have one thing in common, that is, the signal is always the functions with one or more independent variables, generally contains some informations of one or more certain phenomena. Signal can be divided into two categories, stationary signals and non-stationary signals. In practical engineering, the signals are almost always non-stationary, and their frequency-domain characteristics is changed over time.For a long time, limited to the development of the theory, people had to simplify non-stationary signal into stationary signal, then used Fourier transform to process the signal. Fourier analysis plays an important role in mathematics and engineering techniques, but Fourier transform is essentially only applicable to stationary signal, for non-stationary signal, it is not applicable. In order to deal with non-stationary signal analysis, in the the theoretical basis of Fourier analysis, people made a series of signal processing methods, such as short-Time Fourier Transform, windowed Fourier Transform, Gabor transform, wavelet analysis and so on. In 1946, Gabor proposed a short-time Fourier transform theory. Short-time Fourier transform is a single-resolution signal analysis methods,. The idea is to choose a time-frequency localization of the window function, and assume that the window function in a very short time interval is stable, by moving the window function, making the signal and its product in a different limited time is a smooth signal. Many of the non-stationary signal de-noising methods are the use of short-time Fourier transform to filtering de-noising. The time-frequency resolution of short-time Fourier transform is decided by the area of the time-frequency window.For the approximate stationary signal, we can use short-time Fourier analysis method. But for non-stationary signals, when the signal of rapid change, require the window function and window function has a high time resolution, when the waveform changes in slow (mainly low-frequency components), then the window function required a higher frequency resolution. According to Heisenberg principle (uncertainty principle), the signal time resolution and frequency resolution is a contradiction volume, they can not simultaneously achieve arbitrary high time resolution and frequency resolution. The analysis of high-frequency signals require a narrow time window, the analysis of low-frequency signals require a wide time window, that requires time-frequency window size can change with the change of frequency. Short-time Fourier transform can not take care of time resolution and frequency resolution simultaneously. The emergence of wavelet transform resolve this contradiction well, it has a very good localization properties in the time domain and frequency domain, that makes it a powerful signal analysis and processing tools.