The Design And Application Of A New Wavelet Basis

Wavelet transform has a close relationship with Fourier transform and its related theories. The traditional Fourier transform theory is a pure analytical method in frequency domain and has no resolving power in time domain, it can't collect the information in frequency domain and in time domain , so after Fourier transform the frequency characteristic of local time domain of signals cannot be obtained. In order to extract the local information of Fourier transform, Gabor introduced Gabor transform,and then, the short time Fourier transform, namely windowed Fourier transform, is appeared. To some extent, the short time Fourier transform improved the shortcoming of the traditional fourier transform. But the resolution of the short time Fourier transform in the time-frequency plane is changeless. So, wavelet transform was appeared. The wavelet transform conquer the constant of the resolusion in the short time Fourier transform, it could adjust the form of the time-frequency window automatically, thus, it has great time-frequency localization capability.Along with the rapid development of digital signal processing theory since the middle 1960, especially the gradual maturity of filter banks theory, the internal relationship between wavelet transform and filter banks theory is discovered. Consequently, we consider studying wavelet transform and related problems form the perspective of filter banks theory.The main content of this thesis is as follows:(1) Basic theory of wavelet transform. The thesis analyses the basic theoryof wavelet transform and presents the graph which interpret the relationship between scale space and wavelet space lucidly and directly. It also gives the sketch map of the multi-resolusion analysis of signal ,which interpret the properties of the multi-resolusion analysis of signal. The thesis analyzes the implementation of Mallat algorithm.(2) The thesis summarizes discrete time signal and Fourier transform, discrete time system and Z transform respectively. Discrete time signal and discrete time system are not only the foundation of the digital signal processing theory, but also have close relationship with filter banks theory. The thesis also discusses digital filter and basic conceptions of mult-irate system, such as decimation and interpolation, present three types polyphase decomposition. The thesis analyses properties of perfect reconstruction filter banks. By using factorization method of product filter, it constructs a new class of biorthogonal wavelet based on PR filter banks, and obtains graphs of scaling function, wavelet function, dual scaling function and dual wavelet function on compact supported interval,and analyses their frequency response.it uses particle swarm optimization, obtained waveform matching biorthogonal wavelet with rational coefficients based on established parameterized filter banks.(3) The thesis discusses basic method of threshold value signal denoising based on wavelet transform. A new threshold value function with two parameters is constructed, Numerical value experiment contrapose the given singal is carried out. The effect of different wavelets and different threshold value function are compared, and gives the analyses of the results.