Research Of Wavelets Transform Based On Filter Banks

The first generation 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, so after Fourier transform the frequency characteristic of local time domain of signals cannot be obtained. In order to extract the part information of Fourier transform, Gabor introduced Gaussian function as window function of localized time domain. The window function of short time Fourier transform, or windowed Fourier transform which is developed from Gabor transform, is not restricted to Gaussian function and therefore the effectiveness of computing and convenience of realization are ensured. Wavelet transform overcomes the deficiency of constant resolution of short time Fourier transform in time-frequency domain and is thus of 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. For instance, when the whole frequency band is equally divided into two bands, the computation formula of the subband coding decomposition process is the same as the Mallat decomposition formula of biorthogonal discrete wavelet transform, and computation formula of the synthesis procedure is the same as the Mallat reconstruction formula of biorthogonal discrete wavelet transform. Consequently, we consider studying wavelet transform and related problems form the perspective of filter banks theory.The main innovation of this thesis is as follows:(1) Evolutionary optimizing of wavelet system. Based on the analysis of wavelet system performance indices, the author chooses the inter-level redundancy as a performance index for optimizing and optimizes a kind of generalized interpolating wavelet system. After parametrization of the inter-level redundancy of the generalized interpolating wavelet is obtained, the author defines the parametrized expression asthe fitness function and use genetic algorithm to minimize the fitness function. The inter-level redundancy of the generalized interpolating wavelet obtained by evolutionary optimization is less, so the wavelet system is of better performance.(2) Wavelets construction based on filter banks. The author analyses biorthogonality of perfect reconstruction filter banks. The row of 2x2 synthesis modulated matrix and analysis modulated matrix are biorthogonal to the column of their inverse respectively, and the biorthogonality is retained after multiplying order of matrices is changed. Using the constraints of vanishing moments, the author 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.(3) Parametrization of the lifting scheme used for constructing filter banks. The author presents two design rules of constructing two-channel filter banks, that are, the least square error rule of magnitude-frequency responses of high pass filters and the least square error rule of magnitude-frequency responses of low pass filters. The author deduces four types parametrized expression of lifting scheme for constructing linear phase PR filter banks, that are, EE case and 00 case under the first design rule and EE case and 00 case under the second design rule.