Multiwavlet Methods For Signal Processing

All real-valued scalar (urn-) wavelets (with only one scaling function and one mother wavelet) can never possess orthogonality, compact support, linear phase, and high approximation order of the basis functions simultaneously. In order to overcome these drawbacks, more then one scaling function and mother wavelet can be used, hence, it is called multiwavelets. Multiwavelets increasingly have attracted much attention in the research community. Therefor, study of some mutliwavelet basic problems is becoming more urgent and more important. The focus in this dissertation are on discussion of some basic problems in the application of (multi-) wavelets to signal processing, The primary contributions and original ideas included in this dissertation are summarized below: We discussed FIR prefilter with approximation condition to overcome the drawback of the Mallat抯 algorithm and analyzed its approximation property. Then, we proposed single parameter FIR adaptive prefiltering algorithm with approximation condition. For discrete and continual signals, we gave the particular process for prefilter design, respectively. Its main advantages are: (1) the structure and parameters are simple and it is easy to implement; (2) approximation accuracy and rate are satisfactory. Because scaling functions in multiwavelet do not possess lowpass property, Unlike single wavelet, fast discrete multiwavelet decomposition algorithm is not easily achieved. This is a main reason to restrict multiwavelet applications. So we presented an adaptive multiwavelet prefilter. First, we need get uniform simples of signal and translate them into vectors, then, the error between vectors and multiwavelet coefficients in theory is analyzed. We optimize this error and get the filter. We know that if the prefilter is nonorthogonal, the discrete orthogonal multiwavelet transform will not preserve the orthogonality. After analyzing some properties of multiwavelets, we combine single wavelet transform and multiwavelet transform to construct a new function^t from multiscaling I Iv ABSTRACT functions such that ^t has the lowpass property, orthogonality and Strang-Fix condition. So the approximation rate of this prefilter is nonlinear as to scaling factor. Approximation property analysis of the algorithm and the results of simulation in multiwavelet neural network and data compression show its efficiency and practicality. 2-band interpolating wavelet transform has the adavantage that wavelet series transform coefficient is same as binary uniform samples in multiresolution subspace. Thus, prefilter operator of signals in some scaling space is simplified to 1. However, there does not exist the orthogonal 2-band interpolating wavelet with compact support except Haar wavelet. Because multiwavelet has more design parameters than single wavelet, the design of orthogonal compact interpolating multiwavelet become possible. We presented a method for designing orthogonal compact interpolating multiwavelet based on transform scheme. With this transform scheme, we can adaptive become a general multiwavelet into an orthogonal compact interpolating multiwavelet. So pre and post filter are simplified to unit operator. The ability of wavelet packets to decompose high frequency channels can be employed to impr...