| Wavelet theory is the elite of harmonic analysis. It is a milestone in the development history of Fourier analysis. Successful applications have demonstrated its superiorities in signal processing, image processing, quantum field, seismic exploration, speech recognition, music, radar, EF imaging, color copy, celestial body recognition, machine vision, machine fault diagnosis and monitoring, fractal and digital television etc.. In the theory, Fourier analysis can be instead of wavelet analysis in where it is applied. This paper mainly deals with the construction theory of the wavelet bases and wavelet analysis in the signal processing and prediction. It includes several aspects:In the beginning, the basic theory of wavelet analysis is introduced. Depicting synthetically about the research status, application achievements and the problems that still existed now. In addition, the similarities and differences of wavelet analysis and Fourier transform are analyzed.In the signal processing of wavelet analysis, the asymmetry of the wavelet base results in irregular positions of maxima points that corresponding to the singular points in each layer, which causes difficulty in the search of maxima lines. Enlightened by the construction method of spline wavelet with compactly support, a construction approach is proposed to gain a series of semi-orthogonal wavelet bases that have the properties of compactly support, symmetry and linear phase.The real-value scalar wavelet cannot possess desired properties such as symmetry, orthogonality, and shorter support for a given approximation order simultaneously, which are possible in the multi-wavelet. Based on the references, a simple multiwavelet construction approach is proposed. The high pass sequence of the multiwavelet with multiplicity r and 4 coefficients is constructed based on the low pass sequence of orthogonal multi-scale function with multiplicity r and 4 coefficients. In addition, an orthogonal multi-scale low pass sequence with multiplicity 2r and 3 coefficients and its corresponding high pass sequence are constructed.Singularity analysis is an important approach in de-noising. But the calculation formula of singularity exponent did not given in the singularity's definition, which is inconvenience in the applications. In this section, a brief approach to calculate the singularity exponent based on the wavelet decomposion of the noisy signal is proposed. Then a coefficient point can be judged that if it is a signal point or not. On the other hand, an improved wavelet threshold method is proposed for the signal de-noising and parameters estimation.The training algorithm of the wavelet neural network (WNN) is researched. We use the original PSO algorithm and the advanced PSO algorithm to train the WNN. Aiming at the convergent rate, the initialization problems of the translation and scaling parameters are proposed. In order to illuminate the advantage of the advanced PSO, the simulation results of the BP algorithm, original PSO and the advanced PSO with parameters initialization are given in the non-linear function approximation. Furthermore, a wavelet threshold neural network(WTNN) is proposed. It integrates the wavelet threshold de-noising and the Neural Network to form a new network. The structure, training algorithm and selection of the threshold function are represented detailedly. Finally, the simulation results of LFM signal de-noising and prediction with different noise types display that the proposed model outmatches the multilayered perceptron with the wavelet denoising based on a statistical criterion(P+MLP).At last, we sum up and point out the existed problems of my research work. Furthermore, the further research directions are proposed such as the construction of the bior-orthogonal multi-wavelet, the structure and application problems of wavelet threshold neural network etc..
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