Theory Research For Haar Wavelet And Multiwavelets

Wavelet transform, which is a new powerful mathematical tool appeared in recent years, is regarded as the crystal work of pure mathematical field and the great breakthrough in signal-picture analysis and quantum physics, and is thought as the new trend of applied mathematics. Because of its good time-frequency localization and flexible time-frequency windows, wavelet transform has been widely applied in such as signal study, picture process, quantum mechanics, radar, computer identification, earthquake exploration, so on and so forth. Multiwavelets have been studied recently. Where several mother wavelet functions were used to expand a function, it also can be seen as vector-valued wavelets that satisfy conditions in which matrics are involved.The main works of this paper are as follows:Firstly, although Haar function has bad vanish property in frequency domain, it is the only normal orthonormal basis with symmetry and real short-support property. We study the design of Haar wavelet for scale=a (a2) and present a decomposition and reconstruction algorithm in Chapter 3.Secondly, In Chapter 4 we study the design of orthonormal mutiwavelets of multiplicity r with scale=a (a2) .By the factorization theory ,we give parametric expressions for orthonormal causal FIR multifilter banks of r=2 and scale=4,and we found the length of scaling function can be controlled by the parameters.Finally, we provide the error analysis between discrete multiwavelet transform coefficients and continue multiwavelet transform coefficients.