The Theory And Construction Of M-Band Wavelets

The typical wavelet theory is researched around two-band wavelets.In fact,two-band wavelets process well these signals whose energy is concentrated in low frequency or high frequency,but badly those signals whose energy is mainly in middle frequency or distributed into multi-bands frequency.Therefore,in order to acquire satisfactory outcoming with analyzing and processing the signal precisely, it is not enough to use only two-band wavelets, and M-band wavelets is needed.Besides, M-band wavelets have some good properties that two-band wavelets don't have,such as being orthogonal and liner phase simultaneously,and these properties are quite useful in practical applications. So, it has been a hotspot in information fields to research M-band wavelets' theory and construction.This paper generalizes the fruits of the typical two-band wavelet theory to the arbitrary M-band wavelets,which deeply studies three important questions about the scaling functions and wavelets of M-band wavelets:their accuracy of approximation and the number of vanishing moments,the existence and the construction of the scaling functions,and their smoothness.Based on the above investigation into M-band wavelets,the paper expatiates the relation between M-band wavelets and PR filter banks,and offers sufficient and necessary conditions for M-band PR filter banks being wavelet filter banks.There are two main approachs in the construction of M-band orthogonal wavelets with PR filter banks. The first one is based on lattice structure and constraints of regularity, which is realized by solving a constrained optimization; the second one is achieved by orthogonalizing the polyphase matrix with first gaining the scaling filter. The algorithms of the first method are becoming more and more difficult to be fulfilled because the constraints are too complicated with the number of regularity growing large;the existing algorithms belonging to the second approach don't keep the linear phase property. This paper, based on the secong approach, presents a new efficient algorithm to design symmetric bi-orthogonal M-band wavelets with arbitrary regularity by using Grobner basis and syzygy module algorithm in computing algebra, which bi-orthogonalizes the polyphase matrix line by line. Compared with the existing algorithms ,the algorithm overcomes the drawback that the arbitrary regularity and the linear-phase property can not be achieved simultaneously, and has upper computational efficiency.