Construction Of M-Band Wavelets Based On PR Filter Bank

M(M≥3) -band wavelet has been a new research hotspot in the information field. M-band wavelet can simultaneously have compact support, orthogonality, symmetry or anti-symmetry and so on, and these properties are quite useful in practical applications. Besides, M-band wavelet can precisely process partial information of signal distributed into multi-bands frequency. These conquer the deficiencies of two-band wavelet. However, for any positive integer M≥3, there exists no uniform method to construct M-band wavelet. In practice, we usually construct filter, but not wavelet, in the construction of wavelet. Generally speaking, the filter which can be constructed wavelet is called perfect reconstruct filter (PR) .Scaling filter and wavelet filters are called perfect reconstruct filter bank.In this paper, supposing the scaling filter of PR filter bank has been known, based on the characteristic of polyphase matrix of M-band orthogonal or biorthogonal wavelet, we construct the corresponding M-band orthogonal or biorthogonal wavelet filters by matrix extension.This dissertation consists of six chapters.In chapter one, we first introduce the development status and application actuality of the theory of wavelet. Then we introduce the research significance of the theory of M-band wavelet and the general situation of construction to M-band wavelet.In chapter two, we introduce some basal knowledge, such as matrix theory, the basic theory of M-band wavelet and so on. These are the theory foundation for the latter chapters.In chapter three, an approach to construct M-band orthogonal wavelet filters by matrix extension is presented. If the M-band orthogonal scaling filter is known, based on the fact that the polyphase matrix of M-band orthogonal wavelet is a paraunitay matrix, we provide a method of paraunitay matrix extension, which can construct the corresponding M-band wavelet filters, and we give examples to verify the validity of this method.In chapter four, when M is even, we research the approaches to construct M-band orthogonal symmetric wavelet filters by matrix symmetric extension. If the M-band orthogonal scaling filter is known, based on the fact that the polyphase matrix of M-band orthogonal wavelet is a paraunitay matrix, we provide two methods of paraunitay matrix symmetric extension, which can construct the corresponding M-band symmetric wavelet filters, and we give examples to verify the validity of the two methods.In chapter five, when M is odd, we research the approaches to construct M-band orthogonal symmetric wavelet filters by matrix symmetric extension. First we introduce the theory of multiwavelet, then provide two methods to construct multiwavelet filters by matrix symmetric extension. We put two uniform M-band orthogonal symmetric scaling filters to compose a multiwavelet scaling filter, using above methods we can construct two sets of M-band orthogonal symmetric wavelet filters, and we give examples to verify the validity of the two methods.In chapter six, an approach to construct M-band biorthogonal wavelet filters by matrix extension is presented. If the M-band orthogonal scaling filters of analysis and synthesis are known, based on the correlation of polyphase matrix of analysis and synthesis, we provide a method of paraunitay matrix extension, which can construct the corresponding M-band wavelet filters of analysis and synthesis, and we give examples to verify the validity of this method.Finally, a brief summary of the dissertation is given. The suggestion for future research related to the construction of M-band wavelet is brought forward.