Construction Of M-Band Perfect Reconstruction Filter Banks

A filter banks is a structure which decomposes the signal into a set of sub signals.It has been studied in the field of digital signal processing for many years.The perfect reconstruction filter banks can ensure the lossless transmission of information.In recent years,the theory of perfect reconstruction filter banks have been developed rapidly since they are widely used in many fields such as signal and image processing,data mining,compression sensing,and data feature extraction.The construction for perfect filter banks is the basic problem of filter banks theory,and the effective algorithms is still in investigation.The establishment of multi-resolution analysis and wavelet analysis theory provides a unified framework for this.In this thesis,based on the wavelet analysis theory,the construction problem of perfect reconstruction filter banks is studied.The complete decomposition theory of paraunitary polyphase matrix with multiple centers of symmetry is established,and the Euclid algorithm for matrix expansion to construction of perfect reconstruction filter banks is obtained.The thesis is organized as follows.In Chapter 1,we introduce the basic concepts and properties of the filter bank theory,summarize the theory of multi-resolution analysis,explain two different ways to construct the filter bank: the lattice decomposition of paraunitary polyphase matrix and the extension of Laurent polynomial matrix,and elaborate the background and significance of this study.In Chapter 2,by studying the representation of the M-band compactly supported symmetric scaling filter,using the balance vector and orthogonal matrix,two simple construction methods are obtained.As an application,a class of 4-band compactly supported symmetric wavelet frame system with beautiful structure are obtained.The number of free variables in the whole system is less than 4,which effectively reduces the computational complexity.Then,a type of symmetric wavelet system with length 16 is parameterized.The entire system is determined by the three angular parameters ,,and which is conducive to the design of a filter banks with excellent performance.Finally,two numerical examples are given to illustrate this construction method.The lattice decomposition of paraunitary polyphase matrix is an effective method for construction filter banks,however,the filter banks constructed by this method often has only one symmetry center,and the filter bank with multiple symmetry centers usually also has good performance.In view of this,in Chapter 3,we study the construction method of filter banks with multiple symmetry centers.A space decomposition of an orthogonal projection matrix is studied.This decomposition plays a key role in a new complete factorization theory.In addition,the concept of a minimal starting block matrix is proposed and is used to establish a new factorization of a 2m-band paraunitary polyphase matrix with multiple centers of symmetry.This factorization has the completeness property.The different possible forms of the minimal starting block matrix,which lead to the different types of filter banks,are obtained.Through different combinations of minimal starting block matrices and orthogonal projection matrices,the general solutions of a 28)-band paraunitary system with multiple centers are obtained theoretically.The four-band issue is discussed in detail as an example.Its general solution is given.For different situations,by adding regularity conditions,specific numerical examples are given,and the coding gains of these filters are calculated.According to the theory of multiresolution analysis,the construction of wavelet basis(frame)is identical with the construction of perfect reconstruction filter banks.The construction of wavelet system is reduced to solving the matrix satisfying the condition of perfect reconstruction.In Chapter 4,we study the division in the Laurent polynomial ring,and establish the Euclidean algorithm of Laurent polynomial matrix extension.By adding different uniqueness conditions,we establish several Euclidean division theories in the Laurent polynomial ring.Based on this theory,the Laurent polynomial matrix extension algorithms for -band filter banks are obtained.Finally,we give some numerical examples to illustrate our construction method.Compared with the existing matrix extension algorithm,this algorithm is easy to understand and use,and different kinds of filter banks can be obtained by using different Laurent polynomial division.The conclusions and innovations of this thesis,the deficiencies of the research and the problems of further research are summarized in Chapter 5.