Research Of Multiwavelets Theory

A multiwavelet is a wavelet generated by a scaling vector function that consists of several functions. Multiwavelet is associated with multiresolution analysis (MRA). A vector function is called a multiscaling function of multiplicity r MRA if the functions generate MRA. More precisely, an MRA of multiplicity r is a nested sequence of closed subspaces Vj in L2(R) satisfying conditions:â…°),â…±),â…²),â…³),â…´)There exist r functions such that the collection of integer translates forms a Resiz basis of V0, where . Let Wj is the orthogonal complement of Vj in Vj+1, if the integer translates of a set of functions form an orthonormal basis of W0, then are called a set of orthogonal multiwavelets. The multiscaling functions of a MRA satisfy a matrix refinable equation: ,and the multiwavelets satisfy:. Multiscaling functions and multiwavelets naturally generalize the scalar scaling functions and scalar wavelets. They can possess all properties simultaneously such as short support, orthogonality, symmetry and vanishing moments or higher approximation order which are superior to scalar wavelet. This is the reason why research and application of multiwavelets are more and more important in science, technology and engineering area.This paper investigates construction of orthogonal multiwavelets and a condition for approximation order of M-band Multiscaling functions in frequency domain, introduces M-band scale similarity transformation (MST). One can increase the approximation order of multiscaling function with help of MST. In practical application the scalar data needs transforming to vector data, this paper describes prefiltering for multiwavelets, and investigates balanced multiwavelets that can avoid prefiltering. We focus on conditions for p-order balance of multiwavelets in time domain, detail theinterrelation between balance order and approximation order, and demonstrate the sampling property of multiwavelets. A construction algorithm for balanced multiwavelets with symmetric multiscaling functions is obtained, and a set of coefficients of symmetric balanced multiwavelets filter bank are given.