Study On The Balanced Multiwavelet And The Multiwavelet Network

AbstractThe good time-fiequency locality of the wavelet makes it be apopulal tool in many fields, especially the fields of numerical analysisand signal processing. According to the traditional single wavelet theorywe can not consimct the wavelets satisfying comPact support,orthogonal, symmCtYic and high order aPproximating prope1ties at thesame time, Which limit their aPPlication largely The multiwaveletsystem overcomes above limitations and makes a new way of the theoryand application study of wavelet theory. However, it usually lacks a kindof property which traditional single wavelet has, that is the zeromoments of the wavelet W,(t)onR do not necessarily imply zeromoments of the wavelet filter on Z, i.e., balancing property Whichleads to a mixing of the coarse resolution and details coefficientscreating strong oscillations in the signal reconsmicted from the coarseresolution only In the first part of this paPer, given the parametricexpressions of balanced multiwaveIet systems, we discuss theirconstttiction and propeYties, especially the relation betWeen theparameters and the balancing and aFIproximation properties. Then we geta multiwavelet bank making the construction of given multiwaveleteasy.Because of the powerful abilit/ to solve the nonlinear problems ofthe feedforward neural netWork, it has become a popu1ar tool in severalfields. Nevertheless, for the multila3/er structure and the inconvex of theerror surface, it always settles in 1zndesirable minor and we have nosystematical way to determine either the parameters of neurons or itsstructure. Noticing the similarity of the sttuctufe of waveletdecomposition and the feedforwai.d neural netWork, some scientistsconstrUcted the wavelet neural netWorkS (WNN) which not only solvethe constructive problem of the neural netWork but also make it notsettle in the undesirable minor. But tfhe higher order of the problems, themore slowly of the conve1ging ratI3, i.e. the "dimension disaster" hasbecome the obstacle of its aPplication. In the second part of this papeT,the author replaces the traditional wavelet system with the multiwaveletsystem to get the multiwavelet neural netWork (MWN'N). It not onlykeeps most of the good properties c'f tl1e W'NN, but also brings anothergood properties, such as quicker converging rate, more powerfullearning ability. Especially, we prove that the MWNN can solve the 揹imension disaster? which gives the theory basis of its application. Experiments show the superiority of it.