Construction Of Parameterization For3-band Tight Wavelet Frames Base On MRA

The traditional sense of wavelet frames are always based on2-bandcondition in the field of wavelet analysis. But the2-band condition can notmeet all the requirements in practical application. However M-band waveletframes can get better results in some condition. M-bands condition canovercome some shortcomings and cover the shortage, especially make thedivision of frequency band more carefully, and the energy is more centralizedcorresponding to the signal. There are some special properties and advantagesin the symmetry and smoothness.2-band wavelet frame is just take a specialcase of M-band wavelet frames, and it has universal significance and moreprofound academic value to take serious study of M-band.The unitary extension principle is put forward, then more and morewavelet frames related theories have been given and perfect. C.K.Chui givethe examples in practical application, and in the after study construct M-bandwavelet frame with vanishing moment condition. Lihong Cui and ZhengxingCheng give the method of explicit Construction of Wavelet Tight Frames.Parameterizations is one of the important way to design waveletsystems with special properties. Jiang gives the examples of2-band waveletframes for parameterizations. Base on that, this paper pay attention to theparameterizations for construction of3-band MRA wavelet frames which havethe symmetry properties. First let the3-band wavelet frames satisfies theminimum filter condition by restrictive conditions, as a low pass filter andthree high-pass filter for standard, and design each filters have symmetry oranti symmetry properties. Then using unitary extension principle, the filtermatrix can expansion as a square matrix. By symmetry transformation and orthogonal transformation, the filter matrix can be factorized. After that, usingthe method of parameterizations for each factor. We elimination parameters byusing the basic conditions of3-band wavelet frames, sum rules condition andvanishing moments condition. At last, the wavelet frames can be constructedby taking back the solution of parameters.