Interpolating Multiwavelets And Multiwavelets Packet For Signal Processing

Multiwavelets constitute a new chapter which has been added to wavelet theory inrecent years. Since multiwavelets have more than one scaling function and motherwavelet, there is more freedom in the design of multiwavelets than scalar wavelet, andas a result, multiwavelets can simultaneously posses many desired properties such asshort support, orthogonality, symmetry, and high vanishing moments, which a singlewavelet cannot posses simultaneously. Multiwavelets increasingly have attracted muchattention in the research community. The focus in this dissertation is on discussion ofsome basic problems of multiwavelets for signal processing, including interpolatingmultiwavelets, sampling theorems in multiwavelets subspace, orthogonal multiwaveletspacket and biorthogonal multiwavelets packet, the application of wavelet packet andmultiwavelets packet to CDMA system. 1. Since 2-band interpolating wavelet transform has the advantage that waveletseries transform coefficient is same as binary uniform samples in multiresolutionsubspace, Prefilter operator of signal in some scaling space is simplified to one.However, there does not exist the orthogonal 2-band interpolating wavelet with compactsupport except Haar wavelet. Because multiwavlets have more design parameters thansingle wavelet, the design of compact interpolating multiwavelets becomes possible. Weuse the definition of interpolating multiwavelets, and convert the problem of designinginterpolating multiwavelets of multiplicity two into designing a corresponding scalarfilter. Therefore, they are easily designed. Moreover, we discuss some importantproperties of interpolating multiwavelets. In addition to sharing some commonadvantages of the interpolating scalar wavelet, such as equality of the flatness degree ofthe low-pass filters with the approximation order of the scaling function and equalitybetween the uniform samples of signal and its projection coefficients for a given scalar,our multiwavelets possess its own characteristic. One is that there is a simplerelationship between the multiscaling functions and the multiwavelets. The other is thatany continuous signal in a multiresolution subspace can be reconstructed quickly via asimple recursive algorithm without errors induced by truncation of the synthesisfunction. This suggests that prefiltering is not necessary for interpolating discreteorthogonal multiwavelet transform. 2. The sampling theorem plays a crucial role in signal processing and 雷达信号处理重点实验室IV 插值多子波和多子波包信号处理方法研究communication, as it establishes equivalence between discrete signals and analoguesignals. It is well known that classical Shannon sampling theorem, applied toband-limited functions, has several problems. Firstly, real world signals are neverexactly band-limited. Secondly, Shannon's reconstruction formula is rarely used inpractice because of the slow decay of the sinc function. Walter found that sinc functionwas virtually a scaling function of multiresolution analysis and presented the samplingtheorem for wavelet subspaces under very mild condition on the decay and regularity ofscaling function. Then, for any signal of scaling subspace, it can be reconstructed usingthe uniform integer sampling. Following Walter's work, Janssen extended Walter'sresult to the uniform non-integer sampling which was also called shift sampling by Zaktransform. With multiwavelets have attracted much attention, the sampling theorem forseveral generating functions in shift-invariant subspaces and multiwavelets subspaceswere considered. The exact reconstruction formula was introduced by Selesnick.However, Selesnick only focused on multiwavelets with interpolating property. In thefourth chapter, we extend the Walter's wavelet sampling theorem to multiwaveletssubspace. By reproducing kernel Hilbert space, we give the regular...