Wavelet Transform Prefilter Design Based On Hermite Interpolation

Designing prefilters is an important step in the initialization of wavelet transform when discrete wavelet transforms are used to process continuous-time signals. Its aim is obtaining the projection coefficients of a continuous-time signal onto a fine multi -resolution subspace from the samples of the signal. The importance of the initialization comes from two respects. On the one hand, most signals and scaling functions are no explicit expression. Only scaling filters and samples of the continuous-time signals are known. Therefore, analytical calculation is impossible. On the other hand, the calculation based upon numerical integration is time-consuming and requires high sampling rate to continuous-time signals.The thesis presents a novel method based on the Hermite interpolation to design prefilters. Different from traditional prefiltering problem, here we assume that a continuous-time signal and its derivative are sampling with the same rate. It is derived that, in the case of uniform sampling, the coefficients of prefilter are linear combination of the moments of the analysis scaling function that can be obtained from the Hermite interpolation. Moreover, the approximate order of the obtained prefilters linearly depends on the support length of the analysis scaling function. Aslo, the prefilters in the case of non-uniform sampling are discussed. The experimental results and theoretical analysis show that the prefilters using the Hermite interpolation can achieve better performance than other prefilters.