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“On Signal Sampling and Wavelet Coefficient Computation with Biorthogonal Wavelets”
by Xiang-Gen Xia, C.-C. Jay Kuo, and Zhen Zhang
September 1992
Several issues on signal sampling and wavelet coefficient computation for a continuous time signal with biorthogonal wavelet bases are studied in this research. Discrete wavelet transform (DWT) is often used to approximate wavelet series transform (WST) and continuous wavelet transform (CWT), since they can be computed numerically. We first estimate the approximation error of computed DWT coefficients by using the Mallat and Shensa algorithms, and show a procedure to design optimal FIR prefilters used in the Shensa algorithm to reduce the approximation error. Then, we examine a specific prefiltering process by which the computed DWT coefficients can be made exactly the same as the WST coefficients for signals belonging to the wavelet subspaces. A formula characterizing the aliasing error resulted from general signals with such a prefilter is also derived. Finally, numerical examples are presented to show the performance of the optimal FIR prefilters.