While bandlimited wavelets and associated IIR filters have shown serious potential in areas of pattern recognition and communications, the dyadic Meyer wavelet is the only known approach to construct bandlimited orthogonal decomposition. The sine scaling function and wavelet are a special case of the Meyer. Previous works have proposed a M - Band extension of the Meyer wavelet without solving the problem. One key contribution of this thesis is the derivation of the correct bandlimits for the scaling function and wavelets to guarantee an orthogonal basis. In addition, the actual construction of the wavelets based upon these bandlimits is developed. A composite wavelet will be derived based on the M scale relationships from which we will extract the wavelet functions. A proper solution to this task is proposed which will generate associated filters with the knowledge of the scaling function and the constraints for Mband orthogonality.

This dissertation develops a new wavelet design technique that produces a wavelet that matches a desired signal in the least squares sense. The Wavelet Transform has become very popular in signal and image processing over the last 6 years because it is a linear transform with an infinite number of possible basis functions that provides localization in both time (space) and frequency (spatial frequency). The Wavelet Transform is very similar to the matched filter problem, where the wavelet acts as a zero mean matched filter. In pattern recognition applications where the output of the Wavelet Transform is to be maximized, it is necessary to use wavelets that are specifically matched to the signal of interest. Most current wavelet design techniques, however, do not design the wavelet directly, but rather, build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing multiresolution analysis or design a multiresolution analysis that is generated by a scaling function which has a specific corresponding wavelet. In this dissertation, an algorithm for finding both symmetric and asymmetric matched wavelets is developed. It will be shown that under certain conditions, the matched wavelets generate an orthonormal basis of the Hilbert space containing all finite energy signals. The matched orthonormal wavelets give rise to a pair of Quadrature Mirror Filters (QMF) that can be used in the fast Discrete Wavelet Transform. It will also be shown that as the conditions are relaxed...