Summary: It has recently been shown that there is a strong connection between multirate finite impulse response filter banks and orthonormal compactly supported wavelet bases. The filter banks can be used, not just to give a discrete time version of the wavelet transform, but to define continuous time wavelets as well. We briefly review the existing work on the relations between multiresolution analysis, orthonormal bases of compactly supported wavelets, and orthogonal multirate filter banks. We then show how more general filter banks may be used to generate biorthogonal bases of compactly supported wavelets. An important advantage of these designs is that it becomes possible to have symmetric wavelets. We present new results on the algebraic structure of perfect reconstruction filter banks, and show how they may be used to derive general biorthogonal systems of wavelets. Further we show that it is possible to achieve both orthogonality and symmetry in filter banks which use infinite impulse response filters which generate infinitely supported orthogonal wavelets. Design examples of the various methods are presented.
For the entire collection see [Zbl 0809.00021].