Wavelet transforms provide a new technique for time-scale analysis of non-stationary signals. Wavelet analysis uses orthonormal bases in which computations can be done efficiently with multirate systems known as filter banks. This thesis develops a comprehensive set of tools for (multidimensional) multirate signal analysis and uses them to investigate two multirate systems: filter banks and transmultiplexers. Several results in filter bank theory are obtained: a new parameterization of unitary filter banks, a theory of modulated filter banks, a theory of filter banks with symmetry restrictions, reduction of the multidimensional rational sampling rate filter bank problem to the uniform sampling rate filter bank problem, solution to the completion problem for filter banks (by reducing it to the (YJBK) parameterization problem in control theory) etc. Perfect reconstruction filter banks are shown to give structured decompostions of separable Hilbert spaces. Filter banks are used to construct several classes of wavelet bases: multiplicity M wavelet tight frames and frames, regular multiplicity M orthonormal bases, modulated wavelet tight frames etc. The thesis describes the design of optimal wavelets for signal representation and the wavelet sampling theorem. Application of wavelets in signal interpolation and in the approximation of linear-translation invariant operators is investigated
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