The wavelet transforms and time-scale analysis of signals

Orthonormal wavelet bases provide an alternative technique for the analysis of non-stationary signals. Unlike the Gabor representation, the basis functions in the wavelet representation all have the same band width on a logarithmic scale. This thesis develops a general framework for the time-scale analysis of signals. In this context, the ON wavelets form a subclass of DWT wavelets. Efficient algorithms for the computation of the wavelet transforms are also developed. As an application, we discuss the problem of detection of (wideband) signals subjected to scale-time perturbations. The probable unknown parameters for scale-time perturbed signals are the gain, and the scale and time perturbations. This problem is set in the context of classical composite hypothesis testing with unknown parameters, and depending on what the unknown parameters are, one of the wavelet transforms, developed is shown to naturally lead to a detector

Wavelets and filter banks: New results and applications

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