A state space approach to the analysis of frames and wavelets

Abstract

The thesis presents the author's research work on a state space approach to analysis frames and wavelets. Frames are usually represented by linear operators, which have state space representations. The author develops a direct approach to compute the frame bounds of mixed causal-anticasual linear systems, which includes linear time-invariant case and linear time-varying (LTV) case. The advantage of this approach is that it avoids converting the mixed casual-anticausal representation into unstable casual realization, hence saving many computation steps. The causality properties of the cascaded casual-anticausal LTV systems are investigated. They can be applied to the inner-coprime factorizations, which are used to obtain the pseudo-inverse system (dual frame system). Frames are often associated with wavelet frames. One class of wavelet frames is realized in the discrete wavelet transform. The state space realizations of this class wavelet frames and dual systems can be represented. Hence the frame bounds can be obtained with the causal-anticausal state space representations.