Sparse representation for detection of transients using a multi-resolution representation of the auto-correlation of wavelets

This thesis seeks to detect damped sinusoidal transients, specifically capacitor switching transients, buried in noise and to answer the following questions: (1) Can the transient s(t; q) be sparsely represented from sdelta( t) = s(t; q)+epsilon(t) using sparsity methods, where epsilon(t) is white Gaussian noise? (2) Does computing the local auto-correlation of the signal around the transient improve detection? (3) How does the auto-correlation shell representation compare to the wavelet representation? (4) Which basis is "best"? (5) Which method and representation is best? This thesis explores detection schemes based on classical methods and newer sparsity methods. Classical methods considered include reconstruction via wavelets and reconstruction in the novel multi-resolution representation based on the autocorrelation functions of compactly supported wavelets. For simplicity, only four bases are considered: Haar, Daubechies 2, Daubechies 4, and Symlets 2. Sparsity methods include the iterative soft, hard, and combined thresholding algorithms.