<p> The focus of this thesis is on the utilization of the sparsity concept in solving some challenging problems, e.g., finding a unique solution to the under-determined linear system of equations in which the number of equations is less than the number of unknowns. This concept is extended to the problem of solving the under-determined blind source separation (BSS) problem in which the number of source signals is greater than the number of sensors and both the mixing matrix and the source signals are unknowns. In this respect we study three problems: </p> <p> 1. Developing some algorithms for solving the under-determined linear system of equations for the case of a sparse solution vector. In this thesis we develop a new methodology for minimizing a class of non-convex (concave on the non-negative orthant) functions for solving the aforementioned problem. The proposed technique is based on locally replacing the original objective function by a quadratic convex function which is easily minimized. For a certain selection of the convex objective function, the existing class of algorithms called Iterative Re-weighted Least Squares (IRLS) can be derived from the proposed methodology. Thus the proposed algorithms are a generalization and unification of the previous methods. In this thesis we also propose a convex objective function that produces an algorithm that can converge to a sparse solution vector in significantly fewer iterations than the IRLS algorithms.</p> <p> 2. Solving the under-determined BSS problem by developing new clustering algorithms
for estimating the mixing matrix. The under-determined BSS problem is usually solved by following a two-step approach, in which the mixing matrix is estimated in the first step, then the sources are estimated in the second step. For the case of sparse sources, the mixing matrix is usually estimated by clustering the columns of the observation matrix. In this thesis we develop three novel clustering algorithms that can efficiently estimate the mixing matrix, as well as the number of sources, which is usually unknown. Numerical simulations verify the efficiency of the proposed algorithms compared to some well known algorithms that are usually used for solving the same problem.</p> <p> 3. Extraction of a desired source signal from a linear mixture of hidden sources when prior information is available about the desired source signal. There are many situations in which one is interested in extracting a specific source signal. The a priori available information about the desired source signal could be temporal, spatial, or both. In this thesis we develop new algorithms for extracting a desired sparse source signal from a linear mixture of hidden sources. The information available about the desired source signal, as well as its sparsity, are incorporated in an optimization problem for extracting this source signal. Four different algorithms have been developed for solving this problem. Numerical simulations show that the proposed algorithms can be used successfully for removing different kind of artifacts from real electroencephalographic (EEG) data and for estimating the event related potential (ERP) signal from synthesized EEG data.</p> / Thesis / Doctor of Philosophy (PhD)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/17345 |
| Date | January 2009 |
| Creators | Mourad, Nasser |
| Contributors | Reilly, James P., Electrical and Computer Engineering |
| Source Sets | McMaster University |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
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