Sparsity plays an instrumental role in a plethora of scientific fields, including statistical
inference for variable selection, parsimonious signal representations, and solving
under-determined systems of linear equations - what has led to the ground-breaking result
of compressive sampling (CS). This Thesis leverages exciting ideas of sparse signal
reconstruction to develop sparsity-cognizant algorithms, and analyze their performance.
The vision is to devise tools exploiting the ‘right’ form of sparsity for the ‘right’ application
domain of multiuser communication systems, array signal processing systems,
and the emerging challenges in the smart power grid.
Two important power system monitoring tasks are addressed first by capitalizing on
the hidden sparsity. To robustify power system state estimation, a sparse outlier model
is leveraged to capture the possible corruption in every datum, while the problem nonconvexity
due to nonlinear measurements is handled using the semidefinite relaxation
technique. Different from existing iterative methods, the proposed algorithm approximates
well the global optimum regardless of the initialization. In addition, for enhanced
situational awareness, a novel sparse overcomplete representation is introduced to capture
(possibly multiple) line outages, and develop real-time algorithms for solving the
combinatorially complex identification problem. The proposed algorithms exhibit nearoptimal
performance while incurring only linear complexity in the number of lines, which
makes it possible to quickly bring contingencies to attention.
This Thesis also accounts for two basic issues in CS, namely fully-perturbed models
and the finite alphabet property. The sparse total least-squares (S-TLS) approach is
proposed to furnish CS algorithms for fully-perturbed linear models, leading to statistically
optimal and computationally efficient solvers. The S-TLS framework is well
motivated for grid-based sensing applications and exhibits higher accuracy than existing
sparse algorithms. On the other hand, exploiting the finite alphabet of unknown signals
emerges naturally in communication systems, along with sparsity coming from the low
activity of each user. Compared to approaches only accounting for either one of the
two, joint exploitation of both leads to statistically optimal detectors with improved
University of Minnesota Ph.D. dissertation. August 2012. Major: Electrical/Computer Engineering. Advisor: Georgios B. Giannakis. 1 computer file (PDF0; x, 138 pages, appendices A-C.
Sparsity-cognizant algorithms with applications to communications, signal processing, and the smart grid..
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