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dc.contributor.advisorArıkan, Orhan
dc.contributor.authorPilancı, Mert
dc.date.accessioned2016-01-08T18:13:43Z
dc.date.available2016-01-08T18:13:43Z
dc.date.issued2010
dc.identifier.urihttp://hdl.handle.net/11693/15116
dc.descriptionAnkara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 2010.en_US
dc.descriptionThesis (Master's) -- Bilkent University, 2010.en_US
dc.descriptionIncludes bibliographical references leaves 72-79.en_US
dc.description.abstractIn this thesis, new theoretical and practical results on linear equations with various types of uncertainties and their applications are presented. In the first part, the case in which there are more equations than unknowns (overdetermined case) is considered. A novel approach is proposed to provide robust and accurate estimates of the solution of the linear equations when both the measurement vector and the coefficient matrix are subject to uncertainty. A new analytic formulation is developed in terms of the gradient flow to analyze and provide estimates to the solution. The presented analysis enables us to study and compare existing methods in literature. We derive theoretical bounds for the performance of our estimator and show that if the signal-to-noise ratio is low than a treshold, a significant improvement is made compared to the conventional estimator. Numerical results in applications such as blind identification, multiple frequency estimation and deconvolution show that the proposed technique outperforms alternative methods in mean-squared error for a significant range of signal-to-noise ratio values. The second type of uncertainty analyzed in the overdetermined case is where uncertainty is sparse in some basis. We show that this type of uncertainty on the coefficient matrix can be recovered exactly for a large class of structures, if we have sufficiently many equations. We propose and solve an optimization criterion and its convex relaxation to recover the uncertainty and the solution to the linear system. We derive sufficiency conditions for exact and stable recovery. Then we demonstrate with numerical examples that the proposed method is able to recover unknowns exactly with high probability. The performance of the proposed technique is compared in estimation and tracking of sparse multipath wireless channels. The second part of the thesis deals with the case where there are more unknowns than equations (underdetermined case). We extend the theory of polarization of Arikan for random variables with continuous distributions. We show that the Hadamard Transform and the Discrete Fourier Transform, polarizes the information content of independent identically distributed copies of compressible random variables, where compressibility is measured by Shannon’s differential entropy. Using these results we show that, the solution of the linear system can be recovered even if there are more unknowns than equations if the number of equations is sufficient to capture the entropy of the uncertainty. This approach is applied to sampling compressible signals below the Nyquist rate and coined ”Polar Sampling”. This result generalizes and unifies the sparse recovery theory of Compressed Sensing by extending it to general low entropy signals with an information theoretical analysis. We demonstrate the effectiveness of Polar Sampling approach on a numerical sub-Nyquist sampling example.en_US
dc.description.statementofresponsibilityPilancı, Merten_US
dc.format.extentxiv, 79 leaves, illustrationsen_US
dc.language.isoEnglishen_US
dc.rightsinfo:eu-repo/semantics/openAccessen_US
dc.subjectStatistical Signal Processingen_US
dc.subjectSource Polarizationen_US
dc.subjectPolar Codesen_US
dc.subjectInformation Theoryen_US
dc.subjectCompressed Sensingen_US
dc.subjectSparse Signal Processingen_US
dc.subjectErrors in Variables Modelen_US
dc.subjectLeast Squares Estimationen_US
dc.subjectLinear Algebraen_US
dc.subject.lccTK5102.9 .P55 2010en_US
dc.subject.lcshSignal processing--Statistical methods.en_US
dc.subject.lcshSignal processing--Digital techniques--Mathematics.en_US
dc.subject.lcshLinear systems.en_US
dc.subject.lcshEstimation theory.en_US
dc.subject.lcshAlgebras, Linear.en_US
dc.subject.lcshDifferential equations, Linear.en_US
dc.subject.lcshEquations.en_US
dc.titleUncertain linear equationsen_US
dc.typeThesisen_US
dc.departmentDepartment of Electrical and Electronics Engineeringen_US
dc.publisherBilkent Universityen_US
dc.description.degreeM.S.en_US
dc.identifier.itemidB122219


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