Browsing by Subject "rank"

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Open Access
Novel solutions to classical signal processing problems in optimization framework
(2014) Alp, Yaşar Kemal
Novel approaches for three classical signal processing problems in optimization framework are proposed to provide further flexibility and performance improvement. In the first part, a new technique, which uses Hermite-Gaussian (HG) functions, is developed for analysis of signals, whose components have non-overlapping compact time-frequency supports. Once the support of each signal component is properly transformed, HG functions provide optimal representations. Conducted experiments show that proposed method provides reliable identification and extraction of signal components even under severe noise cases. In the second part, three different approaches are proposed for designing a set of orthogonal pulse shapes for ultra-wideband communication systems with wideband antennas. Each pulse shape is modelled as a linear combination of time shifted and scaled HG functions. By solving the constructed optimization problems, high energy pulse shapes, which maintain orthogonality at the receiver with desired timefrequency characteristics are obtained. Moreover, by showing that, derivatives of HG functions can be represented as a linear combination of HGs, a simple optimal correlating receiver structure is proposed. In the third part, two different methods for phase-only control of array antennas based on semidefinite modelling are proposed. First, antenna pattern design problem is formulated as a non-convex quadratically constraint quadratic problem (QCQP). Then, by relaxing the QCQP formulation, a convex semidefinite problem (SDP) is obtained. For moderate size arrays, a novel iterative rank refinement algorithm is proposed to achieve a rank-1 solution for the obtained SDP, which is the solution to the original QCQP formulation. For large arrays an alternating direction method of multipliers (ADMM) based solution is developed. Conducted experiments show that both methods provide effective phase settings, which generate beam patterns under highly flexible constraints.
Open Access
Ramanujan's congruences for the partition function
(2009) Aygin, Zafer Selçuk
In this thesis, we study Ramanujan’s congruences for the partition function and some of their combinatorial interpretations. Our main tools are from the theory of theta function.
Open Access
Random codes and matrices
(2009) Özen, İbrahim
Results of our study are two fold. From the code theoretical point of view our study yields the expectations and the covariances of the coefficients of the weight enumerator of a random code. Particularly interesting is that, the coefficients of the weight enumerator of a code with random parity check matrix are uncorrelated. We give conjectures for the triple correlations of the coefficients of weight enumerator of random codes. From the random matrix theory point of view we obtain results in the rank distribution of column submatrices. We give the expectations and the covariances between the ranks (q −rank) of such submatrices over Fq. We conjecture the counterparts of these results for arbitrary submatrices. The case of higher correlations gets drastically complicated even in the case of three submatrices. We give a formula for the correlation of ranks of three submatrices and a conjecture for its closed form.