Novel solutions to classical signal processing problems in optimization framework

Abstract

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.