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Learn about the most recent theoretical and practical advances in radar signal processing using tools and techniques from compressive sensing. Providing a broad perspective that fully demonstrates the impact of these tools, the accessible and tutorial-like chapters cover topics such as clutter rejection, CFAR detection, adaptive beamforming, random arrays for radar, space-time adaptive processing, and MIMO radar. Each chapter includes coverage of theoretical principles, a detailed review of current knowledge, and discussion of key applications, and also highlights the potential benefits of using compressed sensing algorithms. A unified notation and numerous cross-references between chapters make it easy to explore different topics side by side. Written by leading experts from both academia and industry, this is the ideal text for researchers, graduate students and industry professionals working in signal processing and radar.
In this chapter, we study specific rank-1 decomposition techniques for Hermitian positive semidefinite matrices. Based on the semidefinite programming relaxation method and the decomposition techniques, we identify several classes of quadratically constrained quadratic programming problems that are polynomially solvable. Typically, such problems do not have too many constraints. As an example, we demonstrate how to apply the new techniques to solve an optimal code design problem arising from radar signal processing.
Introduction and notation
Semidefinite programming (SDP) is a relatively new subject of research in optimization. Its success has caused major excitement in the field. One is referred to Boyd and Vandenberghe  for an excellent introduction to SDP and its applications. In this chapter, we shall elaborate on a special application of SDP for solving quadratically constrained quadratic programming (QCQP) problems. The techniques we shall introduce are related to how a positive semidefinite matrix can be decomposed into a sum of rank-1 positive semidefinite matrices, in a specific way that helps to solve nonconvex quadratic optimization with quadratic constraints. The advantage of the method is that the convexity of the original quadratic optimization problem becomes irrelevant; only the number of constraints is important for the method to be effective. We further present a study on how this method helps to solve a radar code design problem. Through this investigation, we aim to make a case that solving nonconvex quadratic optimization by SDP is a viable approach.
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