Convex Problems in Signal Processing and Communications

In this project we investigated applications of convex optimization in communications and signal processing. In general, many engineering problems (many more than normally recognized) are convex which allows them to be solved globally and very efficiently. It is possible to include a wide range of constraints in the problem, e.g. fairness in resource allocation etc. In our work we investigate several different applications including 1. Robust estimation and equalization in linear systems. This has useful applications in systems where e.g. data detection is based on imperfect channel knowledge. 2. Approximate maximum-likelihood (ML) estimation using semi-definite programming. Semi-definite programming has recently been proposed as a promising technique to obtain provable good bounds on certain difficult combinatorial problems. We investigate the application of semi-definite relaxations for ML estimation in connection with e.g. multiuser detection. 3. Joint optimization of analysis and synthesis windows for Gabor analysis. We design tight frames that are optimally localized under the Heisenberg uncertainty measure. 4. A characterization of the solution to general indefinite trust-region problems. In many engineering problems we minimize a quadratic function subject to a single quadratic constraint, e.g., a Least-Squares problem over a sphere. We completely characterize the solution to such problems, and give a closed form solution whenever the solution exists. For references to this work consult [J. Dahl 2003] (Joachim Dahl, Søren Vang Andersen, Bernard H. Fleury, Søren Holdt Jensen, Jakob Stoustrup, Lieven Vandenberghe (University of California Los Angeles))