Sparse Equalizer Filter Design for Multi-path Channels

In this thesis, sparse Finite Impulse Response (FIR) equalizers are designed for sparse multi-path channels under a pre-defined Mean Squared Error (MSE) constraint. We start by examining the intrinsic sparsity of the Zero Forcing equalizers and the FIR Minimum MSE (MMSE) equalizers. Next the equalization MSE is formulated as a quadratic function of the equalizer coefficients. Both the Linear Equalizer (LE) and the Decision Feedback Equalizer (DFE) are analyzed. Utilizing the quadratic form, designing a sparse equalizer under a single MSE constraint becomes an 10-norm minimization problem under a quadratic constraint, as described in [2]. Three previously developed methods for solving this problem are applied, namely the successive thinning algorithm, the branch-and-bound algorithm, and the simple linear programming algorithm. Simulations under various channel specifications, equalizer specifications and algorithm specifications are conducted to show the dependency of the sparsity on these factors. The channels include the ideal discrete multipath channels and the Vehicular A multi-path channels in both the Single-Input-SingleOutput (SISO) and the Multiple-Input-Multiple-Output scenarios. Additionally, the sparse FIR equalizer is designed for MIMO channels under two MSE constraints. This is formulated as an 10-norm minimization problem under two quadratic constraints. A sub-optimal solution by decoupling the two constraints is proposed. Thesis Supervisor: Alan V. Oppenheim Title: Ford Professor of Engineering Acknowledgments First, I am most grateful for my research advisor, Professor Alan Oppenheim. I am fortunate to have had the privilege of being mentored by Al. Al's guidance and support are crucial for the development of this thesis. At the beginning of this path, Al's vision helped me orient myself on the right path. Encouraging me to unconventional thinking and to creativity, stimulating me, and providing me with unlimited freedom, he has made this journey enjoyable and rewarding. I have benefited tremendously from his dedication not only to my intellectual growth, but also my personality development. I deeply appreciate Al's help in shaping me a more organized person. I would like to express my sincere thanks to Dennis Wei for his helpful contribution to this thesis. Dennis has been a close research collaborator and friend over the last two years. Part of this thesis is based on the algorithm that Dennis developed in his PhD thesis. I wish him all the best in his future endeavors. It is a privilege to have been part of the Digital Signal Processing Group (DSPG) at MIT. I would like to thank past and present members of DSPG including Tom Baran, Ballard Blair, Petros Boufounos, Sefa Demirtas, Dan Dudgeon, Zahi Karam, Tarek Aziz Lahlou, Shay Maymon, Joseph McMichael, Martin McCormick, Milutin Pajovic, Charles Rohrs, Guolong Su, and Dennis Wei. The intellectual atmosphere of the group as well as the willingness to share ideas and to collaborate on research problems has made it a very exciting and enriching experience. I also want to thank my fellow students Xun Cai, Atulya Yellepeddi and Feng Gao for their intellectual help in developing this thesis. I gained a lot from the stimulating discussions and I feel very lucky to have such inspiring and encouraging friends. Finally, I am deeply grateful to my parents for their ever present love, their support throughout my education, and their encouraging me to strive for the best. I sincerely thank my whole family.