Maximally sparse convex estimation and equalization

Abstract

Sparse multipath channels are wireless links commonly found in communication systems such as High Frequency radio channels, horizontal and vertical underwater acoustic channels and terrestrial broadcasting channels for High Definition Television. Their impulse responses are characterized by a few significant terms that are widely separated in time. With high speed transmission, the length of a sampled sparse channel can reach hundreds of symbol interval. Thus, the amount of Intersymbol Interference (ISI) at the receiver is very high. Consequently, the presence of an ISI mitigating structure at the receiver, such as the Decision Feedback Equalizer (DFE) is essential. Due to the sparse impulse responses of these channels, traditional estimation techniques such as Least Squares (LS) result in over-parameterization and thus poor performance of the estimator. Also, classical equalizers become too complex for tackling these channels. The problem of estimating and equalizing sparse multipath channels is considered in this thesis. We formulate the sparse channel estimation and the computation of the sparse DFE filters as sparse approximation problems. A usual approach in sparse approximation problems is regularization with an l_1 norm penalty term and usage of convex optimization techniques in order to acquire a solution. Other sparsity promoting penalty functions are available, but the l_1 norm has the advantage to be a convex function, making the l_1 norm regularized approximation problem a convex one. When a problem is formulated as a convex optimization problem, it can be solved by very fast, efficient and reliable algorithms. In order to achieve sparser solutions and still gain from the benefits of the convex optimization theory, the Maximally Sparse Convex (MSC) algorithm utilizes a non-convex regularization term, that promotes sparsity more strongly than the l_1 norm, but chosen such that the total cost function remains convex.