ADMM based algorithm for spike and smooth signal separation using over-complete dictionary

In signal processing, many a time people deal with smooth stationary signals mixed with sharp spikes and most of the time their analysis demands separation of the smooth and spike elements. In this paper, we propose a methodology for this kind of separation based on the well-known notion of using an over complete dictionary to define an underdetermined system of linear equations and picking out its sparsest solution. The Alternating Direction Method of Multipliers (ADMM) framework is proposed for formulating and solving this optimization problem. The study of the performance of the algorithm with respect to certain signal parameters is also included. Performance of the algorithm was tested for different sparsity values at different signal energies and the results are reported. Cases involving group sparsity is also studied.