Sparse Decomposition Algorithms for Underdetermined Blind Source Separation

In signal processing applications, we often need to extract target signals from mixed observations. Underdetermined blind source separation represents a commonly employed approach where sparse decomposition algorithms have gained significant traction. These algorithms operate by representing signals as linear combinations of atoms selected from an overcomplete dictionary or basis set. The key advantage of sparse decomposition lies in its ability to identify sparsity patterns within data while capturing essential signal characteristics, simultaneously achieving dimensionality reduction. This makes sparse decomposition particularly valuable for both signal separation tasks and data compression applications. From an implementation perspective, algorithms like Matching Pursuit (MP) or Orthogonal Matching Pursuit (OMP) iteratively select the most correlated dictionary atoms through correlation computations, while Basis Pursuit (BP) formulations utilize l1-norm optimization techniques to achieve sparsity. The core computational steps typically involve constructing an overcomplete dictionary, computing sparse coefficients via optimization methods, and reconstructing source signals through linear combinations.