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In this work, an effective adaptive block-sparsity log-sum least mean square (BSLS-LMS) algorithm is proposed to improve the convergence performance of cluster sparse system identification. The main idea of the proposed scheme is to add a new block-sparsity induced term into the cost function of the LMS algorithm. We utilize the <inline-formula> <tex-math notation="LaTeX">${l}_{{1}}$ </tex-math></inline-formula> norm of adaptive tap weights and log-sum as a mixed constraint, via optimizing the cost function through the gradient descent method, the proposed adaptive filtering method can iteratively move the identified signals towards the optimal solutions, and finally identify the unknown system accurately. The cluster-sparse system response, with block length and arbitrary average sparsity, is generated by a Markov-Gaussian (M-G) model. For the white Gaussian input data, the theoretical formulas of the steady-state mis-adjustment and convergence behaviors of the BSLS-LMS are derived in a general sparse system and a block sparse system, respectively. Numerical experiments demonstrate that the proposed adaptive BSLS-LMS algorithm achieves much better convergence behavior than conventional sparse signal recovery solutions. The experimental study also verifies the consistency between the simulation results and the theoretical analysis.

Block-Sparsity Log-Sum-Induced Adaptive Filter for Cluster Sparse System Identification