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**Convergence analysis of sparse LMS algorithms with \(l_{1}\)-norm penalty based on white input signal.**
*(English)*
Zbl 1197.94124

Summary: The zero-attracting LMS (ZA-LMS) algorithm is one of the recently published sparse LMS algorithms. It usesan \(l_{1}\)-norm penalty in the standard LMS cost function. In this paper, we perform convergence analysis of the ZA-LMS algorithm based on white input signals. The stability condition is examined and the steady-state mean square deviation (MSD) is derived in terms of the system sparsity, system response length, and filter parameters (step size and zero-attractor controller). In addition, we propose a criterion for parameter selection such that the ZA-LMS algorithm outperforms the standard LMS algorithm. The results are demonstrated through computer simulations.

### MSC:

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

### Keywords:

least mean square (LMS) algorithm; sparsity; zero attracting; \(l_{1}\) norm; mean square deviation (MSD)
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\textit{K. Shi} and \textit{P. Shi}, Signal Process. 90, No. 12, 3289--3293 (2010; Zbl 1197.94124)

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