The previously proposed mixed-norm least-mean-square (MN-LMS) algorithm has shown outstanding performance compared to that of the conventional LMS algorithm. In this paper, the convergence analysis of the MN-LMS algorithm is derived. Based on that, two algorithms that exploit the sparsity of the system have been derived. The first algorithm is proposed by adding $$l_{1}$$ l 1 -norm penalty to the cost function of the MN-LMS algorithms. This term enables us to attract the zero and/or near-to-zero filter coefficients to the zero value faster. However, when the system is near or exactly non-sparse, the algorithm almost fails. To overcome this limitation, we propose another algorithm that uses an approximation of $$l_{0}$$ l 0 -norm penalty term in the cost function of the MN-LMS algorithm. This provides high performance even with completely non-sparse systems. The performances of the proposed algorithms are compared to those of the LMS and MN-LMS algorithms in an acoustic sparse system identification setting. The proposed algorithms provide significant performances compared to the other algorithms under different sparsities and signal-to-noise ratios.

中文翻译：

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混合范数LMS和稀疏系统识别两个版本的收敛性分析

与传统的 LMS 算法相比，先前提出的混合范数最小均方 (MN-LMS) 算法表现出出色的性能。本文导出了MN-LMS算法的收敛性分析。在此基础上，推导出了两种利用系统稀疏性的算法。第一种算法是通过将 $$l_{1}$$l 1 -norm 惩罚添加到 MN-LMS 算法的成本函数中提出的。该术语使我们能够更快地将零和/或接近零的滤波器系数吸引到零值。然而，当系统接近或完全非稀疏时，算法几乎失败。为了克服这个限制，我们提出了另一种算法，该算法在 MN-LMS 算法的成本函数中使用 $$l_{0}$$l 0 -norm 惩罚项的近似值。即使对于完全非稀疏的系统，这也能提供高性能。在声学稀疏系统识别设置中，将所提出算法的性能与 LMS 和 MN-LMS 算法的性能进行比较。在不同的稀疏度和信噪比下，所提出的算法与其他算法相比具有显着的性能。