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

*(English)*Zbl 1197.94124Summary: 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)
PDF
BibTeX
XML
Cite

\textit{K. Shi} and \textit{P. Shi}, Signal Process. 90, No. 12, 3289--3293 (2010; Zbl 1197.94124)

Full Text:
DOI

##### References:

[1] | Schreiber, W. F.: Advanced television systems for terrestrial broadcasting: some problems and some proposed solutions, Proceedings of the IEEE 83, 958-981 (1995) |

[2] | Duttweiler, D. L.: Proportionate normalized least-mean-squares adaptation in echo cancellers, IEEE transactions on speech audio processing, 508-518 (2000) |

[3] | F. Souza, O. Tobias, R. Seara, D. Morgan, Alternative approach for computing the activation factor of the PNLMS algorithm, in: Proceedings of the EUSIPCO, Glasgow, Scotland, August 2009, pp. 2633–2637. |

[4] | O.A. Noskoski, J. Bermudez, Wavelet-packet-based adaptive algorithm for sparse impulse response identification, in: Proceedings of the ICASSP, Honolulu, Hawaii, April 2007, pp. 1321–1324. |

[5] | Tibshirani, R.: Regression shrinkage and selection via the lasso, Journal of royal statistical society series B 58, 267-288 (1996) · Zbl 0850.62538 |

[6] | Y. Chen, Y. Gu, A.O. Hero, Sparse LMS for system identification, in: Proceedings of the IEEE ICASSP, Taipei, Taiwan, April 2009, pp. 3125–3128. |

[7] | Gu, Y.; Jin, J.; Mei, S.: L0 norm constraint LMS algorithm for sparse system identification, IEEE signal processing letters 16, 774-777 (September 2009) |

[8] | Jin, J.; Gu, Y.; Mei, S.: A stochastic gradient approach on compressive sensing signal reconstruction based on adaptive filtering framework, IEEE journal of selected topics in signal processing 4, 409-420 (April 2010) |

[9] | Sayed, A. H.: Fundamentals of adaptive filtering, (2003) |

[10] | Allen, J. B.; Berkley, D. A.: Image method for efficiently simulating small-room acoustics, Journal of acoustic society America 65, 943-950 (1979) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.