### Abstract

The convergence analysis of the Least Mean Square (LMS)
algorithm has been conventionally based on stochastic
signals and describes thus only the average behavior of
the algorithm. It has been shown previously that a
periodic-reference LMS system can be regarded as a linear
time-periodic system whose stability can be determined
from the monodromy matrix. Generally, the monodromy
matrix can only be solved numerically and does not thus
reveal the actual factors behind the dynamics of the
system. This paper derives an estimator for the
eigenvalues of the monodromy matrix. The estimator is
easy to calculate, and it also reveals the underlying
reason for the bad convergence of the LMS algorithm in
some special cases. The estimator is confirmed by
comparing it to the precise eigenvalues of the monodromy
matrix. The estimator is found to be accurate for the
eigenvalues close to unity

Original language | English |
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Title of host publication | Proceedings of 2012 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2012 |

Publisher | IEEE Institute of Electrical and Electronic Engineers |

Pages | 3777-3780 |

ISBN (Electronic) | 978-1-4673-0046-9 |

ISBN (Print) | 978-1-4673-0045-2, 978-1-4673-0044-5 |

DOIs | |

Publication status | Published - 2012 |

MoE publication type | Not Eligible |

Event | IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2012 - Kyoto, Japan Duration: 25 Mar 2012 → 30 Mar 2012 |

### Conference

Conference | IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2012 |
---|---|

Abbreviated title | ICASSP 2012 |

Country | Japan |

City | Kyoto |

Period | 25/03/12 → 30/03/12 |

### Keywords

- LMS algorithm
- monodromy matrix
- estimator for the eigenvalues
- convergence rate

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## Cite this

Haarnoja, T., Tammi, K., & Zenger, K. (2012). An estimator for the eigenvalues of the system matrix of a periodic-reference LMS algorithm. In

*Proceedings of 2012 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2012*(pp. 3777-3780). IEEE Institute of Electrical and Electronic Engineers. https://doi.org/10.1109/ICASSP.2012.6288739