### Abstract

The analysis of the Filtered-x Least Mean Square (FxLMS)
algorithm can be based either on a stochastic or
deterministic approach. One of the drawbacks of the
stochastic analysis is that it relies on several
assumptions which are not justified if the reference
signal is time periodic. To overcome these limitations,
several deterministic methods have been suggested.
However, useful results have been reported only for
setups with certain filter length, secondary path model,
or reference signal. In this paper, an exact Linear
Time-Periodic (LTP) representation is proposed assuming
only that the reference is synchronously-sampled. The
representation is derived for a general FxLMS system,
which also covers the multichannel topology of several
parallel filters and its common narrowband modification.
The representation is found to have a Linear
Time-Invariant (LTI) form if the filter length is
suitably chosen. The usability of the method is
demonstrated in a numerically computed example by
comparing it to an approximate LTI model.

Original language | English |
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Pages (from-to) | 121-130 |

Number of pages | 10 |

Journal | IEEE Transactions on Signal Processing |

Volume | 62 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- active noise control (ANC)
- convergence analysis
- convergent control (CC)
- filtered-x least mean square (FxLMS)
- higher harmonic control (HHC)
- multichannel FxLMS
- narrowband FxLMS
- state-space representation

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## Cite this

Haarnoja, T., Tammi, K., & Zenger, K. (2014). Exact LTP representation of the generalized periodic-reference FxLMS algorithm.

*IEEE Transactions on Signal Processing*,*62*(1), 121-130. https://doi.org/10.1109/TSP.2013.2285513