### Abstract

Here another particular problem is analyzed: the long filter, i.e. a tapped-delay line structure with a large number of taps. For the limiting case of an infinitely long filter, exact closed-form solutions are derived for the steady-state weight-error correlations and the associated “misadjustment”, again valid for any colouring of the input signal and the additive noise, and now also for any stepsize guaranteeing stability.

The analysis is based upon a feedback approach, with a forward branch generating the above-mentioned solution for vanishing stepsize and a peculiar feedback branch responsible for higher-order corrections. As in any feedback structure, instability can occur beyond a critical value of the feedback parameter. In our case an experimentally supported maximum stepsize is found, beyond which spontaneous oscillations might occur.

Original language | English |
---|---|

Pages (from-to) | 690-701 |

Number of pages | 12 |

Journal | Signal Processing |

Volume | 91 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2011 |

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### Cite this

*Signal Processing*,

*91*(4), 690-701. https://doi.org/10.1016/j.sigpro.2010.07.015

}

*Signal Processing*, vol. 91, no. 4, pp. 690-701. https://doi.org/10.1016/j.sigpro.2010.07.015

**Steady-state analysis of the long LMS adaptive filter.** / Butterweck, H.J.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Steady-state analysis of the long LMS adaptive filter

AU - Butterweck, H.J.

PY - 2011

Y1 - 2011

N2 - For the well-known LMS adaptive algorithm no general analytic solutions are available for the steady-state weight-error statistics under stationary stochastic excitation. Only approximate tools have been developed using certain assumptions (like the “independence assumption”) and producing more or less reliable results in practical situations. It is only for the case of a vanishingly small stepsize that such assumptions are not required. There a closed-form solution can be determined for any colouring of the input signal and the additive noise.Here another particular problem is analyzed: the long filter, i.e. a tapped-delay line structure with a large number of taps. For the limiting case of an infinitely long filter, exact closed-form solutions are derived for the steady-state weight-error correlations and the associated “misadjustment”, again valid for any colouring of the input signal and the additive noise, and now also for any stepsize guaranteeing stability.The analysis is based upon a feedback approach, with a forward branch generating the above-mentioned solution for vanishing stepsize and a peculiar feedback branch responsible for higher-order corrections. As in any feedback structure, instability can occur beyond a critical value of the feedback parameter. In our case an experimentally supported maximum stepsize is found, beyond which spontaneous oscillations might occur.

AB - For the well-known LMS adaptive algorithm no general analytic solutions are available for the steady-state weight-error statistics under stationary stochastic excitation. Only approximate tools have been developed using certain assumptions (like the “independence assumption”) and producing more or less reliable results in practical situations. It is only for the case of a vanishingly small stepsize that such assumptions are not required. There a closed-form solution can be determined for any colouring of the input signal and the additive noise.Here another particular problem is analyzed: the long filter, i.e. a tapped-delay line structure with a large number of taps. For the limiting case of an infinitely long filter, exact closed-form solutions are derived for the steady-state weight-error correlations and the associated “misadjustment”, again valid for any colouring of the input signal and the additive noise, and now also for any stepsize guaranteeing stability.The analysis is based upon a feedback approach, with a forward branch generating the above-mentioned solution for vanishing stepsize and a peculiar feedback branch responsible for higher-order corrections. As in any feedback structure, instability can occur beyond a critical value of the feedback parameter. In our case an experimentally supported maximum stepsize is found, beyond which spontaneous oscillations might occur.

U2 - 10.1016/j.sigpro.2010.07.015

DO - 10.1016/j.sigpro.2010.07.015

M3 - Article

VL - 91

SP - 690

EP - 701

JO - Signal Processing

JF - Signal Processing

SN - 0165-1684

IS - 4

ER -