Statistical-Mechanics Approach to Theoretical Analysis of the FXLMS Algorithm

Seiji MIYOSHI; Yoshinobu KAJIKAWA

doi:10.1587/transfun.E101.A.2419

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Statistical-Mechanics Approach to Theoretical Analysis of the FXLMS Algorithm

Seiji MIYOSHI, Yoshinobu KAJIKAWA

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Summary :

We analyze the behaviors of the FXLMS algorithm using a statistical-mechanical method. The cross-correlation between a primary path and an adaptive filter and the autocorrelation of the adaptive filter are treated as macroscopic variables. We obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables under the condition that the tapped-delay line is sufficiently long. The obtained equations are deterministic and closed-form. We analytically solve the equations to obtain the correlations and finally compute the mean-square error. The obtained theory can quantitatively predict the behaviors of computer simulations including the cases of both not only white but also nonwhite reference signals. The theory also gives the upper limit of the step size in the FXLMS algorithm.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E101-A No.12 pp.2419-2433

Publication Date: 2018/12/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E101.A.2419

Type of Manuscript: PAPER

Category: Digital Signal Processing

Authors

Seiji MIYOSHI
Kansai University
Yoshinobu KAJIKAWA
Kansai University

Keyword

FXLMS algorithm, adaptive filter, active noise control, statistical-mechanical informatics, long-filter assumption

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Seiji MIYOSHI, Yoshinobu KAJIKAWA, "Statistical-Mechanics Approach to Theoretical Analysis of the FXLMS Algorithm" in IEICE TRANSACTIONS on Fundamentals, vol. E101-A, no. 12, pp. 2419-2433, December 2018, doi: 10.1587/transfun.E101.A.2419.
Abstract: We analyze the behaviors of the FXLMS algorithm using a statistical-mechanical method. The cross-correlation between a primary path and an adaptive filter and the autocorrelation of the adaptive filter are treated as macroscopic variables. We obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables under the condition that the tapped-delay line is sufficiently long. The obtained equations are deterministic and closed-form. We analytically solve the equations to obtain the correlations and finally compute the mean-square error. The obtained theory can quantitatively predict the behaviors of computer simulations including the cases of both not only white but also nonwhite reference signals. The theory also gives the upper limit of the step size in the FXLMS algorithm.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/transfun.E101.A.2419/_p

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@ARTICLE{e101-a_12_2419,
author={Seiji MIYOSHI, Yoshinobu KAJIKAWA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Statistical-Mechanics Approach to Theoretical Analysis of the FXLMS Algorithm},
year={2018},
volume={E101-A},
number={12},
pages={2419-2433},
abstract={We analyze the behaviors of the FXLMS algorithm using a statistical-mechanical method. The cross-correlation between a primary path and an adaptive filter and the autocorrelation of the adaptive filter are treated as macroscopic variables. We obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables under the condition that the tapped-delay line is sufficiently long. The obtained equations are deterministic and closed-form. We analytically solve the equations to obtain the correlations and finally compute the mean-square error. The obtained theory can quantitatively predict the behaviors of computer simulations including the cases of both not only white but also nonwhite reference signals. The theory also gives the upper limit of the step size in the FXLMS algorithm.},
keywords={},
doi={10.1587/transfun.E101.A.2419},
ISSN={1745-1337},
month={December},}

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TY - JOUR
TI - Statistical-Mechanics Approach to Theoretical Analysis of the FXLMS Algorithm
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2419
EP - 2433
AU - Seiji MIYOSHI
AU - Yoshinobu KAJIKAWA
PY - 2018
DO - 10.1587/transfun.E101.A.2419
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E101-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2018
AB - We analyze the behaviors of the FXLMS algorithm using a statistical-mechanical method. The cross-correlation between a primary path and an adaptive filter and the autocorrelation of the adaptive filter are treated as macroscopic variables. We obtain simultaneous differential equations that describe the dynamical behaviors of the macroscopic variables under the condition that the tapped-delay line is sufficiently long. The obtained equations are deterministic and closed-form. We analytically solve the equations to obtain the correlations and finally compute the mean-square error. The obtained theory can quantitatively predict the behaviors of computer simulations including the cases of both not only white but also nonwhite reference signals. The theory also gives the upper limit of the step size in the FXLMS algorithm.
ER -