An efficient normalized LMS algorithm
The task of adaptive estimation in the presence of random and highly nonlinear environment such as wireless channel estimation and identification of non-stationary system etc. has been always challenging. The least mean square (LMS) algorithm is the most popular algorithm for adaptive estimation and...
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| Published in | Nonlinear dynamics Vol. 110; no. 4; pp. 3561 - 3579 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.12.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0924-090X 1573-269X |
| DOI | 10.1007/s11071-022-07773-0 |
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| Abstract | The task of adaptive estimation in the presence of random and highly nonlinear environment such as wireless channel estimation and identification of non-stationary system etc. has been always challenging. The least mean square (LMS) algorithm is the most popular algorithm for adaptive estimation and it belongs to the gradient family, thus inheriting their low computational complexity and their slow convergence. To deal with this issue, an efficient normalization of the LMS algorithm is proposed in this work which is achieved by normalizing the input signal with an intelligent mixture of weighted signal and error powers which results in a variable step-size type algorithm. The proposed normalization scheme can provide both significant faster convergence in initial adaptation phase while maintaining a lower steady-state mean-square-error compared to the conventional normalized LMS (NLMS) algorithm. The proposed algorithm is tested on adaptive denoising of signals, estimation of unknown channel, and tracking of random walk channel and its performance is compared with that of the standard LMS and NLMS algorithms. Mean and mean-square performance of the proposed algorithm is investigated in both stationary and non-stationary environments. We derive the closed-form expressions of various performance measures by evaluating multi-dimensional moments. This is done by statistical characterization of required random variables by employing the approach of Indefinite Quadratic Forms. Simulation and experimental results are presented to corroborate our theoretical claims. |
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| AbstractList | The task of adaptive estimation in the presence of random and highly nonlinear environment such as wireless channel estimation and identification of non-stationary system etc. has been always challenging. The least mean square (LMS) algorithm is the most popular algorithm for adaptive estimation and it belongs to the gradient family, thus inheriting their low computational complexity and their slow convergence. To deal with this issue, an efficient normalization of the LMS algorithm is proposed in this work which is achieved by normalizing the input signal with an intelligent mixture of weighted signal and error powers which results in a variable step-size type algorithm. The proposed normalization scheme can provide both significant faster convergence in initial adaptation phase while maintaining a lower steady-state mean-square-error compared to the conventional normalized LMS (NLMS) algorithm. The proposed algorithm is tested on adaptive denoising of signals, estimation of unknown channel, and tracking of random walk channel and its performance is compared with that of the standard LMS and NLMS algorithms. Mean and mean-square performance of the proposed algorithm is investigated in both stationary and non-stationary environments. We derive the closed-form expressions of various performance measures by evaluating multi-dimensional moments. This is done by statistical characterization of required random variables by employing the approach of Indefinite Quadratic Forms. Simulation and experimental results are presented to corroborate our theoretical claims. |
| Author | Ahmad, Jawwad Al-Saggaf, Ubaid M. Zerguine, Azzedine Zoubir, Abdelhak M. Moinuddin, Muhammad |
| Author_xml | – sequence: 1 givenname: Azzedine orcidid: 0000-0002-2621-4969 surname: Zerguine fullname: Zerguine, Azzedine email: azzedine@kfupm.edu.sa organization: Department of Electrical Engineering and the Center for Communication Systems and Sensing, King Fahd University of Petroleum and Minerals – sequence: 2 givenname: Jawwad surname: Ahmad fullname: Ahmad, Jawwad organization: Department of Electrical Engineering, Usman Institute of Technology – sequence: 3 givenname: Muhammad surname: Moinuddin fullname: Moinuddin, Muhammad organization: Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Electrical and Computer Engineering Department, King Abdulaziz University – sequence: 4 givenname: Ubaid M. surname: Al-Saggaf fullname: Al-Saggaf, Ubaid M. organization: Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Electrical and Computer Engineering Department, King Abdulaziz University – sequence: 5 givenname: Abdelhak M. surname: Zoubir fullname: Zoubir, Abdelhak M. organization: Signal Processing Group, Technical University of Darmstadt |
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| Cites_doi | 10.1109/TAC.1967.1098599 10.1109/78.218137 10.1109/TSP.2010.2052359 10.1109/TASSP.1987.1165197 10.1109/TASSP.1986.1164814 10.1016/S0165-1684(03)00044-6 10.1016/j.sigpro.2007.09.015 10.1139/f54-039 10.1109/TSP.2011.2181505 10.1109/TCOMM.2015.2496592 10.1109/PROC.1976.10286 10.1016/j.ijleo.2016.11.152 10.1109/TSP.2002.808108 10.1109/ICASSP.2006.1660598 10.1002/9780470374122 10.1016/j.sigpro.2008.07.013 10.1109/TIT.1984.1056886 10.1109/78.558478 10.1109/78.143435 10.1016/j.sigpro.2008.10.022 10.1109/ICTTA.2008.4530045 10.1016/0270-0255(83)90030-1 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer Nature B.V. 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| References_xml | – reference: CostaMBermudezJA noise resilient variable step-size LMS algorithmSignal Process.200888373374810.1016/j.sigpro.2007.09.0151186.94097 – reference: WidrowBMcCoolJMLarimoreMGJohnsonCRStationary and Nonstationary Learning Characteristics of the LMS Adaptive FilterIEEE Proceedings1976641151116242181410.1109/PROC.1976.10286 – reference: SulymanAIZerguineAConvergence and steady-state analysis of a variable step-size NLMS algorithmSignal Process.2003831255127310.1016/S0165-1684(03)00044-61144.93377 – reference: Al-NaffouriTYMoinuddinMExact performance analysis of the $\epsilon $-NLMS algorithm for colored circular Gaussian inputsIEEE Trans. Signal Process.2010581050805090275987710.1109/TSP.2010.20523591392.94065 – reference: RickerWEStock and recruitmentJ. Fish. Res. Board Can.19541155962310.1139/f54-039 – reference: WalachEWidrowBThe least mean fourth (LMF) adaptive algorithm and its familyIEEE Trans. Inf. Theory1984IT–3027528310.1109/TIT.1984.1056886 – reference: GradshteynISRyzhikIMTable of Integral, Series, and Products, Corrected and Enlarged Edition1980New YorkAcademic Press, INC0521.33001 – reference: KwongRHJohnstonEWA variable step size LMS algorithmIEEE Trans. Signal Process.1992401633164210.1109/78.1434350850.93839 – reference: BelaziAAbd El-LatifAAA simple yet efficient S-box method based on chaotic sine mapOPTIK20171301438144410.1016/j.ijleo.2016.11.152 – reference: MethewsVJXieZA stochastic gradient adaptive filter with gradient adaptive step sizeIEEE Trans. Signal Process.1993412075208710.1109/78.218137 – reference: HuangH-CLeeJA new variable step-size NLMS algorithm and its performance analysisIEEE Trans. Signal Process.201260420552060293421910.1109/TSP.2011.21815051393.94268 – reference: HaykinSAdaptive Filter Theory19963Upper-Saddle RiverPrentice-Hall0723.93070 – reference: Al-NaffouriTYSayedAHTransient analysis of adaptive filters with error nonlinearitiesIEEE Trans. Signal Process.200351365366310.1109/TSP.2002.808108 – reference: BershadNJBehavior of the $\epsilon $-normalized LMS algorithm with Gaussian inputsIEEE Trans. Acoust. Speech Signal Process.1987ASSP–35563664410.1109/TASSP.1987.1165197 – reference: RogersTDWhitleyDCChaos in the cubic mappingMath. Model.1983492571172210.1016/0270-0255(83)90030-10531.58033 – reference: ZerguineAChanMKAl-NaffouriTYMoinuddinMCowanCFNConvergence and tracking analysis of a variable normalized LMF (XE-NLMF) algorithmSignal Process.200889577879010.1016/j.sigpro.2008.10.0221161.94373 – reference: Chan, M.K., Cowan, C.F.N.: Using a normalised LMF algorithm for channel equalisation with co-channel interference. In: EUSIPCO 2002, pp. 48–51 (2002) – reference: NagumoJINodaAA learning method for system identificationIEEE Trans. Autom. Control19671228228710.1109/TAC.1967.1098599 – reference: Al-NaffouriTYMoinuddinMAjeebNHassibiBMoustakasALOn the distribution of indefinite quadratic forms in Gaussian random variablesIEEE Trans. Commun.201664115316510.1109/TCOMM.2015.2496592 – reference: SayedAHAdaptive Filters2008New YorkJohn Wiley & Sons10.1002/9780470374122 – reference: Costa, M., Bermudez, J.: A robust variable step size algorithm for LMS adaptive filters. In: IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP, vol. 3 (2006) – reference: AboulnasrTMayyasKA robust variable step-size LMS type algorithm: analysis and simulationIEEE Trans. Signal Process.19974563163910.1109/78.558478 – reference: Hosseini, K., Montazeri, A., Alikhanian, H., Kahaei, M.H.: New classes of LMS and LMF adaptive algorithms. In: 2008 3rd International Conference on Information and Communication Technologies: From Theory to Applications, pp. 1–5 (2008) – reference: ShanTJKailathTAdaptive algorithms with an automatic gain control featureIEEE Trans. Acoust. Speech Signal Process.198835122127 – reference: HarrisRWChabriesDMBishopFAA variable step size (VS) algorithmIEEE Trans. Acoust. Speech Signal Process.19863449951010.1109/TASSP.1986.1164814 – reference: Asad, S.M., Moinuddin, M., Zerguine, A.: On the convergence of a variable step-size LMF algorithm for quotient form. In: IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2010, Dallas, Texas, USA, March 14–19, (2010) – reference: ZhaoSManZKhooSWuHVariable step-size LMS algorithm with a quotient formSignal Process.200889677610.1016/j.sigpro.2008.07.0131151.94441 – volume: 12 start-page: 282 year: 1967 ident: 7773_CR14 publication-title: IEEE Trans. Autom. 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| SubjectTerms | Adaptive algorithms Algorithms Automotive Engineering Classical Mechanics Control Convergence Dynamical Systems Engineering Mechanical Engineering Nonstationary environments Original Paper Performance evaluation Quadratic forms Random variables Random walk Vibration |
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| Title | An efficient normalized LMS algorithm |
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