Mean Weight Behavior of the NLMS Algorithm for Correlated Gaussian Inputs

Al-Naffouri, Tareq Y.; Moinuddin, Muhammad; Sohail, Muhammad Saqib

doi:10.1109/lsp.2010.2090142

Cited by 15 publications

(7 citation statements)

References 16 publications

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“…Note that ||u i || 2 = ||ū i || 2 Λ . Thus, from (11), it can be easily deduced that the entries of the moment matrices Ā, B, and C are completely determined by the expectation 4 Table I contains entries of the form…”

Section: Our Methodologymentioning

confidence: 99%

“…Other prior work has attempted to evaluate these moments but the corresponding analyses do not result in closed form performance expressions [1], [3], [4], or rely on strong assumptions. Examples of these assumptions include the separation principle [5], [6], approximations [7], white input [4], [7], [8], specific structure of input regressor's distribution [5], [10], small step size [5], long filters [7], approximate solutions using Abelian integrals [5] and partial evaluation of moments [11].…”

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confidence: 99%

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Mean-square Analysis of the NLMS Algorithm

Al-Naffouri,

Moinuddin,

Ali

2021

Preprint

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This work presents a novel approach to the mean-square analysis of the normalized least mean squares (NLMS) algorithm for circular complex colored Gaussian inputs. The analysis is based on the derivation of a closed-form expression for the Cumulative Distribution Function (CDF) of random variables of the form (||u|| 2 D1 )(||u|| 2 D2 ) −1 where u is an isotropic vector and D 1 and D 2 are diagonal matrices and using that to derive some moments of these variables. These moments in turn completely characterize the mean-square behavior of the NLMS algorithm in explicit closed-form expressions.Specifically, the transient, steady-state, and tracking mean-square behavior of the NLMS algorithm is studied. I. INTRODUCTIONThe most widely used algorithm for adaptive filters is the Least Mean Squares (LMS) [1] algorithm which provides a solution to the optimal Weiner Filter criterion minimizing the mean square value of the error in a stochastic approximation sense. Since the LMS algorithm belongs to such a class of gradient-type algorithms whose convergence properties depend on the input correlation, it inherits slow convergence, especially when operating on highly correlated signals like speech. This dependency is removed by the input normalization in the NLMS algorithm [2] which results in a great improvement in convergence compared to the LMS algorithm [1].However, the normalization complicates the performance analysis of the algorithm. Many researchers have attempted to study the performance of the NLMS [1], [3]-[9] and to understand why it has superior performance over that of the LMS. In [3], closed-form expressions for the transient analysis and the steady-state mean-square error (MSE) of the NLMS

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Section: Our Methodologymentioning

confidence: 99%

mentioning

confidence: 99%

Mean-square Analysis of the NLMS Algorithm

Al-Naffouri,

Moinuddin,

Ali

2021

Preprint

Self Cite

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“…However, the transient behavior of LLA is not as widely studied. Moreover, theoretical transient behavior analysis is extremely important for understanding the algorithm performance under the framework of system identification in practical applications [25,26,27,28,29,30,31,32] and the existing method for analyzing the transient behavior of adaptive filtering algorithms with nonlinearities are derived under Gaussian noise environment. In this paper, the theoretical model to predict the mean weight behavior and the transient EMSE behavior of the LLA is proposed, analyzed, simulated and discussed.…”

Section: Introductionmentioning

confidence: 99%

Transient Convergence Analysis of Least Lncosh Algorithm

Sun¹,

Li²

2022

Preprint

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Lncosh function has been used for constructing a cost function for devising an adaptive filter algorithm to provide a desired performance under non-Gaussian noise, naming as least Lncosh algorithm (LLA). However, its transient behavior analysis has not been analyzed and investigated. As an important characteristic for the LLA, the transient behavior affects its performance in real-world applications. Herein, we present the theoretical model for evaluating the mean weight behavior and the transient excess mean square error (EMSE) behavior of the LLA. In addition, the proposed models can also be used to predict the transient behavior of the adaptive filtering algorithms with error nonlinearities {under non-Gaussian noises}. The experiment results have verified the validity of the presented models.

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“…Com respeito à determinação de tais momentos, para o caso de sinal de entrada real, diversas abordagens podem ser encontradas na literatura considerando diferentes tipos de aproximações e, consequentemente, resultando em modelos com diferentes níveis de precisão [4]- [11]. Por outro lado, pouco tem sido desenvolvido em se tratando da modelagem do algoritmo NLMS operando com sinais de entrada complexos, podendo-se citar apenas dois recentes artigos [12], [13]. Entretanto, os resultados descritos em tais trabalhos apresentam certas inconsistências.…”

Section: Introductionunclassified

“…Contudo, devido à presença da integral exponencial generalizada, inerente ao fator de regularização considerado, tal afirmação não é válida. Em [13], os mesmos autores propõem uma solução para o cálculo da matriz de autocorrelação normalizada (requerida na expressão que descreve o comportamento médio do vetor de coeficientes), desconsiderando agora o parâmetro de regularização. Porém, é possível mostrar que essa solução diverge quando utilizada em filtros de ordem superior a dois [14].…”

Section: Introductionunclassified

Modelo Estocástico do Algoritmo NLMS para Sinais de Entrada Gaussianos Complexos

Kuhn¹,

Seara²

2012

Anais De XXX Simpósio Brasileiro De Telecomunicações

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Resumo-Este artigo apresenta a modelagem estocástica de um consagrado algoritmo adaptativo da literatura, o algoritmo NLMS (normalized least-mean-square). Considerando um problema de identificação de sistema com planta estacionária e sinal de entrada gaussiano complexo, um modelo preciso obtido analiticamente é aqui derivado. Tal precisão se deve principalmente à forma como o cálculo dos momentos normalizados é efetuado. Resultados de simulação para diferentes cenários de operação são apresentados, mostrando que o modelo proposto prediz satisfatoriamente o comportamento do algoritmo tanto em regime transitório quanto em regime permanente. Palavras-chave-Algoritmo NLMS, filtragem adaptativa, modelagem estocástica.

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Mean Weight Behavior of the NLMS Algorithm for Correlated Gaussian Inputs

Cited by 15 publications

References 16 publications

Mean-square Analysis of the NLMS Algorithm

Mean-square Analysis of the NLMS Algorithm

Transient Convergence Analysis of Least Lncosh Algorithm

Modelo Estocástico do Algoritmo NLMS para Sinais de Entrada Gaussianos Complexos

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