Nonlinear FXLMS algorithm for active noise control systems with saturation nonlinearity

Sahib, Mouayad A.; Kamil, Raja; Marhaban, Mohammad Hamiruce

doi:10.1002/tee.21778

Cited by 15 publications

(24 citation statements)

References 22 publications

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“…Even though the values of the estimated secondary path may not necessarily converge to their true values, the controller will compensate this effect by adjusting its gain during adaptation. This gain swapping effect has been observed in the previous works .…”

Section: Proposed Thf‐nlfxlms Algorithmsupporting

confidence: 84%

“…It was shown that the degree of nonlinearity in can be related to the degree of nonlinearity in by :

α = \sqrt{2 η^{2}}

β = \frac{1}{\sqrt{2 η^{2}}}

These relations are applicable to the Hammerstein and Wiener THF‐NLFXLMS algorithms, and are also applicable to the Wiener‐Hammerstein THF‐NLFXLMS algorithm.…”

Section: Wiener‐hammerstein Nlfxlms Contol Algorithmmentioning

confidence: 92%

“…The proposed secondary path modeling method is given in Subsection 3.2. Substituting for the estimated filters' values, the proposed Wiener‐Hammerstein NLFXLMS algorithm is given:

\begin{matrix} right W (n + 1) & left = W (n) & right \\ right & left + μ e (n) \sum_{j = 0}^{M - 1} \hat{K_{j}} \hat{f^{'}} [y_{s} (n - j)] {\hat{x}}_{f} (n - j) \end{matrix}

where

{\overset{bold-italicx}{true}}_{bold-italicf} (n) = {falsefalse}_{\sum}^{i = 0} \overset{S_{i}}{true} x (n - i)

In the original derivation of the NLFXLMS the secondary path nonlinearity f [·] is represented using scaled error function (SEF) given by :

f_{S E F} (y) = \int_{0}^{y} e^{\frac{- z^{2}}{2 η^{2}}} d z

where η 2 is the degree of nonlinearity. Note that

\lim_{η^{2} \to 0} = η \sqrt{\frac{π}{2}} s g n (y)

and

\lim_{η^{2} \to \infty} = y

, thus by manipulating η 2 , the SEF can represent a broad range of saturation nonlinearity from a hard limiter to a linear device .…”

Section: Wiener‐hammerstein Nlfxlms Contol Algorithmmentioning

confidence: 99%

“…In practical applications this prior information is not available and must be approximated. In the recent works by , the saturation nonlinearity represented by SEF was modeled using THF. The THF function is given by

f_{T H F} (y) = α t a n h (β y)

where α and β are two scaling parameters that determine the strength of the saturation nonlinearity.…”

Section: Wiener‐hammerstein Nlfxlms Contol Algorithmmentioning

confidence: 99%

“…From , in the case of perfect modeling, i.e. e s ( n )=0, the following relationship is satisfied:

\begin{matrix} right & left \sum_{j = 0}^{m - 1} K_{j} α t a n h [β \sum_{i = 0}^{l - 1} S_{i} v (n - i - j)] & right \\ right & left - \sum_{j = 0}^{M - 1} {\hat{K}}_{j} (n) \hat{α} (n - j) t a n h [\hat{β} (n - j) \\ right & left \sum_{= 0}^{L - 1} Ŝ_{i} (n - j) v (n - i - j)] = 0 \end{matrix}

such that

α \overset{K}{true} = \overset{α}{true} (n) \overset{bold-italicK}{true} (n) β Ŝ = \overset{β}{true} (n) \overset{bold-italicS}{true} (n)

Similar to the Hammerstein and the Wiener NLFXLMS, the linear paths

\overset{bold-italicS}{true} (n)

\overset{bold-italicK}{true} (n)

, and the scaling parameters

\overset{α}{true} (n)

and

\overset{β}{true} (n)

do not necessarily have to converge to their true values, as long as (30) is satisfied. The proposed THF‐NLFXLMS algorithm is summarized in Table .…”

Section: Proposed Thf‐nlfxlms Algorithmmentioning

confidence: 99%

See 4 more Smart Citations

NLFXLMS and THF‐NLFXLMS Algorithms for Wiener‐Hammerstein Nonlinear Active Noise Control

Srazhidinov

Kamil

Noor

2017

Asian Journal of Control

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Nonlinear Filtered‐X LMS (NLFXLMS) is an indirect adaptive control algorithm for nonlinear active noise control (NANC) system. The algorithm has been developed for both Hammerstein and Wiener secondary paths where the nonlinearity is represented by scaled error function (SEF) and tangential hyperbolic function (THF). NLFXLMS algorithm is limited in practical application because the degree of nonlinearity has to be known in advance. This limitation leads to the development of the THF‐NLFXLMS algorithm where the degree of nonlinearity is estimated by modelling the secondary path. In this work, the NLFXLMS and THF‐NLFXLMS are extended to Wiener‐Hammerstein system. The performance of the proposed Wiener‐Hammerstein THF‐NLFXLMS is compared with NLFXLMS algorithm which is considered as the benchmark and second order Volterra algorithm of comparable computational complexity. Simulation results show that the THF‐NLFXLMS has a similar performance to NLFXLMS and outperforms the second order Volterra algorithm as the system becomes more nonlinear.

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Section: Proposed Thf‐nlfxlms Algorithmsupporting

confidence: 84%

“…It was shown that the degree of nonlinearity in can be related to the degree of nonlinearity in by :

α = \sqrt{2 η^{2}}

β = \frac{1}{\sqrt{2 η^{2}}}

These relations are applicable to the Hammerstein and Wiener THF‐NLFXLMS algorithms, and are also applicable to the Wiener‐Hammerstein THF‐NLFXLMS algorithm.…”

Section: Wiener‐hammerstein Nlfxlms Contol Algorithmmentioning

confidence: 92%

“…The proposed secondary path modeling method is given in Subsection 3.2. Substituting for the estimated filters' values, the proposed Wiener‐Hammerstein NLFXLMS algorithm is given:

\begin{matrix} right W (n + 1) & left = W (n) & right \\ right & left + μ e (n) \sum_{j = 0}^{M - 1} \hat{K_{j}} \hat{f^{'}} [y_{s} (n - j)] {\hat{x}}_{f} (n - j) \end{matrix}

where

{\overset{bold-italicx}{true}}_{bold-italicf} (n) = {falsefalse}_{\sum}^{i = 0} \overset{S_{i}}{true} x (n - i)

In the original derivation of the NLFXLMS the secondary path nonlinearity f [·] is represented using scaled error function (SEF) given by :

f_{S E F} (y) = \int_{0}^{y} e^{\frac{- z^{2}}{2 η^{2}}} d z

where η 2 is the degree of nonlinearity. Note that

\lim_{η^{2} \to 0} = η \sqrt{\frac{π}{2}} s g n (y)

and

\lim_{η^{2} \to \infty} = y

, thus by manipulating η 2 , the SEF can represent a broad range of saturation nonlinearity from a hard limiter to a linear device .…”

Section: Wiener‐hammerstein Nlfxlms Contol Algorithmmentioning

confidence: 99%

f_{T H F} (y) = α t a n h (β y)

where α and β are two scaling parameters that determine the strength of the saturation nonlinearity.…”

Section: Wiener‐hammerstein Nlfxlms Contol Algorithmmentioning

confidence: 99%

“…From , in the case of perfect modeling, i.e. e s ( n )=0, the following relationship is satisfied:

\begin{matrix} right & left \sum_{j = 0}^{m - 1} K_{j} α t a n h [β \sum_{i = 0}^{l - 1} S_{i} v (n - i - j)] & right \\ right & left - \sum_{j = 0}^{M - 1} {\hat{K}}_{j} (n) \hat{α} (n - j) t a n h [\hat{β} (n - j) \\ right & left \sum_{= 0}^{L - 1} Ŝ_{i} (n - j) v (n - i - j)] = 0 \end{matrix}

such that

α \overset{K}{true} = \overset{α}{true} (n) \overset{bold-italicK}{true} (n) β Ŝ = \overset{β}{true} (n) \overset{bold-italicS}{true} (n)

Similar to the Hammerstein and the Wiener NLFXLMS, the linear paths

\overset{bold-italicS}{true} (n)

\overset{bold-italicK}{true} (n)

, and the scaling parameters

\overset{α}{true} (n)

and

\overset{β}{true} (n)

do not necessarily have to converge to their true values, as long as (30) is satisfied. The proposed THF‐NLFXLMS algorithm is summarized in Table .…”

Section: Proposed Thf‐nlfxlms Algorithmmentioning

confidence: 99%

See 3 more Smart Citations

NLFXLMS and THF‐NLFXLMS Algorithms for Wiener‐Hammerstein Nonlinear Active Noise Control

Srazhidinov

Kamil

Noor

2017

Asian Journal of Control

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Nonlinear Thf‐Fxlms Algorithm For Active Noise Control With Loudspeaker Nonlinearity

Ghasemi¹,

Kamil²,

Marhaban³

2015

Asian Journal of Control

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Adaptive algorithms are prevalently applied in the design of nonlinear active noise control (ANC) systems. The most important nonlinearity in ANC is the saturation effect produced by the electro‐acoustical sensors and transducers. The dominant saturation nonlinearity is in the transducers, which can be represented by a Wiener model. An effective solution to mitigate such nonlinear distortion is to employ the Nonlinear Filtered‐X Least Mean Square (NLFXLMS) algorithm. The controller compensates the nonlinearity using a model of the saturation effect represented by the scaled error function (SEF). However, the NLFXLMS is limited by two practical issues such that the degree of nonlinearity has to be known in advance and the SEF cannot be evaluated in real time. In this work, the NLFXLMS algorithm is modified by incorporating the tangential hyperbolic function (THF) to model the saturation effect of the loudspeaker. The proposed THF‐NLFXLMS algorithm, models the nonlinear secondary path and applies the estimated degree of nonlinearity in the control algorithm design. The results show that the Wiener secondary path, with saturation nonlinearity represented by SEF, can be modelled by THF with a certain degree of accuracy and can yield a reasonable estimate of the degree of nonlinearity. The performance of the proposed algorithm is comparable with the benchmark NLFXLMS and is superior to the conventional FXLMS as well as the second order Volterra algorithm of similar computational complexity with the proposed algorithm.

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Comb-partitioned frequency-domain constraint adaptive algorithm for active noise control

et al. 2021

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Nonlinear FXLMS algorithm for active noise control systems with saturation nonlinearity

Cited by 15 publications

References 22 publications

NLFXLMS and THF‐NLFXLMS Algorithms for Wiener‐Hammerstein Nonlinear Active Noise Control

NLFXLMS and THF‐NLFXLMS Algorithms for Wiener‐Hammerstein Nonlinear Active Noise Control

Nonlinear Thf‐Fxlms Algorithm For Active Noise Control With Loudspeaker Nonlinearity

Comb-partitioned frequency-domain constraint adaptive algorithm for active noise control

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