# MOV-Modified-FxLMS algorithm with Variable Penalty Factor in a Practical
Power Output Constrained Active Control System

Chung Kwan Lai, Dongyuan Shi, , Bhan Lam, , Woon-Seng Gan, All authors are
with the School of Electrical & Electronic Engineering, Nanyang Technological
University (NTU), Singapore. Email: {chungkwan.lai, dongyuan.shi, bhanlam,
<EMAIL_ADDRESS>
###### Abstract

Practical Active Noise Control (ANC) systems typically require a restriction
in their maximum output power, to prevent overdriving the loudspeaker and
causing system instability. Recently, the minimum output variance filtered-
reference least mean square (MOV-FxLMS) algorithm was shown to have optimal
control under output constraint with an analytically formulated penalty
factor, but it needs offline knowledge of disturbance power and secondary path
gain. The constant penalty factor in MOV-FxLMS is also susceptible to
variations in disturbance power that could cause output power constraint
violations. This paper presents a new variable penalty factor that utilizes
the estimated disturbance in the established Modified-FxLMS (MFxLMS)
algorithm, resulting in a computationally efficient MOV-MFxLMS algorithm that
can adapt to changes in disturbance levels in real-time. Numerical simulation
with real noise and plant response showed that the variable penalty factor
always manages to meet its maximum power output constraint despite sudden
changes in disturbance power, whereas the fixed penalty factor has suffered
from a constraint mismatch.

###### Index Terms:

Active noise control (ANC), Output power constraint, Minimum output variance
(MOV), Modified filtered reference least mean square(Modified-FxLMS)
algorithm.

##  I Introduction

Active Noise Control (ANC) often utilizes adaptive algorithms to reduce
acoustic pressure at a desired location [1, 2, 3, 4, 5]. The widely used
filtered-reference least mean square (FxLMS) algorithm uses stochastic
steepest gradient descent to iteratively update control filter coefficients
and minimize the sum of squared error signals over time. When suppressing loud
disturbances, however, the control signal amplitude may exceed the rated
system output power, resulting in output saturation that introduces non-
linearities [6, 7]. Thus, a practical ANC system should regulate its output
within the rated power to prevent overdriven loudspeakers, which introduce
distortion and may lead to mechanical failure. Moreover, signal clipping at
the audio power amplifier can cause distortion, divergence in the FxLMS
algorithm due to overflow in the controller coefficients and its non-linear
behavior [8, 9], which should be considered in realistic adaptive models [10,
11, 12, 13], though often ignored for simplicity [14]. Clipping at the
microphone pre-amplifier will limit the algorithm’s convergence rate, but will
have minimal impact on control performance [15].

Various approaches exist to limit the output signal during adaptive control
[16, 17, 9, 18, 19, 20, 21, 22, 23]. The rescaling [9, 18] and two-gradient
direction FxLMS [19] algorithms adjust the control filter coefficients when
the output exceeds a certain threshold. While effective, constraining the
output signal amplitude is less practical than constraining its output power
as the peak amplitude varies with different noise types. It is thus difficult
to pre-determine an amplitude threshold, which we usually set as the rated
power, whereas a reliable power constraint threshold can be obtained from an
audio equipment datasheet (e.g. amplifier). Therefore, employing a power
constraint is a more practical approach to ensure a safe and reliable
operation of an amplifier. The optimal leaky FxLMS (OLFxLMS) algorithm [20,
21] uses an optimal regularisation parameter to effectively constrain the
output power, but this comes with a trade-off of high computational cost [21].

In contrast, the minimum output variance FxLMS (MOV-FxLMS) algorithm [22]
limits the output variance by including a penalty factor in the cost function
at a lower computational complexity. The analytical formulation of the optimal
penalty factor for optimal control, under a given output variance constraint,
can be derived through a quadratically constrained quadratic program problem
(QCQP) and estimated based on the adaptive inverse modeling technique [23].
This optimal value requires prior knowledge of the noise type, in particular,
the variance of the disturbance signal, which can only be measured offline
before the control stage. Additionally, it assumes constant primary noise
variance during measurement and control stages, making it sensitive to any
mismatch in primary noise variance, which will be illustrated in Section V.

These limitations can be mitigated if the disturbance signal is fed back to
the control system, which allows insight into the disturbance variance during
the control stage. Updated knowledge of the disturbance could be leveraged
from the widely known modified FxLMS algorithm (MFxLMS), where the disturbance
is re-estimated by removing the estimated anti-noise signal from the ‘error’
sensors in every iteration [1, 24]. This letter presents the minimum output
variance MFxLMS (MOV-MFxLMS) algorithm in Section II, along with the optimal
and time-varying penalty factor for MOV-MFxLMS in Section III and Section IV,
respectively, which allows the optimal penalty factor to be obtained in real-
time. This method is verified through numerical simulation in Section V,
before concluding its overall effectiveness in Section VI.

##  II Minimum Output Variance Modified-FxLMS Algorithm

Figure 1 shows a single channel adaptive feedforward ANC system using a
modified-FxLMS arrangement [24, 1]. The control filter
$\mathbf{w}(n)=\left[w_{0}(n)\>w_{1}(n)\>\cdots\>w_{I-1}(n)\right]^{\mathrm{T}}$
containing $I$ FIR coefficients processes the reference signal $x(n)$ to
generate a control signal $y(n)$, which then propagates through the secondary
path $\mathbf{s}$ to reach the error sensor forming an anti-noise signal
$y^{\prime}(n)$. The error signal $e(n)$ is then the difference between
$y^{\prime}(n)$ and the disturbance $d(n)$. The cost function is therefore
formulated based on the minimum output variance to constrain the power output
as in [1]

Figure 1: Modified-FxLMS (MFxLMS) block diagram.

$\displaystyle
J=\mathbb{E}\left[e_{m}^{2}(n)\right]+\alpha\mathbb{E}\left[y^{2}(n)\right],$
(1)

where $\mathbb{E}[\cdot]$ and $\alpha\;(\alpha\geq 0)$ represent the
expectation operator and penalty factor, respectively. The modified error
signal $e_{m}(n)$, where subscript ‘$m$’ is short for ‘modified’, is expressed
as

$\displaystyle e_{m}(n)$ $\displaystyle=\hat{d}(n)-\hat{y}^{\prime}_{m}(n),$
(2)

where the estimated disturbance signal $\hat{d}(n)$ and modified anti-noise
signal $\hat{y}^{\prime}_{m}(n)$ are given respectively by

$\displaystyle\hat{d}(n)$
$\displaystyle=e(n)+\sum_{l=0}^{L-1}\hat{s}_{l}\mathbf{w}^{\mathrm{T}}(n-l)\mathbf{x}(n-l)$
(3)

and

$\displaystyle\hat{y}^{\prime}_{m}(n)=\sum_{i=0}^{I-1}\sum_{l=0}^{L-1}w_{i}(n)\hat{s}_{l}x(n-i-l),$
(4)

while $\mathbf{{s}}=[{s}_{0}\>{s}_{1}\>\cdots\>{s}_{L-1}]^{\mathrm{T}}$,
$\mathbf{\hat{s}}=[\hat{s}_{0}\>\hat{s}_{1}\>\cdots\>\hat{s}_{L-1}]^{\mathrm{T}}$
denote the secondary path and its estimation containing $L$ FIR coefficients.
The control signal $y(n)$ and the filtered reference signal $x^{\prime}(n)$
are given by

$\begin{cases}y(n)&=\mathbf{w}^{\mathrm{T}}(n)\mathbf{x}(n),\\\
x^{\prime}(n)&=\sum_{l=0}^{L-1}\hat{s}_{l}x(n-l).\end{cases}$ (5)

By setting the derivative of the cost function in (1) to 0, the optimal
control filter is derived as

$\displaystyle\mathbf{w}_{\mathrm{opt}}=(\mathbf{R}_{x^{\prime}x^{\prime}}+\alpha\mathbf{R}_{xx})^{-1}\mathbf{r}_{x^{\prime}\hat{d}},$
(6)

where the auto-correlation matrices and correlation vector are given by

$\begin{cases}\mathbf{R}_{xx}&=\mathbb{E}[\mathbf{x}(n)\mathbf{x}^{\mathrm{T}}(n)],\\\
\mathbf{R}_{x^{\prime}x^{\prime}}&=\mathbb{E}[\mathbf{x}^{\prime}(n)\mathbf{x^{\prime}}^{\mathrm{T}}(n)],\\\
\mathbf{r}_{x^{\prime}\hat{d}}&=\mathbb{E}[\mathbf{x^{\prime}}(n)\hat{d}(n)].\end{cases}$
(7)

According to the stochastic gradient descent method to minimize the
instantaneous value of (1), we can use the recursive formula for the control
filter as

$\displaystyle\mathbf{w}(n+1)=\mathbf{w}(n)+\mu\left[\mathbf{x^{\prime}}(n)e_{m}(n)-\alpha\mathbf{x}(n)y(n)\right],$
(8)

which is henceforth known as the minimum output variance modified-FxLMS (MOV-
MFxLMS) algorithm, with $\mu$ as the step-size. Choosing the right value of
penalty factor $\alpha$ from (6) is crucial as it affects the attenuation
performance. A large $\alpha$ will overemphasize the minimization of the
$\mathbb{E}\left[y^{2}(n)\right]$ term in (1), resulting in a gain reduction
in the control filter and a decrease in attenuation performance. On the other
hand, a small $\alpha$ will not effectively constrain the power output,
leading to output saturation.

##  III Optimal penalty factor for MOV-MFxLMS

The optimal penalty factor that satisfies a power output constraint can be
derived by relating the optimal control filter from (6) and the optimal
control filter derived from the Lagrangian optimization problem [23]. The
optimization problem under power output constraint is formulated as

$\displaystyle\min_{\mathbf{w}}\;\;$ $\displaystyle\xi(\mathbf{w})$
$\displaystyle=\mathbb{E}\left[e_{m}^{2}(n)\right]$ (9a) $\displaystyle\;s.t.$
$\displaystyle g(\mathbf{w})$
$\displaystyle=\mathbb{E}\left[y^{2}(n)\right]-\rho^{2}\leq 0$ (9b)

where $\rho^{2}$ is the maximum control signal output power allowed. The
Lagrangian function [25] can thus be formulated to give

$\displaystyle\mathcal{L}(\mathbf{w},\lambda)=\mathbb{E}\left[e_{m}^{2}(n)\right]+\lambda\left\\{\mathbb{E}\left[y^{2}(n)\right]-\rho^{2}\right\\},$
(10)

where $\lambda\;(\lambda\geq 0)$ denotes the Lagrange multiplier. By setting
the derivative of the Lagrangian function to be zero, i.e.
$\nabla\mathcal{L}(\mathbf{w},\lambda)=0$, the optimal control filter can be
derived as

$\displaystyle\mathbf{w}_{\mathrm{o}}=(\mathbf{R}_{x^{\prime}x^{\prime}}+\lambda_{\mathrm{o}}\mathbf{R}_{xx})^{-1}\mathbf{r}_{x^{\prime}\hat{d}},$
(11)

where the optimal Lagrange factor $\lambda_{\mathrm{o}}$ yields solutions
respective to the constrained and unconstrained conditions, that is

$\displaystyle\lambda_{\mathrm{o}}=\frac{\mathbb{E}\left[\hat{d}(n)\hat{y}^{\prime}_{m\mathrm{o}}(n)\right]-\mathbb{E}\left[\hat{y}^{\prime
2}_{m\mathrm{o}}(n)\right]}{\rho^{2}}\;\text{or}\;0,$ (12)

with $\hat{d}(n)$ and $\hat{y}^{\prime 2}_{m\mathrm{o}}(n)$ from (12) replace
the $d(n)$ and $y^{\prime 2}_{\mathrm{o}}(n)$ terms from the MOV-FxLMS
formulation, respectively. Notably, the optimal control filter in (6) is
comparable to the Lagrangian formulation in (11) when
$\mathbf{w}_{\mathrm{opt}}=\mathbf{w}_{\mathrm{o}}$, which in turn relates to
the optimal penalty factor $\alpha_{\mathrm{o}}=\lambda_{\mathrm{o}}$. To
remove the cross-correlation
$\mathbb{E}[\hat{d}(n)\hat{y}^{\prime}_{m\mathrm{o}}(n)]$ from (12), we impose
an additional condition suitable in an ANC system where $\hat{d}(n)$ and
$\hat{y}^{\prime}_{m\mathrm{o}}(n)$ are positively correlated [21, 23]:

$\displaystyle\frac{\mathbb{E}\left[\hat{d}(n)\hat{y}^{\prime}_{m\mathrm{o}}(n)\right]}{\sqrt{\mathbb{E}[\hat{d}^{2}(n)]\mathbb{E}[\hat{y}^{\prime
2}_{m\mathrm{o}}(n)]}}=1,$ (13)

where
$\mathbb{E}[\hat{d}(n)]=\mathbb{E}[\hat{y}^{\prime}_{m\mathrm{o}}(n)]=0$. The
optimal penalty factor is then derived as

$\displaystyle\alpha_{\mathrm{o}}=\lambda_{\mathrm{o}}=G_{s}\left(\sqrt{\frac{\sigma_{\hat{d}}^{2}}{\rho^{2}G_{s}}}-1\right),$
(14)

where $G_{s}$ signifies the power gain of the secondary path [23] given by

$\displaystyle
G_{s}=\frac{\sigma^{2}_{\hat{y}^{\prime}_{m\mathrm{o}}}}{\rho^{2}},$ (15)

and $\sigma_{\hat{d}}^{2}=\mathbb{\mathbb{E}}[\hat{d}^{2}(n)]$,
$\sigma^{2}_{\hat{y}^{\prime}_{m\mathrm{o}}}=\mathbb{E}[\hat{y}^{\prime
2}_{m\mathrm{o}}(n)]$ denote the variance of the estimated disturbance and
optimally constrained anti-noise, respectively.

As $\hat{d}(n)$ is known in the modified arrangement, the optimal penalty
factor (14) can be predicted over time and therefore enabling the controller
to attain the output power constraint, even if the noise type or amplitude
changes throughout the control process. A variable penalty factor should
theoretically outperform the pre-estimated factor in the MOV-FxLMS algorithm,
which is sensitive to the variation in the disturbance.

##  IV Time-varying penalty factor for MOV-MFxLMS

To forecast the ideal variable penalty factor, we need to retreive the varying
$G_{s}$ parameter. However, it is impossible to calculate $G_{s}$ through (15)
as the constrained optimal anti-noise signal
$\hat{y}^{\prime}_{m\mathrm{o}}(n)$ is unobtainable in practice. Additionally,
the value of $G_{s}$ depends on the type of noise. One solution is to estimate
$G_{s}$ through the inverse modelling technique [21, 26], but this method is
not suited for real-time implementation due to its intensive computations and
slow convergence in finding the inverse path.

Alternatively, we can estimate $G_{s}$ using the filtered reference signal and
the reference signal, as the reference signal contains the same frequency
components as the disturbance in most real-world ANC applications. A more
practical estimate of the secondary path’s power gain is thus given by

$\displaystyle\hat{G}_{s}(n)=\frac{\sigma_{x^{\prime}}^{2}(n)}{\sigma_{x}^{2}(n)}\approx
G_{s}.$ (16)

With the inclusion of $x(n)$ and $x^{\prime}(n)$, $\hat{G}_{s}(n)$ updates
accordingly to changes in the disturbance. Furthermore, as $x^{\prime}(n)$ and
$x(n)$ must be obtained to perform the MFxLMS algorithm, no additional
convolution is required to obtain $\hat{G}_{s}(n)$ as opposed to the
previously proposed inverse modelling technique. By assuming local
stationarity in $x(n)$ and $\hat{d}(n)$ within $K$ samples, $G_{s}$ can be
estimated in a real-time implementation as

$\displaystyle\hat{G}_{s}(n)=\frac{\max\left\\{\sum_{k=0}^{K-1}x^{\prime
2}(n-k),\varepsilon_{1}\right\\}}{\max\left\\{\sum_{k=0}^{K-1}x^{2}(n-k),\varepsilon_{2}\right\\}},$
(17)

followed by the variable penalty factor given by

$\displaystyle\alpha(n)=\max\left\\{\hat{G}_{s}(n)\left(\sqrt{\frac{\sum_{k=0}^{K-1}\hat{d}^{2}(n-k)}{K\rho^{2}\hat{G}_{s}(n)}}-1\right),0\right\\},$
(18)

where $\varepsilon_{1}$ and $\varepsilon_{2}$ are the minimal threshold
parameters ensuring that numerator and denominator are non-zero. To prevent
$\alpha(n)$ from becoming negative during the control stage, a minimum
threshold of 0 will be required in (18). Thus, substituting (18) into (8) will
formulate the MOV-MFxLMS algorithm with a time-varying optimal penalty factor
to allow robustness against variations in disturbance noise power, which is

$\displaystyle\mathbf{w}(n+1)=\mathbf{w}(n)+\mu\left[\mathbf{x^{\prime}}(n)e_{m}(n)-\alpha(n)\mathbf{x}(n)y(n)\right].$
(19)

##  V Numerical Simulation

In this section, we conducted a numerical simulation comparison between the
proposed variable penalty factor MOV-MFxLMS and other algorithms under
different conditions.

###  V-A Convergence path due to a sudden change in disturbance

In this simulation, we compared MOV-MFxLMS with the proposed variable penalty
factor, MOV-FxLMS with the predetermined fixed penalty factor, the rescaling
algorithm [9] with maximum output magnitude parameter of $y_{max}=3.46$
obtained through trial and error, and the conventional FxLMS algorithm when
dealing with a sudden varying disturbance. The reference signal containing a
bandlimited white noise (800–$7200\text{\,}\mathrm{Hz}$) undergoes a sudden
change in its signal power from $\sigma_{x}^{2}=0.305$ to
$\sigma_{x}^{2}=0.540$ at the $30\text{\,}\mathrm{s}$ mark. The optimal
penalty factors for the two disturbances are calculated as $\alpha_{1}=0.0461$
and $\alpha_{2}=0.3255$, respectively. The sampling rate, step-size, primary
and secondary path are set as $F_{s}=$16\text{\,}\mathrm{kHz}$$, $\mu=0.0002$,
$\mathbf{p}=[1.62\;0.41]^{\mathrm{T}}$ and
$\mathbf{s}=[0.03\;0.87]^{\mathrm{T}}$. Fig. 2 shows the convergence path of
the control filter compared between the MOV-MFxLMS, MOV-FxLMS, and the
unconstrained FxLMS method.

As shown in Fig. 2(a), FxLMS converges to the optimal control filter
$\mathbf{w}_{\text{opt}}=[1.62\;0.41]^{\mathrm{T}}$ while ignoring the
subjected power constraint of $\rho^{2}=1$, giving the optimal power output
for the control signal of $\sigma_{y_{\mathrm{o}}}^{2}=1.1214$ for the first
$30\text{\,}\mathrm{s}$ noise and $\sigma_{y_{\mathrm{o}}}^{2}=1.9769$ for the
next $30\text{\,}\mathrm{s}$. However, MOV-FxLMS presents an interesting case
as only one of the two optimal penalty factors was used throughout the
simulation, as shown in Fig. 2(b). It managed to converge to the first optimal
constrained filter $\mathbf{w}_{\mathrm{o,1}}=[1.52\;0.38]^{\mathrm{T}}$ in
the first $30\text{\,}\mathrm{s}$ when the corresponding $\alpha_{1}$ was used
but failed to converge to the optimal constrained filter in the subsequent
$30\text{\,}\mathrm{s}$ with a louder disturbance, violating the power output
constraint by having $\sigma_{y_{\mathrm{o}}}^{2}=1.7630$. On the other hand,
the fixed penalty factor of $\alpha_{2}$ yielded a similar observation where
convergence occurred only in the corresponding final $30\text{\,}\mathrm{s}$
where $\mathbf{w}_{\mathrm{o,2}}=[1.14\;0.29]^{\mathrm{T}}$. Although the
constraint was not violated in the first $30\text{\,}\mathrm{s}$ with lower
amplitude disturbance, it led to an over-constrained output of
$\sigma_{y_{\mathrm{o}}}^{2}=0.5673$. While the rescaling algorithm in Fig.
2(c) showed a change in its convergence path after the $30\text{\,}\mathrm{s}$
mark, both paths do not converge to their respective optimal constrained
filter as it confines the maximum amplitude of the control signal, which is
difficult to translate into a precise output-power constraint for a non-
periodic noise.

(a)

(b)

(c)

(d)

Figure 2: Two-weight convergence paths of 2(a): FxLMS, 2(b): MOV-FxLMS with
optimal penalty factor, 2(c): Rescaling algorithm, and 2(d): the variable
penalty factor MOV-MFxLMS algorithm when dealing with the sudden varying
disturbance. Figure 3: The time history of the variable penalty factor, where
the power of disturbance happens to change at $t=$30\text{\,}\mathrm{s}$$.

In contrast, the MOV-MFxLMS algorithm with the variable penalty factor
$(K=256)$ can mitigate this problem. As illustrated in Fig. 2(d), it initially
converges to the first optimal constrained filter (red path) and shifts to the
second optimal constrained filter (blue path) when the second noise is
present, thereby satisfying the power-output requirement for the duration of
the simulation. The same phenomena can be observed in Fig. 3, where the
calculated penalty factor changes at $t=$30\text{\,}\mathrm{s}$$ due to a
change in disturbance power. Due to the moving-average operation from
(17)–(18), the variable penalty factor fluctuates around $\alpha_{1}$ at the
first $30\text{\,}\mathrm{s}$ and around $\alpha_{2}$ at the next
$30\text{\,}\mathrm{s}$. Therefore, this simulation demonstrates that the
proposed variable penalty factor MOV-MFxLMS algorithm is capable to achieve
optimal constraint control even if the disturbance has a sudden change in
amplitude. The same conclusion can be extended to the variation of the primary
noise.

###  V-B Active control on real noise based on measured paths

To further verify its effectiveness, the proposed varying MOV-MFxLMS algorithm
is utilized to attenuate a compounding noise on the primary and secondary path
measured from an air duct [21]. Fig. 4 depicts the error signal and its moving
average output power undergoing four distinct noise stages. The first
$30\text{\,}\mathrm{s}$ contains a 400–$600\text{\,}\mathrm{Hz}$ broadband
noise, and a larger amplitude of 400–$600\text{\,}\mathrm{Hz}$ broadband noise
is compounded to the existing noise for the next $30\text{\,}\mathrm{s}$.
Similarly, a measured construction noise, having a dominant
70–$400\text{\,}\mathrm{Hz}$ noise, and aircraft flyover noise, having a
dominant 400–$3500\text{\,}\mathrm{Hz}$ noise, is compounded to its previous
noise for the 60–$90\text{\,}\mathrm{s}$ and 90–$120\text{\,}\mathrm{s}$ time
frame respectively. The output saturation effect in the control signal is not
considered in this simulation when exceeding the output power constraint.

The FxLMS algorithm has the most noise reduction as shown in Fig. 4(a) while
ignoring its output power constraint shown in Fig. 4(b), especially in the
fourth stage where the output power increases tremendously to an approximate
peak value of $\sigma_{y}^{2}(n)\approx 10$ due to the non-stationary aircraft
noise. For MOV-FxLMS with a fixed penalty factor calibrated for the
30–$60\text{\,}\mathrm{s}$ noise ($\alpha_{0}=0.1944$), it only managed to
meet the power output constraint during the 30–$60\text{\,}\mathrm{s}$ period
but has exceeded the output constraint during the 60–$120\text{\,}\mathrm{s}$
period due to the additional construction noise and aircraft noise. In
contrast, the power output for the 0–$30\text{\,}\mathrm{s}$ period has been
over-constrained as it showed a lower output power than that of the FxLMS
algorithm. This issue can be mitigated by the varying penalty factor MOV-
MFxLMS algorithm that managed to maintain the power output constraint for the
30–$120\text{\,}\mathrm{s}$ period as shown in Fig. 4(b). It has the same
convergence behaviour as FxLMS for the first $30\text{\,}\mathrm{s}$ as the
estimated moving averaged disturbance power is less than
$K\rho^{2}\hat{G}_{s}(n)$, leading to the calculated penalty factor from (18)
to be 0. This verifies the moving average estimation method from (17) and
(18), which satisfy the output power constraint due to a sudden change in
noise type.

(a)

(b)

Figure 4: The (a): disturbance and error signal, (b): 1024 samples moving
average output power $\sigma_{y}^{2}(n)$ during control through various
adaptive algorithms with $\rho^{2}=0.8$, $F_{s}=$16\text{\,}\mathrm{kHz}$$,
$I=256$, $L=320$, $\mu=0.00005$ and $K=1024$.

##  VI Conclusion

In practical ANC applications, a power output constraint is required to
prevent undesired output saturation that leads to a degradation in ANC
performance. Although the recently proposed MOV-FxLMS algorithm is capable of
carrying out this task, it requires prior knowledge of the type of noise and
is therefore unsuitable to implement in a real-world scenario. This paper
proposes a variable penalty factor that can aid the MOV-MFxLMS algorithm in
achieving optimal control under output constraints even with an abruptly
changing noise. Numerical simulations based on measured paths and recorded
real noise demonstrate the effectiveness of the proposed method in meeting the
maximum power output constraint in ANC systems. Since this computational-
efficient estimation of this variable penalty factor only requires information
on the filtered reference and disturbance, it can be readily realized in real-
time ANC applications.

## References

  * Elliott [2000] S. Elliott, _Signal Processing for Active Control_. Academic Press, 2000.
  * Kuo and Morgan [1996] S. M. Kuo and D. R. Morgan, _Active Noise Control Systems: Algorithms and DSP Implementations_ , ser. Wiley Series in Telecommunications and Signal Processing. New York: Wiley, 1996.
  * Kajikawa et al. [2012] Y. Kajikawa, W.-S. Gan, and S. M. Kuo, “Recent advances on active noise control: Open issues and innovative applications,” _APSIPA Transactions on Signal and Information Processing_ , vol. 1, no. 1, 2012.
  * Chang et al. [2016] C.-Y. Chang, A. Siswanto, C.-Y. Ho, T.-K. Yeh, Y.-R. Chen, and S. M. Kuo, “Listening in a Noisy Environment: Integration of active noise control in audio products,” _IEEE Consumer Electronics Magazine_ , vol. 5, no. 4, pp. 34–43, Oct. 2016.
  * Qiu [2019] X. Qiu, _An Introduction to Virtual Sound Barriers_. Boca Raton: Taylor & Francis, a CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa, plc, 2019.
  * Jardin and Baudoin [2007] P. Jardin and G. Baudoin, “Filter Lookup Table Method for Power Amplifier Linearization,” _IEEE Transactions on Vehicular Technology_ , vol. 56, no. 3, pp. 1076–1087, May 2007.
  * Heinle et al. [1998] F. Heinle, R. Rabenstein, and A. Stenger, “A measurement method for the linear and nonlinear properties of electro-acoustic transmission systems,” _Signal Processing_ , vol. 64, no. 1, pp. 49–60, Jan. 1998.
  * Shi et al. [2017] D. Shi, C. Shi, and W.-S. Gan, “Effect of the audio amplifier’s distortion on feedforward active noise control,” in _2017 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC)_. Kuala Lumpur: IEEE, Dec. 2017, pp. 469–473.
  * Qiu and Hansen [2001] X. Qiu and C. H. Hansen, “A study of time-domain FXLMS algorithms with control output constraint,” _The Journal of the Acoustical Society of America_ , vol. 109, no. 6, pp. 2815–2823, Jun. 2001.
  * Sahib et al. [2012] M. A. Sahib, R. Kamil, and M. H. Marhaban, “Nonlinear FXLMS algorithm for active noise control systems with saturation nonlinearity,” _IEEJ Transactions on Electrical and Electronic Engineering_ , vol. 7, no. 6, pp. 598–606, Nov. 2012.
  * Ghasemi et al. [2016] S. Ghasemi, R. Kamil, and M. H. Marhaban, “Nonlinear Thf-Fxlms Algorithm For Active Noise Control With Loudspeaker Nonlinearity: THF-NLFXLMS for ANC with Loudspeaker Nonlinearity,” _Asian Journal of Control_ , vol. 18, no. 2, pp. 502–513, Mar. 2016.
  * Russo and Sicuranza [2007] F. Russo and G. L. Sicuranza, “Accuracy and Performance Evaluation in the Genetic Optimization of Nonlinear Systems for Active Noise Control,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 56, no. 4, pp. 1443–1450, Aug. 2007.
  * Tobias and Seara [2002] O. J. Tobias and R. Seara, “Performance comparison of the FXLMS, nonlinear FXLMS and leaky FXLMS algorithms in nonlinear active control applications,” in _2002 11th European Signal Processing Conference_ , 2002, pp. 1–4.
  * Costa et al. [2001] M. Costa, J. Bermudez, and N. Bershad, “Stochastic analysis of the LMS algorithm with a saturation nonlinearity following the adaptive filter output,” _IEEE Transactions on Signal Processing_ , vol. 49, no. 7, pp. 1370–1387, Jul. 2001.
  * Kuo et al. [2004] S. Kuo, H.-T. Wu, F.-K. Chen, and M. Gunnala, “Saturation Effects in Active Noise Control Systems,” _IEEE Transactions on Circuits and Systems I: Regular Papers_ , vol. 51, no. 6, pp. 1163–1171, Jun. 2004.
  * Elliott and Back [1996] S. Elliott and K. Back, “Effort constraints in adaptive feedforward control,” _IEEE Signal Processing Letters_ , vol. 3, no. 1, pp. 7–9, Jan. 1996.
  * Ahmed et al. [2021] S. Ahmed, M. Tufail, M. Rehan, T. Abbas, and A. Majid, “A Novel Approach for Improved Noise Reduction Performance in Feed-Forward Active Noise Control Systems With (Loudspeaker) Saturation Non-Linearity in the Secondary Path,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 29, pp. 187–197, 2021.
  * Hui Lan et al. [2002] Hui Lan, Ming Zhang, and Wee Ser, “A weight-constrained FxLMS algorithm for feedforward active noise control systems,” _IEEE Signal Processing Letters_ , vol. 9, no. 1, pp. 1–4, Jan. 2002.
  * Shi et al. [2018] D. Shi, W.-S. Gan, B. Lam, and C. Shi, “Two-gradient direction FXLMS: An adaptive active noise control algorithm with output constraint,” _Mechanical Systems and Signal Processing_ , vol. 116, pp. 651–667, Apr. 2018\.
  * Shi et al. [2019] D. Shi, B. Lam, W.-S. Gan, and S. Wen, “Optimal Leak Factor Selection for the Output-Constrained Leaky Filtered-Input Least Mean Square Algorithm,” _IEEE Signal Processing Letters_ , vol. 26, no. 5, pp. 670–674, May 2019.
  * Shi et al. [2021] D. Shi, W.-S. Gan, B. Lam, S. Wen, and X. Shen, “Optimal Output-Constrained Active Noise Control Based on Inverse Adaptive Modeling Leak Factor Estimate,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 29, pp. 1256–1269, 2021.
  * Taringoo et al. [2006] F. Taringoo, J. Poshtan, and M. H. Kahaei, “Analysis of Effort Constraint Algorithm in Active Noise Control Systems,” _EURASIP Journal on Advances in Signal Processing_ , vol. 2006, no. 1, p. 054649, Dec. 2006.
  * Shi et al. [2022] D. Shi, W.-S. Gan, B. Lam, and X. Shen, “Optimal Penalty Factor for the MOV-FxLMS Algorithm in Active Noise Control System,” _IEEE Signal Processing Letters_ , vol. 29, pp. 85–89, 2022.
  * Rupp and Sayed [1996] M. Rupp and A. Sayed, “Modified FxLMS algorithms with improved convergence performance,” in _Conference Record of The Twenty-Ninth Asilomar Conference on Signals, Systems and Computers_ , vol. 2. IEEE Comput. Soc. Press, 1996, pp. 1255–1259. [Online]. Available: http://ieeexplore.ieee.org/document/540900/
  * Boyd and Vandenberghe [2004] S. P. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge, UK ; New York: Cambridge University Press, 2004.
  * Widrow and Stearns [1985] B. Widrow and S. D. Stearns, _Adaptive signal processing_ , ser. Prentice-Hall signal processing series. Prentice-Hall, 1985.