社団法人 電子情報通信学会 THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS
信学技報 IEICE Technical Report CAS2005-3,VLD2005-14,SIP2005-27(2005-6)
A Technique for Active Noise Control Systems with Online Secondary Path Modeling Using Additive Noise Power Scheduling Muhammad TAHIR AKHTAR† , Masahide ABE† , and Masayuki KAWAMATA† † Graduate School of Engineering, Tohoku University, 6-6-05 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980–8579, Japan Email:
[email protected] Abstract This paper proposes a new method for online secondary path modeling (OSPM) in active noise control (ANC) systems. The proposed method comprises two adaptive filters: a noise control filter and an OSPM filter. The noise control filter is adapted by the FxLMS algorithm, and using the same error signal as used in the adaptation of the OSPM filter. The OSPM filter is adapted using a variable step size (VSS) LMS algorithm already proposed by authors. Furthermore, the OSPM filter is excited by an additive random noise with a variable power. Initially, when the disturbance signal is large, a small step size and large additive random noise are used in the OSPM filter. Later, when ANC system converges, step size is increased and the variance of the additive random noise is decreased. The computer simulations show that the proposed method gives best performance among the existing methods. Key words active noise control, online secondary path modeling, FxLMS algorithm, variable step size, noise power scheduling.
1. Introduction The block diagram of ANC system using the filtered-x LMS (FxLMS) algorithm [1] is shown in Fig. 1. Here P (z) is the primary path between the reference microphone and error microphone, and S(z) is the secondary path following the ANC (adaptive) filter W (z). The output of W (z), y(n) is used to generate the antinoise signal y 0 (n), which combines with the primary noise d(n) to reduce noise pressure around
Fig. 1. Block diagram of ANC systems using FxLMS algorithm.
the error microphone. The stability of the LMS algorithm requires the filtering of the reference signal x(n) through the ˆ ˆ secondary path modeling filter S(z). The S(z) can be obtained using offline modeling during an initial training stage for ANC applications [1]. For some applications, however, the secondary path may be time varying, and it is desirable to estimate the secondary path during the online operation of ANC systems [2]. The basic additive random noise technique (Fig. 2), proposed by Eriksson et. al. [3], adds another adaptive filter to model S(z) during online operation of ANC system. The main problem with this system is that the white random noise, v(n), injected into the ANC system for the online secondary path modeling (OSPM), appears in the residual error signal e(n). Thus e(n) comprises two parts: a part required for the control filter W (z) and a part required for the OSPM
ˆ filter S(z). Since e(n) is used in both the control process and modeling process, the part required for one acts as a disturbance for the other. Due to this intrusion between the control process and modeling process, the overall performance of the ANC system is further degraded. In order to improve the performance of Eriksson’s method, the methods proposed in [4]∼[6] add another third adaptive filter. In [4], [5], the third filter removes the interference from the modeling filter, and the modeling filter therefore converges fast. Here no effort is made to improve the performance of the control filter. In [6], the control filter, the modeling filter, and the third filter are cross-updated to reduce the mutual interference between the control process and modeling process (Fig. 3). Simulation results presented in [6] show that
—13—
their proposed method gives the best performance for ANC systems with the OSPM. The main problem is that the three adaptive filter are working simultaneously, and it increases the design complexity. In order to improve the performance of basic Eriksson’s method, without introducing the third adaptive filter, authors have proposed a method in [7]. This method adapts W (z) by using modified-FxLMS (MFxLMS) [8] algorithm and uses a variable step size (VSS) LMS algorithm in adaptˆ ing the OSPM S(z). It avoids using the third adaptive filter, however, its computational complexity is even higher than the three-adaptive-filter-based methods. The increased computational complexity is due to the structure of the MFxLMS algorithm. In order to achieve improved OSPM at reduced computational complexity, authors have proposed a method in [9]. Here W (z) is adapted by the FxLMS algorithm using ˆ the same error signal as used in the modeling filter, and S(z) is adapted by the VSS LMS algorithm as proposed in [7]. Thus basic structure of this method [9] is same as that of Eriksson’s method. It is shown in [9] that the performance of this method is same as that of [7] and at much reduced computational cost. ˆ In all above methods, the OSPM filter, S(z), is excited by an additive random noise with fixed amplitude. If a large additive random noise is used, then the convergence speed of the OSPM filter can be increased. This random noise,
Fig. 2. Eriksson’s method for OSPM in ANC systems.
vector, x L (n) = [x(n)x(n − 1) · · · x(n − L + 1)]T is the Lsample reference signal vector, and x(n) is the reference signal. An internally generated zero-mean white Gaussian noise signal, v(n), uncorrelated with the reference noise x(n), is injected at the output y(n) of the control filter. The residual noise signal is given as e(n) = d(n) − y 0 (n) + v 0 (n), where d(n) = p(n) ∗ x(n) is the primary disturbance signal, y 0 (n) = s(n) ∗ y(n) is the secondary canceling signal, v 0 (n) = s(n) ∗ v(n) is the modeling signal, ∗ denotes the convolution operation, and p(n) and s(n) are impulse responses of P (z) and S(z), respectively. The residual noise signal e(n) is used as an error signal for W (z), and as a desired response ˆ for S(z), i.e., g(n) = ds (n) = e(n). Thus the error signals ˆ for W (z) and S(z) are, respectively, given as
however, eventually appears at the error microphone and de-
g(n) = [d(n) − y 0 (n)] + v 0 (n),
(2)
grades the noise reduction performance of ANC system. It
f (n) = [d(n) − y 0 (n)] + [v 0 (n) − vˆ0 (n)].
(3)
is, therefore, highly desirable to keep it as small as possible. One solution to this problem is to use a variable-power additive random noise in the modeling process. The idea is to
The coefficients of W (z) are updated by the FxLMS algorithm:
use initially a large signal for fast convergence of the model-
w (n + 1) = w (n) + µw g(n)ˆ x 0 (n)
ing filter. Later, when ANC system converges the power of
0
modify the previous method as proposed in [9]. The organization of the rest of this paper is as follows. Section 2 presents an overview of the existing methods, and Section 3 describes the proposed method, for OSPM in ANC systems. A few comments on computational complexity are also presented. Section 4 presents the simulation results, and concluding remarks are presented in Section 5.
Consider Eriksson’s method as shown in Fig. 2. Assuming that W (z) is an FIR filter of tap-weight length L, the secondary signal y(n) is expressed as xL (n) y(n) = w T (n)x
0
ˆ 0 (n) = [ˆ where µw is the step size for W (z), x x0 (n)ˆ x0 (n − 1) · · · x ˆ0 (n − L + 1)]T , and x ˆ0 (n) is the reference signal x(n) ˆ filtered through S(z). We see that W (z) is perturbed by an ˆ 0 (n)v 0 (n). undesired term µwx ˆ Assuming that S(z) is represented by an FIR filter of tapweight length M , the filtered-reference signal x ˆ0 (n) is obtained as xM (n) x ˆ0 (n) = sˆT (n)x
2. Existing Methods: An Overview
(4) 0
ˆ (n)v (n) ˆ (n)[d(n) − y (n)] + µwx = w (n) + µwx
the additive random noise is reduced to a low level. In this paper we propose a new method to realize this solution, and
0
(5)
where sˆ(n) = [ˆ s0 (n)ˆ s1 (n)ˆ s2 (n) · · · sˆM −1 (n)]T is the impulse ˆ response of S(z) and x M (n) = [x(n)x(n−1) · · · x(n−M +1)]T is the M -sample reference signal vector. The LMS update ˆ equation for S(z) is given as sˆ(n + 1) = sˆ(n) + µs f (n)vv (n)
(1)
where w (n) = [w0 (n)w1 (n) · · · wL−1 (n)]T is the tap-weight
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(6)
= sˆ(n) + µsv (n)[v 0 (n) − vˆ0 (n)] + µsv (n)[d(n) − y 0 (n)]
ˆ where µs is the step size for S(z), vˆ0 (n) = sˆ(n) ∗ v(n) is an estimate of v 0 (n) obtained from the modeling filter, and v (n) = [v(n)v(n − 1) · · · v(n − M + 1)]T . Equation (6) shows ˆ that the convergence of S(z) is degraded by an undesired term µsv (n)[d(n) − y 0 (n)]. The Bao’s method [4] introduces a third adaptive to the Eriksson’s method. As shown in Fig. 3, the third adaptive filter B(z), excited by x(n), cancels the interference introduced by [d(n) − y 0 (n)] in the estimation of S(z). This greatly improves the convergence rate of the OSPM filter as compared with the Eriksson’s method. It is important to note that here the error signal used for the control filter is
Fig. 3. Bao’s method for OSPM in ANC systems.
same as residual error signal of the system, i.e. g(n) = e(n), and hence no effort is made to improve the performance of the control process. The Zhang’s method [6] (Fig. 4 without components in dashed box), is one step modification of the Bao’s method. ˆ Here the output of S(z), vˆ0 (n), is subtracted from e(n) to get a modified error signal for W (z): g(n) = [d(n) − y 0 (n)] + [v 0 (n) − vˆ0 (n)]
(7)
This error signal g(n) is also used as a desired response for the third filter H(z). Since H(z) is driven by x(n), and in g(n), [d(n)−y 0 (n)] is correlated with x(n), so its output u(n) converges to [d(n) − y 0 (n)]. This output signal u(n) of H(z) is subtracted from e(n), to get the desired response for the ˆ modeling filter S(z), i.e., ds (n) = e(n) − u(n). Therefore, the ˆ error signal for LMS equation of S(z) is given as f (n) = [d(n) − y 0 (n)] − u(n) + [v 0 (n) − vˆ0 (n)].
（ 1 ） In steady state the additive random noise v(n) is
reduced to such a low level that it cannot track the varia(8)
tions in S(z). Infact, v(n) is reduced to such a low level that
ˆ ˆ When S(z) converges, then (ideally) S(z) ≈ S(z) ⇒ [v 0 (n) − 0
Fig. 4. Zhang’s method for OSPM in ANC systems.
0
vˆ (n)] → 0, and hence g(n) ≈ [d(n) − y (n)]. When H(z)
Zhang’s original formulation [6] [Fig. 4 without components in dashed box] can reduce modeling error to a lower level.
converges, u(n) ≈ g(n) ≈ [d(n) − y 0 (n)] ⇒ f (n) → 0. Hence
（ 2 ） The computational complexity of the Zhang’s
appropriate signals are generated and mutual interaction beˆ tween W (z) and S(z) is greatly reduced.
method is larger as compared with the basic method [3], due
In order to further improve the performance of this ANC system, Zhang et. al. has proposed following noise power scheduling strategy in [10]. ( p c · vm (n) Pg /Pvm , v(n) = p c · vm (n) Px /Pvm ,
to three adaptive filters. （ 3 ） It is a difficult task to find an optimal set of param-
eters for three cross-updated adaptive filters. In [11], the noise power is adjusted on the basis of amplitude of the residual error signal e(n). This strategy, however,
Pg (n) < Px (n) Pg (n) > = Px (n)
introduces a positive feedback to the ANC system [10], and
(9)
ANC system may become unstable. The simulation results
where c is a positive constant, vm (n) is a zero mean white
presented in [10] show that the Zhang’s method [Fig. 4] gives
random noise, and Px (n), Pg (n) and Pvm denote power of
stable performance as compared with the method in [11]. The main objective in this paper is to modify the basic
x(n), g(n) and vm (n), respectively. Initially at n = 0, Pg (n), ˆ the power of g(n), is very large, and when W (z) and S(z)
method [3], so that an improved performance can be realized
converge, Pg (n) converges to zero. Hence, initially the vari-
using two adaptive filters only.
ance of the additive random noise v(n) is adjusted to a large
3. Proposed Method
value and it reduces to (ideally) zero, when the ANC system
Consider Fig. 5, which shows a method for ANC sys-
converges. There are the following problems with the Zhang’s method:
tems with OSPM, previously proposed by authors in [9]. As
—15—
ˆ shown, this method adapts the OSPM filter S(z) by a VSS LMS algorithm. The idea of this VSS LMS algorithm can be explained with reference to (6), which shows that the modeling filter is perturbed by a term µs v(n)[d(n) − y 0 (n)]. Initially, this disturbance may be very large [note that initially canceling signal y 0 (n) is zero], and the modeling filter may diverge. As the ANC system converges, y 0 (n) would converge to d(n) and (ideally) [d(n) − y 0 (n)] → 0. This observation suggests that initially we can use a small value for the step size parameter µs , and later, when the disturbance signal [d(n) − y 0 (n)] is reduced, the step size can be increased
Fig. 5. Previous method for OSPM in ANC systems [9].
accordingly. In [9] following procedure is used to track variations in [d(n) − y 0 (n)], and hence vary the step size µs of ˆ S(z). We define a ratio ρ(n) = Pf (n)/Pe (n)
(10)
where Pf (n) and Pe (n) are power of the modeling error signal f (n) and the residual error signal e(n), respectively. These powers can be estimated as Pγ (n) = λPγ (n − 1) + (1 − λ)γ 2 (n)
(11)
where λ is the forgetting factor (0.9 < λ < 1). It has been shown in [7] that ρ(0) ≈ 1 and limn→∞ ρ(n) → 0. In order to make sure that ρ(n) is with in the limits of zero and one, we constraint it as follows: ( ρ(n) =
Fig. 6. Proposed method for OSPM in ANC systems.
of this method [9], here we propose to varying the power of the additive random noise v(n). When the ANC system is started, e(n) is very large dur-
1,
ρ(n) > 1
0,
ρ(n) < 0.
ing the transient period of adaptation. During this time, a ˆ large additive random noise v(n) can be used so that S(z)
(12)
converges fast. When ANC system converges, the power of
ˆ Now the step size for S(z) is calculated as: µs (n) = ρ(n)µsmin + (1 − ρ(n))µsmax
v(n) can be reduced to realize noise reduction at the error (13)
microphone. We have already found a parameter ρ(n) which
where µsmin and µsmax are the experimentally determined values for lower and upper bounds of the step size. These values are selected so that neither adaptation is too slow nor it becomes unstable. In Eriksson’s method (Fig. 2), the residual error signal, e(n) = [d(n) − y 0 (n)] + v 0 (n), is used as an error signal for adapting W (z). Here [d(n) − y 0 (n)] is the desired component
can track variations in [d(n) − y 0 (n)], and hence can indicate the convergence status of the ANC system [7]: ρ(n) ≈ 1 ⇒ [d(n) − y 0 (n)] is large, and ρ(n) → 0 ⇒ ANC system has converged and [d(n) − y 0 (n)] is reduced. Similar to (13), the variance of the additive noise v(n) can be adjusted as follows: q (14) v(n) = (1 − ρ(n))σv2min + ρ(n)σv2max · vm (n)
and v 0 (n) acts as a disturbance. As compared with e(n), the
where σv2min and σv2max are minimum and maximum values
modeling filter error signal, f (n) = [d(n) − y 0 (n)] + [v 0 (n) −
for the variance of v(n), respectively. The block diagram
0
vˆ (n)], appears to be a better error signal for W (z), because ˆ |v 0 (n) − vˆ0 (n)| < |v 0 (n)| and when S(z) converges then (ide0
0
0
0
ally) v (n) ≈ vˆ (n) ⇒ [v (n) − vˆ (n)] → 0. Therefore, in previous method [9] (Fig. 5), f (n) is used as an error signal ˆ for both the control filter W (z) and the modeling filter S(z),
of the proposed method is shown in Fig. 6. In comparison with the Zhang’s method, the main features of the proposed method are summarized below: （ 1 ） It comprises only two adaptive filters and hence has
reduced computational complexity. （ 2 ） By choosing appropriate values for σv2min and σv2max ,
0
i.e., g(n) = f (n) = e(n) − vˆ (n). The simulation results presented in [9] show that this
it gives better control over the variance of the additive ran-
method gives the best OSPM performance among the ex-
dom noise v(n). The minimum variance can be selected so
isting methods. In order to further improve the performance
that the modeling filter can track small variations in steady
—16—
Magnitude (dB)
state. A computational complexity comparison of the various methods is given below: Num b er of Multiplications/Iteration Eriksson’s method [3]: 2L + 3M + 2 Zhang’s method [10]: 2L + 3M + 2N + 11 Previous method [9]: 2L + 3M + 10 Prop osed method: 2L + 3M + 14
20 0 −20 −40 −60 0
500
1000
Frequency (Hz)
1500
2000
1500
2000
Magnitude (dB)
(a)
ˆ Here L and M is tap weight length of W (z) and S(z), respectively, and N is tap weight length of the third filter H(z) in Zhang’s method. We see that the computational complex-
20 0 −20 −40 −60 0
500
1000
Frequency (Hz)
ity of the proposed method is much smaller than that of the
(b)
Zhang’s method. This difference is due to the third adap-
Fig. 7. The magnitude response of acoustic paths. (a) Primary path
tive filter, H(z), in Zhang’s method, which requires 2N + 1
P (z). (b) Secondary path S(z). (Sold line: Original path, Dashed
multiplications per iteration.
line: Changed path at n = 25000)
4. Simulation Results
the parameter c is selected as 0.75. In proposed method
This section presents the simulation experiments per-
the maximum and minimum variance of the modeling sig-
formed to verify the effectiveness of the proposed method.
nal is selected as σv2max = 1.0, and σv2min = 0.01, respec-
The following methods are considered: 1) Zhang’s method-
tively. The step size parameters for various adaptive filters
a [6], 2) Zhang’s method-b [10], 3) previous method [9] pro-
are adjusted, by trial-and-error, for fast and stable conver-
posed by authors, and 4) proposed method. The perfor-
gence.
mance comparison is done on the basis of the relative mod-
selected as: µw = 5 × 10−5 , µs (method-a) = 1 × 10−2 ,
eling error, ∆S(dB), being defined as: (P ) M −1 ˆi (n)]2 i=0 [si (n) − s ∆S(dB) = 10 log10 . PM −1 2 i=0 [si (n)]
µs (method-b) = 5 × 10−3 , µh = 2 × 10−3 . A small value
In Zhang’s methods the step size parameters are
for µs is used in Zhang’s method-b method, as this method (15)
is excited by a large level modeling signal v(n). In previous and proposed methods, step size parameters are adjusted to
For the primary and secondary paths the experimental data provided by [1] is used. Using this data, offline modeling is used to obtain FIR representation of tap-weight length 128
same values, as: µw = 5 × 10−5 , µsmin = 1 × 10−3 , and µsmax = 1 × 10−2 . All the results shown below are averaged over 10 experiments.
for P (z), and of tap-weight length 64 for S(z). The mag-
The simulation results are presented in Fig. 8. The curves
nitude response of P (z) and S(z) is shown in Fig. 7. The ˆ control filter W (z) and modeling filter S(z) are FIR filters
Fig. 8(a). Here, at n = 25000, the paths change as shown
of tap-weight length L = M = 64, respectively. The third
by dashed lines in Fig. 7. Among Zhang’s methods, the
adaptive filter in Zhang’s method [10], H(z), is selected as
method-b gives a fast initial convergence, but higher steady
an FIR filter of tap-weight length N = 48. The delay ∆ in
state value as compared with the method-a.
Zhang’s method is 30. The control filter is initialized by a
method-b, the degraded modeling performance is due to the
null vector w(0). The third filter H(z) in the Zhang’s method ˆ is also initialized by a null vector h(0). To initialize S(z), of-
fact that the variance of the modeling excitation signal v(n)
fline secondary path modeling is performed which is stopped
gives the best OSPM performance in both the convergence
when the modeling error [as defined in (16)] has been reduced
speed and steady state value. Fig. 8(b) shows that that the
to −5 dB. The resulting weights are used for ˆ s(0) when the
parameter ρ(n) varies between 1 and 0, with the dynamics of
ANC system is started.
the ANC system. The curves for the noise power variation,
for relative modeling error, ∆S(dB) (Eq. 15), are shown in
In Zhang’s
reduces to a very low level. We see that the proposed method
A sampling frequency of 4 kHz is used. The reference
and the modeling filter step size are shown in Fig. 8(c) and
signal x(n) is a narrowband signal comprising frequencies
(d), respectively. In the Proposed method, initially the mod-
of 50 Hz, 100 Hz, 200 Hz and 300 Hz. Its variance is ad-
eling process is excited by a high level random noise with
justed to 2 and a zero-mean white noise is added with SNR
variance σv2max and modeling filter step size is adjusted to
of 30 dB. In Zhang’s method-a and in previous method, the modeling excitation signal v(n) is a zero-mean white Gaus-
µsmin . Later, when ANC system converges and disturbance d(n) − y 0 (n) is reduced, the noise variance is decreased and
sian noise of constant variance 0.05. In Zhang’s method-b
—17—
modeling filter step size is increased accordingly. The pro∆S(dB)
posed method gives the best performance due to this online tuning of the parameters. The corresponding curves for residual noise signal are shown in Fig. 8(e). We see that the Zhang’s method-b gives
Zhang’s method−a Zhang’s method−b Previous method Proposed method
−5 −15 −25 −35 0
1
2
3
Iteration Time
best performance. The reason is that it reduces the model-
4
5 4
x 10
(a)
ing excitation signal to a very low level as compared with the 1
formance of the Zhang’s method-b, as described earlier. The
0.8
ρ (n)
other methods. This, however, degrades the modeling perrest of the three methods have almost similar performance. Thus the Proposed method gives an overall improved perfor-
Previous method Proposed method
0.6 0.4 0.2
mance, and at much reduced computational cost.
0 0
1
2
3
Iteration Time
5. Concluding Remarks
4
5 4
x 10
(b) 0.6
Varaince of v(n)
This paper proposes a new method for ANC systems with online secondary path modeling. The proposed method is a modified version of author’s previous work [9], and incorporates a noise power scheduling strategy. The proposed method initially uses a large additive noise for secondary
Zhang’s method−b Proposed method
0.4 0.2 0.0 0
1
path modeling, and later when ANC system converges, the
2
3
Iteration Time
4
5 4
x 10
(c)
power of additive random noise is reduced to a low level. This .010
ations in steady state. In the existing methods [10], [11], on
.008
µs(n)
low level can be adjusted so the modeling filter can track varithe other hand, the modeling signal is reduced to a very low level which degrades the modeling performance. Another ad-
.006 .004 Previous method Proposed method
.002 0 0
vantage of the proposed method is its reduced computational
1
complexity. Infact, the structure of the proposed method is
2
3
4
Iteration Time
5 4
x 10
(d)
similar to the basic method [3], and comprises only two adap10
MSE (dB)
tive filters. References [1] S. M. Kuo, and D. R. Morgan, Active Noise Control Systems-Algorithms and DSP Implementation. New York: Wiley, 1996. [2] N. Saito, and T. Sone, “Influence of modeling error on noise reduction performance of active noise control systems using filtered-x LMS algorithm,” J. Acoustic. Soc. Jpn. (E), vol. 17,no. 4, pp. 195–202, 1996. [3] L. J. Eriksson, and M. C. Allie, “Use of random noise for on-line transducer modeling in an adaptive active attenuation system,” J. Acoust. Soc. Am., vol. 85, issue 2, pp. 797–802, 1989. [4] C. Bao, P. Sas, and H. V. Brussel, “Adaptive active control of noise in 3-D reverberant enclosure,” J. Sound Vibr., vol. 161, no. 3, pp. 501–514, 1993. [5] S. M. Kuo, and D. Vijayan,“A secondary path modeling technique for active noise control systems,” IEEE Trans. Speech Audio Processing, vol. 5, no. 4, pp. 374–377, 1997. [6] M. Zhang, H. Lan, and W. Ser, “Cross-updated active noise control system with online secondary path modeling,” IEEE Trans. Speech, Audio Proc., vol. 9, no. 5, pp. 598–602, 2001. [7] M. T. Akhtar, M. Abe, and M. Kawamata,“A New Variable Step Size LMS Algorithm-Based Method for Improved Online Secondary Path Modeling in Active Noise Control Systems,” IEEE Trans. Speech Audio Processing. (in press) [8] S. J. Elliot, Signal Processing for Active Control. London, U.K.: Academic Press, 2001.
0
Zhang’s method−a Previous method Proposed method
Zhang’s method−b
−10 −20 0
1
2
3
Iteration Time
4
5 4
x 10
(e) Fig. 8. (a) The relative modeling error, ∆S(dB), (b) The variation of the parameter ρ(n), (c) The power of the modeling excitation signal ˆ v(n), (d) The time varying step size µs (n) for modeling filter S(z), (e) The power of the residual error signal e(n).
[9] M. T. Akhtar, M. Abe, and M. Kawamata, “A Method for Online Secondary Path Modeling in Active Noise Control Systems,” in Proc. IEEE 2005 Intern. Symp. Circuits Systems (ISCAS2005), Kobe, Japan, July 23–26, 2005. (accepted) [10] M. Zhang, H. Lan, and W. Ser, “A robust online secondary path modeling method with auxiliurary power scheduling strategy and norm constraint manipulation,” IEEE Trans. Speech, Audio Proc., vol. 11, no. 1, pp. 45–53, Jan. 2003. [11] D. Cornelissen, and P. C. W. Sommen, “New online secondary path estimation in a multipoint filtered-X algorithm for acoustic noise cancellation,” in Proc. IEEE ProRIS 99, 1999, pp. 97–103.
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