Digital Signal Processing 19 (2009) 241–249
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
An efficient online secondary path estimation for feedback active noise control systems Hamid Hassanpour a,∗ , Pooya Davari b a b
School of Information Technology and Computer Engineering, Shahrood University of Technology, P.O. Box 316, Shahrood, Iran Department of Electrical and Computer Engineering, Mazandaran University, Shariatee Av, Babol, Iran
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online 3 July 2008
In practical cases for active noise control (ANC), the secondary path has usually a time varying behavior. For these cases, an online secondary path modeling method that uses a white noise as a training signal is required to ensure convergence of the system. The modeling accuracy and the convergence rate are increased when a white noise with a larger variance is used. However, the larger variance increases the residual noise, which decreases performance of the system and additionally causes instability problem to feedback structures. A sudden change in the secondary path leads to divergence of the online secondary path modeling filter. To overcome these problems, this paper proposes a new approach for online secondary path modeling in feedback ANC systems. The proposed algorithm uses the advantages of white noise with larger variance to model the secondary path, but the injection is stopped at the optimum point to increase performance of the algorithm and to prevent the instability effect of the white noise. In this approach, instead of continuous injection of the white noise, a sudden change in secondary path during the operation makes the algorithm to reactivate injection of the white noise to correct the secondary path estimation. In addition, the proposed method models the secondary path without the need of using off-line estimation of the secondary path. Considering the above features increases the convergence rate and modeling accuracy, which results in a high system performance. Computer simulation results shown in this paper indicate effectiveness of the proposed method. © 2008 Elsevier Inc. All rights reserved.
Keywords: Active noise control Adaptive filter Feedback structure FxLMS Secondary path On-line modeling
1. Introduction In recent years, by the wide spread use of industrial equipments, acoustic noises become more evident. An active noise control (ANC) is a technique that efficiently attenuates low frequencies unwanted noises where passive methods are either ineffective or tend to be very expensive or bulky. An ANC system is based on a destructive interference of an anti-noise, which have equal amplitude and opposite phase replica primary noise, with unwanted noise (primary noise d(n) in Fig. 1). Following the superposition principle, the result is cancellation or reduction of both noises [1]. The block diagram of feedback ANC system [1] is shown in Fig. 1. Here, P ( z) is the primary path consists of the acoustic response from the reference noise source to the error sensor, and S ( z) is the secondary path. The effect shown by the secondary path transfer function, the path leading from the noise controller output to the error sensor measuring the residual noise, generally causes instability to the standard least mean square (LMS) algorithm. Resolv-
*
Corresponding author. E-mail address:
[email protected] (H. Hassanpour).
1051-2004/$ – see front matter doi:10.1016/j.dsp.2008.06.007
© 2008
Elsevier Inc. All rights reserved.
242
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
Fig. 1. Block diagram of a feedback ANC system.
ing the instability problem requires using FxLMS algorithm [1,2]. The FxLMS algorithm uses estimation of the secondary path to compensate the above-mentioned problem. In many applications the secondary paths are usually time varying, which leads to a poor performance or system instability. Hence, online modeling of secondary path is required to ensure convergence of the ANC algorithm [3,4]. Most of online secondary path modeling techniques involve injection of additional random noise, as a training signal, to provide an appropriate estimation of the secondary path. White noise is usually used as an ideal training signal in modeling the secondary path [5–7]. White noise with larger amplitude results in a better modeling accuracy and convergence rate [3], however, with the cost of an increased residual noise. Thus, existing online secondary path modeling techniques use white noise with a low variance to model the secondary path in order to maintain a low residual noise in steady state. Additionally, injection of the white noise may cause the feedback structure algorithm to diverge, ruining the control system performance. The proposed system is based on an FxLMS algorithm as the main part, and a variable step size (VSS) LMS algorithm [5,6] is used to adapt the modeling filter of the secondary path. The proposed algorithm uses white noise with a large amplitude in modeling the secondary path. However, to increase performance of the algorithm and to prevent the instability effect of the white noise in feedback structures, we stop the VSS-LMS algorithm at the optimum point. Stopping at the optimum point allows benefiting advantages of white noise with larger variance. A sudden change in the secondary path leads to divergence of the online secondary path modeling filter. To mitigate this problem, the proposed algorithm is designed in the way that it controls the secondary path changes. A sudden change in secondary path during the operation makes the algorithm to reactivate injection of the white noise to correct the secondary path estimation. Common secondary path modeling methods use off-line estimation of the secondary path prior to online modeling [5, 6,8]. Off-line estimation can be obtained when the primary noise is absent. However, in many practical cases the primary noise exists even during the off-line modeling [3]. This adversely affects the accuracy of the modeling filter. Therefore, the proposed method models the secondary path without the need of off-line estimation. Considering the above features in the proposed method assists obtaining a better convergence rate and modeling accuracy, which results in a robust system. The rest of the paper is organized as follows. In Section 2, the feedback ANC system is briefly described. Section 3 introduces our proposed method. In Section 4, we illustrate our simulation results, and finally in Section 5 conclusions are drawn. 2. Feedback ANC systems As it was mentioned before, transfer function of the secondary path has a crucial role in generating anti-noise in ANC applications as it introduces delay causing instability problem to the standard LMS algorithm. The instability problem can be resolved using the FxLMS algorithm [1,2] as it uses estimation of the secondary path [1]. This algorithm can be applied to both feedback and feedforward structures [1]. The block diagram of a feedback FxLMS ANC system is shown in Fig. 2 [9]. Here, P ( z) is the primary path, the acoustic response from the reference noise source to the error sensor; and S ( z) represents the secondary path. In this figure Sˆ ( z) is estimation of the secondary path S ( z). Unlike in feedforward structure where a reference sensor is available to pick up the reference signal x(n), the main problem in implementing the feedback ANC system is to estimate the primary noise (d(n)) and using it as a reference signal for the ANC filter (W ( z)) [9]. As Fig. 1 shows, the reference signal x(n) is a sum of two signals e (n) and yˆ (n) [2]: x(n) ≡ dˆ (n) = e (n) +
M −1
sˆm y (n − m),
(1)
m=0
where sˆm represents coefficients of the Mth-order FIR filter Sˆ ( z). In this figure, the secondary signal y (n) is generated as y (n) = w T (n)x L (n),
(2)
where w(n) = [ w 0 (n), w 1 (n), . . . , w L −1 (n)] and x L (n) = [x(n), x(n − 1), . . . , x(n − L + 1)] are the coefficient and signal vectors of length L, the order of the FIR filter W ( z), at time n. These coefficients are updated by the FxLMS algorithm as follows: T
w(n + 1) = w(n) + μ w xˆ (n)e (n),
0 < μ w < 1,
T
(3)
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
243
Fig. 2. Block diagram of feedback ANC system using FxLMS algorithm [9]. x(n) represents the reference signal.
Fig. 3. Block diagram of the proposed feedback ANC system.
where xˆ (n) = [ˆx (n), xˆ (n − 1), . . . , xˆ (n − L + 1)] T and
ˆ
μ w is the step size, and
x (n) = sˆ (n)x M (n) T
(4)
is the filtered reference signal. Here x M (n) = [x(n), x(n − 1), . . . , x(n − M + 1)] , and sˆ (n) = [ˆs0 (n)ˆs1 (n) . . . sˆ M −1 (n)] is the impulse response of the modeling filter Sˆ ( z). For a deep study on feedback FxLMS algorithm the reader may refer to [1,10]. T
T
3. Proposed method Fig. 3 shows block diagram of the proposed ANC system. The residual error signal e (n) of this algorithm is expressed as e (n) = d(n) − y (n) + v (n), y (n) = s(n) ∗ y (n),
v (n) = s(n) ∗ v (n),
(5)
where v (n) is an internally generated white Gaussian noise, which is injected at the output of the control filter W ( z). In this figure Sˆ ( z) is the modeling FIR filter with length M that generates vˆ (n) expressed below: vˆ (n) = sˆ T (n)v M (n).
(6)
As the figure shows, vˆ (n) generates the error signal for both the modeling filter Sˆ ( z) and the control filter W ( z) by subtracting from e (n):
f (n) = d(n) − y (n) + v (n) − vˆ (n).
(7)
Coefficients of the modeling filter Sˆ ( z) are updated as follows: sˆ (n + 1) = sˆ (n) + μs (n) f (n)v(n),
(8)
where μs (n) is the step-size parameter of the VSS-LMS algorithm which will be explained later. The output signal yˆ (n) is used to define estimation of the primary noise dˆ (n): dˆ (n) = f (n) + yˆ (n),
(9)
where x(n) ≡ dˆ (n) is our reference signal, and yˆ (n) is expressed as follows: yˆ (n) = sˆ T (n)y M (n).
(10)
244
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
Finally coefficients of the control filter W ( z) are updated as w(n + 1) = w(n) + μ w xˆ (n) f (n).
(11)
The input to the LMS algorithm is derived by filtering the reference signal through Sˆ ( z): xˆ (n) = sˆ T (n)x M (n).
(12)
The VSS-LMS algorithm introduced in [5] is used to update modeling filter Sˆ ( z) coefficients. For more detail on theory of this algorithm reader may refer to [5,6]. As we mentioned before, the modeling filter in Eq. (8) is updated using the step-size parameter (μs (n)) of VSS-LMS algorithm and this parameter is calculated using the following three steps [6]:
• Initially, the power of error signals e (n) and f (n) are computed: P e (n) = λ P e (n − 1) + (1 − λ)e 2 (n), P f (n) = λ P f (n − 1) + (1 − λ) f 2 (n),
0.9 < λ < 1.
(13)
• Then, the ratio of the estimated powers is obtained:
ρ (n) = P f (n)/ P e (n), ρ (0) ≈ 1,
lim
n→∞
ρ (n) → 0.
• Finally, the step size is calculated as follows: μs (n) = ρ (n)μsmin + 1 − ρ (n) μsmax , 0 < μsmin , μsmax < 1,
(14)
(15)
where μsmin , μsmax and λ are experimentally determined. These values are selected so that the adaptation is neither too slow nor it becomes unstable. Injection of additional random noise into the ANC system in modeling the secondary path causes instability problem especially in feedback structure. To prevent this problem, the injection of white noise is required to be stopped at the optimum point. Here, the VSS-LMS algorithm is briefly described to show the way the optimum point is obtained. The VSS-LMS algorithm is initially set to a small step size. During the process of this algorithm, μs is increased as the error signal f (n) decreases, and vice versa. It needs to be noted that any increase of the step size corresponds to a faster convergence of the adaptive algorithm. Consequently, once W ( z) is slow in reducing e (n), the step size remains small which results in a lower convergence rate. Hence, the modeling filter, Sˆ ( z), converges to a good estimation when f (n) decreases. This happens when μs increases as high as μsmax . Thus, the injection of the white noise is stopped at the optimum point which is measured using
μsmax − μs (n) < α , 1 × 10−5 < α 1 × 10−3 .
(16)
The α parameter should be in the range of 0 < α < μsmax − μsmin . Experimental results for different noise sources showed that it must be in the range of 1 × 10−5 < α 1 × 10−3 as given in Eq. (16). As can be seen from Fig. 3, this condition validity is monitored at the performance monitoring stage. By setting α to a lower value the modeling error as well as the convergence rate are decreased, which results in an accurate modeling of the secondary path. In some practical cases the secondary path may suddenly change. This event derives system to diverge. To prevent this effect Sˆ ( z) needs to be updated. The proposed algorithm is designed in the way that it monitors the secondary path changes by the following expression:
20 log10 f (n) < 0.
(17)
This equation is also obtained by studying behavior of the VSS-LMS algorithm during the process. We tracked the f (n) signal and find out that once the modeling accuracy increases sufficiently, Eq. (17) degrades below zero. Any changes in the secondary path during the operation make this equation to suddenly increase above zero. Thus, using the above equation give the ability to the proposed system to perfectly control the secondary path changes. If the validity of the above equation does not satisfy, the system reactivates the VSS-LMS algorithm and injects white noise to remodel Sˆ ( z). The same as before, the injection is stopped at the optimum point using Eq. (16). The above procedure is repeated during the system operation to adapt the algorithm with characteristics of the environment. Estimation of the secondary path can be obtained by using the off-line modeling method followed by on-line modeling. However, as mentioned before, for some applications the primary noise exists even during the off-line modeling and this will adversely affect the accuracy of the modeling filter. Therefore, there is no need of using off-line estimation of the secondary path in the proposed approach as it is required in the existing methods [5,6,8].
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
245
(a)
(b) Fig. 4. Impulse response of the acoustic paths: (a) Impulse response of the primary path P ( z), (b) impulse response of the secondary path S ( z).
4. Simulations and performance results In this section the proposed ANC system is simulated using Matlab version 7.1. In this simulation, we have used the primary path P ( z) and secondary path S ( z) of the experimental data provided in [11]. The impulse responses of the primary and secondary paths are shown in Fig. 4. Using these data, P ( z) and S ( z) are considered as FIR filters with tap—weight lengths 48 and 16, respectively. Rate of the sampling frequency in this simulation was 2 kHz. Extensive experiments have been performed to find suitable values for a fast and stable performance of the ANC system. Length of FIR filter Sˆ ( z) for modeling the secondary path, and length of the adaptive filter W ( z) used for the noise cancellation have been chosen 16 and 32, respectively. We have performed simulations for four separate cases. In Case 1, we evaluate performance of the proposed system with two different narrowband noises. Comparative results of modeling accuracy and convergence rate for the system are illustrated in Case 2 using three different white noises. Case 3 indicates effectiveness of the proposed algorithm in maintaining its performance against sudden changes of the secondary path behavior. Finally, Case 4 shows the effect of the W ( z) filter length in noise reduction. All the results shown in each case have been obtained as an average 10 different experiments. 4.1. Case 1 In this case, two experiments have been performed using different noises to evaluate performance of the proposed system. In these experiments, the original noise has several narrowband periodic components, which is usual in ANC system applications like those produced by engines, compressors and fans [12]. As the first experiment, the reference signal is a narrowband signal comprising frequencies of 100, 200, 300 and 400 Hz. Its variance is adjusted to 2, and a white noise with SNR of 30 dB is added. The modeling training signal has a zero-mean white Gaussian noise of variance 0.8. Fig. 5 shows the power spectrum of the narrowband noise. In this figure, the solid line depicts the residual noise after cancellation. As can be seen from Fig. 5, the noise cancellation levels for harmonic components 100, 200, 300 and 400 Hz are 62.584, 77.757, 67.915 and 70.081 dB, respectively. In the next experiment, performance of the proposed method is evaluated using engine noise at 3700 rpm. Fig. 6 shows the power spectrum of the engine noise at 3700 rpm and the residual noise after cancellation. In this experiment, as the figure shows, the noise cancellation levels obtained for harmonic components at 185, 247, 308, 370, 431, 493, 555, 617 and 678 Hz are 47.81, 79.505, 79.331, 92.664, 68.064, 19.33, 52.55, 47.78 and 32.89 dB, respectively. The high noise cancellation results in these experiments indicate the accuracy of the proposed system. In fact, the good modeling of the secondary path has yielded this accuracy. It should be noted that a wrong selection of α in Eq. (16) may lead to an inefficient model for Sˆ ( z), which drives the system to an instable situation.
246
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
Fig. 5. Power spectra for the original narrowband noise (dotted line) and the residual noise (solid line) after noise cancellation in Case 1.
Fig. 6. Power spectra for the original engine noise at 3700 rpm (dotted line) and the residual noise (solid line) after noise cancellation in Case 1.
4.2. Case 2 As mentioned before, in order to increase convergence rate and modeling accuracy, the proposed method uses a white noise with a larger variance. Common existing online secondary path modeling methods use white noise with variance 0.05 to model the secondary path, in order to maintain the residual noise low. Here performance of the proposed method is indicated in Fig. 7 by using the first narrowband signal with three white noises of 0.05, 0.22 and 0.8 variances on the basis of the relative modeling error defined as
S (dB) = 10 log10
M −1 2 ˆ i =0 [s i (n) − s i (n)] M −1 2 i =0 [s i (n)]
.
(18)
As can bee seen, the convergence rate and the modeling accuracy are increased by using the white noise with a larger variance. For more studies on the effect of white noise variances, the performance of the three white noises is compared in Fig. 8 on the basis of noise reduction defined below:
2 e (n) R = −10 log10 2 . d (n) The large positive value of R indicates that more noise reduction is achieved.
(19)
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
247
Fig. 7. Relative modeling error versus iteration time n obtained by white noises with 0.05, 0.22 and 0.8 variance in Case 2.
Fig. 8. Noise reduction achieved by white noises with 0.05, 0.22 and 0.8 variance versus time (s) in Case 2.
4.3. Case 3 In this case we assume that the secondary path transfer function suddenly changes during the operation. Fig. 9 shows the magnitude response of the original and changed secondary path. In this figure, the solid line represents secondary path at start point, n = 0, and the dashed line represents the changed path at iteration n = 16,000. Fig. 10 shows the curve for relative modeling error, S (dB) (Eq. (18)). As can be seen from the figure, the proposed method automatically re-estimates the secondary path model to retain performance of the system in reducing noise. 4.4. Case 4 Length of the controller filter, W ( z), always acts as an important parameter in the performance of the system. Increasing the filter length correspondingly increases the ANC system accuracy in noise attenuation. In hardware implementations, however, it is desirable to select the minimum possible filter length in order to decrease the system complexity. Here we represent comparative results using three different filter lengths of 32, 64 and 128 on the basis of noise reduction (19). As Fig. 11 shows, the noise reduction performance is improved as the filter length increased. The resulted simulation parameters in these experiments are summarized in Table 1.
248
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
Fig. 9. Magnitude response of secondary path (solid line: original path, dashed line: changed path at n = 16,000) in Case 3.
(a)
(b) Fig. 10. Simulation results in Case 3: (a) Relative modeling error S (dB) versus iteration time n, (b) modeling filter step size
μs (n) versus iteration time n.
Table 1 Simulation parameters for the proposed approach Simulations
μw
μsmax , μsmin
λ, α
Cases 1, 2 Case 3 Case 4
3 × 10−5 3 × 10−5 5 × 10−6
1 × 10−2 , 15 × 10−4 1 × 10−2 , 15 × 10−4 1 × 10−2 , 15 × 10−4
0.99, 4 × 10−5 0.99, 4 × 10−4 0.99, 1 × 10−4
5. Conclusions This paper proposed a new technique for on-line secondary path modeling in feedback ANC systems. This technique has four distinct capabilities. First, it could resolve the problem of feedback structure with the injection of white noise. Second, the system uses the advantages of white noise with a large variance in secondary path modeling. Third, the proposed technique is robust against sudden changes of secondary path. Finally, there is no need for off-line estimation of the secondary path in this approach. Computer simulations have been conducted for a single-channel feedback ANC system. Simulation results on different noise sources demonstrate that the proposed method has achieved a high performance in noise attenuation.
H. Hassanpour, P. Davari / Digital Signal Processing 19 (2009) 241–249
249
Fig. 11. Noise reduction achieved on 32, 64 and 128 filter length of W ( z) versus time (s) in Case 4.
References [1] S.M. Kuo, D.R. Morgan, Active noise control: a tutorial review, Proc. IEEE 8 (6) (1999) 943–973. [2] Design of Active Noise Control Systems with the TMS320 Family: Application Report, literature number SPRA042, Texas Instruments, 1996. [3] S.M. Kuo, D. Vijayan, Optimized secondary path modeling technique for active noise control systems, in: Proc. IEEE Asia–Pacific Conf. on Circuits and Systems, Taipei, Taiwan, 1994, pp. 370–375. [4] S.M. Kuo, D. Vijayan, A secondary path modeling technique for active noise control systems, IEEE Trans. Speech Audio Process. 5 (4) (1997) 374–377. [5] M.T. Akhtar, M. Abe, M. Kawamata, Modified-filtered-xLMS algorithm based active noise control system with improved online secondary path modeling, in: Proc. IEEE 2004 Int. Mid. Symp. Circuits Systems (MWSCAS 2004), Hiroshima, Japan, 2004, pp. I-13–I-16. [6] M.T. Akhtar, M. Abe, M. Kawamata, A method for online secondary path modeling in active noise control systems, in: Proc. IEEE 2005 Int. Symp. Circuits Systems (ISCAS 2005), May 23–26, 2005, pp. I-264–I-267. [7] M. Zhang, H. Lan, W. Ser, A robust online secondary path modeling method with auxiliary noise power scheduling strategy and norm constraint manipulation, IEEE Trans. Speech Audio Process. 11 (1) (2003) 45–53. [8] W.S. Gan, S. Mitra, S.M. Kuo, Adaptive feedback active noise control headsets: Implementation, evaluation and its extensions, IEEE Trans. Consumer Electron. 5 (3) (2005) 975–982. [9] W.S. Gan, S.M. Kuo, Integrated active noise control communication headset, in: Proc. of the International Symposium on Circuits and Systems (ISCAS 2003), May 25–28, 2003, pp. IV353–356. [10] S.M. Kuo, X. Kong, W.S. Gan, Applications of adaptive feedback active noise control system, IEEE Trans. Contr. Syst. Technol. 11 (2) (2003) 216–220. [11] S.M. Kuo, D.R. Morgan, Active Noise Control Systems—Algorithms and DSP Implementation, Wiley, New York, 1996. [12] A.Q. Hu, X. Hu, S. Cheng, A robust secondary path modeling technique for narrowband active noise control systems, in: Proc. IEEE Conf. on Neural Networks and Signal Processing, vol. 1, December 2003, pp. 818–821.
Hamid Hassanpour received the B.S. degree in computer engineering from Iran University of Science and Technology, Tehran, Iran, in 1993, the M.S. degree in computer engineering from Amirkabir University of Technology, Tehran, Iran, in 1996, and the Ph.D. from the Queensland University of Technology, Brisbane, Australia, in 2004. His research interests include biomedical signal processing, time-frequency signal processing and analysis, new architectures in computer design, text syntax analyzing and image processing.
Pooya Davari received the B.S. and M.S. degrees from Mazandaran University, Babol, Iran, in 2004 and 2008, respectively, both in electronic engineering. His research interests include active noise control, adaptive signal processing, intelligent signal processing, image processing and machine vision.
