A Study on Improving the Robustness of Virtual Sensing Methods in ANC Systems

Shota TOYOOKA 1

Kansai University, Japan 3-3-35 Yamatecho, Suita-shi, Osaka-fu 564-8680, Japan

Yoshinobu KAJIKAWA 2

Kansai University, Japan 3-3-35 Yamatecho, Suita-shi, Osaka-fu 564-8680, Japan

ABSTRACT In this paper, we propose a new online identification method for auxiliary filter-based virtual sensing that can cope with variations in secondary paths, which requires not only online identification of secondary path models but also updating of auxiliary filters when secondary paths change. Therefore, the proposed online identification method has the capability of online identification of auxiliary filters in addition to the conventional online identification of secondary paths. In this paper, we first show theoretically that re-identification is necessary when the secondary path changes in auxiliary filter-based virtual sensing. We then demonstrate through simulations that the noise reduction can be maintained by applying the proposed online identification method even when the secondary path changes, achieving an overall reduction of approximately 8 dB.

1. INTRODUCTION

Active noise control (ANC) system reduces unwanted acoustic noise by generating anti-noise with the same amplitude and opposite phase based on the superposition of acoustic waves [1]– [3]. In general, ANC can generate a zone of quiet (ZoQ) around the error microphone position, but the size of the ZoQ is determined by the frequency of the controlled noise. Specifically, a spherical or crescent-shaped zone with a diameter of 1 / 10 of the wavelength in a di ff use sound field is said to provide 10 dB of noise reduction [4]– [6]. If the noise to be controlled is 1000 Hz, the ZoQ is limited to a space with a diameter of 3.4 cm and if a microphone cannot be placed at the desired location, su ffi cient reduction may not be achieved. In such a case, we need to use the virtual sensing method to move the ZoQ to the desired position. Two main virtual sensing methods for ANC systems have been proposed for ANC systems. One is the Auxiliary Filter based Virtual Sensing (AFVS) method [7]– [15] and the other is the Remote Microphone based Virtual Sensing (RMVS) method [16]– [18]. In these systems, it is known that

1 k927766@kansai-u.ac.jp

2 kaji@kansai-u.ac.jp

eee i 21-24 AUGUST SCOTTISH cea! ahd

when the secondary path is varied, the ANC system diverges due to modeling errors in the secondary path model. To resolve this problem, it is necessary to introduce online modeling techniques [1], [19]– [21] into virtual sensing methods. Since RMVS requires a model of the secondary path to the desired location, re-estimation of the secondary path is not possible [22]. On the other hand, AFVS does not require a model of the secondary path to the desired location, so online identification of the secondary path to the error microphone location is su ffi cient. However, the auxiliary filter required by AFVS includes information on the secondary path to the error microphone location at the tuning stage, so the auxiliary filter must be re-estimated when the secondary path changes. Therefore, this paper proposes a configuration of AFVS that is robust to fluctuations in secondary paths by introducing online modeling of secondary paths into AFVS and adding the function of re-estimating auxiliary filters.

2. VIRTUAL SENSING MOTHOD

This chapter describes the principle of AFVS and then explains the proposed configuration to deal with variations in the secondary path to the error microphone location in AFVS. The virtual sensing method consists of a tuning stage and a control stage. In the tuning stage, the microphone is physically placed at the desired location. Then, an auxiliary or compensation filter is estimated based on the signals from the microphone placed at the desired position (virtual microphone) and the error microphone. In the control stage, the virtual microphone is removed and noise reduction at the desired position is achieved using the error microphone (placed at a distance from the desired position) and the auxiliary or compensation filter obtained in the tuning stage.

2.1. Auxiliary filter based virtual sensing (AFVS) method Figure 1(a) shows the block diagram of the AFVS method during the tuning stage. In the tuning stage, the virtual microphone is placed at the desired position and the optimal noise control filter W ( z ) is estimated to minimize the power of the virtual error signal E v ( z ). The auxiliary filter H ( z ) is then used to estimate the overall path from the noise source to the error microphone position. The virtual error signal E v ( z ) is written as

E v ( z ) = D v ( z ) + Y ′ v ( z ) = { P v ( z ) + S v ( z ) W ( z ) R ( z ) } X ( z ) , (1)

where the virtual primary path P v ( z ) and the virtual secondary path S v ( z ) are the transfer functions from the noise source and secondary source to the virtual microphone, respectively. The reference path R ( z ) is the transfer function from the noise source to the reference microphone, and X ( z ) is the reference signal. These functions are denoted by z-transforms. From (1), the optimal noise control filter W o ( z ) at virtual microphone becomes

W o ( z ) = − P v ( z ) S v ( z ) R ( z ) . (2)

Similarly, the error signal used to update the auxiliary filter H ( z ), including the error signal at the error microphone position, is written as

E h ( z ) E m ( z ) + F ( z ) D m ( z ) + Y ′ m ( z ) + H ( z ) R ( z ) X ( z ) { P m ( z ) + S m ( z ) W ( z ) R ( z ) + H ( z ) R ( z ) } X ( z ) , (3)

where the primary path P m ( z ) and secondary path S m ( z ) are the transfer functions from the noise source and secondary source to the error microphone, respectively. From (3), the optimal auxiliary

filter H o ( z ) yields

H o ( z ) = − P m ( z )

R ( z ) − P v ( z ) S m ( z )

S v ( z ) R ( z )

= − P m ( z )

R ( z ) − S m ( z ) W o ( z ) . (4)

Figure 1(b) shows the block diagram of the AFVS method during the control stage. In the control stage, the noise control filter W ( z ) is updated by adding the output f ( n ) from H ( z ) to the error signal e m ( n ) to achieve noise reduction at the desired position. The error signal e ′ v ( n ) is written as

E ′ v ( z ) = E m ( z ) + F ( z ) = { P m ( z ) + S m ( z ) W ( z ) R ( z ) + H ( z ) R ( z ) } X ( z ) . (5)

From (5), the optimal auxiliary filter W AF ( z ) at the control stage becomes

W AF ( z ) = − P m ( z ) S m ( z ) R ( z ) + H ( z )

(6)

S m ( z )

Since the AFVS method requires a secondary path model ˆ S m ( z ), if S m ( z ) is varied, the noise reduction performance of the ANC system is expected to be reduced because ˆ S m ( z ) has modeling errors [23]– [25].

ZoQ

P v ( z ) d v ( n )

x ( n )

e v ( n )

+ P v ( z ) d v ( n )

x ( n )

+

Σ

ZoQ

+

Σ y ( n ) yʹ v ( n )

R ( z ) r ( n )

+

W ( z ) S v ( z )

yʹ v ( n )

S v ( z )

rʹ v ( n ) e v ( n ) ^

d m ( n )

S v ( z ) NLMS

P m ( z )

yʹ m ( n )

S m ( z )

+ +

y ( n )

R ( z ) r ( n )

S m ( z ) Σ yʹ m ( n )

+ +

W ( z )

d m ( n )

P m ( z )

Σ

eʹ v ( n )

rʹ m ( n )

f ( n )

e m ( n )

+ +

^

S m ( z ) NLMS

H ( z )

Σ

f ( n )

e m ( n )

e h ( n )

+ +

H ( z )

Σ

NLMS

(b) Control stage

(a) Tuning stage

Figure 1: Block diagram of the feedforward ANC system with AFVS.

2.2. AFVS with online modeling technique To solve the problems caused by such modeling errors, this paper proposes an AFVS method that introduces online modeling technique [1], [19]– [21]. Figure 2 shows a block diagram of the AFVS method in the control stage introducing the on-line modeling technique. The block diagram of the tuning stage is the same as in Figure 1(a). In online modeling, the internally generated zero mean white noise v ( n ) is added to the output y ( n ) from W ( z ) to drive the secondary source. The adaptive filter ˆ S m ( z ) is then connected in parallel with the secondary path S m ( z ), and its transfer function is estimated online using the random noise v ( n ). Let us assume that the secondary path varies from S m ( z ) to S ′ m ( z ) during the operation of the ANC system. In this case, the secondary path S ′ m ( z ) is re-estimated by the online modeling technique and W ′ ( z ) converges as follows

W ′ AF ( z ) = − P m ( z ) S ′ m ( z ) R ( z ) + H ( z )

(7)

S ′ m ( z )

By substituting (4) into (7),

W ′ AF ( z ) P m ( z ) S ′ m ( z ) R ( z ) + 1 S ′ m ( z )

− P m ( z )

R ( z ) − S m ( z ) W o ( z )

S m ( z ) S ′ m ( z ) W o ( z ) , (8)

which does not match the optimal noise control filter W o ( z ) at the virtual microphone. This is due to the fact that H ( z ) has no information about S ′ m ( z ). Therefore, when the secondary path varies, not only the secondary path model ˆ S m ( z ) but also the auxiliary filter H ( z ) must be re-estimated online. Therefore, the proposed system in Figure 2 is configured to re-estimate H ( z ) online as well by using the information on W o ( z ) obtained in the tuning stage. Here, when S m ( z ) varies, modeling of S m ( z ) is performed first, and modeling of H ( z ) is started after its convergence. Note that the noise control filter W ( z ) is not updated during the modeling of H ( z ). When H ( z ) has converged, the updating of W ( z ) is restarted.

e v ( n )

x ( n ) P v ( z ) d v ( n )

+

Σ

+

yʹ v ( n )

S v ( z )

d m ( n ) e m ( n )

+

P m ( z )

Σ

+

R ( z )

yʹ m ( n )

+

y ( n )

r ( n )

W ( z )

S m ( z )

Σ

+

+ −

v ( n ) vʹ ( n ) ^

S m ( z ) ^

rʹ m ( n )

eʹ v ( n ) Σ

^

S m ( z ) NLMS

g ( n )

NLMS

y o ( n )

e m ( n ) ^ Σ +

+

Wo ( z )

R andom Noise Generator

+

f ( n ) ^

+

+

Σ H ( z ) ^

Σ

+

f ( n )

e h ( n ) ^

H ( z )

NLMS

Figure 2: Block diagram of ANC system using AFVS with online modeling technique.

3. SIMULATION RESULT

To demonstrate the e ff ectiveness of the proposed method, the noise reduction is calculated through computer simulations for a single channel feedforward ANC. Table 1 shows the simulation conditions. Figure 3 also shows the measurement arrangement for measuring each transfer path used in the simulation. At the tuning stage and the beginning of the control stage, the secondary path S m ( z ) at the position of the error microphone indicated by the solid line in Fig. 3 is used, and after 250000 iterations, the secondary path S m ( z ) is varied by moving the error microphone to the dashed line position in Fig. 3. In AFVS, which does not have online modeling capabilities, the system diverges under the conditions of Figure 3 due to modeling errors caused by secondary path variations. Figure 4 shows the time waveform of the error signal observed at the desired location and the frequency characteristic. It can be seen that the proposed method maintains a noise reduction of about 8 dB even when the secondary path varies. Figure 5 shows the relative modeling error of the secondary

path, defined as

[ || S m ( z ) − ˆ S m ( z ) || 2

∆ S m ( n ) = 10 log 10

(9)

|| S m ( z ) || 2

From Fig. 5, it can be seen that the proposed method can follow changes in the secondary path. However, compared to the normal AFVS, there is a problem that the noise reduction performance at the desired location is degraded due to the white noise generated internally to identify the secondary path online. Figure 6 shows the time waveform of the error signal observed at the desired location and frequency characteristic of the nomal AFVS. From Fig. 6(a), the normal AFVS diverges after secondary path variations due to modeling errors. Comparing Figs. 4 and 6, it can be seen that the noise reduction performance of the proposed method is degraded due to introduces online modeling technique.

Table 1: Simulation conditions ． Noise white noise (1-2000 Hz)

Signal-Noise ratio 12

Iteration 500000

Tap length of W and H 200

Tap length of S and P 200

Tap length of ˆ S m 200

Update algorithm of W and H and ˆ S m NLMS

Step size parameter of W and H (tuning) 0.01

Step size parameter of W and ˆ S m (control) 0.01

Step size parameter of H (control) 0.1

Regularization parameter β 1 . 0 × 10 − 6

Sampling frequency 8000 Hz

S v ( z )

P m ( z )

Error microphone Noise source

Reference microphone

Virtual microphone

Secondary source

W ( z ) x ( n ) r ( n ) y ( n ) e m ( n ) e v ( n )

S m ( z )

R ( z )

10 cm

P v ( z )

10 cm 20 cm

10 cm 100 cm

Figure 3: Mesurement arrangement.

0.15 ANC off on ANC on tion a0?

(a) Time waveform of error signal at the desired position before and after ANC.

(b) Frequency characteristic at desired position.

Figure 4: Simulation results of AFVS with online modeling technique.

Power Spectrum [4B] ANC off, ANC on —ANC on(Sm(z) varied) 500 1000 1500-2000 Frequency [Hz]

Figure 5: The relative modeling error of the secondary path.

10° Iteration

03 0.2 ANC of ANC on 05 1 1.5 Iteration 2

(a) Time waveform of error signal at the desired position before and after ANC.

(b) Frequency characteristic at desired position before the system diverges.

Figure 6: Simulation results of the AFVS without online modeling technique.

Power Spectrum [dB] —ANC of -50 —ANC on -60 -70 -80 -90 -100 0 500 1000 1500-2000 Frequency [Hz]

4. CONCLUSIONS

In this paper, we proposed an AFVS that uses online modeling techniques to deal with secondary path variations. The proposed method is robust against secondary path variations by online re-estimation of both the secondary path model and auxiliary filter. However, the noise reduction performance is degraded by internally generated white noise, so the introduction of a power scheduling technique is a future challenge.

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