Robust estimation of open aperture active control systems using virtual sensing

Chung Kwan Lai School of Electrical and Electronic Engineering, Nanyang Technological University 50 Nanyang Ave, 639798

Bhan Lam 1

School of Electrical and Electronic Engineering, Nanyang Technological University 50 Nanyang Ave, 639798

Jing Sheng Tey School of Electrical and Electronic Engineering, Nanyang Technological University 50 Nanyang Ave, 639798

Woon Seng Gan School of Electrical and Electronic Engineering, Nanyang Technological University 50 Nanyang Ave, 639798

ABSTRACT For active noise control systems in an open aperture application, virtual sensing system is often needed to overcome design constraint in terms of microphone placement. The virtual sensing system, however, make assumptions to the acoustical field and thus is highly sensitive to any changes in the primary field. Taking the window aperture ANC application for example, incoming disturbance signals could impinge from multiple locations, altering the spatial correlation between the physical and virtual microphones. For instance, in the context of a high-rise apartment window, aircraft noise would propagate downwards from the sky, whereas tra ffi c noise propagate upward from the ground. This paper considers the estimation performance of the remote microphone technique, an example of a virtual sensing method, in an open aperture application when faced with changes in the primary noise source. Additionally, this paper introduces a new method for estimating the spectral density due to multiple sources through the incoherent decomposition method. It has been shown experimentally that this algorithm is able to improve the nearfield estimation of the system.

1. INTRODUCTION

The use of virtual sensing technique in an active sound control application has become increasingly popular to tackle its physical limitation in positioning its error sensors at its desired location [1]. When

1 bhanlam@ntu.edu.sg

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

applied to the open aperture active sound control application [2–5], virtual sensing involves estimating the sound pressure inside a room using pressure signals obtained from an array of static monitoring microphones installed at the edge of the window frame. This process can be achieved through the remote microphone technique, described in Section 2, via a fixed observation filter illustrated in Figure 1. This technique, however, made assumptions about the acoustic environment that will only result a good estimation if the acoustic field remain unchanged [6]. In the open aperture application, these noise sources are non-stationary and could arise from unknown directions. For example, it could be an aircraft noise that arises from the top; tra ffi c noise that propagates from the bottom; or a combination of both in which for either case, degrades the estimation accuracy of the virtual sensing system. This paper, while solely focused on the virtual sensing system, investigates the performance of the remote microphone technique when subject to uncertainties to di ff erent noise sources, followed by proposing a new calibration method, as described in Section 4, in obtaining the observation filter in which improve its robustness.

2. VIRTUAL SENSING SYSTEM USING REMOTE MICROPHONE TECHNIQUE

Figure 2 shows the block diagram arrangement of a virtual sensing system using the remote microphone technique [1, 7]. The observation filter, O , can be designed either as a filter in the frequency domain or a causal impulse response in time domain to minimise the expected mean square error between the estimated disturbance signal at the virtual microphone, ˆ d e , and the actual disturbance signal, d e . Thus, the optimal matrix of filter in the frequency and time domain is expressed as [8, 9]

O opt ( j ω ) = S d m d e S d m d m + β I
− 1 = P e S vv P H m  P m S vv P H m + β I  − 1 (1)

and

O opt [ n 0 ] = R T d m d e [ n 0 ] R d m d m [0] + β I ′
− 1 (2)

respectively, where E [ · ] is the expectation operator and β is a regularisation parameter, S d m d e , S d m d m and S vv are spectral density matrices and R d m d e , R d m d m are the correlation matrices, each defined with a general notation of S xy ( ω ) = E h y ( ω ) x ( ω ) H i and R xy [ n 0 ] = E h x [ n ] y [ n − n 0 ] T i respectively. In this paper, it is assumed that no delay parameter is used, i.e. n 0 = 0, as the causality in the open aperture application is generally ensured by assuming that all noise source only propagate from one side of the aperture. The overall estimation error of the remote microphone technique will be used to evaluate

Figure 1: The schematic of the adaptive feedforward control in an open aperture active control system, using virtual sensing based on the remote microphone technique.

the accuracy of the virtual sensing system, given by

trace { S ϵϵ } trace  S d e d e

(3)

L ϵ = 10 log 10

where ϵ = d e − ˆ d e , assuming stationary noise source throughout the evaluation period. The regularisation parameter β from Equation 1 and Equation 2 improves robust stability and performance of the active control system when subject to uncertainties in the acoustic environment [7, 10], though it is assumed to be 0 here as feedforward adaptive control filter was not implemented in this study.

de In Pm Ge Monitoring Virtual Error microphones microphones w Gm Gn

Figure 2: Block diagram of the virtual sensing control algorithm using remote microphone technique [7].

3. NEARFIELD ESTIMATION USING ARRAYS OF MONITORING MICROPHONES

To estimate the e ff ect of the virtual sensing in an open aperture when subject to changing noise source, three monitoring microphones were placed at the aperture frame with dimensions 0 . 6m × 0 . 2m and five virtual error microphones were placed inside the padded box that represents an anechoic room, shown in Figure 3a. Four primary loudspeakers, as shown in Figure 3b, were driven with uncorrelated white noise and were placed outside the open aperture at di ff erent position detailed in Figure 4, representing the primary source from di ff erent directions, such as aircraft and tra ffi c noise. The observation filter matrix O with FIR filter length of J = 400 was calibrated from a predefined set of noise groups, defined as a combination of primary loudspeakers that play simultaneously. Once the calibration stage is finished, the same observation filter is then used to evaluate the accuracy of the nearfield estimation when a di ff erent set of noise group is played instead.

(a)

(b)

Figure 3: Arrangement of (a): monitoring and error microphones at a sampling frequency of 10kHz inside the box and (b): primary loudspeaker 1–4 outside the box.

Figure 4: Diagram of the physical arrangement made in the room.

To clearly see the e ff ect of di ff erent noise sources impacting the nearfield estimation, frequency response is measured. Figure 5 shows the frequency responses from di ff erent primary loudspeakers to one of the error microphones. It can be seen that the changes in both magnitude and phase of frequency response when di ff erent loudspeakers were used are large for the entire frequency spectrum. If the observation filter is designed from using only one of the loudspeakers, the nearfield estimation accuracy of the virtual sensing system will be a ff ected when the noise sources were changed.

0

10 log 10 | P e | (dB)

− 10

− 20

0 100 200 300 400 500 600 700 800 900 1 , 000

Frequency (Hz)

4

2

∠ P e (rad)

0

− 2

0 100 200 300 400 500 600 700 800 900 1 , 000 − 4

Frequency (Hz)

Loudspeaker 1 Loudspeaker 2 Loudspeaker 3 Loudspeaker 4

Figure 5: Magnitude and phase in radians of the frequency response from di ff erent primary loudspeakers to error microphone 3 illustrated in Figure 4.

Figure 6 shows the overall estimation error spectra when all four loudspeakers are played simultaneously, with one using the optimal observation filter calibrated from the same scenario and the mismatched observation filters calibrated when only one loudspeaker is present. When the

optimal observation filter is used, the estimation error showed a good prediction at around -18dB at the frequency around 300Hz and an average of -10dB at frequency below 600Hz. As expected, the estimation performance degrades as frequency increases due to the reduced error zone under the influence of di ff use noise field [7], causing the estimation error to increase up to -2dB at a frequency above 600Hz.

O 1 , 2 , 3 , 4 opt ˆ O 1

Estimation Error (dB)

10

ˆ O 2

0

ˆ O 3

ˆ O 4

− 10

200 300 400 500 600 700 800 900 1 , 000 − 20

Frequency (Hz)

Figure 6: The estimation error spectra when all four primary sources are present, using the optimal observation filter O 1 , 2 , 3 , 4 opt and the mismatched observation filter ˆ O n v obtained when only the n v -th primary source is present.

When a di ff erent observation filter is chosen, however, a positive estimation error is shown for most of its frequency range. Depending on which mismatched observation filter is use, the estimation error increases by a range between 10dB to 20dB as compared to the optimal observation filter at frequency below 600Hz. It is clear that the nearfield estimation of the virtual sensing system is sensitive under the presence of additional source.

Figure 7 shows the estimation error spectra when only loudspeaker 1 and 2 were played simultaneously, with one using the optimal observation filter calibrated from the same scenario and the other using the observation filter calibrated when all loudspeakers were present. While the di ff erence is not large at low frequency around 200Hz to 300Hz, the di ff erence becomes more apparent at higher frequency. The observation filter when only 2 loudspeakers were used allows a better prediction at higher frequencies in contrast to 4 loudspeakers due to the larger error zone [7].

O 1 , 2 opt , Loudspeaker 1,2 ˆ O 1 , 2 , 3 , 4 , Loudspeaker 1,2

Estimation Error (dB)

0

− 10

− 20

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1 , 000

Frequency (Hz)

Figure 7: The estimation error spectra when Loudspeaker 3 and 4 were absent, using the optimal observation filter O 1 , 2 opt and the mismatched observation filter ˆ O 1 , 2 , 3 , 4 calibrated when all four loudspeakers are present.

When the same observation filter, calibrated when all four loudspeakers are present, is used under di ff erent loudspeaker combination shown in Figure 8, the di ff erence in estimation error is small as compared to Figure 7. There is, in fact, a slight improvement in estimation when only 2 loudspeakers were used at the frequency region between 650Hz to 800Hz. While using the observation filter calibrated with all possible noise source is one way to ensure minimal degradation in nearfield estimation subject to changes in noise source, it is still underperformed, compared to the use of optimal filter seen in Figure 7.

Estimation Error (dB)

0

− 10

O 1 , 2 , 3 , 4 opt , Loudspeaker 1,2,3,4 ˆ O 1 , 2 , 3 , 4 , Loudspeaker 1,2

− 20

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1 , 000

Frequency (Hz)

Figure 8: The estimation error spectra when the observation filter calibrated under four loudspeakers is used against di ff erent noise combination.

Figure 9 shows the estimation error spectra when the same obesrvation filter calibrated form loudspeaker 1, ˆ O 1 , is used for di ff erent loudspeaker scenario. This reflects to the actual noise source such as aircraft noise that is often modelled as a moving source. While the observation filter managed to have an optimal estimation when loudspeaker 1 is played, as expected, the nearfield estimation degrades when loudspeaker 3 is played by an average of 10dB at low frequencies and 20dB at high frequencies respectively. This concludes the study for the remote microphone technique when subject to change in primary field, stressing the importance in choosing the correct observation filter that reflects to the current scenario.

10

Estimation Error (dB)

0

O 1 opt , Loudspeaker 1 ˆ O 1 , Loudspeaker 3

− 10

− 20

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1 , 000 − 30

Frequency (Hz)

Figure 9: The estimation error spectra when di ff erent loudspeakers were used, using the observation filter calibrated when only loudspeaker 1 is present.

4. SUPERPOSITION IN SPECTRAL DENSITY MATRIX TO COMBINE THE EFFECT OF MULTIPLE NOISE SOURCE

In designing robust control systems in practice a number of primary sources from di ff erent location would need to be considered. In the open aperture ANC application, there might be multiple sources that will propagate through the window such as aircraft noise that arises from the top, and tra ffi c noise from the bottom. Theoretically, all the combination of sources from di ff erent locations would need to be simulated or measured in order to provide an ensemble of possible observation filter to ensure optimal nearfield estimation.

By assuming that the primary source are incoherent, S vv from Equation 1 becomes a diagonal matrix expressed as





| v 1 | 2 0 · · · 0

0 | v 2 | 2 · · · 0 ... ... ... ...

S vv =

(4)

0 0 · · · | v N v | 2

where the diagonal element variable | v n v | 2 is the primary source strength for the n v -th primary source. Thus, the spectral density matrix S d m d e and S d m d m , obtained when all primary sources are present, can be mathematically decomposed into a series of spectral density matrix when only one source is present, expressed as

S d m d e = S 1 d m d e + S 2 d m d e + · · · + S N v d m d e , (5)

S d m d m = S 1 d m d m + S 2 d m d m + · · · + S N v d m d m , (6)

where S n v xy denotes the spectral density matrix between x and y when only n v -th primary source is present. In the scenario where there is a change in source strength in the primary source after the calibration stage, assuming that the new source strength for each primary source is known, the spectral density matrix can be reconstructed to reflect the change in source strength without any further recalibration, that is

S d m d e = r 2 1 S 1 d m d e + r 2 2 S 2 d m d e + · · · + r 2 N v S N v d m d e (7)

S d m d m = r 2 1 S 1 d m d m + r 2 2 S 2 d m d m + · · · + r 2 N v S N v d m d m (8)

with r n v as the scalar source strength ratio for the n v -th primary source, defined as the ratio of the current source strength to the initial source strength obtained from calibration. Therefore, the optimal observation filter from Equation 1 can be calibrated based on the spectral density obtained from individual primary source, giving

− 1









 + β I



N v X

N v X

n v = 1 r 2 n v S n v d m d e

n v = 1 r 2 n v S n v d m d m

O opt ( j ω ) =

. (9)

As the correlation matrix in the time domain is directly related to the spectral density matrix in the frequency domain, the correlation matrix term R d m d e , R d m d m from Equation 2 can be realised from Equation 7 and Equation 8 by performing inverse fourier transform:

R d m d e = r 2 1 R 1 d m d e + r 2 2 R 2 d m d e + · · · + r 2 N v R N v d m d e , (10)

R d m d m = r 2 1 R 1 d m d m + r 2 2 R 2 d m d m + · · · + r 2 N v R N v d m d m , (11)

assuming that the source strength ratio r n v is constant throughout frequency. This finally reflects to the observation filter in the causal time domain from Equation 2 to be

− 1

T  







 + β I ′ 

N v X

N v X

n v = 1 r 2 n v R n v d m d e [ n 0 ]

n v = 1 r 2 n v R n v d m d m [0]

O opt [ n 0 ] =

. (12)

To observe on how the correlation plot changes under di ff erent noise source configuration, Figure 10 and 11 show the cross-correlation and co-correlation plot respectively under di ff erent noise source configuration. The cross-correlation plot for loudspeaker 4 from Figure 10 received an earlier impulse at around 1ms, compared to loudspeaker 1, 2 and 3 that recieve its initial impulse at around 2ms. This can be explained from the location of the loudspeaker 4 that allow the wavefront to pass through the two microphone quicker than the rest of the loudspeaker. In the combined loudspeaker scenario, it can be seen that it has the combination of impulse of τ = 1ms and τ = 2ms mainly dominated from loudspeaker 3 and 4.

Error Microphone 3 - Monitoring Microphone 1

· 10 − 3

R 1 d e d m R 2 d e d m R 3 d e d m R 4 d e d m R 1 , 2 , 3 , 4 d e d m

4

2

R d e d m

0

− 2

0 2 4 6 8 10 12 14

τ (ms)

Figure 10: The cross-correlation plot between the error and monitoring microphone for each n v - th loudspeaker, R n v em , followed by the cross-correlation plot when all noise groups were played simultaneously.

Monitoring Microphone 1 - Monitoring Microphone 1

8 · 10 − 2

R 1 d m d m R 2 d m d m R 3 d m d m R 4 d m d m R 1 , 2 , 3 , 4 d m d m

6

R d m d m

4

2

0

0 2 4 6 8 10 12 14 16 18 20

τ (ms)

Figure 11: The co-correlation plot of monitoring microphones for each n v -th loudspeaker, R n v mm , followed by the co-correlation plot when all noise groups were played simultaneously.

While the co-correlation plot from Figure 11, as expected, shows the initial impulse at 0ms, the amplitude at τ = 0ms is di ff erent for di ff erent noise source configuration, which is due to the unequal source strength for di ff erent loudspeakers. In a combined loudspeaker scenario, the initial impulse amplitude is much larger than the individual loudspeaker scenario. The remaining impulse

for τ > 0ms in Figure 11 caused by sound reflection is small due to the padded box e ff ect. Overall, it can be seen that the cross-correlation plot changes abruptly with di ff erent loudspeaker configuration.

The correlation matrix obtained from each set of noise groups were added numerically through Equation 10–11 with scalar source strength ratio for each primary source set to r = 1, followed by comparing to the correlation matrix when all noise groups were played simultaneously. Figure 12 and 13 show the comparison between the estimated and actual correlation plot when all four loudspeakers are played simultaneously. Both graphs are similar, showing a good approximation when the correlation matrix under all loudspeakers is estimated through superposition of correlation matrix obtained from each individual loudspeakers.

Error Microphone 3 - Monitoring Microphone 1

4 · 10 − 3

R d e d m

ˆ R d e d m

2

R d e d m

0

− 2

0 2 4 6 8 10 12 14

τ (ms)

Figure 12: The comparison between the estimated and actual cross-correlation plot between error and monitoring microphone, labelled as ˆ R d e d m and R d e d m respectively.

Monitoring Microphone 1 - Monitoring Microphone 1

8 · 10 − 2

R d m d m

ˆ R d m d m

6

R d m d m

4

2

0

0 2 4 6 8 10 12 14 16 18 20

τ (ms)

Figure 13: The comparison between the estimated and actual co-correlation plot on a monitoring microphone, labelled as ˆ R d m d m and R d m d m respectively.

To apply the superposition theory into the remote microphone technique, the estimated correlation matrix is then used to calculate the observation filter through Equation 12, which is then used to compare against the actual observation filter calibrated from measurements. Figure 14 shows the estimation error spectra when all four loudspeaker is present by comparing the actual observation filter, calibrated from all four loudspeakers, and the estimated observation filter, calibrated from individual loudspeakers and derived through Equation 12. While the estimated observation filter has a slight 3dB increase in estimation error at frequencies between 200Hz and 400Hz, the prediction is similar at frequencies above 600Hz. As the virtual sensing technique is performed in real-time, it is expected to have a slight di ff erence in the estimation error spectra per acquisition. It is thus

concluded that the use of superposition in incoherent sources is suitable to improve the virtual sensing performance towards optimum.

0

O opt

Estimation Error (dB)

ˆ O

− 5

− 10

− 15

200 300 400 500 600 700 800 900 1 , 000 − 20

Frequency (Hz)

Figure 14: The estimation error spectra when all four primary sources are present, using the optimal observation filter O opt and the predicted observation filter ˆ O from Equation 12.

5. CONCLUSIONS

A virtual sensing arrangement can be implemented into the open aperture feedforward active control system, to provide active control of sound at the desired location. Such an arrangement can overcome design constraints in which its desired location is awkward to implement, but its estimation performance is sensitive to variations in the primary noise source, as demonstrated experimentally in this paper.

The robust design process in the virtual sensing system thus requires an estimate of all combination of possible primary sources at a list of possible locations, which would be very time-consuming to acquire if there are multiple primary sources. Considering the open aperture application where di ff erent noise sources such as aircraft or tra ffi c noise are incoherent, it is possible to only acquire the correlation matrix from individual primary source at a list of possible direction, and then to combine these individual correlation matrices numerically to estimate the correlation matrix due to di ff erent combination of noise source. It has been shown in real-time experiments that the nearfield estimation in virtual sensing can be significantly improved through superposition of correlation matrices.

ACKNOWLEDGEMENTS

This research is supported by the Singapore Ministry of National Development and the National Research Foundation, Prime Ministers O ffi ce under the Cities of Tomorrow (CoT) Research Programme (CoT Award No. COT-V4-2019-1). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of the Singapore Ministry of National Development and National Research Foundation, Prime Ministers O ffi ce, Singapore.

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