PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Virtual sensing technique for a multi-reference and multi-error active
noise control system
Rong XIE; Chuang SHI; Huiyong LI
University of Electronic Science and Technology of China, Chengdu, China
Corresponding email: shichuang@uestc.edu.cn

ABSTRACTThis paper investigates the virtual sensing technique for a multi-reference and multi-error active noise control(ANC) system. Our previous study has validated the effectiveness of virtual sensing technique for themulti-channel feedforward ANC system with only one reference signal. In this paper, two reference signals
are implemented. The numbers of control filters and auxiliary filters are doubled accordingly. A comparison
is carried out between the condition when 4 physical error microphones are used to model 8 virtual error
microphones and the condition when 8 physical error microphones are used to model 4 virtual error
microphones. It is demonstrated that when the number of physical error microphones are not less than thenumber of secondary loudspeakers, the virtual sensing technique remains effective.
Key word: Feedforward active noise control; Multi-channel active noise control; Virtual sensing

1. INTRODUCTION
Generally, there are two basic ways to abate noise. They are passive noise control (PNC) and active
noise control (ANC) (1). PNC can reduce high-frequency noise efficiently. However, the efficiency ofPNC decreases when the noise frequency is low, due to the relatively high cost and complexity indeployment. ANC utilizes loudspeakers to generate an anti-noise wave, which has the same amplitude
and opposite phase of the noise wave. By doing so, ANC is ideally suited to reduce low-frequency
noise, complementing PNC (2).
The use of microphones in ANC systems classifies them into two categories. The ANC systems
using microphones to collect both reference and error signals are called feedforward ANC systems. Incontrast, the ANC system using microphones to provide merely the error signals is called a feedback
ANC system. In the feedback ANC system, reference signals are thought to be calculated based onerror signals and internal models (3). Feedforward ANC systems are better at reducing broadband
noise and easier to be stabilized than feedback ANC systems (4).
ANC systems are also categorized by the number of loudspeakers, namely secondary loudspeakers.
When one secondary loudspeaker is deployed, often together with one reference microphone and one
error microphone at the same time, the ANC system is called a single-channel ANC system. The active
noise control headphone is the most successful application of the single-channel ANC system,
whereby one ANC system is in charge of each individual ear drum. When the targeted zone of quiet(ZoQ) is relatively large as compared with the noise wavelength, multi -channel ANC systems arenecessary, which may include multiple secondary loudspeakers as well as multiple microphones. Thus,
this paper will be focusing on the multi-channel feedforward ANC system.
The filtered-x least mean squares (FxLMS) is the most widely used adaptive algorithm in ANC
systems (5). The multi-channel version of FxLMS algorithm is also called the multiple error algorithm(6). The goal of FxLMS algorithm is to minimize 2-norm of error signal vector, which includes instantsamples from all the error microphones. This is to ensure that ZoQ is formed around error microphones.
When error microphones cannot be placed permanently in the targeted ZoQ, i.e. the targeted ZoQ is far
away from error microphones, remote sensing or virtual sensing technique becomes necessary (7, 8).
The virtual sensing technique was originally proposed for a single-channel feedback ANC systemand subsequently developed for a single-channel feedforward ANC system (9, 10). In 2018, it wasextended to run with a multi-channel feedforward ANC system (11, 12). However, the validation wascarried out with just one reference signal. In this paper, multiple reference microphones are considered6891

to further understand the performance of virtual sensing technique for feedforward ANC systems.

2. VIRTUAL SENSING TECHNIQUE
There are two stages when virtual sensing technique is applied. The first stage is called the training
stage. In this stage, temporal microphones, which are also called virtual error microphones, can beplaced in the targeted ZoQ in order for adaptive control filters to converge. Meanwhile, auxiliary
filters are trained to estimate error signals of microphones that are placed far from the ZoQ. Those
error microphones are called physical error microphones.
The second stage of virtual sensing technique is called the control stage. Without any microphones
placed in the targeted ZoQ, adaptive control filters converge again by minimizing the difference
between error signals and estimates of error signals calculated with reference signals and pre -trained
auxiliary filters. Compared with fix-coefficient control filters, the virtual sensing technique is likely toretain some adaptivity to the changes in the operating environment.
x(n|i)

pv(i,k)wv(n|i,j)ŝv(j,k)rv(n|i,j,k)LMSyv(n|j)sv(j,k) 6

ev(n|k)Figure 1 – Block diagram of multi-channel FxLMS algorithm
Figure 1 shows the block diagram of multi-channel FxLMS algorithm. In general, a feedforward
multi-channel ANC system consists of ‫ ܫ‬reference microphones, ‫ ܬ‬secondary loudspeakers, and ‫ܭ‬error microphones. There are ‫ ܫ‬ൈ ‫ ܭ‬primary paths, ‫ ܫ‬ൈ ‫ ܬ‬control filters, and ‫ ܬ‬ൈ ‫ ܭ‬secondary paths
and secondary path models. ࢖࢜ ሺ݅ǡ ݇ሻ, ‫ ࢜ܛ‬ሺ݆ǡ ݇ሻ, and ࢙ො࢜ ሺ݆ǡ ݇ሻ denote the primary path, secondary path,and secondary path model, respectively. ݅, ݆, and ݇ indicate indices of the reference microphone,secondary loudspeaker, and error microphone, respectively.
At a discrete time n, the control signal is calculated as a dot product between the reference signal
vector and the control filter coefficient vector, i.e.
(1)‫ݕ‬௩ ሺ݊ȁ݆ሻ ൌ σூ௜ୀଵ ࢞ሺ݊ȁ݅ሻ ή ࢝࢜ ሺ݊ȁ݅ǡ ݆ሻ,
where ή is the dot product operator; ‫ݕ‬௩ ሺ݊ȁ݆ሻ is the output of the ݆-th secondary loudspeaker; ࢞ሺ݊ȁ݅ሻ ൌሾ‫ݔ‬ሺ݊ȁ݅ሻǡ ‫ݔ‬ሺ݊ െ ͳȁ݅ሻǡ ǥ ǡ ‫ݔ‬ሺ݊ െ ‫ܮ‬௪ ൅ ͳȁ݅ሻሿ is the reference signal vector of the ݅ -th reference
microphone; and ࢝࢜ ሺ݊ȁ݅ǡ ݆ሻ is the control filter coefficient vector for the ݆-th secondary loudspeaker
and the ݅-th reference microphone. Moreover, ‫ܮ‬௪ is the tap length of the control filter.
The so-called filtered-x signal is calculated as a dot product between the reference signal vector andthe secondary path model, i.e.
(2)‫ݎ‬௩ ሺ݊ȁ݅ǡ ݆ǡ ݇ሻ ൌ σ௅௦௟ୀଵ ‫ݏ‬Ƹ௩ ሺ݈ȁ݆ǡ ݇ሻ‫ݔ‬ሺ݊ െ ݈ ൅ ͳȁ݅ሻ,where ‫ݏ‬Ƹ௩ ሺ݆ǡ ݇ሻ is the secondary path model for the ݆-th secondary loudspeaker and the ݇-th error
microphone. ‫ܮ‬௦ is the tap length of the secondary path model. The filtered-x signal vector is therefore
given by ࢘࢜ ሺ݊ȁ݅ǡ ݆ǡ ݇ሻ ൌ ሾ‫ݎ‬௩ ሺ݊ȁ݅ǡ ݆ǡ ݇ሻǡ ‫ݎ‬௩ ሺ݊ െ ͳȁ݅ǡ ݆ǡ ݇ሻǡ ǥ ǡ ‫ݎ‬௩ ሺ݊ െ ‫ܮ‬௪ ൅ ͳȁ݅ǡ ݆ǡ ݇ሻሿ.
According to the FxLMS algorithm, the control filter coefficient vector at the discrete time ݊ ൅ ͳ
is updated by(3)࢝࢜ ሺ݊ ൅ ͳȁ݅ǡ ݆ሻ ൌ ࢝࢜ ሺ݊ȁ݅ǡ ݆ሻ െ ߤ σ௄௞ୀଵ ݁௩ ሺ݊ȁ݇ሻ ࢘࢜ ሺ݊ȁ݅ǡ ݆ǡ ݇ሻ,
where ߤ is the step size and ݁௩ ሺ݊ȁ݇ሻ is the error signal of the ݇-th error microphone.
In the training stage of virtual sensing technique, ࢖࢜ ሺ݅ǡ ݇ሻ ‫ ࢜ܛ‬ሺ݆ǡ ݇ሻ, and ࢙ො࢜ ሺ݆ǡ ݇ሻ are defined withrespect to the virtual error microphones. Therefore, they are referred to as the virtual primary path,virtual secondary path, and virtual secondary path model in the latter part of this paper.
Figure 2 highlights the block diagram of the training stage for physical error microphones. ࢖ሺ݅ǡ ݉ሻෝ ࢜ ሺ݅ǡ ݆ሻand ࢙ሺ݆ǡ ݇ሻ are the primary and secondary paths with respect to physical error microphones. ‫ܟ‬is the fixed-coefficient control filter. The adaptive filter ࢎሺ݊ȁ݅ǡ ݉ሻ is called the auxiliary filter. The tap
length of ࢎሺ݊ȁ݅ǡ ݉ሻ is denoted as ‫ܮ‬௛ .

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p(i,m)x(n|i)

ŵv(i,j) 6s(j,m)h(n|i,m)yh(n|m) 6 e(n|m)eh(n|m)LMS

Figure 2 – Training stage of virtual sensing technique for a multi-channel feedforward ANC system
Error signals of physical error microphones are estimated by auxiliary filters based on referencesignals. The output of an auxiliary filter is calculated as(4)‫ݕ‬௛ ሺ݊ȁ݉ሻ ൌ σூ௜ୀଵ ࢞ሺ݊ȁ݅ሻ ή ࢎሺ݊ȁ݅ǡ ݉ሻ,where ࢎሺ݊ȁ݅ǡ ݉ሻ is the auxiliary filter coefficient vector for the ݅-th reference microphone and the݉-th physical error microphone.
According to the LMS algorithm, the auxiliary filter coefficient vector at the discrete time ݊ ൅ ͳ isupdated byࢎሺ݊ ൅ ͳȁ݅ǡ ݉ሻ ൌ ࢎሺ݊ȁ݅ǡ ݉ሻ ൅ ߤ݁௛ ሺ݊ȁ݉ሻ࢞ሺ݊ȁ݅ሻ,(5)where ݁௛ ሺ݊ȁ݉ሻ ൌ ‫ݕ‬௛ ሺ݊ȁ݉ሻ ൅ ݁ሺ݊ȁ݉ሻ; and ݁ሺ݊ȁ݉ሻ is the error signal of the ݉ -th physical error෡ and usedmicrophone. When the training stage is complete, the trained auxiliary filter is denoted as ࢎ
as fixed-coefficient filters in the control stage (see Figure 3).
p(i,m)

x(n|i)

w(n|i,j)ŝ(j,m)

r(n|i,j,k)LMSy(n|j)

ed(n|m)s(j,m)6
6e(n|m)ĥ(i,m)

Figure 3 – Control stage of virtual sensing technique for a multi-channel feedforward ANC system
In Figure 3, the desired error signal ݁ௗ ሺ݊ȁ݉ሻ to update the control filter is constructed by෡ ሺ݅ǡ ݉ሻ.
(6)݁ௗ ሺ݊ȁ݉ሻ ൌ ݁ሺ݊ȁ݉ሻ ൅ σூ௜ୀଵ ࢞ሺ݊ȁ݅ሻ ή ࢎAnd the filtered-x signal vector ࢘ሺ݊ȁ݅ǡ ݆ǡ ݉ሻ is calculated by,(7)࢘ሺ݊ȁ݅ǡ ݆ǡ ݉ሻ ൌ σ௅௦௟ୀଵ ‫ݏ‬Ƹ ሺ݈ȁ݆ǡ ݉ሻ‫ݔ‬ሺ݊ െ ݈ ൅ ͳȁ݅ሻ.
With the desired error signal, the control filter coefficient vector at the discrete time ݊ ൅ ͳ is updatedby(8)࢝ሺ݊ ൅ ͳȁ݅ǡ ݆ሻ ൌ ࢝ሺ݊ȁ݅ǡ ݆ሻ െ ߤ σ௄௞ୀଵ ݁ௗ ሺ݊ȁ݉ሻ ࢘ሺ݊ȁ݅ǡ ݆ǡ ݉ሻ.

3. EXPERIMENTAL VALIDATIONThe primary and secondary paths are provided by a multiple-channel ANC experiment system
shown in Figure 4. Two primary noise loudspeakers are placed in a cube. The two primary broadband
noises are uncorrelated and with frequency bands from 300Hz to 1300Hz and from 500Hz to 1100Hz,
respectively. The secondary loudspeakers are fixed on the opening of the cube. Distance between two
neighboring secondary loudspeakers is 12.5 cm. Three layers of microphones are placed outside thecube. They can be flexibly set up as either physical or virtual error microphones. Each layer consists offour microphones. The distance between two neighboring microphones is 12.5 cm too. The sampling

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frequency is set to 48kHz. Tap lengths of control filter ࢝ሺ݅ǡ ݆ሻ and auxiliary filter ࢎሺ݅ǡ ݉ሻ are set to800 taps. The secondary models have a length of 200 taps. The duration of each adaptation is limited to30 seconds.
ReferenceMicrophones

Secondary Loudspeakers

Error Microphones
Primary Noise
Loudspeakers

8cm
13cm
18cm
1st Layer 2nd Layer 3rd Layer

Figure 4 – Multiple-channel ANC experiment system

Figure 5 – Comparison of noise reduction performance when there are 4 to 8 physical error
microphones and 4 virtual error microphones.
Firstly, microphones in one layer is selected as virtual error microphones. Microphones in the othertwo layers are wholly or separately used as physical error microphones. The baseline performance is6894

provided by the FxLMS algorithm when error microphones are placed in the targeted ZoQ. The noise
reductions of virtual sensing technique are compared among different selections of physical error
microphones.
Figure 5(a) demonstrates the case when microphones in the first layer are set up as virtual error
microphones. The noise reduction is improved when more physical error microphones are used, i.e.
microphones in both the second and third layers. The same trend is also observed in Figures 5(b) and5(c), when microphones in the second and the third layers are configured as virtual error microphones,respectively.
In general, Figure 5 confirms the effectiveness of virtual sensing technique when two reference
signals are involved and the positions of physical error microphones have a significant influence onthe final noise reduction performance. Using more physical error microphones in the virtual sensing
technique can achieve noise reductions closer to the baseline level with the trade-off of increased
computational cost. Furthermore, stopping the adaption earlier may result in the observation thatvirtual sensing technique is more effective in noise reduction than the FxLMS algorithm.
Figure 6 evaluates the cases when there are 8 virtual error microphones. As there are only 4
secondary loudspeakers in the experiment setup, the noise reduction is expected ly lower. The virtualsensing technique is proved to result in similar levels of noise reduction as compared to the standard
FxLMS algorithm with error microphones placed in the ZoQ.

Figure 6 – Comparison of noise reduction performance when there are 4 physical error microphones
and 8 virtual error microphones.

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Furthermore, Figure 7 shows the comparative results when virtual sensing technique is carried out
with different numbers of physical microphones. Figure 7(a) presents the case when microphones inthe third layer are considered as virtual error microphones and microphones in the first layer areused wholly or partially as physical error microphones. If one microphone is removed from the firstlayer, the virtual sensing technique results in a reduced noise reduction level. If more physicalmicrophones are removed, the noise reduction performance worsens. Figure 7(b) exhibits the casewhen microphones in the first layer are set up as virtual error microphones and microphones in thethird layer are used wholly or partially as physical error microphones. In this case, when less than 4physical error microphones are used, the noise reduction performance of virtual sensing technique is
not satisfactory. Lastly, when microphones in both the second and third layers are configured as
virtual error microphones, the number of physical error microphones in the first layer barely affects
the noise reduction as shown in Figure 7(c). This is due to the fact that the baseline level of noise
reduction is relatively low.

Figure 7 – Comparison of noise reduction performance when there are 2 to 4 physical error
microphones and 4 virtual error microphones.

4. CONCLUSIONSThis paper validates the effectiveness of virtual sensing technique for multi-channel feedforward
ANC systems. It has been proved that the virtual sensing technique works compatibly with multiplereference signals. Moreover, the increased number of virtual error microphones has no adverse effect6896

on the performance of virtual sensing technique. However, the number of physical error microphones
has to be greater than the number of secondary loudspeakers. When there are more secondary
loudspeakers than physical error microphones, the multi-channel ANC becomes an underdetermined
optimization problem with arbitrary solutions.

ACKNOWLEDGEMENTS
This work is jointly supported by the National Science Foundation of China (Grant No. 61671137
and Grant No. 61701090) and Sichuan Science and Technology Program (Project No. 2018JY0218 and2019YJ0182).

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