Comparisons of Two Virtual Sensing Methods for Broadband Noise
Reo MAEDA(1) , Yoshinobu KAJIKAWA(2)(1) Kansai University, Japan, leo0756.kansai@gmail.com(2) Kansai University, Japan, kaji@kansai-u.ac.jp

Abstract
Conventional ANC systems can reduce a target noise around the error microphone. Hence, the users for ANC
systems may not get sufﬁcient noise reduction if the user’s ears are apart from the error microphones. Thisis because the size of the Zone of Quite (ZoQ) is 1/10 of wavelength of the target noise. To solve this issue,
some virtual sensing techniques have been already proposed and examined. These techniques can create themaximum noise reduction at the desired positions. In this paper, we introduce two methods, which are called"Remote Microphone" and "Auxiliary Filter based Virtual Sensing" methods to an ANC headrest in order toexamine the effectiveness. In various situations, we implement the two virtual sensing techniques on the ANC
headrest and compare the results through some simulations in MATLAB
Keywords: Active Noise Control, Virtual sensing technique, Remote Microphone method, Auxiliary Filter
based Virtual Sensing method, Headrest

1 INTRODUCTION
Active Noise Control (ANC) is a means of reducing the unwanted noise. ANC is a technique based on
the principle of superposition, i.e., an antinoise with the same amplitude and opposite phase is generated andcombined with an unwanted noise, thus resulting in the cancellation of both noises (1-4). Various products of
ANC have been already introduced in order to improve the interference of the acoustic noise. In this paper, we
focus on ANC headrest system and compare the two virtual sensing techniques.
The conventional ANC can get the maximum noise reduction only around error microphones and the size of
the Zone of Quite (ZoQ) is limited depending on the target noise’s frequency. Moreover, some applications of
ANC (e.g. ANC headrest (5-8)) requires that the error microphones have to be located apart from the desired
locations (e.g. user’s ears) because the error microphones must be installed on the headrest for headrest ANC.
To handle the issue, the virtual sensing techniques (5-13) are usually applied to the ANC headrest. In addition,
the desired locations (user’s ears) may be moved due to user’s head movements (it is difﬁcult to ﬁx the headin natural sitting situations). However, if the virtual sensing techniques can work well at every position and inevery situation, the ANC headrest can be put into practical use by combining it with physical sensors.
We apply two virtual sensing techniques to the ANC headrest for reducing the broadband noise. One is called
Remote Microphone (RM) method; it is valid especially for reducing the narrow band noise (5,6). However,there are two main issues in the RM method. When estimating the transfer function between microphones, somedips often arise at particular frequencies in general acoustic environments. Furthermore, the noise reduction
performance strongly depends on the geometric arrangements of primary noise source and error and virtual
microphones, that is, the error microphones have to be located closer to the primary noise source than the virtual
microphones because the transfer functions from the error microphones to the virtual microphones cannot be
obtained if the virtual microphones are closer to the noise source than the error microphones (causality reason).
On the other hand, the other method, which is called Auxiliary Filter based Virtual Sensing (AF-VS), is more
effective especially for reducing the broadband noise and does not require any geometrical constraint (7-13). In
this paper, we compare the two methods on the noise reduction performances for broadband noise to clarify theeffectiveness.
The paper is organized as follows. Section 2 explains the limitation of ZoQ and what kind of problems the6883

conventional headrest ANC has. Section 3 introduces the two virtual sensing methods. In Section 4, we show
the noise reduction performance of the two method in some situations. Section 5 summarizes the obtained
results.

2 LIMITATION OF ZOQ IN THE GENERAL ANC SYSTEM
Zone of Quite (ZoQ) in general ANC systems is formed only around the error microphones. Hence, in the
ANC headrest, the ZoQs are created at the back of the headrest. Additionally, the size of ZoQ is 1/10 of
the wavelength of the target noise in general ANC systems (2). If the target noise contains high frequency
components and the error microphones are located away from desired locations, adequate noise reduction may
not be obtained. For instance, when the target noise is 1000 Hz, the ANC system creates the ZoQ whose
diameter is 3.4 cm. When one uses the ANC headrest, the positions of the ears may not be located in the
ZoQs because people tend to lean forward where the head is not in contact with the headrest. Naturally, theerror microphones can not be physically placed close to the user’s ears in the ANC headrest. A solution tothis limitation of ZoQ is to utilize the virtual sensing techniques. We therefore examine a headrest ANC with
virtual sensing techniques in order to validate the effectiveness in various situations.

3 VIRTUAL SENSING TECHNIQUES
3.1

Remote Microphone (RM) Method
x(n)

Error
microphone

Virtualmicrophone

dv(n)

dp(n)

C(z)

y(n)

x(n)

+

Σ

P(z)

ec(n)

W(z)

-

dp(n)
y(n)

S(z)
Ŝ(z)

Ŝv(z)

Ŝv(z)
r(n)

NLMS

(a) Tuning stageNLMSeˆ v(n)

+

Σ

ep(n)

yp(n)

+

ŷp(n)
+

Σ

ŷv(n)
-

dˆ p(n)

C(z)

Σ +ˆ

dv(n)

(b) Control stage

Figure 1. Block diagram of feedfoward ANC system with RM.
One way to reduce the target noise at the desired positions is the Remote Microphone (RM) method. The
block diagram of the RM method is shown in Fig. 1. In Fig.1, P(z) is the primary path from the reference
microphone to the error microphone, S(z) and Sv (z) are the secondary paths from the secondary source tothe error and virtual microphones, respectively. In this method, the path from the error microphone to themicrophone at the desired position (virtual microphone), C(z) = V (z) / P(z) is estimated in advance as shownin Fig. 1(a). By using the path, the RM method reduces the target noise at the virtual microphone as shown in
Fig. 1(b). However, this method must estimate the path between the microphones which are located a bit far
from the noise source. It may cause some dips at some particular frequencies according to the distance, whichmakes the noise reduction worse. Additionally, depending on the direction of the noise arrival, the causality is
not satisﬁed and C(z) cannot be estimated exactly.

3.2

Auxiliary Filter Based Virtual Sensing (AF-VS) Method

Among virtual sensing techniques, another way is the Auxiliary Filter based Virtual Sensing (AF-VS) method,which is more effective for reducing the broadband noise. The block diagram of the AF-VS method is shownin Fig. 2. As shown in Fig. 2(a), the auxiliary ﬁlter H(z) converges P(z) / S(z)Wo (z). Hence the auxiliary ﬁlter
includes the information of the optimal noise control ﬁlter at the desired position Wo (z). In the control stage,

6884

the microphone can not be placed at the desired position and the error signal at the desired position can not
be used to update the noise control ﬁlter. By using the auxiliary ﬁlter H(z) estimated in the tuning stage, thismethod can reduce the target noise at the desired position. The AF-VS method does not need to estimate thepath between the microphones and does not degrade the noise reduction performance unlike the RM method.
x(n)

dv(n)

V(z)

y(n)

W(z)
Ŝv(z)

r(n)

+

Sv(z)

NLMS

S(z)

x(n)

V(z)

+

W(z) y(n)

Σ

r(n)

Ŝ(z)

eh(n)

+

Σ

ev(n)
-

ep(n)

y´(n)

S(z)

eh(n)

NLMS

H(z)

NLMS

Σ

yv(n)

dp(n)

P(z)

+ Σ
e (n)
+ p
-

dv(n)

Sv(z)

-

yh(n)

H(z)

ev(n)

y´(n)

dp(n)

P(z)

Σ

yv(n)

-

yh(n)

Σ

+

(b) Control stage(a) Tuning stage

Figure 2. Block diagram of single channel feedfoward ANC system with AF-VS.

4 MULTIPLE CHANNEL FEEDFORWARD ANC HEADREST SYSTEM WITH AF-VS
NLMS

H1(z)
P1(z)

eh1(n)-

yh1(n)

ΣS11(z)r (n)Ŝv11(z) 11

NLMS

Ŝv21(z) r (n)S21(z)

21

W1(z)

y1(n)

V1(z)

V2(z)

Ŝv12(z)

Σ

W1(z)

y (n)

P1(z)

-

Σ

ev1(n)

x(n)

V1(z)

P2(z)
H2(z)

dv1(n)

ev2(n)V2(z)y´11(n)

-

y (n)

-

y21(n)-

Σ e (n)Sv21(z)p1

ev1(n)

Σ-dv2(n)

ev2(n)

Σ

y (n)

-

y12(n)

Sv12(z) 12

-

y (n)

Sv22(z) 12

y (n)

P2(z)

y´12(n)W2(z)dp2(n)y´21(n)-Sv22(z) 22

S22(z)

S11(z)

dp1(n)-

ΣS12(z)

r22(n)y1(n)Sv11(z) 12

y21(n)NLMS

Ŝv22(z)S21(z)y´21(n)dv2(n)

r12(n)- Σ
eh1(n)

NLMS
21

-

y2(n)

yh1(n)

r11(n)

Ŝ21(z) r (n)y´11(n)dv1(n)

Sv12(z)W2(z)-Ŝ11(z)-

Sv11(z) 11
Sv21(z)

x(n)

H1(z)

ep1(n)

dp1(n)y´22(n)yh2(n)-

Ŝ12(z)-

ΣΣ

eh2(n)NLMS-

dp2(n)y2(n)r12(n)

ΣS22(z)
S12(z)

ep2(n)-y´22(n)y´12(n)NLMS
ep2(n)

Ŝ22(z)

r22(n)H2(z)yh2(n)

eh2(n)-

Σ(b) Control stage(a) Tuning stage
Figure 3. Block diagram of multi channel Case (1,2,2) feedfoward ANC system with AF-VS.
In the ANC headrest, there are two secondary sources and two error microphones. When sitting a chairwith headrest, the user’s ears are generally apart from the headrest part. In this case, the crosstalk paths (e.g.
left secondary source to right error microphone) must be considered in the ANC headrest. In this section, we
introduce a multi-channel feedforward ANC headrest system with the AF-VS method. Figure 3 shows block
diagrams of the 2 × 2 adaptive feedforward ANC with the AF-VS method. There are two stages in this6885

method. The ﬁrst one is called the tuning stage, and the second one is called the control stage. In Fig. 3 (a),the noise control ﬁlters and the auxiliary ﬁlters on tuning stage are updated by
2αce (n)rmk (n),
2 vm
β
+
r
mk (n)
m=1 c

wk (n + 1) = wk (n) + ∑hk (n + 1) = hk (n) +αhβh + x(n)2ehm (n)x(n),

(1)
(2)

ehm (n) = e pm (n) − yhk (n),(3)

rmk (n) = ŝTvmk x(n),(4)

where, k = 1, 2 and m = 1, 2 express the two secondary sources and the error microphones, respectively. smk
and svmk express the coefﬁcient vectors of the secondary paths from the secondary sources to the error or
virtual microphones. wk (n) = [wk1 (n), wk2 (n), · · · , wkN (n)]T and hk (n) = [hk1 (n), hk2 (n), · · · , hkK (n)]T express the
coefﬁcient vectors of the noise control ﬁlters Wk and auxiliary ﬁlters Hk whose tap lengths are N and K,
respectively. e pm (n) and evm (n) are the error signals at the error and virtual microphones, respectively. yhk (n)expresses the output signals from the auxiliary ﬁlter Hk (z) ŝvmk is the secondary path model of svmk and rmk (n) =[rmk (n), rmk (n − 1), · · · , rmk (n − N + 1)]T is the ﬁltered reference vector in the FXNLMS algorithm. αc and αhare the step size parameters of the noise control and the auxiliary, respectively. βc and βh are the regularization
factors.
In Fig. 3 (b), the error signals at the virtual microphones are given as
Ev1 (z) = {V1 (z) − Sv11 (z)W1 (z) − Sv12 (z)W2 (z)}X(z),(5)

Ev2 (z) = {V2 (z) − Sv22 (z)W2 (z) − Sv21 (z)W1 (z)}X(z).

(6)

Hence, the gradients of the noise control ﬁlters W1 (z) and W2 (z) are expressed as
2 (z)
∂ Ev1
=2{Sv11 (z)W1 (z) + Sv12 (z)W2 (z) −V1 (z)}Sv11 (z)X 2 (z),∂ W1 (z)
2 (z)
∂ Ev1
=2{Sv11 (z)W1 (z) + Sv12 (z)W2 (z) −V1 (z)}Sv12 (z)X 2 (z),∂ W2 (z)
2 (z)
∂ Ev2
=2{Sv21 (z)W1 (z) + Sv22 (z)W2 (z) −V2 (z)}Sv21 (z)X 2 (z),∂ W1 (z)
2 (z)
∂ Ev2
=2{Sv21 (z)W1 (z) + Sv22 (z)W2 (z) −V2 (z)}Sv22 (z)X 2 (z).
∂ W2 (z)

(7)
(8)
(9)
(10)

From (7), (8), (9), and (10), the optimum noise control ﬁlters W1o (z) and W2o (z) given by
Sv12V2 (z) − Sv22 (z)V1 (z),Sv12 Sv21 (z) − Sv11 (z)Sv22 (z)Sv21V1 (z) − Sv11 (z)V2 (z)W2o (z) =.
Sv12 Sv21 (z) − Sv11 (z)Sv22 (z)W1o (z) =

(11)
(12)

Assuming the noise control ﬁlters are converged, the error signals for updating the auxiliary ﬁlters H1 (z) andH2 (z) are given by6886

Eh1 (z) =E p1 (z) − H1 (z)X(z)={P1 (z) − S11 (z)W1o (z) − S12 (z)W2o (z) − H1 (z)}X(z),Eh2 (z) =E p2 (z) − H2 (z)X(z)(13)={P2 (z) − S22 (z)W2o (z) − S21 (z)W1o (z) − H2 (z)}X(z).

(14)

Therefore, H1 (z) and H2 (z) are expressed as
2 (z)
∂ Eh1
=2{H1 (z) − P1 (z) + S11 (z)W1o (z) + S12 (z)W2o (z)}X 2 (z),∂ H1 (z)
2 (z)
∂ Eh2
=2{H2 (z) − P2 (z) + S22 (z)W2o (z) + S21 (z)W1o (z)}X 2 (z).
∂ H2 (z)

(15)
(16)

The optimum auxiliary ﬁlters converge to
H1o (z) =P1 (z) − S11 (z)W1o (z) − S12 (z)W2o (z)Sv12V2 (z) − Sv22 (z)V1 (z)Sv21V1 (z) − Sv11 (z)V2 (z)− S12 (z),=P1 (z) − S11 (z)Sv12 Sv21 (z) − Sv11 (z)Sv22 (z)Sv12 Sv21 (z) − Sv11 (z)Sv22 (z)H2o (z) =P2 (z) − S22 (z)W2o (z) − S21 (z)W1o (z)Sv21V1 (z) − Sv11 (z)V2 (z)Sv12V2 (z) − Sv22 (z)V1 (z)− S21 (z).
=P1 (z) − S22 (z)Sv12 Sv21 (z) − Sv11 (z)Sv22 (z)Sv12 Sv21 (z) − Sv11 (z)Sv22 (z)

(17)

(18)

We can see that H1o (z) and H2o (z) in (17) and (18) keep the information of the optimum noise control ﬁlters forthe virtual microphones including the crosstalk components, respectively.
Whereas, in the tuning stage, the error signals at the virtual microphones can not be used for updating thenoise control ﬁlters. Instead of (1), the noise control ﬁlters in the tuning stage are updated as2αce (n)rmk (n),2 hm
m=1 βc + rmk (n)wk (n + 1) = wk (n) + ∑(19)ehm (n) = e pm (n) − yhk (n),(20)

rmk (n) = ŝTmk x(n).

(21)

The error signals for updating the noise control ﬁlters are expressed as
Eh1 (z) =E p1 (z) − H1o (z)X(z)=P1 (z)X(z) − S11 (z)W1 (z)X(z) − S12 (z)W2 (z)X(z) − {P1 (z) − S11 (z)W1o (z) − S12 (z)W2o (z)}X(z),Eh2 (z) =E p2 (z) − H2o (z)X(z)(22)=P2 (z)X(z) − S22 (z)W2 (z)X(z) − S21 (z)W1 (z)X(z) − {P2 (z) − S22 (z)W2o (z) − S21 (z)W1o (z)}X(z).

(23)

Therefore,6887

2 (z)∂ Eh1= 2[S11 (z){W1 (z) −W1o (z)} + S12 (z){W2 (z) −W2o (z)}]S11 (z)X 2 (z),∂ W1 (z)
2 (z)
∂ Eh1
= 2[S11 (z){W1 (z) −W1o (z)} + S12 (z){W2 (z) −W2o (z)}]S12 (z)X 2 (z),∂ W2 (z)
2 (z)
∂ Eh2
= 2[S21 (z){W1 (z) −W1o (z)} + S22 (z){W2 (z) −W2o (z)}]S21 (z)X 2 (z),∂ W1 (z)
2 (z)
∂ Eh2
= 2[S21 (z){W1 (z) −W1o (z)} + S22 (z){W2 (z) −W2o (z)}]S22 (z)X 2 (z).
∂ W2 (z)

(24)
(25)
(26)
(27)

From (24), (25), (26) and (27), every noise control ﬁlter converges to the corresponding optimal ones at thevirtual microphones (the desired positions), W1o (z) and W2o (z) Thus, it can be proved to realize the maximum
noise reduction at the desired position without putting the physical microphones there.

5 SIMULATIONS
5.1

Simulation Conditions

In this section, we examine the noise reduction performance at the desired positions (virtual microphones)
to compare the effectiveness of the two virtual sensing techniques through some simulations in MATLAB. The
measurement arrangements and conditions are shown in Figs. 4, 5 and Table 1.
As shown in Fig. 4, the position of the noise source is located in front of and the back of the ANC headrestin order to investigate the capability for the causality. Moreover, we recorded the noise at the every microphones
at the three positions as shown in Fig. 5 in order to examine the robustness for movements of the user’s ears
(desired positions).
130 cm130 cmSecondary sources
5 cm

Error microphone
Virtual microphone
5 cm

Secondary sources

Error microphone Virtual microphone Reference microphone
5 cm

Reference microphone

5 cm
10 cm

Noise source
Headrest

10 cm

5 cm
Virtual microphone
Error microphone

Headrest(a) Causality

5 cm
Error microphone Virtual microphone(b) Non-causality
Figure 4. Measurement arrangements.

(a) Left(b) Forward
Figure 5. Three head directions for comparison.

6888(c) Right

Noise sourceTable 1. Experimental conditions.
Noise
Update algorithm of ﬁlters
Tap length of ﬁlters W , C and H
Step size parameter of ﬁlter α
Regularization parameter βSampling frequency
Cut-off frequency of low-pass ﬁlter

5.2

White noise (500 - 2000 Hz)
NLMS
500
0.05
0.00001
12000 Hz
2500 Hz

Simulations Results

Due to the limitations of space, the results at the right virtual microphone are shown in this paper. Figure 6
shows the spectra of the error signal at the desired position using the RM and AF-VS methods when the user
looks forward like Fig. 5 (b). Next, Fig. 7 illustrates the noise reductions of the two virtual sensing methods
in the ANC headrest for three directions of the user’s head and two different noise source locations.
As indicated in Fig. 6, the AF-VS can reduce the noise stably between 500 - 2000 Hz regardless of the noise
source directions. On the other hand, the RM cannot reduce the target noise especially between 1500 - 2000 Hz
and ampliﬁes over 2000 Hz. This is because it is difﬁcult for ANC systems to reduce the high frequency noise.
Furthermore, the RM in the non-causality arrangement can reduce the target noise less than on the causality one
between 500 - 1500 Hz. This is because C(z) cannot be estimated exactly in the non-causality arrangement.
We summarized the average noise reduction level for three head direction in Table 2. From Fig. 7 and Table
2, the AF-VS gets the almost same noise reduction levels regardless of the noise source position and headdirections. On the other hand, the RM ampliﬁes the target noise at the desired position in the non-causality
arrangement. Besides, on the causality arrangement, the AF-VS is more effective than the RM for reducing
the target noise in the causality arrangement. Additionally, as shown in Fig. 7 (a) and Table 2 (a), the noise
reduction performance for the left direction is not very high totally comparing with the other directions because
the anti-noise cannot arrive at the desired position directly due to the user’s head, which interrupts the pathfrom the secondary source to the desired position.
20

20

15
5
0
-5
-10
-15
-20

5
0
-5
-10
-15
-20

-25
-30

ANC OFF
RM
AF-VS

10
Amplitude [dB]10
Amplitude [dB]15

ANC OFF
RM
AF-VS

-25
0

500

1000

1500
2000
Frequency [Hz]

2500

-30

3000

0

(a) Causality

500

1000

1500
2000
Frequency [Hz]

2500

3000

(b) Non-causality

Figure 6. Comparison of error spectra between the two virtual microphone methods.

20

6

15

4

RM (Non-causal)
RM (Causal)
AF-VS (Non-causal)
AF-VS (Causal)

2
0
-2
-4

Reduction [dB]25

20
Reduction [dB]25

8Reduction [dB]1010
RM (Non-causal)
RM (Causal)
AF-VS (Non-causal)
AF-VS (Causal)

5
0
-5

0

0.5

1.0

1.5
2.0
2.5
Iteration [×106](a) Left

3.0

3.5

4.0

-10

15
10
RM (Non-causal)
RM (Causal)
AF-VS (Non-causal)
AF-VS (Causal)

5
0
-5

-10
0

0.5

1.0

1.5 2.0 2.5
Iteration [×106](b) Forward

3.0

3.5

4.0

-15

0

0.5

1.0

1.5 2.0 2.5
Iteration [×106]

3.0

3.5

4.0

(c) Right

Figure 7. Comparisons of noise reduction between the two virtual microphone methods.

6889

Table 2. Average noise reduction levels for three head directions..
(a) Left(c) Right(b) Forward

MethodNoise source location

RM

AF-VS

MethodNoise source location

RM

AF-VS

MethodNoise source location

Causality

5.00

8.60

Causality

8.03

23.2

Non-causality

-2.65

7.78

Non-causality

-5.44

23.0

RM

AF-VS

Causality

9.92

23.9

Non-causality

-9.60

18.7

6 CONCLUSIONSIn order to compare the effectiveness of the two virtual sensing techniques, the RM and AF-VS methods,
we adopted the two methods to the ANC headrest. In the simulations, the two methods were compared on
the noise reduction performance for broadband noise. As a result, the AF-VS is valid for overall frequency to
reduce the target noise regardless of the noise source directions. By contrast, the RM hardly reduces the noise,
especially on the non-causality arrangement. Moreover, the AF-VS has higher noise reduction performance thanthe RM for the three head directions and the two noise source locations.
In the future, we should check whether the ANC headrest system works when the reference microphone islocated near the headrest ot chair for the convenience. Moreover, with adding one more secondary loudspeaker
and applying Case (1,3,2) or selection type Case (1,2,2), we will try to get the better result for larger space.

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