Limits of noise control over space
Thushara D. ABHAYAPALA(1) , Jihui ZHANG(1) , Prasanga SAMARASINGHE(1) and Wen ZHANG(1,2)(1) Audio & Acoustic Signal Processing Group, College of Engineering & Computer Science, Australia, {Thushara.Abhayapala,Jihui.Zhang, Prasanga.Samarasinghe, Wen.Zhang}@anu.edu.au(2) 1Center of Intelligent Acoustics and Immersive Communications, School of Marine Science and Technology, Northwestern

Polytechnical University, Xi’an Shaanxi 710072, China

Abstract
This paper investigates the best possible performance of noise cancellation over a region of space for a ﬁxed setof secondary sources within a given acoustic environment. We ﬁrst formulate the spatial active noise control(ANC) problem in a 3-D acoustic enclosure (room). Secondly, we derive a wave-domain least square method bymatching the secondary sound ﬁeld to the primary noise ﬁeld in wave domain. Thirdly, we propose a subspacemethod by matching the secondary ﬁeld coefﬁcients to the projection of primary noise ﬁeld in the subspace.
Simulation results compare between the wave-domain least square method and the subspace method, in termsof energy of the loudspeaker driving signals, noise reduction inside the region, and residual noise ﬁeld outside
the region. The subspace matching method provides the limits of the ANC system for the given constraints andthe acoustic characteristics of the room.
Keywords: Sound, Insulation, Transmission

1

INTRODUCTION

Spatial Active noise control (ANC) is a challenging problem as the aim is to create a large quiet zone formultiple listeners in three-dimensional (3-D) space. In spatial ANC applications, such as noise cancellation inaircraft [1] and automobiles [2, 3, 4, 5], multichannel ANC systems equipped with multiple sensors and multiple secondary sources are adopted [6]. In literature, both time-domain [7, 8] and frequency-domain [9, 10]algorithms have been implemented in multichannel ANC systems, which can cancel the noise at error sensor
positions and their close surroundings [10]. Recently, ANC over space has been approached via cylindrical/spherical harmonic-based wave-domain algorithms [11, 12, 13, 14, 15, 16], with which the noise over entire
region of interest can be cancelled directly. Here onwards, we use the terminology ‘wave-domain ANC’ to referto harmonics-based wave-domain ANC.

2

PROBLEM FORMULATION

Let the desired quiet zone be a spherical region with a radius R1 located inside a room and let there by L
loudspeakers as secondary sources (see Fig. 1). The goal of a spatial noise control system is to cancel the
spatial noise ﬁeld within the region of interest by a secondary sound ﬁeld generated by secondary loudspeakers.
Let ν(xx, k) be the noise ﬁeld at a point x ≡ (r, θ , φ ) with respect to an origin within the desired zone of interest
where k = 2π f /c is the wavenumber, c is the speed of sound and f is the frequency in Hz. Similarly, let s(xx, k)be the secondary sound ﬁeld generated by the loudspeakers and e(xx, k) be the residual noise ﬁeld such thate(xx, k) = ν(xx, k) + s(xx, k).

(1)

The secondary sound ﬁeld generated by the loudspeaker array can be written asLs(xx, k) = ∑ dl (k)G (xx, k)

(2)

=1

where d (k) is the driving signal of the th loudspeaker and G (xx, k) represents the acoustic transfer function
from the th loudspeaker to the point x which includes room reﬂections and scattering by other objects in the
room.

47

Figure 1. ANC system in a 3-D room.
The noise ﬁeld at a point x can be expressed using modal decomposition [17, 18] asN

ν(xx, k) = ∑n∑ βnm (k) jn (kr)Ynm (θ , φ ),(3)n=0 m=−nwhere jn (·) is the spherical Bessel function of order n, Yum (·) denotes the spherical harmonics, βnm (k) are thespherical harmonic coefﬁcients of the sound ﬁeld and the truncation limit N = ekR1 /2 [19, 20, 21]. Similarly, the secondary sound ﬁeld s(xx, k) and the residual sound ﬁeld e(xx, k) within the quiet zone can also be
represented by∞ns(xx, k) = ∑∑ γnm (k) jn (kr)Ynm (θ , φ )(4)e(xx, k) = ∑∑ αnm (k) jn (kr)Ynm (θ , φ )(5)n=0 m=−nn∞n=0 m=−nwhere γnm (k) and αnm (k) are the spherical harmonic coefﬁcients of the secondary and residual ﬁelds, respectively. The acoustic transfer function in (2) can also be parameterised in the spherical harmonic domain[22]asNG(xx, k) = ∑n()∑ ηnm (k) jn (kr)Ynm (θ , φ ).

(6)n=0 m=−n()where ηnm (k) is the harmonic coefﬁcient of the acoustic transfer function of the th loudspeaker.
We substitute (4) and (6) into (2) and decompose into spherical harmonic domain to obtainL()

γnm (k) = ∑ dl (k)ηnm (k) for n = 0, . . . , N, m = −n, . . . , n.

(7)

=1

Similarly, by substituting (5), (3) and (5) into (1) and decomposing into spherical harmonic domain, we obtain
αnm (k) = βnm (k) + γnm (k)(8)

for n = 0, . . . , N, m = −n, . . . , n. We combine (7) and (8) to obtain a relationship between modal components ofresidual ﬁeld, noise ﬁeld and the room modes of the acoustic transfer functions of loudspeakers as
α (k) = β (k) + H (k)dd (k)

48

(9)

where α (k) = [α00 , . . . , αNN ], β (k) = [β00 , . . . , βNN ],
⎡ (1)
(2)
η00 (k) η00 (k)
⎢ (1)
(2)
⎢η−11 (k) η−11 (k)
H =⎢
..
⎢ ..
⎣ .
.
(1)

ηNN (k)(2)

ηNN (k)

···
···
..
.

⎤
(L)
η00 (k)⎥(L)
η−11 (k)⎥
.. ⎥
⎥,
. ⎦

···

ηNN (k)(10)(L)

and d (k) = [d1 (k), . . . , dL (k)]T .
In a complete spatial ANC system, there are error microphones located within or around [] the desired quite
zone to obtain e(xx, k) as well as reference microphones to obtain a clean copy of noise signal to be fed into an
adaptive algorithm to drive secondary loudspeaker signals. For the analysis in this paper, we assume that theerror signal e(xx, k) and hence the mode coefﬁcients matrix α (k) can be estimated accurately. The goal of thispaper is to analyse how much of the incident spatial noise ﬁeld β (k) can be cancelled by a given loudspeakerarray geometry within a given acoustic environment (i.e., by a given H ).

3

WAVE-DOMAIN LEAST SQUARE METHOD

One method for deriving the loudspeaker driving signal d (k) is to minimise error coefﬁcients, i.e.,α (k)2 for each kmin αd (k)(11)

where  · 2 is 2-norm operator. The above minimisation problem can be written as
β (k) + H (k)dd (k)2 for each kmin βd (k)(12)

In practical applications, the number of loudspeakers available are less than the number of active modes of thedesired region of interest. Thus, (12) can be solved by least square method which results in minimum meansquared errors [23] solution. The optimal solution of this minimization problem can be written as
H † (k)ββ (k),
d (k) = −H

(13)where (·)† denotes the pseudoinverse of a matrix. We denote this method as ‘wave-domain least square method’(WDLS) as the problem is formed in spherical harmonic (wave-domain) and solved using least squares. To thismethod to work, one has to obtain the acoustic transfer function coefﬁcients between each loudspeaker and theregion of interest. Also, it is assumed that the these coefﬁcients are ﬁxed and invariant with time.

4

SUBSPACE METHOD

In the second method, we construct the subspace spanned by the transfer function coefﬁcient vectors of theloudspeaker array. Then, we project the noise ﬁeld coefﬁcient vector onto the subspace spanned by the loudspeakers. It can be infer that we can only cancel noise which lie in the loudspeaker subspace but the noise in
the null space of the loudspeakers can not be cancelled and constitute the residual error.
4.1 Principal component analysis of the secondary pathLet η () (k) be the wave-domain secondary-path coefﬁcient vector of the th loudspeaker. The matrix H (k)deﬁned in (10) for the entire loudspeaker array represents the secondary paths, whereη (1) (k), . . . , η (L) (k)].
H (k) = [η(14)In an arbitrary loudspeaker array setup in an acoustic environment, the columns of matrix H (k) are not necessarily orthogonal to each other. We use the principal component analysis (PCA) of the correlation matrix of49

H to obtain an orthonormal eigen-basis for the space of the secondary path of loudspeakers in the sphericalharmonic (wave) domain. Here onwards, the frequency dependent k is omitted for notational simplicity.
H ∗ H }, and then decompose this matrix into a set of orthonormal eigenvectors
We take the correlation matrix E{Hand their corresponding eigenvalues, as follows:H ∗ H } = U ΛV ,
E{H

(15)

where U = [uu1 , . . . , u i , . . . , u L ] are the eigenvectors of the correlation matrix, V = U T , the ith column of U
corresponds to the eigenvalue λi , and the eigenvalue are in the matrix⎤⎡
λ1 0 · · · 0⎢ 0 λ2 · · · 0 ⎥
⎥
⎢
(16)
Λ=⎢ .
.. . .
.. ⎥ .
⎣ ..
.
.
.⎦
0

0

···

λL

Here, the vectors u i are written in order of descending eigenvalues λi [24]. Depending on the acoustic environment and the loudspeaker placement not all eigen values are signiﬁcant. Let there are Q signiﬁcant eigen
values, then the corresponding eigenvectors are given by the matrix U  = [uu1 , . . . , u Q ].
We can now construct a subspace O spanned by the loudspeaker secondary-path coefﬁcients H deﬁned as
O = HU ,(17)where the dimension of U  is L × Q, and Q ≤ L. By normalizing each column of matrix O, the orthonormal
vectors o1 , . . . , oB can be obtained. These vectors generate a subspace, which represents the loudspeaker arrayand the acoustic environment. The dimensions of basis O are (N + 1)2 × Q.
For the th loudspeaker, the average acoustic transfer coefﬁcients can be represented in this space asQη () = ∑ κa() oq ,
η̄

(18)

q=1

()

where κq

are the projection coefﬁcients. In vector form, (18) can be written byη () = O κ () ,
η̄
()

()

(19)

()

where κ () = {κ1 , . . . , κq , . . . , κB }T and()κq = η () , o q(20)

where ·, · represents inner product operator between two two vectors, Q × L matrix κ () is the secondary-path
coefﬁcients of the th loudspeaker in the subspace O.
4.2 Projection from the primary noise ﬁeld into the subspace
Below we project the wave-domain coefﬁcients of the primary noise ﬁeld into the subspace O . For a primarynoise ﬁeld represented by vector β , by projecting β into the subspace O , we obtain
Q

ProjO β = ∑ β , oq oq

(21)

q=1

where ProjO β denotes the projection of vector β into subspace O . The matrix form of the projection is represented by(22)
ProjO β = O y ,50

where y = {y1 , y2 , . . . , yQ }T are the primary noise ﬁeld coefﬁcients in the subspace, and yq = β , o q .
Therefore, the primary noise ﬁeld can be separated into two parts: the projected part and the remaining part,β ),β = ProjO β + R(β(23)β ) is the orthogonal complement of the subspace O . The projected part indicates the primary noise
where R(βﬁeld which can be cancelled in this system setup, and the orthogonal complement indicates the primary noise
ﬁeld which can not be cancelled in this system.
β ) = 0, β lies in the subspace, then the primary noise ﬁeld can be completely cancelled by the loudspeaker
If R(ββ ) = 0. This indicates the limitation of noise cancellation over the region ofarray. In more general cases, R(βinterest, under the particular loudspeaker placement and acoustic environment.
4.3 Noise control in the subspace
In this section, we design the driving signal of loudspeaker d (k) to cancel the primary noise ﬁeld projected
into the subspace (ProjO β ). In the subspace, matching the secondary sound ﬁeld coefﬁcients to the projectedprimary noise ﬁeld coefﬁcients, the optimal solution of the secondary sound ﬁeld coefﬁcients can be written byγ = −ProjO β .

(24)

The projection from the primary noise ﬁeld into the loudspeaker subspace ProjO β can be calculated by (22).
In a given loudspeaker setup, the representation of secondary sound ﬁeld coefﬁcients can be rewritten by
γ = Oκ d ,

(25)

Oy .
O κ d = −O(26)

κ (1) , . . . , κ (L) }.
where d = {d1 , . . . , dL }T , κ = {κSubstituting (25) and (22) into (24), we can get

Multiplying the left inverse of O on both sides, (26) becomesO† O y .
O † O κ d = −O

(27)

Further simplifying (27), the ﬁnal equation to design the driving signal can be written as
κ d = −yy.

(28)

The loudspeaker driving signals can be calculated by solving the system of linear equations described by (28).
The number of principal components speciﬁes whether the linear system (28) can be solved exactly. For Q < L,where we use only the largest Q components to generate the subspace, the driving signals d can be derived byκ )† y ,d = −(κ(29)κ )† is the pseudoinverse of the secondary-path coefﬁcients in the subspace. Loudspeaker driving signals
where (κd are dependent on the secondary-path information in the subspace κ and the primary noise ﬁeld coefﬁcientsin the subspace y , as shown in (29).

5SIMULATION RESULTS

Simulation setup: The reverberant environment is modelled as a cuboid room of 6 m ×6 m ×5 m with wall
reﬂection coefﬁcients a [0.75, 0.8, 0.77, 0.85, 0.1, 0.1] and we use the image source method with the image orderof 5. The origin of the room is on the left bottom corner. The region of interest is a sphere with a radius of 0.5m, and the center of the region is (3, 3, 1.5) with respect to the room origin. In the following investigations, the

51

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Figure 2. Energy of Case 1 (a) of the primary noise ﬁeld (pink point is the location of the primary source
projected on to the x-y plane, blue points are the loudspeaker locations projected on the x-y plane, and thered dashed circle is the boundary of the region of interest, (b) of the residual noise ﬁeld using WDLS methodand (c) the subspace method. Energy of Case 2 (d) the primary noise ﬁeld (e) of the residual noise ﬁeld using
WDLS method and (f) the subspace method.
primary noise ﬁeld is a single frequency (200 Hz) spherical wave originating from a point source located within
the room, with a constant magnitude of 10. For the the given parameters, we have N = ekR1 /2 = 3 modes.
Thus, at least (N + 1)2 = 16 microphones must be placed on the boundary to capture the the residual noise ﬁeld.
In this simulation, we place 32 microphones on the spherical boundary, following the Gauss-Legendre sampling
method. White Gaussian noise (60 dB SNR1 ) is added to each microphone recording. To control all modes, 16
loudspeakers are required to be placed outside the control region. To emulate a practical scenario, however, in
this simulation, only 12 loudspeakers are placed outside the region with 8 loudspeakers placed on the x-y planewith two different geometries. The loudspeaker positions for each case are shown in Table ??.
The noise reduction inside the region of interest over Z points Nrin can be written by
Nrin  10 log10∑z E{|ein_z |2 },∑z E{|ein_z (0)|2 }(30)

where E{|ein_z (0)|2 } is the energy of the primary noise ﬁeld at the zth sample point, and E{|ein_z |2 } is theenergy of the residual sound ﬁeld at the zth sample point. To evaluate the loudspeaker energy consumption,
we compare the total energy of all the loudspeakers Ed . The loudspeaker energy consumption is given by
Ed = d T d .
1 Here, the SNR level is with respect to the primary noise ﬁeld level on virtual microphone in the center of the region.

52

5.1 Noise cancellation comparison
We ﬁrst compare cancellation performance using the subspace method and the WDLS method in two different
noise source positions: Case 1: (2, 315◦ , 45◦ ), Case 2: (2, 315◦ , 90◦ ). The energy of the primary noise ﬁeld isshown in Figure 2.
Figure 2 depicts the energy of the residual noise ﬁeld on the x-y plane. As we expected, since the numberof loudspeakers (12) can-not cover all the modes (16) in the region, in all four ﬁgures, the primary noise ﬁeld
in the region of interest can not be fully cancelled. In case 2, compared with the primary noise ﬁeld (Figure2(d)), both the WDLS method and the subspace method can achieve signiﬁcant noise reduction in the region ofinterest, which are dark areas in the middle of Figure 2(c) and Figure 2(f). In case 1, since the noise source islocated in a different hemisphere as the loudspeaker array, compared with Figure 2(a), cancellation performanceinside the region is fairly limited for both WDLS and the subspace method, as shown in Figure 2(b) and Figure2(e). Compared with Figure 2(b) and Figure 2(c), in Figure 2(e) and Figure 2(f), the subspace method results inlower energy of the residual noise ﬁeld outside the region of interest. The WDLS method increases the soundenergy outside the region, especially when the noise reduction level is fairly limited inside the region. Using thesubspace method, we reduce the sound ampliﬁcation to people sitting outside the control area. A comprehensive
simulation study using the theory developed here is reported in [].

6

CONCLUSIONSThis paper developed a subspace based method to obtain the best possible performance of noise cancellation
over a region of space for a ﬁxed set of secondary sources within a given acoustic environment. The subspace
matching method provides the limits of the ANC system for the given constraints and the acoustic characteristicsof the room.

ACKNOWLEDGEMENTS
This work is supported by Australian Research Council Grant DP180102375.

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