GENERATION OF MULTIPLE SOUND ZONES BY SPATIAL FILTERING
IN WAVENUMBER DOMAIN USING A LINEAR ARRAY OF LOUDSPEAKERS
Takuma Okamoto
National Institute of Information and Communications Technology
3-5, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0289, Japanokamoto@nict.go.jp
ABSTRACT
Novel signal processing is proposed for generating acoustically bright and dark zones at arbitrary horizontal positions
using a linear array of loudspeakers. Most conventional
methods are based on the numerical calculation of the inverseof the spatial correlation matrix between control points andthe positions of the loudspeakers. However, such methods
are quite unstable because the acoustic inverse problem isvery ill-conditioned. On the other hand, in the proposed
method, spatial filters in the wavenumber domain are analytically derived by introducing the spectral division method andmodeling sound pressures at the control line as a rectangular
window corresponding to bright and dark zones. Computer
simulation results show that the proposed method can generate acoustically bright and dark zones more accurately thanthe conventional acoustic energy difference maximization
method.
Index Terms— Multizone, linear array of loudspeakers,
spatial Fourier transform, spectral division method, spatial filtering
1. INTRODUCTION
Generating acoustically bright and dark zones [1] using multiple loudspeakers (Fig. 1(a)) is one important and attractive
acoustic communication technique. By the superposition ofvarious bright and dark zones, multiple sound zones can also
be generated (Fig. 1(b)). These techniques are useful not
only for personal sound systems [2–4] but also for multilingual guide services and other virtual reality applications. This
paper focuses not on beamforming methods [2, 5, 6] but on
methods to generate acoustically bright and dark zones using
multiple loudspeakers.
Many methods have been proposed that control bothacoustic contrast or the energy between two spaces [1, 3,4, 7–11] and reproduce multiple sound fields [12–17] using
multiple loudspeakers.
Most conventional methods are based on the numericalcalculation of the inverse of the spatial correlation matrix between control points and the positions of loudspeakers [1,3,4,S2 (ω)F2 (x0 , ω)
Bright zone..
.

S2 (ω)
Dark zone+
S1 (ω)F1 (x0 , ω)D(x0 , ω)Sound zone..
.

S2 (ω)
Sound zone

S1 (ω)
Dark zone
..
.

(b)

Bright zone

S1 (ω)(a)Fig. 1. Generation: (a) acoustically bright and dark zones; (b)multiple sound zones using a linear array of loudspeakers.

8–10, 12–16]. However, such methods are quite unstable because the acoustic inverse problem is very ill-conditioned [18,19]. This problem is the same in sound field reproduction
based on the least squares method [20].
The acoustic energy difference maximization (EDM) [7]
is based on the numerical calculation of the eigenvector of
the spatial correlation matrix instead of the inverse. Thismethod can generate acoustically bright and dark zones moreaccurately than the conventional acoustic contrast maximization [1]. However, it needs to calculate the reproduction results over again to select the optimal tuning factor.
To solve these problems, novel signal processing is proposed for generating acoustically bright and dark zones at arbitrary horizontal positions using a linear array of loudspeakers. The proposed method is based on analytically derived filters by introducing the spectral division method (SDM) [21],which is one sound field reproduction method for planer orlinear arrays of loudspeakers. In the proposed method, an efficient spatial filter at the wavenumber domain is analyticallyderived by modeling sound pressures at the control line as arectangular window corresponding to bright and dark zones.

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2. GENERATION OF ACOUSTICALLY BRIGHTAND DARK ZONES BY SPATIAL FILTERING IN
WAVENUMBER DOMAIN

Secondary source

P (x, yb ) = 1

2.1. Spectral division method in horizontal plane
The proposed method is based on the spectral division methodin the horizontal plane for continuous linear distributions ofsecondary sources [21].
Acoustic pressure P (x, ω) synthesized at position x =[x, y, 0]T with y > 0 by secondary sources at linear array(y = 0) is given as
Z ∞P (x, ω) =D(x0 , ω)G3D (x − x0 , ω)dx0 , (1)−∞where ω = 2πf denotes the radial frequency, f denotes thetemporal frequency, D(x0 , ω) denotes the secondary source(driving signal) at position x0 = [x0 , 0, 0]T , G3D (x − x0 , ω)denotes the transfer function of the secondary source placed
at x0 to point x, and dx0 stands for a surface element for integration. Under the free-field assumption, G3D (x − x0 , ω)is the three-dimensional free-field Green’s function [22], defined as
G3D (x − x0 , ω) =exp(−jk|x − x0 |),4π|x − x0 |(2)√
where j = −1, k = ω/c denotes the wavenumber andc is the speed of sound. When applying the spatial Fouriertransform to (1) with respect to the x-axis, the convolution
along it is performed by the convolution theorem:P̃ (kx , y, 0, ω) = D̃(kx , ω) · G̃(kx , y, 0, ω),(3)where kx denotes the spatial frequency in the direction of xand G̃(kx , y, 0, ω) is the spatial Fourier transpose of G3D (x−x0 , ω) with respect to the x-axis calculated in [23], given as(2) p 2
− 4j H0k − kx2 yfor 0 ≤ |kx | ≤ |k|
G̃(kx , y, 0, ω) =(4)

p


122

kx − k y

2π K0



for 0 < |k| ≤ |kx |,(2)

where H0 denotes the zero-th order Hankel function of thesecond kind and K0 is the zero-th order modified Bessel function of the second kind [22].
When the continuous receiver line is located at y = yref ,the secondary source driving function in the wavenumber domain is directly given byD̃(kx , ω) =P̃ (kx , yref , 0, ω),G̃(kx , yref , 0, ω)

in the spectral division method.

x(5)BrightP (x, yb ) = 0

Dark

Dark

y = yblb
xbyShiftFig. 2. P (x, yb ) modeled by rectangular window for generating acoustically bright and dark zones.
2.2. Spatial filtering in wavenumber domain
In this subsection, spatial filters in the wavenumber domain
are proposed to generate acoustically bright and dark zones.
For the reproduction of each sound source Si (ω) at
each zone using a linear array of loudspeakers, each filter
Fi (x0 , ω) for each sound source Si (ω) at each zone is calculated. (i = 1, 2, · · · , M ). When the number of sound sources
is M , secondary source driving function D(x0 , ω), which
represents the superposition of each Si (ω)Fi (x0 , ω), is givenasMX
Si (ω)Fi (x0 , ω).
(6)
D(x0 , ω) =i=1

The case of M = 2 is shown in Fig. 1.
Next the case of M = 1 and S1 (ω) = 1 is considered. This is equivalent to generating an acousticallybright zone using a linear array of loudspeakers. In thefollowing equations, M is omitted for simplicity. ThenD(x0 , ω) = F (x0 , ω), and (5) is represented asF̃ (kx , ω) =P̃ (kx , yref , 0, ω).
G̃(kx , yref , 0, ω)(7)From (7), filter F̃ (kx , ω) is calculated as the spatial filter inthe wavenumber domain for generating an acoustically brightzone using a linear array of loudspeakers.
In sound field reproduction, P (x, ω) at position x =[x, yref , 0] is the actual acoustic pressure received at the continuous receiver line located at y = yref . In the proposed
method, on the other hand, to generate acoustically brightand dark zones at the continuous receiver line, P (x, ω) iscontinuously set to 1 or 0 at all temporal frequencies. The
positions at P (x, ω) = 1 and 0 correspond to the bright anddark points, respectively.
For generating the bright zone of length lb centeredaround x = 0 at y = yb , P (x, yb ) is modeled by a rect-

3. COMPUTER SIMULATIONS

angular window Π(x/lb ) shown in Fig. 2 and given as
P (x, yb ) = Π



xlb



=



1, for |x| ≤ lb /2
0, elsewhere.

(8)

Then the Fourier transform of P (x, yb ) with respect to x is[22]
kx lblb sin(kx lb /2)= lb sinc.
(9)P̃ (kx ) =(kx lb /2)
2π
In addition, for shifting the center of the bright zone fromx = 0 to x = xb , as shown in Fig. 2, the shift theorem [22]with respect to x is applied to (9) and the Fourier transformof shifted rectangular window P̃shift (kx ) is represented asP̃shift (kx )= P̃ (kx ) exp(jkx xb )
kx lbexp(jkx xb ).
= lb sinc2π(10)Then P̃shift (kx ) is assigned to (7), and the spatial filter in thewavenumber domain for generating bright zone F̃ (kx , ω) is
analytically derived aslb sinc (kx lb /2π) exp(jkx xb ).
F̃ (kx , ω) =G̃(kx , yb , 0, ω)(11)

As a result, a bright zone of arbitrary length lb can be generated at arbitrary horizontal position [xb , yb ] by the proposed
spatial filter in the wavenumber domain F̃ (kx , ω) using a continuous linear secondary monopole source distribution.
2.3. Practical implementation using a linear array ofloudspeakers
In a practical implementation, (11) must be discretized andtruncated [22]. When the channel number of loudspeakers
is N and the distance between adjacent loudspeakers is ∆x,
the length of the array is L = N ∆x and the spatial inverse
discrete Fourier transform (IDFT) of F̃ (kx , ω) is calculated
asFIDFT (x, ω) =1
L

N/2−1Xm=−N/2F̃ (kx , ω) exp(−2πjmn/N ),(12)
where x = n∆x, kx = 2πm/N ∆x, and −N/2 ≤ n ≤N/2 − 1 [22].
In the proposed method, the number of IDFT points in(12) is set to more than 2 × N to reduce the truncation error. This strategy is the same in a practical implementationof nearfield acoustical holography [22]. More than 2 × Nchannels of FIDFT (x, ω) are calculated, and their centered N
channels are finally used for filters in the temporal frequency
domain.

Computer simulations compared the proposed method withthe conventional EDM [7].
3.1. Acoustic energy difference maximization
EDM is briefly introduced here. When using K channels of control points at xco and N channels of loudspeakers at xsp , the spatial averaged correlation matrix
between control points and thePpositions of loudspeakers
Kis calculated as R(ω) = 1/K i=1 GHi (ω)Gi (ω), whereGi (ω) = [G3D (xco,i −xsp,1 , ω) · · · G3D (xco,i −xsp,N , ω)].
Rb (ω) is the spatial correlation matrix between the brightpoints and the positions of the loudspeakers and Rd (ω) isthat between the dark points and the positions of the loudspeakers. The filters of loudspeakers F (ω) are obtained
from the eigenvector of matrix [Rb (ω) − αRd (ω)] that corresponds to the largest eigenvalue of this matrix, where α is atuning factor [7].
3.2. Simulation condition
A sound field is assumed in a three-dimensional free field
where a spherical wave of a single frequency is radiated fromeach loudspeaker. The speed of sound is 343.36 m/s. A linear array of loudspeakers is set along the x-axis and centeredaround x = 0. The channel number of the linear array ofloudspeakers N is 64 and the distance between adjacent loudspeakers ∆x is 0.05. The length of array L is 3.2 m. The position and the length of the bright zone are [xb , yb ] = [−0.8, 2]
and lb = L/2 = 1.6 m, respectively.
In the EDM, a linear array of 64 control points is set parallel to an array of loudspeakers apart from yb along the y-axis.
32 control points at x < 0 are bright points xb , and the otherhalf are dark points xd . Turning factor α is set to 0.9999 for
maximizing the acoustic energy difference between two zones
from the pre-simulations.
In the proposed method, the number of IDFT points inEq. (12) is N × 2 = 128, and the 64 centered channels ofFIDFT (x, ω) are used as filters of the 64 loudspeakers.
The reproduced sound pressure level at position x is defined as
2

PSPL (x, ω) = 10log10 P̂ (x, ω) ,(13)where P̂ (x, ω) is the reproduced sound pressure at positionx.
Furthermore, for evaluating the sound pressure level between bright zone xb and dark zone xd , bright to dark ratio
BDR(ω) is also defined as2
Pxb P̂ (xb , ω)BDR(ω) = 10log10 P
2.
P̂
(x,ω)
dxd(14)

40

1.5

40

1.5

35

35

30

25

020
15

−0.5

0.5

25

015

−0.5

10

10
−1−155−1.5
123Y axis [m]

−1.5

0

1[dB]

23Y axis [m]

0[dB]

4040

1.520

Spatial Nyquist Frequency

11300.52502015−0.5

X axis [m]300.525020
15−0.51010−1−155−1.5
123Y axis [m]

−1.5

0

1[dB]

34040

1.535
130

25020
15−0.5

X axis [m]300.50.525020
15−0.51010−1−155−1.5123

Y axis [m](e) EDM (f = 3 kHz)

0[dB]

−1.5
12Y axis [m]3123
Temporal Frequency [kHz]45Fig. 4. Experimental results of bright to dark ratio BDR(ω)
defined in (14).

035
1100[dB](d) Proposed (f = 1 kHz)(c) EDM (f = 1 kHz)
1.5

2Y axis [m]EDM
Proposed

15

35

35

X axis [m]25(b) Proposed (f = 500 Hz)(a) EDM (f = 500 Hz)
1.5

30

20

BDR [dB]

0.5

X axis [m]X axis [m]30X axis [m]351

10[dB](f) Proposed (f = 3 kHz)

Fig. 3. Experimental results of reproduced sound pressurelevel PSPL (x, ω) defined in (13). White circles parallel with
y = 2 are control points.
For calculating (14), both zones are centered around control
line y = yb . The length of each zone is L/2 = 1.6 m, and thewidth of each zone is 0.4 m.
3.3. Simulation results
The results of the reproduced sound pressure level PSPL (x, ω)and the bright to dark ratio BDR(ω) are shown in Figs. 3 and4, respectively. Both results show that the proposed method
generated bright zones more accurately than the conventional
EDM, especially at more than 1 kHz. The results of Fig. 3also show that the length of the bright zone generated by theEDM at each frequency is different. In contrast, the lengths
in the proposed method are constant at all frequencies. Consequently, the proposed spatial filtering can efficiently control
sound pressures at the target line.

4. CONCLUSION
A method was proposed to generate acoustically bright anddark zones at arbitrary horizontal positions using a linear array of loudspeakers. In the proposed method, the spatial filterin the wavenumber domain was analytically derived by introducing the spectral division method in the horizontal planefor continuous linear distributions of secondary sources andmodeling sound pressures at the control line as a rectangular window corresponding to the bright and dark zones. Thecomputer simulation results show that the proposed method
generated bright and dark zones more accurately than conventional acoustic energy difference maximization.
5. RELATION TO PRIOR WORKThe work presented in this paper focused on a method for generating acoustically bright and dark zones using many loudspeakers. Most conventional methods are based on the numerical calculation of the inverse of the spatial correlation
matrix between control points and the positions of loudspeakers [1,3,4,8–10,12–16]. However, they are quite unstable because the acoustic inverse problem is very ill-conditioned [18,19]. On the other hand, the proposed method is based on analytically derived filters and introduces the spectral division
method [21]. An efficient spatial filter at the wavenumber domain is analytically derived by modeling sound pressures atthe control line as a rectangular window for generating acoustically bright and dark zones at arbitrary horizontal positions
using a linear array of loudspeakers.
6. ACKNOWLEDGEMENT
This study was partly supported by Grant-in-Aid for Young
Scientists B (No. 25871208) from the JSPS.

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