2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 20-23, 2019, New Paltz, NY

3D LOCALIZED SOUND ZONE GENERATIONWITH A PLANAR OMNI-DIRECTIONAL LOUDSPEAKER ARRAY
Takuma Okamoto
National Institute of Information and Communications Technology
3-5, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0289, Japanokamoto@nict.go.jp
ABSTRACTThis paper provides a 3D localized sound zone generation method
using a planar omni-directional loudspeaker array. In the proposed
method, multiple co-centered circular arrays are arranged on thehorizontal plane and an additional loudspeaker is located at the array’s center. The sound field produced by this center loudspeakeris then cancelled using the multiple circular arrays. A localized 3D
sound zone can thus be generated inside a sphere with a maximum
radius of that of the circular arrays because the residual sound field
is contained within the sphere. The resulting sound fields are decomposed into spherical harmonic spectra and the driving functionof the array is then obtained. Compared with the conventional approach that uses monopole pairs to control the even and odd spherical harmonic spectrum components, the proposed method can be
simply realized with a practical planar omni-directional array because it is sufficient to control the 0-th order component. Computer
simulations confirm the effectiveness of the proposed approach.
Index Terms— Sound field synthesis, localized sound zone,
circular loudspeaker array, spherical harmonic expansion, Ambisonics.
1. INTRODUCTION
Generation of a personalized sound zone using loudspeakers is an
attractive and important acoustic application. Many methods torealize personalized sound zones have been investigated and canbe broadly categorized into two main schemes. The first scheme
controls either the acoustic contrast or the energy between two
spaces [1–8]. In contrast, the second approach controls multiple
sound fields [9–18] such that they are simultaneously synthesizedfor different zones in space. These approaches typically control
sound zones located away from the loudspeakers.
In an alternative approach, a localized sound zone is generated
near loudspeakers such that the audible sound pressure only propagates close to the loudspeakers, while it propagates at very low
amplitudes beyond a reference distance. This approach is called
localized sound zone generation [19–21]. This technique is better
suited to actual implementation than the other methods mentioned
above, which control sound zones located away from the loudspeakers, because the sound pressures are not radiated at the walls and nosound reflections occur. Several localized sound zone generation
methods have been proposed.
One such method is based on evanescent wave reproduction [22] using linear or circular loudspeaker arrays [23, 24]. In
This study was partly supported by JSPS KAKENHI Grant Number
JP18K11387.

these methods, however, the distance attenuation properties athigh frequencies are too short, and the listening area becomes too
small because the evanescent waves decay in less than one wavelength [22].
Additionally, analytical approaches using linear, circular or
spherical arrays along with an additional loudspeaker have also
been proposed [19–21]. In these methods, the sound field producedby the additional loudspeaker is cancelled by the sound fields ofthe linear, circular or spherical arrays and a localized sound zone
can thus be generated as a residual sound field. However, undesired
sound pressures propagate toward the vertical angles in the methods
using the linear or circular arrays [19, 21] because they are basedon 2.5D sound field synthesis [25, 26] and only sound pressures on
the horizontal plane can be controlled. While the method using thespherical array can realize 3D localized sound zone generation, thismethod is impractical in terms of both array configuration and listening area because large numbers of loudspeakers are required toform the surrounding spherical array and the listening area is located inside this spherical array [20].
To address the aforementioned problems, this paper provides a
3D localized sound zone generation method with a practical planaromni-directional loudspeaker array. The proposed method is basedon 3D exterior sound field synthesis using a planar array [27, 28].
In [28], undesired sound pressures caused by acoustic noise
sources are cancelled using multiple co-centered circular arrays located on the horizontal plane. In this case, only the sound pressure
outside a sphere with a maximum radius of that of the circular arrays
can be cancelled in principle; however, the residual sound pressuresremain inside the sphere as in [28]. Using this property, an additional loudspeaker is introduced in place of noise sources in [28]and the sound field produced by the loudspeaker is then cancelledusing the circular arrays in the proposed approach, as in [19–21].
3D localized sound zone generation can thus be realized inside a
sphere with a maximum radius of that of the circular arrays.
In [27,28], however, both omni-directional and first-order loudspeakers are required to control B̌nm for n + |m| even and odd
spherical harmonic spectrum components of the 3D exterior soundfield. For actual implementation, first-order loudspeakers are approximately realized using monopole pairs [27, 28] as 3D soundfield recording with a planar microphone array [29]. In the approach
proposed here, the additional loudspeaker is located on the horizontal plane and the sound field produced by this loudspeaker contains
only even components, as noted in [28, 30]. This means that it issufficient to use omni-directional circular arrays. Additionally, by
locating the additional loudspeaker at the array’s center, the soundfield produced by the loudspeaker only contains the 0-th order component and the driving function for the circular arrays can then be© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including
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2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 20-23, 2019, New Paltz, NY

rQ

+

D0

jk
B̌00 = √

4π

D0
D

jk
−B̌00 = − √for r > 04π(a) Monopole sound field

D
Residual sound field
for r < rQ

for r > rQ(b) Exterior sound field synthesis(c) Proposed method

Figure 1: Scheme for proposed 3D localized sound zone generation method using planar omni-directional loudspeaker array: (a) sound field
produced by monopole loudspeaker located at origin with driving function D0 that only contains 0-th order spherical harmonic spectrum
component B̌00 for r > 0; (b) −B̌00 for r > rQ can be synthesized using multiple circular arrays with maximum radius rQ and driving
function D; (c) 3D localized sound zone that can be generated as a residual sound field for r < rQ by simultaneously driving D0 and D.
simplified. Compared with the existing approaches using monopole
pairs [27, 28], the proposed method can be simply realized using apractical planar omni-directional array with half the typical numberof loudspeakers. Additionally, the array configuration and listening
area in the proposed method are more practical than those of theconventional evanescent and residual approaches [19–21, 23, 24].
The scheme for the proposed method is shown in Fig. 1.
2. 3D EXTERIOR SOUND FIELD PRODUCED BY POINT
SOURCE ON HORIZONTAL PLANE
In this paper, the spherical coordinates relative to Cartesian coordinates are defined as used in [22].
The exterior expansion of the 3D sound field for regions exterior to any sound sources is given as:
S(r, θ, φ) =

∞
n
X
X

B̌nm hn (kr)Ynm (θ, φ),(1)n=0 m=−nwhere B̌nm and hn are the spherical harmonic spectra of the exteriorsound field and the n-th order spherical Hankel function of the firstkind, respectively, and k is the wavenumber [22];s(2n + 1) (n − |m|)! |m|m
Yn (θ, φ) =Pn (cos θ) ejmφ(2)4π(n + |m|)!
|
{z
}
|m|

Pn

(cos θ)|m|represents the spherical harmonics; Pn is the associated Legendre function [31]. Under the free-field assumption, the transfer
function of a point source located at r 0 = [r, θ0 , φ0 ]T to a pointr = [r, θ, φ]T is given by the 3D free-field Green’s function [22]:ejk|r −r 0 |4π|r − r 0 |
∞
n
X X
= jk
jn (kr0 )hn (kr)Ynm (θ0 , φ0 )∗ Ynm (θ, φ), (3)G(r, r 0 ) =n=0 m=−n√for r > r0 , where j = −1 and jn is the n-th order sphericalBessel function [22]. When the point source is located on the hori-zontal plane with θ0 = π/2, (3) is represented by:G(r, r 0 )|θ0 =π/2
= jk

∞
n
X
X

jn (kr0 )hn (kr)Pn|m| (0)e−jmφ0 Ynm (θ, φ). (4)n=0 m=−nFrom (4), the sound field produced by the point source located on
the horizontal plane only includes the spherical harmonic spectra for|m|n + |m| even because Pn (0) = 0 when n + |m| is odd [27–30].
Therefore, to synthesize a complete 3D exterior sound field using
loudspeakers on the horizontal plane, both monopole and vertical
dipole components are required. These vertical dipole components|m|include the vertical derivatives of Pn (0) and can control the odd
spectra. To approximately implement the vertical dipole components using actual monopole loudspeakers, monopole pairs are introduced [27, 28]. Therefore, the conventional approach requires
double the number of loudspeakers.
In the proposed approach, the sound field produced by a point
source located at the origin only including the 0-th order sphericalharmonic spectrum component (Fig. 1(a)), is cancelled using multiple co-centered circular loudspeaker arrays on the horizontal plane(Fig. 1(b)). As a result, the proposed method can be simply realized using a practical planar omni-directional array with half thenumber of loudspeakers required for the conventional method using
monopole pairs [27, 28].
3. PROPOSED 3D LOCALIZED SOUND ZONEGENERATION
From (3), the sound field produced by a monopole loudspeaker withdriving function D0 located at the origin (Fig. 1(a)) is given as:
S0 (r, θ, φ) = D0 G(r, r0 = 0)
= D0

0
n
X
X

jk
ejkr
√ hn (kr)Ynm (θ, φ) = D0,4πr4πn=0 m=−n(5)√where j0 (0) = 1, jn (0) = 0 for n > 0, P00 (0) = 1/ 4π andh0 (kr) = ejkr /jkr [22]. Subsequently, D0√is set at 1 for simplicity. From (1), S0 only has the B00 = jk/ 4π component for2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 20-23, 2019, New Paltz, NY∆z(a) Method with monopole pairs(b) Proposed method

Figure 2: Arrangements of loudspeakers with three radii r1,2,3 =
0.1, 0.15 and 0.2 m, where these circular arrays contain five, sevenand nine loudspeakers, respectively: (a) conventional method using
monopole pairs (∆z = 0.05 m) with 43 loudspeakers, where theloudspeaker for D0 is located at [0, 0, − 0.1]T in Cartesian coordinates; (b) proposed method with planar array of 22 loudspeakers.

(b) 20 log10 |S0+1 (x, y = 0, z)|(a) S0+1 (x, y = 0, z)

Figure 3: Results of conventional method using monopole pairs forf = 500 Hz: (a) residual sound field S0+1 (x, y = 0, z) producedby D0 and D even+odd ; (b) sound pressure level of residual soundfield 20 log10 |S0+1 (x, y = 0, z)|.
r > 0. In the proposed approach, S0 is cancelled using multiple
co-centered circular arrays.
A sound field synthesized using a continuous circular soundsource with radius rq on the horizontal plane is given as:
Z 2π
S1 (r, θ, φ) =D(rq , φ0 ) G(r, r q )|θ0 =π/2 dφ0 ,(6)(a) S1 (x, y = 0, z)(b) S0+1 (x, y = 0, z)(c) 20 log10 |S0+1 (x, y, z = 0)|(d) 20 log10 |S0+1 (x, y = 0, z)|Figure 4: Results of proposed method for f = 500 Hz: (a)sound field S1 (x, y = 0, z) produced by D using 21 loudspeakers; (b) residual sound field S0+1 (x, y = 0, z) produced using D0
and D; (c), (d) sound pressure levels for the residual sound fields
20 log10 |S0+1 (x, y, z = 0)| and 20 log10 |S0+1 (x, y = 0, z)|, respectively.
is obtained from (1) and (8). When the maximum order in (8) is N ,(9) can be represented in matrix form as:
even
1
even
B̊ m ' U |m| Ď m ,
2πjk

where

0

where D(rq , φ0 ) is the circular source driving function and r q =[rq , θ0 = π/2, φ0 ]T . As in [27, 28], D(rq , φ0 ) is decomposed into
the 2D cylindrical harmonic spectra as follows:D(rq , φ0 ) =∞X

D̊m,q ejmφ0 .

(7)

m=−∞

From (4) and (7), and by introducing multiple radii (i.e., q =1, 2, · · · , Q), (6) can be represented as:
Q Z 2πX
S1 (r, θ, φ) =D(rq , φ0 ) G(r, r q )|θ0 =π/2 dφ0
q=1= 2πjk

0

Q ∞
n
X
X X

q=1 n=0 m=−nD̊m,q Pn|m| (0)jn (krq )hn (kr)Ynm (θ, φ).
(8)

In [28], the primary sound field Bnm is cancelled using the multiple

circular arrays. In this case, the relationshipB̌nm + 2πjk

Q
X
q=1

D̊m,q jn (krq )Pn|m| (0) = 0

(9)

(10)



 m
even
m
m T
B̌ m = − B̌|m|, B̌|m|+2, · · · , B̌N,|m|

U|m| (r1 ) |m|
 U|m|+2 (r1 )U |m| = 
..


.
|m|

UN (r1 )|m|

U|m| (r2 )|m|

U|m|+2 (r2 )..
.
|m|
UN (r2 )

···

···
..
.
···

|m|

U|m| (rQ )

(11)



|m|
U|m|+2 (rQ ) 
,
..


.
|m|
UN (rQ )(12)

Un|m| (rq ) = jn (krq )Pn|m| (0),

(13)

h
iT
even
D̊ m = D̊m,1 , D̊m,2 , · · · , D̊m,Q ,

(14)

and

in the case where both N and m are either odd or even; otherwise,
replace N in (11) and (12) with N − 1. Then, the driving functionof the circular arrays to synthesize the sound field −Bnm for n+|m|even can be obtained as [28]:
−1
even
1  T
even
U |m| U |m|
UT(15)
D̊ m =|m| B̌ m .
2πjk

2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics(a) Point source(b) Method with monopole pairs

October 20-23, 2019, New Paltz, NY(c) Proposed method

Figure 5: Sound pressure level distributions for simple point source, the conventional method using monopole pairs, and the proposed method.
In the proposed approach, the synthesized sound field is not(11) but (5). Bnm thus only contains the 0-th order component as:

T
jk
even
√
B̌ 0 = −, 0, · · · , 0 .
(16)
4πThen, the driving function also only contains the 0-th order 2D
cylindrical harmonic component and can be simply obtained as:
−1
1 
J 0,
(17)
D̊ 0 = D = − 2 U T
0 U0
8π
where

D = [D(r1 , φ0 ), D(r2 , φ0 ), · · · , D(rQ , φ0 )]T ,
jk
1
jk
even
UT
= − √ √ J 0 = − J 0,
0 B̌ 04π
4π 4π
J 0 = [j0 (kr1 ), j0 (kr2 ), · · · , j0 (krQ )]T ,

(18)
(19)
(20)

and D is obviously independent of φ0 and can be represented using real number coefficients. The proposed approach is thus a special case of√the conventional method [28] when only cancelling
B00 = jk/ 4π. Compared with S0 , however, the −B00 produced
by D in (17) only holds for r > rQ , where rQ is the maximum circular array radius because the sound sources are located in r ≤ rQ .
Consequently, by simultaneously driving D0 and D, a 3D localized sound zone can be generated inside a sphere with rQ as theresidual sound field between S0 , given by (5), and S1 , given by(17) (Fig. 1(c)). In the proposed method, the listening area is entirely inside the sphere, except for the horizontal plane on which
the loudspeakers are arranged.
4. COMPUTER SIMULATIONS
Computer simulations were performed to evaluate the proposed
approach and compare it with the conventional method using
monopole pairs [28]. In all simulations, a 3D free-field was assumed and the speed of sound c was 343.36 m/s. Three co-centeredomni-directional circular loudspeaker arrays were introduced on thehorizontal plane and their radii were set to 0.1, 0.15, and 0.2 m. The
target frequency band ranged up to 800 Hz, as in [28]. The numbers of loudspeakers on these circular arrays were calculated to be
five, seven and nine, respectively, using an array design procedure
given in [28,29]. The total numbers of loudspeakers for the conventional and proposed methods were thus 43 and 22, respectively. In
the conventional method using monopole pairs, both the B even
m and
B odd
m components were controlled, as in [28]. In the simulationsof the conventional method, the loudspeaker for D0 was located at[0, 0, −0.1]T in Cartesian coordinates. In this case, the sound field

produced by D0 contains only B evenand B odd
0 components and the0
driving function for the conventional method, D even+odd , was also
independent of φ0 . The maximum order N in (12) was set at 100.
The loudspeaker arrangements for both the conventional and proposed methods are depicted in Fig. 2.
The results for sound fields S1 (x, y, z) and S0+1 (x, y, z), andtheir sound pressure levels, 20 log10 |S0+1 (x, y, z)|, produced using the conventional approach with monopole pairs and using the
proposed method with a planar omni-directional array at a frequency of f = 500 Hz are shown in Figs. 3 and 4, respectively.
Additionally, the results for the sound pressure levels on the z-axis(0.1 m ≤ z ≤ 1.0 m) for 50 Hz ≤ f ≤ 800 Hz produced using asimple point source and the conventional and proposed approaches
are plotted in Fig. 5. In Figs. 3 to 5, 20 log10 |S0+1 (x = 0, y =0, z = 0.1)| was set at 0 dB for all frequencies.
These results indicate that both the conventional and proposed
methods can generate 3D localized sound zones near the loudspeakers. Specifically, the proposed method’s listening area ranges up to
approximately rQ = 0.2 m and the proposed method can also realize steeper sound pressure attenuation than the conventional approach at all frequencies. This means that S0 can be more accurately cancelled when using the proposed method as comparedto the conventional approach because S0 in the proposed method
only includes the 0-th order spherical harmonic spectrum component and is easily synthesized. Additionally, the proposed method
does not require monopole pairs that include the approximation errors of dipole components.
Consequently, the proposed method can be simply realized using a practical planar omni-directional array with half the number of loudspeakers of the conventional method using monopole
pairs [28]. Additionally, the results indicate that the listening area
provided by the proposed method is more practical than those of theconventional evanescent and residual approaches [19–21, 23, 24].
Future work will include a detailed analysis of the relationships
among the number of circular arrays, their radii and the listening
area shape, and experiments using practically implemented arrays.
5. CONCLUSIONSThis paper has proposed a 3D localized sound zone generation
method using a planar omni-directional loudspeaker array. In thisapproach, the sound field produced by the central loudspeaker iscancelled using multiple circular arrays and a 3D localized soundzone can then be generated. The resulting sound fields were decomposed into their spherical harmonic spectra and the driving functionof the array was derived. The results of computer simulations validated the effectiveness of the proposed approach compared with theconventional method using monopole pairs.

2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

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