2.5D LOCALIZED SOUND ZONE GENERATIONWITH A CIRCULAR ARRAY OF FIXED-DIRECTIVITY LOUDSPEAKERS
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 an analytical approach to generating localizedsound zone close to loudspeakers in the horizontal plane using a circular array of fixed-directivity loudspeakers and an additional loudspeaker based on sound source dimension mismatch. A circular
loudspeaker array and an additional loudspeaker located on the array
center are simultaneously produced in mutual opposite phase. As aresult, the propagated sound pressure at the reference circle for 2.5dimensional sound field synthesis is completely cancelled, but thatnear the circular array is not compensated and localized sound zone
generation is realized. The driving signals of the circular array of
fixed-directivity loudspeakers and the produced sound pressures areanalytically derived. The results of computer simulations indicate
the effectiveness of the proposal. By taking the acoustic sourcereceiver reciprocity into account, the proposed filter coefficients ofloudspeakers are also directly used for close-talking microphone array with directional microphones in the horizontal plane.
Index Terms— 2.5D sound field control, localized sound zone,
circular loudspeaker array, spherical harmonic expansion, Ambisonics.
1. INTRODUCTION
Generating a personalized sound zone by using loudspeakers is
an important and attractive acoustic communication technique. To
realize a personalized sound zone, many approaches have been
proposed, and are broadly categorized into two main approaches.
The first controls the acoustic contrast or the energy between two
spaces [1–8]. In addition, an extended approach controls multiple
sound fields [9–17] so that they are simultaneously synthesized for
different zones in space. These methods typically control soundzones away from the loudspeakers.
In another approach, a localized sound zone is generated near
loudspeakers so that audible sound pressure is only propagated close
to them but at very low amplitudes beyond the reference distance. In
this paper, this approach is called localized sound zone generation.
Several methods for localized sound zone generation have been
proposed. One is based on evanescent wave reproduction [18] using linear or circular arrays of loudspeakers [19, 20]. An analytical method using a linear and circular loudspeaker array combination has been proposed for 3-dimensional (3D) localized sound field
propagation [21, 22]. In addition, an analytical approach using a linear loudspeaker array and an additional monopole directivity loudspeaker based on sound source dimension mismatch has also beenThis study was partly supported by JSPS KAKENHI Grant Number
JP18K11387.

provided [23]. These methods, however, require a large number ofloudspeakers since they are based on higher-order modal control in
the angular and cylindrical harmonic spectrum domains. To address
the problem, this paper extends a previous method based on soundsource dimension mismatch with a linear array [23] into an analytical approach with a circular array of fewer loudspeakers.
Sound field synthesis systems are frequently simplified for synthesis in the horizontal plane. The sound sources are then arranged
on a line or a circle. In actual implementations, 3D monopole
sources instead of 2D line sources are usually employed for thesound sources. Such approaches are called 2.5D sound field synthesis [7, 24–29]. 2.5D sound field synthesis with a linear or a circularsound source can only synthesize correct sound pressure at the reference line or circle. In a previous method with a linear array [23], thesound pressure produced by a point source is cancelled by a linear
array at the reference line and localized sound zone generation in thehorizontal plane near the linear array can be realized by the residual
sound pressure between the point source and linear array.
In the proposed approach, the sound pressure produced by aloudspeaker is cancelled by a circular loudspeaker array and localized sound zone generation in the horizontal plane near the circulararray can also be realized.
In exterior sound field control with single-layer monopole
source distributions, however, a forbidden frequency (Bessel-zero)problem occurs [7, 18]. For considering actual loudspeaker directivity [27, 30, 31] and avoiding the forbidden frequency problem, acircular array of fixed directivity loudspeakers [30] is introduced in
the proposal as in 2D sound field recording with a circular radial
directional microphone array [32, 33]. By locating an additional
fixed-directivity loudspeaker on the array center, the proposal cangenerate a localized sound zone in the horizontal plane by controlling only the 0-th and first order components of 2D cylindricalharmonic spectrum and the number of loudspeakers for the circular array can be reduced compared with conventional higher-ordermodal control methods [20–23].
In addition, by taking the acoustic source-receiver reciprocity [22]into account, the proposed filter coefficients of loudspeakers are directly employed for close-talking microphone array [34–36] withdirectional microphones [32] in the horizontal plane.
2. PROPOSED ANALYTICAL FORMULATION
2.1. Sound field produced by circular fixed-directivity soundsourceA continuous circular fixed-directivity sound source distribution
with radius r1 centered at the origin on the x-y plane is consideredinstead of a circular array of fixed-directivity loudspeakers (Fig. 1).

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y

A sound field synthesized by a continuous circular sound source isgiven asS1 (r, θ, ϕ) =Z 2π

D1 (ϕ1 )T (r, r 1 )dϕ1 ,<latexit sha1_base64="rf5WtD2FO1bFDQSrh96tL06skOQ=">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</latexit>

(1)

rref
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0D1

where D1 (ϕ1 ) is the driving function of the circular source andT (r, r 1 ) is the transfer function from sound source position r 1 =[r1 , π/2, ϕ1 ] to receiver position r. Similar to previous works [27,31], T (r, r 1 ) is modeled as a weighted combination of a monopoleand a dipole with the main lobe pointing away from the center. Under the free-field assumption, T (r, r 1 ) is given as<latexit sha1_base64="kECeVdONZkREA05auDq+YDTMi20=">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</latexit>

in

g

zo

ne



j
cos α ,
k|r − r 1 |
(2)

en



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x

D0
r1

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T (r, r 1 ) =<latexit sha1_base64="0jEde/BFIENs7k9a32fBI5gsmnU=">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</latexit>√
where j = −1, k is the wavenumber, a is the first-order weighting
parameter and α is the angle from the loudspeaker axis. As in previous works [27, 30], (2) for r > r1 is represented by the sphericalharmonic expansion [18] and given as
T (r > , r 1 ) = jk

∞
n
X
X

n=0 m=−nhn (kr) {ajn (kr1 )−j(1 − a)jn′ (kr0 ) Ynm (θ, ϕ)Ynm (π/2, ϕ1 )∗ ,(3)

where jn and jn′ are the n-th order spherical Bessel function and its
radial derivative, hn is the n-th order spherical Hankel function of
the first kind [18], andYnm (θ, ϕ) =s

(2n + 1) (n − |m|)! |m|Pn (cos θ)ejmϕ
4π(n + |m|)!

(4)

is the |m|-th order associated Legendre polynomial of the n-th degree [37].
As in analytical 2.5D higher-order Ambisonics [24,25,27], when
the cylindrical harmonic expansion is applied to (1) with θ = π/2,
the circular convolution theorem holds and (1) is represented as(5)

In addition, the 2.5D spherical harmonic expansion of transfer function T (r, r 1 ) for r > r1 is derived from (3) and given asT̊m (r> , r1 ) = jk

∞
X

|m|
2
hn (kr)jn,1 (a, r1 )Qmn Pn (0) ,

(6)

n=|m|

where
jn,1 (a, r1 ) = ajn (kr1 ) + j(1 − a)jn′ (kr1 ),Qmn =2n + 1 (n − |m|)!
.
4π (n + |m|)!

(7)
(8)

In (7), the radial derivative of the n-th order spherical Bessel function is analytically obtained from the following relationship [18]:jn′ (x) = jn−1 (x) −n+1
jn (x).
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Fig. 1. Arrangements of a circular array of fixed-directivity loudspeakers with radius r1 and driving signals D1 , an additional loudspeaker located on the array center with driving signal D0 , and listening and quiet zones. rref is a reference radius for deriving D1 in
2.5-dimensional sound field control.
2.2. Sound field produced by fixed-directivity sound source atcircular source center
A sound field produced by a fixed-directivity sound source locatedat r 0 = [r0 , θ0 , ϕ0 ] is given as
S0 (r, θ, ϕ) = D0 T (r, r 0 ),|m|
is the m-th order spherical harmonics of the n-th degree, and Pn

S̊m,1 (r) = 2π D̊m,1 T̊m (r, r1 ).

Quiet zone

(9)

(10)

where D0 is the source driving function. After that, D0 is set to 1
for simplicity. When r 0 is located on the origin (Fig. 1), r0 = 0 and
S0 (r, θ, ϕ) is represented as




j
ejkr
a + (1 − a) 1 +cos α
S0 (r, θ, ϕ) = T (r, 0) =
4πr
kr
jk
k
= a h0 (kr) − (1 − a) h1 (kr) cos α,
(11)
4π
4π
where the relationships [18]h0 (x) =

ejx
jx

and

h1 (x) = −(j + x)ejx
x2

are introduced.
When the mainlobe of the dipole component is along the xaxis (Fig. 1), α = ϕ and the 2D cylindrical harmonic expansionof S0 (r, θ, ϕ) on the x-y plane is obtained as
Z 2π
1S̊m,0 (r) =S0 (r, π/2, ϕ)e−jmϕ dϕ.
(12)
2π 0
From (11), S̊m,0 (r) obviously has only the 0-th and first order components and represented asjk
h0 (kr),4πkS̊±1,0 (r) = −(1 − a) h1 (kr),8πS̊0,0 (r) = a

(13)
(14)

where12πZ 2πcos ϕe±jϕ dϕ =

0

1
.
2

(15)

2.3. Driving function of circular source
To cancel the sound pressure produced by the fixed-directivity soundsource with D0 located at the origin on the x-y plane at a reference
radius rred by using the circular sound source, driving function of
the circular source D1 is analytically derived (Fig. 1).
From (5), (13) and (14), D1 also has only the 0-th and first ordercomponents and given asD̊0,1 = −S̊0,1 (rref )jkh0 (krref )= −a,
2π T̊0 (rref , r1 )8π 2 T̊0 (rref , r1 )D̊±1,1 = −kh1 (krref )S̊±1,1 (rref )= (1 − a). (17)
2π T̊±1 (rref , r1 )16π 2 T̊±1 (rref , r1 )When r = rref , the residual sound pressure is completely 0. In the
proposed approach, a horizontal localized sound zone is generated
by residual sound field
1
X

S̊m,0+1 (r)ejmϕ

m=−1

(
)
T̊0 (r, r1 )jk=ah0 (kr) − h0 (krref )4π
T̊0 (rref , r1 )()
T̊±1 (r, r1 )k
h1 (kr) − h1 (krref )cos ϕ− (1 − a)4π
T̊±1 (rref , r1 )(20)
for r < rref where T̊−1 (r, r1 ) = T̊1 (r, r1 ).
2.4. Driving signals of circular array of fixed directivity loudspeakers
A continuous circular source is finally discretized into a circular array of fixed-directivity loudspeakers (Fig. 1). The driving signal ofeach loudspeaker at ϕl in the temporal frequency domain is analytically obtained as
D1 (ϕl ) =1
XD̊m,1 ejmϕl

m=−1

= −a

(b) S1 (r, ϕ)(c) S0+1 (r, ϕ)(d) 20 log10 |S0+1 (r, ϕ)|(16)

The 2D cylindrical harmonic spectrum of the residual soundpressure produced by D0 and D1 (ϕ1 ) is also analytically derivedas()
T̊0 (r, r1 )
jk
,
(18)
S̊0,0+1 (r) = ah0 (kr) − h0 (krref )4π
T̊0 (rref , r1 )()
T̊±1 (r, r1 )k
S̊±1,0+1 (r) = −(1 − a)h1 (kr) − h1 (krref ).
8π
T̊±1 (rref , r1 )(19)

S0+1 (r, ϕ) =(a) S0 (r, ϕ)

jkh0 (krref )kh1 (krref )+ (1 − a)cos ϕl .
8π 2 T̊0 (rref , r1 )8π 2 T̊1 (rref , r1 )(21)

When the number of loudspeakers is L, order m of the 2D cylindrical harmonic spectrum can be calculated up to M = ⌊(L−1)/2⌋,

Fig. 2. Results: (a) sound field S0 (r, ϕ) produced by D0 with cardioid directivity (a = 0.5), (b) sound field produced S1 (r, ϕ) by D1with L = 8 loudspeakers, r1 = 0.1 m, a = 0.5 and rref = 0.5 m,(c) residual sound field S0+1 (r, ϕ) produced by D0 and D1 , and (d)sound pressure level of residual sound field 20 log10 |S0+1 (r, ϕ)|produced by D0 and D1 . f = 1 kHz and black circles are loudspeakers.
where ⌊·⌋ is the floor function [7, 24, 27]. In the proposed approach,M = 1 and the minimum number of L is 3 and the proposal canbe theoretically implemented by 3 + 1 = 4 fixed-directivity loudspeakers. The proposed approach, therefore, can be realized by using
fewer loudspeakers compared with conventional higher-order modal
control methods [20–23].
3. COMPUTER SIMULATIONS
Computer simulations evaluated the proposed approach. In all thesimulations, a three-dimensional free field was assumed. The speedof sound c was 343.36 m/s. For taking actual loudspeaker size into
account, a circular array of L = 8 (rather than 3) cardioid directivity
loudspeakers (a = 0.5) with radius r1 = 0.1 m and an additional
cardioid directivity loudspeaker (a = 0.5) located on the array center were introduced. The spatial Nyquist frequency of the circulararray was about fNyq = 2.19 kHz. n in (6) was truncated to 100.
The results of sound field in the x-y plane produced by D0 , D1
and D0 + D1 with rref = 0.5 m for temporal frequency f = 1 kHz
are shown in Fig. 2. In addition, the results of sound pressure level
20 log10 |S(r, ϕ)| produced by D0 and D0 + D1 with rref = 0.5 m,
and rref = 0.25, 0.5, and 1 m for ϕ = 0, z = 0 m and f =1 kHz where 20 log10 |S(r, ϕ = 0)| with r = 0.2 m was set to
0 dB for all conditions are plotted in Fig. 3. As in a previous methodwith a linear loudspeaker array [23], the result of Fig. 3 shows thatthe produced sound pressure is completely cancelled at set reference
radius rref . If rref is small (= 0.25 m), the distance attenuation
property is precipitous, but the produced sound pressure level at r >Fig. 3. Results of sound pressure level 20 log10 |S(r, ϕ)| producedby simple cardioid loudspeaker with a = 0.5 and proposal with a =0.5, L = 8 loudspeakers, r1 = 0.1 m, and rref = 0.25, 0.5, and 1 m
for ϕ = 0, z = 0 m and f = 1 kHz where 20 log10 |S(r, ϕ = 0)|
with r = 0.2 m is set to 0 dB for all conditions.

Monopole (a = 1)

Cardioid (a = 0.5)

Fig. 4. Results of sound pressure level distribution for monopole(a = 1) and cardioid (a = 0.5) directivity conditions with L = 8
loudspeakers, r1 = 0.1 m, rref = 0.5 m, ϕ = 0 and z = 0 m where
spatial Nyquist frequency of circular array is about 2.19 kHz.

rref is increased again because the distance attenuation properties
between sound sources D0 and D1 are quite different. rref must
be set to an optimal value (= 0.5 or 1.0 m). From the results ofFigs. 2 and 3, the proposed method with an optimal reference radius
can then generate a localized sound zone compared with a simple
cardioid loudspeaker.
To compare the proposed approach using cardioud directivity
loudspeakers (a = 0.5) with that using simple monopole directivity ones (a = 1), Figure 4 shows the results of sound pressurelevel distribution for monopole and cardioid directivity conditions
with rref = 0.5 m, ϕ = 0 and z = 0 m for 100 Hz ≤ f ≤3 kHz and 0.2 m ≤ r ≤ 1.0 m. Although the proposal with cardioud directivity can stably realize localized sound zone up to spatial
Nyquist frequency fNyq = 2.19 kHz, that with simple monopole
directivity is quite unstable because of the forbidden frequencies
similar in a previous work [7]. Figure 5 plotts the results of 0-th
order 20 log10 |T̊0 (rref , r1 )| in (6) and 0-th order Bessel function
J0 (kr1 ) [18] for monopole (a = 1) and cardioid (a = 0.5) directivity conditions with r1 = 0.1 m and rref = 0.5 m. The result
indicates that the instability with monopole directivity loudspeakers
just corresponds to J0 (kr1 ) = 0 and T̊0 (rref , r1 ) with cardioud directivity (a = 0.5) is more stable than that with monopole directivity(a = 1).
Consequently, the proposed analytical approach with fixed-

Fig. 5. Results of 20 log10 |T̊0 (rref , r1 )| and J0 (kr1 ) for monopole(a = 1) and cardioid (a = 0.5) directivity conditions with r1 =0.1 m and rref = 0.5 m.
directivity loudspeakers and an optimal reference radius can effectively generate a localized sound zone in the horizontal plane withoutforbidden frequencies by using fewer loudspeakers compared withconventional higher-order modal control methods [20–22]. Future
work includes the evaluation of the sound pressure radiated to thevertical direction as conducted in a previous method with a linear
array [23].
4. CONCLUSIONSThis paper proposed an analytical method for generating localizedsound zone near loudspeakers in the horizontal plane with a circulararray of fixed-directivity loudspeakers and an additional loudspeakerbased on sound source dimension mismatch. The driving signalsof the circular array of fixed-directivity loudspeakers and the produced sound pressures were analytically derived. The proposal caneffectively generate a localized sound zone in the horizontal plane byusing fewer loudspeakers compared with conventional higher-ordermodal control methods. The results of computer simulations suggested the effectiveness of the proposal.
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