2015 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 18-21, 2015, New Paltz, NY

ANALYTICAL METHODS OF GENERATING MULTIPLE SOUND ZONESFOR OPEN AND BAFFLED CIRCULAR LOUDSPEAKER ARRAYS
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 proposes analytical methods for generating acousticallybright and dark zones on the horizontal plane using circular loudspeaker arrays without and with a cylindrical rigid baffle. Bothsound fields on the horizontal plane produced by continuous circular monopole source distributions without and with a rigid baffle
are analytically calculated using 2.5D sound field representation derived from the 3D cylindrical harmonics expansion. Efficient spatial
filters in the spatial Fourier series domain are analytically derived bymodeling sound pressures at the control circle as a rectangular window that corresponds to bright and dark zones. The computer simulation results show that the proposed methods can generate brightand dark zones near the loudspeakers more accurately than conventional least squares and beamforming methods.
Index Terms— Multiple sound zones, personal audio system,
circular loudspeaker array, beamforming, cylindrical harmonics expansion
1. INTRODUCTION
Generating acoustically bright and dark zones [1] and multiple
sound zones are important and challenging acoustic communication techniques. They are useful not only for personal sound systems [2–8] but also for multilingual guide services and other virtual reality applications. Many methods have been proposed that
control both the acoustic contrast and the energy between two
spaces [1,3–6,9–14] and reproduce multiple sound fields [8,15–19]
using multiple loudspeakers. For realizing personal audio systems
and multiple sound zones, these approaches are more effective thanbeamforming methods [2, 20–22], which maximize the energy to
the target direction with the given input source power.
Most existing methods are based on the least squares solution numerically calculated using the control points and the loudspeaker positions [1, 3–6, 11, 17]. Such methods, however, arequite unstable because the acoustic inverse problem is very illconditioned [23]. This problem is the same in the least squares
based pressure-matching method [24].
Analytical approaches, on the other hand, have been proposed
for generating bright and dark zones at arbitrary horizontal positions
with arbitrary width using a linear sound source [10, 12]. Thesemethods are based on the spectral division method (SDM) [25],which is a sound field reproduction scheme, and spatial filtering in
the wavenumber domain. This method [12] can generate bright anddark zones more efficiently than the conventional energy differenceThis study was partly supported by JSPS KAKENHI Grant Numbers
25871208 and 15K21674.

maximization method [9]. However, as sound field reproduction using linear loudspeaker arrays [25, 26], this method is based on thecontinuous linear distributions of sound sources, and many loudspeakers and a large production space are required.
To generate bright and dark zones using a compact loudspeakerarray suited for practical implementations, this paper proposes new
analytical methods for open and baffled circular loudspeaker arraysthat reduce the number of loudspeakers and the production space.
Pressure-matching and mode-matching beamforming methods
using a circular loudspeaker array mounted on a cylindrical rigid
baffle have been proposed [20, 21, 27–29]. These methods, however, cannot precisely control sound field near the loudspeakers because the formulations in [27–29] include the far field approximation used in [23] and a horizontal far field is assumed in [20,21]. To
control sound field near the loudspeakers effectively, the proposed
methods derive analytical solution without far field approximation.
Moreover, they can control sound field using an open circular loudspeaker array (without a baffle) suited to be combined with visualapplications.
The proposed methods are based on 2.5D sound field representation [25, 30] analytically derived from the 3D cylindrical harmonics expansion [23] for a continuous circular sound source, andbright and dark zones are generated outside a circular source on thehorizontal plane. Firstly, both sound fields on the horizontal plane
produced by continuous circular sound sources without and with acylindrical baffle are analytically derived. After that, as a method
based on SDM using a linear sound source [12], efficient spatial filters in the spatial Fourier series domain are also analytically derived
by modeling sound pressures at the control circle as a rectangular
window that corresponds to bright and dark zones.

2. 2.5D SOUND FIELD REPRESENTATION FOR
CONTINUOUS CIRCULAR MONOPOLE SOURCE
DISTRIBUTIONS
For actual implementations, sound field reproduction systems are
frequently simplified to be reproduced in the horizontal plane. Thenthe secondary sources are arranged on a line or a circle. In actual
implementations, monopole sources instead of line sources are usually employed for secondary sources. Such approaches are called
2.5D sound field reproduction [25, 30]. In this section, 2.5D soundrepresentation for continuous circular sound sources on the horizontal plane is analytically derived from the 3D cylindrical harmonics
expansion.

© 2015 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
reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, orreuse of any copyrighted component of this work in other works.

2015 IEEE Workshop on Applications of Signal Processing to Audio and Acousticszr = [r, φ, z]T
y

φ

r

x



x2 + y 2
y!
φ = tan−1xr=Figure 1: Definition of cylindrical coordinates r = [r, ϕ, z]T relative to Cartesian coordinates x = [x, y, z]T .
2.1. Sound field due to a cylindrical sound source
To describe precise horizontal sound field produced by a continuous
circular monopole source distribution, the sound field produced bya continuous cylindrical sound source is introduced as in [31].
As in [23], cylindrical coordinates r = [r, ϕ, z]T relative toCartesian coordinates x = [x, y, z]T are defined in Fig. 1. Asin [31, 32], the sound pressure P (r, ω) synthesized at position r =[r, ϕ, z]T by a continuous cylindrical sound source with radius r0
is given as
Z 2πZ ∞
P (r, ω) =D(r 0 , ω)G(r, r 0 , ω)r0 dz0 dϕ0 ,
(1)
0

−∞

where ω = 2πf denotes the radial frequency, f represents the temporal frequency, D(r 0 , ω) stands for the driving function of a cylindrical sound source at r 0 = [r0 , ϕ0 , z0 ]T and G(r, r 0 , ω) is thetransfer function from a sound source position r 0 to a receiver position r.
2.2. Sound field due to a cylindrical source without a baffle
When a continuous cylindrical sound source is assumed to be acoustically transparent and under the free-field assumption, G(r, r 0 , ω)in (1) is the three-dimensional free-field Green’s function [23].
When r > r0 , using the two-dimensional spatial Fourier transform of the three-dimensional free-field Green’s function with respect to ϕ and z derived in [23, 31], the 3D cylindrical harmonics
expansion of G(r, r 0 , ω) is represented as
Gopen (r, r 0 , ω) =
∞
X

1
=
2π
n=−∞

Z ∞

ejk|r −r 0 |4π|r − r 0 |October 18-21, 2015, New Paltz, NY

assumed, precise representation of three-dimensional sound fieldmust be required for describing sound field on the horizontal plane
as conventional 2.5D sound field reproduction methods [25, 30].
In this subsection, the 3D cylindrical harmonics expansion of thetransfer function Gbaffled (r, r 0 , ω) from a sound source mounted
on a rigid cylinder r 0 to a receiver r(r > r0 ) is analytically derived.
As in [23, 32], given a rigid cylinder of radius r0 , the incident field Pi (r, ω) and scattered field Ps (r, ω) are respectively described as
Z ∞
∞
X
1
Pi (r, ω) =Cn (kz , ω)Jn (kr r)ejnϕ ejkz z dkz ,
2π −∞
n=−∞
(4)

Ps (r, ω) =

∞
X

1
2π
n=−∞

Z ∞

An (kz , ω)Hn(1) (kr r)ejnϕ ejkz z dkz ,

−∞

(5)

where Cn (kz , ω) and An (kz , ω) are coefficients that depend on
the incident field. The sound pressure gradient on the surfaceof the rigid baffle with radius r0 becomes zero and the relation(1)′
An (kz , ω) = −Jn′ (kr r0 )Cn (kz , ω)/Hn (kr r0 ), is obtained as
(1)′
(1)
′
in [32], where Jn and Hn are the derivatives of Jn and Hn , respectively. If r = r0 , the total sound field P (r, ω) = Pi (r, ω) +Ps (r, ω) is given by the sum of the incident and scattered fields as
P (r 0 , ω) =

∞
X

1
2π
n=−∞

Z ∞

2jCn (kz , ω)(1)′−∞ πkr r0 Hn (kr r0 )ejnϕ0 ejkz z0 dkz .
(6)

Whereas a plane wave is impinged in [20], for the case of a sphericalwave impinging from r (r > r0 ), the incident sound field Pi (r 0 , ω)is obtained from (3) and represented as
ejk|r 0 −r |4π|r 0 − r|
Z ∞
∞
X
1
j
=
Jn (kr r0 )Hn(1) (kr r)ejn(ϕ0 −ϕ) ejkz (z0 −z) dkz .
2π
4
−∞
n=−∞

Pi (r 0 , ω) =

(7)

(2)
(1)

j
Jn (kr r0 )Hn(1) (kr r)ejn(ϕ−ϕ0 ) ejkz (z−z0 ) dkz ,
−∞ 4
(3)

√
(1)
where j = −1, k = ω/c, Hn and Jn denote the wavenumber,the n-th order Hankel function of the first kind and the n-th order
Bessel function of√the first kind, respectively [23]. c is the speed ofsound and kr = k2 − kz2 .
2.3. Sound field due to a cylindrical source with a rigid baffle
In the proposed method, a continuous circular sound source
mounted on the surface of a cylindrical rigid baffle with radius r0
and infinite length to the z-axis is assumed as in [20]. In [20], however, sound field on the horizontal plane due to a continuous circular source (z = 0) was only considered with the far field assumption. When a continuous circular monopole source distribution is

From (4) and (7), Cn (kz , ω) = (j/4)Hn (kr r)e−jnϕ e−jkz z . Finally, using the principle of reciprocity between source and receiver
as in [20], the 3D cylindrical harmonics expansion of the transfer
function Gbaffled (r, r 0 , ω) from a sound source on a rigid cylinderr 0 to a receiver r(r > r0 ) is analytically derived from (6) and given
asGbaffled (r, r 0 , ω)
Z ∞
∞
(1)
X
−Hn (kr r)
1=ejn(ϕ−ϕ0 ) ejkz (z−z0 ) dkz .
2π −∞ 2πkr r0 Hn(1)′ (kr r0 )
n=−∞

(8)

2.4. 2.5D sound field representation for continuous circularsound sources without and with a rigid baffle
Sound pressure P (r, ω) synthesized at position r by a continuous
circular monopole source distribution with radius r0 centered at the2015 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

origin on the x-y plane (Fig. 2) is also given as
Z 2π
P (r, ω) =D(r0 , ϕ0 , z = 0, ω)G(r, r 0 , ω)r0 dϕ0 .

October 18-21, 2015, New Paltz, NY

Bright zone
P (rref , φ, 0, ω) = 1

(9)

Φ

0

Just as in [30], when the spatial Fourier series expansion [23] is applied to (9) with z = 0, the circular convolution theorem holds andthe driving function of a circular sound source is directly derived asD̊n (r0 , ω) =P̊n (r, ω).
2πr0 G̊n (r, r0 , ω)

These transfer functions can precisely describe horizontal soundfields produced by continuous circular monopole source distributions without and with a rigid baffle.
3. GENERATION OF MULTIPLE SOUND ZONES USINGA CIRCULAR SOUND SOURCE
3.1. Spatial filter modeling
For producing the sound signal S(ω) at a zone using a circularsource, optimal filter F (r 0 , ω) of a circular source is derived here.
The sound source driving function is then given as D(r 0 , ω) =S(ω)F (r 0 , ω). S(ω) is independent on F (r 0 , ω) and set to 1.
In the proposed method, bright and dark zones are generated outside a circular source on the horizontal plane. Then D(r 0 , ω) =F (r 0 , ω), and F̊n (r0 , ω) = P̊n (r, ω)/2πr0 G̊n (r > r0 , ω).
As the SDM-based method [10,12], to generate bright and darkzones at continuous receiver circle r = rref on the x-y plane,
P (rref , ϕ, 0, ω) is also continuously set to 1 or 0 at all the temporalfrequencies. The positions at P (r, ω) = 1 and 0 correspond to thebright and dark points, respectively. For generating the bright zone
of width Φ centered around ϕ = ϕS at r = rref as shown in Fig. 2,
P (r, ϕ, 0, ω) is also modeled by a rectangular window and given as 1, for ϕS − Φ/2 ≤ ϕ ≤ ϕS + Φ/2
P (rref , ϕ, 0, ω) =
 0, elsewhere.
(13)
The spatial Fourier series expansion is applied to (13) with respect
to ϕ, described as a sinc function [23], and represented asnΦ
e−jnϕs .
P̊n (Φ, ϕS , ω) = Φsinc(14)
2π

The spatial filter in the spatial Fourier series domain for generatingbright and dark zones is then analytically derived asΦsinc(nΦ/2π)e−jnϕs.
2πr0 G̊n (rref > r0 , ω)Circular sound source

φs

rref
y
r0

(10)

As in [30], the Fourier series coefficient functions of the horizontal transfer function from a circular sound source with radius r0 to a reference circle r is analytically derived. For bothcases of circular sound sources without and with a cylindricalrigid baffle, G̊open,n (r > r0 , ω) and G̊baffled,n (r > r0 , ω) are
derived from Gopen (r, r0 , ϕ, ϕ0 , z = z0 = 0, ω) in (3) and
Gbaffled (r, r0 , ϕ, ϕ0 , z = z0 = 0, ω) in (8), and given as
Z ∞1
jG̊open,n (r > r0 , ω) =Jn (kr r0 )Hn(1) (kr r)dkz , (11)2π −∞ 4Z ∞
(1)
−Hn (kr r)
1
G̊baffled,n (r > r0 , ω) =dkz . (12)2π −∞ 2πkr r0 Hn(1)′ (kr r0 )F̊n (r0 , rref , Φ, ϕs , ω) =Reference circlez(15)

Dark zone
P (rref , φ, 0, ω) = 0

xFigure 2: Produced sound pressure P (rref , ϕ, 0, ω) modeled byrectangular window at rref centered ϕs with width Φ.
Consequently, similar to the SDM-based method [12], a bright zoneof arbitrary width Φ can be generated at arbitrary horizontal direction ϕs with rref > r0 by the proposed spatial filter in spatial
Fourier series domain F̊n (r0 , rref , Φ, ϕs , ω) using a continuous circular monopole source distribution.
3.2. Actual implementation using a circular loudspeaker array
For an actual implementation, a circular loudspeaker array insteadof a continuous circular source is used and (15) must be discretized.
When the number of loudspeakers is L and the array’s radius is r0 ,the order n of the spatial Fourier series in (15) can be calculated
up to N = ⌊(L − 1)/2⌋, where ⌊·⌋ is the floor function. However, the spatial Fourier series at |n| > |kr0 | is the evanescent component and is quite large because of the inverse propagation whenrref > r0 [23]. To calculate a stable filter, n is up to N = ⌊|kr0 |⌋if ⌊|kr0 |⌋ < ⌊(L − 1)/2⌋. Finally, the filter coefficient of each
loudspeaker at ϕl in the temporal frequency domain is obtained as
F (r0 , ϕl , ω) =NXF̊n (r0 , ω)ejnϕl ,n=−Nl = 1, 2, · · · L.

(16)

The proposed methods can be realized by fewer loudspaekers
than in the SDM-based method using a linear loudspeaker array [12]if the circular array’s radius is small.
4. EXPERIMENTS
Computer simulations were performed to evaluate the proposed
methods compared with conventional least squares (LS) basedpressure-matching methods [24] and beamforming (BF) methods
using circular loudspeaker arrays with and without a rigid baffle.
For the case of a circular array mounted on a rigid baffle, the analytical beamforming solution proposed in [20] was used and combined with the spatial filtering in (14) and given as
FBF,baffled (r0 , ϕl , ω)=NX
kr0 Hn′ (kr0 )Φsinc(nΦ/2π)ejn(ϕl −ϕs ).
4j (1−n)
n=−N

(17)

Cn (kz , ω) = j −n e−jnϕs as in [20] and kz = 0 in (4), the beamforming filter combined with the spatial filtering in (14) for an open

2015 IEEE Workshop on Applications of Signal Processing to Audio and Acousticsz = 0 [m]z = 0 [m]z = 0 [m]

1.5

0

1

-0.5

0.5
-20

0
-0.5

-30

-10

Y axis [m]

-20

0

0
X axis [m]1LS (open)

-1
-40

-1.5
-1

[dB]

0
X axis [m]

1

-1

-20

0
-0.5

1

-20

0
-0.5

-30

-40

-1

[dB]

0
X axis [m]

1

0

0

1

2
3
Temporal Frequency [kHz]

4

5

-40

-1.5
-1

[dB]

0
X axis [m]

1

[dB]

Beamforming (baffled) Proposed (baffled)

LS (baffled)

5

0.5

-1

-1.5

-40

0
X axis [m]

0

-30

-1

-1.5

LS method (open)
LS method (baffled)
Beamforming (open)
Beamforming (baffled)
Proposed (open)
Proposed (baffled)

10

-10

Y axis [m]

0.5

-30

-1

15

z = 0 [m]-10

Y axis [m]

-20

0

[dB]

1

-10

-0.5

1

1.5

0

1

0.5

0
X axis [m]Proposed (open)z = 0 [m]
1.5

0

1

-1

-40

-1.5

[dB]

Beamforming (open)z = 0 [m]
1.5

20

-20

0
-0.5

-30

-1
-40

-1

0.5

-30

-1
-1.5

25

-10

Y axis [m]Y axis [m]0.5

Spatial Nyquist Frequency

0

1

-10

Y axis [m]

1.5

0

1

BDR [dB]

1.5

October 18-21, 2015, New Paltz, NY

Figure 4: Experimental results of bright to dark ratio BDR(ω) defined in (19) with Φb = π/2 and Φd = 3π/2.

Φ = π/4
z = 0 [m]z = 0 [m]1.5z = 0 [m]

1.5

0

1

0.5
-20

0
-0.5

-30

-10

Y axis [m]

-20

0
-0.5

-10

Y axis [m]

-1
-40

-1

0
X axis [m]

1

-1
-40

-1.5
-1

[dB]

LS (open)0
X axis [m]1-1

z = 0 [m]

0.5
-20

0
-0.5
-1

-1
-40

1

LS (baffled)

-10

0.5
-20

0
-0.5

-30

-30

-1.5

0

-10

Y axis [m]

-20

0

-30

-1
-40

-1.5
-1

[dB]

[dB]

1

-10

-0.5

1

1.5

0

1

0.5

0
X axis [m]Proposed (open)z = 0 [m]
1.5

0

1

0
X axis [m]

-40

-1.5

[dB]

Beamforming (open)z = 0 [m]
1.5

-1

-20

0
-0.5

-30

-1

-1.5

0.5

-30

Y axis [m]Y axis [m]

0.5

0

1

-10

Y axis [m]

1.5

0

1

0
X axis [m]

1

[dB]

-40

-1.5
-1

0
X axis [m]

1

[dB]

Beamforming (baffled) Proposed (baffled)Φ = π/2

Figure 3: Results of produced sound pressure level with r0 =0.25 m, f = 3 kHz, ϕs = π/4 and rref = 1.0 m. Bluecircles are loudspeakers with L = 32. Sound pressure level at
r = [r0 + 0.1, ϕs , 0]T is set to 0 dB.
circular array is also analytically derived as
FBF,open (r0 , ϕl , ω) =NXΦsinc(nΦ/2π)ejn(ϕl −ϕS ). (18)
2πj −n Jn (kr0 )n=−NIn all the calculations, a three-dimensional free field was assumed. The speed of sound c was 343.36 m/s. (8), (11) and(12) were discretized as dkz → ∆kz = 0.01 and truncated as−100 ≤ kz ≤ 100. A circular loudspeaker array with radiusr0 = 0.25 m was centered at the origin and located on the x-yplane. The number of loudspeakers L was 32, which is half of previous experiments [12]. The reference circle’s radius for the proposed methods was set to rref = 1.0 m. 32 control points were set
on rref for the LS methods. FLS,open and FLS,baffled were calculated using (2) and (8) in which n was also truncated within ±20,respectively.
Similar to the previous work [12], for evaluating the sound pressure level 20log10 |P (r, ω)| between bright zone r b and dark zone

r d , the bright to dark ratio BDR(ω) was also defined asP
r |P (r b , ω)| /Φb
BDR(ω) = 20log10 P b,
r d |P (r d , ω)| /Φd(19)

where Φb and Φb are the widths of the bright and dark zones, respectively. For calculating (19), both were set around control circle
rref −0.2 ≤ r ≤ rref +0.2, the bright zone’s width was Φb = π/2,
and the dark zone’s width was Φd = 3π/2.
Fig. 3 shows the results of each sound pressure level producedby conventional LS, beamforming and proposed methods for f =3 kHz and ϕs = π/4, with Φ = π/4 and π/2, respectively. The
result of bright to dark ratio BDR(ω) with Φ = π/2 is depicted
in Fig. 4. These results indicate that both proposed methods generate bright and dark zones more accurately than conventional LS andbeamforming methods. Moreover, it is obvious that the BDR(ω)
result for the beamforming method using an open circular array isquite unstable at forbidden frequencies where Jn (kr0 ) = 0 [23].
The proposed method for an open circular array, on the other hand,
introduces (11) based on 2.5D sound field representation instead ofJm (kr0 ). The BDR(ω) result, however, suggests that the forbidden frequencies are not a problem in the proposed method for an
open circular array. Detailed analysis and discussion are required
as future work.
Consequently, the proposed spatial filtering based on 2.5D
sound field representation can efficiently control sound pressurenear the loudspeakers than conventional least squares and beamforming methods. The transfer functions derived in (11) and (12)can also expect to be efficiently used for sound field reproduction
using circular loudspeaker arrays without and with a rigid baffle.
5. CONCLUSION
This paper proposed analytical methods to generate acousticallybright and dark zones on the horizontal plane using circular soundsources without and with a rigid baffle. The spatial filters in the
Fourier series domain were analytically derived by introducing
2.5D sound field representation for circular monopole source distributions. The proposed methods can be realized by fewer loudspeakers than in the SDM-based method for a linear array. The
computer simulation results show that the proposed methods generate bright and dark zones near the loudspeakers more accurately
than conventional least squares and beamforming methods.

2015 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

6. REFERENCES[1] J.-W. Choi and Y.-H. Kim, “Generation of an acoustically bright zone with an illuminated region using multiple
sources,” J. Acoust. Soc. Am., vol. 111, no. 4, pp. 1695–1700,
Apr. 2002.
[2] W. F. Druyvesteyn and J. Garas, “Personal sound,” J. Audio
Eng. Soc., vol. 45, no. 9, pp. 685–701, Sept. 1997.
[3] J.-H. Chang, C.-H. Lee, J.-Y. Park, and Y.-H. Kim, “A realization of sound focused personal audio system using acoustic contrast control,” J. Acoust. Soc. Am., vol. 125, no. 4, pp.
2091–2097, Apr. 2009.
[4] S. J. Elliott, J. Cheer, J.-W. Choi, and Y. Kim, “Robustness andregularization of personal audio systems,” IEEE Trans. Audio,
Speech, Lang. Process., vol. 20, no. 7, pp. 2123–2133, Sept.
2012.
[5] Y. Cai, M. Wu, and J. Yang, “Sound reproduction in personal
audio systems using the least-squares approach with acoustic contrast control constraint,” J. Acoust. Soc. Am., vol. 135,
no. 2, pp. 734–741, Feb. 2014.

October 18-21, 2015, New Paltz, NY[17] N. Radmanesh and I. S. Burnett, “Generation of isolated wideband sound fields using a combined two-stage Lasso-LS algorithm,” IEEE Trans. Audio, Speech, Lang. Process., vol. 21,
no. 2, pp. 378–387, Feb. 2013.
[18] W. Jin, W. B. Kleijn, and D. Virette, “Multizone soundfield reproduction using orthogonal basis expansion,” in Proc.
ICASSP, May 2013, pp. 311–315.
[19] W. Jin and W. B. Kleijn, “Multizone soundfield reproduction
in reverberant rooms using compressed sensing techniques,”
in Proc. ICASSP, May 2014, pp. 4728–4732.
[20] F. M. Fazi, M. Shin, F. Olivieri, S. Fontana, and L. Yue, “Comparison of pressure-matching and mode-matching beamforming methods for circular loudspeaker arrays,” in Proc. 137th
Conv. Audio Eng. Soc., Oct. 2014.
[21] F. M. Fazi, M. Shin, F. Olivieri, and S. Fontana, “Low frequency performance of circular loudspeaker arrays,” in Proc.
138th Conv. Audio Eng. Soc., May 2015.
[22] M. R. Bai and Y.-H. Hsieh, “Point focusing using loudspeaker arrays from the perspective of optimal beamforming,”
J. Acoust. Soc. Am., vol. 137, no. 6, pp. 3393–3410, June 2015.

[6] P. Coleman, P. J. B. Jackson, M. Olik, and J. A. Pedersen,“Personal audio with a planar bright zone,” J. Acoust. Soc.
Am., vol. 136, no. 4, pp. 1725–1735, Oct. 2014.

[23] E. G. Williams, Fourier Acoustics: Sound Radiation and
Nearfield Acoustic Holography.
London, UK: Academic
Press, 1999.

[7] T. Betlehem, W. Zhang, M. Poletti, and T. Abhayapala, “Personal sound zones: Delivering interface-free audio to multiple
listeners,” IEEE Signal Process. Mag., vol. 32, no. 2, pp. 81–
91, Mar. 2015.

[24] T. Betlehem and T. D. Abhayapala, “Theory and design ofsound field reproduction in reverberant rooms,” J. Acoust. Soc.
Am., vol. 117, no. 4, pp. 2100–2111, Apr. 2005.

[8] M. A. Poletti and F. M. Fazi, “An approach to generating two
zones of silence with application to personal sound systems,”
J. Acoust. Soc. Am., vol. 137, no. 2, pp. 598–605, Feb. 2015.

[25] J. Ahrens and S. Spors, “Sound field reproduction using planar and linear arrays of loudspeakers,” IEEE Trans. Audio,
Speech, Lang. Process., vol. 18, no. 8, pp. 2038–2050, Nov.
2010.

[9] M. Shin, S. Q. Lee, F. M. Fazi, P. A. Nelson, D. Kim, S. Wang,
K. H. Park, and J. Seo, “Maximization of acoustic energy difference between two spaces,” J. Acoust. Soc. Am., vol. 128,
no. 1, pp. 121–131, July 2010.

[26] T. Okamoto, S. Enomoto, and R. Nishimura, “Least squares
approach in wavenumber domain for sound field recording
and reproduction using multiple parallel linear arrays,” Appl.
Acoust., vol. 86, pp. 95–103, Dec. 2014.

[10] K. Helwani, S. Spors, and H. Buchner, “Spatio-temporal signal preprocessing for multichannel acoutic echo cancellation,”in Proc. ICASSP, May 2011, pp. 93–96.

[27] M. Kolundžija, C. Faller, and M. Vetterli, “Baffled circular
loudspeaker array with broadband high directivity,” in Proc.
ICASSP, Mar. 2010, pp. 73–76.

[11] P. Coleman, P. J. B. Jackson, M. Olik, M. Møller, M. Olsen,
and J. A. Pedersen, “Acoustic contrast, planarity and robustness of sound zone methods using a circular loudspeaker array,” J. Acoust. Soc. Am., vol. 135, no. 4, pp. 1929–1940, Apr.
2014.

[28] ——, “Design of a compact cylindrical loudspeaker array for
spatial sound reproduction,” in Proc. 130th Conv. Audio Eng.
Soc., May 2011.

[12] T. Okamoto, “Generation of multiple sound zones by spatial
filtering in wavenumber domain using a linear array of loudspeakers,” in Proc. ICASSP, May 2014, pp. 4733–4737.

[29] M. Poletti and T. Betlehen, “Design of a prototype variable directivity loudspeaker for improved surround sound reproduction in rooms,” in Proc. 52nd Int. Conf. Audio Eng. Soc., Sept.
2013.

[13] M. Shin, F. M. Fazi, P. A. Nelson, and F. C. Hirono, “Controlled sound field with a dual layer loudspeaker array,” J.
Sound Vib., vol. 333, no. 16, pp. 3794–3817, Aug. 2014.

[30] J. Ahrens and S. Spors, “An analytical approach to sound field
reproduction using circular and spherical loudspeaker distributions,” Acta Acust. Utd. With Acust., vol. 94, no. 6, pp. 988–
999, Nov./Dec. 2008.

[14] S. Zhao and X. Qiu, “Acoustic contrast control in an arcshaped area using a linear loudspeaker array,” J. Acoust. Soc.
Am., vol. 137, no. 2, pp. 1036–1039, Feb. 2015.

[31] T. Okamoto, “Near-field sound propagation based on a circular and linear array combination,” in Proc. ICASSP, Apr. 2015,
pp. 624–628.

[15] M. Poletti, “An investigation of 2D multizone surround soundsystems,” in Proc. 125th Conv. Audio Eng. Soc., Oct. 2008.

[32] S. Koyama, K. Furuya, Y. Hiwasaki, Y. Haneda, and
Y. Suzuki, “Wave field reconstruction filtering in cylindricalharmonic domain for with-hight recording and reproduction,”
IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 22,
no. 10, pp. 1546–1557, Oct. 2014.

[16] Y. J. Wu and T. D. Abhayapala, “Spatial multizone soundfield
reproduction: theory and design,” IEEE Trans. Audio, Speech,
Lang. Process., vol. 19, no. 6, pp. 1711–1720, Aug. 2011.