ANALYTICAL APPROACH TO 2.5D SOUND FIELD CONTROLUSING A CIRCULAR DOUBLE-LAYER 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 a regularization-free analytical approach to 2.5Dinterior and exterior sound field control using a circular double-layer
array of fixed-directivity loudspeakers not only to provide a desired
sound field inside the array but also to reduce the sound energy outside of it in the horizontal plane. The proposed method analytically derives the driving functions of the inner and outer circular arrays based on 2.5D spherical harmonic expansion to simultaneously
control both sound fields inside and outside the array by a modematching framework. The results of computer simulations show thatthe proposed method is certainly effective compared with the conventional least squares approach that requires regularization schemesin terms of synthesis accuracy inside the listening zone over a wideband frequency range and the acoustic contrast between the quiet
zone and the synthesis center at low frequencies.
Index Terms— Interior and exterior sound field control, circular loudspeaker array, fixed-directivity loudspeaker, 2.5D sound field
synthesis, Ambisonics
1. INTRODUCTION
In typical sound field synthesis approaches, such as wave field synthesis [1, 2] and higher-order Ambisonics (HOA) [3–5], only the desired synthesis region is considered [6], and undesired sound pressures are radiated to other regions. In applying sound field synthesisin actual environments, it is crucial not only to provide a sound field
at a desired region but also to reduce the sound energy in other regions.
Multizone sound field synthesis approaches have been investigated to simultaneously control multiple sound fields in multiple
regions [7–14]. In these approaches, however, undesired regions
remain.
To synthesize a desired sound field inside a loudspeaker arrayand completely reduce the undesired sound energy outside of it, 2Dinterior and exterior sound field control approaches with circular arrays of first-order [15] and higher-order [16] line sources have been
proposed. An analytical driving function of a circular first-order line
source has been derived [15], based on the Kirchhoff-Helmholtz integral [17] and the Fourier series expansion [17].
For actual implementations, sound field synthesis systems are
frequently simplified for synthesis in the horizontal plane. The soundsources are then arranged on a line or a circle. In actual implementations, 3D monopole sources (instead of 2D line sources) are usuallyThis study was partly supported by JSPS KAKENHI Grant Number
15K21674.

employed for the sound sources. Such approaches are called 2.5D
sound field synthesis [5, 18–21].
Since implementing line sources, especially first- and higherorder line sources, is difficult in actual 3D environments, a 2.5D interior and exterior sound field control approach in the horizontal plane
using a circular double-layer array of fixed-directivity loudspeakers [22] has been proposed [23] and investigated [24–26]. Experimental validation with an actually implemented array has also beenpresented [27]. This method is based on the least squares (LS) solution, which is numerically calculated using control points and loudspeaker positions. However, the LS-based approach is quite unstablebecause the acoustic inverse problem is very ill-conditioned [17]. To
stably calculate the well-conditioned inversion and driving signals,
regularization schemes are required, such as the truncated singularvalue decomposition (SVD) method or Tikhonov regularization [28].
In the LS method, repeated calculations of the inversion and driving signals are needed to select the optimal regularization parameters [29].
To avoid the regularization problem in the conventional LS
method, this paper provides a regularization-free analytical approach to 2.5D interior and exterior sound field control using acircular double-layer array of fixed-directivity loudspeakers. Theinterior and exterior sound fields produced by the circular doublelayer array of fixed-directivity loudspeakers are represented as 2.5D
spherical harmonic expansion and the driving functions of the inner and outer arrays are analytically derived by extending 2.5D
HOA [5, 18–21].
2. FOURIER SERIES AND SPHERICAL HARMONICEXPANSIONS OF SOUND FIELDSIn this section, we briefly introduce the Fourier series expansion of a
2D sound field and the spherical harmonic expansion of a 3D soundfield produced by a fixed-directivity loudspeaker.
2.1. Fourier series expansion of 2D sound fields
Spherical coordinates relative to Cartesian coordinates are defined inFig. 1.
The interior expansion of a 2D sound field (θ = π/2) in a region
that is homogeneous and free of sources is given as
S(r, φ) =∞X

Åm Jm (kr)ejmφ ,

(1)

m=−∞

where Åm and Jm are the interior sound field expansion coefficientsand the m-th order Bessel function, k is the wavenumber [17], and© 2017 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.

z

]T
p
r = x2 + y 2 + z 2px2 + y 2
y ✓ = tan 1
z
1 y
= tan
x

r = [r, ✓,
✓
r

x

r2

rb

r1

ri
Listeningzone

Fig. 1. Definition of spherical coordinates relative to Cartesian coordinates.

∞X

B̊m Hm (kr)ejmφ ,(2)Quiet zonem=−∞where B̊m and Hm are the exterior sound field expansion coefficients and the m-th order Hankel function of the first kind [17].
2.2. Spherical harmonic expansion of 3D sound field produced
by a fixed-directivity loudspeaker
As in previous research [22], the sound pressure at r, produced by afixed-directivity loudspeaker located at r 0 , is modeled as




j
ejk|r −r 0 |a − (1 − a) 1 +cos α ,
S(r, r 0 ) =4π|r − r 0 |k|r − r 0 |(3)
where a is the first-order weighting parameter and α is the angle
from the loudspeaker axis.
When the main lobe of S(r, r 0 ) points toward the origin, thespherical harmonic expansions of (3) for r < r0 and r > r0 are
given asS(r, r 0 ) = jk

∞
n
X
X

n=0 m=−njn (kr) {ahn (kr0 )−j(1 − a)h0n (kr0 ) Ynm (θ, φ)Ynm (θ0 , φ0 )∗ ,∞nX
X
S(r, r 0 ) = jk
hn (kr) {ajn (kr0 )n=0 m=−n−j(1 − a)jn0 (kr0 ) Ynm (θ, φ)Ynm (θ0 , φ0 )∗ ,r > r0 , (5)where jn and jn0 are the n-th order spherical Bessel function and its
derivative, hn and h0n are the n-th order spherical Hankel function of
the first kind and its derivative, and Ynm is the m-th order sphericalharmonics of the n-th degree [17].

3. ANALYTICAL DRIVING FUNCTION OF A CIRCULAR
DOUBLE-LAYER SOUND SOURCE
In the proposed approach, we first consider a continuous circular
double-layer sound source distribution with radii r1 and r2 (r1 < r2 )centered at the origin on the x-y plane instead of a circular doublelayer loudspeaker array (Fig. 2). A sound field synthesized by acontinuous circular double-layer sound source is given as
Z 2πS(r, θ, φ) =D1 (φ0 )T1 (r, r 1 ) + D2 (φ0 )T2 (r, r 2 )dφ0 , (6)0Fig. 2. Arrangements of circular double-layer array of fixeddirectivity loudspeakers and listening and quiet zones.
where D1 (φ0 ) and D2 (φ0 ) are the driving functions of the innerand outer circular sources and T1 (r, r 1 ) and T2 (r, r 2 ) are the transfer functions from sound source positions r 1 = [r1 , π/2, φ0 ] andr 2 = [r2 , π/2, φ0 ] to receiver position r. Similar to a previous
work [23], these transfer functions are modeled as a weighted combination of a monopole and a dipole with the main lobes of T1 (r, r 1 )and T2 (r, r 2 ) pointing toward and away from the center, respectively. Under the free-field assumption, T1 (r, r 1 ) and T2 (r, r 2 ) aredescribed as




j
ejk|r −r 1 |a − (1 − a) 1 +cos α ,T1 (r, r 1 ) =4π|r − r 1 |k|r − r 1 |(7)
T2 (r, r 2 ) =r < r0 , (4)

rq
rq

√
j = −1. The corresponding expansion for regions exterior to any
sound sources is given as
S(r, φ) =roejk|r −r 2 |4π|r − r 2 |
a + (1 − a) 1 +jcos α .
k|r − r 2 |(8)

As in [5], when the spatial Fourier series expansion is applied to (6)with θ = π/2, the circular convolution theorem holds and (6) isrepresented asnoS̊m (r) = 2π D̊m,1 T̊m,1 (r, r1 ) + D̊m,2 T̊m,2 (r, r2 ) . (9)Similar to 2.5D HOA [5, 18, 20, 21], the 2.5D spherical harmonic
expansions of transfer functions T1 (r, r 1 ) and T2 (r, r 2 ) are derived
from (4) and (5) and given asT̊m,1 (r< , r1 ) = jk

∞
X

|m|
2
jn (kr)hn,1 (a, r1 )Qmn Pn (0) , (10)n=|m|T̊m,2 (r< , r2 ) = jk

∞
X

|m|
2
jn (kr)hn,2 (a, r2 )Qmn Pn (0) , (11)n=|m|T̊m,1 (r> , r1 ) = jk

∞
X

|m|
2
hn (kr)jn,1 (a, r1 )Qmn Pn (0) , (12)n=|m|T̊m,2 (r> , r2 ) = jk

∞
X

n=|m|

|m|
2
hn (kr)jn,2 (a, r2 )Qmn Pn (0) , (13)|m|where Pn is the |m|-th order associated Legendre polynomial of
the n-th degree [30] and
(14)

Cm =

(15)

Um (a, r1 , r2 , ro ) =jn,2 (a, r2 ) = ajn (kr2 ) + j(1 − a)jn0 (kr2 ),

(17)

Qm
n =

2n + 1 (n − |m|)!
.
4π (n + |m|)!

(16)

(18)

Even though r is completely canceled in the driving functionsin the 2D and 3D cases, the driving function of 2.5D HOA depends
on r and accurate sound pressures are synthesized at the reference
circle [5, 18–21].
In the proposed approach, to simultaneously control the soundfields both inside and outside the circular source, the reference distances for the listening and quiet zones inside and outside of it are
set to ri (0 < ri < r1 ) and ro (ro > r2 ), respectively. From (1)and (2), the sound pressures inside and outside the circular source
are then set toS̊m (ri ) = Åm Jm (kri ),(19)S̊m (ro ) = B̊m Hm (kro ) = 0.

(20)(19) and (20) are substituted into (9) and represented in matrix form:T̊m,1 (ri< , r1 ) T̊m,2 (ri< , r2 )
D̊m,1
2π
D̊m,2
T̊m,1 (ro> , r1 ) T̊m,2 (ro> , r2 )Åm Jm (kri )
=
.
(21)
0
Then driving functions D̊m,1 and D̊m,2 are directly derived asD̊m,1 =

T̊m,2 (ro> , r2 )Åm Jm (kri ),2π T̊m (r1 , r2 , ri , ro )D̊m,2 = −T̊m,1 (ro> , r1 )Åm Jm (kri ),2π T̊m (r1 , r2 , ri , ro )

(22)
(23)

where
T̊m (r1 , r2 , ri , ro )= T̊m,1 (ri< , r1 )T̊m,2 (ro> , r2 ) − T̊m,2 (ri< , r2 )T̊m,1 (ro> , r1 ).
(24)
The driving function of 2.5D HOA with reference distancerref = 0 is derived [18] and can synthesize a more accurate soundfield than with rref > 0. To extend the driving functions of the proposed method into those with ri = 0, L’Hôpital’s rule [31] and thefollowing approximations for small arguments, i.e., kr → 0 [17],(kr)njn (kr) ≈,(2n + 1)!!

sgn(m)|m| (kr)|m|and Jm (kr) ≈,2|m| |m|!

are also applied in (22) and (23). The driving functions with ri = 0
are then obtained asD̊m,2sgn(m)|m| (2|m| + 1)!!
,2|m| |m|!

hn,1 (a, r1 ) = ahn (kr1 ) − j(1 − a)h0n (kr1 ),hn,2 (a, r2 ) = ahn (kr2 ) + j(1 − a)h0n (kr2 ),jn,1 (a, r1 ) = ajn (kr1 ) − j(1 − a)jn0 (kr1 ),

D̊m,1

where

ri =0

ri =0

=

T̊m,2 (ro> , r2 )Åm Cm,|m|
m
2πjkQ|m| P|m| (0)2 Um (a, r1 , r2 , ro )=−T̊m,1 (ro> , r1 )Åm Cm,|m|
m
2πjkQ|m| P|m| (0)2 Um (a, r1 , r2 , ro )

(25)
(26)

(27)

h|m|,1 (a, r1 )T̊m,2 (ro> , r2 ) − h|m|,2 (a, r2 )T̊m,1 (ro> , r1 ),(28)
and sgn in (27) is the sign function.
A continuous circular double-layer source is finally discretized
into a circular double-layer loudspeaker array. When the number ofloudspeakers of each layer is L, order m of the spatial Fourier seriesin (25) and (26) can be calculated up to M = b(L − 1)/2c, whereb·c is the floor function. The driving signal of each loudspeaker at
φl in the temporal frequency domain is obtained asD(φl ) =

M
X

m=−M

D̊m

ri =0

ejmφl ,

l = 1, 2, · · · L.

(29)

4. COMPUTER SIMULATIONS
Computer simulations compared the proposed approach with theconventional LS method [23]. In all the simulations, a threedimensional free field was assumed. The speed of sound c was343.36 m/s.
As in a previous work [23], the radii of the inner and outer circular arrays of the fixed-directivity loudspeakers, those of the listening
and quiet zones were r1 = 0.9 m, r2 = 1.0 m, rb = 0.2 m, andrq = 2.0 m, respectively. The quiet zone width was ∆rq = 1.0 m.
The control distances in the proposed method were set to ri = 0 m
and ro = 2.5 m. The arrangement of these radii is depicted in Fig. 2.
In the LS method, the listening and quiet zones were sampled at
discrete points with 0.05 m spacing between adjacent points. The
weighting factor to determine the balance between the potential energy in the quiet zone by the acoustic contrast approach [32] andthe mean square error in the listening zone by the pressure matchingapproach [33] was κ = 0.5, and truncated SVD regularization [28]and the discrepancy principle with the same parameter in [23] wereused to calculate the stable driving signals in the LS method. The
total number of loudspeakers was 64 and L = 32. The maximum
order in (29) was M = 15. n in (12), and (13) was truncated to 100.
The desired sound field was a plane wave with an amplitude of unity
propagating from direction Φ = π/4 and Åm = j m e−jmΦ .
To estimate the synthesized sound field, the synthesis error at position r and the acoustic contrast between position r and the centerof the array (r = 0) were defined as
E(r) = 10 log10
C(r) = 10 log10|Sdes (r) − Ssyn (r)|2,|Sdes (r)|2|Ssyn (r)|2,|Ssyn (r = 0)|2

(30)
(31)

where Sdes (r) and Ssyn (r) were the desired and synthesized soundpressures at position r, respectively.
Figure 3 shows the results of the synthesized sound field, thepreviously defined synthesis error (30), and the previously defined
acoustic contrast (31) for both the conventional LS and proposed
methods using 64 cardioid directivity loudspeakers (a = 0.5) at a
temporal frequency of f = 400 Hz. These results indicate that the
proposed approach effectively controlled both the interior and exterior sound fields as well as the conventional LS method.

z = 0 [m]

0

3

1

2

1

2
0.5

0.5

1
0

0

y [m]

y [m]

1

-1

0

0
-1

-0.5

-0.5

-2
-3
-3

-2
-1

-2

-1

0

1

2

-3
-3

3

-1

-2

-1

x [m]

0

1

2

3

Averaged synthesis error [dB]z = 0 [m]
3

x [m]

-10
-20
-30

-50

(a) Synthesized sound field
3

0

2

-5

2

-5

1

-10

1

-10

0

-15

0

-15

-1

-20

-1

-20

-2

-25

-2

-25

-3
-3

-30

-3
-3

-30

-1

0

1

2

3 [dB]

x [m]

-2

-1

0

1

2

x [m]

3 [dB](b) Synthesis errorz = 0 [m]

3

30

3

30

2

20

2

20

1

10

1

10

y [m]

y [m]

z = 0 [m]

0

0

0

0

-1

-10

-1

-10

-2

-20

-2

-20

-30

-3
-3

-3
-3

-2

-1

0

1

2

x [m]

3 [dB]

1

1.5

2

10
Conventional LS (cardioid)
Conventional LS (monopole)
Proposed (cardioid)
Proposed (monopole)

0
-10
-20
-30
-40
-50
-60
-70

0

0.5

1

1.5

2

Temporal frequency [kHz]Fig. 5. Results of averaged acoustic contrast between quiet zone andcenter of array for LS and proposed methods with cardioid directivity (a = 0.5) and omni-directional (a = 1) loudspeakers.

-30

-2

-1

0

1

x [m]

2

3 [dB](c) Acoustic contrast
Conventional LS

0.5

Fig. 4. Results of averaged synthesis error inside listening zone (r ≤rb ) for LS and proposed methods with cardioid directivity (a = 0.5)
and omni-directional (a = 1) loudspeakers.
Averaged acoustic contrast [dB]z = 0 [m]0-20Temporal frequency [kHz]

3

y [m]

y [m]

z = 0 [m]Conventional LS (cardioid)
Conventional LS (monopole)
Proposed (cardioid)
Proposed (monopole)-40

Proposed

Fig. 3. Results: (a) synthesized sound field, (b) synthesis error defined in (30), and (c) acoustic contrast defined in (31) for conventional LS and proposed methods using 64 cardioid directivity loudspeakers (a = 0.5) at temporal frequency f = 400 Hz. Black circles
in (a) and (b) are 64 loudspeakers.
In addition, the averaged synthesis error inside the listening zone(r ≤ rb ) and the acoustic contrast between the quiet zone with width∆rq and the center of the array up to f = 2 kHz are plotted in Figs. 4
and 5, respectively.
As shown in Fig. 4, the results of the averaged synthesis error suggest that the proposed approach clearly outperformed the LS
method in terms of synthesis accuracy inside the listening zone at
all of the temporal frequencies. This is because it is based on mode
matching just like HOA and truncated order M = 15 is sufficient tocontrol the sound pressures inside the listening zone up to f = 2 kHz
whose corresponding order is krb ≈ 7.3 < M .
The results of the averaged acoustic contrast shown in Fig. 5
indicate that the proposed approach efficiently controlled the quiet
zone with an averaged acoustic contrast below −40 dB up to aroundfNyq,2 ≈ 819 Hz, which is the spatial Nyquist frequency of theouter array that corresponds to kr2 = M . This limitation is thesame as the conventional 2D analytical approach with first-order line

sources [15]. In the proposed method, the averaged acoustic contrast
performance around fNyq,2 was slightly delegated since only thesound pressures on the circle with radius ro are accuratly controlled
and the wavelength corresponding to around fNyq,2 was shorter thanquiet zone width ∆rq .
As previously examined [23], the results in Figs. 4 and 5 withomni-directional loudspeakers (a = 1) are also almost the same as
those with cardioid directivity loudspeakers below about 1.2 kHz.
Consequently, the proposed approach based on an analytical solution without regularization is certainly effective compared withthe conventional LS method that requires regularization schemes interms of synthesis accuracy inside the listening zone over a wideband frequency range and the acoustic contrast between the quiet
zone and the synthesis center at low frequencies.
5. CONCLUSIONSThis paper proposed a regularization-free analytical method to control 2.5D interior and exterior sound fields using a circular doublelayer array of fixed-directivity loudspeakers. The driving functionsof the inner and outer circular arrays were analytically derived basedon the 2.5D spherical harmonic expansion of the sound field produced by fixed-directivity loudspeakers. The results of computer
simulations showed that the proposed method is certainly effective
compared with the conventional LS method with regularization interms of synthesis accuracy inside the listening zone over a wideband frequency range and the acoustic contrast between the quiet
zone and the synthesis center at low frequencies.

6. REFERENCES[1] A. J. Berkhout, D. de Vries, and P. Vogel, “Acoustic control bywave field synthesis,” J. Acoust. Soc. Am., vol. 93, no. 5, pp.
2764–2778, May 1993.
[2] S. Spors, R. Rabenstein, and J. Ahrens, “The theory of wave
field synthesis revisited,” in Proc. 124th Conv. Audio Eng. Soc.,
May 2008.
[3] J. Daniel, “Spatial sound encoding including near field effect: Introducing distance coding filters and a viable, new Ambisonic format,” in Proc. 23rd Int. Conf. Audio Eng. Soc., May
2003.
[4] M. A. Poletti, “Three-dimensional surround sound systems
based on spherical harmonics,” J. Audio Eng. Soc., vol. 53,
no. 11, pp. 1004–1025, Nov. 2005.
[5] J. Ahrens and S. Spors, “An analytical approach to sound field
reproduction using circular and spherical loudspeaker distributions,” Acta Acust. Acust., vol. 94, no. 6, pp. 988–999, Nov.
2008.
[6] F. Schultz and S. Spors, “Comparing approaches to the spherical and planar single layer potentials for interior sound field
synthesis,” Acta Acust. Acust., vol. 100, no. 5, pp. 900–911,
Sept. 2014.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] W. Jin and W. B. Kleijn, “Theory and design of multizone
soundfield reproduction using sparse methods,” IEEE/ACM
Trans. Audio, Speech, Lang. Process., vol. 23, no. 12, pp.
2343–2355, Dec. 2015.
[12] N. Radmanesh, I. S. Burnett, and B. D. Rao, “A Lasso-LS
optimization with a frequency variable dictionary in a multizone sound system,” IEEE/ACM Trans. Audio, Speech, Lang.
Process., vol. 24, no. 3, pp. 583–593, Mar. 2016.
[13] M. A. Poletti and F. M. Fazi, “Generation of half-space soundfields with application to personal sound systems,” J. Acoust.
Soc. Am., vol. 139, no. 3, pp. 1294–1302, Mar. 2016.
[14] W. Zhang, T. D. Abhayapala, T. Betlehem, and F. M. Fazi,“Analysis and control of multi-zone sound field reproduction
using modal-domain approach,” J. Acoust. Soc. Am., vol. 140,
no. 3, pp. 2134–2144, Sept. 2016.
[15] M. A. Poletti and T. D. Abhayapala, “Interior and exteriorsound field control using general two-dimensional first-ordersources,” J. Acoust. Soc. Am., vol. 129, no. 1, pp. 234–244,
Jan. 2011.

[16] M. A. Poletti, T. D. Abhayapala, and P. Samarasinghe, “Interior and exterior sound field control using two dimensional
higher-order variable-directivity sources,” J. Acoust. Soc. Am.,
vol. 131, no. 5, pp. 3814–3823, May 2012.
[17] E. G. Williams, Fourier Acoustics: Sound Radiation and
Nearfield Acoustic Holography, Academic Press, London, UK,1999.
[18] W. Zhang and T. D. Abhayapala, “2.5D sound field reproduction in higher order Ambisonics,” in Proc. IWAENC, Sept.
2014, pp. 342–346.
[19] T. Okamoto, “Analytical methods of generating multiple soundzones for open and baffled circular loudspeaker arrays,” inProc. WASPAA, Oct. 2015.
[20] T. Okamoto, “2.5D higher-order Ambisonics for a sound field
described by angular spectrum coefficients,” in Proc. ICASSP,
Mar. 2016, pp. 326–330.
[21] F. Winter, J. Ahrens, and S. Spors, “On analytic methods for
2.5-D local sound field synthesis using circular distributions ofsecondary sources,” IEEE/ACM Trans. Audio, Speech, Lang.
Process., vol. 24, no. 5, pp. 914–926, May 2016.
[22] M. A. Poletti, F. M. Fazi, and P. A. Nelson, “Sound-field reproduction systems using fixed-directivity loudspeakers,” J.
Acoust. Soc. Am., vol. 127, no. 6, pp. 3590–3601, June 2010.
[23] J.-H. Chang and F. Jacobsen, “Sound field control with a circular double-layer array of loudspeaker,” J. Acoust. Soc. Am.,
vol. 131, no. 6, pp. 4518–4525, June 2012.
[24] J.-H. Chang and F. Jacobsen, “The effect of scattering on soundfield control with a circular double-layer array of loudspeakers,” in Proc. 132nd Conv. Audio Eng. Soc., Apr. 2012.
[25] J.-H. Chang, F. Agerkvist, and M. Olsen, “Arrangements of apair of loudspeakers for sound field control with double-layer
arrays,” in Proc. 52nd Int. Conf. Audio Eng. Soc., Sept. 2013.
[26] Z. Tu, J. Lu, and K. Chen, “Evaluation of robustness of soundfield control with a circular double-layer array of loudspeakers,” in Proc. ICSV21, July 2014.
[27] J.-H. Chang and F. Jacobsen, “Experimental validation ofsound field control with a circular double-layer array of loudspeakers,” J. Acoust. Soc. Am., vol. 133, no. 4, pp. 2046–2054,
Apr. 2013.
[28] P. C. Hansen, Rank Deficient and Discrete Ill-posed Problems:
Numerical Aspects of Linear Inversion, SIAM, Philadelphia,
1998.
[29] 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.
[30] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic
Scattering Theory, Springer, Berlin, 1998.
[31] M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions: with Formulas, Graphs, and Mathematical Tables,
Dover Publications, New York, 1965.
[32] J.-W. Choi and Y.-H. Kim, “Generation of an acousticallybright zone with an illuminated region using multiple sources,”
J. Acoust. Soc. Am., vol. 111, no. 4, pp. 1695–1700, Apr. 2002.
[33] O. Kirkeby and P. Nelson, “Reproduction of plane wave soundfields,” J. Acoust. Soc. Am., vol. 94, no. 5, pp. 2992–3000, Nov.
1993.