2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 17-20, 2021, New Paltz, NY

2D MULTIZONE SOUND FIELD SYNTHESIS WITH INTERIOR-EXTERIOR AMBISONICS
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 presents a two-dimensional multizone sound field synthesis method based on sound field separation and interior-exterior
higher-order Ambisonics (HOA) using two circular loudspeaker arrays. In the conventional methods using a circular loudspeaker array, multiple target local sound zones are represented as a global
interior sound field. However, the control accuracy of the conventional methods is strongly dependent on the positional relationshipbetween the wavefront direction in a bright zone and other darkzones. This is known as the occlusion problem. On the other hand,
the global sound field is defined as a mixture of the interior and exterior sound fields based on sound field separation in the proposed
method. The separated global interior and exterior sound fields arethen simultaneously synthesized via interior-exterior HOA using
two circular loudspeaker arrays with a cylindrical baffle to avoid
the forbidden frequency problem in exterior HOA. The results ofcomputer simulations demonstrate that the proposed method using
the two circular arrays can successfully avoid the occlusion problem
and realize higher control accuracy than the conventional methods
using a single circular array.
Index Terms— Multizone sound field synthesis, occlusion
problem, interior-exterior sound field separation, interior-exteriorsound field synthesis, higher-order Ambisonics
1. INTRODUCTION
Compared with standard sound field synthesis methods, e.g.,higher-order Ambisonics (HOA) [1,2], and acoustic contrast control
methods [3,4], multizone sound field synthesis can not only synthesize a desired sound field in a bright zone but also maximize theacoustic contrast between the bright zone and the remaining darkzones simultaneously [5]. Both pressure-matching-based [6–12]
and mode-matching-based [13–21] methods are investigated.
To synthesize all wavefront directions in a bright zone, a circular loudspeaker array is typically used in both 2D and 2.5D multizone sound field synthesis [6, 7, 9, 13–18, 20, 21]. However, thecontrol accuracy of the conventional methods using a circular array is strongly dependent on the positional relationship between the
wavefront direction in a bright zone and the remaining dark zones.
This is called the occlusion problem [6, 7, 16].
To alleviate the occlusion problem in multizone sound field synthesis, this paper proposes a two-dimensional multizone sound field
synthesis method based on a combination of sound field separation [22, 23] and interior-exterior HOA [24–26] using two circular
loudspeaker arrays with a cylindrical baffle, as illustrated in Fig. 1.
This study was partly supported by JSPS KAKENHI under Grant Number JP18K11387.

R
Zone 1Rl

Zone q
re

ri

Rl

Rq
CylindricalbaffleZone Q
Rl

Interior Ambisonics

Globalregion of interest

Exterior Ambisonics

Figure 1: Geometry of proposed multizone sound field control using
two circular loudspeaker arrays with a cylindrical baffle.
By introducing exterior sound field decomposition and synthesis,
the proposed method is expected to improve the global sound field
representation and the control accuracy for multizone sound field
synthesis.
In the conventional methods [13, 16, 17, 21], multiple target local sound fields are represented by a global interior sound field forinterior mode-matching-based synthesis using a single circular array. However, this representation, which only considers interiorsound fields, has limitations in principle and causes the occlusion
problem [16]. This problem is also encountered in the pressurematching-based methods [6,7]. In contrast, the global sound field isdefined as a mixture of the interior and exterior sound fields basedon sound field separation [22, 23] in the proposed method. The separated global interior and exterior sound fields are then simultaneously synthesized through interior-exterior HOA [24–26] using two
circular loudspeaker arrays.
Although the proposed method is formulated in a 2D heightinvariant sound field using circular loudspeaker arrays based on 2D
cylindrical harmonic expansion for simplicity [16,26], the proposed
method can also be easily extended to form height-variant 3D soundfields using spherical loudspeaker arrays based on spherical harmonic expansion for real world implementation [19].
To ensure the reproducibility of this study, the MATLAB codeused in the simulations conducted in Section 3 is available online.1
1 http://www.okamotocamera.com/waspaa2021.html© 2021 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.

2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics(q)

2. PROPOSED METHOD

is a (2M +1)Q×(2Mi +1) matrix and Tl

2.1. Global interior and exterior sound fields based on soundfield separation
Although the multiple target local interior sound fields are represented as a global interior sound field in the conventional methods [13,16,17,21], the global sound field in the proposed method isdefined as a mixture of the interior and exterior sound fields basedon sound field separation [22, 23].
The 2D cylindrical decomposition of a 2D sound field at a pointr = [r, φ] due to a sound source located at r s = [rs , φs ] is described as [22, 27, 28]:∞
X

Si (r) =Åm Jm (kr)ejmφ

for rs > r,

(1)

B̊m Hm (kr)ejmφ

for rs < r,

(2)

m=−∞
∞
X

Se (r) =m=−∞√

where j =−1, k is the wavenumber, Åm and B̊m are the2D cylindrical harmonic spectra of the interior and exterior soundfields, respectively. Jm and Hm are the Bessel function and the
Hankel function of the first kind, respectively [27]. The 2D interiorexterior mixed sound field is then defined as [22]:Si+e (r) ≃MiXÅm Jm (kr)ejmφ +

m=−Mi

Me
X

B̊m Hm (kr)ejmφ ,

m=−Me

(3)

where Mi = ⌈ekr/2⌉ and Me = ⌈ekrs /2⌉ represent the truncation
limits of the 2D cylindrical harmonic orders, and ⌈·⌉ is the ceilingfunction [29].
When Q local sound zones are assumed and the radius of eachzone is Rl (Fig. 1), the truncation limit of the 2D cylindrical harmonic order for the Q local interior sound fields is M = ⌈ekRl /2⌉.
Each local interior sound field is centered at r q = [Rq , φq ]. Although the 2D cylindrical and 3D spherical harmonic spectra of
the local interior sound fields are obtained from sound pressures
recorded using 2D and 3D higher-order microphones in sound field
analysis and separation [23, 28], the 2D cylindrical harmonic spec(q)tra for each interior local sound field Åm can be given directly, e.g.,
as plane and cylindrical waves, in multizone sound field synthesis.
By applying the addition theorem for Bessel and Hankel functions [30] to (3) as 3D sound field separation [23], each local interior(q)sound field Åm can also be represented as:
Å(q)
m ≃

Mi
X

Ån Jm−n (kRq )ej(n−m)(π+φq )

n=−Mi

+

Me
X

October 17-20, 2021, New Paltz, NY

B̊n Hm−n (kRq )ej(n−m)(π+φq ) .

(4)

n=−Me

(4) can then be rewritten in matrix form as a = T b, where
(1)
(1)
(Q)
(Q)
a = [Å−M , · · · , ÅM , · · · , Å−M , · · · , ÅM ]T is a (2M +1)Q×1
vector for the 2D cylindrical harmonic spectra of the local interiorsound fields, T = [T i T e ],

 (1)
(1)
T(−M +Mi ) · · · · · · · · · T(−M −Mi )


..
..
..
..
..

Ti = 


.
.
.
.
.
(Q)
(Q)
T(M +Mi ) · · · · · · · · ·T(M −Mi )′(1)

T(−M +Me )..
Te = 
.
′(Q)
T(M +Me )

···
..
.
···

···
..
.
···

= Jl (kRq )e−jl(π+φq ) ,

···
..
.
···
′(q)


′(1)
T(−M −Me )

..


.
′(Q)
T(M −Me )= Hl (kRq )e−jl(π+φq ) ,is a (2M + 1)Q × (2Me + 1) matrix, Tl
and b = [Å−Mi , · · · , ÅMi , B̊−Me , · · · , B̊Me ]T is a 2(Mi + Me +1) × 1 vector for the 2D cylindrical harmonic spectra of the global
interior-exterior mixed sound field. The Q local interior soundfields can then be separated into global interior and exterior soundfields as b = T + a [23], where T + is the pseudoinverse of T .
In [16], three objective performance measures are defined for
multizone sound field synthesis:
1. Acoustic contrast: the ratio of the averaged acoustic energy
densities of the bright and dark zones;
2. Bright zone error: the averaged error between the desired
and synthesized sound fields in the bright zone;
3. Array effort: a measure proportional to the loudspeaker
power consumption required to synthesize the desired multizone sound field.
These objective performance measures can also be applied to the
proposed method with the global interior-exterior mixed soundfield. An optimal solution for multizone sound field synthesis canthen be derived as [16, 17]:bopt = [T ∗b T b + λ1 T ∗d T d + λ2 I]−1 T ∗b Ab ,(5)

where T b and T d are the translation matrices for the local brightand dark zones from the global system, respectively, Ab is the matrix for the 2D cylindrical harmonic spectra for the local brightzones, (·)∗ denotes the Hermitian transpose, I is an identity matrix
with dimension 2(Mi + Me + 1), and λ1 and λ2 are the positive Lagrange multipliers. Using (5), the Q local interior sound fields are
optimally separated into global interior and exterior sound fields for
multizone sound field synthesis via interior-exterior HOA.
2.2. Analytical driving functions for interior-exterior HOA using two circular loudspeaker arrays with a cylindrical baffle
In the proposed method, the separated global interior and exterior sound fields bopt derived in (5) are simultaneously synthesized using two circular arrays of omnidirectional loudspeakers
with radii of ri and re centered at the origin for interior and exterior
HOA [24–26], respectively, as shown in Fig. 1.
Under the free-field assumption, as in the conventional
method [13], the driving function for a continuous circular sourcewith radius ri centered at the origin for the interior 2D HOA is analytically obtained as [1, 2]:D̊m (ri ) = −2j Åm.
πHm (kri )(6)

Although the driving function of a continuous circular source ofradius re for the exterior HOA is also analytically derived as−2j B̊m /πJm (kre ), this function is sometimes unstable becauseof the forbidden frequency problem, where Jm (kre ) = 0 [26, 27].
To avoid the forbidden frequency problem for the exterior HOA
and thus derive a stable driving function, a cylindrical baffle with

2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 17-20, 2021, New Paltz, NY

radius re centered at the origin is introduced [31] and a circularsource for the exterior HOA is mounted on the baffle, as shown in
Fig. 1. When the cylindrical baffle is introduced, the sound pressure
at a point r synthesized using a continuous circular source withradius r0 centered at the origin is represented by:
Z 2πS(r) =
D(r 0 )GB (r, r 0 )dφ0 ,(7)0where D(r 0 ) is the driving function for the circular source andGB (r, r 0 ) is the transfer function from the sound source positionr 0 = [r0 , φ0 ] to the receiver position r. Under the free-field assumption, GB (r, r 0 ) is analyticall given as [27]:GB (r < , r 0 ) =GB (r > , r 0 ) =∞XjΓm (r, re )Hm (kr0 )e−jmφ0 ejmφ ,
4
m=−∞

(8)

∞
X
j
Γm (r0 , re )Hm (kr)e−jmφ0 ejmφ , (9)4m=−∞Γm (r, re ) = Jm (kr) −′Jm(kre )
H (kr),′ (kr ) m
Hm
e

(10)

′
′
where Jmand Hm
are the derivatives of Jm and Hm , respectively.
When the 2D cylindrical harmonic expansion is applied to (7)and (8), the circular convolution theorem then holds, [1, 25, 26, 31]
and the sound pressure produced by the interior HOA with D̊m (ri )is represented by:jπD̊m (ri )Γm (r, re )Hm (kri )2
J ′ (kre )H (kr).
= Åm Jm (kr) − Åm m′ (kr ) m
HmeS̊m (r< , ri ) =′Jm(kre ).
′ (kr )
Hm
e

(12)

(13)

Because (5) does not include the scattered components C̊m ,these components are undesired. To cancel out the undesired components C̊m , the driving function for the exterior HOA is derived
to synthesize the exterior component B̊m and also to cancel out theundesired scattered components C̊m produced by the interior HOA
with D̊m (ri ). Similar to (11), using (7) and (9), the sound pressure
produced by the exterior HOA with D̊m (re ) is given by:jπD̊m (re )Γm (re , re )Hm (kr)2
= (B̊m − C̊m )Hm (kr).

S̊m (r> , re ) =

(14)
(15)

From (14) and (15), D̊m (re ) is then analytically derived as2 :D̊m (re ) =2(B̊m − C̊m )jπΓm (re , re )′′= −kre (B̊m Hm(kre ) + Åm Jm(kre )).

Finally, the two continuous circular sources are discretized into
two circular loudspeaker arrays (Fig. 1). When the numbers of loudspeakers for the interior and exterior arrays are Li and Le , the orderm in (6) and (16) can be calculated up to Mi = ⌊(Li − 1)/2⌋ and
Me = ⌊(Le − 1)/2⌋, where ⌊·⌋ represents the floor function. For
actual implementation, Li > Le because ri ≫ re . Therefore, thescattered components C̊m for Me < |m| ≤ Mi cannot be canceled. However, most of the scattered spectra are included in thelower order components and can be sufficiently canceled using only
the |m| ≤ Me components, as shown in Fig. 2.
When Le = 0 and no exterior sound field is considered in (5)and (16), (5) and (6) then correspond to the decomposed global interior sound field and driving function for the conventional methodwith the single circular array. Therefore, the proposed method is a
generalized formulation that includes the conventional method.

(11)

In this case, not only the propagated components but also the scattered components are simultaneously synthesized due to the cylindrical baffle. Using (3) and (12), the radius-independent scatteredcomponent C̊m produced by D̊m (ri ) is then obtained as:
C̊m = −Åm

Figure 2: Scattered components 20 log10 |C̊m | averaged over 0 ≤Φ ≤ π.

(16)

2 The Wronskian relation J (kr)H ′ (kr) − J ′ (kr)H (kr) =mm
mm
2j/kr is introduced here [24, 27].

3. COMPUTER SIMULATIONS
The proposed method using two circular arrays was comparedwith the conventional method using a single circular array [13, 16]
through computer simulations. A 2D free-field was assumed andthe speed of sound c was 343.36 m/s. The target frequency f wasin the 100 to 1,000 Hz range and the wavenumber was k = 2πf /c.
The total number of loudspeakers used for both methods was 64.
The numbers of loudspeakers and the radii of the circular arrays forthe interior and exterior HOA in the proposed method were Li = 53
and Le = 11, and ri = 2.0 m and re = 0.2 m, respectively. Twolocal sound zones with radius Rl = 0.5 m centered at r 1 = [1.0, 0]
for a bright zone and r 2 = [1.0, π] for a dark zone were assumed
and M = ⌈ekRl /2⌉ in (5). The desired sound field in the brightzone was a simple plane wave with a direction of 0 ≤ Φ ≤ π and(1)Åm = j m e−jmΦ . The radius of the global sound field was then
set to R = 1.5 m. The maximum order Mi in (5) and (6) was
Mi = ⌈ekR/2⌉. In the conventional method with Li = 64 loudspeakers, if Mi > ⌊(Li − 1)/2⌋ = 31, then it was truncated to
31. In the proposed method, if Mi > ⌊(Li − 1)/2⌋ = 26, thenit was truncated to 26. The maximum order Me in (5) and (16)was Me = ⌈ekre /2⌉. From the results of preliminary investigations, if Me > 3, then Me in (5) was truncated to 3 for greater
control accuracy. If the maximum order of (16) for C̊m was higher
than ⌊(Le − 1)/2⌋ = 5, it was then truncated to 5. The positive
Lagrange multipliers λ1 and λ2 in (5) were set at 0.2 and 0.001,
respectively, for both methods as in [17]. In the proposed method,
the truncation order used to calculate (8) and (9) was set at 50.
To estimate the control accuracy, the synthesis error
in the bright zone Db of ε(r) = 10 log10 (|Sdes (r) −Ssyn (r)|2 /|Sdes (r)|2 ),and the acoustic contrast between the bright zone Db and the dark zone Dd of2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics(i) Synthesized field(ii) Synthesis error(iii) Sound pressure level

October 17-20, 2021, New Paltz, NY(i) Synthesized field(ii) Synthesis error(iii) Sound pressure level(a) Conventional method (Li = 64, Le = 0) : Φ = π/4(c) Conventional method (Li = 64, Le = 0) : Φ = π(b) Proposed method (Li = 53, Le = 11) : Φ = π/4(d) Proposed method (Li = 53, Le = 11) : Φ = π

Figure 3: Results for (i) synthesized sound field S(r), (ii) synthesis error ε(r), and (iii) sound pressure level 20 log10 |S(r)| (r 1 = [1.0, 0]
was set at 0 dB) with frequency f = 500 Hz and plane wave directions of Φ = π/4 and π. Blue circles represent loudspeakers.

(a) Interior field(a) Averaged for 0 ≤ f ≤ 1, 000(b) Averaged for 0 ≤ Φ ≤ π(b) Exterior field(c) Interior-exterior field

Figure 6: Reconstructed sound fields for proposed method calculated using (5), (1), (2), and (3) with f = 500 Hz and Φ = π.

Figure 4: Synthesis error in bright zone.

(a) Averaged for 0 ≤ f ≤ 1, 000(b) Averaged for 0 ≤ Φ ≤ π

Figure 5: Acoustic contrast χ.
RRχ = 10 log10 ( D |Ssyn (r)|2 dr/ D |Ssyn (r)|2 dr) were inb
dtroduced, where Sdes (r) and Ssyn (r) were the desired andsynthesized sound fields, respectively.
The results for the scattered components 20 log10 |C̊m | in (13)averaged over 0 ≤ Φ ≤ π are plotted in Fig. 2. These results
indicate that most of the scattered spectra up to 1000 Hz can be
sufficiently canceled using the proposed method, even if the maximum order in (16) ranged up to 5. The results for the synthesizedsound field, the synthesis error, and the synthesized sound pressurelevel for both methods with f = 500 Hz and Φ = π/4 and π areshown in Fig. 3. Additionally, the results for the synthesis error inR
the bright zone (1/πRi2 ) D ε(r)dr and the acoustic contrast χ avb
eraged for 100 ≤ f ≤ 1, 000 and 0 ≤ Φ ≤ π are plotted in Figs. 4
and 5, respectively. These results show that the proposed method
can avoid the occlusion problem and significantly improve both thesynthesis accuracy and the acoustic contrast compared with the conventional method, which included the occlusion problem, as shownin Figs. 3(c), 4(a) and 5(a). To illustrate the physical impact of the
proposed method, the reconstructed interior, exterior, and interiorexterior mixed sound fields with f = 500 Hz and Φ = π are shownin Fig. 6. The sound pressures propagating into the dark zone via theinterior field (Fig. 6(a)) can be successfully canceled using the pressures with the opposite phase from the exterior field (Fig. 6(b)) in
the total mixed field (Fig. 6(c)). Consequently, the proposed method
can avoid the occlusion problem and provide improved control accuracy by introducing interior-exterior sound field separation andsynthesis.
4. CONCLUSIONSThis paper proposed a 2D multizone sound field synthesis method
using two circular arrays with a cylindrical baffle. The global
sound field is defined as a mixture of the interior and exterior soundfields based on sound field separation. The separated global interior and exterior sound fields are then simultaneously synthesized
via interior-exterior HOA. Computer simulations validated the effectiveness of the proposed method using two circular arrays.

2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

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