HORIZONTAL 3D SOUND FIELD RECORDING AND 2.5D SYNTHESISWITH OMNI-DIRECTIONAL CIRCULAR ARRAYS
Takuma Okamoto
National Institute of Information and Communications Technology
3-5, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0289, Japanokamoto@nict.go.jp
ABSTRACTA 3D interior sound field can be recorded with a spherical microphone array mounted onto a spherical rigid baffle [1] and synthesizedwith a surrounding spherical loudspeaker array based on 3D higherorder Ambisonics (HOA) [2–4]. 3D HOA is based on the sphericalharmonic expansion of a 3D sound field [5]. However, in 3D HOA, alarge amount of microphones and loudspeakers more than (N + 1)2
elements are required for higher accuracy recording and synthesis
with a maximum spherical harmonic order of N [6]. Therefore, it is
not easy to implement these spherical arrays with a large number ofmicrophones and loudspeakers in an actual environment.
To reduce the number of microphones and loudspeakers and realize practical applications, circular microphone and loudspeaker arrays, instead of spherical ones, are often introduced in the horizontal
plane. When using 2D microphone and loudspeaker arrays, heightinvariant 2D sound fields, both in recording and synthesis environments with height-invariant 2D line receivers and sources, must be
considered. However, the actual environments are 3D sound fields,
and the 3D omni-directional receivers and sources are often used incircular microphone and loudspeaker arrays, instead of 2D line receivers and sources.
Sound field synthesis methods with 2D linear and circular loudspeaker arrays in 3D sound fields are known as 2.5D sound field
synthesis [4, 7–18]. Although there is a restriction that the correct

sound pressures are only synthesized on a reference line or circle
due to the dimension mismatch between the 2D array geometry andthe 3D loudspeaker’s propagation [15, 17], 2.5D sound field synthesis with 2D linear and circular loudspeaker arrays can be used in a
3D sound field.
On the other hand, a sound field recording with a circular array of 3D omni-directional microphones cannot be used in a 3D
sound field because the recorded sound field is typically decomposed
into 2D cylindrical harmonic spectrums that cannot represent the 3D
sound field. Because of this problem, in existing sound field recording and synthesis methods with 2D circular and linear microphoneand loudspeaker arrays, a 2D sound field is assumed in the primary
recording field, and the recorded 2D sound field is synthesized by
2D [19–21] and 2.5D synthesis [8, 10, 12]. To avoid this problem,a spherical microphone array is introduced in the recording of a 3D
sound field for 2.5D synthesis [16].
To record a 3D sound field with 2D planar microphone arrays,a method with multiple co-centered circular microphone arrays inthe horizontal plane has been proposed [22–24].1 By using thesearrays, spherical harmonic spectrums Ǎmn can be estimated and therecorded primary sound field can be reconstructed. However, thismethod must record not only the sound pressures, but also their vertical derivatives, because the recorded sound pressures can only estimate Ǎmn for n+|m| even, and their vertical derivatives can estimate
those for n + |m| odd. Therefore, this approach requires differential
microphones or twice the number of omni-directional microphones
for measuring both the sound pressures and their vertical derivatives.
To realize sound field recording and synthesis with the 2D microphone and loudspeaker arrays used in 3D sound fields, this study
first considered that only a sound pressure measurement with multiple co-centered circular microphone arrays is sufficient for 2.5D synthesis with a circular loudspeaker array, because the 2.5D synthesis
also includes only spherical harmonic spectrums Ǎmn for n + |m|even. This paper presents a horizontal 3D sound field recording and2.5D synthesis method used in 3D sound fields with multiple cocentered omni-directional circular microphone arrays and a circular
loudspeaker array. The processing flowchart of the proposed approach is illustrated in Fig. 1. In comparison with the conventional
2D sound field recording and 2.5D synthesis methods [8, 10, 12], theproposed approach can be entirely used in 3D sound fields. Additionally, the proposed method uses half the numbers of microphones
in comparison with the original planar array method [22–24]. Moreover, the proposed method can estimate the spherical harmonic spectrums with a maximum order of N in the horizontal plane by using
fewer than (N + 1)2 microphones used in a spherical array.

This study was partly supported by JSPS KAKENHI Grant Number
JP18K11387.

1 This approach is extended to 3D sound field control with 2D planar loudspeaker arrays [25, 26] based on the acoustic source and receiver reciprocity.

Although 2.5D sound field synthesis with a circular loudspeaker array can be used in a 3D sound field, a 2D sound field, instead of a
3D sound field, is assumed for a sound field recording with a circular microphone array. This paper presents a horizontal 3D soundfield recording and 2.5D synthesis method used in 3D sound fields
with multiple co-centered omni-directional circular microphone arrays and a circular loudspeaker array without vertical derivative measurements. The spherical harmonic spectrums used for 2.5D synthesis are extracted from the recorded horizontal sound pressures andthe driving function of a circular sound source is analytically derived based on 2.5D higher-order Ambisonics. The results of computer simulations indicate the effectiveness of the proposed method,in comparison with the conventional least squares-based pressure
matching, and 2D cylindrical harmonic analysis-based approaches.
Index Terms— Horizontal 3D sound field recording, 2.5D
sound field synthesis, circular array, spherical harmonic analysis,
Ambisonics
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Estimation

A00
A1 1 A11
A22
A2 2 A02
A3 3 A3 1 A13 A33(b) Amn for n + m even(a) recording S(r, ⇡/2, )<latexit sha1_base64="ZBaPq56in3VJal6g+UtGvaq7eXU=">AAACiXichVFNLwNBGH6sr6qv4iJxaRSpRJiVJhonSS+OqCJRaXbXaCe2u5vZaaMaf8AfcHAiEREXV64u/oCDnyCOlbg4eLvdRBC8k5l55pn3eeeZGdOzha8Ye2rT2js6u7ojPdHevv6BwdjQ8IbvVqTFc5Zru3LLNHxuC4fnlFA23/IkN8qmzTfN/Uxzf7PKpS9cZ13VPL5TNoqO2BOWoYgqxCbyih+oetKYjktuuXJXOMX4UTYpZ/KemJunsSSmC7EEm2VBxH8CPQQJhLHixi6Rxy5cWKigDA4HirANAz61behg8IjbQZ04SUgE+xxHiJK2QlmcMgxi92ks0mo7ZB1aN2v6gdqiU2zqkpRxTLJHdsUa7IFds2f2/mutelCj6aVGs9nScq8weDyafftXVaZZofSp+tOzwh7SgVdB3r2Aad7CaumrhyeN7OLaZH2KnbMX8n/Gntg93cCpvloXq3ztFFH6AP37c/8EG/OzOuHVVGIpFX5FBGMYR5LeewFLWMYKcnTuMW5wizutV9O1tLbYStXaQs0IvoSW+QA8FZZr</latexit>(c) 2.5D synthesis with D(r 0 )<latexit sha1_base64="4+PPGmsftDSr+89kP3iDJgP1RIA=">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</latexit>

F g 1 P ocess ng flowcha o p oposed ho zon a 3D sound fie d eco d ng and 2 5D syn hes s scheme a eco d ng ho zon a 3D soundfie d S(r π/2 φ) w h mu p e co cen e ed omn d ec ona c cu a a ays b es ma ng sphe ca ha mon c spec ums Amn o n+ meven om eco ded ho zon a sound p essu es c conve ng es ma ed sphe ca ha mon c spec ums n o d v ng unc on D(r 0 ) andsyn hes z ng ho zon a 3D sound fie d w h a c cu a oudspeake a ay based on 2 5D h ghe o de Amb son cs
2 3D SOUND FIELD IN HORIZONTAL PLANE
The n e o expans on o a 3D sound fie d n a homogeneous eg on
ee o sou ces s g ven as
S(r θ φ) =

∞
n
X
X

m
Ǎm
n jn (kr)Yn (θ φ)1n=0 m=−nwhe e Ǎmn and jn a e he sphe ca ha mon c spec ums o he ne o sound fie d and he n h o de sphe ca Besse unc on k s
he wavenumbe 5s(2n + 1) (n − m ) mm
Yn (θ φ) =Pn (cos θ) ejmφ24π(n + m )
|
{z
}
m
Pn

co θ

ha mon c spec ums 5 and ep esen ed as
Z 2π
1S(r π/2 φ)e−jmφ dφS̊m (r) =
2π 0
∞
X
m
=
Ǎm
n jn (kr)Pn (0)S(r π/2 φ) =

∞
n
X
X

m
jmφ
Ǎm
n jn (kr)Pn (0)eWhen he max mum sphe ca ha mon c o de s N and he ad us ohe c c e s Rq 5 s ep esen ed asS̊m (Rq ) 'mbecause Pn (0) = 0

whe e on y Ǎmn o n + m even s p eseno n + m odd 22 To ob a n Ǎm
n o n + m odd componen smhe ve ca de va ves o Pn (0) a e a so equ ed and can be mea
su ed by d e en a m c ophones o m c ophone pa s n he ho
zon a p ane 22 Thus no on y omn d ec ona bu a so ve
ca d e en a m c ophones a e equ ed o es ma e he en e 3D
sound fie ds 22 Howeve as shown n 17 he d v ng unc ono he 2 5D syn hes s w h a c cu a oudspeake a ay a so nc udesm
Pn (0) and on y has Ǎmn o n + m even componen s The eo e on y omn d ec ona componen s a e su fic en n he eco dng s age when he eco d ng ob ec ve ega ds he 2 5D syn hes s
3 PROPOSED METHOD
3 1 Hor zon a 3D sound fie d record ng w h mu p e cocen ered omn d rec ona c rcu ar m crophone arrays
F om 3 a con nuous c cu a sound p essu e d s bu on cen e eda he o g n o he x y p ane θ = 0 s conve ed n o 2D cy nd ca

N
X

m
Ǎm
n jn (kRq )Pn (0)

6

n= m

By n oduc ng mu p e ad
ep esen ed n ma x o m

q = 1 2Q o 6

can be7S̊ m = U m Ǎm
whe ehS̊ m = S̊m (R1 ) S̊m (R2 )

3n=0 m=−n

5

n= m

m

s he sphe ca ha mon cs Pns he assoc a ed Legend e unc
on 27
To cons de a 3D sound fie d n he ho zon a p ane w h θ =π/2 1 s ep esen ed as

4m

U m (kR1 ) U m (kR )
2

m
Um =


m

U m (kRQ )S̊m (RQ )

m

iT

m

U m +2 (kR1 )UN (kR1 )8m
UN (kR2 ) 





m

U m +2 (kR2 )mm

U m +2 (kRQ )UN (kRQ )9

Unm (kR) = jn (kR)Pnm (0)

10

and

Ǎm = Ǎmm

Ǎmm +2

Ǎm
N

T

11

n he case whe e bo h N and m a e e he odd o even o he w se
ep ace N n 9 and 11 by N − 1 Then 3D sphe ca ha mon cspec ums Ǎmn o n + m even can be ob a ned om mu p e cocen e ed omn d ec ona c cu a sound p essu e d s bu ons 22
−1
Ǎm = U Tm U m
U Tm S̊ m
12
as desc bed n F gs 1 a and b3.2. 2.5D sound field synthesis with a circular loudspeaker array

4. COMPUTER SIMULATIONS

A sound field synthesized by a continuous circular sound source distribution with a radius r0 centered at the origin of the x-y plane isgiven as:
Z 2π
S(r, θ, φ) =D(r 0 )G3D (r, r 0 )r0 dφ0 ,(13)Computer simulations evaluated the proposed approach and compared it with the conventional least squares-based pressure matching(PM) method [29, 30] and 2D cylindrical harmonic analysis-based
recording and 2.5D synthesis approach [10]. In all the simulations,a 3D free field was assumed and speed of sound c was 343.36 m/s.
In the recording stage, five co-centered omni-directional circular microphone arrays were introduced in the horizontal plane, andthese radii were set to 0.4, 0.35, 0.3, 0.25, and 0.2 m, respectively.
The target frequency band was up to 1 kHz. The maximum order
was obtained as N = dekRq,max /2e = 10 [6] and the number ofmicrophones on each circular array was calculated as 21, 19, 17,
15, and 11, according to an array design procedure provided in [22].
Then, the total number of microphones was 83, which is fewer than(N + 1)2 used in a spherical microphone array.
In the synthesis stage, the radius of a circular loudspeaker array
was r0 = 0.75 m and the number of loudspeakers was 21, according
to maximum order N = 10. To evaluate the estimated sphericalharmonic spectrums, the reconstructed sound field in the horizontal
plane was calculated by (3) up to a maximum order of N = 10.
To compare the proposed method with the conventional 2D
cylindrical harmonic analysis-based recording and 2.5D synthesis
approach, the 2D sound field recording with multiple circular arrays [20] and the 2.5D HOA for 2D sound field synthesis [9, 13]
were directly combined. The 2D cylindrical harmonic spectrums
were estimated from the recorded sound field S̊m (Rq ), as follows:0

where G3D (r, r 0 ) is the transfer function from a sound source position r 0 = [r0 , π/2, φ0 ]T to a receiver position r. Under the freefield assumption, G3D (r, r 0 ) is the 3D free-field Green’s function
and given as [5]:ejk|r −r 0 |G3D (r, r 0 ) =.
(14)
4π|r − r 0 |When assuming an interior sound field, G3D (r, r 0 ) can be represented by the spherical harmonic expansion and given as [5]:G3D (r, r 0 ) = jk

∞
n
X
X

jn (kr)hn (kr0 )Ynm (θ, φ)Ynm (θ0 , φ0 )∗ ,n=0 m=−n(15)where hn is the n-th order spherical Hankel function of the firstkind [5]. By applying the 2D cylindrical harmonic expansion to (13),the circular convolution theorem holds and (13) can be represented
as:
(16)

S̊m (rref ) = 2π D̊m G̊m (rref , r0 ),where rref is the reference radius for 2.5D synthesis with a circularsource. From (15), G̊m (rref , r0 ) is obtained as:G̊m (rref , r0 ) = jk∞Xjn (krref )hn (kr0 )Pn|m| (0)2 .

(17)

n=|m|

This equation indicates that the driving function of 2.5D synthesis|m|with a circular sound source also includes Pn (0), and only has
Ǎm
for
n
+
|m|
even
components.
Then,
the
cylindrical harmonicn
spectrums of the driving function D̊m for synthesizing a horizontal
sound field Ǎnm for n + |m| even estimated in (12) were analyticallyderived from (5), (16), and (17) [4]:P∞|m|mn=|m| Ǎn jn (krref )Pn (0)
.
(18)
D̊m =
P
|m|
2
2π ∞
n=|m| jkjn (krref )hn (kr0 )Pn (0)The driving function of 2.5D HOA with a reference radiusrref = 0 has been derived [4, 9, 13] and can synthesize a more accurate sound field in comparison with rref > 0. To extend the driving
function of the proposed method into that with rref = 0, L’Hôpital’srule [28] is also applied in (18). The driving function with rref = 0
is then obtained [4]:D̊mrref =0

=

Ǎm
|m|

Åm =

S̊m (Rq ),Jm (kRq )(20)

where Jm is the m-th order Bessel function, and Rq was selectedfrom the radii of the multiple circular microphone arrays for the
maximum value of |Jm (kRq )|, so as to avoid forbidden frequencies [20,31]. Then, the driving function of the 2.5D synthesis for theestimated 2D sound field was obtained [9, 13]:D̊m,cyl =2π

P∞Åm Jm (krref )|m|2
n=|m| jkjn (krref )hn (kr0 )Pn (0)

.

(21)

Similar to (19), the L’Hôpital’s rule was also applied in (21) [9, 13]:D̊m,cylrref =0=Åm sgn(m)|m| (2|m| + 1)!!
,
|m|
|m|
2 |m|!2πjkh|m| (kr0 )P|m| (0)2(22)

where sgn is the sign function. Additionally, to evaluate the estimated 2D cylindrical harmonic spectrums, the reconstructed soundfield in the horizontal plane was calculated as [5]:S(r, π/2, φ)) =N
X

Jm (kr)Åm ejmφ .

(23)

m=−N
|m|

2πjkh|m| (kr0 )P|m| (0)

.

(19)

(19) is obviously simpler than (18) because there are no summation
terms and it is thus calculated only from Ǎm|m| components.
The continuous multiple co-centered omni-directional circularsound pressure distributions and a circular source are finally discretized into multiple co-centered omni-directional circular microphone arrays and a circular loudspeaker array [20]. Using the proposed approach, a horizontal 3D sound field recorded with multiple
co-centered omni-directional circular microphone arrays can be synthesized with a circular loudspeaker array based on 2.5D HOA, as
shown in Fig. 1.

In all the simulations, rref was set to 0 m and the driving functions of (19) and (22) were used. In the PM method, the truncatedsingular value decomposition [32] with 40 dB was introduced to calculate the stable driving signals. The maximum order of the 2.5D
HOA for the 2D sound field synthesis was also obtained and used up
to N = 10. The two sound source locations in the recording soundfield were s = [2, 2, 0]T and [2, 2, 2]T in Cartesian coordinates.
To estimate the reconstructed and synthesized sound field, thereconstruction / synthesis error at position r is defined as
E(r) = 10 log10|Sorg (r) − Srec/syn (r)|2,|Sorg (r)|2(24)

(a) original field (b) reconstructed (c) reconstructed (d) synthesized
by PM method
S(r, ⇡/2, )from Ǎmfrom Åmn<latexit sha1_base64="Ff9FJ4Vqo60GqaNMHtvIgSZwbPU=">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</latexit><latexit 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sha1_base64="XSW/cFHsyJ7poFd9/Au8Iqo6LQw=">AAACgnichVG7SgNBFD2u7/iK2ggiBIMiFjKrgiIWio2lr6hgNOyOk7hkX+xOgnHZys4fsLBSEAl2+gk2/oCFnyCWCjYW3mwWREW9w8ycOXPPnTMzumsavmTssUFpbGpuaW1rT3R0dnX3JHv7Nn2n5HGR4Y7peNu65gvTsEVGGtIU264nNEs3xZZeXKrtb5WF5xuOvSErrti1tIJt5A2uSaJyyaGsFIcyyHuOlQqz/EDwYrAY5gI73AusMJdMswkWReonUGOQRhwrTvIKWezDAUcJFgRsSMImNPjUdqCCwSVuFwFxHiEj2hcIkSBtibIEZWjEFmks0GonZm1a12r6kZrTKSZ1j5QpjLAHVmUv7J5dsyf2/mutIKpR81KhWa9rhZvrORlYf/tXZdEscfCp+tOzRB6zkVeDvLsRU7sFr+vLR6cv63NrI8Eou2DP5P+cPbI7uoFdfuWXq2LtDAn6APX7c/8Em5MTKuHV6fTCdPwVbRjEMMbovWewgGWsIEPnHqOKG9wqTcq4oipT9VSlIdb040so8x81H5XE</latexit>(f) synthesizedwith Ǎm|m|<latexit sha1_base64="9CPh8YnhORsghgOkdO0f5/7gqJs=">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</latexit>

with Åm<latexit sha1_base64="xIZSgJkBiz1yHXljWjgxudx8VIA=">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</latexit>s = [2, 2, 0]T<latexit sha1_base64="uEkIOjbMNOSIwXhqBZi7smMUQqE=">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</latexit>s = [2, 2, 2]T<latexit sha1_base64="sscyQSsBMHc3a9Bufz0PpW/ycaE=">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</latexit>

Fig. 2. Results of (a) original sound field; (b)-(f) reconstruction and synthesis errors in horizontal plane for two primary sound source locationsof s = [2, 2, 0]T and [2, 2, 2]T and frequency f = 500 Hz (b) reconstructed from estimated Åm ; (c) reconstructed from estimated Ǎmn ;(d) synthesized by pressure matching (PM) method; (e) synthesized by 2.5D HOA for 2D sound field Åm ; (f) synthesized by 2.5D HOA forhorizontal 3D sound field Ǎm|m| . White circles in (a) and black circles in (d)-(f) represent microphones and loudspeakers, respectively.

(a) s: 2D plane wave
<latexit sha1_base64="cm0EPHqq6LlBssVdsZ1O5sdF9pc=">AAACuXichVE9b9RAEJ2Yr3B85AINEs2KS1BoonGEBEqaSFBQ5uuSSOfTae2b5DZZ71r23iXB8h/gD1BQgYQQ4h/Q0vAHKNLRBsog0VAw9llCEAFjeffN23mzb3fDRKvMIR5PeOfOX7h4afJy48rVa9enmtM3NjM7TCNqR1bbdDuUGWllqO2U07SdpCTjUNNWuP+oXN8aUZopazbcUULdWO4ataMi6ZjqNTcCQweRjWNp+nkQqqLjd/MgDu0hZ1b3Y+kGYqblzxRFI3B06PI5eU8UoqwVWTGmFsXCY5FoaUgcyBEVvWYL57EKcRb4NWhBHSu2+QYC6IOFCIYQA4EBx1iDhIy/DviAkDDXhZy5lJGq1gkKaLB2yFXEFZLZfR53OevUrOG87JlV6oh30fynrBQwi5/wLZ7iR3yHJ/jjr73yqkfp5YjncKylpDf17Nb69/+qYp4dDH6p/unZwQ48rLwq9p5UTHmKaKwfPX1+ur64NpvfxVf4lf2/xGP8wCcwo2/R61VaewENfgD/z+s+CzYX5n3Gq/dby1g/xSTchjswx/f9AJbhCaxAm/d9D5/hBL54S570Bt7euNSbqDU34bfwsp8L5aqD</latexit>(b) s = [2, 2, 0]T<latexit sha1_base64="UBbxl5BOTrvFn5oc5QR5jxr/2lk=">AAACv3ichVHLahRBFD1pH4njI6NuBDeNk0gECbeDoAhCxI3LvCYJTo9Dd09lUqSrqumuGTM2/QP+gAtXCiLiTwhu/AEX+QFBXChEcOPC2z0NokG9TVede+qeW6eqwiSWmSU6mHKOHT9xcnrmVOP0mbPnZpvnL2xmZphGoh2Z2KTbYZCJWGrRttLGYjtJRaDCWGyFe/fK9a2RSDNp9IYdJ6KrgoGWOzIKLFO95gNfi0eRUSrQ/dwPZdHxurmvQrPPmYn7KrC77lzLmyuKhm/Fvs0XwmtuUZa6WXGns3Q9991iMlL3IUtZkKp8oyh6zRYtUhXuUeDVoIU6VkzzFXz0YRBhCAUBDcs4RoCMvw48EBLmusiZSxnJal2gQIO1Q64SXBEwu8fjgLNOzWrOy55ZpY54l5j/lJUu5ukDvaZDek9v6BP9+GuvvOpRehnzHE60IunNPrm0/v2/KsWzxe4v1T89W+zgVuVVsvekYspTRBP96PHTw/Xba/P5VXpBn9n/czqgd3wCPfoWvVwVa8/Q4Afw/rzuo2BzadFjvHqjtUz1U8zgMq5gge/7JpZxHyto875v8RFf8NW56wwc7SSTUmeq1lzEb+GMfwIxqa0E</latexit>(c) s = [2, 2, 2]T<latexit sha1_base64="aivHKQZsSFT1U9zx85F0Wu0aOqY=">AAACv3ichVHLahRBFD1pH4njI6NuBDeNk0gECdVBUAQh4sZlXpMEp8ehu6YyKdJV1XTXjBmb+gF/wIUrBRHxJwQ3/oCL/IAgLhQiuHHh7Z4G0aDepqvOPXXPrVNVcZrI3DJ2MOUdO37i5PTMqcbpM2fPzTbPX9jMzTDjos1NYrLtOMpFIrVoW2kTsZ1mIlJxIrbivXvl+tZIZLk0esOOU9FV0UDLHckjS1Sv+SDU4hE3SkW6X4SxdJ2gW4QqNvuUmaSvIrvrz7WCOecaoRX7tljg13xXlvq5u9NZul6EvqvH7kOSkiBTxYZzvWaLLbIq/KMgqEELdayY5iuE6MOAYwgFAQ1LOEGEnL4OAjCkxHVREJcRktW6gEODtEOqElQREbtH44CyTs1qysueeaXmtEtCf0ZKH/PsA3vNDtl79oZ9Yj/+2quoepRexjTHE61Ie7NPLq1//69K0Wyx+0v1T88WO7hVeZXkPa2Y8hR8oh89fnq4fnttvrjKXrDP5P85O2Dv6AR69I2/XBVrz9CgBwj+vO6jYHNpMSC8eqO1zOqnmMFlXMEC3fdNLOM+VtCmfd/iI77gq3fXG3jaSyel3lStuYjfwhv/BDfsrQc=</latexit>

Fig. 3. Results of averaged synthesis error in horizontal plane for r ≤ 0.2 m and 100 Hz ≤ f ≤ 1 kHz: (a) primary recording sound field is
2D plane wave; (b) source position in primary sound field is s = [2, 2, 0]T ; (c) source position in primary sound field is s = [2, 2, 2]T .
where Sorg (r) and Srec/syn (r) are the original and reconstructed /synthesized sound pressures at position r, respectively. Additionally, the averaged synthesis error for r ≤ 0.2 m was also calculated
to evaluate the synthesis accuracy around the array center, including
an additional case where the primary recording sound field was aheight-invariant 2D plane wave with unity amplitude.
Figures 2 and 3 present the results of the reconstruction and synthesis errors for a frequency of f = 500 Hz, and the averaged synthesis error for 100 Hz ≤ f ≤ 1 kHz.
These results suggest that the proposed method can more accurately synthesize the horizontal 3D sound fields recorded by multiple circular microphone arrays compared with the conventional PM
and 2D analysis-based approaches, particularly in the case where theprimary sound source is not located in the horizontal plane. This isbecause the proposed approach can correctly estimate 3D sphericalmharmonic spectrums Ǎmn for n + |m| even, and extract only Ǎ|m|components. When the primary field is a 2D plane wave, the conventional 2D analysis-based method can be used to achieve high accuracy. However, the PM and 2D analysis-based approaches cannot
correctly synthesize the horizontal 3D sound field, particularly in thecase of s = [2, 2, 2]T because the 3D sound fields include not only

horizontal but also vertical components. However, these methods
cannot extract the horizontal components from the 3D sound fields.
Consequently, the proposed approach can effectively record ahorizontal 3D sound field and synthesize it using omni-directional
circular microphone and loudspeaker arrays without vertical derivative measurements. Additionally, the proposed method can estimate
the spherical harmonic spectrums with a maximum order of N using
fewer than (N + 1)2 microphones used in a spherical array.
5. CONCLUSIONSThis paper proposed a horizontal 3D sound field recording and 2.5D
synthesis method with multiple co-centered circular microphone arrays and a circular loudspeaker array without vertical derivative measurements. The spherical harmonic spectrums used for 2.5D synthesis were extracted from the recorded horizontal sound pressuresand the driving function of a circular sound source was analyticallyderived. The results of computer simulations validated the effectiveness of the proposed method compared with the conventional PM
and 2D cylindrical harmonic analysis-based approaches.

6. REFERENCES[1] E. Fisher and B. Rafaely, “Near-field spherical microphonearray processing with radial filtering,” IEEE Trans. Audio,
Speech, Lang. Process., vol. 19, no. 2, pp. 256–265, Feb. 2011.
[2] 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.
[3] 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.
[4] 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.
[5] E. G. Williams, Fourier Acoustics: Sound Radiation and
Nearfield Acoustic Holography, Academic Press, London, UK,
1999.
[6] R. A. Kennedy, P. Sadeghi, T. D. Abhayapala, and H. M. Jones,“Intrinsic limits of dimensionality and richness in random multipath fields,” IEEE Trans. Signal Process., vol. 55, no. 6, pp.
2542–2556, June 2007.
[7] 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.
[8] S. Koyama, K. Furuya, Y. Hiwasaki, and Y. Haneda, “Analytical approach to wave field reconstruction filtering in spatiotemporal frequency domain,” IEEE Trans. Audio, Speech,
Lang. Process., vol. 21, no. 4, pp. 685–696, Apr. 2013.
[9] W. Zhang and T. D. Abhayapala, “2.5D sound field reproduction in higher order Ambisonics,” in Proc. IWAENC, Sept.
2014, pp. 342–346.
[10] S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and
H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circulararrays with a spherical baffle,” J. Acoust. Soc. Am., vol. 139,
no. 3, pp. 1024–1036, Mar. 2016.
[11] 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.
[12] T. Okamoto, “2.5D higher-order Ambisonics for a sound field
described by angular spectrum coefficients,” in Proc. ICASSP,
Mar. 2016, pp. 326–330.
[13] T. Okamoto, “Analytical approach to 2.5D sound field control
using a circular double-layer array of fixed-directivity loudspeakers,” in Proc. ICASSP, Mar. 2017, pp. 91–95.
[14] G. Firtha, P. Fiala, F. Schultz, and S. Spors, “Improved referencing schemes for 2.5D wave field synthesis driving functions,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol.
25, no. 5, pp. 1117–1127, May 2017.
[15] T. Okamoto, “Horizontal local sound field propagation basedon sound source dimension mismatch,” J. Inf. Hiding Multimed. Signal Process., vol. 8, no. 5, pp. 1069–1081, Sept.
2017.

[16] T. Okamoto, “Angular spectrum decomposition-based 2.5D
higher-order spherical harmonic sound field synthesis with alinear loudspeaker array,” in Proc. WASPAA, Oct. 2017, pp.
180–184.
[17] T. Okamoto, “2.5D localized sound zone generation witha circular array of fixed-directivity loudspeakers,” in Proc.
IWAENC, Sept. 2018, pp. 321–325.
[18] G. Firtha, P. Fiala, F. Schultz, and S. Spors, “On the general relation of wave field synthesis and spectral division method forlinear arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 26, no. 12, pp. 2393–2043, Dec. 2018.
[19] M. A. Poletti, “A unified theory of horizontal holographic
sound systems,” J. Audio Eng. Soc., vol. 48, no. 12, pp. 1155–
1182, Dec. 2000.
[20] T. Okamoto, “Mode-matching-based sound field recording andsynthesis with circular double-layer arrays,” Appl. Sci., vol. 8,
no. 7, 1048, July 2018.
[21] W.-H. Liao, Y. Mitsufuji, K. Osako, and K. Ohkuri, “Microphone array geometry for two dimensional broadband soundfield recording,” in Proc. 145th Conv. Audio Eng. Soc., Oct.
2018.
[22] H. Chen, T. D Abhayapala, and W. Zhang, “Theory and design of compact hybrid microphone arrays on two-dimensional
planes for three-dimensional soundfield analysis,” J. Acoust.
Soc. Am., vol. 138, no. 5, pp. 3081–3092, Nov. 2015.
[23] P. N. Samarasinghe, H. Chen, A. Fahim, and T. D Abhayapala, “Performance analysis of a planar microphone array forthree dimensional soundfield analysis,” in Proc. WASPAA, Oct.
2017, pp. 249–253.
[24] H. Zuo, P. N. Samarasinghe, and T. D Abhayapala, “Exteriorinterior 3D sound field separation using a planar array of differential microphones,” in Proc. IWAENC, Sept. 2018, pp. 216–
220.
[25] P. Chen, P. N. Samarasinghe, and T. D Abhayapala, “3D exterior soundfield reproduction using a planar loudspeaker array,”in Proc. ICASSP, Apr. 2018, pp. 471–475.
[26] B. Bu, C. Bao, and M. Jia, “Design of a planar first-orderloudspeaker array for global active noise control,” IEEE/ACM
Trans. Audio, Speech, Lang. Process., vol. 26, no. 11, pp.
2240–2250, Nov. 2018.
[27] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic
Scattering Theory, Springer, Berlin, 1998.
[28] M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions: with Formulas, Graphs, and Mathematical Tables,
Dover Publications, New York, 1965.
[29] O. Kirkeby and P. Nelson, “Reproduction of plane wave soundfields,” J. Acoust. Soc. Am., vol. 94, no. 5, pp. 2992–3000, Nov.
1993.
[30] 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.
[31] I. Balmages and B. Rafaely, “Open-sphere designs for spherical microphone arrays,” IEEE Trans. Audio, Speech, Lang.
Process., vol. 15, no. 2, pp. 727–732, Feb. 2007.
[32] P. C. Hansen, Rank Deficient and Discrete Ill-posed Problems:
Numerical Aspects of Linear Inversion, SIAM, Philadelphia,
1998.