CLOSE-TALKING RECORDING WITH PLANARLY DISTRIBUTED MICROPHONES
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 close-talking recording method with microphones in arbitrary planar distributions based on sound pressure interpolation. In this method, sound pressures recorded by microphones located near the center of multiple co-centered circular microphone arrays are interpolated using sound pressures recorded bythese arrays via multiple 2D cylindrical harmonic analyses. If thesound sources are outside the spherical boundary formed by the maximum radius of the arrays, the interpolation is completed. However,if the sound sources are inside the spherical boundary, the interpolation is not completed in principle. Using this principle, close-talking
recording is realized as a residual response between recorded andinterpolated sound pressures. The recording radial sensitivity canbe controlled by varying the maximum array radius. By introducing a kernel ridge regression-based sound field interpolation method,
the approach can be extended to the microphones in arbitrary planar
distributions. Simulations confirm the effectiveness of these closetalking recording methods.
Index Terms— Close-talking microphone, sound pressure interpolation, circular microphone array, planar microphone distribution,
kernel ridge regression
1. INTRODUCTION
Microphone arrays are typically used for beamforming to record
only sound signals that propagate from desired directions and many
approaches to this type of recording have been widely investigated [1–5]. For recording of only sound sources located nearmicrophones, however, close-talking recording methods with microphone arrays have also been developed. When compared withclose-talking recording methods that are dependent on the source
directions [6–13], source localization-free approaches are useful because source localization and tracking are not required. Two source
localization-free close-talking methods have been proposed to date.
First, a method that used an open spherical double-layer microphone array combined with a microphone located at the center of
the array was proposed [14]. In this method, the sound pressure at
the array’s center is interpolated from the sound pressures recordedby the array based on the Kirchhoff–Helmholtz integral [15]. If thesound sources are located far from the array, the residual between
the sound pressures recorded by the microphone at the array center and interpolated by the array is small. In contrast, if the soundsources are located close to the array, the residual is large. Using thisprinciple, close-talking recording can be realized.
This study was partly supported by JSPS KAKENHI Grant Number
JP18K11387.

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Fig. 1. Conceptual diagram of proposed close-talking recording withmicrophones in planar distribution rotated to vertical angle for practical implementation.

To reduce the number of microphones required, this approachis realized using an open spherical single-layer microphone arrayrather than a double-layer array based on sound field interpolation in
the spherical harmonic spectrum domain [16]. This method interpolates the sound pressure at the array center from the sound pressures
recorded by the array based on the 0-th order spherical harmonic expansion and can realize close-talking recording as per the method
using the double-layer array [14] while also reducing the number ofmicrophones by half. This method has been applied to near- and farfield sound source separation [17]. In this approach, a spherical arraywith a small radius or a dual-sphere microphone array should be introduced [16, 18] to avoid the forbidden frequency problem where
the spherical Bessel function becomes zero [15]. However, the principle of close-talking used in these methods is based on incomplete
interpolation of the sound field through use of a small number ofmicrophones. Therefore, it is difficult to control the radial sensitivity of the recording. It is also difficult to reduce the number ofmicrophones from a near-equidense distribution on a sphere that isdependent on the target frequency.
To solve this problem, this paper proposes an alternative
residual-based close-talking approach using microphones in a planar
distribution. In the proposed method, the sound pressure at the center of the array is also interpolated using sound pressures recordedby multiple co-centered circular microphone arrays via multiple 2D
cylindrical harmonic analyses [19–21]. If the sound sources arelocated outside the spherical boundary formed by the maximum
radius of the arrays, the interpolation is completed. In contrast,
if the sound sources are located inside the spherical boundary,
the interpolation is not completed in principle. Using this principle, close-talking recording can be realized as a residual response
between the recorded and interpolated sound pressures. When compared with previous methods using spherical arrays [14, 16], therecording radial sensitivity is defined by the maximum radius of© 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
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the arrays and can be controlled by varying the maximum radius.
Additionally, by introducing a kernel ridge regression [22]-based
sound field interpolation method [23], the proposed approach canbe extended to the method using microphones in arbitrary planar
distributions. For actual implementation, the microphones in theplanar distribution are rotated to vertical angles, as shown in Fig. 1,
where close-talking recording can be realized to only record thevoice of speaker (A) because the other sources (speakers (B) and(C), reverberations) are located outside the boundary and the soundpressures from these sources can be completely interpolated.
The proposed method was inspired by localized sound zone generation methods using an open spherical loudspeaker array [24] ormultiple co-centered circular loudspeaker arrays [25]. In [24], thesound field radiated from a point source located at the center of
the array is cancelled using the open spherical array and a localized sound zone can then be generated inside the array. Similarly,
a 3D sound field propagating from a point source located at thecenter of the array can be cancelled using multiple co-centered arrays [26, 27] and a 3D localized sound zone was also generated inside a sphere with the maximum radius of the arrays in [25]. Interestingly, the filter coefficients of the array used for the method
in [24] are equivalent to those used for the close-talking method with
an open spherical microphone array [16] based on acoustic sourcereceiver reciprocity [28]. Application of this reciprocity to the localized sound zone generation method with multiple co-centered circular loudspeaker arrays [25] helped to inspire the proposed closetalking method with multiple co-centered microphone arrays.
To ensure the reproducibility of this study, the MATLAB codeused in the computer simulations conducted in Sec. 4 is available
online. 1
2. HORIZONTAL 3D SOUND FIELD RECORDING

Using (3), a continuous circular sound pressure distribution centeredat the origin of the x-y plane (θ = π/2) can be converted into 2D
cylindrical harmonic spectra [15] and represented by:
Z 2π
1S̊m (r) =S(r, π/2, φ)e−jmφ dφ
(4)
2π 0
∞
X
|m|
=
Ǎm
(5)
n jn (kr)Pn (0).
n=|m|When the maximum spherical harmonic order is N and the radius of
the circle is Rq , (5) can be given asS̊m (Rq ) 'mǍmn jn (kr)Yn (θ, φ),(1)where Ǎmn and jn are the spherical harmonic spectra of the interiorsound field and the n-th order spherical Bessel function, respectively,
and k is the wavenumber [15];s(2n + 1) (n − |m|)! |m|m
Yn (θ, φ) =Pn (cos θ) ejmφ (2)4π(n + |m|)!
|
{z
}
|m|

endre function [29].
To consider a 3D sound field in the horizontal plane with θ =π/2, (1) is represented by:S(r, π/2, φ) =

|m|
jmφ
Ǎm
,
n jn (kr)Pn (0)e(3)

n=0 m=−n|m|where only Ǎmn for n+|m| even is present because Pn (0) = 0 for
n + |m| odd [19]. To obtain Ǎm
n for the n + |m| odd components,
1 https://codeocean.com/capsule/6915360

(6)

n=|m|

S̊ m = U |m| Ǎm ,

(7)

h
iT
S̊ m = S̊m (R1 ), S̊m (R2 ), · · · , S̊m (RQ ) ,

(8)



|m|

|m|

U|m| (kR1 ) U |m| (kR )2
 |m|
U |m| = 
..


.
|m|

U|m|+2 (kR1 )|m|

U|m| (kRQ )(cos θ)|m|represents the spherical harmonics, and Pn is the associated Leg-

∞
n
X
X

|m|
Ǎm
n jn (kRq )Pn (0).

where

n=0 m=−n

Pn

N
X

even

The interior expansion of a 3D sound field in a homogeneous region
that is free from sources is given by:S(r, θ, φ) =2.2. Horizontal 3D sound field recording with multiple cocentered omni-directional circular microphone arrays

Introduction of multiple radii (q = 1, 2, · · · , Q) to (6) allows it tobe represented in matrix form as:

2.1. 3D sound field in horizontal plane

∞
n
X
X

|m|

the vertical derivatives of Pn (0) are required and can be measured
using differential microphones or vertical microphone pairs in thehorizontal plane [19]. Therefore, both omni-directional and vertical differential microphones are required to estimate the entire 3D
sound field [19]. However, the proposed close-talking approach can
set the positions of the interpolated sound pressures freely and these
positions can also be located on the horizontal plane. Then, the interpolated sound pressures also contain only Ǎmn for n + |m| evencomponents and the proposed method does not require the vertical
microphone pairs as used in horizontal 3D sound field recording witha planar microphone array [30] for 2.5D higher-order Ambisonics(HOA) [31, 32].

U|m|+2 (kR2 )..
.
|m|
U|m|+2 (kRQ )

···

···
..
.
···

|m|

UN (kR1 )|m|
UN (kR2 ) 

,
..


.
|m|

UN (kRQ )Un|m| (kR) = jn (kR)Pn|m| (0),

(9)

(10)

and

even
m
m T
Ǎm = Ǎm,
|m| , Ǎ|m|+2 , · · · , ǍN(11)in the case where both N and m are either odd or even; otherwise,
replace N in both (9) and (11) with N − 1. The 3D spherical harmonic spectra Ǎmn for n + |m| even can then be obtained from multiple co-centered omni-directional circular sound pressure distributions as [19]:evenǍm = U +|m| S̊ m ,where U +|m| is the generalized inverse of U |m| [21].

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recorded using a vertical microphone pair located on the z-axis withr = d, θ = 0 and π, as shown in Fig. 2(b). Using this arrangement, the recording remains symmetrical with respect to the horizontal plane and the z-axis. The sum of the recorded sound pressures is thus also symmetrical with respect to the horizontal planeand can be interpolated using the sound pressures recorded by the array, although the vertical pair is not located on the horizontal plane.
Additionally, it is sufficient to calculate only the spherical harmonic
spectra Ǎ0n because the recording is symmetrical with respect to thez-axis. Therefore, the residual can be derived from (3) and given as
N r
X
2n + 1 0
D±d,circ = S(d, 0, 0) + S(d, π, 0) −Ǎn jn (kd),
π
n=0
(14)

where
<latexit sha1_base64="GAlke8P8FBb1rGj6ej5TJRgrBQs=">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</latexit>(b)<latexit sha1_base64="0Dg5OtbMbjCwOinJfEQVmCtnmOU=">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</latexit>(c)Fig. 2. Proposed close-talking recording method with (a) multiple
circular arrays and one additional microphone; (b) multiple circular arrays and a vertical monopole pair; (c) microphones in planar
distribution and a vertical monopole pair.
3. PROPOSED CLOSE-TALKING RECORDING METHOD
3.1. Close-talking recording using multiple co-centered circulararraysIn the proposed method, an additional microphone is also set at thecenter of the multiple circular arrays, as used in the close-talking
methods with an open spherical array [14, 16]; this is shown in
Fig. 2(a). In this case, all microphones are located on the horizontal
plane. The sound pressure recorded using the microphone at thecenter of the array can then be interpolated from the sound pressures
recorded by the multiple circular arrays using (12) and (3). Specifically, by locating the additional microphone at the origin, (3) canbe simplified with only the 0-th order component and the residual
D0,circ between the recorded sound pressure and the interpolatedpressure is also calculated as [16]:Ǎ0(13)D0,circ = S(0, π/2, 0) − √ 0 ,4π√
where j0 (0) = 1 and P00 (0) = 1/ 4π [15].
As described in Section 2.1, (1) and (3) only hold for an interior sound field in which all sound sources are located outside thespherical recording boundary [15]. In this case, the boundary is asphere with the maximum radius of the multiple circular arrays, as
illustrated in Fig. 2(a). Therefore, if the sound sources are locatedoutside the spherical boundary, D0,circ is small. In contrast, if thesound sources are inside the boundary, (1) and (3) no longer holdand D0,circ is large. In the proposed approach, close-talking recording can be realized using this principle. By varying the maximum
radius of the array, the recording radial sensitivity can be controlled
in principle.
However, when the sound sources are inside but close to theboundary, D0,circ becomes small because all the microphones arelocated on √the same horizontal plane and S(0, 0, 0) can be predictedusing Ǎ00 / 4π to some extent, even though the sound sources are
inside the boundary. To expand the recording radial sensitivity inside the boundary while maintaining the simplicity of the approach,Pn0 (1) + Pn0 (−1) =r

2n + 1.
π(15)3.2. Close-talking recording with microphones in arbitrary planar distributions
The proposed close-talking recording approach can be extended touse arbitrary distributions of microphones in the horizontal planerather than multiple co-centered circular arrays. Although the estimated spherical harmonic spectra Ǎmn for n + |m| even are requiredfor 2.5D HOA in [30], the proposed close-talking method ultimately
requires the sound pressures S(0, 0, 0) or S(d, 0, 0) + S(d, π, 0)rather than Ǎm
n . In this case, the kernel-induced sound field interpolation method [23], which is a special case of the harmonic
analysis-based method using omni-directional microphones [33],can be applied. The sound pressures at position r in the horizontal plane are then directly interpolated from the sound pressures
recorded by the microphones in the planar distribution located at
r 1 , r 2 , · · · , r v , · · · , r V without the truncated spherical harmonicorder N as
T
S(r) = (Ψ + λI V )−1 S κ(r),
(16)

where

S = [S(r 1 ) · · · S(r V )]T ,(Ψ)v,v0 = j0 (k||r v − r v0 ||),

κ(r) = [j0 (k||r − r 1 ||), · · · , j0 (k||r − r V ||)]T ,

(17)
(18)
(19)

I V is the identity matrix, and λ is a regularization parameter [23].
The residual when using an additional microphone on the origin isthen given by
T
D0,dist = S(r 0 ) − (Ψ + λI V )−1 S κ(r 0 ),(20)

where r 0 = (0, π/2, 0). Additionally, the residual when using thevertical microphone pair is derived as
D±d,dist =

2 n
o
X
T
S(r i ) − (Ψ + λI V )−1 S κ(r i ) ,

(21)

i=1

where r 1 = (d, 0, 0) and r 2 = (d, π, 0). Using (20) and (21), theproposed close-talking recording method with the microphones in aplanar distribution on the horizontal plane can be realized without
the truncated spherical harmonic order N , as shown in Fig. 2(c).

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Fig. 3. Results for the residual pressure level 20 log10 |D| used as the recording radial sensitivity for the proposed approaches. (a)-(f) wereobtained using three layers of circular arrays; (a)-(b) D0,circ ; (c) D±d,circ ; (d) D0,circ with large radii; (e) D0,dist ; (f) D±d,dist ; (g)-(h)D±d,dist obtained using a rectangular array. Blue circles represent microphones.
4. COMPUTER SIMULATIONS
4.1. Simulation conditions
To evaluate the proposed close-talking recording methods, computer
simulations were performed. A 3D free field was assumed in all
simulations and the speed of sound c was 343.36 m/s. All microphones were omni-directional. In the experiments, three co-centeredcircular microphone arrays were introduced into the horizontal plane
and their radii were set at 0.3, 0.2, and 0.1 m. The number of microphones in each circular array was 16, 14, and 10, respectively,
as shown in Fig. 3(a)-(c), (e), and (f). Additionally, to estimate theeffects of the radii of the arrays, arrays with radii of 0.5, 0.3, and
0.1 m and the same number of microphones were also introduced,
as shown in Fig. 3(d). Therefore, the total numbers of microphones
were 41 for D0,circ and D0,dist , and 42 for D±d,circ and D±d,dist .
Additionally, a rectangular array containing 40 microphones in thehorizontal plane was also introduced as shown in Fig. 3(g) and (h).
In the method with the vertical microphone pair, d in (15) and (21)was set at 0.1 m. For the method based on multiple 2D cylindricalharmonic analyses, the maximum order N in (9) and (11) was set at
6. The regularization parameter λ in (20) and (21) was 10−5 .
To evaluate the recording radial sensitivity, a monopole source
was arranged at [x, y, z]T in Cartesian coordinates, where −1 ≤ x ≤1, −1 ≤ y ≤ 1, and −1 ≤ z ≤ 1. Then, 20 log10 |D| was evaluatedusing the sound pressures received via the microphones.
4.2. Simulation results
The results of the recording radial sensitivity for the frequency f =500 Hz are plotted in Fig. 3. All results indicated that the proposed
approaches based on both multiple 2D cylindrical harmonic analyses and kernel-induced sound field interpolation on the horizontal

plane can realize close-talking recording inside the spherical boundary with the maximum radius of the arrays.
From Fig. 3(b), (c), (e) and (f), the recording radial sensitivities of the proposed methods based on both multiple 2D cylindricalharmonic analyses and kernel-induced sound field interpolation withmultiple circular arrays were almost identical. By introducing a vertical microphone pair or arrays with larger radii, the recording radial
sensitivity can be expanded compared with the methods using a microphone located at the origin, as shown in Fig. 3(c), (d) and (f). The
results in Fig. 3(e) and (h) show that the proposed method based onkernel-induced sound field interpolation can be realized not only using multiple circular arrays but also using a rectangular array on thehorizontal plane.
Consequently, the effectiveness of the proposed close-talking
recording methods based on both multiple 2D cylindrical harmonic
analyses and kernel-induced sound field interpolation on the horizontal plane was validated by the computer simulation results.
Future work will include detailed analysis of the relationships
among the number of circular arrays, their radii, and the recording
area shape, and experiments with practically implemented arrays.
5. CONCLUSIONSThis paper proposed a close-talking recording method with microphones in arbitrary planar distributions based on sound pressure interpolation. In the proposed method, sound pressures recorded using microphones located at the center of the array are interpolatedusing sound pressures recorded using the microphones in the planar distribution. Close-talking recording can then be realized as aresidual response between the recorded and interpolated sound pressures. Computer simulations verified the effectiveness of the proposed close-talking recording method.

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