[Pressure-matching-based 2D sound field synthesis
with equivalent source array]
Izumi TSUNOKUNI(1) , Kakeru KUROKAWA(2) , Yusuke IKEDA(3)(1) Tokyo Denki University, Japan, 19fmi23@ms.dendai.ac.jp(2) Tokyo Denki University, Japan, 19fmi13@ms.dendai.ac.jp(3) Tokyo Denki University, Japan, yusuke.ikeda@mail.dendai.ac.jp

Abstract
Pressure-matching(PM) method is one of well-known effective methods for physical sound field synthesis using
a massive loudspeaker array. In the PM method, all transfer functions between loudspeakers and matching points
have to be measured so as to control the sound pressure at each corresponding matching point. Thus, as thenumber of loudspeakers and matching points increases, it becomes more difficult to measure transfer functions
using a microphone. In addition, it is difficult to achieve high accurate placement of a microphone without arobot or microphone array. In this paper, we proposed an efficient method of local sound field synthesis withpressure matching. This method does not require large-scale measurements as it estimates the transfer functions
between loudspeakers and dense matching points from transfer functions between the loudspeakers and a small
number of points. By sparse optimization, each set of equivalent sources of a loudspeaker is selected from adictionary of mono-pole sources placed near each loudspeaker based on a small number of transfer functions.Thedriving functions of loudspeakers can be derived from transfer functions between numerous virtual matching
points and all equivalent sources. To evaluate our proposed method, two-dimensional simulation experiments
are conducted. Keywords: Sparse modeling, Impulse response, Transfer function

1INTRODUCTION

Pressure Matching(PM) is one of sound field synthesis techniques to physically reproduce a sound field by
matching a sound pressure at each matching point[1]. Generally, numerous matching points with higher density
are required to control sound field at higher frequency. As the number of matching points increases, it takes
more time and effort to obtain all transfer functions. In most of physical sound field synthesis(SFS) methods,
transfer functions between loudspeakers and control points are also required, especially when compensating thecharacteristics of loudspeakers and the sound reflections in the room.
To measure the sound field, measurement methods with a moving microphone have been proposed[2, 3]. To
move a microphone accurately, a robot or a moving instruments is required. In most cases, the size of instruments to move a microphone is not small enough to ignore the sound reflections by itself. In addition, moving
instruments must be sufficiently silent to measure the transfer functions. In [4], a method of interpolation forroom impulse responses in a whole room has been proposed under the assumption of sparsity in the early partof reverberation. However, the large number and complex arrangement of microphones are required. In mostcases, transfer functions in a whole room is not required to control sound field in a specified area.
In this paper, we propose local sound field control method with an estimation for the direct-sound transfer
functions between loudspeakers and arbitrary points in a local area. Assuming sparsity of the sound source,
the wave field can be represented by superposition of the sparse point sources[5, 6]. In our proposed method,
sparse equivalent sources and its weight coefficients are derived to estimate the transfer functions between aloudspeaker and a specified area from impulse responses measured at a small number of points. In this paper,
estimated transfer function are utilized for PM method to synthesis sound field in a local area. In simulation
experiment, we evaluate the errors of the estimated transfer functions in the specified area. Then, we evaluate
reproduction errors of PM using the estimated transfer functions by 2D simulation.

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2PROPOSED METHOD

mth Mic.

Transfer function
from the thmonopole source

nth monopolesource

Transfer functions Loudspeakerfrom the thloudspeaker

Matching points

Figure 1. Concept of proposed reproduction method with equivalent sources. yl is a transfer function vector
from l-th loudspeaker to measurement points.
To reduce the cost of sound field measurement for SFS, we will obtain the transfer functions at an arbitrary
point in the local region from the measurement at a small number of points. Especially, the direct soundincluding frequency characteristics and directivity of loudspeaker is important to control sound field. Now,
consider modeling the direct sound component of the transfer functions from a loudspeaker with equivalentsources. Note that the direct sound includes reflections from a loudspeaker by itself, since it is difficult todivide them.
Figure 1 shows the concept of our proposed reproduction method with equivalent sources. Firstly, it is assumed
that the approximate positions of the loudspeakers xl relative to the microphones are known.
Consider that measured transfer functions are represented by the superposition of multiple point sources xl,n ,which are placed in the area surrounding the expected position of a loudspeaker. In the frequency domain, thetransfer function vector y(∈ CM×1 ) at the M measurement points is given byyl = Gl wl (l = 1, 2, . . . , L).

(1)

Gl = [G(l) m,n ](∈ CM×N ) denotes a Green’s function matrix between the M measurement points and N point
sources surrounding the position of lth loudspeaker, and wl (∈ CN×1 ) denotes the weight coefficient vector for
N point sources. Since each group of point sources includes just one loudspeaker, sparsity of the point sources
can be assumed. Thus, the weight coefficient vector wl can be derived by solving the following optimization
problem:1minimize||yl − Gl wl ||22 + λ ||wl ||1 ,(2)wl
2where λ and ||·||1 are the penalty parameter and L1 -norm, respectively. When the elements of weight coefficient
wl is sufficiently small, the value is considered to be zero. Subsequently, the coefficient is compensated byusing the transfer function at a single measurement point. If the transfer function of direct sound from theloudspeaker is well-modeled by the non-zero weight coefficient ws and the corresponding point sources xs , the2702

0.05 myLine source3ch[2, 0]

1ch 2ch[0, 0]
4ch

xLoudspeakerEvaluation area
0.10 m0.40 m(b) Position of a loudspekaer
and line sources(a) Arrangement ofthe microphone

Figure 2. Arrangement in single loudspeaker simulation. A loudspeaker is positioned at [2, 0]. The dictionaryof line sources in the area with a radius of 0.10 m is placed along with the loudspeaker. The number of line
sources is 300.
transfer function at any points in the area A is estimated as follows,Ĝ(xs , x, ω) = −1 S
(2)∑ ws H0 (k|x − xs |) (x ∈ A),
4π s=1(3)(2)where k denotes the wave number, H0 denotes the Hankel function of 0-th order and second kind. S is thenumber of sparse point sources, which has a non-zero corresponding coefficient. In the following section,2D
simulation was conducted by using 2D Green’s function as shown in Eq.(??).
In PM method, a sound pressures P(xm , ω) by line sources at xl in 2D situation is represented with driving
function of l-th loudspeaker D(xl , ω) and a transfer function G(xl , x p , ω) from l-th loudspeaker xl to matching
points x p as following[1],LP(x p , ω) = ∑ D(xl , ω)G(xl , x p , ω),(4)

l=1
(2)

G(xl , x p , ω) = H0 (k|xl − x p |)(5)

. By using the modeled transfer function Ĝ instead of the transfer function G, Eq.(4) is changed as follows,LP(x p , ω) = ∑ D(xl , ω)Ĝ(xl , x p , ω),

(6)

l=1

The desired sound field P(xs , x p , ω) from a line source xsrc is given by(2)

P(xsrc , x p , ω) = H0 (k|xl − x p |).

(7)

The vector of solution of driving function vector d = [D(xl )] ∈ CL×1 can be derived by least squares method asfollowing,d = [ĜH Ĝ + ρI]−1 ĜH p(8)2703where ρ is a regularization parameter, Ĝ describes the matrix of transfer functions between equivalent sources
and matching points. I is identity matrix. [·]H is complex conjugate transposition.
The number of matching points can be freely increased with higher dense to obtain driving functions. Inaddition, this method is able to apply to sound field synthesis techniques matching the sound pressure at thepoints such as BoSC.

3SIMULATION EXPERIMENTS

00.5-10

B. Unidirectional0(a) Desired line source(b) Superposition ofline source-20-0.5-30-1-40Error [dB]

1Amplitude

A. Omnidirectional

3.1 Conditions
We conducted simulation of a sound field in a 2D free-field condition. Firstly, the errors of estimated and desired transfer functions were compared. Secondly, the accuracy of synthesized sound field by estimated transferfunction and the desired Green’s function was compared. The directivity of the loudspeaker is unidirectional i.e.,
toward the origin, which is given by (1 + cos θ )/2 where θ is angle between the directions of the loudspeakerand measurement position.
To model each loudspeaker, 300 line sources were placed in a circle with a radius of 0.05 m. The penalty
parameter is λ = 10−4 for sparse optimization.
To evaluate reproduction accuracy, the average of reproduction error in the area A with a radius of 0.20 m wasdefined by∑I |Pd (xi , ω) − Ps (xi , ω)|2(9)Error(ω) = 10 log10 i=1 I∑i=1 |Pd (xi , ω)|2
where Pd and Ps denote the desired complex sound pressure and the estimated complex sound pressure, respectively. I is the number of evaluation points inside a circular area A.

(c) Error

Figure 3. The wave front represented by sparse line source, f = 1000 Hz. In the upper figures, original sourceis an omnidirectional line source in horizontal plane. In the lower figures, original source is a unidirectionalline source.

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Table 1. The errors of estimated transfer function and desired transfer function
Frequency [kHz]Error [dB]
The number of line source

1-54.5
192-56.5
203-57.8
184-59.985-59.7
96-55.6
77-58.6
68-58.47

3.2 Estimation of transfer function
The microphones were placed in the interval of 0.05 m at around the origin, and a loudspeaker was placedat [2, 0]. Fig.2 shows an arrangement of instruments for simulation experiment. The number of line sources
as the dictionary is 300. Table 1 shows the errors of estimated transfer function of direct sound using fourmicrophones. The errors were very small at about −55–−60 dB for 1–8 kHz. Thus, direct sound in the areaaround the origin were accurately modeled by superposition of sparse line sources.
Figure3 shows the wave front and the errors of estimated transfer function. As shown in Fig.3, omnidirectionalline source can be accurately modeled. In contrary, when the unidirectional line source was modeled, the sizeof area that transfer functions can be accurately estimated becomes smaller. Thus, the desired sound sourceis closer to line source (or point source), the accuracy improves. Thus, when the source has a directivity, theaccuracy of estimated transfer function is lower.
3.3 Synthesized sound field using estimated transfer function

ySecondary sources

0.40 m=[0, 0]x

Reproduction area

4.0 m

Figure 4. Condition of sound field synthesis by PM method. Sound field is synthesized by a circular array withradius of 2 m. The number of loudspeaker is 84.

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(c) 8 kHz (zoom)(d) 8 kHz (zoom)
0-10(e) 1 kHz
B. Synhtesized sound fieldby proposed method(b) 1 kHz (zoom)(f) 1 kHz (zoom)-20(g) 8 kHz (zoom)(h) 8 kHz (zoom)

Error [dB]

A. Synhtesized sound fieldby green’ s function(a) 1 kHz-30-40

Figure 5. Errors of synthesized sound field : (a)–(d) sound field was synthesized by desired transfer function,and (e)–(h) sound field by estimated transfer function. Figs (a), (c), (e) and (g) show the errors of synthesizedsound field. Figs (b), (d), (f) and (h) are expansion of left figure.

Figure 6. Comparison of errors with PM method: Red circle shows the errors of synthesized sound field byusing modeled transfer functions when the center of reproduced area is at xc = [0, 0]. Red dashed line showsthe errors at xc = [0, 0] by using desired transfer functions. Blue cross and dashed line show the errors when
the center of reproduction area is at xc = [0, 1].
Sound field was reproduced by PM method by using a circular loudspeaker array with an interval of 0.15
m. Sound field was controlled at 500 matching points. In this simulation, the plane wave propagating in
direction negative of y-axis was synthesized in circle of radius 0.2 m (Fig.4). The error maps of synthesizedsound field were illustrated in Fig.5. From the figure, the accuracy of synthesized sound field using modeled2706

transfer function was extremely close to that using Green’s function. Figure 6 shows the errors of synthesizedsound field by modeled transfer function for 1–8 kHz. Comparing sound fields reproduced at xc = [0, 0] by the
proposed method and desired transfer functions, the difference is less than about 5 dB at 4 kHz or more. For
1–3 kHz, the differences are about 5–10 dB. When the center of reproduction area is at xc = [0, 1], the errors
becomes larger. However, this results was caused by lower accuracy of PM method.

4CONCLUSIONSIn this paper, we proposed sound field synthesis method using estimated transfer function to reduce time andeffort for implementation. The transfer functions between loudspeaker and any points in local area can be wellmodeled by superposition of sparse line sources, which are calculated from a few measurement points. By
our proposed method, the number of matching points to control the sound field can be arbitrarily increased.
Thus, a small number of measurement points are only required, even when many matching points are required.
However, when the sound source with directivity is modeled, the accuracy is lower than Omnidirectional soundsource. In future works, we improve modeling a sound source with directivity.

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant Number 17KT0141.

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