PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Reproduction of multichannel audio by frame loudspeaker array
Yuta KAKUZAKI1; Akio ANDO2
1

University of Toyama, Japan

2

University of Toyama, JapanABSTRACT
Multichannel audio reproduces a sound field by loudspeakers surrounding a listener, which necessarily
occupies the living environment in the home. The purpose of this study is to reproduce the sound by
loudspeakers placed around the image display, which is referred to as “frame loudspeaker array”. In thispaper, we focused on the reproduction of the frontal channels of 22.2 multichannel by the array. Each channel
signal of the multichannel audio was distributed into loudspeakers with the weighting coefficients thatminimized the square error between the original sound field by the channel and the reproduced sound field by
the array. The sound fields were represented by the tenth order spherical functions. The signal was passed
through the auditory band pass filter and changed its amplitude and delay according to the weighting
coefficient calculated for the center frequency of the filter. The band pass signals were then summed up togenerate the full band input signal for loudspeakers. The informal listening test showed that this method
could preserve the spatial feature of the original multichannel audio.
Keywords: Sound field synthesis, Loudspeaker array, Multichannel audio

1. INTRODUCTION
Multichannel audio has increased the number of loudspeakers to provide a higher sense of reality. It
usually requires loudspeakers surrounding a listener. For example, 22.2 multichannel uses three layersof loudspeakers to reproduce a three-dimensional sound field [1]. However, the number ofloudspeakers must be limited in order to avoid occupying the living environment in the home. Thisproblem has become serious because the regular broadcasting of 22.2 multichannel has bee n started
with 8K video from December 2018 in Japan. One of the methods to solve this problem may be a u seof loudspeaker array set up in the frontal direction.
Over the past few decades, a considerable number of studies have been made on synthesizing asound field by the loudspeaker array. A wave field synthesis (WFS) is known as a method to recreate asound filed [2-4]. Particularly, the WFS can recreate a 3D sound field using a line loudspea ker arraywith a 2.5D Rayleigh integral [4], which models 3D sound propagation on the basis of the stationary
phase approximation. However, it may not reproduce the vertical sound stage of 22.2 multichannel. On
the other hand, a plane loudspeaker array is not available by the existence of the image display.
In this study, we used a loudspeaker array set on the frame of the image display, which is referred toas “frame loudspeaker array” [5]. We focused on the reproduction of the frontal eleven channels of
22.2 multichannel by the array in this paper. A sound field created by each frontal loudspeaker of 22.2
multichannel (hereinafter referred to as “target field”) was synthesized by the frame loudspeaker arrayusing the spherical control area centering on the listener. In the synthesis, the square error of soundpressures of both sound field, target field and synthesized field, was minimized over the control area
using the least squares method. The sound fields were represented by the tenth order sphericalfunctions. The resultant weighting coefficient for each loudspeaker of the array was divided into theamplitude and delay information, where the delay was obtained after eliminating the discontinuities of
the arctangent function. Each channel signal of 22.2 multichannel was passed through the auditory
band pass filter and changed its amplitude and delay according to the weighting coefficient calculated
for the center frequency of the filter. The band pass signals were then summed up to generate the fullband input signal for loudspeakers.
1
2m1971017@ems.u-toyama.ac.jp
andio@eng.u-toyama.ac.jp1093

2. SOUND FIELD CONTROL
2.1 Frame Loudspeaker Array
The frame loudspeaker array used in this study was designed so that an image display whose size
was 1.78m (W)×1.00m (H) could be installed at the center. The number of loudspeakers was twenty
considering the size of loudspeaker. The numbers were seven in horizontal and five in verticalarranged at equal intervals. Figure 1 shows the experimental frame loudspeaker array. In this paper, it
is assumed that loudspeakers are arranged as shown in Fig. 2.

Figure 1 – Frame loudspeaker array

Figure 2 – Frame loudspeaker array for simulation

2.2 Experimental setup
The listening position was set to ݀ [m] in front of the array. The control area is a spherical area
centered on the listening point and its radius was set to ܴ [m]. In this study, assuming that the volumeof the spherical control area is ͳଷ, the radius R was set to ͲǤ͸ʹ. The relationship between thearray and the control area is shown in Fig. 3.

Figure 3 – Experimental setup
2.3 Sound Field Control Function
Each sound source is assumed to be a monopole sound source and is represented by sphericalfunctions. The sound pressure of the target sound field ‫݌‬௦ ሺ঒ሻ, which is a sound field by one of thechannels of the multichannel audio, and the sound pressure of the synthesized sound field ‫݌‬௥ ሺ঒ሻ areshown as஽௡௡ୀ଴௠ୀି௡݁ ି௜௞ȁ঒ି঒ೞȁሺଶሻ௠തതതതതതതതതതതത‫݌‬௦ ሺ঒ሻ ൌ ܳ؆ ݅݇ܳ ෍ ݆௡ ሺ݇‫ݎ‬ሻ݄௡ ሺ݇‫ݎ‬௦ ሻ ෍ ܻ௡௠ ሺߠ௦ ǡ ߮௦ ሻܻ௡ ሺߠǡ ߮ሻǡͶߨȁ঒ െ ঒௦ ȁ1094(1)௅௡஽௅݁ ି௜௞ȁ঒ି঒೗ ȁሺଶሻ௠തതതതതതതതതതതത‫݌‬௥ ሺ঒ሻ ൌ ෍ ‫ݓ‬௟؆ ݅݇ ෍ ‫ݓ‬௟ ෍ ݆௡ ሺ݇‫ݎ‬ሻ݄௡ ሺ݇‫ݎ‬௟ ሻ ෍ ܻ௡௠ ሺߠ௟ ǡ ߮௟ ሻܻ௡ ሺߠǡ ߮ሻǡͶߨȁ঒ െ ঒௟ ȁ௟ୀଵ௡ୀ଴௟ୀଵ

(2)

௠ୀି௡

respectively, where ‫ݓ‬௟ ሺ݈ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܮ‬ሻ are weighting coefficients that should be solved so as tominimized the error function [6]‫ܧ‬ሺগሻ ൌ මሼ‫݌‬௥ ሺ঒ሻ െ ‫݌‬௦ ሺ঒ሻሽଶ ܸ݀ Ǥ

(3)௏Note that ঒௦ ൌ ሺ‫ݎ‬௦ ǡ ߠ௦ ǡ ߮௦ ሻ is a position of a target sound source, ঒௟ ൌ ሺ‫ݎ‬௟ ǡ ߠ௟ ǡ ߮௟ ሻ is a position of the݈th-channel loudspeaker of the array, ঒ ൌ ሺǡ Ʌǡ ɔሻ is a position of the observation, ݇ is wave number,‫ ܦ‬is the maximum order of the spherical functions, ‫ ܮ‬is the number of sound sources in the array, ܳሺଶሻis the amplitude of the target sound source, ݆௡ is a spherical Bessel function of the first kind, ݄௡ is a௠spherical Hankel function of the second kind, and ܻ௡ is a spherical harmonics function.
Using the least squares method, the weighting coefficients can be obtained by a linear equation of
the following form:ॼগ ൌ ওǡ(4)

where গ is a vector of the weighting coefficient:‫ݓ‬ଵߙଵ ൅ ݅ߚଵ
‫ڭ‬গൌ൥ ‫ ڭ‬൩ൌ൥
(5)൩ǡ‫ݓ‬௅ߙ௅ ൅ ݅ߚ௅and ॼ is a matrix whose elements are spatial parameters of the synthesized field. Note that the vector
গ is frequency dependent, meaning that it should be calculated for each frequency. The synthesizedsound field is controlled by the weighting coefficient calculated byগ ൌ ॼିଵ ওǤ

(6)

Figure 4 shows the expected sound field controlled by this method.

Figure 4 – Sound field control by frame loudspeaker array
Since the weighting coefficients calculated by Eq. (6) depends on the wave number, it changes withthe frequency of the target sound source. An example of the frequency characteristic of the coefficientis shown in Fig. 5. It was found to be unstable behavior (1 to 6) in the low frequency band and stablebehavior (rotating 7 to 8) in the high frequency band. This is caused by the fact that the condition
number of the matrix in Eq. (6) increases in the low band because the long wavelength brings similarlows and columns in the matrix ॼ. Section 3 describes how to improve it. The weighting coefficient
vector গ controls the synthesized sound field. Therefore, it will be referred to as the sound field
control function hereinafter.

1095

Figure 5 – Behavior of sound field control function
2.3.1 Amplitude Estimation and Control
Sound field control function determines the amplitude of each sound source. Denoting the real andimaginary parts of the function of ݈th-channel loudspeaker by ߙ௟ and ߚ௟ , respectively, the amplitude
is obtained by‫ܣ‬௟ ൌ ටߙ௟ ଶ ൅ ߚ௟ ଶ Ǥ

(7)

Amplitude control is performed by applying this equation to the corresponding sound source.
2.3.2 Delay Estimation and Control
Sound field control function determines the delay of each sound source. The phase angle for݈th-channel may be given using an inverse tangent function as߮௟ ൌ ‫ି݊ܽݐ‬ଵ

ߚ௟
Ǥ
ߙ௟

(8)

This may cause phase inversion due to the nature of the arctangent function. To avoid the phaseinversion, we decided the phase using the equationߚ௟‫ି݊ܽݐۓ‬ଵ  ൫ߙ௟ ൒ Ͳ ‫ߚ ת‬௟ ൒ Ͳࠊߙ௟ ൒ Ͳ ‫ߚ ת‬௟ ൑ Ͳ൯ߙ௟ۖۖߚ௟߮௟ ൌ‫ି݊ܽݐ‬ଵ െ ߨ ሺߙ௟ ൏ Ͳ ‫ߚ ת‬௟ ൏ Ͳሻ

(9)
ߙ‫۔‬
௟
ߚ௟
ۖ
ۖ
‫ି݊ܽݐ‬ଵ ൅ ߨ ሺߙ௟ ൏ Ͳ ‫ߚ ת‬௟ ൐ ͲሻǤ‫ە‬ߙ௟
Furthermore, the relationship between the signal phase and the target source frequency should beconsidered. The range of possible values of the phase angle defined by Eq. (9) is Ȃ ߨ ൏ ߮௟ ൏ ߨ.
However, according to the sound field control function shown in Fig. 5, the locus rotates around theorigin, meaning that its angle should be beyond the interval ൣȂ ߨǡ ߨ൧. Therefore, we propose a method
to extend the possible range of the phase angle using the phase shift ofʹ݊ߨሺ݊ ൌ ‫ ڮ‬ǡ െʹǡ െͳǡ Ͳǡ ൅ͳǡ ൅ʹǡ ‫ ڮ‬ሻ in order to add the rotation to the phase angle. The resultant phaseangle is converted into a delay time by the equation߮௟߬௟ ൌ െǤ(10)ʹߨ݂

3. INPROVEMENT OF CONDITION NUMBER OF MATRIXSound field control function is sometimes unstable at low frequency as mentioned in section 2. The
reason is that the condition number of the matrix ॼ (shown in the Eq. (6)) becomes large in the low
frequency band. In this section, we explain the reason and how to fix it.
Since the wave length of the sound wave is much longer than the size of the control area in the low

1096frequency, the phase of the wave was approximately equal over the control area. As the result, eachelement of the matrix ॼ approaches the similar value, and the condition number of the matrix
becomes large. To avoid such instability of the matrix inversion, we propose a method that improves
the condition number by properly determining the radius of the control area.
We changed the radius of the control area from 0.62m to 2.0m and observed the condition numberof the matrix, and the amplitude and the delay. Figure 6 shows a reproduction set up used for thesimulation. In the simulation, the distance from the center of the loudspeaker array to the center of thecontrol area was set to 2.0m, and the imaginary source was set to 0.5m behind the array. The center ofthe control area, the center of the loudspeaker array, and the imaginary sound source are set on astraight line.

Figure 6 – Listening environment for the simulation

Figure 7 – Change of condition number of matrix ॼ with respect to frequency when radius of control
area ܴ is changed

Figure 8 – Change of gain with respect to frequency when radius of control area ܴ is changed

1097

Figure 9 – Change of delay with respect to frequency when radius of control area ܴ is changed
Figures 7, 8 and 9 show the condition number of the matrix ॼ, the amplitude of the weighting
coefficient of the third loudspeaker ‫ܣ‬ଷ, the delay of the loudspeaker ߬ଷ . The reason for choosing thethird loudspeaker is because it has the largest amplitude in the simulation conditions. According to Fig.
7, as the radius of the control area increases, the condition number of the matrix ॼ decreases
particularly in low frequency band. Therefore, the calculation of the sound field control function
becomes stable if the radius of the control area is increased. As shown in Fig. 8, when the radius of thecontrol area increases, the frequency at which the gain becomes constant is lower. This frequency isalmost the same as the frequency at which the condition number becomes constant as shown in Fig. 7.
According to Fig. 9, when the radius of the control area is increased, the delay time becomes constant
not only in the high band but also in the low band. From the above results, it was found that the soundfield control function can be calculated more accurately by increasing the radius of the control area.
However, it is necessary to set the radius of the control area so that the loudspeaker array and the video
screen do not enter the area in consideration of the singularity of the sound pressure of the synthesizedsound field.

4. MULTICHANNEL AUDIO REPRODUCTION
In this section, a method of reproducing the multichannel audio by creating imaginary soundsources at the channel positions is described. Signal processing for creating imaginary sound source isshown in Fig. 10.

Figure 10 – Signal processing for creating imaginary sound source1098

An imaginary sound source can be created at any position by applying a sound field control
function to the source. The function changes according to the frequency of the target sound source
because it depends on the wave number. Since an audio signal has various frequency components, it isdifficult to apply one function to all components. Therefore, we propose a method of dividing an audio
signal into an auditory band based on MPEG-2 AAC [7], and applying a sound filed control function
corresponding to the center frequency of each band to each band signal. In this study, the 69 auditory
bands defined by MPEG-2 AAC is reconstructed into 23 bands such that the three consecutive
subbands were merged into one.
4.1 Signal Processing for Creating Imaginary Sound SourceThis section describes the signal processing procedure shown in Fig. 10. First, the audio signal is
divided into band signals using the MDCT (Modified Discrete Cosine Transform), the auditory band
pass filter and IMDCT (Inverse MDCT). Second, the imaginary sound source positions are set
according to the loudspeaker positions of the multichannel audio. The sound field control function is
calculated based on the source position. The amplitude and the delay are estimated for the centerfrequency of each subband. Third, each band signal is distributed into all loudspeakers of the arraythrough applying the corresponding amplitude and delay. Finally, all band signals for each loudspeaker
are summed up to generate the input signal for the loudspeaker. A desirable sound field may be createdwithin the control area by reproducing the audio signal with the loudspeaker array.
4.2 Multichannel Audio by Imaginary Sound Source
In this section, we propose a method to reproduce the multichannel audio by multiple imaginary
sound sources. The imaginary sound source is generated at the channel position of the multichannel
audio in the manner of section 4.1. A multichannel sound field is synthesized by simultaneously
reproducing all source signals. Figure 11 shows this situation for two-channel case.

Figure 11 – Multichannel synthetic sound field

5. DISCUSSION
In this paper, the control method for a frame loudspeaker array was proposed. However, the comparison
between the plane loudspeaker array, which has loudspeakers also inside the frame, and the frame
loudspeaker array has not been studied yet. Such study may reveal the possibility and limitation of theframe loudspeaker array.
The method of expanding the control area was introduced to reduce the condition number of the matrix
ॼ. Expanding the control area may degrade the sound field in the original area. Although this possible
degradation should be studied by both the subjective and objective evaluations, it still remains unsolved.
The informal listening test showed that this method could preserve the spatial feature of the original
multichannel audio. However, the formal listening test should be conducted. Such investigation will be
carried out in the future.

1099

6. CONCLUSION
This paper describes a method of reproducing a target sound field by controlling a frame
loudspeaker array using a sound field control function optimized by the least squares method. The
function varies depending on the target sound source frequency and has inaccuracies in the low band.
The latter is due to the fact that the condition number of the matrix ॼ used in the calculation becomeslarge. To reduce the condition number, we proposed a method to increase the radius of the sphericalcontrol area according to the source frequency. This method suppressed the condition number andimproved the accuracy of the source field control function. In addition, decreasing the condition
number brought the phenomenon that the amplitude and delay of the sound field control function
converged to the fixed value. We also proposed a method to reproduce the multichannel audio signal
with the frame loudspeaker array. The reproduction of channels other than the frontal channels of 22.2
multichannel audio will be studied in the next step.

ACKNOWLEDGEMENTS
This work was supported by Hoso Bunka Foundation 2017 Grant.

REFERENCES
1. Hamasaki K, Nishiguchi T, Okumura R, Nakayama Y and Ando A. A 22.2 Multichannel Sound System
for Ultrahigh-Definition TV, SMPTE Motion Imaging J., pp. 40-49, 2008.
2. Berkhout A J. A holographic approach to acoustic control, J. Audio Eng. Soc., vol. 36, no. 12, pp.
977-995, 1988.
3. Berkhout A J, de Vries D, Vogel P. Acoustic control by wave field synthesis, J. Acoust. Soc. Am., vol. 93,
no. 5, pp. 2764-2778, 1993.
4. de Vries D. Wave Field Synthesis, AES monograph. Audio Engineering Soc. 2009.
5. Ando A, Tokioka A. Control of loudspeaker array by minimizing fluctuation of frequency response andsynthesized wave front, Proc. ICA2013, 055001, 2013.
6. Yamabayashi D, Yamoto T and Ando A. Control of frame loudspeaker array using spherical harmonic
expansion, Proc. ASJ 2016 Autumn meeting, pp. 409-412, 2016.
7. ISO/IEC JTC1/SC29/WG11 standard. Text of ISO/IEC 13818-7:2005 (MPEG-2 AAC 4th edition),
ISO/IEC JTC1/SC29/WG11, N7126, 2005.

1100