Sound Field Reproduction Based on Pressure Matching with Transfer Functions Modeled by Equivalent Sources and Image Sources

Yukika Suzuki 1

Tokyo Denki university 5 Senju-Asahi-Cho, Adachi-ku, Tokyo, JAPAN

Haruka Matsuhashi 2

Tokyo Denki university 5 Senju-Asahi-Cho, Adachi-ku, Tokyo, JAPAN

Izumi Tsunokuni 3

Tokyo Denki university 5 Senju-Asahi-Cho, Adachi-ku, Tokyo, JAPAN

Yusuke Ikeda 4

Tokyo Denki university 5 Senju-Asahi-Cho, Adachi-ku, Tokyo, JAPAN

ABSTRACT Pressure matching (PM) is an e ff ective method for sound field reproduction. In the PM method, the controlled area is discretized as control points, and the sound pressures at the control points are controlled using the transfer functions between the control points and loudspeakers. However, room impulse responses at several control points from many loudspeakers are required. Previously, we developed the reproduction method based on the modeling of the transfer functions of only the direct sounds generated by the loudspeakers. By modeling transfer functions, many transfer functions can be obtained from a few microphones for PM. However, in actual environments, the sound is reflected by the enclosures of the loudspeakers. In this study, we devised a sound field reproduction method based on the modeling of the transfer functions of direct sounds and primary reflective sounds. We conducted measurement experiments using a 96-ch rectangular loudspeaker array. The proposed method was compared with the PM method that does not consider primary reflective sounds while modeling room impulse responses.

1 22fmi26@ms.dendai.ac.jp

2 21fmi18@ms.dendai.ac.jp

3 21udc02@ms.dendai.ac.jp

4 yusuke.ikeda@mail.dendai.ac.jp

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION

Various sound field synthesis and reproduction / control methods with multiple loudspeakers have been proposed [1] [2] [3] [4]. The pressure matching (PM) method is one of the e ff ective methods of sound field control. It controls the sound field by matching the sound pressures at discretized control points inside the reproduction area with desired sound pressures [5]. Although the PM method can control the sound field with high precision by compensating loudspeaker characteristics and room reverberation, the transfer functions between the loudspeakers and control points are required to be measured in advance. However, handling a large number of measurement points becomes challenging as the numbers of loudspeakers and control points increase. Previously [6], we proposed a PM method based on the estimation of transfer functions based on a few measurements obtained using the equivalent source method (ESM) [7]. Bokai et. al. ( [8]) proposed a multizone sound field reproduction method by controlling the dark zone using the ESM. In the previous studies based on the ESM, only the direct sounds from the loudspeakers were considered. However, the transfer functions of loudspeakers generally include sound reflections such as the sound reflected by the enclosures of the loudspeakers. To estimate the sound field including reflected sounds, a modeling method based on the sparse ESM and image source method have been proposed [9]. In this study, we developed a PM method based on the ESM and image source method by considering the sounds reflected by the enclosures of loudspeakers. For evaluation, we compared the reproduction accuracies of the proposed method with and without the consideration of reflected sounds for sound fields that include direct sound and primary sound reflections.

2. METHOD

2.1. Modeling transfer functions using image source and equivalent source methods To obtain the transfer functions between loudspeakers and control points, we model the transfer functions considering the direct and reflected sounds by using the measurements obtained via a small number of microphones. For simplicity, sound reflections that include only primary reflective sounds are considered. First, we assume that the approximate positions of the loudspeaker and microphones are known. In the frequency domain, the transfer function from a loudspeaker to the m -th microphone ( m = 1 , . . . , M ), including the primary reflective sound, is represented by

y m = y (0) m + · · · + y ( l ) m + · · · + y ( L ) m , (1)

where y (0) m is the direct sound component and y ( l ) m is the component of the primary sound reflected by the l -th wall. Based on the ESM, the transfer function of the direct sound y (0) m is represented by the sum of the transfer functions of point sources placed around the loudspeaker. Thus, the transfer function of the m -th microphone at x ′ m from the loudspeaker, including the primary sound reflected by the l -th wall, is represented as

y ( l ) m z ( l ) m w ( l ) ( l = 0 , 2 , ..., L ) , (2)

z ( l ) m h Z ( x ′ m , x ( l ) 1 ) , . . . , Z ( x ′ m , x ( l ) N ) i , (3)

w ( l ) [ w ( l ) 1 , . . . , w ( l ) N ] T , (4)

where z ( l ) m ( ∈ C 1 × N ) is the transfer function vector from the equivalent source ( n = 1 , . . . , N ) placed around the loudspeaker to the m -th microphone. This can be calculated using the Green’s function under free-field condition. w ( l ) m is the weight coe ffi cient vector of the equivalent source, and l = 0 indicates the direct sound.

Equation (2) is valid in case of the direct sound and all primary reflections, for all microphones. Thus, it can be represented in the matrix form as follows:

y Zw (5) y [ y 1 , . . . , y M ] T (6)

z (0) 1 · · · , z ( L ) 1 ... ... ...





Z

(7)

z (0) M · · · z ( L ) M

w [ w (0)T , . . . , w ( l )T , . . . , w ( L )T ] T (8) (9)

where y denotes the vector of the microphone signals, Z ( ∈ C M × ( N × ( L + 1)) ) is the matrix of the transfer functions from the equivalent sources to the microphones, and w ( ∈ C ( N × ( L + 1)) × 1 ) denotes the vector of the weight coe ffi cients. As the sound and image sources in the space are sparse, the sparsity is assumed for the weight coe ffi cient vector w . Therefore, w is obtained by solving the following optimization problem:

minimize w 1 2 ∥ y − Zw ∥ 2 + λ ∥ w ∥ 1 , (10)

where λ is the penalty parameter and ∥· ∥ 1 and ∥· ∥ 2 denote ℓ 1-norm and ℓ 2-norm, respectively. Using the obtained weight coe ffi cients of the equivalent sources, the estimated transfer function ˆ G from the s -th loudspeaker ( s = 1 , . . . , S ) to an arbitrary position x in the control region A is defined as follows:

n = 1 w ( l ) n e − ik | x − x ( l ) n |

L X

N X

ˆ G ( x spk , s , x , ω ) = 1

| x − x ( l ) n | ( x ∈ A ) , (11)

4 π

l = 0

where i is the imaginary unit, k is the wavenumber, and ω is the frequency.

2.2. Pressure matching using modeled transfer functions In sound field reproduction, the sound pressure P ( x pm , p , ω ) at the control point x pm , p can be represented using the driving functions of the loudspeakers and transfer functions G between the loudspeakers and the control points, as follows:

S X

P ( x pm , p , ω ) =

s = 1 G ( x pm , p , x spk , s , ω ) d ( x spk , s , ω ) (12)

where d ( x spk , s , ω ) is the driving function of the loudspeaker at x spk , s , and G ( x pm , p , x spk , s , ω ) is transfer function G from the s -th loudspeaker x spk , s to the control point x pm , p . The conventional PM method requires to measure all the transfer functions G , whereas the proposed method can obtain the transfer functions ˆ G estimated using Eq.(11). As Eq.(12) is valid for all the control points, the following equation is obtained:

p = ˆ Gd , (13)

where

p [ P ( x pm , 1 , ω ) , . . . , P ( x pm , M pm , ω )] T (14)

d [ d ( x spk , 1 , ω ) , . . . , d ( x spk , S , ω )] T (15)

ˆ G ( x pm , 1 , x spk , 1 ) · · · , ˆ G ( x pm , 1 , x spk , S ) ... ... ... ˆ G ( x pm , M pm , x spk , 1 ) · · · ˆ G ( x pm , M pm , x spk , S )





ˆ G

(16)

(17)

The driving function vector d is obtained using the regularized least squares method ( [10]).

d = [ ˆ G H ˆ G + ρ I ] ˆ G H p , (18)

where ρ is the regularization parameter, [ · ] H is the complex conjugate transpose, and I is the unit matrix.

3. EXPERIMENT

3.1. Conditions

B. Control region

A. Experimental arrangement

Control point

Loudspeaker

Measurement point 0.15 m

Control area

3.6 m

0.6 m

0.6 m

0.05 m

1.5 m

0.6 m

Y-axis

X-axis

0.6 m

3.6 m

Figure 1: Experimental arrangement. (A) Arrangement of the loudspeaker array and controlled region. (B) Arrangement of the control points and measurements for modeling in the controlled region.

Figure 2: Measurement in an anechoic chamber. Room impulse responses were measured at 144 points by moving the 12-ch linear microphone array and using 96 loudspeakers.

In this study, we evaluated the reproduction accuracy of the proposed method by comparing it with the conventional method that does not consider the reflected sounds for modeling the room impulse responses (RIRs) at the control points. Figure 1-A illustrates the experimental arrangement. The shape of the loudspeaker array is a square; the length of each side is 3 . 6 m, 96 loudspeakers are used. Measurement points were placed at an interval of 0 . 05 m. The RIRs at 144 points on the grid centered at the origin were measured by moving the linear microphone array by using a linear servo motor

\

(Fig.2). The measurement signal exhibited a swept-sine signal with a duration of 2 . 56 s. Equivalent sources (900 in number) were randomly placed within a sphere with a radius of 0 . 06 m around each loudspeaker. Twenty measurement points were used to model the transfer functions at the control points. The origin of the coordinates was at the center of the loudspeaker array. To model the transfer functions at the control points, the measured RIRs were considered in a cross-shaped array (Fig.1-B). Other measured RIRs were used for evaluation.

3.2. Results

(a) Desired (b) Conventional (c) Proposal (g) Conventional (h) Proposal

0.3

20

1

2.5 [kHz] 5 [kHz]

0

Normalized amplitude

SNR [dB]

-0.3

Y [m]

10

0

(i) Conventional (j) Proposal

(d) Desired (e) Conventional (f) Proposal

0.3

0

0 X [m] Figure 3: Comparison of desired and reproduced wave fronts and signal-to-noise ratio (SNR) distributions in the proposed and conventional methods.(a)–(c) and (d)–(f) are wave fronts at 2.5 and 5 kHz, respectively.(g)–(h) and (i)–(j) are SNR distributions at 2.5 and 5.0 kHz, respectively. In (a)–(f), the phase of the wave front is set to zero.

-1

-0.3

-0.3 0.3 0 -0.3 0.3 0

-0.3 0.3 0 -0.3 0.3 0 -0.3 0.3 0

X [m]

15

Proposed method Conventional method

14

13

12

i

SNR[dB]

11

10

9

8

7

6

1 1.5 2 2.5 3 3.5 4 4.5 5 Center frequency[kHz]

Figure 4: Mean of SNRs over each 1-kHz frequency band. The resolution of the evaluated frequency is 200 Hz. Moreover, each mean SNR is obtained by averaging over all evaluation positions.

Figures 3 and 4 depict the experimental results. The desired sound field is a plane wave with a propagation direction of (0, 1, 0). Figure 3 shows the results of the sound field synthesis at 2 . 5 and 5 kHz and its SNR distributions. Figure 4 shows the means of SNRs at the 1-kHz frequency bands of which the central frequency is 1–5 kHz. As shown in Figs.3(b) and (e), with regard to the conventional method, the wave front

disturbances have been observed over the entire reproduction area at 2.5 and 5 kHz. As shown in Figs.3(g) and (i), with regard to the conventional method, the SNR values are 0 dB over the entire controlled region. However, as shown in Figs. 3(c) and (f), the wave fronts with regard to the proposed method are less disturbed compared with those of the conventional method. As shown in Figs.3(h) and (j), the reproduction accuracies have significantly improved by 20 dB around the center of the reproduction area. Figure 4 depicts the mean of SNRs at each 1-kHz frequency band. The reproduction accuracies at 1–5 kHz have improved by 1–3 dB. Thus, the experimental results indicate that the modeling of transfer functions by considering first-order reflected sounds in the PM method improved the reproduction accuracy.

4. CONCLUSIONS

We devised a pressure matching method based on the modeling of room impulse responses considering primary reflected sounds by employing the equivalent source method and image source method. The experimental results of the actual measurements showed that the reproduction accuracies of the proposed method improved by approximately 1–3 dB at 1–5 kHz as compared with those of the conventional method. In future studies, we will improve the reproduction accuracies at higher frequencies while considering higher-order reflected sounds.

ACKNOWLEDGEMENTS

This work was partially supported by JSPS KAKENHI under Grant Numbers 19K12049 and 22K12099.

REFERENCES

[1] A. Berkhout, Diemer Vries, and P VOGEL. Acoustic control by wave field synthesis. J.Acoust.Soc.Am. , 93:2764–2778, 1993. [2] A. Berkhout, Diemer Vries, and P VOGEL. An analytical approach to sound field reproduction using circular and spherical loudspeaker distributions. Acta Acustica united with Acustica , 94:988–999, 2008. [3] Mark A. Poletti. Three-dimensional surround sound systems based on spherical harmonics. J.Acoust.Soc.Am. , 53(11):1004–1025, 2005. [4] Terence Betlehem and Thushara Abhayapala. Theory and design of sound field reproduction in reverberant rooms. The Journal of the Acoustical Society of America , 117:2100–11, 2005. [5] M. Miyoshi and Y. Kaneda. Inverse filtering of room acoustics. IEEE Transactions on Acoustics, Speech, and Signal Processing , 36(2):145–152, 1988. [6] Izumi Tsunokuni et al. Pressure-matching based 2d sound field synthesis with equivalent source array. Proc. ICA2019 , pages 2701–2707, 2019.9. [7] G. H. Koopmann et al. A method for computing acoustic fields based on the principle of wave superposition. The Journal of the Acoustical Society of America , 86(6):2433–2438, 1989. [8] Bokai Du, Xiangyang Zeng, and Michael Vorländer. Multizone sound field reproduction based on equivalent source method. Acoustics Australia , 49:317–329, 2021. [9] Izumi Tsunokuni et al. Spatial extrapolation of early room impulse responses in local area using sparse equivalent sources and image source method. Applied Acoustics , 179(108027), 2021.8. [10] Mark A. Poletti. Improved methods for generating focused sources using circular arrays. In Audio Engineering Society Convention 133 , 2012.