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

Vibro-acoustic rendering methods to radiate a uniform sound fieldfrom a panel speaker
Jung-Han Woo1,2; Jeong-Guon Ih1
12

Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea

Currently at Korea Institute of Machinery and Materials (KIMM), Daejeon, Korea

ABSTRACTThe sound radiated from the usual panel speaker employing the single point excitation is distorted due to themulti-modal response of the regular panel, or the effective range of radiation is limited to very low
frequencies. Such problem can be alleviated if an array of actuators is used in the periphery of the panel tocontrol the vibration response for the uniform radiation in a wide frequency range. To achieve the uniform
acoustic field, the two inverse vibro-acoustic rendering techniques can be adopted: first, a vibration patterncomposed of a virtual speaker and baffle is being created, and, second, the spatial distribution of the vibration
is being controlled to radiate the desired sound. Each method is called indirect or direct method. Numericalsimulations are conducted on a thin, simply-supported plate, for which the high frequency limit is 300 Hz.
The indirect method results the rendering error of -20 dB and the direct method -50 dB. Two methods areafflicted with the instability, particularly in using the direct one. Regularization technique is used to overcome
the ill-posedness, that yields the change of condition number from O(1017) to O(104). Input gains are also
reduced by 35 dB for both methods.
Keywords: Panel speaker, actuator array, vibro-acoustic rendering, vibro-acoustic inverse technique

1. INTRODUCTION
Audio systems embedded in various machines or devices often concerns about the acoustic power
and low frequency contents because they are usually constrained in the spatial usage for a compact
layout to have the aesthetic shape and geometric efficiency [1-4]. The generated sounds from them
are usually distorted due to the multi-modal characteristic of the plate, and the radiation efficiency islow. To overcome such problems, a method of using an array of multiple actuators located at theperiphery of a thin plate or surrounding the main actuator is proposed [5].
Two methods converting a panel into an effective sound radiator using the actuator array are
considered: one is to create the circular speaker and tranquil baffle zones simultaneously, and theother is to specify the vibration distribution on a panel that can eventually realize the desired soundradiation. In this work, the characteristics of two inverse rendering methods are compared.

2. DIRECT & INDIRECT INVERSE RENDERING METHODS
2.1 Indirect inverse rendering method (IRM)
The vibration pattern composed of virtual speaker and baffle zones can be created to achieve thedesired sound radiation. The created speaker and baffle can be considered as the virtual speaker system.
Similar to the real loudspeaker, an in-phase back-and-forth piston motion with the amplitude vspk is
expected in the speaker zone. Ideally, there should be no phase difference within the speaker zone andno vibration at the baffle zone. In the steady state, the relation between the velocity responses at the12

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jhwoo@kimm.re.kr
J.G.Ih@kaist.ac.kr

M observation points on the plate and the excitation forces at the actuators is given by
GM N eN1   vM 1 ,(2)where N is the number of actuators, e the input force vector at the actuator positions, v the velocityvector at the observation points on the plate, and G the transfer function between actuators andobservation points [6]. A least-squares solution minimizing the error between the rendered vibration
pattern and the created vibration field can be obtained by the Moore-Penrose inverse as follows:min vt  vˆ subject to vˆ  Geˆ1 ,eˆ1 N 1  G N  M  vt M 1   G *G  G * v t .
†1(3, 4)Here, 𝐞 means the estimated input force vectors obtained by applying the IRM, 𝐯𝐭 the renderedvibration field, 𝐯 the generated vibration velocity field, ||…|| the Euclidean norm, and G† , G* indicate
the pseudo inverse and the conjugate transpose of the matrix G, respectively [6,7].
2.2 Direct inverse rendering method (DRM)
Another method for achieving the desired sound radiation can be thought of. The panel can be set
into the vibration by the actuator array to directly radiate the desired sound pattern without imposing
any constraint on the source vibration pattern. The complex gain for the input force of the actuator
array is estimated from the direct inverse transfer relationship between the desired acoustic field andthe actuator array. The input forces for control actuators can be obtained asmin pt  pˆ subject to pˆ  Heˆ 2 ,eˆ 2 N 1   H N Q pt M 1   H * H  H *p t .
†1(5,6)Here, H indicates the Green’s function between plate vibration and radiated sound pressure, Q thenumber of observation points in acoustic field, 𝐞 the estimated input force vector using the DRM,
𝐩𝐭 the desired sound field, and 𝐩 the actually generated acoustic field.

3. COMPARISON OF THE TWO RENDERING METHODSTo test the two inverse rendering methods, a simply supported aluminum plate with a size of 700
mm x 400 mm x 2 mm is used. A total of 196 actuators are located near the plate edges in 10-mm
uniform spacing to avoid the spatial aliasing. Vibration observation points are distributed at 2079
points in the panel, and the radiated sound field on a hemispherical surface are observed at 3751 points,
which are 1 m away from the plate center. In this work, the error in the acoustic field, ε p, to quantify
the difference between desired and generated fields, and the input power, We, for generating therendered vibration or acoustic field are defined as
 N

(7,8)
 p  20 log10   pˆ  pt  / pt / Q , We  20log10   eˆ / eref  .
q n1

3.1 Results of inverse rendering
To generate a uniform hemispherical sound radiation pattern, the condition kR≪ 1 should besatisfied. To adopt the IRM, a virtual speaker zone with a radius of 75 mm is predetermined on theplate center. The test result with IRM is illustrated in Fig. 1. One can see that the estimated error of
the generated sound field increases with frequency in general except at 180 Hz, which approximately
corresponds to the first longest bending wavelength matching the distance between speaker center andplate corner [5]. Created vibration magnitude and phase fulfilling the rendered sound radiation are
shown in Fig. 2. When the DRM is used, the system ill-conditioning influences the created vibration
field a lot. The average condition number of DRM is about 1017, which means that the use ofregularization is essential in solving the instability problem.

Figure 1 – The reconstruction error by using the two inverse rendering methods:, IRM;, DRM.

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(a)

(c)

(b)

(d)

Figure 2 – Vibration field as virtual speaker and baffle at 160 Hz:(a)(c) magnitude, (b)(d) phase; (a)(b) IRM, (c)(d) DRM.
3.2 Effect of regularization
Among various regularization techniques for obtaining the stable inverse solution of an illconditioned matrix [6-8], Tikhonov method [9] is employed modified separately for IRM and DRM.
The cost functions for suppressing the instability in using the IRM and DRM, respectively, arecomposed of the error between rendered target fields, vibration velocity vt or acoustic pressure p t, andreconstructed fields, G𝐞 or H𝐞 , and the weighted input power. Here, 𝐞 and 𝐞 imply theregularized input force signal. In this work, the optimal regularization parameter β is obtained byusing the L-curve method [10]. To consider the noise-contaminated condition, the normally distributed
random noise is added to the initial rendered field varying the SN ratio of the target field. Figure 3
exhibits the simulation results illustrating the required input energy for the actuators versus thereconstruction error of the acoustic field. One can find that the DRM results are strongly influenced
by the included noise. This coincides with the fact that the condition of the system using the DRM ismuch more ill-posed than that of the IRM. Figures 4 and 5 compare the created vibration field and theestimated input gains of actuators at 160 Hz with the severely contaminated field condition with noise.
After regularization, the initial condition number 6.6x1016 is reduced to 1.1x105 , which is a similarvalue with the system of IRM. One can find that the input gains of actuators are much reduced after
the regularization, in particular, the input gain magnitude of DRM is reduced by about 35 dB.

4. Conclusions
To radiate the desired sound using actuator array located at the periphery of a thin plate surface,
two possible rendering methods are considered to compare their capability in radiating a uniform
sound field. It is shown that the control by using DRM requires too large amount of input power. To
overcome the ill-posed problem in the inverse rendering process, the regularization is adopted, which
results in much reduced and stable input gains even under a highly contaminated signal condition.
Decision on what method is to be used can be made by considering both the acceptable reconstruction
error to be achieved and the input power needed for the control actuators for practical applications.

Figure 3 – Required input energy of the actuators related to the reconstruction error of the acoustic field(160 Hz):(a), IRM;(b), DRM;, SNR=5 dB;(c), SNR=10 dB;, SNR=20 dB.

(d)

Figure 4 – Created vibration field on the plate after the regularization (160 Hz, SNR =10 dB):(a)(c) magnitude, (b)(d) phase; (a)(b) IRM, (c)(d) DRM.

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(b)

(a)

(c)

(d)

Figure 5 – Calculated input gains of actuators after the regularization (160 Hz, SNR =10 dB):(a)(c) magnitude, (b)(d) phase; (a)(b) IRM, (c)(d) DRM.

ACKNOWLEDGEMENTS
This work was partially supported by the BK21-Plus Project and the NRF grants (No.
2016K2A9A2A12003779 and No. 2017R1A2B4007133).

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