paper ID: 299/p.1
On the use of double layer beamforming antenna for industrial applications Jean-Claude Pascala and Jing-Fang Lib a
Laboratoire d'Acoustique de l'Université du Maine (UMR CNRS 6613) and Ecole Nationale Supérieure d'Ingénieurs du Mans (ENSIM) Université du Maine, rue Aristote, 72000 Le Mans, France,
[email protected] b
Visual VibroAcoustics, 51 rue d'Alger, 72000 Le Mans, France Email:
[email protected]
Beamforming microphone antennas have now been widely used in the field of the automotive industry to locate sound sources on transport systems and to build acoustic models. The fields of application are the analysis of the pass-by noise of vehicles, the aero-acoustic noise identification and the source localization in cockpits. The analysis of the mechanical systems results in using more and more these antennas in near fields. In this article, a double layer microphone antenna is proposed and examined. Based on the numerical simulations, analysis is made in order to establish a source identification model using wave fronts of the sound pressure and the particle velocity, which are determined by finite difference between the two layers of microphones. The objective of this processing is to obtain a better localization and identification of the mechanisms of generation of the noise when the antenna is employed in near field (for example, ten centimetres from source). The other advantage of the double layer antenna is the possible attenuation of the back wave. This characteristic can particularly be useful for measurements in cockpit. Its interest is evaluated by simulation in a multi-sources environment.
1. INTRODUCTION Recently the additive beamforming antennas are largely used for applications of industrial acoustics [1,2]. Compared to the technique of Near-field Acoustic Holography (NAH) they have less constraints and are easier to be performed in practice. In particular, they require less number of microphones (several tens instead of several hundreds for NAH) and can operate at a more important distance making it possible to apply them to, for example, the analysis of pass-by noise in automotive and railway fields. However the NAH technique can be used to construct a sound field whereas the additive antennas require a model of acoustical sources that we wish to locate. For example, if one wants to build representative model for engine in the form of a whole of source volume flux. A set of elementary sources consisting of monopoles are often used for this goal. In the application of this type of antenna to engineering acoustics, difficulties can be occurred because of perturbation sources or reflections on walls. Indeed, if the antenna is focalised on a point located in front of it, the system will have the same sensitivity for a point located behind symmetrically at the plane of the microphone antenna. This aspect is particularly unexpected if one wants to apply these techniques for localization of sources in a closed space like a cockpit [6]. In order to find a solution to this problem, the use of double-layer antenna is studied in this paper. Acoustic particle velocity is employed to identify the source. The calculations of the velocity is based on the finite-difference approximation as it is used in sound intensity. The double-layer antennas have already been used in industrial acoustics based on Trott’s antenna
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paper ID: 299/p.2
principle. However the objective of this system was to extract in a near-field, the source components that propagate to the far field [7,8]. Here, the use of the particle velocity has advantages in the localization of the sources: the nearfield component in term of 1 / r 2 increases the effectiveness of focalisation in small distance and it is possible to differentiate the position from the sources on both sides of the antenna. To make the main processing to be understood, first the principle of simple additive antennas is presented. Then it is shown that complementary algorithms can be used to remove the sources at back of the antenna plane.
2. FOCALIZING BEAMFORMING PRINCIPLE The localization of industrial sources by acoustic antenna method uses, generally, a source model consisting of a set of point sources, such as monopoles. The pressure p i (ω) measured by a microphone placed at a distance Ri from source center is expressed in the following form pi (ω) =
A(ω) e − jkRi , Ri
k=
ω c
(1)
The amplitude A depends on the volume flux strength Q that characterizes the source, such that Q = 4π A /( jω ρ0 ) . An estimation of source amplitude Ap is obtained by focalizing the antenna of N microphones on its position, N
Ap (ω) = ∑ i =1
wi pi (ω) e jkRi N wi ∑ i =1 R i
(2)
Following are some well-known bias errors and distortions occurred in the estimation of Ap : - Separation of the sources depends on the width of the main lobe which is inversely proportional to the aperture of the antenna. - The sources out of from the focalisation point can cause a bias error in the estimation of Ap through the side lobes. The weighting coefficients wi make it possible to apply apodisation window to reduce the amplitude of these side lobes but meanwhile resulting in the detriment of the sharpness of the main lobe. - The microphone spacing ∆L can result in the presence of grating lobes at high frequencies if the distance ∆L sin θ j seen by the sources located apart from the focal point is larger than the half-wavelength. The amplitude of these grating lobes is not attenuated by the apodisation coefficients and is in the same magnitude as that of the main lobe. - The response is the same one for the focal points located symmetrically at the plan of the antenna. In other words, the antenna does not allow one to differentiate the parasitic contributions located at the back of antenna.
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paper ID: 299/p.3
d front side
back side
front side
back side
∆L Ri
θi
pi
Ri p 2i
p1 i Q
Figure 1. Scheme of simple (left) and double layers(right) antennas. The expression of particle velocity at the distance Ri from the monopole is given as ui (ω) =
A(ω) e − jkRi ρ0 c Ri
j 1 − . kRi
(3)
This equation can also be used to estimate the amplitude A . By using Euler equation and the finite-difference approximation as it is done in the computation of the sound intensity from measurements, the following approximation can be obtained ρ0 c u i (ω) =
1 ∂pi (ω) p 2 i (ω) − p1 i (ω) ≈ − jk ∂Ri − jk ∆Ri
(4)
where ∆Ri = d cosθi is the aperture spacing (see Figure 1). The difference of the pressure measured by a pair of microphones i is written as p 2 i (ω) − p1i (ω) = − jk∆Ri A(ω)
e − jkRi Ri 1 − (∆Ri 2 Ri )2
(
)
sin (k ∆Ri 2) j cos (k ∆Ri 2) − k Ri k ∆Ri 2
(5)
From equation (5), it is possible to deduce from a way similar to equation (2), an estimation of the amplitude of the source located at the focusing point of an antenna composing of 2 × N microphones N
Au (ω) = ∑ wi [ p 2 i (ω) − p1 i (ω)] i =1
e jkRi
wi k∆Ri
N
cos(k ∆Ri 2 ) sin (k ∆Ri 2 ) + j k Ri k ∆Ri 2
∑ R (1 − (∆R i =1
i
i
2 Ri ) 2
)
(6)
At high frequencies, the bias error due to the finite-difference-approximation technique used to compute the pressure gradient is corrected in Eq. (6). However the procedure can introduce
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paper ID: 299/p.4
important distortions around the zero of sin (k∆Ri 2) , when the near-field term becomes small. For this reason, the spacing between the two layers of the antenna d should not be larger than λ/ 3 . 0.5
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Figure 2. Estimation of source distribution on a plane located at 0.5 m from the antenna plane with a single layer antenna (left) an a double layer antenna (right). One source is located at the back side of antenna. ( f = 1250 Hz, ∆L = 0.1 m). Figure 2 shows the results of two estimations for a set of 4 point sources with a distance d = 0.5 m from the antenna plane. For the single layer antenna, the number of microphones is N = 49 and 2 × N microphones for the double layer antenna. The quadratic values of the estimated amplitudes Ap
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are similar. From this processing one cannot see one of the four source that is located behind the antenna. This one contributes in the same way as the other three ones and disturbs the analysis appreciably in the case of measurements in close space.
3. FRONT/BACK SOURCE SEPARATION The use of a double-layer technique described by Eq. (6) is not sufficient to separate the sources which are on both sides of the antenna plane. However the radial component of the particle velocity used for this estimation contains this information. An estimator can be defined as the real part of product of the conjugate of the estimation of Au by the estimation of Ap on the double layer antenna according to Eq. (2) estimator =
1 Re{Ap Au* } 2
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Figure 3. Localisation of the sources by using a double-layer antenna and the estimator Eq. (7). The positive values (red) are attributed by the sources located in front of the antenna and the negative ones (blue) by the source located at the back of antenna.
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Figure 3 shows the estimator computed by Eq. (7). The negative values are attributed by the source located behind the antenna. In principle, this technique is quite similar to the sound intensity measurement and presents the same characteristics. The result is due to various contributions. The contributions from two sources with the identical strength located symmetrically with respect to the antenna plane are cancelled. 0.5
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Figure 4. Localisation of the sources by using a double-layer antenna and the estimator (8) of 2 the gradient of pressure: estimator Au / 2 (left) and estimator Re Ap Au* / 2 (right).
{
}
To remove the contributions of the generally undesirable sources placed behind the antenna an approximation different from Eq. (4) of the gradient of pressure by finite difference is used [8] ∂p i (ω) p 2 i (ω)e − jk∆ R i / 2 − p1 i (ω)e jk∆ R i / 2 ≈ ∂R i 2∆Ri
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paper ID: 299/p.6
Under these conditions the new estimator obtained for Au is not sensitive to the component of particle velocity produce by the source located at the back of the antenna, if ∆Ri corresponds to the apparent spacing between the ith pair of microphones. Figure 4 shows the estimator computed 2 by taking Eq. (8) into consideration for the quadratic amplitudes Au / 2 and Re Ap Au* / 2 .
{
}
4. CONCLUSIONS The use of double-layer antennas makes it possible to separate the contributions from the sources located on both sides of the antenna plane. Since a technique of finite-difference approximation is employed, the method is sensitive to the phase errors between sensors at low frequencies. However this sensitivity is less than that for sound intensity measurement because these errors are averaged over all microphone pairs used in the antenna. For a resolution equivalent to a simple layer, the number of microphones is doubled, but the contributions of the sources located at the back of the antenna plane can be removed. This advantage can open the way with applications in cockpits of vehicles. The antennas based on the particle velocity can also be used with the high-resolution techniques [9].
REFERENCES 1.
P. Bertrand, A. Jacques, “Applications de la formation de voies en temps réel à la localisation et au diagnostic de source de bruit”, Journées Imagerie Acoustique, Lyon (France), 1-2 mars 1994, pp. 99-104.
2.
G. Elias, "Source localization with a two-dimensional focused array : optimal signal processing for a crossshaped array", Inter-Noise 95, Newport-Beach (USA), 10-12 July 1995, 1175-1178.
3.
M.-A. Pallas, "The focussed vehicle array for the description of the noise sources on a moving car", InterNoise 2000, Nice (France), 27-31 August. 2000, Paper N°326.
4.
F. Poisson, J.C. Valière, P. Herzog, "High speed sources localization using bilinear time-frequency transformation", Applied Acoustics 53, 1-13 (1998).
5.
H. Kook, G.B. Moebs, P. Davies, J.S. Bolton, "An efficient procedure for visualizing the sound field radiated by vehicles during standardized passby tests", Journal of Sound and Vibration 233, 137-156 (2000).
6.
K. Haddad, V. Benoit, "Use of an acoustical imaging system for the study of the acoustic transmission of sealing system in a moving car", Proceedings of ISMA 2002, Leuven (Belgium), 16-18 September 2002, 2329-2336.
7.
L. Gaudriot, M. Mercusot, B. Escudié, "Techniques de champ proche et lointain pour l'analyse spectrale des mécanismes de création du bruit", Revue d'Acoustique 54, 176-186 (1980).
8.
J.-C. Pascal, "Intensimétrie et antennes acoustiques" (Ch. 10) in Rayonnement acoustique des structures (C. Lesueur, Ed.), Eyrolles, 1988.
9.
M. Frikel, S. Bourennane, "High-resilution methods without eigendecomposition for locating the acoustic sources", Applied Acoustics 52, 139-154 (1997).
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