Bayesian parameter estimation of microphone positions, sound speed and dissipation for impedance tube measurements

Ziqi Chen 1 , Cameron J. Fackler, and Ning Xiang 2

Rensselaer Polytechnic Institute 110 8th street, Troy, NY 12180, United States

ABSTRACT With tube measurement widely used for acoustic measurements, calibration plays an important role in verifying and validating the measurement. This work applies a Bayesian method based on an air layer reflectance model to estimate the microphone positions, and sound speed in consideration of environmental e ff ects on uncertainties of the normal incident impedance tube measurements. Bayesian theorem is applied to estimate the microphone positions and the sound speed given the experimental data obtained from the transfer function method (TFM) in the tube measurements. With a hypothetical air layer treated as material under test in front of a rigid backing, a parametric model is established for the TFM tube measurement to estimate the microphone positions using Bayesian inference. With the microphone positions accurately estimated, the sound speed and losses due to tube interior boundary e ff ects are also estimated within the same Bayesian framework. Bayesian analysis results show that Bayesian parameter estimation based on the air layer model is well suited in estimating the sound speed, the microphone positions, and other parameters to ensure highly accurate tube measurements.

1. INTRODUCTION

In building acoustics, the acoustic properties of the wall surface are of great importance for the room-acoustic research. Many e ff orts have been put into the improvement of the measurement for the boundaries, such as impedance tube measurements [1]. Chung [2] proposed a two-microphone transfer-function method for impedance tube measurements. This method is then widely used to measure the acoustic property of materials [3, 4]. Due to the phase mismatches and the microphone positional errors, the accuracy of the two-microphone transfer function method in the impedance tube is undermined. This work measures the sound pressure at two positions in a sequential manner suggested by Chu [5] to avoid the microphone mismatches. In addition to the microphone positions [6], the sound speed and the tube wall dissipation influence the measurement’s accuracy. Unlike the microphone positions, which only need to be calibrated once, the sound speed in the medium air changes from time to time, depending on the ambient environment. The dissipation coe ffi cient is mainly caused by the boundary e ff ect of the tube interior walls and the

1 slanduo0@gmail.com

2 xiangn@rpi.edu

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

Figure 1: Two-microphone impedance tube measurement setup using the transfer function method.

thermal e ff ect in the medium air. This work parameterizes the dissipation, the sound speed, and the microphone positions and applies Bayesian inference to estimate these parameters.

2. MODELS

In this section, the reflectance of the air layer terminated with a rigid backing is theoretically calculated. A rigid termination is directly connected to the tube end instead of any materials under test. A hypothetical air layer in front of the rigid backing is treated as the material under test.

2.1. Reflectance measurement The experimental data is obtained from the two-microphone impedance tube measurement. With the measured responses, the complex reflectance can be expressed as

P i = H 12 − e − j β s

R D = P r

e j β s − H 12 e j2 β L , (1)

where L is the distance from the position ’1’ to the surface of the hypothetical air layer, s is the separation between two microphone positions, P i is the incident sound pressure in the frequency domain, and P r is the reflected sound pressure in the frequency domain. H 12 = P 2 / P 1 is the transfer function between the sound pressures in the frequency domain at position ’1’ and ’2’ in Figure 1. β is the propagation (phase) coe ffi cient of the sound wave in the air. Equation 1 represents the reflectance obtained from the two-microphone impedance tube measurement, but it is only valid in lossless media. In reality, the lossless media does not exist, and the dissipation due to the boundary e ff ect of the tube walls or damping in the air is unavoidable. To incorporate the dissipation, the phase coe ffi cient β in Equation 1 can be replaced with the complex propagation coe ffi cient γ ,

R D = H 12 − e − γ s

e γ s − H 12 e 2 γ L , (2)

with

γ = α ζ + j β = α ζ c + j ω

c , (3)

where α ζ is the damping coe ffi cient. Here α ζ is a function of frequency, and it can be calculated through a ’wide tube’ dissipation model [7],

α ζ = 6 . 7 × 10 − 6 · ( U / A ) ζ √ ω = 9 . 48 × 10 − 6 · ζ

r · √ ω, (4)

where ζ is an unknown factor, U is the perimeter of the tube, A is the cross-sectional area of the tube, and r is the inner radius of the circular tube. The unknown factor ζ in Equation 4 represents the potential variations because of the tube materials or the changing environment.

2.2. Air layer model A hypothetical air layer in front of the rigid backing is assumed with the thickness d . In the similar fashion shown in the previous section, the surface impedance Z s can be written [7] as

Z s = ρ c e γ d + e − γ d

e γ d − e − γ d . (5)

So the surface reflectance R M of the air layer can be expressed as

R M = Z s − ρ c

Z s + ρ c = e − 2 γ d , (6)

where ρ c is the characteristic impedance of the air. Equation 6 presents the theoretical surface reflectance of the air layer in front of the rigid termination, which is considered as the predicted model in the following sections.

3. BAYESIAN PARAMETER ESTIMATION

This work applies a Bayesian method, based on the theoretical reflectance model of the air layer in Equation 6 and the calculated reflectance from the two-microphone transfer function method in Equation 2, to estimate the values of the parameters. The models contain a specific set of parameters, which is collectively denoted as θ , including the separation between microphones s , the distance L from microphone position ’1’ to the surface of the material under test, the sound speed c , and the dissipation coe ffi cient factor ζ . Given the measured reflectance R D and the theoretical reflectance R D , the estimation applying Bayesian theorem yields,

prior z}|{ p ( θ | I ) p ( R D , R M | I ) , (7)

likelihood z }| { p ( R D , R M | θ , I ) ×

posterior z }| { p ( θ | R D , R M , I ) =

where I is the background information known prior to the estimation. One significant information included in I is that ’ the prediction model is able to describe the data well ’. θ = [ s , L , c , ζ ] collectively represents the unknown parameters before the estimation. All the initial knowledge about the parameters before the estimation is encoded into the prior distributions of the parameters. The equivalent acoustic microphone centers are somewhere on the membrane of the microphone. Although influenced by the environment, the sound speed is known within a specific range. Applying the principle of maximum entropy assigns uniform distributions over each parameter ranges [8]. According to the principle of maximum entropy, the likelihood function is assigned as a Student’s t -distribution [8],

− K / 2

K X

 π



p ( R D , R M | θ , I ) = Γ ( K / 2)

k = 1 ϵ 2 k

, (8)

2

with ϵ 2 k = Re 2 ( R D , k − R M , k ) + Im 2 ( R D , k − R M , k ) , (9)

where Γ ( . . . ) is the standard Gamma function, and ϵ k is the residual error at the datum k . The likelihood function p ( R D , R M | θ , I ) describes the probabilities of the residual errors ϵ k . In this work, the residual error is determined by the di ff erence between the theoretical reflectance R M and the measured reflectance R D .

Figure 2: Estimation results of the reflectance and residual absorption errors for d = 1 . 5 cm. (a) Reflectance without dissipation; (b) Reflectance with dissipation; (c) Residual absorption error.

4. EXPERIMENTAL RESULTS

The measurement is carried out in a long tube whose inner diameter is 3.81 cm (1.5 in) and wall thickness is 0.64 cm (0.25 in). The position ’1’ is 11.4 cm (4.5 in) away from the rigid termination, and the position ’2’ is 10.2 cm (4 in) away from the rigid termination. The thickness of the air layer is assumed to be 1.5 cm. The prior distribution of each parameter is listed in Table 1.

Table 1: Prior probability assignment for the four parameters.

Π ( s ) = uniform(1, 1.4) cm Π ( c ) = uniform(340, 350) m / s

Π ( L ) = uniform(10.5, 12.5) cm Π ( ζ ) = uniform(3, 8)

Figure 2 presents the reflectance and absorption coe ffi cients with and without the dissipation. As shown in the figure, incorporating the dissipation leads to the more accurate match of the measured reflectance R D and the modeled reflectance R M , especially the imaginary part. Figure 2 (c) demonstrates that the residual absorption error significantly decreased after incorporating the dissipation coe ffi cient. Incorporating the dissipation coe ffi cient makes the residual absorption error decrease from 0.13 to 0.013.

5. CONCLUSIONS

The current work presents the Bayesian parameter estimation based on an air layer reflectance model and the two-microphone transfer function method for the impedance tube. A hypothetical air layer in front of the rigid termination is assumed as the material under test, and its theoretical reflectance is calculated as the prediction model. The result indicates that the model-based Bayesian parameter estimation performs well in estimating the dissipation coe ffi cient, the sound speed, and the microphone positions. Incorporating the accurate estimated parameters, the measured reflectance accurately matches the modeled reflectance within the 2.5 - 4.2 kHz frequency range. This accurate estimation can serve as the calibration for the impedance tube. The future e ff ort will focus on the wider frequency ranges and varied environmental factors influencing the sound speed, such as temperature and humidity. The application of this framework in the three- or four-microphone methods for the impedance tube also needs to be explored.

REFERENCES

[1] Andrew F Seybert and David F Ross. Experimental determination of acoustic properties using a two-microphone random-excitation technique. J. Acoust. Soc. Am. , 61(5):1362–1370, 1977.

-1 Reflectance o o (a) _ — Model(Imaginary) - - Model(Real) .0 2.0 25 30 3.5 40 4.5 Frequency (kHz) 25 3.0 3.5 4.0 4.5 Frequency (kHz) Absorption error 0.05 (C) sz Without dissipation s S ve ment Aree! Sepegenetel unt yen aene ee —With dissipation 2.0 2.5 3.0 3.5 4.0 4.5 Frequency (kHz)

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