Derivation of time-domain surface impedance boundary conditions based on
in-situ surface measurements and model fitting
Baltazar BRIERE DE LA HOSSERAYE(1) , Huiqing WANG(1) , Fotis GEORGIOU(1) , Maarten HORNIKX(1) , Philip W.
ROBINSON(2)(1) Eindhoven University of Technology, The Netherlands, b.g.j.briere.de.la.hosseraye@tue.nl(5) Facebook & Oculus, The United States, philrob22@fb.com

Abstract
In order to achieve accurate time-domain wave-based simulations of a real room, the acoustical properties ofthe locally reacting materials within the room have to be implemented as time-domain impedance boundary
conditions (TDIBC), in such a way that the behavior of the materials is well-simulated within the wave-based
solver. This paper presents the implementation of such TDIBCs of two materials: a porous absorber and an
acoustic carpet. Firstly, the material properties were measured both in the impedance tube and in-situ witha pressure-velocity sensor. Advantages and drawbacks experienced with both methods in this context will bepresented. Next, the measurement results were fitted to broadband impedance models to extend impedance
data to the lower frequency range (20 Hz – 300 Hz), resulting in broadband impedance data (20 Hz - 4000
Hz). Finally the TDIBCs, in the form of complex reflection coefficients, were fitted as discrete sums of rational
functions.
Keywords: Time-domain boundary condition; In situ measurement

1

INTRODUCTION

The accurate modelling of the acoustical properties of surfaces used as BCs is one of the challenges inherent
to room acoustics simulations, as the accuracy of the results will depend on the proper modelling of the BCs,
regardless of the algorithm used. The required boundary conditions data differs depending on the simulation
method. For geometrical acoustics algorithms, the random incidence absorption coefficient is normally used.
For wave-based methods in the time-domain, which is of interest in this paper, data of the surface impedance(or reflection coefficient) is needed. Moreover, if the surface impedance is obtained in the frequency domain,
as it usually results from measurements, it is required to translate it into time-domain boundary conditions(TDBCs).
This paper proposes a three-steps methodology to implement such TDBCs from measurement data, for the
Discontinuous Galerkin (DG) simulation of a reverberant room treated with two materials that can be treated as
locally reacting: a 40 mm thick porous panel and a 8 mm acoustic carpet.
In Section 2 of this paper, the measurements of the two materials from two different methods is described
and the results are discussed. In Section 3, impedance models are derived and fitted to the measured datato yield broadband impedance results. Finally, in Section 4, the reflection coefficients from the models areapproximated as discrete sums of infinite impulse response (IIR) filters to serve as TDBCs. These TDBCs
where subsequently used to simulate impulse responses and compare them with measured IRs in (8), of which
this paper is a companion.

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2

IMPEDANCE MEASUREMENTS

2.1 Methods
Two distinct approaches were used to measure the surface impedances of the acoustic materials: a Kundt’s tube
method, as laboratory approach of small materials samples, and a pressure-velocity (PU) measurement device,as an in situ approach of the material applied in the room.
2.1.1 Kundt’s tube
The Kundt’s tube measurements were realised using the so-called transfer function method. The microphones
and the speaker used for the measurement are connected to amplifiers which in their turn are connected to a
Creative EMU 0202 USB audio device. Impuse responses are acquired from a desktop running Dirac 7841 v6.0
room acoustic software. The working frequency range of the setup used is 100-2500 Hz, the upper value being
limited by the relatively large microphone spacing. Further description of the measurement setup can be foundin (2).
A 40 mm diameter sample is cut from the material to be placed at one end of the tube. After the sample wasplaced and the tube carefully sealed, an exponential sweep sine signal is sent through the loudspeaker locatedat the opposite end of the tube. The impulse response was then captured at 5 different locations within thetube, with one single microphone per measurement in order to avoid microphone calibration errors. The sweep
signal used had a duration of 2.73 seconds, and the measurement performed with 8 pre-averages to increase
the signal-to-noise ratio. The frequency responses were acquired with a sampling frequency of 192000 Hz andde-convolved into impulse responses by Dirac.
In the post processing part, the impulse responses captured were aligned in time using the measured speakerto-microphone distances, and the surface impedance was retrieved from the correlated IRs using the transfer
functions technique.
2.1.2 In situ PU probe setup
The second setup is an in situ measurement setup making use of a pressure-velocity (PU) sensor, which captures
the pressure and particle velocity in one single direction. It is attached to a small spherical loudspeaker, with aspeaker-to-sensor distance of 26 cm. The pressure and velocity signal are then connected to a signal conditioner
and a front-end. The controls are done from a laptop running the Velo 4.A acquisition software.
The measurement were realized "in situ", directly at the materials mounted in the room. Before measuring,the device was carefully oriented to capture the velocity in the direction of the loudspeaker-probe axis, anda first calibration recording realized with the sensor far from any reflecting plane (“free field measurement”).
Subsequently, the device was mounted on a tripod to capture the velocity normal to the sample surface, witha probe-to-sample distance of 10 mm, and the pressure and velocity were captured. The signal used was aninternal sweep sine (80 to 12000 Hz) of duration 10 seconds and the measurement was repeated 5 times atdifferent locations on the samples.
The surface impedance was retrieved from the recordings using the so called Mirror-Source method (see in (1)),
and a moving average filter was applied on the spectra to reduce the influence from unwanted reflection on the
results. The result were then averaged over the various measurement locations. The working frequency range is
300-8000 Hz.
2.2 Results and discussion
As can be seen in Figure 1 (a), the complex surface impedance resulting from the tube and from in situ
measurements are in good agreement for the porous panel, particularly above 1 kHz. The wiggles observedin the in situ results below 1 kHz are thought to stem from edge diffraction of the sample measured in situ
(see (3)).
However, the surface impedance of the carpet as measured in the tube differs from the in situ measurement, as

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10

Re(Z) [-]

Re(Z) [-]

10
5
0

5

-5
0

-10

103
Frequency [Hz]0Im(Z) [-]

Im(Z) [-]

10
5
0

103

Frequency [Hz]

-5
-10

-5
-10

-15

103
Frequency [Hz]

(a)

103

Frequency [Hz](b)

Figure 1. Comparisons of measured impedances with two different methods. (a): panel, (b): carpet. Plain line:
impedance tube, dashed line: PU in situ
seen in Figure 1 (b). Particularly the peak in absorption coefficient, not shown here, was measured by the tube
method at a much higher frequency (around 1300 Hz) that what was expected from the commercial data (600
Hz). The reason for this mismatch is likely that the cutting of the sample and/or the mounting in the tube,
required for the Kundt’s tube measurement, modifies the acoustical properties of the carpet. In the room, carpet
tiles of 0.96x0.96 m tiles were measured, while it was cut into a 40 mm diameter disk to fit in the tube. As
the surface impedance data needed should be close enough to the one of a sample of infinite dimensions, it
seems that the results from the tube yielding irrelevant data.
It appears from the measurement campaign that both measurement methods have drawbacks in capturing acoustical properties of surfaces: The accuracy of the PU-based in situ technique relies much more on factors difficult
to control: measurement environment, reflectiveness of the sample under consideration, etc. Whereas the Kundt’s
tube method allows to measure the lower frequency range more accurately, it also can suffer from the cuttingof the sample and the mounting in the tube, that can result in a deterioration of the acoustical properties, as it
seems to be the case for the carpet measured here.

3

Model fitting

In order to obtain full-range impedance data of the two materials, broadband impedance models were fitted
to the measurement results. This section describes the impedance models used and presents the full-range
impedance data obtained after fitting.
3.1 Impedance models
3.1.1 Wall panel
The results from the measurements of the porous panel is compared to the impedance model by Miki forporous media, described in (4). This model takes in account two parameters: the air flow resistivity σ and
The thickness of the sample d. In this model the normalized characteristic impedance Zc and the propagation
constant γ of the medium are calculated by empirical laws, that will not be described here for sake of brevity.

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3.1.2 Carpet
A 5-parameters model for the carpet has been developed for the resonant carpet. This model assumes that thecarpet impedance is a combination of two impedance types (5):• a porous material impedance, fitted with the aforementioned Miki’s model. The fitted impedance is referred to as Z porous ,• a ideal membrane resonator impedance. The equivalent resonator is described with: the thickness of thebacking air layer dres , the mass per unit area of membrane m, and the damping factor r. The complex
surface impedance of the resonator is calculated as 2π frj+
2π f m − ρc cot
dres .
(1)
Zresonator ( f ) =ρc ρcc
The surface impedance of the carpet is then modelled as the parallel circuit equivalent of the two impedances:
Zcarpet =Zresonator Z porous.
Zresonator + Z porous(2)

3.2 Results and Discussion
Fitting of the Miki model to the panel’s measurement was realized on the in situ results over the frequency
range 300-4000 Hz. The fitted parameters chosen are presented in Table 1 and the comparison of the resulting
impedance with the in situ measurements is showed in Figure 2 (a). The model is in agreement with the insitu measurement in the frequency range 300 Hz-4 kHz. The full-range (20 - 20000 Hz) normal incidence
absorption coefficient derived from the model is physically admissible over the full range (0 < α < 1), as seenin Figure 2 (b).
Parameter [unit]
σ [Pa · s/m2 ]d [mm]

Fitted value
63000
37

Table 1. Parameters of fitted Miki models

0.8

0Model fit
Measurement-50

Im(Z) [-]

1

103

104

[-]

Re(Z) [-]

5

0.6
0.4

-2
Model fit
Measurement

-4

0.2
0

103
Frequency [Hz]

(a)

104

10

2

10

3

10

4

Frequency [Hz](b)

Figure 2. Porous panel fitted model. (a): Impedance compared to measurements, (b): full range absorption.

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Figure 3 (a) shows the fitted model of impedance for the carpet against measurement, as well as the full range,
normal incidence absorption coefficient of the model. It can be seen that the agreement is good only above 400
Hz. This could be because the carpet is a more reflective surface than the panel, which decreases the accuracyof the in situ technique at low frequency. Furthermore, the absorption coefficient showed in Figure 3 (b) derived
from the model shows to be admissible over the full audible frequency range.

4

Model fit
Measurement

200

Im(Z) [-]

1

0.6
103

104

0.4

-10
-20

0.8

[-]

Re(Z) [-]

6

Model fit
Measurement
103
Frequency [Hz]

104

0.2
0

102

103
Frequency [Hz]

(a)

104

(b)

Figure 3. Carpet fitted model. (a): Impedance compared to measurements, (b): full range absorption.

Parameterσ [Pa · s/m2 ]d [mm]
dres [mm]m [g/m2 ]r [Pa · s/m]

Fitted value
6500
8.5
2.6
2300
5000

Table 2. Parameters of fitted carpet impedance model4

Derivation of TDBCs

This section describes a method used to derive the TDBC for the two acoustic materials, by approximating thecomplex reflection coefficient with discrete sums of IIR filters. The approximation covers the range from 20 Hz
to 5 kHz.
4.1 TDBC models
To be physically admissible, the frequency-domain representation of the acoustic reflection coefficient must
fulfill the conditions of causality, reality and passivity, as described in (6). To obtain such a representation, it ischosen to approximate the spectra of the reflection coefficients (R) with a finite sum of rational functions withsingle poles, either real valued (λk ) or pairs of complex conjugates (αl ± iβl , with αl and βl being the the realparts and imaginary parts respectively). This frequency domain approximation of R is written as:T1
Bl + jCl
Bl − jClAk+∑+,
αl − jβl + jω
l=1 2 αl + jβl + jω
k=1 λk + jωSRapprox (ω) = ∑

(3)

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This form fulfills the reality condition (Rapprox (−ω) = Rapprox (ω)), and with a proper choice of coefficients, thepassivity condition (|Rapprox (ω)| ≤ 1). The causality conditon is met when the poles must have a positive realpart, which imposes that λk ≥ 0 and αl ≥ 0. The time-domain form of the approximated reflection coefficient,rT DBC , to be used as TDBC, can then be analytically computed from the previously obtained coefficient as:
S

T

k=1

l=1

rT DBC (t) = ∑ Ak e−λk t H(t) + ∑ e−αl t [Bl cos(βl t) +Cl sin(βl t)]H(t),(4)

where H(t) is the Heaviside step function. This approach to derive TDBC functions for the reflection coefficient
was taken from (6), where this method was developed and applied on impedance functions.
4.2 Optimization in the frequency domain
In order to approximate the physical models of reflection coefficients with the appropriate mathematical forms
described above, a least-squares minimization in the frequency domain was performed in Matlab. The function
to minimize g was chosen to be the norm of the difference between the physical model and the approximation.
g(x) = ||Rapprox (ω, x) − Rmodel (ω)||.

(5)

For this procedure, the reflection coefficients are compared in the frequency range 20 Hz–5 kHz with a frequency resolution of 1 Hz. The TDBC function of the porous panel was chosen to be a sum of real-polesfunctions only (T = 0), and that of the carpet’s a sum of complex-poles functions only (S = 0). The reason forthis is that the absorption mechanism of the porous panel is mostly dissipative and the absorption mechanism of
the carpet mostly resonant in the frequency range of interest. The minimization of f yields a set of coefficient
xmin that allows to construct an approximation of the physical model over the considered frequency range.
4.3 Optimization constraints and parameters
The absolute value of the time-related coefficients of the TDBC functions (in this case: λk , αl and βl , which
are factors of the time variable in equation 3 and 4) should not be too large compared to 1/∆t (where ∆t is thetime step of the simulation, estimated to 1e-6s) in order to minimize errors. Similarly to what is done in (7), a
maximum value λmax is thus set. This parameter is set relatively to the time resolution intended ∆t by using achosen threshold
L = λmax ∆t = 5.
(6)
The optimization constraints applied to the set of coefficients are finally expressed by the following equations:∀k,0 < λk ∆t ≤ L∀l,0 < αl ∆t ≤ L
∀l,

(7)

|βl |∆t ≤ L

The number of terms in the sum S (for the panel) and T (for the carpet) is kept as small as possible byperforming several tryouts with increasing values, until no improvement of the error is observed. The starting
coefficients of the optimizations were chosen randomly in [0 λmax ] intervals for a set of ten runs, and theresult that was minimizing the fitting error was then selected.
4.4 Results
The coefficients and poles of the approximated reflections coefficients for the porous panel and the carpet are
displayed in Tables 3 and 4 respectively. The numbers of terms in the resulting IIR sums were S = 7 and T = 8
for the panel and the carpet respectively.

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k
Ak
λk

1
-3.3e46.9e3

2-1.8e4
2.7e3

3
1.2e4
1.2e4

4
1.2e4
4.6e3

5
1.2e4
1.2e4

6
1.2e4
4.6e3

7
1.2e4
2.4e3

Table 3. Coefficients of the porous panel’s TDBC function.

l
Bl
Cl
αl
βl

1
-2.5e4-1.9e32.8e3-5.1e32-5.8e4-1.4e4
5.2e3
2.6e3

3
3.3e4
3.7e4
1.2e4
-4.5e3

4
1.3e4
6.3e3
2.5e3
5.5e3

5
5.3e5
3.2e4
9.8e3
4.3e3

6
-1.7e47.8e3
1.2e4-3.8e47-7.2e3-1.8e49.2e3-4.5e4

8
3.1e3
4.1e4
5.9e3
6.3e3

Table 4. Coefficients for the carpet’s TDBC function
As seen in Figure 4 and Table 5, the method used yielded a very close approximation of the reflection coefficients, with a maximum amplitude error smaller than 0.003 and a phase error smaller than 0.006 rad for bothmaterials. Visually, curves from the models and the approximations are nearly indiscernible for both materials, cf. Figure 4. It should also be noted that the approximations functions were also fulfilling the passivity
conditions in both case.
Table 5. Maximum fitting error for IIR approximations of R
ε(|R|) [-]ε(φ (R)) [rad]Panel0.00200.00441

1
0.95|R| [-]|R| [-]

Model
Approx.
0.5

0.9
0.85

0

Carpet0.00290.0056102

ModelApprox.

0.8

103

102

Frequency [Hz]0
ModelApprox.

-0.1
-0.2
-0.3
102

103

angle(R) [rad]angle(R) [rad]

0

-0.4

103
Frequency [Hz]

-0.5
-1
-1.5
-2

Model
Approx.
102

Frequency [Hz]Frequency [Hz]

(a)

(b)

103

Figure 4. TDBC approximations of reflection coefficient. (a) panel, (b) carpet.

5

CONCLUSIONS

Physically admissible TDBCs were derived successfully from measurements of a porous panel and a resonant
carpet. Surface impedance measurement were carried out, using two different methods, the Kundt’s tube ap-

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proach and an in situ approach, and the results compared. The results from both approaches were in fairly
good agreement for the porouse panel but not for the carpet. Broadband impedance models were then fitted
to the measurement results to yield physically admissible, full-range impedance data. The full-range reflection
coefficient was subsequently approximated by a sum of IIR filters to provide time-domain boundary conditions.
In the case of the two materials in this study, the fitted impedance models were in fairly good agreement withthe measurement data in the 400 Hz-4000 Hz frequency range, while showing physically admissible data in thelower frequency range. The approximation of the resulting full-range data as sum of IIR filters was successful,showing a close agreement between the full-range models and the approximations of the complex reflection
coefficient in the 20 Hz-4000 Hz frequency range.

ACKNOWLEDGEMENTS
This work was supported in part by funding from Facebook Technologies, LLC.

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