# Design and Calibration Tests of an Active Sound Intensity Probe

Authors: Thomas Kletschkowski; Delf Sachau
Journal: Advances in Acoustics and Vibration
 (2008)

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## Abstract

The paper presents an active sound intensity probe that can be used for sound source localization in standing wave fields. The probe consists of a sound hard tube that is terminated by a loudspeaker and an integrated pair of microphones. The microphones are used to decompose the standing wave field inside the tube into its incident and reflected part. The latter is cancelled by an adaptive controller that calculates proper driving signals for the loudspeaker. If the open end of the actively controlled tube is placed close to a vibrating surface, the radiated sound intensity can be determined by measuring the cross spectral density between the two microphones. A one-dimensional free field can be realized effectively, as first experiments performed on a simplified test bed have shown. Further tests proved that a prototype of the novel sound intensity probe can be calibrated.


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## Body

## 1. Introduction

Noise
pollution of  enclosed interiors (e.g., aircraft cabins) results  in a limitation
of human comfort. If  noise reduction techniques will be applied in such
situations with success, the sound sources at the interior wall that surrounds
a standing wave field have to be identified. Especially the localization of low-frequency
hot spots in weakly damped enclosures requires the application of advanced
measurements techniques. Several noise source localization methods have been
established. The methods range from simple sound pressure or sound intensity
measurements to more sophisticated methods as beamforming [1] and acoustic
holography [2, 3]. A drawback in applying these techniques is that free-field
conditions are required. Even though the free field requirement does not have
to be met using the inverse boundary element method [4] or the inverse finite
element method [5], a well-validated numerical model of the interior is needed,
if these techniques will be applied with success.To
avoid the time and cost consuming processes of realizing artificial free-field conditions by introducing
passive damping to the interior (which in addition changes the global
characteristic of the investigated enclosure) or the need of sophisticated
numerical models, a prototype of a new mechatronic sound intensity probe with
an active free field (SIAF), see [7], that only influences the local impedance
was developed. The fundamental idea is to generate acoustic free-field
conditions by active noise control. As shown in Figure 1 (left), the three-dimensional
sound field in front of the interior wall is reduced to one dimension by a
mechanical device with sound-hard walls. This device is terminated by a
loudspeaker. The integrated microphone pair is used to separate the remaining
one-dimensional sound field inside the device into its incident and the
reflected components. The reflected wave is cancelled by active noise control,
as shown in Figure 1 (right). In contrast to common active noise control (ANC) techniques
for ducts (e.g., [8, 9]) or common sound tubes with standing wave fields that
are used to determine material properties, see [10], the application of the SIAF-approach
allows sound intensity measurements in (global) standing wave fields using the
local free field of the device.Figure 1
(left) Application and (right) functional principle of an intensity probe with an
active free field [6].In
a sound field consisting of a free field and a reverberant part, the real part
of the time averaged sound intensity determines the free field part. This part
vanishes in a purely diffuse, reverberant sound field, and in a plane standing
wave propagating inside a rigidly terminated enclosure, as shown
in [11]. If a
spherical  wave impinges on the sound soft boundary of a common (passive) sound
tube, parts of its energy are reflected in amplitude and phase. In this case, a
source is not characterized correctly. A conventional sound tube (without
active control) that is placed directly in front of the interior wall would
change the impedance of the area of investigation. The resulting impedance
could be determined by standard methods, see [10], but this value would vary
from point to point. In contrast to this unwanted situation, the SIAF-approach
enables both the energy transport from the source into the mechanical device as
well as its quantification under comparable boundary conditions inside the
mechanical device. Because of the free field inside the probe, the SIAF acts as
a local sound absorber. For this reason, the effect of the device on a source
would be reduced. A SIAF can therefore be applied for sound source
localization, especially in weakly damped interior noise fields at low
frequencies. This would—in general—not be possible, if a standing wave field without
any energy transport would remain inside the mechanical device. Only the active
free field ensures that every acoustic hot spot on the investigated interior
wall may radiate into the local free field.If
a SIAF would be used in a standing wave field, but not directly in front of the
interior wall, it would still act as a local sound absorber enabling an energy
flow into the mechanical device due to active control. This situation is
illustrated in Figure2. Here, the (qualitative) sound pressure distribution as
well as the intensity flow (symbolized by black arrows) was determined by
two-dimensional time-harmonic (f=300Hz) finite element simulations for an enclosure
with sound hard boundaries. The sound source was placed in the center of the cavity.
The SIAF is placed top right. The results shown in Figure 2(a) prove that no
intensity flow can be measured if the SIAF is inactive. If the inner termination
of the SIAF is described by a free-field impedance boundary condition (Z=428.75kgm−2s−1),
see Figures 2(b) and 2(c), an energy flow into the mechanical device is
enabled. The direction of this energy flow varies with the spatial orientation
of the probe, as shown in Figure 2. Therefore, inside the mechanical device it
would only be possible to measure the local intensity in longitudinal direction
which is—in general—not the magnitude of the intensity vector. Obviously,
the localization of a single acoustic hot spot would be impossible, if the SIAF
acts as a local absorber in the middle of the sound field. To identify a local
source, the SIAF has to be used close to the interior boundary as shown in Figure
1 (left). The minimum working distance should be chosen in such a way that contact
between the SIAF and the interior wall is excluded. The maximum working
distance has to be small enough to ensure that an energy flow into the SIAF
results only from a source that is located on the investigated part of the
boundary. Assuming an upper frequency limit of 1kHz, due to the area of
application of ANC, the maximum working distance should be lower than ten
percent of the minimum wave length (e.g., 3 cm).(a) Standing wave field with inactive SIAF and (b) and (c) intensity flow into
active SIAF.
(a)(b)(c)This
paper is divided into four parts. The first explains the calculation of the
reflected wave and the control concept. Furthermore, experimental results of
tests that were performed on a simplified test bed are reported. The second
part presents the results of calibration tests that were performed on a
SIAF-prototype. A short summary is given in the third part of the paper.


## 2. Control Strategy and Proof of Concept

To
realize free-field conditions, the sound field inside the mechanical device has
to be separated into its incident and reflected components. This
separation can be
performed in frequency as well as in time domain. As wave separation
in frequency domain requires a frequency domain control algorithm, it was not
applied. Instead, a time domain approach was used for time-discrete adaptive control,
as
illustrated in Figure3.Figure 3
Block-diagram for SIAF-approach based on wave separation in time domain.As shown in [12],
the reflected wave component of a one-dimensional standing wave field can be
determined by employing two microphones and a time-delay. One of the SIAF microphones
has to be placed in front of the canceling loudspeaker at x=0,
the other at a distance Δx.
Assuming plane wave propagation, the total pressure at the discrete time step n picked up at the microphones is given by(1)p1(n)=pi(0,n)+pr(0,n),p2(n)=pi(Δx,n)+pr(Δx,n)=pi(0,n+N)+pr(0,n−N), where the continuous
time delay τ=TsN=Δx/c is given by the separation distance Δx,
the sample time Ts,
the number of delayed time steps N, and the speed of sound c.
If p2(n) is delayed by τ, the delayed sound pressure is defined as(2)p2τ(n)=p2(n−N)=pi(0,n)+pr(0,n−2N),and the error signal e(n) that is calculated as follows(3)e(n)=p2τ(n)−p1(n)=pr(0,n−2N)−pr(0,n) represents the reflected wave only. As shown in [12], the calculation of
the error signal fails, if the distance between the microphones equals a
multiple of one half of the wave length, if (3) is analyzed for tonal
excitation pr(0,n)=Asin[2πf(nTs−x/c)].
As shown in [6], the time delay number N is limited by Shannon's law 1≤N<Tmin/2TS, there Tmin represents the periodic time of the highest
frequency of interest fmax.As shown in Figure3, the filtered reference least mean square (FxLMS) algorithm was used to
update the coefficients of the adaptive finite impulse response (FIR) filter W(z).
These coefficients are used to generate the driving signal y(n) for the canceling loudspeaker. The acceleration
of the interior wall that in the simplest case is represented by the membrane
on the noise source was used as reference signal x(n) that is required for adaptive feed foreword
control. A feedback controller would be independent of a reference signal. But,
in contrast to an adaptive control scheme based on FIR filters, a feedback
controller based on infinite impulse response (IIR) filters is not
unconditionally stable, see [13]. Offline plant modeling based on the common
least mean square algorithm was applied to identify the secondary path model S^(z). As shown in [13],
the single-channel leaky FxLMS algorithm can be summarized as follows.(1)
Adaptive filtering(4)y(n)=wT(n)x(n), where y(n) is the controller output at the discrete
time step n,  w(n) is the L×1 column matrix of the filter
coefficients for a FIR filter of length L,
and x(n) is the L×1 column matrix of the buffered reference
signal x(n).(2)
Single-channel prefiltering(5)x′(n)=∑i=0I−1s^i(n)x(n−i), where  I is the order of the FIR filter S^(z).(3)
Weight vector update(6)w(n+1)=vw(n)+μx′(n)e(n), where v is the leakage factor, μ is the convergence factor, and e(n) is the error signal at the discrete time step n.The choice of
a proper filter lengthL depends on parameters such as the
characteristics of the error signal (e.g., tonal or broadband), the impulse
response of the secondary path, and the computing power of the applied signal
processor. For tonal excitation, a filter length of L=2 would be sufficient to determine amplitude and
phase of the control signal as well as to model the secondary path. In practice,
however, a filter length of L=4 can be advantageous, because of the nonideal
transfer behavior of many electro-dynamic loudspeakers at very low frequencies.
If broadband disturbances have to be controlled using a certain sampling
frequency, the number of filter taps must be high enough to model the impulse
response of the secondary path in the analyzed frequency band but small enough
to guarantee causality.To test the performance of the SIAF approach, the algorithm was
implemented on a real time processor (type: dSpace DS1103). The time delay
number was set toN=2.
As motivated by Figure 3, a sound tube with a square cross-section
(edge length a=92mm) and upper frequency limit fu≈1864Hz (according to [10, equation (2)]) terminated by
electro-dynamical loudspeakers (type: SPEAKA MT 60/80) was used as a simplified
test bed for first experiments. Two microphones (type: Ono Sokki MI-1233) were
used. Microphone 1 was placed in front of the canceling loudspeaker and microphone
2 in a distance of Δx=13.7cm.
In compliance with [10, equation (4)], the upper frequency limit was reduced to fured≈1152Hz for this reason. An accelerometer (type:
B&K 4374) that was amplified by a signal conditioner (type: B&K Nexus
2692) was used to detect the reference signal. Furthermore, several analogue
high- and low-pass filters, respectively, (type: Kemo VBF21) were applied to
avoid aliasing. A multichannel FFT analyzer (type: Ono Sokki DS2100) was used
for data analysis.The reduction of the reflected
waves was analyzed for a broadband disturbance withfmax<fured.
The system was excited by frequency-banded white noise (50Hz≤f≤1KHz). Therefore, a filter length of L=128 was used for the adaptive weight vector w(n) as well as for the secondary path model S^(z).
The sampling frequency was set to fS=5kHz.
The results shown in Figure 4 prove that the controller cancels a reflected
wave, if and only if the associated frequency is detected by the reference
sensor. A total reduction of −16.2 dB was achieved in the analyzed frequency band.
The SIAF-control-strategy was applied successfully. Sound intensity was not
determined during this test.Figure 4
Broadband noise reduction obtained by SIAF approach applied to a sound tube.


## 3. Design of a Prototype and Calibration Tests

A first realization of a SIAF that is based on
commercial components is shown in Figure5(a). It consists of a sound source
(type: B&K  4295) and a self-made cylindrical adapter (inner diameter d=38 mm). Two phase-matched
microphones (type: B&K 4295) were integrated into this adapter. A portable
FFT analyzer (type: B&K 3560B with a phase match of ±0.017∘
at 50 Hz using B&K sound intensity probes) was used for data processing. The
microphone spacing was set to Δx=10 cm. According to [10, equation (4)], the
upper frequency is given by furedSIAF≈1543 Hz.(a) SIAF prototype, see [6], and (b)
experimental setup for calibration
tests.
(a)(b)An
anechoic chamber was used to test the possibility of calibrating the SIAF prototype.
The experimental setup is shown in Figure5(b). An electro-dynamical
loudspeaker (type: PAB-8MK2) was used as the acoustic
source, and a free-field microphone (type: B&K 4188) was applied to measure
the sound pressure level at a fixed reference point. An accelerometer (type:
B&K 4374), amplified by a signal conditioner (type:
B&K Nexus 2692), was used to assure a precise reproduction of the
excitation and to detect the reference signal. The radiated sound intensity was measured for the
third-octave band center
frequencies between 80 Hz and 500 Hz. First reference data were collected by a
conventional sound intensity probe (type: B&K 3595 with microphone pair
4197 using a spacer of 12 mm). The measurements were then repeated using the
SIAF prototype. Signal processing was performed on a power computer
(type: dSpace DS1103). Tonal disturbances were used to test the SIAF prototype.
Therefore, a filter length of L=4 was used for the adaptive weight vector w(n) as well as for the secondary path model S^(z) which was remodeled for every frequency. The
time delay number was set to N=2,
and a sampling rate of fS=10kHz was used for these tests.The
results shown in Figure6(b) prove that the controller was capable of reducing
the reflected wave for all analyzed frequencies. Only minor sound pressure
deviations between the SIAF microphones were measured. The results presented in
Figure 6(b) also confirm sound propagation in a plane wave field, because the
same amount of energy was determined at the two measurement points. For this
reason and because of the fact that the distance between the loudspeaker membrane
and the front side microphone of the SIAF was smaller than the triple diameter
of the tube, compare with
[10], additional corrections for the losses along the boundaries of the tube (needed
for higher frequencies, f>500Hz) were not taken into account.Sound pressure at the SIAF microphones: (a) uncontrolled and (b) controlled.
(a)(b)As proposed in [11], the sound
intensity was calculated from the imaginary part of the cross-spectral density G12(f) between the two SIAF microphones (p-p probe):(7)I(f)=−12πfρΔxIm[G12(f)],where ρ is the density of the fluid. An alternative
approach is given by the application of the combined sound-pressure,
particle-velocity probe (p-u probe) that
is described in [14]. This probe consists of a small electret condenser
microphone and a particle velocity transducer. The latter is based on the
technique of the acoustic intensity meter that was proposed in [15]. Particle
velocity measurements using transducers that were analyzed in [16] were only
possible by using a small fan to generate a permanent air flow. Because of the
nonlinearities of these transducer types, the permanent air flow is needed to
define the working point of the sensor. The p-u probe, presented in [14], is independent of a permanent air flow, and its
application, especially in plane standing wave fields, ensures that at least
one of the transducers is not located at a nodal point. If, however, the sound
pressure level exceeds the upper sound level—110 dB, see [17]—of the electret
microphone, the pressure sensor of the p-u probe can cause nonlinearities, as shown in [18]. This operating mode must be
avoided, if a linear controller is applied. As shown in [14], the application
of a p-u probe is advantageous, if
the influence of background noise—coming from sources outside the measurement
plane—on the phase mismatch between the sensors has to be reduced. In
contrast to the phase mismatch between the two microphones of a p-p probe, the p-u phase mismatch is exacerbated in
strongly reactive sound fields (e.g., in a plane standing wave), see [14]. Furthermore,
it is more difficult to calibrate a p-u probe than a p-p probe, as concluded
in [14].The
SIAF approach requires a wave separation in the time domain that is based on
two microphones. Therefore, sound intensity measurement was performed according
to (7). This procedure enables a direct comparison between the sound intensity
that is measured using the SIAF approach and the sound intensity that is
determined by a conventional p-p probe. The measurement limitations of this approach for sound intensity
measurement are summarized in [11]. Neglecting errors coming from off-axis
measurements, the quality of the results is determined by the phase mismatch
error, the finite difference approximation error, and the near field error.The
phase mismatch between the two channels in the analyzing system determines the “low-frequency
limit,” see [11]. As written in [11], the maximum phase mismatch might be ±0.3∘ for a good probe and analyzer combination. The
calibration tests were performed using phase matched microphone-analyzer
combinations. In this case, the phase mismatch at low frequencies depends only
on the microphone spacing. Using a 12 mm spacer for the conventional sound
intensity probe and a spacing of 10 cm for the SIAF, the change of phase over
the spacer is given by ±1∘ and ±8.4∘ for the SIAF and for the lowest analyzed frequency, respectively.
Hence, phase mismatch errors were not taken into account during the calibration
tests. If the SIAF concept is applied using a microphone-analyzer combination
that is not perfectly calibrated in phase, phase calibration (e.g., by a
procedure based on switching the microphones as described in [10]) is required. As
also outlined in [11], the finite difference approximation error of an ideal
two-microphone sound intensity probe in a plane wave of axial incidence is
given by(8)EFD(f)=10log10sinkΔxkΔx,withk=2πfc. This
error determines the “high-frequency limit” of a p-p probe. It has to be taken into account that, because the
pressure gradient that is needed to calculate the particle velocity is
approximated by a simple finite difference scheme every time, the intensity is
measured. If a p-u probe is used for
sound intensity measurements, EFD can be avoided, because the particle velocity
is measured directly. A finite difference approximation of the pressure
gradient is not needed in this case. The “high-frequency limit” as well as the
“low-frequency limit” of a p-u probe
is determined by the frequency response curves of the microphone and the
particle-velocity transducer.Using
the calculations for a two microphone probe in a sound field of a point source
radiating into a free field, as presented in [11], the near field error can be
described as follows:(9)ENF(Δxx)=−10log10(1−14(Δxx)2).   The
near field error described by (9) is a function of the separation distance
between the two microphones Δx and the distance from the source to the middle
point between the microphones x.
A negligible near field error of 0.063 dB was determined for the conventional sound
intensity probe with the 12 mm spacer (Δx=1.2cm, x=5cm), but it was found that an overestimation of ENF=2.4dB has to be taken into account for a SIAF (Δx=10cm, x=7.6cm) without a surrounding tube.In
addition to these measurement limitations, and because of the tube that
surrounds the SIAF-microphone pair, the deviations between plane wave
propagation in one dimension inside the SIAF and spherical wave propagation in
three dimensions have to be taken into account by RMS values of the measured sound pressure:(10)EPW(f)=20log10(p˜22⋅10−5Pa)−20log10(p˜Ref2⋅10−5Pa). This
error is in the following called plane wave error. It was calculated as the
difference between the sound pressure level measured at SIAF-microphone 2 and
the sound pressure level measured at the reference microphone. The latter was
placed above the actively controlled SIAF, as shown in
Figure 5(b).The
frequency dependencies of the plane wave error (that was derived from
measurements) as well as of the finite difference approximation error, given by
(8), are shown in
Figure 7(a). It can be seen that the first results in an
overestimation of the measured sound pressure level. The second leads to an
underestimation of the measured sound intensity level.Effect of errors (a) on sound intensity measurement and (b) deviation in
sound intensity.
(a)(b)The
deviationΔI=ISIAF−IConv between the sound intensity that was measured using
the SIAF prototype and the sound intensity that was determined by the
conventional probe is
shown in Figure 7(b). It was found that these deviations can be reduced, if (a)
the near field error, (b) the finite difference approximation error, and (c) the
plane wave error are used to calculate a corrected deviation ΔIcorr=ΔI−(EFD+ENF+EPW).
The corrected values are also shown in Figure 7(b). Without correction, the
measurement error was at least 2.1 dB. Using the correction based on the three
relevant error types, the mean deviation in the analyzed frequency band, calculated
as(11)ΔI¯:=10log10(1Nf∑i=1Nf10|ΔIi|/1010−12Wm−2),where Nf (representing the number of analyzed
frequencies) could be reduced from 3.24 dB down to 0.61 dB. The maximum deviation
was reduced from 4.2 dB down to −1.5 dB. A minimum deviation of 0.1 dB was found
at 200 Hz after correction. The results prove that the SIAF prototype can be
calibrated successfully under well-defined boundary conditions. For practical
use, however, a calibrator that can easily be attached and coupled to a well-defined
sound source (e.g., a pistonphone) would be required.The
experimental setup shown in Figure5(b) was a simple example of a sound source that
is located directly in front of the entrance of the tube. To study the SIAF concept
for other arrangements, time-harmonic finite element simulations (f=300Hz) were carried out in two dimensions using
free field conditions for the outer boundary of the analyzed area. It was
found, that a SIAF placed in a certain distance to the surface of the source would
again act as a local absorber that changes the direction of the energy flow as
motivated by Figure 8(a). If two uncorrelated sources are present, and the SIAF
would again not be collocated to one of these sources, the time average of the
local intensity in longitudinal probe direction would be measured, as shown in
Figure 8(b). This quantity would be determined by the sound field generated by
these interfering sources. If the SIAF is placed between two sources with equal
strength, as illustrated by Figure 8(c), the uni-directional intensity
component of the source facing the open end of the sound tube would be
measured.Pressure distribution and intensity flow in a free field
with (a) one and (b)
and (c) two sources.
(a)(b)(c)Using
the arrangement consisting of loudspeaker, accelerometer, and SIAF, compare
with Figure5(b), the prototype
was also tested in a typical lab. The reverberation time of this room varies
between 0.85 second at
125 Hz and 0.5  second at
400 Hz. The arithmetic mean in the frequency band between 100 Hz and 5 kHz is
given by 0.64 second and corresponds to values desired for rooms of medium size
(250m3–5000m3), see [19]. In
this situation, a positive sound intensity was measured for every tested
frequency, if active noise control based on wave separation in time domain was
applied. The data shown in Table 1 indicates that a SIAF can also be used to
detect acoustic hot spots in weakly damped interiors as proposed in [7].Table 1
Sound intensity measured in an enclosure
using the SIAF-prototype.
f/Hz100125250500ISIAF/Wm−2
without control−0.043−0.018−0.002−0.004ISIAF/Wm−2  with control+0.056+0.048+0.018+0.006IConv/Wm−2  without control+0.536+0.107+0.292+0.348The
sound intensities that were measured directly in front of the loudspeaker using
the conventional sound intensity probe—without the surrounding sound tube and
without active control—are also listed in Table1. The comparison of the
results proves that the SIAF was indeed capable of detecting the loudspeaker as
an acoustic source. In contrast to the calibration test that was performed
under idealized conditions, the magnitudes of ISIAF
(with control) and
IConv
cannot be compared, because the first quantifies
the energy transport of travelling plane waves that are absorbed inside the
SIAF, whereas the second quantifies the circulation of energy in the (partly
active and partly reactive) near field of an acoustic source that acts in a
standing wave field.It
is obvious that it is more sophisticated to calculate the plane wave error for
sound fields excited by more complicated sources than used in the calibration
tests. Nevertheless, to identify a sound source by application of the SIAF approach
that works under conditions in which conventional methods cannot be used, it is
sufficient to detect a positive energy flow, even if the absolute values are
not equal to values that can be measured with sensors of top-level quality.


## 4. Summary and Outlook

The
functioning principle as well as a prototype of a new sound intensity probe was
presented. It was found that a free field can be realized inside this probe by
active noise control, if a microphone pair in combination with a wave
separation approach is used to determine the reflected wave components. The
FxLMS algorithm was used for the signal control task using an accelerometer as a
nonacoustic reference sensor. This approach will be sufficient, if access to a
signal which is linearly related to the disturbance is given (e.g., in an
acoustic test center or during acoustic ground tests of new aircrafts). It was
shown that a sound intensity probe with an active free field can be calibrated
successfully inside an anechoic chamber, if the finite difference approximation
errors that appear during numerical calculation of the pressure gradient based
on a two microphone technique, the near field error, and the deviations between
plane wave propagation in one dimension and spherical wave propagation in three
dimensions are taken into account. But, at this stage of development, the
accuracy of the SIAF is of course not as high as the accuracy of well-established
conventional probes. It was also found that, because of the active-free field
inside the probe, a SIAF is capable of detecting an acoustic source inside standing
wave fields that typically appear in weakly damped interiors. Future research
will be focussed on the development of a calibration procedure that allows
calibration outside an anechoic chamber. To be independent of a reference
signal that is well correlated to the disturbance, the application of a stable
feedback controller is intended. Furthermore, it is planned to redesign the
shape of the open end of the SIAF in order to smooth the change of impedance
along the inlet of the probe by conical or exponential horns.



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*Source: 574806-2008-05-15.xml*