International
Carpathian Control
Conference ICCC’ 2002
MALENOVICE,
CZECH REPUBLIC
May 27-30, 2002
ACOUSTICAL HOLOGRAPHY FOR VISUALIZATION OF SOUND
FIELDS
Petr GRÄTZ
Department of Control and Instrumentation,
VUT – Brno University of Technology,
Brno, Czech Republic, gratz@feect.cz
Abstract: This project deals with simulation, comparison and optimisation of different
methods for near field array visualisation of the sound fields. These methods are based on
principles of the Near Field Acoustical Holography (NAH). All simulations are realised in
MATLAB environment. For the good comparison and optimisation of the methods, a
realistic model of vibrating surface has been developed. The main interest is focused on the
precision of the sound pressure field estimation, calculating time and sensitivity to the
measurement errors for different NAH methods and algorithms. Realistic conditions are
used for the simulations. The vibrating surface and its radiation is described at the first. The
detailed description of different methods and their optimisation follows, and after this all
presented methods are compared.
Key words: microphone array, acoustics, holography, sound fields.
1 Introduction: general instructions
This text deals with a description, simulation and optimization of the Near Field
Acoustical Holography algorithms. There are several methods using for near-field
visualization of sound fields. One of the most powerful is Near Field Acoustical
Holography (NAH). With this method it is possible to reconstruct 3D sound pressure field
from in-plane measured data. For online visualization of sound fields, use of a microphone
array is required. In-plane measured data are back-propagated to an estimation plane using
NAH algorithm. Four different NAH algorithms have been tested, optimized and the results
obtained with these algorithms have been compared. For the good comparison and
optimization, a realistic model of sound source has been developed. This is realized with
simply supported vibrating plate. All simulations are realized in MATLAB environment.
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2 Visualization of sound fields
Visualization of sound fields is especially used to localize sound sources. There are
several methods using for visualization of sound fields. One of the most useful and
powerful methods is Near-Field Acoustical Holography (NAH). With this technique it is
possible to reconstruct 3D sound field from in-plane measured data. From data obtained in
the measurement plane (the holography plane) it is also possible to calculate other
quantities (particle velocity vector, sound intensity vector, radiated sound power).
For NAH visualization of sound fields there are a few methods to measure the
sound field in measurement plane. For stationary sound sources it is possible to measure the
sound pressure with a single microphone and a reference signal or to measure the normal
part of the sound intensity using a sound intensity probe and a reference signal. For nonstationary sound sources it is necessary to measure the complete sound field in the
holography plane at one moment and therefore microphone arrays are used for this purpose.
From these holography (measured) data it is then possible to calculate the sound field in
other planes using the NAH technique.
3 Simply supported vibrating plate
To obtain as realistic as possible simulation results, the simply supported vibrating
plate model has been developed. Next figure presents the simulation conditions.
Figure 1. Simulation conditions
The surface displacement of the simply supported vibrating plate is done by :
w( x, y,ω ) = −
F ∞ ∞ Φ mn ( xo , yo )Φ mn ( x, y )
[m]
∑∑
ρh m=1 n=1
ω 2 − ω 2 mn
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(1)
where: F is the excitation force amplitude [N]
ρ is the plate density [kg/m3]
h is the plate thickness [m]
ω is the excitation force angular frequency [rad/s]
ωmn is the modal eigenfrequency above-mentioned [rad/s]
xo, yo are x and y positions of point force source [m]
Φ mn ( x, y) =
-1
2
sin(mπx / Lx ) sin(nπy / L y ) [m ]
Lx L y
(2)
where: Lx, Ly are the vibrating plate dimensions [m]
m, n =1, 2, 3, …. plate modes
Figure 2. Plate surface displacement for source frequency 50 Hz and 500 Hz
The sound pressure radiated by the vibrating plate is calculated using the following
formula:
pr =
− iωρ 0
2π
∑v S
n
n
e ikrn − ω 2 ρ 0
=
rn
2π
∑w S
n
n
e ikrn [Pa]
rn
where: rn is the distance between point number n on the plate and above the plate [m]
k is the wave number k=2πf/c [m-1]
vn is the point surface velocity [ms-1]
wn is the point surface displacement [m]
Sn is the surface point area (dx x dy, distances between imaginary point sources)
ω is the angular frequency [rad/s]
ρo is the density of the air [m/s]
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(3)
Figure 3. Sound pressure field radiated by vibrating plate driven at 50 Hz and 500 Hz
(0,05 m above plate)
A simply supported plate is driven by a harmonic force placed near the plate corner.
Radiation from the plate is simulated using 1024 point sources. Above the plate is placed a
microphone array with 256 microphones. For simulations data for Bruel & Kjaer array
microphones of Type 4935 have been used.
4 3D sound field reconstruction using NAH
The simulation is based on comparison of the sound pressure field calculation in zx
plane using one of the NAH algorithms. The reference sound pressure has been calculated
using formula (3). With this formula, the sound pressure field has been calculated in the
estimation plane zx and in microphone array plane zh. From this measurement (holography)
plane zh is then calculated the sound pressure field in the estimation plane zx (calculated
with NAH) and the results are then compared with the reference calculated directly (3).
Figure 4. NAH algorithm testing
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Four different NAH algorithms have been used for back propagating the sound field
from the holography plane to the estimation plane:
1.
2.
3.
4.
Spatial transform of sound field NAH (STSF NAH) with K-space filter
STSF NAH with Wiener filter
Iteration NAH
Statistically optimal NAH (SONAH)
These algorithms have been optimized for conditions described above and simulation
results for plate driven at 50 and 500 Hz are compared for different NAH algorithms. The
accuracy is the difference between directly and NAH calculated sound field in estimation
plane.
As an example next figures presents the sound pressure field calculated directly, using
Iterative NAH and the error of the Iterative NAH algorithm for a plate driven at 500 Hz :
Figure 5. Sound pressure field for plate driven at 500 Hz. Top left – direct calculation, top
right – Iterative NAH calculation, bottom - difference (error) between direct and Iterative
NAH calculation. Note that the dashed lines show plate dimensions and a cross shows the
position of the plate excitation force.
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STSF
Wiener
Iterative
SONAH
STSF
Wiener
Iterative
SONAH
5
0
dB
-5
-10
-15
-20
10
0
dB
-10
-20
-30
-40
-50
Figure 6. Calculation error range for different NAH methods over vibrating plate area
(top – 50 Hz, bottom – 500 Hz)
5 Conclusion
From above presented simulation results is clear that the best estimations of the sound
pressure can be obtained using Iterative STSF NAH and SONAH. It is possible to see that
the estimation of the sound pressure is very accurate at the low and the middle frequencies.
The disadvantage of the SONAH algorithm is quite long calculating time.
References
WILLIAMS, E. G. 2000. Fourier Acoustics. San Diego, USA : Academic Press, 2000, 306
pp. ISBN 0-12-753960-3.
RANDAL, R. B. 1987. Frequency analysis. Naerum, Denmark : Bruel & Kjaer, 1987, 344
pp. ISBN 8787355 078.
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