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TRANSCRIPT
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TECHNICAL REVIEW
Surface Microphone
NAH and Beamforming using the same Array
SONAH
No.1 2005
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Previously issued numbers of
Brel & Kjr Technical Review
1 2004 Beamforming1 2002 A New Design Principle for Triaxial Piezoelectric Acc
Use of FE Models in the Optimisation of AccelerometeSystem for Measurement of Microphone Distortion andMedium to Very High Levels
1 2001 The Influence of Environmental Conditions on the PresMeasurement MicrophonesReduction of Heat Conduction Error in Microphone Pre
CalibrationFrequency Response for Measurement Microphones ConfidenceMeasurement of Microphone Random-incidence and PResponses and Determination of their Uncertainties
1 2000 Non-stationary STSF1 1999 Characteristics of the vold-Kalman Order Tracking Filt1 1998 Danish Primary Laboratory of Acoustics (DPLA) as Pa
Metrology OrganisationPressure Reciprocity Calibration Instrumentation, ReMP.EXE, a Calculation Program for Pressure ReciprocMicrophones
1 1997 A New Design Principle for Triaxial Piezoelectric AccA Simple QC Test for Knock SensorsTorsional Operational Deflection Shapes (TODS) Meas
2 1996 Non-stationary Signal Analysis using Wavelet Transfor
Fourier Transform and Wigner-Ville Distribution1 1996 Calibration Uncertainties & Distortion of Microphones
Wide Band Intensity Probe. Accelerometer Mounted R2 1995 Order Tracking Analysis1 1995 Use of Spatial Transformation of Sound Fields (STSF)
Automative Industry
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TechnicalReview
No. 1 2005
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Contents
Acoustical Solutions in the Design of a Measurement Micropho
Mounting......................................................................................
Erling Sandermann Olsen
Combined NAH and Beamforming Using the Same Array .......J. Hald
Patch Nearfield Acoustical Holography Using a New Statistica
Method ........................................................................................
J. Hald
TRADEMARKS
Falcon Range is a registered trademark of Brel& Kjr Sound& Vibration Measu
PULSE is a trademark of Brel&Kjr Sound&Vibration Measurement A/S
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Acoustical Solutions in the Design Measurement Microphone for SurfMounting
Erling Sandermann Olsen
AbstractThis article describes the challenges encountered, and the sol
design of surface microphones for measurement of sound pres
of aircraft and cars. Given the microphones outer dimension
cavity shape should be found, together with the best possible p
solution. The microphones surface should be smooth so as to
ated noise. Since the microphones are intended to be used on th
and cars, they must work in a well documented way in a temp
55C up to +100C and in a static pressure range from one a
one or two tenths of an atmosphere. The static pressure even ch
on the surface of aircraft and cars due to the aerodynamically
RsumCet article traite des difficults quil a fallu surmonter lors d
microphones de surface utiliss pour les mesures de pression
face des automobiles et des aronefs, et des solutions qui ont des cotes extrieures minuscules du microphone, il fallait trouv
pour la cavit arrire et la meilleure solution possible pour l
sion. Il fallait aussi que la surface soit suffisamment lisse po
gnr par le vent. Comme ces capteurs sont destins des mes
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ZusammenfassungDieser Artikel beschreibt Problemstellungen und Lsungen beivon Oberflchenmikrofonen fr Schalldruckmessungen auf de
Flugzeugen und Autos. Bei gegebenen Auenabmessungen de
die optimale Form fr den rckwrtigen Hohlraum gefunden wer
besten geeignete Lsung fr die Druckausgleichsffnung. Die O
krofons sollte mglichst glatt sein, um Windgerusche zu verm
krofone an der Auenflche von Flugzeugen und Autos eingesemssen sie im Temperaturbereich von 55C bis +100C und b
drcken von 1 atm bis hinab zu 0,1 oder 0,2 atm in dokumentier
Der aerodynamisch erzeugte Druck bewirkt berdies, dass sich d
mit der Position auf der Oberflche von Flugzeugen und Autos n
IntroductionAt the end of the year 2000, a large-scale aircraft manufac
Brel & Kjr to find out if we could design and produce a me
phone capable of being mounted on aircraft surfaces. At that tim
group in Brel& Kjrs R&D department was looking at new
measurement condenser microphones. If successful, this new prowould allow us to produce the required flat microphone design, a
decided to carry on with the development. The microphone sh
20 kHz pressure field microphone of normal measurement micro
not more than 2.5 mm in height. It should not interfere with the ai
wings and it should work under normal conditions for aircraft s
large temperature and static pressure variations, de-icing, etc.Two microphone types have been developed, Brel& Kjr Su
Type 4948 and Type 4949. Both are pressure field measurement
built-in preamplifier, 20mm in diameter and 2.5 mm high. Th
article is to present some of the challenges in the acoustic des
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First, since a part of the stiffness in the diaphragm system o
due to the mechanical stiffness of the air in the cavity behindsince the stiffness of the air is proportional to the static pressur
depends on the static pressure. The static pressure dependency
expressed in dB/kPa, is given by:
where x is the change in pressure, cd is the mechanical com
phragm, c is the ratio of specific heats of air,ps is static pressuof the cavity and Veq,d is the equivalent volume at 1 atmosphe
compliance.
Second, if the static pressure is different outside and inside
static force will displace the average position of the diaphragm,
response of the microphone. Therefore, the cavity must be ve
have a certain cut-off frequency. Below the cut-off frequencyinsensitive to pressure variations. Above the cut-off frequen
works as intended. Assuming that the vent is a narrow tube bet
the surroundings and ignoring the influence of heat conductio
the cut-off frequency [1] is given by:
where cc is the mechanical compliance of the cavity, Rv is ac
Sps,dB
d
dx------ 20 1
cdx
cdps Vc+-------------------------
logd
dx------ 20 log 1
1
ps-----
Ve------
=
8,6861
ps-----
Veq,d
Veq,d Vc+------------------------
NG1
2------ cd cc+
Rvcdcc----------------- 1
2------pscd Vc+
RvcdVc------------------------- 1
2------ a4
8l--------- pscd Vc+
cdVc-------------------------= = =
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Boundary Element Model (BEM) calculations of the acoustic
cavity.The BEM method used for the calculations was the direct co
a formulation for axisymmetric bodies [3] with an improved ca
near-singular integration [4] and using a cosine expansion of t
bles in order to calculate non-axisymmetric sound fields [5]. N
into consideration in the calculations. Since the condition numb
matrix of the model presents maxima at eigenfrequencies [6], iplotted in order to identify the eigenmodes of the microphone
was the same as used for the calculations for Intensity Calibrat
Plots of the condition number as a function of frequency are
the cavity configurations mentioned in this article. Measured r
configurations are shown in Fig. 4.
Fig.2. Condition number plots for the first four terms in the cosine efields in the three cavity configurations mentioned in the text:
a) all three volumes included; b) blocked between inner volume and
ring volume blocked
: m = 0, axisymmetrical modes : m = 1, one nodeli
: m = 2, two nodelines : m = 2, three node
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In the first prototype, a) in Fig. 2, a cross-sectional mode is
11.2KHz and the lowest axisymmetrical mode is present at aroparing with the measured response, a) in Fig. 4, the cross sectio
present itself whereas the axisymmetrical mode creates a large di
response. The axisymmetrical mode is clearly the expected reso
mass of the air in the narrow part and the volumes of the cavity. T
Fig. 3 where the phase of the sound pressure is shown. The sou
outer ring is in counterphase with the rest of the sound field in th
It may be possible to remove the axisymmetrical mode from th
of the microphone without reducing the total cavity volume by bl
part of the cavity between the inner volume and the ring volumnumbers are shown for this situation in b) ofFig. 2. As compared
ple, the lowest cross-sectional mode has moved down in frequ
lowest axisymmetrical mode has moved up in frequency, to jus
phones frequency range. This situation is also shown in cu
Fig.3. Phase at 16 kHz of the calculated sound field in the cavity wi
included. The frequency is that of the axisymmetrical mode identified in a
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large the active volume could be made without having axisymclose to the microphones active frequency range. In c) ofFig. 2
are shown for a calculation where the narrow part of the gap is
side of the openings to the inner ring volume. That is, in these
ume below the backplate and the inner ring volume are coupled
part of the cavity. Now, the lowest axisymmetrical mode is aro
ably well above the frequency range of interest. The lowest croaround 17 kHz. The response of a later prototype with a sim
shown as c) in Fig. 4.
This calculation showed that although the rear cavity had
smaller than the total internal volume of the microphone, the in
some of the narrow sections could still be included in the rear c
volume of the rear cavity is around 90 mm3
. With the equivalenphragm compliance of around 7.5mm3, the resulting static pre
the microphone is around 0.007 dB/kPa. This is within the r
for normal measurement microphones [7] and it was founmicrophone. However, if versions of the microphone have to
Fig.4. Frequency responses of prototypes with three cavity configura
text: a) all three volumes included; b) blocked between inner volume an
c) outer ring volume blocked
050112
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form the basis of this article. Since the microphone is intended f
ing, the surface which will be mounted flush is likely to be thmicrophone housing that is exposed to the same static pressure
Furthermore, in the presence of airflow, the static pressure can
face due to aerodynamic lift forces and turbulence. Therefore, th
zation should represent the average static pressure on the surface
as closely as possible.An ingenious pressure equalization solutio
A groove is incorporated around the entire diaphragm perimet
equalization tube is connected to the bottom of the groove. The
tion system is illustrated in Fig. 5.
The groove is wide enough to prevent any significant acousti
yet it is narrow enough to dampen standing waves and not allow
into the groove (under normal circumstances). Of course, in the
phone icing-over, or if the surface of the microphone is tempor
water, the pressure equalization system will be inoperative, b
groove is open, the equalization system has proven to work as in
situations as well as in tests.
The pressure equalization solution did present some enginrelated to the temperature range of the microphone. Due to chang
the materials in the microphone, it is difficult to maintain conta
sealings perfectly airtight at extremely low temperatures, especi
i ti f th di i t b ll d f i th d
Fig.5. Sketch showing the principle of the static pressure equalization
Groove
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reverts to within the operating temperature range, the lower
reverts to its initial value.Influence of AirflowObviously, a microphone for sound pressure measurements
rapid airflow must not, by itself, produce any noise due to the
of the microphone must be as flat as possible and have no re
that can generate noise in the presence of the airflow. The su
phone should also be flush with the surface it is mounted in. aircraft surfaces the microphone must be embedded into the s
applications with more moderate wind speeds such as autom
microphone does not necessarily have to be embedded in the
avoid wind generated noise, as long as there are no sharp edge
The flatness of the microphone is achieved by welding the di
top of its carrying surface. In this way, the diaphragm can be tmicrophone housing. In order to avoid accidental destruction
however, it is recessed a few hundredths of a millimeter relati
surface. The groove for pressure equalization is positioned ju
ing, so using this method, the microphone is unlikely to have an
flow.
For applications where the microphone does not have to be eface, different mounting flanges have been designed that form
from the surface of the microphone to the surrounding surface
made slightly flexible so that they can be mounted easily on mo
faces.
ConclusionsIn this article the solutions to some acoustical challenges in th
a special microphone have been presented with special empha
of static pressure variations.
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lations and common sense considerations have led to a succes
microphone for surface mounting.
AcknowledgementsThe author wishes to thank all his colleagues at Brel& Kjr
pated in the development of the Surface Microphone, espec
Anders Eriksen, Johan Gramtorp, Jens Ole Gullv, Bin Liu and of the microphone development department whose combined w
successful design of the Surface Microphone.
References
[1] See, for example, Beranek L.L., Acoustics, Acoust. SoAcoustic Elements, (1996).
[2] Olsen E.S., Cutanda V., Gramtorp J., Eriksen A., Calc
Field in an Acoustic Intensity Probe Calibrator a Prac
Boundary Element Modeling, Proceedings of the 8th
Conference on Sound and Vibration, Hong Kong (2001).
[3] Seybert A.F., Soenarko B., Rizzo F.J., Shippy D.J., A
Equation Formulation for Acoustic Radiation and Scat
metric Bodies and Boundary Conditions, J. Acoust. So
1247 (1986).
[4] Cutanda V., Juhl P.M., Jacobsen F., On the Modeling of N
the Standard Boundary Element Method, J. Acoust. Soc1303 (2001).
[5] Juhl P.M., An Axisymmetric Integral Equation Formulat
Non Axisymmetric Radiation and Scattering of a Known
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Combined NAH and BeamformingSame Array
J. Hald
AbstractThis article deals with the problem of how to design a microp
forms well for measurements using both Nearfield Acoustical
and Beamforming (BF), as well as how to perform NAH pro
array measurements. NAH typically provides calibrated sou
while BF provides unscaled maps. The article also describes a
sound intensity scaling of the BF maps in such a way that a
vides a good estimate of the sub-area sound power. Results
speaker measurements are presented.
Rsum
Cet article traite de la difficult de concevoir une antenne mitout la fois aux mesures dimagerie acoustique par holograph
(NAH) et beamforming (BF) et au traitement NAH des m
moyen dantennes de gomtrie irrgulire. La technique NA
ment une cartographie calibre de lintensit acoustique tandis
fournit des cartes dpourvues dchelle. Cet article dcrit gal
de mise lchelle des cartes dintensit acoustique BF de tellegration surfacique conduise une juste estimation de la puiss
lment de surface. Avec une prsentation des rsultats dun m
un jeu de haut-parleurs.
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beschreibt auch eine Methode zur Skalierung der Schallintensi
rungen, bei der die Integration ber die Flche eine gute Abscschallleistungen ergibt. Es werden Ergebnisse von eine
Lautsprechern vorgestellt.
Introduction
Fig. 1 shows a rough comparison of the resolutions on the soand RNAH that can be obtained with Beamforming (BF) an
Acoustical Holography (NAH), respectively.
The resolution is defined here as the smallest distance betweemonopoles of equal strength on the source plane that allows the
in a source map produced with the method under consideration.
the near-axial resolution is roughly:
Fig.1. Resolution of Holography (NAH) and Beamforming (BF)
Log (Resolution)
Beamforming
HolographyR
NAH~ L
RNAH
~
RBF
~ (L/D)
Log (Freque
050
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L. Since typically, the focusing capabilities of Beamforming r
microphones be exposed almost equally to any monopole on tmeasurement distance required is normally equal to, or gre
diameter. As a consequence, the resolution cannot be better th
(approximately), which is often not acceptable at low frequenc
For NAH, the resolutionRNAHis approximately half the wav
quencies, which is only a bit better than the resolution of Beam
frequencies it never gets poorer than approximately the meas
By measuring very near the source using a measurement gri
spacing, NAH can reconstruct part of the evanescent waves t
tially away from the source, [2]. This explains the superior low
tion of NAH.
However, NAH requires a measurement grid with less th
spacing at the highest frequency of interest, covering at least th
to build up a complete local model of the sound field. This req
method impractical at higher frequencies because too many m
are needed. To get a comparable evaluation of the number of m
needed for BF we notice that usually the smallest possible me
LD is applied to get the highest spatial resolution. Since fudeteriorates quickly beyond a 30 angle from the array axis, th
area is only slightly larger than the array area, [1]. Fortunatelymised irregular array geometries, good suppression (at least 10
can be achieved up to frequencies where the average elemen
wavelengths, typically 34 wavelengths. So to map a quadrat
dimension of four wavelengths, NAH requires more than 64
tions, whereas Beamforming can achieve the same results wit
This often makes BF the only feasible solution at high frequencA combined measurement technique using NAH at low freq
forming at high frequencies therefore seems to provide the b
However, traditional NAH requires a regular grid array that co
sound source while Beamforming provides optimal high freq
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The principle of the combined measurement technique is ill
using a new so-called Sector Wheel Array design, which will bein the following chapter. Based on two recordings taken with the
different distances (a nearfield SONAH measurement and a BF m
intermediate distance), a high-resolution source map can be ob
wide frequency range. The measurement distance shown for Be
small a bit larger than half the array diameter. Simulations and
ments described in this article show that with, for example, th
Wheel Array ofFig. 2, Beamforming processing works well dow
Array Designs for the Combined Measuremen
Fig.2. Principle of the combined SONAH and Beamforming technique
urements with the same array
Source
Irregula
Uniform
1 metre d
60 elem
Source
1
2
Holography (SONAH) 12 cm distance
501200 Hz
Resolution ~ 12 cm
Beamforming
50 60 cm distance
10008000Hz
Resolution ~ 0.7
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microphone positions in such a way that the Maximum Sidel
minimised over a chosen frequency range. The MSL is definedthe so-called Array Pattern, i.e., in connection with a Delay-A
ing method focused at infinite distance, see Appendix and refe
ically the MSL has many local minima when seen as a fun
variables, an iterative optimisation algorithm will usually stop
close to the starting point. Many starting points are therefore ne
solution. Such starting points can, for example, be gener
number generators to scan a certain space of geometries.
In references [4] and [1] the optimised array geometries w
Wheel Arrays consisting of an odd number of identical line
spokes in a wheel, see Fig. 3.
The odd number of spokes is chosen to avoid redundant spatia
vals. The optimisation for low MSL ensures good suppressi
Fig. 3. Typical Spoke Wheel Array geometry with 66 microphones optim
applications
050060
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.6 -0.5-0.4 -0.3 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
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When the same array has also to be used for near-field holograp
at very small measurement distances, a more uniform density is tant. This will be covered in more detail in the following text.
Various irregular array designs have been published that exhib
density of the microphones over the array area and still maintain
over a wide frequency band, for example, the spiral array of [
Logarithmic Spiral array, [6]. These arrays, however, lack the ro
of the Wheel Array that allows a modular construction and tha
very efficiently in a numerical optimisation to minimise the MS
Sector Wheel Array geometry was developed. Fig. 4 shows a Pa
Spiral array with 60 elements, a Sector Wheel Array with 60 elem
Wheel Array with 84 elements. For all three arrays the diameter
meter.
The Sector Wheel Arrays maintain the rotational symmetry of
Arrays, but angularly limited sectors replace the small line arr
E h f th id ti l t t i i thi 12 l
Fig. 4. Three different irregular array geometries with uniform element d
circle around all three arrays has a diameter of 1.2 meters, so the arr
around 1 meter
SecSector Wheel (60)Packed Log. Spiral (60)
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very low sidelobe level at frequencies below approximately
free focusing angle would be 1500Hz. With the array very
source, as required for holography processing, free focusing an
ered, because waves will be incident from all sides. The 1500
be just a little bit below the frequency, where the average spaci
ments of the array is half a wavelength. The average element
mately 10 cm.
Optimisation of the Sector Wheel Array geometries in Fig. 4
by adjusting (using a MiniMax optimisation program) the co
ments in a single sector in such a way that the maximum MSLthe frequency range of interest. In this process a limit was pu
1500 Hz for the 84-element array and up to 1200 Hz for the 60-
turned out, this helped maintain the uniform element distributi
possibility of using the array for holography at frequencies
Fig.5. Maximum Sidelobe Level (MSL) for the three different array gethe Spoke Wheel ofFig. 3. The focusing of the array is restricted here
array axis
Packed Log
Sector Whe
Sector WheSpoke Whe
0 1000 2000 3000 4000 5000 6000
-18
-15
-12
-9
-6-3
0
-21
-24
Frequency (Hz)
MSL(dB)
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responding 60-element Sector Wheel over the Beamforming fre
the Sector Wheel is significantly better over a rather wide range o
where it applies for SONAH holography.
Simulation of Beamforming Measurements at Source Distance
Some simulated measurements were performed to investigate huniform arrays ofFig. 4 would perform with Beamforming fro
distance of 0.6 m, i.e., a bit more than half the array diamete
shown in Fig. 6 for the case of 5 uncorrelated monopoles of
8000 Hz.
The Beamforming calculations have been performed using
algorithm (with exclusion of Auto-spectra) described in referen
the source plane at 0.6 m distance. Compared to Fig. 5 the Auto-
Fig.6. Simulated measurements on 5 monopoles at 8 kHz and at a mea
60 cm with the three array designs shown in Fig. 4. The displayed range
0.5
0.5
0.4
0.3
0.2
0.1
0
0
-0.1-0.2
-0.3
-0.4
-0.5
-0.5 0.5
Packed Log. Spiral (60) Sector Wheel (60) Se
0-0.5 -0.5
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distance than for the infinite focus distance represented in Fig.
2 dB higher. The 8 kHz data presented in Fig. 6 are not entirel
the relative performance of the three arrays over the full freq
look instead at 3kHz, then according to Fig. 4 the 60-element
has approximately 6dB lower sidelobe level than the 60-eleme
mic Spiral.
The following consideration illustrates the advantage of
NAH for source location at high frequencies. If the maps in F
duced with traditional NAH, then a measurement grid with
1.2 1.2 m would have had to have been used, with a grid spac
this would have meant approximately 3600 measurement posit
Numerical Simulations to Clarify the SuitabilArrays for HolographyAnother series of simulations were performed to investigate th
over which the three arrays ofFig. 4 and the Wheel Array ofF
SONAH holography measurements. In SONAH (and other typ
plete reconstruction of the entire near field is attempted over
the measurement area. This is possible only if the spatial safield taken by the array microphones provide at least a comple
the pressure field over the area covered by the array. So from
samples it must be possible to reconstruct (interpolate) the so
the measurement area. This can be done by the SONAH algori
The problem of reconstructing a (2D) band-limited signal fro
has been covered quite extensively in the literature; see for ex
that the reconstruction can be performed in a numerically stable
that the distribution of the sampling (measurement) points exh
uniform density across the sampling area. Such a criterion was
the Sector Wheel Arrays.
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pressure from the same monopole, and the Relative Average Erro
each frequency as the ratio between a sum of squared errors an
sum of squared true pressure values. The summation was, in b
interpolation points and all sources.
Fig. 7 gives a comparison of the Relative Average Interpolatio
with the four different arrays.
Clearly, the 84-element optimised Sector Wheel Array can refield over the array area up to the previously mentioned 1500H
while the 60-element Sector Wheel Array provides acceptable a
around 1200 Hz. This actually means that the two Sector Wheel
the same frequency ranges as regular arrays with the same aver
Fig.7. Comparison of Relative Average Interpolation Error for the three
the Wheel Array ofFig. 3. The error is averaged over a set of monopol
distance of 30cm
Packed Log. S
Sector Wheel Sector Wheel
Spoke Wheel
0 500 1000 1500 2000 2500
-30
-25
-20
-15
-10
-5
0
-35
-40
Frequency (Hz)
RelativeAverageerr
or(dB)
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Intensity Scaling of Beamformer Output
When combining low-frequency results obtained with SONquency results obtained with Beamforming it is desirable to ha
in the same way. This is not straightforward, however, as will b
following description of the basic output from SONAH and Be
Based on the measured pressure data, SONAH builds a soun
within a 3D region around the array, and using that model it is
sound field parameter. Typically the sound intensity normal tcalculated to get the information about source location and
measurement is taken very near the sources, the energy radia
within a 2p solid angle will be captured and included with the
sound power estimates.
Beamforming, on the other hand, is based on a measurement
mediate distance from the sources where only a fraction of t
covered by the array. Rather than estimating sound field param
region, directional filtering is performed on the sound field i
array. As a result only the relative contributions to the sound p
position from different directions is obtained. Reference [8] de
the output that allows the contribution at the array position f
areas to be read directly from the Beamformer maps. This is, of
only if the pressure distribution across the array area from
sources is fairly constant, which will be true if the array cove
solid angle as seen from the sources.
But in the context of this article, we wish to take BF measu
possible to the source area, in order to obtain the best possible s
a consequence, the radiation into a rather large fraction of th
measured. We should therefore be in a better position to get infexample) the sound power radiated through the source plane.
the Beamformer output in such a way that the scaled map re
strength (in some way) it seems logical to scale it as active soun
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performed looking at a single monopole point source in the fa
assuming that the array provides a good angular resolution, i.e., a
lobe covers only a small solid angle. An evaluation is then given
duced by the far-field assumption and the assumption of a narrow
done both for Delay-And-Sum processing and for the Cross-s
with exclusion of Auto-spectra. The main conclusions are for th
tor Wheel Array ofFig. 4 and for frequencies above 1200 Hz:
1) The error is less than 0.4dB when using a measurement d
than the array diameter.
2) At smaller measurement distances the error increases, but
approximately 0.6dB when the distance is larger than 0
diameter.
In the Appendix it is argued that if the scaling works for a singl
source, then it holds also for a set of incoherent monopole so
plane. If sources are partially coherent and/or if single sources ational, then because of the limited angular coverage of the arra
power estimation cannot possibly be obtained. Fortunately,
sound sources tend to have low spatial coherence in the frequ
Beamforming will be used in the combined NAH/BF method.
The derivation of the scaling is based on matching the area-in
the known sound power for a monopole sound source. In the degration was performed only over the hot spot corresponding to th
Beamformer. At high frequencies many sidelobes will typically b
ping area, and it turns out that area-integration over a large nu
will typically contribute significantly to the sound power. T
avoided in practice by the use of a finite dynamic range during th
typically around 10 dB. A frequency dependent adjustment of thto match the resolution is not practical.
The measurement results to be presented in the following sec
influence of measurement distance, size of the power integrati
presence of more than a single source Also the sound power
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Measurements
In order to test the performance of the 60-element Sector Whmeasurements were taken at 12 cm distance from two sma
SONAH processing, and at 55 cm and 100cm distance for Be
ing. The microphones used in the array were Brel & Kjr Typ
distances, measurements were taken with coherent and inco
Fig.8. 1/3-octave sound intensity maps for the measurements with o
right excited by broadband random noise. The four rows represent Bments from 100 and 55 cm distance, SONAH from 12cm distance and
sound intensity probe at 7 cm distance. The 1/3-octave centre frequen
top of the columns. Dynamic range is 15 dB
500 Hz 1 kHz200 Hz 2 kHz
Beamforming from 55 cm
Beamforming from 100 cm
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excitation of the two speakers and also with only one speaker ex
these three excitations, a scan was performed approximately 7
two loudspeakers with a Brel& Kjr sound intensity probe Ty
speakers were identical small PC units with drivers of diameter 7
mounted with 17 cm between the centers of the drivers. The Beam
ing was performed with the Cross-spectral algorithm with exclu
tra, [1].
Fig. 8 shows 1/3-octave sound intensity maps for the measu
the speaker on the right excited. The arrangement of the speak
some of the contour plots.
The four rows of contour plots represent the Beamforming m
from a distance of 100 cm and 50 cm, the SONAH measurements
distance and the measurements taken with an intensity probe f
7 cm. For the first three rows (representing Beamforming and SO
sound intensity has been estimated in the source plane over an arsize 80 cm 80 cm, while the last row shows the sound intens
from the plane of the speakers over an area of size 36cm 21 cm
15 dB dynamic range from the maximum level, with 1.5 dB step
ours. Yellow/orange/green colours represent outward intensity an
resent inward intensity. The absolute levels will be presented sub
area integrated sound power data.The resolution obtained with Beamforming and SONAH is i
with the expectations as shown in Fig. 1. The bend on the re
SONAH is in this case at approximately 1500 Hz, being deter
whereL is the measurement distance and k is the wavelength. C
quencies the Beamforming resolution is very poor, while abo
1.5 kHz it is approximately as good as that obtained with the souSONAH provides good resolution over the entire frequency
approximately 1200 Hz, the average spacing of the microphone
reconstruct the sound pressure variation across the measuremen
distortions will slowly appear as frequency increases and mo
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for the small plots in Fig. 8). In addition, many other types of
formed based on the same data, such as transient analysis of rad
As mentioned previously, the sound intensity scaling of the
forming is defined in such a way that area integration over the
provide a good estimate of the sound power from a monopole
depicts the 1/3-octave sound power spectra for the single speak
scan with the sound intensity probe and from the Beamform
Fig.9. 1/3-octave sound power spectra for the single speaker meas
probe map has been integrated over the entire mapping area shown in
ing measurement, taken at a distance of 55 cm, has been integratedarea and over the mainlobe area only
Intensity Probe
BF, 55 cm, Full area
BF, 55 cm, Small area
160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3
55
60
65
70
75
80
50
45
Frequency (Hz)
SoundPower(dB)
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D is the array diameter of approximately 1 m, refer to equation (A
dix. At low frequencies the mainlobe is larger than the entire map
by 0.8 m, and therefore the two Beamforming spectra are identica
power is underestimated, because the power outside the ma
included, and also the assumptions made for the sound intensity s
refer to the Appendix. At high frequencies the power estimated b
too high, even when the integration covers the mainlobe area on
because the loudspeaker is no longer omni-directional as assum
but concentrates the radiation in the axial direction, towards the adiameter of the driver unit is approximately one wavelength. A
the over-estimation could be the tendency of the intensity scalin
when the measurement distance is very small, see Fig. A3. Loo
power obtained by integration over the entire mapping area, it is
high frequencies. The reason is that sidelobes (ghost images)
cantly when the integration area is much larger than the mainlobthe array has good sidelobe suppression as the present Sector Wh
Fig.10. 1/3-octave sound power spectra for the single speaker meas
intensity probe result is included. But now the results from Beamforming
distance of 55cm and 100cm are included. For both of these, the sou
covers the entire mapping area
Intensity Probe
BF, 55 cm, Full area
BF, 100 cm, Full area
60
65
70
75
80
SoundPow
er(dB)
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Fig. 10 shows results similar to those ofFig. 9, but instead
influence of the size of the power integration area, the influe
ment distance is now investigated. For both the Beamforming m
at different distances, the power integration has been perform
mapping area. At low frequencies, the biggest underestimati
measurement taken at the longest distance, because the resol
consequently a larger part of the power falls outside the mappi
quencies the measurement at 55 cm distance produces the bigg
There are several reasons for that. One is that the sidelobes becmeasurement distances smaller than the array diameter. Anothe
resolution: a narrower mainlobe means that the ratio between th
the mainlobe-area increases significantly. Finally, the scaling
mate the sound power when used with measurements taken at v
as can be seen in Fig. A3.
Fig. 11 shows the 1/3-octave sound power spectra for theobtained with intensity probe, SONAH and Beamforming.
Fig. 11. 1/3-octave sound power spectra for the single speaker mea
obtained with Intensity Probe, SONAH and Beamforming are compare
55
60
65
70
75
80
Soun
dPower(dB)
Intensity ProbeSONAH
BF, 55 cm, 10dB Range
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measurement at 55cm distance has been chosen, and for that
sound power integration has been performed over the full sound
Fig. 8), but using only a 10 dB range of intensity data (i.e., dat
level is less than 10 dB below Peak level are ignored). The resu
that obtained with integration over the mainlobe area only,
500 Hz this leads to a good estimate of the sound power, apart fr
discussed overestimation at the highest frequencies. SONAH pro
power estimates up to approximately 1.6 kHz, apart from a sm
(which could be due to the small measurement area that is useintensity probe). But above 1.6 kHz the sound power is incre
mated with SONAH.
As expected, the results with equal but incoherent excitation o
are very similar to the results with only one loudspeaker excited
spectra all increase by approximately 3dB over the major par
range, but the differences between the spectra remain unchangresults are shown here.
Fig.12. 1/3-octave sound power spectra for the case of the two speake
the same white noise signal. Results obtained with Intensity Probe, SON
ing are compared
Intensity Probe
SONAH
BF, 55 cm, 10dB Range
55
60
65
70
75
80
SoundP
ower(dB)
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Equal but coherent in-phase excitation of the two loudspeake
hand, cause the radiation to deviate more from being omni-dir
lates the assumptions on which the intensity scaling of Bea
based.
Fig. 12 depicts the 1/3-octave sound power spectra obtai
probe, SONAH and Beamforming with identical excitation o
The SONAH spectrum follows the intensity probe spectrum in
as for the case of only a single speaker being excited. But the so
from the scaled Beamformer map shows additional deviatiorange from 1kHz to 2 kHz. In that frequency range the distan
speakers is between half a wavelength and one wavelength, w
radiation in the axial direction. But the deviation remains w
2 dB from the power spectrum obtained with the sound intensit
ConclusionsA new combined array measurement technique has been pr
Near-field Acoustical Holography and Beamforming to be
same array. This combination can provide high-resolution no
over a very broad frequency range based on two recordings w
different distances from the source. The key elements in the prthe use of SONAH for the holography calculation, sound inte
Beamformer output and the use of a specially designed irreg
form element density. The optimised Sector Wheel Array i
applicable array with very high performance, particularly fo
part. Numerical simulations and a set of measurements conf
the combined method and of the Sector Wheel array design. Ttionality is all supported in PULSE Version 9.0 from Brel & K
A di S d I t it S li f B f
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Here, x is the temporal angular frequency, kk is the wavefictitious plane wave incident from the direction in which
(see Fig. A1) and k= x/c is the wave number. In equation (A
factor equal to ejxtis assumed.Through our choice of time delays m(), or the equivalen
wave number vector kk, we have tuned the beamformdirection . Ideally we would like to measure only signals arriv
tion, in order to get a perfect localisation of the sound source.
much leakage we will get from plane waves incident from
assume now a plane wave incident with a wave number vectorkfrom the preferred kk. The pressure measured by the miideally be:
which according to equation (A.3) will give the following ou
former:
Here, the function W
( , )
1
M----- wmPm
m 1=
M
( ) ejm ( ) 1
M----- wm
m 1=
M
Pm(= =
Pm ( ) P0ejk0 rm
=
B ( , )P0
M------ wme
j k k0( ) rm
m 1=
P0W k k0( )=
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microphone positions rm have z-coordinates equal to zero, the
independent ofKz. We therefore consider the Array Pattern Wo
plane, and when it is used, as in equation (A.5), the 3D wavenum
jected onto the (Kx , Ky) plane. In that plane, W has an area
around the origin with a peak value equal to 1 at (Kx ,Ky) = (0, 0
According to equation (A.5), this peak represents the high s
waves coming from the direction , in which the array is focused
an illustration of that peak, which is called the mainlobe. Other
which are calledsidelobes, will cause waves from such directiomeasurement of the mainlobe direction , creating so-called
ghost images. The Maximum Sidelobe Level (MSL) is def
between the highest sidelobe and the mainlobe for a given freque
In the expression (A.5) for the response to a plane wave, notic
exactly equal to the amplitude P0 of the plane wave, when th
towards the direction of incidence of the plane wave, i.e., when kFor stationary sound fields it is natural to operate with the matr
between the microphones, which provides a better average rep
stationary phenomena. Exclusion of the auto-spectra offers the po
ing the influence of noise in the individual measurement channe
that it also often reduces the sidelobe level, [1]. For the deriva
intensity scaling we will, however, not use the Cross-spectral foscaling holds for the Cross-spectral formulation as well, as lon
such a way that the response to an in-focus incident plane wa
squared amplitude of the wave. The formulation in reference [1]
The validity of the intensity scaling in combination with the Cro
former is investigated both through simulations in this appendi
practical measurements.From the literature it is known that the size and shape of th
array pattern is determined almost entirely by the size and ov
array, [9], [1], while the sidelobes are highly affected by the actu
microphones The shape of the mainlobe is usually close to the
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whereD is the diameter of the aperture (or of the array), J1 is
of order 1, and is the projection ofKonto the (Kx ,Ky) p
achieved is a general approximation for the shape of the mainpendent of the specific positioning of the microphones,
for
Here,K1 is the first null of the aperture array pattern,
3.83
being the first null of the Bessel function of the first order.
Derivation of the ScalingFor the derivation we now assume a single monopole point
axis at a distanceL that is so large that the amplitude and pha
practically constant across the array area. Thus, for the array
plane wave with amplitudeP0 incident with wave number
where is the unit vector in thez-direction. The sound powe
monopole is then:
W K( ) 2
J11
2
---KD
1
2---KD
----------------------- K K,=
K
W K( ) W K( ) K K1
W K1
( )
1
2---K1D 1=
1
z
Pa 4L2
I 4L2 P0
2
2c----------- 2L
2 P02
c-----------= = =
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where the known values of the two wave number vectors have
order to use the approximation (A8) for the mainlobe of the arra
to project the wave number vectors onto thexy-plane, which lea
for
h being the angle from the array axis (thez-axis) to the focus dir
The Beamformer is now used to create a source map in the plan
tion on this source plane is described by its distanceR to thez-ax
angle . Assuming relatively small angles from thez-axis we ca
mation:
where h is still the angle to thez-axis. Use of equation (A.13) in
following approximate expression for the mainlobe of the be
the source plane:
for
By the use of equation (A.9), we get for the radius R1 of the
source plane
B ( ) P0W k k0( ) P0W k kz+( )= =
B ( ) P0W k ( )sin( ) k ( )sin K1
R L ( )tan L ( )sin=
B R ( , ) P0WkR
L------
R
K1L
k---------- R1
R1
K1L
k---------- 2
L
kD-------1 1,22
L
D------=
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is now defined in such a way that the integral ofBI(R, ) over half of the radiated sound powerPa , i.e., the power radiated
containing the array:
Use of equation (A.7), substitution with the variable
for R in equation (A.17) and application of the relation (A.15)
with
The scaling factor can finally be obtained through use of the ex
the sound power in equation (A.19):
1
2---Pa B R ( , )
2R Rd d
0
2
0
R1
2P02
W2
0
R1
kR
L------
Rd= =
ukR
L------
D
2----
kD
2L-------R=
1
2---Pa 2P0
22J1 u( )
u-------------
0
1
2
2L
kD-------
2u du 32
P0L
kD-----------
= =
J1 u( )
u-------------
2
u ud
0
1
0,419
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geometry, but the width of the mainlobe is inversely proport
diameter measured in wavelengths (refer to equation A.15). To
integrated power with increasing array diameter, the scaling facproportionality mentioned above.
Evaluation of ErrorsThe major principle of the scaling is that area integration of the s
provide a good estimate of the sub-area sound power. For that r
to use the term Sound Intensity Scaling about the method. The
for a single omni-directional point source in such a way that area
peak created by the mainlobe equals the known radiated pow
source. So by this definition the total power will be within th
from the source position, and integration over a larger area will
mation of the sound power. One reason for choosing this definitimainlobe has a form that depends only on the array diameter and
phone positions. Other choices would be somewhat arbitrary, w
gration over a larger area to get the total power and would need
to depend on the particular set of microphone positions. But th
sidelobes on the power integration is a drawback if the mainlob
and sound power integration is performed over an area much laof the mainlobe on the source plane, then the level of sidelobe
beamforming can contribute significantly to the power integratio
nificant over-estimation of the sound power. The solution ado
significant over-estimation is to use only a finite dynamic range
in the area integration, typically around 10 dB. The applied dyn
ever, should depend on the MSL of the array.The scaling was derived for a single omni-directional point s
axis. Beyond that we have assumed the monopole to be so far aw
that its sound field has the form of a plane wave across the arra
d h b i h f fi ld i l i h
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1) A Delay-And-Sum beamformer focused at the finite
without any amplitude/distance compensation, [1].
2) The Cross-spectral beamformer with exclusion of Auto-reference [1]. This method compensates for the amplitu
the array of the sound pressure from a monopole on the
The output has then been scaled as sound intensity through mu
scaling factora of equation (A.21), and finally the sound p
mated by integration over a circular area with radius equal toR
A.15) around the array axis.Fig. A2 shows the ratio between the estimated and the trues
bels for the case of the Delay-And-Sum beamformer. At 10
(and therefore the hot spot generated around the source positio
covers an angle of approximately 24 from the array axis. This
Fig.A2. Difference in decibels between estimated and true Sound
value is from an Intensity scaled Delay-And-Sum Beamformer. The so
the array axis
0 0.5 1 1.5 2 2.5 3 3.5 4
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0.1
0
-0.1
-0.2
Distance (m)
Pow
erError(dB)
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region relative to the array certainly does not hold. But fortuna
not get worse than 0.6dB (approximately) for distances down
diameter. To achieve the best possible resolution it is desirable measurement distances as small as this.
Fig. A3 shows the difference between the estimated and the tru
decibels for the case of the Cross-spectral beamformer with e
spectra.
This algorithm is implemented in Brel & Kjrs Stationary an
Beamforming calculation software, and therefore it has been use
ments presented in this article. Comparison ofFig. 3 and Fig. 4eral the Cross-spectral algorithm produces smaller errors then th
algorithm, except at the very short measurement distance of 0.5
It is, of course, also important to consider, how the sound inten
Fig.A3. Difference in decibels between estimated and true Sound Po
value is from an Intensity scaled Cross-spectral Beamformer with exclusThe source is a monopole on the array axis
0 0.5 1 1.5 2 2.5 3 3.5 4
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0.1
0
-0.1
-0.2
Distance (m)
PowerError(dB
)
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Patch Nearfield Acoustical Hologr
Using a New Statistically Optimal M
J. Hald
AbstractThe spatial FFT processing used in Near-field Acoustical H
makes the method computationally efficient, but it introduces s
dowing effects, unless the measurement area is significantly larg
A Statistically Optimal NAH (SONAH) method is introduced w
plane-to-plane calculations directly in the spatial domain. Theref
representation in the spatial frequency domain and for zero pasignificantly reducing the spatial windowing effects. This art
SONAH algorithm and presents some results from numerica
practical measurements.
RsumLe traitement FFT spatial utilis dans lholographie acoustique(NAH) rend la mthode efficace sur le plan computationnel m
deffets de fentrage inopportuns, sauf dans le cas o la surface m
cativement plus grande que la source. La mthode faisant inter
Statistically Optimal NAH (SONAH) est ici prsente. Les calc
plan plan dans le domaine spatial, elle vite le besoin dune repr
domaine de frquence spatial et dun calage du zro, rduisant dcative les effets de fentrage spatial. Cet article dcrit lalgorithm
sente plusieurs rsultats obtenus par simulation numrique
pratiques.
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Es wird eine statistisch optimierte Methode der akustischen
(SONAH) vorgestellt, die Berechnungen von Ebene zu Ebene d
Bereich ausfhrt. Damit entfllt die Notwendigkeit fr eine Dachen Frequenzbereich und fr Zero Padding, wodurch rum
wesentlich reduziert werden. Dieser Artikel beschreibt den SO
und stellt Ergebnisse numerischer Simulationen und praktische
IntroductionA plane-to-plane propagation of a sound field away fromdescribed mathematically as a 2D spatial convolution with a pr
2D spatial Fourier transform reduces this convolution to a simp
a transfer function. In Near-field Acoustical Holography (NAH
form is implemented as a spatial FFT of the pressure data me
area.
The use of spatial FFT and multiplication with a transfer fu
frequency domain is computationally very efficient, but it intr
The discrete representation in the spatial frequency domain
replica in the spatial domain, causing wrap-around errors
plane. A standard way of spacing the replica away from the rea
is to use zero padding, introducing, however, a sharp spatial w
dow causes spectral leakage in the spatial frequency domain [1
the measurement area must be significantly larger than the sour
turbing window effects. This is a problem, for example, in co
Domain NAH, [2], and Real-time NAH, which do not allow the
measurement area through scanning. The new Statistica
(SONAH) method performs the plane-to-plane transformation
tial domain rather than going via the spatial frequency domain,
Theory of SONAH
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homogeneous, i.e., the sources of the sound field are forz< da
The array measurements are performed in the planez= 0.
From the theory of NAH, [1], for example, it is well-known th
forzdcan be written as an infinite sum of plane propagating
cent waves:
Here, K (kx , ky) is a wave number vector,P(K) is the Plane Wa
Fig.1. Geometry
Measurement
plane
y
z
d
Source
region
050075
r( ) 1
2( )2
-------------- P
K( )K r( )dK=
Kx y z , ,( ) ej k
xx k
yy k
zz d+( )+ +( )
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Notice that the elementary wave functions K have identicalone on the source planez= d. The evanescent wave function
tion Circle, i.e., for , are decaying exponentially awSince equation (1) has the form of an inverse spatial Fourier t
Wave Spectrum P is a representation of the sound field in th
domain.
We assume that the complex sound pressurep(rn) has been
tions rn (xn,yn, 0) on the measurement plane. We wish to e
p(r) at an arbitrary position r (x,y,z) in the source free regionto estimatep(r) as a linear combination of the measured sound
In order that equation (4) can provide good estimates for al
sources forzd, it must in particular provide good estimate
plane wave functions . If, on the other hand, equation (4)
mates for all , then it provides good estimates for any soun
forzd.
We therefore require formula (4) to provide good estimationof these elementary wave functions:
Solution of this set of linear equations in a least squares se
cients cn, means that we obtain the estimator (4) that is optim
containing only the chosen function sub-set, and with approxim
of each function, i.e., with equal content of a set of spatial fr
K k>
r( ) cn
n 1=
r( ) p rn( )
KK
Km
r( ) cn
n 1=
r( ) Km rn( ) m 1M=,
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This allows (5) to be written as follows:
The regularised least squares solution to (7) is:
where A is the conjugate transpose ofA, I is a unit diagonalregularisation parameter. We now let the numberM, of elementa
used to determine the estimation coefficients increase towards i
the distribution of these wave functions in the Kdomain appr
distribution:
Here, * represents complex conjugate and the integration is owave spectrum domain. Notice that the switch in equations (9) a
representation introduces an identical re-scaling of the matric
This implies a re-scaling of the regularisation parameterh of equ
The matrix AA can be seen as an Auto-correlation matrix fo
urement positions, while A can be seen as containing cross co
the measurement points and the calculation position.The integrals in equations (9) and (10) can be reduced analytic
ofKto polar co-ordinates: K= (kx, ky) = (Kcos(w),Ksin(w)). W
position vectorR (x,y) and let Rn be thexy-component ofrn. F
r( ) Ac r( )
c r( ) AA 2I+( )
1A r( )=
AA[ ]nn' Km*
m rn( )Km rn'( ) 1k2-------- K
*
rn( )= m
A[ ]n Km
*
m
rn( )Km r( ) 1
k2
-------- *
K rn( )K= m
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and further by polar angle integration and use of (3):
where . Equation (10) can be treated in a simClearly, all diagonal elements of the Autocorrelation matri
becauseRnn = 0 for all n, and the value can be shown to be:
To solve for the vector c of prediction coefficients in equat
choose the regularisation parameterh. It is shown in reference
assumptions the optimal value is given by:
where SNR is the effective Signal-to-Noise-Ratio in Decibels
i l ki i ll f
AA[ ]nn'1
k2
--------2 ej k
*z kz( )d
0
J0 KRnn'( )KdK=
2k
2
J00
k
KRnn'( )KdK 2+ k
2
e
2 K2
k2
d
J0k
KRnn'( )KdK=
2J1 kRnn'( )
kRnn'----------------------- 2k
2e
2 K2
k2
d
k
J0 KRnn'( )Kd+=
Rnn' Rn Rn'
AA[ ]nn 11
2 kd( )2
----------------+=
2
1 12 kd( )
2----------------+ 10
SNR
10-----------
=
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Here, p is a vector containing the measured pressure signals,
equation (8). Notice that the vectorpT(AA+h2I)-1 of de-correl
over the microphone positions needs to be calculated only one can be used for calculation of the pressure at many other posit
cross correlation vectors A(r).
The particle velocity can be obtained in the same way as a line
the measured pressure signals. To derive the required estimatio
start with an equation equivalent to (5), but with the particle velo
tary wave functions on the left-hand side. As a result, we obexpression for the particle velocity:
where A is a vector of correlations between the pressure apositions and the particle velocity at the calculation position. Not
pT(AA+h2I)-1 of de-correlated measured pressure data fro
applies also in equation (16).
Based on the sound pressure and the particle velocity, the soun
calculated.
Numerical SimulationsA set of measurements was simulated with the set-up illustrated
The grid represents an 8 8 element microphone array with the microphones being at the corners of the grid. Two coherent i
point sources of equal strength are positioned 6cm below the a
tance that is twice the grid spacing. The positions of the point souin Fig. 2 by black dots. Clearly, the array does not cover the ent
NAH will introduce severe spatial window effects. SONAH calc
formed in the measurement plane (z= 0) and in a plane half way b
l d th t l ( 3 ) Th l l ti
uz r( ) pT
AA 2I+( )
1A r( )
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central section covering the rest. For each section/area the re
level was calculated from the formula:
where the summations are both over the relevant section. A c
definition is that a section with a low level of particle velocity
high relative error level. Fig. 3 shows the relative error levelsplane for the central area, for the edge and for the total area.
the average relative error is seen to be lower than 18 dB over
frequency range from 500 Hz to 5 kHz.
i 4 h h di d f h l l i
Fig.2. Microphone grid and point sources. The grid spacing is 3cm
point sources are 6cm below the array. The left source is 6 cm to the le
050
Lerr 10 log10ui
true
uiui
true 2--------------------------------------
=
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around 3.5 kHz. Above that frequency the estimated power slow
ably because the number of microphones is too small to uniquely
Fig.3. Relative average error level for SONAH calculation of particle vel
ment plane, z=0
Fig. 4. Relative average error level for SONAH calculation of particle ve
tion plane, z= 3 cm
0 500 1000 1500 2000 2500 3000 3500 4000
0
-5
-10
-15
-20
-25
-30
Frequency (Hz)
RelativeError(dB)
0 500 1000 1500 2000 2500 3000 3500 4000
5
0
-5
-10
-15
-20
-25
Frequency (Hz)
RelativeError(dB
)
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simultaneously at all measurement positions. The holography formed through an FFT transform of the full time-section to b
quency domain, followed by NAH or SONAH calculation fo
finally inverse FFT transform back to the time domain, [2]. In
calculation time for the SONAH calculations, matrix interpola
along the frequency axis on the correlation matrices AA, A
few further efforts to reduce computation time, the SONAH caa few times longer than traditional NAH (based on spatial FFT
applied 120-element array.
The example to be presented here was a measurement on
track of a large Caterpillar track-type tractor. The main source
were around the areas where the track passes over the sprocket
and front idlers. We took a measurement with a 10 cm spaced 1positioned over a small Carrier Roller with a relatively low leve
Fig. 6 shows a picture of the measurement area and plots of th
averaged (RMS) particle velocity maps for the frequency band
b d ) Cl l SONAH h h b bili
Fig. 5. True and estimated sound power for the central and the full sec
area at z = 3 cm
Full a
Full a
Centr
Centr
0 500 1000 1500 2000 2500 3000 3500 4000
41
40
39
38
37
36
Frequency (Hz)
Soundpower(dB)
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ConclusionsThe new Statistically Optimal NAH (SONAH) method has been
method performs the plane-to-plane transformation directly in t
avoiding the use of spatial FFT. Careful numerical programmin
tion times only slightly longer than FFT based NAH. Numerica
practical results demonstrate that SONAH opens up a poss
acoustical holography measurements with an array that is smalland still keep errors at an acceptable level.
References
[1] Maynard J. D., Williams E. G., Lee Y., Near-field Acous
I. Theory of Generalized Holography and the DevelopmeJ. Acoust. Soc. Am. 78 (4), 13951413, October 1985.
[2] Hald J., Time Domain Acoustical Holography and
Sound & Vibration 1625 February 2001
Fig.6. Averaged Particle Velocity maps for the 1/12-octave bands 2051
SONAH calculationMeasurement area
i l i d b f
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1 1989 STSF A Unique Technique for Scan Based Near-FieHolography Without Restrictions on Coherence
2 1988 Quantifying Draught Risk1 1988 Using Experimental Modal Analysis to Simulate Struct
ModificationsUse of Operational Deflection Shapes for Noise Contro4 1987 Windows to FFT Analysis (Part II)
Acoustic Calibrator for Intensity Measurement Systems3 1987 Windows to FFT Analysis (Part I)2 1987 Recent Developments in Accelerometer Design
Trends in Accelerometer Calibration
1 1987 Vibration Monitoring of Machines4 1986 Field Measurements of Sound Insulation with a BatteryAnalyzerPressure Microphones for Intensity Measurements withImproved Phase PropertiesMeasurement of Acoustical Distance between IntensityWind and Turbulence Noise of Turbulence Screen, Nos
Intensity Probe with Wind Screen3 1986 A Method of Determining the Modal Frequencies of StCoupled ModesImprovement to Monoreference Modal Data by Addingof Freedom for the Reference
2 1986 Quality in Spectral Match of Photometric TransducersGuide to Lighting of Urban Areas
1 1986 Environmental Noise Measurements
Special technical literature
Previously issued numbers of
Brel & Kjr Technical Review(Continued from cover page 2)
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