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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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