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Page 1
TECHNICAL REVIEW
BV 0056 – 11
ISSN 0007–2621
Beamforming
HEADQUARTERS: DK-2850 Nærum · Denmark
Telephone: +45 4580 0500 · Fax: +45 4580 14 05
www.bksv.com · [email protected]
No.1 2004
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Previously issued numbers of
Brüel & Kjær Technical Review
1 – 2002 A New Design Principle for Triaxial Piezoelectric Accelerometers
Use of FE Models in the Optimisation of Accelerometer Designs
System for Measurement of Microphone Distortion and Linearity from
Medium to Very High Levels
1 – 2001 The Influence of Environmental Conditions on the Pressure Sensitivity of
Measurement Microphones
Reduction of Heat Conduction Error in Microphone Pressure Reciprocity
Calibration
Frequency Response for Measurement Microphones – a Question of
Confidence
Measurement of Microphone Random-incidence and Pressure-field
Responses and Determination of their Uncertainties
1 – 2000 Non-stationary STSF
1 – 1999 Characteristics of the Vold-Kalman Order Tracking Filter
1 – 1998 Danish Primary Laboratory of Acoustics (DPLA) as Part of the National
Metrology Organisation
Pressure Reciprocity Calibration – Instrumentation, Results and Uncertainty
MP.EXE, a Calculation Program for Pressure Reciprocity Calibration of
Microphones
1 – 1997 A New Design Principle for Triaxial Piezoelectric Accelerometers
A Simple QC Test for Knock Sensors
Torsional Operational Deflection Shapes (TODS) Measurements
2 – 1996 Non-stationary Signal Analysis using Wavelet Transform, Short-time
Fourier Transform and Wigner-Ville Distribution
1 – 1996 Calibration Uncertainties & Distortion of Microphones.
Wide Band Intensity Probe. Accelerometer Mounted Resonance Test
2 – 1995 Order Tracking Analysis
1 – 1995 Use of Spatial Transformation of Sound Fields (STSF) Techniques in the
Automative Industry
2 – 1994 The use of Impulse Response Function for Modal Parameter Estimation
Complex Modulus and Damping Measurements using Resonant and Nonresonant Methods (Damping Part II)
1 – 1994 Digital Filter Techniques vs. FFT Techniques for Damping Measurements
(Damping Part I)
2 – 1990 Optical Filters and their Use with the Type 1302 & Type 1306
Photoacoustic Gas Monitors
1 – 1990 The Brüel & Kjær Photoacoustic Transducer System and its Physical
Properties
2 – 1989 STSF — Practical Instrumentation and Application
Digital Filter Analysis: Real-time and Non Real-time Performance
1 – 1989 STSF — A Unique Technique for Scan Based Near-Field Acoustic
Holography Without Restrictions on Coherence
(Continued on cover page 3)
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Technical
Review
No. 1 – 2004
bv005611.book Page iii Tuesday, March 23, 2004 12:46 PM
Contents
Beamforming .......................................................................................................... 1
J.J. Christensen and J. Hald
Copyright © 2004, Brüel & Kjær Sound & Vibration Measurement A/S
All rights reserved. No part of this publication may be reproduced or distributed in any form, or by any
means, without prior written permission of the publishers. For details, contact: Brüel & Kjær
Sound & Vibration Measurement A/S, DK-2850 Nærum, Denmark.
Editor: Harry K. Zaveri
bv005611.book Page 1 Tuesday, March 23, 2004 12:46 PM
Beamforming
by J.J. Christensen and J. Hald
Abstract
This article explains the basic principles of Beamforming, including the main performance parameters Resolution and Sidelobe Level. Special attention is given to
the influence of array design and to cross-spectral beamforming. Different array
designs, including Brüel & Kjær’s newly patented wheel array design, are described
and compared, and the basic principle of Brüel & Kjær’s geometry optimisation
method is outlined. A new, improved version of cross-spectral beamforming used
in Beamforming Software Type 7768 is introduced and its benefits are verified. The
article also provides some guidelines for performing good measurements and
finally, describes a set of measurements representing typical applications.
Résumé
Cet article traite succintement du concept d’imagerie par formation de faisceaux, et
notamment des principaux paramètres essentiels aux performances de l’antenne
que sont la Résolution et le Niveau de lobe latéral. Une attention toute particulière
est portée sur l’influence de la forme de l’antenne et sur la formation de faisceaux
par approche interspectrale. Diverses conceptions d’antennes, dont l’antenne circulaire Brüel & Kjær nouvellement brevetée, y sont présentées et comparées; les principes fondamentaux de la méthode propriétaire d’optimisation géométrique y sont
soulignés. Une nouvelle version amendée de l’approche interspectrale implémentée dans le Logiciel Beamforming Software Type 7768 est également présentée et
ses avantages sont vérifiés. Cet article inventorie par ailleurs les points contribuant
à la réalisation de mesures de qualité, pour conclure par la description d’une série
de mesures se rapportant à des applications typiques.
Zusammenfassung
Dieser Artikel erläutert die Grundprinzipien des Beamforming einschließlich der
Hauptparameter Auflösung und Nebenmaxima (Sidelobe Level). Besondere Aufmerksamkeit wird dem Einfluss der Array-Konstruktion und dem Beamforming
nach dem Kreuzspektrum-Verfahren gewidmet. Es werden verschiedene Array1
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Konstruktionen beschrieben und verglichen, darunter Brüel & Kjærs patentiertes
Wheel Array. Außerdem wird das Grundprinzip der Geometrieoptimierung von
Brüel & Kjær skizziert. Eine neue verbesserte Version des in der Beamforming
Software Typ 7768 verwendeten Beamforming nach dem Kreuzspektrum-Verfahren wird vorgestellt und dessen Vorteile nachgewiesen. Der Artikel enthält auch
Richtlinien zur Durchführung guter Messungen und beschreibt eine Serie von Messungen, die typische Anwendungen repräsentieren.
Introduction
Planar Near-field Acoustical Holography (NAH) is an established technique for
efficient and accurate noise source location [1, 2]. NAH can provide high-resolution source maps on a planar source surface from measurements taken over a regular rectangular grid of points close to the source. The measurement grid must
capture the major part of the sound radiation into a half space and therefore completely cover the noise source plus approximately a 45º solid angle. The grid spacing must be less than half a wavelength at the highest frequency of interest. Thus,
the number of measurement points gets very high when the source is much larger
than the wavelength, which always occurs at sufficiently high frequencies. The
same problem arises when for some reason it is not possible to measure close to
the source. Then, because of the required 45º coverage angle, the measurement
area must be very large. In these cases, beamforming is an attractive alternative.
Beamforming is an array-based measurement technique for sound-source location from medium to long measurement distances. Basically, the source location is
performed by estimating the amplitudes of plane (or spherical) waves incident
towards the array from a chosen set of directions. The angular resolution is
inversely proportional to the array diameter measured in units of wavelength, so
the array should be much larger than wavelength to get a fine angular resolution. At
low frequencies, this requirement usually cannot be met, so here the resolution will
be poor. Unlike NAH, beamforming does not require the array to be larger than the
sound source. For typical, irregular array designs, the beamforming method does
not allow the measurement distance to be much smaller than the array diameter. On
the other hand, the measurement distance should be kept as small as possible to
achieve the finest possible resolution on the source surface.
An important difference between beamforming and NAH is that beamforming
can use irregular array geometries, for example, random array geometries. The use
of a discrete set of measurement points on a plane can be seen as a spatial sampling
2
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of the sound field. NAH requires a regular, rectangular grid of points in order to
apply a 2D spatial DFT. Outside the near-field region, such a regular grid will suppress spatial aliasing effects very well, if the grid spacing is just less than half a
wavelength. When the grid spacing exceeds half a wavelength, spatial aliasing
components quickly get very disturbing. Irregular arrays on the other hand can
potentially provide a much smoother transition: spatial aliasing effects can be kept
at an acceptable level up to a much higher frequency with the same average spatial
sampling density. This indicates why beamforming can measure up to high frequencies with a fairly low number of microphones.
Theory
Delay-And-Sum Beamforming for Infinite Focus Distance
The principle of Beamforming is best introduced through a description of the basic
Delay-and-Sum beamformer. As illustrated in Fig. 1, we consider a planar array of
M microphones at locations rm (m = 1, 2, …, M) in the xy-plane of our coordinate
system. When such an array is applied for Delay-and-Sum Beamforming, the
measured pressure signals pm are individually delayed and then summed [3]:
Fig. 1. (a) A microphone array, a far-field focus direction, and a plane wave incident from the
focus direction. (b) A typical directional sensitivity diagram with a main lobe in the focus direction and lower sidelobes in other directions
Plane wave
0º
Sidelobe – 30º
30 dB
30º
Main lobe
20 dB
– ␬
– 60º
10 dB
60º
␬
–90º
(a)
90º
(b)
040009
3
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M
b ( ␬, t ) =
∑ wm pm ( t – ∆m ( ␬ ) )
(1)
m=1
where wm are a set of weighting or shading coefficients applied to the individual
microphone signals. The individual time delays ∆m are chosen with the aim of
achieving selective directional sensitivity in a specific direction, characterised
here by a unit vector ␬. This objective is met by adjusting the time delays in such a
way that signals associated with a plane wave, incident from the direction ␬, will
be aligned in time before they are summed. Geometrical considerations (Fig. 1)
show that this can be obtained by choosing:
␬⋅r
(2)
∆ m = -------------mc
where c is the propagation speed of sound. Signals arriving from other far-field
directions will not be aligned before the summation, and therefore they will not
add up coherently. Thus, we have obtained a directional sensitivity, as illustrated
in Fig. 1(b).
The frequency domain version of eq. (1) for the Delay-and-Sum beamformer
output is:
B ( ␬, x ) =
M
∑
m=1
w m P m ( x )e
– jx∆ m ( ␬ )
M
=
w m P m ( x )e
∑
m=1
jk ⋅ r m
(3)
Here, x is the temporal angular frequency, k ≡ –k␬ is the wave number vector of a
plane wave incident from the direction ␬ in which the array is focused (see Fig. 1),
and k = x/c is the wave number. In eq. (3) an implicit time factor equal to e jxt is
assumed. Because k ≡ –k␬, we can write B(k, x) instead of B(␬, x).
Through our choice of time delays ∆m(␬), or equivalently of the “preferred”
wave number vector k ≡ –k␬, we have “tuned” the beamformer on the far-field
direction ␬. Ideally, we would like to measure only signals arriving from that direction, in order to get a perfect localisation of the sound sources. To investigate, how
much “leakage” we will get from plane waves incident from other directions, we
now assume a plane wave incident with a wave number vector k0 different from the
preferred k ≡ –k␬, Fig. 2. The pressure measured by the microphones will then be:
Pm ( x ) = P0 e
– jk 0 ⋅ r m
(4)
which, according to eq. (3), will give the following output from the beamformer:
4
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Fig. 2. A plane wave, with wave number vector k0, incident from a direction different from the
focus direction ␬. For a planar array the beamformer output, eq. (5), is a function of the difference K of the projections kˆ 0 and kˆ of the wave number vectors k0 and k onto the plane
defined by the array
–
≡– ␬
␬
^
^
B ( ␬, x ) = P 0
M
∑ wm e
m=1
j ( k – k0 ) ⋅ rm
^
≡ P0 W ( k – k0 )
040010
(5)
Here, the function W,
W(K) ≡
M
∑ wm e
m=1
jK ⋅ r m
(6)
is the so-called Array Pattern. It has the form of a generalised spatial DFT of the
weighting function w, which equals zero outside the array area. In the case of uniform shading, wm ≡ 1, the array pattern, eq. (6), depends only on the array geometry. In the following we shall mainly be concerned with uniform shading and will
5
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consequently omit the wm term from the equations. The effect of non-uniform
shading is discussed in “Regular Arrays” on page 19.
Because the microphone positions rm have z-coordinate equal to zero, the Array
Pattern, eq. (6), is independent of Kz. We shall therefore consider the Array Pattern
W only in the (Kx , Ky) plane, that is, we consider the projections of the wave
number vectors onto that plane, Fig. 2. There, W has an area with high values
around the origin, with a peak value equal to M at (Kx , Ky) = (0, 0). According to
eq. (5), this peak represents the high sensitivity to plane waves coming from the
direction ␬, in which the array is focused. Fig. 1 contains an illustration of that
peak, which is called the main lobe. Other directional peaks are called sidelobes.
Fig. 1 contains information about only a single frequency and a single focus direction. Eq. (5) shows that the array pattern, eq. (6), contains information about the
sidelobe structure for all focus directions and all frequencies. Fig. 3 depicts a section of the array pattern corresponding to the array geometry depicted in Fig. 4.
Fig. 3. Contour plot of the Array Pattern, eq. (6), corresponding to the array depicted in Fig. 4.
For a given upper frequency, xmax, of the array’s intended use, all sidelobes between the
90°
0
main lobe radius Kmin
and the circle Kmax = 2xmax/c must be taken into account. When beamformer focusing is restricted to be within 30° of the array axis, only sidelobes for wave num30°
bers below Kmax = (3/2)xmax/c are relevant
– 20
– 40
–
6
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Fig. 4. An example of a planar 66-channel beamformer array and its equivalent circular
aperture (---). The microphone positions (●) are randomly distributed over the circular
aperture
A sidelobe (a peak in the array pattern different from the peak at the origin) will,
according to eq. (5), have the effect that a single incident plane wave will be measured with a significant level when focusing in a specific direction not equal to the
direction from which the plane wave is actually incident. The sidelobes will therefore create so-called Ghost Images in measured directional source maps. The level
of the sidelobes relative to the main lobe (the dynamic range) defines the ability of
the beamformer to suppress ghost images.
Normalisation
In the case of perfect focus, k = k0, the delay-and-sum beamformer, eq. (5), amplifies the input signal by the number of channels: B(–k0/k, x) = MP0. For this reason
the delay-and-sum beamformer is often normalised by the number of channels:
1
B ( ␬, x ) = ---M
M
P m ( x )e
∑
m=1
jk ⋅ r m
(7)
7
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Resolution
The resolution of a beamformer describes its ability to distinguish waves incident
from directions close to each other. When focusing on sources in the far field, resolution is the smallest angular separation between two plane waves that allows
them to be separated, and for sources at a finite distance a practical definition of
resolution is the minimum distance between two sources such that they can be separated.
Consider two plane waves with wave number vectors k1and k2, |k1| = |k2| = k,
incident on a beamformer array with array pattern W. Assuming unity amplitude for
both plane waves, the beamformer output is a superposition of the form:
Fig. 5. The curves show the beamformer output, B(␬,x), cf. eq. (8) resulting from two plane
waves with wave number vectors k1 and k2 incident on a planar array. The Rayleigh criterion
for resolution states that the two directions can be resolved when the peak of the shifted array
pattern W(k – k2) corresponding to the plane wave with angle of incidence, h, falls on the first
zero (or minimum) of W(k – k1)
h
(
–
(␬, x)
h
h
h
8
h
)
bv005611.book Page 9 Tuesday, March 23, 2004 12:46 PM
B(␬, x) = W(k – k1) + W(k – k2)
(8)
cf. eq. (5). The Rayleigh criterion [3] states that the two directions can be just
exactly resolved when the peak of W(k – k2) falls on the first zero of W(k – k1), cf.
Fig. 5. Assuming that the required angular separation between k1and k2 is small, it
can be shown (see “Appendix: Resolution” on page 47) that at finite distance, z,
the minimum resolvable source separation in the radial direction, R(h), is given
by:
zR
1
R ( h ) = --------K- ------------k cos 3 h
(9)
where RK is the main lobe width in the array pattern and h is the off-axis angle.
The value of RK is, according to the Rayleigh criterion, given by the first null
(minimum), K0min, of the array pattern: RK = K0min . The exact value depends on the
positions of all the array microphones through eq. (6), but a good general estimate
can be calculated by considering the limiting cases where we have an infinite
number of transducers uniformly distributed over a line segment of length D or a
circular disc with radius D/2. In other words, we imagine we are able to sample the
sound field at all points within an area (aperture) instead of only at a few discrete
positions. In this continuous case we should use an integral expression for the
array pattern, eq. (6), the aperture smoothing function:
jK ⋅ r d
1
W ( K ) = -------------d r
w ( r )e
d
( 2p ) r < D ⁄ 2
∫
(10)
where d = 1 for the line segment, d = 2 for the circular aperture and w(r) is now a
continuous shading function. In the case of uniform shading, eq. (10) can be evaluated to, J1 being the Bessel function of order 1 [3]:
sin ( K x D ⁄ 2 )
W ( K x ) = ------------------------------,
Kx ⁄ 2
d = 1
(11)
2
2
K = Kx + Ky ,
d = 2
(12)
W ( K ) = pD
-------- J 1 ( KD ⁄ 2 ),
K
From eq. (11) and eq. (12) we find that the first zero in the array pattern corresponding to the line segment and the circular aperture occurs at:
0
2p
K min = a -----D
(13)
9
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where a =1 for the linear aperture and a ≈ 1.22 for the circular aperture. Now,
using the fact that the wave number k is related to the wavelength, k, by k = 2p/k
we obtain by insertion into eq. (9) the desired expression for beamformer resolution:
a z
(14)
R ( h ) = ------------- ---- k
3
cos h D
For on-axis incidence, h = 0, the resolution is given by:
z
R Axis = a  ---- k
 D
(15)
We notice that the resolution is proportional to the wavelength and becomes better with larger aperture size, but worse with increasing array to object distance.
This relation is not limited to acoustics; the reader may be familiar with the fact that
the ability of an optical camera to resolve details depends on the lens diameter and
the distance to the object.
Comparing the on-axis and general off-axis resolution, eq. (15) and eq. (14), we
notice that the ratio between them is given by:
1
R(h)
(16)
-------------- = ------------3
R Axis
cos h
This ratio is depicted in Fig. 6 and we observe that for angles of incidence more
than 30° off-axis, the resolution becomes more than 50% greater than the on-axis
resolution. For this reason the useful beamformer opening angle is in practice
restricted to 30°.
h
Fig. 6. The variation of the ratio between off-axis and on-axis resolution as given by eq. (16)
h
10
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Maximum Sidelobe Levels
The presence of sidelobes in the directivity pattern (Fig. 1) or, correspondingly, in
the array pattern, will cause waves from non-focus directions to leak into the
measurement of the main lobe direction ␬. This will produce false peaks/sources
in a measured directional source map. A good phased array design can therefore
be characterised by having low Maximum Sidelobe Level (MSL), measured relative to the main lobe level. We define the radial profile of the array pattern, Wp(K),
by:
2
W p ( K ) ≡ 10 ⋅ log 10 max W ( K ) ⁄ M
2
(17)
K = K
and based on this profile we define the Maximum Sidelobe Level function as:
MSL ( K ) ≡
max
0
< K′ ≤ K
K
min
W p ( K′ ) = 10 ⋅ log 10
max
K
2
W(K) ⁄ M
2
(18)
0
< K ≤K
min
where K0min is given by eq. (13). A comparison of the radial profile, Wp, for a particular discrete array and its corresponding circular aperture is given in Fig. 7. The
MSL function for the same discrete array is shown in Fig. 8.
For practical use of the array pattern, its radial profile and the MSL function, it is
important to know which part of the array pattern is “active”, when the array is
used at a specific frequency. With reference to eq. (5), the argument to the array
pattern is the difference vector k(␬) – k0 between the wave number vector k of the
in-focus plane wave and the wave number vector k0 of the incident plane wave.
Both k and k0 have length equal to the wave number k, and the difference vector
has maximum length when the two plane waves have opposite directions – pointing, for example, in the positive and negative x-axis directions – meaning that
|k – k0| ≤ 2k (see Fig. 2). Therefore, at a given frequency x, only the section
|K| ≤ 2k = 2x/c of the Array Pattern will be “visible”. If we restrict the focus direction ␬ to be within an angle h from the array axis, then the maximum length of the
difference vector reduces, so only the section K ≤ K h ( x ) of the array pattern
max
will be visible with:
h ( x ) ≡ [ 1 + sin ( h ) ] x
(19)
K max
---c
If the Array Pattern has low MSL for |K| ≤ Kmax, then a beamformer application
will provide accurate directional source maps with a low degree of false images up
to the frequency xmax given by (Fig. 3):
11
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Fig. 7. Comparison of the aperture smoothing function, eq. (12), for a circular aperture and
the radial array pattern profile of eq. (18) for a “circular” 66-channel discrete array with the
same diameter (Fig. 4). The aperture smoothing function of the uniformly shaded circular
aperture represents the, theoretically, best sidelobe suppression attainable. Due to the finite
number of points sampled with the discrete array, the sidelobe levels are much higher. The
main lobe widths of the discrete array and the equivalent circular aperture are nearly identical
as this quantity is determined by the aperture size
K max c
(20)
x max ( h ) = -----------------------1 + sin ( h )
As an example, if the beamformer will be focused on directions ␬ not more than
30° off-axis, then the upper limiting frequency becomes xmax(30º) = 2/3Kmaxc
(Fig. 3). We notice the general relation:
K h ( x max ) = K max
max
⇒
4
x max ( 30° ) = --- x max ( 90° )
3
12
(21)
bv005611.book Page 13 Tuesday, March 23, 2004 12:46 PM
Fig. 8. The sidelobe level profile, Wp, eq. (17) and the maximum sidelobe level function
MSL(K), eq. (18), for the array depicted in Fig. 4. The functions are extracted from the corresponding array pattern of Fig. 3
Eq. (19) is a linear relation between frequency x and the highest array pattern
h ( x ) that is active at that frequency. The argument K of
wave number K = K max
the MSL function, eq. (18), is exactly an upper limiting wave number in the array
h ( x ) as argument in the
pattern, so therefore it is straightforward to use K = K max
MSL function. Thereby we have expressed the MSL as a function of frequency x:
h ( x )] . In order to calculate the MSL at a given frequency, one needs to
MSL[K max
specify the maximum focusing angle h. We have chosen to always show or specify
90° ( x )] , that is, no restriction on focus direction.
the worst case MSL[K max
Cross-spectral Formulation with Exclusion of Autospectra
For stationary sound fields it is natural to operate with averaged cross- and autospectra, and it turns out that exclusion of autospectra in beamforming calculations
13
bv005611.book Page 14 Tuesday, March 23, 2004 12:46 PM
is advantageous in several respects. The average power output from the Delayand-Sum beamformer can be derived from eq. (3):
V ( ␬, x ) ≡ B ( ␬, x )
2
M
=
∑ Pm ( x )Pn * ( x )e
m, n = 1
M
=
∑
C nm ( x ) e
jk ⋅ ( r m – r n )
jk ⋅( r m – r n )
(* indicates complex conjugate)
(22)
m, n = 1
where we have introduced the cross-spectrum matrix:
C nm ( x ) ≡ P m ( x )P n * ( x )
We may split eq. (22) into an autospectrum part and a cross-spectrum part:
V ( ␬, x ) =
M
M
∑
m=1
C mm +
∑
m≠n
C nm e
jk ⋅( r m – r n )
(23)
Here, the autospectra Cmm will contain self-noise from the individual channels,
such as wind-noise and electronic noise from the data acquisition hardware. For
that reason it would be desirable to omit the first sum in eq. (23). Ideally, the crossspectra Cnm, m ≠ n, are not affected by the self-noise, because the self-noise in one
channel is generally incoherent with the self-noise in any other channel. Under
that condition, averaging will suppress contributions from self-noise in the crossspectra. We can assess the effect of excluding the autospectra by considering the
plane wave response of the cross-spectral beamformer. For a plane wave with
wave number vector k0 and amplitude P0 the pressure recorded by the mth microphone is Pm = P0exp(–jk0rm). Insertion of this in eq. (22) leads to the following
expression for the beamformer power output:
V ( ␬, x ) = P 0
= P0
M
∑e
m, n = 1
2
M
∑e
m, n = 1
2
– j k 0 ⋅( r m – r n ) jk ⋅ ( r m – r n )
e
j ( k – k0 ) ⋅ ( rm – rn )
2
= P0 U ( k – k0 )
where we have introduced the Power Array Pattern:
14
(24)
bv005611.book Page 15 Tuesday, March 23, 2004 12:46 PM
U(K) ≡ W(K)
2
M
=
∑ e
m, n = 1
jK ⋅ ( r m – r n )
(25)
In a similar way we see that the self-term-free versions of the power array pattern, eq. (25), and the cross-spectral beamformer response, eq. (23):
U′ ( K ) ≡
M
∑
m≠n
e
jK ⋅ ( r m – r n )
and
V′ ( ␬, x ) ≡
M
∑ Cnm e
m≠n
jk ⋅ ( r m – r n )
(26)
are for plane waves related by:
V′(␬, x) = |P0|2U′(k – k0)
(27)
Thus, removal of the autospectral terms from the cross-spectral beamformer,
eq. (23), corresponds to omitting the self-terms from the definition of the power
array pattern, eq. (25). Provided the reduced array pattern U′ has lower sidelobe
level than U, we can therefore reduce the level of ghost images in cross-spectral
beamformer output by omitting the autospectra.
Comparing the definitions of the array pattern U and the reduced version U′ we
find that:
U′(K) = U(K) – M
(28)
M2
M2
The main lobe is therefore reduced from
(for U) to
– M (for U′), and the
highest sidelobe is reduced from M 2·10MSL/10 to M 2·10MSL/10 – M. Assuming first
that U′ does not become negative, this leads to the following Maximum Sidelobe
Level for U′:
MSL ⁄ 10
 M 2 ⋅ 10 MSL ⁄ 10 – M
M ⋅ 10
–1
MSL′ = 10 ⋅ log 10  ----------------------------------------------- = 10 ⋅ log 10  ------------------------------------------


2
M
–
1


M –M
which is easily shown to be always smaller (better) than MSL. As an example, for
the 66-channel array depicted in Fig. 4, the MSL equals –9.5 dB over a wide frequency range. Over that frequency range MSL′ equals –10.1 dB, meaning that the
highest sidelobe has been reduced by 0.6 dB (Fig. 8). At lower frequencies the gain
is bigger. If the power array pattern U contains values less than M, then the
reduced array pattern U′ will have areas with negative values. The worst case is
when U has a null. In that case the minimum value of U′ equals –M, which will
have the same effect as a sidelobe with amplitude equal to M. Such a sidelobe will
not affect MSL′ as long as M is smaller than M2·10MSL/10 – M. This condition has
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bv005611.book Page 16 Tuesday, March 23, 2004 12:46 PM
been fulfilled for all the arrays that we have been designing. Additionally, this
worst-case condition will not occur, when only array geometries without redundant spacing vectors are used.
Finite Focus Distance
Up to now we have only considered the resolution of incoming plane waves, corresponding to point sources at infinite distance. The time delays ∆m of eq. (2) were
chosen with the aim of aligning in time the signals of a plane wave arriving from
the far-field direction ␬ before the summation of the Delay-And-Sum beamforming, eq. (1). Use of a plane wave for calculation of the delays corresponds to focusing of the array at infinite distance in the chosen direction. To focus on a point
source at a finite distance, the delays should align in time the signals of a spherical
wave radiated from the focus point.
The expression for Delay-And-Sum beamforming for focusing at a point r at a
finite distance becomes:
B ( r, x ) =
M
∑ Pm ( x )e
m=1
– j x∆ m ( r )
(29)
Here we have replaced the delays, eq. (2), with the form:
r – rm ( r )
(30)
∆ m ( r ) = -----------------------c
where rm(r) ≡ |r – rm| is the distance from microphone m to the focus point, Fig. 9.
The near-field version of eq. (23) for the beamformer power output appears as:
V ( r, x ) =
M
M
∑
m=1
C mm +
∑ Cnm e
m≠n
jx [ ∆ n ( r ) – ∆ m ( r ) ]
(31)
Cross-spectral Imaging Function
Eq. (31) for finite-distance beamforming contains no compensation for the fact
that different positions on the assumed source plane have different distances to the
array transducers and therefore are attenuated by different amounts. For a single
source at ri, a possible correction could be to replace the cross-spectrum matrix by
the scaled version Cnmrm(ri)rn(ri). The introduction of a scaled cross-spectrum
matrix is, however, an ad-hoc correction with uncontrolled effects. A sound
approach can be achieved by assuming a model where the recorded sound field is
generated by a monopole distribution. For each position on the source plane the
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bv005611.book Page 17 Tuesday, March 23, 2004 12:46 PM
Fig. 9. In near-field focusing, spherical waves emitted by a monopole source at the focus
point r are assumed. Signal delays are computed according to eq. (30)
estimated source strength reflects how well the sound field from a monopole point
source at that position fits the sound field measured by the array.
Let rm, m = 1, …, M, be the transducer coordinates and let r be the position of a
monopole. The pressure, Pm, recorded by the mth transducer is then given by
Pm(r) ≡ P0vm(r) = P0v(rm – r), where P0 is the source strength and v(r) is the steering vector given by:
v(r) = e–jk|r|/|r|
(32)
and the cross-spectrum,
C mod
nm ,
between channel m and n is:
*
*
C mod
(33)
nm = Pn P m = avn ( r )v m ( r )
where a is a real amplitude coefficient. Then we define an error function, E(a, r),
between the model cross-spectra and the measured cross-spectra, Cnm:
E ( a, r ) =
M
∑
m, n = 1
C nm – C mod
nm
2
M
=
C nm – avn* ( r )v m ( r )
∑
m, n = 1
2
(34)
As shown in “Appendix: The Cross-spectral Imaging Function” on page 43, the
minimisation of this error function corresponds to the maximisation of the Crossspectral Imaging Function defined by:
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bv005611.book Page 18 Tuesday, March 23, 2004 12:46 PM
M
C nm ( x )v n ( r )vm* ( r )
∑
m, n = 1
2
1
I ( x, r ) ≡ ---- -------------------------------------------------------------------M
M
2
2
vn ( r ) vm ( r )
(35)
∑
m, n = 1
In practice I 2(x, r) is computed over a discrete mesh covering the focus area. In
the resulting map, peaks are interpreted as areas with a high probability of finding
a source. See Fig. 17 and Fig. 18. As also discussed in “Appendix: The Crossspectral Imaging Function” on page 43, this interpretation is justified by the fact
that in the far-field limit I2(x, r) is identical to the mean square value of the delayand-sum expression, eq. (7). Due to this connection with the plane wave case, we
can expect improved sidelobe levels from the self-term-free version of the imaging function, eq. (35):
M
C nm ( x )v n ( r )vm* ( r )
∑
m≠n
2
1
J ( x, r ) ≡ --------------------------- -------------------------------------------------------------M(M – 1)
M
2
2
vn ( r ) vm ( r )
(36)
∑
m≠n
where the modified normalisation factor reflects the omission of the diagonal
terms in the cross-spectral matrix. This modified normalisation factor ensures that
the imaging functions I and J will be identical when focusing on a far-field point
source on the array axis. The benefit of autospectra exclusion in the cross-spectral
imaging function is illustrated in Fig. 10.
Array Design
The performance of a beamformer array is to a very large extent determined by the
array geometry because this defines the beamformer response through the array
pattern, cf. eq. (5). From the array pattern we can extract the maximum sidelobe
level profile, which defines the ability to suppress ghost images as a function of
frequency. This enables us to investigate the properties of a given array design.
The reverse problem: how to design an array with a desired usable frequency
range and resolution is more complicated. In this section we will review a number
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Fig. 10. Comparison of the output of three different beamforming algorithms for a configuration with two incoherent 3 kHz monopole sources of equal strength. The data were generated
using the array shown in Fig. 4. In the legend I refers to the full cross-spectral imaging function, eq. (35), J is the cross-spectral imaging function, eq. (36), which excludes the autospectra, and B is the delay-and-sum algorithm, eq. (29). All curves are normalised to 0 dB at
maximum
of array designs including both previously published, regular and irregular array
designs and novel, numerically optimised array geometries, Fig. 11.
As we shall see below, the maximum sidelobe levels can be improved by shading, i.e., application of a smooth spatial window function. This improvement is,
however, obtained at the expense of decreased resolution ability.
Regular Arrays
The simplest example of a regular array is the uniform line array (ULA), which is
a one-dimensional linear array with equidistant microphone spacing. Though we
are mainly interested in planar arrays, this array is well suited to demonstrate a
number of important features of regular beamformer arrays. The microphone coor19
bv005611.book Page 20 Tuesday, March 23, 2004 12:46 PM
Fig. 11. Examples of regular and irregular array configurations. (a) 65-ch. cross-array, (b) 64ch. grid array, (c) 66-ch. optimised random array, (d) 66-ch. Archimedean spiral array, (e) 66ch. optimised wheel array, (f) 66-ch. optimised half-wheel array
(a)
(b)
(c)
0.5
0.4
0
y [m]
0.2
y [m]
y [m]
0.2
0
0
−0.2
−0.2
−0.4
−0.2
0
x [m]
0.2
−0.5
−0.5
(d)
0
x [m]
0.5
0
x [m]
0.5
(f)
(e)
0.4
1
0.5
0.8
y [m]
0
y [m]
0.2
y [m]
−0.5
0
−0.2
0.6
0.4
0.2
−0.4
−0.4 −0.2
0
0.2
x [m]
0.4
−0.5
−0.5
0
x [m]
0.5
0
−0.5
0
x [m]
0.5
040019
dinates, xm, of a ULA with microphone spacing d and M ≡ 2M½+1 microphones
can be written as:
xm = (m – M½)d,
m = 0, …, M – 1
(37)
In the case of uniform shading, the corresponding array pattern [3] can be evaluated
to:
sin ( MKd ⁄ 2 )
W ( K ) = -------------------------------sin ( Kd ⁄ 2 )
(38)
We notice that eq. (38) is a periodic function of K, the period equalling 2p/d. In
addition to the main lobe at K = 0, the ULA array pattern exhibits repetitions of the
main lobe, so-called grating lobes, at the positions K = p(2p/d), p = ±1, ±2, …,
Fig. 12(a).
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Fig. 12. Graph (a) illustrates the side-lobe structure of a regular array with grid spacing d. In
addition to the main lobe at K = 0, grating lobes occur at Kx = ± 2pKN, KN = p/d, where p is a
positive integer. Also shown is the wave number vector k0 at incidence angle 30°, and the
focus direction wave number vector k in the direction –30°. In any case |k| = |k0| = k and we
choose a frequency such that k = 2KN. The projection kx0 of k0 onto the x-axis equals the
Nyquist wave number KN, and for this reason both the main lobe and the grating lobe at 2KN
contributes when beamforming is performed in the entire visible region (indicated by the
green line) – KN < kx < 3KN, where kx is the projection of k on to the x-axis. In the resulting
directional source map (b) a ghost source is seen at h = –30° in addition to the true source at
h = 30°. The directional source map (c) illustrates the situation for on-axis incidence at the
same frequency
(a)
–
2
0
–
0º
(b)
– 30º
0º
(c)
30º
– 30º
30º
30 dB
– 60º
– 90º
60º
10 dB
20 dB
– 60º
30 dB
90º
040021
20 dB 60º
10 dB
– 90º
90º
040022
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Plots of the ULA array pattern are depicted in Fig. 13 for both uniform shading
and shading with a Hamming window. Comparing the two curves we see that the
effect of the windowing is to lower the sidelobe levels and broaden the main lobe,
that is, the dynamic range of the beamformer is improved at the cost of resolution.
Fig. 13. Plot of the array pattern for a uniform linear array with uniform shading and with a
Hamming window applied. The effect of the window is to lower the sidelobes at the expense
of widening the main lobe
(m – 1)
Spatial Aliasing
When sampling a time domain signal with a constant sampling rate fs = 1/Ts, the
highest frequency that can be unambiguously reconstructed is given by the
Nyquist frequency, fN = fs/2 = 1/(2Ts), or the corresponding angular frequency,
xN = 2pfN = p/Ts with a period of TN = 2Ts [4]. Similarly, when spatially sampling
a signal with a sampling interval equal to d, then the spatial Nyquist angular frequency (the Nyquist wave number) is KN = p/d with a period length equal to 2d.
Plane waves with wavelength shorter than kmin = 2d therefore cannot be unambig22
bv005611.book Page 23 Tuesday, March 23, 2004 12:46 PM
uously reconstructed from the spatial samples, implying that the highest frequency
fmax with sufficient sampling density is:
c
c
f max = ---------- = -----2d
k min
(39)
In general, time signals with frequencies f ± pfs, p = 1, 2, …, cannot be distinguished when sampled with a sampling frequency fs = 2fN, and therefore they will
all contribute when we estimate the content of, for example, the frequency f. Similarly, plane waves with angular frequencies K ± pKs, p = 1, 2, …, on the measurement plane cannot be distinguished when sampled with a spatial angular sampling
frequency Ks = 2KN, and consequently they will all contribute when we estimate
the content of any one of them in a spatially sampled sound field.
Typically, when aliasing occurs in the processing of a time domain signal, then
there is a frequency component f > fN that is under-sampled and will contribute at a
single frequency f + pfs in the “visible frequency range”, –fN ≤ f + pfs ≤ fN, p being
an integer. The term “visible” means that only frequency components in that range
will be estimated and used. Aliasing in beamforming happens in the same way, but
here the “visible wave number range” is controlled by the beamforming algorithm
and will not be restricted in the same way to –KN ≤ K + pKs ≤ KN. As stated in
eq. (19), the visible range will be up to –2k ≤ K + pKs ≤ 2k, and therefore there may
be several aliased components. The effect is that an angle h of plane wave incidence will contribute at several aliased angles ha. This is illustrated in Fig. 12(a)
where we consider measurement with a regular array with grid spacing d at a temporal frequency equal to twice the maximum frequency, eq. (39), for the array. The
wave number vector k0 of a plane incident wave is therefore twice as long as the
Nyquist wave number: |k0| = k = 2KN, KN = p/d. We consider a plane wave incident
in the xz-plane at an angle 30° from the array axis, meaning that the projection of k0
on the array plane has length equal to the Nyquist wave number: kx0 = –KN.
According to eq. (5), the output from the beamformer will be given by
B(kx, x) = P0W(kx – kx0) when we focus on a plane wave incident with wave number
vector k with x-component kx. When focusing is scanned over all possible incident
plane waves with wave number vectors k in the xz-plane, |k| = k, then kx will scan
the interval from –k = –2KN to k = 2KN, meaning that the argument Kx = kx – kx0 to
the array pattern will scan the interval from –KN = p/d to 3KN = 3p/d. By inspection
of the array pattern in Fig. 12(a) we see that maximum output will be obtained both
at the main lobe (Kx = 0) for kx = kx0 = – KN and at the first grating lobe (Kx = 2KN)
for kx = –kx0 = KN. In the resulting directional source map, Fig. 12(b), the plane
wave incident at h = +30° contributes also as an aliasing component at ha = –30°.
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For on-axis incidence at the same frequency, the resulting directional source map is
even more confusing since in addition to the main lobe, two grating lobes of the
array pattern at Kx = ±2KN are included, Fig. 12(c).
The Grid Array
A straightforward realisation of a two-dimensional array is the grid array with
microphone spacing d. Fig. 11(b) shows an example of an 8 × 8 grid array with
microphone spacing d = 1/7 m, and hence the highest frequency is fmax ≈ 1.2 kHz.
The maximum sidelobe level for this grid array is depicted in Fig. 14 as a function
of frequency. The sharp cut-off at the maximum frequency is a characteristic of
regular arrays. When such an array is applied for beamforming at frequencies
above fmax, the grating lobes may show up in the source map as false sources, socalled ghost images. The problem is illustrated in Fig. 17(c) for a single 5 kHz
monopole and in Fig. 18(c) for a four-source configuration of 5 kHz monopoles.
Clearly, the grating lobes introduce disturbing ghost images in the source map.
Fig. 14. Comparison of the Maximum Sidelobe Levels for the 1 m arrays depicted in Fig. 11.
The Maximum Sidelobe Level function MSL(K), cf. eq. (18), is here shown as a function of frequency, f, which is related to the wave number K by f = Kc/4p, cf. eq. (19)
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bv005611.book Page 25 Tuesday, March 23, 2004 12:46 PM
The Cross-array
The grid array can only be designed for higher frequencies by decreasing the grid
spacing. For a fixed aperture size this is costly in terms of transducers and the
resulting array may lose its acoustic transparency. With a given number of transducers and desired aperture size, an efficient way of constructing a regular array
with large usable bandwidth is the cross-array, Fig. 11(a), which is a combination
of two uniform linear arrays. If D is the aperture size and M is the number of transducers, the microphone spacing is approximately D/( M – 1) in the grid array and
approximately 2D/(M –1) in the cross-array. Thus the maximum frequency is a
factor ( M + 1)/2 higher for the cross-array than for the grid array.
The cross-array array pattern exhibits high sidelobes along the directions defined
by the constituting line arrays and good sidelobe suppression in all other directions,
Fig. 17(a). In beamforming source reconstruction using the cross-array, the
“ridges” protruding from the position of each source may interfere constructively
and produce ghost images, Fig. 18(a).
Cross-array Beamforming Algorithm
The ghost-image problem caused by the structure of the array pattern can to some
extent be circumvented by processing each line array separately and then combining the results [5]. Formally, the source mapping is obtained by taking the geometric mean BX-array(r, x) of the beamformer outputs B1(r, x), B2(r, x) from the two
Hanning weighted line arrays:
B X-array ( r, x ) =
B 1 ( r, x )B 2 ( r, x )
(40)
The result is a much improved sidelobe structure for single-source focusing,
Fig. 17(b). Problems with ghost images for multi-source configurations cannot,
however, be avoided, Fig. 18(b). Resolution is degraded compared to standard
Delay-and-Sum or Cross-spectral beamforming because of the applied Hanning
weighting.
Irregular Arrays
The major limitation of regular arrays is the previously described aliasing problem
introduced by the repeated sampling spacing. This severe aliasing, producing
ghost images of the same level as the true sources, can be avoided when the array
geometry is totally non-redundant, that is, no difference vector between any two
transducer positions is repeated. For non-redundant arrays, which typically have
25
bv005611.book Page 26 Tuesday, March 23, 2004 12:46 PM
an irregular or random geometry, the sidelobe structure does not exhibit the sharp
cut-off frequency we have encountered for the regular arrays. Instead, the array
patterns of irregular arrays have, in general, gradually increasing maximum
sidelobe levels, Fig. 14.
In general, irregular (non-redundant) arrays outperform traditional regular array
designs, but it is difficult to find out how the design should me made (or modified)
to obtain high performance. Therefore, when designing irregular arrays for a given
frequency range, one often resorts to a tedious trial and error cycle. The performance of parametric irregular arrays, for example, array geometries based on one or
several concentric logarithmic spirals [6], or on an Archimedean spiral [7],
Fig. 11(d), are easier to investigate because the range of a single or a few parameters can be tested systematically. In many cases, though, the MSL as a function of
the relevant design parameter(s) exhibits highly erratic behaviour.
Another complication with irregular arrays is that, due to their complicated
geometry, the transducer support structure can be difficult to realise. Both the support structure and the cabling are complicated and, as a consequence, operation in a
practical measurement situation is difficult or tedious. Also, the need for high resolution at large measurement distances can only be met with relatively large dimensions of the arrays. Thus, an array with a diameter of several metres is often
required. In connection with outdoor applications it is therefore of practical importance that the array construction allows for easy assembly and disassembly at the
site of use, and for easy transport.
Optimised Arrays
An alternative approach to the problem of designing irregular arrays with a wellcontrolled performance is to use numerically optimised array geometries. Specifically, an array can be optimised to a given frequency range by adjusting the transducer coordinates so that the maximum sidelobes are minimised over the
frequency range of the array’s intended use:
minimise MSL ( K )
{ rm }
for
K < K max
(41)
where Kmax is determined by the desired upper frequency and applied opening
angle, cf. “Maximum Sidelobe Levels” on page 11. In the optimisation process the
transducer coordinates are subject to certain geometrical constraints. Naturally, the
transducer positions must not overlap, and we must also require that the positions
26
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are confined within an area of linear dimension D, for example, a disc with diameter D or a box with side length D.
The simplest example of an optimised planar array geometry is the optimised
random array. Fig. 11(c) shows the result of optimising the random array of Fig. 4
according to eq. (41) with a Kmax corresponding to fmax = 5 kHz, Kmax =
90°
( 2p⋅fmax), cf. eq. (19). The geometry of the resulting array still looks ranKmax
dom, but when comparing the sidelobe levels before and after optimisation,
Fig. 14, it is noticed that the sidelobe levels are reduced by several dB for frequencies below fmax. The optimisation method thus provides an efficient and well-controlled method for decreasing the sidelobe levels over a given frequency range.
Brüel & Kjær Wheel Array
By optimising a random array, a beamformer with excellent performance can be
achieved. From a practical point of view the random array is, however, difficult
both to manufacture and operate, due to its complicated geometry. Also, the optimisation of a random array is numerically very demanding because of the large
number of free variables.
Fig. 15 shows an example of a patented [8] Wheel Array design that is the result
of an optimisation with the geometrical constraint that the transducers are confined
to a set of tilted linear spokes. The patented design consists of typically an odd
number N of identical line arrays arranged around a centre as spokes in a wheel,
with identical angular spacing between the spokes. All spokes are tilted the same
angle away from radial direction. The geometry is invariant under a rotation
n⋅360°/N around the centre, n being any integer.
Fig. 15. Wheel Array. All spokes are tilted the same angle away from radial direction, here
illustrated by a lateral offset d
X
d
040025
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Fig. 16. Example of 90-channel wheel array with integrated cabling
The mechanical design of the array shown in Fig. 16 is fully modular: all spokes
can easily be detached from the supporting inner and outer rings which themselves
can be disassembled. Thus the whole array structure can be disassembled and
transported in a standard-sized flight case. Regarding the cabling, each spoke
mounts 6 microphones, which through integrated cabling are connected to a common plug (LEMO-connector). Each spoke is then connected to a 6-channel (or 12channel) input module through a single cable. Despite its irregularity the wheel
array has its sensors grouped in easily identifiable logical units. In a practical meas28
bv005611.book Page 29 Tuesday, March 23, 2004 12:46 PM
urement situation, which requires channel detection, calibration and occasionally
detection of hardware faults, this is a great advantage.
A variant of the wheel array is the half-wheel array, intended for measurements
above a fully reflective ground/floor. In the half-wheel array, the spoke tilt angle is
zero because the array design is required to be symmetric with respect to the
ground floor, Fig. 11(f).
Comparison of the MSL-curves of a wheel array with other arrays with the same
channel count and diameter shows that the wheel array performs better than the traditional regular arrays, Fig. 14. The optimised random array performs slightly better than the wheel array but, as mentioned above, this type of array is difficult to
operate. Summarising, the wheel array combines high performance with easy operation and manufacturing.
Instrumentation
A complete PULSE-based beamforming system consists of the following main
components:
• A beamforming array structure with transducers, cabling and optional Web
camera.
• A PULSE front-end system
• PULSE data acquisition software including Data Recorder Type 7701 controlled by the Acoustic Test Consultant Type 7761
• A beamforming calculation module Type 7768
• For display of the results, Noise Source Identification Type 7752 is used
A complete configuration list is given in the Product Data for Type 7768.
Beamformer Arrays
There are several possibilities for beamformer array structures, including patented
wheel arrays, traditional cross and grid arrays as well as other irregular array types
(logarithmic spirals, Archimedean spirals). For optimal performance, we recommend one of our patented wheel types of array: see “Brüel & Kjær Wheel Array”
on page 27.
The structures can be full arrays or “half arrays”, where mirror-ground conditions can be utilised. A special possibility is flush-mounted microphones, which
can be beneficial in wind-tunnel applications or at high frequencies to avoid diffraction in the microphones and in the support structure.
29
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Fig. 17. Contour plots showing simulation results of source reconstruction using the arrays
depicted in Fig. 11 applied with different beamforming algorithms. All data sets are normalised
to 0 dB at maximum and the dynamic range is 15 dB. In the simulations, a 5 kHz monopole
was placed in front of the array centre at 1 m distance. (a) and (b) show the results for the
cross array, Fig. 11(a), using delay-and-sum and Hanning weighted cross-array beamforming,
respectively [“Cross-array Beamforming Algorithm” on page 25]. (c) – (f) show the results of
delay-and-sum beamforming for the grid array, the optimised random array, the Archimedean
spiral array and the wheel array depicted in Fig. 11(b) – (e). (g) – (i) illustrate the outcome of
applying cross-spectral beamforming with autospectra exclusion to the optimised random
array, the Archimedean spiral array and the wheel array, respectively
Transducers
Two types of array microphones are ideal for beamforming measurements. Array
Microphone Type 4935 is ideal for measurement below 5 kHz, Type 4935-W001
extends the frequency range up to 10 kHz, and finally Array Microphone Type
4944 A is ideal for measurements up to 20 kHz and for array types with flushmounted microphones. Both microphone types support IEEE 1451.4 Transducer
30
bv005611.book Page 31 Tuesday, March 23, 2004 12:46 PM
Fig. 18. Contour plots showing simulation results of source reconstruction using the arrays
depicted in Fig. 11 applied with different beamforming algorithms. All data sets are normalised
to 0 dB at maximum and the dynamic range is 15 dB. In the simulations, four 5 kHz monopoles
were placed in front of the array at 1m distance. Green asterisks indicate the source positions. (a) and (b) show the results for the cross array, Fig. 11(a), using delay-and-sum and
Hanning weighted cross-array beamforming, respectively [“Cross-array Beamforming Algorithm” on page 25]. (c) shows delay-and-sum beamforming with the grid array, and (d) – (f)
give the results of using cross-spectral beamforming with the optimised random array, the
Archimedean spiral array and the wheel array. The last row illustrates a mirror-ground situation. (g) represents delay-and-sum beamforming using the cross-array, while (h) and (i) show
the corresponding results for the half-wheel array, Fig. 11(f), using mirror-ground delay-andsum beamforming (h) and mirror-ground cross-spectral beamforming (i)
Electronic Data Sheet, which allows automatic transfer of the transducers’ serial
numbers and sensitivity data to be used directly in the application. Calibration can
be performed on six microphones in parallel using the special pistonphone adaptor
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WA 0728, and the system automatically detects which channels are being calibrated.
Other transducers such as hydrophones for underwater applications can also be
used.
Data Acquisitions System and Beamforming Calculations
Data acquisition is performed using PULSE hardware and software. The measurement process is controlled by PULSE Acoustic Test Consultant (ATC) Type 7761
and involves use of PULSE Data Recorder Type 7701. ATC provides fast and easy
setup of multichannel array systems, including automatic channel detection, parallel multichannel calibration, real-time level monitoring and on-line determination
of channel status. The measured time data is stored in a PULSE database (based on
Microsoft® SQL Server™), from which it can be retrieved for beamforming calculations using Type 7768.
When searching in large databases, measurements can be identified according to
user-defined criteria based on meta-data stored with the measurement data.
From each measurement it is possible to perform multiple calculations, for
example, by focusing on specific parts of the test object or on specific frequency
bands. Also, it is possible to choose whether the calculation is performed with the
free-field or mirror-ground algorithm (see Fig. 22 for a screen dump of the Calculation Setup dialog). Three different types of calculations are available: Stationary,
Quasi-stationary and Non-stationary. The first two methods are based on the crossspectral imaging function described in “Cross-spectral Imaging Function” on
page 16. When “Stationary” is selected, the cross-spectra are averaged over the
entire selected time interval before being applied in the cross-spectral imaging
function with autospectrum exclusion, eq. (36). When “Quasi-stationary” is
selected, the same procedure is performed in a number of subintervals where
approximate stationarity can be assumed. This method is useful for the analysis of,
for example, slow run-ups where the sound field can be assumed approximately
stationary in narrow RPM intervals. With “Non-Stationary” selected, a timedomain beamforming calculation is performed in the following way: First an FFT
is performed on the full selected time record length. Then the near-field Delay-andSum algorithm eq. (29) (normalised by the channel count) is applied for each FFT
line. Finally, the frequency domain results are converted into time-domain using
inverse FFT. The resulting time data can then be averaged in time intervals or, if a
tachometer signal has been recorded, in angle-, tacho-, or RPM-intervals.
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To display the result of a calculation using Noise Source Identification software
Type 7752, simply drag and drop it into a display window. The display window
contains both a map and a spectral view of the result.
These views are aligned, so the map always represents the frequency range
selected by a delta-cursor in the spectrum, and the spectrum always shows the data
for the selected cursor point on the map. Additionally, extensive display management tools are available, including zoom, scroll, tilt, rotate, and animation. Different calculations can be displayed in separate display windows for comparison, or in
the same display for a complete 3D result.
Practical Aspects of Designing and Using Beamformer Arrays
When designing an array for beamforming and when performing practical measurements, a number of practical aspects must be taken into account. These include
the lower and upper frequency limitations of the array design, the MSL of the array
defining its dynamic range, the array diameter, the measurement distance, the spatial resolution and the size of the mapping area. As will be explained below, these
quantities are highly interconnected. The relations are summarised and illustrated
in Table 1.
The Maximum Sidelobe Level (MSL) defines the dynamic range in the ability of
the array to separate sources in different directions (see “Maximum Sidelobe Levels” on page 11). If, for example, the MSL is equal to –12 dB for frequencies up to
5 kHz and focusing within an angle of 30° from the axis, then within these limitations ghost images of a source in a single direction will always be suppressed by at
least 12 dB relative to the strength of the real source. Other sources, which are not
12 dB weaker than the first source, will therefore not be hidden by the ghost images
of the first source. The MSL is an important parameter when choosing an array
design. Conversely, when interpreting results obtained by a given array, it is important to be aware of its MSL.
The acoustical environment, such as reflections and disturbing sources, must be
considered when choosing an appropriate array design. A fully reflective floor can
be exploited beneficially by using an array designed for the mirror-ground situation, for example, a half wheel, Fig. 11(f), and by applying a mirror-ground algorithm to the recorded signals. Other reflections will significantly disturb the
measurement if the reflected contributions are within the dynamic range of the
array, that is, if they are dampened relative to the direct contribution by less than
the array’s MSL. Strong disturbing sources or reflections behind a planar array
should be avoided, as the array cannot distinguish a source behind the array from
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Table 1. Beamformer properties at 30º opening angle
Frequency range for chosen
threshold T
T
f max ( 30° ) = 4--- f T
3
cf min ( 30° ) = --D
040030
Resolution at distance z
Array
R = 1,22 ---z- k
D
Area covered at distance z
L = 1.15z
30
30
z
040029
its mirror image in the array plane. Sources behind the array will show up in the
source map as image sources located in front of the array.
For many array designs there is no strict upper limit on the usable frequency
range, because the MSL is slowly increasing with frequency for a given applied
opening angle. In the case of regular arrays, however, the presence of grating lobes
imposes a rather strict upper frequency limit. We can illustrate this by considering
the MSL of the grid array, Fig. 14, with grid spacing d. This MSL-curve shows a
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sharp cut-off at the spatial angular sampling frequency Ks = 2KN = 2p/d, which for
a 90° maximum off-axis focusing angle corresponds to a maximum frequency
equal to fmax(90°) = Kmaxc/4p, see eq. (20). According to eq. (21), the maximum
frequency for 30º off-axis angle is then given by fmax(30°) = (4/3)fmax(90°). For the
array in question, with grid spacing d = 1/7 m, this means that when applied with its
maximum 90º opening angle, we must restrict ourselves to frequencies below
fmax(90°) = 1.2 kHz, whereas the same array can be used up to fmax(30°) = 1.6 kHz
if the maximum off-axis angle is reduced to 30º. The MSL-curve for irregular
arrays does not have a sharp cut-off at a well-defined frequency. Instead the MSL
deteriorates gradually with increasing frequency, Fig. 14. In this case, choosing a
threshold level T can identify the upper usable frequency fT (Table 1). The threshold level, T, should be chosen as the acceptable overall MSL, and in practical measurement situations a threshold level of T = –10 dB or lower should be preferred.
The threshold frequency fT is then the upper usable frequency at 90º off-axis angle,
and at 30º maximum off-axis angle we have:
fmax(30°) = (4/3)fT
(42)
The lower usable frequency of a beamformer array cannot be inferred from the
arrays MSL-curve. Instead, considerations about the obtainable resolution at finite
distance can give a useful number. A beamformer array relies on the phase differences between the signals recorded by its transducers to determine the angle of
incidence of an incoming sound field. For wavelengths larger than the array aperture the phase differences become too small for the beamformer to effectively
identify the angle of incidence, and as a consequence the ability to resolve different sources will be poor. We can determine a minimum frequency, fmin, by the
requirement that when applying the beamformer array with a 30º maximum offaxis angle it must be possible to resolve two maximally separated monopoles with
frequency fmin. Here, maximum separation means that the distance between the
two sources is 2tan(30°)z ≈ 1.15z which is the linear size of the focus area at distance z for an opening angle of 30º around the axis. Then, referring to eq. (15) with
a = 1.22, the lower frequency for an array with diameter D is determined from
1.15z = 1.22(z/D)[c/fmin(30°)], or
fmin(30°) ≈ c/D
(43)
In many cases, however, the resolution will be too poor at frequencies as low as
given by eq. (43).
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Application Examples
The application examples given in here represent noise source location problems,
where beamforming is an attractive measurement method.
A Vehicle in a Test Hall
The first example is measurement of the noise radiation from a complete vehicle.
At high frequencies a complete vehicle will be much larger than the wavelength,
meaning that Near-field Acoustical Holography (NAH) will require a huge
number of measurement points. For the cases when the vehicle is operated in a
steady state on a dynamometer drum in a test hall, a scanning technique – such as
STSF – could be used to measure all these positions with a realistic microphone
array. The measurement time will, however, be significant, and there will be problems such as stationarity and reference selection. For the case of run-up operating
conditions (simulated pass-by), a scanning technique is not easy to use: RPM
interval averaging will typically have to be used, and quasi-stationarity will have
to be assumed in each interval, which is a problematic assumption. The main
advantages of NAH (STSF) are:
• High resolution on the source plane – also at low frequencies
• Possibility to simulate source modifications on the source plane. For example, this enables contribution analysis in any far field position
• Source ranking based on calibrated intensity maps
Beamforming, on the other hand, provides the possibility of doing a broad-banded,
one-shot measurement with an irregular array at some intermediate measurement
distance, typically 3 –7 m. The directional contribution maps obtained from such a
measurement will show the positions of the noise-radiating regions with the highest relative contributions to the noise at the array position. No calibrated source
descriptor such as sound power can be achieved, however.
Since a Test Hall has a reflecting ground plane, it is obvious to measure with a
half-wheel array.
A Vehicle in a Wind-tunnel
For measurements on a vehicle in a wind-tunnel, there are many similarities with
measurements in a test hall: there is a reflecting ground plane allowing the use of a
half-wheel array, and the source will be much larger than the wavelength at high
frequencies. An additional condition is that measurements have to be taken at
quite long distances to stay out of flow. This is natural for beamforming, but
would further increase the size of the measurement area for NAH (STSF), and at
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such large measurement distances NAH loses its ability to provide good low-frequency resolution. To maintain good low-frequency resolution with NAH, a more
difficult in-flow scanning of a microphone array is required.
The 90-channel wheel array of Fig. 16 has been used to perform a measurement
on a car in a wind tunnel. This array has a diameter of 2.4 m and provides low MSL
(–14 dB) up to 3 kHz with 90° maximum off-axis angle and 4 kHz with 30° maximum off-axis angle. An alternative NAH measurement, also covering the frequency range up to 3 kHz, would require in-flow scanning of a microphone array
over a grid covering the car side with a grid spacing not larger than 5 cm. The
beamforming measurement is a simple “one-shot” recording taken with the 90-element array outside the flow region.
The vehicle was centred in the flow section, facing the wind, and the wind speed
was set to 130 km/h. The wheel array was placed parallel to the side of the car at a
distance of 3.3 m. The stationary beamforming calculation shown in Fig. 19 clearly
Fig. 19. Car in a wind-tunnel at 130 km/h wind speed
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reveals noise radiation from the front wheel, the side mirror, the A-pillar and the
door handle in the frequency interval from 2.1 kHz to 2.6 kHz. The spectrum plot
represents the contour cursor position on the door handle, and the contour plot represents the frequency interval selected by the delta cursor in the spectrum plot. For
this application, a low MSL (a large dynamic range) is very important to prevent
ghost images from the strong radiation around the wheel to mask the other sources.
A Large Source (Crane)
Beamforming is a very powerful technique for noise source location on large
sources up to rather high frequencies, requiring only a single recording with a
microphone array. In this example, a 42-channel wheel array with a diameter of
1 m was positioned 7 m from a crane hoisting at maximum load, see Fig. 20. The
array, the geometry of which is shown in Fig. 15, has MSL below –10.6 dB up to
4.8 kHz for 90° maximum off-axis angle and up to 6.4 kHz for 30° maximum offFig. 20. Mobile crane hoisting at maximum load
38
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axis angle. The rather long measurement distance was chosen in order to get a
large mapping area, at the sacrifice of resolution. According to the overview in
Table 1, the size of the mapping area can be up to 1.15 × 7 m ≈ 8 m within the 30°
maximum off-axis angle, but the resolution will be only 1.22 × 7/1 ≈ 8.5 wavelengths.
Fig. 20 shows the source location for a frequency band around 2.05 kHz, where
the spectrum shows a peak. The spectrum represents an area of high-level radiation
over a cover plate that is probably resonating within the selected frequency band.
An Engine at High Frequencies
This example is a measurement on a car engine with a 66-element, 1 meter diameter wheel array. This wheel array has MSL below –10.4 dB up to 16 kHz with 90°
maximum off-axis angle. The array was hanging approximately 0.9 m over an
open engine compartment, with the engine running at 3500 RPM, and a single
time history recording was made with a frequency bandwidth of 12.8 kHz. Fig. 21
contains the result of a Stationary beamforming calculation for the 6.3 kHz, 1/3octave band, the contour interval being 1 dB.
Fig. 21. Averaged 6.3 kHz, 1/3-octave band for a car engine
39
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A NAH measurement covering the frequency range up to, for example, 8 kHz
would require measurement over a regular rectangular grid of points with spacing
around 2 cm. To cover the entire engine compartment would require approximately
2500 measurement positions, which should be compared with the 66 positions used
by the beamformer. One major difference between the two techniques is that NAH
can provide calibrated maps of Sound Intensity, Pressure and Particle Velocity
close to the source, while with beamforming one can only get contour plots showing relative contributions to the sound field at the array position. No calibrated
absolute levels near the source surface are obtained with beamforming.
Transient Analysis of Engine Noise Radiation
Another measurement was performed on the same engine at a constant speed of
2000 RPM using the same array at a distance of 81 cm from the nearest point on
the engine. But now, in addition to the microphone signals, a tacho signal was also
recorded to provide RPM and crank angle information, and a Non-stationary type
of calculation was performed. Fig. 22 shows the Calculation Setup used to define
the parameters in the calculation. Here, the time interval for processing, the size
and resolution of the mapping area and the distance to the mapping area has been
Fig. 22. Calculation Setup
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defined. Under the tab page named Processing Parameters, the frequency range
and type of averaging have been chosen. In this case we have used 1/3-octave
band filters and performed (crankshaft) Angle Interval averaging in intervals of 5°.
Thus, for each 1/3-octave band we get 720/5 = 144 maps of the radiation during an
engine cycle consisting of two revolutions. For each angle interval, averaging has
been performed over all rotations within the selected time interval.
When doing angle interval averaging, one has to be aware of the time smearing
performed by the impulse response of the frequency band-pass filters used. With a
frequency bandwidth equal to B, the impulse response will have duration around
1/B. This should be related to the time for one rotation, which is 60 s/RPM. An
impulse will therefore be smeared over an angle of approximate width 6° · RPM/B.
For the present measurement we shall be looking at data for the 3.15 kHz, 1/3octave band, and the smearing angle turns out to be approximately 15°.
A typical result is shown in Fig. 23. The angle meter shows a crank angle interval
from 615° to 620° and the spectrum plot shows that only the 3.15 kHz, 1/3-octave
band has been selected for mapping. The spectrum represents the cursor position
on the peak of the contour plot. Clearly, the 3.15 kHz, 1/3-octave band is very
strong at the chosen peak position. To the left of the Angle (and RPM) meter, the
variation with angle at the cursor position is plotted. The sharp peaks indicate that
the radiation at the peak position is very impulsive and concentrated at a few discrete shaft angles.
Another typical type of analysis would be averaging in RPM intervals for run-up
measurements.
For the present measurement, we used only a single tacho signal providing a single impulse per cycle (of two revolutions). This provides a unique identification of
a reference angular position during each cycle, but it does not necessarily allow an
accurate estimation of the angle during the entire cycle. For this, the system supports the use of a second high-resolution tacho signal.
Conclusions
Beamforming is a noise source location method based on measurements with a
planar array of microphones or hydrophones at an intermediate distance from the
source. The beamforming calculation can then basically resolve the relative contributions from different directions to the sound field seen by the array. Pure directional resolution would correspond to focus at infinite distance. Our
implementations always focus on a source plane parallel with the array plane, i.e.,
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Fig. 23. Non-Stationary beamforming. Stationary 2000 RPM. Shaft angle averaging in intervals of 5°
at finite distance, but still the method maps only relative contributions: no calibrated maps of sound field parameters near the source can be obtained. On the
other hand, a single measurement with an array of typically 42 – 90 transducers
can map a large source area up to high frequencies. The map can cover directions
up to approximately 30° from the array axis. Typically, a resolution of around one
wavelength can be obtained, which is almost as good as holography at high fre-
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quencies, but at low frequencies holography can do much better. At very high frequencies, on the other hand, where holography is not feasible because of the
required half-wavelength microphone grid spacing, beamforming can provide
high resolution with relatively few measurement points.
The basic principle of beamforming has been outlined, and the performance of
different array designs has been analysed and compared. The wheel array design,
which is patented by Brüel & Kjær, has been shown to offer some distinct advantages: a combination of high performance and ease of handling. A new, self-termfree, cross-spectral beamforming algorithm has been described and shown to offer:
(i) suppression of noise in the individual measurement channels, (ii) suppression of
sidelobes (and thus of ghost images) and (iii) a natural distance correction for the
individual array transducers when the array is used at a finite distance. The measurement system has been described together with guidelines for design and use of
the system, and finally some typical application examples have been presented.
Appendix: The Cross-spectral Imaging Function
In this appendix we derive the Cross-spectral Imaging function applied in PULSE
Stationary Beamforming. For each position on the source plane, the estimated
source strength reflects how well the sound field from a monopole point source at
that position fits the sound field measured by the array. The approach is inspired
by reference [9].
The error function defined in eq. (34):
E ( a, r ) =
M
∑
m, n = 1
C nm – C mod
nm
2
M
=
C nm – avn* ( r )v m ( r )
∑
m, n = 1
2
(A.1)
can be written in a form that is convenient for the derivations, if we stack all the
columns of the cross-spectral matrix [Cnm] in a single-column matrix, and arrange
the elements [vn*vm] in a similar matrix:
g
= [ C nm ]
h ( r ) = v*n ( r )v m ( r )
(A.2)
For each position r, we first determine the monopole strength â that minimises
the error function, eq. (A.1). To do this we notice that the error function is mini-
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mised by least squares solution of g ≈ ah, which upon multiplication from the left
with h† leads to:
†
†
â ( r ) = h g ⁄ h h
(A.3)
(† transposed complex conjugate)
Appealing to the fact that the cross-spectral matrix is Hermitian and to the definition (A.2) of h and g, we see that h†g is real and equal to g†h. Therefore â is also
real.
Use of eq. (A.2), eq. (A.3) and the relation g†h = h†g in eq. (A.1) leads to the following expression for the error function:
E ( â, r ) = g – âh ( r )
2
†
2
(h g)
= g g – â ( h g + g h ) + â h h = g g – ---------------- (A.4)
†
h h
†
†
†
2 †
†
Minimising the error function over all r thus corresponds to maximising the Imaging Function, I(x, r),
M
†
2
∑
2
C nm ( x )v n ( r )vm* ( r )
(h g)
m, n = 1
I ( x, r ) ≡ ---------------- = ----------------------------------------------------------------------†
M
2
h h
v*n ( r )v m ( r )
4
(A.5)
∑
m, n = 1
over all r (we choose the definition I 4 since eq. (A.5) has units of power squared).
In practice I(x, r) is computed over a discrete mesh covering the focus area. In the
resulting map, peaks are interpreted as areas with a high probability of finding a
source. This interpretation can be justified if we compare the imaging function in
the far field with the corresponding expression for the Delay-and-Sum beamformer, eq. (7). For large R ≡ |r| the approximation |rm – r| ≡ Rm ≈ R is valid in connection with the amplitude of vm(r) ≡ v(rm – r). In the far-field limit eq. (A.5) can
therefore be approximated by:
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R
M
∑
4
2
M
∑
C nm v n vm*
C nm e
– jkR n jkR m
2
e
m, n = 1
m, n = 1
I = --------------------------------------------------- ≈ ------------------------------------------------------------4
R
4
M
∑
m, n = 1
v*n v m
M
∑ e
m, n = 1
2
– jk ( R n – R m )
1
C nm e
= ------2
M m, n = 1
M
∑
– jkR n jkR m 2
e
2
(A.6)
Expanding the square of eq. (7) and taking the average we find:
B
2
– jk␬ ( r m – r n )
– jk ( R n – R m )
1
1
= ------= ------P*n P m e
C nm e
2
2
M m, n = 1
M m, n = 1
M
M
∑
∑
(A.7)
where we have used that the difference in travel paths Rn – Rm equals the projection difference ␬⋅(rm – rn), Fig. A1. Obviously, we have:
I 2/M = B
2
which shows us that the imaging function in the far field, when normalised by the
channel count, equals the output of the Delay-and-Sum beamformer. This observation justifies the chosen interpretation. The normalisation factor has been included
in eq. (35).
Fig. A1. For a source in the far field, the difference, Rn – Rm, in the propagation path length to
the transducers at rn and rm can be calculated from the diagram
Far field source
␬·
␬
␬·
040028
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It could seem obvious to map the monopole strength â(r) of eq. (A.3) instead of
the imaging function I 2(r) that represents the reduction of the cross-spectral modelling error across the array. The main problem with â(r) is that the amplitude will
increase when the distance to the array is increasing. So at angles far away from the
array axis, â(r) will tend to predict unrealistically high levels.
The summations in eq. (A.5) can be rewritten in matrix form as:
M
C nm v n vm* = v
∑
m, n = 1
M
∑ vn
m, n = 1
2
* 2
vm
T
Cv
*
(A.8)
T
= w 1w
*
where v and w are the column vectors v = [vm] and w = [|vm|2], and where 1 is an
M by M unity matrix with all elements equal to 1. The exclusion of autospectra can
be formulated by introducing modified versions C′ and 1′ of the cross-spectral
matrix and the unity matrix, respectively, where zeros have replaced all diagonal
elements. Then the imaging function, eq. (36), can be expressed as:
T
*
2
v C′v
1
J ( x, r ) = --------------------------- ----------------------M ( M – 1 ) w T 1′w *
(A.9)
Mirror-Ground Implementation
In the case of a fully reflective surface perpendicular to the array plane, we can
alter eq. (A.9) to make use of the fact that beyond the real sources there will seem
to be equally strong image sources. The presence of a “mirror ground” is equivalent to a free-field situation, where we have no mirror ground, but instead image
sources in addition to the real sources. The resulting sound field will be image
symmetric in the (original) ground plane, so therefore we can assume a mirror
array that will measure exactly the same as the real array. If we let v̂ m ( r ) be the
steering vector component from the mth transducer in the mirror array to the
source at position r we can obtain a mirror-ground version of eq. (A.9) by using
the expanded column vectors ṽ , w̃ and the expanded cross-spectral matrix C̃
defined by:
ṽ = v ,
v̂
46
w̃ = w ,
ŵ
˜ = C′ C′
C
C′ C′
(A.10)
bv005611.book Page 47 Tuesday, March 23, 2004 12:46 PM
The cross-spectral imaging function for the mirror-ground situation is then:
T˜ *
2
ṽ
1
ṽ C
J˜ ( x, r ) = ----------------------------------- ---------------------2M ( 2M – 2 ) w̃ T 1˜ w̃ *
(A.11)
The normalising factor is here 2M(2M – 2), since now 4M elements out of 4M 2 in
˜ equal zero, and 1̃ is constructed from 1′ exactly as C
˜ is constructed from C′.
C
Appendix: Resolution
With reference to Fig. 5, we can evaluate the resolution as follows. As stated in
“Resolution” on page 8, two incident plane waves of equal amplitude with wave
number vectors k1and k2, |k1| = |k2| = k, can be resolved only if the projections k̂ 1
0
and k̂ 2 of k1 and k2 on the array plane are not within RK = Kmin
from each other.
0
Here, RK is the main-lobe width in the array pattern W(K), and Kmin
is the position of the first minimum of the array pattern. In Fig. 5 we have
kˆ 2 – kˆ 1 = R K
so the two plane waves are just exactly resolvable.
In the focus plane, we shall look only at the resolution in the radial direction, that
is, in the direction away from the array axis. We choose to look at the resolution
along the x-axis. The plane wave with wave number vector k1 is incident at an
angle h from the array axis, and the exactly resolvable plane wave with wave
number vector k2 is incident at an angle h +Rh from the axis, Rh being the angular
resolution in the radial direction. These two directions span the resolution R along
the x-axis at the distance z from the array plane (on the focus plane).
In the derivation we will now assume a very fine resolution, meaning that we can
consider the resolutions in K, h and x to be differential: RK = dK, Rh = dh and R =
dx. From K = ksin(h) we get dK = kcoshdh and therefore:
dK
dh = -------------k cos h
From the relation x = z tg(h) we get:
z dh
dx = ------------2
cos h
and combination of the two above relations leads to:
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z dK 1
dx = ----------- ------------k cos 3 h
Finally, we replace the differential resolutions with the real finite resolutions and
get:
z RK 1
R ≈ ---------- ------------k cos 3 h
References
[1]
Maynard J.D., Williams E.G. and Lee Y., “Nearfield acoustical holography:
I. Theory of generalized holography and the development of NAH”, J.
Acoust. Soc. Am. 78(4), 1985, pp. 1395 – 1413.
[2]
Hald J., “STSF – a unique technique for scan-based Near-field Acoustic
Holography without restrictions on coherence”, Brüel & Kjær Technical
Review No. 1, 1989, pp. 1 – 50.
[3]
Johnson D. H. and Dudgeon D. E., “Array Signal Processing: Concepts and
Techniques”, Prentice Hall, New Jersey, 1993.
[4]
Proakis J.G. and Manolakis D. G., “Digital Signal Processing, Principles,
Algorithms, and Applications”, 3rd Edition, Prentice Hall, New Jersey,
1996.
[5]
Yasushi Takano, “Two-dimensional Microphone Array System for Measuring Distribution of Aerodynamic Noise-sources on High-speed Trains”,
International Symposium on Simulation, Visualization and Auralization for
Acoustic Research and Education, Tokyo, 1997
[6]
Underbrink J. R. and Dougherty R. P., “Array Design for Non-intrusive
Measurement of Noise Sources”, Proceedings of Noise-Con 96, pp. 757 –
762, 1996.
[7]
Nordborg A., Wedemann J. and Willenbrink L., “Optimum Array Microphone Configuration”, Proceedings of Internoise 2000.
[8]
Danish patent No. PA 2002 00412.
[9]
Elias G., Proceedings of Internoise 1995, pp.1175 – 1178.
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bv005611.book Page 49 Tuesday, March 23, 2004 12:46 PM
Previously issued numbers of
Brüel & Kjær Technical Review
(Continued from cover page 2)
2 – 1988 Quantifying Draught Risk
1 – 1988 Using Experimental Modal Analysis to Simulate Structural Dynamic
Modifications
Use of Operational Deflection Shapes for Noise Control of Discrete Tones
4 – 1987 Windows to FFT Analysis (Part II)
Acoustic Calibrator for Intensity Measurement Systems
3 – 1987 Windows to FFT Analysis (Part I)
2 – 1987 Recent Developments in Accelerometer Design
Trends in Accelerometer Calibration
1 – 1987 Vibration Monitoring of Machines
4 – 1986 Field Measurements of Sound Insulation with a Battery-Operated Intensity
Analyzer
Pressure Microphones for Intensity Measurements with Significantly
Improved Phase Properties
Measurement of Acoustical Distance between Intensity Probe Microphones
Wind and Turbulence Noise of Turbulence Screen, Nose Cone and Sound
Intensity Probe with Wind Screen
3 – 1986 A Method of Determining the Modal Frequencies of Structures with
Coupled Modes
Improvement to Monoreference Modal Data by Adding an Oblique Degree
of Freedom for the Reference
2 – 1986 Quality in Spectral Match of Photometric Transducers
Guide to Lighting of Urban Areas
1 – 1986 Environmental Noise Measurements
Special technical literature
Brüel & Kjær publishes a variety of technical literature which can be obtained from
your local Brüel & Kjær representative.
The following literature is presently available:
•
•
Catalogues (several languages)
Product Data Sheets (English, German, French,)
Furthermore, back copies of the Technical Review can be supplied as listed above.
Older issues may be obtained provided they are still in stock.
BV0056-11_TR_Cover2004.qxd
23-03-2004
13:33
Page 1
TECHNICAL REVIEW
BV 0056 – 11
ISSN 0007–2621
Beamforming
HEADQUARTERS: DK-2850 Nærum · Denmark
Telephone: +45 4580 0500 · Fax: +45 4580 14 05
www.bksv.com · [email protected]
No.1 2004
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