null  null
No. 2 • 1989
STSF—Practical instrumentation and application
Digital Filter Analysis: Real-time and Non Real-time
Performance
Previously issued numbers of
Brliel & Kjaer Technical Review
1-1989 STSF - A Unique Technique for scan based Near-field Acoustic
Holography without restrictions on coherence
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 F F T Analysis (Part II).
Acoustic Calibrator for Intensity Measurement Systems
3-1987 Windows to F F T 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
4-1985 Validity of Intensity Measurements in Partially Diffuse Sound Field
Influence of Tripods and Microphone Clips on the Frequency Response
of Microphones
3-1985 The Modulation Transfer Function in Room Acoustics
RASTI: A Tool for Evaluating Auditoria
2-1985 Heat Stress
A New Thermal Anemometer Probe for Indoor Air Velocity
Measurements
1-1985 Local Thermal Discomfort
4-1984 Methods for the Calculation of Contrast
Proper Use of Weighting Functions for Impact Testing
Computer Data Acquisition from Briiel & Kjaer Digital Frequency
Analyzers 2131/2134 Using their Memory as a Buffer
3-1984 The Hilbert Transform
Microphone System for Extremely Low Sound Levels
Averaging Times of Level Recorder 2317
2-1984 Dual Channel F F T Analysis (Part II)
(Continued
on cover page 3)
No. 2 • 1989
Contents
STSF-Practical instrumentation and applications
1
by Kevin B. Ginn & Jorgen Hald
Digital Filter Analysis: Real-time and Non Real-time Performance
by Svend Gade
28
STSF-Practical instrumentation
applications
and
Kevin B. Ginn & Jorgen Hald
Abstract
The Spatial Transformation of Sound Fields (STSF) technique involves a
scan using an array of transducers over a (planar) surface close to the
source under investigation. From cross spectra measured during the scan,
a principal component representation of the sound field is extracted. Any
power descriptor of the near field (intensity, sound pressure, etc.) can then
be investigated by means of Near-field Acoustic Holography (NAH), while
the more distant field can be determined by application of Helmholtz' integral equation.
The theoretical foundation of the cross spectrum principal component
technique implemented in STSF has been discussed by J. Hald in [1] and
the advantages of STSF compared to a number of acoustical holography
techniques have been demonstrated. In this article, the practical implementation of STSF is discussed and some applications are described.
Sommaire
La technique de transformation spatiale des champs sonores (Spatial
Transformation of Sound Fields) est fondee sur le balayage d'une surface
plane pres de la source etudiee. Une representation de la composante principale est extraite de l'interspectre mesure pendant le balayage. Tout descripteur de puissance du champ proche (intensite acoustique, pression sonore, etc.) est etudie par holographie acoustique du champ proche (Nearfield acoustic holography), alors que le champ plus distant est determine
par Inequation integrale de Helmholtz.
1
Le fondement theorique de la technique de composante principale
interspectrale employee en STSF a ete couvert par J. Hald [1], et les avantages de la STSF sur diverses techniques holographiques ont ete demontres. Cet article decrit sa mise en application et en discute quelques
exemples.
Zussammenfassung
Fur die raumliche Transformation von Schallfeldern (Spatial Transformation of Sound Fields — STSF) wird mit einer Reihe von Aufnehmern
eine ebene Flache nahe der Schallquelle abgetastet. Das gemessene
Kreuzspektrum reprasentiert die Hauptkomponenten des Schallfelds.
Alle Leistungsparameter des Nahfelds (Intensitat, Schalldruck usw.) konnen mit der akustischen Nahfeldholographie (NAH) untersucht werden.
Fiir das fernere Feld wird die Helmholtz'sche Integralgleichung benutzt.
Die Theorie der Kreuzspektrum-Hauptkomponententechnik fiir STSF
sowie die Vorteile von STSF gegeniiber anderen akustischen HolographieVerfahren werden in [1] von J. Hald diskutiert. Der Artikel beschaftigt
sich mit der praktischen Anwendung von STSF und beschreibt einige
Beispiele.
1. Principle of STSF
The basic principle of the Spatial Transformation of Sound Fields (STSF)
technique is illustrated in Fig. 1. Based on cross spectra measured over a
planar surface close to the source under investigation, all parameters of
the sound field can be mapped over a three dimensional region extending
from the surface of the source to infinity. The near field is predicted from
the scan data using Near-field Acoustic Holography (NAH), while the
more distant field is calculated using Helmholtz' Integral Equation (HIE).
In addition to the 3D mapping capability based on 2D measurements,
STSF also provides the possibility of doing Partial Source Attenuation
Simulation (PSAS). This simulation is done by backwards propagation of
the cross spectral representation of the sound field from the measurement
surface to (ideally) the source surface, then modifying the representation
at the source surface according to an attenuation model and finally predicting the sound field from the modified representation.
2
I
■
Fig. 1. Principle of STSF
2. Instrumentation
The instrumentation for STSF can vary greatly for different applications.
Two standard systems, based on a 2-channel F F T analyzer and a dual
channel, real-time frequency analyzer, are depicted in Fig. 2.
The main parts of the STSF instrumentation are:
1. Transducers
2. Cross spectrum analyzer
3. Controller, data storage and post processing.
2.1.
Transducers
The planar scan over the measurement area involves the measurement of
pressure signals or particle velocity signals. The pressure signals can be
measured using an array of ordinary B & K microphones of the same type,
3
Fig. 2. STSF System Type 9606 with 2032 or Type 9607 with 2133
no phase matching between the individual microphones being necessary.
The particle velocity signals can be determined from a finite difference
approximation using an array of phase matched pairs of microphones. The
difference between the signals from the front and the rear microphones of
the pairs is determined directly from the analogue signals by one of the
scan multiplexers.
If one is only interested in the far field in a horizontal plane then a far
quicker measurement can be performed by summing the outputs from
each column of microphones in real time thus effectively treating the column of scan microphones as a single transducer. This summation is performed by the scan multiplexer. The measurement time is reduced by a
factor equal to the number of scan microphones.
The reference microphones should be placed between the sound source
and the scan plane and should be well distributed over the scan area so
that from every scan position there is always at least one reference microphone within one correlation length [4]. This distance is set by the bandwidth being used. For third octaves a correlation length is about two wavelengths. For twelfth octaves it is about eight wavelengths. A greater
distance than the correlation length will result in a poorer representation
of the sound field over the scan area, i.e. a poorer representation of the
large cross spectrum matrix. Preferably the correlation length should be
sufficiently large to include the entire scan area from a single reference.
In the systems depicted in Fig. 2 the cross spectra are measured using a
column of scan microphones and an array of reference microphones. The
4
number of microphones in a practical set-up need not be excessively large.
In fact the minimum number of transducers required is two, one reference
and one scan transducer. A practical application using S T S F in situ with
only two transducers to investigate the noise radiated by a seismic survey
lorry is described in Ref. [2].
Instead of reference microphones other transducers could be employed
e.g. accelerometers mounted on the source, pressure transducers inside an
engine's cylinder, an electrical signal from say a vibration exciter. The advantages of using a transducer such as an accelerometer on the test object
itself are firstly that the influence of background noise on the measurement can be considerably reduced and secondly one can position the accelerometers directly on the suspected source. For delicate or very hot
sources a non-contact laser vibrometer could be used. A disadvantage is
that an accelerometer placed close to a node line would give little or no
signal. A sound pressure field rarely has corresponding node lines. Thus
relatively more reference accelerometers than microphones need to be
used to completely represent the sound field radiated from a structure [3].
2.2. Cross spectrum
analyzer
In theory any cross spectrum analyzer could be employed for STSF. Three
systems have been developed by Briiel & Kjaer, based on three analyzers:
• Sound Intensity Analyzer 2134
• Dual Channel Real-time Frequency Analyzer 2133
Table 1. Frequency ranges of the three STSF
Systems
5
• Dual Channel Signal Analyzer 2032 high speed version. The 2034 standard version could also be used but is not to be recommended as the
measurement time would become excessively long.
The main features of the three systems are given in Table 1.
2.3. Controller, data storage and
post-processing
For the original STSF system based on a modified Sound Intensity Analyzer 2134, the measurement is controlled by the Graphics Recorder Type
Fig. 3. STSF system based on Sound Intensity Analyzer Type 2134 modified to measure
cross spectra. The instrumentation
here is set-up to measure particle velocity during the
scan
6
2313 fitted with a dedicated Application Package BZ 7007. Data is stored
on a Digital Cassette Recorder 7400 then transferred to a VAX/micro
VAX for holographic post-processing (Fig. 3).
For STSF Systems Type 9606 and 9607 based on analyzers 2032/34 and
2133 respectively, a micro VAX II or VAX station II is used as the controller and post-processor. A full STSF software package for VAX based systems consists of two parts:
1. Control and data acquisition
2. Calculation and plot.
The menu-driven control part includes a number of inspection functions which enables the validity of the data to be checked before proceeding to the STSF calculations. Measurement errors such as too few or poorly positioned reference transducers, too short averaging times, or spurious
data, can thus be detected. If necessary, new measurements can be made
and substituted into the original data set.
Fig. 4. Flow diagram for STSF
measurements
7
A series of prompts, warnings, menus and a recurring overview of the
hard key functions guide the user through the measurement and calculation procedures.
In the standard calculation and plot program the following possibilities
are available.
1. Sound pressure level along a line
2. Sound pressure spectrum at a specified position
3. Sound power spectrum
4. Acoustical holography
5. Contour plots of: pressure, active intensity, reactive intensity, particle
velocity, radiation pattern
6. Simulation of source attenuation.
In addition, a number of supplementary programs can be offered
including:
1. Drawing the test object on the contour plots.
2. Plotting vector intensity diagrams.
3. Measuring and plotting programs using separable co-ordinate systems
other than Cartesian.
A simplified flow diagram of the processing of STSF measurements is
shown in Fig. 4.
3. Measurement Procedure
Good results require well defined measurement conditions. The measurement set-up is described by a set of parameters (e.g. type of probe used,
number of references, measurement grid size, environmental conditions,
presence of reflective plane) which are keyed into the computer by the
user. A typical set-up is shown in Fig. 5.
The most important decisions for the user concern the geometry of the
measurement. The starting point is the frequency range of interest. The
spacing between the scan microphones is closely related to the upper frequency of interest. There must be at least two microphones per wavelength
at this upper frequency in order to avoid spatial aliasing. The size of the
scan area then has to be set. This area must adequately cover the source as
shown in Fig. 6, otherwise the basic hypothesis that practically all the energy of the sound field radiated into the measurement half-space passes
through the measurement window, will no longer be valid.
The distance from the scan microphones to the source should be approximately equal to but not less than, one inter-scan-microphone distance in
order to avoid spatial aliasing. At this distance, the amplitudes of the un8
Fig. 5. Typical STSF measurement
set-up
Fig. 6. Determination of the size of the scan area for a) free field conditions
ground conditions. The distances a ^ Ax and b ^ 3 Ax. Also b ^ A/8
b) mirror
9
der sampled evanescent waves have decayed by approximately 20 dB from
the source surface. If measurements were attempted at significantly greater distances from the source then the exponentially decaying evanescent
waves (the waves necessary to describe the details of the sound field) would
be swamped by the background noise level.
If a reflecting ground surface is present, then measurements must be
made right down to the ground. The positioning of the scan microphones is
critical because for the calculations an image source is created using the
floor as the plane of symmetry. Thus unless the first scan microphone is
positioned exactly one half an inter-scan-microphone spacing above the
floor, the plots resulting from the subsequent calculations will be
distorted.
3.1. Inspection of measured data
The Inspect function of the STSF program packages is used to verify and
obtain plots of the measured data and also to check the validity of the
data. The validity check is performed using the following three functions:
1) Validation:
To check the representation of the field at
scan and reference positions.
2) Eigenvalue spectrum:
To plot the eigenvalue spectrum for the reference signal cross spectrum matrix.
3) Stationarity check:
To check the stationarity of a selected reference signal during the scan.
In the Validation function the field representation is checked by comparing the measured sound pressure level to the level calculated from the
measured cross spectra. With one reference, the difference between the
two pressure levels expresses the coherence between the array microphone
signal and the reference signal. With several references, the difference
represents a multiple coherence. Provided a sufficient set of references is
employed, the multiple coherence is close to one or equivalently, the difference between the measured and the represented sound pressure level is
small at all scan positions.
There will always be some discrepancy between the measured and calculated spectra at the scan microphone positions. In general at low frequency
this deviation will be mainly due to too short an averaging time, and the
high frequency deviation will be mainly due to an insufficient number of
references or to a poor positioning of the references. The deviation will be
10
smallest in regions close to the references. Non-stationarity of the source
signals will tend to create discrepancies at all frequencies.
In practice the validation procedure should be performed before a complete scan is made. This can be accomplished by initially taking measurements only at the set of array positions where the measured and the represented pressure levels are to be compared. By doing so, one avoids
spending time taking complete measurements only to obtain bad data.
Good measurements may be obtained in environments with severe uncorrelated background noise by positioning the references very close to the
source. This means that the cross spectra and thus all STSF calculations
will be unaffected by the background noise. Since, however, the measured
autospectra (pressure) will be affected, the above validation procedure
does not apply.
The Eigenvalue spectrum function shows how many significant independent partial sources the reference measurement (cross spectra between
every pair of references) has identified for each frequency band. Each eigenvalue represents the autopower of a virtual independent partial source.
Thus, all eigenvalues should be non-negative; the occurrence of negative
eigenvalues is due to non-stationarity of the source during the reference
measurement or use of too short an averaging time. If large negative eigenvalues are observed, the source has probably been very non-stationary during the reference measurement, which is not acceptable. If nothing can be
done to improve the stationarity of the source then the reference measurement should be repeated using a longer averaging time.
Stationarity during the scan is also important, although not as critical as
during the reference measurement. To be able to check the stationarity
during scan, the autospectrum at a specified reference is measured and
stored for each position of the array. After the scan, the variation of this
reference autospectrum during the scan, can be compared with the value
obtained during the reference measurement by means of the Stationarity
Check function.
4. Applications
Practical applications of STSF can be found in many branches of industry
and in the automotive industry in particular. Here the engine group, transmission group and whole vehicle group could all apply STSF in their work.
The engine group for example, would be particularly interested in the
acoustical holography capability for source location during the development of new products. The whole vehicle group would be interested in the
11
calculation of radiation patterns as a way of quantifying the noisiness of a
vehicle. Tyre manufacturers are interested in the radiation characteristics
of various tyre profiles. The aeronautic and building industries are interested in the transmission paths via elements such as windows and doors.
Some practical measurements are described below.
4.1. Measurement on a Motorcycle
The measurement was performed using the narrowband system Type
9606, over the right-hand side of an idling motorcycle situated on a quiet
road. Eleven microphones were scanned over 14 measurement positions,
the distance between each scan position being 7 cm. Thus the total scan
area was 98 cm x 77 cm, Fig. 7. The scan plane was situated at 10 cm from
the crankshaft cover (Fig. 8). Five reference microphones were employed.
The total measurement time was 30 min. As the motorcycle was inclined
during the measurement and as the engine was relatively far from the
ground, it was judged that reflections from the ground had little influence
Fig. 7. Position of STSF measurement
12
area on the
motorcycle
Fig. 8. Position of the microphone array relative to the
motorcycle
on the noise which propagated from the engine and through the measurement window. Thus in the subsequent calculations free-field propagation
was assumed. Figures 9 to 13 show contour maps of pressure, active intensity and reactive intensity as calculated by the near-field holography technique. All results are for a 100 Hz band centred on 1600 Hz. The parameters listed to the right of the plots are for conditioning the data by means
of windows, spatial filters and dynamic ranges before performing the calculations. A comparison of Figs. 9 and 10 shows that active intensity is a
better indicator than pressure for locating the principal noise generating
regions. The reactive intensity plot of Fig. 11 is even more revealing, pinpointing the crankshaft cover. The regions of negative reactive intensity
(indicated by dotted contour lines on Fig. 11) indicate that the pressure
gradient in these regions decreases as the motorcycle is approached i.e. as
one passes through the radiation cone of the crankshaft cover. When trying to interpret such contour plots it should be remembered that regions
13
Fig. 9. Pressure contour map in the scan plane (z = 0 m)
14
Fig. 10. Active intensity
contour map in the scan plane (z — 0 m)
Fig. 11. Reactive intensity
Fig. 12. Active intensity
contour map in the scan plane (z = 0 m)
contour map 10 cm closer to the motorcycle (z - -0,1 m) 15
Fig. 13. Pressure contour map 10 cm closer to the motorcycle (z = -OJ
m)
of high reactive intensity do not necessarily indicate the presence of a
sound source [4]. Plots of the various acoustical parameters should be used
to complement one another when performing a noise survey. Figs. 12 and
13 show the contour plots of active intensity and pressure at 10 cm closer
to the motorcycle, clearly indicating the ability of the S T S F technique to
"focus" on the sound source. It is interesting to note that in the scan plane
the pressure-intensity index in front of the crankshaft box cover is about
2 dB whereas 10 cm closer to the engine the index has increased as one
would expect to about 3,5 dB. In general, the closer one gets to a sound
source the more complex the acoustical field becomes and the greater the
pressure-intensity index.
4.2. Measurement
on a Whole Vehicle
A measurement on a complete vehicle was performed using the 2134 third
octave band system. Eight microphones were scanned over 32 positions
over the right-hand side of an idling car situated in a semi-anechoic chamber. The total measurement time was 80 min. A calculation line was posi16
Fig. 14. Position of car in semi-anechoic
tion/control line as indicated
test chamber with scan position and
calcula-
tioned at 5 m from the scan plane, 1,16 m above the ground (Fig. 14). For
the calculations it was assumed that the floor acted as a perfect reflector
i.e. a mirror ground. The sound pressure level was calculated in the 80 Hz
third octave band and the results compared with the level measured directly at 11 points along the line. The direct measurement was possible
because the test chamber had extremely large dimensions, approximately
12 x 20 x 8 m, with a lower limiting frequency of 60 Hz. The agreement
within a region of ± 2,5 m of the axis is within ± 0,5 dB (Fig. 15). Agreement in the ± 5,0 m range is within ± 2 dB. Higher accuracy is to be expected on axis whilst at the extremities of the line, errors due to a finite
scan area and finite room dimensions start to influence the results. For a
discussion of the effect of bandwidth on the measured results see [4] & [5].
17
Fig. 15. Comparison between calculated and measured sound pressure level along a line.
The source was a passenger car in a semi-anechoic
chamber
18
4.3. Measurement of tyre noise
The tyres under test were mounted on a car which was run in free gear on a
dynamometer. The surface of the rotating road was gravelled and the simulated speed was 60 km/hr. The measurements were performed using the
narrowband STSF system Type 9606. The scan plane was 7 cm from the
wheel. Twelve microphones were scanned over 20 positions. Three reference microphones were used. The total measurement time was less than 30
minutes. The results were stored in the form of synthesized 100 Hz bands.
Figs. 16 and 17 show the active and reactive intensity distribution at the
surface of the wheel for the 100 Hz band centred at 400 Hz. The phenomenon of circulating acoustical energy can be seen in the active intensity plot.
The main noise sources are in the vicinity of the tyre/roller contact area.
At the higher frequency of 2000 Hz, the active intensity plot shows how
virtually all noise generation is related to the tyre/roller contact area
(Fig. 18). The reactive intensity plot (Fig. 19) suggests that there is a periodic series of pressure maxima around the rim of the tyre perhaps associ-
Fig. 16. Active intensity at the wheel surface (z = -0,07 m) at 400 Hz. The dotted
indicate regions of negative
intensity
curves
19
Fig. 17. Reactive intensity
20
Fig. 18. Active intensity
at the wheel surface (z = -0,07 m) at 400 Hz
at the wheel surface (z = -0,07 m) at 2000 Hz
Fig. 19. Reactive intensity
Fig. 20. Active intensity
at the wheel surface (z = -0,07 m) at 2000 Hz
in the scan plane (z = 0,0 m) at 2800 Hz
21
Fig. 21. Active intensity
at the wheel surface (z = -0,07 m) at 2800 Hz
Fig. 22. Pressure along a line at 7,5 m from the car at 2800 Hz. Scale along x axis
± 50 m
22
Fig. 23. The overall A-weighted
2800 Hz at 7,5 m from the car
sound pressure level in the frequency
range 400 Hz to
ated with the tyre's vibration. The 2800 Hz active intensity plot clearly
shows the presence of two equally large sources (Fig. 20). Translation of
the data in Fig. 20 to the surface of the wheel (Fig. 21) shows how the active intensity plot is focussed. Fig. 22 is the calculated sound pressure level
along a line situated at 7,5 m from the car, at 1,5 m above a reflecting
ground in the 100 Hz band centred around 2800 Hz. Notice the very sharp
minimum on axis which is due to the dipole effect. The line length has
been chosen to be from - 5 0 m to + 50 m for the sake of clarity. The calculation is most accurate in the region around the axis i.e. ± 45° corresponding in this case to ± 7,5 m. The overall sound pressure level at 7,5 m from
the car in the A-weighted frequency range from 400 Hz to 2800 Hz is
shown in Fig. 23.
4A. Measurements
on a work station
Two S T S F measurements were performed over the front of a free standing
workstation (0,15 x 0,6 x 0,8 m) at a distance of 7 cm using firstly two reference microphones and secondly one reference accelerometer. The refer23
ence microphones were positioned close to the noise sources of interest: the
fan and the disc drive. The accelerometer was fastened with wax directly
to the disc drive's casing. Figs. 24 to 26 show three active intensity plots
obtained from the measurement using reference microphones. Fig. 24
shows the total field. Figs. 25 and 26 show the partial fields no. 1 and no. 2
respectively. It should be noted that the term partial field as used here
means in fact virtual partial field i.e. a linear combination of the actual
partial fields. These calculated partial fields correspond well with the position of the disc drive and the ventilation grills and other openings in the
cabinet. Fig. 27 was obtained from the measurement which used a reference accelerometer. It represents the part of the total sound field which is
coherent with the vibration of the disc drive's casing. This part of the field
is seen to be very similar to the partial field no. 1. Thus S T S F can be used
to assess the relative importance of the various partial fields on the total
sound field. Alternatively a particular noise source can be "isolated" by
placing an accelerometer directly on its surface. For delicate or lightweight
structures a non-contact laser vibrometer could be used.
Fig. 24. Active intensity:
24
1450 Hz, total field, two reference
microphones
Fig. 25. Active intensity:
1450 Hz, partial field no.l, two reference
microphones
Fig. 26. Active intensity:
1450 Hz, partial field no.2, two reference
microphones 25
Fig. 27. Active intensity: 1450 Hz, total field, one reference accelerometer on the disc
drive
Conclusions
The S T S F technique employs an efficient principal component measurement technique to achieve a cross spectral representation of the sound
field. Three practical instrumentation systems together with menu driven
programs have been developed which enable the user to obtain a wealth of
information about the noise fields under investigation. Examples of applications of the technique have been described to illustrate how S T S F can be
used by noise control engineers.
References
[1]
26
Hald, J.:
" STSF-a unique technique for scan-based near-field
acoustic holography without restrictions on coherence", Technical Review No. 1, 1989, B & K
publication
[2]
Astrup, T.:
"Acoustic intensity and spatial transformation
used
to describe the sound field around a seismic vibrator",
Proceedings of Nordic Acoustic Meeting, 1986,
pp.413-415
[3]
Ginn, K.B. &
Hald, J.:
"Source location using accelerometers as reference
transducers for the STSF technique", Proceedings of
ICA, 1989
[4]
Hald, J.:
"Development of STSF with emphasis on the influence of bandwidth; Part I, Background and Theory",
Noise Con. Proceedings, 1988
[5]
Ginn, K.B. &
Hald, J.:
"Development of STSF with emphasis on the influence of bandwidth; Part II, Instrumentation
and computer simulation", Noise Con. Proceedings, 1988
27
Digital Filter Analysis: Real-time and
Non Real-time Performance
Svend Gade
Abstract
This article demonstrates how the upper frequency limit for a digital realtime filter analyzer can be extended without need for increased processing
capacity or increased sampling frequency.
A consequence of this kind of signal processing is that some of the filters
will be operating in non real-time in such measurement mode.
Sommaire
Cet article demontre comment la limite haute frequence d'un analyseur en
temps reel a filtrage numerique peut etre amelioree sans recourir a l'augmentation des capacites de traitement ou de la frequence
d'echantillonnage.
L'inconvenient de ce type de traitement de signal est que quelques
filtres ne fonctionnent plus en temps reel dans un tel mode de mesure.
Zussammenfassung
Die obere Frequenzgrenze eines Echtzeit-Analysators mit digitalen Filtern laBt sich erhohen, ohne daB groBere Verarbeitungskapazitat oder hohere Abtastfrequenz notig sind. Einige Filter arbeiten dann jedoch nicht
in Echtzeit.
28
Introduction
x
In acoustics there is a long tradition for using /\ octave and V3 octave
analysis i.e. constant percentage bandwidth analysis by means of analogue
filters. The use of digital filters operating in real-time up to 20 kHz was
introduced in the mid seventies Ref. [1]. The interest in having a more
narrow yet relative frequency resolution has resulted in the introduction
of a new generation of digital constant percentage bandwidth filter analyzers with the choice of Vi, V3, V12 and V24 octave analysis, namely the
Briiel & Kjaer Real-time Frequency Analyzer Type 2123 and Dual Channel
Real-time Frequency Analyzer Type 2133.
Using V12 octave and V24 octave analysis, the upper real-time frequency
limit is restricted to respectively 4 kHz and 2 kHz (octave centre frequency in single channel mode) due to the requirement of calculation capacity
in the signal processor. It will be shown how the upper frequency limit can
be extended by two octaves at the cost of losing real-time capacity in the
three highest octaves.
It should be noted that the signal processing described in this article
only applies to analyzers without the High Frequency Expansion Unit,
ZT0318.
Digital Filters
Digital filters, which work on a sample to sample basis, consist of adders,
multipliers and delay units Ref. [2]. The two-pole building blocks found in
2123 and 2133 (using software implementation as shown in Fig. 1), consist
of two adders, two multipliers and two delay units.
Multiple-pole filters (generally required to achieve steep filter roll-off)
can be formed by cascading two-pole sections. By appropriate choice of
the coefficients (B1 and B2) it is possible to generate digital equivalents of
all well-known filter types e.g. Butterworth, Chebyshev, etc. Ref. [3]. Thus
these digital filters have an infinite impulse response (IIR-filters), which
ensures consistency with data bases obtained using analogue filters.
One main advantage of a digital filter is that the same software (or hardware) can be used to generate any filter shape, e.g. Vi octave, Vs octave, V12
octave etc., just by changing the filter coefficients used in the calculations.
As shown later, digital filters are best adapted to logarithmic frequency
scales and constant percentage bandwidth analysis, in contrast to
D F T / F F T analysis which intrinsically yields constant bandwidth and is
best presented on a linear frequency scale.
29
Fig. 1. Block Diagram of two-pole digital filter unit used in 2123 and 2133. Z
of one sampling period
l
is a delay
Parallel Filtering
A parallel bank of Vi octave digital filters is obtained in the following
manner:
Each sample coming from the ADC (Analogue to Digital Converter) is
first bandpass filtered in the highest octave of interest and lowpass filtered (see Fig. 2).
The cut-off frequency of the lowpass filtering is one octave lower than
the previous maximum frequency content. It is now possible to discard
every second sample in the lowpass filtered signal without losing any further information, i.e. once the highest octave in frequency is filtered away
it is quite valid to use half the previous sample rate while still complying
with Shannon's Sampling Theoremf.
These lowpass filtered samples with half the sampling frequency can
now be fed back to the bandpass filter section, and since - for all digital
signal analysis - the filter characteristics are defined only in relation to the
sampling frequency, the same filter coefficients will now give a new octave
bandpass filter one octave lower in frequency.
t Shannon's Sampling Theorem states that a sampled time signal must not contain components at frequencies above half the sampling rate (the so-called Nyquist frequency)
30
Fig, 2. Simplified
Block Diagram for a Digital Filter
Analyzer
In a similar manner, the lowpass filtered samples can be fed back to the
lowpass filter and again filtered one octave lower, once again allowing each
second sample to be discarded, and so on. Hence, a filter with the same
characteristics, but with a centre frequency one octave lower, is created by
halving the sampling rate.
In 2123 and 2133 this process is continued 17 times covering 18 octaves
with octave centre frequencies from 16 kHz down to 0,125 Hz.
Considering the number of samples to be processed in each octave and
calling the number in 16 kHz octave M , the number in the 8 kHz octave is
M
M
/29 in the 4 kHz octave A, and so on. Thus the total number of samples
l
l
to be processed in all octaves below the highest is M( k + k + V8 ) = M,
i.e. the same number of samples as in the highest octave.
Consequently it is only necessary for the bandpass filtering to process
one data value from the highest octave plus one other in one of the lower
octaves in each sampling period to achieve real-time parallel filtering.
This decimation principle yields constant percentage bandwidth analysis which is best presented on a logarithmic frequency scale. The filter coefficients are slightly changed between every second decimation to achieve
Decade filters rather than Octave filters, as specified in international filter
standards.
Octave filters show the same relative filter characteristics when we
move either one octave up or down in frequency, while Decade filters show
the same relative filter characteristics when we move either one decade up
or down in frequency. This means for example, that the octave bandwidth
(3n/10)
filters in single channel mode have centre frequencies given by io
where n is an integer between - 3 and 14.
31
Real-time Analysis
16
Using a sampling frequency of 2 Hz = 65536 Hz the sampling period is
15,3 )us.
To obtain V3 octave single channel real-time analysis up to 22,4 kHz
(16 kHz octave centre frequency) six bandpass and two low pass filiations within one sampling period must be performed (see Fig. 3).
Highest Octave
1
1
BP
A lower Octave
1
BP
11
BP
1
BP
LP
1
—
BP
BP
2
3
LP
1
|
2
|
2
3
I
1
|
|
2
|
|
881718
881718
Fig. 3. Six bandpass and two lowpass filters must be processed within one sampling
1
period to obtain single channel real-time J3 octave analysis
All V3 octave, V12 octave and V24 octave bandpass filters are six-pole filters achieved by cascading three two-pole filter units. The Vi octave bandpass filters are 14-pole filters except for the dual-channel mode with
16 kHz upper frequency limit, in which case six-pole filters are used. The
lowpass filters are ten-pole filters using five two-pole filter units in cascade. Since one mathematical operation in the signal processor (multiplication or summation) takes approximately 100 ns one six-pole bandpass
filtration takes 1,2 jus and one five-pole lowpass filteration 2^s (see Fig. 1).
Thus, the filtering done in Fig. 3 takes about 11,2 ^s, is well within the
sampling period and therefore in real-time.
Except for the order of the filters, the real-time frequency also depends
on the number of measurement channels and the number of filters per
octave (i.e. bandwidth) as shown in Fig. 4.
32
Fig. 4. Real-time frequency depends on the number of channels, the number of filters per
octave (i.e. bandwidth) and the order of the filter
Frequency Extension in Non Real-time
For a single channel V12 octave analysis the real-time frequency limit is up
l
to 4 kHz octave centre frequency. That is four times lower than for /z
octave analysis, because four times more filters have to be processed compared to V3 octave analysis. A maximum of six six-order bandpass filters
including lowpass filtration can be calculated within one sampling period
of 15,3 ^ s .
33
1
Fig. 5. Choice between real-time /i2 octave analysis up to 4 kHz and partly non realtime analysis up to 16 kHz in single channel mode
Since 50% of the calculation time is used for processing the 12 filters in
the 4 kHz octave in real-time, we could utilize this calculation capacity to
process the filters in the three octaves 4 kHz, 8 kHz and 16 kHz but of
course not in real-time.
This gives the choice between V12 octave analysis in real-time up to and
including 4 kHz, or V12 octave analysis up to 16 kHz where the filters up to
2 kHz are processed in real-time, as shown in Fig. 5.
Non Real-time Filtering
In Fig. 6 it is shown for V12 octave single channel analysis how the V12
octave filters are working part-time for the three highest octaves in the
signal processor. The figure shows in which octave and which partial octave group each individual sample will be processed. A partial octave
group consists of three V12 octave filters in this case.
This "Parallel-Serial" processing means that the non real-time filtered
octaves will not be updated simultaneously and with the same speed as the
real-time filtered octaves. In Fig. 7 it is shown in which order the 36 non
real-time Vi2 octave filters are updated in a cycle and displayed on the
screen.
34
Fig. 6. Non real-time filtering in the signal processor. All samples are used except those
for the dummy octave, which are discarded
{
Fig. 7. The order at which the /i2 octave filters are updated on the screen
First the three highest V12 octave filters in the 16 kHz octave are updated. Then the six highest filters in the 8 kHz octave are updated. The last
filters updated in this sequence, except for the dummy octave, are the
three lowest filters in the 16 kHz octave.
35
As can be seen from Fig. 6 the time the filters are active or working in a
cycle, the partial filtering time, T F I L T is one eighth of the cycle time, T T 0 T
^FILT
=
"77 ' ^ T O T
(1)
As will be shown later we also have a partial averaging time, T AVG which is
the effective time the signal from the filters is fed to the detector. This is
one sixteenth of the cycle time.
^AVG
=
~~T7~~ ' ^ T O T
lb
(2)
This means that if in the measurement setup a chosen linear averaging
time, TA is a multiple of the cycle time, then the effective averaging time in
the highest three octaves is only Yi6 of the averaging time indicated in the
measurement setup,
Thus the BT-product achieved in the 4 kHz, 8 kHz and 16 kHz octaves
will be the same as for the 250 Hz, 500 Hz and 1 kHz octaves respectively.
For exponential averaging the achieved BT product will be in agreement
with that in the measurement setup chosen averaging time, when the
warning line beneath the non real-time filters has disappeared.
Partial Filtering Time
Non real-time filtering corresponds to applying a rectangular time window on the signal to be filtered. If this window is too short compared to the
effective duration of the impulse response of the filter, this windowing will
produce a smearing (leakage) of the frequency response of the filter. In
other words, the filters have to "settle" before averaging can take place.
Especially for very narrow filters a long partial filtering time, T F I L T is
required.
Fig. 8 shows a typical impulse response function of a filter. From this
picture it seems reasonable to choose a partial filtering time which is at
least four to six times longer than the rise time, TR of the filter. It is also
shown that the rise time is approximately equal to the reciprocal of the
filter bandwidth, B.
36
Fig. 8. Typical filter impulse response, where TR is the filter rise time and B is the filter
bandwidth
The largest effect of the windowing is found on the partial octave filter
with the lowest centre frequency i.e. for single channel V12 octave analysis
a filter with a centre frequency of fc = 2900 Hz.
Thus for this filter we have
B = fc • ( 2
Tr
1/24
~ ~ir ~
-2"
6 ms
1/24
)^166Hz
(3)
(4)
and a partial filtering time required is at least somewhat longer than
24 - 35 ms.
37
J
Fig. 9. Envelope of impulse response for the /i2 octave filter with a centre frequency of
2900 Hz. Two rectangular windows with a length of 15,625 ms and 62,5 ms are shown
In Fig. 9 two examples of rectangular windows are shown on the envex
lope of the impulse response of the /i2 octave filter with a centre frequency of 2900 Hz.
In the first example a window length of 15,625 ms corresponding to 256
data samples is used (the apparent sampling frequency is
14
fs = 2 Hz = 16384 Hz for the 4 kHz octave).
Fig. 10 shows how the envelope of the filter characteristic of the rectanx
gular window is much broader than the filter characteristic of the /i2
octave filter. Thus the resulting filter characteristics will be highly affected and mainly determined by the rectangular window.
In the second example found in Fig. 9, a window length of 62,50 ms corresponding to 1024 data samples is used.
38
j
Fig. 10. Frequency response of the /i2 octave filter with a centre frequency of 2900 Hz
and the envelope of the frequency response of a rectangular window with a duration of
15,625 ms
Fig. 11 shows the overall filter characteristics, which in this case are
identical to the filter characteristics of the Vi2 octave filter alone as
shown in Fig. 10. Clearly, the windowing in the second example produces
no smearing effect. Thus 62,50 ms is a proper choice of partial filtering
time for the filter to be able to settle before an averaging of the output
from the filter can take place.
For dual channel V12 octave analysis, the lowest non real-time octave
band when using the non real-time frequency extension is 2 kHz. This
means that the required partial filtering time for the filters to settle in this
measurement mode is twice the partial filtering time for single channel
operation, namely 125 ms.
39
1
Fig. 11. Overall frequency response of the fv2 octave filter with a centre frequency of
2900 Hz using a partial filtering time of 62,5 ms
Consequently for single channel Y24 octave analysis the required partial
filtering time will be 250 ms. To avoid making the total averaging time,
T T 0 T (see Fig. 6) or the cycle time (the interval between each screen update) too long, a partial filtering time of 125 ms for the V24 octave single
channel analysis has been chosen. As a consequence a minor smearing effect of the filter characteristics, below - 7 0 dB is found for this measurement mode, see Fig. 12.
Partial Averaging Time
The process of part time averaging also corresponds to applying a time
window on the signal. The signal in this case is the squared output from
the filters. The shortest possible partial averaging time in the signal pro40
}
Fig. 12. /24 octave filter centred at 1433 Hz using a 125 ms partial filtering time. A minor smearing effect is found at low amplitudes due to a "too short" windowing.
cessing that can be chosen, without lack of confidence in the results will be
shown. A short partial averaging time is also necessary to achieve a short
total averaging time.
In principle we could start the part-time averaging of the filter output
simultaneously with the start of the part-time filtering. On the other hand
this would require an unreasonably long averaging time due to the finite
rise time of the filter. Therefore, it is better to wait and start the averaging
when the filter output has settled. In other words, the part-time averaging
should start some time after the start of the part-time filtering, as indicated in the previous section.
For stationary_bandlimited random signals y(t), it can be shown that
the mean value, y and standard deviation, a are given by
41
750220/1
Fig. 13. The ripple components for linear and exponential
sinewaves. TA = 2RC = 2r (twice the time constant)
2
7 =^
2
f y (t)
dt
(RC) averaging of squared
(5)
4 34
a«
ZZ1_
[dB]
(6)
VB • T
where B is the filter bandwidth and T is the total effective averaging time
(Ref. [2]).
This means that it is possible to choose a very short partial averaging
time, T A V G as long as the averaging takes place over many part intervals.
For stationary deterministic signals we have a very different situation.
The partial averaging time, T AVG has to be so long that the ripple on the
partial averaged result is within the specification of accuracy of the analyzer which is ± 0,1 dB.
42
This is because if we average a worst case sinusoidal signal using n noncontiguous time intervals with a partial averaging time, TAVG there is no
guarantee that the result will be any better than the result after the first
partial average.
Let us assume here that a sinusoidal signal with a frequency coinciding
with the centre frequency of the filter is analyzed. The output of the filter
is squared and averaged using an averaging time of TA. For linear averaging we will only get a correct result if the averaging time is a multiple of the
period time of the squared signal, that is, if half an integer number of
periods of the original signal are analyzed (see Fig. 13).
For exponential averaging, a reasonably accurate result is first obtained
after many periods (and not necessarily an integer number) of the signal
have been analyzed (see Fig. 13).
Since the ripple of the filters in the analysis is specified to be within
± 0,05 dB, the ripple of the averaged sinusoidal signal also has to be within a maximum of ± 0,05 dB after one partial averaging time to fulfil the
overall specification of ± 0,1 dB.
For exponential averaging we have the following relationship between
ripple and averaging time (Ref. [2]):
Ripple (dB) = 10 log 1 ±
—
dB
(7)
Using equation (7) we can calculate the minimum partial averaging times
for single and dual channel operations as well as V12 and V24 octave analysis. The results are found in Table 1.
From the results it is seen that the minimum required partial averaging
times are considerably shorter than the minimum required partial filtering times.
In the Digital Filter Analyzers it has been decided to use partial averaging times, T A V G equal to the minimum partial filtering times required.
This decision has been taken in order to avoid making the relative averaging time too small. This yields total partial filtering times that are two
times of that indicated in the previous section.
Thus, for single channel V12 octave analysis we have that T F I L T =
125 ms, TAVG = 62,5 ms and T T 0 T = 1 s (see Fig. 6). This means that all
the non real-time filters are updated on the screen once per second in this
measurement mode (see Fig. 7). The results are summarized in Table 2.
43
™
,
Channels
Filter
F
ilter
B a n d w i d tthh
Centre
F r e q u e n ccyy
Minimum Partial
A v e r a g i nn gg T i m e
1
V12
V12 octaves
2900
2900 Hz
4,74
4,74 ms
ms
2
V12 octaves
octaves
1450 Hz
9,48 ms
1
V24
V24 octaves
1433
1433 Hz
9,59
9,59 ms
ms
T02001GBO
T02001
GBO
Table 1. Minimum
modes
partial
Octave
fsamp
16 kHz
kHz
16
16
8 kHz
4 kHz
2 Hz
1
155
2 Hz
Hz
14
14
2 Hz
averaging
^ "FFIILLT
T
125 ms
125 ms
125 ms
times for the three non real-time
^AVG
^"AVG
62,5 ms
62,5 ms
62,5 ms
Samples/T
TFFIILLTT
2
T
^ TT O TT
13
= 8192
8192
1Iss
12
12
= 4096
Is
1s
11
U
= 2048
Is
1s
2
2
measurement
T02002GB0
Table 2. Sampling frequency, total partial filtering time, partial averaging time and the
number of processed samples for each filter and the cycle time, for the highest three
1
octaves for non real-time single channel /i2 octave analysis
Octave
fsamp
fsamp
^* F FILT
ILT
^^ A
G
AV
VG
Samples/T
TFFIILLTT
T TOT
TOT
8 kHz
1
155
250 ms
125 ms
8192
2 ss
14
14
250 ms
125 ms
4096
2 ss
13
13
250 ms
125 ms
2048
2 ss
4 kHz
kHz
2 kHz
kHz
2 Hz
Hz
2 Hz
2 Hz
T02003GB0
Table 3. Sampling frequency, total partial filtering time, partial averaging time, and
the number of processed samples for each filter for the highest three octaves for non real1
1
time single channel /24 octave and dual channel /i2 octave analysis
The results for V12 octave dual channel and V24 octave single channel
analysis are summarized in Table 3.
If an upper frequency limit of, for example an 8 kHz octave centre frequency for single channel V12 octave analysis is chosen, the above mentioned filtering process is also carried out. The result from the 16 kHz octave band, however, is not displayed or stored.
It should be noted that once the filtering process is initiated it is in progress all the time, whereby the starting point for an averaging can be anywhere within the cycle. Thus the three highest V12 octave filters in the
16 kHz octave are not necessarily the first to be updated on the screen.
44
Examples
The results of single channel V12 octave analysis of a white noise random
signal are shown in Fig. 14. Only in the case where an averaging time of 1 s
(or greater) is used will all of the filters be updated. For a correct scaling of
the non real-time filters, an averaging time which is an integer number of
l
the cycle time must be used. For example when using TA = k s (Fig. 14a)
the non real-time filters have a level which is 4 times (6 dB) too high. This
bias error depends on when in the cycle time the filters/detectors are updated. Worst case is a factor of 8 (9 dB). This means that this bias error, € ^
will have a maximum value of
eb = 10 x log [ n/(n-l
+ 0,125 ) ] dB
(8)
where n is the number of cycles chosen in the measurement setup.
This also means that at least 10 cycles must be chosen as the linear averaging time to ensure an error of less than 0,5 dB. This problem can be
overcome by setting Special Parameter no. 24 in the General Setup, Dis-
1
Fig. 14. A stationary white noise random signal analyzed using single channel /i2 octave analysis, a) Averaging time of 250 ms, b) Averaging time of 500 ms, c) Averaging
time of 750 ms, d) Averaging Time of 1 s. Power Spectral Density scaling is used
45
play of Lin. Leq to Only Final Values. In this case all filters are updated at
the end of the selected linear averaging time.
For running exponential averaging an averaging time of at least two
times the cycle time is sufficient.
Also note that Power Spectral Density (PSD) has been chosen for data
representation to give a "horizontal" spectrum rather than the well-known
spectrum showing a level increase of 3 dB per octave (see Ref. [4]).
The results of the analysis of 10 short impulses are shown in Fig. 15. The
10 impulses which are regarded here as being one transient were recorded
1
Fig. 15. Analysis of a transient using single channel /i2 octave analysis up to 16 kHz
octave centre frequency
a) 10 impulses to be analyzed
1
b) One impulse expanded. The period time is approximately J4 ms indicating that most
of the energy content is found around 4 kHz
c) The transient analyzed using a Is averaging time from 0s to Is
d) The transient analyzed using a 1 s averaging time from 0,1s to 1,1s
46
in time mode (Fig. 15a). One impulse is shown in Fig. 15b. The transient
was then analyzed from Os to I s (Fig. 15c) and from 0,1s to 1,1 s
(Fig. 15 d). Notice how the results in the three highest octaves above the
2,74 kHz cursor position are different due to the lack of real-time
operation.
Conclusion
It has been shown how the upper frequency limit for Vi2 octave and V24
octave digital filter analysis can be extended by two octaves at the cost of
losing real-time capacity in the three highest octaves. This mode should
only be used when analyzing stationary signals. Furthermore, to ensure
correct scaling of all filters, one should either (a) only display the final
values for linear averaging or (b) choose an averaging time which is a multiple (at least a factor of 10) of the analysis cycle time. For exponential
averaging at least a factor of two times the cycle time has to be chosen as
the averaging time. The cycle time is either one or two seconds depending
on the measurement mode. The cycle time is also the time interval between each update of the screen for the non real-time filters. It should also
be noted that the effective averaging time for these three octaves is Vi6 of
the averaging time chosen in the measurement set-up for linear averaging
only.
Here it should be noted that with High Frequency Expansion Unit,
ZT 0318 installed, the signal processing is always in real-time independent
of chosen frequency range and filter bandwidth. The upper frequency limit both in single and dual channel operation with this option is 63 kHz
octave centre frequency for both octave and third octave bandwidth analysis. For V12 and V24 octave bandwidth analysis the upper frequency limits
are 16 kHz and 8 kHz octave centre frequencies respectively.
References
[1]
Roth, O.:
"Digital Filters in Acoustic Analysis Systems ", B & K
Technical Review, No. 1, 1977
[2]
Randall R.B.
"Frequency Analysis". B & K, 1987
47
[3]
Rabiner, L.R.,
Gold, B.
"Theory and Application of Digital Signal
ing" Prentice-Hall, N.J., 1975
[4]
Gade, S.,
Herlufsen, H.:
"Signal Types and Units" B & K Technical Review,
No.3, 1987
48
Process-
Previously issued numbers of
Briiel & Kjaer Technical Review
(Continued from cover page 2)
1-1984 Dual Channel F F T Analysis (Part I)
4-1983 Sound Level Meters - The Atlantic Divide
Design principles for Integrating Sound Level Meters
3-1983 Fourier Analysis of Surface Roughness
2-1983 System Analysis and Time Delay Spectrometry (Part II)
1-1983 System Analysis and Time Delay Spectrometry (Part I)
4-1982 Sound Intensity (Part II Instrumentation and Applications)
Flutter Compensation of Tape Recorded Signals for Narrow Band
Analysis
3-1982 Sound Intensity (Part I Theory).
2-1982 Thermal Comfort.
1-1982 Human Body Vibration Exposure and its Measurement.
4-1981 Low Frequency Calibration of Acoustical Measurement Systems.
Calibration and Standards. Vibration and Shock Measurements.
3-1981 Cepstrum Analysis.
2-1981 Acoustic Emission Source Location in Theory and in Practice.
1-1981 The Fundamentals of Industrial Balancing Machines and Their
Applications.
4-1980 Selection and Use of Microphones for Engine and Aircraft Noise
Measurements.
3-1980 Power Based Measurements of Sound Insulation.
Acoustical Measurement of Auditory Tube Opening.
2-1980 Zoom-FFT
1-1980 Luminance Contrast Measurement.
4-1979 Prepolarized Condenser Microphones for Measurement Purposes.
Impulse Analysis using a Real-Time Digital Filter Analyzer.
3-1979 The Rationale of Dynamic Balancing by Vibration Measurements.
Interfacing Level Recorder Type 2306 to a Digital Computer.
2-1979 Acoustic Emission.
1-1979 The Discrete Fourier Transform and F F T Analyzers.
Special technical literature
Briiel & Kjaer publishes a variety of technical literature which can be obtained
from your local Briiel & Kjaer representative.
The following literature is presently available:
O
O
O
O
O
O
Mechanical Vibration and Shock Measurements (English), 2nd edition
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Catalogues (several languages)
Product Data Sheets (English, German, French, Russian)
Furthermore, back copies of the Technical Review can be supplied as shown in
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