TECHNICAL REVIEW 1999 Characteristics of the Vold-Kalman Order Tracking Filter

TECHNICAL REVIEW 1999 Characteristics of the Vold-Kalman Order Tracking Filter
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TECHNICAL REVIEW
BV 0052 – 11
ISSN 0007–2621
Characteristics of the Vold-Kalman
Order Tracking Filter
1999
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Previously issued numbers of
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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
Non-resonant 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
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1 – 1990 The Brüel & Kjær Photoacoustic Transducer System and its Physical
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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
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
(Continued on cover page 3)
Technical
Review
No. 1 – 1999
Contents
Characteristics of the Vold-Kalman Order Tracking Filter ....................... 1
6. Gade, H. Herlufsen, H. Konstantin-Hansen, H. Vold
Copyright © 1999, 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
Photographer: Peder Dalmo
Characteristics of the Vold-Kalman Order
Tracking Filter
S. Gade, H. Herlufsen, H. Konstantin-Hansen, H. Vold*
Abstract
In this article the filter characteristics of the Vold-Kalman Order Tracking Filter are presented. Both frequency response as well as time response and their
time-frequency relationship have been investigated. Some guidelines for optimum choice of filter parameters are presented.
The Vold-Kalman filter enables high performance simultaneous tracking of
orders in systems with multiple independent shafts. With this new filter and
using multiple tacho references, waveforms, as well as amplitude and phase
may be extracted without the beating interactions that are associated with
conventional methods. The Vold-Kalman filter provides several filter shapes
for optimum resolution and stop-band suppression. Orders extracted as waveforms have no phase bias, and may hence be used for playback, synthesis and
tailoring.
Résumé
Cet article présente les caractéristiques du filtrage Vold-Kalman applicable
pour les suivis d’ordres. Y sont considérées la réponse en fréquence, la réponse
temporelle et les relations fréquence-temps. Quelques conseils et idées directives sur le paramétrage optimal du filtre y sont également donnés.
Le filtrage Vold-Kalman permet d’effectuer un suivi d’ordres simultané et
très performant pour l'analyse de systèmes multi-axiaux. Associé à l’utilisation de multi-références tachymétriques, ce nouvel outil permet d’extraire les
données module, phase et formes d’onde sans les perturbations et les
phénomènes de battement observés avec les méthodes conventionnelles.
Diverses formes de filtre permettent l'obtention d’une résolution toujours opti-
* Vold Solution Inc., USA
1
male et la suppression de la bande d’arrêt. Extraits comme formes d’onde sans
erreur sur la phase, les ordres peuvent être réutilisés pour la relecture, la synthèse et l’optimisation des données.
Zusammenfassung
Dieser Artikel stellt die Filtercharakteristiken des Vold-Kalman-Filters für
Ordnungsanalyse vor. Untersucht wurden Frequenzgang und Zeitcharakteristik sowie die Zeit-Frequenz-Beziehung. Es werden Hinweise zur optimalen
Auswahl von Filterparametern gegeben.
Das Vold-Kalman-Filter erlaubt eine präzise simultane Ordnungsanalyse in
Systemen mit mehreren voneinander unabhängig rotierenden Wellen. Mit
dem neuen Filter können unter Verwendung mehrerer Referenz-Tachogeber
die Ordnungsverläufe sowie Amplituden und Phasen ohne die mit traditionellen Methoden verbundenen Schwebungseffekte ermittelt werden. VoldKalman liefert verschiedene Filterformen, um optimale Auflösung und Selektivität zu erreichen. Als Wellenformen extrahierte Ordnungen besitzen keine
systematischen Phasenfehler und sind deshalb für Playback, Synthese und
“Tailoring” geeignet.
Introduction
Vold and Leuridan [1] introduced in 1993 an algorithm for high resolution,
slew rate independent order tracking based on the concepts of Kalman filters.
Kalman filters have been employed successfully in control and guidance systems since the sixties, with particular applications to avionics and navigation
[8,9]. The main feature of the filters is accurately tracking of signals of known
structure among noise and other signal components with different structure.
The new algorithm has been successful as implemented in a commercial software system in solving data analysis problems previously intractable with
other analysis methods. At the same time certain deficiencies have surfaced,
prompting the development of an improved formulation, in particular the
capability of being able to control the frequency and the time response of the
filter and to separate close and crossing orders [3].
This article gives an introduction to the new Vold-Kalman algorithm,
presents the frequency and time response of the filters and their time-frequency relationship and gives some examples of their applications using
PULSE, the Brüel & Kjær, Multi-analyzer System Type 3560 [4].
2
Overview of Methods for Analysis of Non-Stationary
Signals
A number of traditional analysis methods can be used for the analysis of nonstationary signals and they can roughly be categorized as follows:
1) Divide into Quasi-stationary segments by proper selection of analysis
window
a) Record the signal in a time buffer (or on disk) and analyze afterwards: Scan Analysis
b) Analyze on-line and store the spectra for later presentation and
post-processing: Multifunction/multibuffer measurements
2) Analyze individual events in a cycle of a signal and average over several
cycles: Gated measurements and analysis
3) Sample the signal according to its frequency/RPM variations (adaptive
data resampling): Order tracking measurements
4) Analysis using Autoregressive Signal Modelling: Maximum Entropy
Spectral Analysis [11].
The procedure used for Vold-Kalman order tracking analysis (adaptive filtering, where the presumed model of the signal is not fixed in time or frequency content, but automatically adapts itself as the speed of the device
under test changes) is as described in 1a) above, i.e. a post-processing procedure based on time data.
Order Tracking Analysis
Order tracking is the art and science of extracting the sinusoidal content of
measurements from acousto-mechanical systems under periodic loading/excitation. Order tracking is used for troubleshooting, design and synthesis [5, 18].
Each periodic loading produces sinusoidal overtones, or orders/harmonics, at
frequencies that are multiples of that of the fundamental tone (RPM). The
orders may be regarded as amplitude and phase modulated carrier waves that
vary in frequency. Many practical systems have multiple shafts that may run
coherently through fixed transmissions, or partially related through belt slippage and control loops, or independently, such as when a cooling fan operates
in an engine compartment.
In order analysis we focus on machine revolutions rather than on time as the
base for the signal analysis. Thus in the spectrum domain the focus is on
orders rather than frequency components.
3
Traditional Analysis Techniques for Run-up/
Run-down Tests
One of the main applications of order tracking is analysis of run-ups and rundowns. Investigation of the system responses and dynamic behaviour at the
various rotational speeds is a key element in design, troubleshooting, product
testing and quality control. Depending on the information required from the
test, different analysis techniques can be applied.
The most simple is determination of the acoustic response in terms of the
overall Linear, A, B or C weighted level as a function of RPM. This is often used
in product testing for comparison with reference (tolerance) curves. No diagnostic information is obtained.
“Next level” of analysis is the inclusion of spectral information. In acoustic
testing 1/1-octave or 1/3-octave spectra are often used in order to get the spectral
content in constant percentage filter bands as a function of rotational speed (constant percentage bandwidth means that the bandwidth is proportional with the
frequency). These can reveal frequency regions with annoying resonances and
they relate to the human perception of the radiated noise. Information about the
individual orders cannot be extracted except for the lowest harmonics with 1/3octave filters. Fig.1 shows the contour plot of the acoustic response during a runup of a motor analyzed in 1/3-octaves. The 1st (and to some degree the 2nd) order
is identified together with regions of high response levels. Fig.2 shows the overall level as well as some individual 1/3-octave bands as a function of RPM. More
information about the lower harmonic order components could be obtained by
going to for example 1/12-octave filter bandwidth. Real-time 1/12-octave filters,
however, require more DSP (Digital Signal Processing) power in the system.
More advanced acoustic processing is to use a Loudness Analyzer and store
non-stationary Specific Loudness as a function of RPM, [14]. Non-stationary
Loudness gives a much better model of the human hearing system and takes
into account that the sensitivity of the human ear is not the same for all frequencies and levels as well as the effect of temporal and frequency masking.
In order to resolve the various orders in the frequency domain, narrowband
frequency spectra (FFT without tracking) could be applied. FFT gives the spectral information in constant bandwidth (same bandwidth at all frequencies and
RPMs) and facilitates identification of harmonic families when presented on a
linear frequency axis. This enables diagnosis and source identification.The
advantage of using FFT without tracking is that it does not cost very much in
terms of DSP power. Another advantage is that one FFT analyzer can relate the
analysis to more tacho signals: multi-shaft analysis. A disadvantage is that
4
Fig. 1. 1/3-octave colour contour plot (maximum interpolation used) of the acoustic
response during run-up
Fig. 2. Selected 1/3-octave bands and overall linear level of acoustic response as functions
of RPM
the smearing of the individual (and especially the higher) order components in
the frequency spectra will appear if the run-up / run-down is fast. This is due
to the fact that each FFT spectrum represents a certain time window and
therefore a certain change of the rotational speed. Another aspect is that
rather large Fourier transforms (many lines in the FFT) are required in cases
where the test is running over a wide speed (RPM) range and higher harmonics have to be included in the analysis. Example: RPM range from 600 RPM
(10 Hz) to 6000 RPM (100 Hz), 16 orders included in the spectra (i.e. min. 16 x
100 Hz = 1.6 kHz frequency span) and a resolution of 2 Hz (5 FFT lines per
5
order at 600 RPM) will require 800 FFT lines in the analysis (2 Hz resolution
with a frequency span of 1.6 kHz).
Fig. 3 shows the contour plot of an FFT analysis of the same acoustic
response as in Fig. 1. The harmonic family of the fundamental periodic loading
is easily identified. Resonance frequencies excited by the various harmonic
orders appear on vertical lines parallel to the RPM axis. Fig. 4 shows the first 4
harmonics and the total level in the selected frequency span, as a function of
RPM. This information is obtained by slicing in the contour plot (post-processed in “real-time” during the test) or by pre-processing, i.e. directly measured
by the FFT analyzer during the run-up/run-down. Both order – as well as frequency-slices can be measured/extracted using these methods.
Fig. 3. Contour plot of the FFT spectra of the acoustic response as function of RPM
Fig. 4. Selected orders and total level (in the frequency span) as functions of RPM
6
The above mentioned disadvantages can be overcome by use of tracking.
The tracking technique uses the instantaneous RPM values for calculation of
the samples referenced to the revolution of the rotating shafts instead of the
time clock, see e.g. [5]. Fourier transform of the revolution based samples
results in order spectra instead of frequency spectra and harmonic orders
related to the measured RPM remains in fixed lines in the order spectrum.
This means that smearing of the order related components is avoided and
order components, which might have been smeared out in a frequency spectrum, can be identified. The number of lines required in the order spectra for a
certain test is less than the number of lines needed if frequency spectra were
used. Example: 20 orders is analyzed with a resolution of 0.2 order (5 lines per
order, giving possibility to identify inter-harmonic components) by use of 100
line order spectra independent of the speed range. The price to pay is that the
real-time tracking algorithms require more DSP power than the “normal”
FFT analysis.
Fig. 5 shows the contour plot of order spectra of the same acoustic response
as in Fig. 1 and Fig. 3. The orders appear on vertical lines parallel to the RPM
axis whereas resonances appear on hyperbolic curves (fixed frequency curves
indicated by the cursor). Slices of the first 10 harmonics and the total level as
a function of RPM is shown (as a waterfall plot) in Fig. 6. The order slice
results obtained by the frequency analysis and the order analysis are the
same.
In product testing there is often a requirement of having the results in realtime (on-line) containing information of both individual orders, overall levels
(Linear, A, B or C weighted) and in some cases 1/1- or 1/3-octave bands as well.
This can only be obtained using a combination of the mentioned techniques.
The key issue, in order to reduce the test time and have consistency of data in
the various analyses is to perform the analyses simultaneously. This is the
case in the results illustrated in Fig. 1 through Fig. 6 where the Brüel & Kjær
PULSE, Multi-analyzer System Type 3560 is used. Four analyzers are used in
parallel: A tachometer (for RPM calculations), a 1/3-octave analyzer (1/3octaves and overall levels), an FFT analyzer (frequency spectra) and an order
analyzer (order spectra) giving all the results simultaneously. A loudness analyzer can be included as well. In this example only one tachometer signal is
used. In multiaxle applications the different tachometer signals should be
applied to the tachometer. The scaling of the RPM axis and the calculation of
the order slices can then be referenced to the different rotation speeds (RPMs).
If tracking is used (as in Fig. 5 and Fig. 6) an order analyzer would have to be
defined for each reference tachometer signal.
7
Fig. 5. Contour plot of real-time order tracking spectra (resampling technique) of the
acoustic response
Fig. 6. First 10 harmonics and the total level (in the span of 20 orders), from real-time
order tracking spectra, as functions of RPM (shown as a waterfall plot)
These analysis techniques can be applied in real-time (on-line) which
means that the results are obtained during the test. They are based upon conventional frequency (Fourier) analysis that means that the resolution in the
time domain is linked to the resolution in the frequency domain and vice
versa. This gives a limitation of how accurate phenomena can be identified in
both domains in the same analysis. This is also referred to as the uncertainty
principle. For FFT analysis the relation is written as ∆f × T =1, where T is the
record length (“resolution” in time domain) and ∆f is the line spacing in the
spectrum (“resolution” in frequency domain). If the changes in the signals are
8
too rapid we will not be able to follow (analyze) these changes using these
techniques. The real-time order tracking technique, mentioned above, is based
on resampling and there is a limitation of how fast the speed (RPM) is allowed
to change (also called slew rate limitation).
These limitations are examples of where the Vold-Kalman tracking filter
can be used, rendering a comprehensive set of analysis tools. It minimizes the
resolution problems and has no slew rate limitations.
Applications of Vold-Kalman Order Tracking
Filtering
The Vold-Kalman algorithm enables simultaneous estimation of multiple
orders, effectively decoupling close and crossing orders, e.g., separating drive
shaft orders from wheel orders in suspension tuning. Decoupling is especially
important for acoustics applications, where order crossings cause transient
beating events. The new algorithm allows for a much wider range of filter
shapes, such that signals with sideband modulations are processed with high
fidelity. Finally, systems subject to radical RPM changes, such as transmissions, are tracked also through the transient events (e.g. gear shifts) associated with abrupt changes in inertia and boundary conditions. The goal of
order tracking is to extract selected orders in terms of amplitude and phase,
called Complex Orders or Phase Assigned Orders, or as waveforms. The order
functions are extracted without time delay (no phase shift), and may hence be
used in synthesis applications for sound quality, e.g. removal of nuisance
orders and laboratory simulations. Other applications include multiplane balancing, and measurements of Operational Deflection Shapes, ODS.
Fundamental Notions and Equations
Mechanical systems under periodic loading, such as those with one or more
rotating shafts will respond in measurements with a superposition of sine
waves whose frequencies are integer multiples of the fundamental excitation
frequencies. Notice that this also applies to the limit cycle response of non-linear systems. As the periodic loadings change their speed, the responses will
also change their frequencies accordingly. Since mechanical systems normally
have transfer characteristics dependent upon frequency, the amplitude and
phase of these sine waves will typically also change as the periodic loadings
change their speed. The sine waves whose frequencies are constant multiples
9
of an underlying periodic loading are said to be harmonics or orders of that
loading, also when the multiple is fractional due to gears or belt drives.
It is often helpful to visualize an order as an amplitude/phase modulated
radio signal. The underlying sine wave whose frequency is a multiple of the
fundamental periodic phenomenon would be the carrier wave, while the
slowly varying amplitude and phase function that modulates the carrier wave
is the radio program. A radio receiver demodulates the signal by removing the
carrier wave and playing the modulation function, also known as the (complex) envelope. Now, in mechanical systems, the carrier wave may continually
change its frequency, making the orders similar to amplitude modulated
spread spectrum radio signals.
The goal of order tracking is to extract selected orders in terms of amplitude
and phase or as real time series. These entities will in general be estimated as
functions of time to allow for any pattern of speed or axle RPM variations.
Phasors and Complex Envelopes
The carrier wave of an order can be visualised as a phasor, which is a complex
oscillator with an instantaneous frequency proportional to a constant multiple
of the underlying axle speed as in the equation
t


Θ k ( t ) = exp  2πki f ( u ) du


0
∫
(1)
where the integral of frequency gives the angle travelled by the axle up to the
current time, k is the order number and i = – 1 . Note that the phasor is
always on the complex unit circle.
The amplitude modulated complex order is now given as the product
Ak(t) Θk(t) where Ak(t) is the complex envelope. These complex orders must
occur in complex conjugate pairs to sum to a real signal, such that the total
superposition X(t) of orders relative to an axle can be written as
X(t) =
∑ Ak ( t )Θk ( t )
(2)
k
where k runs over all positive and negative multiples of the underlying axle
speed.
10
Time Variant Zoom
The expression for the phasor (1) implies the functional relationship
Θ k ( t )Θ j ( t ) = Θ k + j ( t )
(3)
In particular, inspection of equation (1) shows that Θ0(t) ≡ 1. When the axle
speed has been estimated as a function of time, equations (2) and (3) then
show that we can centre a designated order at DC (zero frequency) by the time
variant zoom transform
Θ – j ( t )X ( t ) = A j ( t ) +
∑ Ak ( t )Θk – j ( t )
(4)
k≠j
The low frequency complex envelope Aj(t) has now been straightened, and
may, for example, be extracted in the time domain by any suitable lowpass filter. This super-heterodyning process is similar to the tracking of an amplitude
modulated spread spectrum radio source.
Using traditional tracking algorithms the centre frequency of a bandpass
tracking filter is controlled by the instantaneous RPM. The filter AC output is
then detected (squared and averaged) into a “slowly” varying DC using, for
example, a traditional RMS detector, with all its well known limitations,
among those – averaging time dependent ripple in the output as well as loss of
phase information. For the Vold-Kalman processing, the order of interest is
frequency shifted (or zoomed) to DC, a shift which is controlled by the instantaneous RPM and then followed by a lowpass filtration which replaces the
bandpass filtering process. Thus there is no need for any further detection; i.e.
the Vold-Kalman filtration includes a simultaneous envelope detection of both
magnitude and phase. Order Waveforms are then obtained by remodulating
the complex orders by the order frequency carrier wave (given by the RPM
profile), i.e. the complex orders are heterodyned from DC to its original frequency location.
Vold-Kalman Filter
The basic idea behind the Vold-Kalman filter is to define local constraints that
state that the unknown complex envelopes are smooth and that the sum of
the orders should approximate the total measured signal. The smoothness
11
condition is called the structural equation, and the relationship with the
measured data is called the data equation. The somewhat ambiguous notation
X (n ∆t) = X (n), where ∆t is the sampling time increment, is used in this article
to simplify the mathematical exposition.
Structural Equations
The complex envelope Aj (t) is the low frequency modulation of the carrier
wave Θj (t). Low frequency entails smoothness, and one sufficient condition for
smoothness is that the function locally can be represented by a low order polynomial. This condition can be expressed in continuous time using multiple differentiation as
s
d Aj ( t )
= ε(t)
------------------s
dt
(5)
where ε(t) represents higher order terms. The same constraint can be applied
to sampled data by using the difference operator such that
s
∇ A j ( n ) = ε̃ ( n )
(6)
Note that the difference operator of a given order annihilates all polynomials of one order less. Equation (6) is called the structural equation when the
~
right hand side ε(n) is regarded as noise or error. This complex equation will
be satisfied in an approximate sense for all discrete time points in the data
set.
The structural difference equation in the first generation of the VoldKalman algorithm is a real second order equation for the real modulated
waveform
X j ( n ) = 2Re ( A j ( n )Θ j ( n ) )
(7)
that requires a demodulation through the Vandermonde equations to find the
amplitude and phase, see [1]. This operator also only annihilates the constant
polynomial.
12
Data Equation
The structural equation only enforces the smoothness conditions on the estimates of the complex envelopes, such that we need an equation that relates
the estimates to the measured data. The simplest such condition is to state
that the sum of the orders differs only by an error term from the measured
data as expressed in the equation
X(n) –
∑ Aj ( n )Θj ( n ) = δ̃ ( n )
(8)
j∈J
where the summation is for a desired subset of orders.
The above equation (8) is called the data equation when the right-hand side
δ̃ ( n ) is regarded as noise or error. This equation will also be enforced in an
approximate sense for all sampled data points.
Decoupling
When several orders are estimated simultaneously, equation (8) ensures that
the total signal energy will be distributed between these orders, and together
with the smoothness conditions of the structural equation (6) this enforces a
decoupling of close and crossing orders as is demonstrated in the examples.
The mathematics of this procedure is quite analogous to the repeated root
problem in modal analysis, see, [7]. When orders coincide in frequency over an
extended time segment, the allocation of energy to such orders is poorly
defined, and numerical ill conditioning may ensue. Widening the filter bandwidth is one possible remedy in this case.
Example of Tacho Processing
Any method with high resolution needs proper controls, and for the VoldKalman filter this means a very accurate estimation of the instantaneous
RPM such that the tracking filter will follow the peak of the order functions
instead of extracting data from the foothills. The methodology that has been
chosen for the Vold-Kalman filter is that of fitting cubic splines in a least
squares sense to the table of level crossings from a tacho waveform. Choice of
other possible methods is discussed in [1]. Tacho processing is explained in
more details later.
13
Fig. 7. Corrupted tacho waveform, Run-down from 5200 RPM to 400 RPM
Fig. 8. Raw data and spline fit
14
The spline fit enables automatic rejection of outlying data points with a subsequent refit on a censored table of level crossings. The spline fit and censoring
process capability is illustrated by the corrupted tachometer signal from a
5200 RPM to 400 RPM run-down lasting for about 15 s. There are three dropouts with a total duration of 15% of the total measurement time as shown in
Fig. 7. The raw level crossing table and spline fit without rejection is shown in
Fig. 8. The fitted data is then compared with the raw table, the largest deviants
(in this case 1% of the raw RPM table has been specified for rejection) are censored and a refit is performed, with the clean results depicted in Fig. 9. As a
result the three “wild” RPM points have been removed and the spline fit is sufficiently stiff to straddle the empty sections. This is a fully automatic process,
often obviating the need for manual tacho signal repair. Notice that the raw
RPM profiles have been drawn as curves, i.e. with straight-line connections
between the table values, both without and with rejections shown in Figs.8 & 9.
Fig. 9. Censored Data with Corrected Spline Fit (curves offset for ease of comparison)
15
Vold-Kalman Filter Process, Step by Step – an
Example
To illustrate the ease of use of the Vold-Kalman filter process as implemented
in the Brüel & Kjær PULSE, Multi-analysis System Type 3560, a run-up
measurement on a small single shaft electrical motor has been performed.
Fig. 10 shows the vibration response signal, which was recorded together with
a tacho signal. A 1.6 kHz frequency range and a total recording of 20 s have
been selected using a PULSE Time Capture Analyzer [17]. The number of
samples recorded is 81 920 in each channel. 18 s of the recorded signal was
extracted for Vold-Kalman tracking using a delta cursor.
Fig. 10. Vibration time signal of the run-up
a) Overview of the Event using Fourier Analysis
The first step is to use conventional techniques in order to gain some insight
in the harmonic orders of interest, gearshifts etc. Fig. 11 shows a contour plot
of an STFT (Short Time Fourier Transform) of the vibration signal. The record
length of each transform is set to 125 ms (512 samples) resulting in 200 lines
in the frequency domain (line spacing ∆f of 8 Hz). An overlap of 66% is used
resulting in a multi-buffer of 500 spectra covering the selected 18 s. From the
contour plot it is revealed that the dominating orders are nos. 1, 3, 9 and 10
and as expected, no gearshift is present.
16
Fig. 11. An STFT of the vibration signal
b) Tacho Processing
As mentioned earlier any method with high resolution needs proper controls,
and for the Vold-Kalman filter this means a very accurate estimation of the
instantaneous RPM such that the tracking filter will follow orders correctly.
The method of fitting cubic splines to the table of level crossings from a tacho
waveform gives an analytic expression of the RPM as a continuous function of
time, i.e. an instantaneous or “sample by sample” estimate of RPM, with true
tracking of the shaft rotation angle for phase fidelity. As a consequence VoldKalman filtering will produce a complex order value for every measurement
sample point. Fig. 12 shows the tacho settings, including slope, hysteresis and
gearing that defines the table of RPM values, each of them positioned in time
exactly midway between two consecutive tacho pulses as shown in Fig. 8.
The tacho table is divided into a number of segments, in which a cubic least
squares spline fit, allowing for one local minimum and one local maximum, is
applied in order to smooth the data. Fewer the segments, the stiffer and
smoother the interpolation would be. The more segments are chosen, the closer
the interpolation will fit the data points. In practice one should increase the
17
Fig. 12. Tacho setting property page
number of segments until a reasonable match to the raw RPM is obtained. Too
many segments might fit non-physical variations (noise) in the raw RPM.
Continuity and first derivative continuity constraint requirement is applied
between segments. There is also the option of specifying hinge points (singular events) in the spline fit, such that sudden changes in inertial properties
can be tracked, as in the case of clutching and gearshifts by relaxing the first
derivative continuity at shift points. The operator identifies the exact locations of hinge points by expanding the RPM profiles for detailed visual inspection.
The spline fit also permits automatic rejection of outlying data points
(such as tacho dropouts) with a subsequent refit on a censored table of level
crossings as mentioned earlier. The Curve Fit property page is shown in
Fig. 13.
18
Fig. 13. Tacho table Curve Fit property page
Fig. 14. Comparison of the measured (raw, tabular) and curve fitted RPM profiles
Fig. 14 shows a superimposed graph of the measured and the curve fitted
RPM profiles. The maximum slew rate in this case is seen to be approximately
800 RPM/s and the range is between 1000 RPM and 6000 RPM.
19
c) Vold-Kalman Filtering
Fig. 15. Vold-Kalman filter property page
Orders can now be extracted from the signal in terms of waveforms or as Phase
Assigned Orders (Complex Orders).
The Vold-Kalman filter property page is shown in Fig.15. The computational
complexity of the Vold-Kalman filter is proportional to the number of time samples and to the number of orders to be extracted, but also to some degree
depends on filter type, output, bandwidth and decimation. For example, the
computation time for a three-pole filter is 10% longer than for a two-pole filter.
When decoupling is used, however, the computational cost can be high, since
constraint conditions are being enforced between the order function estimates.
Fig. 16 shows the waveform of the 1st, 3rd, 9th and 10th order extracted
using a two-pole Vold-Kalman filter with a bandwidth of 10% (i.e. 10% of the
fundamental frequency). Extracted waveforms can be played via a sound card
and they can be exported as wave files. Sound Quality application is an example were this is very useful.
20
Fig. 16. Waveforms of the 1st, 3rd, 9th and 10th Order, extracted using a two-pole VoldKalman filter with 10% bandwidth
Fig. 17. Magnitude of the 4 most dominating orders as a function of time, extracted using
a two-pole Vold-Kalman filter with 10% bandwidth
21
Extracted as Phase Assigned Orders means that the orders are determined
in terms of magnitude and phase. Fig. 17 shows the magnitude of the Phase
Assigned Orders of the 1st, 3rd, 9th and 10th order, which were the 4 most
dominating orders. A two-pole Vold-Kalman filter with a bandwidth of 10% is
used. It should be noted that the phase of the orders is highly sensitive to
choice of filter bandwidth for time sections with poor signal to noise ratio. This
is most evident when displaying unwrapped phase.
The Vold-Kalman analysis is a time-based analysis yielding results, Phase
Assigned Orders and Waveforms as a function of time. Often it is desirable to
plot the Phase Assigned Orders as a function of RPM, which can be easily
done by exporting the order of interest and the RPM-profile to a spreadsheet.
The magnitude of the first order is shown in Fig.18 using MS Excel. The decimation features shown in Figs. 12 and 15 are used in this case in order to
reduce the amount of data to a size, which can be handled by a spreadsheet.
Fig. 18. Magnitude of the first order as a function of RPM, extracted using a two-pole
Vold-Kalman filter with 10% bandwidth and plotted via MS Excel spreadsheet
Filter Characteristics in Frequency Domain
Bandwidth selection is done in terms of constant frequency bandwidth or proportional to RPM bandwidth (i.e. constant percentage bandwidth). The bandwidth specification in the Brüel & Kjær Vold-Kalman implementation [16] is
in terms of the half – power points, i.e. 3 dB bandwidth. Bandwidth proportional to RPM is recommended for the analysis of higher harmonic orders or
analysis of wide RPM ranges.
22
The filter shape is measured by sweeping a sine wave through a VoldKalman filter with a fixed centre frequency and fixed bandwidth. A sweep rate
of 1Hz/s is used for measuring the filter shapes, shown in Fig.19, of a VoldKalman filter with a centre frequency of 100 Hz and a bandwidth of 8 Hz. The xaxis, which is a time axis scaled in seconds, can directly be interpreted as a frequency axis scaled in Hz (with a fixed offset). It is seen that a one-pole filter has
very poor selectivity, a two-pole filter has a much better selectivity, whereas a
three-pole filter provides the best selectivity. The 60 dB shape factor, i.e. the
ratio between the 60dB bandwidth and the 3 dB bandwidth is often used for
describing the selectivity of a filter. The 60 dB shape factor has been measured
for the one-, two- and three-pole filters for bandwidths in the range from
0.125 Hz to 16Hz. These tests showed that the 60dB shape factor for a given
pole specification increases slightly as a function of bandwidth. The one-pole filter has a 60dB shape factor of approximately 50 (variation from 48.8 to 50.8).
The two-pole filter has a 60 dB shape factor of approximately 7.0 (variation from
6.80 to 7.07) and the three-pole filter has a 60dB shape factor of approximately
3.6 (variation from 3.58 to 3.68). Thus the selectivity of the three-pole filter is
twice as good as the two-pole filter and 14 times better than the one-pole filter.
Good selectivity is important to avoid interference (or leakage) between orders.
Fig. 19. Comparison of filter shapes for one-, two- and three-pole Vold-Kalman filters with
a bandwidth of 8 Hz
23
Fig. 20. Comparison of the frequency response in the passband for one-, two- and threepole Vold-Kalman filters with the same bandwidth of 8 Hz
Another characteristic of the filter is the frequency response within the
passband. As seen in Fig. 20 the two- and three-pole filters have a much flatter
frequency response in the passband compared to the one-pole filter, with the
three-pole filter having the flattest frequency response. The flatness of the frequency response in the passband is important when analyzing the amplitude
and phase modulation of the harmonic carrier frequency. Amplitude and
phase modulation corresponds in the frequency domain to sidebands centered
around the harmonic carrier frequency component, which means that the
more flat the frequency response is the more correct the modulation will show
up the filter analysis.
The phase of the Frequency Responses for the different filter types is zero in
the full frequency range within the measurable dynamic range. Thus there is
no phase distortion or time delay for tracked orders including modulations
identified as lower and upper sidebands.
24
Filter Characteristics in Time Domain
The time response of Vold-Kalman filters is important to understand when
analyzing transient phenomena and responses to lightly damped resonances
being excited during a run-up or a run-down. The time response has been
investigated by applying a tone burst with a certain duration to a VoldKalman filter with a fixed centre frequency corresponding to the frequency of
the toneburst. In Fig. 21 the magnitude (envelope) of the response of a filter
centered at 100 Hz with a bandwidth of 8 Hz is shown using a logarithmic yaxis. A 100 Hz tone burst with a duration of 1 s is applied to the filter. One
very important feature is that the time response is symmetrical in time, i.e. it
appears to behave like a non-causal filter. This is because Vold-Kalman filtering is implemented as post processing allowing for non-causal filter implementation and extraction of order waveforms with no phase bias or distortion,
i.e. without a time delay. Fig. 21 shows the time response for one-, two- and
three-pole filters with a bandwidth of 8 Hz.
The one-pole filter has, as expected, the shortest decay time and an exponential decay which appears as a straight line when displayed with a logarithmic y-axis, while the two-pole and three-pole filters in addition to the longer
Fig. 21. Comparison of the magnitude of the time responses for one-, two- and three-pole
Vold-Kalman filters with a bandwidth of 8 Hz. The applied signal, a tone burst of 1 s duration, is shown as well
25
Fig. 22. Comparison of Time Responses for one-pole 2 Hz, 4 Hz and 8 Hz bandwidth VoldKalman filters. The applied signal is a tone burst of 1 s duration
decay times also show some lobes. The main lobes of all three filter types show
on the other hand nearly the same progress in the upper 25 dB, i.e. the same
“early decay”, which means their behaviour in terms of how fast they can follow amplitude changes of orders are nearly identical.
Fig. 22 shows the time response of a one-pole filter for 3 different choices of
filter bandwidths. As expected the decay time is inversely proportional to the
bandwidth. Since the slope for a one-pole filter is very similar to the slope of
the early decay for two- and three-pole filters with the same bandwidth we
can extract the following important time-frequency relationship for all three
types of Vold-Kalman filters,
B3dB × τ = 0.2
(9)
where B3dB is the 3 dB bandwidth of the Vold-Kalman filter and τ is the time
it takes for the time response to decay 8.69 dB ( = 10 × log e2). One pole filter is
based on a second order structural equation, i.e. a one degree of freedom resonator (an SDOF system) which has the time-frequency relationship, σ × τ = 1
(see Ref. [19]) corresponding to B3dB × τ = 1/π. Thus we have an optimum time26
frequency relationship close to the Heisenberg’s uncertainty limit using VoldKalman filtering. Due to its symmetry the efficient duration of the time
response has to be considered 2τ rather than τ. If reverberation time T60
instead of time constant τ is preferred, the relation (9) becomes,
B3dB × T60 = 1.4
(10)
When zooming in around the beginning or the end of the tone burst a difference between the three filter types is revealed as seen in Fig. 23. The one-pole
filter has a smooth decay before the stop of the tone burst, whereas the twopole and the three-pole filters show a ripple with a maximum deviation (overshoot) from the steady state response of 0.28 dB and 0.46 dB respectively. This
overshoot phenomenon is only seen in analysis results when analyzing signals
with abrupt amplitude changes (such as in the case of a tone burst) or when a
too narrow filter bandwidth is selected for the analysis (i.e., the time constant,
τ of the filter is too long for the signal to be analyzed).
As an additional observation, all time responses have decayed to – 6dB, irrespective of the chosen filter parameters, at the location where the tone burst stops,
Fig. 23. Detailed picture of the time response of the one-, two- and three-pole filters at the
end of the tone burst
27
see Figs. 21 and 22. (i.e. where the energy of the order signal inside the analyzed time window is reduced by 3 dB). Due to the sudden change in the
nature of the signal, from a sine wave to nothing, a further leakage of the
order, into neighbouring frequencies by 3 dB is seen. A similar effect is
observed using FFT analysis. See Appendix A fore more details.
The phase curve is zero degrees in the full time range where the time burst
exists as expected, except for some small edge effects (less than 1 degree for
the 8 Hz fillter) at the two discontinuities, i.e., the Vold-Kalman filter has no
time delay for a tracked signal.
Phase of Orders and Waveforms
The major question is: What is the reference for the detected phase? For the
Phase Assigned Autospectrum in PULSE, the phase spectrum assigned to the
autospectrum is the phase of the cross-spectrum between the selected signal
and the reference signal selected by the user, e.g. a tacho signal or any other
suitable signal. This is a typical dual channel measurement.
For the Vold-Kalman filtering the situation is slightly different, the phase
is extracted from the vibration/acoustic signal itself, i.e. from one signal only:
In a time domain model the Vold-Kalman filter fits sine waves at the
selected order/carrier frequency to the vibration/acoustic signal. For a onepole filter three data samples are fitted at a time in a recursive manner [2,
20]. For two and three pole filters more samples are used in the curve fit/
filtering process.
The phase of a particular Phase Assigned Order component at time zero
(beginning of time record) will be the phase at time zero for the curve fitted sine
wave result, similar to the phase of a sine wave component in a Fourier Spectrum which is the phase of this component at the beginning of the time record.
The standard convention is as follows: if a sine wave has its maximum amplitude at time zero, then the phase of its Fourier component is per definition 0
degrees. If a sine wave has its minimum amplitude at time zero, then the phase
is 180 degrees. For zero crossings with positive and negative slopes the phase is
minus 90 and plus 90 degrees respectively. Thus the starting phase for a Phase
Assigned Order is the phase of the order signal itself irrespective of the actual
starting phase of the tacho reference / carrier / RPM signal. This is also a consequence of the fact that only the RPM profile (i.e., frequency) and not phase is
derived from the tacho signal. The phase of the tacho reference/carrier/RPM
signal can then be regarded to have zero phase at time zero for any order of
interest, and the phase for any order is thus assigned to the phase of the rotat28
ing shaft, i.e. the tacho signal with an arbitrary offset. The phase at any later
time for the Phase Assigned Orders depends on the transfer properties
between the forcing functions from the rotating shaft to the measurement/
observation position.
When calculating the order waveform from the Phase Assigned Orders and
the RPM, the order waveform will have a phase at time zero which is the combination of the phase of the carrier wave (per definition 0 degrees) and the
phase of the phase assigned order at time zero, which is the measured phase of
the order waveform at time zero. Thus the output order waveform will have no
phase shift, i.e. no time delay, with respect to the measured signal and thus can
be used in synthesis studies as shown later.
Selection of Bandwidth and Filter Type
Selection of the filter bandwidth is basically a compromise between having a
bandwidth which is sufficiently narrow to separate the various order components in the signal and a bandwidth which is sufficiently wide, i.e. giving a
sufficiently short filter response time, in order to follow the changes in the signal amplitude. The contour plot of the STFT analysis can be used for evaluating the separation of the various components. Using the radio signal analogy
this means that not only the carrier frequency located at the centre frequency
of the filter, but also the AM/FM modulation components should be inside the
filter passband, since it is these components that contain the information of
interest. An extremely narrow filter that suppresses all the sidebands would
produce an order with constant magnitude as a function of time and constant
slope of the phase.
Various research tests have shown that when orders pass through a resonance the time constant of the filter, τ, should be shorter than 1/10 of the time,
T3dB , it takes for the particular order to sweep through the 3 dB bandwidth of
the resonance, ∆ f3dB. This ensures an error of less than 0.5 dB of the peak
amplitude at the resonance using a one-pole filter. For two- and three-pole filters the error of the measured peak will be less. If no resonance is observed in
the extracted orders the (minimum) time it takes for the order to increase/
decrease 6 dB may be used instead.
Thus for the time constant of the filter we have that:
τ ≤ (1/10) × T3dB
(11)
or when combining (9) and (11) in terms of the bandwidth of the filter:
29
B3dB = 0.2 /τ ≥ 2 / T3dB
(12)
The time it takes for order number k to sweep through the 3dB bandwidth is
T3dB = ∆ f3dB /(k × SRHz)
(13)
T3dB = ∆f3dB /(k × SRRPM / 60)
(14)
or
where SRHz and SRRPM is the sweep rate in Hz/s and RPM/s, respectively.
This means that the bandwidth, B3dB of the Vold-Kalman filter extracting
order number k should follow,
B3dB ≥ (2 × k × SRHz) /∆ f3dB
(15)
B3dB ≥ (k × SRRPM) / (30 × ∆ f3dB)
(16)
or
i.e. the selected bandwidth should be chosen so that it is proportional to the
order number and the sweep rate, and inversely proportional to the bandwidth of the resonance of interest.
When the sweep rate and bandwidth in (16) are unknown, it is more practical to use equation (12) as shown in example 2.
Example 1
In the first example a linear sweep of a square wave, with a sweep rate of
17200 RPM/s from 12000 RPM to 63000 RPM (286.7 Hz/s from 200 Hz to
1050 Hz), passing through a known resonance is analyzed. The resonance frequency is 795 Hz and the 3 dB bandwidth of the resonance is 16 Hz, corresponding to 1% damping. The first three orders are analyzed. Using (16) we
have for the 1st order:
B3dB ≥ (1 × 17200) / (30 × 16) Hz = 35.8 Hz
for the 2nd order:
30
B3dB ≥ (2 × 17200) / (30 × 16) Hz = 71.6 Hz
and for the 3rd order:
B3dB ≥ (3 × 17200) / (30 × 16) Hz = 107.5 Hz
The Vold-Kalman filter bandwidth can be specified in terms of constant frequency bandwidth or proportional to RPM bandwidth (i.e., constant percentage bandwidth). For proportional bandwidth the value is entered as a
percentage of the basic shaft speed, thus 10% bandwidth gives a resolution of
0.1 order. Proportional bandwidth is the best choice when analyzing over wide
RPM ranges or when analyzing higher orders. A bandwidth of 35.8 Hz for the
1st order at the resonance frequency of 795 Hz corresponds to 4.5% bandwidth, a bandwidth of 71.6 Hz for the 2nd order at 795 Hz corresponds to 18%
bandwidth and a bandwidth of 107.5 Hz for the 3rd order at 795 Hz corresponds to 40% bandwidth. Fig. 24 shows the magnitude of the Phase Assigned
Orders extracted with a two-pole filter with proportional bandwidth of 4.5%,
18% and 40% for the 1st, 2nd and 3rd orders respectively.
Fig. 24. Magnitude of the Phase Assigned Orders of the first three orders extracted with a
two-pole Vold-Kalman filter with bandwidths of 4.5%, 18% and 40% respectively
31
The peak amplitudes measured with one-, two- and three-pole filters with
bandwidths of 4.5%, 18% and 40% for the 1st, 2nd and the 3rd orders respectively are given in Table 1. The correct peak amplitudes were found by widening the filter bandwidth until the amplitude did not increase any more.
Table of measured
peak amplitudes
One-pole filter
4.5%, 18%, 40%
Two-pole filter
4.5%, 18%, 40%
Three-pole filter
4.5%,18%, 40%
Correct
Amplitude
1st order
–5.3 dB
–5.1 dB
–5.0 dB
–5.0 dB
2nd order
–6.3 dB
–6.0 dB
–6.0 dB
–6.0 dB
3rd order
–7.4 dB
–7.2 dB
–7.1 dB
–7.1 dB
Table 1. Peak amplitudes in dB for the 1st, 2nd and 3rd order component extracted with
one-, two-, and three-pole Vold-Kalman filters with bandwidths of 4.5%, 18% and 40%
respectively
Fig. 25. Magnitude of the Phase Assigned Orders of the first three orders extracted with a
one-pole Vold-Kalman filter with bandwidths of 4.5%, 18% and 40% respectively. Notice
the interference due to the limited selectivity of the one-pole filter
32
The peak amplitude errors for the one-pole filter is thus 0.3dB and for the twoand three-pole filters within 0.1dB having a minimum bandwidth given by (16).
A second resonance at 1900 Hz, being excited by the second and the third
orders, is also seen in Fig. 24.
Using a filter with proper selectivity is very important for the analysis. This
is illustrated in Fig. 25, which shows the result of the Vold-Kalman filtering
using the one-pole filter instead of the two-pole filter used in Fig. 24. All other
analysis parameters are kept unchanged. The limited selectivity of the onepole filter causes a lot of interference from the other orders especially at the
positions where these pass through the resonances. The interference is most
dominant for the 3rd order due to the wider bandwidth needed to extract this
order. The interference from the 2nd order can even lead to misinterpretations
of “non-existing” resonances. Decoupling cannot be used to avoid this kind of
interference over a wide time span. Using the two-pole filter (Fig. 24) a small
amount of interference is still seen for the 3rd order in the analysis. The threepole filter will completely suppress the interference from the other orders in
this case.
Fig. 26. Detailed view of the part in the contour plot where the 2nd and 3rd order components excite the first resonance. The free decay of the damped natural frequency of 795 Hz
is clearly seen. A 3200 line analysis, giving ∆ f of 2 Hz, and a step of 10 ms between the
spectra is used
33
The ripples indicated in Fig. 24, on the decaying slope after the orders have
passed the resonance still need some explanation. These ripples are caused by
an interaction between the order component and the free decay of the natural
frequency for the lightly damped resonance. This phenomenon can be investigated by looking at the contour plot of an STFT analysis. Fig. 26 shows a
detailed view of the part in the contour plot where the 2nd and 3rd order components excite the first resonance. A 3200 line analysis, giving a ∆f of 2 Hz,
and a step of 10 ms between the spectra (corresponding to 98% overlap) is
used. The free decay of the resonance after the point in time where the orders
have “crossed” the resonance frequency is clearly seen. When the decaying
oscillations of the damped natural frequency are inside the passband of the filter extracting the given order, the beating interference will occur. The beating
is most severe for the third order because of the wider bandwidth used in the
analysis. Since there is no “natural” tacho signal, which relates to the damped
natural frequency, it is not possible to make decoupling of these components.
The only way to get less beating interaction is to use a narrower filter bandwidth in order to get the free decaying natural frequency faster outside the
passband bandwidth after the resonance crossing of the order. This will, however, cause violation on the requirement of the minimum bandwidth given by
(12), (15) or (16).
Example 2
In this example a fast run-up of a spin - drier is analyzed. A tacho signal giving 12 pulses per revolution is used and the vibration responses in the tangential, radial and axial directions are measured. Fig. 27 shows the contour plot of
the STFT analysis of the radial response. It is seen that the response is dominated by the 1st order (unbalance) and the 22nd order. Each Fourier transform is based upon a record length of 250ms giving a line spacing ∆f of 4 Hz.
The 1st order is dominated by one resonance. The run-up takes approximately 6 seconds and the curve fitted RPM profile is shown in Fig. 28.
The peak value and the time, T3 dB it takes for the 1st order to sweep
through the 3 dB bandwidth of the dominating resonance is found by applying a three-pole filter with wide bandwidth (up to 100%). Using a bandwidth
of more than 100% gives ripples due to beating interference even with the
three-pole filter. From these analyses T3 dB is found to be 464 ms and the peak
of the resonance is found to be 12.6 dB. Using (12) this means that the minimum bandwidth should be 4.31 Hz. The peak of the resonance is at 681 RPM
(11.3 Hz) which means that the minimum bandwidth should be 38%. Using a
bandwidth of 38% gives a peak value of 12.2 dB (i.e., an error of 0.4 dB).
34
Fig. 27. Contour plot of an STFT analysis of a run-up of a spin drier
Fig. 28. Curve-fit of RPM as a function of time used as input for Vold-Kalman filtering
35
Fig. 29. 1st order of the radial, tangential and axial response, during a run-up of the spin
drier, extracted using a three-pole Vold-Kalman filter with 50% bandwidth
The same peak value is found using one-pole and two-pole filters. For the
one-pole filter with 38% bandwidth the extracted order is, however, contaminated by ripples (beating interference), even at the resonance, due to the
limited selectivity. Fig. 29 shows the 1st order of the radial, tangential and
axial response extracted using a three-pole filter with a bandwidth of 50%.
The same resonance is seen in the axial response, whereas the dominating
resonance in the tangential response is at 911 RPM (15.2 Hz). A lower resonance at 375 RPM (6.25 Hz) is seen in the radial and axial response and at
388 RPM (6.47 Hz) in the tangential response. T3 dB for this resonance is
found to be 415 ms for the radial response indicating that the bandwidth
should be at least 76%. Using 50% bandwidth with a three-pole filter gives
an under-estimation of approximately 0.8 dB. A two-pole filter gives a beating interference at this resonance with bandwidth larger than 50%, and
proper measurement of this resonance is not possible with a one-pole filter
due to strong beating interference even for bandwidth as narrow as 20%.
The 22nd order can be extracted using a three-pole filter with a bandwidth
of 60%, which is found to be the minimum bandwidth for the dominating
36
resonance at 933 RPM (15.6 Hz) in the radial response. A two-pole filter gives
a small interference at the resonance with 60% bandwidth, and for a one-pole
filter interference is experienced for bandwidths wider than 40%.
Crossing Orders
To illustrate the effectiveness of the Vold-Kalman filter with decoupling of
close and crossing orders, two signals have been mixed, a 1 kHz signal and a
300 Hz to 2000 Hz swept signal containing several orders as shown in the
STFT contour plot in Fig. 30. The duration of the signal is 6 s. The example
simulates a system with two independent axles. All orders and the 1 kHz sine
wave were generated with constant amplitude.
The first two swept orders and the 1 kHz signal were extracted using 10%
bandwidth (0.1 order resolution) two-pole Vold-Kalman filters without decoupling. The magnitude of the two swept orders is shown in Fig. 31 and the 1 kHz
signal is shown in Fig. 32. As seen in this case the 1 kHz order strongly interacts with the swept 4th order around time 0.1s, the 3rd order around time 0.4s,
Fig. 30. An STFT of a signal mixed from a 1 kHz sine wave and a swept signal containing
several harmonics (orders)
37
Fig. 31. First and second order of the swept signal extracted without decoupling using
two-pole Vold-Kalman filter with a bandwidth of 10%
Fig. 32. 1 kHz signal extracted without decoupling using two-pole Vold-Kalman filter with
a bandwidth of 10%
38
Fig. 33. First and second order of the swept signal extracted using decoupling and twopole Vold-Kalman filter with a bandwidth of 10%
Fig. 34. 1 kHz signal extracted using decoupling and two-pole Vold-Kalman filter with a
bandwidth of 10%
39
the 2nd order around time 1 s and the first order around time 2.7 s respectively, showing strong beating phenomena.
When the two tacho signals are used in a simultaneous estimation (i.e. with
decoupling), but with the same filter parameters as in the single order estimation (i.e. without decoupling), we achieve a dramatic improvement in the quality of estimation, see Fig. 33 and Fig. 34. However, the 1 kHz still interacts
with the swept orders nos. 3 & 4, since they were not included in the calculations.
Crossing orders from two independent motors
In this example the vibration response from two independent rotating motors
is analysed. The response is measured at the support structure in a situation
where one motor is running up and the other is running down resulting in
crossing orders. In Fig. 35 the speed versus time functions (RPM profiles) of
the two motors are shown.
The first order from the two motors is extracted using a two-pole VoldKalman filter with a bandwidth of 20%. The magnitude of the phase assigned
orders is shown in Fig. 36.
A strong interaction between the two orders is seen in the area where the
orders are crossing, i.e. where both orders are inside the filter passband.
Applying the decoupling technique with the filter setting otherwise
unchanged the same two first order components are extracted as shown in
Fig. 37.
In order to verify the fidelity of the decoupling technique an analysis of a
run-up of motor 2, with motor 1 switched off, was performed. The magnitude of
the first order of motor 2 from this test is shown as a function of time in Fig. 38.
The same filter parameters as in the previous tests are used. The shape of the
order is very similar to that calculated using decoupling shown in Fig. 37 verifying the validity of the decoupling technique. The acceleration of motor 2, in
the case with motor 1 switched off, was a little lower compared to the case
where both motors were running. The order, shown as a function of time, in
Fig. 38 therefore seems stretched (delayed) compared to that shown in Fig. 37.
This could also be one of the reasons why the peak amplitude of the dominating resonance (at 1.3 s and 1.4 s respectively) gets a little higher in Fig.38.
Another reason could be that the resonance is rather sensitive to boundary
conditions and these might not be exactly the same in the two cases.
40
Fig. 35. RPM as a function of time (speed profile) for motor 1 (green) and motor 2 (blue)
Fig. 36. First order of the two motors extracted using a two-pole Vold-Kalman filter with a
bandwidth of 20%. Notice the strong interaction (beating) between the two orders in the
area where the orders are crossing
41
Fig. 37. First order of the two motors extracted using decoupling. Vold-Kalman filter setting otherwise as in Fig.36 without the decoupling. Notice that the interaction (beating)
between the orders is avoided
Fig. 38. First order in a run-up of motor 2 without motor 1 running. The Vold-Kalman
filter parameters are the same as those used in Fig. 36 and Fig. 37. The result should be
compared to that of motor 2 in Fig. 37 using decoupling
42
Sound Quality Synthesis
Since the Vold-Kalman filter extracts order waveforms without time delay, i.e.,
the extracted order time signals are coincident with the total signal; these
time signals can be used for synthesis studies. This is especially interesting in
the field of Sound Quality Engineering, where time-, frequency- and orderediting and simulation of acoustic signals is an important tool for product
sound optimisation. In the following simple example the Brüel & Kjær Sound
Quality Software Type 7698 has been used for post-processing of Vold-Kalman
data. For sound perception the most dominating order is the 9th in the example shown in Figs. 10 – 17. The 9th order waveform was subtracted (any
amount of attenuation is possible) from the total waveform using the mixer
editing facility in version 3.0 of Type 7698 software. The time signal with the
9th order removed is shown in Fig. 39, but a comparison with Fig. 10 reveals
no apparent visual difference, due to the fact that the amplitude of the 9th
order (see Fig. 16) is about a factor of two smaller than the amplitude of the
first order.
Fig. 39. Vibration time signal of the run-up with the 9th order removed (The amplitude
scale changed to Pa)
43
Fig. 40. An STFT of the vibration signal with the 9th order removed
Fig. 41. Loudness analysis of the 9th order. Especially the frequency masking effect is
clearly seen
44
It is on the other hand quite evident from the STFT contour plot shown in
Fig. 40 compared to Fig. 11, that the 9th order has been removed as well as
this was audible from a playback via the sound card of the original and edited
signals. Since Type 7698 is a dedicated package for sound measurement, the
amplitude scaling is in SPL or Pa rms amplitudes rather than vibration levels
or vibration rms amplitudes. The SQ application provides the ability to listen
to both the extracted orders and to the residual sound with the orders
removed.
The individual orders may be analysed using STFT, digital filtering (using
standardised filter shapes such as 1/3-octaves), Loudness Analysis or other
techniques such as Wavelets or Wigner-Ville distribution [11, 21]. Fig. 41
shows an example where the 9th order waveform has been analysed using
non-stationary Loudness calculations in order to study both time and frequency masking effects of the specific order.
Gearshift Example
The measurement was done on a light truck with a V-8 engine and an overdrive automatic transmission, which was run on a dynamometer in a semianechoic room. Data was acquired using an eight-channel DAT recorder and
shown here calibrated in volts, i.e. uncalibrated with respect to engineering
units, such as Pa and m/s2. Using PULSE Interface to SONY DAT – Type
7706 the digital data was interpolated and resampled and directly analyzed
by PULSE via a SONY PCIF 500 SCSI interface box without any need for further digital/analogue/digital conversions. For this example a full-throttle runup was performed under light load condition, using a tractive (drag) force of
only 50 lbf (≈ 222 N). Fig. 42 shows the RPM versus time curves from the
tachometers on both the engine and the drive shaft (propeller shaft) for the
complete run using a wide frequency and time range pre-analysis. See also
Ref. [3] for more details.
Microphone signals from a binaural recording using a Brüel & Kjær HATS
(Head and Torso Simulator) Type 4100 in the passenger seat, as well as accelerometer signals at several locations (pinion gear housing, transmission case
at drive shaft side etc.), were recorded. The right ear signal from the HATS
was analyzed in order to illustrate the capability of the Vold-Kalman Order
Tracking Filter for handling gearshifts. It was decided to focus on the shift
from the 2nd gear into the 3rd gear around time 11,5 s shown in Fig. 42. Thus a
second PULSE multi-analysis of the recording was performed using a Tachometer Analyzer for triggering and RPM detection and monitoring purposes, an
FFT analyzer for real-time FFT-order processing and a Time Capture Analyzer for capturing data for Vold-Kalman order tracking analysis.
45
Fig. 42. RPM profiles of run-up, with gear shift events
Fig. 43. Contour plot of Sound recording at right ear. Orders 4, 8 and 12, the engine firing
harmonics are indicated
For the FFT analysis a 400 Hz baseband, 100 line analysis (T = 250 ms and
∆f = 4 Hz) with Maximum overlap was performed. Maximum overlap was in
this case 99,5% since each FFT calculation including averaging etc. took
1,4 ms. Exponential averaging with 1 average was used. Spectra as well as preprocessed order slices were stored into a multi-buffer, using the 1900 RPM of
46
the drive shaft as a start reference, i.e. used as time zero. Update interval was
10 ms with a total of 551 spectra corresponding to a duration of 5,5 s.
The Time Capture Analyzer was also triggered to start at 1900 RPM making
a 5 s recording up to approximately 2800 RPM using a frequency range of
1600 Hz corresponding to a total of 20 480 time samples per channel.
Fig. 43 presents a contour plot of the microphone (time captured) signal from
the right ear. 400 lines FFT with 80% overlap and a total of 96 spectra are
shown. As expected the contour plot indicates that the dominant frequency
content is found along the half-order lines of the engine, especially also at
orders 4.0, 8.0 and 12.0, the V-8 engine firing harmonics. As seen the 4th order
is the dominating order both before and after the gearshift.
The data represents a typical case of high slew rates, especially the engine
RPM at the shift points (up to 2800 RPM/s). Fig. 44 displays the engine RPM
curve near the shift from 2nd into 3rd gear, showing the curvefit RPM
(smoothed) curve overlaid with the raw estimate (RPM table). The hinge point
was found from the raw RPM by using the “display zoom” facility and visual
inspection as well as the standard cursor readings, Maximum and Minimum
Values (in this case found by PULSE at 1,758 s and at 2,280 s respectively).
These numbers were then keyed into the Vold-Kalman RPM Curve Fit property page (see Fig. 13). In the actual calculation the Number of Segments was
set to 10 even though fewer segments could have been used successfully.
Fig. 45 shows the magnitude of engine order 4.0, extracted using a three pole
Vold-Kalman filter with a relative bandwidth of 20% (≈ 10 Hz). Fig. 46 shows
the magnitude of engine order 4.0 using real-time FFT processing. Also an
order slice bandwidth of 20% corresponding to approximately 2,5 FFT lines
was used for comparison purposes. Due to the record length of 250 ms, all FFT
based events are displaced, compared to the Vold-Kalman extracted order,
with a delay of 125 ms corresponding to ½ the FFT record length.
When comparing the two 4th order slices the two methods agree extremely
well for the slowly changing amplitudes, but the FFT is not able to accurately
track the two rapid level changes which are seen in the Vold-Kalman results
around the shift point between time 2,0 s and 2,4 s. This is due to the fact that
the Vold-Kalman filtering has no slew rate limitations although the real-time
FFT was actually able to track the RPM changes in this case. On the other
hand the Vold-Kalman filtering provides a better time resolution as well as
more data points compared to the FFT technique as explained in the following.
The FFT multi-buffer settings result in 501 data points along the 5 s long Zaxis (25 multibuffer entries per FFT record-length, one entry per 7 FFT calculations). Each FFT record represents approximately 94 ms (= 250 ms × 37,5%)
47
Fig. 44. Detail of the Raw RPM (red colour) and the Curve Fit RPM (blue colour) at the
shift point
Fig. 45. Engine order 4.0 extracted using Vold-Kalman order tracking filtering. X-axis is
displayed from 0 s to 5s
when using a Hanning weighting function. The effective duration of the Hanning weighting is defined in Ref. [5]. The Vold-Kalman technique results in
this case in 20480 data points (as mentioned earlier, for export purposes decimation of the data is possible). The selected frequency bandwidth of 10 Hz corresponds to a length of the IRF of 2τ = (0,2 × 2 / 10 Hz) = 40 ms.
48
Fig. 46. Engine order 4.0 extracted using real-time FFT processing. X-axis is displayed
from 0,125 s to 5,125 s
In order to verify that the 4.0 order was extracted correctly, the order waveform was subtracted from the original signal using the PULSE Sound Quality
application. A contour plot of the signal with the 4th order removed was successfully produced using a procedure as described earlier, see Figs. 11 & 40.
In Ref. [3] also a file of the residual sound was generated by extracting the 14
most significant engine orders, and then subtracting this file from the original
file. This was done using the MTS* Systems Corporation IDEAS Sound Quality software. The fact that this residual sound has consistently lower levels
(about 30 dB) indicates that both the magnitude and phase of the extracted
orders were processed with high fidelity, even passing through the gearshift.
When listening to the residual sound produced by removing the Vold-Kalman
filtered orders, only very little trace of the engine sound was heard through the
gearshift. This fact is also a confirmation that the amplitude and phase tracking is very accurate, and it also illustrates the practical value of the VoldKalman methods for harmonic extraction and editing in Sound Quality applications involving high slew rates as found at gear shifts.
* Acknowledgement
The tape data was provided by courtesy of MTS Systems Corporation.
49
Conclusion
The Vold-Kalman filter enables order tracking without tracking-ability
slew rate limitations, with only speed limitation due to the filter response
time. Phase assigned orders (shown as real, imaginary, magnitude, phase and
Nyquist plots) as well as order time waveforms (playback using sound cards)
are available. Abrupt changes of the RPM, such as in gear shifts, and tacho
drop-outs can be handled and finally decoupling of close and crossing orders is
possible. The only disadvantages of the technique is non-real time processing,
“longer” calculation time, no information between orders and some prior
knowledge of the contents of the signal is required.
The characteristics of the one-pole, two-pole and three-pole Vold-Kalman
order tracking filters have been investigated in the time and the frequency
domain. The three-pole filter has the best selectivity and therefore the best
ability to suppress ripples due to beating interference from the other order
components in the signal. In the time response to a tone burst the two- and
three-pole filters exhibit small ripples (overshoot). This will, however, only
contaminate the results when the signal contains abrupt changes in the
amplitude or when the bandwidth of the filter is selected too narrow for the
signal.
The time-frequency relationship of the three filter types shows an optimum
relationship close to the Heisenberg’s uncertainty limit and is given by B3dB × τ
= 0.2, where B3dB is the 3 dB bandwidth of the Vold-Kalman filter and τ is the
time it takes for the time response to decay 8.69 dB (a factor of e).
Selection of the bandwidth of the filter should follow B3dB ≥ 2 / T3dB, where
T3dB is the time it takes for an order to sweep through the 3 dB bandwidth of a
resonance (or the time it takes for an order to change 6 dB in amplitude). In
almost all cases the three-pole filter is the best choice due to its better selectivity in the frequency domain. Today the main use of single pole filter is to be
able to duplicate processing done in earlier implementation of the VoldKalman filtering.
In situations where different orders related to different rotating shafts
(tacho signals) are close or crossing each other, decoupling can be used to separate the orders without beating interference.
50
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
Vold, H., Leuridan, J., “Order Tracking at Extreme Slew Rates, Using
Kalman Tracking Filters”, SAE Paper Number 931288, 1993
Vold, H., Mains, M., Blough, J., “Theoretical Foundations for High Performance Order Tracking with the Vold-Kalman Tracking Filter”, SAE
Paper Number 972007, 1997
Vold, H., Deel, J., “Vold-Kalman Order Tracking: New Methods for Vehicle Sound Quality and Drive Train NVH Applications”, SAE Paper
Number 972033, 1997
Brüel & Kjær, “PULSE, the Multi-Analyzer System - Type 3560”, Product Data, BP 1611 - 14, BP 1795
Gade, S., Herlufsen, H., Konstantin-Hansen, H., Wismer, N.J., “Order
Tracking Analysis”, Technical Review No.2, 1995, Brüel & Kjær
Blough, J., Brown, D., Vold, H., “The Time Variant Discrete Fourier
Transform as an Order Tracking Method”, SAE Paper Number 972006,
1997
Vold, H., Kundrat, J., Rocklin, G., Russell, R., “A Multi-Input Modal
Estimation Method for Mini-Computers”, SAE Paper Number 820194,
1982
Kalman, R. E., “A new approach to linear filtering and prediction problems”, Trans. Amer. Soc. Mech. Eng., J. Basic Engineering, 82, 32 – 45,
1960
Kalman, R. E., Bucy, R. S., “New results in linear filtering and prediction
theory”, Trans. Amer. Soc. Mech. Eng., J. Basic Engineering, 83, 95 –
108, 1961
Vold, Herlufsen, Mains, Corwin-Renner, “Multiple Axle Order Tracking
with the Vold-Kalman Tracking Filter”, Sound and Vibration Magazine,
30 – 34, May 1997
Leuridan, J., Van der Auweraer, H., Vold, H., “The Analysis of Nonstationary Dynamic Signals”, Sound & Vibration Magazine, 14 – 26,
August 1994
Randall, R.B., “Frequency Analysis”, Brüel & Kjær, 1987
Brigham, E.O., “The Fast Fourier Transform”, Prentice-Hall, Inc.,Englewood Cliffs, N.J., 1974
Gade S., Herlufsen. H., Konstantin-Hansen, H., Ladegaard, P., “Simultaneous Zwicker Loudness and One Third Octave Measurements”, Internoise Proceedings, Christchurch, NZ, 1998
Gade S., Herlufsen. H., “Use of weighting Functions in DFT/FFT Analysis”, Brüel & Kjær Technical Reviews Nos. 3 & 4, 1987
51
16) Brüel & Kjær, “Vold-Kalman Order Tracking Filter – Type 7703”, Product Data, BP 1760
17) Brüel & Kjær, “Time Capture – Type 7705”, Product Data, BP 1762
18) Brüel & Kjær, “Order Analysis – Type 7702”, Product Data, BP 1634
19) Gade S., Herlufsen. H., “Digital Filter Techniques vs. FFT Techniques
for Damping Measurement”, Brüel & Kjær Technical Review No. 1, 1994
20) Leuridan, J., Kopp, G.E., Moshrefi, N., Vold, H., “High Resolution Order
Tracking Using Kalman Tracking Filters – Theory and Applications”,
SAE Paper Number 951332, 1995
21) Gade, S., Gram-Hansen, K., “Non-stationary Signal Analysis using
Wavelet Transform, Short-time Fourier Transform and Wigner-Ville Distribution”, Brüel & Kjær Technical Review Nos. 2, 1996
52
Appendix A
We have observed that all time response curves are crossing at −6 dB at the
two points in time where the applied tone burst is started and where it is discontinued, Figs. 21 & 22. At theses points in time we have that ½ the filter
time response contains the signal while the rest of the filter time response
contains no signal. Thus the overall time response has decayed by 3 dB (half
power) while the signal inside the tracking filter has decayed by 6 dB. A similar phenomena is observed using FFT analysis and is explained in details in
the following.
As an illustration, an FFT analysis using 400 lines, frequency range of fspan
= 1600 Hz, resolution of ∆f = 4 Hz and a record length of 250 ms is used. A sinusoidal signal of 1Vrms, 800 Hz (i.e., 200 periods) is analyzed as shown in Fig. A1.
Cursor readings show in Fig. A2 the level to be 0 dB at 800 Hz as well as the
total power is 0 dB. No leakage is seen.
Fig. A1. Time signal of a 1Vrms , 200 period, 800 Hz sine wave analyzed using a rectangular weighting function
53
Fig. A2. Frequency spectrum of a 200 period, 800 Hz sine wave analyzed using a rectangular weighting function, no leakage is seen
Using a rectangular weighting with an amplitude of A and time length of T
results in a spectrum weighting of
H(f) = AT × sin(πTf) / (πf )
(A1)
which is convolved with the spectrum of the 800 Hz signal and then sampled
(in this case) at frequencies which is multiples of ∆f = 4 Hz. The weighting
function and its corresponding spectrum are shown in Fig. A3. A main lobe
with a maximum amplitude of AT and a width of 2∆f is shown, and all side
lobes have a width of ∆f. In this case the spectrum is sampled at the centre of
the main lobe and at all zero crossings between the lobes resulting in a calculated/displayed FFT spectrum consisting of one line only, which is scaled to an
amplitude value of 1 corresponding to 0 dB. See Ref. 12 (Appendix A) and Ref.
13 (Chapter 2).
54
Frequency
a
c
Time
T
AT
A
b
d
Time
T /2
2 ∆f
Frequency
A
AT/2
4 ∆f
990182
Fig. A3. Time and frequency scaling of a rectangular weighting function
Now the second half of the record is zero padded, corresponding to using a
rectangular weighting function with same amplitude, A, but of half the
length, T/2. See Fig. A4. This weighting results in a spectrum
H(f) = AT/2 × sin(πTf/2) / (πf )
(A2)
which is shown in Fig. A3. All the lobes have double width compared to the
non-zero padded case, i.e. the main lobe has a width of 4∆f and the side lobes
have a width of 2∆f. The amplitude has decreased by a factor of 2, corresponding to – 6dB. Thus the corresponding FFT spectrum of the 800 Hz sine wave
will show a line spectrum where the main lobe is sampled 3 times and furthermore sampled at the centre of all side lobes as well as at the zero crossings
between all lobes. See Fig. A5. The amplitudes/levels of the individual FFT
lines follow the well known sequence, 1, 2/π, 2/3π, 2/5π, 2/7π, … , see Fig. 2.15
in Ref. [12].
55
Fig. A4. Time signal of a 200 period, 800 Hz sine wave analyzed using half a rectangular
weighting function, i.e. only 100 periods are within the window
Fig. A5. Frequency spectrum of a 200 period, 800 Hz sine wave analyzed using half a rectangular weighting function, i.e. only 100 periods are within the window
56
The cursor readings show a level of –6 dB at 800 Hz as expected according to
the discussion above. The total level is –3 dB (half power), which is also
expected since the time window contains data in only half the record in this
case.
This is also the case using other time weighting functions as long as they are
symmetrical in time. Figs. A6 & A7 show the results using Hanning weighting.
In this case the resulting spectrum is most easily understood by convolving
the FFT spectrum shown in Fig. A5, with the FFT spectrum of a Hanning
weighting function, taking amplitude as well as phase into account. The
phase of the spectrum shown in Fig. A5 is +90 degrees for all non-zero amplitude values at frequencies lower than the centre frequency, 0 degrees at the
centre frequency and –90 degrees for all non-zero amplitude values at frequencies higher than the centre frequency, see Fig. 2.24 Ref. [12]. The Fourier
spectrum of a Hanning (scaled to an amplitude of 2 in the centre of the record)
consists of three values, 1 at the centre frequency, f0 and –½ (or ½ and a phase
of 180o) at the two neighbouring frequencies, f0 – ∆f and f0 + ∆f. From the
mean square values the Noise Bandwidth of the Hanning weighting is found
to be 12 + ½2 + ½2 = 1.5 or 1.76 dB. See ref. [15].
Fig. A6. Time signal of a 200 period, 800 Hz sine wave analyzed using half a Hanning
weighting function, i.e. only 100 periods are within the window
57
Fig. A7. Frequency spectrum of a 200 period, 800 Hz sine wave analyzed using half a Hanning weighting function, i.e. only 100 periods are within the window
Convolution of the spectrum in Fig. A5 with the spectrum of a Hanning
results in the spectrum shown in Fig. A7 as explained in the following.
When one of the zero values are convolved with the Hanning the result is an
amplitude value which is the mean value of the two adjacent lines but with
opposite phase.
When one of the non-zero side lobe values are convolved with the spectrum
of a Hanning, the original value is unchanged due to the fact that the two adjacent lines have zero amplitude values.
When the centre frequency value at f0 is convolved with the spectrum of a
Hanning the original is unchanged due to the fact that the two adjacent lines
have equal amplitude but opposite phase.
To convolve the two lines adjacent to the centre frequency with the spectrum
of the Hanning requires a little more complex calculation, resulting in a slight
increase in amplitude from 0.32 (–9.9 dB) to 0.40 (–7.9 dB) and a phase change
from respectively 90° to 128° for the left, (f0 – ∆f) and –90° to –128° for the
right, (f0 – ∆f) FFT line.
Thus for the resulting spectrum shown in Fig. A7 the centre line will also in
this case show a level of –6 dB and the total power of the spectrum –3 dB, after
58
compensating the total reading for the Noise Bandwidth (1.76 dB) of the Hanning weighting.
Conclusion
When the FFT time record contains a stationary sinusoidal signal in only the
first or the last half of the record the total power of the spectrum is decreased
by 3 dB irrespective of choice of weighting function as long as this function is
symmetrical in the record. Half of the power/energy of this signal (–6 dB) is
found at the expected frequency while the rest (– 6 dB) is found as leakage.
Using a Vold-Kalman order tracking filter, which also has a symmetrical time
response, the leakage components are normally not seen, while the –6 dB level
of the tracking frequency is observed in the time response as discussed above
as well as shown in Fig. 22.
59
Previously issued numbers of
Brüel & Kjær Technical Review
(Continued from cover page 2)
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
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:
❍
❍
❍
❍
❍
Acoustic Noise Measurements (English), 5th. Edition
Noise Control (English, French)
Frequency Analysis (English), 3rd. Edition
Catalogues (several languages)
Product Data Sheets (English, German, French,)
Furthermore, back copies of the Technical Review can be supplied as shown in the
list above. Older issues may be obtained provided they are still in stock.
BV0052-11_omslag.qxd
10/02/00
13:53
Page 1
TECHNICAL REVIEW
BV 0052 – 11
ISSN 0007–2621
Characteristics of the Vold-Kalman
Order Tracking Filter
1999
HEADQUARTERS: DK-2850 Nærum · Denmark
Telephone: +45 45 80 05 00 · Fax: +45 45 80 14 05
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