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The Fundamentals of
Signal Analysis
Application Note 243
2
Table of Contents
Chapter 1 Introduction 4
Chapter 2 The Time, Frequency and Modal Domains:
A matter of Perspective 5
Section 1: The Time Domain 5
Section 2: The Frequency Domain 7
Section 3: Instrumentation for the Frequency Domain 17
Section 4: The Modal Domain 20
Section 5: Instrumentation for the Modal Domain 23
Section 6: Summary 24
Chapter 3 Understanding Dynamic Signal Analysis 25
Section 1: FFT Properties 25
Section 2: Sampling and Digitizing 29
Section 3: Aliasing 29
Section 4: Band Selectable Analysis 33
Section 5: Windowing 34
Section 6: Network Stimulus 40
Section 7: Averaging 43
Section 8: Real Time Bandwidth 45
Section 9: Overlap Processing 47
Section 10: Summary 48
Chapter 4 Using Dynamic Signal Analyzers 49
Section 1: Frequency Domain Measurements 49
Section 2: Time Domain Measurements 56
Section 3: Modal Domain Measurements 60
Section 4: Summary 62
Appendix A The Fourier Transform: A Mathematical Background 63
Appendix B Bibliography 66
Index 67
3
Chapter 1
Introduction
The analysis of electrical signals is In Chapter 3 we develop the proper- Because of the tutorial nature of this
a fundamental problem for many ties of one of these classes of analyz- note, we will not attempt to show
engineers and scientists. Even if the ers, Dynamic Signal Analyzers. These detailed solutions for the multitude of
immediate problem is not electrical, instruments are particularly appropri- measurement problems which can be
the basic parameters of interest are ate for the analysis of signals in the solved by Dynamic Signal Analysis.
often changed into electrical signals range of a few millihertz to about a Instead, we will concentrate on the
by means of transducers. Common hundred kilohertz. features of Dynamic Signal Analysis,
transducers include accelerometers how these features are used in a wide
and load cells in mechanical work, Chapter 4 shows the benefits of range of applications and the benefits
EEG electrodes and blood pressure Dynamic Signal Analysis in a wide to be gained from using Dynamic
probes in biology and medicine, and range of measurement situations. The Signal Analysis.
pH and conductivity probes in chem- powerful analysis tools of Dynamic
istry. The rewards for transforming Signal Analysis are introduced as Those who desire more details
physical parameters to electrical sig- needed in each measurement on specific applications should look
nals are great, as many instruments situation. to Appendix B. It contains abstracts
are available for the analysis of elec- of Agilent Technologies Application
trical signals in the time, frequency This note avoids the use of rigorous Notes on a wide range of related
and modal domains. The powerful mathematics and instead depends on subjects. These can be obtained free
measurement and analysis capabili- heuristic arguments. We have found of charge from your local Agilent
ties of these instruments can lead to in over a decade of teaching this field engineer or representative.
rapid understanding of the system material that such arguments lead to
under study. a better understanding of the basic
processes involved in the various
This note is a primer for those who domains and in Dynamic Signal
are unfamiliar with the advantages of Analysis. Equally important, this
analysis in the frequency and modal heuristic instruction leads to better
domains and with the class of analyz- instrument operators who can intelli-
ers we call Dynamic Signal Analyzers. gently use these analyzers to solve
In Chapter 2 we develop the concepts complicated measurement problems
of the time, frequency and modal with accuracy and ease*.
domains and show why these differ-
ent ways of looking at a problem
often lend their own unique insights.
We then introduce classes of instru-
mentation available for analysis in
these domains.
* A more rigorous mathematical justification for the
arguments developed in the main text can be found
in Appendix A.
4
Chapter 2
The Time, Frequency and
Modal Domains:
A Matter of Perspective Section 1: The Time Domain This electrical signal, which repre-
sents a parameter of the system, can
In this chapter we introduce the be recorded on a strip chart recorder
The traditional way of observing
concepts of the time, frequency and as in Figure 2.2. We can adjust the
signals is to view them in the time
modal domains. These three ways of gain of the system to calibrate our
domain. The time domain is a record
looking at a problem are interchange- measurement. Then we can repro-
of what happened to a parameter of
able; that is, no information is lost duce exactly the results of our simple
the system versus time. For instance,
in changing from one domain to direct recording system in Figure 2.1.
Figure 2.1 shows a simple spring-
another. The advantage in introducing
mass system where we have attached
these three domains is that of a Why should we use this indirect
a pen to the mass and pulled a piece
change of perspective. By changing approach? One reason is that we are
of paper past the pen at a constant
perspective from the time domain, the not always measuring displacement.
rate. The resulting graph is a record
solution to difficult problems We then must convert the desired
of the displacement of the mass
can often become quite clear in the parameter to the displacement of the
versus time, a time domain view of
frequency or modal domains. recorder pen. Usually, the easiest way
displacement.
to do this is through the intermediary
After developing the concepts of each of electronics. However, even when
Such direct recording schemes are
domain, we will introduce the types measuring displacement we would
sometimes used, but it usually is
of instrumentation available. The normally use an indirect approach.
much more practical to convert
merits of each generic instrument Why? Primarily because the system in
the parameter of interest to an
type are discussed to give the reader Figure 2.1 is hopelessly ideal. The
electrical signal using a transducer.
an appreciation of the advantages and mass must be large enough and the
Transducers are commonly available
disadvantages of each approach. spring stiff enough so that the pen's
to change a wide variety of parame-
ters to electrical signals. Micro- mass and drag on the paper will not
phones, accelerometers, load cells,
conductivity and pressure probes are
just a few examples.
Figure 2.1 Figure 2.2
Direct Indirect
recording of recording of
displacement - displacement.
a time domain
view.
5
affect the results appreciably. Also Figure 2.3
the deflection of the mass must be Simplified
large enough to give a usable result, oscillograph
operation.
otherwise a mechanical lever system
to amplify the motion would have to
be added with its attendant mass
and friction.
With the indirect system a transducer
can usually be selected which will not
significantly affect the measurement.
This can go to the extreme of com-
mercially available displacement
transducers which do not even con-
tact the mass. The pen deflection can
be easily set to any desired value by
controlling the gain of the electronic
amplifiers.
Figure 2.4
This indirect system works well Simplified
until our measured parameter begins oscilloscope
operation
to change rapidly. Because of the (Horizontal
mass of the pen and recorder mecha- deflection
nism and the power limitations of circuits
its drive, the pen can only move omitted for
at finite velocity. If the measured clarity).
parameter changes faster, the output
of the recorder will be in error. A
common way to reduce this problem
is to eliminate the pen and record on
a photosensitive paper by deflecting
a light beam. Such a device is
Another common device for display- The strip chart, oscillograph and
called an oscillograph. Since it is
ing signals in the time domain is the oscilloscope all show displacement
only necessary to move a small,
oscilloscope. Here an electron beam is versus time. We say that changes
light-weight mirror through a very
moved using electric fields. The elec- in this displacement represent the
small angle, the oscillograph can
tron beam is made visible by a screen variation of some parameter versus
respond much faster than a strip
of phosphorescent material. time. We will now look at another
chart recorder.
It is capable of accurately displaying way of representing the variation of
signals that vary even more rapidly a parameter.
than the oscillograph can handle.
This is because it is only necessary to
move an electron beam, not a mirror.
6
Section 2: The Frequency Figure 2.5
Any real
Domain waveform
can be
It was shown over one hundred years produced
ago by Baron Jean Baptiste Fourier by adding
sine waves
that any waveform that exists in the
together.
real world can be generated by
adding up sine waves. We have illus-
trated this in Figure 2.5 for a simple
waveform composed of two sine
waves. By picking the amplitudes,
frequencies and phases of these sine
waves correctly, we can generate a
waveform identical to our Figure 2.6
desired signal. The relationship
between the time
Conversely, we can break down our and frequency
domains.
real world signal into these same sine a) Three-
waves. It can be shown that this com- dimensional
bination of sine waves is unique; any coordinates
real world signal can be represented showing time,
frequency
by only one combination of sine
and amplitude
waves. b) Time
domain view
Figure 2.6a is a three dimensional c) Frequency
graph of this addition of sine waves. domain view.
Two of the axes are time and ampli-
tude, familiar from the time domain.
The third axis is frequency which
allows us to visually separate the
sine waves which add to give us our
complex waveform. If we view this
three-dimensional graph along the
frequency axis we get the view in
Figure 2.6b. This is the time domain
view of the sine waves. Adding them
together at each instant of time gives However, if we view our graph along sine wave, we have uniquely
the original waveform. the time axis as in Figure 2.6c, we characterized our input signal in the
get a totally different picture. Here frequency domain*. This frequency
we have axes of amplitude versus domain representation of our signal
frequency, what is commonly called is called the spectrum of the signal.
the frequency domain. Every sine Each sine wave line of the spectrum
wave we separated from the input is called a component of the
appears as a vertical line. Its height total signal.
represents its amplitude and its posi-
tion represents its frequency. Since
we know that each line represents a
* Actually, we have lost the phase information of the sine
waves. How we get this will be discussed in Chapter 3.
7
The Need for Decibels
Since one of the major uses of the frequency Figure 2.8
domain is to resolve small signals in the The relation-
presence of large ones, let us now address ship between
the problem of how we can see both large decibels, power
and small signals on our display and voltage.
simultaneously.
Suppose we wish to measure a distortion
component that is 0.1% of the signal. If we set
the fundamental to full scale on a four inch
(10 cm) screen, the harmonic would be only
four thousandths of an inch (0.1 mm) tall.
Obviously, we could barely see such a signal,
much less measure it accurately. Yet many
analyzers are available with the ability to
measure signals even smaller than this.
Since we want to be able to see all the Figure 2.9
components easily at the same time, the Small signals
only answer is to change our amplitude scale. can be measured
with a logarithmic
A logarithmic scale would compress our large
amplitude scale.
signal amplitude and expand the small ones,
allowing all components to be displayed at the
same time.
Alexander Graham Bell discovered that the
human ear responded logarithmically to power
difference and invented a unit, the Bel, to help
him measure the ability of people to hear. One
tenth of a Bel, the deciBel (dB) is the most
common unit used in the frequency domain
today. A table of the relationship between
volts, power and dB is given in Figure 2.8.
From the table we can see that our 0.1%
distortion component example is 60 dB below
the fundamental. If we had an 80 dB display
as in Figure 2.9, the distortion component
would occupy 1/4 of the screen, not 1/1000
as in a linear display.
8
It is very important to understand Figure 2.7
that we have neither gained nor lost Small signals a) Time Domain - small signal not visible
information, we are just represent- are not hidden
in the frequency
ing it differently. We are looking at domain.
the same three-dimensional graph
from different angles. This different
perspective can be very useful.
Why the Frequency Domain?
Suppose we wish to measure the
level of distortion in an audio oscilla-
tor. Or we might be trying to detect
the first sounds of a bearing failing on
a noisy machine. In each case, we are
trying to detect a small sine wave in b) Frequency Domain - small signal easily resolved
the presence of large signals. Figure
2.7a shows a time domain waveform
which seems to be a single sine wave.
But Figure 2.7b shows in the frequen-
cy domain that the same signal is
composed of a large sine wave and
significant other sine wave compo-
nents (distortion components). When
these components are separated in
the frequency domain, the small
components are easy to see because
they are not masked by larger ones.
The frequency domain's usefulness
is not restricted to electronics or
mechanics. All fields of science and
engineering have measurements like The Frequency Domain: sounds out of loud background noise
these where large signals mask others A Natural Domain thanks in part to its frequency
in the time domain. The frequency domain capability. A doctor listens
domain provides a useful tool in At first the frequency domain may to your heart and breathing for any
analyzing these small but important seem strange and unfamiliar, yet it unusual sounds. He is listening for
effects. is an important part of everyday life. frequencies which will tell him
Your ear-brain combination is an something is wrong. An experienced
excellent frequency domain analyzer. mechanic can do the same thing with
The ear-brain splits the audio spec- a machine. Using a screwdriver as a
trum into many narrow bands and stethoscope, he can hear when a
determines the power present in bearing is failing because of the
each band. It can easily pick small frequencies it produces.
9
So we see that the frequency domain Figure 2.10
is not at all uncommon. We are just Frequency
not used to seeing it in graphical spectrum
examples.
form. But this graphical presentation
is really not any stranger than saying
that the temperature changed with
time like the displacement of a line
on a graph.
Spectrum Examples
Let us now look at a few common sig-
nals in both the time and frequency
domains. In Figure 2.10a, we see that
the spectrum of a sine wave is just a
single line. We expect this from the
way we constructed the frequency
domain. The square wave in Figure
2.10b is made up of an infinite num-
ber of sine waves, all harmonically
related. The lowest frequency present
is the reciprocal of the square wave
period. These two examples illustrate
a property of the frequency trans-
form: a signal which is periodic and
exists for all time has a discrete fre-
quency spectrum. This is in contrast
to the transient signal in Figure 2.10c
which has a continuous spectrum.
This means that the sine waves that
make up this signal are spaced
infinitesimally close together.
Another signal of interest is the
impulse shown in Figure 2.10d. The
frequency spectrum of an impulse is
flat, i.e., there is energy at all frequen-
cies. It would, therefore, require
infinite energy to generate a true
impulse. Nevertheless, it is possible
to generate an approximation to
an impulse which has a fairly flat
spectrum over the desired frequency
range of interest. We will find signals
with a flat spectrum useful in our
next subject, network analysis.
10
Network Analysis Figure 2.11
One-port
If the frequency domain were network
restricted to the analysis of signal analysis
spectrums, it would certainly not be examples.
such a common engineering tool.
However, the frequency domain is
also widely used in analyzing the
behavior of networks (network
analysis) and in design work.
Network analysis is the general
engineering problem of determining
how a network will respond to an
input*. For instance, we might wish
to determine how a structure will
behave in high winds. Or we might
want to know how effective a sound
absorbing wall we are planning on
purchasing would be in reducing
machinery noise. Or perhaps we are
interested in the effects of a tube of
saline solution on the transmission of
blood pressure waveforms from an
artery to a monitor.
All of these problems and many more
are examples of network analysis. As
you can see a "network" can be any
system at all. One-port network
analysis is the variation of one
parameter with respect to another,
both measured at the same point
(port) of the network. The impedance
or compliance of the electronic
or mechanical networks shown in
Figure 2.11 are typical examples of
one-port network analysis.
* Network Analysis is sometimes called Stimulus/Response
Testing. The input is then known as the stimulus or
excitation and the output is called the response.
11
Two-port analysis gives the response Figure 2.12
at a second port due to an input at Two-port
the first port. We are generally inter- network
analysis.
ested in the transmission and rejec-
tion of signals and in insuring the
integrity of signal transmission. The
concept of two-port analysis can be
extended to any number of inputs
and outputs. This is called N-port
analysis, a subject we will use in
modal analysis later in this chapter.
We have deliberately defined network
analysis in a very general way. It
applies to all networks with no
limitations. If we place one condition
on our network, linearity, we find
that network analysis becomes a
very powerful tool. Figure 2.13
Linear network.
Figure 2.14 Figure 2.15
Non-linear Examples of
system non-linearities.
example.
2
1
1
2
12
When we say a network is linear, we Figure 2.16
mean it behaves like the network A positioning
in Figure 2.13. Suppose one input system.
causes an output A and a second
input applied at the same port causes
an output B. If we apply both inputs
at the same time to a linear network,
the output will be the sum of the
individual outputs, A + B.
At first glance it might seem that all
networks would behave in this fash-
ion. A counter example, a non-linear
network, is shown in Figure 2.14.
Suppose that the first input is a force
that varies in a sinusoidal manner. We
Other forms of non-linearities are The second reason why systems are
pick its amplitude to ensure that the
also often present. Hysteresis (or linearized is to reduce the problem
displacement is small enough so that
backlash) is usually present in gear of nonlinear instability. One example
the oscillating mass does not quite hit
trains, loosely riveted joints and in would be the positioning system
the stops. If we add a second identi-
magnetic devices. Sometimes the shown in Figure 2.16. The actual
cal input, the mass would now hit the
non-linearities are less abrupt and are position is compared to the desired
stops. Instead of a sine wave with
smooth, but nonlinear, curves. The position and the error is integrated
twice the amplitude, the output is
torque versus rpm of an engine or the and applied to the motor. If the gear
clipped as shown in Figure 2.14b.
operating curves of a transistor are train has no backlash, it is a straight-
two examples that can be considered forward problem to design this
This spring-mass system with stops
linear over only small portions of system to the desired specifications
illustrates an important principal: no
their operating regions. of positioning accuracy and
real system is completely linear. A
response time.
system may be approximately linear
The important point is not that all
over a wide range of signals, but
systems are nonlinear; it is that However, if the gear train has exces-
eventually the assumption of linearity
most systems can be approximated sive backlash, the motor will "hunt,"
breaks down. Our spring-mass system
as linear systems. Often a large causing the positioning system to
is linear before it hits the stops.
engineering effort is spent in making oscillate around the desired position.
Likewise a linear electronic amplifier
the system as linear as practical. This The solution is either to reduce the
clips when the output voltage
is done for two reasons. First, it is loop gain and therefore reduce the
approaches the internal supply
often a design goal for the output of a overall performance of the system,
voltage. A spring may compress
network to be a scaled, linear version or to reduce the backlash in the gear
linearly until the coils start pressing
of the input. A strip chart recorder train. Often, reducing the backlash
against each other.
is a good example. The electronic is the only way to meet the
amplifier and pen motor must both be performance specifications.
designed to ensure that the deflection
across the paper is linear with the
applied voltage.
13
Analysis of Linear Networks Figure 2.17
Linear network
As we have seen, many systems are response to a
designed to be reasonably linear to sine wave input.
meet design specifications. This
has a fortuitous side benefit when
attempting to analyze networks*.
Recall that an real signal can be
considered to be a sum of sine waves.
Also, recall that the response of a
linear network is the sum of the
responses to each component of the
input. Therefore, if we knew the
response of the network to each of
the sine wave components of the
input spectrum, we could predict
the output.
It is easy to show that the steady-
state response of a linear network
to a sine wave input is a sine wave Figure 2.18
The frequency
of the same frequency. As shown in response of
Figure 2.17, the amplitude of the a network.
output sine wave is proportional to
the input amplitude. Its phase is
shifted by an amount which depends
only on the frequency of the sine
wave. As we vary the frequency of
the sine wave input, the amplitude
proportionality factor (gain) changes
as does the phase of the output.
If we divide the output of the
network by the input, we get a
* We will discuss the analysis of networks which
have not been linearized in Chapter 3, Section 6.
14
normalized result called the frequen- Figure 2.19
cy response of the network. As Three classes
shown in Figure 2.18, the frequency of frequency
response.
response is the gain (or loss) and
phase shift of the network as a
function of frequency. Because the
network is linear, the frequency
response is independent of the input
amplitude; the frequency response is
a property of a linear network, not
dependent on the stimulus.
The frequency response of a network
will generally fall into one of three
categories; low pass, high pass,
bandpass or a combination of these.
As the names suggest, their frequency
responses have relatively high gain in
a band of frequencies, allowing these
frequencies to pass through the
network. Other frequencies suffer a
relatively high loss and are rejected
by the network. To see what this
means in terms of the response of a
filter to an input, let us look at the
bandpass filter case.
15
In Figure 2.20, we put a square wave Figure 2.20
into a bandpass filter. We recall from Bandpass filter
Figure 2.10 that a square wave is response to a
square wave
composed of harmonically related input.
sine waves. The frequency response
of our example network is shown in
Figure 2.20b. Because the filter is
narrow, it will pass only one compo-
nent of the square wave. Therefore,
the steady-state response of this
bandpass filter is a sine wave.
Notice how easy it is to predict
the output of any network from its
frequency response. The spectrum of
the input signal is multiplied by the
frequency response of the network
to determine the components that
appear in the output spectrum. This
frequency domain output can then
be transformed back to the time
domain.
In contrast, it is very difficult to
compute in the time domain the out-
put of any but the simplest networks.
A complicated integral must be evalu-
ated which often can only be done
numerically on a digital computer*. If
we computed the network response
by both evaluating the time domain
integral and by transforming to the
frequency domain and back, we
would get the same results. However,
it is usually easier to compute the
output by transforming to the
frequency domain. Figure 2.21
Time response
Transient Response of bandpass
filters.
Up to this point we have only dis-
cussed the steady-state response to a
signal. By steady-state we mean the
output after any transient responses
caused by applying the input have
died out. However, the frequency
response of a network also contains
all the information necessary to
predict the transient response of the
network to any signal.
* This operation is called convolution.
16
Let us look qualitatively at the tran- Figure 2.22
sient response of a bandpass filter. If Parallel filter
a resonance is narrow compared to analyzer.
its frequency, then it is said to be a
high "Q" resonance*. Figure 2.21a
shows a high Q filter frequency
response. It has a transient response
which dies out very slowly. A time
response which decays slowly is said
to be lightly damped. Figure 2.21b
shows a low Q resonance. It has a
transient response which dies out
quickly. This illustrates a general
principle: signals which are broad in
one domain are narrow in the other.
Narrow, selective filters have very
long response times, a fact we will
find important in the next section.
Section 3:
Instrumentation for the
Frequency Domain
Just as the time domain can be
measured with strip chart recorders,
oscillographs or oscilloscopes,
the frequency domain is usually
Network analyzers are optimized to The Parallel-Filter
measured with spectrum and
give accurate amplitude and phase Spectrum Analyzer
network analyzers.
measurements over a wide range of
network gains and losses. This design As we developed in Section 2 of
Spectrum analyzers are instruments this chapter, electronic filters can be
difference means that these two
which are optimized to characterize built which pass a narrow band of
traditional instrument families are
signals. They introduce very little frequencies. If we were to add a
not interchangeable.** A spectrum
distortion and few spurious signals. meter to the output of such a band-
analyzer can not be used as a net-
This insures that the signals on the pass filter, we could measure the
work analyzer because it does not
display are truly part of the input power in the portion of the spectrum
measure amplitude accurately and
signal spectrum, not signals passed by the filter. In Figure 2.22a
cannot measure phase. A network
introduced by the analyzer. we have done this for a bank of
analyzer would make a very poor
spectrum analyzer because spurious filters, each tuned to a different
responses limit its dynamic range. frequency. If the center frequencies
of these filters are chosen so that
In this section we will develop the the filters overlap properly, the
properties of several types of spectrum covered by the filters can
analyzers in these two categories. be completely characterized as in
Figure 2.22b.
* Q is usually defined as:
Q = Center Frequency of Resonance
Frequency Width of -3 dB Points
** Dynamic Signal Analyzers are an exception to this rule,
they can act as both network and spectrum analyzers.
17
How many filters should we use to Figure 2.23
cover the desired spectrum? Here we Simplified
have a trade-off. We would like to be swept spectrum
able to see closely spaced spectral analyzer.
lines, so we should have a large
number of filters. However, each
filter is expensive and becomes more
expensive as it becomes narrower,
so the cost of the analyzer goes up
as we improve its resolution. Typical
audio parallel-filter analyzers balance
these demands with 32 filters, each
covering 1/3 of an octave.
Swept Spectrum Analyzer
One way to avoid the need for such
a large number of expensive filters is
to use only one filter and sweep it
Figure 2.24
slowly through the frequency range Amplitude
of interest. If, as in Figure 2.23, we error form
display the output of the filter versus sweeping
the frequency to which it is tuned, too fast.
we have the spectrum of the input
signal. This swept analysis technique
is commonly used in rf and
microwave spectrum analysis.
We have, however, assumed the input
signal hasn't changed in the time it
takes to complete a sweep of our
analyzer. If energy appears at some
frequency at a moment when our If we sweep the filter past a signal is fast, but has limited resolution and
filter is not tuned to that frequency, too quickly, the filter output will not is expensive. The swept analyzer
then we will not measure it. have a chance to respond fully to the can be cheaper and have higher
signal. As we show in Figure 2.24, resolution but the measurement
One way to reduce this problem the spectrum display will then be in takes longer (especially at high
would be to speed up the sweep error; our estimate of the signal level resolution) and it can not analyze
time of our analyzer. We could still will be too low. transient events*.
miss an event, but the time in which
this could happen would be shorter. In a parallel-filter spectrum analyzer Dynamic Signal Analyzer
Unfortunately though, we cannot we do not have this problem. All the
filters are connected to the input In recent years another kind of
make the sweep arbitrarily fast
signal all the time. Once we have analyzer has been developed
because of the response time of
waited the initial settling time of a which offers the best features of the
our filter.
single filter, all the filters will be parallel-filter and swept spectrum
settled and the spectrum will be valid analyzers. Dynamic Signal Analyzers
To understand this problem,
and not miss any transient events. are based on a high speed calculation
recall from Section 2 that a filter
routine which acts like a parallel
takes a finite time to respond to
So there is a basic trade-off between filter analyzer with hundreds of
changes in its input. The narrower the
parallel-filter and swept spectrum filters and yet are cost-competitive
filter, the longer it takes to respond.
analyzers. The parallel-filter analyzer with swept spectrum analyzers. In
* More information on the performance of swept
spectrum analyzers can be found in Agilent
Application Note Series 150.
18
addition, two channel Dynamic Signal Figure 2.25
Analyzers are in many ways better Gain-phase
network analyzers than the ones we meter
operation.
will introduce next.
Network Analyzers
Since in network analysis it is
required to measure both the input
and output, network analyzers are
generally two channel devices with
the capability of measuring the ampli-
tude ratio (gain or loss) and phase Figure 2.26
difference between the channels. Tuned net-
All of the analyzers discussed here work analyzer
measure frequency response by using operation.
a sinusoidal input to the network
and slowly changing its frequency.
Dynamic Signal Analyzers use a
different, much faster technique for
network analysis which we discuss
in the next chapter.
Gain-phase meters are broadband
devices which measure the amplitude
and phase of the input and output
sine waves of the network. A sinu-
soidal source must be supplied to
stimulate the network when using a
gain-phase meter as in Figure 2.25.
The source can be tuned manually
and the gain-phase plots done by
hand or a sweeping source, and an
x-y plotter can be used for automatic
frequency response plots.
The primary attraction of gain-phase
meters is their low price. If a
sinusoidal source and a plotter are
already available, frequency response
measurements can be made for a very
low investment. However, because
gain-phase meters are broadband,
they measure all the noise of the
network as well as the desired sine
wave. As the network attenuates the
Tuned network analyzers minimize By minimizing the noise, it is also
input, this noise eventually becomes a
the noise floor problems of gain- possible for tuned network analyzers
floor below which the meter cannot
phase meters by including a bandpass to make more accurate measure-
measure. This typically becomes a
filter which tracks the source fre- ments of amplitude and phase. These
problem with attenuations of about
quency. Figure 2.26 shows how this improvements do not come without
60 dB (1,000:1).
tracking filter virtually eliminates the their price, however, as tracking
noise and any harmonics to allow filters and a dedicated source must
measurements of attenuation to be added to the simpler and less
100 dB (100,000:1). costly gain-phase meter.
19
Tuned analyzers are available in the Figure 2.27
frequency range of a few Hertz to The vibration
many Gigahertz (109 Hertz). If lower of a tuning fork.
frequency analysis is desired, a
frequency response analyzer is often
used. To the operator, it behaves
exactly like a tuned network analyzer.
However, it is quite different inside.
It integrates the signals in the time
domain to effectively filter the signals
at very low frequencies where it is
not practical to make filters by more
conventional techniques. Frequency
response analyzers are generally lim-
ited to from 1 mHz to about 10 kHz.
Section 4:
The Modal Domain
In the preceding sections we have
developed the properties of the time
and frequency domains and the
instrumentation used in these
domains. In this section we will
develop the properties of another
domain, the modal domain. This
change in perspective to a new
domain is particularly useful if we are
interested in analyzing the behavior
of mechanical structures.
To understand the modal domain let Figure 2.28
us begin by analyzing a simple Example
mechanical structure, a tuning fork. vibration modes
If we strike a tuning fork, we easily of a tuning fork.
conclude from its tone that it is pri-
marily vibrating at a single frequency.
We see that we have excited a
network (tuning fork) with a force
impulse (hitting the fork). The time
domain view of the sound caused by
the deformation of the fork is a
lightly damped sine wave shown
in Figure 2.27b.
In Figure 2.27c, we see in the
frequency domain that the frequency
response of the tuning fork has a
major peak that is very lightly
damped, which is the tone we hear.
There are also several smaller peaks.
20
Each of these peaks, large and small, Figure 2.29
corresponds to a "vibration mode" Reducing the
of the tuning fork. For instance, we second harmonic
by damping the
might expect for this simple example second vibration
that the major tone is caused by the mode.
vibration mode shown in Figure
2.28a. The second harmonic might
be caused by a vibration like
Figure 2.28b
We can express the vibration of any
structure as a sum of its vibration
modes. Just as we can represent an
real waveform as a sum of much sim-
pler sine waves, we can represent any
vibration as a sum of much simpler
vibration modes. The task of "modal"
analysis is to determine the shape
and the magnitude of the structural
deformation in each vibration mode.
Once these are known, it usually
becomes apparent how to change
the overall vibration.
Figure 2.30
Modal analysis
For instance, let us look again at our of a tuning fork.
tuning fork example. Suppose that we
decided that the second harmonic
tone was too loud. How should we
change our tuning fork to reduce the
harmonic? If we had measured the
vibration of the fork and determined
that the modes of vibration were
those shown in Figure 2.28, the
answer becomes clear. We might
apply damping material at the center
of the tines of the fork. This would
greatly affect the second mode which
has maximum deflection at the center
while only slightly affecting the
desired vibration of the first mode.
Other solutions are possible, but all
depend on knowing the geometry of
each mode.
The Relationship Between the Time,
Frequency and Modal Domain
To determine the total vibration
of our tuning fork or any other
structure, we have to measure the
vibration at several points on the results like Figure 2.30b. We measure
structure. Figure 2.30a shows some frequency response because we want
points we might pick. If we to measure the properties of the
transformed this time domain data to structure independent of the
the frequency domain, we would get stimulus*. * Those who are more familiar with electronics might
note that we have measured the frequency response of
a network (structure) at N points and thus have performed
an N-port Analysis.
21
We see that the sharp peaks Figure 2.31
(resonances) all occur at the same The relationship
frequencies independent of where between the
frequency and
they are measured on the structure. the modal
Likewise we would find by measuring domains.
the width of each resonance that the
damping (or Q) of each resonance
is independent of position. The
only parameter that varies as we
move from point to point along the
structure is the relative height of
resonances.* By connecting the
peaks of the resonances of a given
mode, we trace out the mode shape
of that mode.
Experimentally we have to measure
only a few points on the structure to
determine the mode shape. However,
to clearly show the mode shape in
our figure, we have drawn in the
frequency response at many more
points in Figure 2.31a. If we view this
three-dimensional graph along the
distance axis, as in Figure 2.31b, we
get a combined frequency response.
Each resonance has a peak value cor-
responding to the peak displacement
in that mode. If we view the graph
along the frequency axis, as in Figure
2.31c, we can see the mode shapes of
the structure.
We have not lost any information by
this change of perspective. Each However, the equivalence between the modal domain to minimize the
vibration mode is characterized by its the modal, time and frequency effects of noise and small experimen-
mode shape, frequency and damping domains is not quite as strong as tal errors. No information is lost in
from which we can reconstruct the that between the time and frequency this curve fitting, so all three domains
frequency domain view. domains. Because the modal domain contain the same information, but not
portrays the properties of the net- the same noise. Therefore, transform-
work independent of the stimulus, ing from the frequency domain to the
transforming back to the time domain modal domain and back again will
gives the impulse response of the give results like those in Figure 2.32.
structure, no matter what the stimu- The results are not exactly the same,
lus. A more important limitation of yet in all the important features, the
this equivalence is that curve fitting frequency responses are the same.
is used in transforming from our This is also true of time domain data
frequency response measurements to derived from the modal domain.
* The phase of each resonance is not shown for clarity of
the figures but it too is important in the mode shape. The
magnitude of the frequency response gives the magnitude
of the mode shape while the phase gives the direction of
the deflection.
22
Section 5: Figure 2.32
Curve fitting
Instrumentation for removes
the Modal Domain measurement
noise.
There are many ways that the modes
of vibration can be determined. In our
simple tuning fork example we could
guess what the modes were. In simple
structures like drums and plates it is
possible to write an equation for the
modes of vibration. However, in
almost any real problem, the solution
can neither be guessed nor solved
analytically because the structure is
too complicated. In these cases it is
necessary to measure the response
of the structure and determine
the modes.
There are two basic techniques for
determining the modes of vibration in
complicated structures: 1) exciting
only one mode at a time, and 2) Figure 2.33
computing the modes of vibration Single mode
from the total vibration. excitation
modal analysis.
Single Mode Excitation
Modal Analysis
To illustrate single mode excitation,
let us look once again at our simple
tuning fork example. To excite just
the first mode we need two shakers,
driven by a sine wave and attached
to the ends of the tines as in Figure
2.33a. Varying the frequency of the
generator near the first mode reso-
nance frequency would then give us
its frequency, damping and mode
shape.
In the second mode, the ends of the
tines do not move, so to excite the
second mode we must move the
shakers to the center of the tines. If
we anchor the ends of the tines, we
will constrain the vibration to the
second mode alone.
23
In more realistic, three dimensional Figure 2.34
problems, it is necessary to add many Measured
more shakers to ensure that only one mode shape.
mode is excited. The difficulties and
expense of testing with many shakers
has limited the application of this
traditional modal analysis technique.
Modal Analysis From Total Vibration
To determine the modes of vibration
from the total vibration of the
structure, we use the techniques
developed in the previous section.
Basically, we determine the frequency
response of the structure at several
points and compute at each reso-
nance the frequency, damping and
what is called the residue (which From the above description, it is
represents the height of the reso- Section 6: Summary
apparent that a modal analyzer
nance). This is done by a curve-fitting requires some type of network
routine to smooth out any noise or In this chapter we have developed
analyzer to measure the frequency the concept of looking at problems
small experimental errors. From response of the structure and a
these measurements and the geome- from different perspectives. These
computer to convert the frequency perspectives are the time, frequency
try of the structure, the mode shapes response to mode shapes. This can
are computed and drawn on a CRT and modal domains. Phenomena that
be accomplished by connecting a are confusing in the time domain are
display or a plotter. If drawn on a Dynamic Signal Analyzer through
CRT, these displays may be animated often clarified by changing perspec-
a digital interface* to a computer tive to another domain. Small signals
to help the user understand the furnished with the appropriate soft-
vibration mode. are easily resolved in the presence of
ware. This capability is also available large ones in the frequency domain.
in a single instrument called a Struc- The frequency domain is also valu-
tural Dynamics Analyzer. In general, able for predicting the output of any
computer systems offer more versa- kind of linear network. A change to
tile performance since they can be the modal domain breaks down
programmed to solve other problems. complicated structural vibration
However, Structural Dynamics problems into simple vibration
Analyzers generally are much easier modes.
to use than computer systems.
No one domain is always the best
answer, so the ability to easily change
domains is quite valuable. Of all the
instrumentation available today, only
Dynamic Signal Analyzers can work
in all three domains. In the next
chapter we develop the properties
of this important class of analyzers.
* GPIB, Agilent's implementation of
IEEE-488-1975 is ideal for this application.
24
Chapter 3
Understanding Dynamic
Signal Analysis
We saw in the previous chapter that Figure 3.1
the Dynamic Signal Analyzer has the The FFT samples
speed advantages of parallel-filter in both the time
and frequency
analyzers without their low resolution domains.
limitations. In addition, it is the only
type of analyzer that works in all
three domains. In this chapter we will
develop a fuller understanding of this
important analyzer family, Dynamic
Signal Analyzers. We begin by pre-
senting the properties of the Fast
Fourier Transform (FFT) upon which
Dynamic Signal Analyzers are based.
No proof of these properties is given,
but heuristic arguments as to their va-
lidity are used where appropriate. We
then show how these FFT properties
cause some undesirable characteris-
tics in spectrum analysis like aliasing
and leakage. Having demonstrated a
potential difficulty with the FFT, we
then show what solutions are used
to make practical Dynamic Signal
Analyzers. Developing this basic
knowledge of FFT characteristics
makes it simple to get good results
with a Dynamic Signal Analyzer in a
wide range of measurement problems.
Section 1: FFT Properties Figure 3.2
A time record
is N equally
The Fast Fourier Transform (FFT) spaced samples
is an algorithm* for transforming of the input.
data from the time domain to the fre-
quency domain. Since this is exactly
what we want a spectrum analyzer to
do, it would seem easy to implement
a Dynamic Signal Analyzer based
on the FFT. However, we will see
that there are many factors which
complicate this seemingly frequency domain in a continuous samples closer together. Later in
straightforward task. manner, but instead must sample and this chapter, we will consider what
digitize the time domain input. This sample spacing is necessary to
First, because of the many calcula- means that our algorithm transforms guarantee accurate results.
tions involved in transforming digitized samples from the time do-
domains, the transform must be main to samples in the frequency
implemented on a digital computer if domain as shown in Figure 3.1.**
the results are to be sufficiently accu-
rate. Fortunately, with the advent of Because we have sampled, we no
microprocessors, it is easy and inex- longer have an exact representation
pensive to incorporate all the needed in either domain. However, a sampled
computing power in a small instru- representation can be as close to * An algorithm is any special mathematical method of
ment package. Note, however, that ideal as we desire by placing our solving a certain kind of problem; e.g., the technique
we cannot now transform to the you use to balance your checkbook.
** To reduce confusion about which domain we are in,