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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: 5
A matter of Perspective
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 In Chapter 3 we develop the
is a fundamental problem for properties of one of these classes Because of the tutorial nature of
many engineers and scientists. of analyzers, Dynamic Signal this note, we will not attempt to
Even if the immediate problem Analyzers. These instruments are show detailed solutions for the
is not electrical, the basic param- particularly appropriate for the multitude of measurement prob-
eters of interest are often changed analysis of signals in the range lems which can be solved by
into electrical signals by means of of a few millihertz to about a Dynamic Signal Analysis. Instead,
transducers. Common transducers hundred kilohertz. we will concentrate on the fea-
include accelerometers and load tures of Dynamic Signal Analysis,
cells in mechanical work, EEG Chapter 4 shows the benefits of how these features are used in a
electrodes and blood pressure Dynamic Signal Analysis in a wide wide range of applications and
probes in biology and medicine, range of measurement situations. the benefits to be gained from
and pH and conductivity probes in The powerful analysis tools of using Dynamic Signal Analysis.
chemistry. The rewards for trans- Dynamic Signal Analysis are
forming physical parameters to introduced as needed in each Those who desire more details
electrical signals are great, as measurement situation. on specific applications should
many instruments are available look to Appendix B. It contains
for the analysis of electrical sig- This note avoids the use of rigor- abstracts of Hewlett-Packard
nals in the time, frequency and ous mathematics and instead Application Notes on a wide
modal domains. The powerful depends on heuristic arguments. range of related subjects. These
measurement and analysis capa- We have found in over a decade can be obtained free of charge
bilities of these instruments can of teaching this material that such from your local HP field engineer
lead to rapid understanding of the arguments lead to a better under- or representative.
system under study. standing of the basic processes
involved in the various domains
This note is a primer for those and in Dynamic Signal Analysis.
who are unfamiliar with the Equally important, this heuristic
advantages of analysis in the instruction leads to better instru-
frequency and modal domains ment operators who can intelli-
and with the class of analyzers gently use these analyzers to
we call Dynamic Signal Analyzers. solve complicated measurement
In Chapter 2 we develop the con- problems with accuracy and
cepts of the time, frequency and ease*.
modal domains and show why
these different ways of looking
at a problem often lend their own
unique insights. We then intro-
duce classes of instrumentation
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: This electrical signal, which
The Time Domain represents a parameter of the
In this chapter we introduce the system, can be recorded on a strip
concepts of the time, frequency The traditional way of observing chart recorder as in Figure 2.2. We
and modal domains. These three signals is to view them in the time can adjust the gain of the system
ways of looking at a problem are domain. The time domain is a to calibrate our measurement.
interchangeable; that is, no infor- record of what happened to a Then we can reproduce exactly
mation is lost in changing from parameter of the system versus the results of our simple direct
one domain to another. The time. For instance, Figure 2.1 recording system in Figure 2.1.
advantage in introducing these shows a simple spring-mass
three domains is that of a change system where we have attached Why should we use this indirect
of perspective. By changing per- a pen to the mass and pulled a approach? One reason is that we
spective from the time domain, piece of paper past the pen at a are not always measuring dis-
the solution to difficult problems constant rate. The resulting graph placement. We then must convert
can often become quite clear in is a record of the displacement of the desired parameter to the
the frequency or modal domains. the mass versus time, a time do- displacement of the recorder pen.
main view of displacement. Usually, the easiest way to do this
After developing the concepts of is through the intermediary of
each domain, we will introduce Such direct recording schemes electronics. However, even when
the types of instrumentation avail- are sometimes used, but it usually measuring displacement we
able. The merits of each generic is much more practical to convert would normally use an indirect
instrument type are discussed to the parameter of interest to an approach. Why? Primarily be-
give the reader an appreciation of electrical signal using a trans- cause the system in Figure 2.1 is
the advantages and disadvantages ducer. Transducers are commonly hopelessly ideal. The mass must
of each approach. available to change a wide variety be large enough and the spring
of parameters to electrical sig- stiff enough so that the pens
nals. Microphones, accelerom- mass and drag on the paper will
eters, load cells, conductivity
and pressure probes are just a
few examples.
Figure 2.1 Figure 2.2
Direct record- Indirect recording
ing of displace- of displacement.
ment - a time
domain view.
5
not affect the results appreciably. Figure 2.3
Also the deflection of the mass Simplified
must be large enough to give a oscillograph
operation.
usable result, otherwise a me-
chanical lever system to amplify
the motion would have to be
added with its attendant mass
and friction.
With the indirect system a trans-
ducer can usually be selected
which will not significantly affect
the measurement. This can go to
the extreme of commercially
available displacement transduc-
ers which do not even contact the
mass. The pen deflection can be Figure 2.4
Simplified
easily set to any desired value oscilloscope
by controlling the gain of the operation
electronic amplifiers. (Horizontal
deflection
circuits
This indirect system works well omitted for
until our measured parameter be- clarity).
gins to change rapidly. Because of
the mass of the pen and recorder
mechanism and the power limita-
tions of its drive, the pen can only
move at finite velocity. If the mea-
sured parameter changes faster, tive paper by deflecting a light capable of accurately displaying
the output of the recorder will be beam. Such a device is called signals that vary even more rap-
in error. A common way to reduce an oscillograph. Since it is only idly than the oscillograph can
this problem is to eliminate the necessary to move a small, handle. This is because it is only
pen and record on a photosensi- light-weight mirror through a necessary to move an electron
very small angle, the oscillograph beam, not a mirror.
can respond much faster than a
strip chart recorder. The strip chart, oscillograph and
oscilloscope all show displace-
Another common device for dis- ment versus time. We say that
playing signals in the time domain changes in this displacement rep-
is the oscilloscope. Here an resent the variation of some pa-
electron beam is moved using rameter versus time. We will now
electric fields. The electron beam look at another way of represent-
is made visible by a screen of ing the variation of a parameter.
phosphorescent material. It is
6
Section 2: The Frequency Figure 2.5
Domain Any real
waveform
can be
It was shown over one hundred produced
years ago by Baron Jean Baptiste by adding
Fourier that any waveform that sine waves
together.
exists in the real world can be
generated by adding up sine
waves. We have illustrated this in
Figure 2.5 for a simple waveform
composed of two sine waves. By
picking the amplitudes, frequen-
Figure 2.6
cies and phases of these sine
The relationship
waves correctly, we can generate between the time
a waveform identical to our and frequency
domains.
desired signal.
a) Three
dimensional
Conversely, we can break down coordinates
showing time,
our real world signal into these
frequency and
same sine waves. It can be shown amplitude
that this combination of sine b) Time
domain view
waves is unique; any real world
c) Frequency
signal can be represented by only domain view
one combination of sine waves.
Figure 2.6a is a three dimensional
graph of this addition of sine
waves. Two of the axes are time
and amplitude, 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 However, if we view our graph represents a sine wave, we have
waveform. If we view this three along the time axis as in Figure uniquely characterized our input
dimensional graph along the 2.6c, we get a totally different signal in the frequency domain*.
frequency axis we get the view picture. Here we have axes of This frequency domain represen-
in Figure 2.6b. This is the time amplitude versus frequency, what tation of our signal is called the
domain view of the sine waves. is commonly called the frequency spectrum of the signal. Each sine
Adding them together at each domain. Every sine wave we wave line of the spectrum is
instant of time gives the original separated from the input appears called a component of the
waveform. as a vertical line. Its height repre- total signal.
sents its amplitude and its posi-
tion represents its frequency.
Since we know that each line
* 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
domain is to resolve small signals in the
presence of large ones, let us now address Figure 2.8
the problem of how we can see both large The relation-
and small signals on our display ship between
simultaneously. decibels, power
and voltage.
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. (.1mm) 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
components easily at the same time, the only Figure 2.9
answer is to change our amplitude scale. A Small signals
logarithmic scale would compress our large can be measured
signal amplitude and expand the small ones, with a logarithmic
amplitude scale.
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
a) Time Domain - small signal not visible
that we have neither gained nor Small signals
are not hidden
lost information, we are just in the frequency
representing it differently. We domain.
are looking at 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 os-
b) Frequency Domain - small signal easily resolved
cillator. Or we might be trying to
detect the first sounds of a bear-
ing failing on a noisy machine. In
each case, we are trying to detect
a small sine wave in 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 frequency domain that the
same signal is composed of a
large sine wave and significant
other sine wave components
(distortion components). When
these components are separated
in the frequency domain, the The Frequency Domain: to its frequency domain capabil-
small components are easy to see A Natural Domain ity. A doctor listens to your heart
because they are not masked by and breathing for any unusual
larger ones. At first the frequency domain may sounds. He is listening for
seem strange and unfamiliar, yet frequencies which will tell him
The frequency domains useful- it is an important part of everyday something is wrong. An experi-
ness is not restricted to electron- life. Your ear-brain combination enced mechanic can do the same
ics or mechanics. All fields of is an excellent frequency domain thing with a machine. Using a
science and engineering have analyzer. The ear-brain splits the screwdriver as a stethoscope,
measurements like these where audio spectrum into many narrow he can hear when a bearing is
large signals mask others in the bands and determines the power failing because of the frequencies
time domain. The frequency present in each band. It can easily it produces.
domain provides a useful tool pick small sounds out of loud
in analyzing these small but background noise thanks in part
important effects.
9
So we see that the frequency Figure 2.10
domain is not at all uncommon. Frequency
We are just not used to seeing it spectrum ex-
amples.
in graphical form. But this graphi-
cal 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
signals in both the time and fre-
quency 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 con-
structed the frequency domain.
The square wave in Figure 2.10b
is made up of an infinite number
of sine waves, all harmonically
related. The lowest frequency
present is the reciprocal of the
square wave period. These two spectrum. This means that the fore, require infinite energy to
examples illustrate a property of sine waves that make up this generate a true impulse. Never-
the frequency transform: a signal signal are spaced infinitesimally theless, it is possible to generate
which is periodic and exists for close together. an approximation to an impulse
all time has a discrete frequency which has a fairly flat spectrum
spectrum. This is in contrast to Another signal of interest is the over the desired frequency range
the transient signal in Figure impulse shown in Figure 2.10d. of interest. We will find signals
2.10c which has a continuous The frequency spectrum of an with a flat spectrum useful in our
impulse is flat, i.e., there is energy next subject, network analysis.
at all frequencies. It would, there-
10
Network Analysis Figure 2.11
One-port
If the frequency domain were network
analysis
restricted to the analysis of signal examples.
spectrums, it would certainly not
be such a common engineering
tool. However, the frequency
domain is also widely used in
analyzing the behavior of net-
works (network analysis) and
in design work.
Network analysis is the general
engineering problem of determin-
ing 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 purchas-
ing would be in reducing machin-
ery noise. Or perhaps we are
interested in the effects of a tube
of saline solution on the transmis-
sion 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 net-
work 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 re- Figure 2.12
sponse at a second port due to an Two-port
input at the first port. We are gen- network
analysis.
erally interested in the transmis-
sion and rejection 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 net-
work 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 Figure 2.13
analysis becomes a very powerful Linear network.
tool.
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, Figure 2.16
we mean it behaves like the net- A positioning
work in Figure 2.13. Suppose one system.
input 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
Other forms of non-linearities are The second reason why systems
all networks would behave in this
also often present. Hysteresis (or are linearized is to reduce the
fashion. A counter example, a
backlash) is usually present in problem of nonlinear instability.
non-linear network, is shown
gear trains, loosely riveted joints One example would be the posi-
in Figure 2.14. Suppose that the
and in magnetic devices. Some- tioning system shown in Figure
first input is a force that varies in
times the non-linearities are less 2.16. The actual position is com-
a sinusoidal manner. We pick its
abrupt and are smooth, but non- pared to the desired position and
amplitude to ensure that the
linear, curves. The torque versus the error is integrated and applied
displacement is small enough so
rpm of an engine or the operating to the motor. If the gear train
that the oscillating mass does not
curves of a transistor are two has no backlash, it is a straight
quite hit the stops. If we add a
examples that can be considered forward problem to design this
second identical input, the mass
linear over only small portions of system to the desired specifica-
would now hit the stops. Instead
their operating regions. tions of positioning accuracy and
of a sine wave with twice the
response time.
amplitude, the output is clipped
The important point is not that all
as shown in Figure 2.14b.
systems are nonlinear; it is that However, if the gear train has ex-
most systems can be approxi- cessive backlash, the motor will
This spring-mass system with
mated as linear systems. Often hunt causing the positioning
stops illustrates an important
a large engineering effort is spent system to oscillate around the
principal: no real system is
in making the system as linear as desired position. The solution
completely linear. A system may
practical. This is done for two is either to reduce the loop gain
be approximately linear over a
reasons. First, it is often a design and therefore reduce the overall
wide range of signals, but eventu-
goal for the output of a network performance of the system, or to
ally the assumption of linearity
to be a scaled, linear version of reduce the backlash in the gear
breaks down. Our spring-mass
the input. A strip chart recorder train. Often, reducing the back-
system is linear before it hits the
is a good example. The electronic lash is the only way to meet the
stops. Likewise a linear electronic
amplifier and pen motor must performance specifications.
amplifier clips when the output
both be designed to ensure that
voltage approaches the internal
the deflection across the paper
supply voltage. A spring may com-
is linear with the applied voltage.
press linearly until the coils start
pressing against each other.
13
Analysis of Linear Networks Figure 2.17
Linear network
As we have seen, many systems response to a
sine wave input.
are designed to be reasonably lin-
ear to 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. There-
fore, if we knew the response of
the network to each of the sine
Figure 2.18
wave components of the input The frequency
spectrum, we could predict the response of a
output. network.
It is easy to show that the steady-
state response of a linear network
to a sine wave input is a sine
wave of the same frequency. As
shown in Figure 2.17, the ampli-
tude of the output sine wave is
proportional to the input ampli-
tude. 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 propor-
tionality factor (gain) changes as
does the phase of the output.
If we divide the output of the
* We will discuss the analysis of networks
which have not been linearized in
Chapter 3, Section 6.
14
network by the input, we get a Figure 2.19
normalized result called the fre- Three classes
quency response of the network. of frequency
response.
As shown in Figure 2.18, the fre-
quency response is the gain (or
loss) and phase shift of the net-
work as a function of frequency.
Because the network is linear, the
frequency response is indepen-
dent of the input amplitude; the
frequency response is a property
of a linear network, not depen-
dent on the stimulus.
The frequency response of a net-
work 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 fre-
quencies 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 Figure 2.20
wave into a bandpass filter. We Bandpass filter
recall from Figure 2.10 that a response to a
square wave
square wave is composed of input.
harmonically related 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 com-
ponent of the square wave. There-
fore, 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 re-
sponse of the network to deter-
mine the components that appear
in the output spectrum. This fre-
quency domain output can then
be transformed back to the time
domain.
In contrast, it is very difficult to
compute in the time domain the
output of any but the simplest
networks. A complicated integral
must be evaluated which often
can only be done numerically on a
digital computer*. If we computed
the network response by both
Figure 2.21
evaluating the time domain inte- Time response
gral and by transforming to the of bandpass
frequency domain and back, we filters.
would get the same results. How-
ever, it is usually easier to com-
pute the output by transforming
to the frequency domain.
Transient Response
Up to this point we have only
discussed the steady-state re-
sponse 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
* This operation is called convolution.
16
information necessary to predict Figure 2.22
the transient response of the net- Parallel filter
work to any signal. analyzer.
Let us look qualitatively at the
transient response of a bandpass
filter. If a resonance is narrow
compared to its frequency, then
it is said to be a high Q reso-
nance*. Figure 2.21a shows a
high Q filter frequency response.
It has a transient response which
dies out very slowly. A time re-
sponse 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.
Network analyzers are optimized The Parallel-Filter
Section 3: to give accurate amplitude and Spectrum Analyzer
Instrumentation for the phase measurements over a
Frequency Domain wide range of network gains and As we developed in Section 2 of
losses. This design difference this chapter, electronic filters can
Just as the time domain can means that these two traditional be built which pass a narrow band
be measured with strip chart instrument families are not of frequencies. If we were to add
recorders, oscillographs or interchangeable.** A spectrum a meter to the output of such a
oscilloscopes, the frequency analyzer can not be used as a bandpass filter, we could measure
domain is usually measured with network analyzer because it does the power in the portion of the
spectrum and network analyzers. not measure amplitude accurately spectrum passed by the filter. In
and cannot measure phase. A net- Figure 2.22a we have done this
Spectrum analyzers are instru- work analyzer would make a very for a bank of filters, each tuned to
ments which are optimized to poor spectrum analyzer because a different frequency. If the center
characterize signals. They intro- spurious responses limit its frequencies of these filters are
duce very little distortion and few dynamic range. chosen so that the filters overlap
spurious signals. This insures that properly, the spectrum covered
the signals on the display are In this section we will develop the by the filters can be completely
truly part of the input signal properties of several types of characterized as in Figure 2.22b.
spectrum, not signals introduced analyzers in these two categories.
by the analyzer.
* Q is usually defined as:
Center Frequency of Resonance
Q=
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 Figure 2.23
to cover the desired spectrum? Simplified
Here we have a trade-off. We swept spectrum
analyzer.
would like to be able to see
closely spaced spectral lines, so
we should have a large number
of filters. However, each filter is
expensive and becomes more ex-
pensive as it becomes narrower,
so the cost of the analyzer goes
up as we improve its resolution.
Typical audio parallel-filter ana-
lyzers balance these demands
with 32 filters, each covering
1/3 of an octave.
Swept Spectrum Analyzer Figure 2.24
Amplitude
error form
One way to avoid the need for sweeping
such a large number of expensive too fast.
filters is to use only one filter and
sweep it slowly through the fre-
quency range of interest. If, as in
Figure 2.23, we display the output
of the filter versus the frequency
to which it is tuned, we have the
spectrum of the input signal. This
swept analysis technique is com-
monly used in rf and microwave changes in its input. The narrower limited resolution and is expen-
spectrum analysis. the filter, the longer it takes to sive. The swept analyzer can be
respond. If we sweep the filter cheaper and have higher resolu-
We have, however, assumed the past a signal too quickly, the filter tion but the measurement takes
input signal hasnt changed in the output will not have a chance to longer (especially at high resolu-
time it takes to complete a sweep respond fully to the signal. As we tion) and it can not analyze
of our analyzer. If energy appears show in Figure 2.24, the spectrum transient events*.
at some frequency at a moment display will then be in error; our
when our filter is not tuned to estimate of the signal level will be Dynamic Signal Analyzer
that frequency, then we will not too low.
measure it. In recent years another kind of
In a parallel-filter spectrum ana- analyzer has been developed
One way to reduce this problem lyzer we do not have this prob- which offers the best features
would be to speed up the sweep lem. All the filters are connected of the parallel-filter and swept
time of our analyzer. We could to the input signal all the time. spectrum analyzers. Dynamic Sig-
still miss an event, but the time in Once we have waited the initial nal Analyzers are based on a high
which this could happen would be settling time of a single filter, all speed calculation routine which
shorter. Unfortunately though, we the filters will be settled and the acts like a parallel filter analyzer
cannot make the sweep arbitrarily spectrum will be valid and not with hundreds of filters and yet
fast because of the response time miss any transient events. are cost competitive with swept
of our filter.
So there is a basic trade-off
* More information on the performance of
To understand this problem, between parallel-filter and swept swept spectrum analyzers can be found in
recall from Section 2 that a filter spectrum analyzers. The parallel- Hewlett-Packard Application Note Series
takes a finite time to respond to filter analyzer is fast, but has 150.
18
spectrum analyzers. In addition, Figure 2.25
two channel Dynamic Signal Gain-phase
Analyzers are in many ways better meter
operation.
network analyzers than the ones
we will introduce next.
Network Analyzers
Since in network analysis it is
required to measure both the in-
put and output, network analyzers
are generally two channel devices
with the capability of measuring Figure 2.26
Tuned net-
the amplitude ratio (gain or loss) work analyzer
and phase difference between the operation.
channels. All of the analyzers dis-
cussed here measure frequency
response by using a sinusoidal
input to the network and slowly
changing its frequency. Dynamic
Signal Analyzers use a different,
much faster technique for net-
work analysis which we discuss
in the next chapter.
Gain-phase meters are broadband
devices which measure the ampli-
tude and phase of the input and
output sine waves of the network.
A sinusoidal 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 typically becomes a problem virtually eliminates the noise
made for a very low investment. with attenuations of about and any harmonics to allow
However, because gain-phase 60 dB (1,000:1). measurements of attenuation to
meters are broadband, they mea- 100 dB (100,000:1).
sure all the noise of the network Tuned network analyzers mini-
as well as the desired sine wave. mize the noise floor problems of By minimizing the noise, it is also
As the network attenuates the gain-phase meters by including a possible for tuned network ana-
input, this noise eventually bandpass filter which tracks the lyzers to make more accurate
becomes a floor below which source frequency. Figure 2.26 measurements of amplitude and
the meter cannot measure. This shows how this tracking filter phase. These improvements do
19
not come without their price, Figure 2.27
however, as tracking filters and a The vibration
dedicated source must be added of a tuning fork.
to the simpler and less costly
gain-phase meter.
Tuned analyzers are available
in the frequency range of a
few Hertz to many Gigahertz
(109 Hertz). If lower 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 limited to from
1 mHz to about 10 kHz.
Section 4:
The Modal Domain
In the preceding sections we have Figure 2.28
developed the properties of the Example
time and frequency domains and vibration modes
of a tuning fork.
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 us begin by analyzing a simple
mechanical structure, a tuning
fork. If we strike a tuning fork, we
easily conclude from its tone that
it is primarily vibrating at a single
frequency. We see that we have lightly damped sine wave shown response of the tuning fork has a
excited a network (tuning fork) in Figure 2.27b. major peak that is very lightly
with a force impulse (hitting damped, which is the tone we
the fork). The time domain In Figure 2.27c, we see in the fre- hear. There are also several
view of the sound caused by quency domain that the frequency smaller peaks.
the deformation of the fork is a
20
Each of these peaks, large and Figure 2.29
small, corresponds to a vibration Reducing the
mode of the tuning fork. For in- second harmonic
by damping the
stance, we might expect for this second vibration
simple example that the major mode.
tone is caused by the 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 simpler sine waves,
we can represent any vibration as
a sum of much simpler vibration
modes. The task of modal analy-
sis is to determine the shape and
the magnitude of the structural
deformation in each vibration Figure 2.30
mode. Once these are known, it Modal analysis
of a tuning fork.
usually becomes apparent how to
change the overall vibration.
For instance, let us look again at
our 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 affect-
ing the desired vibration of the
first mode. Other solutions are
possible, but all depend on know-
ing the geometry of each mode.
vibration at several points on the measure the properties of the
structure. Figure 2.30a shows structure independent of the
The Relationship Between
some points we might pick. If stimulus*.
The Time, Frequency and
we transformed this time domain
Modal Domain
data to the frequency domain, * Those who are more familiar with
we would get results like Figure electronics might note that we have
To determine the total vibration
2.30b. We measure frequency measured the frequency response of a
of our tuning fork or any other network (structure) at N points and thus
response because we want to
structure, we have to measure the have performed an N-port Analysis.
21
We see that the sharp peaks Figure 2.31
(resonances) all occur at the The relationship
same frequencies independent between the
frequency and
of where they are measured on the modal
the structure. Likewise we would domains.
find by measuring the width of
each resonance that the damping
(or Q) of each resonance is inde-
pendent 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 mea-
sure 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 re-
sponse 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 re-
sponse. Each resonance has a
peak value corresponding to the
peak displacement in that mode.
If we view the graph along the
frequency axis, as in Figure 2.31c, However, the equivalence modal domain to minimize the
we can see the mode shapes of between the modal, time and effects of noise and small experi-
the structure. frequency domains is not quite mental errors. No information is
as strong as that between the time lost in this curve fitting, so all
We have not lost any information and frequency domains. Because three domains contain the same
by this change of perspective. the modal domain portrays the information, but not the same
Each vibration mode is character- properties of the network inde- noise. Therefore, transforming
ized by its mode shape, frequency pendent of the stimulus, trans- from the frequency domain to the
and damping from which we can forming back to the time domain modal domain and back again will
reconstruct the frequency domain gives the impulse response of give results like those in Figure
view. the structure, no matter what 2.32. The results are not exactly
the stimulus. A more important the same, yet in all the important
limitation of this equivalence is features, the frequency responses
that curve fitting is used in trans- are the same. This is also true of
forming from our frequency re- time domain data derived from
sponse measurements to the 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
Instrumentation for Curve fitting
the Modal Domain removes
measurement
noise.
There are many ways that the
modes of vibration can be deter-
mined. In our simple tuning fork
example we could guess what the
modes were. In simple structures
like drums and plates it is pos-
sible 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.
Figure 2.33
There are two basic techniques Single mode
excitation
for determining the modes of modal analysis.
vibration in complicated struc-
tures; 1) exciting only one mode
at a time, and 2) computing the
modes of vibration from the total
vibration.
Single Mode Excitation
Modal Analysis
To illustrate single mode excita-
tion, 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 gen-
erator 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 dimen- Figure 2.34
sional problems, it is necessary to Measured mode
add many more shakers to ensure shape.
that only one mode is excited.
The difficulties and expense of
testing with many shakers has
limited the application of this tra-
ditional modal analysis technique.
Modal Analysis From
Total Vibration
To determine the modes of vibra-
tion from the total vibration of the
structure, we use the techniques
developed in the previous section.
Basically, we determine the fre- From the above description, it is Section 6: Summary
quency response of the structure apparent that a modal analyzer
at several points and compute at requires some type of network In this chapter we have developed
each resonance the frequency, analyzer to measure the frequency the concept of looking at prob-
damping and what is called the response of the structure and lems from different perspectives.
residue (which represents the a computer to convert the fre- These perspectives are the time,
height of the resonance). This is quency response to mode shapes. frequency and modal domains.
done by a curve-fitting routine to This can be accomplished by Phenomena that are confusing in
smooth out any noise or small connecting a Dynamic Signal the time domain are often clari-
experimental errors. From these Analyzer through a digital inter- fied by changing perspective to
measurements and the geometry face* to a computer furnished another domain. Small signals
of the structure, the mode shapes with the appropriate software. are easily resolved in the pres-
are computed and drawn on a This capability is also available ence of large ones in the fre-
CRT display or a plotter. If drawn in a single instrument called a quency domain. The frequency
on a CRT, these displays may be Structural Dynamics Analyzer. In domain is also valuable for pre-
animated to help the user under- general, computer systems offer dicting the output of any kind of
stand the vibration mode. more versatile performance since linear network. A change to the
they can be programmed to solve modal domain breaks down com-
other problems. However, Struc- plicated structural vibration prob-
tural Dynamics Analyzers gener- lems into simple vibration modes.
ally are much easier 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 avail-
able 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.
* HP-IB, Hewlett-Packards implementation
of IEEE-488-1975 is ideal for this
application.
24
Chapter 3
Understanding Dynamic
Signal Analysis
We saw in the previous chapter Figure 3.1
that the Dynamic Signal Analyzer The FFT samples
has the speed advantages of paral- in both the time
and frequency
lel-filter analyzers without their domains.
low resolution 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 presenting the properties
of the Fast Fourier Transform
(FFT) upon which Dynamic Sig-
nal Analyzers are based. No proof
of these properties is given, but
heuristic arguments as to their
validity are used where appropri-
ate. We then show how these FFT
properties cause some undesir-
able characteristics in spectrum
analysis like aliasing and leakage.
Having demonstrated a potential
difficulty with the FFT, we then
show what solutions are used to Figure 3.2
make practical Dynamic Signal A time record
Analyzers. Developing this basic is N equally
spaced samples
knowledge of FFT characteristics of the input.
makes it simple to get good
results with a Dynamic Signal
Analyzer in a wide range of
measurement problems.
Section 1: FFT Properties
The Fast Fourier Transform
(FFT) is an algorithm* for sufficiently accurate. Fortunately, Because we have sampled, we no
transforming data from the time with the advent of microproces- longer have an exact representa-
domain to the frequency domain. sors, it is easy and inexpensive to tion in either domain. However,
Since this is exactly what we incorporate all the needed com- a sampled representation can be
want a spectrum analyzer to do, it puting power in a small instru- as close to ideal as we desire by
would seem easy to implement a ment package. Note, however, placing our samples closer to-
Dynamic Signal Analyzer based that we cannot now transform to gether. Later in this chapter,
on the FFT. However, we will see the frequency domain in a con- we will consider what sample
that there are many factors which tinuous manner, but instead must spacing is necessary to guarantee
complicate this seemingly sample and digitize the time accurate results.
straight-forward task. domain input. This means that our
* An algorithm is any special mathematical
algorithm transforms digitized method of solving a certain kind of
First, because of the many calcu- samples from the time domain to problem; e.g., the technique you use
lations involved in transforming samples in the frequency domain to balance your checkbook.
domains, the transform must be as shown in Figure 3.1.** ** To reduce confusion about which domain
implemented on a digital com- we are in, samples in the frequency domain
puter if the results are to be are called lines.
25
Time Records Figure 3.3
The FFT works
A time record is defined to be on blocks
of data.
N consecutive, equally spaced
samples of the input. Because it
makes our transform algorithm
simpler and much faster, N is
restricted to be a multiple of 2,
for instance 1024.
As shown in Figure 3.3, this
time record is transformed as a
complete block into a complete
block of frequency lines. All the
samples of the time record are
needed to compute each and
every line in the frequency do-
main. This is in contrast to what
one might expect, namely that a
single time domain sample trans-
forms to exactly one frequency Figure 3.4
domain line. Understanding this A new time
record every
block processing property of the sample after
FFT is crucial to understanding the time record
many of the properties of the is filled.
Dynamic Signal Analyzer.
For instance, because the FFT
transforms the entire time record
block as a total, there cannot be
valid frequency domain results
until a complete time record has
been gathered. However, once
completed, the oldest sample
could be discarded, all the
samples shifted in the time
record, and a new sample added
to the end of the time record as
in Figure 3.4. Thus, once the time
record is initially filled, we have
a new time record at every time wait for the filters to respond, too much information, too fast.
domain sample and therefore then we can see very rapid This would often give you thou-
could have new valid results in changes in the frequency domain. sands of transforms per second.
the frequency domain at every With a Dynamic Signal Analyzer Just how fast a Dynamic Signal
time domain sample. we do not get a valid result until Analyzer should transform is a
a full time record has been gath- subject better left to the sections
This is very similar to the behav- ered. Then rapid changes in the in this chapter on real time band-
ior of the parallel-filter analyzers spectra can be seen. width and overlap processing.
described in the previous chapter.
When a signal is first applied to a It should be noted here that a new
parallel-filter analyzer, we must spectrum every sample is usually
26
How Many Lines are There? Figure 3.5
The relationship
We stated earlier that the time between the time
and frequency
record has N equally spaced domains.
samples. Another property of
the FFT is that it transforms
these time domain samples to
N/2 equally spaced lines in the
frequency domain. We only get
half as many lines because each
frequency line actually contains
two pieces of information, ampli-
tude and phase. The meaning of
this is most easily seen if we look
again at the relationship between
the time and frequency domain.
Figure 3.5 reproduces from Chap-
ter II our three-dimensional graph
of this relationship. Up to now we
have implied that the amplitude Figure 3.6
and frequency of the sine waves Phase of
frequency domain
contains all the information nec- components is
essary to reconstruct the input. important.
But it should be obvious that the
phase of each of these sine waves
is important too. For instance, in
Figure