The Fundamentals ofSignal Analysis

Application Note 243

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3Table 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

4The analysis of electrical signals is a fundamental problem for manyengineers and scientists. Even if theimmediate problem is not electrical,the basic parameters of interest areoften changed into electrical signalsby means of transducers. Commontransducers include accelerometersand load cells in mechanical work,EEG electrodes and blood pressureprobes in biology and medicine, andpH and conductivity probes in chem-istry. The rewards for transformingphysical parameters to electrical sig-nals are great, as many instrumentsare available for the analysis of elec-trical signals in the time, frequencyand modal domains. The powerfulmeasurement and analysis capabili-ties of these instruments can lead torapid understanding of the systemunder study.

This note is a primer for those whoare unfamiliar with the advantages ofanalysis in the frequency and modaldomains and with the class of analyz-ers we call Dynamic Signal Analyzers.In Chapter 2 we develop the conceptsof the time, frequency and modaldomains and show why these differ-ent ways of looking at a problemoften lend their own unique insights.We then introduce classes of instru-mentation available for analysis inthese domains.

In Chapter 3 we develop the proper-ties of one of these classes of analyz-ers, Dynamic Signal Analyzers. Theseinstruments are particularly appropri-ate for the analysis of signals in therange of a few millihertz to about a hundred kilohertz.

Chapter 4 shows the benefits ofDynamic Signal Analysis in a widerange of measurement situations. Thepowerful analysis tools of DynamicSignal Analysis are introduced asneeded in each measurement situation.

This note avoids the use of rigorousmathematics and instead depends onheuristic arguments. We have foundin over a decade of teaching thismaterial that such arguments lead toa better understanding of the basicprocesses involved in the variousdomains and in Dynamic SignalAnalysis. Equally important, thisheuristic instruction leads to betterinstrument operators who can intelli-gently use these analyzers to solvecomplicated measurement problemswith accuracy and ease*.

Because of the tutorial nature of thisnote, we will not attempt to showdetailed solutions for the multitude ofmeasurement problems which can besolved by Dynamic Signal Analysis.Instead, we will concentrate on thefeatures of Dynamic Signal Analysis,how these features are used in a widerange of applications and the benefitsto be gained from using DynamicSignal Analysis.

Those who desire more details on specific applications should lookto Appendix B. It contains abstractsof Agilent Technologies ApplicationNotes on a wide range of related subjects. These can be obtained freeof charge from your local Agilentfield engineer or representative.

Chapter 1 Introduction

* A more rigorous mathematical justification for the arguments developed in the main text can be found in Appendix A.

5A Matter of Perspective

In this chapter we introduce the concepts of the time, frequency andmodal domains. These three ways oflooking at a problem are interchange-able; that is, no information is lost in changing from one domain toanother. The advantage in introducingthese three domains is that of achange of perspective. By changingperspective from the time domain, thesolution to difficult problems can often become quite clear in thefrequency or modal domains.

After developing the concepts of eachdomain, we will introduce the typesof instrumentation available. Themerits of each generic instrumenttype are discussed to give the readeran appreciation of the advantages anddisadvantages of each approach.

Section 1: The Time Domain

The traditional way of observing signals is to view them in the timedomain. The time domain is a recordof what happened to a parameter ofthe system versus time. For instance,Figure 2.1 shows a simple spring-mass system where we have attached a pen to the mass and pulled a pieceof paper past the pen at a constantrate. The resulting graph is a recordof the displacement of the mass versus time, a time domain view ofdisplacement.

Such direct recording schemes aresometimes used, but it usually ismuch more practical to convert the parameter of interest to an electrical signal using a transducer.Transducers are commonly availableto change a wide variety of parame-ters to electrical signals. Micro-phones, accelerometers, load cells,conductivity and pressure probes arejust a few examples.

This electrical signal, which repre-sents a parameter of the system, canbe recorded on a strip chart recorderas in Figure 2.2. We can adjust thegain of the system to calibrate ourmeasurement. Then we can repro-duce exactly the results of our simpledirect recording system in Figure 2.1.

Why should we use this indirectapproach? One reason is that we arenot always measuring displacement.We then must convert the desiredparameter to the displacement of therecorder pen. Usually, the easiest wayto do this is through the intermediaryof electronics. However, even whenmeasuring displacement we wouldnormally use an indirect approach.Why? Primarily because the system inFigure 2.1 is hopelessly ideal. Themass must be large enough and thespring stiff enough so that the pensmass and drag on the paper will not

Chapter 2 The Time, Frequency and Modal Domains:

Figure 2.2Indirect recording of displacement.

Figure 2.1Direct recording of displacement - a time domain view.

6affect the results appreciably. Alsothe deflection of the mass must belarge enough to give a usable result,otherwise a mechanical lever systemto amplify the motion would have tobe added with its attendant mass and friction.

With the indirect system a transducercan usually be selected which will notsignificantly affect the measurement.This can go to the extreme of com-mercially available displacementtransducers which do not even con-tact the mass. The pen deflection canbe easily set to any desired value bycontrolling the gain of the electronicamplifiers.

This indirect system works well until our measured parameter beginsto change rapidly. Because of themass of the pen and recorder mecha-nism and the power limitations of its drive, the pen can only move at finite velocity. If the measured parameter changes faster, the outputof the recorder will be in error. Acommon way to reduce this problemis to eliminate the pen and record ona photosensitive paper by deflecting a light beam. Such a device is called an oscillograph. Since it is only necessary to move a small, light-weight mirror through a verysmall angle, the oscillograph canrespond much faster than a stripchart recorder.

Another common device for display-ing signals in the time domain is theoscilloscope. Here an electron beam ismoved using electric fields. The elec-tron beam is made visible by a screenof phosphorescent material. It is capable of accurately displayingsignals that vary even more rapidlythan the oscillograph can handle. This is because it is only necessary tomove an electron beam, not a mirror.

The strip chart, oscillograph andoscilloscope all show displacementversus time. We say that changes in this displacement represent thevariation of some parameter versustime. We will now look at anotherway of representing the variation of a parameter.

Figure 2.3Simplified oscillograph operation.

Figure 2.4Simplified oscilloscope operation (Horizontal deflection circuits omitted for clarity).

7Section 2: The FrequencyDomain

It was shown over one hundred yearsago by Baron Jean Baptiste Fourierthat any waveform that exists in thereal world can be generated byadding up sine waves. We have illus-trated this in Figure 2.5 for a simplewaveform composed of two sinewaves. By picking the amplitudes, frequencies and phases of these sinewaves correctly, we can generate awaveform identical to our desired signal.

Conversely, we can break down ourreal world signal into these same sinewaves. It can be shown that this com-bination of sine waves is unique; anyreal world signal can be representedby only one combination of sinewaves.

Figure 2.6a is a three dimensionalgraph of this addition of sine waves.Two of the axes are time and ampli-tude, familiar from the time domain.The third axis is frequency whichallows us to visually separate the sine waves which add to give us ourcomplex waveform. If we view thisthree-dimensional graph along the frequency axis we get the view inFigure 2.6b. This is the time domainview of the sine waves. Adding themtogether at each instant of time givesthe original waveform.

However, if we view our graph alongthe time axis as in Figure 2.6c, we get a totally different picture. Herewe have axes of amplitude versus frequency, what is commonly calledthe frequency domain. Every sinewave we separated from the inputappears as a vertical line. Its heightrepresents its amplitude and its posi-tion represents its frequency. Sincewe know that each line represents a

sine wave, we have uniquely characterized our input signal in thefrequency domain*. This frequencydomain representation of our signal is called the spectrum of the signal.Each sine wave line of the spectrumis called a component of the total signal.

Figure 2.6The relationship between the time and frequency domains.a) Three- dimensional coordinates showing time, frequency and amplitudeb) Time domain viewc) Frequency domain view.

Figure 2.5Any real waveform can be produced by adding sine waves together.

* Actually, we have lost the phase information of the sinewaves. How we get this will be discussed in Chapter 3.

8The Need for Decibels

Since one of the major uses of the frequencydomain is to resolve small signals in the presence of large ones, let us now address the problem of how we can see both large and small signals on our display simultaneously.

Suppose we wish to measure a distortion component that is 0.1% of the signal. If we setthe fundamental to full scale on a four inch (10 cm) screen, the harmonic would be onlyfour 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 components easily at the same time, the only answer is to change our amplitude scale.A logarithmic scale would compress our largesignal amplitude and expand the small ones,allowing all components to be displayed at thesame time.

Alexander Graham Bell discovered that thehuman ear responded logarithmically to powerdifference and invented a unit, the Bel, to helphim measure the ability of people to hear. Onetenth of a Bel, the deciBel (dB) is the most common unit used in the frequency domaintoday. 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 belowthe 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.

Figure 2.8The relation-ship between decibels, power and voltage.

Figure 2.9Small signals can be measured with a logarithmic amplitude scale.

9It is very important to understandthat we have neither gained nor lostinformation, we are just represent-ing it differently. We are looking atthe same three-dimensional graphfrom different angles. This differentperspective can be very useful.

Why the Frequency Domain?

Suppose we wish to measure thelevel of distortion in an audio oscilla-tor. Or we might be trying to detectthe first sounds of a bearing failing ona noisy machine. In each case, we aretrying to detect a small sine wave inthe presence of large signals. Figure2.7a shows a time domain waveformwhich seems to be a single sine wave.But Figure 2.7b shows in the frequen-cy domain that the same signal iscomposed of a large sine wave andsignificant other sine wave compo-nents (distortion components). Whenthese components are separated inthe frequency domain, the small components are easy to see becausethey are not masked by larger ones.

The frequency domains usefulness is not restricted to electronics ormechanics. All fields of science andengineering have measurements likethese where large signals mask othersin the time domain. The frequency domain provides a useful tool in analyzing these small but importanteffects.

The Frequency Domain: A Natural Domain

At first the frequency domain mayseem strange and unfamiliar, yet it is an important part of everyday life.Your ear-brain combination is anexcellent frequency domain analyzer.The ear-brain splits the audio spec-trum into many narrow bands and determines the power present in each band. It can easily pick small

sounds out of loud background noisethanks in part to its frequencydomain capability. A doctor listens to your heart and breathing for anyunusual sounds. He is listening forfrequencies which will tell him something is wrong. An experiencedmechanic can do the same thing witha machine. Using a screwdriver as astethoscope, he can hear when abearing is failing because of the frequencies it produces.

Figure 2.7Small signals are not hidden in the frequency domain.

a) Time Domain - small signal not visible

b) Frequency Domain - small signal easily resolved

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So we see that the frequency domainis not at all uncommon. We are justnot used to seeing it in graphicalform. But this graphical presentationis really not any stranger than sayingthat the temperature changed withtime 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 frequencydomains. In Figure 2.10a, we see thatthe spectrum of a sine wave is just asingle line. We expect this from theway we constructed the frequencydomain. The square wave in Figure2.10b is made up of an infinite num-ber of sine waves, all harmonically related. The lowest frequency presentis the reciprocal of the square waveperiod. These two examples illustratea property of the frequency trans-form: a signal which is periodic andexists for all time has a discrete fre-quency spectrum. This is in contrastto the transient signal in Figure 2.10cwhich has a continuous spectrum.This means that the sine waves thatmake up this signal are spaced infinitesimally close together.

Another signal of interest is the impulse shown in Figure 2.10d. Thefrequency spectrum of an impulse isflat, i.e., there is energy at all frequen-cies. It would, therefore, require infinite energy to generate a trueimpulse. Nevertheless, it is possibleto generate an approximation to an impulse which has a fairly flatspectrum over the desired frequencyrange of interest. We will find signalswith a flat spectrum useful in ournext subject, network analysis.

Figure 2.10Frequency spectrum examples.

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Network Analysis

If the frequency domain were restricted to the analysis of signalspectrums, it would certainly not besuch a common engineering tool.However, the frequency domain isalso widely used in analyzing thebehavior of networks (network analysis) and in design work.

Network analysis is the general engineering problem of determininghow a network will respond to aninput*. For instance, we might wishto determine how a structure willbehave in high winds. Or we mightwant to know how effective a soundabsorbing wall we are planning onpurchasing would be in reducing machinery noise. Or perhaps we areinterested in the effects of a tube ofsaline solution on the transmission ofblood pressure waveforms from anartery to a monitor.

All of these problems and many moreare examples of network analysis. Asyou can see a network can be anysystem at all. One-port networkanalysis is the variation of oneparameter with respect to another,both measured at the same point(port) of the network. The impedanceor compliance of the electronic or mechanical networks shown inFigure 2.11 are typical examples ofone-port network analysis.

Figure 2.11One-port network analysis examples.

* Network Analysis is sometimes called Stimulus/ResponseTesting. The input is then known as the stimulus or excitation and the output is called the response.

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Two-port analysis gives the responseat a second port due to an input atthe first port. We are generally inter-ested in the transmission and rejec-tion of signals and in insuring theintegrity of signal transmission. Theconcept of two-port analysis can beextended to any number of inputsand outputs. This is called N-portanalysis, a subject we will use inmodal analysis later in this chapter.

We have deliberately defined networkanalysis in a very general way. Itapplies to all networks with no limitations. If we place one conditionon our network, linearity, we find that network analysis becomes a very powerful tool.

Figure 2.12Two-port network analysis.

Figure 2.14Non-linear system example.

Figure 2.15Examples ofnon-linearities.

Figure 2.13Linear network.

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When we say a network is linear, wemean it behaves like the network in Figure 2.13. Suppose one inputcauses an output A and a secondinput applied at the same port causesan output B. If we apply both inputsat 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 allnetworks would behave in this fash-ion. A counter example, a non-linearnetwork, is shown in Figure 2.14.Suppose that the first input is a forcethat varies in a sinusoidal manner. Wepick its amplitude to ensure that the displacement is small enough so thatthe oscillating mass does not quite hitthe stops. If we add a second identi-cal input, the mass would now hit thestops. Instead of a sine wave withtwice the amplitude, the output isclipped as shown in Figure 2.14b.

This spring-mass system with stopsillustrates an important principal: noreal system is completely linear. Asystem may be approximately linearover a wide range of signals, buteventually the assumption of linearitybreaks down. Our spring-mass systemis linear before it hits the stops. Likewise a linear electronic amplifierclips when the output voltageapproaches the internal supply voltage. A spring may compress linearly until the coils start pressingagainst each other.

Other forms of non-linearities arealso often present. Hysteresis (orbacklash) is usually present in geartrains, loosely riveted joints and inmagnetic devices. Sometimes thenon-linearities are less abrupt and aresmooth, but nonlinear, curves. Thetorque versus rpm of an engine or theoperating curves of a transistor aretwo examples that can be consideredlinear over only small portions oftheir operating regions.

The important point is not that allsystems are nonlinear; it is that most systems can be approximatedas linear systems. Often a large engineering effort is spent in makingthe system as linear as practical. Thisis done for two reasons. First, it is often a design goal for the output of anetwork to be a scaled, linear versionof the input. A strip chart recorder is a good example. The electronicamplifier and pen motor must both bedesigned to ensure that the deflectionacross the paper is linear with theapplied voltage.

The second reason why systems arelinearized is to reduce the problem of nonlinear instability. One examplewould be the positioning systemshown in Figure 2.16. The actual position is compared to the desiredposition and the error is integratedand applied to the motor. If the geartrain has no backlash, it is a straight-forward problem to design this system to the desired specificationsof positioning accuracy and response time.

However, if the gear train has exces-sive backlash, the motor will hunt,causing the positioning system tooscillate around the desired position.The solution is either to reduce theloop gain and therefore reduce theoverall performance of the system, or to reduce the backlash in the geartrain. Often, reducing the backlash is the only way to meet the performance specifications.

Figure 2.16A positioning system.

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Analysis of Linear Networks

As we have seen, many systems aredesigned to be reasonably linear tomeet 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 theinput. Therefore, if we knew the response of the network to each ofthe sine wave components of theinput 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 of the same frequency. As shown inFigure 2.17, the amplitude of the output sine wave is proportional tothe input amplitude. Its phase is shifted by an amount which dependsonly on the frequency of the sinewave. As we vary the frequency of the sine wave input, the amplitudeproportionality factor (gain) changesas does the phase of the output. If we divide the output of the network by the input, we get a

Figure 2.17Linear network response to a sine wave input.

Figure 2.18The frequency response of a network.

* We will discuss the analysis of networks which have not been linearized in Chapter 3, Section 6.

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normalized result called the frequen-cy response of the network. As shown in Figure 2.18, the frequencyresponse is the gain (or loss) andphase shift of the network as a function of frequency. Because thenetwork is linear, the frequencyresponse is independent of the inputamplitude; the frequency response isa property of a linear network, notdependent on the stimulus.

The frequency response of a networkwill generally fall into one of threecategories; low pass, high pass, bandpass or a combination of these.As the names suggest, their frequencyresponses have relatively high gain ina band of frequencies, allowing thesefrequencies to pass through the network. Other frequencies suffer arelatively high loss and are rejectedby the network. To see what thismeans in terms of the response of afilter to an input, let us look at thebandpass filter case.

Figure 2.19Three classes of frequencyresponse.

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In Figure 2.20, we put a square waveinto a bandpass filter. We recall fromFigure 2.10 that a square wave iscomposed of harmonically relatedsine waves. The frequency responseof 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 thisbandpass filter is a sine wave.

Notice how easy it is to predict the output of any network from its frequency response. The spectrum ofthe input signal is multiplied by thefrequency response of the network to determine the components thatappear in the output spectrum. Thisfrequency 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 donenumerically on a digital computer*. Ifwe computed the network responseby both evaluating the time domainintegral and by transforming to thefrequency domain and back, wewould get the same results. However,it is usually easier to compute theoutput by transforming to the frequency domain.

Transient Response

Up to this point we have only dis-cussed the steady-state response to asignal. By steady-state we mean theoutput after any transient responsescaused by applying the input havedied out. However, the frequencyresponse of a network also containsall the information necessary to predict the transient response of thenetwork to any signal.

Figure 2.20Bandpass filter response to a square wave input.

Figure 2.21Time response of bandpass filters.

* This operation is called convolution.

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Let us look qualitatively at the tran-sient response of a bandpass filter. Ifa resonance is narrow compared toits frequency, then it is said to be ahigh Q resonance*. Figure 2.21ashows a high Q filter frequencyresponse. It has a transient responsewhich dies out very slowly. A time response which decays slowly is saidto be lightly damped. Figure 2.21bshows a low Q resonance. It has atransient response which dies outquickly. This illustrates a general principle: signals which are broad inone domain are narrow in the other.Narrow, selective filters have verylong response times, a fact we willfind important in the next section.

Section 3: Instrumentation for theFrequency Domain

Just as the time domain can be measured with strip chart recorders,oscillographs or oscilloscopes, the frequency domain is usuallymeasured with spectrum and network analyzers.

Spectrum analyzers are instrumentswhich are optimized to characterizesignals. They introduce very little distortion and few spurious signals.This insures that the signals on thedisplay are truly part of the input signal spectrum, not signals introduced by the analyzer.

Network analyzers are optimized togive accurate amplitude and phasemeasurements over a wide range ofnetwork gains and losses. This designdifference means that these two traditional instrument families are not interchangeable.** A spectrumanalyzer can not be used as a net-work analyzer because it does notmeasure amplitude accurately andcannot measure phase. A networkanalyzer would make a very poorspectrum analyzer because spuriousresponses limit its dynamic range.

In this section we will develop theproperties of several types of analyzers in these two categories.

The Parallel-Filter Spectrum Analyzer

As we developed in Section 2 of this chapter, electronic filters can bebuilt which pass a narrow band offrequencies. If we were to add ameter to the output of such a band-pass filter, we could measure thepower in the portion of the spectrumpassed by the filter. In Figure 2.22awe have done this for a bank of filters, each tuned to a different frequency. If the center frequencies of these filters are chosen so that the filters overlap properly, the spectrum covered by the filters canbe completely characterized as inFigure 2.22b.

Figure 2.22Parallel filter analyzer.

* 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.

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How many filters should we use tocover the desired spectrum? Here wehave a trade-off. We would like to beable to see closely spaced spectrallines, so we should have a large number of filters. However, each filter is expensive and becomes moreexpensive as it becomes narrower, so the cost of the analyzer goes up as we improve its resolution. Typicalaudio parallel-filter analyzers balancethese demands with 32 filters, eachcovering 1/3 of an octave.

Swept Spectrum Analyzer

One way to avoid the need for such a large number of expensive filters isto use only one filter and sweep itslowly through the frequency range of interest. If, as in Figure 2.23, wedisplay the output of the filter versusthe frequency to which it is tuned, we have the spectrum of the inputsignal. This swept analysis techniqueis commonly used in rf andmicrowave spectrum analysis.

We have, however, assumed the inputsignal hasnt changed in the time ittakes to complete a sweep of our analyzer. If energy appears at somefrequency at a moment when our filter is not tuned to that frequency,then we will not measure it.

One way to reduce this problemwould be to speed up the sweep time of our analyzer. We could stillmiss an event, but the time in whichthis could happen would be shorter.Unfortunately though, we cannotmake the sweep arbitrarily fastbecause of the response time of our filter.

To understand this problem, recall from Section 2 that a filtertakes a finite time to respond tochanges in its input. The narrower thefilter, the longer it takes to respond.

If we sweep the filter past a signaltoo quickly, the filter output will nothave a chance to respond fully to thesignal. As we show in Figure 2.24, the spectrum display will then be inerror; our estimate of the signal levelwill be too low.

In a parallel-filter spectrum analyzerwe do not have this problem. All thefilters are connected to the input signal all the time. Once we havewaited the initial settling time of asingle filter, all the filters will be settled and the spectrum will be validand not miss any transient events.

So there is a basic trade-off betweenparallel-filter and swept spectrumanalyzers. The parallel-filter analyzer

is fast, but has limited resolution andis expensive. The swept analyzer can be cheaper and have higher resolution but the measurement takes longer (especially at high resolution) and it can not analyzetransient events*.

Dynamic Signal Analyzer

In recent years another kind of analyzer has been developed which offers the best features of theparallel-filter and swept spectrumanalyzers. Dynamic Signal Analyzersare based on a high speed calculationroutine which acts like a parallel filter analyzer with hundreds of filters and yet are cost-competitivewith swept spectrum analyzers. In

* More information on the performance of swept spectrum analyzers can be found in Agilent Application Note Series 150.

Figure 2.24Amplitude error form sweeping too fast.

Figure 2.23Simplified swept spectrum analyzer.

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addition, two channel Dynamic Signal Analyzers are in many ways betternetwork analyzers than the ones wewill introduce next.

Network Analyzers

Since in network analysis it is required to measure both the inputand output, network analyzers aregenerally two channel devices withthe capability of measuring the ampli-tude ratio (gain or loss) and phasedifference between the channels. All of the analyzers discussed heremeasure frequency response by usinga sinusoidal input to the network and slowly changing its frequency.Dynamic Signal Analyzers use a different, much faster technique fornetwork analysis which we discuss in the next chapter.

Gain-phase meters are broadbanddevices which measure the amplitudeand phase of the input and outputsine waves of the network. A sinu-soidal source must be supplied tostimulate the network when using again-phase meter as in Figure 2.25.The source can be tuned manuallyand the gain-phase plots done byhand or a sweeping source, and an x-y plotter can be used for automaticfrequency response plots.

The primary attraction of gain-phasemeters is their low price. If a sinusoidal source and a plotter arealready available, frequency responsemeasurements can be made for a verylow investment. However, becausegain-phase meters are broadband,they measure all the noise of the network as well as the desired sinewave. As the network attenuates theinput, this noise eventually becomes afloor below which the meter cannotmeasure. This typically becomes aproblem with attenuations of about 60 dB (1,000:1).

Tuned network analyzers minimizethe noise floor problems of gain-phase meters by including a bandpassfilter which tracks the source fre-quency. Figure 2.26 shows how thistracking filter virtually eliminates thenoise and any harmonics to allow measurements of attenuation to 100 dB (100,000:1).

By minimizing the noise, it is alsopossible for tuned network analyzersto make more accurate measure-ments of amplitude and phase. Theseimprovements do not come withouttheir price, however, as tracking filters and a dedicated source mustbe added to the simpler and less costly gain-phase meter.

Figure 2.26Tuned net-work analyzeroperation.

Figure 2.25Gain-phase meter operation.

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Tuned analyzers are available in thefrequency range of a few Hertz tomany Gigahertz (109 Hertz). If lowerfrequency analysis is desired, a frequency response analyzer is oftenused. To the operator, it behaves exactly like a tuned network analyzer.However, it is quite different inside. It integrates the signals in the timedomain to effectively filter the signalsat very low frequencies where it isnot practical to make filters by moreconventional techniques. Frequencyresponse analyzers are generally lim-ited to from 1 mHz to about 10 kHz.

Section 4: The Modal Domain

In the preceding sections we havedeveloped the properties of the timeand frequency domains and theinstrumentation used in thesedomains. In this section we willdevelop the properties of anotherdomain, the modal domain. Thischange in perspective to a newdomain is particularly useful if we areinterested in analyzing the behaviorof mechanical structures.

To understand the modal domain letus begin by analyzing a simplemechanical structure, a tuning fork. If we strike a tuning fork, we easilyconclude 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 forceimpulse (hitting the fork). The timedomain 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 frequencyresponse of the tuning fork has amajor peak that is very lightlydamped, which is the tone we hear.There are also several smaller peaks.

Figure 2.27The vibration of a tuning fork.

Figure 2.28Example vibration modes of a tuning fork.

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Each of these peaks, large and small,corresponds to a vibration mode of the tuning fork. For instance, wemight expect for this simple examplethat the major tone is caused by thevibration mode shown in Figure2.28a. The second harmonic might be caused by a vibration like Figure 2.28b

We can express the vibration of anystructure as a sum of its vibrationmodes. Just as we can represent anreal waveform as a sum of much sim-pler sine waves, we can represent anyvibration as a sum of much simplervibration modes. The task of modalanalysis is to determine the shapeand the magnitude of the structuraldeformation in each vibration mode.Once these are known, it usuallybecomes apparent how to change the overall vibration.

For instance, let us look again at ourtuning fork example. Suppose that wedecided that the second harmonictone was too loud. How should wechange our tuning fork to reduce theharmonic? If we had measured thevibration of the fork and determinedthat the modes of vibration werethose shown in Figure 2.28, theanswer becomes clear. We mightapply damping material at the centerof the tines of the fork. This wouldgreatly affect the second mode which has maximum deflection at the centerwhile only slightly affecting thedesired vibration of the first mode.Other solutions are possible, but alldepend on knowing the geometry ofeach 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 thevibration at several points on thestructure. Figure 2.30a shows somepoints we might pick. If we transformed this time domain data tothe frequency domain, we would get

results like Figure 2.30b. We measurefrequency response because we wantto measure the properties of thestructure independent of the stimulus*.

Figure 2.29Reducing the second harmonic by damping the second vibration mode.

Figure 2.30Modal analysis of a tuning fork.

* 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 performedan N-port Analysis.

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We see that the sharp peaks (resonances) all occur at the same frequencies independent of wherethey are measured on the structure.Likewise we would find by measuringthe width of each resonance that thedamping (or Q) of each resonance is independent of position. The only parameter that varies as wemove from point to point along thestructure is the relative height of resonances.* By connecting thepeaks of the resonances of a givenmode, we trace out the mode shapeof that mode.

Experimentally we have to measureonly a few points on the structure todetermine the mode shape. However,to clearly show the mode shape inour figure, we have drawn in the frequency response at many morepoints in Figure 2.31a. If we view thisthree-dimensional graph along thedistance axis, as in Figure 2.31b, weget a combined frequency response.Each resonance has a peak value cor-responding to the peak displacementin that mode. If we view the graphalong the frequency axis, as in Figure2.31c, we can see the mode shapes ofthe structure.

We have not lost any information bythis change of perspective. Eachvibration mode is characterized by itsmode shape, frequency and dampingfrom which we can reconstruct thefrequency domain view.

However, the equivalence betweenthe modal, time and frequencydomains is not quite as strong as that between the time and frequencydomains. Because the modal domainportrays the properties of the net-work independent of the stimulus,transforming back to the time domaingives the impulse response of thestructure, no matter what the stimu-lus. A more important limitation ofthis equivalence is that curve fitting is used in transforming from our frequency response measurements to

the modal domain to minimize theeffects of noise and small experimen-tal errors. No information is lost inthis curve fitting, so all three domainscontain the same information, but notthe same noise. Therefore, transform-ing from the frequency domain to themodal domain and back again willgive results like those in Figure 2.32.The results are not exactly the same,yet in all the important features, thefrequency responses are the same.This is also true of time domain dataderived from the modal domain.

Figure 2.31The relationship between the frequency and the modal domains.

* The phase of each resonance is not shown for clarity of the figures but it too is important in the mode shape. Themagnitude of the frequency response gives the magnitudeof the mode shape while the phase gives the direction ofthe deflection.

23

Section 5: Instrumentation for the Modal Domain

There are many ways that the modesof vibration can be determined. In oursimple tuning fork example we couldguess what the modes were. In simplestructures like drums and plates it ispossible to write an equation for themodes of vibration. However, inalmost any real problem, the solutioncan neither be guessed nor solvedanalytically because the structure istoo complicated. In these cases it isnecessary to measure the response of the structure and determine the modes.

There are two basic techniques fordetermining the modes of vibration incomplicated structures: 1) excitingonly one mode at a time, and 2) computing the modes of vibrationfrom the total vibration.

Single Mode Excitation Modal Analysis

To illustrate single mode excitation,let us look once again at our simpletuning fork example. To excite justthe first mode we need two shakers,driven by a sine wave and attached to the ends of the tines as in Figure2.33a. Varying the frequency of thegenerator near the first mode reso-nance frequency would then give usits frequency, damping and modeshape.

In the second mode, the ends of thetines do not move, so to excite thesecond mode we must move theshakers to the center of the tines. Ifwe anchor the ends of the tines, wewill constrain the vibration to the second mode alone.

Figure 2.32Curve fitting removes measurement noise.

Figure 2.33Single mode excitation modal analysis.

24

In more realistic, three dimensionalproblems, it is necessary to add manymore shakers to ensure that only onemode is excited. The difficulties andexpense of testing with many shakershas limited the application of this traditional modal analysis technique.

Modal Analysis From Total Vibration

To determine the modes of vibrationfrom the total vibration of the structure, we use the techniquesdeveloped in the previous section.Basically, we determine the frequencyresponse of the structure at severalpoints and compute at each reso-nance the frequency, damping andwhat is called the residue (which represents the height of the reso-nance). This is done by a curve-fittingroutine to smooth out any noise orsmall experimental errors. Fromthese measurements and the geome-try of the structure, the mode shapesare computed and drawn on a CRTdisplay or a plotter. If drawn on aCRT, these displays may be animatedto help the user understand the vibration mode.

From the above description, it isapparent that a modal analyzerrequires some type of network analyzer to measure the frequency response of the structure and a computer to convert the frequencyresponse to mode shapes. This can be accomplished by connecting aDynamic Signal Analyzer through a digital interface* to a computer furnished with the appropriate soft-ware. This capability is also available in a single instrument called a Struc-tural Dynamics Analyzer. In general,computer systems offer more versa-tile performance since they can beprogrammed to solve other problems.However, Structural DynamicsAnalyzers generally are much easierto use than computer systems.

Section 6: Summary

In this chapter we have developed the concept of looking at problemsfrom different perspectives. Theseperspectives are the time, frequencyand modal domains. Phenomena thatare confusing in the time domain areoften clarified by changing perspec-tive to another domain. Small signals are easily resolved in the presence oflarge ones in the frequency domain.The frequency domain is also valu-able for predicting the output of anykind of linear network. A change tothe modal domain breaks down complicated structural vibrationproblems into simple vibrationmodes.

No one domain is always the bestanswer, so the ability to easily changedomains is quite valuable. Of all theinstrumentation available today, onlyDynamic Signal Analyzers can workin all three domains. In the next chapter we develop the properties of this important class of analyzers.

Figure 2.34Measured mode shape.

* GPIB, Agilents implementation of IEEE-488-1975 is ideal for this application.

25

We saw in the previous chapter thatthe Dynamic Signal Analyzer has thespeed advantages of parallel-filteranalyzers without their low resolutionlimitations. In addition, it is the onlytype of analyzer that works in allthree domains. In this chapter we willdevelop a fuller understanding of thisimportant analyzer family, DynamicSignal Analyzers. We begin by pre-senting the properties of the FastFourier Transform (FFT) upon whichDynamic Signal Analyzers are based.No proof of these properties is given,but heuristic arguments as to their va-lidity are used where appropriate. Wethen show how these FFT propertiescause some undesirable characteris-tics in spectrum analysis like aliasingand leakage. Having demonstrated apotential difficulty with the FFT, wethen show what solutions are used to make practical Dynamic Signal Analyzers. Developing this basicknowledge of FFT characteristicsmakes it simple to get good resultswith a Dynamic Signal Analyzer in awide range of measurement problems.

Section 1: FFT Properties

The Fast Fourier Transform (FFT) is an algorithm* for transformingdata from the time domain to the fre-quency domain. Since this is exactlywhat we want a spectrum analyzer todo, it would seem easy to implementa Dynamic Signal Analyzer based on the FFT. However, we will see that there are many factors whichcomplicate this seemingly straightforward task.

First, because of the many calcula-tions involved in transformingdomains, the transform must beimplemented on a digital computer ifthe results are to be sufficiently accu-rate. Fortunately, with the advent ofmicroprocessors, it is easy and inex-pensive to incorporate all the neededcomputing power in a small instru-ment package. Note, however, thatwe cannot now transform to the

frequency domain in a continuousmanner, but instead must sample anddigitize the time domain input. Thismeans that our algorithm transformsdigitized samples from the time do-main to samples in the frequencydomain as shown in Figure 3.1.**

Because we have sampled, we nolonger have an exact representationin either domain. However, a sampledrepresentation can be as close toideal as we desire by placing our

samples closer together. Later in this chapter, we will consider whatsample spacing is necessary to guarantee accurate results.

Chapter 3 Understanding DynamicSignal Analysis

Figure 3.1The FFT samplesin both the time and frequency domains.

Figure 3.2A time record is N equally spaced samples of the input.

* An algorithm is any special mathematical method of solving a certain kind of problem; e.g., the technique you use to balance your checkbook.

** To reduce confusion about which domain we are in, samples in the frequency domain are called lines.

26

Time Records

A time record is defined to be N consecutive, equally spaced samplesof the input. Because it makes ourtransform algorithm simpler andmuch faster, N is restricted to be a multiple of 2, for instance 1024.

As shown in Figure 3.3, this timerecord is transformed as a completeblock into a complete block of frequency lines. All the samples of the time record are needed to compute each and every line in thefrequency domain. This is in contrastto what one might expect, namelythat a single time domain sampletransforms to exactly one frequencydomain line. Understanding this blockprocessing property of the FFT is crucial to understanding many of the properties of the Dynamic Signal Analyzer.

For instance, because the FFT transforms the entire time recordblock as a total, there cannot be valid frequency domain results until a complete time record has beengathered. However, once completed,the oldest sample could be discarded,all the samples shifted in the timerecord, and a new sample added tothe end of the time record as inFigure 3.4. Thus, once the time recordis initially filled, we have a new timerecord at every time domain sampleand therefore could have new validresults in the frequency domain atevery time domain sample.

This is very similar to the behavior ofthe parallel-filter analyzers describedin the previous chapter. When a signalis first applied to a parallel-filter ana-lyzer, we must wait for the filters torespond, then we can see very rapidchanges in the frequency domain.With a Dynamic Signal Analyzer wedo not get a valid result until a fulltime record has been gathered. Thenrapid changes in the spectra can be seen.

It should be noted here that a newspectrum every sample is usually toomuch information, too fast. Thiswould often give you thousands oftransforms per second. Just how fast

a Dynamic Signal Analyzer shouldtransform is a subject better left tothe sections in this chapter on realtime bandwidth and overlap processing.

Figure 3.3The FFT works on blocks of data.

Figure 3.4A new time record every sample after the time record is filled.

27

How Many Lines are There?

We stated earlier that the time record has N equally spaced 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 eachfrequency line actually contains twopieces of information, amplitude andphase. The meaning of this is mosteasily seen if we look again at therelationship between the time andfrequency domain.

Figure 3.5 reproduces from Chapter 2our three-dimensional graph of thisrelationship. Up to now we haveimplied that the amplitude and frequency of the sine waves containsall the information necessary to reconstruct the input. But it shouldbe obvious that the phase of each of these sine waves is important too.For instance, in Figure 3.6, we haveshifted the phase of the higher frequency sine wave components of this signal. The result is a severedistortion of the original wave form.

We have not discussed the phaseinformation contained in the spectrum of signals until nowbecause none of the traditional spectrum analyzers are capable ofmeasuring phase. When we discussmeasurements in Chapter 4, we shallfind that phase contains valuableinformation in determining the cause of performance problems.

What is the Spacing of the Lines?

Now that we know that we have N/2equally spaced lines in the frequencydomain, what is their spacing? Thelowest frequency that we can resolvewith our FFT spectrum analyzer mustbe based on the length of the timerecord. We can see in Figure 3.7 that

if the period of the input signal islonger than the time record, we haveno way of determining the period (orfrequency, its reciprocal). Therefore,the lowest frequency line of the FFTmust occur at frequency equal to thereciprocal of the time record length.

Figure 3.5The relationship between the time and frequency domains.

Figure 3.6Phase of frequency domain components is important.

28

In addition, there is a frequency lineat zero Hertz, DC. This is merely theaverage of the input over the timerecord. It is rarely used in spectrumor network analysis. But, we havenow established the spacing betweenthese two lines and hence every line;it is the reciprocal of the time record.

What is the Frequency Range of the FFT?

We can now quickly determine thatthe highest frequency we can measure is:

fmax = l

because we have N/2 lines spaced bythe reciprocal of the time recordstarting at zero Hertz *.

Since we would like to adjust the fre-quency range of our measurement,we must vary fmax. The number oftime samples N is fixed by the imple-mentation of the FFT algorithm.Therefore, we must vary the period ofthe time record to vary fmax. To dothis, we must vary the sample rate sothat we always have N samples in ourvariable time record period. This isillustrated in Figure 3.9. Notice thatto cover higher frequencies, we mustsample faster.

* The usefulness of this frequency range can be limited bythe problem of aliasing. Aliasing is discussed in Section 3.

Figure 3.7Lowest frequency resolvable by the FFT.

Time

Time

Ampl

itude

Ampl

itude

Period of input signal longer than the time record.Frequency of the input signal is unknown..

b)

Time Record

Time Record

Period of input signal equals time record.Lowest resolvable frequency.

a)

??

Figure 3.8Frequencies of all the spectrallines of the FFT.

Figure 3.9Frequency range of Dynamic Signal Analyzers is determined by sample rate.

N 1

2 Period of Time Record

29

Section 2*: Sampling and Digitizing

Recall that the input to our DynamicSignal Analyzer is a continuous analog voltage. This voltage might be from an electronic circuit or couldbe the output of a transducer and be proportional to current, power, pressure, acceleration or any numberof other inputs. Recall also that theFFT requires digitized samples of theinput for its digital calculations.Therefore, we need to add a samplerand analog to digital converter (ADC)to our FFT processor to make a spec-trum analyzer. We show this basicblock diagram in Figure 3.10.

For the analyzer to have the highaccuracy needed for many measure-ments, the sampler and ADC must bequite good. The sampler must samplethe input at exactly the correct timeand must accurately hold the inputvoltage measured at this time untilthe ADC has finished its conversion.The ADC must have high resolutionand linearity. For 70 dB of dynamicrange the ADC must have at least 12 bits of resolution and one halfleast significant bit linearity.

A good Digital Voltmeter (DVM) willtypically exceed these specifications,but the ADC for a Dynamic SignalAnalyzer must be much faster thantypical fast DVMs. A fast DVM mighttake a thousand readings per second,but in a typical Dynamic Signal Analyzer the ADC must take at least a hundred thousand readingsper second.

Section 3: Aliasing

The reason an FFT spectrum analyzer needs so many samples persecond is to avoid a problem calledaliasing. Aliasing is a potential prob-lem in any sampled data system. It isoften overlooked, sometimes with disastrous results.

A Simple Data Logging Example of Aliasing

Let us look at a simple data loggingexample to see what aliasing is andhow it can be avoided. Consider theexample for recording temperatureshown in Figure 3.12. A thermocoupleis connected to a digital voltmeterwhich is in turn connected to a print-er. The system is set up to print thetemperature every second. Whatwould we expect for an output?

If we were measuring the tempera-ture of a room which only changesslowly, we would expect every reading to be almost the same as theprevious one. In fact, we are samplingmuch more often than necessary todetermine the temperature of theroom with time. If we plotted theresults of this thought experiment,we would expect to see results likeFigure 3.13.

Figure 3.10Block diagram of dynamic Signal Analyzer.

Figure 3.11The Sampler and ADC must not introduce errors.

Figure 3.13Plot of temperature variation of a room.

Figure 3.12A simple sampled data system.

* This section and the next can be skipped by those notinterested in the internal operation of a Dynamic SignalAnalyzer. However, those who specify the purchase ofDynamic Signal Analyzers are especially encouraged toread these sections. The basic knowledge to be gainedfrom these sections can insure specifying the best analyzerfor your requirements.

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The Case of the Missing Temperature

If, on the other hand, we were measuring the temperature of a small part which could heat and coolrapidly, what would the output be?Suppose that the temperature of our part cycled exactly once everysecond. As shown in Figure 3.14, ourprintout says that the temperaturenever changes.

What has happened is that we havesampled at exactly the same point onour periodic temperature cycle withevery sample. We have not sampledfast enough to see the temperaturefluctuations.

Aliasing in the Frequency Domain

This completely erroneous result isdue to a phenomena called aliasing.*Aliasing is shown in the frequencydomain in Figure 3.15. Two signalsare said to alias if the difference oftheir frequencies falls in the frequen-cy range of interest. This differencefrequency is always generated in theprocess of sampling. In Figure 3.15,the input frequency is slightly higherthan the sampling frequency so a lowfrequency alias term is generated. Ifthe input frequency equals the sam-pling frequency as in our small partexample, then the alias term falls atDC (zero Hertz) and we get the constant output that we saw above.

Aliasing is not always bad. It is called mixing or heterodyning in analog electronics, and is commonlyused for tuning household radios andtelevisions as well as many othercommunication products. However,in the case of the missing tempera-ture variation of our small part, wedefinitely have a problem. How canwe guarantee that we will avoid thisproblem in a measurement situation?

Figure 3.16 shows that if we sampleat greater than twice the highest frequency of our input, the alias products will not fall within the frequency range of our input.Therefore, a filter (or our FFT processor which acts like a filter)after the sampler will remove thealias products while passing thedesired input signals if the samplerate is greater than twice the highestfrequency of the input. If the samplerate is lower, the alias products willfall in the frequency range of theinput and no amount of filtering will be able to remove them from the signal.

Figure 3.14Plot of temperature variation of a small part.

Figure 3.15The problem of aliasing viewed in the frequency domain.

* Aliasing is also known as fold-over or mixing.

31

This minimum sample rate requirement is known as the NyquistCriterion. It is easy to see in the timedomain that a sampling frequencyexactly twice the input frequencywould not always be enough. It is lessobvious that slightly more than twosamples in each period is sufficientinformation. It certainly would not be enough to give a high quality timedisplay. Yet we saw in Figure 3.16 thatmeeting the Nyquist Criterion of asample rate greater than twice themaximum input frequency is suffi-cient to avoid aliasing and preserveall the information in the input signal.

The Need for an Anti-Alias Filter

Unfortunately, the real world rarelyrestricts the frequency range of itssignals. In the case of the room temperature, we can be reasonablysure of the maximum rate at whichthe temperature could change, but we still can not rule out stray signals.Signals induced at the powerline frequency or even local radio stationscould alias into the desired frequencyrange. The only way to be really certain that the input frequency rangeis limited is to add a low pass filterbefore the sampler and ADC. Such afilter is called an anti-alias filter.

An ideal anti-alias filter would looklike Figure 3.18a. It would pass all the desired input frequencies with noloss and completely reject any higherfrequencies which otherwise couldalias into the input frequency range.However, it is not even theoreticallypossible to build such a filter, muchless practical. Instead, all real filterslook something like Figure 3.18b witha gradual roll off and finite rejectionof undesired signals. Large input signals which are not well attenuatedin the transition band could still aliasinto the desired input frequency

Figure 3.16A frequency domain view of how to avoid aliasing - sample at greater than twice the highest input frequency.

Figure 3.18Actual anti-alias filters require higher sampling frequencies.

Figure 3.17Nyquist Criterion in the time domain.

32

range. To avoid this, the sampling fre-quency is raised to twice the highestfrequency of the transition band. Thisguarantees that any signals whichcould alias are well attentuated bythe stop band of the filter. Typically,this means that the sample rate isnow two and a half to four times themaximum desired input frequency.Therefore, a 25 kHz FFT SpectrumAnalyzer can require an ADC thatruns at 100 kHz as we stated withoutproof in Section 2 of this Chapter*.

The Need for More Than One Anti-Alias Filter

Recall from Section 1 of this Chapter,that due to the properties of the FFTwe must vary the sample rate to varythe frequency span of our analyzer.To reduce the frequency span, wemust reduce the sample rate. From our considerations of aliasing, wenow realize that we must also reducethe anti-alias filter frequency by thesame amount.

Since a Dynamic Signal Analyzer is a very versatile instrument used in a wide range of applications, it isdesirable to have a wide range of frequency spans available. Typicalinstruments have a minimum span of1 Hertz and a maximum of tens tohundreds of kilohertz. This fourdecade range typically needs to becovered with at least three spans perdecade. This would mean at leasttwelve anti-alias filters would berequired for each channel.

Each of these filters must have very good performance. It is desirablethat their transition bands be as

narrow as possible so that as manylines as possible are free from aliasproducts. Additionally, in a two channel analyzer, each filter pair must be well matched for accuratenetwork analysis measurements.These two points unfortunately meanthat each of the filters is expensive.Taken together they can add signifi-cantly to the price of the analyzer.Some manufacturers dont have a low enough frequency anti-alias filteron the lowest frequency spans to savesome of this expense. (The lowestfrequency filters cost the most of all.)But as we have seen, this can lead toproblems like our case of the missing temperature.

Digital Filtering

Fortunately, there is an alternativewhich is cheaper and when used inconjunction with a single analog anti-alias filter, always provides aliasingprotection. It is called digital filteringbecause it filters the input signal afterwe have sampled and digitized it. Tosee how this works, let us look atFigure 3.19.

In the analog case we already discussed, we had to use a new filter every time we changed the sample rate of the Analog to DigitalConverter (ADC). When using digitalfiltering, the ADC sample rate is leftconstant at the rate needed for thehighest frequency span of the analyz-er. This means we need not changeour anti-alias filter. To get thereduced sample rate and filtering we need for the narrower frequencyspans, we follow the ADC with a digital filter.

This digital filter is known as a decimating filter. It not only filters the digital representation of the signalto the desired frequency span, it alsoreduces the sample rate at its outputto the rate needed for that frequencyspan. Because this filter is digital,there are no manufacturing varia-tions, aging or drift in the filter.Therefore, in a two channel analyzerthe filters in each channel are identi-cal. It is easy to design a single digitalfilter to work on many frequencyspans so the need for multiple filtersper channel is avoided. All these factors taken together mean that digital filtering is much less expen-sive than analog anti-aliasing filtering.

Figure 3.19Block diagrams of analog and digital filtering.

* Unfortunately, because the spacing of the FFT linesdepends on the sample rate, increasing the sample ratedecreases the number of lines that are in the desired frequency range. Therefore, to avoid aliasing problemsDynamic Signal Analyzers have only .25N to .4N linesinstead of N/2 lines.

33

Section 4: Band Selectable Analysis

Suppose we need to measure a smallsignal that is very close in frequencyto a large one. We might be measur-ing the powerline sidebands (50 or 60 Hz) on a 20 kHz oscillator. Or wemight want to distinguish betweenthe stator vibration and the shaft imbalance in the spectrum of a motor.*

Recall from our discussion of the properties of the Fast FourierTransform that it is equivalent to a set of filters, starting at zero Hertz,equally spaced up to some maximumfrequency. Therefore, our frequencyresolution is limited to the maximumfrequency divided by the number of filters.

To just resolve the 60 Hz sidebandson a 20 kHz oscillator signal wouldrequire 333 lines (or filters) of theFFT. Two or three times more lineswould be required to accuratelymeasure the sidebands. But typicalDynamic Signal Analyzers only have200 to 400 lines, not enough for accurate measurements. To increasethe number of lines would greatly increase the cost of the analyzer. Ifwe chose to pay the extra cost, wewould still have trouble seeing theresults. With a 4 inch (10 cm) screen,the sidebands would be only 0.01 inch(.25 mm) from the carrier.

A better way to solve this problem is to concentrate the filters into thefrequency range of interest as inFigure 3.20. If we select the minimumfrequency as well as the maximumfrequency of our filters we can zoomin for a high resolution close-up shotof our frequency spectrum. We nowhave the capability of looking at theentire spectrum at once with low resolution as well as the ability tolook at what interests us with muchhigher resolution.

This capability of increased resolution is called Band SelectableAnalysis (BSA).** It is done by mixingor heterodyning the input signaldown into the range of the FFT span

selected. This technique, familiar toelectronic engineers, is the processby which radios and televisions tune in stations.

The primary difference between theimplementation of BSA in DynamicSignal Analyzers and heterodyneradios is shown in Figure 3.21. In aradio, the sine wave used for mixingis an analog voltage. In a DynamicSignal Analyzer, the mixing is doneafter the input has been digitized, sothe sine wave is a series of digitalnumbers into a digital multiplier. This means that the mixing will bedone with a very accurate and stabledigital signal so our high resolutiondisplay will likewise be very stableand accurate.

* The shaft of an ac induction motor always runs at a rateslightly lower than a multiple of the driven frequency, aneffect called slippage.

** Also sometimes called zoom.

Figure 3.20High resolution measurements with Band Selectable Analysis.

Figure 3.21Analyzer block diagram.

34

Section 5: Windowing

The Need for Windowing

There is another property of the FastFourier Transform which affects itsuse in frequency domain analysis. We recall that the FFT computes thefrequency spectrum from a block ofsamples of the input called a timerecord. In addition, the FFT algorithmis based upon the assumption thatthis time record is repeated through-out time as illustrated in Figure 3.22.

This does not cause a problem withthe transient case shown. But whathappens if we are measuring a contin-uous signal like a sine wave? If thetime record contains an integral number of cycles of the input sinewave, then this assumption exactlymatches the actual input waveform as shown in Figure 3.23. In this case,the input waveform is said to be periodic in the time record.

Figure 3.24 demonstrates the difficulty with this assumption when the input is not periodic in thetime record. The FFT algorithm iscomputed on the basis of the highlydistorted waveform in Figure 3.24c.

We know from Chapter 2 that the actual sine wave input has a frequency spectrum of single line.The spectrum of the input assumedby the FFT in Figure 3.24c should be

Figure 3.24Input signal not periodic in time record.

Figure 3.22FFT assumption - time record repeated throughout all time.

Figure 3.23Input signal periodic in time record.

35

very different. Since sharp phenome-na in one domain are spread out inthe other domain, we would expectthe spectrum of our sine wave to bespread out through the frequencydomain.

In Figure 3.25 we see in an actualmeasurement that our expectationsare correct. In Figures 3.25 a & b, wesee a sine wave that is periodic in thetime record. Its frequency spectrum is a single line whose width is deter-mined only by the resolution of our Dynamic Signal Analyzer.* On theother hand, Figures 3.25c & d show a sine wave that is not periodic in the time record. Its power has beenspread throughout the spectrum aswe predicted.

This smearing of energy throughoutthe frequency domains is a phenome-na known as leakage. We are seeingenergy leak out of one resolution lineof the FFT into all the other lines.

It is important to realize that leakageis due to the fact that we have takena finite time record. For a sine waveto have a single line spectrum, it mustexist for all time, from minus infinityto plus infinity. If we were to have an infinite time record, the FFTwould compute the correct single line spectrum exactly. However, sincewe are not willing to wait forever tomeasure its spectrum, we only look at a finite time record of the sinewave. This can cause leakage if thecontinuous input is not periodic inthe time record.

It is obvious from Figure 3.25 that theproblem of leakage is severe enoughto entirely mask small signals close to our sine waves. As such, the FFTwould not be a very useful spectrumanalyzer. The solution to this problemis known as windowing. The prob-lems of leakage and how to solve

them with windowing can be themost confusing concepts of DynamicSignal Analysis. Therefore, we willnow carefully develop the problemand its solution in several representa-tive cases.

* The additional two components in the photo are the harmonic distortion of the sine wave source.

Figure 3.25Actual FFT results.

b)

a) & b) Sine wave periodic in time record

d)

c) & d) Sine wave not periodic in time record

a)

c)

36

What is Windowing?

In Figure 3.26 we have again repro-duced the assumed input wave formof a sine wave that is not periodic inthe time record. Notice that most ofthe problem seems to be at the edgesof the time record, the center is agood sine wave. If the FFT could be made to ignore the ends and con-centrate on the middle of the timerecord, we would expect to get muchcloser to the correct single line spectrum in the frequency domain.

If we multiply our time record by a function that is zero at the ends of the time record and large in themiddle, we would concentrate theFFT on the middle of the time record.One such function is shown in Figure3.26c. Such functions are called window functions because they force us to look at data through a narrow window.

Figure 3.27 shows us the vast improvement we get by windowingdata that is not periodic in the timerecord. However, it is important torealize that we have tampered withthe input data and cannot expect perfect results. The FFT assumes theinput looks like Figure 3.26d, some-thing like an amplitude-modulatedsine wave. This has a frequency spectrum which is closer to the correct single line of the input sinewave than Figure 3.26b, but it still isnot correct. Figure 3.28 demonstratesthat the windowed data does nothave as narrow a spectrum as anunwindowed function which is periodic in the time record.

Figure 3.26The effect of windowing in the time domain.

Figure 3.27Leakage reduction with windowing.

a) Sine wave not periodic in time record b) FFT results with no window function

c) FFT results with a window function

37

The Hanning Window

Any number of functions can be usedto window the data, but the mostcommon one is called Hanning. Weactually used the Hanning window inFigure 3.27 as our example of leakagereduction with windowing. TheHanning window is also commonlyused when measuring random noise.

The Uniform Window*

We have seen that the Hanning window does an acceptably good job on our sine wave examples, bothperiodic and non-periodic in the timerecord. If this is true, why should wewant any other windows?

Suppose that instead of wanting thefrequency spectrum of a continuoussignal, we would like the spectrum of a transient event. A typical tran-sient is shown in Figure 3.29a. If wemultiplied it by the window functionin Figure 3.29b we would get thehighly distorted signal shown inFigure 3.29c. The frequency spectrum of an actual transient with and with-out the Hanning window is shown inFigure 3.30. The Hanning window hastaken our transient, which naturallyhas energy spread widely through thefrequency domain and made it lookmore like a sine wave.

Therefore, we can see that for transients we do not want to use the Hanning window. We would liketo use all the data in the time recordequally or uniformly. Hence we willuse the Uniform window whichweights all of the time record uniformly.

The case we made for the Uniformwindow by looking at transients can be generalized. Notice that ourtransient has the property that it iszero at the beginning and end of thetime record. Remember that we intro-duced windowing to force the input

to be zero at the ends of the timerecord. In this case, there is no needfor windowing the input. Any func-tion like this which does not require awindow because it occurs completely

within the time record is called a self-windowing function. Self-windowingfunctions generate no leakage in theFFT and so need no window.

* The Uniform Window is sometimes referred to as aRectangular Window.

Figure 3.28Windowing reduces leakage but does not eliminate it.

b) Windowed measurement - input not periodicin time record

a) Leakage-free measurement - input periodicin time record

Figure 3.29Windowing loses information from transient events.

Figure 3.30Spectrums of transients.

b) Hanning windowed transientsa) Unwindowed trainsients

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There are many examples of self-windowing functions, some of whichare shown in Figure 3.31. Impacts,impulses, shock responses, sinebursts, noise bursts, chirp bursts andpseudo-random noise can all be madeto be self-windowing. Self-windowingfunctions are often used as the exci-tation in measuring the frequencyresponse of networks, particularlyif the network has lightly-damped

resonances (high Q). This is becausethe self-windowing functions gener-ate no leakage in the FFT. Recall thateven with the Hanning window, someleakage was present when the signalwas not periodic in the time record.This means that without a self-win-dowing excitation, energy could leakfrom a lightly damped resonance intoadjacent lines (filters). The resultingspectrum would show greater damping than actually exists.*

The Flat-top Window

We have shown that we need a uniform window for analyzing self-windowing functions like transients.In addition, we need a Hanning window for measuring noise and periodic signals like sine waves.

We now need to introduce a thirdwindow function, the flat-top window,to avoid a subtle effect of theHanning window. To understand

this effect, we need to look at theHanning window in the frequencydomain. We recall that the FFT actslike a set of parallel filters. Figure3.32 shows the shape of those filterswhen the Hanning window is used.Notice that the Hanning functiongives the filter a very rounded top.If a component of the input signal is centered in the filter it will bemeasured accurately**. Otherwise,

the filter shape will attenuate thecomponent by up to 1.5 dB (16%)when it falls midway between the filters.

This error is unacceptably large if we are trying to measure a signalsamplitude accurately. The solution isto choose a window function whichgives the filter a flatter passband.Such a flat-top passband shape isshown in Figure 3.33. The amplitudeerror from this window function doesnot exceed .1 dB (1%), a 1.4 dBimprovement.

Figure 3.33Flat-top passband shapes.

* There is another way to avoid this problem using BandSelectable Analysis. We will illustrate this in the nextchapter.

** It will, in fact, be periodic in the time record

Figure 3.31Self-windowing function examples.

Figure 3.32Hanning passband shapes.

Figure 3.34Reduced resolution of the flat-top window.

Flat-top

Hanning

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The accuracy improvement does not come without its price, however.Figure 3.34 shows that we have flat-tened the top of the passband at theexpense of widening the skirts of thefilter. We therefore lose some abilityto resolve a small component, closelyspaced to a large one. Some DynamicSignal Analyzers offer both Hanningand flat-top window functions so thatthe operator can choose between increased accuracy or improved frequency resolution.

Other Window Functions

Many other window functions arepossible but the three listed aboveare by far the most common for general measurements. For specialmeasurement situations other groupsof window functions may be useful.We will discuss two windows whichare particularly useful when doingnetwork analysis on mechanicalstructures by impact testing.

The Force and Response Windows

A hammer equipped with a forcetransducer is commonly used to stimulate a structure for responsemeasurements. Typically the forceinput is connected to one channel of the analyzer and the response ofthe structure from another transduceris connected to the second channel.This force impact is obviously a self-windowing function. Theresponse of the structure is also self-windowing if it dies out withinthe time record of the analyzer. Toguarantee that the response does goto zero by the end of the time record,an exponential-weighted windowcalled a response window is some-times added. Figure 3.35 shows a response window acting on the re-sponse of a lightly damped structurewhich did not fully decay by the endof the time record. Notice that unlikethe Hanning window, the responsewindow is not zero at both ends of

the time record. We know that the response of the structure will be zeroat the beginning of the time record(before the hammer blow) so there is no need for the window function to be zero there. In addition, most ofthe information about the structuralresponse is contained at the begin-ning of the time record so we makesure that this is weighted most heavi-ly by our response window function.

The time record of the exciting forceshould be just the impact with thestructure. However, movement of thehammer before and after hitting thestructure can cause stray signals inthe time record. One way to avoidthis is to use a force window shownin Figure 3.36. The force window is

unity where the impact data is validand zero everywhere else so that theanalyzer does not measure any straynoise that might be present.

Passband Shapes or Window Functions?

In the proceeding discussion wesometimes talked about windowfunctions in the time domain. Atother times we talked about the filterpassband shape in the frequencydomain caused by these windows. Wechange our perspective freely towhichever domain yields the simplestexplanation. Likewise, some DynamicSignal Analyzers call the uniform,Hanning and flat-top functions win-dows and other analyzers call those

Figure 3.36Using the force window.

Figure 3.35Using the response window.

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functions pass-band shapes. Usewhichever terminology is easier for the problem at hand as they arecompletely interchangeable, just asthe time and frequency domains arecompletely equivalent.

Section 6: Network Stimulus

Recall from Chapter 2 that we canmeasure the frequency response atone frequency by stimulating the network with a single sine wave andmeasuring the gain and phase shift atthat frequency. The frequency of thestimulus is then changed and themeasurement repeated until alldesired frequencies have been measured. Every time the frequencyis changed, the network responsemust settle to its steady-state valuebefore a new measurement can betaken, making this measurementprocess a slow task.

Many network analyzers operate inthis manner and we can make themeasurement this way with a twochannel Dynamic Signal Analyzer. Weset the sine wave source to the centerof the first filter as in Figure 3.37. The analyzer then measures the gain and phase of the network at this frequency while the rest of theanalyzers filters measure only noise.We then increase the source frequen-cy to the next filter center, wait for

the network to settle and then meas-ure the gain and phase. We continuethis procedure until we have measured the gain and phase of the network at all the frequencies of the filters in our analyzer.

This procedure would, within experimental error, give us the sameresults as we would get with any ofthe network analyzers described inChapter 2 with any network, linear or nonlinear.

Noise as a Stimulus

A single sine wave stimulus does not take advantage of the possiblespeed the parallel filters of a Dynamic Signal Analyzer provide. Ifwe had a source that put out multiplesine waves, each one centered in a filter, then we could measure the frequency response at all frequenciesat one time. Such a source, shown inFigure 3.38, acts like hundreds of sinewave generators connected together.Although this sounds very expensive,

Figure 3.37Frequency response measurements with a sine wave stimulus.

Figure 3.38Pseudo-random noise as a stimulus.

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just such a source can be easily generated digitally. It is called a pseudo-random noise or periodic random noise source.

From the names used for this sourceit is apparent that it acts somewhatlike a true noise generator, exceptthat it has periodicity. If we addtogether a large number of sinewaves, the result is very much likewhite noise. A good analogy is thesound of rain. A single drop of watermakes a quite distinctive splashingsound, but a rain storm sounds likewhite noise. However, if we add together a large number of sinewaves, our noise-like signal will periodically repeat its sequence.Hence, the name periodic randomnoise (PRN) source.

A truly random noise source has aspectrum shown in Figure 3.39. It isapparent that a random noise sourcewould also stimulate all the filters atone time and so could be used as anetwork stimulus. Which is a betterstimulus? The answer depends uponthe measurement situation.

Linear Network Analysis

If the network is reasonably linear,PRN and random noise both give thesame results as the swept-sine test ofother analyzers. But PRN gives thefrequency response much faster. PRNcan be used to measure the frequencyresponse in a single time record.Because the random source is truenoise, it must be averaged for severaltime records before an accurate fre-quency response can be determined.Therefore, PRN is the best stimulusto use with fairly linear networksbecause it gives the fastest results*.

Non-Linear Network Analysis

If the network is severely non-linear,the situation is quite different. In thiscase, PRN is a very poor test signaland random noise is much better. Tosee why, let us look at just two of thesine waves that compose the PRNsource. We see in Figure 3.40 that

if two sine waves are put through a nonlinear network, distortion products will be generated equallyspaced from the signals**. Unfortu-nately, these products will fall exactlyon the frequencies of the other sinewaves in the PRN. So the distortionproducts add to the output and there-fore interfere with the measurement

Figure 3.39Random noise as a stimulus.

Figure 3.40Pseudo-random noise distortion.

* There is another reason why PRN is a better test signalthan random or linear networks. Recall from the last section that PRN is self-windowing. This means that unlike random noise, pseudo-random noise has no leakage.Therefore, with PRN, we can measure lightly damped (highQ) resonances more easily than with random noise.

** This distortion is called intermodulation distortion.

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of the frequency response. Figure3.41a shows the jagged response of a nonlinear network measured withPRN. Because the PRN sourcerepeats itself exactly every timerecord, this noisy looking trace neverchanges and will not average to thedesired frequency response.

With random noise, the distortioncomponents are also random and willaverage out. Therefore, the frequencyresponse does not include the distor-tion and we get the more reasonableresults shown in Figure 3.41b.

This points out a fundamental problem with measuring non-linearnetworks; the frequency response isnot a property of the network alone,it also depends on the stimulus.Each stimulus, swept-sine, PRN andrandom noise will, in general, give adifferent result. Also, if the amplitudeof the stimulus is changed, you willget a different result.

To illustrate this, consider the mass-spring system with stops thatwe used in Chapter 2. If the massdoes not hit the stops, the system is linear and the frequency responseis given by Figure 3.42a.

If the mass does hit the stops, the output is clipped and a large number of distortion components aregenerated. As the output approachesa square wave, the fundamental com-ponent becomes constant. Therefore,as we increase the input amplitude,the gain of the network drops. We get a frequency response like Figure3.42b, where the gain is dependent on the input signal amplitude.

So as we have seen, the frequencyresponse of a nonlinear network isnot well defined, i.e., it depends onthe stimulus. Yet it is often used inspite of this. The frequency responseof linear networks has proven to be avery powerful tool and so naturallypeople have tried to extend it to non-linear analysis, particularly sinceother nonlinear analysis tools haveproved intractable.

If every stimulus yields a differentfrequency response, which oneshould we use? The best stimuluscould be considered to be one whichapproximates the kind of signals youwould expect to have as normalinputs to the network. Since any largecollection of signals begins to looklike noise, noise is a good test signal*.As we have already explained, noiseis also a good test signal because itspeeds the analysis by exciting all thefilters of our analyzer simultaneously.

But many other test signals can beused with Dynamic Signal Analyzersand are best (optimum) in othersenses. As explained in the beginningof this section, sine waves can beused to give the same results as othertypes of network analyzers althoughthe speed advantage of the DynamicSignal Analyzer is lost. A fast sinesweep (chirp) will give very similarresults with all the speed of DynamicSignal Analysis and so is a better test signal. An impulse is a good testsignal for acoustical testing if the net-work is linear. It is good for acousticsbecause reflections from surfaces at different distances can easily beisolated or eliminated if desired. Forinstance, by using the force windowdescribed earlier, it is easy to get thefree field response of a speaker byeliminating the room reflections fromthe windowed time record.

Band-Limited Noise

Before leaving the subject of networkstimulus, it is appropriate to discussthe need to band limit the stimulus.We want all the power of the stimulusto be concentrated in the frequencyregion we are analyzing. Any power

* This is a consequence of the central limit theorem. As anexample, the telephone companies have found that whenmany conversations are transmitted together, the result islike white noise. The same effect is found more commonlyat a crowded cocktail party.

Figure 3.42Nonlinear system.

Figure 3.41Nonlinear transfer function.

a) Pseudo-random noise stimulus b) Random noise stimulus

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outside this region does not contribute to the measurement and could excite non-linearities. This can be a particularly severeproblem when testing with randomnoise since it theoretically has thesame power at all frequencies (whitenoise). To eliminate this problem, Dynamic Signal Analyzers often limitthe frequency range of their built-innoise stimulus to the frequency spanselected. This could be done with anexternal noise source and filters, butevery time the analyzer span changed,the noise power and filter would haveto be readjusted. This is done auto-matically with a built-in noise sourceso transfer function measurementsare easier and faster.

Section 7: Averaging