Fundamentals of Signal Analysis Series

UnderstandingDynamic Signal AnalysisApplication Note 1405-2

Table of Contents

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Section 1: FFT Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

Section 2: Sampling and Digitizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

Section 3: Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Section 4: Band-Selectable Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Section 5: Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Section 6: Network Stimulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

Section 7: Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Section 8: Real-Time Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

Section 9: Overlap Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

This application note is a primerfor those who are unfamiliar withthe class of analyzers we calldynamic signal analyzers. Theseinstruments are particularlyappropriate for the analysisof signals in the range of a fewmillihertz to about a hundredkilohertz.

In this note, we avoid usingrigorous mathematics and insteaddepend on heuristic arguments.We have found in over a decadeof teaching this material thatsuch arguments lead to a betterunderstanding of the basicprocesses involved in dynamicsignal analysis. Equally important,this heuristic instruction leads tobetter instrument operators whocan intelligently use these analyzersto solve complicated measurementproblems with accuracy andease.*

In Application Note 1405-1,Introduction to Time, Frequencyand Modal Domains, we introducedthe concepts of the time, frequencyand modal domains and thetypes of instrumentation availablefor making measurements ineach domain. We discussed the

advantages and disadvantages ofeach generic instrument type andnoted that the dynamic signalanalyzer has the speed advantagesof parallel-filter analyzers withouttheir low-resolution limitations.In addition, it is the only type ofanalyzer that works in all threedomains.

In this application note, we willdevelop a fuller understandingof dynamic signal analyzers.We begin by presenting theproperties of the Fast FourierTransform (FFT), upon whichdynamic signal analyzers arebased. We then show how theseFFT properties cause someundesirable characteristics inspectrum analysis like aliasingand leakage. Having demonstrateda potential difficulty with the FFT,we then show what solutions areused to make practical dynamicsignal analyzers. Developingthis basic knowledge of FFTcharacteristics makes it simpleto get good results with a dynamicsignal analyzer in a wide range ofmeasurement problems.

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Introduction

* A more rigorous mathematical justification for thearguments developed in the main text can be found inApplication Note 1405-4

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The Fast Fourier Transform (FFT)is an algorithm* for transformingdata from the time domain to thefrequency domain. Since this isexactly what we want a spectrumanalyzer to do, it would seemeasy to implement a dynamicsignal analyzer based on the FFT.However, we will see that thereare many factors that complicatethis seemingly straightforwardtask.

First, because of the many calcu-lations involved in transformingdomains, the transform must beperformed on a computer if theresults are to be sufficientlyaccurate. Fortunately, with theadvent of microprocessors, it iseasy and inexpensive to incorporateall the needed computing powerin a small instrument package.Note, however, that we cannotnow transform to the frequencydomain in a continuous manner,but instead must sample anddigitize the time domain input.This means that our algorithmtransforms digitized samples fromthe time domain to samples inthe frequency domain as shownin Figure 1.1.**

Because we have sampled, we nolonger have an exact representationin either domain. However, asampled representation can beas close to ideal as we desire byplacing our samples closer together.Later, we will consider whatsample spacing is necessaryto guarantee accurate results.

Time RecordsA time record is defined to beN consecutive, equally spacedsamples of the input. Because itmakes our transform algorithmsimpler and much faster, N isrestricted to be a multiple of 2,for instance 1024.

As shown in Figure 1.3, this timerecord is transformed as a completeblock into a complete block offrequency lines. All the samplesof the time record are needed tocompute each and every line inthe frequency domain. This is incontrast to what one might expect,

Section 1: FFT Properties

Figure 1.1. The FFT samples in both the time and frequency domains

Figure 1.2. A time record is N equally spaced samples of the input.

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

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

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namely that a single time domainsample transforms to exactly onefrequency domain line. Under-standing this block processingproperty of the FFT is crucialto understanding many of theproperties of the dynamic signalanalyzer.

For instance, because the FFTtransforms the entire time recordblock as a total, there cannot bevalid frequency domain resultsuntil a complete time record hasbeen gathered. However, oncecompleted, the oldest samplecould be discarded, all the samplesshifted in the time record, and anew sample added to the end ofthe time record as in Figure 1.4.Thus, once the time record is

initially filled, we have a newtime record at every time domainsample and therefore could havenew valid results in the frequencydomain at every time domainsample.

This is very similar to the behaviorof the parallel-filter analyzersdescribed in AN1405-1, Section 3.When a signal is first applied toa parallel-filter analyzer, we mustwait for the filters to respond,then we can see very rapid changesin the frequency domain. With adynamic signal analyzer we do notget a valid result until a full timerecord has been gathered. Thenrapid changes in the spectra canbe seen.

It should be noted here that a newspectrum every sample is usuallytoo much information, too fast.This would often give youthousands of transforms persecond. In later sections on real-time bandwidth and overlapprocessing, we discuss just howfast a dynamic signal analyzershould transform.

Figure 1.3. The FFT works on blocks of data. Figure 1.4. A new time record every sample after the time record is filled

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How Many Lines are There?We stated earlier that the timerecord has N equally spacedsamples. Another property ofthe FFT is that it transformsthese time domain samples toN/2 equally spaced lines in thefrequency domain. We only gethalf as many lines because eachfrequency line actually containstwo pieces of information, amplitudeand phase. The meaning of this ismost easily seen if we look againat the relationship between thetime and frequency domain.

Figure 1.5 shows a three-dimen-sional graph of this relationship.Up to now, we have implied thatthe amplitude and frequencyof the sine waves contains allthe information necessary toreconstruct the input. But itshould be obvious that the phaseof each of these sine waves isimportant too. For instance, inFigure 1.6, we have shifted thephase of the higher frequency sinewave components of this signal.The result is a severe distortionof the original waveform.

We have not discussed the phaseinformation contained in thespectrum of signals until nowbecause none of the traditionalspectrum analyzers are capable ofmeasuring phase. In ApplicationNote 1405-3, you will seethat phase contains valuableinformation in determining thecause of performance problems.

Figure 1.5. The relationship between the time and frequency domains

Figure 1.6. Phase of frequency domain components is important.

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What is the Spacing of the Lines?Now that we know that we haveN/2 equally spaced lines in thefrequency domain, what is theirspacing? The lowest frequencythat we can resolve with our FFTspectrum analyzer must be basedon the length of the time record.We can see in Figure 1.7 that if theperiod of the input signal is longerthan the time record, we have noway of determining the period(or frequency, its reciprocal).Therefore, the lowest frequencyline of the FFT must occur atfrequency equal to the reciprocalof the time record length.

In addition, there is a frequencyline at zero Hertz, dc. This is merelythe average of the input over thetime record. It is rarely used inspectrum or network analysis.But, we have now established thespacing between these two linesand hence every line; it is thereciprocal of the time record.

What is the Frequency Range of the FFT?We can now quickly determinethat the highest frequency we canmeasure is:

fmax =N 1 2 Period of Time Record

because we have N/2 lines spacedby the reciprocal of the timerecord starting at zero Hertz.*

Since we would like to adjust thefrequency range of our measure-ment, we must vary fmax. Thenumber of time samples N is fixedby the implementation of the FFTalgorithm. Therefore, we mustvary the period of the time recordto vary fmax. To do this, we mustvary the sample rate so that wealways have N samples in our

variable time record period. This isillustrated in Figure 1.9. Notice

that to cover higher frequencies,we must sample faster.

Figure 1.8. Frequencies of all the spectral lines of the FFT

Figure 1.9. Frequency range of dynamic signal analyzers is determined by sample rate.

* The usefulness of this frequency range can be limitedby the problem of aliasing. Aliasing is discussed inSection 3.

Figure 1.7. Lowest frequency resolvable by the FFT

( )

Recall that the input to our dynamicsignal analyzer is a continuousanalog voltage. This voltage mightbe from an electronic circuit or itcould be the output of a transducerand be proportional to current,power, pressure, accelerationor any number of other inputs.Recall also that the FFT requiresdigitized samples of the input forits digital calculations. Therefore,we need to add a sampler andanalog-to-digital converter (ADC)to our FFT processor to make aspectrum analyzer. We show thisbasic block diagram in Figure 2.1.

For the analyzer to have thehigh accuracy needed for manymeasurements, the sampler andADC must be quite good. Thesampler must sample the input atexactly the correct time and mustaccurately hold the input voltagemeasured at this time until theADC has finished its conversion.The ADC must have high resolutionand linearity. For 70 dB of dynamicrange the ADC must have at least12 bits of resolution and one halfleast-significant-bit linearity.

A good digital multimeter (DMM)will typically exceed thesespecifications, but the ADC fora dynamic signal analyzer mustbe much faster than typical fastDMMs. A fast DMM might take athousand readings per second,but in a typical dynamic signalanalyzer the ADC must take atleast a hundred thousand readingsper second.

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Section 2:* Sampling and Digitizing

* You can skip this section and the next if you are notinterested in the internal operation of a dynamic signalanalyzer. However, if you specify the purchase ofdynamic signal analyzers, you are especially encouragedto read these sections. The basic knowledge you gainfrom these sections can insure you specify the bestanalyzer for your requirements.

Figure 2.1. Block diagram of a dynamic signal analyzer

Figure 2.2. The sampler and ADC must not introduce errors.

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Section 3: Aliasing

The reason an FFT spectrumanalyzer needs so many samplesper second is to avoid a problemcalled aliasing. Aliasing is apotential problem in any sampleddata system. It is often overlooked,sometimes with disastrous results.

A Simple Data LoggingExample of AliasingLet us look at a simple data loggingexample to see what aliasing isand how it can be avoided. Considerthe example for recording temp-erature shown in Figure 3.1. Athermocouple is connected to adigital voltmeter that is in turnconnected to a printer. The systemis set up to print the temperatureevery second. What would weexpect for an output? If we weremeasuring the temperature of aroom that only changes slowly, wewould expect every reading to bealmost the same as the previousone. In fact, we are samplingmuch more often than necessaryto determine the temperature ofthe room with time. If we plottedthe results of this “thoughtexperiment,” we would expectto see results like Figure 3.2.

The Case of theMissing TemperatureIf, on the other hand, we weremeasuring the temperature of asmall part that could heat andcool rapidly, what would theoutput be? Suppose that thetemperature of our part cycled

exactly once every second. Asshown in Figure 3.3, our printoutsays that the temperature neverchanges.

What has happened is that wehave sampled at exactly the samepoint on our periodic temperaturecycle with every sample. We havenot sampled fast enough to see thetemperature fluctuations.

Figure 3.1. A simple sampled data system

Figure 3.3. Plot of temperature variation of a small part

Figure 3.2. Plot of temperature variation of a room

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Aliasing in theFrequency DomainThis completely erroneous resultis due to a phenomena calledaliasing.* Aliasing is shown in thefrequency domain in Figure 3.4.Two signals are said to alias if thedifference of their frequencies fallsin the frequency range of interest.This difference frequency is alwaysgenerated in the process of sampling.In Figure 3.4, the input frequencyis slightly higher than the samplingfrequency so a low frequency aliasterm is generated. If the inputfrequency equals the samplingfrequency as in our small partexample, then the alias term fallsat DC (zero Hertz) and we get theconstant output that we saw above.

Aliasing is not always bad. It iscalled mixing or heterodyning inanalog electronics, and is commonlyused for tuning household radiosand televisions as well as manyother communication products.However, in the case of the missingtemperature variation of our smallpart, we definitely have a problem.How can we guarantee that wewill avoid this problem in ameasurement situation?

Figure 3.5 shows that if we sampleat greater than twice the highestfrequency of our input, the aliasproducts will not fall within thefrequency range of our input.Therefore, a filter (or our FFTprocessor, which acts like a filter)after the sampler will remove thealias products while passing thedesired input signals if the samplerate is greater than twice thehighest frequency of the input. Ifthe sample rate is lower, the aliasproducts will fall in the frequencyrange of the input and no amountof filtering will be able to removethem from the signal.

This minimum sample raterequirement is known as theNyquist Criterion. It is easy tosee in the time domain that asampling frequency exactly twicethe input frequency would notalways be enough. It is lessobvious that slightly more thantwo samples in each period is

sufficient information. It certainlywould not be enough to give ahigh-quality time display. Yet wesaw in Figure 3.5 that meeting theNyquist Criterion of a sample rategreater than twice the maximuminput frequency is sufficient toavoid aliasing and preserve all theinformation in the input signal.

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

Figure 3.4. The problem of aliasing viewed in the frequency domain

Figure 3.5. A frequency domain view of how to avoid aliasing - sample at greater than twice the highest inputfrequency

Figure 3.6. Nyquist Criterion in the time domain

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The Need for an Anti-Alias FilterUnfortunately, the real worldrarely restricts the frequencyrange of its signals. In the case ofthe room temperature, we can bereasonably sure of the maximumrate at which the temperaturecould change, but we still cannot rule out stray signals. Signalsinduced at the power-line frequencyor even local radio stations couldalias into the desired frequencyrange. The only way to be reallycertain that the input frequencyrange is limited is to add a lowpass filter before the sampler andADC. Such a filter is called ananti-alias filter.

An ideal anti-alias filter wouldlook like Figure 3.7a. It would passall the desired input frequencieswith no loss and completely rejectany higher frequencies whichotherwise could alias into theinput frequency range. However,it is not even theoretically possibleto build such a filter, much lesspractical. Instead, all real filterslook something like Figure 3.7bwith a gradual roll off and finiterejection of undesired signals.Large input signals that are notwell attenuated in the transitionband could still alias into thedesired input frequency range.To avoid this, the samplingfrequency is raised to twice thehighest frequency of the transitionband. This guarantees that anysignals that could alias are wellattentuated by the stop band of thefilter. Typically, this means thatthe sample rate is now two-and-a-half to four times the maximumdesired input frequency. Therefore,a 25 kHz FFT spectrum analyzercan require an ADC that runs at100 kHz*.

The Need for More Than One Anti-Alias FilterYou may recall from Section 1 thatdue to the properties of the FFT,we must vary the sample rate tovary the frequency span of ouranalyzer. To reduce the frequencyspan, we must reduce the samplerate. From our considerations ofaliasing, we now realize that wemust also reduce the anti-aliasfilter frequency by the sameamount.

Since a dynamic signal analyzeris a very versatile instrument usedin a wide range of applications, itis desirable to have a wide rangeof frequency spans available.Typical instruments have aminimum span of 1 Hertz and amaximum of tens to hundreds ofkilohertz. This four-decade rangetypically needs to be covered withat least three spans per decade.This would mean at least twelveanti-alias filters would be requiredfor each channel.

Each of these filters must havevery good performance. Theirtransition bands should be asnarrow as possible, so that asmany lines as possible are freefrom alias products. Additionally,in a two-channel analyzer, eachfilter pair must be well matchedfor accurate network analysismeasurements. These two points,unfortunately, mean that each ofthe filters is expensive. Takentogether they can add significantlyto the price of the analyzer. To cutexpenses, some manufacturersdon’t use a low enough frequencyanti-alias filter on the lowest-frequency spans. (The lowestfrequency filters cost the most ofall.) But as we have seen, this canlead to problems like our “case ofthe missing temperature.”

Figure 3.7. Actual anti-alias filters require higher sampling frequencies.

* Unfortunately, because the spacing of the FFT linesdepends on the sample rate, increasing the sample ratedecreases the number of lines that are in the desiredfrequency range. Therefore, to avoid aliasing problemsdynamic signal analyzers have only .25N to .4N linesinstead of N/2 lines.

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Digital FilteringFortunately, there is analternative that is cheaper, andwhen used in conjunction witha single analog anti-alias filter,always provides aliasing protection.It is called “digital filtering,”because it filters the input signalafter we have sampled and digitizedit. To see how this works, let uslook at Figure 3.8.

In the analog case we alreadydiscussed, we had to use a newfilter every time we changed thesample rate of the analog-to-digitalconverter (ADC). For digitalfiltering, the ADC sample rate isleft constant at the rate needed forthe highest-frequency span of theanalyzer. This means we need notchange our anti-alias filter. Toget the reduced sample rate andfiltering we need for the narrowerfrequency spans, we follow theADC with a digital filter.

This digital filter is known as adecimating filter. It not only filtersthe digital representation of thesignal to the desired frequencyspan, it also reduces the samplerate at its output to the rateneeded for that frequency span.Because this filter is digital, thereare no manufacturing variations,aging or drift in the filter. Therefore,in a two-channel analyzer, thefilters in each channel are identical.It is easy to design a single digitalfilter to work on many frequencyspans so the need for multiplefilters per channel is avoided.All these factors taken togethermean that digital filtering ismuch less expensive thananalog anti-aliasing filtering.

Figure 3.8. Block diagrams of analog and digital filtering

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Section 4: Band-Selectable Analysis

Suppose we need to measure asmall signal that is very close infrequency to a large one. We mightbe measuring the power-line side-bands (50 or 60 Hz) on a 20 kHzoscillator. Or we might want todistinguish between the statorvibration and the shaft imbalancein the spectrum of a motor.*

Recall from our discussion of theproperties of the Fast FourierTransform that it is equivalentto a set of filters, starting at zeroHertz, equally spaced up to somemaximum frequency. Therefore,our frequency resolution is limitedto the maximum frequencydivided 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 foraccurate measurements. To increasethe number of lines would greatlyincrease the cost of the analyzer.If we chose to pay the extra cost,we would still have trouble seeingthe results. With a 4-inch (10 cm)screen, the sidebands would beonly 0.01 inch (.25 mm) from thecarrier.

A better way to solve this problemis to concentrate the filters intothe frequency range of interestas in Figure 4.1. If we select theminimum frequency as well as themaximum frequency of our filterswe can “zoom in” for a highresolution close-up shot of ourfrequency spectrum. We now

have the capability of looking atthe entire spectrum at once withlow resolution, as well as theability to look at what interestsus with much higher resolution.

This capability of increasedresolution is called band-selectableanalysis (BSA).** It is done bymixing or heterodyning the inputsignal down into the range of theFFT span selected. This technique,familiar to electronic engineers, isthe process by which radios andtelevisions tune in stations.

The primary difference betweenthe implementation of BSA indynamic signal analyzers andheterodyne radios is shown inFigure 4.2. In a radio, the sinewave used for mixing is an analogvoltage. In a dynamic signalanalyzer, the mixing is done afterthe input has been digitized, sothe “sine wave” is a series ofdigital numbers into a digitalmultiplier. This means that themixing will be done with a veryaccurate and stable digital signalso our high-resolution displaywill likewise be very stable andaccurate.

Figure 4.1. High-resolution measurements with band-selectable analysis

Figure 4.2. Analyzer block diagram

* The shaft of an ac induction motor always runs at arate slightly lower than a multiple of the drivenfrequency, an effect called slippage.

** Also sometimes called “zoom.”

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Section 5: Windowing

The Need for WindowingThere is another property ofthe Fast Fourier Transform thataffects its use in frequency domainanalysis. We recall that the FFTcomputes the frequency spectrumfrom a block of samples of theinput called a time record. Inaddition, the FFT algorithm isbased upon the assumption thatthis time record is repeatedthroughout time, as illustratedin Figure 5.1.

This does not cause a problemwith the transient case shown.But what happens if we aremeasuring a continuous signallike a sine wave? If the timerecord contains an integralnumber of cycles of the inputsine wave, then this assumptionexactly matches the actual inputwaveform as shown in Figure 5.2.In this case, the input waveformis said to be periodic in the timerecord.

Figure 5.3 demonstrates thedifficulty with this assumptionwhen the input is not periodic inthe time record. The FFT algorithmis computed on the basis of thehighly distorted waveform inFigure 5.3c. We know fromChapter 2 that the actual sine waveinput has a frequency spectrumof single line. The spectrum ofthe input assumed by the FFT inFigure 5.3c should be very different.Since sharp phenomena in onedomain are spread out in theother domain, we would expectthe spectrum of our sine wave tobe spread out through the frequencydomain.

Figure 5.1. FFT assumption — time record repeated throughout all time

Figure 5.2. Input signal periodic in time record

Figure 5.3. Input signal not periodic in time record

In Figure 5.4 we see in an actualmeasurement that our expectationsare correct. In Figures 5.4a and b,we see a sine wave that is periodicin the time record. Its frequencyspectrum is a single line whosewidth is determined only by theresolution of our dynamic signalanalyzer. On the other hand,Figures 5.4c and d show a sinewave that is not periodic in thetime record. Its power has beenspread throughout the spectrumas we predicted.

This smearing of energy through-out the frequency domains is aphenomena known as leakage.We are seeing energy leak out ofone resolution line of the FFTinto all the other lines.

It is important to realize thatleakage is due to the fact that wehave taken a finite time record.For a sine wave to have a singleline spectrum, it must exist for alltime, from minus infinity to plusinfinity. If we were to have aninfinite time record, the FFTwould compute the correct singleline spectrum exactly. However,since we are not willing to waitforever to measure its spectrum,we only look at a finite time recordof the sine wave. This can cause

leakage if the continuous input isnot periodic in the time record.

It is obvious from Figure 5.4 thatthe problem of leakage is severeenough to entirely mask smallsignals close to our sine waves. Assuch, the FFT would not be a veryuseful spectrum analyzer. Thesolution to this problem is known

as windowing. The problems ofleakage and how to solve themwith windowing can be the mostconfusing concepts of dynamicsignal analysis. Therefore, we willnow carefully develop the problemand its solution in severalrepresentative cases.

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

Figure 5.4. Actual FFT results

a) and b) Sine wave periodic in time record

a) b)

c) d)

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What is Windowing?In Figure 5.5 we have againreproduced the assumed inputwave form of a sine wave that isnot periodic in the time record.Notice that most of the problemseems to be at the edges of thetime record; the center is a goodsine wave. If the FFT could bemade to ignore the ends andconcentrate on the middle of thetime record, we would expect toget much closer to the correctsingle-line spectrum in thefrequency domain.

If we multiply our time record bya function that is zero at the endsof the time record and large in themiddle, we would concentrate theFFT on the middle of the timerecord. One such function is shownin Figure 5.5c. Such functions arecalled window functions becausethey force us to look at datathrough a narrow window.

Figure 5.6 shows us the vastimprovement we get by windowingdata that is not periodic in the timerecord. However, it is importantto realize that we have tamperedwith the input data and cannotexpect perfect results. The FFTassumes the input looks likeFigure 5.5d, something like anamplitude-modulated sine wave.This has a frequency spectrumwhich is closer to the correctsingle line of the input sine wavethan Figure 5.5b, but it still is notcorrect. Figure 5.7 demonstratesthat the windowed data does nothave as narrow a spectrum as anunwindowed function which isperiodic in the time record.

Figure 5.5. The effect of windowing in the time domain

c) FFT results with a window function

Figure 5.6. Leakage reduction with windowing

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

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The Hanning WindowAny number of functions can beused to window the data, butthe most common one is calledHanning. We actually used theHanning window in Figure 5.6 asour example of leakage reductionwith windowing. The Hanningwindow is also commonly usedwhen measuring random noise.

The Uniform Window*We have seen that the Hanningwindow does an acceptably goodjob on our sine wave examples,both periodic and non-periodicin the time record. If this is true,why should we want any otherwindows?

Suppose that instead of wantingthe frequency spectrum of acontinuous signal, we would likethe spectrum of a transient event.A typical transient is shown inFigure 5.8a. If we multiplied it bythe window function in Figure5.8b we would get the highlydistorted signal shown in Figure5.8c. The frequency spectrum ofan actual transient with andwithout the Hanning window isshown in Figure 5.9. The Hanningwindow has taken our transient,which naturally has energy spreadwidely through the frequencydomain and made it look morelike a sine wave.

Therefore, we can see that fortransients we do not want to usethe Hanning window. We wouldlike to use all the data in the timerecord equally or uniformly. Hencewe will use a uniform windowwhich weights all of the timerecord uniformly.

The case we made for the uniformwindow by looking at transientscan be generalized. Notice that ourtransient has the property that it

is zero at the beginning and end ofthe time record. Remember thatwe introduced windowing to forcethe input to be zero at the ends ofthe time record. In this case, thereis no need for windowing theinput. Any function like this

which does not require a windowbecause it occurs completelywithin the time record is calleda self-windowing function. Self-windowing functions generate noleakage in the FFT and so need nowindow.

a) Leakage-free measurement — input periodic intime record

Figure 5.7. Windowing reduces leakage but does not eliminate it.

b) Windowed measurement — input not periodic intime record

Figure 5.8. Windowing loses information from transient events.

a) Unwindowed trainsients b) Hanning windowed transients

* The uniform window is sometimes referred to as a“rectangular window.”

Figure 5.9. Spectrums of transients

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There are many examples of self-windowing functions, some ofwhich are shown in Figure 5.10.Impacts, impulses, shock responses,sine bursts, noise bursts, chirpbursts and pseudo-randomnoise can all be made to be self-windowing. Self-windowingfunctions are often used as theexcitation in measuring thefrequency response of networks,particularly if the network haslightly-damped resonances(high Q). This is because the self-windowing functions generateno leakage in the FFT. Recall thateven with the Hanning window,some leakage was present whenthe signal was not periodic in thetime record. This means thatwithout a self-windowing excitation,energy could leak from a lightlydamped resonance into adjacentlines (filters). The resultingspectrum would show greaterdamping than actually exists.*

The Flat-top WindowWe have shown that we need auniform window for analyzingself-windowing functions liketransients. In addition, we need aHanning window for measuringnoise and periodic signals likesine waves.

We now need to introduce a thirdwindow function, the flat-topwindow, to avoid a subtle effectof the Hanning window. Tounderstand this effect, we need tolook at the Hanning window in thefrequency domain. We recall thatthe FFT acts like a set of parallelfilters. Figure 5.11 shows theshape of those filters when theHanning window is used. Noticethat the Hanning function givesthe filter a very rounded top. If a

component of the input signal iscentered in the filter it will bemeasured accurately.** Otherwise,the filter shape will attenuatethe component by up to 1.5 dB(16 percent) when it falls midwaybetween the filters.

This error is unacceptably large ifwe are trying to measure a signal’samplitude accurately. The solutionis to choose a window functionwhich gives the filter a flatterpassband. Such a flat-top passbandshape is shown in Figure 5.12. Theamplitude error from this windowfunction does not exceed .1 dB(1%), a 1.4 dB improvement.

The accuracy improvementdoes not come without its price,however. Figure 5.13 shows thatwe have flattened the top of the

* There is another way to avoid this problem using band-selectable analysis. We illustrate this in AgilentApplication Note 1405-3.

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

Figure 5.12. Flat-top passband shapes

Figure 5.13. Reduced resolution of the flat-top window

Figure 5.11. Hanning passband shapes

Figure 5.10. Self-windowing function examples

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passband at the expense ofwidening the skirts of the filter.We therefore lose some abilityto resolve a small component,closely spaced to a large one.Some dynamic signal analyzersoffer both Hanning and flat-topwindow functions so that theoperator can choose betweenincreased accuracy or improvedfrequency resolution.

Other Window FunctionsMany other window functions arepossible but the three listed aboveare by far the most common forgeneral measurements. For specialmeasurement situations othergroups of window functions maybe useful. We will discuss twowindows that are particularlyuseful when doing network analysison mechanical structures byimpact testing.

The Force andResponse WindowsA hammer equipped with a forcetransducer is commonly used tostimulate a structure for responsemeasurements. Typically the forceinput is connected to one channelof the analyzer and the responseof the structure from another

transducer is connected to thesecond channel. This force impactis obviously a self-windowingfunction. The response of thestructure is also self-windowing ifit dies out within the time recordof the analyzer. To guarantee thatthe response does go to zero bythe end of the time record, anexponential-weighted windowcalled a response window issometimes added. Figure 5.14shows a response window actingon the response of a lightlydamped structure which did notfully decay by the end of the timerecord. Notice that unlike theHanning window, the responsewindow is not zero at both ends ofthe time record. We know that theresponse of the structure will bezero at the beginning of the timerecord (before the hammer blow)so there is no need for the windowfunction to be zero there. Inaddition, most of the informationabout the structural response iscontained at the beginning of thetime record so we make sure thatthis is weighted most heavily byour response window function.

The time record of the excitingforce should be just the impactwith the structure. However,movement of the hammer before

and after hitting the structure cancause stray signals in the timerecord. One way to avoid this isto use a force window shown inFigure 5.15. The force window isunity where the impact data isvalid and zero everywhere elseso that the analyzer does notmeasure any stray noise thatmight be present.

Passband Shapes orWindow Functions?In the proceeding discussion wesometimes talked about windowfunctions in the time domain.At other times we talked aboutthe filter passband shape in thefrequency domain caused bythese windows. We change ourperspective freely to whicheverdomain yields the simplestexplanation. Likewise, somedynamic signal analyzers callthe uniform, Hanning and flat-topfunctions “windows” and otheranalyzers call those functions“pass-band shapes.” Use whicheverterminology is easier for theproblem at hand, as they arecompletely interchangeable,just as the time and frequencydomains are completelyequivalent.

Figure 5.14. Using the response window

Figure 5.15. Using the force window

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As we explained in AgilentApplication Note 1405-1, we canmeasure the frequency responseat one frequency by stimulatingthe network with a single sinewave and measuring the gain andphase shift at that frequency.The frequency of the stimulusis then changed and the measure-ment repeated until all desiredfrequencies have been measured.Every time the frequency is changed,the network response must settleto its steady-state value before anew measurement can be taken,making this measurement processa slow task.

Many network analyzers operatein this manner and we can makethe measurement this way witha two-channel dynamic signalanalyzer. We set the sine wavesource to the center of the firstfilter as in Figure 6.1. Theanalyzer then measures the gainand phase of the network at thisfrequency while the rest of theanalyzer’s filters measure onlynoise. We then increase the sourcefrequency to the next filter center,wait for the network to settle andthen measure the gain and phase.We continue this procedure untilwe have measured the gain andphase of the network at all thefrequencies of the filters in ouranalyzer.

This procedure would, withinexperimental error, give us thesame results as we would get withany of the network analyzersdescribed in Agilent ApplicationNote 1405-1, with any network,linear or nonlinear.

Noise as a StimulusA single sine wave stimulus doesnot take advantage of the possiblespeed the parallel filters of adynamic signal analyzer provide.If we had a source that put outmultiple sine waves, each onecentered in a filter, then we couldmeasure the frequency responseat all frequencies at one time.Such a source, shown in Figure6.2, acts like hundreds of sinewave generators connectedtogether. Although this soundsvery expensive, just such a sourcecan be easily generated digitally.It is called a pseudo-random noiseor periodic random noise source.

From the names used for thissource it is apparent that it actssomewhat like a true noisegenerator, except that it hasperiodicity. If we add together alarge number of sine waves, theresult is very much like whitenoise. A good analogy is the soundof rain. A single drop of watermakes a quite distinctive splashingsound, but a rain storm soundslike white noise. However, if weadd together a large number ofsine waves, our noise-like signalwill periodically repeat itssequence. Hence, the nameperiodic random noise (PRN)source.

Section 6: Network Stimulus

Figure 6.1. Frequency response measurements with a sine wave stimulus

Figure 6.2. Pseudo-random noise as a stimulus

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A truly random noise source has aspectrum shown in Figure 6.3. Itis apparent that a random noisesource would also stimulate allthe filters at one time and socould be used as a networkstimulus. Which is a betterstimulus? The answer dependsupon the measurement situation.

Linear Network AnalysisIf the network is reasonablylinear, PRN and random noiseboth give the same results as theswept-sine test of other analyzers.But PRN gives the frequencyresponse much faster. PRN canbe used to measure the frequencyresponse in a single time record.Because the random source istrue noise, it must be averagedfor several time records before anaccurate frequency response canbe determined. Therefore, PRN isthe best stimulus to use with fairlylinear networks because it givesthe fastest results.*

Non-Linear Network AnalysisIf the network is severely non-linear, the situation is quitedifferent. In this case, PRN is avery poor test signal and randomnoise is much better. To see why,let us look at just two of the sinewaves that compose the PRNsource. We see in Figure 6.4 thatif two sine waves are put througha nonlinear network, distortionproducts will be generatedequally spaced from the signals.**Unfortunately, these products willfall exactly on the frequencies ofthe other sine waves in the PRN.So the distortion products add tothe output and therefore interferewith the measurement of thefrequency response. Figure 6.5ashows the jagged response of anonlinear network measured withPRN. Because the PRN source

repeats itself exactly every timerecord, this noisy-looking tracenever changes and will not averageto the desired frequency response.

With random noise, the distortioncomponents are also random andwill average out. Therefore, thefrequency response does notinclude the distortion and weget the more reasonable resultsshown in Figure 6.5b.

This points out a fundamentalproblem with measuring non-linear networks; the frequencyresponse is not a property of thenetwork alone, it also dependson the stimulus. Each stimulus,swept-sine, PRN and randomnoise will, in general, give adifferent result. Also, if theamplitude of the stimulus ischanged, you will get a differentresult.

Figure 6.3. Random noise as a stimulus

∆

∆

∆∆

Figure 6.4. Pseudo-random noise distortion

* There is another reason why PRN is a better test signalthan random or linear networks. Recall from the lastsection that PRN is self-windowing. This means thatunlike random noise, pseudo-random noise has noleakage. Therefore, with PRN, we can measure lightlydamped (high Q) resonances more easily than withrandom noise.

** This distortion is called intermodulation distortion.

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

Figure 6.5. Nonlinear transfer function

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To illustrate this, consider themass-spring system with stopsthat we used in Agilent ApplicationNote 1405-1. If the mass does nothit the stops, the system is linearand the frequency response isgiven by Figure 6.6a.

If the mass does hit the stops,the output is clipped and a largenumber of distortion componentsare generated. As the outputapproaches a square wave, thefundamental component becomesconstant. Therefore, as we increasethe input amplitude, the gainof the network drops. We get afrequency response like Figure6.6b, where the gain is dependenton the input signal amplitude.

So, as we have seen, the frequencyresponse of a nonlinear networkis not well defined, i.e., it dependson the stimulus. Yet it is oftenused in spite of this. The frequencyresponse of linear networks hasproven to be a very powerful too,and so naturally people have triedto extend it to non-linear analysis,particularly since other nonlinearanalysis tools have provedintractable.

If every stimulus yields a differentfrequency response, which oneshould we use? The “best” stimuluscould be considered to be one thatapproximates the kind of signalsyou would expect to have asnormal inputs to the network.Since any large collection ofsignals begins to look like noise,noise is a good test signal.* As we

have already explained, noise isalso a good test signal because itspeeds the analysis by excitingall the filters of our analyzersimultaneously.

But many other test signals can beused with dynamic signal analyzersand are “best” (optimum) in othersenses. As explained in the begin-ning of this section, sine wavescan be used to give the sameresults as other types of networkanalyzers although the speedadvantage of the dynamic signalanalyzer is lost. A fast sine sweep(chirp) will give very similarresults with all the speed ofdynamic signal analysis, and sois a better test signal. An impulseis a good test signal for acousticaltesting if the network is linear.It is good for acoustics becausereflections from surfaces atdifferent distances can easily beisolated or eliminated if desired.For instance, by using the “force”window described earlier, it iseasy to get the free field responseof a speaker by eliminating theroom reflections from thewindowed time record.

Band-Limited NoiseBefore leaving the subject ofnetwork stimulus, it is appropriateto discuss the need to band limitthe stimulus. We want all thepower of the stimulus to beconcentrated in the frequencyregion we are analyzing. Anypower outside this region does notcontribute to the measurementand could excite non-linearities.This can be a particularly severeproblem when testing with randomnoise since it theoretically has thesame power at all frequencies(white noise). To eliminate thisproblem, dynamic signal analyzersoften limit the frequency range oftheir built-in noise stimulus to thefrequency span selected. Thiscould be done with an externalnoise source and filters, but everytime the analyzer span changed,the noise power and filter wouldhave to be readjusted. This is doneautomatically with a built-in noisesource so transfer functionmeasurements are easier andfaster.

Figure 6.6. Nonlinear system

* This is a consequence of the central limit theorem. Asan example, the telephone companies have found thatwhen many conversations are transmitted together, theresult is like white noise. The same effect is found morecommonly at a crowded cocktail party.

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Section 7: Averaging

To make it as easy as possible todevelop an understanding ofdynamic signal analyzers we havealmost exclusively used exampleswith deterministic signals, i.e.,signals with no noise. However,the real world is rarely so obliging.The desired signal often must bemeasured in the presence ofsignificant noise. At other timesthe “signals” we are trying tomeasure are more like noisethemselves. Common examplesthat are somewhat noise-likeinclude speech, music, digitaldata, seismic data and mechanicalvibrations. Because of these twocommon conditions, we mustdevelop techniques both tomeasure signals in the presence ofnoise and to measure the noiseitself.

The standard technique in statisticsto improve the estimates of a valueis to average. When we watch anoisy reading on a dynamic signalanalyzer, we can guess the averagevalue. But because the dynamicsignal analyzer contains digitalcomputation capability, we canhave it compute this average valuefor us. Two kinds of averagingare available, RMS (or “power”averaging) and linear averaging.

RMS AveragingWhen we watch the magnitude ofthe spectrum and attempt to guessthe average value of the spectrumcomponent, we are doing a crudeRMS* average. We are trying todetermine the average magnitudeof the signal, ignoring any phasedifference that may exist betweenthe spectra. This averaging

technique is very valuable fordetermining the average power inany of the filters of our dynamicsignal analyzers. The moreaverages we take, the better ourestimate of the power level.

In Figure 7.1, we show RMSaveraged spectra of random noise,digital data and human voices.Each of these examples is a fairlyrandom process, but whenaveraged we can see the basicproperties of its spectrum.

If we want to measure a smallsignal in the presence of noise,RMS averaging will give us a goodestimate of the signal plus noise.We can not improve the signal-to-noise ratio with RMS averaging;we can only make more accurateestimates of the total signal-plus-noise power.

a) Random noise b) Digital data

Figure 7.1. RMS averaged spectra

c) Voices. Traces were separated 30 dB for clarityUpper trace: female speakerLower trace: male speaker

* RMS stands for “root-mean-square” and is calculatedby squaring all the values, adding the squares together,dividing by the number of measurements (mean) andtaking the square root of the result.

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Linear AveragingHowever, there is a technique forimproving the signal-to-noise ratioof a measurement, called linearaveraging. It can be used if atrigger signal is available that issynchronous with the periodicpart of the spectrum. Of course,the need for a synchronizingsignal is somewhat restrictive,although there are numeroussituations in which one isavailable. In network analysisproblems, the stimulus signalitself can often be used as asynchronizing signal.

Linear averaging can beimplemented many ways, butperhaps the easiest to understandis where the averaging is done inthe time domain. In this case, thesynchronizing signal is used totrigger the start of a time record.Therefore, the periodic part of theinput will always be exactly thesame in each time record we take,whereas the noise will, of course,vary. If we add together a series ofthese triggered time records anddivide by the number of recordswe have taken, we will computewhat we call a linear average.

Since the periodic signal will haverepeated itself exactly in eachtime record, it will average to itsexact value. But since the noise isdifferent in each time record, itwill tend to average to zero. Themore averages we take, the closerthe noise comes to zero and wecontinue to improve the signal-to-noise ratio of our measurement.

Figure 7.2 shows a time recordof a square wave buried in noise.The resulting time record after128 averages shows a markedimprovement in the signal to noiseratio. Transforming both results tothe frequency domain shows howmany of the harmonics can nowbe accurately measured becauseof the reduced noise floor.

a) Single record, no averaging b) Single record, no averaging

c) 128 linear averages d) 128 linear averages

Figure 7.2. Linear averaging

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Until now we have ignored thefact that it will take a finite timeto compute the FFT of our timerecord. In fact, if we could computethe transform in less time thanour sampling period we couldcontinue to ignore this computa-tional time. Figure 8.1 shows thatunder this condition we could geta new frequency spectrum withevery sample. As we have seenfrom the section on aliasing, thiscould result in far more spectrumsevery second than we couldpossibly comprehend. Worse,because of the complexity of theFFT algorithm, it would take avery fast and very expensivecomputer to generate spectrumsthis rapidly.

A reasonable alternative is to adda time record buffer to the blockdiagram of our analyzer (Figure8.2). In Figure 8.3 we can see thatthis allows us to compute thefrequency spectrum of the previoustime record while gathering thecurrent time record. If we cancompute the transform before thetime record buffer fills, then weare said to be operating in realtime.

To see what this means, let uslook at the case where the FFTcomputation takes longer than thetime to fill the time record. Thecase is illustrated in Figure 8.4.Although the buffer is full, wehave not finished the last transform,so we will have to stop takingdata. When the transform isfinished, we can transfer the timerecord to the FFT and begin totake another time record. Thismeans that we missed some inputdata and so we are said to be notoperating in real time.

Recall that the time record is notconstant but deliberately varied tochange the frequency span of theanalyzer. For wide frequencyspans, the time record is shorter.Therefore, as we increase thefrequency span of the analyzer,we eventually reach a span wherethe time record is equal to the FFTcomputation time. This frequencyspan is called the real-timebandwidth. For frequency spansat and below the real-timebandwidth, the analyzer doesnot miss any data.

Real-TimeBandwidth RequirementsHow wide a real-time bandwidthis needed in a dynamic signalanalyzer? Let us examine a fewtypical measurements to get afeeling for the considerationsinvolved.

Adjusting DevicesIf we are measuring the spectrumor frequency response of a devicethat we are adjusting, we need towatch the spectrum change inwhat might be called psychologicalreal time. A new spectrum everyfew tenths of a second is sufficientlyfast to allow an operator to watchadjustments in what he or shewould consider to be real time.However, if the response time ofthe device under test is long, thespeed of the analyzer is immaterial.We will have to wait for the deviceto respond to the changes beforethe spectrum will be valid, nomatter how many spectrums wegenerate in that time. This iswhat makes adjusting lightlydamped (high Q) resonancestedious.

Section 8: Real-Time Bandwidth

Figure 8.1. A new transform every sample Figure 8.2. Time buffer added to block diagram

Figure 8.3. Real-time operation Figure 8.4. Non-real-time operation

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RMS AveragingA second case of interest indetermining real-time bandwidthrequirements is measurementsthat require RMS averaging. Wemight be interested in determiningthe spectrum distribution of thenoise itself or in reducing thevariation of a signal contaminatedby noise. There is no requirementin averaging that the recordsmust be consecutive with no gaps.Therefore, a small real-timebandwidth will not affect theaccuracy of the results.

However, the real time bandwidthwill affect the speed with whichan RMS averaged measurementcan be made. Figure 8.5 showsthat for frequency spans above thereal-time bandwidth, the time tocomplete the average of N recordsis dependent only on the time tocompute the N transforms. Ratherthan continually reducing the timeto compute the RMS average as weincrease our span, we reach afixed time to compute N averages.

Therefore, a small real-timebandwidth is only a problem inRMS averaging when large spansare used with a large number ofaverages. Under these conditionswe must wait longer for the answer.Since wider real-time bandwidthsrequire faster computations andtherefore a more expensiveprocessor, there is a straight-forward trade-off of time versusmoney. In the case of RMSaveraging, higher real-timebandwidth gives you somewhatfaster measurements at increasedanalyzer cost.

TransientsThe last case of interest indetermining the needed real-timebandwidth is the analysis oftransient events. If the entiretransient fits within the timerecord, the FFT computation timeis of little interest. The analyzercan be triggered by the transientand the event stored in the timerecord buffer. The time tocompute its spectrum is notimportant.

However, if a transient eventcontains high-frequency energyand lasts longer than the time

record necessary to measure thehigh-frequency energy, then theprocessing speed of the analyzeris critical. As shown in Figure8.6b, some of the transient will notbe analyzed if the computationtime exceeds the time recordlength.

In the case of transients longerthan the time record, it is alsoimperative that there is some wayto rapidly record the spectrum.Otherwise, the information will belost as the analyzer updates thedisplay with the spectrum of thelatest time record. A special

Figure 8.5. RMS averaging time

Figure 8.6. Transient analysis

display which can show morethan one spectrum (“waterfall”display), mass memory, a high-speed link to a computer or ahigh-speed facsimile recorder isneeded. The output device mustbe able to record a spectrum everytime record or information will belost. Fortunately, there is an easyway to avoid the need for anexpensive wide real-time bandwidthanalyzer and an expensive, fastspectrum recorder. One-timetransient events like explosionsand pass-by noise are usuallydigitally recorded for lateranalysis because of the expense ofrepeating the test. Continuouslysampled time data can be recordedinto a large time-capture memoryor a high-speed through-put disk.This allows you to analyze thedata later with no informationloss.

So we see that there is no clear-cut answer to what real-timebandwidth is necessary in adynamic signal analyzer. Exceptin analyzing long transient events,the added expense of a wide real-time bandwidth gives littleadvantage. It is possible to analyzelong transient events with anarrow real-time bandwidthanalyzer, but it does require therecording of the input signal. Thismethod is slow and requires someoperator care, but you can avoidpurchasing an expensive analyzerand fast spectrum recorder. It isa clear case of speed of analysisversus dollars of capitalequipment.

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In Section 8 we considered thecase where the computation of theFFT took longer than the time ittook to collect the time record.In this section we will look at atechnique, overlap processing,which can be used when the FFTcomputation takes less time thangathering the time record.

To understand overlap processing,let us look at Figure 9.1a. We seea low-frequency analysis wheregathering the time record takesmuch longer than the FFTcomputation time. Our FFTprocessor is sitting idle much ofthe time. If instead of waiting foran entirely new time record weoverlapped the new time recordwith some of the old data, wewould get a new spectrum asoften as we computed the FFT.This overlap processing isillustrated in Figure 9.1b. Tounderstand the benefits of overlapprocessing, let us look at the samecases we used in the last section.

Adjusting DevicesWe saw in the last section that weneed a new spectrum every fewtenths of a second when adjustingdevices. Without overlap processingthis limits our resolution to a fewHertz. With overlap processingour resolution is unlimited. Butwe are not getting something fornothing. Because our overlappedtime record contains old datafrom before the device adjustment,it is not completely correct. It doesindicate the direction and theamount of change, but we mustwait a full time record after thechange for the new spectrum tobe accurately displayed.

Nonetheless, by indicating thedirection and magnitude of thechanges every few tenths of asecond, overlap processing doeshelp in the adjustment of devices.

RMS AveragingOverlap processing can givedramatic reductions in the timeto compute RMS averages with agiven variance. Recall that windowfunctions reduce the effects ofleakage by weighting the ends ofthe time record to zero. Overlappingeliminates most or all of the timethat would be wasted taking thisdata. Because some overlappeddata is used twice, more averages

must be taken to get a givenvariance than in the non-overlapped case. Figure 9.2shows the improvements thatcan be expected by overlapping.

TransientsFor transients shorter than thetime record, overlap processingis useless. For transients longerthan the time record, the real-timebandwidth of the analyzer andspectrum recorder is usually alimitation. If it is not, overlapprocessing allows more spectra tobe generated from the transient,usually improving resolution ofresulting plots.

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Section 9: Overlap Processing

Figure 9.1. Understanding overlap processing

Figure 9.2. RMS averaging speed improvements with overlap processing

SummaryIn this application note, we havedeveloped the basic properties ofdynamic signal analyzers. Wefound that many properties couldbe understood by consideringwhat happens when we transforma finite, sampled time record. Thelength of this record determineshow closely our filters can bespaced in the frequency domainand the number of samplesdetermines the number of filtersin the frequency domain. We alsofound that unless we filtered theinput we could have errors due toaliasing, and that finite time recordscould cause a problem calledleakage that we minimized bywindowing. We then addedseveral features to our basicdynamic signal analyzer toenhance its capabilities. Band-selectable analysis allows us tomake high-resolution measure-ments even at high frequencies.Averaging gives more accuratemeasurements when noise ispresent and even allows us toimprove the signal-to-noise ratiowhen we can use linear averaging.Finally, we incorporated a noisesource in our analyzer to act as astimulus for transfer functionmeasurements.

BibliographyBendat, Julius S. and Piersol,Allan G., “Random Data: Analysisand Measurement Procedures,”Wiley-Interscience, New York,1971.

Bendat, Julius S. and Piersol,Allan G., “EngineeringApplications of Correlationand Spectral Analysis,”Wiley-lnterscience, New York, 1980.

Bracewell, R., “The FourierTransform and its Applications,”McGraw-Hill, 1965.

Cooley, J.W. and Tukey, J.W.,“An Algorithm for the MachineCalculation of Complex FourierSeries,” Mathematics ofComputation, Vo. 19, No. 90,p. 297, April 1965.

McKinney, W., “Band SelectableFourier Analysis,” Hewlett-Packard Journal, April 1975,pp. 20-24.

Otnes, R.K. and Enochson, L.,“Digital Time Series Analysis,”John Wiley, 1972.

Potter, R. and Lortscher, J.,“What in the World is OverlapProcessing,” Hewlett-PackardSanta Clara DSA/Laser Division“Update,” Sept. 1978.

Ramse , K.A., “EffectiveMeasurements for StructuralDynamics Testing,” Part 1,Sound and Vibration Magazine,November 1975, pp. 24-35.

Roth, P., “Effective MeasurementsUsing Digital Signal Analysis,”IEEE Spectrum, April 1971,pp. 62-70.

Roth, P., “Digital FourierAnalysis,” Hewlett-PackardJournal, June 1970.

Welch, Peter D., “The Use of FastFourier Transform for theEstimation of Power Spectra: AMethod Based on Time AveragingOver Short, ModifiedPeriodograms,” IEEETransactions on Audio andElectro-acoustics, Vol. AU-15,No. 2, June 1967, pp. 70-73.

Related Agilent LiteratureAgilent Application Note —Introduction to Time, Frequencyand Modal Domains, pub. no.1405-1

Agilent Application Note —Using Dynamic Signal Analysers, pub. no. 1405-3

Agilent Application Note —The Fourier Transform: AMathematical Background, pub. no. 1405-4

Product Overview — Agilent 35670A DynamicSignal Analyzer, pub. no. 5966-3063E

Product Overview — Agilent E1432/33/34VXI Digitizers/Source, pub. no. 5968-7086E

Product Overview — Agilent E9801B DataRecorder/Logger, pub. no. 5968-6132E

29

GlossaryAliasing — a phenomenon that

can occur when a signal is notsampled at greater than twicethe maximum frequencycomponent; high-frequencysignals appear as low-frequencycomponents; avoided byfiltering out signals greater than1/2 the sample rate

Anti-alias filter — a low pass filterinstalled before the samplerand analog-to-digital converterto limit the input frequencyrange of a signal to preventaliasing; designed to filter outfrequencies greater than 1/2 thesample rate (typically 1/2.56 toallow for filter rolloff)

Band-selectable analysis — ananalysis capability that allowsyou to “zoom in” for a high-resolution close-up shot of thefrequency spectrum byconcentrating filters in thefrequency range of interest.

Digital filter — a decimating filterthat filters the digitalrepresentation of the inputsignal (after it has beensampled and digitized) to thedesired frequency span. It alsoreduces the sample rate at itsoutput to the rate needed forthat frequency span.

Fast Fourier Transform (FFT) —an algorithm used in computersand DSAs to compute discretefrequency components fromsampled time data; invented byCooley and Tukey

Flat-top window — a windowingfunction that minimizesamplitude error for off-centerinput-signal components

Force window — a windowingfunction that eliminates straysignals; used for the excitationsignal in impact test to improvesignal-to-noise ratio

Hanning window — a windowingfunction used to reduce leakagewhen measuring noise andperiodic signals like sine waves

Leakage — the spreading ofenergy throughout thefrequency domain; energy leaksout of one resolution line of anFFT into other lines, sometimesmasking small signals close to asine wave. This happens whenthe signal is not periodic withina time record. Applying anappropriate window functionwill minimize the amplitudeerror due to the leakage.

Linear averaging — a techniquefor improving the signal-to-noise ratio of a measurement;can be used if a trigger signal isavailable that is synchronouswith the periodic part of thespectrum

Lines — To reduce confusionabout which domain we are in,samples in the frequencydomain are called lines.

Nyquist Criterion — the minimumtheoretical sample rate for abaseband signal to bereproduced in sampled form,equal to twice the highestfrequency of the input signal. Inmost cases, you will want to usea higher sample rate torepresent the signal accurately.

Periodic random noise (PRN) —a kind of pseudo-random noisethat periodically repeats itssequence; the best stimulus fortesting linear networks.

Pseudo-random noise — amathematically generatedrandom noise whose period ismatched to time record length,thus eliminating leakage

Random noise — true noise

Real time — If we can compute thetransform (FFT) before the timerecord buffer fills, we are saidto be operating in real time.

Rectangular window — anotherterm for uniform window

Response window — anexponential-weighted windowthat guarantees that theresponse dies out (goes to zero)within the time record; used inimpact test to avoid leakageerror; also called “exponentialwindow”

RMS averaging — a technique formeasuring small signals in thepresence of noise. RMSaveraging is useful forprocessing stationary signals.Each data block can beoverlapped to achieve themaximum number of averages.RMS averaging producesamplitude information only.You lose phase information.

Self-windowing functions — afunction that does not require awindow because it occurscompletely within the timerecord or its period is matchedto time-record length (itgenerates no leakage in theFFT)

Time record — a block of Nconsecutive, equally spacedsamples of the input; this blockis the basic unit transformed byan FFT.

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Transfer function — a ratio of theoutput over the input, both inthe Laplace domain;sometimes usedinterchangeably with“frequency response function”

Uniform window — a windowingfunction that weights all of thetime record uniformly; used fortransient signals

Windowing — a way to reduceleakage by forcing an FFT tolook at data through a narrowwindow where the input is zeroat both ends of the time record.Many different functions can beused to window the data,depending on the type ofmeasurement you are making.

Zoom — another term for band-selectable analysis

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