Noise and Vibration Book

8/6/2019 Noise and Vibration Book

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FREQUENCY ANALYSIS

4 FREQUENCY ANALYSISDuring frequency analysis, three different types of signals must theoretically and practicallybe handled in different ways. These signals are

periodic signals, e.g. from rotating machines random signals, e.g. vibrations in a car caused by the road-tire interaction transient signals, e.g. shocks arising when a train passes rail joints

4.1 Periodic signals - Fourier seriesJean Baptiste Joseph Fourier, who was a French mathematician around the start of the 19 thcentury, discovered that all periodic signals can be split up into a (potentially infinite) sum ofsinusoids, where each sinusoid has an individual amplitude and phase, see Figure 4.1. Aperiodic signal is thus distinguished by that fact that it contains (sinusoids with) discretefrequencies. These frequencies are l/Tp, 2/Tp, 3/Tp, etc., where Tp is the period of the signal.

Figure 4.1. Using Fourier's theory, every periodic signal can be split into a (potentially infinite) number ofsinusoidal signals. Shown in the figure is a periodic signal which consists of the sum of the threefrequencies liTf" 2ITf" 3ITf" where Tf' is the period of the signal.The Fourier series is mathematically formulated so that every periodic signal xp( t ) can bewritten as

(4.1)

The coefficients ak and bk can be calculated as

for k = 0,1,2, ...(4.2)

for k = 1, 2, 3, ...

where the integration occurs over an arbitrary period of xp( t ). To make the equation easier tointerpret physically, one can also describe Equation (4.1) as a sinusoid at each frequency,where the phase angle for each sinusoid is described by the variable fjJk. We then obtain

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(4.3)

where Go is the same as in Equation (4.1). Comparing Equations (4.1), (4.2), and (4.3) we seethat the coefficients in Equation (4.3) can be obtained from Gk and bk in Equation (4.2)througha: = ~ a : ' + b,29, = arctan (!: 1 (4.4)By making use of complex coefficients, Ck, instead of Gk and bk, the Fourier series canalternatively be written as a complex sum as in Equation (4.5).

'X. J27f!: Ix (t) = """" c c/' k

k= - x

In this equationac =_11

II 2'I + ' [ ~ ./2"k

1 1 J --IC = -(a - b ) = - x (t)c I;, dtk 2 k k T /'11

(4.5)

(4.6)

and the integration occurs over an arbirtrary period of the signal xp( t ) as before. Note inEquation (4.5) that the summation occurs over both positive and negative frequencies, that is,k=0, 1, 2, '" Since the left side of the equation is real (we assume that the signal xp is anordinary, real measured signal), the right side must also be real. Because the cosine function isan even function and the sine function is odd, then the coefficients Ck must consequentlycomply with

Re[c_kl = Re[cJIm[c k 1= - Im[c!: 1 (4.7)

for alI k > nd where * represents complex conjugation. Hence, the real part of eachcoefficient Ck are even and the imaginary part odd. For real signals, which are usually bandlimited, the Fourier series summation can be done over a smaller frequency interval,k = 0, 1, 2, .. . , N, where the coefficients for k > N are negligible when N is sufficientlyhigh.Note also that each coefficient Ck is half as large as the signals amplitude at the frequency k,which is directly apparent from Equation (4.6). Thus, the fact that we introduce negativefrequencies implies that the physical frequency contents are split symmetricalIy, with half as

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(true) positive frequencies, and half as (virtual) negative frequencies. The same thing occursfor the continuous Fourier transform, which will shortly be described.

4.2 Spectra of periodic signalsTo describe a periodic signal, either a linear spectrum or a power spectrum is used. The mostintuitive spectrum for periodic signals is the linear spectrum of Figure 4.2, which basicallyconsists of a specification of the coefficients for amplitude and phase angle according toEquation (4.3). We will later see that when estimating spectra for periodic signals, many timesone cannot simply compute this spectrum since, due to superimposed noise, it requiresaveraging, see Section 6.1. Therefore, the so-called power spectrum is more common in FFTanalyzers. This spectrum consists generally of the squared RMS-value for each sinusoid in theperiodic signal, and is obtained by squaring the coefficients a; in Equation (4.4) and dividingby 2. Phase angle is thus missing in the power spectrum.

Amplitude Spectrum, V

lITp 21TpPhase Spectrum, Degrees

f

Figure 4.2. Amplitude spectrum of a periodic signal. The spectrum contains only the discrete frequencies /IT,,,2/Tpo 31Tp, etc., where Tp is the signal period.

4.3 Frequency and timeTo understand the difference between the time and frequency domains, that is, the informationthat can be retrieved from the different domains, the time domain and the frequency domain,we can study the illustration in Figure 4.3. We see in the time domain the sum of all includedsine waves, while in the frequency domain we see each isolated sine wave as a spectralcomponent. Therefore, if we are interested in, for example, the time signal's minimum ormaximum value, we must consider the time domain! However, if we want to see whichspectral components exist in the signal, we should rather look to the frequency domain.Remember that it is the same signal we see in both cases, that is, all signal information iscontained in both domains. Various elements of this information, however, can be more easilyidentified in one or the other domain.

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Frequency

Amplitude

Time

Figure 4.3. The time and frequency domains can be regarded as the signal appearance from different angles.Both planes contain all the signal information (actually, for this to be completely accurate, the phase spectrummust also be included along with the above amplitude spectrum).

4.4 Random processesRandom processes are signals that vary randomly with time. They do, however, have onaverage constant characteristics such as mean value, RMS value, and spectrum. The signalswe shall study are assumed to be both stationary and ergodic. A stationary signal is a signalwhose statistical characteristics, the mean value, RMS value, and higher order moments, areindependent of time. An ergodic signal is a subclass of the stationary signals, for which thetime average of one signal in an ensemble is equal to the average of all signals in theensemble at a single time. These concepts are central to an understanding of the upcominganalysis, and therefore they will be described in a bit more detail.In statistics a random process is called a stochastic process. A stochastic process x( t ) impliesa conceptual (imagined, abstract) function, which, for example, could be "the voltage whicharises due to thennal noise in a carbon film resistor of type XXX at a temperature of20 C" orthe like. This time function is random in that all real resistors like those in the example, in anexperiment where one measures voltage, will exhibit different functions xl t). For one suchprocess one can, for example, calculate the expected value, E[x(t)], which is defined as

'X.

f-l, ( ) = E [x( t)1= J ( t) . p, (x)dx (4.8)where px( x) is the statistical density/unction of the process. Another common measure is thevariance, o J ), which is defined as

' -a;(t) = E[(x(t) - f-l,(t)tj = J x - f-l,f .p,(x)dx (4.9)55

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The variance can, for our voltage example, be interpreted as proportional to the powerdeveloped in the resistor when the voltage is supplied over it. Variance is rarely used inpractice, but the square root of the variance, the standard deviation, is more common, mostlybecause this quantity has the same units as the measured signal.Besides expected value and variance, we also define the more general central moments, Mi , as}vI" [x(t)] = E[(x(t) -IiJ'] (4.10)The variance is consequenctly equal to M2. It is important to understand that all of the abovevalues are time-dependent, implying, possibly against the intuitive understanding of many, notwhat we normally mean by average. If we experimentally want to estimate the expectedvalue according to the definition, we would (in our example) have to measure a large numberof resistors and then average the voltage over all measurements at each time instant. Thistype of averaging is called ensemble averaging, since we carry out a computation for anensemble of resistors, see Figure 4.4.

_Time _.... - . - - - - . - ~

, Ensemble

Figure 4.4. The difference between ensemble values and time average values. The basic definitions fromstatistics are based on ensemble values. For ergodic signals these values can be replaced by time averages.Physical signals are always ergodic, if hey are stationary.A stationary stochastic process is a process for which the above statistical measures areindependent of ime. For example, to measure the expected value of our voltage, if the signalwas stationary, we would only need to measure each resistor at a single common time, to. Butin this example, for an exclusively stationary signal we must measure a (large) number ofdifferent resistors.For those signals we normally measure in physical situations, there is another restriction thatapplies in relation to the above, namely that the signals are ergodic. For ergodic signals theabove ensemble values are equal to the corresponding time values, implying that thedefinitions can be replaced by time averages. For example, the expected value can be replacedby

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TE[x(t)] = J-L. = lim _1J (t)dt1 'f'-"X- 2T

-T(4.11)

Similarly, the moment ofEquation (4.10) can, for an ergodic signal, be calculated by usingT

AI" [x(t)j = E[(x - J-LJ"] = J . ~ ~ J x - IL,)" dt-T

(4.12)

If in our example we assume that the voltage is ergodic, it implies that it would suffice tomeasure one resistor during a certain time, and then use these time values according toEquation (4.11) to calculate the expected value. This is what is done in reality, but at the sametime it is important to understand the fundamental difference between the above definitionswhen reading literature in the field. For further reading about statistical concepts, consider[Bendat & Piersol, 1986; Bendat & Piersol, 1993]. It should also be mentioned thatstationarity is very important to check before analysis of a random signal. Methods for thischeck are given in [Bendat & Piersol, 1986; Brandt, 2000].When the above statistical measures are calculated from experimental data, the followingestimations are usually used. If we assume that the signal is stationary and ergodic, theexpected value is estimated with the mean value

(4.13)

where we have simplified the term x l1=x( n). This notation will often be used below as itsimplifies the reading of the equations. The mean value is often denoted by a line above thevariable name. For other variables we use the '"hat" symbol /\ to show that we are dealing withan estimate and not a theoretical expected value. For an arbitrary variable, the expected valueis estimated by the above formula. The standard variation, O ' ~ , is estimated using, ax' definedby

a, = ~ _ I _ I . ( x -x fN -1 =1 (4. 14)where N - 1 in the denominator is used so that the estimator becomes consistent, i.e. with anincreased number of averages the estimation should approach the true value, without biaserror. This concept is not so important in practice where we nonnally use more than 20 valuesin the averaging.A common value related to the standard deviation is the Root lYfean Square (RMS) value of asignal. The name directly implies how the value is calculated, namely as

1 N ,RMSr = - I . ,:N n=1 (4.15)The RMS value, as is evident from the equation, is equal to the standard deviation when themean value of the signal is zero. The value corresponds also, for any dynamic signal, to theDC voltage which the signal could be replaced by in order to cause the same power

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dissipation in a resistor. More generally, the value of a dynamic signal is used as a firstmeasure of the "size" of the signal. For a sinusoidal signal it is easy to show that the RMSvalue corresponds to the peak value divided by the square root of2.

4.5 Spectra of random processesAs opposed to periodic signals, random signals have continuous spectra, that is, they containall frequencies and not only discrete frequencies. Hence we cannot display the amplitude orRMS value for each incoming frequency, but we must instead describe the signal with adensity-type spectrum (compare, for example, with discrete and continuous probabilityfunctions). The unit for noise spectra is therefore, for example, (m/s2)2/Hz, if the signal is anacceleration measured in m/s 2 This spectrum is called Power Spectral Density, PSD. Anexample is shown in Figure 4.5.

0 200 Averages, 75% Overlap, 4(=2.5 Hz

N -5.....'"-"il -lOrr 5[t il -20p..ci'.9-! -25II )"iiuu


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stochastic, ergodic functions, x( t ) and y( t ), where x( t) is seen as the input and y( t) theoutput, is defined as

(4.17)...For correlation functions, the following relationships hold for real signals x( t) and}{ t ).R[C (-T) = R[[ (T) even fUllction (4.18)

(4.19)Now we define the two-sided power spectral density, denoted S ~ . l f ) , as the Fourier transformof the autocorrelation function

"-5,)!) = S' {R,,(T)} = JR,,(T)e- J27r!T dT (4.20)and analogous to this function we define the two-sided cross-spectral density, Sy.lf) as

x5. (1) = S' {R (T)} = JR (T)e- J27r !TdT,II" ,1J.l: .11.1' (4.21 )The negative frequencies in the Fourier transform act such that half of the physical frequencycontent appears as positive frequencies, and the other half as negative frequencies. Therefore,experimentally we never measure the two-sided functions, but instead define the single-sidedspectral densities, Auto (Power) Spectral Density, PSD or G . ~ . l f ) , and Cross-SpectralDensity, CSD or Cvx(f), asG,.,(J) = 25,J!) for 1 > 0G,,,(O) = 8 n :(0) (4.22)G,1,,(!) = 25yr (J) for 1 > 0G!I"(O) = 8!11'(0)

(4.23)

For the single-sided spectral densities, CxxC f ) and CyxC f ) , it follows directly from thecharacteristics of correlation functions according to Equations (4.18) and (4.19) thatCu(f) is real

4.6 Transient signals

(4.24)(4.25)

Finally, in addition to the previously mentioned periodic and random signals, we havetransient signals. Like the random processes they have continuous spectra. However, asopposed to random signals, transient signals do not continue indefinitely. It is therefore not

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possible to scale their spectra by the power in the signal (power is of course energy/time).I n ~ t e a d , transient signals a r ~ ~ e n e r a l l y scaled by their energy, and thus such.spectra can haveumts of for example (m/s-ts/Hz. The spectrum most commonly used IS called EnergySpectral Density, ESD. Because energy is power times time, we obtain the definition of theESDESD = TPSD (4.26)where T is the time it takes to collect one time block, see Section 5.6. The ESD shall beinterpreted through the area under the curve, which corresponds to the energy in the signal. InSection 6.7 below there is more on spectrum estimation for transient signals.An alternative linear, and therefore more physical, spectrum for a transient signal is obtainedby using the continuous Fourier transform without further scaling. The transient spectrum of asignal x( t ) is consequently defined asTJf) = :s{x(t)} (4.27)This is a two-sided spectrum and we will return to a discrete approximation of this spectrumin Section 6.7.

4.7 Interpretation of spectraWhat is the usefulness of defining these different spectra? The motivation is naturally thatthrough studying the spectrum we will hopefully gain some insight into how the signal insome sense behaves. In order to understand the signal, we first need an understanding of whatcan be read from the spectrum. For periodic signals, this is relatively simple, as they basicallyconsist only of a sum of individual sinusoids. By knowing what these signals are, that is theiramplitude, phase and frequency, we can also recreate the measured signal at any specific time,if we so choose.We may also want to know for example the RMS value for the signal, in order to know howmuch power the signal generates. This can be done using a formula called Parseval'sTheorem, see Appendix D. For a periodic signal, which has of course a discrete spectrum, weobtain its total RMS value by summing the included signals using Equation (4.28),

(4.28)where I xkl is the RMS value of each sinusoid for k= I, 2, 3... The RMS value of a signalconsisting of a number of sinusoids is consequently equal to the square root of the sum of theRMS values. I xkl corresponds consequently to the value of each peak in the linear spectrum.For a noisy signal we cannot interpret the spectrum in the same way. This signal contains allfrequencies, which makes it a bit tedious to count them! Instead, we interpret the area underthe PSD in a specific frequency range, see Figure 4.6, which also follows from Parseval'sTheorem. To calculate the noise signal's RMS value from the PSD we use Equation (4.29),

R:\IS" = JGil (f)df = .Jarea under the curve (4.29)60

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5 EXPERIMENTAL FREQUENCY ANALYSISIn practice, frequency analysis in the field of noise and vibration analysis generally makes useof the discrete Fourier transform to estimate spectra. In this chapter we shall show how thistransform is used in practice. Before we begin, however, we need to learn a bit aboutestimation of random (stochastic) variables, ~ n d therefore we begin with a briefdescription ofthe error measures we will use from now on.

5.1 Errors in stochastic variable estimationsWhen calculating statistical errors, one differentiates between two types: bias (systematic)error, and random error [Bendat & Piersol, 1986]. We assume that we shall estimate (that is,measure and calculate) a parameter, , which can be for example the power density spectrumof a random signal, G.u . In the theory of statistical variables, the "hat" symbol, /\, is usuallyused for a variable estimate, so we denote our estimate 9 (G r r ). We now define the biaserror, b , as,)b. = [9]-1>

r)(5.1)

that is, the difference between the expected value of our estimate and the "true" value . Wegenerally divide this error by the true value to obtain the normali:ed bias error, Ch, as

b.r)E=-Ii

(l + 2c:,) 9-5% (5.5)

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5.3 Octave and third-octave band spectraThe original way of measuring a spectrum was to have an adjustable band-pass filter andvoltmeter for AC voltage, as was shown in Figure 5.1. To be able to compare spectra fromdifferent measurements, the frequencies and bandwidths used were standardized at an earlystage. It was at that time natural to choose a constant relative bandwidth, so that thebandwidth increased proportionally with the center frequency. Thus, if we denote the centerfrequency by /m and the width of the filter by B, we then have that. IB- = constant1m (5.8)

The chosen frequencies were distributed into octaves, meaning that each center frequency waschosen as 2 times the previous one, and the width of each band-pass filter was twice as largeas that of the previous filter. The lower and upper limits were chosen using the geometricalaverage, that is

(5.9)where.ll is the lower frequency lim it and j;J the upper limit. The resulting relationship betweenthe lower and upper frequency lim its for octave bands is

(5.10)_ .1 /2!" - !,,, 2In some instances the octave bands give too coarse a spectrum for a signal, in which case afiner frequency division can be used. The most common division used is the third-octaveband, where every octave is split into three frequency bands, so that!, = f , ) - l / ( i1 J", (5.11 )f = f 21/( iJ Il Jm

More generally, one can split each octave into n parts. These frequency bands are generallylin-octave bands and their frequency limits are given byf = f 2- 112 "J[ Jm (5.12)

The center frequencies for octave and third-octave bands are standardized in [ISO 266],among other places. The standard center frequencies for third-octave bands are ... 10, 12.5,16, 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 250 ... where boldface stands for center frequenciescommon to the octave bands. It is clear that these frequencies are not exact doublings, butrather rounded values from Equation (5.13) below. In Equation (5.13), p is a negative orpositive integer number.f = 1000.10,,/ 10Jm (5.13)

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5.4 Time constantsIf one measures a band-pass filtered signal that is not stationary, a signal which varies as afunction of time is obtained. There is, however, a limit to how fast this signal can change,even if the input varies, because of the band-pass filter's time constant. This value describeshow quickly the signal rises to (l-e- I ) or about 63% of the final value, when the level of theinput signal is suddenly altered. For a band-pass filter with bandwidth B the time constant, r,is approximately I

1T= -B (5.14)For octave and third-octave band measurements, the different frequency bands consequentlyhave different time constants, that is, longer time constant for lower frequency band. To theright in Figure 5.1 a typical octave band spectrum for a vibration signal can be seen. Note thatto the far right the so-called total signal level, that is, the signal's RMS value (within a givenfrequency range) is shown. The position of this bar differs depending on analyzermanufacturer, but it is usually shown on either side of the octave bands.

5.5 Real-time versus serial measurementsTo measure an acoustic signal's spectral contents using octave bands, one can in the simplestcase use a regular sound level meter with an attached filter bank, that is, a set of adjustablefilters which, often automatically, steps through the desired frequency range and stores theresult for each frequency band. This type of measurement is called serial since the frequencybands are measured one after the other. Naturally, this method only works when the signal(sound) is stationary.In order for the measurement to go faster, or if the signal is not stationary, one can instead usea real-time analy=er, which is designed with all of the third-octave bands in parallel, so thatthe same time data can be used for all frequency bands.

5.6 The Discrete Fourier Transform (OFT)In an FFT analyzer, as evident from the name, the FFT is used to calculate the spectrum. FFTis an abbreviation for Fast Fourier Transform, which is a computation method (algorithm)which actually calculates the Discrete Fourier Transform (OFT), only in a faster way thandirectly applying the OFT. We shall therefore begin by studying the OFT and follow withhow it is used in an FFT analyzer.Let us assume that we have a sampled signal x( n )=x(nLlt ) where x( n) is written insimplified notation. We further assume that we have collected N samples of the signal whereN is usually an integer power of 2, that is, N = 21' where p is an integer number. There arealgorithms for the FFT that do not require this assumption, but they are slower than thosewhich assume an integer power of2.The (finite) discrete Fourier transform, X( k) = X(k4f ) , of the sampled signal x( n) ISusually defined as

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.V-I -./2rrl-"X(k) = Lx(n)c - .- for k = 0,1,2, ... , N - 1 (5.15),,=0

which we call the forward transform of x(n). To calculate the time signal from the spectrumX(k), we use the inverse Fourier transform,1 N -I ./2rrllkx(n) = - L X (k)c.v for n = 0,1,2 ... , N - 1N k=o

I

(5.16)

It should be pointed out that the definition of the OFT presented in Equation (5.15) is notunique. One may find definitions with different scaling factors in front of the sum in theliterature. When confronted with new software, one should therefore test a known signal firstto find out which definition of the OFT is used. A simple way to test is to create a signal withan integer number of periods and with an N of, say, 1024 samples. See Section 5.7 on how tocreate such periodicity. By checking the result of an FFT and comparing with the fonnulaeabove, the definition used can be identified. The definition according to Equations (5.15) and(5.16) is common, and is the one used, for example, by MATLAB.The spectrum obtained from the above definition of the OFT is not scaled physically. This isclearly seen by studying the value for k = 0. The frequency 0 corresponds to the OCcomponent in the signal, that is, the average value. But, according to Equation (5.15) abovewe have

.'V-I

X(O) = Lx(n) = N x (5.17)where we let x denote the mean value of x(n). It can thus be concluded that Equation (5.15)must be divided by N in order to be physically meaningful. which is done when using onlyX(k). As a rule, however, we cannot measure only a time block of data because we have noisein our measurements. Therefore we need to average the signal, and to that end other scalingfactors are needed, which will be described below.It should also be noted here, that the discrete Fourier transfonn in Equation (5.15) differssubstantially from the analog Fourier transfonn, see also Appendix O. First of all, the OFT iscomputed from a finite number of samples. Secondly, the DFT is not scaled in the same unitsas the analog Fourier transfonn, since the differentiator dt is missing. The analog Fouriertransfonn of a signal with unit of m/s 2 would have unit mis, whereas the OFT will have unitsof m/s 2 In Chapter 6 this will be clear as we present how to compute scaled spectra from theOFT results.

5.7 Periodicity of the Discrete Fourier TransformAs evident from Equation (5.15) above, the discrete Fourier transfonn X( k) is periodic withperiod N, that is,X( k )=X( k+N) (5.18)This result arises because we have sampled the signal, which, according to the samplingtheorem, implies that we make it periodic on the frequency axis, so that it repeats at every

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S.B Oversampling in FFT-analysisIf we use N time samples, which we usually call the block size or frame size, the DFT results-n half as many, that is, N 12 positive (useable) frequencies, as seen in Figure 5.2. Thesefrequency values are usually calledfrequency lines or spectral lines.Block size Corresponding(# of time samples) # of spectral lines

256 101512 201

1024 4012048 8014096 16018192 3201

Table 5.1. Typical block sizes and correspondingnumbers of useable spectral lines when applyingFFT.

Because the analog anti-aliasing filter is not ideal, but has some slope after the cutofffrequency, as seen in Figure 5.3, we cannot sample with a samling frequency which is only2'Bman the bandwidth of the signal. In the FFT analyzer, a "standard" oversampling factor of2.56 has been established. Thus, we can only use the discrete frequency values up tok = N 12.56. Typical values for the block size and corresponding number of spectral lines aregiven in Table 5.1. The frequency here which corresponds to k = N 12.56 is called thebandwidth, BI17(u .

co 0"0. ; -20,c&. -40," -60Co>c" -80:s0-" -1000 100 200 300 400 500 600 700 800 900 1000

0.,"tb -200"-400

..c!:l... -600-800 0 100 200 300 400 500 600 700 800 900 1000

Frequency,Hz

Figure 5.3. Typical anti-aliasing filter. Because of the filter's non-ideal characteristics, the cutoff frequency, f.,needs to be set lower than half of the sampling frequency. It is typically set to Ix 12.56 in FFT analyzers. forhistorical reasons, which approximately corresponds to 0,8 j/2, In the figure the cutoff frequency is 800 Hz.which gives a sampling frequency of 2.56'800=2048 Hz. Note the non-linear phase characteristic, which will bediscussed in Section 7.7.

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I1,, N 12 we can no longer (easily) calculate for example an impulseresponse. In that case, however, the following qualities of the Fourier transform may be used.See also Appendix E.For a real measured signal, the real part of the Fourier transform is an even function and theimaginary part an odd function, that is,

Re{X(-k)} = Re{X(k)} (5.21 )\

1m {X(-k)} = -1m {X(k)} (5.22)These qualities, called the Fourier transform symmetry properties, are valid exactly, even forthe OFT, provided that there does not exist any aliasing distortion. Naturally, these attributesare valid when we "shift down" the upper N 12 spectral lines so they lie to the left of k = 0, sothat we have a two-sided spectrum X( k), k = 0, 1, 2, ... Thus, according to Equations(5.21) and (5.22), the negative frequencies can be "filled in" before inverse transformation.Close study of the OFT result will show that the value X(k = NI2+ 1), for example valuenumber 513 if the block size is 1024 samples, will be a real-valued number, which is notequal to X(O). It is not the NI2 number because of the "skew" in the periodic repetitiondiscussed in Section 5.7. Furthermore, this value (for k = NI2+1) cannot be discarded if anexact reproduction of the time signal x( n ) is to be computed. This is often overlooked andonly the first NI2 values stored. The correct number of values to store in order to be able tocompute back the original time signal is instead N12+ 1, in our example with 1024 block size,thus 513 frequency values should be stored. Then all negative frequencies can be filled inaccurately.

5.10 LeakageWhat happens if we, for example, compute the DFT with a frequency increment of 41= 2 Hz,but the measured signal is a sinusoid of 51 Hz, so that the signal frequency lies right betweentwo spectral lines in the DFT (50 and 52 Hz)? The result is that we get one peak at 50 Hz andone at 52 Hz. However, both peaks are lower than the true value, see Figure 5.4. To easilyobserve this error in the figure, we have scaled the OFT by dividing by N and taking theabsolute value of the result, see also Section 6.3. Thus the correct value should be11 J2 == 0.7 , the RMS value of the sine wave.

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0.50.80.6 0.40.40.2 0.3

0-0.2 0.2-0.4-0.6 0.1-0.8

-1 00 0.1 0.2 0.3 0.4 0.5 30 40 50 60 70Time, s Frequency, (Hz)

Figure 5.4. Time block (left) and linear spectrum (right) of a 51 Hz sinusoid. 256 time samples have been used,giving 128 spectral lines. The frequency increment is 2 Hz. Instead of the expected value 0.7, that is, the RMSvalue of a sinusoid with amplitude of I, we get one peak much too low (in this case 40% too low). There are alsomore non-zero frequency values to the left and right of the 50 Hz and 52 Hz values. This phenomenon is calledleakage since the frequency content in the signal has "leaked" out to the sides.As seen in the figure the resulting peak is far too low, by as much as 40%! Furthermore, itlooks like the frequency content has "leaked" away on either side of the true frequency of 51Hz. This phenomenon is therefore called leakage.One way to explain the leakage effect is by studying what happens in the frequency domainwhen we limit the measurement time to a finite time, which corresponds to multiplying ouroriginal, continuous signal by a time window which is 0 outside the interval t E (-TI2,TI2),and 1 within this same interval. A multiplication with this function, w( t ), in the time domainis analogous to a convolution with the corresponding Fourier transform, W( f ) . We thusobtain the weighted Fourtier transform of x(t) w(t) , denoted X,(f) , as

' -X",(f) = X(f) * lV(f) = JX(u)lV(f - u)du (5.23)where * denotes convolution. W(f) is the transfonn of a rectangular time window, in our caselV(f) = T sin(7rfT) = T sine (.tT)(7r fT)To make the convolution easier we exchange the two functions, and make use of

.....

X",(.t) = X(f) * lV(f) = lV(f) *X(f) = J V(u)X(f -u)du71

(5.24)

(5.25)


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-30

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-l ) -8 -7 -6 -5 -4 -3 -2 -I 0 2 3 4 5 6 7 8 9 k/1)Figure 5.5. OFT of a sinusoid which coincides with a spectral line. The convolution between the transform of the(rectangular) time window, WU), and the sinusoid's true spectrum, DUo), results in a single spectral line.

o

10 . . , .. ."

,, , . , , .' ,. , ,.' , ,.' .. .' " " '. .. .' '. " ".' '. " ".'. '. " ".' '. " r..' '. " "" '. " ".' '. ".' '. '..' '. ".' '. "'. "

.,: " ". " "" '. " '. "" '. '.. '.. .' " ". . , . . .' " '. , .' ., . ; " '.. , , , .: .' " '.'. . . " '., . .' '. . , . " " '. . '. ' . .' '..' " '. " .' '..' '. " " .' '..' '. " '. .' '..' '. " " ., '..' '; '" ". ., '.

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20 , , , : , , , , '" ' . " " .'.' "" .' .'" " "" .' .'" " .'" .' .'" .' ".' ".' ".' .'.' .'" ".' "" .'" .''

30

40

:: '. '.: :'. '. " " :: .5 0 L L ~ ~ U L L L ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 I 1 2 3 4 5 6 7 8 9.Ii)

Figure 5.6. Leakage. The frequency of the sine wave is located at/o, exactly mid way between k=0 and k=l,corresponding to an integer number of periods plus one half period in the time window. When a periodic signaldoes not have an integer number of periods in the measurement window, then due to the finite measurement timethe convolution results in too low a frequency peak. At the same time the power seems to "leak" into nearbyfrequencies; The total power in the spectrum is still the same.The convolution between the Fourier transfonn of our sine wave and that of the time windowthus implies that we allow the latter, W( I), to sit at the frequency II), that is, we constructW(I-fo). Then we shift the transfonn of the (continuous) sinusoid, which is a single spectralline, all the way to the left ( k = 0), and multiply the two. At each k for which we wish tocompute the convolution, we center the sinusoid spectral line at that same k, mUltiply the two

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and sum all the values (all frequencies). In some cases (see Figure 5.5), for example wherefo = ko ',cjf, the sinusoid may move up the time window such that for all integer numbers k weobtain only one single value from the convolution, since for all k except for k = ko the spectralline of the sinusoid corresponds to a zero in W(f-fo).Illustrated in Figure 5.6 is the result of the convolution as described above, for the case wherethe sinusoid lies between two spectral lines (we have an integer number plus one half periodin the time window). The Fourier transform of the time window is instead centered at afrequency fo which is not a multiple of the frequency increment !::.f We see in the picture thatwe obtain several frequency lines which slowly decrease to the left and right, and we get twopeaks which are the same height, although much lower than the sinusoidal RMS value. It canbe shown that if the RMS values of all spectral lines are summed according to Equation (4.28)the result is equal to the RMS value of the sinusoid. Hence, the power in the signal seems to"leak" out to nearby frequencies, giving the name leakage.

I5.11 The picket-fence effectAn alternative way to look at the discrete spectrum X(k) we obtain from the OFT is to seeeach spectral line as the result of a band-pass filtering, followed by an RMS computation ofthe signal after the filter. This process is often illustrated as in Figure 5.7 with a number ofparallel band-pass filters, where each filter is centered at the frequency k (or k l1f if we thinkin Hz). This method of looking at the OFT is reminiscent of viewing the true spectrumthrough a picket fence and therefore it is called the picket-fence effect. Note that the picketfence effect is also analog with the method for measuring a spectrum with octave bandanalysis that we discussed in Section 5.3. As was mentioned in Section 5.2 this principle isthe only way to measure or compute spectral content.

k - -+ - - . . . - - - 4 .... ,...

Figure 5,7. The picket-fence effect. Each value in the discrete spectrum corresponds to the signal's RMS valueafter band-pass filtering. If we study a tone lying between two frequencies we will obtain too Iowa value.

5.12 Windows for periodic signalsAs we saw above, we obtain an amplitude error when estimating a sinusoid with a non-integernumber of periods in the observed time window. This error is caused by the fact that wetruncate the true, continuous signal. By using a weighting function other than the rectangularone used above in the leakage discussion, we can reduce this amplitude error. This process iscalled time-windowing and is illustrated in Figure 5.8.

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EXPERI.\4ENTAL FREQUENCY ANALYSIS

dB0

-10

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'." '1', 1," , 1,

,, , ", , ,, ' , ,, , ,.'" , ," ..," .," .," "" "" ,.,,. ,." ." "" "" """",,"4 5

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, , ," " " "., " ... '.',.' '.' .. ",! " " ""

r, r,r' " "" " ,,"

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Figure 5.5. OFT ofa sinusoid which coincides with a spectral line. The convolution between the transform of the(rectangular) time window, W(f), and the sinusoid's true spectrum, O(fo), results in a single spectral line.

o

10

20

30

40

50

, ", ',""'.'"""""""::

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, I I,

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, 'I I I II I 1 I '" I 1, I \ I \ I \I t I I I" I, I I" " I r" " II,I " II,I " II,I II II" I, ' II,I 't II/' " 11" I , I I" II II" 'I H" 'I II:: :: u

7 8 9

,, ,, ,""" ,""""""""

Figure 5.6. Leakage. The frequency of the sine wave is located atfo, exactly mid way between k=0 and k=l,corresponding to an integer number of periods plus one half period in the time window. When a periodic signaldoes not have an integer number of periods in the measurement window, then due to the finite measurement timethe convolution results in too low a frequency peak. At the same time the power seems to "leak" into nearbyfrequencies; The total power in the spectrum is still the same.The convolution between the Fourier transfonn of our sine wave and that of the time windowthus implies that we allow the latter, W( f ) , to sit at the frequency fo, that is, we constructW(f-fo). Then we shift the transfonn of the (continuous) sinusoid, which is a single spectralline, all the way to the left ( k = 0), and multiply the two. At each k for which we wish tocompute the convolution, we center the sinusoid spectral line at that same k, mUltiply the two

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Anders Brandt, Introductory Noise & Vibration A n a ~ v s i s

and sum all the values (all frequencies). In some cases (see Figure 5.5), for example wherefo = ko '.d.f, the sinusoid may move up the time window such that for all integer numbers k weobtain only one single value from the convolution, since for all k except for k = ko the spectralline of the sinusoid corresponds to a zero in W(f-fo).Illustrated in Figure 5.6 is the result of the convolution as described above, for the case wherethe sinusoid lies between two spectral lines (we have an integer number plus one half periodin the time window). The Fourier transform of the time window is instead centered at afrequency fo which is not a multiple of the frequency increment We see in the picture thatwe obtain several frequency lines which slowly decrease to the left and right, and we get twopeaks which are the same height, although much lower than the sinusoidal RMS value. It canbe shown that if the RMS values of all spectral lines are summed according to Equation (4.28)the result is equal to the RMS value of the sinusoid. Hence, the power in the signal seems to'"leak" out to nearby frequencies, giving the name leakage.

5.11 The picket-fence effectAn alternative way to look at the discrete spectrum X(k) we obtain from the DFT is to seeeach spectral line as the r e ~ l t of a band-pass filtering, followed by an RMS computation ofthe signal after the filter. This process is often illustrated as in Figure 5.7 with a number ofparallel band-pass filters, where each filter is centered at the frequency k (or k Ilf if we thinkin Hz). This method of looking at the DFT is reminiscent of viewing the true spectrumthrough a picket fence and therefore it is called the picket-fence effect. Note that the picketfence effect is also analog with the method for measuring a spectrum with octave bandanalysis that we discussed in Section 5.3. As was mentioned in Section 5.2 this principle isthe only way to measure or compute spectral content.

k\--"""T"--4--

Figure 5.7. The picket-fence effect Each value in the discrete spectrum corresponds to the signal's RMS valueafter band-pass filtering. Ifwe study a tone lying bet\veen two frequencies we will obtain too Iowa value.

5.12 Windows for periodic signalsAs we saw above, we obtain an amplitude error when estimating a sinusoid with a non-integernumber of periods in the observed time window. This error is caused by the fact that wetruncate the true, continuous signal. By using a weighting function other than the rectangularone used above in the leakage discussion, we can reduce this amplitude error. This process iscalled time-windowing and is illustrated in Figure 5.8.

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The time window used in Figure 5.8 is called a Hanning window and is one of the mostcommon windows used in FFT analyzers. The effect of the window is that it eliminates the"jump" in the periodic repetition of the time signal, but it is not very intuitive that it wouldimprove the result. It can be shown, however, that we can estimate the amplitude much betterthan with the rectangular window.

' . .. . . . '. . . . . .,' "

..

'.

a)

x b)

c)

" "'.' . '.. . .. .. ....',: .:

" "" .,

'. .

0.7d)

06

05

04

OJ

020.1

00 SO 100 150 200 250 300

Figure 5.8. Illustration of time-windowing with a Hanning window. The window lessens the jump at the ends ofthe repeated signal. In a) is shown the periodic repetition (dotted line) of the actual measured signal (solid line).In b) is shown the Hanning window and in c) is shown the result of the multiplication of the two. In d) is shownthe result of calculating the spectrum with the Hanning window (solid) and without (dotted). Note that \vhen thewindow is used, both the amplitude is closer to the true value (0.7), and the leakage has decreased. There doesexist an amplitude error of up to 16 %.To obtain a better estimate of the amplitude of a pure sinusoid, we need to create a windowwith a Fourier transform that is flatter and wider than that of the rectangular window. Throughthe years, a large collection of windows has been developed. Many FFT analyzers thereforehave a large number of different windows from which to choose. We shall here examine twowindows, the Hanning and the Flattop window.The Hanning window is probably the most common window used in FFT analysis. It ISdefined by half a period of a cosine, or alternatively one period of a squared sine, such that

() ..) (1fn) 1 [ (21fn))W If n = sm- N = '2 1 - cos N for n=O, 1, 2, .. . , N - 1 (5.26)The Hanning window's Fourier transform has a main lobe that is wider than the rectangularwindow, so that the maximum error decreases to 16%. This error is of course still too large inmany cases, for example when one desires to measure the amplitude of a sinusoidal signal. Inthat case, the Flattop window may be utilized, which yields a maximum error in amplitude of0.1 %, a bit more acceptable.The flattop window is not actually a uniquely defined window, but a name given to a group ofwindows with similar characteristics. When we use flattop windows in this book, we use a

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Anders Brandt. Introductory Noise & Vibration Ana(vsis

window called Potter 301 [Potter, 1972]. In Figure 5.9 and Figure 5.10 the three windows,rectangular, Hanning, and flattop, with their Fourier transfonns are shown for comparison.

0.5

O ~ O ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ T ~

1

0.5

O ~ ~ = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ ~ ~ o T0.5o t - - - - - ~ - - -

- 0 . 5 ~ ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ o Time TFigure 5.9. Time windows. Time-domain plots of the rectangular (top), Hanning (middle) and flattop (bottom)windows.There is a price to pay for the decreased amplitude uncertainty when we use time windows.The price is in the fonn of increased frequency uncertainty, which occurs because the betterthe amplitude uncertainty, the wider the main lobe of the spectrum of the window. Therefore,if we measure a sinusoid with a frequency that matches one of our spectral lines, then thepeak will become wider than if we had used the rectangular window. The flattop window,which has the best amplitude uncertainty, also has the widest main lobe. This trade-off isrelated with the bandwidth-time product which is explained more in relation to errors in PSDcomputation with windowing in Section 6.11. Figure 5.11 shows what the DFT of a sinusoidwhich exactly matches a spectral line looks like, both after windowing with the Hanningwindow and with the flattop window. As shown, the Hanning window results in 3 spectrallines which are not zero, while the flattop window gives a whole 9 non-zero spectral lines.Even with windows other than the rectangular, we get leakage when the sinusoid's frequencydoes not match up with a spectral line, as seen in Figure 5.12. What detennines the leakage ishow the window's side lobes fall off to each side of the center. The faster the fall-off of theside lobes, the wider the main lobe, which gives yet another compromise. The decreasing ofthe side lobes is usually measured as an approximate slope per octave. The flattop window,because of its large main-lobe width is only used when it is known that the spectrum does notcontain many neighboring frequencies. The Hanning window is often used therefore as astandard window since it gives a reasonable compromise between time accuracy andfrequency resolution.

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O.S

0.7\ . / Flattop0.6 .5 Hanning .' ,/ ,.0.4

0.3 .2 , 0.1 .. .. .0 '"-s -6 -4 -2 0 2 4 6 S

k

Figure 5.11. The widening of the frequency peak is the price we pay to get a more accurate amplitude. In thefigure is shown the linear spectrum of a sinusoid with amplitude I and frequency that matches the spectral linemarked "0", for both Hanning (solid) and t1attop (dashed) windowing, With the tlattop window the peak is muchwider than with the Hanning window. For clarity, the two values at k=I for the spectrum after Hanningwindowing are shown with black dots.

O . s ~ - - ~ - - ~ - - ~ - - - - ~ - - ~ - - ~ - - - - ~ - - ~ - - - , - - - - .

0.7

0.6

0.5

0.4

0.3

0.2

0.1

-s -6

, ' ---\I \

I \" " FlattopI \: ': \ Hanning

I 'I 'I 'I I: \ RectangularI 'I '

I 'I 'I '/ 'I ',

/ .," .'/ '/ '/ '/ '/ '/ '/ '/ '/ '

./ '/ '

-4 -2 ok

2 4 6 s

Figure 5.12. The spectrum of a sinusoid with frequency right between the frequencies k = -I and k = O. Threedifferent windows have been used: rectangular (solid), Hanning (dotted) and t1attop (dashed). From the figureone can see the compromise between amplitude and frequency uncertainties.

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EXPERIAIENTAL FREQUENCY ANALYSIS

5.13 Windows for noise signalsThe window's influence on a random signal is a bit different than that described in Section 5.7since noise signals have continuous spectra, as opposed to periodic signals which havediscrete spectra. The result of the convolution between the continuous Fourier transform ofthe window and the noise signal is therefore more complicated to understand. Ifwe recall thatconvolution implies that the qualities of both signals are "mixed," we can understand that thewindow will introduce a "ripple" in the noise signals spectral density. At the same time, weget a smoothing of the spectral density, due to the influence of the main lobe. For narrowfrequency peaks, for example if we measure resonant systems with low damping as discussedin Chapter 2 we get an undesired widening of the resonance peaks. More on these bias errorsare discussed in Section 6.7.The qualities most important to the influence of the window when determining spectraldensities are the width of the main lobe and the height of the side lobes. The narrower theseside lobes are, the less influence we get from nearby frequency content during convolution.Th flattop window is never used for random signals, since its main lobe is too wide. The mostcommon is tlfe Hanning window and many FFT analyzers have no other windowimplemented for noise analysis, although even the so-called Kaiser-Bessel window can besuitable to use.

5.14 Frequency resolutionFrom the above discussion about widening of frequency peaks, it is clear that with a certainfrequency increment, ,1J, one may not, after the OFT computation, discern between twosinusoids, separated in frequency by only one spectral line. For this reason we shoulddifferentiate between frequency increment and frequency resolution. Frequency resolutionusually implies the smallest frequency difference that is possible to discern between twosignals, while the frequency increment is the distance between two frequency values in theOFT computation, that is, ,1[ Frequency resolution depends upon the window, while thefrequency increment depends only on the measurement time, T.There is no exact frequency resolution for a particular window. How close two sinusoids canbe in frequency, in order for the spectrum still to show two peaks, depends. on the width of thewindow's main lobe, but also on where between the spectral lines the two sine waves arelocated.

5.15 Summary of the OFTIt is not so easy to keep clear all these concepts and their influence on the discrete Fouriertransfonn. Therefore, to make things easier we will finish this chapter with a summary[afterN. Thrane, 1979]. In Figure 5.13 the different steps in the OFT are shown and thefollowing text explains the different steps.We start with a continuous time signal as in Figure 5.13 (A. 1). In Figure 5.13 (8.1) is shownthe Fourier transform of this continuous (infinite) signal, which is of course also continuous,but band-limited so that we fulfil the sampling theorem. For the sake of simplicity we (andThrane) have used a time function which is a Gaussian function, and it has the same shape intime and frequency.

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TimeA.I B.I

x(t)

A.2 I I1'1I' I' , I' I I' I' B.2

A.3 B.3

A.4 B.4

,"I 'I 'I 'I ', '

Frequency

, /

X(f)

,"I '," \ X(f)*SlU)I ,

WU)

A.5 B.5 /\./lV\Y(f)'SIU)'WU)

B.6 'I' I I' I' I' 1 I I'

B.7

Figure 5.13. Summary of the OFT. See text for explanation. [After Thrane, 1979]

I' I'

, ,I \ X(k),,.

The discrete sampling we then carry out is equivalent to multiplying the signal by an idealtrain of pulses with unity value at each sampling instant and zeros between, see Figure 5.13(A.2) and (A.3). In the frequency domain, this operation corresponds with a convolution withthe equivalent Fourier transfonn, which is a train of pulses at multiples of the samplingfrequency, f,. We consequently obtain a repetition of the spectrum at each k . f, .This isactually a proof of the sampling theorem, since if the bandwidth of the original spectrumwould be wider than f,./2, the periodic repetition of the spectra will overlap, see Figure 5. I3(B.2) and (B.3).The next step is measuring only during a finite time, which in the time domain is equivalent tomultiplying by a rectangular window as in Figure 5.13 (AA) and (A.5). In the frequencydomain this operation is equivalent to the convolution with a sinc f u n c t i ~ n as in (BA) and

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Noise and Vibration Book