14correspondingtothefrequencyfTransientorrandomstationarydatacannotgenerallybeassumedtobeperiodic withinthe observed interval [o,T].However,thepreviousFourierseriesrepresentations can be extended by considering that what occurs within the discrete intervalTtoapproach infinity.This leads totheFourier integral,(8)x(f) =x(t)e-2mdt-8f<8where X(f) will exist if,(9)Ix(t) / dt < 80Sincex(t) ismeasured overthe finite interval [0,T] then its Fouriertransformexists and isreferredtoas thefiniteFouriertransformgivenby:(10)[x(t)e-mdtX.()=X(fT)=Such finiteFouriertransforms will always existforfinite lengthrecords of stationarydataEquations8and9demonstratethatatall discretefrequencies f,=(k/m),thefiniteFouriertransformyields(11)X(f.T)=TAk±1,±2,±3,..Henceiff isrestrictedtotake ononlythesediscretefrequencies,thenthefiniteFouriertransformwo&ddwill actuallyproduceaFourierseriesofperiodT.Whenx(t)is sampledatpointstapart,therecord lengthbecomesT=NAt,whereNisthesamplesize.This inducesaNyquistcutofffrequencyf=1/2At.Also,thecomputationstreatthedataasifitwereperiodic data withperiod T.The continuous function x(t) is replaced bythe discrete series X,=x(nt)forn=1,2,3...N,andtheFouriertransformX(f)is replacedbythediscreteFouriertransformX,=X(k△f)fork=1,2,3,..N.ThediscreteFouriertransform is symmetric aboutk=N/2,and theFouriertransformpair is given by.Nknj2nk=1,23,.,NX=Atx,expNn-1(12)j2m.knx,=AfxexpK=1,2,3, ...NN1-1
15TheFourieramplitude spectrum obtained fromEquation 12 is the same as thatgiven in Equation7.TwoFourieramplitude spectratypicalofroughan smoothhighwayprofiles are illustrated inFigure 4.These spectra indicate thatthe significant differences in theFourieramplitudesbetweenthetwo surfaces occur inthewavelengthsgreaterthan approximately8feet.Thishassignificant consequences on the interpretation ofparameters using the amplitude spectrum as ameasureofroughness.This is discussedinthe summaryof this chapter.Correlation functionsIfx(t)andy(t)representtworandomstationaryprocesses,the covariancefunctionbetweenx(t)andy(t)foranytimedelayrisgivenby:C()-E[(x()-u)(y(t+t)-u)](13)lim1"[(x) -(y+)-)dt-R,(-,where:R.f0-x0yd(14)Forthegeneral case where x(t)andy(t)represent differentdata,R,(r) in Equation14 is calledthecross-correlation functionbetweenx(t)andy(t).Forthe special casewherex(t)=y(t),(15)Ca(0)-Ra()-where:lim 1x()xt+) t(16)R()"T-TJ。is calledtheauto-correlationfunctionofx(t)and C,(t)theauto-covariancefunction.Thevalueof the auto-correlation function at=O is the mean square value of the data,which is the sumof thevarianceand the square of the mean value of thedata,where,Ra(0)-o+(17)Ro(o)"u
16EWooe082119=ea20oaaSIOe电■-LO咖+十十T+SO'00+000S10FOLO-S0'000e
17Theautocorrelationfunctionalso convergestoaconstant valueequal tothesquareof themeanasrincreasesto infinity.Figure5illustratestheautocorrelationfunctionsofeight128ft.sectionsmakingup1024ft.oftheroughhighwaysection.Traditionallythese functions were used as an indicatorof the degree of accuracy to which the current time history can be used to predicteventsbeyondtheperiodofobservationT.Forexamplethecorrelationfunctionofasinewaveisacosinewavewithanamplitude equal to the mean square value of the original sine wave and of the same singular frequency and wavelength.Thesinewave correlation function remains constant over all timedelays T,suggesting that one can predictfuture values of the datapreciselybased on pastvalues.Fora sinusoidal function this is indeedthecase.Anothertypeof informationwhich couldbe interpreted fromautocorrelationfunctions is thedominantfrequency contentoftheoriginaldata.However,this infomation is more clearly interpreted fromautospectral densityfunctionsorFourieramplitudespectra.Inthe caseofthe sinusoid,thisfrequency isreadilyobtainedas thereciprocalofthedecorrelation time (ordistance)which isgivenbythevalueofthelagT.whenthatfunctionfirstreacheszero,and isequalto/4As thedatafunction resembles less and less a pure sinusoid and becomesmore ofa wide band random signal (as inthecaseofroadprofiles)theinterpretationofthedecorrelationdistancebecomeslessclearandisrelatedonlytothebandwidthB,of dominantfrequencies oftheautospectral densityfunction and thedecorrelation distance T=1(2B)(Bendat and Plersol1980)SpectraldensityfunctionsThe spectral density functionbetween two timehistories x(t) and y(t)oftwo random stationary processes may bedefined as theFouriertransformof thecorrelation functionas follows:S, (f)=R,(t)e-12mf dr(18)Forthe general case where x(t) and y(t) are different SxyY(f)l is called the cross-spectral density function or more simply thecross spectrum.Forthe special caseofx(t)andy(t)beingequal,Sy (f)=R,(t)e =12mf dr(19)
17 The autocorrelation function also converges to a constant value equal to the square of the mean as r increases to infinity. Figure 5 illustrates the autocorrelation functions of eight 128 ft. sections making up 1024 ft. of the rough highway section. Traditionally these functions were used as an indicator of the degree of accuracy to which the current time history can be used to predict events beyond the period of observation T. For example the correlation function of a sine wave is a cosine wave with an amplitude equal to the mean square value of the original sine wave and of the same singular frequency and wavelength. The sine wave correlation function remains constant over all time delays T, suggesting that one can predict future values of the data precisely based on past values. For a sinusoidal function this is indeed the case. Another type of information which could be interpreted from autocorrelation functions is the dominant frequency content of the original data. However, this information is more clearly interpreted from autospectral density functions or Fourier amplitude spectra. In the case of the sinusoid, this frequency is readily obtained as the reciprocal of the decorrelation time (or dist ance), which is given by the value of the lag T0, when that function first reaches zero, and is equal to /4. As the data function resembles less and less a pure sinusoid and becomes more of a wide band random signal (as in the case of road profiles) the interpretation of the decorrelation distance becomes less clear and is related only to the bandwidth B, of dominant frequencies of the autospectral density function and the decorrelation distance T0= 1 (2B) (Bendat and Plersol 1980) Spectral density functions The spectral density function between two time histories x(t) and y(t) of two random stationary processes may be defined as the Fourier transform of the correlation function as follows: = - -j2mfr Sxy (f) Rxy (t) e dr (18) For the general case where x(t) and y(t) are different SxyY(f)l is called the cross-spectral density function or more simply the cross spectrum. For the special case of x(t) and y(t) being equal, = - -j2mfr Sxy (f) Rxy (t) e dr (19)