NIST Technical Note 1337: Characterization of Clocks and Oscillators

We can use the convolution theorem to simplyi11ustrate the existence of atiases. Thh theoremstates that multiplication in the time domaincorresponds to convolution in the frequency domain,and the time domain end frequency domain representationsare Fourier transform pairs) TheFourier tran"form of yet) In figure lO.l(a} isdanotod by Y(f); thus:Y(fl convolvad with Af ;s danotad by Y(f)'a(f) and is shown in figura 10.2(c). Wo sao thatthe transform ....(f) is repeated with origins atf =y. Conversely, high frequency data with informationaround f = f will fold into thQ data aroundthe origin bil!tween -f and "'f ' In the computations sof pow~r spectra, we encounter error5 as shown infiguro (10.3).andTheY(f) : j-y(t).j2rrftdt (10.5)=y(t) : -l f Y(f)oj2rrftdf (10.6)Zn-=function Y(f) is depicted tn f1gure lO. Z(a).The Fourier transform of ~(t) ;s shown in figure10.2(b) and is gillen by .6.(f) wh@re applying thediscrete transform yields:(10.7)recall1ng that---'". '.,=Mt): E 6(t -nT). (10.8)n=fromeq (10.2).L ~..I...-.-+" f+.-.J\.--,----+-r'"'. '. +--1C4) {I.}FIGURE 10.2Is'-~FIGURE 10.3Al lased power spectra dtle to folding.Spactra, (b) Aliasod Spoctr._(a) TrueTwo pionil!ers in infonnation theory, HaroldNyquist and Claude Shannon, developed designcriteria for discrete-continuous processing sys~tems. Given a specified accuracy, 'We can conveytime-domain process Yet) through a finite bandwidthwhose up~er limit f is the highest ~ignificantspectral component of y(t). ForNdiscretecontinuousprocess Yk(t) Iideally the input signalspectrum should not extend beyond f ' ors(10.9)where f is givE'!:n by eq 00.4). Equatio" (10.9)sis refered to as the lIShannon limit. /IIn practice, there is never a case in whichthere is absolutely no signal or noise compon@ntabove f,.. Filtars are used befor!' the ADC inorder to suppress components above f which foldNinto tho lower bandwidth of intarost. rhis 00­called anti-aliasing fi1ter usually must be quiteSOphisticated in order to have low ripple in thepassband, constan't phase delay in the passband?and steep rolloff characteristics. In examiningthe rollotf requirements of the anti-aliasingfilter, we Can apply a fundamental filter propertythat the output spectrum is equal to the input28TN-41