NIST Technical Note 1337: Characterization of Clocks and Oscillators

spectrum multiplied by the sqllare of the frequencyresponse function; that is,5(f) [H(f)]' • 5(f) (10.10)outThe filter response must be flat to f and at~Nt@n~ate aliased nois~ compone~ts at f % f =2nf %sf. In digitizing the data. the observedspectra will be the sum of the baseband sp@ctrum(to f N) and all spectra '!oIhich are folded into thebaseband spectrum5(f) • 5 0(f) + 5. 1 (2f, - f) + 5., (2f, + f)Observed.5.> (4f, - f).". + S; (2; U, • t f))(f, + f))where M ;s an appropriate finite limit.(10.11)For a ghen rejettion at an upper freQuency.clearly the cutoff freqllency f for the anti~c~liasing filter Should be as low as possible torelax the rolloff requirements.Recall that annth order lClw"'pass filter has frequency responsefunctionand output spectrumS fS(f) = --Tff~c"f)"2n:-out 1 +(10.12)(10.13)and ~f~er sampling, we haye (applying eQ (10.11))5(t)observed(10.14)If fcis cnosen ta be higher than f then theN ,first tenn (baseband spectrum) is negl igiblyaffected by the filter, which is our hope.It;sthe second term (the sum of the folded in spectra)which causes an error.As an example of the rol10ff requirement.consider the measurement of noise process net) atf ~ 400 Hz in a 1 Hz. bandwidth on a digital spectrumanalyzer. Suppose net) 15 white; that is,k constantO =Suppose further that '!ole(10.15 )w; sh to on ly measure thenoi 51!" from 10 Hz to 1 kHl.; thus f N = 1 kHz.. Letus assume a sampling frequency of f s = 2f Nor2 k.Hz. If we impose a 1 dB error limit inSobserved and have 60 dB of dynaralc range, then wecan tolerate an error limit of 10~6 du~ to aliasingeffects in this measurement. and the second term; neQ (10.14) must be reduced to this lelJel.choose f =1.5kHz and obtainc5( f)observedWe can2 (f + ~f) 2n1· --...-;';---'-'-­f c(10.15)The term in the series which contributes mas':. isat i :: -1. the nearest fold-in.The denominatormust be 10 6 or more to real;!e the allowable error11mit and at n > a this condition is met.Thenext most contributing term is i = +: at whlch theerror is < 10- 1 for n = 8, a negfigible contribution.The error increases as f increases for afixed n because the Dearest fold-in (i = ~1)coming dO>tnin frequency (note fig. 10.2(c)) andpower there is filtered less by the anti-aliasingfilter. Let us look. at the worst case (f ::: 1kHz)to determine a design criteria for this example.At f =1 kHz, we must haye n ~ 10.10~poleThus the requirement in this example is for alo",-pass filter (60 dB/octave rol1off).10.4 Some History of Spectrum Analysis Leading tothe Fast Fourier TransformNewton in his Principia (1687) documented thefirst mathematical treatment of wave motion al­is29TN-42