NIST Technical Note 1337: Characterization of Clocks and Oscillators

estimation increasingly attracti~e. Today thechoice between digital or analog methods dependsmore on the objectives of the analysis rather thanon technical 1imitation!ii. However, many aspectsof digital spectrum analysis are not well known bythe casual user io the laboratory while the analoganalysi5 methods and their limitations are understoodto a greater l!xtent.Digital spectrum analysis is realized usingthe di5cre-te Fourier transform (OFT). a IIOdifiedversion of the cont1nuous transform depicted ineq. (10.5) and (10.6). By sallplinq the inputwaveform yet) at discrete intervals of t1~e t •1~t representing the sampled waveform by eq (10.2)and integrating eo (10.5) yields~Y(f) = l:t=~.y(lt)e'J20flT (10.20)Equation (10.20) is a Fourier series expansion.Becayse t(t) ;s specified as being bandlimited,the Fourier transform as calculated by eq (10.20)is as accurate as eq (lO~ 5); however, it cannotextend beyond the Nyquist fr!quency, eq (10.4).In practice ..,e cannot compute the Fouriertransform to an infinite extent, and we are restrictedto some observation time T consisting ofnAt intervals.This produces a spectrum which isnat eont1 nuolJs in f but rather is computed ",ithresolution Af ",hereM=.1..= 1 (10.21)nAt 'fWith this change. we g.t the discrlte finitetransforllN-1 .Y{mAf) = l: y (t)e·JZroo.l.fnt (10.22)","0 1The OFTcomputes a sampled Fourier series,end eq (10.22) assumes that the function y(t)repeats ibelt with period T. Y(mAf) is calledthe "1 fne spectrum." A cmnparison of the OFT withthe continuous Fourier transform 15 shown later inpart 10.7.The fast Fourier tr.nsform (FFT) is ,n algorithm""hi ch efficientiy computes the 1i ne spectrumby reducing the number of adds and 1I1l1itipl iesinvolved in eq (10.22). If we choose T/At toequal a rational power of 2, ttlen a symmetricmAtrix can be derived through which YR,(t) passesand quickly yieldS YClMf). An M-point tronsformationby the direct method requires a processingtime proportional to N2 whereas the FFT requires atime proportional to N 10g2 N.The approximateratio of FFT to direct comput1ng time is given bywhere H = 2 Y ,Mlog. N log, HHZ =--N- = if (10.23)For e.ample, if H = 2'0, the FFTrequires less than 1/100 of the norlllal processingtime.Wemust calculate both the magnitude andphase of a frequency in the line spectrum, L e. ,the real and imaginary part at the given frequency,N poi nts in the time damaina11 ow ~/2 comp Texquantities in the frequency domain.The power spectrWl of y(t) is computed bysquaring the real and imaginary components, addingthe two together and dividing by the total time T.We haveS (mAf) = R[Y(mAf)]2 .. I[Y(mAfll' (10,24)yfThis qua.ntity is the sampled power spectrumand again assumes periodicity in proces5 yet) withtot.l period T.1010. 5 LeakagoSampledinvolvesdigitaltransformingContlnucus precess )I(t)through a data windowdescribed byspectrum analysis alwaysa finite bloc~of data.15 "looked at" for T timewhich can functiona11y bey' (t) = w(t)'y{t) (10,25)The time­where wet) is thlf time domain window.discrete counterpart to eq (10.25) is31TN-44