NIST Technical Note 1337: Characterization of Clocks and Oscillators

Some noi se wi 11 cause thE!! instantaneous fre"quency to "jitter" aro ....nd u o • with probabl1ity ofbeing higher or lower than ""0' We thus usuallyfind a I/lJedestal " associated with "0 as shown infigurE!' 8.4..,--·0FIGURE a.4ihe orocess of break.ing 110wna signal intoall of its various components of frequency iscalled Fourier exoansion (see 'Sec. X). Inother wordS,the. addition of all the frequencycomponent.s, called Fourier frequency components,produr:es the original signal. ihe value of aFouri er frequency i.sthe di fference between thefrequency component and the fundamental fr@quency.The power spectrum can be normalized to unity suchthat the tota1 area under the curve equa 1sane.The power spectrum normalized in this way is thepower spectral density,The power spectrum. often call ed the RFspectrum, of Vet) is ",ery useful in many appl i­cations~Unfortunately, if one is given the RFspectrum, it ;5 impossible to determine whetherthe power at different Fourier frequencies is aresult of amplitude fluctuations lIe(t)1I or phasefluctlJations '1¢J(t).The RF spectrum can be separatedinto two independent spectra, one being the'Spectral densi..!l of ~ often called the AMpower spectra) density and the other being thespectral densi~ of ~.For the purposes here~the phase-f1uctuationcomponents are the ones of interest. The spectraldensity of phase- fluctuations is denoted by S~(f)wherE!! IIf" is Fourier frequency. for the fre"quently encountered case where the AM power spec~tral d@nsity is negligibly small and the t.otalmOdulation of the phase fluctuations is small(mean-square value is much less than one rad2),the RFspectrum has approximately the same shapeas the phase spectral density_However, a maindifference in the representation is that the RFspectrum includes the fundamental $ignal (carrier),and the phase spectral density does not.Another major di fference is that the RFspectrumis a power spectral density and is measured inunits of watts/hertz.The phase spectral densityinvolves no II powerl' measurement of the electricalsignal. The units are radians·/hertz. It iste'"Pting to think. of S~(f) as a IIpowe~' spectraldens i ty because in practice it is measured bypassing Vet) through a phase detector and measuringthe detector's output power spectrum.The measure~ment technique makes use of the relation that forsmall deviations (6~ « 1 radian),5 (f) = (-,Vrm~s_(f_) )' (8.3)$ V,Where Vrms(f) is the root-mean"square noise.voltageper ./HZ at a Four1 er frequency II fll. and V ; s thessensitivity (volts per radian) at the phase quadratureoutput of a phase detector ~hlch15 comparingthe two oscillators. In the next section, we willlook at a scheme for directly measuring 5$(f).One question 'Wemight ask, is. IIMow do fre"quency changes relate to phase fluctuations?lIAfter all it's the frequency stability of anoscillator that ;s a major consideration in manyapplications.The frequency is equal to a rate ofchange. in the phase of a 'Sine wave.11'11'5 tells usthat fluctuations in an oscillator's output frequencyare related to phase fluctuations since wemust change the rate of ".(t)" to accomplish ashift in "V(t)ll, the frequency at time t. A rateof change of total u4'J~e have thenr(t)U is denoted by "*T(t)".lm>(t) ~ ~T(t) (8.4)The dot denotes the mathematical operation ofdifferentiation on the function 4'J with respect torits independent variable t.;II From eq (8.4)• As an analogy, the same operation relates theposition of an object with its velocity.20TN-33