NIST Technical Note 1337: Characterization of Clocks and Oscillators

192 SAMUEL R STEINLIST OF SYMBOLSdf Number of degrees of freedomfFourier (cycle) frequencyG•.uw) Open-loop transfer function of a phase-locked loopf\sl Transfer function of the loop filter of a phase-locked loopH(J - fol Transfer function of a filterK. Sensitivity of a linear phase detector in volu per radianpNumber of counts accumulated by a time-interval counterQ Quality factor of a resonance. equal to the ratio of the stored energy to theenergy lost in one cycle5 Independent variable of the Laplace transform527Estimate of the two-sample or Allan varianceS,IJ) One-sided power spectral density of the fractional-frequency deviationsS.l!l One-sided power spectral density of the phase deviationsS~SU) Two-sided power spectral density of the phase deviationsS~~,U) Two-sided power spectral density of the phase deviations of the carrierS~pU) Two-sided power spectral density of the phase deviations of the pedestalSrU) Two-sided power spectral density of the instantaneous voltageV o Peak voltage of a signal generatorv.. Output voltage of a phase detectorVI Voltage applied to the tuning element of the voltage-controlled oscillator of aphase-locked loopV(rl Instantaneous voltage of a signal generator or other deviceX(I) Time deviation required by a signal generator operating at nominal frequency Voto accumulate phase equal to 4>(f).Xt!l Fourier transform of x(f).11 I) Instantaneous fractional-frequency offset from nominal)\ Mean fractional-frequency offset over the kth interval7 Exponent off for a power-law spectral densityal'Q Frequency uncertainty due to the quantization of measurements~ Damping constant of a phase-locked loopP. Exponent of r for a power-law Allan variance'it) Instantaneous frequencyVaNominal frequency of a signal generatorV(r ,: ( 2) Mean frequency over the interval f\ ~ f ~ f 1V H Heterodyne frequency or dill'ercnce frequency between two oscillatorsPI/' Probability density of the chi-squared distributiona;(N, T. r) Sample variance of N fractional-frequency deviations. each averaged over aperiod r spaced at intervals Ta;(r) The two-sample or Allan variance of the fractional-frequency deviationsmod a;frJ Modified Allan variancerAveraging timePeriod of the time base of a counter.plr) Instantaneous phase deviationlChi-squared distributionwFourier (angular) frequencyNatural frequency of a phase-locked loopTN-62