NIST Technical Note 1337: Characterization of Clocks and Oscillators

MRNES et al.: CHARACTERIZATION OF FREQGENCY ST.,BILITYFrom the analog voltage one may use analog spectrumanalyzers to determine S~ (f), the frequency stability. Byconverting to digital data, other analyses are possibleon a computer.F. Common Hazards1) Errors Caused by Signal-Processing Equipment: Theintent of most frequency stability measurements is toevaluate the source and not the measuring equipment.Thus, one must know the performance of the measuringsystem. Of obvious importance are such aspects of themeasuring equipment as noise level, dynamic range,resolution (dead time), and frequency range.It has been pointed out that the noise bandwidth fl isvery essential for the mathematical convergence of certainexpressions. Insofar as one wants to measure the signalsource, one must know that the measuring system is notlimiting the frequency response. At the very least, onemust recognize that the frequency limit of the measuringsystem may be a very important, implicit parameter foreither .,.:(t) or S.(j). Indeed, one must account for anydeviations of the measuring system form ideality such asa "nonflat" frequency response of the spectrum analyzeritself.Almost any electronic circuit that processes a signalwill, to some extent, convert amplitude fluctuations at theinput terminals into phase fluctuations at the output.Thus, AM noise at the input will cause a time-varyingphase (or FM noise) at the output. This can impose importantconstraints on limiters and automatic gain control(AGe) circuits when good frequency stability is needed.Similarly, this imposes constraints on equipment used forfrequency stability measurements.2) Analog Spectrum Analyzers (Frequency Domain) :Typical analog spectrum analyzers are very similar indesign to radio receivers of the superheterodyne type, andthus certain design features are quite similar. For example,image rejection (related to predetection bandwidth)is very important. Similarly, the actual shape of theanalyzer's frequency window is important since this affectsspectral resolution. As with receivers, dynamicrange can be critical for the analysis of weak signals inthe presence of substantial power in relatively narrowbandwidths (e.g., 60 Hz).The slewing rate of the analyzer must be consistentwith the analyzer's frequency window and the post-detectionbandwidth. If one has a frequency window of 1 Hz,one cannot reliably estimate the intensity of a brightline unless the slewing rate is much slower than 1 Hz/s.Additional post~detection filtering will further reduce themaximum usable slewing rate.S) Spectral Density Estimation from Time DomainData: It is beyond the scope of this paper to present acomprehensive list of hazards for spectral density estimation;one should consult the literature [2]-[5J. There115are a few points, however, which are worthy of specialnotice: a) data aliasing (similar to predetection bandwidthproblems); b) spectral resolution; and c) COnfidenceof the estimate.4) Variances of Frequency Fluci:ual.ions "':(T): It is notuncommon to have discrete frequency modulation of asource such as that associated with the power supplyfrequencies. The existence of discrete frequencies in S.(j)can cause .,.:(.,.) to be a very rapidly changing functionof T. An interesting situation results when T is an exactmultiple of the period of the modulation frequency (e.g.,one makes T = 1 s and there exists 6O-Hz frequencymodulation on the signal). In this situation, .,.:(T = 1 s)can be very optimistic relative to values with slightlydifferent values of T.One also must be concerned with the convergenceproperties of "':(T) since not all noise processes will havefinite limits to the estimates of "':(T) (see Appendix I).One must be as critically aware of any "dead time" in themeasurement process as of the system bandwidth.5) SigruJ Source and Loading: In measuring frequencystability one should specify the exact location in thecircuit from which the signal is obtained and the natureof the load used. It is obvious that the transfer characteristicsof the device being specified will depend on the loadand that the measured frequency stability might beaffected. If the load itself is not constant during themeasurements, one expects large effects on frequencystability.8) Confidence of the Estim.a.u: As with any measurementin science, one wants to know the confidence to assign tonumerical results. Thus, when one measures S.U) or ,,:(T),it is important to know the accuracies of these estimates.a) The Allan Variance: It is apparent that a singlesample variance u:(4, T, T) does not have good confidence,but, by averaging many independent samples, one canimprove the accuracy of the estimate greatly. There is akey point in this statement, "independent samples." Forthis argument to be true, it is important that one samplevariance be independent of the next. Since cr:(2, 1", T) isrelated to the first difference of the frequency (11),it is sufficient that the noise perturbing Yet) have "independentincrements," i.e., that y(t) be a random walk.In other words, it is sufficient that S.(j) r-J ,-2 for lowfrequencies. One can show that for noise processes thatare more divergent at low frequencies than r J , it isdifficult (or impossible) to gain good confidence onestimates of o?(T). For noise processes that are lessdivergent than r J , no problem exists.It is worth noting that if we were interested in.,.:(N = CD, 1", T), then the limit noise would becomeS.(j) r-J f instead of ,-2 as it is for 0-:(2, 1", .,.). Since mostreal signal generators possess low-frequency divergentnoises, (cr:(2, T, T» is more useful than cr:(N = co, T, T).Although the sample variances ,,:(2, T, 1") will not benormally distributed, the variance of the average of 11'1.TN-156