NIST Technical Note 1337: Characterization of Clocks and Oscillators

Table 1 organizes the measurement methods in the above manner giving performancelimits, advantages and disadvantages for each. Figure 1 following the table provides limitationsfor phase noise measurements as a function of Fourier frequency. The reader is again remindedthat the table is highly simplified giving nominal levels that can be achieved. Exceptions can befound to almost every entry.A.6 RELATIONSHIP OF THE MODIFIED ALLAN VARIANCE TO THE ALLAN VARIANCEIn sorting through this set of papers and other published literature on the Allan Variance,we were stimulated to further consider the relationship between the modified Allan variance andthe Allan variance. The ideas which were developed in this process have not been published, sowe include them here.Paper D.6, "Characterization of Frequency Stability: Analysis of theModified Allan Variance and Properties of Its Estimate," by Lesage and Ayi adds new insightsand augments the Allan and Barnes paper (D.5), "A Modified Allan Variance with IncreasedOscillator Characterization Ability." In this section we extend the ideas presented in these twopapers and provide further clarification of the relationship between the two variances, both ofwhich are sometimes referred to as two-sample variances.Figure 2 shows the ratio [mod 0ir)/oy(r)]2 as a function of n, the number of time orphase samples averaged together to calculate mod Oy( r). This ratio is shown for power-law noisespectra (indexed by the value of ex) running from f to f2. These corrected results have a somewhatdifferent shape for ex = -1 than those presented in either paper D.5 or paper D.6. Furthermore,this figure also shows the dependence on bandwidth for the case where ex = L For allother values of ex shown, there is no dependence on bandwidth. Table 2 gives explicit values forthe ratio as a function of n for low n as well as the asymptotic limit for large n. For ex = 1, theasymptotic limit of the ratio is considerably simplified from that given in papers D.5 and D.6.With these results it is possible to easily convert between mod air) and oir) for any of thecommon power-law spectra.The information in figure 2 and table 2 was obtained directly from the basic definitions ofaye r) and mod 0/ r) using numerical techniques. The results of the numerical calculations werechecked against those obtained analytically by Allan and Barnes (D.5) and Lesage and Ayi (D.6).For ex = 2, 0 and -2 the results agree exactly. For ex = 1 and -1 the analytical expressions arereally obtained as approximations. The numerical calculations are obviously more reliable.Details of our calculations can be found in a NIST report [1]. A useful integral expression (notcommonly found in the literature) for modo~( f) is2 2 !lJA sylf) sin 6 ( 1t 1;/)mooa,('t) - -- ~----d/.n41t2't~ 0 f2 sin 2 (1t't o f)Figure 2 and table 2 show that, for the fractional frequency fluctuations, mod Oy( r)always yields a lower value than Oy(.,.). In the presence of frequency modulation (FM) noisewith ex ~ 0, the improvement is very significant for large n. This condition has been examined indetail by Bernier [2]. For white FM noise, a = 0, the optimum estimator for time interval .,. isto use the value of the times or phases separated by r to determine frequency. This is analogousto the algorithm for calculating Oy(r) which yields the one-sigma uncertainty in the estimateTN-9