NIST Technical Note 1337: Characterization of Clocks and Oscillators

AppendixWith reference to figure 1, the frequency sampling window has an equivalentphase sampling window.The intent is to evaluate the variance, S(M), of thesampled phase function in terms of the phase autocorrelation function, R(r).The process here is to correctly account for terms and cross-terms coming fromsquaring and averaging the samples for each M. The B 3(2,M,r,p.) function canthen be obtained from the relation,S(M)S(l) .W'+2for appropriate M, r, and p.. The denominator is just the two-sample variancewith dead time for MT and Mr (in accordance with the definition ofB 3 (2,M,r,p.». The factors common to the numerator and denominator are ignoredin the following.For MI, the variance S(l) is justS(l)4·R(O) - 4'R(r) - 4·R(T) + 2·R(T+r) + 2R(T-r),where use has been made of the definition of the autocorrelation function,R(T)E(¢>(t)· ¢>(t+T)].It is convenient to define a function G(T) asG(T)2·R(T) - R(T+r) - R(T-r).Similarly, S(2) can now be written in the form,S(2) = 8'R(O) - 8·R(r) + 2'G(T) - 4'G(2T) - 2·G(3T).Following this procedure, we can verify that the general S(M) is just17TN-312