Characterization of Frequency Stability

Consider a signal generator whose instantaneous output voltage V(t) may be written as V(t) = [V 0 + ??(t)] sin [2??v 0 t + s(t)] where V 0 and v 0 are the nominal amplitude and frequency, respectively, of the output. Provided that ??(t) and ??(t) = (d??/(dt) are sufficiently small for all time t, one may define the fractional instantaneous frequency deviation from nominal by the relation y(t) - ??(t)/2??v o A proposed definition for the measure of frequency stability is the spectral density S y (f) of the function y(t) where the spectrum is considered to be one sided on a per hertz basis. An alternative definition for the measure of stability is the infinite time average of the sample variance of two adjacent averages of y(t); that is, if y k = 1/t ??? tk+r = y(t k ) y(t) dt where ?? is the averaging period, t k+1 = t k + T, k = 0, 1, 2 ..., t 0 is arbitrary, and T is the time interval between the beginnings of two successive measurements of average frequency; then the second measure of stability is ?? y 2 (??) ??? (y k+1 - y k ) 2 /2 where denotes infinite time average and where T = ??. In practice, data records are of finite length and the infinite time averages implied in the definitions are normally not available; thus estimates for the two measures must be used. Estimates of S y (f) would be obtained from suitable averages either in the time domain or the frequency domain.