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.