Highly stable clocks (e.g., cesium clocks, hydrogen maser clocks, or quartz crystal oscillators) have angular frequencies of ticking that tend to wander so much

Highly stable clocks (e.g., cesium clocks, hydrogen maser clocks, or quartz crystal oscillators) have angular frequencies ω of ticking that tend to wander so much over very long timescales that their variances diverge. For example, a cesium clock has random-walk noise on very long timescales (low frequencies)

and correspondingly,

For this reason, clockmakers have introduced a special technique for quantifying the frequency fluctuations of their clocks. Using the phase

they define the quantity

where ω̅  is the mean frequency. Aside from the √2, this Ф_τ(t) is the fractional difference of clock readings for two successive intervals of duration τ . (In practice the measurement of t is made by a clock more accurate than the one being studied; or, if a more accurate clock is not available, by a clock or ensemble of clocks of the same type as is being studied.)

(a) Show that the spectral density of τ (t) is related to that of ω(t) by

(b) The Allan variance of the clock is defined as

Show that

where α is a constant of order unity that depends on the spectral shape of S_ω(f) near f = 1/(2τ). Explain why, aside from the factor α, the right-hand side of Eq. (6.67) is the rms fractional fluctuation of ω at frequency 1/(2τ) in bandwidth 1/(2τ).

(c) Show that, if ω has a white-noise spectrum, then the clock stability is better for long averaging times than for short; if ω has a flicker-noise spectrum, then the clock stability is independent of averaging time; and if ω has a random-walk spectrum, then the clock stability is better for short averaging times than for long. (See Fig. 6.11.)

 

Fig. 6.11

S(f) x 1/f at low f; (6.63a)