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=== Precision timekeeping === {{Main page|Allan variance}} The ubiquitous 1/f noise poses a "noise floor" to precision timekeeping.<ref name="Press-1978" /> The derivation is based on.<ref>{{Cite book |last=Voss |first=R.F. |title=33rd Annual Symposium on Frequency Control |chapter=1/F (Flicker) Noise: A Brief Review |date=May 1979 |chapter-url=https://ieeexplore.ieee.org/document/1537237 |pages=40–46 |doi=10.1109/FREQ.1979.200297|s2cid=37302662 }}</ref> [[File:AllanDeviation.svg|right|thumb|300x300px|A clock is most easily tested by comparing it with a ''far more accurate'' reference clock. During an interval of time ''τ'', as measured by the reference clock, the clock under test advances by ''τy'', where ''y'' is the average (relative) clock frequency over that interval.]] Suppose that we have a timekeeping device (it could be anything from [[Crystal oscillator|quartz oscillators]], [[Atomic clock|atomic clocks]], and [[Hourglass|hourglasses]]<ref>{{Cite journal |last1=Schick |first1=K. L. |last2=Verveen |first2=A. A. |date=October 1974 |title=1/f noise with a low frequency white noise limit |url=https://www.nature.com/articles/251599a0 |journal=Nature |language=en |volume=251 |issue=5476 |pages=599–601 |doi=10.1038/251599a0 |bibcode=1974Natur.251..599S |s2cid=4200003 |issn=1476-4687|url-access=subscription }}</ref>). Let its readout be a real number <math>x(t)</math> that changes with the actual time <math>t</math>. For concreteness, let us consider a quartz oscillator. In a quartz oscillator, <math>x(t)</math> is the number of oscillations, and <math>\dot x(t)</math> is the rate of oscillation. The rate of oscillation has a constant component <math>\dot x_0</math>and a fluctuating component <math>\dot x_f</math>, so <math display="inline">\dot x(t) = \dot x_0 + \dot x_f(t)</math>. By selecting the right units for <math>x</math>, we can have <math>\dot x_0 = 1</math>, meaning that on average, one second of clock-time passes for every second of real-time. The stability of the clock is measured by how many "ticks" it makes over a fixed interval. The more stable the number of ticks, the better the stability of the clock. So, define the average clock frequency over the interval <math>[k\tau, (k+1)\tau]</math> as<math display="block">y_k = \frac{1}{\tau}\int_{k\tau}^{(k+1)\tau}\dot x(t)dt = \frac{x( (k+1 ) \tau) - x(k\tau)}{\tau}</math>Note that <math>y_k</math> is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock{{NoteTag|Though in practice, since there are no ideal clocks, <math>t</math> is actually the ticks of a much more accurate clock.}}. The [[Allan variance]] of the clock frequency is half the mean square of change in average clock frequency:<math display="block">\sigma^2(\tau) = \frac 12 \overline{(y_{k} - y_{k-1})^2} = \frac{1}{K}\sum_{k=1}^K \frac 12 (y_{k} - y_{k-1})^2</math>where <math>K</math> is an integer large enough for the averaging to converge to a definite value. For example, a 2013 atomic clock<ref>{{Cite journal |last1=Hinkley |first1=N. |last2=Sherman |first2=J. A. |last3=Phillips |first3=N. B. |last4=Schioppo |first4=M. |last5=Lemke |first5=N. D. |last6=Beloy |first6=K. |last7=Pizzocaro |first7=M. |last8=Oates |first8=C. W. |last9=Ludlow |first9=A. D. |date=2013-09-13 |title=An Atomic Clock with 10 –18 Instability |url=https://www.science.org/doi/10.1126/science.1240420 |journal=Science |language=en |volume=341 |issue=6151 |pages=1215–1218 |doi=10.1126/science.1240420 |pmid=23970562 |arxiv=1305.5869 |bibcode=2013Sci...341.1215H |s2cid=206549862 |issn=0036-8075}}</ref> achieved <math>\sigma(25000\text{ seconds}) = 1.6 \times 10^{-18}</math>, meaning that if the clock is used to repeatedly measure intervals of 7 hours, the standard deviation of the actually measured time would be around 40 [[Femtosecond|femtoseconds]]. Now we have<math display="block">y_{k} - y_{k-1} = \int_\R g(k\tau - t) \dot x_f(t) dt = (g\ast \dot x_f)(k\tau) </math>where <math>g(t) = \frac{-1_{[0, \tau]}(t) + 1_{[-\tau, 0]}(t)}{\tau}</math> is one packet of a [[Square wave (waveform)|square wave]] with height <math>1/\tau</math> and wavelength <math>2\tau</math>. Let <math>h(t)</math> be a packet of a square wave with height 1 and wavelength 2, then <math>g(t) = h(t/\tau)/\tau</math>, and its Fourier transform satisfies <math>\mathcal F[g](\omega) = \mathcal F[h](\tau\omega)</math>. The Allan variance is then <math>\sigma^2(\tau) = \frac 12 \overline{(y_{k} - y_{k-1})^2} = \frac 12 \overline{(g\ast \dot x_f)(k\tau)^2} </math>, and the discrete averaging can be approximated by a continuous averaging: <math>\frac{1}{K}\sum_{k=1}^K \frac 12 (y_{k} - y_{k-1})^2 \approx \frac{1}{K\tau}\int_0^{K\tau} \frac 12(g\ast \dot x_f)(t)^2 dt</math>, which is the total power of the signal <math>(g\ast \dot x_f)</math>, or the integral of its [[Spectral density|power spectrum]]: [[File:Illustration for Allan variance of 1-f noise.png|thumb|315x315px|<math>\sigma^2(1)</math> is approximately the area under the green curve; when <math>\tau</math> increases, <math>S[g](\omega) </math> shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis; when <math>S[\dot x_f](\omega) \propto \omega^{-\alpha}</math>, the combined effect of both is that <math>\sigma^2(\tau) \propto \tau^{\alpha-1}</math>]] <math display="block">\sigma^2(\tau) \approx \int_0^\infty S[g\ast \dot x_f](\omega) d\omega = \int_0^\infty S[g](\omega) \cdot S[\dot x_f](\omega) d\omega = \int_0^\infty S[h](\tau \omega) \cdot S[\dot x_f](\omega) d\omega</math>In words, the Allan variance is approximately the power of the fluctuation after [[Band-pass filter|bandpass filtering]] at <math>\omega \sim 1/\tau</math> with bandwidth <math>\Delta\omega \sim 1/\tau </math>. For <math>1/f^\alpha</math> fluctuation, we have <math>S[\dot x_f](\omega) = C/\omega^\alpha</math> for some constant <math>C</math>, so <math>\sigma^2(\tau) \approx \tau^{\alpha-1} \sigma^2(1) \propto \tau^{\alpha-1}</math>. In particular, when the fluctuating component <math>\dot x_f</math> is a 1/f noise, then <math>\sigma^2(\tau)</math> is independent of the averaging time <math>\tau</math>, meaning that the clock frequency does not become more stable by simply averaging for longer. This contrasts with a white noise fluctuation, in which case <math>\sigma^2(\tau) \propto \tau^{-1}</math>, meaning that doubling the averaging time would improve the stability of frequency by <math>\sqrt 2</math>.<ref name="Press-1978" /> The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.<ref>{{Citation |last=Vessot |first=Robert F. C. |title=5.4. Frequency and Time Standards††This work was supported in part by contract NSR 09-015-098 from the National Aeronautics and Space Administration. |date=1976-01-01 |url=https://www.sciencedirect.com/science/article/pii/S0076695X08607103 |work=Methods in Experimental Physics |volume=12 |pages=198–227 |editor-last=Meeks |editor-first=M. L. |access-date=2023-07-17 |series=Astrophysics |publisher=Academic Press |doi=10.1016/S0076-695X(08)60710-3 |language=en|url-access=subscription }}</ref>
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