Fundamentals of Time and Frequency

2 Japan Standard Time and its Related Research 2-1 Fundamentals of Time and Frequency

KAJITA Masatoshi, KOYAMA Yasuhiro, and HOSOKAWA Mizuhiko

Time (frequency) is one of the fundamental physical quantities, that can be measured most accurately among them. In this paper, the principles for the measurement of the time & frequency and the estimations of its uncertainties are summarized. Keywords Time and frequency measurement, International system units, Frequency measurement, Allan variance

1 History and signifi cance of addition to these, the standard for acoustic fre- frequency quency has been provided since a tuning fork was invented in the 18th century. Following Frequency is how many times a periodic this, improvements of frequency precision phenomenon is repeated per unit time. It has made two major leaps. The fi rst was the devel- been known from antiquity that using an ap- opment of crystal oscillators originating with propriate natural phenomenon enables us to the discovery of the piezoelectric effect of maintain this quantity extremely accurately. quartz crystal by Paul-Jacques Curie and Pierre What human beings fi rst recognized as such Curie in 1880. The second was the develop- would be the periodicity and accuracy of revo- ment of atomic frequency standards starting lutions of celestial bodies such as the sun, the with the utilization of inversion transition of moon and stars. Utilizing this, the clock time ammonia molecules in 1949. The accuracy of was determined and the calendar was orga- atomic frequency standards has reached 16 dig- nized. However, these are extremely large- its until today since it was fi rst developed with scaled movements that are beyond human con- an exceptional accuracy of 10 digits about 60 trol. While one could enumerate pitch control years ago. Furthermore, it can be said that with stringed instruments and wind instruments atomic frequency standards in the optical re- from such ancient times as one of what humans gion, whose accuracy has been dramatically created by themselves and could control, it is improved from the outset of the 21st century, is considered that full-fl edged art started from the the third leap. For the history of frequency discovery of the isochronism of the pendulum standards, refer to [1] listed in the bibliography by Galileo from the 16th century that the fre- below. quency of a pendulum depends only on its In this manner, fairly high-accuracy fre- length regardless of bob weights. In the 17th quency signals are easily available today due to century, Huygens invented pendulum clocks highly accurate oscillators. Now, how can we and furthermore mechanical clocks utilizing measure and evaluate its precision? Phases and the accurate frequency of machinery vibration amplitudes are the basic elements of frequency caused by a hairspring and a balance wheel. In and noise is further added to real signals. As

KAJITA Masatoshi et al. 3 will be discussed below, since amplitude fl uc- Soon after the advent of atomic clocks, the tuations not usually problematic with good fre- accuracy of clocks was improved by about 3 quency signals, it is necessary to measure and digits compared with the crystal clocks avail- evaluate accuracy, stability, noise etc., for the able up until that time. Atomic clocks take the phase as a function of time with appropriate frequency of electromagnetic waves absorbed measures. This paper will give an explanation and emitted by atoms (neutral, ions) as a stan- of the measuring methods for frequency sig- dard. Originally called “atomic frequency stan- nals obtained from high-accuracy frequency dards,” atomic clocks do not function as clocks oscillators etc. and evaluating methods for until they are used together with crystal clocks these measures. 2 provides an overview the de- etc. and feedback errors. The differences be- gree of the accuracy of frequency oscillators tween atomic clocks and conventional clocks and defi nes quantity as the basis for frequency are detailed below: evaluation. 3 discusses the principle and basis (i) Electrons ranging around a nucleus can for frequency measurement and 4 explains its only acquire energy intermittently. The energy evaluating methods. 5 offers commentary on conditions are unique since they can only be classifi cations of noise and spectral expansion determined by Coulomb force between elec- and 6 presents a conclusion. trons and a nucleus. An atom can absorb and emit electromagnetic waves with frequency (v) 2 Accuracy of time/frequency corresponding to difference in each level ener-

standards gy (Ei) (ν = Ei − Ek / h. Here, h stands for Planck constant). 2.1 Accuracy of frequency standards (ii) An atomic clock, unlike conventional (Why are atomic clocks accurate?) clocks, takes a single atom (in conditions of Along with technological developments for gas) as a standard and accordingly its condition oscillators discussed in 1, the accuracy of time is unique (interactions between neighboring at- displayed on a clock has been improved. The oms and molecules etc., are not unique in a frequency of the pendulum of a pendulum solid state). clock developed in the 17th century can be ex- In short, atomic clocks are the adaptation of pressed by: microcosmic phenomena to macro world. As a result, even commercially available atomic （1） clocks ensure an accuracy of more than 12 digits. This means that it will take several tens of Here, l stands for the length of the pendulum thousands of years to generate an error of one and g for gravitational acceleration. Still, this second. The advent of atomic clocks has en- period cannot be always stable as (a) the length abled us to experimentally demonstrate the of the pendulum varies at a rate of about 10−6 relativistic effect that time passes more slowly with a change in temperature by 1K, and (b) by force of movements and gravity. gravitational acceleration varies at a rate of However, also atomic clocks have limited about 10−7 with a locational change on the accuracy. What was shown in (i) above is not earth, especially with a change in altitude by completely correct since there is a width of Δν 1m. It holds true also with crystal clocks, whose = 1/2πτ in accordance with a limited time of τ oscillating frequencies depend on temperature because of uncertainty principle for energy and etc., that clocks gain or lose as temperature time (a fundamental principle of quantum me- changes. However, crystal clocks are more sta- chanics). τ is sometimes limited by the interac- ble than pendulum clocks as the rate of vibration time between atoms and electromagnetic tion frequency to frequency width (called Q waves and sometimes determined by the life- value) for the former is higher than that for the span of quantum state with a ﬁ xed phase (lim- latter by 5–6 digits. ited by spontaneous emission, and collision

4 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 etc.). Also, electron orbits can be distorted by um and reaches 18 digits only from cumulative external electric fi elds and magnetic fi elds to uncertainty factors [2]. cause transition frequency shifts (one caused by electric fi elds is called Stark shift and one 2.2 Defi nition of frequency caused by magnetic fi elds Zeeman shift). What As the accuracy of clocks improves, it be- was shown in (ii) is also not completely accu- comes necessary to provide a defi nition for de- rate; even gaseous atoms collide against one termining certain length of time. Time is one of another or the wall of a container. Consequent- base quantities as well as length, mass, electric ly, the transition frequency shift at that moment current, thermodynamic temperature, amount appears as collision shift. There are also fre- of substance and luminous intensity that are de- quency shifts caused by relativistic effects from fi ned in the International System of Units (SI), the movements and gravity of atoms that freely and its basic unit is the second (s) [3]. For ex- fl y around (second-order Doppler shift and pressing time units longer than a second though, gravitational shift). the kilosecond (ks: 103 sec.) etc., could be used Once laser cooling could decrease the tem- but conventional units such as minute (min: perature of atoms as far as several μK, they be- 60sec.), hour (hour: 3600sec.) etc. are used in came sluggish and could be made to stand still most cases. Time units shorter than a second, at one place, which enabled us to take longer millisecond (ms: 10−3 sec.) and microsecond measuring time τ. In addition, with further im- ( μs: 10−6sec.), etc., are used as defi ned in SI. provements of the accuracy of clocks, the sec- Originally, one second was defi ned as ond-order Doppler shift decreased below 18 “1/86400 of the average rotation period of the digits, cesium fountain atomic clocks have earth”. The short-term variation at that time been able to measure with an accuracy of 16 was approximately 10−8. digits. Consequently, following the new defi ni- Previous clocks took microwave frequency tion: “one second shall be 1/331556925.9747 1‒50GHz as a standard that required a mea- of the revolution period of the earth” by the In- surement period of more than 10 days in order ternational Committee of Weights and Mea- to gain an accuracy of 16 digits. With the fre- sures in 1956, the Measurement Act of Japan quency shift and uncertainty on the same level, was also revised [4]. The accuracy of observa- taking high frequencies as a standard is more tion at this time was approximately 10−9. advantageous for shortening measuring time, The world’s fi rst atomic clock was devel- and many atomic quantum transitions in the oped with reference to the inversion transition optical region have narrow line width and are frequency of the ammonia molecule in 1949. accordingly suitable for highly accurate fre- Subsequently in 1955, the fi rst Cs atomic clock quency standards. Furthermore, because of im- was developed at the National Physical Labo- provements of laser technology, the possibility ratory (NPL) [5]. Furthermore, after a collabora- of determining time and frequency at a higher tive research by NPL and the U.S. National accuracy has been arising. However, there was Observatory (USNO), Cs hyperfi ne structure a major hurdle to measure frequency in the op- transition frequency in zero magnetic fi eld was tical region without impairing the accuracy of measured at 9 192 631 770±20 Hz by the sec- frequency standards in the microwave range. ond on the mean solar day [6]. Thereafter, Cs Frequency measurement in the optical region atomic clocks were developed at several other had not been performed until the development research institutions until 1960, and thus the of femtosecond pulse laser with stable cyclic utility of atomic clocks as a time standard has frequencies enabled us to measure optical fre- become to be recognized. In 1967, the plenary quencies in around 2000. At present, the accu- session of the General Conference on Weights racy of optical transition frequency of Al+ ion and Measures adopted the defi nition: “one sec- is, for instance, beyond the defi nition by cesi- ond shall be the time equivalent to 9192631770

KAJITA Masatoshi et al. 5 times as long as the radiation period corre- that evaluates: sponding to the transition between two hyperfi ne structures of the ground state of cesium-113 （2） atom” and accordingly the Measurement Act of Japan was revised in 1972 [7]. as the average frequency for τ seconds when However, as was discussed in 2.1, transi- opening the gate with clock signals for τ sec- tion frequencies can shift by force of electric onds, wherein input signals of N period pass fi elds, magnetic fi elds and atomic movements, through. etc. Therefore, the defi ned frequency of the ce- (2) Period Measurement Method sium atom is “frequency with both the electric There is also a method for evaluating fre- and magnetic fi elds are zero and on the condi- quency by measuring period as its reciprocal. tion of the atom standing still on the earth’s ge- When opening the gate only for the time of nd oid”. What are known as primary frequency period of input signals and measuring n times standards are atomic clocks made to be able to the clock pulse of τu interval, evaluate all the uncertainty factors for themselves so that the defi ned frequency as such can （3） be presented without compared with other clocks. Since the cesium atom cannot be actu- expresses the average period of nτu seconds. ally measured in the above-mentioned ideal (3) Time Interval Measurement Method (TIC) condition, primary frequency standards present The time interval measurement is per- the defi ned values of time and frequency by formed by assigning the opening and closing monitoring factors causing frequency shifts operations to two input signals and then mea- (the magnitude of electric and magnetic fi elds suring the time during that span as the number etc.) and making corrections with the estima- of clock pulses. Suppose that the frequencies of tion of frequency shifts included in measured the two signals are almost the same and their frequencies [8]‒[10]. signal waveforms are sin (2πνt + φ1) and sin

(2πνt + φ2) respectively. Then, time difference

3 Basis for measurement Td for zero crossing of those two signals can be measured as: 3.1 Frequency measurement principles （4） In this Chapter, we will discuss the measurement methods for frequencies or periods of (4) Measurement Errors alternating current signals. Since frequency is a The most major factor for measurement er- reciprocal of period and vice versa, either can rors is an error of ±1 count (±1 error) on the be determined by measuring the other. number of pulses that pass the gate time. Count- 3.1.1 Measurement by electric ing accuracy is the reciprocal of counted value. counters [11][12] As a result, the period measurement is more (1) Frequency Measurement Method advantageous for measuring low frequencies First, AC input signals are attenuated and and so is the frequency direct measurement for ampliﬁ ed as far as appropriate magnitude to be high frequencies. Therefore, recent counters modiﬁ ed into pulse forms. Meanwhile, output adopt the period for frequencies lower than frequencies of 1, 5, 10MHz etc. that are consti- those whose error ranges by the frequency di- tuted by a high stability oscillator are divided rect measurement and the period measurement by powers of 10 and selected by the dividing cross and frequency direct measurement for ratio selecting switch to be made into clock sig- higher frequencies. Such frequency counters nals of 1, 10, 100ns etc. The frequency direct are called reciprocal counters. count method of input signals is the method Various ways to minimize ±1 error by mea-

6 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 If signal noise is smaller than trigger voltage and rise time is short, trigger error is small. (b) Time Base Errors Errors caused by ﬂ uctuations and drift of the frequency to be a standard for clock signals. While it is often the case that clock signals go by crystal oscillators, time base errors can be constrained if using atomic clocks as the base via an external input ter- minal. (c) Trigger Level Timing Errors Fig.1 The fundamental concept of the frequen- These are measurement errors arising at cy, periodicity, and time interval measurements time of measuring time intervals, which oc- At the periodicity measurement, the input sig- cur on the occasion that the trigger response nals 1 and 2 are given by a common signal time differs when opening and closing the resorce. gate. For performance example, a model an- nounced by Tektronix, Inc. in September, 2010 suring a span shorter than pulse interval have exerts a time resolution of 50ps and a frequen- been contrived. Major examples for that are the cy resolution of 12 digits per second [14]. time expansion system that expands short time 3.1.2 Time difference measurement intervals immeasurable by an internal clock to method with mixers (Dual Mixer measure by the same clock, the time vernier Time Difference: DMTD) [15] principle that synchronizes an oscillator whose The Dual Mixer Time Difference (DMTD) frequency is slightly different from that of the is a device that measures zero crossing time in- reference clock with input signals to calculate tervals for two signals with the almost same the relationship between the oscillator and frequency of v more accurately than usual TIC. clock signals, the multi-phase clock system Equation（4） shows that the lower ν becomes, that uses multiple clocks with a slight phase the smaller ±1 error becomes. This holds true shift to improve resolution, and the time-volt- for trigger errors, time base errors and trigger age conversion system that converts short time level timing errors. Thus, convert beat signals intervals less than pulse intervals into voltage for the both input signals and bias signal sin to measure time from that voltage values. (2πν0t) into the forms of sin [2π (ν − ν0) t +φ 1]

Among these, the time expansion system and and sin [2π (ν − ν0) t +φ 2] to input into TIC. the time vernier principle have the drawback Here, a phase shifter makes an adjustment to that they need a long pause for measurement, hold φ1 −φ 2 constant. The principle for that is and the multi-phase system has another that it shown in Fig. 2. Time difference interval Tdmtd requires a large circuit size. Therefore the time- measured by this equipment is: voltage conversion system is advantageous to obtain high speed and high resolution. （5） Sources of error other than ±1 error are as follows [13]. Since the error range for Tdmtd is as same as the

(a) Trigger Errors case of measuring Td directly by TIC, the error

When input voltage signals have noise, it is accuracy of Td to be determined by DMTD im-

converted into time error in the form of: proves by ν/(ν − ν0). Trigger error = [Signal noise/Trigger voltage] × (Trigger pulse rise time).

KAJITA Masatoshi et al. 7 However, frequency components observed when phase noise is actually evaluated by SA are electric power within the ﬁ nite bandwidth of Δf, which can be expressed by:

（7）

When linewidth Δf0 for carrier frequency components is adequately narrower than Δf, they are expressed by:

（8） Fig.2 The block diagram of the Dual Mixer Time Difference (DMTD) And since values observed as:

3.1.3 Noise measurement for frequency [16] [17] （9） Although the frequency of an ideal frequency oscillator must be expressed by a single sine wave, an actual oscillator generates variations in oscillating output. Among these, the vary according to a bandwidth of measurement, short-term amplitude variation is called AM the bandwidth must be speciﬁ ed. Thus, since noise, which is not usually brought question this is inconvenient for comparing measure- very much as it itself does not connect to fre- ments with different bandwidths, it is common quency variations. On the other hand, the short- to evaluate phase noise by using a power den- term phase variation is called phase noise, sity of E( f0 + fm)/Δf per 1 Hz bandwidth. In real- which can directly connect to frequency ﬂ uc- ity, it is much more common to display what tuations and accordingly an important factor to expresses: determine characteristics of frequency sources. The phase noise contained in an oscillator appears as the jitter in the time-domain mea- （10） surement with oscilloscopes. Meanwhile, in the frequency-domain measurement with a spectrum analyzer (SA), the phase noise is observed to spread toward the bottom in the based on the unit of dBc/Hz together with off- neighborhood of the carrier frequency of oscil- set frequencies. lator. This can be considered that it is spreading The drawbacks at time of the measurement oscillating output energy also toward adjacent of phase noise with SA are that phase noise frequencies other than carrier output frequency. lower than SA’s phase noise cannot be mea- It is a fundamental rule for evaluating phase sured and that phase noise and amplitude noise noise that the intensity ratio of the offset fre- are indistinguishable. The Phase Lock Loop quency component with a gap of fm from carrier (PLL) method is superior to measurements frequency f0 to the carrier is evaluated by dBc with SA in that since it constrains carrier sig- as follows: nals by phase-locking of the reference oscillator for a signal source to be measured by a （6） phase difference of π/2, it can measure pure phase noise with amplitude noise removed in

8 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 high dynamic ranges. One can ﬁ nd a correla- Thus, the ratio of Standard Deviation to aver- tion between measured results of two signal age value becomes: sources to reduce phase noise of measuring in- strument itself. Still, there is a drawback for （15） PLL method that it can measure ﬁ nite offset frequencies. which proves to vary inversely with the square root of N (the number of samples). 3.2 Uncertainty of frequency The least square method is often used to es- measurement timate a most probable value from the results 3.2.1 Average and variation for of repeated measurements. Consider the case measurement samples that there is parameter set λ to be determined

If one wants to obtain high-accuracy mea- and most probable value Xj to jth measurement sured values, they should repeat measurements is given by function Xj(λ) of λ. Then, evaluate as often as possible and apply averaging opera- parameter λ that makes the following least: tions to obtained data to raise accuracy. In this case, not only average values but also varia- （16） tions in measured data are important factors. The ideas of average and variation will be 3.2.2 Uncertainty of time/frequency demonstrated with a simple example below. In general, there are factors that cause gaps Let probabilities in the binomial distribution of measured values from true values: ones that taking conditions A and B be p and 1−p respec- cause statistical dispersion in spit of zero aver- tively. Assuming that probability for the num- age and ones that cause systematic gaps from ber of samples taking A condition among N true values for certain reasons. Uncertainty items of samples to be n is P(n), the average when estimating true values from measured value of n is expressed as: values can be estimated from uncertainty uA re- sulted from statistical factors (Type A) and sys-

（11） tematic uncertainty uB(Type B) as:

Here, when using: （17）

（12） In the above, uA denotes the ratio of ﬂ uctuation in measured values to average values. Suppose the following can be obtained: that ﬂ uctuation haven’t been estimated by Standard Deviation; then the ratio becomes small if the number of measured data is in- （13） creased as shown in Equation（15） . In order to

make uB small, it is necessary to identify factors to be causes for gaps from true values and Next, Standard Deviation σ that is often used as add corrections per measurement condition. the parameter for variations in samples is ex- The measured value distribution for the case pressed as: that uA and uB are large and small will be shown in Fig. 3.

An important factor for estimation of uA when really comparing time/frequency be- （14） tween two atomic clocks is the dispersion of the difference in time/frequency between the two clocks and it does not matter whether or not their average values agree.

KAJITA Masatoshi et al. 9 Meanwhile, what determines uB is whether direction over time is also partly discussed. the gaps from true values caused by electric Output signal V(t) when discussing fre- ﬁ elds, magnetic ﬁ elds, relativistic effects etc., quency stability is expressed as follows: can be made small or precisely estimated to correct. （18）

In case of uB, how much the average of measured values of two clocks each agrees Here, V0 and v0 are amplitudes of output serves as a standard for estimating gaps from signals and averages of frequencies, and ε and true values. In order to minimize uB, it is neces- φ denote amplitude fl uctuations and phase sary to perform frequency measurements under fl uctuations. Since the actually obtained ampli- various conditions and determine parameters to tude can be mostly regarded as a fairly small be shift factors by means of the least square fl uctuation for estimating frequency changes, method. Specifi cally, it can be pointed out that V0 can be usually deemed as invariable. Sup- one measures variations in measured frequen- pose instantaneous phase Φ(t) as: cies by changing magnetic fi eld sizes in order to identify the infl uence of Zeeman effects and （19） estimates frequency in zero magnetic fi eld, and so on. then instantaneous frequency v(t) is evaluated by: 4 Basis for evaluation of frequency stability [11] （20）

In this Chapter, we will discuss what pa- Here, assume: rameters one should use to estimate the frequency changes necessary to estimate statisti- （21） cal uncertainty uA for time/frequency, which is useful when comparing and discussing charac- Then, defi ne dimensionless parameter y(t) teristics of clocks measured under different standing for frequency changes as: conditions. While frequency changes to be discussed below mainly cover frequency fl uctua- （22） tions that randomly change over time, the frequency drift that frequency changes in a single Also, defi ne time change x(t) as:

（23）

As shown in 2, y(t) cannot actually measure a value for that moment but one averaged by frequency counter’s gate time τ:

（24）

In order to estimate the size of frequency changes, it would be natural to ﬁ rst evaluate the dispersion:

（25） Fig.3 The distribution of measurements with large (or small) statistical uncertainty uA and systematic uncertainty uB of frequency changes:

10 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 represents the frequency components of frequency ﬂ uctuations. Using the power spectral density leads to: （26）

（32）

Here, deﬁ ne the autocorrelation function for phase and frequency: Furthermore, consider that temporal differenti- ation in the time domain corresponds to multi- plication by 2πf in the frequency domain, which （27） leads to:

（33） 2 Rx (τ) is based on the unit of s and Ry (τ) is non- dimensional. Then, one can readily understand: then Equation（32） can be expressed by:

（28）（34）

Autocorrelation function is a parameter used to In reality, the power spectral density of fre- indicate how much effect remains after gaps quency changes can be expressed by the poly- from average values of phase and frequency at nomial in f: a certain time change only by time τ. By using this parameter, Equation（26） can be modiﬁ ed （35） into: It is often the case from the past measured data （29） that discussions go on in the range of a = −2, −1, 0, 1, 2. Here, each term denotes as follows: As will be discussed below, speed for the time a = −2 Random walk FM noise change of phase should be considered with fre- −1 Flicker FM noise quency components to be more understand- 0 White FM noise able. For that, consider power spectral density 1 Flicker PM noise

Sx,y ( f ) that is the application of Fourier trans- 2 White PM noise form to the autocorrelation function of phase White PM and FM noise and fl icker noise will change: be discussed in 5 below. Now, when viewing Equation（31） , it turns out that this integration diverges with a = −2, −1 in f → 0 range. It does not actually reach an （30） infi nite value as the residual of frequency changes is evaluated from fi nite measured data following Equation（22） . Still, since acquired 2 Sx ( f ) is based on the unit of s /Hz and Sy ( f ) on values depend on how to draw samples, this in- the unit of /Hz. The case of τ = 0 leads to: tegration is not preferable as a stability scale. This divergence of integral values represents （31） that frequency is endlessly changed while sam- pling is repeated with random walk and fl icker which indicates that the power spectral density FM noises over an extended time period. When

KAJITA Masatoshi et al. 11 evaluating the residual of frequency changes, the average frequency to be a standard cannot be obtained until the entire measurement is completed, which is also problematic. （38） Then, samples for frequency changes of a ﬁ nite time period are taken by N sample variance that can be deﬁ ned below:

This result can be easily understood from a −(a+1) that f max ×Δf is proportional to τ as H( f ) peaks at around f = 3/(2πτ) and integral width （36） max Δf is approximately 1/(πτ). To express this further in different words, it corresponds to dividing the magnitude of ﬂ uctuations with frequency

component fmax by the sample size proportional Here, T denotes repeated frequencies for mea- to τ. The above-mentioned infi nite integral will surement of τ seconds and tk+1 = tk + T (k = 0, 1, be infi nite with a = 1, 2. Then, consider the case 2, ,,,,) is effected. It means that each measure- of using a low-pass fi lter that shields frequency ment involves a blank time of T−τ. Of these, N components higher than maximum frequency sample variance under conditions of T = τ and component fh. In this case, Equation（37） must 2 N = 2 is especially called Allan variance σy : be replaced with the fi nite integral of:

（39）

（37） It results in: a = 1

（40）

H( f ) will be shown in Fig. 4. It implies that a = 2 there is no divergence in f → 0 range for the Allan variance even with a = −2, −1. This is （41） also the measured result of frequency changes for fi nite measuring time τ, which corresponds Thus, since the case of either a = 1 (fl icker PM to the fact that there is no problem with its sam- noise) or 2 (white PM noise) approximately pling for over an infi nite time period. shows τ−2-dependence, the two kinds of noise Allan variance’s τ-dependence will be dis- will be indistinguishable. This result is caused cussed in the following way: by the fact that since H( f ) reaches a local maximum in all frequencies expressed by f = 3/ (2πτ) × (2m+1) (m is an integer) (ref. Fig. 4) and all the frequency components for a = 2 get involved evenly with the Allan variance, the in- terpretation that the Allan variance stands for the frequency components for 3/(2πτ) of frequency fl uctuations turns out to be invalid. What was contrived to differentiate be-

12 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 2 3 tween the cases of a = 1 (ﬂ icker PM noise) and a = 2. Modσy (τ) becomes proportional to τ0/τ a = 2 (white PM noise) was the Modiﬁ ed Allan to show τ-dependence that is largely different 2 variance Modσy (τ). Let τ be the integral mul- from the case of a = 1. This can be interpreted

tiple of least-time unit τ0 that is τ = nτ0, then in a way that while all frequency components 2 Modσy (τ) is defi ned: for f = 3/(2πτ) × (2m+1) make an even contribu- 2 tion in the Allan variance, Modσy (τ) becomes proportional to τ − (a+1) also in case of a = 2 as in case of a<1 as only frequency components for f = 3/(2πτ) contribute in the Modifi ed Allan （42） variance. In Fig. 5, τ-dependence of the Allan variance and the Modifi ed Allan variance is shown. Other than frequency fl uctuations of the Calculate Equation（42） to lead to [18]: kinds that were considered in the above, there is frequency drift. Frequency drift has properties in proportion to τ2 both in the Allan variance and the Modifi ed Allan variance. Mea- sured drift is usually considered separately （43） from random noise, and yet it corresponds to f −3 FM noise as long as it is not removed. Both the Allan variance and the Modifi ed Allan variance that are evaluated respectively from Equa- Looking at the above, it turns out that M ( f ) tions（37） and（43） will diverge for this noise. attenuates as (πfτ) grows while it is approximate Frequency fl uctuations also do not neces- a to 1 in (πfτ)<1 range. Figure 4 shows H10–mod ( f ) sarily take forms that can be expressed by f .

(Hn–mod ( f ) of n = 10) to compare with H( f ). An example of this is in the case that the phase Since contribution to the integral of Equation is modulated with the sinusoidal wave: （43） is limited within (πfτ)<1 range with a<1, 2 2 Modσy (τ) is about the same as σy (τ). How- （44） ever, a large difference arises when the part 2 of (πfτ)≫ 1 largely contributes and Modσy (τ)/ In this case, from: 2 σy (τ) is inversely proportional to n in case of

π τ π τ Fig.4 H( f ) and H10-mod( f ) as a function of Fig.5 The dependence of Allan variance and πfτ modiﬁ ed-Allan variance on the measurement time τ

KAJITA Masatoshi et al. 13 low. （45） FM noise occurs due to noise within the oscillating loop and PM noise due to that outside of the loop. This corresponds to the fact that （46） phase changes are observed as phase changes as they are since there is no feedback outside of the following is derived: the oscillating loop, but they are observed as frequency changes to offset phase changes that can occur within the loop. The higher speed （47） phase changes within the loop, the smaller oscillating frequency changes become. Thus, the

This becomes with τ ≪ 1/(2 fm): dependence of PM noise on f differs from that of FM noise. Suppose that the maximum width of phase noise Δθ is 2π and the maximum fre- （48） quency change is spectral width Δν of the oscillator; then the correlation between frequency and is consequently indistinguishable from fre- ﬂ uctuations (y = δν/ν0) and phase changes quency drift in this range. However, the Allan (x = Δθ/ν0) is estimated as: variance peaks at around τ = 1/2 fm and decreases when τ exceeds it. This holds with the Modi- ﬁ ed Allan variance as well. （51） In order to really measure the stability of time and frequency, it is necessary to compare more than two frequencies and measure frequency differences between them. Variance σAB Consequently, the following is derived: of frequency difference between frequencies A and B can be evaluated from variances σ and A （52） σB of frequencies A and B each by:

（49） Meanwhile, phase noise (PM noise) outside of the loop becomes:

In case of σA ≫ σB, measured σAB can be safely regards as σA. However, when the stabilities of each frequency are comparable (when （53） measuring the highest level of stability in par- ticular), it is necessary to estimate stabilities of each frequency by using three frequencies in Now, as was shown in 4, phase noise is of the form of: two kinds: one called white noise (without dependence on f ) and the other ﬂ icker noise (in （50） proportion to f −1). White noise is thermal noise

when v0 is in the microwave region and quantum noise (ﬂ uctuations in photon number) 5 Spectrum for oscillating when it is in the optical region. This difference

frequency changes [11] owing to the regions of v0 is caused by different impacts that the discontinuity of light energy The dependence of noise that determines makes on ﬂ uctuations according to whether uppermost limits of frequency stability on fre- one-photon energy (hv0) is larger or smaller quency source properties will be discussed be- than kBT0 (T0 is the absolute temperature of the

14 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010 oscillator). Flicker noise occurs through phase lan variance due to ﬂ icker FM noise become modulation with the nonlinearity of circuit ele- small when Q-values of the oscillator rise. ments and such. Express the power spectral When Q-values become high, the Allan vari- density of phase modulation within and outside ance noise can be made small even in the re- the loop as: gion where white FM noise is dominant. If taking measuring time τ even longer, the Allan variance adversely becomes large. This holds either due to random walk or frequency drifts. The dependence of the Allan variance on τ Then, the following is derived: does not necessarily take as simple a form as described above. For instance, when the phase is sinusoidally modulated, as shown in Equa- tion（47） , the Allan variance becomes an in- creasing function of τ in a short τ-range and a （54） decreasing function in a long τ-range.

6 Conclusion

Furthermore, consider random walk noise Time/frequency are physical quantities that (∝f −2) and apparent noise due to linear fre- are measurable with the highest degree of ac- quency drift (∝f −3); then they can be expressed curacy and play an important role for contem- by: porary science and technology. Their measurement and evaluation has a close connection with the passage of time and averaging time, （55） and thus has aspects that cannot be evaluated only by average and Standard Deviation as the The dependence of the Allan variance on case of other qualities. The stability scale of the measuring time τ was shown in Fig. 5. The Al- Allan variance is a typical instance; it forms an lan variance decreases in short-range τ as tak- important basis for measurement and evalua- ing longer τ. It is often the case that white FM tion of frequencies to understand how it differs noise is dominant for Cs atomic clocks etc. In from normal Standard Deviation and relates to such a case, the Allan variance can be kept the power spectral density and types of noise. down by raising Q-values. For hydrogen ma- For understanding of this, the present paper of- sers etc., mostly PM noise is more dominant fered a general consideration of measurement than white FM noise as (v0/Q) is low. If taking techniques for time/frequency and provided an τ longer than a certain time, ﬂ icker FM noise outline of crucial ideas for estimating its uncer- (∝f −1) becomes conspicuous. Thus, the Allan tainty. It will be the author’s pleasure if it helps variance becomes constant to τ and accordingly relevant parties work on measurement and sets a limit on frequency stability. As was evaluation of frequency. shown in Equation（54） , limit values of the Al-

KAJITA Masatoshi et al. 15 References 1 M. Hosokawa, “Physics Education at University (Phys. Soc. of Japan),” Vol. 14, No. 3, p. 125, 2008. (in Japanese) 2 C.W. Chou, D. B. Hume, J. C. Koelmeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett., Vol. 104, pp. 070802(1–4), 2010. 3 http://www-lab.ee.uec.ac.jp/text/misc/si_units.html 4 Japanese law 61st , 3-3 (April 15th 1958). (in Japanese) 5 L. Essen and J. V. L. Parry, Nature, Vol. 176, pp. 280–282, 1955. 6 W. Markovitz, R. G. Hall, L. Essen, and J. V. L. Parry, Phys. Rev. Lett., Vol. 1, pp. 105 –107, 1958. 7 Japanese law 27th , 3-3 (May 9th, 1972). (in Japanese) 8 M. Kumagai et al., “Caesium Atomic Fountain Primary Frequency Standard NICT-CsF1,” Special issue of this NICT Journal, 2-3, 2010. 9 K.Matsubara et al., “40Ca+ Ion Optical Frequency Standard,” Special issue of this NICT Journal, 3-2, 2010. 10 A. Yamaguchi et al., “A Strontium Optical Lattice Clock,” Special issue of this NICT Journal, 3-3, 2010. 11 K. Yoshimura, Y. Koga, and N. Oura, “Frequency and Time – Fundamental of Atomic clock & System of Atomic Time,” The Inst. of Electro., Info. and Comm. Eng., 1989. (in Japanese) 12 http://www.yokogawa.co.jp/tm/TI/keimame/torjikan/kouzou.htm 13 http://www.yokogawa.co.jp/tm/TI/keimame/torjikan/kakudo.htm 14 http://ednjapan.cancom-j.com/news/2010/9/7224 15 F. Nakagawa, M. Imae, Y. Hanado, and M. Aida, IEEE Trans. Inst. Meas. Vol. 54, pp. 829–832, 2005. 16 http://em-ﬁ eld.jp/reduce-pn/l.html 17 http://gate.ruru.ne.jp/rfdn/TechNote/SpaCN.asp 18 J. Vanier and C. Audoin, “The Quantum Physics of Atomic Frequency Standard,” p. 251, Adam Hilger, Bristol and Phyladelphia.

(Accepted Oct. 28, 2010)

KAJITA Masatoshi, Ph.D. KOYAMA Yasuhiro, Ph.D. Senior Researcher, Space-Time Group Leader, Space-Time Standards Standards Group, New Generation Group, New Generation Network Network Research Center Research Center Atomic Molecular Physics, Frequency Space Geodesy, Radio Science Standard

HOSOKAWA Mizuhiko, Ph.D. Executive Director, New Generation Network Research Center Atomic Frequency Standards, Space-Time Measurements

16 Journal of the National Institute of Information and Communications Technology Vol.57 Nos.3/4 2010