IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

vol. 62, no. 5,

May

2015

791

The Dynamic Allan Variance IV: Characterization of Atomic Clock Anomalies Lorenzo Galleani, Senior Member, IEEE, and Patrizia Tavella, Senior Member, IEEE Abstract—The number of applications where precise clocks play a key role is steadily increasing, satellite navigation being the main example. Precise clock anomalies are hence critical events, and their characterization is a fundamental problem. When an anomaly occurs, the clock stability changes with time, and this variation can be characterized with the dynamic Allan variance (DAVAR). We obtain the DAVAR for a series of common clock anomalies, namely, a sinusoidal term, a phase jump, a frequency jump, and a sudden change in the clock noise variance. These anomalies are particularly common in space clocks. Our analytic results clarify how the clock stability changes during these anomalies.

I. Introduction

P

recise clocks are employed in a growing number of applications. In global navigation satellite systems (GNSSs), such as GPS and Galileo, the user position is obtained from the time of flight of the signals traveling from the satellite to the receiver, and the positioning error is ultimately bounded by the stability of the precise clocks on board the satellites and in the ground stations. Unfortunately, stability can change with time due, for instance, to temperature, radiations, aging, and sudden breakdowns. This variation with time of the clock stability can be represented with the dynamic Allan variance (DAVAR) [1]–[3]. The DAVAR is a function of time and of the observation interval. When the clock follows the specifications, the DAVAR is stationary with time and changes with the observation interval only. When an anomaly occurs, the DAVAR changes also with time, its shape depending on the type of anomaly. In this article we analytically derive the DAVAR for some of the most common anomalies of precise clocks, namely, a sinusoidal term on the clock frequency deviation, a phase jump, a frequency jump, and a change of variance of the white frequency noise (WFN) component of the clock noise. These anomalies play a crucial role in GNSSs, and their characterization is hence fundamental. A deterministic sinusoidal term whose period corresponds to the satellite orbital period is in fact a typical residual of the orbit estimation algorithm. A phase jump is a sudden variation in the time deviation of the clock, and it correManuscript received October 10, 2014; accepted February 21, 2015. L. Galleani is with the Politecnico di Torino, 10129 Torino, Italy (email: [email protected]). P. Tavella is with the Istituto Nazionale di Ricerca Metrologica, 10135 Torino, Italy. DOI http://dx.doi.org/10.1109/TUFFC.2014.006733

sponds to a delta function on the frequency deviation. It is typically due to an anomaly or to a scheduled synchronization. A frequency jump is an anomaly whose physical causes have not been fully understood yet. It is a critical anomaly for a GNSS, because it affects the clock predictability. Finally, a change in the variance of the WFN component of the clock noise is the least common of the considered anomalies, but experimental data show that it is potentially correlated to major clock failures. As an example, in Fig. 1 we show the behavior of a clock onboard the GPS satellite SVN30 for a few months of 2011. After the clock noise variance increases suddenly at approximately t = 130 days, the clock is turned off and four days later a new one is used, as confirmed by the corresponding GPS notice advisory to Navstar users (NANU). The reasons for characterizing these anomalies are manyfold. First, from a fundamental point of view, we want to understand how stability changes during these anomalies. Similarly to what is done for other representations, such as the Fourier and Laplace transforms, we want to build a table of the DAVAR for the most interesting behaviors of the clock noise. Second, the analytic DAVAR is not affected by the fluctuations naturally arising when we estimate it from measured data. This knowledge makes detecting and identifying anomalies easier for the operator that monitors the clock health status by inspecting the DAVAR. Finally, knowledge of the analytic DAVAR can be helpful in designing anomaly detectors that track changes in the clock stability. We note that the results presented here belong to a research line on the characterization of precise clock anomalies [4]–[6]. The article is organized as follows. In Section II we review the Allan variance and the DAVAR. Then, in Section III we obtain the analytic DAVAR for the considered anomalies. II. The Allan Variance and Its Dynamic Version We define the time deviation of a clock as [7]

x(t) = h(t) − h 0(t), (1)

where h(t) is the clock reading and h0(t) is a time reference. The corresponding normalized frequency deviation is given by

y(t) =

d x(t) . (2) dt

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σ y2(τ) =



1 1 2 Tlim T →∞

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T /2

∫−T /2(y (t + τ) − y (t))2 d t. (7)

The Allan variance can be computed also in presence of long periods of missing data [11]. It is convenient to rewrite the Allan variance as σ y2(τ) =



1 1 lim 2 T →∞ T

T /2

∫−T /2∆ 2(t, τ) d t, (8)

where ∆(t, τ) = y (t + τ) − y (t), (9)



and Δ2(t, τ) = Δ(t, τ)2. This quantity can be written through the convolution

Fig. 1. Change of variance of a GPS clock. The plot shows the frequency deviation of a clock onboard a GPS satellite. After a sudden increase in variance at approximately t = 130 days, the clock is turned off and a new one is used.

∆(t, τ) =



where hτ(t) is defined as h τ(t) =

In practice, we typically measure the discrete-time time deviation x[n] = x(nTs), where Ts is the sampling time. The corresponding discrete-time average frequency deviation y[n] is obtained as

y[n ] =

x[n ] − x[n − 1] . (3) Ts

By inverting (2), we can rewrite y[n] as

1 y[n ] = Ts

nTs

∫(n −1)T y(t ′) d t ′. (4) s

This result shows that y[n] is obtained by averaging the corresponding continuous-time version y(t). The experimental observations show that, aside from a typical deterministic drift, the time and frequency deviation have the structure of noise, and therefore they are referred to as clock noise [7]. The stability of the clock depends on the size of the fluctuations of the clock noise. The standard measure of clock stability is the Allan variance [8]–[10],

σ y2(τ) =

1 (y (t + τ) − y (t))2 , (5) 2

where τ is the observation interval, the average frequency deviation y (t) is given by

y (t) =

1 τ

and the averaging operator

t

∫t −τy(t ′) d t ′, (6)

+∞

∫−∞ h τ(t − t ′)y(t ′) d t ′, (10)

1 [P (t + τ/2) − Pτ(t − τ/2)], (11) τ τ

and Pτ(t) is the rectangular window Pτ(t) =



{1,0,

−τ/2 ≤ t ≤ τ/2, (12) t > τ/2.

The advantage of (10) is that the convolution often simplifies analytic calculations. If Δ2(t, τ) is an ergodic random process, we can rewrite the Allan variance (8) as σ y2(τ) =



1 E[∆ 2(t, τ)], (13) 2

where E is the expected value obtained as an ensemble average over t. The clock stability can change with time due to several physical factors. This time-varying stability can be represented with the DAVAR, defined as σ y2(t, τ) =



1 2(T w − 2τ)

t +T w/2 − τ

∫t −T /2+ τ E[∆ 2(t ′, τ)] d t ′, (14) w

where Tw is the length of the analysis window and 0 < τ < Tw/2. When y(t) is deterministic, the DAVAR becomes σ y2(t, τ) =



1 2(T w − 2τ)

t +T w/2 − τ

∫t −T /2+ τ ∆ 2(t ′, τ) d t ′. (15) w

We note that (14) is a simplified version of the original definition of the DAVAR given in [1],

σ y2(t, τ) =

is defined as where

1 2(T w − 2τ)

t +T w /2 − τ

∫t −T

w /2 + τ

E[∆ 2(t, t ′, τ)] d t ′, (16)

galleani and tavella: the dynamic allan variance iv: characterization of atomic clock anomalies

∆(t, t ′, τ) = yTw(t, t ′ + τ) − yTw(t, t ′), (17)



σ y2(τ) = A 2



with yTw(t, t ′) =



1 τ

t′

∫t′−τ

yTw(t, t ′′) d t ′′, (18)

yTw(t, t ′) = y(t ′)PTw(t − t ′). (19)

The quantity yTw(t, t ′) is a windowed version of the frequency deviation. We note that

yTw(t, t ′) = y (t ′), (20)



∆(t, t ′, τ) = ∆(t ′, τ), (21)

for t − (Tw/2 − τ) ≤ t′ ≤ t + Tw/2 − τ, which represents the integration interval in (16). Replacing this result in (16) gives (14). III. Dynamic Stability Analysis of Clock Anomalies We analytically derive the Allan variance and DAVAR for a sinusoidal term, a phase jump, a frequency jump, and a sudden change in the clock noise variance. All the obtained results are exact because no approximation method is used. We illustrate our results with a series of examples, where we use dimensionless quantities for simplicity.

σ y(τ) = A



We consider the sinusoid y(t) = A cos(2π f 0t + ϕ), (22)

where A is the amplitude, f0 is the frequency, and φ is the phase. To obtain the Allan variance, we first note that, from (6),

y (t) = A

sin π f 0τ cos[π f 0(2t − τ) + ϕ] . (23) π f 0τ

Substituting t with t + τ gives

To obtain the DAVAR, we replace the deterministic term Δ(t, τ) in (15), and after a few calculations, we obtain sin 4 π f 0τ  sin 2π f 0(T w − 2τ)  cos(4π f 0t + 2ϕ)  . 1 − 2π f 0(T w − 2τ)  π 2f 02τ 2  (28)

y2(t, τ) = A 2 σ

We note that the first term is the Allan variance (26) of a sinusoid, hence sin 2π f 0(T w − 2τ)   σ y2(t, τ) = σ y2(τ)  1 − cos(4π f 0t + 2ϕ)  . 2π f 0(T w − 2τ)   (29) The corresponding dynamic Allan deviation (DADEV) is sin 4π f 0(τ − T w/2) cos(4π f 0t + 2ϕ). 4π f 0(τ − T w/2) (30) σ y(t, τ) = σ y(τ) 1 −

To analyze this result, we rewrite it as σ y(t, τ) = σ y(τ) 1 − α(τ) cos(4π f 0t + 2ϕ), (31)



y (t + τ) = A

sin π f 0τ cos[π f 0(2t + τ) + ϕ] . (24) π f 0τ

sin 2 π f 0τ ∆(t, τ) = −2A sin(2π f 0t + ϕ). (25) π f 0τ

Finally, from (8), we obtain the well-known Allan variance [12]

α(τ) =



sin 4π f 0(τ − T w/2) . (32) 4π f 0(τ − T w/2)

We first note that the time dependence in the DADEV (31) is due solely to the sinusoidal term cos(4πf0t + 2φ), which has twice the oscillation frequency and phase of the original sinusoid (22). We also point out that, in general, we are interested in the observation interval values in the range 0 < τ ≤ Tw/4, because for larger τ values the estimates of the Allan deviation and of the DADEV are affected by large statistical fluctuations due to the limited number of samples. We now consider different behaviors of the DAVAR for a sinusoidal term depending on the oscillation frequency f0. 1) Large f0: If

Replacing in (9) returns

sin 2 π f 0τ . (27) π f 0τ

where

A. Sinusoidal Term



sin 4 π f 0τ . (26) π 2f 02τ 2

The corresponding Allan deviation is

and

793



f0 ≥

10 , (33) πT w

then

σ y(t, τ) ≈ σ y(τ) (34)

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

Fig. 2. DADEV of a high-frequency sinusoidal term. When a sinusoid has a large oscillation frequency f0 according to (33), its DADEV is stationary with time in the interval 0 < τ < Tw/4, where it approximately equals the Allan deviation of the sinusoid.

for 0 < τ ≤ Tw/4. This result can be obtained by considering that 1 1 ≤ , π f 0T w 4π f 0(τ − T w/2) (35) α(τ) cos(4π f 0t + 2ϕ) ≤ α(τ) ≤

where 1/(πf0Tw) is obtained by setting τ = Tw/4 in |4πf0(τ − Tw/2)|. The approximation in (34) holds when 1/(πf0Tw) ≪ 1, for instance if 1 ≤ 0.1, (36) πf 0T w



which is (33). In Fig. 2 we show the DADEV with window length Tw = 100, computed for the sinusoid with parameters f0 = 50/(πTw), A = 1, φ = 0. The picture clearly shows that the DADEV is stationary with time, and changes with τ according to the Allan deviation of a sinusoid given in (27). The observation interval spans the values 0 < τ < Tw/4.

then

σ y(t, τ) ≈

2σ y(τ) sin(2π f 0t + ϕ) . (38)

Therefore, for small f0 values, the DADEV has large oscillations in time with period 1/(2f0) between the approximate values

0 < σ y(t, τ) ≤

2σ y(τ). (39)

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2015

Fig. 3. DADEV of a low-frequency sinusoidal term. When a sinusoid has small oscillation frequency f0 according to (37), its DADEV oscillates in time like the magnitude of a sine function with period 1/(2f0), and increases with time with a linear behavior.

To obtain this result, we note that the first zero of α(τ) is located at τ0 = Tw/2 − 1/(4f0). If τ0 is far from Tw/2, for instance Tw/2 − τ0 ≥ 10Tw/2, which is (37), then 1 − ε ≤ α(τ) ≤ 1 for 0 < τ ≤ Tw/4, where 0 < ε ≪ 1 is a small positive constant. Therefore, for 0 < τ ≤ Tw/4, we can take α(τ) ≈ 1 and approximate the DADEV as

σ y(t, τ) ≈ σ y(τ) 1 − cos(4π f 0t + 2ϕ).

By using the trigonometric identity

1 − cos(4π f 0t + 2ϕ) = 2 sin 2(2π f 0t + ϕ), (40)

we obtain (38). In Fig. 3 we show the DADEV with window length Tw = 100, computed for the sinusoid with parameters f0 = 1/(100Tw), A = 1, φ = 0. We note that the behavior of the DADEV corresponds to the approximation (38). The oscillation period with time is in fact 1/(2f0) = 5000, and the variation with τ is the Allan deviation of a sinusoid given in (27), which behaves linearly for the considered τ range because f0 is small.

2) Small f0: If 1 f0 ≤ , (37) 20T w

vol. 62, no. 5,

3) General Oscillatory Behavior: In general, when

1 10 ≤ f0 ≤ , (41) πT w 20T w

the DADEV mixes the variation with τ of the Allan deviation of a sinusoidal term and the oscillation with time. Specifically, for a given τ value the DADEV oscillates in time with period 1/(2f0) between the values

σ y(τ) 1 − α(τ) ≤ σ y(t, τ) ≤ σ y(τ) 1 + α(τ) . (42)

For a given t value, the DADEV changes with τ as

galleani and tavella: the dynamic allan variance iv: characterization of atomic clock anomalies

Fig. 4. Typical DADEV of a sinusoidal term. In general, when the oscillation frequency f0 of a sinusoid is neither small or large according to (41), its DADEV is a mixture of the oscillations in time with period 1/ (2f0) and of the Allan deviation of a sinusoid.



σ y(t, τ) = σ y(τ) 1 − k α(τ), (43)

where k = cos(4πf0t + 2φ) is the value taken by the sinusoidal term at the specific t value. In Fig. 4 we show the DADEV with window length Tw = 100, computed for the sinusoid with parameters f0 = 2/ Tw, A = 1, φ = 0. As predicted, the DADEV exhibits both the dependence on the Allan deviation of a sinusoid given in (27), and the oscillation in time due to the cosine term. 4) Stationary Observation Interval Values: For

τ = T w/2 −

m , (44) 4f 0

where m = 1, 2, ..., and m < 2f0Tw, it is

5) Stationary Time Sampling: Suppose that we compute the DADEV at the discrete time instants tn = nTs. If

Ts = ϕ=

m , (46) 4f 0

2k + 1 π, (47) 2

where m and k are integer numbers, then

Fig. 5. DADEV of a sinusoidal term with stationary time sampling. When we sample a sinusoid according to (46), and its phase follows (47), the resulting DADEV is stationary with time.

Therefore, the resulting discrete-time DADEV is constant with time and does not show the presence of the sinusoidal term. To prove this result, we see that m 2k + 1   +2 π  , (49) cos(4π f 0nTs + 2ϕ) = cos  4π f 0n 4f 0 2  

σ y(nTs, τ) = σ y(τ). (48)

= 0, (50)



and hence (48) follows. If we change Ts or φ, or both of them, then the discrete-time DAVAR varies with time. In Fig. 5 we show the DADEV with window length Tw = 100, computed for the sinusoid with parameters f0 = 2/ Tw, A = 1, φ = π/2, and with sampling time Ts = 1/(4f0). We see that the DADEV appears to be stationary with time. If we modify the sampling time to Ts = 1/(4f0) + 1, we obtain Fig. 6, where we clearly see that the DADEV changes with time. B. Phase Jump

σ y(t, τ) = σ y(τ). (45)

For these values of τ the DADEV is hence constant with time and equals the Allan deviation of a sinusoid. This result is due to the fact that α(τ) is a sinc function with zeros located in τ = Tw/2 − m/(4f0).

795

We consider the phase jump

x(t) = ∆xu(t), (51)

where Δx is the size of the variation of the time deviation and u(t) is the Heaviside step function, defined as

u(t) =

{1,0,

t ≥ 0, (52) t < 0.

From (2), the corresponding frequency deviation is

y(t) = ∆x δ(t), (53)

where δ(t) is the Dirac delta function. Therefore, the convolution in (10) immediately returns

∆(t, τ) = ∆xh τ(t), (54)

796

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

Fig. 6. DADEV of a sinusoidal term for a quasi-stationary sampling. If we take the conditions (46)–(47) used to generate the stationary discrete-time DADEV of Fig. 5, and we slightly perturb them, for instance by changing the sampling time Ts, the resulting discrete-time DADEV becomes nonstationary with time.

and consequently, ∆x 2 ∆ 2(t, τ) = 2 P2τ(t). (55) τ



Replacing this result in (8) gives σ y2(τ) =



∆x 2 1 lim 2 T →∞ T 2τ

T /2

∫−T /2P2τ(t) d t,

and by using the inequality 1 T →∞ T lim



1

T /2

2τ, ∫−T /2P2τ(t) d t ≤ Tlim →∞ T

we have

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Fig. 7. DADEV of a phase jump. The DADEV of a phase jump is concentrated about the time interval −Tw/2 < t < Tw/2, and it decays with τ. This decrease is due to the fact that the frequency deviation of a phase jump is a delta function, and the delta function is averaged out at large observation interval values, because it is a local variation in frequency.

The τ region is 0 < τ < Tw/2 unless specified. In Fig. 7 we show the DADEV with window length Tw = 100 for a phase jump with Δx = 1. We see that the DADEV is nonzero only in the region −Tw/2 < t < Tw/2. This result is expected because outside this region the frequency deviation is zero. We also note that the DADEV is larger for small τ values. This behavior is due to the fact that the frequency deviation is a delta function, and hence it is averaged for large τ values. We point out that (56) is obtained by using the definition of the Allan variance, which is given for time series with infinite length. In practice, we estimate the Allan variance by using an estimator that operates on a finite number of measurements. Nevertheless, the Allan variance of the finite-length time series does not typically reveal

σ y2(τ) = 0. (56)



Therefore, the Allan variance does not track a phase jump. To compute the DAVAR, we replace Δ2(t, τ) in (15), obtaining

σ y2(t, τ) =

∆x 2 2(T w − 2τ)τ 2

t +T w/2 − τ

∫t −T /2+ τ P2τ(t ′) d t ′. (57) w

The solution to this integral is ∆x 2 2(T w − 2τ)τ 2 t + T w/2, −T w/2 ≤ t < − T w/2 − 2τ ,  − T w/2 − 2τ ≤ t ≤ T w/2 − 2τ ,  2τ,  0 < τ ≤ T w / 4, × − T / − 2 2τ ≤ t ≤ T w/2 − 2τ ,  w T w − 2τ, T w/4 < τ < T w/2,  T w/2 − t, T w/2 − 2τ ≤ t ≤ T w/2. (58) σ y2(t, τ) =

{ {

Fig. 8. WFN with a phase jump. The plot shows the frequency deviation of a WFN with an impulse at time t = 1000, corresponding to a phase jump.

galleani and tavella: the dynamic allan variance iv: characterization of atomic clock anomalies

Fig. 9. Allan deviation of a WFN with a phase jump. The plot shows the Allan deviation estimated from the data shown in Fig. 8. Aside from the fluctuations due to the estimation process, the Allan deviation shows the typical slope of a WFN and does not reveal the presence of a phase jump, despite its large size.

797

Fig. 10. DADEV of a WFN with a phase jump. The plot shows the DADEV estimated from the data shown in Fig. 8. Aside from the fluctuations due to the estimation process, the DADEV clearly reveals the presence of a phase jump at time t = 1000.

therefore the presence of an anomaly. Consider, for instance, the (average) frequency deviation of a WFN shown in Fig. 8. At time t = 1000 an impulse occurs, corresponding to a phase jump. Although the phase jump is large, the Allan deviation shown in Fig. 9 does not reveal it, and aside from the typical fluctuations of the estimation process, it simply shows the slope of a WFN. The DADEV in Fig. 10 gives instead an effective representation of both the WFN component and the phase jump. The random fluctuations of the DADEV are due to the estimation process. Therefore, only by estimating the Allan deviation on a series of overlapping windows of length Tw much smaller than the total duration of the available time series, we can build the DADEV surface and successfully reveal the presence of an anomaly. A similar reasoning holds for the other anomalies considered in the next sections, namely, frequency jumps and changes of variance in the clock noise, as well as, in general, for any anomaly localized in time. C. Frequency Jump We consider the frequency jump

y(t) = ∆yu(t), (59)

where Δy is the size of the variation of the frequency deviation. By replacing y(t) in (10), we obtain

∆(t, τ) = ∆y

t

∫−∞h τ(t ′) d t ′. (60)

The solution to this integral is

t   ∆(t, τ) = ∆y  1 −  P2τ(t), (61) τ 



 t t2  ∆ 2(t, τ) = ∆y 2  1 − 2 + 2  P2τ(t). (62)  τ τ 

From (8), we have

σ y2(τ) = 0. (63)

Therefore, the Allan variance does not track a frequency jump. This anomaly is instead revealed by the DAVAR, obtained by replacing Δ2(t, τ) in the integral (15), whose solution is in (64), see next page. In Fig. 11 we show the DADEV with window length Tw = 100 for a frequency jump Δy = 1. Similarly to the phase jump, the DADEV is concentrated in the region −Tw/2 < t < Tw/2. Contrary to the phase jump, the DADEV for a frequency jump increases with τ. This phenomenon is caused by the fact that a frequency jump is a global variation in the frequency deviation, whereas a delta function, corresponding to the phase jump, is a local variation. Consequently, for large τ values, the averaging of the frequency jump performed by the DADEV does not decay with τ. When the length Tw of the analysis window increases, the DADEV of the frequency jump spreads out in the time domain, and its size decreases. We illustrate this phenomenon in Fig. 12, where the DADEV is computed for Tw = 400. When Tw → +∞, the size of the DADEV reaches zero, as predicted by (63). D. Change of Variance We consider the case of a clock subject to a sudden change of variance of its WFN component. Before we actually derive the Allan variance and DAVAR of this anom-

798

σ y2(t, τ) =

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

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∆y 2 2(T w − 2τ)

  t3 T T2 T3 T T    + w2 t 2 + w2 t + w 2 , − w ≤t ≤− w +τ  2   2 2 3τ 2τ 4τ 24τ    3 2 3 2  t T − T − T T τ τ T T 4 ( 4 ) w w w w w w 2   + t + t + − + T − , − + τ < ≤ − + 2 τ τ t 2( )  w   2 2 2τ 3τ 2 2τ 2 4τ 2 24τ 2    Tw 2 Tw Tw   0 < τ ≤ 4 τ, − + 2τ < t ≤ − 2τ   3 2 2     T w3 T w2 Tw Tw t3 T w − 4τ 2 (T w − 4τ)2    t + − + 2( T − τ ), − 2 τ < t ≤ − τ − + − t w 2 2 2 2  2τ 2 2  4 τ 24 τ τ τ 3 2    3 2 3  t T T T T T  w 2 w w w w   − + t − t + , − τ < t ≤  2 2 2 2  2 2   τ 4 τ 24 τ τ 3 2       t3 T T2 T3 T T    + w2 t 2 + w2 t + w 2 , − w ≤t ≤− w +τ  2  2 2 3τ 2τ 4τ 24τ     Tw t3 T w − 4τ 2 (T w − 4τ)2 T w3 T w2 Tw    − 2τ + t + t+ − + 2(T w − τ), − +τ

The dynamic Allan Variance IV: characterization of atomic clock anomalies.

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