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IEO STATES ITMENT OF AMERCE

LICATION

*n»o» /

NBS TECHNICAL NOTE 626

U.S.

MOMENT

OF

IMMERCE

National

Bureau

C, of

>0 ds

753

folk

7Q

The Generation of an

Accurate and Uniform

Time Scale with

Calibrations and Prediction
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 was established by an act of Congress March 3,

1 901 . The Bureau's overall goal is to strengthen and advance the Nation's science and

technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measure- ment system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Computer Sciences and

Technology, and the Office for Information Programs'.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates

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THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific information generated within NBS and other agencies of

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2 Part of the Center for Radiation Research.

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1 Part of the Center for Building Technology.
GUREAU OF^SIAWUAkJS

JAN 2 9

t loo

- The Generation of an Accurate and

T^l Uniform Time Scale with

_ Calibrations and Prediction

Kazuyuki Yoshimura

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical notes are designed to supplement the

Bureau's regular publications program. They provide

a means for making available scientific data that are of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

tfx- -AJtJL . wuo +4^. h* £>£k.

W N

^.4-^V X

V,„ of *

U.S. DEPARTMENT OF COMMERCE, Peter G. Peterson, Secretary

NATIONAL BUREAU OF STANDARDS, Lawrence M. Kushner, Acting Director

«» Issued November 1972
National Bureau of Standards Technical Note 626

Nat. Bur. Stan d.( U.S.), Tech. Note 626, 64 pages (November 1972)

CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. (Government Printing Office, Washington, D. C. 20402

(Order by SD Catalog No. CI 3.46:626.)
Table of Contents

THE GENERATION OF AN ACCURATE AND UNIFORM TIME SCALE

WITH CALIBRATIONS AND PREDICTION

Page

Abstract iv

Key Words iv

1. INTRODUCTION 1

2. ALGORITHMS TO OBTAIN THE OPTIMUM FILTER

RESPONSE FUNCTIONS 4

A. Approach using calibrated frequency data which are

located before a prediction interval 5

B. Approach using calibrated frequency data which are

located in a prediction interval 12

3. ENSEMBLE TIME USING SEVERAL CLOCKS 18

4. CONSIDERATION OF "ACCURACY" 24

5. COMPARISONS BETWEEN TIME SCALES 28

6. ON AN APPROACH USING TIME DIFFERENCE DATA

REFERRED TO THE ENSEMBLE TIME 30

7. CONCLUSIONS 37

ACKNOWLEDGMENTS 38

REFERENCES 39

FIGURES 40

TABLE I 59

in
THE GENERATION OF AN ACCURATE AND UNIFORM TIME SCALE

WITH CALIBRATIONS AND PREDICTION*

Kazuyuki Yoshimurat

Atomic Frequency and Time Standards Section

National Bureau of Standards

Boulder, Colorado 80302 USA

Abstract

We express a predicted time interval (or frequency) of a single clock as a weighted sum of frequency data obtained by

calibrations against a primary standard, and derive a matrix

equation for the optimum weighting coefficients (called the

optimum filter response function) involving the Allan variances,

Two approaches are used. One of the approaches turns out to

be a generalization of Barnes' approach described in his 1966

IEEE paper.

We solve the matrix equation to get the optimum filter

response functions for white noise frequency modulation (FM),

flicker noise FM and linear combinations of them. Other

important time dispersive mechanisms exist in practice but

are not considered here. We obtain the result that the mean

square time prediction error -would increase as elapsed time

t for the case without intermediate calibrations.

We obtain the result that with a small number of good

clocks one can construct a time scale whose accuracy is

limited by the accuracy of a primary standard. We show that,

over a long time range, linear prediction algorithms based on

frequency calibrations with a primary standard give a time

scale of much better accuracy and stability than when intermittent calibrations are not used, and that (at least for

statistically identical clocks), no advantage is gained by using

the time scale itself as a "primary standard" for intermediate

calibrations.

i Key Words : Accurate and uniform time scale; Allan variance;

Dispersion of time scale; Ensemble time (error);

Prediction interval; Primary standard and clocks.

^Contribution of the National Bureau of Standards, not subject to copyright,

t Guest Worker from the Radio Research Laboratories, Tokyo, Japan.

IV
1. INTRODUCTION

An atomic time scale can provide one of the best time scales in

terms of uniformity, accuracy, reliability, and reasonable accessibility [1]. In the areas of accuracy and uniformity, it has a clear

advantage over astronomical time scales. An atomic time scale can be

actually realized with atomic frequency standards and appropriate

counters; such devices exhibit undesirable phenomena such as frequency

or time jumps, aging, and failures, as well as the signals being contaminated by random noises. All of those phenomena introduce very

important problems in the generation of an atomic time scale. Random

noises, however, which contaminate the signals from the atomic clocks--

commonly white noise frequency modulation (FM) and flicker noise FM

in cesium clocks--are included in the process of generating a time

scale and result in time dispersion such as random -walk and flicker walk.

So these random noise processes can be said to contribute basically to

the time scale error, and are the only processes treated in this paper.

In order to obtain a much better time scale than that generated by

a single clock, a group of clocks is commonly used. There are various

algorithms used, such as a weighted sum over clocks or the acceptance

or rejection of clocks, to get as good a time scale as possible from the

group of clocks. These algorithms involve assumptions about the

behavior of each clock. In practice, laboratories adopt their own

algorithms to generate local time scales and compare these time scales

with each other [2]- [5]. It is common, however, for laboratories to

try to obtain a more uniform time scale instead of a more accurate and

uniform time scale, basically by taking a weighted sum (usually or 1 weights [6]) over several clocks without using a primary standard.

This can result in a significant drift away from the defined atomic frequency and an accumulating time error relative to an ideal time scale [7],
In a recent paper [3], Mungall compares and discusses

differences between time scales which do or do not use a primarystandard, and points out the problems in the latter time scale. It was

also suggested in a recent conference [5] that a study be conducted on

how to have concurrently both an accurate and uniform IAT scale (the

International Atomic Time scale). In order to accomplish this, a primary standard must be tightly related to the generation of the time

scale, namely it must be used to calibrate periodically secondary

standards (or clocks) of lower accuracy. Periodic calibration is

usually necessary because the primary standard can be operated only

for brief periods and because frequent re-evaluations of its accuracy

are necessary.

In this paper, we predict the time change for each clock between

calibration intervals by filtering calibrated frequency data and derive a matrix equation for the optimum filter response functions which make

the mean square predicted time error a minimum. We apply these

results to the generation of a time scale from several clocks. We find

that we can predict the correct time with quite a small error using a small number of clocks (even one), and thus generate a time scale whose

accuracy is limited by the accuracy of the primary standard- -the best

case.

The dispersion of the time scale thus calculated is in a statistical

sense essentially proportional to the square root of the elapsed time,

while the time scale without intermediate calibrations may disperse proi portionally to the elapsed time for the flicker noise FM case which is

usually the dominant noise process after some time (more than 10 days,

for example) has elapsed. So in a short time range, the latter time scale,

which may use a large number of clocks, may be better in uniformity

than the former; but in a very long range, the former may become better

and better in accuracy and uniformity.
An approach in which the time of a secondary standard is predicted

with a primary standard was presented in Barnes' 1966 IEEE paper [8].

One of two approaches described in this paper is a generalization of

Barnes' approach to include any number of calibrations at arbitrary

positions.

We also discuss the application of the linear prediction method to

the case without a primary standard, trying to obtain better uniformity

in a time scale than that due to accuracy of the primary standard for

both short and long time range (where accuracy here is used for only the

random uncor related contributions of the primary standard), and show

that (at least for statistically identical clocks), no advantage is gained

by using the time scale itself as a "primary standard" for intermediate

calibrations.
2. ALGORITHMS TO OBTAIN THE OPTIMUM FILTER

RESPONSE FUNCTIONS

Figure 1(a) shows an ideal case in which the time of a clock is

moving away in a linear manner from the horizontal line- -an ideal or defined time scale. If this slope is determined, the whole movement of

the clock times can be perfectly predicted, so we can construct an ideal

time scale using this clock by subtracting a linear term from its time.

But signals from actual clocks are contaminated by noise- -commonly

by white noise FM (spectral density of frequency fluctuation S (f) cc f ) - 1 y

or flicker noise FM (S (f) cc f )or both (referred to symbolically as 0-1 y

f and f FM noise). In this case we cannot predict perfectly the

movement of the clock time, thus leaving some error of prediction as shown in figure 1(b). But by making the predicted error as small as

possible, one may construct a time scale close to an ideal time scale.

By taking a weighted sum of prediction errors for a group of clocks, we

may obtain a smaller error in the time scale as shown with the dotted

curve in figure 1(c). We call this weighted error the ensemble time

error, or (for simplicity) the ensemble time. The ensemble time thus

calculated can be used as a time reference (taking the place of an ideal

time scale) with respect to which the time of any clock can be determined.

We will describe two approaches, or two prediction methods,

using frequency data for a single clock obtained by calibrations against

a perfect primary frequency standard. Deterministic frequency offset

(or time drift) and frequency drift, however, are* not considered here.

Frequency or time jumps and any kinds of clock aging, -which are also

important phenomena to be considered for the generation of a time scale,

are not dealt with either. With no loss of generality, we will assume

that the noise processes are stationary through all the calculations. We

must determine how best to predict the time slope or frequency.
A. Approach using calibrated frequency- data which are located

before a prediction interval

The first approach assumes the calibration intervals precede the

prediction interval as shown in figure 2. Let the duration of the predic

tion interval be T and the duration of the calibration interval be hT.

Here, hT and the It are not necessarily integral multiples of T. Over a frequency calibration with a primary standard, the

ge fractional frequen

only to noise is written as

th

average fractional frequency offset on the i calibration interval due

t

y. = rr / y(t) dt = */

x(t^) - x(t^ - hT)

I hT / ' w hT

t -hT (1 I

= (x - x„ , )/hT

where x is the clock time referred to an ideal time scale. Let us

define the predicted average fractional frequency over the prediction

interval as a summation of eq (1):

z2w Yl y

= z.

a«v 2^ a

«

= : (2:

Xj — Xj _ Xj — Xj

where the a are weighting factors for y . That is, the numbers a

Xj Xj Xj weight the various measurements made in time on the single clock bein^

considered. In a very real sense the a define a response function in

the time domain for a prediction filter.
We define the predicted time x at the end of the prediction

interval as x = x + Ty . Then the error is e - x - x . Using (2),

we can obtain then

eo = T V<x - x

i

) = ^?a

* Ax*,h" Axo,i (3)

where Ax„ , = x. - x_ , and Ax„ , = x^ - x . je, h A l+h 0,1 1 The mean square prediction error is written as (4)

«o>

= \ [? a

. ux

(hT) + 2 SE, v.' { - u

x

(kT) + H«k+ h)T>

+ |ux

((k-h)r)]l + "^Ea (U(iT) + u((A+h-l)T)

h I

(4)

U (u+h)r) - U ({I - 1)t)} + u (t) X X X

2 2 where (c_) denotes the ensemble average of e , and k = # - I and

UXXX (kr) = 2[R (0) - R (kr)l (see reference [9]). R (T) is the autocorX

relation of time.

Using the relation between the U-function and the Allan variance

with zero dead time [9 ] ,

U (kT) = -k(k- 1) T

2

<CT 2

(k, T)> + k U (T), (5)

x y x

we can obtain
H Jo ^ JS x» ^

*

(6)

where C(k) and Q(#) are functions of the Allan variances given by

C(k) = -k(k- l)(a

2

(k, T)> + j(k+ h)(k+h - 1) <(T

2

(k + h, T)>

y ^ y

+ |(k- h)(k- h - l)<cr

2

(k - h, T)>

2 y

and . (7)

Q(£) = 4 C- *(i - 1) (u

Z

(A, T)> - (£+h - l)(£+h- 2) <CT 2 (£+h- 1, t)>

2 y y

2 2 + (j&+h)(j&+h-l)<Cy (£+ h, T)> + (1 - 1)(1 - 2) <a (I - 1, T)>}.

Equation (6) is the expression for the mean square of the

predicted time error in terms of the Allan variances, frequency stability

measures in the time domain.

In order to obtain the minimum value of eq (6), let us consider a function

where \ is a Lagrange undetermined multiplier. Differentiating F with

respect to a with I = I or £ = I and canceling out X. we get a matrix equation for determining the optimum filter response functions;
[C(4 - ^)-H}a + {H-C(l - ij)}a

1 p

Ea fC(£- £ ) -C(£ - £J,} = h{Q(£ ) - Q(£.)}

£ p 1

J p 1 £^£

P

(£ = £ £ . .., £ ) (8)

p Z 3 L

*l

and T* a = 1 where H s h(h- 1) <cr 2

(h, T)>.

£ y

= £,

In eqs (6) to (8), h and the # need not necessarily be integers as we will see later; the Allan variances in such a case may be interpolated.

The Allan variances for the various noise models are calculated and

given in reference [10].

First, consider the white noise FM case where the Allan variance

takes a simple form because of its independence of k:

<CT 2

(k; T) ) = CT 2

(T), C(k) = hV(T ), Q(£) = h<J

2

(T),

y

V > //

y y y

2 ,2 where 0" (t) = \0" (2, T) ) . From the matrix equation, we get constant

y y

factors for all £,a = 1/L, so the normalized mean square error will

AJ be:

V 1 V 2 , _1_

2 2, = h^ a

£ + l = Lh

T a

y

(r) £

(9)

1

1 as Lh » 1

2 as = 1

1 T 'U as << 1 .
Notice that the optimum a does not include the clock parameter

Ju 2

(j (t), but only the parameter L, the number of calibrations used for

prediction. For large Lh, eq (9) approaches 1, the smallest value.

For flicker noise FM, the Allan variance takes a more

complicated form, but C(k) or Q(&) can be written in a product form of

the Allan variance and other parameters.

(1) k - 1 '

so

C(k) = a

2

(T)c(k)/c(l), Q(j>) = CT 2

(T)q(i)/c(l)

y y

where

c(l) = 2 In 2

c(k) = -k

2

in(k) + ^(k + h)

2

jen(k + h) +^(k - h)

2

j^n(k - h) (10)

1 2 2

q(j&) =^{4 ln(l)-(i+h-l) £n(jfc+h - 1)

2 2 + (A i-h) ln(l+ h) + (& - 1) £n(JL - 1)} .

The mean square error and the matrix equation will be given by

,2

2 2

° - = -5p>*v c<k)+ !? a

.e q(fl -? a^

n,h, (u

T a (T)/c(l) h £< A' 4 I

and
[c(j> - j&)-H}a + {H - c(£ - X.)}a

pr 1 j&. p 1 £

1 r p

A ' P

I) - c(i- A )}==h{q(X ) - q(l.)]

1 p 1 4^41

( x - Jb , H , . p 12 (12)

respectively. Notice that the matrix equation does not contain the Allan

variance, so the a depend only on the calibration parameters.

-1

For f + f FM noise, the Allan variance is given by the sum of

the Allan variance of each process, assuming the processes are

independent;

<CT (k, T)> = <<72

(k, T)> + <CT2

(k, T)> . 7 f Y f"

So the mean square error is given by

< eo>

2 2, , T (T (T)

y f -i

= r 1^2 , c(l "T EEa a ,c(k)

L h

2 *<*' £ X

(13)

and the matrix equation by

10
[c(JL - I) -H}a A + {H - c(i - I )}a

P 1 Aj P 1 P

+ V* a {c(£- A )-c(£- 1J} = h{q(X )- q(l.)}

*L-/ X p 1 pi

X 7^ J^P

(1 = £_, JL, .... jL) (14)

p 1 Z Li

2 2 2 where r = r(T) = CT (T) „/cr (T) , and H = h 4n(h) - rhc(l).

y f o y f -i

Figure 3 is a computed result for a prediction with only one

calibration interval. The horizontal axis shows the time distance of

the calibration interval from the prediction interval. The vertical axis

shows the mean square prediction error normalized by that with calibration position equal to 1. The prediction error of the white noise FM

case does not depend on the position of the calibration, while the prediction error of the flicker noise FM case is getting worse with the

distance. So it is better with flicker noise FM to locate a calibration

interval as close to the prediction interval as possible.

Figure 4 is the case where from two to five calibration intervals

are used for a prediction. The last calibration is always located just

preceding the prediction interval. But positions of others are changeable, which are shown on the horizontal axis as "spacing. " The vertical

2 axis shows the mean square prediction error normalized by T times

the Allan variance.

As expected, with white noise FM, the prediction error does not

depend on the position of calibration, only on how many calibrations are

used for a prediction (actually on the total calibration time). With

flicker noise FM, it depends on the position, as well as the number of

calibrations used. In the two cases for flicker noise FM, there are

optimum spacings which are shown in the diagram with black points where

the mean square error becomes minimum.

11
Figure 5 shows the optimum filter response functions with L

equal to 5. One is for a two -day (t = 1 day) spacing between calibrations

and the other for contiguous calibrations. For white noise FM, a factors are always constant, while for flicker noise FM, the calibration

interval nearest the prediction interval is always given the largest value,

showing that it may be enough to use only one calibration interval nearest

to the prediction interval. With contiguous calibration, the a are

oscillating, perhaps due to an aliasing phenomenon.

B. Approach using calibrated frequency data which are located

in a prediction interval

In this approach using calibrated frequency data, the calibration

intervals are located within the prediction interval as shown in figure 6. M and I are not necessarily intergers. The calculation procedure is

almost the same as that described for approach A.

The average fractional frequency offset due to noise over the

calibration intervals is given by

H= ( V x*-i )/T <* = V V •••• V- (15:

We define an estimated average fractional frequency offset y- & , , OM

during the - M interval by a summation of (15), and a predicted time

interval (x - x ) as follows: ,

*L

y0M Z a „ y„ , V^ a „ = !> K, - x^ = MTy\ _. (16)

I

y I L~i I MO OM

1 = 1

12
The predicted time error is

€0M = "M" XM

= MT% " (XM" X )

= Mj"<-* a„ Ax, - Ax„„

I I OM

(17)

where Ax„ = Ty„ = x„ - x„ , and Ax„,, = x, , - x^.

I

y i l JL-l OM M

Then the mean square of the predicted time error is finally given

by

<€ > = -2M T

2 EE a a ,C(k) + 2Mt2 £ a Q(£) - T M(M - 1) (a (M,r)>

(18)

where k = I -I, and

C(k) = -k(k- l)<CT

2

(k, t)> + 4(k+1 ) k <^(k + 1» T)>

y 2 y

+ -j(k- l)(k-2) <a

2

(k- 1, T)> 2 y

and (18')

Q(£) = 4 {-(M - £)(M -JL-l) <a

2

(M - I, r) > 2 y

+ (M - 4 + 1)(M - 1) (cr 2(M . jj + I, T)> - (X - 1)(A - 2) <CT 2

(£ -1,7))

+ Jt(i - 1)<CT

2

(^ r)>]

y

13
From eq (18), the matrix equation for the optimum a is

C(l.- 1,). -C(i - i

1 ,a

i

1 P

£>£ {cu- y - c(i- x

1 )3 = ^{Q(A

p

) - Q(^)}

i^i1

(i = v v •••' V < 19 >

and V* a . = 1. i

*

For white noise FM, the optimum response functions again

become constant. Then the normalized mean square error will be given

by

< 2) -_-OM .,2^ 2 w M - L

= M E a, - M = -j—

7 CTy

( T) * (20)

2 - M /L. as M »L

- as M = L

- M(M - 1) as L = 1

where a = 1/L.

For flicker noise FM, the mean square error is given by eq (21)

and the matrix equation for the optimum a by eq (22).

Xj

,2 .

= -2M 2 X)Ea/,a /) , c(k) + 2MX;a.q(£) - M

2

Xn(M) . (21)

T (T (T)/c(l) £<£ £

y

14
c(£ - £ )a - c(£ - £ )a

p 1 £. p 1 £

1 p

+ Z) a

^ [cU_ y

c(i_ v ] = m [q( y - q( v ] £tl,

£^£

P

(4 = I I .... I ) (22)

p 2 3 L

where

c(k) = -k

2

£n(k) + ^(k + l)

2

£n(k + 1) + hk - l)

2

£n(k - 1)

and (23) 12 2

q(jfc) = - {-(M - I) 4n(M - £) + (M - £ + 1) £n(M - 4 + 1)

- (£ - l)

2

4n(4 - 1) + £

2 £n(£)} .

Now let us consider the case where there is only one calibration

in a prediction interval in figure 6. Equation (23) will be reduced to

where

<€ 2

) —TT~ = "T77 (2q(Jt.) - M 4n(M)} (24)

c c. c (1 ) 1 T CT (T)

y

q(0 = -r {-(M - £ ) 2

je.n(M - £ ) + (M - 1 + l)

2 £n(M - £ + 1)

x Cj ± J. A J.

- (I - l)

2

£n(£ - 1)+ A £n(£ )] .

15
In eq (24), for example, let M = 3 and locate the calibration

interval in the middle of the prediction interval. Then the result

coincides with eq (31) of Barnes' paper published by IEEE in 1966 [8].

And for another example, let M = 2n + 1, where n is an integer, and

again locate the calibration interval in the middle of the prediction

interval- -namely, I - n + 1; the result again coincides with eq (61) of

Barnes' paper. Thus eq (24), where I and M are not necessarily

integers, seems to be the analytical solution of Barnes' eq (47). Thus

approach B, which can include any number of calibrations, can be said

to be the generalization of Barnes' approach.

For f + f FM noise, the meai

equations will be given by (25) and (2 6)

For f + f FM noise, the mean square error and matrix

<4>

2 2

T (J (T)

y r 1 ^<£*rU) + W) H

2 EEa^,c,k , l<l

+ ZMj^a q(l) - M

Z in(M)l

(25;

2 , 2 where r = r(T) = a (T) _/cr (T) , y fo y f -i

[rc(l) + c(£ - j^)} a -{rc(l) + c(j& - J^)^

. 1 P

+ 2a

£

fc(1 - t >- c ^- V ] = M

[qU

p

)_ q( V }

14 I, a 4 i

lv (l = JL, l., ..., I ) (26)

p 2 3 L

16
Figure 7 shows the mean square prediction error for one

calibration in a week. Calibration duration is supposedly one day.

The horizontal axis is the position of the calibration interval. With

white noise FM, the mean square error does not depend on the position.

With flicker noise FM, locating a calibration in the center gives the

best result. It can be shown that predictions based on calibrations

located symmetrically about the center of the prediction interval give

the same error.

Figure 8 shows the mean square error versus number of

calibrations in a week. Again calibration duration is supposedly one

day. For flicker noise FM, optimum calibration distributions are

selected for calibration number 1 to 3. Quick improvement with the

number of calibrations is shown. The dotted curve shows the improvement due to a number of clocks without calibration, assuming each has

the same noise level, and assuming constant weighting factors for each

clock.

Figure 9 shows the influence of calibration duration. Two

calibrations per week and optimum- -for f noise FM- -locations for

calibrations are selected. The vertical axis is the mean square error

normalized by that of the 24-hour calibration duration. As expected,

longer calibration duration is preferable.

Figure 10 shows the optimum filter response functions a for

Xj flicker noise FM. It is interesting to note that two a factors for

Xj calibrations symmetrically located have the same value, analogous to

the white noise FM case. Figure 11 shows the influence of increasing the size of a single

prediction interval, M , where a rate of two calibrations per week is

used and their positions are fixed. For both white noise FM and flicker

noise FM, the error is almost linearly increasing with the size of the

17
2

prediction interval, instead of increasing with M for flicker noise FM.

(Similar results were obtained by Barnes [8]. ) The accumulated mean

square error over M/7 prediction periods of seven days each lies

essentially on the same curve; we shall discuss this in detail later.

3. ENSEMBLE TIME USING SEVERAL CLOCKS

With approach B, it may be possible to increase the size of a prediction interval and construct a useful ensemble time with only one

Prediction interval. From the predicted time x of each clock at M, M

we construct an ensemble time estimate using weights w^ , for each of b to M

the N clocks:

N . . N

-ens <e^ i „i ^-> i M X—i M M L*i M

i= 1 i = 1

The correct value for this ensemble time is

N

m ens v-^* i i T = > w x M L~i M M

i = 1

so the ensemble time erroro I, is \a

* T

ens *ens

M M " M

N

i i ^ = I>M«M " (2?)

i = 1
The mean square of T in eq (27) is

<*£> = ?>£<&> < 28 > l

since the clocks are independent.

The optimum value of the weighting factors which makes the mean

square of the ensemble time error minimum will be

OM 7 0M;

by the same calculation procedure as described for eq (8). The

minimum value of the mean square error is then:

(?*) ( ^5> : M min

OM

For white noise FM or flicker noise FM, the mean square error

of one clock involves only the Allan variance to describe the clock. So

under the assumption that all the N clocks have the same type of noisewhite noise FM or flicker noise FM, the optimum weighting factors and

the minimum mean square error will be expressed in simple forms as

opt \7 a'W a

l2

(r)

y y

and

-1

(FTfe)

<*£> = K.-T^y —i— (32) M mm M

y

where £L is constant, calculable from eqs (20), (21), and (30)

19
In general, for mixed noises, optimum weighting factors can not

be expressed only with the Allan variances, but include other parameters

related to calibration and prediction methods.

For the general case, with either approach A or B, where manyprediction intervals are connected in series (see fig. 19), the change in

ensemble time over the (M - 1, M) interval can be written;

N

m ens m ens ^-*» i ^-s i T M

" TM-1

= 2^ WM

(XM- XM-1 ) 'LWM

= 1 ' i= 1

and its estimate

N

-ens -ens <r-^ l * „ T -T = > w (x - x ). M M-l Z-r M

v M M-l

i = 1

Then the change in ensemble time error is

N

i i M "M-l £-* "M CM ' i = 1

! i = Z v

:

'

or M N

j=li=l

(T

Q = 0) , (33)

where the c. is the prediction error for the i clock on the (j - 1, j)

J interval, and the initial error is assumed to be zero.

20
The mean square of eq (33) will be

<f

2

> M i J

EiZ,iZ, v-v^ 1 x / x 1 \ W. {€. > + 22-r 2-r W - W -'^ e -' C -'/ i Lj J J J<J

J J J J

(34)

under independence of each clock.

The mean square ensemble error now contains covariance terms,

in addition to the mean square terms we have had before. Equations

for the covariance terms can also be derived by calculation procedure

similar to those already used.

We can also construct a matrix equation for weighting factors

from eq (34)

i2

2w. <€. ) + 2 Y] w., <€.-€.,) + \. =

j J frf. J J J J

j H

(i = 1, 2, . . . , N and j = 1,2, . . . , M]

(35;

and 2J w. = 1 for all j where \. is a Lagrange undetermined multiplier,

i

J J But the size of this matrix is the square of N times M, which will

unlimitedly increase with M, and the -weighting factors will be functions

of M. This seems impractical.

For white noise FM, however, since the covariance term

disappears, the results become simple.

W

.opt U^ ,12, , 12, (? 12. J 12. . J \ 1 <€. >/ <€. > V l a (T)/ CT (T)

(36)

21
and

< *M > = min

In general, certain clocks in the ensemble of clocks may come to

be rejected with increasing time on account of their prediction errors

getting much larger than others. Then it may be preferable to construct

the ensemble of clocks with a smaller number of clocks all of which have

the same type of noise. Approach B, in this sense, seems better than

approach A because it may give a simple form for the weighting factors

even for mixed noises (see fig. 11).

Under the assumption of a fixed weighting factor for each interval

(w. = w ), again we can get simple forms:

<T^> =£wi2

<(lV)

2

> . (38)

i J

So

-1

1 \ 1

i ((Ec

1

) 2

)/ ((Ec

1

) 2

)

J J j

J

(39)

and

(40)

We calculated ((Ic ) ) in (38) in order to obtain some numerical idea of

the mean square error of the ensemble time. For approach B, we already

showed an example of this quantity in figure 11, so here some examples

for approach A will be explained.

22
Figures 12 and 13 show the influence of positions of calibrations.

An almost equally spaced distribution will give the best result for the

flicker case as shown with (1). Figure 12 is for one calibration used

for a prediction and figure 13 is for two calibrations used for a prediction. It should be pointed out that the slope of the mean square errors

* 2

T ,(M) may actually be smaller for flicker noise than for white noise,

although this is not indicated in the figures, where the normalization

influences the slope.

Figure 14 is the accumulated error in one week versus the

number of calibrations in a week. Rapid improvement with increasing

rate is shown for flicker noise FM.

Figure 15 is the accumulated error versus the number of

calibrations used for a prediction. Calibration rate is two calibrations

per week. For -white noise FM, rapid improvement with L is shown

and for flicker noise FM, L = 1 gives a local minimum independent of

M. For f + f FM noise with r = 10, it is apparent that there are

optimum values L. Improvement with larger L, however, seems quite

small, so setting L equal to 1 may be sufficient for flicker noise FM or

white and flicker FM noise in approach A.

Figure 18 shows the simulation results of a time scale using one

clock for mixed noises shown in figures 16 and 17, using a white noise

and flicker noise model [11]. In the diagram, improvement by prediction is clearly shown with curves A and B, which are predicted time

errors for curve S, the time error of the mixed frequency noise given

in figures 16 and 17. We will discuss figure 18 in more detail later.

23
4. CONSIDERATION OF "ACCURACY"

All the calculations so far mentioned took into consideration only

noises of clocks- -white or flicker noise FM. But actually, precision of

measurements and "accuracy" of the primary standard must be also

considered. Here, effects of random uncorrelated noise as it affects

the accuracy of the primary standard, denoted "accuracy, " will be discussed; the imprecision of the measurements will be assumed negligible,

We shall assume that the total average fractional frequency offset

may be expressed as a sum of two independent random variables,

——Oi — , .. ,

'a = yt

+ 6yt. (41)

J J J

where the first term depends on clock noise and the second on noise

(inaccuracy) in the primary standard. Then the predicted average

fractional frequency offset will be

A i --, i _i _ i —Oi _ i — . .

and the predicted time error will be

i ^i A

i «j = Ty

i - Ax

j (43)

Oi .^ i . - . Oi . _ i

J J = «?

i+ ?V(T «V^" + ^

J where

<j - EVTy

i > - Ax

j and 6e

j s £

a

i. (T6y

i.' (44)

24
Then the mean square prediction error, finally, must be written

as a sum of the noise dependent term and an "accuracy" dependent one

as

, i2. , 0i2. , „ i2. . .

_

<€.>=<€. > + <6e. > . (45,

J J J

_2 _2 Assuming (fiy ) = (6y ) and using equation (44),

J

<6€

l2

> = <(£>* TbY^ )

~- E a

i

• (T <6y ». (46)

J *• J J *• J

where covariance components of inaccuracy are neglected for simplicity.

It must be borne in mind that the correlated components of accuracy,

which are in general unknown, may be of primary importance, particularly in long term. Then equation (45) will be written as

<0 = <€°

lZ

> + Ea1;- (T <6y »• (47)

J

J

Some correction must be made for the matrix equations derived

in section 2. But since the second term in equation (47) gives the same

effect as white noise FM, no correction of a factors for white noise FM

i

is necessary. With the particular cases for flicker noise FM shown in

figure 10 where a factors are constant, no correction is necessary

either.

From equation (33), the ensemble time error and its mean square

will be corrected to

\* = EE wlfl = Y,Y wl

(*

01 + s^

1

) (48) M V^ l i 4"^ i j i 1 j

j j x jj j j

25
and

<t6 = <(EE»i f

i

) 2

>+ <(EEwi 6€

i

) 2 V - (iEE»0).+ (iEIw>()')> (49) ijJJ ^jJJ

respectively.

In order to get a rough numerical estimation of the "accuracy,

we assumed constant weighting factors for

w. = 1/N = const, and a„ ~ a

1 I. I

J

used to obtain equation (50) as the mean square error of the ensemble

time. Then:

<T^> = ^"<(E^)

2

> + M'A.r 2(6y

2

> (50)

where

2 A = Va„ ( < 1) = const.

The first term in equation (50) is the noise dependent one and the

second is "accuracy" dependent, assuming constant mean square of the

inaccuracy for each interval. The mean square accumulated error for

one clock will take the form of a product of the Allan variance and some

coefficient, so if

<(Zcb

2

> = r

2

a

2

(r) k-M ,

J

J Y

26
then eq (50) will be simplified as

<t^>/T 2

cr 2

(T) = k-M/N + MA(6y2

)/a

2 My (T) (5i;

y

where k is constant.

For the generation of the most accurate and uniform time scale,

the first term in eq (51) should be negligibly small compared to the

second. The condition for that will be given by

k-M/N «MA(6y2

)/(j

2

(T) (52)

for which

<T^> « MAT 2

(6y

2

) • (53)

Notice that the ratio of mean square frequency error due to

"accuracy" to the Allan variance due to noises is related to this condition. In eq (5 3) the mean square ensemble time error due to

inaccuracy increases linearly with M , as does the prediction error as mentioned before.

Table I gives a numerical comparison between the error dependent

on prediction and error dependent on "accuracy, " related to condition

(52). If one has a clock 10 times as good in stability as a primary standard in accuracy, then in seven days the value of the right hand in eq (52)

will be 700. There is also shown in the table numerical examples of the

coefficient k • M by approach B. Take (2) for instance, namely two

calibrations in a week and 24-hour calibration duration: as may be seen,

those values of k • M are sufficiently small compared to 700. So in this

case, we can construct a good time scale with only one clock, which is

27
very simple for the generation of a time scale. With the value coming

from a condition of two calibrations per week and six-hour calibration

duration, one needs several clocks for condition (52), even using such

good clocks.

5. COMPARISONS BETWEEN TIME SCALES

We have obtained the result that the mean square of the predicted

time error with a primary standard increases with M (or t , elapsed

time) for white noise FM, flicker noise FM, or primary standard

inaccuracy due to random uncorrelated noise. One can see this from

the sample shown in figure 18 by comparing curves A and B with curve

W which is time error due to pure white noise, which also represents

time error due to "inaccuracy": those curves appear to have similar

slopes

It is well-known that the mean square of a simply accumulated

time error (without prediction and calibrations) increases proportionally

2

to elapsed time t for white noise FM and to t for flicker noise FM.

For large value of t (^ 10 days, for example), however, flicker noise

FM will be usually dominant (see fig. 17), and for larger value of t , _2

the dominant spectral density S (f) may be proportional to f for which

Y

3

mean square time error is proportional to t . Curve S in figure 18 which is a time error for one clock can be

compared for that with curve A or B or W. (Notice that even in the

-2

prediction method, if time error due to a certain noise, f for exampl

is not included in the prediction, then it will become dominant and give

3 the same time error (i. e. , t ) as its simply accumulated time error af

long time has elapsed. )

28
Let us consider the ideal case where the predicted time error is

much smaller than that due to inaccuracy; then from eq (5 3), the mean

square time error is given

<T^> ~ t.T<6y2

) (54) M pred 1

}

where t = Mr. The mean square of a simply accumulated time error

averaged over N clocks will be [10]

<T

2

.) . « t 2

a

2

(t )/N (55) M simple 1 y 1

where constant weighting factors over clocks are assumed.

Then the ratio of the mean square errors is

(I2

) J (t

2

) . = r N/M (56) M pred M simple a

where r = <6y )/<y (t_), a v 1 -2,, 2

ry

2

Let r =10 for instance, then the both mean square errors will

a be the same in 300 days (^ 1 year) with N = 3 and 1000 days (^ 3 years)

with N = 10. After that a time scale generated by the prediction method

-with a primary standard will be getting better and better relative to the

other. It also has to be considered that the "accuracy" of primary

standards will continue to be improved [12].

29
6. ON AN APPROACH USING TIME DIFFERENCE DATA

REFERRED TO THE ENSEMBLE TIME

A time scale generated by the prediction method with a primarystandard is restricted in its uniformity by the accuracy of the primary

standard used, which will be commonly worse than the frequency

stability of clocks. The uniformity of a time scale by the method of a simple weighted sum over clocks, namely without prediction and calibration, may be much improved for a short time range by increasing

the number of clocks. (See eq (55). ) So, it will be natural to expect to

obtain a better uniformity and accuracy in a time scale by generating

the time scale with the optimum time prediction and with an initial

calibration, but without intermediate calibrations.

In this case, one might use the ensemble time as the reference,

instead of a primary standard. So the time difference (or frequency)

data of each clock used for the frequency prediction with approach A

would be referred to the ensemble time as shown with notation t in

figure 19. The following discussion on this approach, however, will

show that little improvement can be expected.

Let the predicted frequency or time difference be given as a linear function of time difference data--L in number:

T & =

flM

(AtM-r ••• AtM-L>

L

i . i

L Mi M - I

1 = 1

(57;

T,

where r. is the length of each (prediction) interval, f , , is a linear M B u M

operator on {At.} at the M interval and At are time difference

30
data for i clock referred to the ensemble time at M - I interval

whose duration is T , (see fig. 19);. M - I

At

111= t - t and t 11= x - T M-l M-l M - I - 1 M-J& M-l M-J&-1

so

i i i " A M - I

= (XM - I

' XM - I - l

] ' (1 M - I

' TM - I- 1

Ax1

- AT . (5 8) M- I M - I

v

In the approaches so far mentioned, the a = a in eq (57), so

1VJ. JJ» Xj that f = f (see eq (2)).

The predicted time error may be expressed as

€M

= XM " XM = AxM " TyM = AXM " fM

(AxM - i

+ fM

(ATM - i

'

(59)

The first term in (59) is the real time increase during the M interval

and the second is a prediction term for the first. The third term came

from using ensemble time referred data, and is, in a sense, redundant.

Using eq (59), the ensemble time error will be expressed from eq (33),

assuming fixed weighting factors for each interval as

. M ... t ^yw'rtAx'-f1 ^1 j + f!(Ai. n m Z^ Z^ j j y-i j

v j-r ;

i

j = l

(60)

31
where w = w for all j is assumed. Again the first term is the real

J time increase during the 0-M interval and the rest is a prediction for

the first.

Now let us reduce the third term. Similarly to eq (60), we will

get the following two equations and so an equation for their difference

L y-Jj i_ i. L

i j=l J l

J l

J l

1

J l

T. , V w

1 V {Ax.1 - f.^Ax/ J+f^AT. .)}

and

i J-A-l i i i i

\ J!=l

111 11

SO

^T._

4

= ^''(ix^ - f^i-,/ + i^i,-,/? • < 61)

Applying the linear function f and next Sw , we will obtain

J i 11 A (62)

+ F.- F. (AT. , „

. J 3-1 J-A-^.

where F = T w

1

- f

1 and may be called an average linear function over

J i J clocks.

32
Substitution of eq (62) into eq (60) gives

M

i j=i

) + F.(Ax' )

(63)

M

^•f1 (Ax1 )l+y^F • F (AT

j j

j j j = i

J J 1

which again includes the last term- -a function of ensemble time

difference.

Repeating the above procedure and defining the error function

Af1 = f.1

- F for all k, (64)

K. K. ic

we will finally obtain eq (65),

M

Tw = V w V TAx - Af (Ax J - F.- Af .(Ax1

n n ]

i j =1 i

F.- F. • Af1

n n (Ax1n A n )

F • F • F . . . F • Af1 (Ax

1

) j i-i j-i-ij j-i-ij.....^^ J-*-^..--V- 3 J- £^r---V-2

F • F • F F • f

1 (Ax1

) j m j-i-jj j-i-i,...-^^ J-^-^----V_2 '-"r-ViJ

M

33
where

R

i i

= F

i' Fi-r" Fi it i

(AT

i a i

>-*°- (66)

The last two terms consist of real time difference data belonging

to zero or before zero intervals. The last term R is a function of

initial values of the ensemble time error, so it seems reasonable to

assume this zero. If the linear functions for all the clocks are all the

same, i. e. , if the clocks are statistically identical, then all the error

functions in eq (64) are equal to zero, then all the medium terms in

eq (65) will disappear to give

M

(67)

The important thing to notice in this equation is that the ensemble

* i i error T , does not depend on the clock values (x , . . . , x, , ,

) actually M ^

1 M-l ' occurring in the interval, but only on values predicted for them from

values occurring before j = 1. Thus, no new information is added in

* ens forming the estimate T based on clock frequency estimation using

a ens a ens

the ensemble values (T , ...,T ). If, in fact, the clocks started

at j = 0, (x =0 for m ^ 0), eq (67) is merely the simple weighted m

average without prediction

T = > w (x ,

, - x M jLj v M

i = l

34
In either case, the results of using approach A shown in figure 3 show

that for flicker noise FM, the error T will grow rapidly, at least as

fast as \M/N, as we advance M units from the last calibration.

It is easily shown that this result does not depend on the fact that

we have weights w = w . When the weights are allowed to vary, the

clocks remaining statistically identical, we obtain instead eq (67)

N M

1=1 J = l

F.-F. .. . . F. „ „ f. . Ax1 M-2 j-i )}

M-l

where again the second (correction) term does not depend on clock

values which occur for j > 0, and vanishes for clocks started at j = 0,

In the general case where the clocks are required to be

statistically identical, we obtain by the same procedure

N M

M £-i ^—I I 11 &

1 1 -

.

i=l j=l J J J J F.-Ag1 .(Ax

1 -!

J J-i J

F.-F • Ag

1 (Ax

1

„ ) J j-i ^j-i-ij j-i-Y^^

F.-F. . . . F

J J-i J-i

-F.-F.

j J-i

M

R.

j = l j,i

^gi

M-4

M-3

j-i

i'j-i

(A:

M-3 >i

I

(Ax:

M-2 j-i

-i ) M-2

-LM i»]

(68)

35
where

(69:

N

r = 7 w I . Sf = w f k La k k ' gk k k

i = I

N

A % = w

k

(£k

- E w

k

fk

» s = l

Equation (68) does contain prediction terms dependent on clock

values occurring on the interval j = (1, M - 1), so that new information

is being added. But it does not seem likely that the error T would M

show significantly reduced growth rate from that obtained in the case

with statistically identical clocks. It is important to note that eq (68)

for T is nonlinear (of degree M) in the weights w , so that the

optimum estimation of these weights is very difficult.

The conclusion appears to be that while frequency calibrations

against a primary standard can greatly improve the accuracy and

stability of the time scale, for the noise processes considered in this

paper, no improvement in these quantities can be gained (at least in

the case of statistically identical clocks) by using the time scale itself as a primary standard.

36
7. CONCLUSIONS

We have obtained the result that one can construct with a small

number of clocks a time scale whose accuracy is limited by the accuracy of a primary standard. We have also shown that the time

dispersion of the time scale thus calculated may be proportional to the

2

square root of the elapsed time t in a statistical sense instead of t

for the time scale without intermediate calibrations, so the former

may become much better in both accuracy and uniformity than the latter

after a long time has elapsed. Furthermore, algorithms using a primary standard have the advantage that it is easier to identify frequency drift, jumps, or aging of the clocks used.

It must be emphasized that these conclusions are based on clocks

having only continuous random f and f FM noise with constant

amplitudes. In practice, these amplitudes can vary in time, and many

other anomalous forms of behavior occur, which will strongly influence

the choice of both the clock ensemble and the time scale algorithm to be

used.

We have shown that (at least for statistically identical clocks),

no advantage is gained by using the time scale itself as a "primarystandard" for intermediate calibrations.

It is also true, however, that the time scale averaged over a large number of clocks without intermediate calibrations may have

a much better uniformity for a short time range than that depending

on a primary standard utilized as in the above methods of prediction,

where the short term uniformity of the time scale is limited by

accuracy of the primary standard. Therefore, it may be possible to

generate the two kinds of time scale simultaneously and actually

utilize the former time scale periodically calibrated by the latter.

37
ACKNOWLEDGMENTS

The author is sincerely indebted to Dr. S. Jarvis, Jr. , Mr. D. W. Allan, and Dr. J. A. Barnes of the National Bureau of

Standards for many informative discussions and for significant contribution in making manuscript criticisms. The author also wishes

to acknowledge the help of Dr. H. Akima in the Institute of

Telecommunication Sciences for his kind aid in computer programming and presentation of his valuable sub-programs, and the

assistance kindly given by Mrs. E. Helfrich in manuscript preparation.

38
REFERENCES

[1] Barnes, J. A., Proc. IEEE 55, 822 (1967).

[2] Winkler, G. M.R., Hall, R. G. , Percival, D. B. , Metrologia 6,

126 (1970).

[3] Mungall, A. G. , Metrologia 7, 146(1971).

[4] Allan, D. W. , Gray, J. E. , Metrologia 7, 79 (1971).

[5] Program and Abstracts of International Symposium on Algorithms

used in Calculation of Atomic Time Scales, Boulder, Colorado,

30 June and 1 July 1972.

[6] Guinot, B. , Granveaud, M. , a paper presented at the above

symposium.

[7] Smith, H. M. , Proc. IEEE _60, 479 (1972).

[8] Barnes, J. A., Proc. IEEE 54, 207 (1966).

[9] Allan, D. W. , Proc. IEEE 54, 221 (1966).

[10] Barnes, J. A. et al. , NBS Technical Note 394, October 1970;

also IEEE Trans. IM-20, 105 (1971).

[11] Barnes, J. A., Jarvis, S.,Jr., NBS Technical Note 604,

June 1971.

[12] Hellwig, H. , Metrologia 6, 118 (1970).

39
(a)

^ Ideal or Defined

Time Scale

Ideal Time

Scale

T = 2W^1

i = l

Ideal

Time

Scale

Fig. 1 (a) Linear drift of time. (b) Time error, x , due to random

noises and its prediction error, e . (c) A group of predicted

time errors, e

1

. and the ensemble time, T.

40
Calibration Interval

it hr hr

-** £l #l-\

Prediction

Interval

IT T hH—

I

0z £> 10

*,T

Fig. 2 Approach A.

41
CM

h=l a L=l

** 2

f"

1 Noise FM

f°+f"' Noise FM

f° Noise FM

t_L

2 4 6

Position of Calibration {£)

Fig. 3 Mean square prediction error by approach A as a function of

position of a calibration, f and f~* noise FM refer to white

and flicker noise FM, respectively, and f + f~ noise FM

refers to a mixed noise of both with power ratio r = 1.

42
2.0

h = i

K>

CM

<\J

I- 1.5

E

CM

1.0

L=2 f

L = 5 r

(ex)^s

=2

M I M I M T

I CAL CAL CAL

(FIXED)

/PREDICTION

\ INTERVAL

12 3 4 5 6

Spacing of Calibration

Fig. 4 Mean square prediction error by approach A as a function of

spacing of calibration. fu and f "

* refer to white and flicker

noise FM, respectively. An example of the calibration

spacing is shown for I - 2,

s

43
h=l 8 L=5

*- o _ o

u_ c

-J U_

6

3 a> E

.«— o

Q. Q. O to

a> cr

Two Day Spacing

Calibration (f"

1

)

f

Cont. Calibration

(f- 1

)

O 1 2 3 4 5

Fig. 5 Optimum filter response functions by approach A. f and f refer to white and flicker noise FM, respectively.

44
Calibration Interval

r r r T T

-e*- £\ i>2 £f\ ^L M

j2lt

Mr

Prediction Interval

Fig. 6 Approach B.

45
II

CVJ o

1.0

0.5

M =7 a i Cal./Week

f°

12 3 4 5 6 7

Position of Calibration

(*c)

Fig. 7 Mean square prediction error by approach B as a function of

position of a calibration. Calibration duration is one day.

f and f "

* refer to white and flicker noise FM, respectively.

46
CM O

CM £$

M =7

Optimum Positions of Calibrations

are Selected for f

_1and L = l~3.

Dashed Line ( —) is for

Number of Calibrations ( L)

or Clocks (N)

Fi. Mean square prediction error by approach B as a function of

number of calibrations in a week. Calibration duration is one

day. f and f" refer to white and flicker noise FM, respectively. The dashed continuation for the f curve shows the

case where the optimum positions of calibrations are selected

for L = 4 ~ 6; Saturday and Sunday are used for calibration

days.

47
10

O

CVJ

ii I-

^\

cvj o

8-

6-

v^ 4-

CVJ

M = 7 Days a

2Cals/Week

3 6 9 12 15 18 21 24

Calibration Duration

T( Hours)

Fig. 9 Mean square prediction error by approach B as a function of

calibration duration, f and f~ refer to white and flicker

noise FM, respectively.

48
0.8 r - 0.6h

M = 7

C

k- o ~ =5 0.4-

;— c

<D

if) 3

£ c

•^ o

O w

cr

0.2

f

_l Noise FM

Qr*-

(b)

(O

X X

\_-x~..x'

I 2 3 4 5 6

DateU)

Fig. 10 Optimum filter response functions for flicker noise FM byapproach B. (a) and (b): Two calibrations per week,

(c): Three calibrations per week, (d): Five calibrations per

week. Calibration duration is one day.

49
CM §

5r2-

2Cals/Week

Optimum Positions

of Calibrations are

Selected for f

r

14

Mn or M

M

^

28

v. Mn

j

Fig. 11 Mean square prediction error by approach B as a function of

the size of one prediction interval, Mn, or M/7 periods of

- 1 7 days each, M. Calibration duration is one day. f and f refer to white and flicker noise FM, respectively.

50
CM

h=l L=l a 2Cals/Week

i- 4

2-

uJ I I L

14 21 28

M

Fig. 12 Mean square of accumulated prediction error by approach A

(L = 1). Calibration duration is one day. f and f to white and flicker noise FM, respectively.

refer

51
h=l L=2 a 2Cals/Week

OJ

CM

X/

Fig. 13 Mean square of accumulated prediction error by approach A

(L = 2) . Calibration duration is one day. f and f~ refer

to white and flicker noise FM, respectively.

52
h=l a L=i

1.0rCAL

(l)fad i I I I (2)

j(4)

1 2 3 4 5 6 7

Number of Calibrations

per Week

Fig. 14 Mean square of accumulated prediction error by approach A

as a function of number of calibrations in a week. Calibration

duration is one day. f , f and f + f refer to white noise

FM, flicker noise FM, and a mixed noise of both, respectively.

The dashed continuations of the curves show the case where the

optimum positions of calibrations are selected for the horizontal

axis 4 ~ 7; Saturday and Sunday are used for calibration days.

53
C\J

1.0

*"- 0.5 Wi ! s/

h = l a 2Cals/Week

J I L

1 2 3 4 5

L

6 7 8

T f°+ r

l

(r=io)

M Increasing

M Increasing

Fig. 15 Mean square of accumulated prediction error by approach A

as a function of number of calibrations used for a prediction.

Calibration duration is one day. f , f" and f +f _1 refer to

white noise FM, flicker noise FM, and mixed noise of both,

respectively.

54
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57
Ideal Time

Scale

Fig. 19 Prediction interval (M - 1, M). x

1 is time error of i clock

and x\ r is prediction for x1 at M. tt , , is time error of i M

clock at M mb-.1 1, referred to the ensemble time error T at

i \ Al^A^ , ;th - • .-M-l M - 1 and P is predicted time error of i Ln clock at M,

referred to the ensemble time error T , ,. r is duration

j- u- • 4. i -i • j- ¥ -J*.- M j t

- th

of prediction interval, T v* , is predicted time advance of l

• MM * i clock at M and ei , is prediction error for T, y . M MM

58
<sy

2

>

CT2(T) o.i

2

0.2

2

0.5

2

I

2

2

2

5

2

io

2

Mx

<8 y

2>

MV 0.07 0.14 1.75 7 28 175 700

y

2(r)

K

f -i 6.84

(3)

I6.8

(2)

55.4

(1) 656(2)

'

K

f

o 9.33

(3)

I7.5

(2) 42(1) 364(2)

"

K=<€

2

7>/T 2

(=lday)<T

y

2

(r ),

(1)

(2)

(3)

(2)'

ICal/Week, r

2Cals/Week, T

3 Cal s/Week, T

2Cal s/Week, r

M =7

= 1 Day

= 1 Day

= 1 Day

= 6 Hours

Optimum Positions for Calibrations are Used

Table I A numerical comparison between predicted time error by approach B

shown with K and K and time error due to "accuracy" shown with

/ -2 2

f

~ f M * \6y )/a (T). K or K is equal to k • M in eq (51) for flicker

y

f f noise FM or white noise FM, respectively.

59
FORM NBS-114A (1-71) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS-TN-626

2. Gov't Accession No.

4. TITLE AND SUBTITLE

The Generation of an Accurate and Uniform Time Scale

with Calibrations and Prediction

7. AUTHOR(S)

Kazuyuki Yoshimura

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

3. Recipient's Accession No.

5. Publication Date

6. Performing Organization Code

8. Performing Organization

10. Project/Task/ Work Unit No. 2730245

12. Sponsoring Organization Name and Address

Same as Item 9.

11. Contract/Grant No.

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) We express a predicted time interval (or frequency) of a single clock as a weighted sum of frequency data obtained by calibrations against a primary standard,

and derive a matrix equation for the optimum weighting coefficients (called the optimum filter response function) involving the Allan variances. Two approaches are

^ used. One of the approaches turns out to be a generalization of Barnes' approach in

his 1966 IEEE paper.

We solve the matrix equation to get the optimum filter response functions for

white noise frequency modulation (FM), flicker noise FM and linear combinations of

them. Other important time dispersive mechanisms exist in practice but are not

considered here. We obtain the result that the mean square time prediction error

would increase as elapsed time t 2 for the case without intermediate calibrations.

We obtain the result that with a small number of good clocks one can construct j a time scale whose accuracy is limited by the accuracy of a primary standard. We i show that, over a long time range, linear prediction algorithms based on frequency,

calibrations with a primary standard give a time scale of much better accuracy and

stability than when intermittent calibrations are not used, and that (at least for

statistically identical clocks), no advantage is gained by using the time scale itself as a "primary standard" for intermediate calibrations.

17. key WORDS (Alphabetical order, separated by semicolons) Accurate and uniform time scale; Allan

variance; Dispersion of time scale; Ensemble time (error); Prediction interval;

Primary standard and clocks.

18. AVAILABILITY STATEMENT

fxl UNLIMITED.

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

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64

22. Price

60£

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