**4.3 Algorithms of group time and frequency scale formation**

Grouping of keepers ensures high precision and reliability of autonomous storage and playback of group time and frequency scales in intervals of long duration. This is achieved by application of the corresponding algorithms of processing of results of internal and external comparisons of group keeper's time scales. Principle of algorithms is in parameter estimation of individual keepers according to results of scale and frequency measurements within a group and drift compensation from group time scale nominal value. Changes of points of time of keepers' time scales *kT <sup>i</sup>* and frequency scales *kf <sup>i</sup>* , that form a group, are represented by mathematical models of the kind:

$$T\_i\left(k+1\right) = T\_i\left(k\right) + \int\_{t\_k}^{t\_{k+1}} f\left(\tau\right) \cdot d\tau,\ T\_i\left(0\right) = T\_{i0}\,\tag{4}$$

$$f\_i(k+1) = f\_i(k) + d\_i(k) \cdot \left[t\_{k+1} - t\_k\right] + w(k+1), \\ f\_i(0) = f\_{i0}.\tag{5}$$

Within a group of keepers, differences between points of scales *z k T k Tk v k* 00 0 *i ii* and differences between frequencies *z k f k fk v k f i* 00 0 *i fi* with reference to the *T k* <sup>0</sup> scale and the *f*<sup>0</sup> *k* frequency of the key keeper which has the best characteristics in the group are measured. Estimated time scale point *T*<sup>0</sup> 0 and estimated frequency *f*<sup>0</sup> 0 of the key keeper for the point of time <sup>0</sup>*t* , received as a result of comparison of the key keeper with time and frequency standard, are assumed as initial conditions for group scale formation.

Algorithm of group keeper's time scale formation includes forecast calculation of group keepers frequencies:

$$
\vec{f}\_i(k+1) = \hat{f}\_i(k) + \hat{d}\_i(k) \cdot \left[t\_{k+1} - t\_k\right] \tag{6}
$$

according to calculated estimations of frequencies *fi <sup>k</sup> <sup>f</sup>*0 0 *kz k f i* and calculation of an average estimation of the key keeper frequency for the ensemble of keepers:

$$
\hat{f}\_0\left(k+1\right) = \frac{1}{n-1} \sum\_{i=1}^{n-1} \beta\_i \cdot \left[ \vec{f}\_i\left(k+1\right) - z\_{f0i}\left(k+1\right) \right].\tag{7}
$$

The received estimations of frequencies *f k <sup>i</sup>* and time scale estimations*T k <sup>i</sup>* , calculated according to equations*Tk T k z k i i* 0 0 , are used for calculation of time scales forecast :

$$
\vec{T}\_i(k+1) = \hat{T}\_i(k) + \hat{f}\_i(k) \cdot \left[t\_{k+1} - t\_k\right], \\
\hat{T}\_i(0) = \hat{T}\_{i0} \tag{8}
$$

On the basis of these forecasts and results of scale measurements *z k* <sup>0</sup>*<sup>i</sup>* 1 an average estimation of the key keeper's scale for the ensemble of keepers is calculated:

$$
\hat{T}\_0\left(k+1\right) = \frac{1}{n-1} \sum\_{i=1}^{n-1} \gamma\_i \cdot \left[\vec{T}\_i\left(k+1\right) - z\_{0i}\left(k+1\right)\right].\tag{9}
$$

The given algorithms are used in practical work for formation of main time and frequency standards scales.

#### **5. Synchronization of spaced-apart clocks**

#### **5.1 General information**

Time scale synchronization implies correction of electronic clock that is correction of its indications and of reference generator frequency. After synchronization is carried out, for

estimation of individual keepers according to results of scale and frequency measurements within a group and drift compensation from group time scale nominal value. Changes of points of time of keepers' time scales *kT <sup>i</sup>* and frequency scales *kf <sup>i</sup>* , that form a group, are

<sup>1</sup>

*k*

Within a group of keepers, differences between points of scales *z k T k Tk v k* 00 0 *i ii* and differences between frequencies *z k f k fk v k f i* 00 0 *i fi* with reference to the *T k* <sup>0</sup> scale and the *f*<sup>0</sup> *k* frequency of the key keeper which has the best characteristics in the group are measured. Estimated time scale point *T*<sup>0</sup> 0 and estimated frequency *f*<sup>0</sup> 0 of the key keeper for the point of time <sup>0</sup>*t* , received as a result of comparison of the key keeper with time and frequency

Algorithm of group keeper's time scale formation includes forecast calculation of group

*fi i i kk k fk dk t t* <sup>1</sup> <sup>1</sup> (6)

according to calculated estimations of frequencies *fi <sup>k</sup> <sup>f</sup>*0 0 *kz k f i* and calculation of

The received estimations of frequencies *f k <sup>i</sup>* and time scale estimations*T k <sup>i</sup>* , calculated according to equations*Tk T k z k i i* 0 0 , are used for calculation of time scales forecast :

1 0 1 , 0 *Tk Tk i i i k ki i f kt tT T*

On the basis of these forecasts and results of scale measurements *z k* <sup>0</sup>*<sup>i</sup>* 1 an average

 1 0 0 1

The given algorithms are used in practical work for formation of main time and frequency

Time scale synchronization implies correction of electronic clock that is correction of its indications and of reference generator frequency. After synchronization is carried out, for

<sup>1</sup> 1 11

*ii i*

<sup>1</sup> 1 11

*i i f i*

. (7)

. (8)

. (9)

an average estimation of the key keeper frequency for the ensemble of keepers:

1 *n*

estimation of the key keeper's scale for the ensemble of keepers is calculated:

1 *n*

*n*

**5. Synchronization of spaced-apart clocks** 

*i T k Tk z k*

*n*

*i f k fk z k*

1 0 0 1

standard, are assumed as initial conditions for group scale formation.

*k*

*t i i i i t Tk Tk f d T T* 

<sup>0</sup> 1 , 0 ,

(4)

1 1 1 0 , 0 . *i i i kk i i f k f k d k t t wk f f* (5)

represented by mathematical models of the kind:

keepers frequencies:

standards scales.

**5.1 General information** 

some finite time interval a period of an event at the scale being formed coincides with the required precision with a period of the same event at the standard scale. As time goes, coincidence precision decreases as a result of different disturbing influences on the electronic clock; that is why spaced-apart clocks (comparison of clocks) should be synchronized regularly.

Until recently the most widespread methods of exact time scale synchronization were those that used radio signals of broadcasting stations having transmitted information concerning points and intervals of time at different frequencies. In order to transmit this information to a consumer, different codes and train of impulses can be used; the most widely known among them is the method of 6 points. A transmitting station pings 6 second signals (points) and one extended signal (hyphen) at the beginning of a minute within a system of universal coordinated time *UTC* , and according to the latter by means of receiving equipment time services clocks are synchronized. Precision of synchronization over radio signals is limited by conditions of radio-wave propagation in the earth's atmosphere and by instability of signal delay in receiving and transmitting equipment paths. They are used nowadays mostly in ship navigation, where permissible errors of clocks' indications are several dozens and hundreds of milliseconds. Use of television channels, man-made Earth satellites, as well as natural radiation sources, periodic electromagnetic oscillations from outer space (of pulsars) enables to considerably improve precision of transmission of time and frequency units' sizes.
