Calculation of the period of the phase signal.


Every day the UTC time difference between the zero crossing point of the phase signal obtained in the mathematical modelisation and the series of events: To, To + 23h56m4s, To + 2.[23h56m4s], To + 3.[23h56m4s], etc... is measured , where To is the UTC time of the first zero crossing point.

As after 178 days, the zero crossing point has not been away from the above events of more than 40 minutes (with an algebraic average of +12 minutes relatively to the series of events) , the mean period of the phase signal may be considered in the worst case as being the sidereal day with an accuracy of : [40*60*2]/178 = 26.9 seconds.

This result is in good agreement with the result which is obtained with a standard statistical analysis.

To show that, we must note firstly that the n=179 zero crossing points: To, T1, T2, ....T178 define (n-1)n[1/2] trials of the measurement of the period which are T1-To, T2-T1, T3-T2, etc... for one time the period, T2-T0, T3-T1, T4-T2, etc... for two times the period, etc for three times the period...until 178 times the period with a total of trials of 15931.

Thus as the worst case standard deviation relatively to the average of the measured days ( S = 23h56m12s) is : eq19 , the standard deviation of the average or uncertainty is : eq20.

Unfortunately (but without real importance)a Fourier analysis of the phase signal has not been performed, simply due to a lack of time to program the computer to do it. Because this experiment was not really my job at the Belgian Telephone Company.

Finaly the negative linear drift (see below) of the zero-crossing points relatively to GMT days will convince anybody that the period has nothing to do with the mean solar day.

gmt

I think also important to point out here that in the experiment there is a PLL-follower between each of the two 1.5 km coaxial lines and the corresponding phase comparators which was integrated in the apparatus for an other usage in telephony (test of synchronization of noisy signals from long distance rubidium clocks) and thus difficult to extract The presence of these PLLs , due to its time constant of only 100s, is without influence on the result of the experiment and also because they have been tested with a ceasium clock to be as stable as 2 ns of non-cumulative phase rotation a day.

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