Clock Error with a Wiener Predictor and by Numerical Calculation

This paper gives a closed-form solution for the minimum error one can expect from a linear prediction filter applied to a clock for which the fractional frequency power spectrum consists of white noise and the integral of white noise. Measurement error is also included. Expressing the problem as a Wiener filter rather than a Kalman filter simplifies the solution. The Kalman matrices in the Wiener representation are diagonal. This permits one to derive the optimum linear filter directly from the spectrum. Since the spectrum of flicker noise is not rational, no closed-form Wiener solution is possible. It is demonstrated, however, that all error terms including flicker can be taken into account by a technique of numerical integration in the frequency domain. The technique is valid for any filter for which the integrals, and consequently the error, do not diverge. It is shown that every first-order prediction filter with two poles must have the form of the Wiener filter, except for the position of the poles. A special case of this filter is shown to be the first-order exponential predictive filter. The error can be expressed in powers of the prediction time, with four coefficients, one for each spectral term. The values of these coefficients were calculated and plotted for the first-order exponential filter. These coefficients were used to calculate the error for three clocks. The results are plotted for prediction times of two hours and one day.