Gps-based Adaptive Estimating of Time Errors for Clock Synchronization with a Fir Filter

In this paper, we investigate one of the possibilities to adapt an unbiased moving average (MA) filter cfinite impulse response [FIRIjZter) to the slope of time error function. The linear regression coefficient is used as a statistical estimator of sample slope. We evaluate the error of the slope estimate and present two options for the adapting coefficient determination. To examine them, we generate the time error noisy signal with linear trend and estimate it with a simple MA, the optimally unbiased MA, and the two adapted fiiers. The particular errors of the filters, namely bias, RMSD, RMSE and maximal error are then compared. Finally we mark special features of the linear-regression-based adaptation. INTRODUCTION Fast and extremely accurate “on-line” GPS-based estimating of time errors in timekeeping is probably the noblest example of the optimal filtering application in Time and Frequency. Estimation deals here with the time error function of local clock, which demonstrates some special features, which are: Its deterministic model seems to be quadratic at least, since it is formed with an initial constant error (xo), an initial frequency offset of local oscillator for the reference one (yo), and a linear aging component (D) of local oscillator. Its noise fits well white Gaussian noise (note: the more noise is whitened, the more accurate linear optimal estimate is achieved). Such properties of the time error function allow applying the linear optimal filtering theory (linear Kalman filtering) straightforward to recent studies [l] have showed that for the sake of accuracy, it is not enough just to use the three-space-state Kalman filter totally matched with the above-mentioned clock time model. The known computation problem here is noise produced by the state space discrete time Kalman filter (the more states, the more noise). On the other hand, both the realizable Wiener and the simple moving average (MA) filters inherently produce the bias for the nonstationary processes. In our report [2], we show that of the filters with the same time constant an optimally unbiased MA filter (finite impulse response [FIR] filter) produces noise lower then the three-state Kalman with rather the same bias. However, the noise is bigger than that of the simple MA. Knowing that a simple MA produces the lowest possible noise among all the filters [3] and zero-bias for the stationary process (yo = 0 and D = 0), we then wonder if it is possible to adapt the filter [2] to the slope of the time error function? In other words, can we design the unbiased MAfilter with the same smallest noise as that of a simple MA? The answer, unfortunately, is not exhaustively positive.