Optimal FIR filtering of clock models

In recent years, finite impulse response (FIR) filtering has demonstrated important features making the approach attractive for various applications in timekeeping. In this tutorial, we review methods of optimal FIR filtering of discrete-time clock models. The tutorial is organized to have two parts. In the first part, possible methods for optimal filtering of clock state are observed and it is pointed out that FIR filtering is most natural for solving clock problems. In the second part, we give simple engineering presentations for methods of FIR filtering and discussed a number of useful applications associated with filtering out the measurement noise, clock synchronization by the Global Positioning System (GPS) timing one pulse per second (1PPS) signals, prediction of clock errors, holdover problem, ascertaining the initial clock state by FIR smoothing, best linear fitting of clock errors, and some others. The methods of FIR filtering are supplied with the engineering algorithms. For the comparison, the trade-off between the Kalman algorithm and the optimal FIR one is also discussed in almost each of the applications. We show that the most efficient engineering solution for clocks is the l-degree unbiased FIR filter, predictor, or smoother. Neither of these estimators involves noise and the clock initial state to the algorithm. Herewith, in applications to clocks, each of them demonstrates features often superior to the Kalman algorithm, when a number of measurements on the averaging interval is large.