Application of Kalman Filters and ARIMA Models to Digital Frequency and Phase Lock Loops

摘要

This paper considers various 'optimal' solutions to the slave oscillator system for two physical models. Model A is a reference oscillator whose frequency variations are white noise (i.e., random walk phase fluctuations). The slave oscillator, when free running, has pure random walk noise of the frequency variations (i.e., random walk FM), in contrast to the reference. Thus, the reference signal has a higher level of short-term fluctuations than the free running slave, and the reverse is true in long-term. While this physical model is not adequate for many applications it is qualitatively similar to real situations and it is easy to recognize and understand just what is optimized. Model B is a reference oscillator with negligible frequency noise. The slave oscillator is contaminated with both white frequency noise and random walk frequency noise. This model also describes certain situations encountered in practice. One example is the problem of providing a real time output from an ensemble of oscillators, since the ensemble average is more stable than any of the member oscillators. Asymptotically, Kalman Filters often approach a simple ARIMA model. (Box and Jenkins). As shown below, these ARIMA models in turn are optimal. One of the primary advantages of Kalman filters over ARIMA models is that they easily handle transient responses such as initial turn-on or irregular data sampling (Gelb). Still ARIMA models often are adequate for many real systems.