Stochastic models and statistical analysis for clock noise

Abstract

In this primer we first give an overview of stochastic models that can be used to interpret clock noise. Because of their statistical tractability, we concentrate on fractionally differenced (FD) processes, which we relate to fractional Brownian motion, discrete fractional Brownian motion, fractional Gaussian noise and discrete pure power law processes. We discuss several useful extensions to FD processes, namely, composite FD processes, autoregressive fractionally integrated moving average processes and time-varying FD processes. We then consider the statistical analysis of clock noise in terms of how these models are manifested via the spectral density function (SDF) and the wavelet variance (WV), the latter being a generalization of the well-known Allan variance. Both the SDF and the WV decompose the process variance with respect to an independent variable (Fourier frequency for the SDF, and scale for the WV); similarly, judiciously chosen estimators of the SDF and WV decompose the sample variance of clock noise. We give an overview of the statistical properties of estimators of the SDF and the WV and show how these estimators can be used to deduce the parameters of stochastic models for clock noise.