PTTI-tutorial

Clock Statistics: A TutorialDon PercivalApplied Physics LaboratoryUniversity of Washington, SeattleMotivating Example: I• consider following measurements:30-40-11010 1.160 0-10 -1.160 256 512t (days)− top: X =time (phase) difference between clock 55 andtUSNO time scale at day t (adjusted for systematic drift)(1)− bottom: X = X −X ∝ fractional frequency deviatet t−1taveraged over one day1X - X (ns) X (ns)t t-1 t13parts in 10Motivating Example: II• clock statistics used to summarize performance(1)− if X constant, clock 55 agrees with time scale (essentially)t(1)−X has stochastic (noise-like) fluctuationst− statistics used to quantify fluctuations• sample statistics(1)N−11− mean: µˆ = Xtt=0N(here N =512=# of measurements)(1)N−12 1 2− variance: σˆ = (X −µˆ)t=0 tN−σˆ (standard deviation) is measure of spread• easiest to interpret µˆ &ˆ σ if data taken to be independentsamples from Gaussian (i.e., normal) distribution2Motivating Example: III• Q: is Gaussian assumption reasonable?• comparison of histogram to probability density function:0.20.10.0-10 -5 0 5 10x (ns)− Gaussian assumption seems reasonable3PDFsMotivating Example: IV• Q: is independent assumption reasonable?• under Gaussianity, uncorrelatedness implies independence• sample autocorrelation sequence measures uncorrelatedness:(1) (1)N−τ−1(X −µˆ)(X −µˆ)t=0 t t+τρˆ = ,τ =1, 2,...,N− 1τ(1)N−12(X −µˆ)tt=0• can interpret ρˆ as correlation ...