Simulations of the Hadamard Variance: Probability Distributions and Confidence Intervals

Power-law noise in clocks and oscillators can be simulated by Fourier transforming a modified spectrum of white phase noise. This approach has been applied successfully to simulation of the Allan variance and the modified Allan variance in both overlapping and nonoverlapping forms. When significant frequency drift is present in an oscillator, at large sampling times the Allan variance overestimates the intrinsic noise, while the Hadamard variance is insensitive to frequency drift. The simulation method is extended in this paper to predict the Hadamard variance for the common types of power-law noise. Symmetric real matrices are introduced whose traces-the sums of their eigenvalues-are equal to the Hadamard variances, in overlapping or nonoverlapping forms, as well as for the corresponding forms of the modified Hadamard variance. We show that the standard relations between spectral densities and Hadamard variance are obtained with this method. The matrix eigenvalues determine probability distributions for observing a variance at an arbitrary value of the sampling interval $\tau$, and hence for estimating confidence in the measurements.