Discrete simulation of power law noise

A method for simulating power law noise in clocks and oscillators is presented based on modification of the spectrum of white phase noise, then Fourier transforming to the time domain. Symmetric real matrices are introduced whose traces–the sums of their eigenvalues–are equal to the Allan variances, in overlapping or non-overlapping forms, as well as for the corresponding forms of the modified Allan variance. Diagonalization of these matrices leads to expressions for the probability distributions for observing a variance at an arbitrary value of the sampling or averaging interval $\tau$, and hence for estimating confidence in the measurements. A number of applications are presented for the common power-law noises.