1/f Spectra as a Consequence of the Randomness of Variance

Abstract

It is a general conviction that any measured noise be stochastically continuous and weak stationary. Therefore, standard noise analysis uses the substitution of ensemble averages by time averages, and it considers likewise the autocorrelation function and the sample spectrum as an unbiased and complete characterization of the measured process. However, randomly distributed discontinuities make the constant variance turn into a random one. This contradicts the standard suppositions. We consider the random walk as a typical non-continuous process and derive the influence of the ‘variance of variance’ on the measured spectrum. In contrast to the standard analysis, sums of squares are no longer proportional to the chi-square-distribution, but to a distribution with a larger variance. When decomposing the data into fixed and random variance components, it can be shown that, despite independent increments, the random variance component produces a positive and time dependent expectation of the covariance. This is the source of the typically shaped non-zero autocorrelation function and the 1/f spectrum. The expectation of the autocorrelation at any given time difference is the product of the random variance component and a factor, which depends only on the total number of data and on the number of sampling intervals between the associated pairs of data. Consequently, the 1/f spectrum is no longer to be understood within the meaning of Parseval’s theorem. The larger the ratio of the random to the fixed variance component, the higher the 1/f increase onset frequency. ‘Almost smooth’ processes yield an ‘almost white’ spectrum, larger variance of increments generates a 1/f spectrum over a larger range of frequencies, and if the quotient between random and fixed variance components approaches 1, the 1/f spectrum will appear to extend over the full range of frequencies.