On the Autocorrelation Function of 1/ f Noises

The outputs of many real-world complex dynamical systems are time series characterized by power-law correlations and fractal properties. The first proposed model for such time series comprised fractional Gaussian noise (fGn), defined by an autocorrelation function C ( k ) with asymptotic power-law behavior, and a complicated power spectrum S ( f ) with power-law behavior in the small frequency region linked to the power-law behavior of C ( k ) . This connection suggested the use of simpler models for power-law correlated time series: time series with power spectra of the form S ( f ) ∼ 1 / f β , i.e., with power-law behavior in the entire frequency range and not only near f = 0 as fGn. This type of time series, known as 1 / f β noises or simply 1 / f noises, can be simulated using the Fourier filtering method and has become a standard model for power-law correlated time series with a wide range of applications. However, despite the simplicity of the power spectrum of 1 / f β noises and of the known relationship between the power-law exponents of S ( f ) and C ( k ) , to our knowledge, an explicit expression of C ( k ) for 1 / f β noises has not been previously published. In this work, we provide an analytical derivation of C ( k ) for 1 / f β noises, and we show the validity of our results by comparing them with the numerical results obtained from synthetically generated 1 / f β time series. We also present two applications of our results: First, we compare the autocorrelation functions of fGn and 1 / f β noises that, despite exhibiting similar power-law behavior, present some clear differences for anticorrelated cases. Secondly, we obtain the exact analytical expression of the Fluctuation Analysis algorithm when applied to 1 / f β noises.