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