The mathematical genesis of the phenomenon called “1/f noise ” (10frg132)

“1/f noise ” refers to the phenomenon of the spectral density, S(f), of a signal, having the form S(f) = constant/f α, where f is frequency and α is a signal-dependent parameter. In physical situations the phenomenon is often considered to occur on an interval bounded away from both zero and infinity because these endpoints are not observable. Mathematically, however, the behavior of S(f) near these endpoints, particularly as f → 0, is of considerable interest. 1/f α signals with 0.5 < α < 1.5 are found widely in nature, occurring in physics, biology, astrophysics, geophysics, economics, psychology, language and even music [1, 2] (note this overview closely follows parts of [2]). The case of α = 1, or “pink noise”, is both the canonical case, and the one of most interest; surprisingly, as illustrated in Figure 1, many of the values for α found in nature are very near 1.0. Henceforth the term “1/f noise ” will refer only to this case; the value of α will be specified if it is other than 1.0. Figure 1. Some examples of roughly 1/f power spectra from various empirical domains. Reprinted from [2]. 1 2 1/f noise is an intermediate between white noise (α = 0) with no correlation in time and a random walk (Brownian motion, α = 2) with no correlation between increments. Brownian motion is the integral of white