Generalized Amplitude Truncation of Gaussian 1/f^alpha noise

Abstract

We study a kind of filtering, an amplitude truncation with upper and lower truncation levels x_max and x_min. This is a generalization of the simple transformation y(t)=sgn[x(t)], for which a rigorous result was obtained recently. So far numerical experiments have shown that a power law spectrum 1/f^alpha seems to be transformed again into a power law spectrum 1/f^beta under rather general condition for the truncation levels. We examine the above numerical results analytically. When 1<alpha<2 and x_max = -x_min = a, the transformed spectrum is shown to be characterized by a certain corner frequency f_c which divides the spectrum into two parts with different exponents. We derive f_c depending on a as f_c sim a^(-2/(alpha-1)). It turns out that the output signal should deviate from the power law spectrum when the truncation is asymmetrical. We present a numerical example such that 1/f^2 noise converges to 1/f noise by applying the transformation y(t)=sgn[x(t)] repeatedly.