Generalized Amplitude Truncation of Gaussian 1/f^α 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/ƒ^α appears to be transformed again into a power law spectrum 1/ƒ^β under rather general condition for the truncation levels. We examine the above numerical results analytically. When 1<α<2 and x_max=-x_min=a, the transformed spectrum is shown to be characterized by a certain corner frequency ƒ_c which divides the spectrum into two parts with different exponents. We derive ƒ_c depending on a as ƒ_c ∼ a^-2/(α-1). It turns out that the output signal should deviate from the power law spectrum when the truncation is asymmetric. We present a numerical example such that 1/ƒ² noise converges to 1/ƒ noise by applying the transformation y(t)=sgn[x(t)] repeatedly.