Leviathan

1/f noise in Power spectrum Leads to Long-memory effects

Whilst it is relatively simple to yield power-law (scale-free) relationships in the correlation function (see below), it is of particular interest if the relationship is of the form, $$P(f) \sim f^{-1}$$ .. so called '1/f noise', 'flicker noise', or 'pink noise' (see w: link on parent page). The reasoning is as follows:

  1. We have the temporal correlation function: $$ G(\tau) = \langle N(\tau_0)N(\tau_0+\tau)\rangle_{\tau_0} - \langle N(\tau_0)\rangle_{\tau_0}{2}$$ and the cosine transformed '''power spectrum''' $$ S(f) = 2\int_{0}{\infty} d\tau G(\tau) \cos(2\pi f\tau)\,\,.$$,

  2. Now, suppose $$S(f) \sim f{-\beta}$$ and $$G(\tau) \sim \tau{-\alpha}$$,

  3. Notice that given $$ S(f) = 2\int_{0}{\infty} d\tau G(\tau) \cos(2\pi f\tau)$$ on the LHS we have $1/f{\beta}$ for the power spectrum, whilst on the right hand side we have $G(\tau)$ which is a linear function of $\tau$. Now $\tau$ is the interval of correlation, or in other words the period of correlation. Hence, since we assume $G(\tau)\sim 1/\tau{-\alpha}$, we are actually assuming, $G(\tau)\sim f{\alpha}$;

  4. Combine this with the cosine transform of f in the power spectrum and we have that (LHS ~ RHS):$$1/f{\beta} \sim \frac{f{\alpha}}{f} = 1/f^{1-\alpha}$$

That is, if we find that $\beta \rightarrow 1$ (as in flicker noise), then this implies that $\alpha \rightarrow 0$, which causes the assumed relationship for $G(\tau)$ to break down and instead be replaced by slow logarithmic decay. Very low-frequency signals have significant impact on inter-temporal events. Hence, the interest in 1/f distributions in the power function.

Ref: p.9 of Jensen

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