Relationship between Long and Short-term Correlations

As Voss and Clarke[50] point out many fluctuating signals can be characterized by a single correlation time $\tau_{c}$. In which case, for time scales much smaller than $\tau_{c}$ (which means for frequencies much larger than $1/ \tau_{c}$) the signal is correlated and the power spectrum's slope is close to that of the $1 / {f}^{2}$ line, and in the regions much bigger than $\tau_{c}$ ( $f \ll 1/ \tau_{c}$) the spectrum is similar to that of white noise. However, a signal which behaves like $1/f$ noise cannot be characterized by a single correlation time. In fact a spectrum with a $1/f$ slope implies a scale-invariant correlation between long-term and short-term correlation in the region in which the spectrum is exhibiting the $1/f$ slope.