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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.

Shahrokh Yadegari 2001-03-01