Does a power spectral density that follow a power law imply that the time hisory follows a power law?

Question

This is a question related to this previous question "I cannot find reference (paper) of this relation $u(t)t^{α} ↔^{FT} f^{-(α+1)}$".

The signal that I work with g(t), is the time history of a velocity component flucutuations. This signal comply with the rule:

$\bar{g}=lim_{T->+\infty}\int_{0}^{T}g(t)dt=0$

The Power spectrum of g(t) follows in a frequency range a power law "-5/3", in other frequency ranges the slope can change. (SEE for example, figure 1 [here])2. The power law frequency range, following the rule $u(t)t^{α} ↔^{FT} f^{-(α+1)}$ should imply that also the time history has a power law. The signals I work with does not seems to have a power law. Is this can be due to a masking performed by the behaviour of the other frequency range? The rule

$\bar{g}=lim_{T->+\infty}\int_{0}^{T}g(t)dt=0$

can violate some hypotheses at the base of $u(t)t^{α} ↔^{FT} f^{-(α+1)}$" ?

Answer 1

should imply that also the time history has a power law

No. The PSD is an incomplete description of the signal since it has lost the phase information. There are infinitely many time sequences that have the same PSD. For example a unit impulse, white noise and a linear sweep have the same PSD (roughly speaking) but are completely different time-domain wave forms.

In your case $u(t) \cdot t^{\alpha}$ is one signal that fits your PSD but there are many others that will fit too but have a totally different shape.

This being said, your observation of a conflict has some merit. Your first equation simply states that $g(t)$ is mean free which in turns implies that the PSD at $f=0$ must be zero, but $0^{-5/3}=1$. I'm guessing that your power law applies only to a limited frequency range which doesn't include $f=0$ and hence there is no conflict.