I am Fourier analyzing at a bunch of signals of equal length and find that $$\frac{d\ln({E[P]})}{d\ln({PT'})}$$ is constant across the dataset, where $T$ is a vector of the periods (inverse of frequencies) of each signal, and $P$ is the vector of the power corresponding to each of those periods. $E[P]$ is the mean power of the given signal. I include a graph to illustrate.
Should I expect this relation in any signal? Or does this point to something unique about the signals I am dealing with?
