Property: if $ E_{\infty} $ is finite, then $ P_\infty $ equals to zero.


Proof:

$ E_\infty = \int^{+\infty}_{-\infty}|x(t)|^2 dt $

$ P_\infty = \displaystyle\lim_{T\to\infty} \dfrac{1}{2T} \int^{+T}_{-T}{|x(t)|^2}{dt} $

We see from the equations above that,

$ P_\infty = \displaystyle\lim_{T\to\infty} \dfrac{E_\infty}{2T} $

For $ E_{\infty} < {\infty} $, we got that,

$ P_\infty = \displaystyle\lim_{T\to\infty} \dfrac{E_\infty}{2T} = 0 $

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Recent Math PhD now doing a post-doctorate at UC Riverside.

Kuei-Nuan Lin