(Lu Zhang-show the property) |
(Show the property- Lu Zhang-ECE301summer2009) |
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Proof: | Proof: | ||
| − | <math>E_\infty | + | |
| + | <math>E_\infty = \int^{+\infty}_{-\infty}|x(t)|^2 dt</math> | ||
| + | |||
| + | <math>P_\infty = \displaystyle\lim_{T\to\infty} \dfrac{1}{2T} \int^{+T}_{-T}{|x(t)|^2}{dt}</math> | ||
| + | |||
| + | We see from the equations above that, | ||
| + | |||
| + | <math>P_\infty = \displaystyle\lim_{T\to\infty} \dfrac{E_\infty}{2T}</math> | ||
| + | |||
| + | For <math>E_{\infty} < {\infty}</math>, we got that, | ||
| + | |||
| + | <math>P_\infty = \displaystyle\lim_{T\to\infty} \dfrac{E_\infty}{2T} = 0 </math> | ||
Revision as of 08:22, 21 June 2009
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 $
