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

Revision as of 09:22, 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 $

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