(Lu Zhang-show the property) |
(→if E_{\infty} is finite, then P_\infty equals to zero- Ali Alyoussef: new section) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 2: | Line 2: | ||
---- | ---- | ||
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> | ||
| + | |||
| + | == if E_{\infty} is finite, then P_\infty equals to zero- Ali Alyoussef == | ||
| + | |||
| + | from the formula, it can be seen that | ||
| + | P = the limit of (E/2T) when T goes to infinity. | ||
| + | |||
| + | and if E is a fixed value < infinity | ||
| + | |||
| + | |||
| + | => P = E/infinity which will guarantees that we will have a result of zero for P. | ||
Latest revision as of 17:28, 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 $
if E_{\infty} is finite, then P_\infty equals to zero- Ali Alyoussef
from the formula, it can be seen that
P = the limit of (E/2T) when T goes to infinity.
and if E is a fixed value < infinity
=> P = E/infinity which will guarantees that we will have a result of zero for P.
