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== Signal == | == Signal == | ||
<font size="4"> | <font size="4"> | ||
| − | <math> | + | <math>f(t)=2cos(t)</math> |
</font> | </font> | ||
==Energy== | ==Energy== | ||
| − | |||
| − | <math>E=\int_0^{2\pi}{|2cos(t)|^2dt}</math> | + | According to formula of Energy of a singal,we can get: |
| + | <font size="4"> | ||
| + | |||
| + | <math>E = \int_{t_1}^{t_2} \! |f(t)|^2\ dt</math> | ||
| + | |||
| + | <math>=\int_0^{2\pi}{|2cos(t)|^2dt}</math> | ||
<math>=\int_0^{2\pi}{(2(2cos(t)^2-1)+2)dt}</math> | <math>=\int_0^{2\pi}{(2(2cos(t)^2-1)+2)dt}</math> | ||
| − | <math>=\int_0^{2\pi}{2+cos(2t))dt</math> | + | <math>=\int_0^{2\pi}{2+cos(2t))dt}</math> |
<math>=(2t+sin(2t))|_{t=0}^{t=2\pi}</math> | <math>=(2t+sin(2t))|_{t=0}^{t=2\pi}</math> | ||
| − | <math>= | + | <math>=4\pi+0-0-0</math> |
<math>=4\pi</math> | <math>=4\pi</math> | ||
| + | </font> | ||
| + | |||
| + | ==Power== | ||
| + | |||
| + | According to formula of Power of a singal,we can get: | ||
| + | <font size="4"> | ||
| + | |||
| + | <math>P=\frac{1}{2T}\int_{-T}^{T}\!|f(t)|^2\ dt</math> | ||
| + | |||
| + | <math>=\frac{1}{4\pi}\int_{-2\pi}^{2\pi}{|2cos(t)|^2dt}</math> | ||
| + | |||
| + | <math>=\frac{1}{4\pi}\int_{-2\pi}^{2\pi}{(2(2cos(t)^2-1)+2)dt}</math> | ||
| + | |||
| + | <math>=\frac{1}{4\pi}\int_{-2\pi}^{2\pi}{(2+cos(2t))dt}</math> | ||
| + | |||
| + | <math>=\frac{1}{4\pi}(2t+sin(2t))|_{t={-2\pi}}^{t=2\pi}</math> | ||
| + | |||
| + | <math>=\frac{1}{4\pi}(4\pi+0-(-4\pi)-0)</math> | ||
| + | |||
| + | <math>=\frac{1}{4\pi}(8\pi)</math> | ||
| + | |||
| + | <math>=2</math> | ||
| + | |||
| + | |||
| + | </font> | ||
Latest revision as of 16:01, 3 September 2008
Signal
$ f(t)=2cos(t) $
Energy
According to formula of Energy of a singal,we can get:
$ E = \int_{t_1}^{t_2} \! |f(t)|^2\ dt $
$ =\int_0^{2\pi}{|2cos(t)|^2dt} $
$ =\int_0^{2\pi}{(2(2cos(t)^2-1)+2)dt} $
$ =\int_0^{2\pi}{2+cos(2t))dt} $
$ =(2t+sin(2t))|_{t=0}^{t=2\pi} $
$ =4\pi+0-0-0 $
$ =4\pi $
Power
According to formula of Power of a singal,we can get:
$ P=\frac{1}{2T}\int_{-T}^{T}\!|f(t)|^2\ dt $
$ =\frac{1}{4\pi}\int_{-2\pi}^{2\pi}{|2cos(t)|^2dt} $
$ =\frac{1}{4\pi}\int_{-2\pi}^{2\pi}{(2(2cos(t)^2-1)+2)dt} $
$ =\frac{1}{4\pi}\int_{-2\pi}^{2\pi}{(2+cos(2t))dt} $
$ =\frac{1}{4\pi}(2t+sin(2t))|_{t={-2\pi}}^{t=2\pi} $
$ =\frac{1}{4\pi}(4\pi+0-(-4\pi)-0) $
$ =\frac{1}{4\pi}(8\pi) $
$ =2 $
