(→cos(t-2)) |
(→cos(t-2)) |
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== cos(t-2) == | == cos(t-2) == | ||
| − | A time shift should not effect the energy or power of periodic function over one period. | + | A time shift should not effect the energy or power of periodic function over one period (0 to 2<math>\pi</math> in this case). |
Revision as of 05:54, 5 September 2008
cos(t-2)
A time shift should not effect the energy or power of periodic function over one period (0 to 2$ \pi $ in this case).
$ u = (t-2) $
Energy
$ E=\int_{-2}^{2\pi-2}{|cos(u)|^2du} $
$ E=\frac{1}{2}\int_{-2}^{2\pi-2}(1+cos(2(u)))du $
$ E=\frac{1}{2}((u+\frac{1}{2}sin(2(u)))|_{u=-2}^{u=2\pi-2} $
$ E=\frac{1}{2}(2\pi-2 + .378 -(-2 - .378)) $
$ E=\pi $
Power
$ E=\frac{1}{2\pi-0}\int_{-2}^{2\pi-2}{|cos(u)|^2du} $
$ =\frac{1}{2\pi-0} *{\frac{1}{2}}\int_{-2}^{2\pi-2}(1+cos(2u))du $
$ =\frac{1}{4\pi}((u)+\frac{1}{2}sin(2u))|_{u=-2}^{u=2\pi-2} $
$ =\frac{1}{4\pi}(2\pi-2+.378-(-2+.378)) $
$ =\frac{1}{2} $
