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Latest revision as of 22:14, 4 September 2008
Signal
$ x(t) = e^{jt} $
Energy
$ E_\infty = \int_{-\infty}^\infty |x(t)|^2dt $
$ = \int_{-\infty}^\infty |e^{jt}|^2dt $
$ = \int_{-\infty}^\infty |cos(t) + jsin(t)|^2dt $ (Euler's Formula)
$ = \int_{-\infty}^\infty {\sqrt{cos^2(t) + sin^2(t)}}^2dt $ (Magnitude of a Complex Number)
$ = \int_{-\infty}^\infty dt $
$ = t|_{-\infty}^\infty $
$ = \infty - (-\infty) $
$ = \infty $
Power
$ P_\infty = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T |x(t)|^2dt $
$ = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T |e^{jt}|^2dt $
$ = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T |cos(t) + jsin(t)|^2dt $ (Euler's Formula)
$ = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T {\sqrt{cos^2(t) + sin^2(t)}}^2dt $ (Magnitude of a Complex Number)
$ = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T dt $
$ = \lim_{T \to \infty} \frac{1}{2T} t|_{-T}^T $
$ = \lim_{T \to \infty} \frac{1}{2T} [T - (-T)] $
$ = \lim_{T \to \infty} \frac{1}{2T} (2T) $
$ = \lim_{T \to \infty} 1 $
$ = 1 $
