E infinty & P infinty - Ryne Rayburn (rrayburn) - Rhea

Work By Ryne Rayburn (rrayburn)

$ x(t)=\sqrt{t} $

$ E\infty=\int_{-\infty}^\infty |x(t)|^2\,dt $

   $ E\infty=\int_{-\infty}^\infty |\sqrt{t}|^2\,dt=\int_0^\infty t\,dt $ (due to sqrt limiting to positive Real numbers)
   $ E\infty=.5*t^2|_{-\infty}^\infty $
   $ E\infty=.5(\infty^2-0^2)=\infty $

$ P\infty=lim_{T \to \infty} \ 1/(2T)\int_{-T}^{T} |x(t)|^2\,dt $

   $ P\infty=lim_{T \to \infty} \ \frac{1}{(2T)}\int_{-T}^T |\sqrt{t}|^2\,dt=\int_0^T t\,dt $
   $ P\infty=lim_{T \to \infty} \ \frac{1}{(2T)}*.5t^2|_0^T $
   $ P\infty=lim_{T \to \infty} \ \frac{1}{(2T)}*(.5T^2) $
   $ P\infty=lim_{T \to \infty} \ (.25T)=\infty $

$ x(t)=\cos(t)+\jmath\sin(t) $

$ |x(t)|=|\cos(t)+\jmath\sin(t)|=\sqrt{\cos^2(t)+\sin^2(t)}=1 $

$ E\infty=\int_{-\infty}^\infty |x(t)|^2\,dt $

   $ E\infty=\int_{-\infty}^\infty |1|^2\,dt=t|_{-\infty}^\infty $
   $ E\infty=\infty $

$ P\infty=lim_{T \to \infty} \ 1/(2T)\int_{-T}^{T} |x(t)|^2\,dt $

   $ P\infty=lim_{T \to \infty} \ \frac{1}{(2T)}\int|1|^2dt $
   $ P\infty=lim_{T \to \infty} \ \frac{1}{(2T)}*t|_{-T}^T $
   $ P\infty=lim_{T \to \infty} \ \frac{1}{(2T)}*(T-(-T)) $
   $ P\infty=lim_{T \to \infty} \ 1 $
   $ P\infty=1 $

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