Problem
Given complex signal $ f(t)=e^{jt} = \cos(t) + j \sin(t) $, find $ E_\infty $ and $ P_\infty $.
Background Knowledge
$ E_\infty(x(t)) = \int_{-\infty}^\infty |x(t)|^2\,dt $
$ P_\infty(x(t)) = \lim_{T \to \infty} (\frac{1}{2T} \int_{-T}^T |x(t)|^2\,dt) $
Solution
- $ |x(t)| = |\cos(t) + j \sin(t)| = \sqrt{(\cos(t))^2+(\sin(t))^2} = \sqrt{1} = 1 $
- $ E_\infty(x(t)) = \int_{-\infty}^\infty |x(t)|^2\,dt = \int_{-\infty}^\infty 1\,dt = t|_{-\infty}^{\infty} = \infty $
- $ P_\infty(x(t)) = \lim_{T \to \infty} (\frac{1}{2T} \int_{-T}^T |x(t)|^2\,dt) = \lim_{T \to \infty} (\frac{1}{2T} \int_{-T}^T 1\,dt) = \lim_{T \to \infty} (\frac{1}{2T} t|_{-T}^T) $
$ = \lim_{T \to \infty} (\frac{1}{2T} (2T)) = \lim_{T \to \infty} (1) = 1 $
