Signal Energy
Signal Energy expended from $ t_1\! $ to $ t_2\! $ for CT functions is given by the formula $ E = \int_{t_1}^{t_2} \! |x(t)|^2\ dt $
The total signal energy for a signal can be found by the formula $ E = \int_{-\infty}^{\infty} \! |x(t)|^2\ dt $
For DT signals, the total energy is given by the formula $ E_{\infty} = \sum^{\infty}_{n=-\infty} |x[n]|^2 \! $
Example:
$ x(t) = e^{j (\pi t)}\! $
$ E = \int_{-\infty}^{\infty} \! |e^{j (\pi t)}|^2\ dt $
$ E = \int_{-\infty}^{\infty} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt\! $
$ E = \int_{-\infty}^{\infty} 1 dt \! $
$ E = x|_{-\infty}^{\infty}\! $
$ E = \infty $
Signal Power
For CT functions, the power of a signal from $ t_1\! $ to $ t_2\! $ is given by the function $ P_{avg}=\frac{1}{t_2-t_1} \int_{t_1}^{t_2} |x(t)|^2\ dt \! $
The total signal power is given by the function $ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} |x(t)|^2\ dt \! $
Total signal power for DT signals is given by the formula $ P_{\infty} = \lim_{N \to \infty} \frac{1}{2N+1} \sum^{N}_{n=-N} |x[n]|^2\! $
Example:
$ x(t) = e^{j (\pi t)}\! $
$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} \! |e^{j (\pi t)}|^2\ dt $
$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt $
$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} \! 1 dt $
$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} 2t $
$ P_{\infty} = 1\! $
