Define energy and power signals

• Energy signal: a signal whose total energy is finite and non-zero, $0 < E < \infty$. Its average power is then zero. Example: a single pulse, $e^{-2t}u(t)$.
• Power signal: a signal whose average power is finite and non-zero, $0 < P < \infty$. Its total energy is then infinite. Example: $\cos\omega_0 t$, $u(t)$, any periodic signal.

For a continuous-time signal:

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

For a discrete-time signal:

$$E = \sum_{n=-\infty}^{\infty}|x[n]|^2, \qquad P = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2$$

A signal that has neither finite energy nor finite non-zero power is neither an energy nor a power signal (e.g. $e^{t}$, $t\,u(t)$).

Examples

1. Energy signal: $x(t) = e^{-2t}u(t)$

$$E = \int_0^\infty e^{-4t}\,dt = \frac{1}{4}, \qquad P = 0$$

2. Power signal: $x(t) = A\cos\omega_0 t$

$$P = \frac{1}{T_0}\int_0^{T_0} A^2\cos^2\omega_0 t\,dt = \frac{A^2}{2}, \qquad E = \infty$$

3. DT energy signal: $x[n] = (0.5)^n u[n]$, $E = \sum_{n=0}^\infty 0.25^n = \frac{1}{1-0.25} = \frac{4}{3}$.

4. DT power signal: $x[n] = u[n]$, $P = \lim_{N\to\infty}\frac{N+1}{2N+1} = \frac{1}{2}$.

In general, time-limited (finite-duration) bounded signals are energy signals, and periodic signals are power signals.