Short note: Energy and power signals

Signals are classified by their energy and average power.

• Energy signal: total energy is finite, $0 < E < \infty$, so average power $P = 0$. Usually time-limited or decaying signals. Example: $e^{-at}u(t)$ ($a>0$), $E = 1/(2a)$.
• Power signal: average power is finite and non-zero, $0 < P < \infty$, so $E = \infty$. Usually periodic or everlasting signals. Example: $A\cos\omega t$, $P = A^2/2$; $u(t)$, $P = 1/2$.

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

(For DT signals the integrals become sums over $n$ and $2T$ becomes $2N+1$.)

A signal cannot be both. Some signals are neither, e.g. $e^{2t}$ or the ramp $t\,u(t)$, where both $E$ and $P$ are infinite.