Energy or power: e^(−5t)u(t+2)

• 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. $e^{t}$, $t\,u(t)$).

$x(t) = e^{-5t}u(t+2)$

$u(t+2) = 1$ for $t \ge -2$, so the signal starts at $t = -2$ and then decays.

Energy:

$$\begin{aligned} E &= \int_{-2}^{\infty}e^{-10t}\,dt = \left[\frac{e^{-10t}}{-10}\right]_{-2}^{\infty} \\ &= 0 + \frac{e^{20}}{10} = \frac{e^{20}}{10} \approx 4.85 \times 10^{7} \end{aligned}$$

This is a large but finite value.

Power:

$$P = \lim_{T\to\infty}\frac{1}{2T}\cdot\frac{e^{20}}{10} = 0$$

Answer: $x(t)$ is an energy signal with $E = e^{20}/10 \approx 4.85 \times 10^7$ J (normalised) and $P = 0$.