Energy Signals

Average Energy $E_x$ $x(t)$ is defined as,

E_{x} = \int_{- \infty}^{\infty} | x (t) |^{2} d t

$0<E_x<\infty$ $x(t)$ is an Energy Signal.

Power Signals

Average Power $P_x$ $x(t)$ is defined as,

P_{x} = lim_{T \to \infty} \frac{1}{T} \int_{- \frac{T}{2}}^{\frac{T}{2}} | x (t) |^{2} d t

$0<P_x<\infty$ $x(t)$ is a Power Signal.

NOTE: a signal cannot be both an energy and a power signal. It can, however, be neither.

Energy Spectral Density $\Psi_x(f)$ $x(t)$ is defined as,

Ψ_{x} (f) = | X (f) |^{2}

Power Spectral Density $S_x(f)$ $x(t)$ is defined as,

S_{x} (f) = lim_{T \to \infty} \frac{1}{T} | X_{T} (f) |^{2}

$X_T(f)$ $x_T(t)$ and,

x_{T} (t) = x (t) Π (\frac{t}{T})

$x_T(t)$ $x(t)$ $-\frac{T}{2}<t<\frac{T}{2}$ $0$ $t$ :

\begin{matrix} x_{T} (t) = {\begin{cases} x (t), & i f - \frac{T}{2} < t < \frac{T}{2} \\ 0, & o t h e r w i s e \end{cases} \end{matrix}

$x(t)$ is a periodic signal, the Average Power is defined as,

P_{x} = \sum_{n = - \infty}^{\infty} | X_{n} |^{2}

and the Power Spectral Density is defined as,

S_{x} (f) = \sum_{n = - \infty}^{\infty} | X_{n} |^{2} δ (f - \frac{n}{T})

$y(t)$ $x(t)$ $h(t)$ ,

S_{y} (f) = S_{x} (f) | H (f) |^{2}