Energy and power signal; bandwidth of a system

Energy and power signals

• Energy signal: finite total energy $0 < E = \int_{-\infty}^{\infty}|x(t)|^2dt < \infty$; its average power is zero. Example: a single pulse, $e^{-t}u(t)$.
• Power signal: finite non-zero average power $0 < P = \lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}|x(t)|^2dt < \infty$; its energy is infinite. Example: $A\cos\omega t$, unit step.

Bandwidth of a system

The bandwidth of a system is the range of frequencies over which it passes signals with acceptable gain, usually the range where $|H(f)|$ stays within 3 dB of its maximum (i.e. power falls to half, amplitude to $1/\sqrt2 = 0.707$ of maximum).

• Low-pass system: passes $0$ to $f_c$; bandwidth $B = f_c$.
• Band-pass system: passes $f_L$ to $f_H$; bandwidth $B = f_H - f_L$.

 |H(f)|  Low-pass             |H(f)|  Band-pass
  1 |-----.                    1 |      .---.
0.707|.....\                0.707|.....|.....|....
    |       \                    |    /       \
    +--------\--> f              +---/----+----\--> f
            fc                     fL    f0    fH
    |<- B ->|                       |<-- B -->|

A signal passes through a system without much distortion only if the system bandwidth is at least equal to the signal bandwidth.