Band pass and band limited signals; LTI properties

Band-limited signal

A signal $g(t)$ is band-limited to $W$ Hz if its spectrum is zero beyond $W$:

$$G(f)=0\quad\text{for}\ \lvert f\rvert>W$$

Its energy lies between $0$ and $W$ (it is a low-pass or baseband signal). Examples: telephone speech limited to 3.4 kHz; $g(t)=\text{sinc}(2Wt)$, whose spectrum is a rectangle from $-W$ to $W$; audio after a 20 kHz anti-aliasing filter. Such signals can be sampled without loss at $f_s\ge 2W$.

Band-pass signal

A band-pass signal has its spectrum concentrated in a band of width $2W$ around a carrier frequency $f_c$, with $f_c\gg W$, and is zero near $f=0$:

$$G(f)\ne0\ \ \text{only for}\ f_c-W\le\lvert f\rvert\le f_c+W$$

It can be written as $g(t)=g_I(t)\cos2\pi f_ct-g_Q(t)\sin2\pi f_ct$. Examples: an AM broadcast signal at 1 MHz with 10 kHz bandwidth; an FM station at 100 MHz; $m(t)\cos2\pi f_ct$ (DSB-SC).

 Band-limited (low-pass)        Band-pass
      |G(f)|                       |G(f)|
   +--+--+                  +--+         +--+
   |  |  |                  |  |         |  |
 --+--+--+---> f      ------+--+----+----+--+---> f
  -W  0  W                  -fc     0     fc

Properties of an LTI system

1. Linearity (superposition): $a x_1+b x_2\to a y_1+b y_2$; includes homogeneity and additivity.
2. Time invariance: $x(t-t_0)\to y(t-t_0)$.
3. Complete description by impulse response: $y(t)=x(t)*h(t)$.
4. Frequency response: $Y(f)=H(f)X(f)$; a sinusoid in gives a sinusoid of the same frequency out, with changed amplitude and phase. No new frequencies are created.
5. Commutative, associative and distributive with respect to convolution: cascaded LTI systems give $h_1*h_2$; parallel systems give $h_1+h_2$.
6. Causality: causal if $h(t)=0$ for $t<0$.
7. Stability (BIBO): stable if $\int_{-\infty}^{\infty}\lvert h(t)\rvert dt<\infty$.
8. Power spectra: for random inputs $S_y(f)=\lvert H(f)\rvert^2S_x(f)$.