There is a necessary and sufficient condition for a function $f$ of the real variable
to be "band-limited". It is called Wiener-Paley theorem. $f$ must be a restriction on the real
line of an entire function of exponential type. Of course, here one implicitly assumes that
$f$ belongs to an appropriate space which permits to interpret it as a "signal".
For example, $L^2$ (finite energy), or $L^\infty$ (bounded amplitude), or Schwarz temperate
distributon etc.