Signal Theory paradox
There is no paradox.

Periodic functions have no beginning and no end.  Signals always have a beginning and an end.  Therefore, no signal can be exactly represented by a periodic function.

But, the mathematical techniques that we use to analyze periodic functions are both powerful and useful.  So useful that we can pretend that a small segment of some actual, finite signal is a segment of some infinite periodic function, and we can analyze it as such, and learn something interesting about the signal.