Periodic and Aperiodic Signals

Periodic: A signal is periodic if for some positive constant $T_0$

$$x(t)=x(t+T_0)\text{for all }t$$

The smallest value of $T_0$ that satisfies this is the fundamental period of $x(t)$. A periodic signal remains unchanged when time-shifted by one period; thus, they must start at $-\infty$ and continue forever.

We also have:

$${\int }_a^{a+T_0}x(t)dt={\int }_b^{b+T_0}x(t)dt$$

such that the area under $x(t)$ over any duration $T_0$ is constant.

A signal is aperiodic if it is not periodic.