Periodic and aperiodic signals

Periodic signal

A signal is periodic if it repeats itself exactly after a fixed interval:

$$x(t + T) = x(t) \ \ \forall t, \qquad x[n+N] = x[n] \ \ \forall n$$

The smallest positive $T$ (or integer $N$) for which this holds is the fundamental period; $\omega_0 = 2\pi/T$ is the fundamental frequency.

Examples:

• $x(t) = \sin(2\pi t)$: period $T = 1$ s.
• A square wave or sawtooth wave repeating every $T$ seconds.
• $x[n] = \cos(\pi n/4)$: $\omega_0/2\pi = 1/8$, so $N = 8$.
• Sum $x(t) = \cos 2t + \sin 3t$: periods $\pi$ and $2\pi/3$, ratio $3/2$ is rational, so periodic with $T = 2\pi$.

Aperiodic signal

A signal that does not repeat for any finite $T$ (or $N$) is aperiodic (non-periodic). It can be seen as a periodic signal with $T \to \infty$.

Examples:

• $x(t) = e^{-2t}u(t)$, a single rectangular pulse, the unit step $u(t)$, the impulse $\delta(t)$.
• $x(t) = \cos t + \cos \sqrt{2}\,t$: ratio of periods $\sqrt{2}$ is irrational, so it never repeats.
• $x[n] = \cos(n/6)$: $\omega_0/2\pi = 1/12\pi$ is irrational, so it is aperiodic even though the continuous-time $\cos(t/6)$ is periodic.

Periodic (repeats every T)       Aperiodic (one pulse)
  _    _    _    _                    ____
 | |  | |  | |  | |                  |    |
_| |__| |__| |__| |__ t          ____|    |_____ t
 |<-T->|                             0    2