Periodic signal

In signal theory, a broad class of signals are periodic. A continuous-time (CT) signal is periodic if:

$$x(t)=x(t+T);\forall t$$

by a time shift of $T$. We say that its periodic with period $T$. A discrete-time (DT) signal is periodic if:

$$x[n]=x[n+N];\forall n$$

The fundamental period $T_0$ or $N_0\in \mathbb{Z}$ is the smallest positive value of $T$ or $N$ for which the above holds; how long a periodic signal takes to repeat itself. For example, $\sin (t)$ has $T_0=2\pi$. The inverse is the fundamental frequency.

$${\omega }_0=2\pi f_0=\frac{2\pi }{T_0}$$

The fundamental period is uniquely defined. A constant CT signal is periodic but has no defined fundamental period. A constant DT signal is periodic but with $N_0=1$.

For DT signals, if $N_0\notin \mathbb{Q}$ (i.e., it’s an irrational real number), it’s aperiodic. If $N_0\in \mathbb{Q}$, it is periodic. The numerator indicates the period it takes to “repeat” itself. The denominator indicates how many sampling periods are needed for the signal to actually repeat.

Aperiodic signals are periodic signals with an infinite period.

Theorems and operations

The sum/product of two periodic CT signals are not necessarily periodic. For both cases, by theorem, if one period is a rational multiple of the other, then the two periods will eventually line up, i.e.,¹

$$l{T}_1=k{T}_2$$ $$\frac{T_1}{T_2}=\frac{k}{l}\in \mathbb{Q}$$

i.e., if the ratio is irrational (like $\pi$), the signals will never line up.

To find the fundamental period of composed signals, we take the least common multiple of the two periods. This isn’t always necessarily true — for some signals (usually easy to graph ones), it may be smaller than the LCM.

We can also define periodic signals as the infinite sum of finite duration signals:

$$x_{\text{per}}(t)=\sum _{-\infty }^{+\infty }{x}_{\text{fin}}(t-k{T}_0)$$

Footnotes

1. The exact definition states that $k,l\in {\mathbb{Z}}_{>0}$. Since the period of a signal cannot be negative, this doesn’t really change anything for us. ↩