Sum of periodic signals can be aperiodic

If $x_1(t)$ has period $T_1$ and $x_2(t)$ has period $T_2$, the sum $x_1(t)+x_2(t)$ is periodic only if there is a common period $T = mT_1 = kT_2$ for some integers $m, k$, i.e. only if

$$\frac{T_1}{T_2} = \frac{k}{m} = \text{rational number}$$

If $T_1/T_2$ is irrational, there is no common period and the sum is aperiodic, even though each part is periodic.

Numerical example

$x(t) = \cos 2t + \cos \pi t$

• $\cos 2t$: $T_1 = \frac{2\pi}{2} = \pi$ s
• $\cos \pi t$: $T_2 = \frac{2\pi}{\pi} = 2$ s

$$\frac{T_1}{T_2} = \frac{\pi}{2} \quad\text{(irrational)}$$

No integers $m, k$ satisfy $m\pi = 2k$, so the peaks of the two cosines never line up again after $t = 0$ (at $t=0$ the sum is 2, but it never returns to exactly 2). Hence $x(t)$ is aperiodic.

Contrast (periodic case): $\cos 2t + \cos 3t$ has $T_1 = \pi$, $T_2 = 2\pi/3$, ratio $3/2$ (rational), so the sum is periodic with $T = \text{LCM} = 2\pi$ s.

This justifies the statement: a combination of two CT periodic signals may be aperiodic when the ratio of their periods is irrational.