Periodic signals; condition for cos(2πf₀n + θ)

• Periodic signal: repeats itself after a fixed interval. CT: $x(t+T) = x(t)$ for all $t$; DT: $x[n+N] = x[n]$ for all $n$, $N$ a positive integer. The smallest such $T$ or $N$ is the fundamental period. Examples: $\sin t$, $\cos(\pi n/4)$.
• Aperiodic signal: does not repeat for any finite period. Examples: $e^{-t}u(t)$, $\delta[n]$, $\cos(2n)$.

Condition for $x[n] = \cos(2\pi f_0 n + \theta)$ to be periodic

For period $N$:

$$\begin{aligned} x[n+N] &= \cos(2\pi f_0 n + 2\pi f_0 N + \theta) \end{aligned}$$

This equals $x[n]$ for all $n$ only if the extra angle is a multiple of $2\pi$:

$$2\pi f_0 N = 2\pi k \;\Rightarrow\; f_0 = \frac{k}{N}, \quad k, N \text{ integers}$$

So $x[n]$ is periodic only if $f_0$ is a rational number. If $f_0 = k/N$ in lowest terms, the fundamental period is $N$.

Example: $f_0 = 3/10$ gives $N = 10$. $f_0 = 1/(2\pi)$ (i.e. $\cos n$) is irrational, so the signal is aperiodic.