EECS20N: Signals and Systems

Discrete periodic signals

Consider signals of the form x: DiscreteTime → Reals, where the set DiscreteTime = Integers provides indices for samples of the signal. Such signals are called discrete-time signals. A discrete-time signal is periodic if there is a non-zero integer p ∈ DiscreteTime such that for all n ∈ DiscreteTime,

x(n + p) = x(n).

Note that, somewhat counterintuitively, not all sinusoidal discrete-time signals are periodic. Consider

x(n) = cos(2π f n).

For this to be periodic, we must be able to find a non-zero integer p such that for all integers n,

x(n + p) = cos(2π f n + 2π f p) = cos(2π f n) = x(n).

This can be true only if (2π f p) is a multiple of 2π . I.e., if there is some integer m such that

2π f p = 2π m.

Dividing both sides by 2π p, we see that this signal is periodic only if we can find nonzero integers p and m such that

f = m/p.

In other words, f must be rational. Only if f is rational is this signal periodic.