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.