# 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.