# Course Notes for lecture 4, ECE301 Fall 2008, Prof. Boutin

Note: these were taken by students: they are NOT the official instructor's notes. Watch out for typos and mistakes!

## Periodic Functions

The definition of a periodic function given in class is as follows:
The function x(n) is periodic *if and only if* there exists an integer N such that x(n+N) = x(n). The value of N is called the "period".

As an example, we can use the function $ x(n) = e^{\omega_0 j n} $. To prove this, we do the following:

$ x(n+N) = x(n) $

$ e^{\omega_0 j (n+N)} = e^{\omega_0 j n} $

$ e^{\omega_0 j n} e^{\omega_0 j N} = e^{\omega_0 j n} $

$ e^{\omega_0 j N} = 1 $

$ \cos(\omega_0 N) + j\sin(\omega_0 N) = 1 $

---Which is true if:

$ \omega_0 N = k2\pi $ (where k is an integer)

---at some point.

This leads to the conclusion that if $ {\omega_0 \over 2\pi} = {k \over N} $

or, put another way, $ {\omega_0} \over {2\pi} $ is a rational number, then the function is periodic.

Put yet another way: if the equation is of the form $ e^{\omega_0 j n} $ and $ \omega_0 $ is made up of $ \pi $ and a rational component (contains no irrationals besides $ \pi $) then the function is periodic.