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.

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