Latest revision as of 06:15, 14 April 2010

Periodic versus non-periodic functions (hw1, ECE301)

Read the instructor's comments here.

Periodic Function

The definition of a periodic function is as follows: x[n] is periodic iff there exists an integer N such that x[n+N] = x[n]. The value of N is called the "Period."

An example of a periodic function is $ f(t) = e^{2\pi j} $. To prove this, we do the following:

$ x[n+N] = x[n] $

$ e^{2\pi j (n+N)} = e^{2\pi j n} $

$ {e^{2\pi j n} e^{2\pi j N}} = e^{2\pi j n} $

$ e^{2\pi j N} = 1 $

$ \cos(2\pi N) + j\sin(2\pi N) = 1 $

Which is true if

$ 2\pi N = 2\pi $

at some point.

Since N = 1 yields $ 2\pi = 2\pi $, we can conclude that in fact, $ f(t) = e^{2\pi j} $ is periodic.

Non-Periodic Function

The function shown is $ f(t) = {\sin(t)\over t} $.

Note that the function never repeats itself. It changes constantly over its entire expanse.

The function $ f(t) = {\sin(t)\over t} $ from 0 to 15.