Periodic versus non-periodic functions (hw1, ECE301)

Read the instructor's comments here.

Definition

A function is defined as periodic if it can be moved along the x axis to a place where it exactly matches its original form. In mathematical terms, x(t) is periodic if and only if:

$ \,\! x(t+T)=x(t) $

Examples of periodic and non-periodic functions

Periodic examples:Basically any trigonometric function

$ \,\!cos(t)=cos(t+2\pi) $

$ \,\!sin(t)=sin(t+4\pi) $

This example can be shown to be periodic by drawing a graph, or simply computing values

$ \,\!cos(\pi+2\pi)=cos(\pi)=-1 $

also, any square, triangle, or sawtooth waves are periodic

Non-Periodic examples

any algebraic function:

$ \,\!f(t)=2x+5 $

$ f(t)=\frac{2x^3+5}{4^x-x} $

$ \,\!f(t)=log(x)+e^{x+2} $

any power, exponential or logarithmic function, without a periodic portion, are non-periodic as well.

Periodicity Test for Exponentials

If $ \,\!\frac{\omega_0}{2\pi} $ in $ e^{j\omega_0t} $ is rational, then function is periodic

so $ \,\!e^{2} $ is not periodic because $ \,\!\frac{2}{2\pi}=\frac{1}{\pi} $ is not rational