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| − | + | [[Category:ECE301]] | |
| − | + | [[Category:periodicity]] | |
| − | + | =Periodic versus non-periodic functions ([[Homework_1_ECE301Fall2008mboutin|hw1]], [[ECE301]])= | |
| − | + | <span style="color:green"> Read the instructor's comments [[hw1periodicECE301f08profcomments|here]]. </span> | |
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== Definition == | == Definition == | ||
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== Examples of periodic and non-periodic functions == | == Examples of periodic and non-periodic functions == | ||
| − | Periodic examples:Basically any trigonometric function | + | |
| + | === Periodic examples:Basically any trigonometric function === | ||
<math>\,\!cos(t)=cos(t+2\pi)</math> | <math>\,\!cos(t)=cos(t+2\pi)</math> | ||
<math>\,\!sin(t)=sin(t+4\pi)</math> | <math>\,\!sin(t)=sin(t+4\pi)</math> | ||
| + | |||
| + | This example can be shown to be periodic by drawing a graph, or simply computing values | ||
| + | |||
| + | <math>\,\!cos(\pi+2\pi)=cos(\pi)=-1</math> | ||
also, any square, triangle, or sawtooth waves are periodic | also, any square, triangle, or sawtooth waves are periodic | ||
| − | Non-Periodic examples | + | === Non-Periodic examples === |
any algebraic function: | any algebraic function: | ||
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<math>\,\!f(t)=2x+5</math> | <math>\,\!f(t)=2x+5</math> | ||
| − | <math>f(t)=\frac{2x+5}{4-x}</math> | + | <math>f(t)=\frac{2x^3+5}{4^x-x}</math> |
| + | |||
| + | <math>\,\!f(t)=log(x)+e^{x+2}</math> | ||
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| + | any power, exponential or logarithmic function, without a periodic portion, are non-periodic as well. | ||
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| + | |||
| + | == Periodicity Test for Exponentials== | ||
| + | |||
| + | If <math>\,\!\frac{\omega_0}{2\pi}</math> in <math>e^{j\omega_0t}</math> is rational, then function is periodic | ||
| + | |||
| + | so <math>\,\!e^{2} </math> is not periodic because <math>\,\!\frac{2}{2\pi}=\frac{1}{\pi}</math> is not rational | ||
Latest revision as of 07:25, 14 April 2010
Contents
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
