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[[Category:ECE301]]
 
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[[Category:periodicity]]
4. Give an example of a periodic function (either CT or DT) and demonstrate that this function is periodic. Give an example of a non-periodic function (either CT or DT) and demonstrate that this function is not periodic. Post your answers on Rhea.
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=Periodic versus non-periodic functions ([[Homework_1_ECE301Fall2008mboutin|hw1]], [[ECE301]])=
 
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<span style="color:green"> Read the instructor's comments [[hw1periodicECE301f08profcomments|here]]. </span>
 
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== Periodic and Non-Periodic functions ==
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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:
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=== 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>
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This example can be shown to be periodic by drawing a graph, or simply computing values
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<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:
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=== 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>
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<math>f(t)=\frac{2x^3+5}{4^x-x}</math>
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<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==
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If <math>\,\!\frac{\omega_0}{2\pi}</math> in <math>e^{j\omega_0t}</math> is rational, then function is periodic
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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 06:25, 14 April 2010

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

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood