(→Non-Periodic Function: Decaying Square Wave) |
(→Non-Periodic Function: Decaying Square Wave) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
== Periodic Function: Square Wave == | == Periodic Function: Square Wave == | ||
| − | Given the following square wave <math>x(t)</math>: | + | Given the following square wave <math>\,x(t)\,</math>: |
[[Image:jkubasci_Periodic_square_ECE301Fall2008mboutin.jpg]] | [[Image:jkubasci_Periodic_square_ECE301Fall2008mboutin.jpg]] | ||
| − | The function is periodic since there is a value <math>T</math> such that <math>x(t+T)=x(t)</math>. In this example, the fundamental period is <math>2\pi</math>, which one possible value for <math>T</math>. | + | The function is periodic since there is a value <math>\,T\,</math> such that <math>\,x(t+T)=x(t)\,</math>. In this example, the fundamental period is <math>\,2\pi\,</math>, which one possible value for <math>\,T\,</math>. |
== Non-Periodic Function: Decaying Square Wave == | == Non-Periodic Function: Decaying Square Wave == | ||
| + | Given the following decaying square wave <math>\,y(t)=x(t)e^{\frac{-t}{5}}\,</math> (<math>x(t)</math> is the square wave defined in the previous section): | ||
[[Image:jkubasci_Nonperiodic_decaying_square_ECE301Fall2008mboutin.jpg]] | [[Image:jkubasci_Nonperiodic_decaying_square_ECE301Fall2008mboutin.jpg]] | ||
| + | |||
| + | By inspection, this function is non-periodic, as there is no <math>\,T\,</math> such that <math>\,y(t+T)=y(t)\,</math>. (The function is simply alternating between <math>\,e^{\frac{-t}{5}}\,</math> and <math>\,-e^{\frac{-t}{5}}\,</math> every <math>\,\pi\,</math> units, which neither are periodic.) | ||
Latest revision as of 21:01, 4 September 2008
Periodic Function: Square Wave
Given the following square wave $ \,x(t)\, $:
The function is periodic since there is a value $ \,T\, $ such that $ \,x(t+T)=x(t)\, $. In this example, the fundamental period is $ \,2\pi\, $, which one possible value for $ \,T\, $.
Non-Periodic Function: Decaying Square Wave
Given the following decaying square wave $ \,y(t)=x(t)e^{\frac{-t}{5}}\, $ ($ x(t) $ is the square wave defined in the previous section):
By inspection, this function is non-periodic, as there is no $ \,T\, $ such that $ \,y(t+T)=y(t)\, $. (The function is simply alternating between $ \,e^{\frac{-t}{5}}\, $ and $ \,-e^{\frac{-t}{5}}\, $ every $ \,\pi\, $ units, which neither are periodic.)
