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| − | =Periodic Function= | + | ==Periodic Function== |
A CT periodic function: <math>\,y(t)=sin(t)</math> | A CT periodic function: <math>\,y(t)=sin(t)</math> | ||
| − | ==Proof== | + | ===Proof=== |
| − | == | + | To prove a function is periodic first you have to know the definition of a '''periodic fucntion'''<br> |
| + | A '''periodic function''' is a function that repeats its values after some definite period has been added to its independent variable. This property is called periodicity. | ||
| + | |||
| + | Let's say we have: <br><br><math>\,y(t+T)=sin(t+T)</math>, <math>\,T=2\pi</math> | ||
| + | <br><br> | ||
| + | Therefore, <math>\,sin(t+T)=sin(t+2\pi)</math> | ||
| + | <br><br> | ||
| + | and <math>\,sin(t+2\pi)=sin(t)</math> | ||
| + | <br><br> | ||
| + | The function repeats itself in a certain period which is <math>2\pi</math> so it's a periodic function. | ||
| + | |||
| + | ==Non-periodic Function== | ||
| + | A CT non-periodic function: <br><math>\,y(t)=t</math> | ||
| + | ===Proof=== | ||
| + | If the function is periodic, then there must exist a non-zero ''T'', that makes <math>y(t+T)=y(t)</math>, and there is no such number except 0 to satisfy the function <math>y(t)=t</math>. Therefore, the function is non-periodic. | ||
Latest revision as of 17:37, 5 September 2008
Periodic Function
A CT periodic function: $ \,y(t)=sin(t) $
Proof
To prove a function is periodic first you have to know the definition of a periodic fucntion
A periodic function is a function that repeats its values after some definite period has been added to its independent variable. This property is called periodicity.
Let's say we have:
$ \,y(t+T)=sin(t+T) $, $ \,T=2\pi $
Therefore, $ \,sin(t+T)=sin(t+2\pi) $
and $ \,sin(t+2\pi)=sin(t) $
The function repeats itself in a certain period which is $ 2\pi $ so it's a periodic function.
Non-periodic Function
A CT non-periodic function:
$ \,y(t)=t $
Proof
If the function is periodic, then there must exist a non-zero T, that makes $ y(t+T)=y(t) $, and there is no such number except 0 to satisfy the function $ y(t)=t $. Therefore, the function is non-periodic.
