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A CT non-periodic function: <br><math>\,y(t)=t</math> | A CT non-periodic function: <br><math>\,y(t)=t</math> | ||
===Proof=== | ===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>. | + | 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.
