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== Example of a Linear System == | == Example of a Linear System == | ||
| − | + | The following system is linear: | |
| + | |||
| + | <math>\,s(t)=2x(t+3)\,</math> | ||
| + | |||
| + | |||
| + | '''Proof:''' | ||
| + | |||
| + | We have two functions: <math>\,x_1(t), x_2(t)\,</math>. | ||
| + | |||
| + | After applying the functions to the system <math>\,s(t)\,</math>, we get: | ||
| + | |||
| + | <math>\,y_1(t)=2x_1(t+3)\,</math> | ||
| + | |||
| + | <math>\,y_2(t)=2x_2(t+3)\,</math> | ||
| + | |||
| + | Thus, | ||
| + | |||
| + | <math>\,ay_1(t)+by_2(t)=\,</math> | ||
| + | |||
| + | <math>\,a(2x_1(t+3))+b(2x_2(t+3))=\,</math> | ||
| + | |||
| + | <math>\,2ax_1(t+3)+2bx_2(t+3)\,</math> | ||
| + | |||
| + | |||
| + | Now, apply <math>\,ax_1(t)+bx_2(t)\,</math> to the system <math>\,s(t)\,</math>: | ||
| + | |||
| + | <math>\,2(ax_1(t+3)+bx_2(t+3))=\,</math> | ||
| + | |||
| + | <math>\,2ax_1(t+3)+2bx_2(t+3)\,</math> | ||
| + | |||
| + | |||
| + | Since the two results are equal | ||
| + | |||
| + | <math>\,2ax_1(t+3)+2bx_2(t+3)=2ax_1(t+3)+2bx_2(t+3)\,</math> | ||
| + | |||
| + | the system is linear. | ||
== Example of a Non-Linear System == | == Example of a Non-Linear System == | ||
The following system is non-linear: | The following system is non-linear: | ||
| − | <math>\, | + | <math>\,s(t)=2x(t)+3\,</math> |
| + | |||
| + | |||
| + | '''Proof:''' | ||
| + | |||
| + | We have two functions: <math>\,x_1(t), x_2(t)\,</math>. | ||
| + | |||
| + | After applying the functions to the system <math>\,s(t)\,</math>, we get: | ||
| + | |||
| + | <math>\,y_1(t)=2x_1(t)+3\,</math> | ||
| + | |||
| + | <math>\,y_2(t)=2x_2(t)+3\,</math> | ||
| + | |||
| + | Thus, | ||
| + | |||
| + | <math>\,ay_1(t)+by_2(t)=\,</math> | ||
| + | |||
| + | <math>\,a(2x_1(t)+3)+b(2x_2(t)+3)=\,</math> | ||
| + | |||
| + | <math>\,2ax_1(t)+3a+2bx_2(t)+3b\,</math> | ||
| + | |||
| + | |||
| + | Now, apply <math>\,ax_1(t)+bx_2(t)\,</math> to the system <math>\,s(t)\,</math>: | ||
| + | |||
| + | <math>\,2(ax_1(t)+bx_2(t))+3=\,</math> | ||
| + | |||
| + | <math>\,2ax_1(t)+2bx_2(t)+3\,</math> | ||
| + | |||
| + | |||
| + | Since the two results are not equal | ||
| + | <math>\,2ax_1(t)+3a+2bx_2(t)+3b\not= 2ax_1(t)+2bx_2(t)+3\,</math> | ||
| − | + | the system is non-linear. | |
Latest revision as of 18:06, 11 September 2008
Definition of Linearity
A system is linear if for any inputs $ \,x_1(t), x_2(t)\, $ yielding outputs $ \,y_1(t), y_2(t)\, $, respectively, the response to
$ \,ax_1(t)+bx_2(t)\, $ is
$ \,ay_1(t)+by_2(t)\, $, where $ \,a,b\in \mathbb{C}, a\not= 0 ,b\not= 0\, $.
Example of a Linear System
The following system is linear:
$ \,s(t)=2x(t+3)\, $
Proof:
We have two functions: $ \,x_1(t), x_2(t)\, $.
After applying the functions to the system $ \,s(t)\, $, we get:
$ \,y_1(t)=2x_1(t+3)\, $
$ \,y_2(t)=2x_2(t+3)\, $
Thus,
$ \,ay_1(t)+by_2(t)=\, $
$ \,a(2x_1(t+3))+b(2x_2(t+3))=\, $
$ \,2ax_1(t+3)+2bx_2(t+3)\, $
Now, apply $ \,ax_1(t)+bx_2(t)\, $ to the system $ \,s(t)\, $:
$ \,2(ax_1(t+3)+bx_2(t+3))=\, $
$ \,2ax_1(t+3)+2bx_2(t+3)\, $
Since the two results are equal
$ \,2ax_1(t+3)+2bx_2(t+3)=2ax_1(t+3)+2bx_2(t+3)\, $
the system is linear.
Example of a Non-Linear System
The following system is non-linear:
$ \,s(t)=2x(t)+3\, $
Proof:
We have two functions: $ \,x_1(t), x_2(t)\, $.
After applying the functions to the system $ \,s(t)\, $, we get:
$ \,y_1(t)=2x_1(t)+3\, $
$ \,y_2(t)=2x_2(t)+3\, $
Thus,
$ \,ay_1(t)+by_2(t)=\, $
$ \,a(2x_1(t)+3)+b(2x_2(t)+3)=\, $
$ \,2ax_1(t)+3a+2bx_2(t)+3b\, $
Now, apply $ \,ax_1(t)+bx_2(t)\, $ to the system $ \,s(t)\, $:
$ \,2(ax_1(t)+bx_2(t))+3=\, $
$ \,2ax_1(t)+2bx_2(t)+3\, $
Since the two results are not equal
$ \,2ax_1(t)+3a+2bx_2(t)+3b\not= 2ax_1(t)+2bx_2(t)+3\, $
the system is non-linear.
