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Lets take the system <math>y(t)=8x(t)</math> , so lets get 2 y's and 2 x's out of that: | Lets take the system <math>y(t)=8x(t)</math> , so lets get 2 y's and 2 x's out of that: | ||
| + | |||
<math>y_1(t)=8x_1(t)</math> for <math>x_1(t)=t</math> | <math>y_1(t)=8x_1(t)</math> for <math>x_1(t)=t</math> | ||
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Now testing the theory: | Now testing the theory: | ||
| − | <math>ax_1+bx_2= | + | <math>ax_1(t)+bx_2(t)=at+b2t</math> and |
| − | <math>ay_1+by_2= | + | <math>ay_1(t)+by_2(t)=a8t+b16t</math> , which can be reduced to |
Revision as of 13:55, 11 September 2008
Contents
Linearity
Background
Language Definition
A system is considered linear if 2 separate inputs, multiplied by 2 different constants, can produce 2 separate outputs multiplied by those same constants.
Mathematical Definition
A system is called linear if: For any inputs $ x_1(t) $ and $ x_2(t) $ yielding outputs of $ y_1(t) $ and $ y_2(t) $,
$ ax_1(t)+bx_2(t)=ay_1(t)+by_2(t)\,\! $
Example of Linear system
The easiest way to determine linearity is using standard definition:
Lets take the system $ y(t)=8x(t) $ , so lets get 2 y's and 2 x's out of that:
$ y_1(t)=8x_1(t) $ for $ x_1(t)=t $
$ y_2(t)=16x_2(t) $ for $ x_2(t)=2t $
Now testing the theory:
$ ax_1(t)+bx_2(t)=at+b2t $ and
$ ay_1(t)+by_2(t)=a8t+b16t $ , which can be reduced to
