| Line 22: | Line 22: | ||
<math>y(t)</math>= 5<math>x_1</math> + 3 <math>x_2</math> | <math>y(t)</math>= 5<math>x_1</math> + 3 <math>x_2</math> | ||
| + | |||
| Line 29: | Line 30: | ||
<math>y_2</math>= <math>sin^2 t</math> | <math>y_2</math>= <math>sin^2 t</math> | ||
| + | |||
| + | Let , | ||
| + | |||
| + | <math>x_3</math> = <math>t</math> + <math>sin t</math> | ||
| + | |||
| + | The output is | ||
| + | |||
| + | <math>y(t)</math> = <math>x_3</math> | ||
| + | |||
| + | <math>y(t)</math> = <math>(t + sin t)^ 2</math> | ||
Revision as of 14:41, 12 September 2008
A system is said to be linear if it follows the following conditions
1) The response to $ x_1(t) $ + $ x_2(t) $ is $ y_1(t) $ +$ y_2(t) $.
2) The response to $ ax_1(t) $ is $ ay_1(t) $, where a is any complex constant.
Example for a linear system is
$ x_1 $ = 8$ e^t $
$ x_2 $=8$ t^2 $
Let ,
$ x_3 $ = 5$ e^t $ + 3$ t^2 $
The output is
$ y(t) $ = 8$ x_3 $
$ y(t) $=40 $ e^t $ + 24 $ t^2 $
$ y(t) $= 5$ x_1 $ + 3 $ x_2 $
The example for a linear system is
$ y_1 $ = $ t^2 $
$ y_2 $= $ sin^2 t $
Let ,
$ x_3 $ = $ t $ + $ sin t $
The output is
$ y(t) $ = $ x_3 $
$ y(t) $ = $ (t + sin t)^ 2 $
