| Line 13: | Line 13: | ||
:<math>\alpha y_1(t) + \beta y_2(t) = F \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math> | :<math>\alpha y_1(t) + \beta y_2(t) = F \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math> | ||
for any scalar complex values <math>\alpha \,</math> and <math>\beta \,</math>. | for any scalar complex values <math>\alpha \,</math> and <math>\beta \,</math>. | ||
| + | |||
| + | That is to say that in a linear system the inputs can be shifted and/or scaled and the outputs will reflect those exact changes. | ||
| + | |||
| + | ''Example:'' | ||
| + | :<math>y_1(t) = F \left \{ x_1(t) \right \} </math> & <math>\alpha = 4 \,</math> , <math>\beta = 5 \,</math>. | ||
| + | |||
| + | :<math>y_2(t) = F \left \{ x_2(t) \right \} </math> | ||
Revision as of 14:25, 19 September 2008
Part A: Understanding System's Properties
Linear System
Given any two inputs
- $ x_1(t) \, $
- $ x_2(t) \, $
as well as their respective outputs
- $ y_1(t) = F \left \{ x_1(t) \right \} $
- $ y_2(t) = F \left \{ x_2(t) \right \} $
then to be a linear system,
- $ \alpha y_1(t) + \beta y_2(t) = F \left \{ \alpha x_1(t) + \beta x_2(t) \right \} $
for any scalar complex values $ \alpha \, $ and $ \beta \, $.
That is to say that in a linear system the inputs can be shifted and/or scaled and the outputs will reflect those exact changes.
Example:
- $ y_1(t) = F \left \{ x_1(t) \right \} $ & $ \alpha = 4 \, $ , $ \beta = 5 \, $.
- $ y_2(t) = F \left \{ x_2(t) \right \} $
