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 \} $
