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 \} $
then
- $ 4 y_1(t) + 5 y_2(t) = F \left \{ 4 x_1(t) + 5 x_2(t) \right \} $
Non-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 non-linear system,
- $ \alpha y_1(t) + \beta y_2(t) \neq 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 non-linear system the inputs can be shifted and/or scaled; however, the outputs will not reflect those exact changes.
Example:
- $ y_1(t) = x_1(t)^2 $ & $ \alpha = 5 \, $ , $ \beta = 6 \, $.
- $ y_2(t) = x_2(t)^2 $
then
- $ 5 y_1(t) + 6 y_2(t) \neq (5x_1(t))^2 + (6x_2(t))^2 $
