Difference between revisions of "3.A David Hartmann - Linear System ECE301Fall2008mboutin" - Rhea

Latest revision as of 14:41, 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 \} $

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 $