(New page: == Part A: Understanding System's Properties == '''Linear System''' :<math>x_1(t) \,</math> :<math>x_2(t) \,</math> as well as their respective outputs :<math>y_1(t) = H \left \{ x_1(t)...) |
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== Part A: Understanding System's Properties == | == Part A: Understanding System's Properties == | ||
'''Linear System''' | '''Linear System''' | ||
+ | |||
+ | Given any two inputs | ||
:<math>x_1(t) \,</math> | :<math>x_1(t) \,</math> | ||
:<math>x_2(t) \,</math> | :<math>x_2(t) \,</math> | ||
as well as their respective outputs | as well as their respective outputs | ||
− | :<math>y_1(t) = | + | :<math>y_1(t) = F \left \{ x_1(t) \right \} </math> |
− | :<math>y_2(t) = | + | :<math>y_2(t) = F \left \{ x_2(t) \right \} </math> |
− | then a linear system | + | then to be a linear system, |
− | :<math>\alpha y_1(t) + \beta y_2(t) = | + | :<math>\alpha y_1(t) + \beta y_2(t) = F \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math> |
− | for any | + | 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> | ||
+ | |||
+ | then | ||
+ | |||
+ | :<math>4 y_1(t) + 5 y_2(t) = F \left \{ 4 x_1(t) + 5 x_2(t) \right \} </math> | ||
+ | |||
+ | |||
+ | ---- | ||
+ | |||
+ | '''Non-Linear System''' | ||
+ | |||
+ | Given any two inputs | ||
+ | |||
+ | :<math>x_1(t) \,</math> | ||
+ | :<math>x_2(t) \,</math> | ||
+ | as well as their respective outputs | ||
+ | :<math>y_1(t) = F \left \{ x_1(t) \right \} </math> | ||
+ | :<math>y_2(t) = F \left \{ x_2(t) \right \} </math> | ||
+ | then to be a non-linear system, | ||
+ | :<math>\alpha y_1(t) + \beta y_2(t) \neq F \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math> | ||
+ | for any scalar complex values <math>\alpha \,</math> and <math>\beta \,</math>. | ||
+ | |||
+ | 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:'' | ||
+ | :<math>y_1(t) = x_1(t)^2 </math> & <math>\alpha = 5 \,</math> , <math>\beta = 6 \,</math>. | ||
+ | |||
+ | :<math>y_2(t) = x_2(t)^2 </math> | ||
+ | |||
+ | then | ||
+ | |||
+ | :<math>5 y_1(t) + 6 y_2(t) \neq (5x_1(t))^2 + (6x_2(t))^2 </math> |
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 $