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| − | + | ==Problem 4== | |
| + | A general deterministic system can be described by operator <math>H</math> that maps an input <math>x(t)</math> as a function of <math>t</math> to an output <math>y(t)</math>. | ||
| + | |||
| + | Given two valid inputs | ||
| + | :<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) \right \} </math> | ||
| + | :<math>y_2(t) = H \left \{ x_2(t) \right \} </math> | ||
| + | then a linear system must satisfy | ||
| + | :<math>\alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math> | ||
| + | for any [[scalar (mathematics)_ECE301Fall2008mboutin|scalar]] values <math>\alpha \,</math> and <math>\beta \,</math>. | ||
| + | <!-- Insert picture dipicting the superposition and scaling properties --> | ||
Revision as of 06:19, 12 September 2008
Problem 4
A general deterministic system can be described by operator $ H $ that maps an input $ x(t) $ as a function of $ t $ to an output $ y(t) $.
Given two valid inputs
- $ x_1(t) \, $
- $ x_2(t) \, $
as well as their respective outputs
- $ y_1(t) = H \left \{ x_1(t) \right \} $
- $ y_2(t) = H \left \{ x_2(t) \right \} $
then a linear system must satisfy
- $ \alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} $
for any scalar values $ \alpha \, $ and $ \beta \, $.
