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==Problem 4== | ==Problem 4== | ||
| − | A | + | A linear is system is a system that given two valid inputs: |
| + | :<math>x_1(t) </math> | ||
| + | :<math>x_2(t) </math> | ||
| + | with respective outputs: | ||
| + | :<math>y_1(t) = H*[ x_1(t) ] </math> | ||
| + | :<math>y_2(t) = H*[ x_2(t) ] </math> | ||
| + | will satisfy the equation | ||
| + | :<math>\alpha y_1(t) + \beta y_2(t) = H*[ \alpha x_1(t) + \beta x_2(t) ]</math> | ||
| + | for any <math>\alpha </math> and <math>\beta </math>. | ||
| − | + | ==Example of Linear System== | |
| − | :<math>x_1(t) | + | define |
| − | :<math>x_2(t) | + | :<math>x_1(t) = 4t </math> |
| − | + | :<math>x_2(t) = 3t </math> | |
| − | :<math>y_1(t) = H | + | :<math>H = 87 </math> |
| − | :<math>y_2(t) = H \ | + | therefore |
| − | + | :<math>y_1(t) = H*[ x_1(t) ] = 87*[4t]</math> | |
| − | :<math>\alpha y_1(t) + \beta y_2(t) = H | + | :<math>y_2(t) = H*[ x_2(t) ] = 87*[3t]</math> |
| − | + | ||
| − | + | :<math>\alpha y_1(t) + \beta y_2(t) = 87 * [\alpha (4t)] + 87 *[ \beta (3t)] = 87 * [\alpha (4t) + \beta (3t)] </math> | |
| + | |||
| + | Which satisfies the equation | ||
| + | :<math>\alpha y_1(t) + \beta y_2(t) = H*[ \alpha x_1(t) + \beta x_2(t) ]</math> | ||
| + | |||
| + | ==Example of Non-Linear System== | ||
| + | define | ||
| + | :<math>x_1(t) = t^4 </math> | ||
| + | :<math>x_2(t) = t^3 </math> | ||
| + | therefore | ||
| + | :<math>y_1(t) = [ x_1(t) ]^2 = t^8</math> | ||
| + | :<math>y_2(t) = [ x_2(t) ]^2 = t^6</math> | ||
| + | :<math>\alpha y_1(t) + \beta y_2(t) = \alpha (t^8) + \beta (t^6) \neq [\alpha x_1(t^4) + \beta x_2(t^3) ]^2</math> | ||
Latest revision as of 07:11, 12 September 2008
Problem 4
A linear is system is a system that given two valid inputs:
- $ x_1(t) $
- $ x_2(t) $
with respective outputs:
- $ y_1(t) = H*[ x_1(t) ] $
- $ y_2(t) = H*[ x_2(t) ] $
will satisfy the equation
- $ \alpha y_1(t) + \beta y_2(t) = H*[ \alpha x_1(t) + \beta x_2(t) ] $
for any $ \alpha $ and $ \beta $.
Example of Linear System
define
- $ x_1(t) = 4t $
- $ x_2(t) = 3t $
- $ H = 87 $
therefore
- $ y_1(t) = H*[ x_1(t) ] = 87*[4t] $
- $ y_2(t) = H*[ x_2(t) ] = 87*[3t] $
- $ \alpha y_1(t) + \beta y_2(t) = 87 * [\alpha (4t)] + 87 *[ \beta (3t)] = 87 * [\alpha (4t) + \beta (3t)] $
Which satisfies the equation
- $ \alpha y_1(t) + \beta y_2(t) = H*[ \alpha x_1(t) + \beta x_2(t) ] $
Example of Non-Linear System
define
- $ x_1(t) = t^4 $
- $ x_2(t) = t^3 $
therefore
- $ y_1(t) = [ x_1(t) ]^2 = t^8 $
- $ y_2(t) = [ x_2(t) ]^2 = t^6 $
- $ \alpha y_1(t) + \beta y_2(t) = \alpha (t^8) + \beta (t^6) \neq [\alpha x_1(t^4) + \beta x_2(t^3) ]^2 $
