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 $
