Difference between revisions of "HW2-C Scott Hamilton ECE301Fall2008mboutin" - Rhea

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 $