Revision as of 07:09, 12 September 2008 by Shamilto (Talk)

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^6 $
$ y_2(t) = [ x_2(t) ]^2 = t^5 $
$ \alpha y_1(t) + \beta y_2(t) = \alpha (t^6) + \beta (t^5) \neq [\alpha x_1(t) + \beta x_2(t) ]^2 $

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Mu Qiao