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Finding the output of an LTI system when we know the system function

  1. Aug 1, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose that we have a linear time invariant system with a system function

    [tex]H(s) = \frac{s+1}{s^{2} + 2s + 2} [/tex]

    we want to find the output of the system when the input is [tex] x(t) = e^{\left | t \right |}[/tex]

    3. The attempt at a solution

    what I tried to do is find the laplace transform of x(t), then since we know the laplace transform of the system function and it is known that Y(s) = H(s) X(s)

    I thought that it would be ok to find Y(s) and then using the inverse laplace transform of a rational function(which is easy to find) I can find the given output..

    now, I have a problem with the LT of e^|t|

    we can rewrite this as

    [tex] e^{|t|} = e^{t} u(t) + e^{-t} u(-t) [/tex]

    that's fine.. now it's easier to find the LT of e^|t|

    we have

    [tex]e^{t} u(t) \overset{L}{\rightarrow} \frac{1}{s-1} [/tex] with region of convergence Real{s} > 1

    and

    [tex]e^{t} u(t) \overset{L}{\rightarrow} \frac{1}{s+1} [/tex] with region of convergence Real{s} < - 1

    by linearity the LT of [tex]e^{|t|}[/tex] won't exist because the intersection of the two regions of convergence will be an empty set..

    now what can I say about the output? can I say that the integral won't converge and hence the output will be infinity?

    thanks in advance
     
  2. jcsd
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