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

1. Aug 1, 2011

### Jncik

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

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

$$H(s) = \frac{s+1}{s^{2} + 2s + 2}$$

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

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

$$e^{|t|} = e^{t} u(t) + e^{-t} u(-t)$$

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

we have

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

and

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

by linearity the LT of $$e^{|t|}$$ 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?