# Impulse response function & Laplace transforms

i am given the Laplace transform of an impulse response function, as well as its input. i am supposed to find its output.

H(s) = 1/s2 + s + 1
x(t) = sin2(t-1)U(t-1)

what i have done so far is the following:
i know that Y(s) = H(s)X(s) and from this i can easily find y(t)
so i found X(s) since H(s) is already given...this may be wrong
X(s) = 2e-s/s2 + 4

then i found Y(s)
Y(s) = 2e-s/(s2 + s + 1)(s2 + 4)
then i know that y(t) should be 2y(t-1) because of the e-s time shifting

but i am stuck here... is there a better way to solve this problem using Convolution integral or any other way? i know how to do both the partial fractions method and convolution integral but i keep getting stuck. i think i have set up the problem wrong, perhaps in the X(s) solution

Yes it will be messy all ways!

vela
Staff Emeritus
Homework Helper
If you want to continue with what you started, ignore the exponential factor for now and find the inverse Laplace transform of

$$\frac{2}{(s^2+s+1)(s^2+4)}$$

Start by using partial fractions to get separate terms:

$$\frac{2}{(s^2+s+1)(s^2+4)} = \frac{As+B}{s^2+s+1}+\frac{Cs+D}{s^2+4}$$

The more painful way of finding y(t) would be to find h(t) from H(s) and then convolve x(t) and h(t).