Inverse Laplace transform Signals and systems

[mattbrrtt]

Homework Statement

Compute Y(t).

Homework Equations

Y(s)= (s(2s+11)+ω(8s+4ω^2))/((s^2+ω^2)(s+9)(s+2))

The Attempt at a Solution

(s(2s+11)+ω(8s+4ω^2 ))/((s^2+ω^2)(s+9)(s+2))= A/(s^2+ω^2 )+B/(s+9)+C/(s+2)

Every example I have looked at does not have the ω variable, but I am not sure that is the problem.

This is a problem from a lab for a signals and systems course. The problem starts by giving h(t) =[cos 2t + 4 sin 2t]u(t), and asking for the impulse response. Then an input of x(t) = (5/7)(e^-t)-(12/7)(e^-8t) is provided. The output response is then calculated, and that is where this question picks up.

Thank you for any assistance.
Matt

 

[vela]

The numerator of the first fraction should be of the form As+B because the denominator is a quadratic.

I assume X(s) and H(s) are supposed to be the Laplace transforms for the given x(t) and h(t). If so, you need to recalculate X(s) as it isn't correct.

ω is a constant. It's the frequency in the cosine and sine terms, so you can set it to 2 in H(s).

 

[mattbrrtt]

I am guessing that the mistake made with the X(s) and H(s) terms is that I simplified and ended up with unfavorable numerators. I am going to try using :

H(s)=1/(s+ω^2 )+4/(s^2+ω)
X(s)=1/(s+2)+1/(s+9)

and calculate from there.

Is this correct?

Thank you.

 

[vela]

I am guessing that the mistake made with the X(s) and H(s) terms is that I simplified and ended up with unfavorable numerators. I am going to try using :

H(s)=1/(s+ω^2 )+4/(s^2+ω)
X(s)=1/(s+2)+1/(s+9)

and calculate from there.

Is this correct?

I edited my first post a couple of times after I first submitted it, so you may want to reread it.

Your original H(s) was correct. You just have to set ω=2. It's the angular frequency which appears in the terms of x(t).

X(s), however, remains a mystery. I'm not sure how you're getting 1/(s+2) and 1/(s+9) terms in it.

It turns out if you get X(s) and H(s) correct, the algebra simplifies a lot, and you should find it relatively easy to find y(t).

 

[mattbrrtt]

Thanks.
The X(s) comes from the formula that was given in the assignment.
x(t)=(5/7)(e^-t)-(12/7)(e^-8t)

When I was writing my explanation, I think I found an error. I will post back.