How to take the fourier transform of a function?

[XcKyle93]

Homework Statement

Find the Fourier transform of x(t) = e^-t sin(t), t >=0.

We're barely 3 weeks into my signals course, and my professor has already introduced the Fourier transform. I barely understand what it means, but I just want to get through this problem set.

Homework Equations

I honestly don't know. I know it's an improper integral with bounds -∞ and ∞, that is
∫x(t)e^-j2∏tdt

The Attempt at a Solution

I get something VERY LONG which does not seem right.

 

[rude man]

How can sin(t) expressed in exponential notation?

BTW your integral is missing dt. It is dt, right?

 

[XcKyle93]

Yes, that is correct.

I know that sin(t) = (1/2) * j * (e^-t - e^t), so x(t) = (1/2)*j*(e^t(-j-1) - e^t(j-1)). But then I'd have something really painful to integrate, right?

 

[rude man]

You're right, can't do it that way. Will come back to you in a short while.

 

[rude man]

Looks like convolution in the frequency domain is the way to go. You remember the convolution theorem?

So you'll need F(ω) for f(t) = exp(-at) and f(t) = sin(ωt), both for t > 0 and both = 0 for t < 0. The first one is easy; let me know how you're managing with the second ... remember when you take F(ω) to integrate from 0 to ∞, not -∞ to +∞.

EDIT: never mind convolution. You can do the integral.

The integral is not as bad as you think. Key point is that lim t→∞ {e^{-(a + jb)t}} = 0 providing a > 0 which it is in your case.