How to take the fourier transform of a function?

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.

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rude man
Homework Helper
Gold Member
How can sin(t) expressed in exponential notation?

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

Yes, that is correct.

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

rude man
Homework Helper
Gold Member
You're right, can't do it that way. Will come back to you in a short while.

rude man
Homework Helper
Gold Member
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.

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