Find the Fourier Transform of sin(pi*t) , |t|<t0

[atrus_ovis]

Homework Statement

Solve the above F.T.

Homework Equations

http://en.wikipedia.org/wiki/Euler's_formula

http://en.wikipedia.org/wiki/Fourier_transform

The Attempt at a Solution

I use euler's formula and apply the definition of the F.S. and i get to zero, not surprisingly, as the sine is an odd function.
At a "common F.T. pairs, there's an entry sin(ω₀t) <-> jπ(δ(ω+₀)-δ(ω-ω₀))
However the entry says nothing about t being contained.Any hints? Do i manipulate the signal being odd?

 

[vela]

Are you trying to find the Fourier series, which is what I assume you mean by F.S., or the Fourier transform as in the title of this thread?

 

[atrus_ovis]

The Fourier Transform.
I edited the OP.

 

[vela]

Show us the integral you did. You shouldn't get 0.

 

[atrus_ovis]

1/2j * Integral from -1/3 to 1/3 of (e^jπt-e^-jπt) * e^-jΩtdt

you mean that one, right?

 

[atrus_ovis]

Ah, hold on, i end up at
[1/(jπ-jΩ)] * sin((π-Ω)/3) + [1/(jπ+jΩ)] * sin((π+Ω)/3)

how does that look to you?

 

[vela]

I didn't check every detail, but it looks right.

 

[atrus_ovis]

Thank you.
One last thing: From there, can you manipulate it to end up in a sinc function? (normalized or not)
Or do you need to have it in a form of sin(a*Ω)/aΩ (meaning, having Ω in the argument as a common factor.

 

[vela]

I think you made a sign error somewhere. You can write it as a difference of two sinc functions.

 

[atrus_ovis]

I just used the property sin(-a) = - sin(a)

what would the argument of the sinc function be? You can't factor out Ω.

 

[vela]

The sinc function is defined as $\text{sinc } x := \frac{\sin x}{x}$. The x is a dummy variable in this expression. You can replace it with anything you want as long as you do it everywhere, e.g.,
$$\text{sinc }(meow) = \frac{\sin (meow)}{meow}$$