# Thread: An exponential integral problem coming from a discrete time Fourier transform.

1. ## An exponential integral problem coming from a discrete time Fourier transform.

Hi,

I posted a full question regarding this problem in the "other advanced maths" section of the forum if you would like to see my work thus far. However it seems my problem may have now just come down to a calculus problem.

This is what my solution boils down to. If I can solve this my problem will be gone:

$Y(n) = \int\limits_{0.5}^{0.5} \frac{e^{j2 \pi fn}}{ (1 - \frac{1}{4} e^{-j2 \pi f}) \times (1+ \frac{1}{2}e^{-j2 \pi f} ) } df$

Where can I start in trying to solve this integral?

Any help or advice is greatly appreciated.

I hope the moderators can understand my reasoning for posting the question here as it appears since my progression through my original problem it now boils down to calculus.

Thanks a lot!

2. Originally Posted by Dint
Hi,

I posted a full question regarding this problem in the "other advanced maths" section of the forum if you would like to see my work thus far. However it seems my problem may have now just come down to a calculus problem.

This is what my solution boils down to. If I can solve this my problem will be gone:

$Y(n) = \int\limits_{0.5}^{0.5} \frac{e^{j2 \pi fn}}{ (1 - \frac{1}{4} e^{-j2 \pi f}) \times (1+ \frac{1}{2}e^{-j2 \pi f} ) } df$

Where can I start in trying to solve this integral?

Any help or advice is greatly appreciated.

I hope the moderators can understand my reasoning for posting the question here as it appears since my progression through my original problem it now boils down to calculus.

Thanks a lot!
Putting $u=e^{2\pi i f}$ this becomes a contour integral around the unit circle in the complex plain of a rational function and the residue theorem should allow you to evaluate it.

CB

3. Ugh the residue theorm is way over my head. We haven't done anything like that in class. I've either come to this integral a complicated way or there's some other way.

I'll see if I can wrap my head around the residue theorm. Thanks cb.

4. I've been given a hint that when I find my Y(t), partial fraction decomposition may help. I can't however see PFD being of use.