Finding output eqn using DTFT and inverse DTFT knowing impulse and input eqns!

Hi,

The question is apparently simple, but I am very stuck and have been working on it for way too long, I must be missing something :(

Any help is greatly appreciated.

I am given an impulse response:

$\displaystyle h(n) = (-1/2)^n u(n) $

and input:

$\displaystyle x(n) = (1/4)^n u(n) $

so to calculate $\displaystyle y(n) $

I know that:

$\displaystyle y(n) = x(n) * h(n) $ and $\displaystyle Y(f) = X(f) \times H(f) $

so my strategy (and following the questions instructions) was to find $\displaystyle X(f)$ then multiply it with $\displaystyle H(f)$ to get $\displaystyle Y(f)$ and finally use the inverse DTFT to find $\displaystyle y(n)$.

so...

$\displaystyle X(f) = \sum\limits_{n=0}^\infty x(n) e^{-(j2 \pi fn)} $

$\displaystyle = \sum\limits_{n=0}^\infty (1/4) e^{-(j2 \pi fn)} $

simplifying this summation...

$\displaystyle X(f) = \frac {1}{1 - \frac{1}{4} e^{-(j2 \pi f)}} $

similarly

$\displaystyle H(f) = \frac {1}{1 + \frac{1}{2} e^{-(j2 \pi f)}} $

Now, when I multiply these two together, I have no idea how to find the inverse DTFT.

I have got:

$\displaystyle Y(f) = \frac{1}{(1 - \frac{1}{4} e^{-(j2 \pi f)}) \times (1 + \frac{1}{2} e^{-(j2 \pi f)}) } $

how can I find the DTFT of something like this? I don't know the period of it and I have no idea how to simplify it into something to apply the inverse DTFT which I know as:

http://upload.wikimedia.org/math/7/2...781d98321b.png

like I said, any advice would be greatly appreciated.

I have just found myself completely lost with this problem and doing loops trying to simplify my $\displaystyle Y(f) $ is absolutely killing me.

Thanks a lot!