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

  1. #1
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    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:

     h(n) = (-1/2)^n   u(n)

    and input:

     x(n) = (1/4)^n u(n)

    so to calculate  y(n)

    I know that:

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

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

    so...

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

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

    simplifying this summation...

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

    similarly

     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:

     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:



    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  Y(f) is absolutely killing me.

    Thanks a lot!
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  2. #2
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    Joined
    Apr 2010
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    So looking further at my notes I've found that apparently the period of my functions x(n) and h(n), the period will always be 1.

    So I then find:

     x(n) = \int\limits_{-0.5}^{0.5} X(f) e^{j2 \pi fn} df

    so from my  Y(f) I get:

     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

    I have no idea where to start when evaluating this integral...
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