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I have this problem, and I can't get to the solution. $$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$$

The solution is $$X(e^{j\omega}) = \frac{0.64 − 2(0.6)^{11} \cos(11ω) + 2(0.6)^{12} \cos(10ω)}{1.36 − 1.2 \cos(ω)} $$

But I don't know how to get there, i think it is related to a series(geometric, maybe) and things like Euler formula but still not able to find a way to get there. This is an exercise(3.3.2) from Proakis and Inge book, Digital Signal Processing using Matlab.

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  • $\begingroup$ Homework? I can see three separate properties of the Fourier Transform that you can use to simplify this if you apply them in turn -- can you name one? $\endgroup$
    – TimWescott
    Nov 29, 2021 at 0:31
  • $\begingroup$ Linearity, I think. $\endgroup$
    – Derteck
    Nov 29, 2021 at 7:42
  • $\begingroup$ The very outermost operation going on there is summing $x_2(n) = {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$ from $-\infty$ to $+\infty$. If you knew the Fourier transform of $x_2$, what would be your overall Fourier transform? (Yes, the Socratic Method is named after a guy who was forced by his colleagues to drink poison. None the less, I'm hoping you'll have an "aha" moment here). $\endgroup$
    – TimWescott
    Nov 29, 2021 at 21:41

1 Answer 1

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$$X(e^{j\omega}) = \sum_{n=-10}^{\infty} {(0.6)^{|n|}}e^{-j\omega n} - \sum_{n=11}^{\infty} {(0.6)^{|n|}}e^{-j\omega n}$$

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  • $\begingroup$ This is kinda hard to read ... Use Latex maybe? $\endgroup$
    – Matt L.
    Nov 29, 2021 at 11:29
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    $\begingroup$ Please do not answer obvious homework problems directly. We have a policy here of helping people answer homework problems, by finding where they're stuck and giving that missing bit the prof missed, or helping them separate the problem into tractable bits -- but we do not do homework here. Just an answer with no effort behind it isn't what homework is about: it doesn't serve the original poster, or the engineering community at large. $\endgroup$
    – TimWescott
    Nov 29, 2021 at 16:11
  • $\begingroup$ This is not homework, I'm just solving the exercises of the book. And I have simplified the summation limiting it by the unit step subtraction bounds, and also expressed the modulos of n by separating and tracking when n gets positive, but from that into the final solution is a blurr. $\endgroup$
    – Derteck
    Nov 29, 2021 at 18:30
  • $\begingroup$ If you can link me to an example that can help for me to solve this one, it would be highly appreciated. $\endgroup$
    – Derteck
    Nov 29, 2021 at 20:53
  • $\begingroup$ @Derteckpt: please edit your question with the comment that it's not homework and that you're self-studying. $\endgroup$
    – TimWescott
    Nov 29, 2021 at 21:40

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