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I came across this problem in an exercise for Fourier analysis. I tried solving just $e^{−a|n|}$ to get the Fourier transform of a similar form as seen on the LHS because it looked familiar. But in that case in the denominator, I get $f^2$ (where $f$ is the frequency as given in the equation below) instead of $n^2$ (I'm integrating with respect to n). I used the following relation:

Fourier transform of $x(n)$: $$X(f)=\int_{-\infty}^\infty x(n)e^{-2\pi jfn}dn$$

Solving this gives: $$\frac{2a}{a^2+4 \pi^2 f^2}$$

I think my approach is wrong here.

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    $\begingroup$ in what context are you seeing this problem? what have you tried? $\endgroup$ – gt6989b Feb 26 '19 at 16:03
  • $\begingroup$ I tried solving just $e^{-a \left\lvert n \right\rvert}$ to get the Fourier transform of a similar form as seen on the LHS, but in the denominator, it comes out to be $f^2$ instead of $n^2$ (I'm integrating with respect to n). So I'm at a complete loss in this problem. Am I not even looking at it right? $\endgroup$ – Swapnil Feb 26 '19 at 16:07
  • $\begingroup$ Welcome to MSE. What have you tried? Are you stuck on some concept? Is your calculation wrong? What do you need help with? $\endgroup$ – Andrei Feb 26 '19 at 16:07
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    $\begingroup$ My guess is Poisson Summation. Related: math.stackexchange.com/questions/3123991/… $\endgroup$ – mrtaurho Feb 26 '19 at 16:39
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    $\begingroup$ If you "got it" please fix the post as quid suggested in meta. Having non-fixed badly received question may hamper your use of the site later. We are somewhat strict about a few things. The CEO of StackExchange compared some of the rules that are necessary here to the rules around the Burning Man -event. Newcomers reportedly find those rather unintuitive :-) $\endgroup$ – Jyrki Lahtonen Feb 26 '19 at 18:14

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