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I have been given this problem from a textbook (not homework, trying to study for an exam. The goal is to find the Fourier transform of this function.

$\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$

Can anyone give me a hint or point me in the right direction of how to compute the Fourier Transform? Thanks!

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  • $\begingroup$ Find the Fourier transform of each summand. Then you end up with a geometric series. $\endgroup$ – Stephen Montgomery-Smith Nov 18 '13 at 5:37
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The FT of each individual impulse is

$$\int_{-\infty}^{\infty} dt \, \delta(t-k T) \, e^{i \omega t} = e^{i k \omega T}$$

so that the FT of the sum is a geometric series:

$$\sum_{k=0}^{\infty} \left ( a \, e^{i \omega T}\right )^k = \frac{1}{1-a \, e^{i \omega T}}$$

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  • $\begingroup$ Thank you so much! However, isn't the first integral equal to $e^{-ikwT}$? $\endgroup$ – ArKi Nov 18 '13 at 5:49
  • $\begingroup$ @aikitect: No. Note that $$\int dt \, \delta(t-a) f(t) = f(a)$$ $\endgroup$ – Ron Gordon Nov 18 '13 at 5:50

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