How do you compute the Fourier Transform of this Unit-Impulse Function?

Question

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!

Answer 1

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:

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