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

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!

• Find the Fourier transform of each summand. Then you end up with a geometric series. – Stephen Montgomery-Smith Nov 18 '13 at 5:37

$$\int_{-\infty}^{\infty} dt \, \delta(t-k T) \, e^{i \omega t} = e^{i k \omega T}$$
$$\sum_{k=0}^{\infty} \left ( a \, e^{i \omega T}\right )^k = \frac{1}{1-a \, e^{i \omega T}}$$
• Thank you so much! However, isn't the first integral equal to $e^{-ikwT}$? – ArKi Nov 18 '13 at 5:49
• @aikitect: No. Note that $$\int dt \, \delta(t-a) f(t) = f(a)$$ – Ron Gordon Nov 18 '13 at 5:50