Find FT of periodic Impulse?

Question

How to find a Fourier transform of periodic Impulse?

Hint : use Fourier Series
delta T = sigma of (delta (t-nT)
or its exponential form
..............t₂
Cn=1/T∫ δ(t - n·Τ) · exp(-i·n·ω·t) dt
..............t₁

Answer 1

This is the kind of FT you want(its easy!). Find FS coeffs by taking that integral for Cn, but no need to take the integral as you can use the sifting property which says..

integral( delta(t-to)*exp(i*k*w*t)) = exp(i*k*w*to)

When finding the FS coeffs(Cn in this case) you only need to integrate over one period of the function you are finding the FS coeffs of(Though in this case, as I said above, you need not perform the explicit integration). After using the above sifting property you will end up with:

Cn = 1/T*(exp(-i*n*w*n*T)

now use the fact that the FS of of a function

f(t) = sum(Cn*exp(i*n*w*t))

where the sum is from n = -inf to n = inf

That is the Fourier Series representation of the periodic impulse.