Dirac comb Fourier transform

Question

I have $ h(t) = a \delta (t) + \delta (t-T) + a \delta (t - 2T) $ and I found that the Fourier transform of this impulse response is $ H(f) = a + e^{-i 2 \pi f T } + a e^{-i 2 \pi f 2 T } $ is this transform correct ? because my book obtained the same result but after wrote it as $ (1 + 2a cos ( 2 \pi f T ) ) e^{- i 2 \pi f T } $

Answer 1

Hint.

$$ a+e^{-i\alpha}+a e^{-2 i\alpha} = \left(a e^{i\alpha}+ 1 + a e^{- i\alpha}\right)e^{- i\alpha} = (1+2a\cos(\alpha))e^{- i\alpha} $$