# Fourier transform of Dirac comb perturbed with Gaussian distribution around each Dirac delta function?

Given a finite extent of Dirac comb one can determine its Fourier transform in various ways, for example as convolution of Dirac comb and sinc function or by Dirchilet kernel.

However when I try to introduce quasi-periodicity in finite Dirac comb, meaning each Dirac delta function can be moved around the ideal position, with Gaussain probability distribution, is it possible to Fourier transform such a quasi-periodic function so that a standard deviation in Fourier spectrum can be estimated.

• You meant $\sum_{n=-N}^N \delta(x-n-w_n)$ whose Fourier transform is $\hat{f}(\xi)=\sum_{n=-N}^N W_n e^{-2i \pi \xi n}$ where $W_n=e^{-2i \pi w_n \xi}$ ? Then you can look at things like $\int_{-\infty}^\infty \varphi(\xi) |\hat{f}(\xi)- D_N(\xi)|^2 d\xi$ for $\varphi \in L^1$ Nov 9, 2017 at 0:29
• yes. if $w_n$ has Gaussian distribution around $n$. thanks for writing it in a better way Nov 9, 2017 at 6:32
• I found the answer in a closed form by using moment generating function for Gaussian distribution. Nov 9, 2017 at 18:46