I'm trying to evaluate $$g[x] = f[x] \ast f[x]$$ where * is the convolution operator and
$$f[x] = RECT(\frac{x-2.5}{5}) \cdot exp (+i \pi x^2)$$
I assume the best approach to this equation is: find the fourier transform of $$f(x) = F(\xi)$$ then $$F(\xi) \cdot F(\xi) = G(\xi) $$ Then perform the inverse fourier transform $$g(x) =\mathscr F^{-1} [G(\xi) ] $$
The issue I have is actually calculating the fourier transform of the chirp and then how to multiply that with a rect. Any help or resources to find the answer would be appreciated.
UPDATE: for the Fourier transforms I get: $$F[\xi] = 5 SINC[\frac{x-2.5}{\frac{1}{5}}] \ast e^{\frac{i\pi}{4}}\cdot e^{-i \xi^2} $$
I think that should be right, but then G[$\xi$] would have to equal something messy: $$G[\xi] = [5 SINC[\frac{x-2.5}{\frac{1}{5}}] \ast e^{\frac{i\pi}{4}}\cdot e^{-i \xi^2}] \cdot [5 SINC[\frac{x-2.5}{\frac{1}{5}}] \ast e^{\frac{i\pi}{4}}\cdot e^{-i \xi^2}]$$
does this approach seem right or can I simplify it down somehow?