convolution of gaussian and sinc function

Question

I have some data that I know is the convolution of a sinc function (fourier transform artifact) and a gaussian (from the underlying model). I would like to fit this data to a functional form of the convolution - is there an analytic form of the convolution of a sinc and a gaussian?

Answer 1

Fourier transform of $\operatorname{sinc}(x)$ is $$ \hat{\operatorname{sinc}}(\omega) = \int_{-\infty}^\infty \operatorname{sinc}(x) \mathrm{e}^{i x \omega} \mathrm{d} x = \pi \operatorname{rect}(\omega) = \left\{ \begin{array}{cc} \pi & -1 < \omega < 1 \\ 0 & \text{otherwise} \end{array} \right. $$

Fourier transform of the Gaussian density is it's characteristic function: $$ \phi(\omega) = \exp\left( i \mu \omega - \frac{\sigma^2 \omega^2}{2} \right) $$ The convolution you seek to find is the inverse Fourier transform of the product of Fourier transforms: $$ \begin{eqnarray} f(z) &=& \frac{1}{2 \pi} \int_{-\infty}^\infty \hat{\operatorname{sinc}}(-\omega) \phi(\omega) \mathrm{e}^{-i \omega z} \mathrm{d} \omega \\ &=& \frac{1}{2 } \int_{-1}^1 \phi(\omega) \mathrm{e}^{-i \omega z} \mathrm{d} \omega = \int_0^1 \cos((\mu-z) \omega) \exp\left(-\frac{\sigma^2 \omega^2}{2} \right) \mathrm{d} \omega \\ &=& \frac{1}{2 \sigma} \mathrm{e}^{-\frac{(z-\mu)^2}{2 \sigma^2}} \left( \frac{1}{2} \operatorname{erf}\left(\frac{\sigma}{\sqrt{2}} - i \frac{z-\mu}{\sqrt{2} \sigma}\right) - \frac{1}{2} \operatorname{erf}\left(-\frac{\sigma}{\sqrt{2}} - i \frac{z-\mu}{\sqrt{2} \sigma}\right) \right) \end{eqnarray} $$