I am interested in deriving an analytical expression for the throughput of a pulse having a Gaussian intensity profile along the spatial $x$ direction and a boxcar-like time dependence passing through a slit-shaped shutter. My approach is to take the convolution of the shutter and pulse over $x$ and $t$.
The pulse is defined as
$$ g(x,t)=\begin{cases} \frac{1}{\Delta t\sigma\sqrt{2}}\exp \left (-\frac{x^2}{2\sigma^2} \right ), & \text{ if } t_0-\frac{\Delta t}{2} \le t \le t_0+\frac{\Delta t}{2} \\\\ 0, & \text{ otherwise}\end{cases} $$
and the slit function is defined as
$$ s(x,t) = \begin{cases} \frac{1}{2v|t-t_0|\Delta t}, & \text{ if } t_0-\frac{\delta_x}{v}\le t \le t_0 + \frac{\delta_x}{v} \text{ and }-(t-t_0)v\le x \le (t-t_0)v \\\\ 0, & \text{ otherwise} \end{cases} $$
where $v$ is the velocity at which the shutter opens and closes, $\delta_x$ is the maximum opening of the shutter, and $\sigma$ is the standard deviation of the Gaussian, $t_0$ is the time at which the shutter is at its maximum opening and the Gaussian pulse is centered around the shutter in time as well and $\Delta t$ is the length of the Gaussian pulse.
To find the expression for the pulse after the shutter, I believe I should take the convolution over $x$ and $t$
$$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}s(\chi,\tau)g(\chi-x,\tau-t)\mathrm{d}\chi\mathrm{d}\tau $$
At this point I am not sure how to proceed, mainly because of the $\tau$ dependence of the $x$ domain. Should I integrate over $\chi$ first? Perhaps my whole approach is flawed.