Does anyone know or can achieve a nicer solution for the integral of the form
$$ f(y, \gamma) = \frac{ 1 }{ \sqrt{ 2\pi } } \frac{ 1 }{ 2\sqrt{3} } \int_{ -\sqrt{3} }^{ \sqrt{3} } x e^{ - \frac{1}{2} (y - \sqrt{ \gamma }x )^{2} } dx $$ ?
I thought due to the form of the integral we can obtain a nice solution, however what I obtain is far from a nice solution. I've tried with a change of variable $ z = \frac{1}{ \sqrt{2} } ( y - \sqrt{ \gamma } x ) $ obtaining
$$ f(y, \gamma) = \frac{y}{2\gamma} \left[ \mathrm{erf} \left( \frac{ y + \sqrt{ 3\gamma } }{ \sqrt{2} } \right) - \mathrm{erf} \left( \frac{ y - \sqrt{ 3\gamma } }{ \sqrt{2} } \right) \right] + \frac{ \sqrt{2} }{ \sqrt{\pi} \gamma } e^{ - \frac{1}{2} y^{2} } e^{ - \frac{3}{2} \gamma } \sinh( \sqrt{ 3\gamma } y ) $$
This is the numerator of the non-linear estimator $ \mathrm{E}[x|y] $ for an additive scalar Gaussian channel ( $ y = \sqrt{ \gamma } x + n $ ) with an input uniformly distributed ( $ f(x) $ ) zero mean and variance normalized to unity.
I would like to compute the MMSE using Matlab. Although I know Matlab has the erf function already defined, I'm curious about If there exist a nice expression.
Ignoring the constants, by a linear change of variable you can split the integral in two terms
$$\int ue^{-u^2}du$$ and $$\int e^{-u^2}du,$$ with general bounds.
The first one is elementary, while the second is known to require the error function.
So no better solution.