# Beginner level: deconvolving Gaussian noise to find initial PDF

noob question.

An initial process has an unknown pdf, P(X), which is then subjected to additive Gaussian noise. Repeated sampling of the sum is performed so that

$y=x+noise$

is repeatedly measured and the pdf for y is determined, P(Y). Is there anything that can be said analytically about P(X)?

I'd appreciate any simple references.

Thank you

You have a random variable $X$ whose pdf is $f_X(x)$. when another random variable $N\sim f_N(n)$ is added to $X$, then the pdf of the sum is just $f_Y=f_N*f_X$, where $*$ is convolution. This can be written in the Fourier domain as $F_Y=F_NF_X$. So if you know both $F_Y$ and $F_N$, and your $F_N$ is well-behaved, then $F_X=\frac{F_Y}{F_N}$.
After that, you can acquire $f_X$ by inverse Fourier transform.
Start by looking at the histogram of $y$, by which you can find $f_Y$. I think you can easily do the rest, given the above information.