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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

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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.

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