Hi everyone,


I will be too happy if anybody help me find a solution for the following problem.
In fact, I have a big problem that I could not solve it for weeks.


Assume that we have we have two independent zero mean Gaussian random variables, $X$ and $Z$ and we define the new random variable $Y$ as $Y=X+Z$. I define the sign and the magnitude of this three random variables as $X\_s$, $Y\_s$ and $Z\_s \in \{+1,-1\}$, $X\_M$, $Y\_M$ and $Z\_M \in R^+$, respectively. 
It is clear that the pairs $X\_s$ and $X\_M$ are independent, this is the same for $Y\_s$ and $Y\_M$, $Y\_s$ and $Y\_M$. 


My problem is how to find the conditional distribution $f(y\_M|x\_M)=?$.


I have a solution for this but I could not convince myself that my answer is true.
We have, 
$$f(y\_M|x\_M)= f(y\_M|x\_M,x\_s=+1)f(x\_s=+1|x\_M)+ f(y\_M|x\_M,x\_s=-1)f(x\_s=-1|x\_M)$$
$$=f(y\_m|x\_M,x\_s=+1)f(x\_s=+1)+ f(y\_M|x\_M,x\_s=-1)f(x\_s=-1)$$
$$=0.5\*(f(y\_M|x\_M,x\_s=+1)+ f(y\_M|x\_M,x\_s=-1))$$
then, since both $f(y\_M|x\_M,x\_s=+1)$ and $f(y\_M|x\_M,x\_s=-1)$ have the same distribution, i.e., "Folded normal distribution" then from above equation $f(y\_M|x\_M)$ follows "Folded normal distribution".


If it is correct it means that $f(y\_M|x\_M)= f(y\_M|x\_M,x\_s)$ that implies $X\_s$ and $Y\_M$ condition on $X\_M$ are independent!!!!!!!!!!!


But from $y\_M = \lvert x+z \rvert$, $Y\_M$ depends on both $X\_M$ and $X\_s$ !!!!
I am really confused and I will be too grateful if anybody help me solve the problem.
Thanks a lot in advance,



Conditional distribution of the modulus of the output of AWGN channel given the  modulus of the input
Yes, $Y\_M$ is independent of $X\_s$ - it doesn't matter whether or not you condition on $X\_M$.
You essentially give the answer yourself - conditioning on $X\_M$, you get that $Y | X\_M$ is a symmetric r.v., with $f(Y|X\_M) = \frac{1}{2\sigma\_Z}[\varphi(\frac{Y-X\_M}{\sigma\_z})+\varphi(\frac{Y+X\_M}{\sigma\_z})]$ where $\varphi$ is the standard normal density. Therefore, summing on $Y=\pm Y\_M$ you get $f(Y\_M|X\_M) = \frac{1}{\sigma\_Z}[\varphi(\frac{Y\_M-X\_M}{\sigma\_z})+\varphi(\frac{Y\_M+X\_M}{\sigma\_z})]$ (for any $Y\_M \geq 0$) which is equal to $f(Y\_M|X\_M,X\_s)$. It is true that $Y=X+Z$ depends on $X\_s$ but you can write $f(Y | X\_s=1) = f(-Y | X\_s=-1)$ and when you take the absolute value $Y\_M = |Y|$ becomes independent of $X\_s$.


