If I have a complex signal $$ y = h e^{j\phi} + n $$ where $ h \sim \mathcal C \mathcal N (0, \sigma_h^2) $ and $ n \sim \mathcal C \mathcal N (0, \sigma_n^2) $.
With $ h = |h|e^{j\theta} = |h|\cos \theta + j|h|\sin \theta $ and $ n = n_r + j n_i $, I can rewrite $y$ as $$ y = |h| e^{j(\phi+\theta)} + n $$ and thus the real part of $y$ should be $$ y_r = |h| \cos(\phi + \theta) + n_r $$
Now my question is, how do I find the distribution (pdf) $ p(y_r; \theta)$ with these given information?
Should it be $ \mathcal N (0, \sigma_h^2 + \sigma_n^2) $
or $ \mathcal N (|h| \cos(\phi+\theta), \sigma_n^2)$ or anything else?
note: 1. $h$ and $n$ are independent.
(I am assuming $h$, $\phi$ and $n$ are independent.)
Note that a complex Gaussian random variable $x\sim\mathcal{CN}(0,\sigma^2)$ is circularly symmetric, i.e., for every $\theta \in [-\pi, \pi)$ the law (distribution) of $e^{i\theta}x$ is identical to the law of $x$. This is usually denoted as
$$ e^{i\theta}x \stackrel{d}{=}x, $$
where $\stackrel{d}{=}$ means "equality in distribution".
It follows that
$$ \begin{align} y &= e^{i \phi} h + n\\ &\stackrel{d}{=}h+n\\ \end{align} $$
and the last sum is, of course, distributed as $\mathcal{CN}(0,\sigma_h^2+\sigma_n^2)$. By fundamental properties of complex Gaussian random variables, the real and imaginary parts of $y$ are independent and distributed as $\mathcal{N}(0,(\sigma_h^2+\sigma_n^2)/2)$.