I need some help with the following problem:
Let $X$, $Y$, and $Z$ are independent circularly symmetric complex Gaussian random variable with zero mean and unit variance, i.e., $X$, $Y$, and $Z \sim CN(0, 1)$.
These random variables can be represented in polar form as \begin{equation} X=r_1 e^{i \alpha}\\ Y=r_2 e^{i \beta}\\ Z=r_3 e^{i \gamma} \end{equation} where $r_1$, $r_2$ and $r_3$ are the magnitude (Rayleigh distributed) and $\alpha$, $\beta$ and $\gamma$ are the phase of the variables (Uniformly distributed over $[0, 2\pi]$).
How can I represent the random variable $aX+bY+cZ$ (where a,b, c are constant) in polar form as a function of the variables $a$, $b$, $c$, $r_1$, $r_2$, $r_3$, $\alpha$, $\beta$ and $\gamma$. Particularly I am interested to know the phase of $aX+bY+cZ$ in terms of these variables.
Thank you very much