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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

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  • $\begingroup$ You have not told us the JOINT distribution of $X,Y,Z$. They can have complex Gaussian distributions without the triple $(X,Y,Z)$ having a JOINT Gaussian distribution, and if the triple has a joint Gaussian distribution there is the question of what the covariances are. If $X,Y,Z$ are independent then the covariances are $0.$ If they are JOINTLY Gaussian and the covariances are $0,$ then they are independent. If they are individually Gaussian but not jointly Gaussian the the covariances could be $0$ even though they are not independent. $\endgroup$ – Michael Hardy May 4 '17 at 19:03
  • $\begingroup$ Yes, I forget to mention X, Y, Z are independent complex Gaussian, also a, b and c are constant. Can we find the phase of aX+bY+cZ ? $\endgroup$ – Sabyasachi G May 5 '17 at 1:23

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