This is a completely practical problem in application.
Considering $ A \sin(\phi + \phi_{0}) = \sum_{k=1}^{N} A_k \sin(\phi + \phi_k), \forall \phi \in \mathbb{R} $ where $\phi$ is a deterministic variable on the real line, $ A_k \sim N (\mu_{A}, \sigma_{A}) $ and
$ \phi_k, k=1..N$ have a certain identical distribution (here I further simplified the assumption).
And it is also with further assumption that $ \mu_{A}>0$ and $ \sigma_\phi < \mu_{A}$. Furthermore, random variables $A_k$, $\phi_k$, $k = 1..N$ are jointly independent.
Roughly speaking, the summation represents a superposition of N first harmonics whose amplitude and argument have random behaviors. My interest is to know how the components influence the overall harmonic, in terms of distribution.
Since it holds for every $\phi$, it can be simplified into: $ A e^{i \phi_0} = \sum_{k=1}^{N} A_k e^{i \phi_k} $
So, I would like to know how I can derive the distribution of $ A $ and $\phi_0$ from above equation. If precise distribution cannot be found, an approximate solution would also be grateful. As for approximation, one may consider that most $3\sigma_\phi < \mu_A$, so the negative values can be truncated off, and only the positive values are under consideration.