# Approximate the angle of weighted sum of complex numbers

1. Sep 16, 2013

### changyongjun

1. The problem statement, all variables and given/known data

This is not a homework question, but I'm facing this from my research.
I have N complex numbers defined as $x_{n}=|\alpha_n| \cdot e^{j \theta_n}$ for $n = 1,\ldots,N$
and my observation is the sum of those numbers $r = \sum_{n=1}^{N} x_n$.

From the observation $r$, I want to approximately estimate the weighted average of $\theta_k$ like

$\hat{\theta}=\frac{ \sum |\alpha_n| \theta_n } { \sum |\alpha_n| }$

2. Relevant equations

3. The attempt at a solution

From numerical simulation, I know that

$atan2 ( \sum_{n=1}^{N} |\alpha_n| \cdot e^{j \theta_n} ) \approx \hat{\theta}$ if $|\theta_x - \theta_y| << 1$ for all $x$ and $y$.

Is there any clue how to approximate this estimation theoretically?

Thanks all in advance
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution