Approximate the angle of weighted sum of complex numbers

[changyongjun]

Homework Statement

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

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

Homework Equations

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

 

[mighty2000]

Hello, thank you for bringing this question to our attention. I understand the importance of accurately estimating values in research. From your description, it seems that you are trying to estimate the weighted average of \theta_k from the sum of N complex numbers.

One approach you could take is to use the properties of complex numbers to simplify the problem. For example, you could rewrite each complex number x_n as |\alpha_n| \cdot (\cos \theta_n + j \sin \theta_n). Then, the sum of these numbers can be written as a single complex number with a magnitude and phase angle, which can be used to estimate the weighted average of \theta_k.

Another approach could be to use statistical methods, such as regression analysis, to estimate the relationship between the individual \theta_n values and the weighted average. This could help you derive a more accurate and precise estimation method.

Overall, it would be helpful to have more information about your research and the specific context in which you are trying to estimate \theta_k in order to provide a more tailored solution. I hope this helps and good luck with your research!