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Approximate the angle of weighted sum of complex numbers

  1. Sep 16, 2013 #1
    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 [itex]x_{n}=|\alpha_n| \cdot e^{j \theta_n}[/itex] for [itex] n = 1,\ldots,N [/itex]
    and my observation is the sum of those numbers [itex] r = \sum_{n=1}^{N} x_n [/itex].

    From the observation [itex]r[/itex], I want to approximately estimate the weighted average of [itex] \theta_k [/itex] like

    [itex] \hat{\theta}=\frac{ \sum |\alpha_n| \theta_n } { \sum |\alpha_n| } [/itex]


    2. Relevant equations


    3. The attempt at a solution

    From numerical simulation, I know that

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

    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
     
  2. jcsd
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