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As an electrical engineer I am interested in the phase shift of two signals. At one specific moment I can identify the phasor of a signal as a vector in the complex plane. To transfer the problem into math:

I've got two 2D-vectors $\vec{a}$ and $\vec{b}$, which both change random in time, but are somehow dependend of each other. I am able to measure the average and the maximum absolute value of the length of both vectors for a certain period of time. Additionally, I am able to measure the average and the maximum of the length of the geometric sum $|\vec{a}+\vec{b}|$ of the two vectors.

Now, I want to use the law of cosines $|\vec{a}+\vec{b}|^2 = |\vec{a}|^2+|\vec{b}|^2-2|\vec{a}||\vec{b}|cos(\varphi)$ to tell something of the statistic of the phase $\pi - \varphi$ between both vectors.

I am kind of worried whether the average of the phase is possible to determine in that way since I killed the phase information previously by taking the absolute values. Is it as simple as that?

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If you know $|a+b|$, $|a|$, and $|b|$ at a particular timestep, you can certainly calculate the cosine of the angle between them using (essentially; I think you made a typo) the method you described: $$\cos(\varphi)=-\frac{|a+b|^2-|a|^2-|b|^2}{2|a||b|}.$$ Unfortunately, if all you know are the average (and maximum) lengths of $a$, $b$, and $a+b$, you can't get any distributional information on $\cos(\varphi)$ (or any other function of $\varphi$) -- it's not necessarily true that $$\mathbb E[f(x,y,z)] = f(\mathbb E[x], \mathbb E[y], \mathbb E[z])$$ if $f$ is some "complicated" (non-linear) function.

If, however, you are able to make this calculation at every timestep, then you can compute $\cos(\varphi)$ for every timestep, and thus whatever else you want involving the distribution of $\varphi$.

Carl Schildkraut
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