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The following algorithm takes a set of signals, and return a complex number:

  • For a given set of signals - $\{s_1(n),s_2(n),...,s_M(n)\}$, compute the DFT of each of them - $\{S_1(k),S_2(k),...,S_M(k)\}$
  • Choose a specific frequency, $k=k_f$, and take the corresponding coefficient for each signal
  • Each of these coefficients is a complex number, compute the complex sum and divide by the sum of magnitudes (absolute values, amplitudes) $$ C(k_f) = \frac{\sum_{m=1}^{M}S_m(k_f)}{\sum_{m=1}^{M}\|S_m(k_f)\|} $$

If each coefficient has a different phase the magnitude of the result will be small due to cancellation. But if many coefficients will have the same phase (especially if their magnitude is greater than the magnitude of the rest), the magnitude of the result will be big.

In my case each signal also has a given weight and a weighted sum is used (for both of the summations in the last step), I need to measure the influence of the phase/shift of the chosen frequency on the weights.

If such a relation exists, the magnitude of the result measure its strength, and the angle of the result is the phase that cause an increase in the weights.

I think I read in the past about a similar measure in the context of direction of arrival (DOA) detection using array of sensors but I can't remember the details.

Is there a name for this technique? I seek for a name I can find in research papers

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You can call $C(k_f)$ the "angular coherence", or "angular correlation coefficient" at frequency $k_f$.

Alright, I just made that up if you just need a name for it! If you need some interpretation, here is some detail:

A complex number can be represented by a vector in the polar system. So we have $M$ vectors here. The numerator is a vector sum and the denominator is a normalizing factor. From the triangle inequality, the magnitude of $C$ is always in the interval $[0,1]$. $C$ is zero if the vectors are anti-symmetric pairs or if they completely cancel eachother, and is one if all are equal. So the remaining possibilities represent how close their composition are to either these two extremes.

I suppose you asked for the name to find more resources. Alternatively, you can ask your possible related questions directly, and probably there are some people here to answer.

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    $\begingroup$ "Note that C=0 implies that M is even if all values are nonzero", why the sum of 3 complex numbers can't be 0? $\endgroup$ – dvb Apr 16 '17 at 10:07
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    $\begingroup$ It was a mistake! We can of course have three vectors that cancel eachother. $\endgroup$ – msm Apr 16 '17 at 10:54
  • $\begingroup$ Thanks for the answer, these are precisely the names I had in my mind :) I updated my question to be more specific. I seek at least for a name that describe the coherence in the time domain, maybe "delay coherence" at frequency? $\endgroup$ – D . Apr 16 '17 at 12:14

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