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Assume there are 11 signals. These signals are very similar to each other in both time and frequency domains. (And the similarity between them is not random; they are similar because they have somehow a common source. Actually they are well-correlated.) All these signals have the same number of samples. And I have access to them all.

One of these signals is important for me. (Let's name them S_main & S1~S10)

Now here's the question:

  • How can I combine S1 ~ S10 in order to re-generate S_main theoretically !!?

[Note: S_main itself is available !!! But I want to re-generate it using the combination of S1~S10 , in order to compare the physically-measured S_main to the theoretically-computed S_main.]

Thanks for your help.

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    $\begingroup$ We need more information. What is the relationship of S_main to S1 - S10? $\endgroup$ – Jim Clay May 29 '13 at 14:12
  • $\begingroup$ They are related in terms of amplitude, waveform, frequency, max., min., etc. $\endgroup$ – Omid1989 May 29 '13 at 15:04
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    $\begingroup$ As @JimClay has already pointed out, you need some model how your signals are generated in order to be able to solve your problem. If S_main=-S1, just invert S1, if S_main=aS1+bS2, use some optimality criterion to estimate constants a and b, etc.etc. Do you see how arbitrary these "solutions" are? This comes from the fact that your problem is not well-defined. $\endgroup$ – Matt L. May 29 '13 at 15:26
  • $\begingroup$ Yes, what @MattL said. We need to know more than just "they are related". Are they supposed to be the exact same and they just have different noise? $\endgroup$ – Jim Clay May 29 '13 at 16:12
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    $\begingroup$ If we talk about a relation or a model, we usually mean a parametric model with some unknown parameters. I.e. you need to know how the signals are related in terms of some model. And even knowing this relation/model, it is usually still not so trivial to solve the estimation problem. $\endgroup$ – Matt L. May 29 '13 at 16:42
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If the difference between signals is mostly additive Gaussian noise, you could try averaging together the 10 measured signals to estimate the original. If there is fat-tailed shot noise, you could try applying a median filter on your 10 measured signals. If these assumptions are incorrect, then the results may be garbage.

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  • $\begingroup$ Thanks hotpaw2 for your answer. Actually the difference between the signals is beyond noise. So I need to do a weighted averaging rather than a simple average. And the question is how to find the weights. $\endgroup$ – Omid1989 May 29 '13 at 18:46

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