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Hi everybody

There are 11 signals:

  • S_main : The original signal

  • S1 ~ S10 : 10 signals that are correlated to S_main with different correlation coefficients (coeff1 ~ coeff10)

Now here's the question:

How can I combine signals S1, S2, ..., S10 in order to regenerate S_main !!?

Thanks a billion for your upcoming help.

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  • $\begingroup$ You really ought to determine the distribution of $(S_1,\ldots,S_{10})$ conditioned on $S_\mathrm{main}$. Then you can estimate $S_\mathrm{main}$, for example, according to maximum likelihood. $\endgroup$ – Dustin G. Mixon May 19 '13 at 16:53
  • $\begingroup$ Thanks Dustin for your comment. May you explain a little bit more what you mean !? What should I do !? $\endgroup$ – Omid1989 May 19 '13 at 16:57
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Suppose you know the distribution of $(S_1,\ldots,S_{10})$ conditioned on $S_\mathrm{main}$. In the case of continuous distributions, you'll have $f(s_1,\ldots,s_{10}|s_\mathrm{main})$. This tells you how much of the probability is concentrated around $(s_1,\ldots,s_{10})$ when $S_\mathrm{main}=s_\mathrm{main}$. But you want to go the other way around: You want to be able to guess what $s_\mathrm{main}$ is once you know $(s_1,\ldots,s_{10})$. One method of estimation is maximum likelihood. Here, you define likelihood as $$ \mathcal{L}(s_\mathrm{max};s_1,\ldots,s_{10}) :=f(s_1,\ldots,s_{10}|s_\mathrm{main}). $$ Note that if you know the distribution, you get the likelihood function for free. For maximum likelihood estimation, all you have to do is find the $s_\mathrm{max}$ that maximizes $\mathcal{L}(s_\mathrm{max};s_1,\ldots,s_{10})$. For exponential-type distributions, it's often more convenient to take the log (and get a log-likelihood function), and maximizing this is equivalent since log is an increasing function.

If you also know the distribution $g(s_\mathrm{main})$ of $S_\mathrm{main}$, you should use a maximum a posterior probability (MAP) estimate. This method updates the distributional knowledge of $S_\mathrm{main}$ with the observation of $(s_1,\ldots,s_{10})$ according to Bayes' rule. After determining the posterior distribution of $S_\mathrm{main}$, your estimate is the $s_\mathrm{main}$ which maximizes this distribution, similar maximum likelihood estimation. This form of estimation is the bread and butter of Bayesian inference.

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  • $\begingroup$ Thank you very much Dustin for your answer. Your answer is very well; however it's much related to probability. My problem is in the field of signal processing. I have 11 signals (S_main & S1,_,S10) which are very much similar to each other in terms of max., min., freq., ...; I think you're looking at the problem from the probabilistic view. And that's why you use terms like distribution, likelihood, ... $\endgroup$ – Omid1989 May 20 '13 at 5:51
  • $\begingroup$ Actually I think I have to use some methods like Orthogonalization and Orthonormalization. But I'm not sure!! $\endgroup$ – Omid1989 May 20 '13 at 6:46
  • $\begingroup$ Omid1989, you should think about how to phrase what you want as a specific math problem, i.e., with vectors, matrices, etc. Note that the phrase "correlation coefficient" and the "statistics" tag you chose both led me to believe that your signals are actually random variables! $\endgroup$ – Dustin G. Mixon May 20 '13 at 11:40
  • $\begingroup$ You're right. I used the statistics tag, because I couldn't find any other proper one. Sorry about that. Actually, I want to generate S_main. The simplest way to do so is: (S1+S2+...+S10)/10 ; But definitely this is wrong!! I want to know how to combine S1 , ... , S10 in order to generate S_main. $\endgroup$ – Omid1989 May 20 '13 at 16:08

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