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I know this is probably a simple question, but I haven't been able to find a satisfactory answer anywhere...

Say you have a time series signal of finite length N. Call it $y[n]$. It looks like a sine-gaussian perhaps but with some random effects. The mean of zero, and no trend is present.

Now, say you receive this signal $y[n]$, and you have a set of K other 'candidate' signals, $x_{1}[n], x_{2}[n], ...,x_{K}[n]$. You want to see what the most likely candidate is. Is there a way to do this, and attach probabilities? For example, candidate 1 has a 20% probability of being present, candidate 2 15%, etc. such that the percents add to 100%.

Some notes:

  • I want amplitude to matter. If the candidate signal has a much smaller amplitude than $y[n]$, it should be less likely to be present, than another candidate which is otherwise exactly the same, but with a higher amplitude.

  • The variance of each data point in the signals (either $y[n]$ or $x[n]$) is unknown. All we are given is what is mentioned above. I know that in order to do something like a $\chi^{2}$ goodness of fit test (which has been suggested to me), something must be known about the variance of each data point in the signals.

  • The closest thing I have found is matched filtering, but how do I compute probabilities like I mentioned above? Or is computing probabilities like that sort of the wrong answer to the question?

  • Coherence is related, but its more about how the signals change over time (from my limited understanding). All the signals mentioned have a finite length N, and the signals are already matched in time (we only care about how similar they are at a particular time instant). Time delays between them are irrelevant.

Thanks!! Any thoughts any of you have on this would be greatly appreciated!

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    $\begingroup$ Matched filtering the signal $y[n]$ by the various candidates is an option, as you noted. Unless you have a firmer statistical model for the system, then I don't think you'll be able to assign actual probabilities, but you could generate arbitararily-scaled "scores" that could represent the likelihood of the presence of any of the candidates. Choosing a way to normalize the correlator outputs will be important, as you stated that you want amplitude differences to be taken into account. $\endgroup$
    – Jason R
    Mar 16, 2013 at 16:37

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Though the Matched Filter is the best tool detection of a known signals under AWGN it should work well here as well.

To say something about the probabilities the question is, do you know something about the energy of the received signals?
If you do, you should easily say something about the probabilities.

Pay attention that if the assumption is a signal could be within a different signal you need to apply a generalized Matched Filter (Multi Decision).

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