# How to Tell How Likely a Signal Is Present in Another One (Variance Unknown)?

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!

• 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. – Jason R Mar 16 '13 at 16:37