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Cross correlation with sum of signals?

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Terminator3

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Is it feasible?
For example, we searching signal F1, F2 and F3.
We can do cross correlation of input signal S with known F1, F2 and F3 separately.
What about finding cross correlation of S with (F1+F2+F3)?
(if we really do not want to know F index, just presence).

If it is feasible, what is the name of such technique?
 

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