This looked initially a pretty simple question but it is turning to be something I can't figure out:
We want to detect a pulse (signal) in a given interval of time (say 1us) by simply setting a threshold with a comparator. The noise (added to the signal) is bandwidth limited (BW) and Gaussian (known sigma). An event is defined as the comparator going from zero to one. If the threshold is low enough (set to be able to detect small signals), I'll have false positives.
How do I compute how many times the comparator will trigger in that given interval due to noise (no signal present)?
The temptation is to answer "you integrate the area below the Gaussian pdf (probability distribution function) for all values above the threshold". But that is an answer for discrete systems. I.e., that would tell me something like, if you have 1000 samples, 5% will result on false positive.
Nevertheless, here I am in continuous time domain.
- If I want to go down that path, how many samples I assume I have in the 1us interval?
- And how do we account for the fact that samples are not totally uncorrelated from the neighboring ones (as the noise BW is limited)? I.e., the signal may get above the threshold and stay there for a bit but I only want to account for the 0 to 1 step.