# Number of triggers in a given interval for a given bandwidth limited noise

This looked initially a pretty simple question but it is turning to be something I can't figure out:

Typical problem:

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.
• do you know the height of the pulse? if there is a range of heights, do you know the minimum height of the pulse? – robert bristow-johnson Aug 1 '17 at 4:43
• When you say "pulse" do you mean signal or noise? Maybe a bit more background... This is for lidar. The system fires a laser pulse and this bounces back of the target. By measuring the time between the firing and the return one gets the distance to the target. The return pulse can be very big but as you want to increase the range (see further) the pulse usually becomes weaker and weaker (eventually under the noise). The problem about the optimum detector (how to distinguish a signal pulse from noise) can quickly get very complicated but I am already stuck on the most basic one... thanks! – Eduardo Aug 1 '17 at 18:28
• i meant only what you meant by "pulse" – robert bristow-johnson Aug 1 '17 at 19:03