Beat detection and calculation of start time
Interesting. I would be very tempted to use correlation and given the OP wants to specifically find the occurrence of a first pulse implies that the correlation interval is restrained to be one pulse cycle wide. (Unless we know further information, and specifically how many pulses occur after a start position- if this is known then further processing can be done on the entire &quot;burst&quot;). The first question to determine the applicability of an optimized correlation approach is how correlated each of the pulses are to each other, within a given sequence and from sequence to sequence (repeatability over time). This can be evaluated by cross-correlating various captures to a representative pulse and looking at the statistics on the peak magnitude of the cross correlation function, and how much margin this allows relative to the &quot;noise&quot; correlation everywhere else.  If that were to suggest correlation could be used (by providing a sufficient probability of detection vs probability of false alarm, which such metrics would be clear from this evaluation), I would then determine the mean reference pulse and use that to detect a pulse present or not present condition using a correlation function. This is basically and optimally a matched filter; optimally if the background noise is white. If the noise is not white, then targeted noise rejection filters could also provide significant advantages - this can be determined by capturing the background with no pulses present and evaluating the time and frequency statistics of those captures to see if there are particular details that can be targeted and eliminated (for example 60 Hz spurs from power lines, etc...).
Where a correlation / matched filter approach wouldn't work well is if the content of each pulse in time and frequency is mostly random from pulse to pulse, in which case an optimized receiver would instead consist of a power detector with a pre-selection filter around the dominant frequency content of the pulse prior to detection.  The first cross-correlation testing described above would reveal the optimum strategy based on those statistics. In either case, detecting the first (or any) occurrence limits the observation time for decision to be one pulse interval, which then limits the available SNR to make any such decision for any approach (unless other intelligence is known that make a first pulse unique from any other).