To keep things simple, one thing at a time:
1. Confusing peak detection with envelope detection - the former requires hardware support while the later can be done in software.
I'm not sure I understand what you mean by this. One does not have to sample the peak to determine the magnitude or when it occurs.
The input here is the 100 ps 10 MHz repetition pulse. The scope samples are 1000 ps apart. It's quite clear that the band limited pulse is well captured in dot mode and shows a ~ 1 ns rise time.
So why are these supposed to be different and how?
I should like to note that the small dip in most of the displays just prior to the rise of the pulse is real. The pulse actually starts there, but there is phase dispersion and delay in the analog circuitry. I'll demonstrate that later using a 20 GHz SD-26 sampling head which can more accurately display a 100 ps pulse.
The ringing before the step in average mode is the result of using a zero phase sinc interpolator. The correct interpolator is the minimum phase impulse response of the AFE which is precisely what is shown using persistence and dot mode.
Watching the live display, the distribution of the dots varies with time. I suspect it is simply the result of clock jitter rather than a design feature. However, dot mode and persistence let you get a very good representation of the pulse.
Have Fun!
Reg