Rise and fall timings of a pulse generator
From  Hartley's  Law relating Analog \$ f_{-3dB}\$ and Digital \$f_{Rate}\$ Bandwidth \$f_{BW}\$ (= 0.35/rise time) and number of bits N per sample. Sometimes shown as follows;

\$ f_{Rate} = 2*f_{-3dB_{BW}} \cdot log_2(N) =6.02f_{-3dB_{BW}} *log_{10}\cdot (N)\$ 


let N=10 bits with resolution of 1/1024 as your "precision"
if your message   is a square wave is 1MHz and BW desired is 300MHz or ~1ns rise time then ADC sampling rate must be >>6.02*300MHz*log10= 1.8GHz


A Bessel filter must attenuate >60 dB ~ 900 MHz for aliasing &lt; resolution since N=10 implies 60 dB range and 900MHz is 1/2 sampling rate but has -3dB BW of 300MHz  thus >= 6th order LPF.

**I'm not sure how you wanted to do a Reimann Sum on an exponential signal.