You could benefit from reading the original document a few more times.
The ADC is done via a comparator - it compares the (pulsed) voltage ramp on the DUT (the "ramp" on the dut will be mostly due to the ESR + some charge-up of the capacitor. Due to the very short charge-up time, the 2nd effect is minimized) vs. the voltage ramp on a standard capacitor. Initially, the voltage ramp on the DUT, after being amplified, will be higher than the voltage ramp on the standard capacitor -> the comparator output pulses.
Until the point where the voltage ramps up sufficiently on the standard capacitor -> "missing pulse".
It is a crude form of ADC but necessary due to the very fast charge up time on the dut - necessary to prevent the dut to be materially charged up.