(I've reworded this question from earlier)
I ground the input of an ADC that has 50 Ohms internal termination, and observe a noise waveform. I measure the RMS voltage of this waveform.
Then I remove the ground at the ADC input and terminate the ADC input to 50 Ohms (just a resistor to ground, that's all). I measure exactly the same value for RMS voltage of the resulting waveform.
Assume this ADC noise is Gaussian (random).
Now I connect my signal to the ADC. Will the ADC noise above (the RMS value) add directly to the "true" signal? So, if I measure 2 mV RMS when grounding the input, will the observed signal be the "true" signal plus 2 mV RMS noise?
Or, is the noise generated internal to the ADC flowing through the internal ADC termination such that when I connect my signal source with 50 Ohms termination, the voltage divider that results cuts the observed signal in half as well as the input referred noise from the ADC? (So, I'd observe the "true" signal plus 1 mV RMS noise.)