I'm investigating quantization error.
I have an analogue waveform that looks like this and is of theoretically infinite resolution:-
I've sampled it as (8 bit oscilloscope readings & 0b1) to produce a file of single ones or zeros. The samples are partially random as per quantization theory, but I want to check the correlation with the analogue original. Theory suggests that there should be some correlation.
How can I do that? Clearly I can't compare the 1 bit samples with the 8 bit samples they came from.
Put another way, how can I show that my quantization is entirely uncorrelated to the underlying waveform?
Edit & maybe a solution:-
St is the theoretical and perfect(!) signal.
S1 is my sample set as (scope reading) & 0b1.
S2 is another independent sample set as (scope reading) & 0b1.
It occurs to me that I can't possibly compare S1 with St, as quantisation error will be present in both sets. But, if S1 ~ St, then by inference S2 ~ St. Hence I can test the hypothesis that S1 ~ S2 since they're both derived from common signal St. If there is a correlation, then S1 must be correlated to St too.
So I can just use normal correlation analysis (or Hamming distance, Jaccard index) between S1 and S2 to measure any. Yes/no?