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It can easily be proven that an ideal analogue delta sigma modulator shapes the quantisation noise, as explained for example in this AD note under EQ 3: https://www.analog.com/media/en/training-seminars/tutorials/MT-022.pdf

Is there a similar mathematical high level proof that shows that only the average of the bitstream approximates the input signal rather than instantaneous single values?

PalimPalim
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  • Here's a high level proof: https://electronics.stackexchange.com/questions/576827/what-does-the-integrator-do-in-a-delta-sigma-converter/576833#576833 <--after all, we know they work and we know resistors do what is defined by ohm's law but, do we ask for a proof of ohm's law (we might on physics sites but not on EE). Or, do we ask for a proof that a PID controller does what it says on the tin? – Andy aka Aug 30 '23 at 15:07
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    Isn't this just the same as the proof underlying ensemble averaging of N samples having the same expected value and a variance reduced by $\sqrt{N}$ as compared to the original signal? – Scott Seidman Aug 30 '23 at 15:13
  • We usually look at sigma delta converters in the spectral domain, filters, noise-shaping, all that stuff. In which case, average means the DC component, though we usually generalise that to short term average, which means low frequencies. Instantaneous single values are the sum of all frequencies, including the ones to where the noise has been pushed by shaping, so would expect them to be very noisy indeed. – Neil_UK Aug 30 '23 at 18:53

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