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For example, I want to convert an analog voltage signal to digital. I want to have a signal to noise ratio of at least 50,000 without aliasing. What sampling rate and resolution could I use to achieve this.

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SNR required for your ADC is \$10log_{10}(50000) = 47dB\$. This will be contributed by your thermal and quantization noise. To keep some scope for the thermal noise we need quantization noise better than 47dB. For instance, if you choose signal to quantization noise ratio (SQNR) of 50dB, then the required resolution becomes: $$SQNR = 6.02*N + 1.76 \implies N = 9\ bits$$ The thermal noise is decided by the size of capacitors you choose and is given by: $$NF\frac{kT}{C},$$ where, k = Boltzmann Constant,
NF = noise factor for your system, which includes the contribution of noise from your op-amp and sampling noise etc. It is design dependent factor.
Based on this and your thermal noise margin, you can calculate the size of the capacitors required. But usually at such low resolutions designs are not thermal noise limited.
The bandwidth of the ADC gives you the upper bound on the sampling rate you can achieve, and if you are going for a Nyquist Rate converter the maximum input frequency which you can apply without aliasing would be half of this rate.
The bandwidth, in turn, also depends on the size of the capacitors used, higher the capacitors lower the bandwidth. So, there would be a trade-off between bandwidth and resolution. Higher resolution would require you to have higher capacitors to decrease noise but your bandwidth also goes down, so you have to make some choice based on your requirements.

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Resolution error depends on smallest signal and resolution like 16 bit gives 64k but only if full scale signal, and aliasing error depends on noise band definition and filter band stop rejection ratio to give 50k:1 SNR or almost 100dB. good luck. Better start defining the problem better to accomplish anything. Goals= design specs.

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Here is 94.6dB SNR. Notice the Thermal noise (of the resistor in sensor and resistor inside the ADC in series with charge-sampling capacigtor) contribute a random noise of same size as the quantization "noise" uncertainty.

enter image description here

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