Is there a good rule of thumb for setting the gain of a noisy signal before digitization? Say I have a 16-bit ADC with a ±1V range; to what level should I amplify the noise floor of my signal relative to the resolution of the ADC.
This ADC has nominal resolution of 31μV and my hypothetical small signal lives in 100nVrms noise within the sampling bandwidth. How much amplification should I apply before the ADC so that I can increase my SNR through oversampling.
Your signal has a dynamic range (max to noise level). Your ADC has a dynamic range (full scale to noise level).
Let's say your ADC has a larger dynamic range than your signal. Congratulations, you have a choice of gains to use. The maximum usable gain is the ratio of the max and full scale, use any more and it will overload your ADC. The minimum useful gain is the ratio of your noise levels. Using less gain will further degrade the noise level on your signal. Pick a gain somewhere in that range and rejoice. You may want to think whether there's any way you can improve your signal noise at source.
Let's say your signal has a larger dynamic range than your ADC. The range of gains suggested above has just vanished to nothing. The only thing you can do is set the gain at the ratio of the max signal levels, as an overloaded ADC is a non-working ADC. You have to suffer the fact that the ADC limits your noise, and you may want to think about getting a better one, or averaging readings to improve the noise.
Unless - the signal isn't at its maximum all the time, but only sometimes. Then you may be able to use switched gain to improve things. Set the gain to the ratio of the present signal maximum and your ADC full scale. If it overloads, turn the gain down. If the reading is persistently below some threshhold, say half scale, then turn the gain up to improve the noise. If you can tolerate the downtime of the occasional overload, then this works. There are variations on this, for instance use two ADCs with different gains. Choose the largest signal that isn't overloaded.
How much amplification should I apply before the ADC so that I can increase my SNR through oversampling.
At least enough to read the noise (including internal noise in the ADC). If the total noise is 100nV then you need to amplify it to 31uV. Then you have a 1 bit sample that can be 'oversampled' to extract a signal which is below the resolution of the ADC.
This will provide the greatest dynamic range. If you don't need this much this range then amplify the signal until the largest signal + noise is slightly below 1V. Avoid clipping because this will lose information and distort the signal during noise peaks.
Some years back I implemented magnetic beacon positioning receiver, using 600,000X (116dB gain) in the 80Khz to 150KHz region. With considerable use of "local battery" power to the various gain modules, there was no instability of oscillation. The maxgain output, using 6volt lantern battery was about 1volt RMS "random" noise, with some obvious clipping. Was that a problem? Dunno. For various reasons, the project was put on hold. However...........
The gain was switchable in 6dB steps, from 20X to 600,000X: used 1x/2x/4x/8x upfront module followed by 8x/8x and 64x. Purpose was to keep the ADC loaded between -12dB and -6dB, to always allow for noise spiking so the downstream FFT filtering would get linear version (refrigerator motor noise, etc) of whatever Hfield interferences perturbed the loop antenna.
