External signal noise versus ADC resolution

Question

One of the answers in this question says: “Having an ADC that has better resolution than the external signal noise is a waste of money.”

I want to illustrate and understand what is meant here. So for clarity I tried to ask by using the following figures:

Above in Fig1 random noise in color red imposed on a constant input signal where the peak to peak is 1mV as shown in the figure. And in Fig1 the ADC has a resolution of 1mV. In Fig2 however the same input now goes to an ADC which has a resolution of 0.3mV.

Does that mean for this input any ADC with resolution Δ<1mV is redundant? What could be the minimum adequate resolution?

Imagine the 1mV peak to peak superimposed random noise is always the case. And the signal is coming from any voltage source with a sensitivity of 0.5mV. So there are two things here. One is the sensitivity of the source like force transducer sensitivity and the other is the random noise I showed in red.

Considering all these I cannot make reasoning what limits what along with ADC resolution.

Answer 1

You shouldn't believe every answer you see to questions, though they are often worth thinking about.

Question - Should an ADC have better resolution than the noise of the input signal?

Answer - Yes, but the answer is complicated by the type of noise, and the bandwidth of the signal and the noise. It also leaves open the question of how much better.

Let's assume a noise like the well defined noise you've drawn rather than random noise, it won't make too much difference when handwaving, but it is important when you get down to detailed calculations.

It's probably better to ask the question in a slightly different way, 'when should I stop striving to increase the ADC resolution?'

Obviously if the ADC resolution is 10mV, with 1mV of noise on the signal, then adding an extra bit to the ADC will always yield essentially an extra bit of answer accuracy. The ADC is the limiting factor, not the input noise. We can take a single reading, or take many and average them, that conclusion is unchanged.

If the ADC resolution is 1mV and the noise is 10mV, then adding an extra bit to the ADC will never improve the answer accuracy significantly. The noise is the limiting factor. If we take a single reading, then the uncertainty is limited by the noise. If we take multiple readings and average, then the ADC already has sufficient resolution to reduce the noise.

If the ADC resolution and the noise are comparable, then when taking a single reading, we can't improve the reliability of the answer (the likelihood that it will fall within the limits of our published specification for the equipment) by increasing the ADC resolution. In this circumstance, an extra bit of ADC resolution may not be worth it.

However, if we take multiple readings and average, then we really need more samples of the noise to make that process work well (reliably). We really need to resolve the noise into 4 or 5 levels, with a constant input signal, to get the most benefit from averaging it. Ten levels would be a little better, but not a factor of two better.