Very fundamental question: by what number do you divide your ADC counts?

Question

What number is it correct to divide by your ADC counts as a first step in converting counts to voltage? For example, in a 10 bit ADC, is it correct to divide by 1024 or 1023?

I have always divided by 1024, which means that a full count would yield less than full scale voltage reading. I think this is a more accurate representation of the hardware, though, since there is usually an LSB of error. This is also what I've read in every datasheet, as well as various DSP literature and white papers from ADI and TI. Please prove me right (or wrong) with the detailed reasoning why!

Links to lit: http://www.ti.com/general/docs/litabsmultiplefilelist.tsp?literatureNumber=slyw038b

http://www.analog.com/en/education/education-library/data-conversion-handbook.html

http://www.analog.com/en/education/education-library/scientist_engineers_guide.html

Answer 1

The correct answer depends on implementation and what you decide to call Full Scale; if the end points are included in the conversion mapping used, then the answer would be $2^n-1$ as I depict in the graphic below.

The easy way to see this is to consider a simple 3 bit converter which has 8 levels, for the conversion of a unipolar signal from 0 to 1V, where we specifically have mapped (in the design of the converter) 0V (+/- 1/2 lsb) to 000 and 1V (+/- 1/2 lsb) to 111:

Here we see clearly how with 8 levels the resulting mapping is given as

$$V_{out} = 1V\frac{b}{2^3-1}$$

where b is the digitized value of interest, in this case ranging from 0 to 7.

Because our digital range is inclusive of the end points of our voltage range, it would always be 1 less than the number of levels in this case.

Although not an ADC, but a counter example I could find showing the implementation dependence is A DAC implemented with an R-2R ladder (https://en.wikipedia.org/wiki/Resistor_ladder). This shows a mapping where a division of $2^n$ is used, and the full scale voltage IF based on the reference voltage used is never actually reached as an output of the DAC (but why not simply call "Full Scale" the voltage corresponding to the maximum digital value? - Here you see that it is simply a matter of how you decide to scale it).