This was a genuinely thought provoking article. I had to challenge some personal assumptions.
Coming from an electrical engineering background, I disagree with how the author presented "Two types of quantizers". Mathematically rigorous, but not grounded in practical systems.
In ADCs, there is always an inherent +-1/2 LSB of quantisation uncertainty. The transfer characteristic is always mid-tread sampling, or at least I haven't come across any counter examples. This is true for bipolar or unipolar ADCs.
The lowest code is negative voltage reference, and the highest code the positive reference. The transfer characteristic plot will show what the author has demonstrated, that the highest and lowest bins will effectively be 1/2LSB in width.
In a unipolar system, this has the consequence of not being able to represent the midpoint voltage precisely, or in other words, the gray problem. In a bipolar system, 0V will be mid-tread N/2 value, but that doesn't mean it has "256 ranges".
So, I'll be sticking with (VREF+ - VREV-) * k / (2^N - 1). Or in other words I agree with the normalisation by 255. It's the fence post error all over again, you have N values, but N-1 ranges. If you have less ranges than you do values, you need to distribute 1 of those ranges between two values, hence the 1/2LSB range endpoints.
All ADCs I have looked at document that they can't represent the positive full scale. For instance, for an 8 bit ±1 V ADC, -128 represents -1 V, +127 represents 127/128=0.99219 V. The transition from 126 to 127 happens at 1.5 LSB from the positive full range. 1 LSB difference represents 1/128 = 0.00781 V difference, and not 2 / 255 = 0.00784 V.
But if you actually care about what the voltage (and uncertainty) is, most of this is difference is mostly pointless, you're reference will have a bias, there are linearity errors and so on. 1 LSB will not match either the 1/128 or 2/255, you will need parameters to compensate for it.
Coming from an electrical engineering background, I disagree with how the author presented "Two types of quantizers". Mathematically rigorous, but not grounded in practical systems.
In ADCs, there is always an inherent +-1/2 LSB of quantisation uncertainty. The transfer characteristic is always mid-tread sampling, or at least I haven't come across any counter examples. This is true for bipolar or unipolar ADCs.
The lowest code is negative voltage reference, and the highest code the positive reference. The transfer characteristic plot will show what the author has demonstrated, that the highest and lowest bins will effectively be 1/2LSB in width.
In a unipolar system, this has the consequence of not being able to represent the midpoint voltage precisely, or in other words, the gray problem. In a bipolar system, 0V will be mid-tread N/2 value, but that doesn't mean it has "256 ranges".
So, I'll be sticking with (VREF+ - VREV-) * k / (2^N - 1). Or in other words I agree with the normalisation by 255. It's the fence post error all over again, you have N values, but N-1 ranges. If you have less ranges than you do values, you need to distribute 1 of those ranges between two values, hence the 1/2LSB range endpoints.