# How many bits of information does a voltage of 0-5V , with accuracy of ±50mV convey?

In my intro to logical design class, we were shown this simple question as one of the first examples for the exercises we will have. The question was:

An analog voltage is in the range of 0–5 V. If it can be measured with an accuracy of ±50 mV, at most how many bits of information does it convey?

The answer we were provided stated that:

Accuracy of ±50 mV means that the analog signal is divided every 100mV, so we have 50 discrete measurements from 0 to 5V. Thus $$log_250= 5.64$$ bits. My questions are:

What does accuracy of 50mV really mean?

How did we come up with a division of every 100mV?

If out analog voltage was in range 0 to 6V, then would we say that 6V = 6000mv, so $$\frac{6000}{100} = 60 \text{ discrete measurements, thus } log_260 = ... \text{ ?}$$

And lastly, what am I really being asked here? What do 5.64 bits really show us?

I would appreciate any help given!!

What does accuracy of 50mV really mean?

It's an accuracy of +/- 50 mV and that means there is a spread of 100 mV. Put another way, you can't rely on a measurement being absolutely accurate so, if you measured a value of (say) 1 volt, the real voltage will be somewhere between 0.95 volts and 1.05 volts.

How did we come up with a division of every 100mV?

The +/- 50 mV inevitably sub-divides measurements into blocks having a range of 100 mV. That defines the error-free accuracy when converting the analogue voltage to a digital value.

What do 5.64 bits really show us?

It's a number that you can use to make comparisons. Many ADC specifications do this; an ADC may have 16 bit resolution but, at full scale and taking into account quantization and noise, the effective number of bits might only be 15.5. It's not a real number - it's a comparative guide number that tells you how good one ADC is versus another.

Dover Books sells a fine introduction (various probability examples, and philosophy of information theory) paper back.

On the other hand, in the game of 20_questions, people are challenged to rapidly zero in on the correct answer, using binary_chopping questions if possible.

Such as "thinking of a number between 1 and 100." How many questions are needed to detect that number?

• < 50 YES

• < 25 NO

• < 37 NO

• < 45 NO

• < 48 YES

• Is the number 47 NO

• Is the number 46 YES

Except for the last question (used to verify), these are binary_chopping style of numbers.

In a pure binary_interval, 1----128 for example, the # of questions is log2(size) exactly.

Now --- what if the interval sizes, in our example being exactly ONE, are not uniform? In ADC analog_digital_converters, that non_uniformity is called Differential Nonlinearity.

All these imperfections go into ENOB effective number of bits.

• I can't see what the last two paragraphs have to do with the question(s) asked. Commented Oct 17, 2020 at 10:51
• I see the connection, but it is only half made and doesn't really answer the question. If the rest of it were there it would be OK, but there's too big a gap between "assume a spherical chicken" and "X=1" for this to be useful.
– JRE
Commented Oct 17, 2020 at 11:32