# Electronic – Analog-Digital converter’s LSB formula

What is the correct definition of the LSB voltage for an analog to digital converter?

I have found two formulas:

LSB = FS / (2N)

LSB = FS / (2N – 1)

N … resolution in bits of the ADC.

FS … Full scale input range

Has this anything to do with the location of the first code transition?

Can someone clarify this problem for me?

The definition for the LSB as $$LSB = \frac{FSR}{2^N}$$ can be found in the IEEE Standard 1241-2010 ("IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters") and is commonly used for commercial devices. It can therefore be considered the right definition.

For integrated circuit design, when building an ADC as part of a signal processing chain or as a sub-block of a larger ADC it sometimes can make sense to to define the LSB differently (\$LSB = FSR/(2^N-1)\$).

The reason is that there are two conventions for ADCs depending on the code transition levels used. One is the so-called mid-tread convention where FSR/2 is right in the middle of a code and the first transition occurs at LSB/2. The other is the mid-riser convention where FSR/2 occurs at transition and the first transition occurs at LSB.

The transfer functions of both types are shown below, the dotted lines indicate the range for mid-tread type and the dashed lines are for the mid-riser.

As shown in the graph the last-transition for the mid-tread type occurs 3/2 LSB below FSR while the first transition is at 1/2 LSB. In order to have a symmetric transfer function, the last transition is sometimes made 1/2 LSB below the maximum voltage. So one LSB is removed from the upper end.

In this case the LSB would indeed be FSR/(2^N-1).