Effective Number of Bits - Example

Example

The following are measurements of a 3-bit unipolar D/A converter with reference voltage V_ref = 8 V:

Digital Input	000	001	010	011	100	101	110	111
Analog Output (V)	-0.01	1.03	2.02	2.96	3.95	5.02	6.00	7.08

The offset error in this case is -0.01 V or -0.01 LSB as 1 V = 1 LSB in this example. The gain error is

.

Correcting the offset and gain error, we obtain the following list of measurements:

• (0, 1.03, 2.00, 2.93, 3.91, 4.96, 5.93, 7) LSB

This allows the INL and DNL to be calculated:

• INL = (0, 0.03, 0, -0.07, -0.09, -0.04, -0.07, 0) LSB
• DNL = (0.03, -0.03, -0.07, -0.02, 0.05, -0.03, 0.07, 0) LSB

The absolute and relative accuracy can now be calculated. In this case, the ENOB absolute accuracy is calculated using the largest absolute deviation, in this case 0.08 V:

The effective number of bits relative accuracy is calculated using the largest relative (INL) deviation, in this case 0.09 V.

For this kind of ENOB calculation, note that the effective number of bits can be larger or smaller than the actual number of bits. When the ENOB is smaller than the ANOB, this means that some of the least significant bits of the result are inaccurate. However, one can also argue that the ENOB can never be larger than the ANOB, because you always have to add the quantization error of an ideal converter which is +-0.5 LSB. Different designers may use different definitions of ENOB! Another problem with this calculation is that it does not make a difference if e.g. DNL is large at one point of the transfer curve or at multiple ones. The SNDR depends not only on worst-case DNL but on the DNL distribution. On the other hand, the method is still valuable, because a full SNDR measurement or simulation would take much more time.