## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Differential Error Vector (DEV) as DNL Variant Better Suited for Communication Models  
In analogy with the transition from conventional **INL** to **DNL**, we now introduce a variant of the **DNL** vector, namely the \"Differential Error Vector\", **DEV**, more suitable for communication-oriented eDAC applications than the conventional DNL metric is. Our proposed modification of the DNL definition (287) simply replaces the average-step normalizer \(\overline{\
abla\mathbf{J}}\) (an RV) by its mean: the mean-average-step \(\langle\overline{\
abla\mathbf{J}}\rangle\) (325) (a deterministic scalar characteristic of a batch or lot of eDAC chips):  
**Definition 18**. : _The \"Differential Error Vector\" (DEV) vector, **DEV**, of an eDAC current constellation, \(\mathbf{J}\), is obtained from the steps vector, \(\
abla\mathbf{J}\equiv\{\
abla I_{c}\}_{c=1}^{C-1}=\{I_{c+1}-I_{c}\}_{c=1}^{C-1}\) (22) as follows: first, the deviations of the steps \(\
abla I_{c}\) from the \"mean-average step\", \(\langle\overline{\
abla\mathbf{J}}\rangle\) are evaluated, forming the random \"steps-deviation\" sequence_  
\[ \begin{array}{l}\{\
abla I_{c}-\langle\overline{\
abla\mathbf{J}}\rangle\}_ {c=1}^{C-1}=\{\
abla I_{1}-\langle\overline{\
abla\mathbf{J}}\rangle,\
abla I _{2}-\langle\overline{\
abla\mathbf{J}}\rangle,\ldots,\
abla I_{C-1}-\langle \overline{\
abla\mathbf{J}}\rangle\}\\ =\
abla\mathbf{J}-\langle\overline{\
abla\mathbf{J}}\rangle\mathbf{I}_{C-1} ^{C-1}\end{array}  \tag{352}\]  
_where, as seen in (325),_  
\[ \begin{array}{l}\langle\overline{\
abla\mathbf{J}}\rangle\equiv\frac{1}{C-1 }\sum_{i=1}^{C-1}\langle\
abla I_{i}\rangle=\bigg{\langle}\frac{1}{C-1}\sum_{ i=1}^{C-1}\
abla I_{i}\bigg{\rangle}\\ =\frac{1}{C-1}\langle I_{C}-I_{1}\rangle=\frac{1}{C-1}\{\langle I_{C} \rangle-\langle I_{1}\rangle\}=\frac{\langle\text{FS}\rangle}{C-1}\end{array}  \tag{353}\]  
_is the \"mean-average-step\" of the constellation, a scalar metric used as normalizer in both our IEV and DEV, obtained by taking the (stochastic) mean of the (arithmetic) average step \(\overline{\
abla\mathbf{J}}\) (276) used in the conventional DNL definition (the order of the mean and average may be reversed as in the second equality in (353))._Finally, the \(DEV\) vector is obtained by normalizing (dividing) the \"steps-deviation\" sequence (286) by the mean-average step, \(\langle\overline{\
abla{\bf J}}\rangle\):  
\[ \begin{array}{l}{\bf DEV_{j}}\equiv\frac{\
abla{\bf J}-\langle\overline{\
abla {\bf J}}\rangle{\bf i}_{\rm C-1}^{\rm V}}{\langle\overline{\
abla{\bf J}} \rangle}=\frac{\
abla{\bf J}}{\langle\overline{\
abla{\bf J}}\rangle}-{\bf i}_{ \rm C-1}^{\rm V},\\ {\rm DEV_{j}}[c]\equiv[{\bf DEV}]_{c}\equiv\frac{\
abla_{L}-\langle\overline{ \
abla{\bf J}}\rangle}{\frac{1}{\rm C-1}\langle\overline{\
abla{\bf J}} \rangle}=\frac{\
abla_{L}-\langle\overline{\
abla{\bf J}}\rangle}{\langle \overline{\
abla{\bf J}}\rangle}=\
abla J_{c}/\langle\overline{\
abla{\bf J}} \rangle-1\end{array}  \tag{354}\]  
Note 1: an alternative notation for the \"mean-average-step\" normalizer, \(\langle\overline{\
abla{\bf J}}\rangle\), sometimes referred to as \"LSB\", is \(\langle\overline{\
abla{\bf J}_{c}}\rangle\equiv\langle\overline{\
abla{\bf J }}\rangle\) (reminding us that it is a joint property of all mean steps of the constellation--their arithmetic average). Note 2: although the \({\rm DEV}\) metric is dimensionless, it is said to be measured in \"LSB\" units, e.g., 0.04 LSB. The same language is used for the conventional DNL, but it is more problematic there, as in the DNL definition the \"LSB\" the normalizer, \(\overline{\
abla{\bf J}}=\overline{\
abla{\bf J}_{c}}\), is random, whereas in our \({\rm DEV}\) definition the \"LSB\" normalizer \(\langle\overline{\
abla{\bf J}}\rangle=\langle\overline{\
abla{\bf J}_{c}}\rangle\) is deterministic, being more naturally referred to as \"LSB\".