## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Conventional DNL Review (in Our Notation)  
The DNL is the second well-known eDAC metric complementing the INL. The DNL as an affine transformation of the sequence of constellation steps \(\
abla J_{c}=J_{c+1}-J_{c}\) (for \(c=1\), \(2\), \(\ldots\), C-1), working out the steps deviations with regard to the average step, \(\overline{\
abla\mathbf{J}}\), finally having these deviations expressed in units of the average step (same normalizer reference as used in the conventional INL definition). Comparing the INL and DNL definitions, the INL is a normalized version of the sequence of deviations of the eDAC output levels (with regard to the nominal levels), whereas the DNL is a normalized version of the sequence of deviations of the output steps (with regard to the average step). The conventional INL and DNL definitions are both based on a common normalization: both metrics divide their respective deviations sequences by the average step, \(\overline{\
abla\mathbf{J}}\), measuring both the levels deviations sequence (for the INL) and the steps deviations sequence (for the DNL) in average-step units. **Definition 12**. : _The (conventional) DNL vector of an eDAC constellation, \(\mathbf{J}\), is obtained from the output steps vector, \(\
abla\mathbf{J}\equiv\{\
abla I_{c}\}_{c=1}^{\text{C}-1}=\{J_{c+1}-J_{c}\}_ {c=1}^{\text{C}-1}\). (22) is as follows. First, the deviations of the steps \(\
abla J_{c}\) with regard to their average, \(\overline{\
abla\mathbf{J}}\) are evaluated, forming the \"steps-deviation\" sequence_  
\[\left\{\
abla J_{c}-\overline{\
abla\mathbf{J}}\right\}_{c=1}^{\text{C}-1}= \left\{\
abla J_{1}-\overline{\
abla\mathbf{J}},\
abla J_{2}-\overline{\
abla \mathbf{J}},\ldots,\
abla J_{\text{C}-1}-\overline{\
abla\mathbf{J}}\right\} =\
abla\mathbf{J}-\overline{\
abla\mathbf{J}}\mathbf{I}_{\text{C}-1}^{\vee} \tag{286}\]  
_with \(\overline{\
abla\mathbf{J}}\equiv\frac{1}{\text{C}-1}\sum_{i=1}^{\text{C}-1} \
abla J_{i}\) (276) the \"average-step\" of the constellation, generally an RV._  
_Finally, the steps-deviations vector (286) is normalized (divided) by the average step:_  
\[ \begin{split}&\mathbf{DNL_{j}}=\frac{\
abla\text{J}-\overline{ \
abla\mathbf{I}}\mathbf{I}_{\text{C}-1}^{\vee}}{\overline{\
abla\mathbf{J}}}= \frac{\
abla\mathbf{J}-\mathbf{I}_{\text{C}-1}^{\vee}}{\overline{\
abla \mathbf{J}}}=\
abla J_{c}/\overline{\
abla\mathbf{J}}-1\end{split}  \tag{287}\]The last equality in each of the two lines in (287), is an alternative DNL formulation, dividing each of the steps, \(\
abla J_{c}\), by the average step, \(\
abla\mathbf{J}=\overline{\
abla J_{c}}\), then subtracting unity (as opposed to subtracting off the average-step, then normalizing). Conversely, solving the equation \(\mathrm{DNL}[c]=\
abla J_{c}/\overline{\
abla\mathbf{J}}-1\) for \(\
abla J_{c}\) yields a formula for unnormalized steps of the constellation, as an affine transformation of the DNL samples:  
\[\
abla J_{c}=\left(1+\mathrm{DNL}_{\mathbf{J}}[c]\right)\overline{\
abla \mathbf{J}},\hskip 14.226378 pt\
abla\mathbf{J}=(\mathbf{\mathfrak{I}}_{ \mathrm{C-1}}^{\vee}+\mathbf{DNL}_{\mathbf{J}})\overline{\
abla\mathbf{J}} \tag{288}\]  
This formula highlights the interpretation of the \(c\)-th DNL sample as the fractional deviation of the \(c\)-th step \(\
abla J_{c}\) relative to the average-step \(\
abla\mathbf{J}=\overline{\
abla J_{c}}=\overline{\
abla J_{c}}\). Note 1: all statements made, in the INL context, regarding the construction of \(\mathbf{J}^{\mathrm{nom}}\) as a uniform random sequence interpolation stretching between the LSL and MSL end-levels of the given \(\mathbf{J}\), also apply here, in the DNL context. Note 2: when the intended constellation is evident from the context superscript, \(\mathbf{J}\) is discarded, writing \(\mathbf{DNL}\) for \(\mathbf{DNL}_{\mathbf{J}}\) and \(\mathrm{DNL}_{c}\) for the \(c\)-th sample \(\mathrm{DNL}[c]\) of \(\mathbf{DNL}\). The constellation levels were expressed formula \(J_{c}\ =J_{1}+\sum_{i=1}^{c-1}\
abla J_{i}\) (23) as accumulations of the steps.