## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Conventional DNL Review (in Our Notation)
This relation carries over to obtaining the INL samples as accumulations of the DNL samples. To see this, let us express the levels in terms of the DNL (and the initial level \(J_{1}\)) plugging Formula (288) for the steps into (23) and simplifying:  
\[ \begin{array}{l}J_{c}=J_{1}+\sum_{i=1}^{c-1}\
abla J_{i}=J_{1}+\sum_{i=1}^{c -1}(1+\mathrm{DNL}_{i})\overline{\
abla\mathbf{J}}=J_{1}+\overline{\
abla \mathbf{J}}(c-1)+\overline{\
abla\mathbf{J}}\hskip-2.845276 pt\sum_{i=1}^{c-1} \mathrm{DNL}_{i}\\ =J_{1}+\overline{\
abla\mathbf{J}}(c-1)+\overline{\
abla\mathbf{J}}\hskip-2.845276 pt \sum_{i=1}^{c-1}\mathrm{DNL}_{i}=J_{c}^{\mathrm{nom}}+\overline{\
abla \mathbf{J}}\hskip-2.845276 pt\sum_{i=1}^{c-1}\mathrm{DNL}_{i}\end{array}  \tag{289}\]  
where in the last line above \(J_{c}^{\mathrm{nom}}=J_{1}+\frac{\mathrm{FS}}{C-1}(c-1)\) (278) was used. To recapitulate, (289) expresses the constellation levels in terms (average-step-scaled) of accumulations of the DNL samples:  
\[J_{c}=J_{c}^{\mathrm{nom}}+\overline{\
abla\mathbf{J}}\hskip-2.845276 pt\sum_{i =1}^{c-1}\mathrm{DNL}_{i}, \tag{290}\]  
This equation is equivalent to  
\[(J_{c}-J_{c}^{\mathrm{nom}})/\overline{\
abla\mathbf{J}}=\sum_{i=1}^{c-1} \mathrm{DNL}_{i}, \tag{291}\]  
But the LHS of (291) is identified as \(\mathrm{INL}_{c}=(J_{c}-J_{c}^{\mathrm{nom}})/\overline{\
abla\mathbf{J}}\) (277). Replacing the LSH of (291) by \(\mathrm{INL}_{c}\) (as per (277)) then yields a compact result: \(\mathrm{INL}_{c}=\sum_{i=1}^{c-1}\mathrm{DNL}_{i}\). This result is restated below in a corollary. Now, recalling \(I_{u}\equiv\overline{\
abla\mathbf{J}}\equiv\overline{\
abla J_{c}}\) (58), we observe that: the \(\mathbf{DNL}\) vector of an eDAC constellation could be alternatively expressed as the deviation in the eDAC steps from the (random) unary step, \(I_{u}\), measured in \"mean-unary-current\" \(\langle I_{u}\rangle\) units:  
\[\begin{array}{l}\mathrm{DNL}_{\mathbf{J}}[c]\equiv\big{[}\mathbf{DNL}_{ \mathbf{J}}\big{]}_{c}=\frac{\
abla I_{c}-I_{u}}{I_{u}}=\
abla J_{c}/I_{u}-1 \equiv\overline{\wedge}\hskip-2.845276 pt\widehat{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{ \mathbf{  \mathbf{            }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}\}\}\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\          { {  
**Corollary 6**.: _The DNL and INL metrics are linked by linear transformations: accumulating the DNL yields the INL; conversely, first-differencing the INL yields the DNL:_  
\[\mathrm{INL}_{c}=\sum_{i=1}^{c-1}\mathrm{DNL}_{i} \tag{294}\] \[\mathbf{DNL}=\
abla\mathbf{INL}\Leftrightarrow\mathrm{DNL}_{c}= \
abla\mathrm{INL}_{c} =\mathrm{INL}_{c+1}-\mathrm{INL}_{c}\Leftrightarrow\mathrm{INL}_{c+1}= \mathrm{INL}_{c}+\
abla\mathrm{INL}_{c}\] (295) \[\mathrm{INL}_{C-c} =\mathrm{INL}_{C}-\sum_{i=C-c}^{C-1}\mathrm{DNL}_{i} \tag{296}\]  
Proof of Corollary 6.: Relegated to Appendix B. \(\square\)  
Note 1: the linear transformations (294), relating the INL and DNL vectors, are not invertible (are not inverses of each other), consistent with the \(\mathbf{DNL}\) and \(\mathbf{INL}\) vectors of a given eDAC having different dimensions (\(\mathbf{DNL}\) has \(C\)-1 elements, whereas \(\mathbf{INL}\) has \(C\) elements). Note 2: it is possible to have eDACs sharing the same INL statistics, yet having DNLs of different variances, eDACs are topology-dependent. This is the case for the ThWgt, BiWgt and Seg eDAC, which may be designed for same INL, though their DNLs generally differ.