## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Differential Error Vector (DEV) as DNL Variant Better Suited for Communication Models
Since the \({\rm DEV}\) definition (354) is effectively obtained from the conventional DNL one, (287), simply by replacing \(\overline{\
abla{\bf J}}\) by \(\langle\overline{\
abla{\bf J}}\rangle\) everywhere in the defining formulas for the conventional DNL, the same replacement may be made in all corollaries stemming the original DNL definition, as derived in Section 3.2. In particular formulas (288), (290) and (294) may now be retrofitted with the substitutions \({\rm INL}\):= \({\rm IEV}\), \({\rm DNL}\):= \({\rm DEV}\), \(\overline{\
abla{\bf J}}\) := \(\langle\overline{\
abla{\bf J}}\rangle\), \(\int_{c}^{\rm non}:=\langle f_{c}^{\rm non}\rangle\) and \({\rm FS}:=\langle f{\rm FS}\rangle\), yielding five useful formulas listed next:  
**Corollary 10**. : _Mapping \({\bf DEV}\) vectors to step-vectors, \(\
abla{\bf J}\), and to constellation levels, \(J_{c}\):_  
\[ \begin{array}{l}\
abla{\bf J}_{c}=(1+{\rm DEV}[c])\langle\overline{\
abla{ \bf J}}\rangle=(1+{\rm DEV}[c])\frac{\langle{\rm FS}\rangle}{\overline{\rm C -1}}\\ \
abla{\bf J}=({\bf i}_{\rm C-1}^{\rm V}+{\bf DEV})\langle\overline{\
abla{\bf J }}\rangle=({\bf i}_{\rm C-1}^{\rm V}+{\bf DEV})\frac{\langle{\rm FS}\rangle}{ \overline{\rm C-1}}\end{array}  \tag{355}\]  
\[ \begin{array}{l}J_{c}=\langle I_{c}^{\rm non}\rangle+\langle\overline{\
abla {\bf J}}\rangle\sum_{i=1}^{c-1}{\rm DEV}[i]=\langle I_{c}^{\rm non}\rangle+ \frac{\langle{\rm FS}\rangle}{\overline{\rm C-1}}{\sum_{i=1}^{c-1}{\rm DEV}[i] }\end{array}  \tag{356}\]  
_Back-and-forth transformations between \({\rm IEV}\) and \({\rm DEV}\)_:  
\[ \begin{array}{l}{\rm IEV}[c]=\sum_{i=1}^{c-1}{\rm DEV}[i]\\ {\rm DEV}[c]=\
abla{\rm IEV}[c]{=\rm IEV}[c+1]-{\rm IEV}[c],\end{array}  \tag{357}\]  
**Proof. ** This corollary may be directly derived from the \({\rm DEV}\) definition (354), paraphrasing the derivation of similar formulas pertaining to \({\rm INL}\) and \({\rm DNL}\) in Sections 3.1-3.3. \(\square\)  
At this point, as we have done with the \({\rm IEV}\), we may view the \({\rm DEV}\) as a \"unary-normalization\", providing a useful alternative point of view, simply making the substitutions \(\langle\overline{\
abla{\bf J}}\rangle:=\langle I_{u}\rangle\) in the \({\rm DEV}\) definition (354), yielding the following alternative definition. **Definition 19**. : _The \({\rm DEV}\) of an eDAC constellation, \({\bf J}\), is evaluated from the constellation steps \(\
abla{\bf J}\) in equivalent forms, by having them \"unary-normalized\" (expressed in units of \(\langle I_{u}\rangle\) ) to yield \(\
abla\,{\bf J}\), then subtracting off \({\bf i}_{\rm C-1}^{\rm V}\), i.e., subtracting unity from all \({\rm C}\)-\({\bf i}\) elements of \(\
abla\,{\bf J}\) :_  
\[{\bf DEV_{j}}\equiv\frac{\
abla{\bf J}}{\langle I_{u}\rangle}-{\bf i}_{\rm C-1 }^{\rm V}=\
abla\,{\bf\widehat{J}}-{\bf i}_{\rm C-1}^{\rm V},\qquad{\rm DEV_{j} }[c]=\frac{\
abla J_{c}}{\langle I_{u}\rangle}-1=\
abla\,{\bf\widehat{J}}_{c}-1 \tag{358}\]  
\[\therefore\,{\bf DEV_{j}}\equiv\
abla\,{\bf\widehat{J}}-{\bf i}_{\rm C-1}^{\rm V} \tag{359}\]  
_Conversely, solving (358) for the normalized steps, \(\
abla\,{\bf\widehat{J}}\), yields the useful relation:_  
\[\
abla\,{\bf\widehat{J}}={\bf i}_{\rm C-1}^{\rm V}+{\bf DEV_{j}} \tag{360}\]_In case the eDAC is \"perfect\", then \(\mathbf{DEV_{J}}=0\) and the normalized steps are all unity._  
_Note: it is instructive to verify that this alt def of the_ \(DEV\) _metric is consistent with the alt def_  
\[\mathbf{IEV_{J}}=\widehat{\mathbf{J}}-\left\langle\widehat{\mathbf{J}}^{- \mathbf{nom}}\right\rangle \tag{328}\]  
_(328) of the_ \(LEV\) _metric. Taking the means of (328) and of (359):_  
\[\left\langle\mathbf{IEV_{J}}\right\rangle=\left\langle\widehat{\mathbf{J}} \right\rangle-\left\langle\widehat{\mathbf{J}}^{-\mathbf{nom}}\right\rangle\]  
_Let us verify the consistency of the two equations in (361), based on_ \(\mathbf{DEV}=\
abla\mathbf{IEV}\)_(357).