## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Integral Error Vector (IEV) as INL Variant Better Suited for Communication Modeling
Consistent with \(\mathbf{J}^{\text{nom}}\) having been defined as a uniform random constellation satisfying \(J_{1|\text{C}}^{\text{nom}}=J_{1|\text{C}}\) (Z75), the new reference \(\langle\mathbf{J}^{\text{nom}}\rangle\) satisfies \(\left\langle I_{1|\text{C}}^{\text{nom}}\right\rangle=\left\langle I_{1|\text {C}}\right\rangle\) (323). Hence, the mean-nominal-referred EV, \(\delta_{\langle\text{nom}\rangle}\mathbf{J}\), describes the deviation of the actual constellation from a deterministic reference uniform constellation _fitting the means \((\langle\text{LSL}|\text{MSL}\rangle)\) of the random end-points_ LSL and MSL of the given constellation, \(\mathbf{J}\). Note 2: In fact, both \(\mathbf{J}^{\text{nom}}\) and \(\langle\mathbf{J}^{\text{nom}}\rangle\) are both uniform references, but \(\mathbf{J}^{\text{nom}}\) is random (its sample sequences have random FS and end-points, yet the inner levels of any of its sample sequences are uniformly spaced), whereas \(\langle\mathbf{J}^{\text{nom}}\rangle\) is a deterministic uniform sequence with fixed end-points (323) and fixed FS. Note 3: our IEV is essentially a modification of the conventional INL definition \(\mathbf{INL}\equiv\delta_{\text{nom}}\mathbf{J}/\overline{\mathbf{V}}\) (277), obtained by adopting \(\delta_{\langle\text{nom}\rangle}\mathbf{J}\) in lieu of \(\delta_{\text{nom}}\mathbf{J}\) in the numerator and \(\langle\overline{\mathbf{V}}\rangle\) in lieu of \(\overline{\mathbf{V}}\mathbf{J}\) in denominator, i.e., replacing the \"average-step\" random denominator \(\overline{\mathbf{V}}\overline{\mathbf{J}}=\overline{(\
abla J)_{c}}=\frac{ \text{FS}}{\text{C}-1}\) in the INL definition (277) by a deterministic numerator  
\[\begin{split}&\langle\overline{\mathbf{V}}\rangle\rangle \equiv\langle\overline{\
abla J_{c}}\rangle=\left\langle\frac{1}{\text{C}}  
**Definition 17**.: _The IEV of an eDAC constellation is defined as a unary-normalized version of \(\delta_{\langle\text{nom}\rangle}\mathbf{J}\). The IEV is given by the mean-nominal-referred deviation vector \(\delta_{\langle\text{nom}\rangle}\mathbf{J}\) measured in \"mean-unary-current\" \(\langle I_{u}\rangle\) units:_  
\[\mathbf{IEV_{J}}\equiv\delta_{\langle\text{nom}\rangle}^{\frown}\mathbf{J} \equiv\frac{\delta_{\langle\text{nom}\rangle}\mathbf{J}}{\langle I_{u}\rangle }=\frac{\mathbf{J}-\langle\text{$\mathbf{J}$}^{\text{nom}}\rangle}{\langle I_{ u}\rangle},\quad\text{with }\langle I_{u}\rangle\equiv\frac{1}{C-1}\langle\text{PSJ}\rangle, \tag{327}\]  
**Corollary 9**. : _IEV as mean-nominal referred deviation of the normalized current \(\widehat{\mathbf{J}}\):_  
\[\mathbf{IEV_{J}}\equiv\delta_{\langle\text{nom}\rangle}\mathbf{J}=\widehat{ \mathbf{J}}-\left\langle\widehat{\mathbf{J}}^{\text{nom}}\right\rangle=\delta _{\langle\text{nom}\rangle}\widehat{\mathbf{J}} \tag{328}\]  
_In brief, the deviations operator \(\delta_{\langle\text{nom}\rangle}\) and the unary-normalizer operator \(\frown\) commute:_  
\[\mathbf{IEV_{J}}\equiv\delta_{\langle\text{nom}\rangle}^{\frown}\mathbf{J}= \delta_{\langle\text{nom}\rangle}\widehat{\mathbf{J}} \tag{329}\]  
_The constellation levels expressed in terms of the nominal constellation levels and the_ \(\mathbf{IEV}\)_:_  
\[\mathbf{J}=\langle\text{$\mathbf{J}$}^{\text{nom}}\rangle+\mathbf{IEV_{J}} \cdot\langle I_{u}\rangle \tag{330}\]  
_The constellation levels expressed in terms of the initial level (the LSL) and the IEV:_  
\[ \begin{array}{l}J_{c}=\langle I_{1}\rangle+\big{[}(c-1)+\text{IEV_{J}}[c] \big{]}\langle I_{u}\rangle\\ \widehat{I}_{c}=\left\langle\widehat{I}_{1}\right\rangle+(c-1)+\text{IEV_{J}} [c]\end{array}  \tag{331}\]  
**Proof of Corollary 9. ** Relegated to Appendix B.2. \(\square\)  
Note 1: The second line in (331) is readily obtained from the first one by dividing the first line by \(\langle I_{u}\rangle\). Specializing \(J_{c}=\langle I_{1}\rangle+\big{[}(c-1)+\text{IEV_{J}}[c]\big{]}\langle I_{u}\rangle\) (331) to \(c\):= 1 and taking means:  
\[