## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Integral Error Vector (IEV) as INL Variant Better Suited for Communication Modeling  
We proceed to introduce alternative DAC constellation metrics, namely the IEV and DEV, akin to and inspired by the INL and DNL metrics, yet more suitable for communication systems modeling, optical transmission links modeling in particular. In a nutshell, the modified IEV | DEV metrics may be obtained from their conventional INL | DNL counterparts by replacing \(I_{u}\equiv\overline{\
abla\mathbf{J}}\) in the INL | DNL definitions by its mean \(\langle I_{u}\rangle\equiv\langle\overline{\
abla\mathbf{J}}\rangle\). This simple yet subtle change makes all the difference, as is it presently going to become apparent. Let us start with an auxiliary definition, introducing a modified form of the error vector: _Mean-nominal-referred error vector (EV)_. **Definition 15**. : _\"Mean-nominal-referred error vector (EV)\" \(\delta_{\rm(nom)}\mathbf{J}\) of a \(C\)-levels constellation, \(\mathbf{J}\):_  
\[ \begin{array}{l}\delta_{\rm(nom)}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J }^{\rm nom}\rangle=\left\{\delta_{\rm(nom)}I_{1},\delta_{\rm(nom)}I_{2}, \ldots,\delta_{\rm(nom)}J_{C}\right\}\\ \delta_{\rm(nom)}J_{c}\equiv J_{c}-\langle J_{c}^{\rm nom}\rangle,\ \ \text{ for }c=1,2,\ldots,C-1\end{array}  \tag{322}\]  
_defined as the vector of deviations of the constellation levels \(\left\{J_{c}\right\}_{c=1}^{C}\) with regard to the means \(\left\{\langle I_{c}^{\rm nom}\rangle\right\}_{c=1}^{C}\) of the nominal positions of the levels. The \"mean-nominal\" constellation, \(\langle\mathbf{J}^{\rm nom}\rangle\equiv\left\{\langle I_{c}^{\rm nom}\rangle \right\}_{c=1}^{C}\), is now used as reference Its LSL and MSL coincide with the means of the respective (random) MSL and LSL of the given constellation \(\mathbf{J}\):_  
\[\left\langle I_{1|C}^{\rm nom}\right\rangle=\left\langle I_{1|C}\right\rangle \tag{323}\]  
_Note: Comparing the definition of our original nom-referred \(EV\)\(\delta_{\rm nom}J_{C}\equiv J_{C}-J_{c}^{\rm nom}\) (274) with the new \"mean-nom-referred\" definition \(\delta_{\rm(nom)}J_{C}\equiv J_{C}-\langle I_{c}^{\rm nom}\rangle\) (322), it is apparent that the \"reference\" or \"nominal\" samples \(J_{c}^{\rm nom}\) are now replaced by their means, \(\langle J_{c}^{\rm nom}\rangle\)._  
_\"Integral Error Vector\" (IEV)_  
**Definition 16**. : _\"Integral Error Vector\" (IEV) is our communication-oriented modified INL metric, defined as a normalized version of \(\delta_{\rm(nom)}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\rm nom}\rangle\) (322), obtained by dividing the mean-nominal-referred deviations, \(\delta_{\rm(nom)}\mathbf{J}\), by the \"mean-average step\" \(\left\langle\overline{\mathbf{J}}\right\rangle\) (325) of the constellation:_  
\[ \begin{array}{l}\mathbf{IEV}\equiv\mathbf{IEV}_{\mathbf{J}}=\frac{\delta_{ \rm(nom)}\mathbf{J}}{\langle\overline{\mathbf{J}}\rangle}=\frac{\mathbf{J}- \langle\mathbf{J}^{\rm nom}\rangle}{\frac{1}{\Gamma_{\rm T}}\langle\mathbf{F} _{\mathbf{J}}\rangle}\\ \text{s.t.}\ \left\langle I_{1|C}^{\rm nom}\right\rangle=\left\langle I_{1|C} \right\rangle,\left\langle J_{c}^{\rm nom}\right\rangle=\frac{C-C}{C-1}\cdot \langle I_{1}\rangle+\frac{C-1}{C-1}\cdot\langle J_{C}\rangle,\ \ \ \langle\mathbf{FS}_{\mathbf{J}}\rangle\equiv\langle J_{C} \rangle-\langle I_{1}\rangle\\ \text{IEV}[c]\equiv\text{IEV}_{c}\equiv[\mathbf{IEV}]_{c}\equiv\frac{\delta_{ \rm(nom)}\mathbf{J}}{\langle\overline{\mathbf{J}}\rangle}=\frac{L-\langle I_{c }^{\rm nom}\rangle}{\frac{1}{C-1}\langle\mathbf{FS}_{\mathbf{J}}\rangle} \end{array}  \tag{324}\]  
Note 1: comparing the new \"mean-nominal-referred EV\" definition \(\delta_{\rm(nom)}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\rm nom}\rangle\) (322) with the \"nominal-referred EV\" definition (274) \(\delta_{\rm non}\mathbf{J}\equiv\mathbf{J}-\mathbf{J}^{\rm nom}\) used in the INL, the random reference \(\mathbf{J}^{\rm nom}\) in (274) is now replaced by its mean \(\langle\mathbf{J}^{\rm nom}\rangle\), a deterministic sequence.