## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### EVM of QAM Constellations Readily Evaluated from the IEV Metrics of the IQ eDACs  
In this subsection, we extend the EVM-RMS definition from the real-axis (1D) to the complex plane (2D(, aggregating the EVs and EVMs of the I and Q tributaries into corresponding complex-valued metrics, to enable modeling QAM constellations in wireless! optical communication links, accounting for the impairments of the driver eDACs. EVM for QAM is a known metric, but a new aspect is that the IEV and DEV eDAC metrics are complexified here for the first time and the complex IEV is related to the QAM EVM. For a QAM transmitter nesting a pair of I and Q tributaries, each with its own eDAC driver, we aggregate the EV, IEV and DEV metrics of the two I and Q tributaries into corresponding complex-valued metrics. **Definition 21**. : _Complex-valued EV, IEV, DEV for a QAM Tx IQ-nesting a pair of eDACs:_  
_Let \(\mathbf{J}_{1}\),\(\mathbf{J}_{Q}\) denote the actual constellations of I and Q tributary eDACs, and let \(\left\langle\mathbf{J}_{1}^{\text{nom}}\right\rangle\), \(\left\langle\mathbf{J}_{Q}^{\text{nom}}\right\rangle\) denote their corresponding \"mean-nominal\" reference constellation._  
_Then, the corresponding complex constellations (labeled by over-tildes) are defined as_  
\[\text{complex QAM constel.}:\widetilde{\mathbf{J}}\equiv\mathbf{J}_{1}+j \mathbf{J}_{\text{Q}^{\prime}},\ \therefore\left\langle\widetilde{\mathbf{J}}^{\text{nom}}\right\rangle\equiv \left\langle\widetilde{\mathbf{J}}_{1}^{\text{nom}}\right\rangle+j\left\langle \widetilde{\mathbf{J}}_{Q}^{\text{nom}}\right\rangle \tag{387}\]  
_and the corresponding complex mean-nominal-referred Error Vectors are defined as_  
\[\text{complex EV}:\delta_{(\text{nom})}\widetilde{\mathbf{J}}\equiv\delta_{( \text{nom})}\mathbf{J}_{1}+j\delta_{(\text{nom})}\mathbf{J}_{\text{Q}} \tag{388}\]  
_The IEV I DEV of the complex constellation (in brief complex-\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbfmathbfmathbfmathbfmathbf { }}}}}}}}}}}}}}\)\)\) are defined as_  
\[\text{complex EV}:\ \ \mathbf{IEV_{\widetilde{\mathbf{J}}}\equiv \mathbf{IEV_{\mathbf{J}_{1}}+j\mathbf{IEV_{\mathbf{J}_{Q^{\prime}}}}},\ \ \ \ \mathbf{DEV_{\widetilde{\mathbf{J}}}\equiv\mathbf{DEV_{\mathbf{J}_{1}}+j\mathbf{DEV_{ \mathbf{J}_{Q}}}}} \tag{389}\]  
**Corollary 11**. : _Complex EV and IEV in terms of (normalized) I and Q constellations:_  
\[\text{Complex mean}-\text{nominal referred deviation}:\delta_{(\text{nom})} \widetilde{\mathbf{J}}=\widetilde{\mathbf{J}}-\left\langle\widetilde{\mathbf{J}}^ {\text{nom}}\right\rangle \tag{390}\]  
_\"Complexifying\" the IEV alt def (327) yields_  
\[\mathbf{IEV}_{\widetilde{\mathbf{J}}}=\delta_{(\text{nom})} \widetilde{\mathbf{J}}/\left\langle I_{u}\right\rangle=\delta_{(\text{nom})} \widetilde{\mathbf{J}}=\widetilde{\widetilde{\mathbf{J}}}-\left\langle \widetilde{\mathbf{J}}^{\text{nom}}\right\rangle=\text{IEV}_{\mathbf{J}_{1}}+j \text{IEV}_{\mathbf{J}_{\text{Q}}} \tag{391}\]  
\[\text{with}\ \widetilde{\widetilde{\mathbf{J}}}\equiv\widetilde{\mathbf{J}}/ \left\langle I_{u}\right\rangle=\widetilde{\mathbf{J}}_{I}+j\widetilde{ \mathbf{J}}_{Q} \tag{392}\]  
_\"Complexifying\" the DEV alt def (358) yields_  
\[\mathbf{DEV}_{\widetilde{\mathbf{J}}}=(\underbrace{\
abla\,\widetilde{ \mathbf{J}}_{I}-\mathbf{1}_{\text{C}-1}^{\forall}}_{\text{DEV}_{I_{I}}})+j( \underbrace{\
abla\,\widetilde{\mathbf{J}}_{Q}-\mathbf{1}_{\text{C}-1}^{ \forall}}_{\text{DEV}_{I_{Q}}})=\
abla\,\widetilde{\mathbf{J}}-(\mathbf{1}_ {\text{C}-1}^{\forall}+j\mathbf{1}_{\text{C}-1}^{\forall}) \tag{393}\]  
**Proof. ** The simple derivations of the formulas of this corollary are omitted.