## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### EVM of QAM Constellations Readily Evaluated from the IEV Metrics of the IQ eDACs
\(\square\)  
The complexified EV \(\delta_{(\text{nom})}\widetilde{\mathbf{J}}\) and \(\mathbf{IEV}_{\widetilde{\mathbf{J}}}\) enable the extension of the concept of EVM from a real-valued constellation to a complex-valued constellation, paraphrasing def (381) of real-valued EVM, simply by replacing all real-valued quantities by complex ones:  
**Definition. **_EVM of complex constellation, \(\widetilde{\mathbf{J}}\equiv\mathbf{J}_{I}+j\mathbf{J}_{Q}\)_:_  
\[\text{EVM}_{\widetilde{\mathbf{J}}}\equiv\frac{\text{RAMS}\{\delta_{(\text{ nom})}\widetilde{\mathbf{J}}\}}{\text{RoG}\Big{\{}\left\langle\widetilde{ \mathbf{J}}^{\text{nom}}\right\rangle\Big{\}}}=\sqrt{\frac{\sum_{\text{C}=1}^{ \text{C}}\!\left\langle\left|\widetilde{l}_{\text{C}}-\left\langle\widetilde{ j}_{\text{C}}^{\text{nom}}\right\rangle\right|^{2}\right\rangle}{\sum_{\text{C}=1}^{ \text{C}}\!\left|\left\langle\widetilde{j}_{\text{C}}^{\text{nom}}\right\rangle -\left\langle\widetilde{j}_{\text{C}}^{\text{nom}}\right\rangle\right|^{2}}} \tag{394}\]  
_It is useful to relate the EVM of a complex QAM constellation to the EVMs of the I and Q tributary constellations. Here we assume the statistical \"symmetry\" of the QAM constellation, the technical term being the \"circularity\" of the complex random vector \(\widetilde{\mathbf{J}}\equiv\mathbf{J}_{I}+j\mathbf{J}_{Q}\), amounting here to \(\mathbf{J}_{I},\mathbf{J}_{Q}\) being IID vectors. Thus, per this definition, \(\delta_{(\text{nom})}\widetilde{\mathbf{J}}\) is circular._  
_We shall need the generic properties of the RAMS and RoG for complex (circular) vectors._  
**Lemma 9**. : _For any complex random vector \(\widetilde{\mathbf{R}}\equiv\mathbf{R}+j\mathbf{R}_{Q}\) with independent Re&Im parts \(\mathbf{R}_{I},\mathbf{R}_{Q}\) has its I and Q components adding on an root-sum-of-squares basis:_  
\[\text{RAMS}\Big{\{}\widetilde{\mathbf{R}}\Big{\}}\equiv\sqrt{\Big{(}\,\text{ RAMS}\{\widetilde{\mathbf{R}}_{I}\}\Big{)}^{2}+\Big{(}\,\text{RAMS}\{ \widetilde{\mathbf{R}}_{Q}\}\,\Big{)}^{2}} \tag{395}\]  
_Now assuming the complex random vector is circular (i.e., \(\mathbf{R}_{I},\mathbf{R}_{Q}\) are IID),_  
\[\text{RAMS}\{\widetilde{\mathbf{R}}\}\equiv\sqrt{2}\text{RAMS}\{\widetilde{ \mathbf{R}}_{I|Q}\} \tag{396}\]  
_For a deterministic real vector with identical Re&Im components, \(\widetilde{\mathbf{R}}^{\text{det}}=\widetilde{\mathbf{R}}_{I}^{\text{det}}+j \widetilde{\mathbf{R}}_{I}^{\text{det}}\), the RoG of the complex vector, \(\widetilde{\mathbf{R}}^{\text{det}}\), is \(\sqrt{2}\) times the RoG of the real component:_  
\[\text{complex EV}:\delta_{(\text{nom})}\widetilde{\mathbf{J}}\equiv\delta_{( \text{nom})}\mathbf{J}_{1}+j\delta_{(\text{nom})}\mathbf{J}_{\text{Q}} \tag{388}\]0Proof of Lemma 9.: The derivation is omitted, but it essentially stems from the RAMS (A102) and RoG (A103) definitions. \(\square\)  
**Lemma 10**. : _For a QAM constellation with IID mismatch statistics of its IQ components, the EVM of the complex constellation, equals the EVM of either of the I I Q tributary constellations:_  
\[\text{complex EV}:\delta_{(\text{nom})}\widetilde{\mathbf{J}}\equiv\delta_{( \text{nom})}\mathbf{J}_{1}+j\delta_{(\text{nom})}\mathbf{J}_{\text{Q}} \tag{388}\]1  
_This means that having evaluated the EVM of the I I Q PAM-C tributaries we are done--we immediately have the EVM of the C\({}^{2}\)-QAM complex constellation. Specifically, since the results (382)-(386) for the EVM of a real-valued constellation, in terms of the IEV, carry over to I I I Q component constellation, they also yield the EVM of the complex QAM constellation.