## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### EVM Communication Constellation Metric--Evaluated from Our IEV eDAC Metric
Conventionally, expressions equivalent to those in the second line in (381) are used to define EVM [15; 16; 17], but here we structure the EVM definition in terms of our streamlined notation \(\delta_{\langle\mathrm{nom}\rangle}J_{c}\equiv J_{c}-\langle J_{c}^{\mathrm{ nom}}\rangle\) (322) of this paper. In the second equality in (381) (A103) was used in order to evaluate the RoG of the mean-nominal reference constellation, \(\langle\mathbf{J}^{\mathrm{nom}}\rangle\) as the Root-Average-Square of the distances of the ideal constellation points from their center-of-mass. We now present our key findings positing our IEV metric as a preferred tool for express EVM evaluation. We essentially show that the EVM is nothing but a scaled version of the RAMS of our **IEV** vector. **Theorem 6**. : _The EVM of a 1D PAM-C constellation is proportional to the Root-Average-Mean-Square (RAMS) of the **IEV** vector of the given constellation,_  
\[\mathrm{EVM_{\mathbf{J}}}=\frac{1}{\mathrm{RoG}\Big{\{}\left<\overline{\hat{ \mathbf{J}}^{\mathrm{nom}}}\right>\Big{\}}}\mathrm{RAMS}\{\mathbf{IEV_{ \mathbf{J}}}\}=\frac{1}{\mathrm{RoG}\Big{\{}\left<\overline{\hat{\mathbf{J}} ^{\mathrm{nom}}}\right>\Big{\}}}\sqrt{\frac{1}{C}\overline{\sum_{c=1}^{C} \left<\mathrm{IEV_{\mathbf{J}}[c]^{2}\right>}}} \tag{382}\]  
_with proportionality constant, \(\overline{\mathrm{RoG}\Big{\{}\left<\overline{\hat{\mathbf{J}}^{\mathrm{nom }}}\right>\Big{\}}}\), given by the inverse of the radius-of-gyration of the normalized constellation, parameterized by the constellation size, \(C\) as follows:_  
\[\mathrm{UNIP}|\mathrm{BiP}\ \mathrm{ThWgt}|\mathrm{BiWgt}:\ \ \frac{1}{ \mathrm{RoG}\left<\overline{\hat{\mathbf{J}}^{\mathrm{nom}}}\right>}=\sqrt{ \frac{12}{C^{2}-1}} \tag{383}\]  
_expressing \(\mathrm{EVM_{\mathbf{J}}}\), for any 1D constellations of interest, as proportional to the Root-Average-Mean-Square \(\mathrm{IEV_{\mathbf{J}}^{\mathrm{RAMS}}}\equiv\mathrm{RAMS}\{\mathbf{IEV_{ \mathbf{J}}}\}\):_  
\[ \begin{array}{l}\mathrm{UNIP}|\mathrm{BiP}\ \mathrm{ThWgt}|\mathrm{BiWgt}: \mathrm{EVM_{\mathbf{J}}}=\sqrt{\frac{12}{C^{2}-1}}\cdot\mathrm{IEV_{ \mathbf{J}}^{\mathrm{RAMS}}}\\ =\sqrt{\frac{12}{(C^{2}-1)C}\sum_{c=1}^{C}\left<\mathrm{IEV_{\mathbf{J}}[c]^{ 2}\right>}}\end{array}  \tag{384}\]  
This result is obtained by plugging (383) into (382) and simplifying. For DACs impaired by deterministic (static) bias + mismatch errors, we plug our formula \(\mathrm{IEV_{\mathbf{J}}^{\mathrm{RAMS}}}=\sqrt{\left<\mathbf{IEV_{\mathbf{J }}^{\odot 2}}\right>}=\sqrt{\overline{\mathbf{IEV_{\mathbf{J}}^{\odot 2}}}+\frac{C-1}{2} \sigma_{u}^{2}}\) (349) yielding  
\[\mathrm{EVM_{\mathbf{J}}}=\sqrt{\frac{12}{C^{2}-1}}\cdot\sqrt{ \overline{\mathbf{IEV_{\mathbf{J}}^{\odot 2}}}+\sigma_{u}^{2}\frac{C-1}{2}}\quad \mathrm{for\ \mathrm{BIP}\ BiWgt}|\mathrm{ThWgt}\ \mathrm{or\ for\ UNIP BiWgt} \tag{385}\]  
These formulas for the EVM of non-uniform PAM-C are parameterized by the STD, \(\sigma_{u}\) of the relative mismatch, by the DET IEV bias vector, \(\mathbf{IEV_{\mathbf{J}}^{\odot 2}}\) and by the constellation size, \(C\) of the DAC. In particular, for unbiased DACs (e.g., conventional linear eDACs), the bias component nulls out, \(\mathbf{IEV_{(J)}}=0\), reducing (385) to simple expressions proportional to the mismatch relative STD, \(\sigma_{u}\), with coefficients of proportionality parameterized by \(C\):  
\[\text{unbiased mismatched eDACs}:\text{EVM}_{\text{J}}=\sigma_{u}\sqrt{\frac{6}{C+1}} \text{ for }\text{BIP}|\text{UNIP }\text{BiWgt}|\text{ThWgt} \tag{386}\]  
**Proof of Theorem 6. ** Relegated to Appendix B.2.