## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### EVM Communication Constellation Metric--Evaluated from Our IEV eDAC Metric  
Communication engineers do not typically use INL \(|\)DNL metrics customarily utilized in the eDAC community, to assist in characterizing the transmission performance of BIP \(|\)UNIP PAM transmission constellations. Communication practitioners typically assess and specify the quality of the transmitted constellations based on distinct constellation quality measures tailored to the communication context, such as error vector magnitude (EVM) and Symbol \(|\) Bit Error Rate (SER \(|\) BER). In this subsection we show how IEV \(|\)DEV metrics, as introduced in this paper for eDAC characterization, facilitate immediate evaluation of the EVM in wireless \(|\)optical transmission applications. We start by reviewing the conventional definition of the EVM metric for any given randomly perturbed and/or non-uniform 1D PAM constellation. Note: we distinguish between 1D (UNIP \(|\)BIP) PAM constellations and 2D constellations such as QAM. 1D\(|\)2D then refer to the dimensionality of the real-axis and the IQ-plane. For 1D communication constellations, the \"mean-nominal constellation\" \(\langle\mathbf{J}^{\text{nom}}\rangle\) is taken as the \"perfect\" perturbation-free uniformly-spaced (unbiased) target constellation we ideally aim to have, whereas the actually generated imperfect constellation, perturbed by electronic mismatch and distorted by static bias, is represented by the random vector \(\mathbf{J}\). The conventional EVM formulation [15; 16; 17], characterizing the quality of the eDAC-generated 1D constellation in a communication transmitter, is cast in the formalism of our paper, viewed as a functional extracting a scalar (the EVM) summarizing the random vector \(\delta_{(\text{nom})}\mathbf{J}=\mathbf{J}-\langle\mathbf{J}^{\text{nom}}\rangle\) (322), namely the \"mean-nominal-referred error vector (EV)\". **Definition 20**. : _The error vector magnitude (EVM) of an eDAC output constellation, \(\mathbf{J}\), is given by a functional \(\delta_{(\text{nom})}\mathbf{J}\mapsto\text{EVM_{J}}\) acting on the constellation Error-Vector (EV), \(\delta_{(\text{nom})}\mathbf{J}:\)_  
\[ \begin{split}\text{EVM}_{\mathbf{J}}\equiv\frac{\text{RAMS}\{ \delta_{(\text{nom})}\mathbf{J}\}}{\text{RoG}\{\left\{\mathbf{J}^{\text{nom}} \right\}\}}=\frac{\text{RAMS}\{\mathbf{J}-\langle\mathbf{J}^{\text{nom}} \rangle\}}{\text{RAS}\{\left\{\mathbf{J}^{\text{nom}}\right\}-\overline{ \langle\mathbf{J}^{\text{nom}}\rangle}\mathbf{1}_{C}^{\mathbf{\gamma}}\right\}} \\ =\frac{\sqrt{\frac{1}{C}\overline{\sum_{c=1}^{C}}\Big{\langle} \left|J^{c}-\langle\left|J_{c}^{\text{nom}}\right\rangle\right|^{2}\Big{\rangle} }}{\sqrt{\frac{1}{C}\overline{\sum_{c=1}^{C}}\Big{|}\langle\left|J_{c}^{\text{ nom}}\right\rangle-\overline{\langle\left|J_{c}^{\text{nom}}\right\rangle} \Big{|}^{2}}}=\sqrt{\frac{\overline{\sum_{c=1}^{C}}\Big{\langle}\left|J_{c}^{ \text{nom}}\right\rangle-\overline{\langle\left|J_{c}^{\text{nom}}\right\rangle }\Big{|}^{2}}}\end{split}  \tag{381}\]  
_where the RAMS, RoG and RAS functionals were defined in_ Section 1.1. Note: recall that a functional is a mapping \(\mathbb{R}^{C}\to\mathbb{R}\). In words, the deviations \(\delta_{\langle\mathrm{nom}\rangle}J_{c}\equiv J_{c}-\langle J_{c}^{\mathrm{nom}}\rangle\) (322) of the levels of the actual constellation \(\mathbf{J}\), with regard to the ideal levels of a \"nominal\" uniform constellation, \(\langle\mathbf{J}^{\mathrm{nom}}\rangle\), are concisely represented by a single scalar metric, the Root-Average-Mean-Square (RAMS) of the nominal-referred deviations, albeit normalized by the Radius-of-Gyration (A103) of the \"perfect\" constellation. Note: the normalization by the denominator of the EVM in (381) ensures such that at a given target EVM, larger extent constellations may tolerate larger RAMS-deviations.