## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Modeling of Statistical Properties of the INL \(|\) DNL Metrics for (un)Biased eDACs
An important sub-class of eDACs features zero-mean random perturbation mechanisms with regard to the nominal output constellation, yielding constellation deviations, \(\delta_{\mathrm{nom}}\mathbf{J}\), free of systematic error (zero-bias), thus satisfying \(\langle\delta_{\mathrm{nom}}\mathbf{J}\rangle=0\). In this \"bias-free\" case, taking the mean of the definition \(\delta_{\mathrm{nom}}\mathbf{J}=\mathbf{J}-\mathbf{J}^{\mathrm{nom}}\) (322) yields  
\[\langle\delta_{\mathrm{nom}}\mathbf{J}\rangle=\langle\mathbf{J}-\mathbf{J}^{ \mathrm{nom}}\rangle=\langle\mathbf{J}\rangle-\langle\mathbf{J}^{\mathrm{nom} }\rangle=0\Leftrightarrow\langle\mathbf{J}\rangle=\langle\mathbf{J}^{\mathrm{nom}}\rangle \tag{312}\]  
i.e., the mean \(\langle\mathbf{J}\rangle\) of the given perturbed constellation coincides with mean \(\langle\mathbf{J}^{\mathrm{nom}}\rangle\) of the associated nominal uniform constellation. Note: conversely, a \"biased\", also known as non-uniform constellation features, by definition, \(\langle\mathbf{J}\rangle\
eq\langle\mathbf{J}^{\mathrm{nom}}\rangle\). In fact, the \"bias-free\" eDAC condition pertains to the fundamental physical mechanism of electronic mismatch underlying Proposition 1, which ignored systematic bias errors on the eDAC chip. The fundamental mismatch perturbations \(\delta\mathbf{J}\) at the eDAC output are then zero-mean, which is equivalent to having zero-mean \(\star\)NL (i.e., INL, DNL) vector metrics. Having defined (mismatch-induced) perturbations of the source currents as deviations \(\delta I_{s}\equiv I_{s}-\langle I_{s}\rangle\) of the source currents from their means, it follows that \(\langle\delta I_{s}\rangle\equiv\langle I_{s}-\langle I_{s}\rangle\rangle=0\). Invoking the linear transformation model of source currents to output currents, \(\mathbf{J}=\mathbf{C}\mathbf{I}\), we then have  
\[\delta\mathbf{J}=\mathbf{C}\delta\mathbf{I}\,.\,.\,\langle\delta\mathbf{J} \rangle=\mathbf{C}\langle\delta\mathbf{I}\rangle=\mathbf{C0}=\mathbf{0}. \tag{313}\]  
But since the means \(\langle\mathbf{J}\rangle\) for \(\mathrm{BiWgt}\,\mathrm{I}\mathrm{Wgt}\,\mathrm{UNIP}\,\mathrm{BIP}\) eDACs were shown to be uniform, it is readily shown that \(\langle\delta_{\mathrm{nom}}\mathbf{J}\rangle\equiv\langle\mathbf{J}\rangle- \langle\mathbf{J}^{\mathrm{nom}}\rangle=\langle\mathbf{J}^{\mathrm{nom}} \rangle-\langle\mathbf{J}^{\mathrm{nom}}\rangle=\mathbf{0}\). **Lemma 6**. : _For \"unbiased eDACs\" featuring \"bias-free zero-mean perturbations\",_  
\[\langle\delta_{\mathrm{nom}}\mathbf{J}\rangle=0\Leftrightarrow\langle\mathbf{ J}\rangle=\langle\mathbf{J}^{\mathrm{nom}}\rangle \tag{314}\]  
_It is only for such eDACs that the means of the INL and DNL come out null:_  
\[\langle\mathbf{INL}\rangle=0=\langle\mathbf{DNL}\rangle,\qquad\langle\mathrm{ INL}[c]\rangle=0=\langle\mathrm{DNL}[c]\rangle\;\;\forall c\in\{1,2,\ldots,C-1\} \tag{315}\]  
**Proof. ** Relegated to Appendix B. \(\square\)  
Note: \"unbiased\" eDACs, to which Lemma 6 pertains, are essentially uniform eDACs perturbed by zero-mean current mismatch. If the mismatch were hypothetically nulled out, the resulting eDACs would be uniform. Conversely, eDACs which are non-uniform under the hypothetical mismatch-free assumption are \"biased\". To recapitulate, Lemma 6 may be compactly restated as a chain of equivalences:  
\[\text{unbiased eDACs}:\ \left\langle\delta_{\text{nom}}\mathbf{J}\right\rangle=0 \Leftrightarrow\left\langle\mathbf{J}\right\rangle=\left\langle\mathbf{J}^{ \text{nom}}\right\rangle\Leftrightarrow\left\langle\mathbf{INL}\right\rangle= 0\Leftrightarrow\left\langle\mathbf{DNL}\right\rangle=0. \tag{316}\]  
Often, rather than specifying all components of the INL \(|\) DNL variance vectors, a single characteristic aggregate scalar summarizing the entire \(\sigma_{\text{INL}}^{2}\left|\sigma_{\text{DNL}_{\mathbf{J}}}^{2}\right.\) vector is extracted.