## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### \(DEV\) Statistics of (un)Biused eDACs--Deterministic and Random Zero-Mean components
: \(IEV\,|\,DEV\,DET+ZM\) _decompositions for generally biased eDACs and their first-order and second-order statistics._  
_The IEV and DEV are, respectively, decomposed into \(DET+ZM\) components as follows:_  
\[\mathbf{IEV_{j}}=\mathbf{IEV_{(j)}}+\widehat{\delta\mathbf{J}} \tag{369}\]  
\[\mathbf{DEV_{j}}=\ \mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{(j)}}+\mathbf{DEV_{ \delta\mathbf{J}}}=\mathbf{DEV_{(j)}}+\delta\
abla\widehat{\mathbf{J}}=\ \left\langle\mathbf{DEV_{j}}\right\rangle+\delta\
abla\widehat{\mathbf{J}} \tag{370}\]  
_The IEV and DEV vectors are Gaussian distributed, since \(\mathbf{J}\) is Gaussian distributed. First-order statistics of the \(\mathbf{IEV}\,|\,\mathbf{DEV}\) vectors:_  
\[\left\langle\mathbf{IEV_{j}}\right\rangle=\mathbf{IEV_{(j)}}=\left\langle \widehat{\mathbf{J}}\ \right\rangle-\left\langle\widehat{\mathbf{J}}^{\text{nom}}\right\rangle= \delta_{(\text{nom})}\left\langle\widehat{\mathbf{J}}\ \right\rangle \tag{371}\]  
\[\left\langle\mathbf{DEV_{j}}\right\rangle=\mathbf{DEV_{(j)}}=\
abla\left\langle \widehat{\mathbf{J}}\ \right\rangle-\ \mathbf{1}_{C-1}^{\forall} \tag{372}\]  
_Second-order statistics of the \(\mathbf{IEV}\,|\,\mathbf{DEV}\) vectors,_  
\[\text{MS}_{\mathbf{IEV_{j}}}\equiv\left\langle\mathbf{IEV_{j}^{2}}\right\rangle =\left\langle\mathbf{IEV_{j}}\right\rangle^{\odot 2}+\sigma_{\mathbf{IEV_{j}}}^{2}= \mathbf{IEV_{(j)}^{\odot 2}}+\sigma_{\mathbf{j}}^{2} \tag{373}\]  
\[\text{MS}_{\mathbf{DEV_{j}}}\equiv\left\langle\mathbf{DEV_{j}^{2}}\right\rangle =\left\langle\mathbf{DEV_{j}}\right\rangle^{\odot 2}+\sigma_{\mathbf{DEV_{j}}}^{2}= \mathbf{DEV_{(j)}^{\odot 2}}+\sigma_{\
abla\widehat{\mathbf{J}}}^{2} \tag{374}\]  
_In the case of unbiased eDACs with uniform constellations, consistent with (132) we have:_  
\[\sigma_{\mathbf{IEV_{j}}}^{2}=\sigma_{\mathbf{j}}^{2}=\left\{ \begin{array}{ c}\{0,1,2,\ldots,C-1\}\sigma_{u}^{2}\ \text{ for UNIP ThWgt}|\text{BiWgt}\\ \frac{1}{2}(C-1)\sigma_{u}^{2}\mathbf{1}_{C}^{\vee}\ \text{ for BIIP ThWgt}|\text{BiWgt}\end{array} \right. \tag{375}\]  
_As for the steps statistics as captured in the DEV metric, \(\sigma_{\mathbf{DEV_{j}}}^{2}=\sigma_{\
abla\widehat{\mathbf{J}}}^{2}\), we have,_  
\[\sigma_{\mathbf{DEV_{j}}[c]}^{2}=\sigma_{\
abla\widehat{I}_{c}}^{2}=\left\{ \begin{array}{c}\sigma_{u}^{2}\mathbf{X}^{T}[c]\{2^{B-1},2^{B-2},\ldots,4,2,1 \}\ \text{ for UNIP ThWgt}|\text{BiWgt}\\ 2\sigma_{u}^{2}\mathbf{X}^{T}[c]\{2^{B-1},2^{B-2},\ldots,4,2,1\}\ \text{ for BIIP ThWgt}|\text{BiWgt}\end{array} \right. \tag{376}\]  
\[\text{with }\mathbf{X}[c]\equiv\mathbf{b_{BiWgt}}[c]\ \text{XOR }\mathbf{b_{BiWgt}}[c+1]\,\text{ \emph{per (151)}}\]  
_The max-variance component of \(\sigma_{\mathbf{DEV_{j}}}^{2}\) (using (163)) occurs \"mid-code\" and is given \(b\):_  
\[\max_{c\in\{12,\ldots,C-1\}}\sigma_{\text{DEVj}[2^{B-1-1}]}^{2}=\sigma_{ \
abla\widehat{I}_{2^{B-1-1}}}^{2}=\left\{ \begin{array}{c}\sigma_{u}^{2}(2^{B }-1)\ \text{ for UNIP BiWgt}\\ 2\sigma_{u}^{2}(2^{B}-1)\ \text{ for BIIP BiWgt}\end{array} \right. \tag{377}\]  
_RAMS scalar metrics concisely representing the \(\mathbf{IEV}\,|\,\mathbf{DEV}\) vectors,_\[ \begin{split}\text{IEV}_{\mathbf{J}}^{\text{RAMS}}\equiv\text{RAMS}\{ \mathbf{IEV_{J}}\}\equiv\sqrt{\overline{\left\langle\mathbf{IEV_{J}^{2}}\right\rangle }}=\sqrt{\overline{\left\langle\mathbf{IEV_{J}}\right\rangle^{\odot 2}+\sigma_{\mathbf{IEV_{J}}}^{2}}}= \sqrt{\overline{\left\mathbf{IEV_{J}^{\odot 2}}+\sigma_{\mathbf{J}}^{2}}}\\ \text{DEV}_{\mathbf{J}}^{\text{RAMS}}\equiv\text{RAMS}\{\mathbf{ DEV_{J}}\}\equiv\sqrt{\overline{\left\langle\mathbf{DEV_{J}^{2}}\right\rangle}}=\sqrt{ \overline{\left\langle\mathbf{DEV_{J}}\right\rangle^{\odot 2}}+\sigma_{\mathbf{DEV_{J}}}^{2}}= \sqrt{\overline{\mathbf{DEV_{J}^{\odot 2}}}+\sigma_{\mathbf{\
abla\overline{J}}}^{2}} \end{split}