## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### \(DEV\) Statistics of (un)Biused eDACs--Deterministic and Random Zero-Mean components  
**Lemma 8**: _eDAC (unary-normalized) steps and their perturbations expressed in terms of the constellation \(DEV\):_  
_Consider the normalized and unnormalized eDAC constellation steps:_  
\[\
abla\widehat{\mathbf{J}}=\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{J}} \quad\therefore\
abla\mathbf{J}=(\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{J}}) \langle I_{u}\rangle \tag{364}\]  
_The following random zero-mean perturbations and derived quantities are all equal: the perturbations \(\delta\mathbf{DEV_{J}}\) of the \(DEV\) of the constellation, \(\mathbf{J}\); the perturbations \(\delta\
abla\widehat{\mathbf{J}}\) of the normalized steps \(\widehat{\
abla}\mathbf{J}=\
abla\widehat{\mathbf{J}}\); the steps \(\
abla\widehat{\delta}\mathbf{J}\) of the normalized perturbations, \(\widetilde{\delta}\mathbf{J}\); the \(DEV\), \(\mathbf{DEV_{\delta}}\), of the constellation perturbations, \(\delta\mathbf{J}\)_**:**  
\[\delta\mathbf{DEV_{J}}=\delta\
abla\widehat{\mathbf{J}}=\
abla\widehat{ \delta}\widehat{\mathbf{J}}=\
abla\widehat{\delta}\mathbf{J}=\mathbf{1}_{C- 1}^{\forall}+\mathbf{DEV_{\delta}} \tag{365}\]  
\[\delta\
abla\mathbf{J}=\left(\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{ \delta}}\right)\langle I_{u}\rangle \tag{366}\]  
_where we recall the definition (A97) of the perturbation operator \(\delta\)._  
**Proof of Lemma 8. ** Relegated to Appendix B.2. \(\square\)  
Note: the particular result \(\
abla\widehat{\delta}\widehat{\mathbf{J}}=\mathbf{1}_{C-1}^{\forall}+ \mathbf{DEV_{\delta}}\) in (365) may be justified as equivalent to \(\
abla\widehat{\delta}\mathbf{J}=\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{ \delta}}\), which is a special case of \(\
abla\widehat{\mathbf{J}}=\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{J}}\) (360) under the substitution \(\mathbf{J}:=\delta\mathbf{J}\). For \"biased\" eDACs impaired by systematic biases, in addition to zero-mean mismatch effects we have non-zero \"biases\", \(\left\langle\mathbf{DEV_{J}}\right\rangle=\mathbf{DEV_{J}}\
eq 0\). The second-order statistics of the \(\mathrm{DEV}\) vectors of biased eDACs is worked out by leveraging our earlier results on the statistics of \(\
abla\widehat{\mathbf{J}}\). We start from the IEV result \(\mathbf{IEV_{j}}=\mathbf{IEV_{(j)}}+\widehat{\delta\mathbf{J}}\) (342), applying the \(\
abla\) operator onto both sides of the equality yielding  
\[\
abla\mathbf{IEV_{j}}=\
abla\mathbf{IEV_{(j)}}+\
abla\widehat{\delta\mathbf{J }}\Leftrightarrow\mathbf{DEV_{j}}=\mathbf{DEV_{(j)}}+\
abla\widehat{\delta \mathbf{J}} \tag{367}\]  
where in the last equality \(\mathbf{DEV}=\
abla\mathbf{IEV}\) (357) was used. Now, \(\
abla\widehat{\delta\mathbf{J}}\) in the last expression in (367) may be equated with \(\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{\delta\mathbf{J}}}\) as per (365); thus, \(\
abla\widehat{\delta\mathbf{J}}:=\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{ \delta\mathbf{J}}}\) is substituted into the last equality in (367), yielding:  
\[ \begin{array}{c}\mathbf{DEV_{j}}=\mathbf{DEV_{(j)}}+\
abla\widehat{\delta \mathbf{J}}=\mathbf{DEV_{(j)}}+\left(\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{ \delta\mathbf{J}}}\right)\\ \therefore\mathbf{DEV_{j}}=\mathbf{1}_{C-1}^{\forall}+\mathbf{DEV_{(j)}}+ \mathbf{DEV_{\delta\mathbf{J}}}\end{array}  \tag{368}\]  
Hence, we have obtained a DET + ZM decomposition for the DEV metric, akin to that earlier obtained in (344) for the IEV metric:  
**Theorem 5**.