## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### IEV of Biased eDACs--Decomposed into Deterministic and Random Zero-Mean Components  
Biased eDACs are those either affected by systematic errors or inherently designed to yield non-uniform output constellations. For biased eDACs, the IEV has non-zero mean, and the mean eDAC output constellation is now non-uniform (the mean constellation, \(\langle\mathbf{J}\rangle\), does not coincide with the uniform reference constellation \(\langle\mathbf{J}^{\text{nom}}\rangle\)):  
\[\text{biased eDAC}:\langle\mathbf{J}\rangle\
eq\langle\mathbf{J}^{\text{nom}} \rangle\Leftrightarrow\left\langle\delta_{\text{(nom)}}\mathbf{J}\right\rangle \equiv\langle\mathbf{J}\rangle-\langle\mathbf{J}^{\text{nom}}\rangle\
eq 0 \Leftrightarrow\left\langle\mathbf{IEV}_{\mathbf{J}}\right\rangle\
eq 0 \tag{336}\]  
In biased eDACs, besides the mismatch-induced random errors, there are also present deterministic level offsets (systematic errors), i.e., built-in departures from the uniform nominal grid. For \"biased\" eDACs, it is useful to decompose the total mean-nom deviations vector \(\delta_{\text{(nom)}}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\text{nom}}\rangle\) (322) into the sum of a fixed, deterministic, static component (the bias vector), \(\delta_{\text{(nom)}}\langle\mathbf{J}\rangle\), plus a zero-mean random perturbation, \(\delta\mathbf{J}\). Formally, the mean-nomimal-referred deviation \(\delta_{\text{(nom)}}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\text{nom}}\rangle\) is processed by subtracting and adding \(\langle\mathbf{J}\rangle\), yielding a decomposition of \(\delta_{\text{(nom)}}\mathbf{J}\) into a zero-mean (ZM) component + a deterministic (DET) component:  
\[\delta_{\text{(nom)}}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\text{nom} }\rangle=\underbrace{\mathbf{J}-\langle\mathbf{J}\rangle}_{\delta_{\text{(nom) }}\langle\mathbf{J}\rangle=\langle\delta_{\text{(nom)}}\mathbf{J}\rangle}= \underbrace{\delta\mathbf{J}}_{\text{ZM}}+\underbrace{\delta_{\text{(nom)}} \langle\mathbf{J}\rangle}_{\text{DET}} \tag{337}\]  
where the definition of perturbation,  
\[\delta\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}\rangle \tag{338}\]  
is recalled. The key result (337) is restated in the following  
**Lemma 7**. : \(DET+ZM\) _decomposition of mean-nominal-referred deviations and the IEV:_  
\[\delta_{\text{(nom)}}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\text{nom }}\rangle=\underbrace{\delta_{\text{(nom)}}\langle\mathbf{J}\rangle}_{\text{ DET}}+\underbrace{\delta\mathbf{J}}_{\text{ZM}} \tag{339}\]  
_where the nonlinear DET and random ZM respective additive components are given by_  
\[\text{DET}:\;\delta_{\text{(nom)}}\langle\mathbf{J}\rangle= \langle\mathbf{J}\rangle-\langle\mathbf{J}^{\text{nom}}\rangle=\langle \mathbf{J}-\mathbf{J}^{\text{nom}}\rangle=\langle\mathbf{J}-\langle\mathbf{J} ^{\text{nom}}\rangle\rangle=\left\langle\delta_{\text{(nom)}}\mathbf{J}\right\rangle \tag{340}\] \[\text{ZM}:\;\delta\mathbf{J}=\mathbf{J}-\langle\mathbf{J}\rangle \tag{341}\]  
Proof of Lemma 7.: In (339), the additive component \(\delta\mathbf{J}\) is a ZM perturbation, \(\langle\delta\mathbf{J}\rangle=0\Leftrightarrow\langle\delta\mathbf{J} \rangle\equiv\mathbf{J}-\langle\mathbf{J}\rangle\), whereas the \(\delta_{\text{(nom)}}\langle\mathbf{J}\rangle\) component is DET, essentially the non-zero mean of \(\delta_{\text{(nom)}}\mathbf{J}\), with the first equality in (340) obtained by the substitution \(\delta_{\text{(nom)}}\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}^{\text{nom} }\rangle\Big{|}_{\mathbf{J}:=\langle\mathbf{J}\rangle}\) into the definition of \(\delta_{\text{(nom)}}\mathbf{J}\), and the last equality in (340) obtained by applying the mean operator onto both sides of \(\mathbf{J}-\langle\mathbf{J}^{\text{nom}}\rangle=\delta_{\text{(nom)}} \mathbf{J}\).\(\square\). Unary-normalizing the DET + ZM decomposition (339) of \(\delta_{\text{(nom)}}\mathbf{J}\) yields the following.