## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
\begin{split}\widehat{I}_{s}^{(\cdot)}\equiv\langle\widehat{I}_ {s}\rangle=\langle\widehat{I}_{s}\rangle\equiv\langle I_{s}\rangle/\langle I_ {u}\rangle,\quad\widehat{\mathbf{I}}^{(\cdot)}\equiv\langle\widehat{\mathbf{ I}}\rangle=\langle\mathbf{I}\rangle/\langle I_{u}\rangle\\ \widehat{J}_{c}\equiv\text{J}_{C}/\langle I_{u}\rangle, \qquad\widehat{\mathbf{J}}\equiv\mathbf{J}/\langle I_{u}\rangle\\ \widehat{J}_{c}^{(\cdot)}\equiv\langle\widehat{J}_{c}\rangle= \langle\widehat{J}_{c}\rangle\equiv\langle I_{c}\rangle/\langle I_{u}\rangle, \quad\widehat{\mathbf{J}}^{(\cdot)}\equiv\langle\widehat{\mathbf{J}}\rangle= \langle\mathbf{J}\rangle/\langle I_{u}\rangle\end{split}  \tag{75}\]  
where \(\langle\widehat{I}_{s}\rangle=\langle\widehat{I}_{s}\rangle\) and \(\langle\widehat{J}_{c}\rangle=\langle\widehat{J}_{c}\rangle\) reflect the commutativity of the normalization and mean (expectation) operations. We mainly reserve the alternative notation \(\prescript{\circ}{\circ}{}\) for designating the mean in the context of mean-normalized (or normalized-mean) currents:  
\[\widehat{I}_{s}^{(\cdot)}\equiv\left\langle\widehat{I}_{s}\right\rangle\!, \ \widehat{\mathbf{I}}^{(\cdot)}\equiv\left\langle\widehat{\mathbf{I}}\right\rangle \!,\quad\widehat{J}_{c}^{(\cdot)}\equiv\left\langle\widehat{J}_{c}\right\rangle \!,\ \ \widehat{\mathbf{J}}^{(\cdot)}\equiv\left\langle\widehat{\mathbf{J}}\right\rangle \tag{76}\]  
The normalized mean source currents of UNIP BiWgt eDACs are given by  
\[ \begin{split}\widehat{\mathbf{I}}^{(\cdot)}& \text{UNIP BiWgt}\equiv\left\{\widehat{I}_{s}^{(\cdot)}\text{UNIP BiWgt}\right\}_{s=0}^{B-1}=\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\}\\ \widehat{I}_{s}^{(\cdot)}&\text{UNIP BiWgt}\equiv \left\langle\widehat{I}_{s}^{-\text{UNIP BiWgt}}\right\rangle=2^{s}\end{split}  \tag{77}\]  
generated by simply dividing (69) by \(\langle I_{u}\rangle\). For BIP, using \(\mathbf{I}^{\text{BIP}}=\frac{1}{2}\mathbf{I}^{\text{UNIP}}\), the normalized currents are half those of UNIP:  
\[ \begin{split}\widehat{\mathbf{I}}^{(\cdot)}& \text{BIP BiWgt}\equiv\left\{\widehat{I}_{s}^{(\cdot)}\text{BIP BiWgt}\right\}_{s=0}^{B-1}=\frac{1}{2} \big{\{}2^{B-1},2^{B-2},\ldots,4,2,1\big{\}}\\ \widehat{I}_{s}^{(\cdot)}&\text{BIP BiWgt}\equiv \left\langle\widehat{I}_{s}^{-\text{BIP BiWgt}}\right\rangle=\frac{1}{2}2^{s}=2 ^{s-1},\text{ for }s=B,B-1,B-2,\ldots,1\end{split}  \tag{78}\]For UNIP \(|\) BIP ThWgt eDACs, the corresponding results are  
\[ \begin{split}\widetilde{\mathbf{I}}^{(\backslash\text{ UNIP ThWgt}}&\equiv\bigg{\{}\widetilde{I}_{s}^{(\backslash\text{ UNIP ThWgt}}\bigg{\}}_{s=0}^{S-1}=\underbrace{\{1,1,\dots,1\}}_{S=C-1\text{ ones}}=\mathbf{1}_{S}^{\forall}\\ \widetilde{I}_{s}^{(\backslash\text{ UNIP ThWgt}}&\equiv\bigg{\langle} \widetilde{I}_{s}^{\text{ UNIP ThWgt}}\bigg{\rangle}=1\end{split}  \tag{79}\]  
It is worth reiterating that, in our convention \(\mathbf{I}^{\text{BIP}}=\frac{1}{2}\mathbf{I}^{\text{UNIP}}=\frac{1}{2} \mathbf{I}\), the all-positive source current vectors (be they unnormalized, \(\mathbf{I}\) or normalized, \(\widetilde{\mathbf{I}}\)) do depend on the eDAC polarity (UNIP \(|\) BIP), but they are determined by the eDAC topology (ThWgt \(|\) BiWgt). The eDAC polarity (UNIP \(|\) BIP) determines the code matrices, \(\mathbf{C}\), (having either \(0\,1\) or \(\pm 1\) elements for UNIP \(|\) BIP, respectively). In the case of the BIP BiWgt eDAC, wherein \(\big{\langle}I_{0}^{\text{BIP}}\big{\rangle}=\frac{1}{2}\big{\langle}I_{u}^{ \text{BIP}}\big{\rangle}\) (see (63)), we have  
\[\big{\langle}I_{c}^{\text{BIP BiWgt}[\text{ThWgt}}\big{\rangle}=(2c-C-1)\big{ }\big{\langle}I_{0}^{\text{BIP}}\big{\rangle}=(2c-C-1)\frac{1}{2}\Big{\langle} I_{u}^{\text{BIP}}\Big{\rangle}=[c-\frac{C+1}{2}]\big{\langle}I_{u}^{\text{ BIP}}\big{\rangle} \tag{80}\]  
The normalized output current levels of BIP BiWgt \(|\) ThWgt eDACs are obtained by dividing by \(\big{\langle}I_{u}^{\text{BIP}}\big{\rangle}\):  
\[