## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### BiWgt-Parallelization of a Unary Array of Current Sources
Inspecting the array of seven IID ThWgt UNIP positive-valued sub-sources, this array is now augmented by an array of nominally antipodal sub-sources, consisting of seven IID ThWgt UNIP positive-valued sub-sources (in fact, the two sub-sources arrays are drawn interspersed in Figure 2b,d, such as to reduce the complexities of interconnects upon sub-sources partitioning and parallel-connections). Mathematically, the \(2\times 7=14\) sources are partitioned and grouped as follows, with the lowest (inner-most parenthesized) levels in the hierarchy comprising sub-sources to be electrically parallel-connected, the next higher level comprising pairs of nominally antipodal, nominally dyadic-sized, current sources to be applied to the multiplexers which select either positive or negative source currents:  
\[ \begin{split}\mathbf{I}^{\text{ThWgt-BIP}+-}& \equiv\Big{\{}\Big{\{}\Big{\{}I_{6^{+}}^{\text{ThWgt-BIP}},I_{5^{+}}^{\text{ ThWgt-BIP}},I_{4^{+}}^{\text{ThWgt-BIP}},I_{3^{+}}^{\text{ThWgt-BIP}}\Big{\}}, \Big{\{}I_{3^{-}}^{\text{ThWgt-BIP}},I_{4^{-}}^{\text{ThWgt-BIP}},I_{5^{-}}^{ \text{ThWgt-BIP}},I_{6^{-}}^{\text{ThWgt-BIP}}\Big{\}}\Big{\}},\\ &\Big{\{}\Big{\{}I_{2^{+}}^{\text{ThWgt-BIP}},I_{1^{+}}^{\text{ ThWgt-BIP}}\Big{\}},\Big{\{}I_{1^{-}}^{\text{ThWgt-BIP}},I_{2^{-}}^{\text{ ThWgt-BIP}}\Big{\}}\Big{\}},\quad\Big{\{}\Big{\{}I_{0^{+}}^{\text{ThWgt- BIP}}\Big{\}},\Big{\{}I_{0^{-}}^{\text{ThWgt-BIP}}\Big{\}}\Big{\}} \Big{\}}\end{split}  \tag{255}\]  
Mapping this vector of current sub-sources to the vector of corresponding means yields  
\[ \begin{split}\left\langle\mathbf{I}^{\text{ThWgt-BIP}+-}\right\rangle \equiv\Big{\{}&\Big{\{}\Big{\{}\frac{1}{2}\langle I_{u} \rangle,\frac{1}{2}\langle I_{u}\rangle,\frac{1}{2}\langle I_{u}\rangle, \frac{1}{2}\langle I_{u}\rangle\Big{\}},\Big{\{}-\frac{1}{2}\langle I_{u} \rangle,-\frac{1}{2}\langle I_{u}\rangle,-\frac{1}{2}\langle I_{u}\rangle,-\frac{1}{2}\langle I_{u}\rangle\Big{\}}\Big{\}},\\ &\Big{\{}\Big{\{}\frac{1}{2}\langle I_{u}\rangle,\frac{1}{2} \langle I_{u}\rangle\Big{\}},\Big{\{}-\frac{1}{2}\langle I_{u}\rangle,-\frac{1 }{2}\langle I_{u}\rangle\Big{\}}\Big{\}},\quad\Big{\{}\Big{\{}\frac{1}{2} \langle I_{u}\rangle,\Big{\{}-\frac{1}{2}\langle I_{u}\rangle\Big{\}}\Big{\}} \Big{\}}\end{split}  \tag{256}\]  
where the factor of \(\
icefrac{{1}}{{2}}\) being due to the UNIP vs. BIP scaling. Upon parallel-connecting (summing) the sub-source currents at the inner-most levels in (255), we obtain the array \(\mathbf{I}^{\text{BiWgt-BIP}+-}\) of six current sources entering the three MUXes:\[ \begin{array}{l}\mathbf{I}^{\text{BiWgt-BIP+-}}\equiv\left\{\left\{I_{2^{+}}^{ \text{BiWgt-BIP+-}},\quad I_{2^{-}}^{\text{BiWgt-BIP+-}}\right\},\left\{I_{1^{+} }^{\text{BiWgt-BIP+-}},\quad I_{1^{-}}^{\text{BiWgt-BIP+-}}\right\},\left\{I_{0^ {+}}^{\text{BiWgt-BIP+-}},\quad I_{0^{-}}^{\text{BiWgt-BIP+-}}\right\}\right\} \\ =\left\{\left\{I_{6^{+}}^{\text{ThWgt-BIP}}\parallel I_{5^{+}}^{\text{ThWgt- BIP}}\parallel I_{4^{+}}^{\text{ThWgt-BIP}}\parallel I_{3^{+}}^{\text{ThWgt- BIP}}\parallel I_{3^{-}}^{\text{ThWgt-BIP}}\parallel I_{4^{-}}^{\text{ThWgt- BIP}}\parallel I_{5^{-}}^{\text{ThWgt-BIP}}\parallel I_{6^{-}}^{\text{ThWgt- BIP}}\right\},\\ \left\{I_{2^{+}}^{\text{ThWgt-BIP}}\parallel I_{1^{+}}^{\text{ThWgt-BIP}}, \quad I_{1^{-}}^{\text{ThWgt-BIP}}\parallel I_{2^{-}}^{\text{ThWgt-BIP}} \right\},\quad\left\{I_{0^{+}}^{\text{ThWgt-BIP}},I_{0^{-}}^{\text{ThWgt-BIP}} \right\}\right\}\end{array}  \tag{257}\]  
The correspondence of this mathematical representation to Figure 2b,d is evident, as we have a quad of positive currents and a quad of negative currents, each parallel-connected, to be switched in the left-hand mux, a pair of positive currents and a pair of negative currents each parallel-connected to be switched in the mid mux, a singlet positive currents and a singlet negative current, to be switched in the right-hand mux. The second-order statistics of \(\mathbf{I}^{\text{BiWgt-BIP+-}}\) (257) is readily derived (to be compared with our earlier results for the statistics).