## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### BiWgt-Parallelization of a Unary Array of Current Sources
The mean of \(\mathbf{I}^{\text{BiWgt-BIP+-}}\) is obtained by summing up the identical means within the inner-most parenthesized levels in (256) (corresponding to the summations of 4, 2, 1 currents due to the parallel connections in (257)). The summations of 3, 2, 1 mean currents simply scale up the common mean \(\frac{1}{2}\langle I_{u}\rangle\) by factors of 4, 2, 1, yielding  
\[ \begin{array}{l}\langle\mathbf{I}^{\text{BiWgt-BIP+-}}\rangle\\ =\left\{\left\{\left\{I_{2^{+}}^{\text{BiWgt-BIP+-}}\right\},\,\left\langle I _{2^{-}}^{\text{BiWgt-BIP+-}}\right\}\right\},\left\{\left\langle I_{1^{-}}^{ \text{BiWgt-BIP+-}}\right\rangle,\,\left\langle I_{1^{-}}^{\text{BiWgt-BIP+-}} \right\rangle,\,\left\{\left\langle I_{0^{+}}^{\text{BiWgt-BIP+-}}\right\rangle,\,\left\langle I_{0^{-}}^{\text{BiWgt-BIP+-}}\right\rangle\right\}\right\} \\ \equiv\left\{\left\{4\cdot\frac{1}{2}\langle I_{u}\rangle,4\cdot\left(-\frac{ 1}{2}\langle I_{u}\rangle\right)\right\},\left\{2\cdot\frac{1}{2}\langle I_{u }\rangle,2\cdot\left(-\frac{1}{2}\langle I_{u}\rangle\right)\right\},\left\{1 \cdot\frac{1}{2}\langle I_{u}\rangle,1\cdot\left(-\frac{1}{2}\langle I_{u} \rangle\right)\right\}\right\}\\ \end{array}  \tag{258}\]  
To recapitulate, the six means of the source currents entering the three MUXes are  
\[ \begin{array}{l}\left\langle I_{2^{\pm}}^{\text{BiWgt-BIP+-}}\right\rangle= \pm 2\langle I_{u}\rangle,\quad\left\langle I_{1^{\pm}}^{\text{BiWgt-BIP+-}} \right\rangle=\pm\langle I_{u}\rangle,\quad\left\langle I_{0^{\pm}}^{\text{BiWgt -BIP+-}}\right\rangle=\pm\frac{1}{2}\langle I_{u}\rangle\end{array}  \tag{259}\]  
The corresponding variances are readily evaluated using \(\sigma_{1}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle\langle|\mathbf{I}|\rangle\) (109) (the notation \(\langle|\mathbf{I}|\rangle\) meaning \(\langle|\mathbf{I}|\rangle\equiv\{|I_{S-1}|,\ldots,|I_{2}|,|I_{1}|,|I_{0}|\}\), to account for negative sources):  
\[ \begin{array}{l}\text{var}\left\{I_{2^{\pm}}^{\text{BiWgt-BIP+-}}\right\}= \sigma_{u}^{2}\langle I_{u}\rangle\langle\left\langle\pm 2\langle I_{u} \rangle\right|\rangle=2[\sigma_{u}\langle I_{u}\rangle]^{2}=2\cdot 0.01( \text{mA})^{2}\\ \text{var}\left\{I_{1^{\pm}}^{\text{BiWgt-BIP+-}}\right\}=\sigma_{u}^{2} \langle I_{u}\rangle\langle\left|\pm\langle I_{u}\rangle\right|\rangle=[\sigma_{u }\langle I_{u}\rangle]^{2}=\qquad 0.01(\text{mA})^{2}\\ \text{var}\left\{I_{1^{\pm}}^{\text{BiWgt-BIP+-}}\right\}=\sigma_{u}^{2} \langle I_{u}\rangle\Big{\langle}\left|\pm\frac{1}{2}\langle I_{u}\rangle\right| \Big{\rangle}=\frac{1}{2}[\sigma_{u}\langle I_{u}\rangle]^{2}=\frac{1}{2}\cdot 0.01(\text{mA})^{2}\end{array}  \tag{260}\]  
where the numerical expressions in each of the three lines were obtained by substituting the parameters \(\langle I_{u}\rangle=1\text{mA}\), \(\sigma_{u}=0.1\) (196) for the example at hand. The output current is the summation of the three mux outputs, written as  
\[J_{c}^{\text{BiWgt-BIP+-}}=I_{2^{\pm}}^{\text{BiWgt-BIP+-}}+I_{1^{\pm}}^{\text{ BiWgt-BIP+-}}+I_{0^{\pm}}^{\text{BiWgt-BIP+-}},\text{ for }c=1,2,3,\ldots,8 \tag{261}\]  
where there are \(2\times 2\times 2\) = 8 combinations depending on the signs of \(2^{\pm},1^{\pm},0^{\pm}\). The three RVs \(I_{2^{\pm}}^{\text{BiWgt-BIP+-}},I_{1^{\pm}}^{\text{BiWgt-BIP+-}},I_{0^{\pm}}^{ \text{BiWgt-BIP+-}}\) are mutually independent (each being a summation of parallel-connected sub-source RV, from subsets (comprising 4, 2, 1 elements) of the 14 sub-sources are which are themselves mutually independent (as they are spatially distinct sources on the chip). As \(I_{2^{\pm}}^{\text{BiWgt-BIP+-}},I_{1^{\pm}}^{\text{BiWgt-BIP+-}},I_{0^{\pm}}^{ \text{BiWgt-BIP+-}}\) are independent, their variances are then additive:  
\[