## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### BiWgt-Parallelization of a Unary Array of Current Sources
\begin{array}{l}\text{var}\mathbf{I}^{\text{BiWgt-BIP}}\equiv\Big{\{}\text{ var}I_{2}^{\text{BiWgt-BIP}},\text{var}I_{1}^{\text{BiWgt-BIP}},\text{var}I_{0}^{ \text{BiWgt-BIP}}\Big{\}}\\ \underset{\#2}{=}\Big{\{}\text{var}\Big{\{}I_{6}^{\text{ThWgt-BIP}}+I_{5}^{ \text{ThWgt-BIP}}+I_{4}^{\text{ThWgt-BIP}}+I_{1}^{\text{ThWgt-BIP}}+I_{3}^{ \text{ThWgt-BIP}}\Big{\}},\\ \text{var}\Big{\{}I_{2}^{\text{ThWgt-BIP}}+I_{1}^{\text{ThWgt-BIP}}\Big{\}}, \text{var}\Big{\{}I_{0}^{\text{ThWgt-BIP}}\Big{\}}\Big{\}}\\ \underset{\#3}{=}\Big{\{}\text{4 var}\Big{\{}I_{6|5|43}^{\text{ThWgt-BIP}},\ \text{2 var}\Big{\{}I_{2|1}^{\text{ThWgt-BIP}}\Big{\}},\text{var}\Big{\{}I_{0}^{ \text{ThWgt-BIP}}\Big{\}}\Big{\}}\\ \underset{\#4}{=}\Big{\{}4\cdot\frac{1}{2}\text{var}\Big{\{}I_{3}^{\text{ThWgt- UNIP}}\Big{\}},\ 2\cdot\frac{1}{2}\text{var}\Big{\{}I_{1}^{\text{ThWgt-UNIP}}\Big{\}},1\cdot\frac{1}{2} \text{var}\Big{\{}I_{0}^{\text{ThWgt-UNIP}}\Big{\}}\Big{\}}\\ \underset{\#5}{=}\Big{\{}2,1,\frac{1}{2}\Big{\}}\sigma_{I_{u}}^{2}\underset{\#6} {=}\Big{\{}2,1,\frac{1}{2}\Big{\}}\big{[}\sigma_{u}\langle I_{u}\rangle\big{]}^ {2}\underset{\#7}{=}\Big{\{}2,1,\frac{1}{2}\Big{\}}\big{[}\sigma_{u}\langle I_{u }\rangle\big{]}^{2}=\Big{\{}2,1,\frac{1}{2}\Big{\}}\big{[}\sigma_{u}\langle I_ {u}\rangle\big{]}^{2}=\Big{\{}2,1,\frac{1}{2}\Big{\}}\big{[}\sigma_{u}\langle I_ {u}\rangle\big{]}^{2}=\Big{\{}2,1,\frac{1}{2}\Big{\}}\cdot 0.01\big{[}\text{mA}\big{]}^{2}\end{array}  \tag{251}\]  
where in \(\#2\) (249) was used; in \(\#3\), notation such as \(\text{var}\Big{\{}I_{6|5|4|3}^{\text{ThWgt-BIP}}\Big{\}}\) means that the variances of the sources labeled by \(s=6\), \(5\), \(4\), \(3\) are identical:\[ \begin{split}\text{var}\Big{\{}I_{6\in[5]4|3}^{\text{ThWgt-BIP}}\Big{\}}& =\text{var}\Big{\{}I_{6}^{\text{ThWgt-BIP}}\Big{\}}=\ldots=\text{var} \Big{\{}I_{3}^{\text{ThWgt-BIP}}\Big{\}}=\sigma_{I_{u}}^{2}=\left[\sigma_{u} \langle I_{u}\rangle\right]^{2}\\ \text{var}\Big{\{}I_{2|1}^{\text{ThWgt-BIP}}\Big{\}}& =\text{var}\Big{\{}I_{2}^{\text{ThWgt-BIP}}\Big{\}}=\text{var} \Big{\{}I_{1}^{\text{ThWgt-BIP}}\Big{\}}=\sigma_{I_{u}}^{2}=\left[\sigma_{u} \langle I_{u}\rangle\right]^{2}\\ \text{var}\Big{\{}I_{2|1}^{\text{ThWgt-BIP}}\Big{\}}& =\sigma_{I_{u}}^{2}=\left[\sigma_{u}\langle I_{u}\rangle\right]^{2}\end{split}  \tag{252}\]  
Equalities #4,#5,#6 readily follow from the three variance formulas above. Finally, in #7 and #8, we plugged in the specifications \(\langle I_{u}\rangle=1\text{mA}\), \(\sigma_{u}=0.1\) (196) pertinent to all our 3-bit eDAC examples in Section 2.7-2.9. To recapitulate,  
\[\text{var}^{\text{BiWgt-BIP}}=\left\{2,1,\frac{1}{2}\right\}[\sigma_{u}\langle I _{u}\rangle]^{2}=\left\{2,1,\frac{1}{2}\right\}\cdot 0.01\text{mA}^{2} \tag{253}\]  
The result (253) retrieves our earlier result (216). For completeness, we also recapitulate (250)  
\[\left\langle\mathbf{I}^{\text{BiWgt-BIP}}\right\rangle=\left\{2,1,\frac{1}{2} \right\}\langle I_{u}\rangle=\left\{2,1,\frac{1}{2}\right\}[\text{mA}] \tag{254}\]  
The first- and second-order statistics, (209) and (251) of the source currents vector (245) based on the \"BiWgt-parallelization of a unary array\" synthesis, is consistent with our earlier derived (212) and (216), in Section 2.6, wherein the source current array was simply designed with mean source currents in ratios of 4:2:1 (thus requiring the widths of the three current sources to be in ratios 4:2:1). Proposition 1 was used to predict the variances of the three source currents. This example then shows that BiWgt- parallelization of the unary array synthesis is consistent with Proposition 1, in terms of its resulting second-order statistics. Consider now, still for the 3-bit BiWgt-BIP eDAC current sources array, the \"physical\(\pm\)\" version of the \"BiWgt parallelization of ThWgt unary current sources\" constructions, as depicted in Figure 2b,d.