## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### BiWgt-Parallelization of a Unary Array of Current Sources  
In ASIC layout designs of \(B\)-bits BiWgt UNIP eDACs, the \(B\) current sources to be switched and additively combined are often obtained by a structure implementing an array of \(2^{B}-1\) nominally identical \"unary\" current sources (hence, having IID statistical distributions), referred to here as the \"sub-sources\", all sharing a common mean \(\langle I_{u}\rangle\), as if this is a ThWgt UNIP eDAC with \(S=C-1=2^{B}-1\) levels. However, \(2^{B}-1\) sub-sources of this \"unary array\" are partitioned into \(B\) subsets to form \(B\) effective sources. The \(B\) subsets of unary sub-sources, respectively, comprise \(\left\{2^{B-1},\ldots,4,2,1\right\}\) unary sources (consistent with \(\sum\left\{2^{B-1},\ldots,4,2,1\right\}=2^{B}-1=S\)). The \"unary\" sub-sources within each subset are electrically combined in parallel, to effectively yield the \(B\) binary-weighted current sources, featuring mean currents forming a dyadic sequence,  
\[\left\langle\mathbf{I}^{\text{BiWgt-UNIP}}\right\rangle=\Big{\{}2^{B-1}\langle I _{u}\rangle,\ldots,4\langle I_{u}\rangle,2\langle I_{u}\rangle,\langle I_{u} \rangle\Big{\}}. \tag{236}\]  
We refer to this construction as \"BiWgt-parallelization of a ThWgt unary sources array\". Let us exemplify this construction for 3-bit (eight levels) BiWgt UNIP and BIP eDAC. Staring with the 3-bit BiWgt UNIP eDAC case, the task is to synthesize its three random sources  
\[\mathbf{I}^{\text{BiWgt-UNIP}}\equiv\Big{\{}I_{2}^{\text{BiWgt-UNIP}},I_{1}^{ \text{BiWgt-UNIP}},I_{0}^{\text{BiWgt-UNIP}}\Big{\}} \tag{237}\]  
To this end, an array of seven IID ThWgt UNIP sub-sources is formed  
\[\mathbf{I}^{\text{ThWgt-UNIP}}\equiv\Big{\{}I_{6}^{\text{ThWgt-UNIP}},I_{5}^{ \text{ThWgt-UNIP}},I_{4}^{\text{ThWgt-UNIP}},I_{3}^{\text{ThWgt-UNIP}},I_{2}^{ \text{ThWgt-UNIP}},I_{1}^{\text{ThWgt-UNIP}},I_{0}^{\text{ThWgt-UNIP}}\Big{\}},\]  
and conceptually partitioned into three subsets comprising 4, 2, 1 ThWgt sub-sources:  
\[\Big{\{}\Big{\{}I_{6}^{\text{ThWgt-UNIP}},I_{5}^{\text{ThWgt-UNIP}},I_{4}^{ \text{ThWgt-UNIP}},I_{3}^{\text{ThWgt-UNIP}}\Big{\}},\Big{\{}I_{2}^{\text{ThWgt -UNIP}},I_{1}^{\text{ThWgt-UNIP}}\Big{\}}\Big{\}} \tag{238}\]The sub-sources within each of the three subsets are then parallel-connected (denoted \(\parallel\)):  
\[ \begin{split} I_{0}^{\text{BiWgt-UNIP}}&=I_{0}^{\text{ ThWgt-UNIP}}\\ I_{1}^{\text{BiWgt-UNIP}}&=I_{2}^{\text{ThWgt-UNIP}} \parallel I_{1}^{\text{ThWgt-UNIP}}=I_{2}^{\text{ThWgt-UNIP}}+I_{1}^{\text{ ThWgt-UNIP}}\\ I_{2}^{\text{BiWgt-UNIP}}&=I_{6}^{\text{ThWgt-UNIP}} \parallel I_{5}^{\text{ThWgt-UNIP}}\parallel I_{4}^{\text{ThWgt-UNIP}} \parallel I_{3}^{\text{ThWgt-UNIP}}\\ &=I_{6}^{\text{ThWgt-UNIP}}+I_{5}^{\text{ThWgt-UNIP}}+I_{4}^{ \text{ThWgt-UNIP}}+I_{3}^{\text{ThWgt-UNIP}}\end{split}  \tag{239}\]  
Note: evidently, upon parallel connection of current sources, the currents are added up. The resulting array of three BiWgt BIP current sources is  
\[\mathbf{I}^{\text{BiWgt-UNIP}}\equiv\left\{I_{2}^{\text{BiWgt-UNIP}},I_{1}^{ \text{BiWgt-UNIP}},I_{0}^{\text{BiWgt-UNIP}}\right\} \tag{240}\]  
The means of these three current sources are readily verified to form a dyadic sequence,  
\[ \begin{split}&\left\langle\mathbf{I}^{\text{BiWgt-UNIP}}\right\rangle \equiv\left\{\left\langle I_{2}^{\text{BiWgt-UNIP}}\right\rangle,\left\langle I _{1}^{\text{BiWgt-UNIP}}\right\rangle,\left\langle I_{0}^{\text{BiWgt-UNIP}} \right\rangle\right\}\\ &=\left\{\sum_{s=3}^{6}\left\langle I_{s}^{\text{ThWgt-UNIP}} \right\rangle,\ \sum_{s=1}^{2}\left\langle I_{s}^{\text{ThWgt-UNIP}}\right\rangle,\ \left\langle I_{0}^{\text{ThWgt-UNIP}}\right\rangle\right\}\\ &=\left\{\sum_{s=3}^{6}\left\langle I_{u}\right\rangle,\ \sum_{s=1}^{2}\left\langle I_{u}\right\rangle,\ \left\langle I_{u}\right\rangle\right\}=\left\{4\langle I_{u}\right\rangle,\ 2\langle I_{u}\rangle,\ \left\langle I_{u}\right\rangle\right\}=\left\{4,2,1\right\}\langle I_{u} \rangle\end{split}