## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Example: Analytic Models for 3-Bit (PAM8) eDACs of the ThWgt \(|\)BiWgt UNIP \(|\) BiP Types
\begin{split}&\big{\langle}\text{BiWgt-BIP}+-\big{\rangle}\\ &=\Big{\{}\big{\langle}I_{2}^{\text{BiWgt-BIP}}\big{\rangle},- \big{\langle}I_{1}^{\text{BiWgt-BIP}}\big{\rangle},\big{\langle}I_{1}^{\text{ BiWgt-BIP}}\big{\rangle},-\big{\langle}I_{1}^{\text{BiWgt-BIP}}\big{\rangle}, \big{\langle}I_{0}^{\text{BiWgt-BIP}}\big{\rangle},-\big{\langle}I_{0}^{\text{ BiWgt-BIP}}\big{\rangle}\Big{\}}\\ &=\frac{1}{2}\Big{\{}\big{\langle}I_{2}^{\text{BiWgt-UNIP}}\big{\rangle},-\big{\langle}I_{1}^{\text{BiWgt-UNIP}}\big{\rangle},\big{\langle}I_{1}^{ \text{BiWgt-UNIP}}\big{\rangle},-\big{\langle}I_{1}^{\text{BiWgt-UNIP}} \big{\rangle},\big{\langle}I_{0}^{\text{BiWgt-UNIP}}\big{\rangle},-\big{ \langle}I_{0}^{\text{BiWgt-UNIP}}\big{\rangle}\Big{\}}\\ &=\frac{1}{2}\{4\langle I_{u}\rangle,-4\langle I_{u}\rangle,2 \langle I_{u}\rangle,2\langle I_{u}\rangle,\langle I_{u}\rangle,-\langle I_{u} \rangle\}\\ &=\quad\Big{\{}2\langle I_{u}\rangle,-2\langle I_{u}\rangle, \langle I_{u}\rangle,\langle I_{u}\rangle,\frac{1}{2}\langle I_{u}\rangle,- \frac{1}{2}\langle I_{u}\rangle\Big{\}}\end{split}  \tag{221}\]  
The physical bipolar \(\pm\) array operation, generating the eight output current levels of the BiWgt BIP eDAC, may be modeled by eight equations in six unknowns, compactly expressed as  
\[\begin{split}& \begin{bmatrix}I_{1}^{\text{BiWgt-BIP}}\\ I_{2}^{\text{BiWgt-BIP}}\\ I_{3}^{\text{BiWgt-BIP}}\\ I_{4}^{\text{BiWgt-BIP}}\\ I_{5}^{\text{BiWgt-BIP}}\\ I_{6}^{\text{BiWgt-BIP}}\\ I_{7}^{\text{BiWgt-BIP}}\\ &\underbrace{I_{8}^{\text{BiWgt-BIP}}}_{\text{J}^{\text{BiWgt-BIP}}} \end{bmatrix} &\underbrace{ \begin{bmatrix}I_{2^{+}}^{\text{BiWgt-BIP}}\\ I_{2^{+}}^{\text{BiWgt-BIP}}\\ I_{2^{+}}^{\text{BiWgt-BIP}}\\ I_{0}^{\text{BiWgt-BIP}}\\ \end{bmatrix} }_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I_{1^{+}}^{\text{BiWgt-BIP}}}_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I_{1^{-}}^{\text{BiWgt-BIP}}}_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I_{0^{+}}^{\text{BiWgt-BIP}}}_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I_{0^{+}}^{\text{BiWgt-BIP}}}_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I_{1^{-}}^{\text{BiWgt-BIP}}}_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I_{0^{+}}^{\text{BiWgt-BIP}}}_{\text{C}^{\text{BiWgt-BIP}}}\\ &\underbrace{I  
e.g., the sixth equation represented by this matrix-vector multiplication is  
\[J_{6}^{\text{BiWgt-BIP}}=I_{2^{-}}^{\text{BiWgt-BIP}}+I_{1^{+}}^{\text{BiWgt-BIP}}+ I_{0^{-}}^{\text{BiWgt-BIP}}, \tag{223}\]  
i.e., we switch-in the negative third source, the positive second source, the negative first source. All eight equations may be compactly written as  
\[J_{c}^{\text{BiWgt-BIP}}=I_{2^{\pm}[c]}^{\text{BiWgt-BIP}}+I_{1^{\pm}[c]}^{ \text{BiWgt-BIP}}+I_{0^{\pm}[c]}^{\text{BiWgt-BIP}},\;c=1,2,3,...8 \tag{224}\]  
where \(2^{\pm}[c]\), \(1^{\pm}[c]\), \(0^{\pm}[c]\) are \(\pm 1\)-valued, determined by the \(c\)-index (for \(c=1,2,3,\ldots,8\)). The linear transformation (222) is an alternative description of the BiWgt-BIP eDAC, equivalent to our earlier developed linear transformation \(\mathbf{J}=\mathbf{CI}\left(1\right)\), which, upon being specialized for the 3-bit BiWgt-BIP eDAC, reads  
\[ \begin{bmatrix}J_{1}^{\text{BiWgt-BIP}}\\ J_{2}^{\text{BiWgt-BIP}}\\ J_{3}^{\text{BiWgt-BIP}}\\ J_{4}^{\text{BiWgt-BIP}}\\ J_{5}^{\text{BiWgt-BIP}}\\ J_{6}^{\text{BiWgt-BIP}}\\ J_{7}^{\text{BiWgt-BIP}}\\ J_{8}^{\text{BiWgt-BIP}}\end{bmatrix} = \begin{bmatrix}1&1&1\\ 1&1&-1\\ 1&-1&1\\ 1&-1&-1\\ -1&1&1\\ -1&1&-1\\ -1&-1&-1\\ -1&-1&-1\end{bmatrix}\begin{bmatrix}I_{2}^{\text{BiWgt-BIP}}\\ I_{1}^{\text{BiWgt-BIP}}\\ I_{0}^{\text{BiWgt-BIP}}\end{bmatrix}