## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Matrix-Based Modeling of Current-Steering eDACs: From Current Sources to Output Levels
Finally, the two mux outputs \(\mp\frac{1}{2}\langle I_{1}\rangle\) and \(\mp\frac{1}{2}\langle I_{0}\rangle\) are summed up to generate one of the four signed sums (12) amounting to the linear transformation (11) corresponding to the code matrix \(\mathbf{C}_{\text{DIP}}^{\text{BiWgt}}\) in (10). As the two MUXes are controlled by a pair of bits, \(b_{1},\ b_{0}\), if these two Boolean variables take values in the set \(\{\pm 1\}\) instead of \(\{1,0\}\), then the vector \([b_{1},\ b_{0}]\in\{[-1,\ -1],[-1,\ 1],[1,-1],[1,\ 1]\}\), with the control bit combinations referred to as \"codewords\" coincide with the rows of the code matrix \(\mathbf{C}_{\text{DIP}}^{\text{BiWgt}}\) in (10). As for the behavioral model of Figure 1b for UNIP-PAM4, we now return to using binary codewords \([b_{1},\ b_{0}]\in\{[0,\ 0],[0,\ 1],[1,0],[1,\ 1]\}\) (corresponding to the rows of \(\mathbf{C}_{\text{UNIP}}^{\text{BiWgt}}\) in (10)). For UNIP, the RHS input of each of the two MUXes is null (grounded). Thus, the mean output of the s-th mux is \(\langle I_{1}^{\text{BIF}}\rangle|0\), with the first (second) outcome for \(b_{s}=1\) (\(b_{s}=0\)). Summing up the two mux outputs yields the four possible combinations  
\[\left\langle I_{c}^{\text{UNIP}}\right\rangle=\left\langle I_{1}^{\text{ UNIP}}\right\rangle\!\left|0+\left\langle I_{0}^{\text{UNIP}}\right\rangle\! \left|0\in\left\{0,\left\langle I_{0}^{\text{UNIP}}\right\rangle,\left\langle I _{1}^{\text{UNIP}}\right\rangle,\left\langle I_{0}^{\text{UNIP}}\right\rangle +\left\langle I_{1}^{\text{UNIP}}\right\rangle\right\}\right. \tag{13}\]  
Having justified the correspondence of the behavioral models of Figure 1a,c with our code matrix-based linear transformations, the remaining Figure 1b,d pertaining to BIP and UNIP ePAM16 are evident generalizations, as further detailed in the figure caption. Note: In Figure 1, factors of \(\mathbb{11}\) appear in the BIP formulations; such factors are absent in the UNIP formulations. The factors of \(\mathbb{11}\) are justified in (29)-(32) in Section 2.2 in the following. For now, these factors should be accepted as an ad hoc convention. For higher-order PAM constellations, the coding matrices become larger, e.g., for \(C=8\):  
\[\text{UNIP}-\text{PAM8}:\ \mathbf{C}_{\text{UNIP}}^{\text{BiWgt}}= \begin{bmatrix}0&0&0 \\ 0&0&1\\ 0&1&0\\ 0&1&1\\ 1&0&0\\ 1&0&1\\ 1&1&0\\ 1&1&1\end{bmatrix} ,\  
The \(B\) current sources in our model for BiWgt eDACs (be they UNIP \(|\) BIP) form a so-called \"dyadic sequence\", defined as a geometric sequence with ratio 2,  
\[ \begin{split}&\left\langle I_{s}^{\text{BiWgt}}\right\rangle=2^{s} \left\langle I_{0}^{\text{BiWgt}}\right\rangle,\text{ for }s=0,1,2,..,B-1\\ &\left\langle I^{\text{BiWgt}}\right\rangle=\left\{2^{B-1} \left\langle I_{0}^{\text{BiWgt}}\right\rangle,2^{B-2}\left\langle I_{0}^{ \text{BiWgt}}\right\rangle,\ldots,4\left\langle I_{0}^{\text{BiWgt}}\right\rangle,2\left\langle I_{0}^{\text{BiWgt}}\right\rangle,\left\langle I_{0}^{\text{ BiWgt}}\right\rangle\right\}\end{split}  \tag{15}\]  
where we note that the first element, indexed \(s=0\), \(\left\langle I_{0}^{\text{BiWgt}}\right\rangle\), of the increasing geometric sequence is the least element, and it is also the GCD of the dyadic sequence of elements. In UNIP BiWgt eDACs, the \(B\) current sources are weighted by \(0\,11\) values, to form the linear combination representing the eDAC output, implemented by having the switching matrix either include or exclude (either steer-in or block, divert away) some or all of the current sources in the binary linear combination:  
\[ \begin{split}&\left\langle I_{c}^{\text{BiWgt-UNIP}}\right\rangle=(0 \,|1)2^{B-1}\left\langle I_{0}^{\text{BiWgt-UNIP}}\right\rangle+(0\,|1)2^{B-2 }\left\langle I_{0}^{\text{BiWgt-UNIP}}\right\rangle+\\ &\ldots+(0\,|1)4\left\langle I_{0}^{\text{BiWgt-UNIP}}\right\rangle +(0\,|1)2\left\langle I_{0}^{\text{BiWgt-UNIP}}\right\rangle+(0\,|1)\left\langle I _{0}^{\text{BiWgt-UNIP}}\right\rangle\end{split}