## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Matrix-Based Modeling of Current-Steering eDACs: From Current Sources to Output Levels
The successive codewords \(\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c]\) are 0 1-valued bitstrings corresponding to the base-2 representations of the successive integers 0, 1, 2, \(\ldots\), \(C-1\)). The counting code (5) satisfies  
\[\left(\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c]\right)^{T}\left\{2^{B-1},2^{ B-2},\ldots,4,2,1\right\}=\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\} \cdot\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c]=c-1 \tag{6}\]  
Introducing the notation  
\[\text{d}\text{y}\text{d}\text{y}\text{d}^{[B]}\equiv\left\{2^{B-1},2^{B-2}, \ldots,4,2,1\right\} \tag{7}\]  
for the \(B\)-element dyadic sequence, (a geometric sequence with ratio \(\
u_{2}\) ending in 1), (6) is then rewritten as  
\[\text{d}\text{y}\text{d}^{[B]}\cdot\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c] =\left(\text{d}\text{y}\text{d}\text{y}\text{d}^{[B]}\right)^{T}\mathbf{b}_{ \text{UNIP}}^{\text{BiWgt}}[c]=c-1 \tag{8}\]  
For a BIP eDAC (be it ThWgt 1 BiWgt), (1) is specialized to \(\mathbf{J}^{\text{BIP}}=\mathbf{C}^{\text{BIP}}\mathbf{I}^{\text{BIP}}\); taking the mean (or expectation, denoted \(\left\langle\cdot\right\rangle\) per Section 1.1) of both sides, yields a linear relation between the means of the source currents and of the output current levels:  
\[\mathbf{J}^{\text{BIP}}=\mathbf{C}^{\text{BIP}}\mathbf{I}^{\text{BIP}} \Rightarrow\left\langle\mathbf{J}^{\text{BIP}}\right\rangle=\mathbf{C}^{ \text{BIP}}\left\langle\mathbf{I}^{\text{BIP}}\right\rangle \tag{9}\]  
As an example, consider a BiWgt BIP-PAM4 eDAC with \(C=4\), \(B=2\),  
\[\mathbf{I}\equiv\left\{I_{1},I_{0}\right\}\text{, }\mathbf{J}\equiv\left\{I_{1},I_{2},I_{3},I_{4}\right\}\]  
\[\mathbf{C}_{\text{UNIP}}^{\text{BiWgt}}=\left[ \begin{array}{cc}0&0\\ 0&1\\ 1&0\\ 1&1\end{array} \right]\text{, }\mathbf{C}_{\text{BIP}}^{\text{BiWgt}}=\left[ \begin{array}{cc}-1&-1\\ -1&1\\ 1&-1\\ 1&1\end{array} \right]\text{.} \tag{10}\]  
For BiWgt BIP-PAM4 eDACs, the linear transformation of mean currents, \(\left\langle\mathbf{J}^{\text{BIP}}\right\rangle=\mathbf{C}^{\text{BIP}} \left\langle\mathbf{I}^{\text{BIP}}\right\rangle\) (9) reduces to  
\[\left\langle\mathbf{I}^{\text{BIP}}\right\rangle=\left\{\left\langle I_{1}^{ \text{BIP}}\right\rangle,\left\langle I_{0}^{\text{BIP}}\right\rangle\right\},\ \ \left\langle\mathbf{J}^{\text{BIP}}\right\rangle=\left\{\left\langle I_{1}^{ \text{BIP}}\right\rangle,\left\langle I_{2}^{\text{BIP}}\right\rangle,\left \langle I_{3}^{\text{BIP}}\right\rangle,\left\langle I_{4}^{\text{BIP}}\right\rangle\right\} \tag{11}\]  
Conceptual block diagrams implementing the behavioral model of BIP 1 UNIP-PAM4 16 BiWgt eDACs are depicted in Figure 1. These concrete examples are worth carefully inspecting, to gain insight into our abstract mathematical models of BiWgt eDACs. The behavioral model of Figure 1 need not implement multiplications by matrices with generic elements--for our purposes it suffices to consider code matrices with weights \(\pm 1\) for UNIP and 110 for UNIP, which amounts to sign flips of source currents and inclusion/exclusion of source currents. We now explain how the behavioral model seen in Figure 1a for BIP-PAM4, based on electronic MUXes, implements the signed-sums linear transformation (11), concisely written as  
\[\left\langle J_{c}^{\text{BIP}}\right\rangle=\mp\left\langle I_{1}^{\text{ BIP}}\right\rangle\mp\left\langle I_{0}^{\text{BIP}}\right\rangle \tag{12}\](with distinct combinations of \(\pm\) signs for each of the output level indices, \(c=1\), \(2\), \(3\), \(4\)). Recall the functionality of each of the two MUXes as selector of one of its two inputs, as determined by the control bit, \(b_{s}\), for \(s=0,1\): when \(b_{s}=1\) (\(b_{s}=0\)), the RSH (LHS) input to the mux is passed to the mux output. Since the pair of inputs to each mux have been designed to be antipodal, \(\mp\frac{1}{2}\langle I_{s}\rangle\), then the mux controlled by bit \(b_{s}\) generates \(\mp\frac{1}{2}\langle I_{s}\rangle\) at its output.