## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
Our analysis, as summarized in Table 1, extends those known UNIP results to the case of PAM16 BIP BiWgt eDACs, wherein, as per Table 1, we have  
\[\text{BIP BiWgt eDACs}:\frac{1}{\sigma_{u}^{2}}\sigma_{\
abla\,\widetilde{ \mathsf{I}}}^{2}=\frac{1}{\sigma_{u}^{2}}\frac{1}{\langle I_{u}\rangle^{2}} \sigma_{\
abla\,\widetilde{\mathsf{I}}}^{2}=\left\{2,6,2,14,2,6,2,30,2,6,2,14, 2,6,2\right\}\sigma_{u}^{2} \tag{158}\]  
Most generally, the second-order statistics of the steps for PAM-C BiWgt eDACs may be modeled using Theorem 2. Inspection of Table 1 indicates that the \"worst offender\" (the entry with the highest mismatch variance) is located \"mid-code\" i.e., at the transition of the eighth level to the ninth level. In a similar table, made for a UNIP eDAC, the \"worst offender\" would still occur at the eighth to ninth level transition, from codeword \(\mathbf{b}[8]=0111\) to codeword \(\mathbf{b}[9]=1000\), wherein all bits at corresponding locations flip upon the transition from \(c=8\) to \(c=9\), i.e., \(\mathbf{b}[9]\) is one's-complement of \(\mathbf{b}[8]\). The XOR of these two codewords is \(1111\) (indicating that all corresponding bits differ) and the mismatch variance comes out \(15\) units for UNIP, two times that, i.e., \(30\), for BIP (see the entry at the fourth column and eighth row in Table 1),  
\[\mathbf{X}[8]\equiv\mathbf{b}[8]\ \text{XOR}\ \mathbf{b}[9]=0111\ \text{ XOR}\ 1000=1111 \tag{159}\]reducing (155) to  
\[ \begin{split} 16-\text{level UNIP BiWgt eDAC}:\ \widetilde{\sigma}_{\
abla\text{J}_{8}}^{2} /\sigma_{u}^{2}=\mathbf{X}^{T}[c]\{8,4,2,1\}\\ =\{1,1,1,1\}T\{8,4,2,1\}=8+4+2+1=15\\ 16-\text{level BIP BiWgt eDAC}:\ \widetilde{\sigma}_{\
abla\text{J}_{8}}^{2 }/\sigma_{u}^{2}=2\mathbf{X}^{T}[c]\{8,4,2,1\}=2\cdot 15=30\\ \end{split}  \tag{160}\]  
For UNIP BiWgt eDACs \(\widetilde{\sigma}_{\
abla\text{J}_{8}}^{2}=15\sigma_{u}^{2}\), whereas for BIP BiWgt eDACs \(\widetilde{\sigma}_{\
abla\text{J}_{8}}^{2}=30\sigma_{u}^{2}\). Generalizing the \(C=2^{4}\) example, to a \(C=2^{B}\)-levels BiWgt code, the worst-offender successive UNIP codewords are now \(\mathbf{b}[2^{B-1}]=0\underset{\mathcal{B}-1}{\overset{11...1}{\sum}}, \mathbf{b}[2^{B-1}+1]=\underset{\mathcal{B}-1}{\overset{100...0}{\sum}}\), yielding  
\[\mathbf{X}[2^{B-1}]\equiv\mathbf{b}[2^{B-1}]\ \text{XOR}\ \mathbf{b}[2^{B-1}+1]\ = \underset{\mathcal{B}\text{ones}}{\overset{1}{\sum}}=\mathbf{1}_{B}^{ \vee} \tag{161}\]  
\begin{table}
\begin{tabular}{|c||c c c c c c c c c|} \hline \hline
**Levels Index** & \multicolumn{5}{c|}{**Codeword**} & \multicolumn{5}{c|}{**XOR**} & \multicolumn{5}{c|}{**Steps VAR**} & \multicolumn{1}{c|}{**Steps Index**} \\ \hline \hline
1 & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(0\) & \(0\) & \(0\) & \(1\) & \(2\) & \(1\) \\
2 & \(-1\) & \(-1\) & \(-1\) & \(1\) & \(0\) & \(0\  
Substituting the index \(c=2^{B-1}\) and the XOR \(\mathbf{X}[2^{B-1}]=\mathbf{1}_{B}^{\forall}\) into (155) yields for the inner product \(\mathbf{X}^{T}[c]\big{\{}2^{B-1},2^{B-2},\ldots,4,2,1\big{\}}\) expression,  
\[ \begin{split}& C-\text{level UNIP eDACs : }\ \mathbf{X}^{T}[2^{B-1}]\big{\{}2^{B-1},2^{B-2},\ldots,4,2,1\big{\}}\\ &=\left(\mathbf{1}_{B}^{\forall}\right)^{T}\big{\{}2^{B-1},2^{B- 2},\ldots,4,2,1\big{\}}=\Sigma\big{\{}2^{B-1},2^{B-2},\ldots,4,2,1\big{\}}= \sum_{s=0}^{B-1}2^{s}=2^{B}-1\end{split}  \tag{162}\]  
such that (155) reduces to the following (for \(B=4\) the factors of 15 and 30 correctly arise):  
\[ \begin{split}\text{worst normalized step variance : }\ \ \sigma_{\
abla\widetilde{I}_{2^{B-1}}}^{2}=\widetilde{\sigma}_{\
abla \widetilde{I}_{2^{B-1}}}^{2}=\left\{\begin{array}{ll}\sigma_{u}^{2}(2^{B}-1 )\ \ \text{for UNIP BiWgt}\\ 2\sigma_{u}^{2}(2^{B}-1)\ \ \text{for BIP BiWgt}\end{array}\right.\end{split}  \tag{163}\]  
The UNIP BiWgt result \(\widetilde{\sigma}_{\
abla\widetilde{I}_{2^{B-1}}}^{2}=\sigma_{u}^{2}(2^{B}-1)\) is familiar to eDAC practitioners (barring normalizing factors and notation to reconcile) as it enters into the evaluation of the worst variance of the DNL metric.