## 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
Note: a third alternative derivation of the output variances would be based on \(\sigma_{\mathbf{J}}^{2}=\mathbb{C}^{\odot 2}\sigma_{\mathbf{I}}^{2}\) (119) with \(\mathbb{C}:=\mathbb{C}^{\text{BiWgt-BIP}+-}\), \(\mathbb{J}:=\mathbb{J}^{\text{BiWgt-BIP}}\), \(\mathbb{I}:=\mathbb{I}^{\text{BiWgt-BIP}}\) (see (222)); details are omitted. In Section 2.5, we derived the second-order statistics of the steps \(\
abla\mathbf{J}\) of a BiWgt BIP eDAC. In particular, Table 1 addressed the 4-bit (BiWgt BIP PAM16) case. We now specialize (155) of theorem II to the \(B=3\)-bit (BiWgt BIP-PAM8) eDAC case of interest here:  
\[ \begin{split}\sigma_{\mathbf{VJ}}^{2}=\sigma_{u}^{2}\langle I_{u }\rangle^{2}\cdot 2\mathbf{X}_{\text{BiWgt}}^{\text{BIP}}\{4,2,1\}=\sigma_{u}^{2} \langle I_{u}\rangle^{2}\cdot 2\begin{pmatrix}0&0&1\\ 0&1&1\\ 0&0&1\\ 1&1&1\\ 0&0&1\\ 0&1&1\\ 0&0&1\end{pmatrix}=\sigma_{u}^{2}\langle I_{u}\rangle^{2}\cdot\begin{pmatrix}2 \\ 6\\ 2\\ 14\\ 2\\ 2\\ 6\\ 2\end{pmatrix}\end{split}  \tag{231}\]  
where the \"XOR-steps matrix\" \(\mathbf{X}_{\text{BiWgt}}^{\text{BIP}}\) (154) was evaluated for the 3-bit BiWgt BIP eDAC:  
\[ \begin{split}\mathbb{C}_{\text{BiWgt}}^{\text{BIP}}=\begin{pmatrix}- 1&-1&-1\\ -1&-1&1\\ -1&1&-1\\ 1&-1&1\\ 1&-1&-1\\ 1&1&-1\\ 1&1&1\end{pmatrix}\cdot\
abla\mathbb{C}_{\text{BiWgt}}^{\text{BIP}}=\begin{pmatrix} 0&0&2\\ 0&2&-2\\ 0&0&2\\ 2&-2&-2\\ 0&0&2\end{pmatrix}\cdot\begin{pmatrix}\
abla\mathbb{C}_{\text{BiWgt}}^{\text{BIP }}\\ \sigma^{2}=\begin{pmatrix}0&0&4\\ 0&4&4\\ 4&4&4\\ 0&0&4\end{pmatrix}\cdot\mathbb{X}_{\text{BiWgt}}^{\text{BIP}}=\begin{pmatrix}0&0&1 \\ 0&1&1\\ 1&1&1\\ 0&0&1\\ 0&0&1\end{pmatrix}\\ 0&0&1\end{pmatrix}\end{split}  \tag{232}\]  
Using our specifications \(\langle I_{u}\rangle=1\) mA, \(\sigma_{u}=0.1\) (196) we have \(\sigma_{u}^{2}\langle I_{u}\rangle^{2}=0.01\text{ mA}^{2}\), and the steps variance vector (231) results in  
\[ \begin{split}&\sigma_{\
abla\mathbf{J}}^{2}=0.01\text{ mA}^{2}\{2,6,2,14,2,6,2\}\\ &=\{0.02,\ 0.06,\ 0.02,\ 0.14,\ 0.02,\ 0.06,\ 0.02\}\text{mA}^{2}\end{split}  \tag{233}\]  
The worst-case variance, 0.14 mA\({}^{2}\), occurs mid-code at the fourth-to-fifth level transition. It is also useful to derive the mean output steps \(\langle\
abla\mathbf{J}\rangle\) in this special 3-bits BiWgt BIP case:  
(234)  
Evidently, all mean output constellation steps are equal to \(\langle I_{u}\rangle=1\)mA; thus, the mean constellation points are equispaced by \(\langle I_{u}\rangle=1\)mA. Note that an alternative derivation would take the mean of (140),  
\[ \begin{split}&\langle\
abla\mathbf{J}\rangle=\langle\
abla( \mathbf{CI})\rangle=\langle(\
abla\mathbf{C})\mathbf{I}\rangle=(\
abla \mathbf{C})\left\langle\overline{\mathbf{I}}\ \right\rangle\langle I_{u}\rangle=\langle I_{u}\rangle(\
abla\mathbf{C}) \frac{1}{2}\{4,2,1\}\\ &=\langle I_{u}\rangle\cdot\frac{1}{2}\
abla\mathbf{C}\{4,2,1\} =\langle I_{u}\rangle\cdot\mathbf{1}_{7}^{\forall}\end{split}  \tag{235}\]  
where the last equality above used \(\frac{1}{2}\
abla\mathbf{C}\{4,2,1\}=\mathbf{1}_{7}^{\forall}\) verified using the explicit form of \(\
abla\mathbf{C}_{\text{BiWgt}}^{\text{BIP}}\) in (232). In this subsection, we have analytically obtained the means and variances of the output levels and steps for the example of the 3-bit BiWgt BIP eDAC. These analytic predictions are going to be numerically verified in Section 2.9 by means of Monte Carlo simulations.