## 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
 \tag{200}\]  
The \(C\) = eighth level (the MSL) is the noisiest, having the largest STD:  
\[\sigma_{l_{8}^{\rm ThWgt-UNIP}}=0.1\sqrt{7}\ {\rm mA}=0.265\ {\rm mA} \tag{201}\]  
The Gaussian distributions of the outputs current levels have the following PDFs:  
\[J_{c}^{\rm ThWgt-UNIP}\sim{\rm normal}\Big{[}(c-1)\cdot 1\ {\rm mA},\ (c-1)\cdot(0.1\ {\rm mA })^{2}\Big{]},\ \ {\rm for}\ c=1,2,3,\ldots,8 \tag{202}\]  
We now model a \(B\) = 3-bits (\(C\) = 8 levels) BiWgt UNIP eDAC, with source currents vector  
\[{\bf I}^{\rm BiWgt-UNIP}\equiv\Big{\{}I_{2}^{\rm BiWgt-UNIP},I_{1}^{\rm BiWgt- UNIP},I_{0}^{\rm BiWgt-UNIP}\Big{\}} \tag{203}\]  
The mean source currents are in binary ratios, forming a three-element dyadic sequence:  
\[\left\langle{\bf I}^{\rm BiWgt-UNIP}\right\rangle=\{4\langle I_{u}\rangle,2 \langle I_{u}\rangle,\langle I_{u}\rangle\}=\{4,2,1\}\langle I_{u}\rangle. \tag{204}\]  
In our special case, \(\langle I_{u}\rangle=1{\rm mA}\); thus,  
\[\left\langle{\bf I}^{\rm BiWgt-UNIP}\right\rangle=\{4,2,1\}{\rm mA}. \tag{205}\]  
For a generic \(\langle I_{u}\rangle\), the mean output currents of this BIP 3-bit BiWgt eDAC are given by  
\[\left\langle I_{c}^{\rm BiWgt-UNIP}\right\rangle\in\{(0|1)4\langle I_{u} \rangle+(0|1)2\langle I_{u}\rangle+(0|1)\langle I_{u}\rangle\}=\{0,1,2,3,4,5,6,7\}\langle I_{u}\rangle \tag{206}\]  
thus, in our special case, \(\langle I_{u}\rangle=1{\rm mA}\), the eDAC generates the output constellation  
\[\left\langle I_{c}^{\rm BiWgt-UNIP}\right\rangle=\{0,1,2,3,4,5,6,7\}{\rm mA} \tag{207}\]  
The analytic model for the variances and the STDs of the three BIP BiWgt source currents is obtained using \(\sigma_{\bf I}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle\langle{\bf I}\rangle\) (109) with \(\langle{\bf I}\rangle:=\left\langle{\bf I}^{\rm BiWgt-UNIP}\right\rangle\) as per (204):  
\[ \begin{array}{l}\sigma_{\rm 1 BiWgt-UNIP}^{2}\equiv\Big{\{}\sigma_{ \rm 1 BiWgt-UNIP}^{2},\sigma_{\rm 1 BiWgt-UNIP}^{2},\sigma_{\rm 0 BiWgt-UNIP}^{2}\Big{\}}=\sigma_{u}^{2} \langle I_{u}\rangle\Big{\{}\Big{\langle}I_{2}^{\rm BiWgt-UNIP}\Big{\rangle} \Big{\langle}I_{1}^{\rm BiWgt-UNIP}\Big{\rangle},\Big{\langle}I_{0}^{\rm BiWgt -UNIP}\Big{\rangle}\Big{\}}\\ =\sigma_{u}^{2}\langle I_{u}\rangle\{4\langle I_{u}\rangle,2\langle I_{u} \rangle,\langle I_{u}\rangle\}=\sigma_{u}^{2}\langle I_{u}\rangle^{2}\{4,2,1 \}=(0.1)^{2}{\rm mA}^{2}\{4,2,1\}\\ \sigma_{\rm 1 BiWgt-UNIP}=\sigma_{u}\langle I_{u}\rangle\Big{\{}\sqrt{4},\sqrt{2},1 \Big{\}}=0.1\Big{\{}2,\sqrt{2},1\Big{\}}{\rm mA}\end{array}  \tag{208}\]or finally,  
\[\sigma^{2}_{\mathbf{I}^{\text{BWgt-UNP}}}=\sigma^{2}_{u}\langle I_{u}\rangle^{2} \{4,2,1\}=\{4,2,1\}\cdot 0.01\text{ mA}^{2} \tag{209}\]  
i.e., using (206) and (209), the distribution of the source currents vector is  
\[\mathbf{I}^{\text{BiWgt-UNP}} \sim\Big{\{}\text{normal}[4\langle I_{u}\rangle,2\sigma_{u} \langle I_{u}\rangle],\text{normal}\Big{[}2\langle I_{u}\rangle,\sqrt{2} \sigma_{u}\langle I_{u}\rangle\Big{]},\text{normal}[\langle I_{u}\rangle, \sigma_{u}\langle I_{u}\rangle]\Big{\}} \tag{210}\] \[\sim\Big{\{}\text{normal}[4\text{ mA},0.2\text{ mA}],\text{ normal}\Big{[}2\text{ mA},0.1\sqrt{2}\text{ mA}\Big{]},\text{normal}[1\text{ mA},0.1\text{ mA}]\Big{\}}\]  
Let us now model the case of a 3-bit BiWgt BIP eDAC, with currents array  
\[\mathbf{I}^{\text{BiWgt-BIP}}\equiv\Big{\{}I^{\text{BiWgt-BIP}}_{2},I^{\text {BiWgt-BIP}}_{1},I^{\text{BiWgt-BIP}}_{0},I^{\text{BiWgt-BIP}}_{0}\Big{\}}. \tag{211}\]  
Per our convention for UNIP vs. BIP source currents, we must now scale the mean source currents by factors of \(\frac{1}{2}\), taking BiWgt BIP eDAC source currents half as large as their BiWgt UNIP counterparts:  
\[\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}}\Big{\rangle}=\frac{1}{2}\{4\langle I _{u}\rangle,2\langle I_{u}\rangle,\langle I_{u}\rangle\}=\bigg{\{}2,1,\frac{ 1}{2}\Big{\}}\langle I_{u}\rangle. \tag{212}\]  
The mean output currents generated by the BIP 3-bit BiWgt eDAC are then  
\[