## 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
\begin{split}\Big{\langle}I^{\text{BiWgt-BIP}}_{c}\Big{\rangle} \in\Big{\{}\frac{1}{2}\big{(}\pm 4\langle I_{u}\rangle\pm 2\langle I_{u} \rangle\pm\langle I_{u}\rangle\big{)}\Big{\}}\\ =\{-3.5,\ -2.5,\ -1.5,\ -0.5,\ 0.5,\ 1.5,\ 2.5,\ 3.5\}\langle I _{u}\rangle\end{split}  \tag{213}\]  
Note that the mean full scale (FS) of this eDAC is  
\[\Big{\langle}\text{FS}^{\text{BiWgt-BIP}}\Big{\rangle}=3.5\langle I_{u} \rangle-(-3.5\langle I_{u}\rangle)=7\langle I_{u}\rangle \tag{214}\]  
coinciding with the mean FS of the ThWgt UNIP eDAC simulated above:  
\[\Big{\langle}\text{FS}^{\text{BiWgt-UNIP}}\Big{\rangle}=7\langle I_{u} \rangle-0=7\langle I_{u}\rangle \tag{215}\]  
The variances of the three BIP BiWgt source currents are obtained using \(\sigma^{2}_{\mathbf{I}}=\sigma^{2}_{u}\langle I_{u}\rangle\langle\mathbf{I}\rangle\) (109), yielding the following prediction of our analytic modeling:  
\[\sigma^{2}_{\mathbf{I}^{\text{BWgt-BIP}}}\equiv\Big{\{}\sigma^{2 }_{I^{\text{BWgt-BIP}}},\sigma^{2}_{I^{\text{BiWgt-BIP}}_{1}},\sigma^{2}_{I^{ \text{BWgt-BIP}}_{1}}\Big{\}}=\sigma^{2}_{u  
The resulting Gaussian distributions are then identical for all eight output levels:  
\[ \begin{split}& J_{c}^{\text{BiWgt-BIP}}\sim\text{normal}\Big{[}3.5 \langle I_{u}\rangle,\sigma_{u}\langle I_{u}\rangle\sqrt{3.5}\Big{]}=\text{ normal}\Big{[}3.5\text{ mA},0.1\sqrt{3.5}\text{ mA}\Big{]}\\ &\text{for}\ c=1,2,3,\ldots,8\end{split}  \tag{219}\]  
Using a dyadic array satisfying \(\langle I_{2}\rangle=2\langle I_{1}\rangle=4\langle I_{0}\rangle\) yields the perfect UNIP-PAM8 constellation \(\langle\mathbf{J}\rangle=\{-7-5,-3,-1,1,3,5,7\}\langle I_{0}\rangle\). (d) is functionally equivalent to (c), but in (d), the three currents \(\langle\mathbf{I}\rangle=\{\langle I_{2}\rangle,\langle I_{1}\rangle,\langle I _{0}\rangle\}\) are synthesized using a \"uniform\" array of nominally seven current sources of identical means \(\langle I_{u}\rangle\), partitioned in subsets of 1, 2, 4 elements, respectively, \(\{\{\langle I_{u}\rangle,\langle I_{u}\rangle,\langle I_{u}\rangle,\langle I _{u}\rangle\},\{\langle I_{u}\rangle,\langle I_{u}\rangle\}\}\). The unary sources in the second and third subsets are electrically parallelized as shown in (d), generating the triplet of dyadic currents \(\langle\mathbf{I}\rangle=\{\langle I_{2}\rangle,\langle I_{1}\rangle,\langle I _{0}\rangle\}=\{4\cdot\langle I_{u}\rangle,2\cdot\langle I_{u}\rangle,1\cdot \langle I_{u}\rangle\}=\{4,2,1\}\langle I_{u}\rangle\). It is readily verified, in all four diagrams, that the total of all mean currents of the elements of the sources array (current absolute-values in the BIP case), equals \(\langle I_{\text{tot}}\rangle=7\langle I_{u}\rangle=7\langle I_{0}\rangle\), whereas the mean full scales, \(\langle\text{FS}\rangle\), of the \(\mathbf{J}\)-constellations come out equal, for UNIP \(|\) BIP eDACs, both given by \(\langle\text{FS}\rangle=7\langle I_{u}\rangle=7\langle I_{0}\rangle\). At this point, let us consider an equivalent model of the BiWgt BIP eDAC based on the \"physical\(\pm\)\" bipolar current sources (53) that are arrayed in pairs in each of three current-mirror cells implementing 1-bit controlled MUXEDs switching in either a positive source current or its antipodal counterpart. Thus, taking \(B=3\), the six random source currents (each consisting of its mean + perturbation) are denoted as  
\[\mathbf{I}^{\text{BiWgt-BIP}+-}=\left\{I_{2^{+}}^{\text{BiWgt-BIP}},I_{2^{-}}^ {\text{BiWgt-BIP}},I_{1^{+}}^{\text{BiWgt-BIP}},I_{1^{-}}^{\text{BiWgt-BIP}},I _{0^{+}}^{\text{BiWgt-BIP}},I_{0^{-}}^{\text{BiWgt-BIP}}\right\} \tag{220}\]  
Taking \(B=3\) in (53), the six current sources have means three of which are positive and three of which are negative,  
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