## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Monte Carlo Simulations for PAM8 eDACs Verifying Key Analytic Statistics of This Section  
In this subsection, we validate, by means of Monte Carlo simulations (MC-sims), the statistical analytic model derived heretofore. All simulations refer to the 3-bit 8 level eDAC examples analytically modeled in the last two subsections, the parameters of which were specified in (196), namely \(C=8,\ \ B=3,\ \ \langle I_{u}\rangle=1\)mA, \(\sigma_{u}=0.1\). Specifically, we Monte Carlo simulate ThWgt! BiWgt, UNIP! BIP eDACs with these parameters. Figures 3-6 below verify Proposition 1 and Theorems 1 and 2 of this section, by means of MC-sims. In Figures 3, 4 and 6, histograms of various current RVs are plotted overlaid with the Gaussian analytic PDFs stemming from our analytic models. In all cases, we effectively verify the analytic models over an effective ensemble of 1500 3-bit BiWgt! ThWgt BIP! UNIP eDACs, all sharing the parameters specified in (196). The number of drawn realizations of each of the independent RVs for each of the considered currents or current metrics (e.g., the steps of the output levels of the currents) is then 1500. This is then the total number of points accumulated within each of the displayed histograms. Inspecting the multiple (sub)figures, it is apparent that, for each of the RVs of interest, at this sample size, the histograms well approximate the theoretical PDFs overlayed onto the respective histograms, validating the analytic models by means of the MC-sims presented here, which fit the theory. Figure 3 simulates 3-bit (PAM8) UNIP eDACs, of the ThWgt and BiWgt topologies. The current sources in the ThWgt UNIP case of Figure 3a are drawn IID, all \(S=C-1=7\) of them normally distributed as the unary current, \(I_{u}\), namely (with \(\langle I_{u}\rangle=1\)mA, \(\sigma_{u}=0.1\)):  
\[I_{s}^{\rm ThWgt-UNIP}\ \sim\ {\rm normal}[\langle I_{u}\rangle,\sigma_{u} \langle I_{u}\rangle]={\rm normal}[1\ {\rm mA},0.1\ {\rm mA}],\ \ {\rm for}\ s=0,1,2,3,4,5,6 \tag{263}\]  
The 1500 instances of the randomly drawn 7-vector \(\mathbf{\Gamma}^{\rm ThWgt-UNIP}\) are then propagated via the \(\mathbf{C}\)-matrix, yielding 1500 instances of the output currents vector of the eDAC:  
\[\mathbf{J}^{\rm ThWgt-UNIP}=\mathbf{C}^{\rm ThWgt-UNIP}\mathbf{\Gamma}^{\rm ThWgt -UNIP}. \tag{264}\]  
Figure 5: “Regression-like” linear fits for the (Mean, Var) pairs (the red points) as evaluated for the 1500 randomly drawn values out of the current distributions of a PAM8 UNIP BiWgt eDAC, parameterized by unary current \(\langle I_{u}\rangle=1\) mA and by relative mismatch \(\sigma_{u}=0.10\), overlayed by the linear fits \(\rm Var\propto Mean\) (the blue lines through the origin) numerically confirming key analytic results of this section. (**a**): Verification of Proposition 1 for the three UNIP BiWgt source currents \(\left\{I_{2}^{\rm BiWgt-UNIP},I_{1}^{\rm BiWgt-UNIP}\right\}\). (**b**): Verification of Corollary 4 (132) for the 8 output currents \(\left\{I_{c}^{\rm BiWgt-UNIP}\right\}_{c=1}^{8}\). Equivalently, the resulting eight output level distributions, shown in Figure 3b, are evaluated by accumulating the seven source currents, \(J_{c}^{\text{ThWgt-UNIP}}=\sum_{s=0}^{c-1}I_{s}^{\text{ThWgt-UNIP}}\), \(c=1,2,\ldots,8\):  
\[ \begin{array}{ll}J_{1}^{\text{ThWgt-UNIP}}=0,&J_{2}^{\text{ThWgt-UNIP}}=I_{0}^ {\text{ThWgt-UNIP}},&J_{3}^{\text{ThWgt-UNIP}}=I_{1}^{\text{ThWgt-UNIP}}+I_{0}^ {\text{ThWgt-UNIP}},\\ J_{4}^{\text{ThWgt-UNIP}}=I_{2}^{\text{ThWgt-UNIP}}+I_{1}^{\text{ThWgt-UNIP}}+I_ {0}^{\text{ThWgt-UNIP}},&\ldots,J_{8}^{\text{ThWgt-UNIP}}=\sum_{s=0}^{7}I_{s }^{\text{ThWgt-UNIP}}\end{array}