## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Monte Carlo Simulations for PAM8 eDACs Verifying Key Analytic Statistics of This Section
, \tag{265}\]  
The eight histograms of the 1500 elements (output current levels) \(\{J_{(1)}^{\text{ThWgt-UNIP}}[c],\)\(J_{(2)}^{\text{ThWgt-UNIP}}[c],\ldots,J_{(1500)}^{\text{ThWgt-UNIP}}[c]\}\) for \(c=1,2,\ldots,8\) are displayed in Figure 3b along with the eight analytic Gaussian PDF overlays (202):  
\[J_{c}^{\text{ThWgt-UNIP}}\sim\text{normal}\Big{[}(c-1)\cdot 1\text{mA}, (c-1)\cdot(0\,.1\text{mA})^{2}\Big{]},\ \ \text{for}\ c=1,2,3,\ldots,8 \tag{266}\]  
Figure 6: Monte Carlo verification of the statistics of the BIP PAM8 BiWgt eDAC output current steps \(\{\
abla I_{c}\}_{c=1}^{7}\) (namely the differences between successive output constellation levels), evaluated for the same Monte Carlo simulation as in Figures 2 and 3 (1500 sample values drawn for each of the three source currents). **(a)**: Histograms of the 7 output steps, overlaid with the Gaussian analytic PDFs corresponding to the mean steps \(\langle\
abla\mathbf{J}\rangle=\frac{1}{4}\{0,1,2,\ldots 7\}\langle I_{u}\rangle\) and variances predicted by Theorem 2 for C = 8 constellation points, namely \(\sigma_{\
abla\mathbf{J}}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle^{2}\cdot( \
abla\mathbf{C})^{\otimes 2}\frac{1}{4}\{4,2,1\}\) where the parameters in our Monte Carlo simulation are \(\langle I_{u}\rangle=1\) mA, \(\sigma_{u}=0.1\). **(b)**: Corresponding empirical variances of the output steps (“measured” for the random samples forming the histograms displayed in (top)). The empirical output level variances are represented as the heights of the bars, seen to closely approximate the expected analytic variance values [0.02, 0.06, 0.14]. The variance of the step \(\
abla I_{4}\) at the “mid-code” location is the worst one (\(\sim\)0.14), consistent with the analysis in Section 2.5. The good fit between PDFs and the histogram plots, in all subfigures, verifies the analytics. The visuals of Figure 3b are consistent with the expected results, e.g., the spreads (STDs) of the UNIP ThWgt eDAC output levels are seen to monotonically progress as the square roots of an arithmetic sequence (see last line of (200)). Thus, Figure 3b verifies Theorem 1 for a UNIP ThWgt eDAC. Figure 3c takes up the BiWgt UNIP case, displaying the three empirical distributions (histograms) of the current sources, overlayed by the analytic Gaussian PDFs of the current sources, showing excellent fit with the analytic prediction (210). To obtain Figure 3d, the vector of the three current sources \(\mathbf{I}^{\text{BiWgt}-\text{UNIP}}\equiv\left\{I_{2}^{\text{BiWgt}-\text{ UNIP}},I_{1}^{\text{BiWgt}-\text{UNIP}},I_{0}^{\text{BiWgt}-\text{UNIP}} \right\}\right\}\) (203), drawn 1500 from the Gaussian IID distribution (210), is propagated via the \(8\times 3\) BiWgt-BIP \(\mathbf{C}\)-matrix, yielding 1500 instances of the output currents vector of the eDAC:  
\[\mathbf{J}^{\text{BiWgt}-\text{UNIP}}=\mathbf{C}^{\text{BiWgt}-\text{UNIP}} \mathbf{I}^{\text{BiWgt}-\text{UNIP}}, \tag{267}\]  
Operationally, this amounts to a single matrix multiplication  
\[\left[I_{(1)}^{\text{BiWgt}-\text{UNIP}},I_{(2)}^{\text{BiWgt}-\text{UNIP}}, \ldots,I_{(1500)}^{\text{BiWgt}-\text{UNIP}}\right]=\mathbf{C}^{\text{BiWgt} -\text{UNIP}}\left[\mathbf{I}_{(1)}^{\text{BiWgt}-\text{UNIP}},\mathbf{I}_{ (2)}^{\text{BiWgt}-\text{UNIP}},\ldots,\mathbf{I}_{(1500)}^{\text{BiWgt}- \text{UNIP}}\right] \tag{268}\]  
using the extended \(7\times 1500\) and \(8\times 1500\) matrices formed in terms of their columns. The eight histograms of the 1500 elements (output current levels) \(\{I_{(1)}^{\text{BiWgt}-\text{UNIP}}[c],\)\(J_{(2)}^{\text{BiWgt}-\text{UNIP}}[c],\ldots,J_{(1500)}^{\text{BiWgt}-\text{UNIP}}[c]\}\) for \(c=1,2\ldots,8\) are displayed in Figure 3d, along with their analytic Gaussian PDF overlays.