## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Monte Carlo Simulations for PAM8 eDACs Verifying Key Analytic Statistics of This Section
Comparing Figure 3b,d, it is apparent that the respective distributions of the eight output levels fit the same eight analytic Gaussian PDFs, irrespective of the type of ThWgt source currents (and corresponding \(\mathbf{C}\)-matrix), namely the sources Figure 3a to obtain Figure 3b, vs. the sources of Figure 3c to obtain Figure 3d.Thus, in terms of output level variances, UNIP ThWgt \(\mathbf{I}\) BiWgt eDACs are statistically equivalent. Qualitatively, it is seen that any of the displayed currents in Figure 3b,d have their spreads (STDs) increase from left to right, essentially consistent with Proposition 1, as the means of the distributions monotonically increase. Concurrently, since PDF-like histograms are displayed, the histogram heights are reduced in inverse proportion. Figure 4 addresses a BiWgt BIP eDAC. Figure 4a depicts the histograms of the three \"mathematical\" (all-positive) current sources. Those are akin to those of the BiWgt UNIP eDAC of Figure 3c, except that the means of the three BiWgt BIP sources of Figure 4a are scaled by a factor of \(\
icefrac{{1}}{{2}}\) relative to those of the BiWgt UNIP sources of Figure 3c, as per the conventions in Section 2.2. The means can be read off in Figure 4a at positions {2,1, \(\
icefrac{{1}}{{2}}\)} mA as compared to {4, 2, 1} mA in Figure 3c. The STDs of the three BiWgt BIP current sources of Figure 4a are roughly visualized to correspond to the rule according to which the STDs are proportional to the square-roots of the means. In Figure 4b, the resulting BIP BiWgt eDAC outputs, \(\left\{I_{c}^{\text{BiWgt}-\text{BIP}}\right\}_{c=1}^{8}\) are simulated for each of the 1500 randomly drawn realizations of the sources triplet \(\left\{I_{2}^{\text{BiWgt}},I_{1}^{\text{BiWgt}},I_{0}^{\text{BiWgt}}\right\}\),  
\[J_{c}^{\text{BiWgt}-\text{BIP}}=\pm I_{2}^{\text{BiWgt}-\text{BIP}}\pm I_{1}^{ \text{BiWgt}-\text{BIP}}\pm I_{0}^{\text{BiWgt}-\text{BIP}},\text{ for }c=1,2,\ldots,8 \tag{269}\]  
with the signs selected for each \(c\) according to the c-th row of the \(8\times 3\) BIP-ThWgt code matrix. The eight histograms of Figure 4b indicate that the output currents are identically distributed, fitting the common theoretical PDF \(\sim\text{normal}\left[3.5\text{ mA},0.1\sqrt{3.5}\text{ mA}\right]\) per (219). In fact, this feature of the BIP BiWgt eDACs is surprising, since for a UNIP BiWgt eDAC the output current levels are quite differently distributed than depicted in Figure 3d. This statistical property of the BIP BiWgt eDACs may be traced to the current sources just flipping signs (see (269)), as the output linear combination of statistically independent current sources is formed by the switching electrical network to generate the output constellation. To Monte Carlo simulate vector \(\mathbf{I}^{\text{BiWgt-BIP}}\equiv\left\{l_{2}^{\text{BiWgt-BIP}},l_{1}^{\text{ BiWgt-BIP}},l_{0}^{\text{BiWgt-BIP}}\right\}\) (203) of the three current sources, independently drawn 1500 times 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: \(\mathbf{J}^{\text{BiWgt-BIP}}=\mathbf{C}^{\text{BiWgt-BIP}}\mathbf{I}^{\text{ BiWgt-BIP}}\). Operationally, the repeated random drawing amounts to a single matrix multiplication,  
\[\left[\mathbf{J}_{(1)}^{\text{BiWgt-BIP}},\mathbf{J}_{(2)}^{\text{BiWgt-BIP}}, \ldots,\mathbf{J}_{(1500)}^{\text{BiWgt-BIP}}\right]=\mathbf{C}^{\text{BiWgt- BIP}}\left[\mathbf{I}_{(1)}^{\text{BiWgt-BIP}},\mathbf{I}_{(2)}^{\text{BiWgt- BIP}},\ldots,\mathbf{I}_{(1500)}^{\text{BiWgt-BIP}}\right] \tag{207}\]  
using extended 7 \(\times\) 1500 and 8 \(\times\) 1500 random matrices for the source and output currents, respectively, with these matrices formed by stacking their columns as indicated.