## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Monte Carlo Simulations for PAM8 eDACs Verifying Key Analytic Statistics of This Section
One may compare the visual mean full-scale range (FSR) \(\left[\left\langle\right\rangle_{1}^{\text{BiWgt-BIP}},\left\langle\right\rangle _{8}^{\text{BiWgt-BIP}}\right\rangle\)  
\(=[-3.5,3.5]\), as obtained from Figure 4a with the FSR \(\left[\left\langle\right\rangle_{1}^{\text{BiWgt-BIP}},\left\langle\right\rangle _{8}^{\text{BiWgt-BIP}}\right\rangle]=[0,7]\) obtained from Figure 3d (the mean LSL and MSL may be read-off in the two sub-figures). The mean FSRs are seen as identical, \(3.5-(-3.5)=7-0=7\), consistent with the conventions for UNIP vs. BIP BiWgt eDACs, as introduced in Section 2.2. Figure 4c,d pertain to the alternative BIP BiWgt source currents \"physical\(\pm\)\" model developed in (220)-(222) for the 3-bit BIP BiWgt eDAC. There are now six \"physical\(\pm\)\" sources  
\[\mathbf{I}^{\text{BiWgt-BIP+}-}=\left\{l_{2^{+}}^{\text{BiWgt-BIP}},l_{2^{-}}^ {\text{BiWgt-BIP}},l_{1^{+}}^{\text{BiWgt-BIP}},l_{1^{-}}^{\text{BiWgt-BIP}},l_ {0^{+}}^{\text{BiWgt-BIP}},l_{0^{-}}^{\text{BiWgt-BIP}}\right\} \tag{208}\]  
as depicted in Figure 4c. Comparison with the three \"physical\(\pm\)\" sources Figure 4a indicates that the positive sources of Figure 4c, and the three \"mathematical\" sources of Figure 4a, coincide in their respective distributions. The three new negative sources in (208) have means that are mirror images of the means of the positive sources, whereas the variances of \"\(\pm\)\"-mirror pairs are, respectively, identical. The current sources vector (208) is propagated via a different switching matrix than usual, namely the 8x6 matrix \(\mathbf{C}^{\text{BiWgt-BIP+}-}\) in (222):  
\[ \begin{array}{l}\left[\mathbf{J}_{(1)}^{\text{BiWgt-BIP+}-},\mathbf{J}_{(2 )}^{\text{BiWgt-BIP+}-},\ldots,\mathbf{J}_{(1500)}^{\text{BiWgt-BIP+}-}\right] \\ =\mathbf{C}^{\text{BiWgt-BIP+}-}\left[\mathbf{I}_{(1)}^{\text{BiWgt-BIP+}-}, \mathbf{I}_{(2)}^{\text{BiWgt-BIP+}-},\ldots,\mathbf{J}_{(1500)}^{\text{BiWgt- BIP+}-}\right]\end{array}  \tag{209}\]  
The eight histograms corresponding to resulting 1500 output current vectors, namely  
\[\left\{l_{(1)}^{\text{BiWgt-UNIP}}[c],l_{(2)}^{\text{BiWgt-UNIP}}[c],\ldots,l_{ (1500)}^{\text{BiWgt-UNIP}}[c]\right\}\text{for each of}c=1,2,\ldots,8 \tag{210}\]  
are displayed in Figure 4d along with the analytic Gaussian PDF overlays. Visual inspection of this figure indicates that it features essentially identical respective distributions as those of Figure 4b, verifying that the \"mathematical\" sources of Figure 4a are equivalent to the \"physical\(\pm\)\" sources of Figure 4b. Figure 5 proceeds to directly verify the foundational second-order statistical results of this section, namely Proposition 1 and Corollary 4, respectively, exemplified in parts (a), (b) of Figure 5. The linear fits in Figure 4a empirically corroborates the proportionality of the BiWgt source current variances and their respective means, as per Proposition 1. The linear fit in Figure 5b verifies Corollary 4 by showing that the output mean currents and the output variances form proportional arithmetic sequences, to an excellent approximation, as per (132). The datapoints in Figure 5b were evaluated by passing each of the 1500 realizations of the triplets of UNIP BiWgt sources via the 3 \(\times\) 8 UNIP BiWgt code matrix, to generate the eight output currents realizations (1500 random values each), then evaluating their sample means and variances. Figure 6 verifies the result (155) of Theorem 2 on the second-order statistics of the seven steps \(\left\{\
abla\right\}_{c=1}^{7}\) of the BIP PAM8 BiWgt eDAC. The histograms and PDF fits of the output  
steps are presented Figure 6a, whereas the output variances of the steps are illustrated in the bars diagram of Figure 6b. Additional detail is provided in the caption of Figure 6. Excellent fit is displayed in Figure 6a between the histograms and the theoretical Gaussian PDFs for the steps, validating the statistical analysis of Section 2.3.