## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
** The simple derivation, based on (81) and (82), is omitted.\(\square\)  
Defining the \"total positive source current\" of BIP \(I_{\text{tot}}\) as the sum of the all-positive source currents,  
\[I_{\text{tot}}^{\text{BIP}}=\Sigma\mathbf{I}=\sum_{s=0}^{S-1}I_{s}^{\text{BIP}} \tag{87}\]  
we notice that the physical BIP source current vector, denoted \(\mathbf{I}^{\pm}\), has an \"absolute total\", namely total of the absolute values of its elements (L1-norm), which is double \(I_{\text{tot}}^{\text{BIP}}\) (since the negative antipodal currents feature the same absolute sum as the positive currents do):  
\[I_{\text{tot}}^{+-}\equiv\Sigma\Big{|}\mathbf{I}^{+-}\Big{|}=2\Sigma\mathbf{I }^{\text{BIP}}=2I_{\text{tot}}^{\text{BIP}} \tag{88}\]  
It is useful to normalize the total source current as well, since \(I_{\text{tot}}^{(}\right\rangle\equiv\left\langle I_{\text{tot}}\right\rangle =J_{\text{C}}\) for either UNIP \(\left|\text{BIP BiWgt}\right|\text{ThWgt}\) eDACs, and then dividing by \(\left\langle I_{u}\right\rangle\) yields \(\overset{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text  
Previewing our approach, we assume nothing more than the principle, in [5], according to which the stochastic spread of any current source on the eDAC chip is inversely proportional to the square root of the transistor area that generates it,  
\[\text{STD}\propto\frac{1}{\sqrt{\text{Area}}} \tag{92}\]  
(see Appendix A), augmented by the reasonable assumption that individual source currents are statistically independent (provided the source transistors are sufficiently separated, spatially, on the chip). Mismatch errors, referred to here as (mismatch-induced) perturbations, are defined as the random deviations of the source currents from their means, denoted as follows:  
\[\delta I_{s}\equiv I_{s}-\langle I_{s}\rangle \tag{93}\]  
As per [5], we assume that the source currents, \(I_{s}\), of the current-steering eDAC are statistically independent (hence uncorrelated). To advance, we must relate the individual source current variances to the current source structural (physical) parameters. For all current-steering eDACs of interest, the variances \(\left\{\sigma_{I_{s}}^{2}\right\}_{s=0}^{S-1}\) of the \(S\) source currents are modeled in this paper by the following key. **Proposition 1**. : _The eDAC current sources, \(\left\{I_{s}\right\}_{s=0}^{S-1}\), are statistically independent, have variances that are proportional to their means, \(\operatorname{var}I_{s}\propto\langle I_{s}\rangle\) and are Gaussian distributed_  
\[I_{s}\sim\operatorname{normal}\left[\langle I_{s}\rangle,\sigma_{uu}^{2} \langle I_{s}\rangle\right] \tag{94}\]  
_with \(\sigma_{uu}^{2}>0\) a proportionality factor (in [Ampere] units) relating means and variances for all current sources on the eDAC chip, wafer or batch of wafers, assuming statistical spatial homogeneity of the chip/wafer/batch, barring systematic mismatch errors:_  
\[\bigg{\langle}\widetilde{I}_{c}^{\text{UNIP BiWgt}[\text{ThWgt}}\bigg{\rangle}= c-1,\ \ \ \bigg{\langle}\widetilde{\mathbf{J}}^{\text{ UNIP BiWgt}[\text{ThWgt}}\bigg{\rangle}=\{0,1,2,\dots,C-1\} \tag{83}\]0  
**Proof of Proposition 1. ** Relegated to Appendix A. \(\square\)  
Note: reviewing the proof is highly recommended, as it distills the physical basis of the underlying device physics [5] into a statistical formulation used as \"anchor\" for our entire development of an all-analytic model of eDACs and oDACs.