## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
Note 2: \"non-uniform\" eDAC constellations are equivalently referred to in Section 3.4 as \"biased\" constellations. **Proof of Corollary 4. ** The result (132) was proven in the text above the corollary. To prove (133), namely the surprising property that the variances of all output currents \(\{J_{c}\}_{c=1}^{C}\) attain a common value, and moreover, the variance is the same even for non-uniform BIP output constellations, the most intuitive derivation inspects Formula (3) for BIP eDACs, namely \(J_{c}^{\text{BIP}}\ =\pm\text{I}_{S-1}^{\text{BIP}}\pm\text{I}_{S-2}^{\text{ BIP}}\pm...\pm\text{I}_{0}^{\text{BIP}}\pm\text{I}_{0}^{\text{BIP}}\), takes its mean,  
\[\left\langle J_{c}^{\text{BIP}}\right\rangle\ =\pm\left\langle I_{S-1}^{\text{ BIP}}\right\rangle\pm\left\langle I_{S-2}^{\text{BIP}}\right\rangle\pm...\pm \left\langle I_{1}^{\text{BIP}}\right\rangle\pm\left\langle I_{0}^{\text{ BIP}}\right\rangle \tag{136}\]  
and uses Proposition 1 to evaluate the variance of the \(c\)-th output level as follows:\[ \begin{split}\forall_{c\in\{1,2,...,C\}}:&\quad\text{var} \gamma_{c}^{\text{BIP}}=\text{var}[\pm I_{5-1}^{\text{BIP}}]+\text{var}[\pm I_{5-2 }^{\text{BIP}}]+...+\text{var}[\pm I_{1}^{\text{BIP}}]+\text{var}[\pm I_{0}^{ \text{BIP}}]\\ &\qquad\qquad\qquad=\text{var}I_{5-1}^{\text{BIP}}+\text{var}I_{5 -2}^{\text{BIP}}+...+\text{var}I_{1}^{\text{BIP}}+\text{var}I_{0}^{\text{BIP}} \\ &\qquad\quad=\sigma_{uu}^{2}\langle I_{5-1}^{\text{BIP}}\rangle+ \sigma_{uu}^{2}\langle I_{5-2}^{\text{BIP}}\rangle+...\sigma_{uu}^{2}\langle I _{1}^{\text{BIP}}\rangle+\sigma_{uu}^{2}\langle I_{0}^{\text{BIP}}\rangle\\ &\qquad\quad=\sigma_{uu}^{2}\sum_{i=0}^{S-1}\langle I_{s}^{ \text{BIP}}\rangle=\sigma_{uu}^{2}\langle I_{\text{tot}}^{\text{BIP}}\rangle =\sigma_{uu}^{2}\langle I_{\text{C}}^{\text{BIP}}\rangle=\sigma_{\text{6}}^{2 }\sum_{i=0}^{2}\frac{1}{2}\langle I_{\text{C}}^{\text{UNIP}}\rangle\\ &\qquad\quad=\frac{1}{2}\sigma_{uu}^{2}(C-1)\langle I_{u}\rangle \ \ \equiv\frac{1}{2}\sigma_{u}^{2}\langle I_{u}\rangle(C-1)\langle I_{u}\rangle= \frac{1}{2}\sigma_{u}^{2}(C-1)\langle I_{u}\rangle^{2}\\ &\qquad\quad\text{where factor of $\chi_{2}$ for the means in the BIP case, }\langle I_{\text{tot}}^{\text{BIP}}\rangle=\langle I_{\text{C}}^{\text{BIP}} \rangle=\frac{1}{2}\langle I_{\text{C}}^{\text{UNIP}}\rangle=\frac{1}{2} \langle I_{\text{tot}}^{\text{UNIP}}\rangle,\end{split}  \tag{137}\]  
where factor of \(\chi_{2}\) for the means in the BIP case, \(\langle I_{\text{tot}}^{\text{BIP}}\rangle=\langle I_{\text{C}}^{\text{BIP}} \rangle=\frac{1}{2}\langle I_{\text{C}}^{\text{UNIP}}\rangle=\frac{1}{2} \langle I_{\text{tot}}^{\text{UNIP}}\rangle\), in equality #6, is justified by our convention of halving the mean values of the positive source currents in the case of BIP relative to UNIP, to ensure the physical constraint that the absolute total of the BIP current sources of both signs equals that of the UNIP current sources; in #8 \(\sigma_{uu}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle\) (107) was used. The equalities chain (137) provides the most intuitive derivation of the BIP variance formula \(\forall_{c\in\{1,2,...,C\}}:\ \text{var}\gamma_{c}^{\text{BIP}}=\frac{1}{2} \sigma_{u}^{2}(C-1)\langle I_{u}\rangle^{2}\) in Corollary 4, this derivation holding even for non-uniform constellations, as we note that no assumption has been made on the mean values of the constellation output levels. Next, consider the even more surprising result (134) on the (arithmetic) average of the variances of the \(C\) levels irrespective of the topology of the eDAC (ThWgt! BiWgt), its polarity (UNIP! BIP), the (non-)uniformity of its constellation.