## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### UNIP vs. BIP Source Currents Convention
\begin{split}\left<{\rm I}^{\rm BiWgt-BIP+-}\right>& \equiv\Big{\{}\frac{1}{2}\Big{\langle}I_{B-1}^{\rm BiWgt-BIP}\Big{\rangle},- \frac{1}{2}\Big{\langle}I_{B-1}^{\rm BiWgt-BIP}\Big{\rangle},\frac{1}{2} \Big{\langle}I_{B-2}^{\rm BiWgt-BIP}\Big{\rangle},-\frac{1}{2}\Big{\langle}I_ {B-2}^{\rm BiWgt-BIP}\Big{\rangle},\ldots\\ &\ldots\frac{1}{2}\Big{\langle}I_{1}^{\rm BiWgt-BIP}\Big{\rangle},-\frac{1}{2}\Big{\langle}I_{1}^{\rm BiWgt-BIP}\Big{\rangle},\frac{1}{2} \Big{\langle}I_{0}^{\rm BiWgt-BIP}\Big{\rangle},-\frac{1}{2}\Big{\langle}I_{ 0}^{\rm BiWgt-BIP}\Big{\rangle}\Big{\}}\end{split}  \tag{53}\]  
In fact, the means of the positive sources in the signed physical array \(\langle{\rm I}^{+-}\rangle\) coincide with the elements of \(\frac{1}{2}\langle{\rm I}\rangle\), whereas the negative sources in \(\langle{\rm I}^{+-}\rangle\) coincide with the elements of \(-\frac{1}{2}\langle{\rm I}\rangle\). It is then evident that  
\[\Sigma\Big{|}\Big{\langle}{\rm I}^{\rm BiWgt+-}\Big{\rangle}\Big{|}=\Sigma \bigg{|}\frac{1}{2}\langle{\rm I}\rangle\bigg{|}+\Sigma\bigg{|}-\frac{1}{2} \langle{\rm I}\rangle\bigg{|}=\Sigma\big{|}\langle{\rm I}\rangle|=\Sigma \Big{|}\Big{\langle}{\rm I}^{\rm UNIP}\Big{\rangle}\Big{|} \tag{54}\]thus, we have verified that the \"absolute total\" (meaning the sum of the absolute values of) the physical source currents is the same for UNIP and BIP by virtue of our source currents convention, i.e., \(\Sigma\big{|}\big{\langle}\mathrm{I}^{\mathrm{BIP}+-}\big{\rangle}\big{|}=\Sigma \big{|}\big{\langle}\mathrm{I}^{\mathrm{UNIP}}\big{\rangle}\big{|}\) generally holds for either UNIP \(|\) BIP constellations of the same size, as already formally proven in (35) above. We are about to show, in the discussion around Proposition 1 below, that the source currents (ignoring their signs, i.e., taking absolute values) are proportional to the areas of the MOS transistors implementing the current sources, namely \(\Sigma\mathbf{I}\)\(|\)\(\propto\)\(\Sigma A\), with \(\Sigma A\) the total area of the active sources. The equality of absolute totals of source currents, for BIP \(|\) UNIP, amounts to a corresponding equality of the total areas of the current sources:  
\[\Sigma A^{\mathrm{BIP}}=\Sigma A^{\mathrm{UNIP}} \tag{55}\]  
This convention enables a unified treatment fairly comparing BiWgt \(|\) ThWgt eDAC topologies of both UNIP \(|\) BIP polarities. \(\Sigma A\) is regarded as a key design constraint to be equally assigned to various alternative designs. We then recapitulate the following equivalences:  
\[\Sigma A^{\mathrm{BIP}}=\Sigma A^{\mathrm{UNIP}}\ \ \Leftrightarrow\ \ \Sigma\Big{|}\Big{\langle}\mathrm{I}^{\mathrm{BIP}+-}\Big{\rangle}\Big{|}= \Sigma\Big{|}\Big{\langle}\mathrm{I}^{\mathrm{UNIP}}\Big{\rangle}\Big{|}\ \ \Leftrightarrow\ \ \Big{\langle}\mathrm{FS}^{\mathrm{BIP}}\Big{\rangle}=\Big{\langle}\mathrm{FS}^ {\mathrm{UNIP}}\Big{\rangle} \tag{56}\]  
Note: In communication links the span (full-scale) of the transmitted constellation is a key transmission resource, since over linear transmission channels the received constellation is proportional to the transmitted one. The larger the FSR (span) of the received constellation is, the better the link BER performance. Thus, \(\Big{\langle}\mathrm{FS}^{\mathrm{BIP}}\Big{\rangle}=\Big{\langle}\mathrm{FS}^ {\mathrm{UNIP}}\Big{\rangle}\) is a useful criterion for comparison of communication scenarios using BIP vs. UNIP constellations.