## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### UNIP vs. BIP Source Currents Convention
\begin{split}&\left\langle J_{c}^{\text{BIP}}\right\rangle=\mathbf{b }_{\text{BIP}}^{T}[c]\frac{1}{2}\left\langle\mathbf{I}\right\rangle=\mathbf{b }_{\text{BIP}}[c]\cdot\frac{1}{2}\left\langle\mathbf{I}\right\rangle=2( \mathbf{b}_{\text{UNIP}}[c]-\frac{1}{2}\mathbf{1}_{C}^{\forall})\cdot\frac{1} {2}\left\langle\mathbf{I}\right\rangle\\ &=(\mathbf{b}_{\text{UNIP}}[c]-\frac{1}{2}\mathbf{1}_{C}^{\forall })\cdot\left\langle\mathbf{I}\right\rangle=\left\langle\mathbf{b}_{\text{UNIP }}[c]\cdot\mathbf{I}\right\rangle-\frac{1}{2}\mathbf{1}_{C}^{\forall}\cdot \left\langle\mathbf{I}\right\rangle\end{split}  \tag{43}\]  
Now plugging  
\[\mathbf{b}_{\text{UNIP}}[c]\cdot\mathbf{I}=\mathbf{b}_{\text{UNIP}}[c]\cdot \mathbf{I}^{\text{UNIP}}=J_{c}^{\text{UNIP}},\quad\mathbf{1}_{C}^{\forall} \cdot\left\langle\mathbf{I}\right\rangle=\sum_{s=0}^{S-1}\left\langle I_{s} \right\rangle\equiv\left\langle I_{\text{tot}}^{\text{UNIP}}\right\rangle= \left\langle J_{C}^{\text{UNIP}}\right\rangle \tag{44}\]  
into the last expression in (43) yields  
\[\left\langle J_{c}^{\text{BIP}}\right\rangle=\left\langle J_{c}^{\text{UNIP}} \right\rangle-\frac{1}{2}\left\langle J_{C}^{\text{UNIP}}\right\rangle \tag{45}\]  
Thus, the BIP mean output current constellation is obtained from the UNIP one by a down-shift by half the mean UNIP MSL (or \(1/2\) the total mean current of the UNIP sources):  
\[\text{UNIP - to - BIP shift transformation}:\left\langle J_{c}^{\text{BIP}}\right\rangle=\left\langle J_{c}^{ \text{UNIP}}\right\rangle-\frac{1}{2}\left\langle J_{C}^{\text{UNIP}}\right\rangle =\left\langle J_{c}^{\text{UNIP}}\right\rangle-\frac{1}{2}\left\langle J_{ \text{tot}}^{\text{UNIP}}\right\rangle \tag{46}\]  
Note: More concisely, the last result is viewed as the BIP mean output constellation being obtained by centering the one-sided UNIP mean output constellation, which spans the FSR \(\left[0,\left\langle J_{C}^{\text{UNIP}}\right\rangle\right]\), yielding the FSR \(\left[-\frac{1}{2}\left\langle J_{C}^{\text{UNIP}}\right\rangle,\frac{1}{2} \left\langle J_{C}^{\text{UNIP}}\right\rangle\right]\) for the BIP mean output constellation. Note that both constellations have identical full-scale, \(\left\langle J_{C}^{\text{UNIP}}\right\rangle\). The UNIP \(\,\)I BIP convention amounts to design constraints on eDAC full scales and chip areas. Considering the shift transformation \(\left\langle J_{c}^{\text{BIP}}\right\rangle=\left\langle J_{c}^{\text{UNIP}} \right\rangle-\frac{1}{2}\left\langle J_{C}^{\text{UNIP}}\right\rangle\) (45), plug in \(c:=1\) and use \(\left\langle J_{1}^{\text{UNIP}}\right\rangle=0\), yielding the relation \(\left\langle J_{1}^{\text{BIP}}\right\rangle=-\frac{1}{2}\left\langle J_{C}^{ \text{UNIP}}\right\rangle\) between the LSLs of UNIP vs. BIP; similarly, plug in \(c:=C\) to obtain the relation \(\left\langle J_{C}^{\text{BIP}}\right\rangle=\frac{1}{2}\left\langle J_{C}^{ \text{UNIP}}\right\rangle\) between the MSLs of UNIP vs. BIP. The two relations may be compactly recapped by stating the FSRs of BIP and UNIP, respectively (consistent with the last note above):  
\[\left[J_{1}^{\text{BIP}},J_{C}^{\text{BIP}}\right]=\left[-\frac{1}{2}J_{C}^{ \text{UNIP}},\ \frac{1}{2}J_{C}^{\text{UNIP}}\right] \tag{47}\]\[\left<{\rm I}^{\rm BiWgt-BIP+-}\right>=\frac{1}{2}\Big{\{}2^{B-1}\langle I_{u} \rangle,-2^{B-1}\langle I_{u}\rangle..,4\langle I_{u}\rangle,-4\langle I_{u} \rangle,2\langle I_{u}\rangle,-2\langle I_{u}\rangle,\langle I_{u}\rangle,- \langle I_{u}\rangle\Big{\}} \tag{52}\]  
whereas, per Proposition 1 to be developed in Section 2.3 below, the variances of the source currents in (51) will be shown proportional to the respective means of the source currents. Note: For ThWgt-BIP eDACs, we should set \(S=C-1\) in (40). For BiWgt-BIP eDACs, we should set \(S=B\), using as many positive current sources as the number of bits, \(B\); thus, the total number of positive and negative current sources is now \(2B\):  
\[