## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### UNIP vs. BIP Source Currents Convention  
The last subsection explored the linear transformation mapping the eDAC source currents into the eDAC output current levels. This subsection addresses the relative scaling of the source currents used in UNIP vs. BIP eDACs, such as to facilitate meaningful performance comparison between the ThWgt1 BiWgt polarities for a given eDAC topology. We introduce a convention specifying the ratio of UNIP vs. BIP source currents _under the constraint of a given total of the mean \"physical\" source currents_ (equivalently, for a given chip area of the \"physical\" current sources) for either of the two polarities (UNIP \(|\) BIP):  
\[ \begin{split}\text{UNIP}|\text{BIP}-\text{convention}:& \ \left\langle\mathbf{1}^{\text{BIP}}\right\rangle=\tfrac{1}{2}\left\langle \mathbf{I}^{\text{UNIP}}\right\rangle\\ &\Leftrightarrow\left\langle I_{s}^{\text{BIP}}\right\rangle= \tfrac{1}{2}\left\langle I_{s}^{\text{UNIP}}\right\rangle,\ \text{for}\ s=0,1,2,\ldots,S-1\end{split}  \tag{29}\]  
Note: Recall that, for a UNIP BiWgt eDAC, there as are many effective current sources, \(S=B=\log_{2}C\), as the number of bits, \(B\), determining the \(C=2^{B}\) codewords or levels. In contrast, the UNIP ThWgt eDAC topology uses \(S=C-1=2^{B}-1\) sources. Henceforth, denoting \(\mathbf{I}^{\text{UNIP}}\) simply as \(\mathbf{I}\), then our UNIP \(|\) BIP convention (29) may be equivalently formulated as  
\[\mathbf{I}\equiv\mathbf{I}^{\text{UNIP}}\ \ \therefore\ \left\langle\mathbf{I} \right\rangle\equiv\left\langle\mathbf{I}^{\text{UNIP}}\right\rangle,\ \ \ \left\langle\mathbf{I}^{\text{BIP}}\right\rangle=\frac{1}{2}\left\langle \mathbf{I}^{\text{UNIP}}\right\rangle=\frac{1}{2}\left\langle\mathbf{I}\right\rangle \tag{30}\]  
The corresponding eDAC output currents and their means are then given by  
\[ \begin{split}&\left\langle\mathbf{I}^{\text{UNIP}}\right\rangle= \mathbf{C}^{\text{UNIP}}\left\langle\mathbf{I}^{\text{UNIP}}\right\rangle= \mathbf{C}^{\text{UNIP}}\left\langle\mathbf{I}\right\rangle\\ &\left\langle\mathbf{I}^{\text{BIP}}\right\rangle=\ \ \ \mathbf{C}^{\text{BIP}}\left\langle\mathbf{I}^{\text{BIP}}\right\rangle\ \ =\mathbf{C}^{\text{BIP}}\tfrac{1}{2}\left\langle\mathbf{I}\right\rangle\end{split}  \tag{31}\]  
Note: operationally, a factor of \(\
icefrac{{1}}{{2}}\) appears in the BIP formulations; this factor is absent in the UNIP formulations. Component-wise, we have (with the notation \(\mathbf{I}\equiv\mathbf{I}^{\text{UNIP}}\)) the \"switch & sum\" models,  
\[ \begin{split}&\left\langle I_{c}^{\text{UNIP}}\right\rangle= \mathbf{b}_{\text{UNIP}}^{T}[c]\left\langle\mathbf{I}\right\rangle=(0\!\!\!\!1) \langle I_{S-1}\rangle+(0\!\!\!\!1)\langle I_{S-2}\rangle+....+(0\!\!\!\!1) \langle I_{1}\rangle+(0\!\!\!\!1)\langle I_{0}\rangle\\ &\left\langle I_{c}^{\text{BIP}}\right\rangle=\ \ \ \ \mathbf{b}_{\text{BIP}}^{T}[c]\tfrac{1}{2}\left\langle \mathbf{I}\right\rangle=\pm\tfrac{1}{2}\langle I_{S-1}\rangle\pm\tfrac{1}{2} \langle I_{S-2}\rangle\pm....\pm\tfrac{1}{2}\langle I_{1}\rangle\pm\tfrac{1}{2} \langle I_{0}\rangle\end{split}  \tag{32}\]  
Note: the simple linear combination (32) of source currents model with either \(\pm 1\) or \(0\!\!\!1\) coefficients renders the evaluation of the second-order statistics of \(J_{c}\) relatively straightforward, given the second-order statistics of the source current vector \(\mathbf{I}\equiv\{I_{S-1},I_{S-1},\ldots,I_{1},I_{0}\}\), of which statistics is about to be derived in Section 2.4. At this point, let us clarify what we mean by \"physical\" source currents, as opposed to the \"mathematical\" source currents, \(\mathbf{I}^{\text{BIP}},\mathbf{I}^{\text{UNIP}}\) heretoforerce used in our linear models, assumed to consist of all-positive-valued vectors. In our BIP \"mathematical\" linear model, negative current contributions are generated by means of multiplication the \(-1\) elements of the \(\mathbf{C}\)-matrix, rather than by means of defining current sources of negative polarity.