## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
 \tag{68}\]  
The FS of either UNIP \(|\text{BIP}\) is given by a common expression  
\[\text{UNIP}|\text{BIP}:\ \ \Big{\langle}\text{FS}^{\text{UNIP}|\text{BIP}} \Big{\rangle}=(C-1)\big{\langle}I_{u}^{\text{UNIP}|\text{BIP}}\Big{\rangle} \tag{69}\]  
but from (63) and (67) it seems that the unary currents for UNIP \(|\text{BIP}\) C-level eDACs differ by a factor-of-two (below \(I_{0}\) denotes the least source current)  
\[\Big{\langle}I_{u}^{\text{UNIP}}\Big{\rangle}=I_{0}^{\text{UNIP}},\ \ \ \Big{\langle}I_{u}^{\text{BIP}}\Big{\rangle}=2I_{0}^{\text{BIP}} \tag{70}\]  
but we made the convention that the all-positive source mean currents of BIP and UNIP are in 1:2 ratio as per \(\big{\langle}I_{0}^{\text{BIP}}\big{\rangle}=\tfrac{1}{2}\big{\langle}I_{0}^{ \text{UNIP}}\big{\rangle}\); thus, evidently, so are the least source currents:  
\[\Big{\langle}I_{0}^{\text{BIP}}\Big{\rangle}:=\frac{1}{2}\Big{\langle}I_{0}^{ \text{UNIP}}\Big{\rangle} \tag{71}\]  
It then follows from the last two equations that the respective unary currents of the UNIP \(|\text{BIP}\) constellations come out equal, which is convenient, as so are the corresponding full scales, and so are the two total areas of the source current arrays in the eDAC chips  
\[ \begin{split}&\text{UNIP vs. BIP convention :}\\ &\langle I_{u}\rangle\equiv\big{\langle}I_{u}^{\text{UNIP}}\big{\rangle}= \big{\langle}I_{u}^{\text{BIP}}\big{\rangle}\Leftrightarrow\langle\text{FS} \rangle\equiv\Big{\langle}\text{FS}^{\text{UNIP}}\Big{\rangle}=\Big{\langle} \text{FS}^{\text{BIP}}\Big{\rangle}\Leftrightarrow A\equiv A^{\text{UNIP}}=A^{ \text{BIP}}\end{split}  \tag{72}\]Henceforth, we no longer distinguish between \(\langle I_{u}^{\text{UNIP}}\rangle\), \(\langle I_{u}^{\text{BIP}}\rangle\) denoting the common value of either of these quantities as \(\langle I_{u}\rangle\). Likewise, we no longer distinguish between \(\left\langle\text{F}\text{S}^{\text{UNIP}}\right\rangle\), \(\left\langle\text{F}\text{S}^{\text{BIP}}\right\rangle\) denoting the common value of either of these quantities as \(\langle\text{F}\text{S}\rangle\). Note: the result (72) stems from our \"physical\" BIP source currents array comprising mirror antipodal currents, despite our usage of only all-positive current arrays, \(\mathbf{I}^{\text{BIP}},\mathbf{I}^{\text{UNIP}}\) mathematically. For any eDAC topology (BiWgt! ThWgt) and/or polarity (BIP! UNIP), as soon as the pertinent unary current \(\langle I_{u}\rangle\) is determined, it is useful to have it put to use, in order to obtain normalized representations of all current quantities in our statistical model, simply having all current quantities divided by \(\langle I_{u}\rangle\). This will be seen to render the statistical model formulation simpler and more elegant. All amplitude-domain quantities, be they random variables or deterministic metrics (mean, STD), are normalized by the mean unary current, \(\langle I_{u}\rangle\). Any current RV or amplitude-domain statistical metric of the eDAC is then expressed in units of \(\langle I_{u}\rangle\). Normalized scalars/vectors are denoted by over-crescents \(\sim\). Here are three examples of normalized current quantities:  
\[\widehat{I}_{s}\equiv I_{s}/\langle I_{u}\rangle,\qquad\widehat{\mathbf{I}} \equiv\mathbf{I}/\langle I_{u}\rangle,\qquad\widehat{\sigma}_{I_{s}}=\widehat{ \sigma}_{I_{s}}/\langle I_{u}\rangle=\sigma_{\widehat{I}_{s}}, \tag{73}\]  
where \(\sigma_{X}\equiv\sqrt{\text{var}X}\) denotes the standard-deviation of a random variable (RV), X. At times, we shall also use the alternative compact superscript notation, \(\prescript{\circ}{\circ}{}\) to designate the mean (expectation) for any random object \(\mathbf{R}\) (equivalent to wrapping by \(\langle\rangle\)),  
\[\mathbf{R}^{(\cdot)}\equiv\langle\mathbf{R}\rangle \tag{74}\]  
The following are additional examples of normalized mean currents or current vectors:  
\[