## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Example: Analytic Models for 3-Bit (PAM8) eDACs of the ThWgt \(|\)BiWgt UNIP \(|\) BiP Types
 \tag{225}\]  
The relation between the \"mathematical\" all-positive three source currents (211) and the \"physical\(\pm\)\" source currents (220), respectively, driving the two models (222) and (225) is  
\[\left\{I_{2}^{\text{BiWgt-BIP}},I_{1}^{\text{BiWgt-BIP}},I_{0}^{\text{BiWgt- BIP}}\right\}=\left\{I_{2^{+}}^{\text{BiWgt-BIP}},I_{1^{+}}^{\text{BiWgt-BIP}},I_{0^{+}}^{ \text{BiWgt-BIP}}\right\} \tag{226}\]  
Our \"physical\(\pm\)\" six-vector current array, \(\mathbf{I}^{\text{BiWgt-BIP}+-}\) (220) (to be propagated through the switching linear transformation (222)) is characterized by the stipulation that its six \(\pm\) current sources be statistically independent and have means of forming the dyadic sequence  
\[\left\langle I_{2^{\pm}}^{\text{BiWgt-BIP}}\right\rangle=\pm 2\langle I_{u}\rangle, \;\;\left\langle I_{1^{\pm}}^{\text{BiWgt-BIP}}\right\rangle=\pm\langle I_{u }\rangle,\left\langle I_{0^{\pm}}^{\text{BiWgt-BIP}}\right\rangle=\pm\frac{1}{2 }\langle I_{u}\rangle \tag{227}\]  
and thus have variances proportional to the means, per Proposition 1. The variances may be explicitly evaluated using \(\sigma_{1}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle\langle\left|\mathbf{I} \right|\rangle\) (109), the notation \(\langle\left|\mathbf{I}\right|\rangle\) meaning \(\langle\left|\mathbf{I}\right|\rangle\equiv\left\{\left|I_{S-1}\right|,\ldots, \left|I_{2}\right|,\left|I_{1}\right|,\left|I_{0}\right|\right\}\), to account for negative current sources:  
\[\left\{\sigma_{I_{2^{\pm}}^{\text{BiWgt-BIP}}}^{2},\sigma_{I_{ \pm}^{\text{BiWgt-BIP}}}^{2},\sigma_{I_{0^{\pm}}^{\text{BiWgt}}}^{2}\right\} =\sigma_{u}^{2}\langle I_{u}\rangle\Big{\{}\Big{|}\Big{\langle}I_{2^{\pm}}^{ \text{BiWgt-BIP}}\Big{\rangle}\Big{|}\Big{\langle}I_{1^{\pm}}^{\text{BiWgt-BIP} }\Big{\rangle}\Big{|}\Big{|}\Big{\langle}\Big{\langle}I_{0^{\pm}}^{\text{BiWgt -BIP}}\Big{\rangle}\Big{\rangle}\Big{|}\Big{\}} \tag{228}\] \[=\sigma_{u}^{2}\langle I_{u}\rangle\Big{\{}2\langle I_{u}\rangle, \langle I_{u}\rangle,\frac{1}{2}\langle I_{u}\rangle\Big{\}}=\sigma_{u}^{2} \langle I_{u}\rangle^{2}\Big{\{}2,1,\frac{1}{2}\Big{\}}=(0.1)^{2}(1\;\text{mA} )^{2}\Big{\{}2,1,\frac{1}{2}\Big{\}}\] \[=0.01(\text{mA})^{2}\Big{\{}2,1,\frac{1}{2}\Big{\}}\]i.e., the six \"physical\(\pm\)\" current sources have STDs given by  
\[ \begin{split}\sigma_{\text{\tiny{BiWgt-BIP}}}&=\sigma_ {\text{\tiny{BiWgt-BIP}}}=0.1\sqrt{2}\text{ mA}\\ \sigma_{\text{\tiny{BiWgt-BIP}}}&=\sigma_{\text{ \tiny{BiWgt-BIP}}}=0.1\text{ mA}\\ \sigma_{\text{\tiny{BiWgt-BIP}}}&=\sigma_{\text{ \tiny{BiWgt-BIP}}}=0.1\frac{1}{\sqrt{2}}\text{ mA}^{2}\end{split}  \tag{229}\]  
Once the source current variances are known, as per (228), we are ready to have them propagated via the switching matrix (222). This may be effectively accomplished by considering the individual linear equations corresponding to (222), observing that the summation \(J_{c}^{\text{BiWgt-BIP}}=I_{2^{\pm}[c]}^{\text{BiWgt-BIP}}+I_{1^{\pm}[c]}^{ \text{BiWgt-BIP}}+I_{0^{\pm}[c]}^{\text{BiWgt-BIP}}\) (224) of three independent currents implies that the variance of the \(c\)-th output current level is given by the sum of variances of the three participating current sources that are switched in:  
\[ \begin{split}&\text{var}\Big{\{}I_{c}^{\text{BiWgt-BIP}}\Big{\}}= \text{var}\Big{\{}I_{2^{\pm}[c]}^{\text{BiWgt-BIP}}\Big{\}}+\text{var}\Big{\{}I _{1^{\pm}[c]}^{\text{BiWgt-BIP}}\Big{\}}+\text{var}\Big{\{}I_{0^{\pm}[c]}^{ \text{BiWgt-BIP}}\Big{\}}\\ &=2\sigma_{u}^{2}\langle I_{u}\rangle^{2}+\sigma_{u}^{2}\langle I _{u}\rangle+\frac{1}{2}\sigma_{u}^{2}\langle I_{u}\rangle=3.5\sigma_{u}^{2} \langle I_{u}\rangle=3.5(0.1)^{2}\text{mA}^{2}=0.035\text{ mA}^{2}\end{split}  \tag{230}\]  
This alternative derivation of the (equal) variances of the output levels of 3-bit BiWgt BIP eDAC, pertaining to the \"physical\(\pm\)\" BiWgt BIP eDAC structure, retrieves our earlier obtained result (218), which assumed the \"math model\" of positive-valued current sources modulated by the signs of the code (switching) matrix, \(\mathbb{C}^{\text{BiWgt-BIP}}\).