## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
For BIP eDACs, the elements of \(\mathbf{C}\) are \(\pm 1\); thus, their squares are always \(1\)-s. It follows that  
\[\mathbf{C}_{\text{UNIP}}^{\odot 2}=\mathbf{C}_{\text{UNIP}},\ \ \mathbf{C}_{\text{BIP}}^{ \odot 2}=\mathbf{1}_{\text{C}\times\text{S}}^{\vee} \tag{121}\]  
Specializing (121) to UNIP and BIP eDACs, be they BiWgt| ThWgt, yields  
\[ \begin{split}&\sigma_{\text{UNIP}}^{2}=\mathbf{C}_{\text{ UNIP}}^{\odot 2}\sigma_{\text{UNIP}}^{2}=\mathbf{C}_{\text{UNIP}}\sigma_{ \text{UNIP}}^{2}=\mathbf{C}_{\text{UNIP}}\sigma_{u}^{2}\langle l_{u}\rangle \big{\{}\mathbf{U}^{\text{UNIP}}\big{\}}\\ &=\sigma_{u}^{2}\langle l_{u}\rangle\cdot\big{\langle}\mathbf{C }_{\text{UNIP}}\mathbf{I}^{\text{UNIP}}\big{\rangle}=\sigma_{u}^{2}\langle l _{u}\rangle\cdot\big{\langle}\mathbf{J}^{\text{UNIP}}\big{\rangle}\end{split}  \tag{122}\]  
where \(\sigma_{\text{I}}^{2}=\sigma_{u}^{2}\langle l_{u}\rangle\langle\mathbf{I}\rangle\) (109) and \(\mathbf{J}=\mathbf{CI}\) were specialized to UNIP. In fact, the result above most generally pertains to UNIP eDACs with non-uniform constellations. Dividing (122) by \(\langle l_{u}\rangle^{2}\) yields a result (generally valid for non-uniform constellations) featuring the proportionality of the UNIP normalized variance vector and the normalized current constellation vector:  
\[\sigma_{\text{J}^{\text{UNIP}}}^{2}=\sigma_{u}^{2}\Big{/}\widetilde{\mathbf{ J}}^{\text{UNIP}}\Big{\rangle} \tag{123}\]  
In the special case of a UNIP eDAC with a uniform output constellation, (122) reduces to  
\[ \begin{split}&\sigma_{\text{J}^{\text{UNIP}}}^{2}=\sigma_{u}^{2} \langle l_{u}^{\text{UNIP}}\rangle\cdot\big{\langle}\mathbf{J}^{\text{UNIP}} \big{\rangle}=\sigma_{u}^{2}\langle l_{u}^{\text{UNIP}}\rangle\cdot\{0,1,2,3, \dots,C-1\}\big{\langle}I_{u}^{\text{UNIP}}\big{\rangle}\\ &=\sigma_{u}^{2}\langle l_{u}^{\text{UNIP}}\rangle^{2}\cdot\{0,1,2,3,\dots,C-1\}\end{split}  \tag{124}\]  
Dividing (124) by \(\langle l_{u}\rangle^{2}\) yields a result (valid for uniform constellations):  
\[\sigma_{\text{J}^{\text{UNIP}}}^{2}=\sigma_{u}^{2}\{0,1,2,3,\dots,C-1\} \tag{125}\]  
Moreover, the BIP variance results in (132) hold not just for uniform constellations with \(C\) equispaced mean levels over the FSR \(\left[-\frac{1}{2}(C-1),\frac{1}{2}(C-1)\right]\langle I_{u}\rangle\). More generally, they hold for any BIP mean constellation \(\{\langle I_{c}\rangle\}_{c=1}^{C}\), be it uniform or non-uniform and for either BiWgt \(|\) ThWt BIP eDACs, we have for BiWgt\(|\)ThWgt unif\(|\)non - unif BIP output constellations : \(\sigma_{\text{-BIP}}^{2}=\frac{1}{2}(C-1)\sigma_{u}^{2}\mathbf{1}_{C}^{\forall}\) (133)  
However, the UNIP variance vector stated in (132) exclusively holds for any UNIP eDAC generating a uniform mean constellation: for BiWgt\(|\)ThWgt unif UNIP output constellation, \(\langle\mathbf{J}\rangle=\{0,1,2,\ldots,C-1\}\langle I_{u}\rangle\) : \(\sigma_{\text{-B}}^{2}=\{0,1,2,\ldots,C-1\}\sigma_{u}^{2}\)  
For any BiWgt eDAC, be it UNIP \(|\) BIP, be its output constellation unif\(/\)non-unif, the following result holds for the (arithmetic) average variance of the \(C\) constellation levels (with arithmetic averages denoted by overbars),  
\[\text{for generally non $-$ unif BiWgt UNIP eDACs} \tag{134}\] \[ \begin{split}\frac{\sigma_{\text{-UNE}}^{2}}{\text{J}_{ \text{-BWPt}}^{2}}&\equiv\frac{1}{2}(C-1)\sigma_{u}^{2}\\ \end{split} \]  
To recapitulate for either BIP eDACs of any topology (BiWgt \(|\) ThWgt) or for UNIP BiWgt eDACs, both categories featuring generally non-uniform constellations (uniform constellations as special cases), the average variance is given by  
\[\text{for non $-$ unif, UNIP $-$ BiWgt $|$ BIP $-$ (BiWgt|ThWgt) eDACs}:\frac{\sigma_{\text{-B}}^{2}}{ \text{J}}=\frac{1}{2}(C-1)\sigma_{u}^{2} \tag{135}\]  
Note 1: the eDACs that (135) does not hold for are \"non-unif UNIP-ThWgt\" ones. For all other types of eDACs, (135) does hold (e.g., it holds for unif UNIP eDACs of any topology; it holds for unif/non-unif BIP eDACs of any topology).