## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### UNIP vs. BIP Source Currents Convention
Nevertheless, for the purpose of statistical modeling of variances, which is our specific target, our \"mathematical\" construction, based on \(B\) non-negative source currents, is indistinguishable from a circuit-model based on \(2B\) independent bipolar source currents for BIP eDACs, \(2B-1\) independent unipolar sources currents for UNIP eDACs, though, as discussed earlier, with the proviso of imposing the condition \(\mathbf{I}^{\text{BIP}}=\frac{1}{2}\mathbf{I}^{\text{UNIP}}\) onto the respective source currents. Note 2: The scaling by the factor-of-half relation, \(\left\langle\mathbf{I}^{\text{BIP}}\right\rangle=\frac{1}{2}\big{\langle} \mathbf{I}^{\text{UNIP}}\big{\rangle}\), holds only in-the-mean. Generally, \(\mathbf{I}^{\text{BIP}}\
eq\frac{1}{2}\mathbf{I}^{\text{UNIP}}\), since \(\mathbf{I}^{\text{BIP}}\), \(\mathbf{I}^{\text{UNIP}}\) may be additively decomposed into means + perturbations; it is only the means that are in a factor-of-half relation, the perturbations are not. Writing \(\mathbf{I}^{\text{BIP}}=\frac{1}{2}\mathbf{I}^{\text{UNIP}}\) (without wrapping the means around both sides) may lead to erroneous statistical results, inconsistent with the correct statistical model to be developed in Section 2.4, e.g., \(\mathbf{I}^{\text{BIP}}=\frac{1}{2}\mathbf{I}^{\text{UNIP}}\) implies \(\text{var}\left\{\mathbf{I}^{\text{BIP}}\right\}=\frac{1}{4}\text{var}\left\{ \mathbf{I}^{\text{UNIP}}\right\}\) but this relation between the variances is incorrect. We preview that the actual consequence of Proposition 1 developed in Section 2.4 is \(\text{var}\left\{\mathbf{I}^{\text{BIP}}\right\}=\frac{1}{2}\text{var}\left\{ \mathbf{I}^{\text{UNIP}}\right\}\). Note 3: It will be shown in the sequel that the UNIP \(\,\)I BIP \"fixed-total\" constraint (35) is consistent with having identical chip areas of the source currents aboard the BIP and UNIP eDACs ASICs; thus, the performance comparison of BIP vs. UNIP eDACs is consistently carried out under the constraint of identical investments of the salient design resource of chip area (the current sources chip area is the primary resource determining the quality of the eDAC in the wake of currents mismatch). Moreover, our UNIP \(\,\)I BIP convention renders the mean full scales of corresponding UNIP and BIP eDACs equal, \(\left\langle\text{FS}^{\text{BIP}}\right\rangle=\left\langle\text{FS}^{ \text{UNIP}}\right\rangle\) and we show that the \"unary current\" metrics (defined in Section 2.4) are equal. We now proceed to explore the impact of our UNIP \(\,\)I BIP convention on the comparative characteristics of the output current constellations of BIP vs. UNIP. We first explore the relations between BIP vs. UNIP \(\,\)BiWgt codewords \(\mathbf{b}_{\text{BIP}}[c]\), which are \(B\)-bit bitstrings obtained from \(\mathbf{b}_{\text{UNIP}}[c]\) per the mapping \(0\rightarrow-1\) and \(1\rightarrow+1\), formalized in vector notation as the following affine transformation taking UNIP codewords into BIP ones,  
\[\left\langle\mathbf{I}^{\text{BiWgt}}\right\rangle=\left\{\left\langle I_{1} ^{\text{BiWgt}}\right\rangle,\left\langle I_{0}^{\text{BiWgt}}\right\rangle \right\},\quad\left\langle\mathbf{I}^{\text{BiWgt}+-}\right\rangle=\left\{ \left\langle I_{1}^{\text{BiWgt}}\right\rangle,-\left\langle I_{1}^{\text{ BiWgt}}\right\rangle,\left\langle I_{0}^{\text{BiWgt}}\right\rangle,- \left\langle I_{0}^{\text{BiWgt}}\right\rangle\right\} \tag{33}\]2  
where we recall the all-ones vector notation \(\mathbf{1}_{m}^{\forall}\) introduced in Appendix C. Now, (32) yields  
\[