## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### UNIP vs. BIP Source Currents Convention
Most generally, the physical array of 5 bipolar currents (the generalization of Figure 1) comprises the following bipolar source currents:  
\[\text{physical}+-\text{ model of current sources array for BIP :} \tag{40}\] \[\mathbf{I}^{\text{BIP}+-}\equiv\Big{\{}^{\text{BIP}+-}_{(S-1)^{+},\,^{\text{BIP}+-}}_{(S-1)^{-}},^{\text{BIP}+-}_{(S-2)^{+}},\ldots,^{\text{ BIP}+-}_{1^{+}},^{\text{BIP}+-}_{1^{-}},^{\text{BIP}+-}_{0^{+}},^{\text{BIP}+-}_{0^{-}} \Big{\}}\]  
These are the source currents, each consisting of its mean + perturbation. Thus, in the \"physical\(\pm\) model\", there are now 25 current sources, having means \(S\) of which are positive and \(S\) of which are negative. In contrast, in the \"math model\",  
\[\left\langle\mathbf{I}^{\text{BiWgt}}\right\rangle=\left\{\left\langle I_{1} ^{\text{BiWgt}}\right\rangle,\left\langle I_{0}^{\text{BiWgt}}\right\rangle \right\},\quad\left\langle\mathbf{I}^{\text{BiWgt}+-}\right\rangle=\left\{ \left\langle I_{1}^{\text{BiWgt}}\right\rangle,-\left\langle I_{1}^{\text{ BiWgt}}\right\rangle,\left\langle I_{0}^{\text{BiWgt}}\right\rangle,- \left\langle I_{0}^{\text{BiWgt}}\right\rangle\right\} \tag{33}\]0  
onto which the \(\mathbf{C}^{\text{BIP}}\) matrix is to act in order to generate the output current of the eDAC, as per \(\mathbf{J}^{\text{BIP}}=\mathbf{C}^{\text{BIP}}\mathbf{j}^{\text{BIP}}\), we just have \(S\) current sources, \(I^{\text{BIP}}_{s}\), for \(s=S-1,S-2,...3,2,1,0\), all having positive means (negative currents are effectively provided by \(-1\) elements of the switching matrix \(\mathbf{C}\)). In fact, these S positive-mean current sources \(I^{\text{BIP}}_{s}\) of \(\mathbf{I}^{\text{BIP}}\) have means, respectively, equal to those of the currents in \(\mathbf{I}^{\text{BIP}+-}\) (40) with positive means:  
\[\left\langle\mathbf{I}^{\text{BiWgt}}\right\rangle=\left\{\left\langle I_{1} ^{\text{BiWgt}}\right\rangle,\left\langle I_{0}^{\text{BiWgt}}\right\rangle \right\},\quad\left\langle\mathbf{I}^{\text{BiWgt}+-}\right\rangle=\left\{ \left\langle I_{1}^{\text{BiWgt}}\right\rangle,-\left\langle I_{1}^{\text{ BiWgt}}\right\rangle,\left\langle I_{0}^{\text{BiWgt}}\right\rangle,- \left\langle I_{0}^{\text{BiWgt}}\right\rangle\right\} \tag{33}\]1  
Thus, for UNIP eDACs, the \"mathematical\" and \"physical\(\pm\)\" source arrays coincide, both consist of \(\mathbf{I}\equiv\left\{I_{S-1},I_{S-2},\ldots,\frac{1}{2}I_{1},\frac{1}{2}I_{ 0}\right\}=\mathbf{I}^{\text{UNIP}}\), whereas for BIP, the \"physical\(\pm\)\" array \(\mathbf{I}^{\text{BIP}+-}\) has twice as many sources (25), compared to the \"math model\" array \(\mathbf{I}^{\text{BIP}}\) comprising \(S\) positive-currents. The \"physical\(\pm\)\" array, \(\mathbf{I}^{\text{BIP}+-}\), comprises \(S\) positive-mean currents coinciding in-the-mean with the elements of \(\mathbf{I}^{\text{BIP}}\) and \(S\) positive-mean currents antipodal in-the-mean to the elements of \(\mathbf{I}^{\text{BIP}}\). Note 1: we have seen that \"physical\(\pm\)\" circuits of BIP BiWgt eDACs practically use twice as many sources: viewing the BIP sources as paired up in antipodal pairs (assigning to each current level both signs), we would have their antipodal source currents generated by two separate effective current sources (possibly the two halves of a current mirror cell) rather than being derived from a single source. In contrast, in our mathematically convenient model, \(\mathbf{J}=\mathbf{CI}\), wherein \(\mathbf{I}\) comprises \(B\) non-negative source currents, and this effect is attained by a linear transformation entailing a \(\pm 1\)-valued code matrix, such that the bipolar sign flips \(\pm 1\) in the linear combinations forming the \(J_{c}\) currents be conceptually provided by \(\pm 1\)-valued codewords. As a result, our simple matrix-based model does not correctly predict cross-correlations between eDAC output levels.