## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### UNIP vs. BIP Source Currents Convention
This constraint is formulated for a BiWgt eDAC as  
\[\text{UNIP}\big{|}\text{BIP}-^{\prime}\text{fixed}-\text{total}^{\prime}\text {constraint}:\Sigma\Big{|}\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}+-}\Big{\rangle} \Big{|}=\Sigma\Big{\langle}\mathbf{I}^{\text{BiWgt-UNIP}}\Big{\rangle} \tag{35}\]  
We now verify that our UNIP \(|\) BIP-convention (29) satisfies \"fixed-total\" constraint (35): since the positive-valued elements of \(\left\langle\mathbf{I}^{\text{BiWgt}+-}\right\rangle\) coincide with \(\left\langle\mathbf{I}^{\text{BiWgt}}\right\rangle\), whereas the negative-valued elements of \(\left\langle\mathbf{I}^{\text{BiWgt}+-}\right\rangle\) coincide with \(-\left\langle\mathbf{I}^{\text{BiWgt}}\right\rangle\); therefore, it is evident that  
\[\Sigma\Big{|}\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}+-}\Big{\rangle}\Big{|} =\Sigma\Big{|}\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}}\Big{\rangle}\Big{|} +\Sigma\Big{|}-\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}}\Big{\rangle}\Big{|} =2\cdot\Sigma\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}}\Big{\rangle} \tag{36}\]  
The factor-of-two in the last equality is compensated by the factor of \(\frac{1}{2}\) in our UNIP \(|\) BIP convention \(\left\langle\mathbf{I}^{\text{BIP}}\right\rangle=\frac{1}{2}\langle\mathbf{I}^ {\text{UNIP}}\rangle\) (29):  
\[\Sigma\Big{|}\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}+-}\Big{\rangle}\Big{|} =2\cdot\Sigma\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}}\Big{\rangle}=2\cdot \Sigma\frac{1}{2}\Big{\langle}\mathbf{I}^{\text{BiWgt-UNIP}}\Big{\rangle}= \Sigma\Big{\langle}\mathbf{I}^{\text{BiWgt-UNIP}}\Big{\rangle} \tag{37}\]  
\(\square\). Note: Yet another equivalent, intuitive, statement: the \"physical\" BIP array also uses mirror-image sources; therefore, it comprises twice as many current sources; to have a fixed total current, we must half all source currents upon transitioning from UNIP to BIP. Having twice as many sources for BIP compensates for scaling the mean amplitude of the BIP sources by a factor of \(\frac{1}{2}\), relative to the UNIP case. As a simple example of the UNIP \(|\) BIP \"fixed-total\" constraint (35), consider the BIP PAM4 BiWgt eDAC of Figure 1a, implementing the switch-and-sum operation \(\left\langle I_{c}\right\rangle=\mp\frac{1}{2}\langle I_{1}\rangle\mp\frac{1} {2}\langle I_{0}\rangle\), using four physical current sources having means \(\frac{1}{2}\langle I_{0}\rangle,\frac{1}{2}\langle I_{1}\rangle\), \(-\frac{1}{2}\langle I_{0}\rangle,-\frac{1}{2}\langle I_{1}\rangle\). The sum of the absolute values of the four mean source currents of this BIP-PAM4 BiWgt eDAC equals  
\[\Sigma\Big{\langle}\mathbf{I}^{\text{BiWgt-BIP}+-}\Big{|}\Big{\rangle}=\left| \frac{1}{2}\langle I_{1}\rangle\right|+\left|-\frac{1}{2}\langle I_{1}\rangle \right|+\left|\frac{1}{2}\langle I_{0}\rangle\right|+\left|-\frac{1}{2}\langle I _{0}\rangle\right|=\langle I_{1}\rangle+\langle I_{0}\rangle \tag{38}\]A UNIP-PAM4 eDAC may use just the pair of current sources \(\langle I_{1}\rangle\), \(\langle I_{0}\rangle\), having them combined as per \(\langle I_{c}\rangle=(0|1\rangle\langle I_{0}\rangle+(0|1\rangle\langle I_{1})\). The sum total of all currents here may also yield \(\langle I_{1}\rangle+\langle I_{0}\rangle\)  
\[\Sigma\Big{|}\Big{\langle}\mathbf{I}^{\text{UNIP}}\Big{\rangle}\Big{|}=| \langle I_{0}\rangle|+|\langle I_{1}\rangle|=\langle I_{0}\rangle+\langle I_{ 1}\rangle \tag{39}\]  
Thus, this example validates fixed total current constraint (35). The UNIP I BIP PAM4 example is readily generalized to higher-order BIP PAM constellations utilizing multiple current sources.