## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
Note 2: upon evaluating the \"mean-average\", the orders of arithmetic averaging and stochastic mean (expectation) may be commuted, \(\left\langle\overline{\
abla\mathbf{J}}\right\rangle=\left\langle\overline{ \
abla\mathbf{J}}\right\rangle\). In the special case that the mean output current constellation is uniform (e.g., for UNIP! BIP BiWgt! ThWgt eDACs), the definition above for \(\left\langle I_{u}\right\rangle\) coincides with the fixed nearest-neighbors (NN) separation of the given uniform constellation. For BIP eDACs, be they BiWgt! ThWgt; we have  
\[\mathrm{BIP\ BiWgt}\mathrm{ThWgt}:\ \left\langle I_{c}^{\mathrm{BIP}} \right\rangle=(2c-C-1)\left\langle I_{0}^{\mathrm{BIP}}\right\rangle,\ \ \mathrm{for}\ c=1,2,3,\ldots,C \tag{61}\]  
with \(\left\langle I_{0}^{\mathrm{BIP}}\right\rangle\) the least source current. The fixed NN-spacing of this arithmetic sequence is  
\[\mathrm{BIP\ BiWgt}\mathrm{|ThWgt}:\ \left\langle\overline{\
abla J_{c}^{ \mathrm{BIP}}}\right\rangle=2\left\langle I_{0}^{\mathrm{BIP}}\right\rangle \tag{62}\]  
The BIP unary current is twice the least source current for BiWgt! ThWgt BIP eDACs,  
\[\mathrm{BIP\ BiWgt}\mathrm{|ThWgt}:\ \left\langle I_{u}^{\mathrm{BIP}} \right\rangle\equiv\left\langle\overline{\
abla J_{c}}\right\rangle=2\left\langle I _{0}^{\mathrm{BIP}}\right\rangle \tag{63}\]It is useful to also list the expressions for the LSL and MSL and the FS of the BIP eDAC:  
\[ \begin{split}&\text{BIP BiWgt}\big{|}\text{ThWgt}:\big{\langle}I_{ 1|C}^{\text{BIP}}\big{\rangle}=\mp(C-1)\big{\langle}I_{0}^{\text{BIP}}\big{\rangle},\\ &\Big{\langle}\text{FS}^{\text{BIP}}\Big{\rangle}\equiv\langle J_ {C}\rangle-\langle J_{1}\rangle=2(C-1)\big{\langle}I_{0}^{\text{BIP}}\big{\rangle} =2(C-1)\tfrac{1}{2}\big{\langle}I_{u}^{\text{BIP}}\big{\rangle}=(C-1)\big{ \langle}I_{u}^{\text{BIP}}\big{\rangle}\end{split}  \tag{64}\]  
where in the last equality we used  
(65)  
Note: (64) is consistent with \(\big{\langle}I_{u}^{\text{BIP}}\big{\rangle}=\tfrac{1}{C-1}\big{\langle}\text {FS}^{\text{BIP}}\big{\rangle}\), in fact holding for any eDAC. For UNIP BiWgt \(|\text{ThWgt}:\ \Big{\langle}I_{c}^{\text{BIP}}\big{\rangle}=(c-1)\big{\langle}I_{0}^{ \text{BIP}}\big{\rangle}\), for \(c=1,2,3,\ldots,C\)  
The corresponding results for the UNIP eDACs are then  
\[\text{UNIP BiWgt}|\text{ThWgt}:\ \Big{\langle}\overline{\
abla J_{c}^{\text{ UNIP}}}\Big{\rangle}=\Big{\langle}I_{0}^{\text{UNIP}}\Big{\rangle} \tag{66}\]  
The UNIP unary current equals the least source current for BiWgt \(|\text{ThWgt}\) UNIP eDACs,  
\[\text{UNIP BiWgt}\,|\text{ThWgt}:\Big{\langle}I_{u}^{\text{UNIP}}\Big{\rangle} \equiv\Big{\langle}\overline{\
abla J_{c}^{\text{UNIP}}}\Big{\rangle}= \Big{\langle}I_{0}^{\text{UNIP}}\Big{\rangle} \tag{67}\]  
It is useful to also list the expressions for the LSL and MSL and the FS of UNIP eDAC:  
\[ \begin{split}&\text{UNIP BiWgt}\,|\text{ThWgt}:\ \Big{\langle}I_{1|C}^{\text{ UNIP}}\Big{\rangle}=0\ |\ (C-1)\big{\langle}I_{0}^{\text{UNIP}}\big{\rangle}=0\ |\ (C-1)\big{\langle}I_{u}^{\text{ UNIP}}\big{\rangle}\\ &\Big{\langle}\text{FS}^{\text{UNIP}}\Big{\rangle}\equiv\big{ \langle}I_{C}^{\text{UNIP}}\big{\rangle}-\big{\langle}I_{1}^{\text{UNIP}} \big{\rangle}=(C-1)\big{\langle}I_{0}^{\text{UNIP}}\big{\rangle}=(C-1)\big{ \langle}I_{u}^{\text{UNIP}}\big{\rangle}\end{split}