## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
 \tag{182}\]  
_where_ \(\mathbf{1}_{S}^{\forall}\) _is the all-ones vector (for ThWgt:_ \(S=B=\log_{2}C\)_; for BiWgt:_ \(S=C-1\)_):_  
\[\mathbf{1}_{S}^{\forall}\equiv\underbrace{\left\{1,1,...1\right\}}_{S\ \text{ones}}=(S-1)_{\text{base2}}=\mathbf{b}_{\mathrm{UNIP}|\text{BIP}}[C] \tag{183}\]  
_The second line in (_182_), equivalent to the one's complement relation between the mirror-image codewords._
**(ii)**: _For a BIP ThWgt \(|\) BiWgt eDAC, the two codewords_ \(\mathbf{b}_{\mathrm{BIP}}[c]\) _and_ \(\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]\)_, at mirror indices, are in a zero's-complement relation, i.e., the two codewords sum up to zero; the \"bipolar bits\" of_ \(\mathbf{b}_{\mathrm{UNIP}}[c_{\tau C}]\) _are obtained from those of_ \(\mathbf{b}_{\mathrm{UNIP}}[c]\) _by the substitutions_ \(1\to-1\)_,_ \(-1\to 1\)_. Equivalently, the two mirror-image BIP codewords are antipodal:_  
\[\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]\ \overset{\mathrm{zero^{\prime} complement}}{\leftrightarrow}\ \mathbf{b}_{\mathrm{UNIP}}[c]\ \Leftrightarrow\ \mathbf{b}_{\mathrm{BIP}}[c]\ +\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]=\mathbf{0}\ \Leftrightarrow\ \mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]=-\mathbf{b}_{\mathrm{BIP}}[c] \tag{184}\]
**(iii)**: _For either eDAC topology \(|\) polarity (ThWgt \(|\) BiWgt, LINIP \(|\) BiIP), the sequence of code-word-steps,_ \(\{\
abla\mathbf{b}[c]\}_{c=1}^{c-1}\) _satisfies_  
\[\
abla\mathbf{b}[c_{\tau(C-1)}]=\
abla\mathbf{b}[c]\Leftrightarrow\
abla\mathbf{ b}[C-c]=\
abla\mathbf{b}[c]\ \text{with}\ c_{\tau(C-1)}=\left.C+1-c\right|_{C:=C-1}=C-c \tag{185}\]  
**(iv)**: _The C-levels \(\{j_{c}\}_{c=1}^{\text{C}}\) of the eDAC output constellation, \(\mathbf{J}=\mathbf{CI}\), satisfy, irrespective of the source current vector \(\mathbf{I}=\{I_{s}\}_{s=S-1}^{0}\), the following complementarity properties. For UNIP eDACs (be they BiWgt \(\dagger\) ThWgt):_  
\[J_{c}^{\text{UNIP}}+J_{c_{\tau\text{C}}}^{\text{UNIP}}=I_{\text{tot}}\Leftrightarrow J_{c_{\tau\text{C}}}^{\text{ UNIP}}-\frac{1}{2}I_{\text{tot}}=\frac{1}{2}I_{\text{tot}}-J_{c}^{\text{ UNIP}} \tag{186}\]  
_i.e., the eDAC output current levels \(J_{c}\), \(J_{c_{\tau\text{C}}}\) at mirror-image indices complement each other to the total source current,_  
\[I_{\text{tot}}=\mathbf{\Upsilon}_{S}^{\text{y}}\cdot\mathbf{I}=\sum_{s=0}^{S- 1}I_{s} \tag{187}\]  
_Any UNIP eDAC output current constellation, \(\mathbf{J}^{\text{UNIP}}\), is always raised-skew-centrosymmetric, with center given by half the total current,_  
\[\mathbf{J}^{-\text{UNIP}}=\frac{1}{2}I_{\text{tot}} \tag{188}\]  
_For BIP eDACs (BiWgt \(\dagger\) ThWgt), the output current constellations are skew-centrosymmetric with zero center, i.e., are antisymmetric,_  
\[\mathbf{J}^{-\text{BIP}}=0,\hskip 28.452756 ptJ_{c}^{\text{BIP}}+J_{c_{ \tau\text{C}}}^{\text{BIP}}=0\Leftrightarrow J_{c_{\tau\text{C}}}^{\text{BIP} }=-J_{c}^{\text{BIP}} \tag{189}\]  
\[J_{1}^{\text{BIP}}=-J_{\text{C}}^{\text{BIP}},J_{2}^{\text{BIP}}=-J_{C-1}^{ \text{BIP}},\ldots,J_{c}^{\text{BIP}}=-J_{C+1-c^{*}}^{\text{BIP}},\ldots,J_{C}^ {\text{BIP}}=-J_{1}^{\text{BIP}}\] (v)  
_For either eDAC topology/polarity, and for any source current, \(\mathbf{I}\), the steps vector \(\
abla\mathbf{J}\) is always centrosymmetric, i.e., steps paired to have mirror-image indices are equal:_  
\[\
abla J_{c}=\
abla J_{c_{\tau(C-1)}}\ \Leftrightarrow\ \ \
abla J_{c}=\
abla J_{C-c}\ \ \Leftrightarrow\ \ \
abla\mathbf{J}=\left(\
abla\mathbf{J}\right)^{\text{T}}\ \Leftrightarrow\ \ \
abla\mathbf{J}\text{ is centrosymmetric} \tag{190}\]  
_Note: For a C-level eDAC, the steps vector \(\
abla\mathbf{J}\) is a length-C-1 sequence. Its \"center\" is now \(\frac{1}{2}[(\text{C}-1)+1]=\frac{1}{2}C\) rather that \(\frac{1}{2}(C+1)\) for the original C-levels constellation. The definition of image-index \(c_{\tau\text{C}}\) (167) now changes to \(c_{\tau(C-1)}=C-c\) as specified in (185)._  
**Proof of Theorem 3. **_The derivation is somewhat long; hence, it is relegated to Appendix B.