## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
\(\square\)_  
Note 1: the symmetry properties in the theorem above assume neither that the ThWgt sources are all equal, nor that the BiWgt sources are dyadic, i.e., in ratios 1:2:4;,... The theorem stems from the symmetry of the codewords, albeit requiring no particular symmetry of the current sources array. Definitions (169), (170) of (skew-)centrosymmetry of a vector in \(\mathbb{R}^{\text{C}}\) may be generalized to a sequence of vectors, i.e., an indexed set, of vectors, \(\mathbf{V}\equiv\{\mathbf{v}[m]\}_{m=1}^{M},\ \ \mathbf{v}[m]\in\mathbb{R}^{\text{C}}\):  
\[\mathbf{V}\text{ is centrosymmetric}\Leftrightarrow\mathbf{v}[m_{tM}]= \mathbf{v}[m],\ \ \text{with}\ m_{tM}\equiv M+1-m \tag{191}\] \[\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]\ \overset{\mathrm{zero^{\prime} complement}}{\leftrightarrow}\ \mathbf{b}_{\mathrm{UNIP}}[c]\ \Leftrightarrow\ \mathbf{b}_{\mathrm{BIP}}[c]\ +\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]=\mathbf{0}\ \Leftrightarrow\ \mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]=-\mathbf{b}_{\mathrm{BIP}}[c] \tag{184}\]0  
The vectors \(\mathbf{v}[m]\) need not be (skew-)centrosymmetric themselves. Per this extended definition of (skew)centrosymmetry, the property (185) in Theorem 3, may be compactly expressed as the following equivalent statement:  
The sequence of codeword-steps \(\{\
abla\mathbf{b}[c]\}_{c=1}^{C-1}\) is centrosymmetric: \(\
abla\mathbf{b}[c_{\tau(C-1)}]=\
abla\mathbf{b}[c]\). We now exemplify Theorem 3 for a BiWgt UNIP 16-PAM oDAC (\(C=8\), \(B=3\)), the code matrix \(\mathbf{C}_{\text{UNIP}}^{\text{BiWgt}}\) of which was stated in (14). Abbreviating the one's-complement as \({}^{1\text{C}}\), the key property in **(i)** is readily verified,  
\[\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]\ \overset{\mathrm{zero^{\prime} complement}}{\leftrightarrow}\ \mathbf{b}_{\mathrm{UNIP}}[c]\ \Leftrightarrow\ \mathbf{b}_{\mathrm{BIP}}[c]\ +\mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]=\mathbf{0}\ \Leftrightarrow\ \mathbf{b}_{\mathrm{BIP}}[c_{\tau C}]=-\mathbf{b}_{\mathrm{BIP}}[c] \tag{184}\]1The positions of the 1\({}^{\prime}\)c-pairs are seen as symmetric around the center of the sequence. Pairs with indices that are mirror-images around the center are one's-complementary. As for **(iii)**, in the case of a UNIP BiWgt eDAC, let us evaluate the codeword-steps at indices \(c=1\), \(2\), \(3\), and at the corresponding mirror-image indices \(7=1_{17},\ 6=2_{17},\ 5=3_{17}\):  
\[ \begin{split}\
abla\mathbf{b}[1]&=(0\ 0\ 1)-(0\ 0\ 0)=(1\ 1\ 1)-(1\ 1\ 0)=\
abla\mathbf{b}[7]\\ \
abla\mathbf{b}[2]&=(0\ 1\ 0)-(0\ 0\ 1)=(1\ 1\ 0)-(1\ 0\ 1)= \
abla\mathbf{b}[6]\\ \
abla\mathbf{b}[3]&=(0\ 1\ 1)-(0\ 1\ 0)=(1\ 0\ 1)-(1\ 0\ 0)= \
abla\mathbf{b}[5]\end{split}  \tag{193}\]  
where all six vector differences in the three equalities are seen to be equal \((0\ 0\ 1)\). To validate the complementarity of currents **(iv)** for the UNIP BiWgt eDAC:  
\[ \begin{split}\mathbf{b}[1]+\mathbf{b}[8]&=\mathbf{ 1}_{3}^{\forall}\ \therefore(\mathbf{b}[1]+\mathbf{b}[8])\cdot\mathbf{I}=\mathbf{1}_{3}^{\forall} \cdot\mathbf{I}\ \therefore\mathbf{b}[1]\cdot\mathbf{I}+\mathbf{b}[8]\cdot\mathbf{I}= \mathbf{1}_{3}^{\forall}\cdot\mathbf{I}\ \ \therefore I_{1}+J_{8}=I_{\text{tot}}\\ \mathbf{b}[2]+\mathbf{b}[7]&=\mathbf{1}_{3}^{ \forall}\therefore(\mathbf{b}[2]+\mathbf{b}[7])\cdot\mathbf{I}=\mathbf{1}_{3}^{ \forall}\cdot\mathbf{I}\therefore\mathbf{b}[2]\cdot\mathbf{I}+\mathbf{b}[7] \cdot\mathbf{I}=\mathbf{1}_{3}^{\forall}\cdot\mathbf{I}\therefore J_{2}+J_{7}=I_ {\text{tot}}\\ \mathbf{b}[3]+\mathbf{b}[6]&=\mathbf{1}_{3}^{\forall} \therefore(\mathbf{b}[3]+\mathbf{b}[6])\cdot\mathbf{I}=\mathbf{1}_{3}^{\forall} \cdot\mathbf{I}\ \therefore\mathbf{b}[3]\cdot\mathbf{I}+\mathbf{b}[6]\cdot\mathbf{I}= \mathbf{1}_{3}^{\forall}\cdot\mathbf{I}\ \therefore J_{3}+J_{6}=I_{\text{tot}}\end{split} \]  
As for **(v)**, the centrosymmetry of the steps, we expect,  
\[\
abla J[1]=\
abla J[7],\quad\
abla J[2]=\
abla J[6],\quad\quad\
abla J[3]= \
abla J[5] \tag{194}\]  
readily validated using \(\
abla J[c]=\
abla\mathbf{b}[c]\cdot\mathbf{I}\) for c = 1, 2, 3, as well as (193):  
\[