## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
_Equivalently, if the elements of a vector_ \(\mathbf{v}=\{v_{1},v_{2},\ldots,v_{\mathrm{C}}\}\) _satisfy_ \[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]3 (178) _then_ \(\mathbf{v}\) _is raised-skew-centrosymmetric, and its center is_ \(\overset{-}{\mathbf{v}}=\frac{1}{2}k\)_._ _Thus, for the vector_ \(\mathbf{v}\) _satisfying the averages (centers) of each of the mirror-image elements_ \(v_{c},v_{c_{\mathrm{TC}}}\) _of_ \(\mathbf{v}\) _for any_ \(c\)_, are all equal to the center (average) of_ \(\mathbf{v}\)_:_ \[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]4 (179)
8. _Given a pair of raised-skew-centrosymmetric C-vectors,_ \(\mathbf{u}\)_,_ \(\mathbf{v}\)_, if their FSRs are equal, i.e.,_ \(\left[u_{1},v_{1}\right]=\left[u_{2},v_{2}\right]\) _(in other words, their LSLs are equal and their LSLs are equal) that implies_ \(\overset{-}{\mathbf{u}}=\mathbf{v}\)_, i.e., the two vectors have the same center.__The difference_ \(\mathbf{u}-\mathbf{v}\) _of two raised-skew-centrosymmetric_ \(C\)_-vectors,_ \(\mathbf{u}\)_,_ \(\mathbf{v}\)_, generally yields a raised-skew-centrosymmetric vector._ _More strongly, the difference_ \(\mathbf{u}-\mathbf{v}\) _is skew-centrosymmetric, provided that_ \(\mathbf{u},\mathbf{v}\) _either have equal centers,_ \(\underline{\mathbf{u}}=\mathbf{v}\)_, or have equal full-scale ranges (FSRs),_ \([u_{1},u_{C}]=[v_{1},v_{C}]\) _(i.e.,_ \(\mathbf{u}\)_,_ \(\mathbf{v}\) _have same LSL,_ \(u_{1}=v_{1}\)_, and same MSL,_ \(u_{C}=v_{C}\)_)_ _The first-difference (steps-vector),_ \(\Delta\mathbf{v}\)_, of a raised-skew-centrosymmetric vector,_ \(\mathbf{v}\)_, always yields a centrosymmetric vector:_ \[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]5 (180) _If a_ \(C\)_-vector_ \(\mathbf{v}\) _is centrosymmetric, an affine transformation of the form_ \[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]6 (181) _yields a centrosymmetric vector_ \(\mathbf{v}^{\mathrm{aff}[a,b]}\)_._ _If the_ \(C\)_-vector_ \(\mathbf{v}\) _is skew-centrosymmetric, the affine transformation of the form (_181_) generates a raised-skew-centrosymmetric vector,_ \(\mathbf{v}\)_._  
**Proof of Corollary 5. ** relegated to Appendix B.1\(\square\)  
As in all areas of physics and engineering, symmetries are useful to enable insight into eDACs as well. We proceed to apply the (skew-)centre-symmetry formalism above to explore new properties and points of view for various eDAC parameters and metrics. **Theorem 3**. : _Centrosymmetries of codewords, output levels and steps of possibly non-uniform eDACs having the ThWgt \(|\) BiWgt eDAC topologies and the LINIP \(|\) BiIP polarities:_  
**(i)**: _For a LINIP ThWgt \(|\) BiWgt eDAC, the_ \(c\)_-th row of_ \(\mathbf{C}_{\mathrm{UNIP}}\) _and its_ \(c_{\tau C}\)_-th row, namely the codewords,_ \(\mathbf{b}_{\mathrm{UNIP}}[c]\) _and_ \(\mathbf{b}_{\mathrm{UNIP}}[c_{\tau C}]\) _at mirror-image indices,_ \(c,c_{\tau C}\) _(with_ \(c_{\tau C}\equiv C+1-c\)_), are in a one's-complement relation, i.e., the bits of_ \(\mathbf{b}_{\mathrm{UNIP}}[c_{\tau C}]\) _are related to the corresponding bits of_ \(\mathbf{b}_{\mathrm{UNIP}}[c]\) _by the substitutions_ \(1\to 0\)_,_ \(0\to 1\)_,_  
\[ \begin{split}&\mathbf{b}_{\mathrm{UNIP}}[c_{\tau C}]\ \overset{\mathrm{ones^{\prime} complement}}{\leftrightarrow}\ \mathbf{b}_{\mathrm{UNIP}}[c]\\ &\Leftrightarrow\{\mathbf{b}_{\mathrm{UNIP}}[c]\ \ \text{XOR}\ \ \mathbf{b}_{\mathrm{UNIP}}[c_{\tau C}]\}=\mathbf{1}_{B}^{\forall} \Leftrightarrow\mathbf{b}_{\mathrm{UNIP}}[c]\ +\mathbf{b}_{\mathrm{UNIP}}[c_{\tau C}]=\mathbf{1}_{S}^{\forall}\end{split}