## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
: _Any \(C\)-vector, \(\mathbf{v}\), is expressible as the sum of a balanced vector and a constant vector:_  
\[\mathbf{v}=\mathbf{b}+t\mathbf{\Omega}_{C}^{\vee},\ \text{with}\ \overline{\mathbf{b}}=0. \tag{172}\]  
_Moreover, \(t\) is unique, given by the average of \(\mathbf{v}\), \(t=\overset{\leftarrow}{\mathbf{v}}\), i.e., \(\mathbf{v}=\mathbf{b}+\overset{\leftarrow}{\mathbf{v}}\mathbf{\Omega}_{C}^{\vee}\), and so is the \"balanced component\" \(\mathbf{b}\) unique for any given \(\mathbf{v}\); hence, it may be denoted by \(\mathbf{b}=\mathbf{v}^{\text{bal}}\).__The unique decomposition into additive balanced and constant components is then written in the form_  
\[\mathbf{v}=\mathbf{v}^{\mathrm{bal}}+\overset{-}{\mathbf{v}}\mathbf{1}_{\mathrm{ C}}^{\forall} \tag{173}\]  
_Conversely, any C-vector, \(\mathbf{v}\), may be translated by \(-\mathbf{v}\mathbf{1}_{\mathrm{C}}^{\forall}\) to yield a balanced vector,_  
\[\mathbf{v}^{\mathrm{bal}}\equiv\mathbf{v}-\overset{-}{\mathbf{v}}\mathbf{1}_ {\mathrm{C}}^{\forall} \tag{174}\]  
_uniquely associated with \(\mathbf{v}\), referred to as the \"balancing of \(\mathbf{v}\) \"(also known as the \"centering of \(\mathbf{v}\)\")._  
_If_ \(\mathbf{v}\equiv\mathbf{v}^{\mathrm{bal}}\Leftrightarrow\overset{-}{\mathbf{ v}}=0\) _then_ \(\mathbf{v}\) _is a balanced vector to begin with._  
**Definition 9**. : _A vector \(\mathbf{v}\) is referred to as \"raised-skew-centrosymmetric\", if it may be expressed as a non-zero translation, by a constant vector \(t\mathbf{1}_{\mathrm{C}}^{\forall}\), of a skew-centrosymmetric vector:_  
\[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]0  
**Corollary 5**. : _Some properties of (skew)centrosymmetries._  
1. _Any skew-centrosymmetric vector is balanced._
2. _Any raised-skew centrosymmetric vector is imbalanced._
3. _Any raised-skew centrosymmetric vector may be translated to yield a skew-centrosymmetric vector._
4. _Not all balanced vectors are skew-centrosymmetric._
5. _The elements of a raised-skew-centrosymmetric vector_ \(\mathbf{v}=\{v_{1},v_{2},\ldots,v_{\mathrm{C}}\}\) _satisfy_ \[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]1 (176) _i.e., the centers of all pairs of mirror-image elements coincide with one another and with the center of the given vector. Equivalently, the balanced elements_ \(v_{c}^{\mathrm{bal}}\equiv v_{c}-\overline{\mathbf{v}}\) _satisfy_ \[\mathbf{v}\mapsto\mathbf{v}^{\dagger}\,,\,\text{s.t.}\,\left[\mathbf{v}^{ \dagger}\right]_{c}=\left[\mathbf{v}\right]_{C+1-c} \tag{166}\]2 (177) _Conversely, if the elements of a vector_ \(\mathbf{v}=\{v_{1},v_{2},\ldots,v_{\mathrm{C}}\}\) _satisfy either (_176_) or (_177_), then_ \(\mathbf{v}\) _must be raised-skew-centrosymmetric._
6. _Given a C-vector,_ \(\mathbf{v}\)_, if there exists a translation_ \(\mathbf{v}-t\mathbf{1}_{\mathrm{C}}^{\forall}\) _by a constant vector_ \(-\mathbf{t}=-t\mathbf{1}_{\mathrm{C}}^{\forall}\) _such that_ \(\mathbf{v}-t\mathbf{1}_{\mathrm{C}}^{\forall}\) _is skew-centrosymmetric, then necessarily_ \(t=\overline{\mathbf{v}}\)_; thus, the only translation, by a constant vector, of the given vector into a centrosymmetric one is the translation by the constant vector_ \(-\mathbf{v}\mathbf{1}_{\mathrm{C}}^{\forall}\)_._
7.