## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
Note that, for the BIP eDAC polarity of interest, all code-steps (elements of the \(\
abla\textbf{C}_{\text{BIP}}\) matrix) are tri-level-valued, \(0|2|-2\) (since for BIP eDACs, the elements of the **C** matrix are \(\pm 1\); hence, \((\pm 1)-(\pm 1)=0\,|\pm 2\)), whereas for UNIP polarity the elements of the **C** matrix are \(0\,|1\); thus, the UNIP code-steps (elements of \(\
abla\textbf{C}_{\text{UNIP}}\)) matrix are obtained as \((0\,|1)-(0\,|1)=0\,|\pm 1\)); thus, assume the values \(0|1|-1\). For the purpose of steps variance evaluation, we require the Shur-square of \(\
abla\textbf{C}\) or equivalently the Shur-squares of its rows, \(\Big{(}\
abla\textbf{b}^{T}[c]\Big{)}^{\odot 2}\), readily obtained by XORing the successive eDAC codewords of the eDAC code \(\Big{\{}\textbf{b}^{T}[c]\Big{\}}_{c=1}^{C}\). For the next lemma, we define the XOR of a pair of Boolean vectors, i.e., vectors with elements restricted to one of a pair of values (in our case of interest, the two possible values being \(0\,|1\) for UNIP and \(\pm 1\) for BIP):  
**Definition 5**. : _For a pair of length-\(C\) Boolean vectors, \(\mathbf{b}^{(i)}\equiv\left\{b_{1}^{(i)},b_{2}^{(i)},\ldots,b_{S}^{(i)}\right\},\ i=1,2\), with elements \(b_{s}^{(i)}\in\left\{\beta_{0},\beta_{1}\right\}\) with \(\beta_{0}\
eq\beta_{1}\) (e.g., \(\left\{\beta_{0},\beta_{1}\right\}=\left\{0,1\right\}\ \shortmid\left\{\beta_{0},\beta_{1}\right\}=\left\{-1,1\right\}\) for UNIP \(\mid\) BIP) the XOR of the vectors, \(\mathbf{b}^{(1)},\mathbf{b}^{(2)}\), is defined as the binary-valued \(S\)-vector_  
\[\mathbf{X}\equiv\mathbf{b}^{(1)}\text{XOR}\ \mathbf{b}^{(2)}=\left\{X_{1}, X_{2},\ldots,X_{S}\right\}, \tag{146}\] \[\text{with}\ \ X_{s}\equiv\left\{ \begin{array}{ll}\mathbf{1} \text{ if }b_{s}^{(1)}\
eq\ b_{s}^{(2)}\\ \mathbf{0}\text{ if }b_{s}^{(1)}=\ b_{s}^{(2)}\end{array} \right.\ \equiv b_{s}^{(1)}\text{XOR}\ b_{s}^{(2)},\ \ \forall_{s\in\left\{1,2,\ldots,S\right\}}\]  
_thus, the XOR of two Boolean vectors is defined elementwise and in turn the XOR of two bits, \(b_{s}^{(1)}\text{XOR}\ b_{s}^{(2)}\), is defined as \(1\) if \(b_{s}^{(1)}\
eq b_{s}^{(2)}\), \(0\) if \(b_{s}^{(1)}=b_{s}^{(2)}\) a definition valid even for BIP, in which case \(b_{s}^{(1)},b_{s}^{(2)}\) are not binary-valued but rather \(\pm 1\)-valued._  
**Lemma 3**. : _Shur-squares of the code-steps, expressed as XOR of successive codewords:_  
\[\left(\
abla\mathbf{b}^{T}[c]\right)^{\odot 2}=\left(\mathbf{b}^{T}[c+1]- \mathbf{b}^{T}[c]\right)^{\odot 2}=\left\{ \begin{array}{ll}\mathbf{X}[c]\equiv \mathbf{b}^{T}[c]\text{ XOR}\ \mathbf{b}^{T}[c+1]&\text{for UNIP}\\ \mathbf{4X}[c]=4\Big{\{}\mathbf{b}^{T}[c]\text{ XOR}\ \mathbf{b}^{T}[c+1] \Big{\}}\text{ for BIP}\end{array} \right. \tag{147}\]  
_Thus, the elements of \(\left(\
abla\mathbf{b}^{T}[c]\right)^{\odot 2}\) are \(01\,1\) valued for UNIP and \(01\,4\) valued for BIP._  
For the purpose of evaluating eDAC statistical metrics such as the DNL and our own DEV metric (introduced in Section 3), the variance of the steps vector, \(\
abla\mathbf{J}\), is to be evaluated. Since \(\
abla\mathbf{J}\) is obtained from the vector of independent current sources, \(\mathbf{I}\), by the linear transformation \(\
abla\mathbf{J}=\left(\
abla\mathbf{C}\right)\mathbf{I}\) (140), the variance propagation rule \(\sigma_{\mathbf{TX}}^{2}=\mathbf{T}^{\odot 2}\sigma_{\mathbf{X}}^{2}\) (118) is applicable upon plugging \(\mathbf{T}:=\
abla\mathbf{C},\ \mathbf{X}:=\mathbf{I},\ \ \mathbf{TX}:=\
abla\mathbf{J}\), yielding \(\sigma_{\
abla\mathbf{J}}^{2}\)\(=\left(\
abla\mathbf{C}\right)^{\odot 2}\sigma_{\mathbf{I}}^{2}\). **Theorem 2**.