## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
: _Variance vector of the steps at the current output \(\mathbf{J}=\mathbf{CI}\) of a linear UNIP \(\mid\) BIP eDAC, of the BiNgt \(\mid\) ThWgt type:_  
\[\sigma_{\
abla\mathbf{I}_{\text{UNIP}\mid\text{BIP}}}^{2}=\left( \
abla\mathbf{C}_{\text{UNIP}\mid\text{BIP}}\right)^{\odot 2}\sigma_{\mathbf{I}_{ \text{UNIP}\mid\text{BIP}}}^{2}=\left\{\
abla\mathbf{C}_{\text{UNIP}\mid\text{ BIP}}\right\}^{\odot 2}\overset{\left\langle\right\rangle}{\mathbf{I}}_{\text{UNIP}\mid \text{BIP}}^{\left\langle\right\rangle}\sigma_{u}^{2}, \tag{148}\] \[\therefore\sigma_{\
abla\overset{\left\langle\right\rangle}{\mathbf{ I}}_{\text{UNIP}\mid\text{BIP}}}^{2}/\sigma_{u}^{2}=\left\{\
abla\mathbf{C}_{ \text{UNIP}\mid\text{BIP}}\right\}^{\odot 2}\overset{\left\langle\right\rangle}{ \mathbf{I}}_{\text{UNIP}\mid\text{BIP}}^{\left\langle\right\rangle}\]  
_the variances of the eDAC output steps are expressed component-wise as_  
\[\sigma_{\
abla\overset{\left\langle\right\rangle}{\mathbf{I}}_{c}}^{2} /\sigma_{u}^{2}=\ \overset{\left\langle\right\rangle}{\mathbf{\sigma}}_{\
abla\overset{ \left\langle\right\rangle}{\mathbf{I}}_{c}}^{2}/\sigma_{u}^{2}=\left[\
abla \mathbf{b}_{\text{UNIP}\mid\text{BIP}}^{\left\langle\right\rangle}[c]\right]^ {\odot 2}\overset{\left\langle\right\rangle}{\mathbf{I}}_{\text{UNIP}\mid\text{BIP}}^{ \left\langle\right\rangle}\\ =\left\{\begin{array}{ll}\left\{\mathbf{  
\[\left\langle\widetilde{\mathbf{I}}_{\text{UNIP}}\right\rangle=\left\langle \widetilde{\mathbf{I}}\right\rangle=\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\},\quad\left\langle\widetilde{\mathbf{I}}_{\text{BIP}}\right\rangle=\frac{1}{2 }\left\langle\widetilde{\mathbf{I}}\right\rangle=\frac{1}{2}\left\{2^{B-1},2 ^{B-2},\ldots,4,2,1\right\}\] (150) _and the c-th XOR vector was defined as c-th row of the steps code matrix \(\mathbf{\
abla C}\)_: \[\mathbf{X}_{\text{UNIP}|\text{BIP}}^{T}[c]\equiv\mathbf{b}_{\text{UNIP}| \text{BIP}}^{T}[c]\text{ XOR }\mathbf{b}_{\text{UNIP}|\text{BIP}}^{T}[c+1]\ = \begin{cases}\left[(\mathbf{\
abla C}_{\text{UNIP}})^{ \odot 2}\right]_{c_{i}}\text{ for UNIP}\\ \left[\frac{1}{4}(\mathbf{\
abla C}_{\text{BIP}})^{\odot 2}\right]_{c_{i}}\text{ for BIP}\end{cases} \] (151) Specializing now to the BiWgt eDAC topology (for the UNIP \(|\) BIP cases, respectively), \[\sigma_{\
abla\widetilde{\gamma}_{c}}^{2}\ /\sigma_{u}^{2}=\ \widetilde{\sigma}_{\
abla \widetilde{\gamma}_{c}}^{2}/\sigma_{u}^{2}=\left\{ \begin{array}{ll}\mathbf{X }^{\text{BiWgt}}[c]\cdot\left\langle\widetilde{\mathbf{I}}_{\text{UNIP}}^{ \text{BiWgt}}\right\rangle&\text{for UNIP}\\ \\ 4\mathbf{X}^{\text{BiWgt}}[c]\cdot\left\langle\widetilde{\mathbf{I}}_{\text{ BIP}}^{\text{BiWgt}}\right\rangle&\text{for BIP}\end{array} \right.\] (152) \[=\left\{ \begin{array}{ll}\mathbf{X}^{\text{BiWgt}}[c]\cdot \left\langle\widetilde{\mathbf{I}}^{\text{BiWgt}}\right\rangle&\text{for UNIP}\\ \\ 2\mathbf{X}^{\text{BiWgt}}[c]\cdot\left\langle\widetilde{\mathbf{I}}^{\text{ BiWgt}}\right\rangle&\text{for BIP}\end{array} \right.=\left\{ \begin{array}{ll}\mathbf{X}^{\text{BiWgt}}[c] \cdot\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\}&\text{for UNIP BiWgt}\\ \\ 2\mathbf{X}^{\text{BiWgt}}[c]\cdot\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\}& \text{for BIP BiWgt}\end{array} \right.\] The vector form of (152) consists of matrix-times-column-vector multiplications, \[\frac{1}{\sigma_{u}^{2}\langle\lambda_{u}\rangle^{2}}\sigma_{\text{VI}}^{2}= \left\{ \begin{array}{ll}\mathbf{X}_{\text{BiWgt}}^{\text{UNIP}}\{2^{B-1},2 ^{B-2},\ldots,4,2,1\}&\text{for UNIP BiWgt}\\ \\ 2\mathbf{X}_{\text{BiWgt}}^{\text{BIP}}\{2^{B-1},2^{B-2},\ldots,4,2,1\}&\text{ for BIP BiWgt}\end{array} \right.\] (153) where we introduced a \"steps-XOR matrix\" \(\mathbf{X}\), having \(\mathbf{X}^{T}[c]\) (151) in its c-th row: \[\mathbf{X}_{\text{BiWgt}}^{\text{UNIP}|\text{BIP}}= \begin{cases}\left(\mathbf{ \
abla C}_{\text{UNIP}}^{\text{BiWgt}}\right)^{\odot 2}\text{for UNIP}\\ \\ \frac{1}{4}\left(\mathbf{\
abla C}_{\text{BIP}}^{\text{BiWgt}}\right)^{\odot 2} \text{for BIP}\end{cases}