## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
\begin{split}\widehat{\sigma}_{I_{s}}&\equiv \sigma_{I_{s}}/\langle I_{u}\rangle,\ \ \widehat{\sigma}_{\mathbf{I}}\equiv\sigma_{\mathbf{I}}/\langle I_{u}\rangle= \sigma_{\widehat{\mathbf{I}}}\\ \widehat{\sigma}_{I_{c}}&\equiv\sigma_{I_{c}}/ \langle I_{u}\rangle=\sigma_{\widehat{\mathbf{I}}_{c}},\ \widehat{\sigma}_{\mathbf{J}}\equiv\sigma_{\mathbf{J}}/ \langle I_{u}\rangle=\sigma_{\widehat{\mathbf{J}}}\end{split}  \tag{112}\]  
Correspondingly, all squared-current quantities (in particular, variances) are normalized by the squared mean unary current, \(\langle I_{u}\rangle^{2}\):  
\[ \begin{split}\widehat{\sigma}_{I_{s}}^{2}&\equiv \sigma_{I_{s}}^{2}/\langle I_{u}\rangle^{2}=\sigma_{\widehat{I}_{s}}^{2},\ \ \widehat{\sigma}_{\mathbf{I}}^{2}\equiv\sigma_{\mathbf{I}}^{2}/\langle I_{u} \rangle^{2}=\sigma_{\mathbf{I}}^{2}\\ \widehat{\sigma}_{I_{c}}^{2}&\equiv\sigma_{I_{c}}^{2 }/\langle I_{u}\rangle^{2}=\sigma_{\widehat{\mathbf{J}}_{c}}^{2},\ \widehat{\sigma}_{\mathbf{J}}^{2}\equiv\sigma_{\mathbf{J}}^{2}/\langle I_{u} \rangle^{2}=\sigma_{\widehat{\mathbf{J}}}^{2}\end{split}  \tag{113}\]  
It then follows that the normalized variances are proportional to the corresponding normalized means (with \(\sigma_{I_{s}}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle\langle I_{s}\rangle\) (108) used in the fourth equality below):  
\[\widehat{\sigma}_{I_{s}}^{2}=\sigma_{\widehat{I}_{s}}^{2}\equiv\mathrm{var} \{\langle I_{s}\rangle/\langle I_{u}\rangle\}=\frac{\sigma_{I_{s}}^{2}}{ \langle I_{u}\rangle^{2}}=\frac{\sigma_{u}^{2}\langle I_{u}\rangle\langle I_{s }\rangle}{\langle I_{u}\rangle^{2}}=\frac{\sigma_{u}^{2}\langle I_{s}\rangle}{ \langle I_{u}\rangle}=\sigma_{u}^{2}\widehat{I}_{s}^{\langle\rangle} \tag{114}\]  
**Proof. ** Apply Lemma 2 to the eDAC linear model \(\mathbf{J}=\mathbf{CI}\) (1), by substituting \(\mathbf{T}:=\mathbf{C},\ \mathbf{X}:=\mathbf{I},\ \ \mathbf{Y}:=\mathbf{J}\) in (118), yielding the variance vector (119) at the eDAC output.\(\square\)  
Note: Theorem 1 yields the variance vector \(\sigma_{\mathbf{J}^{\text{BIP}|\text{UNIP}}}^{2}\) at the linear eDAC output, \(\mathbf{J}=\mathbf{CI}\). Evidently, the mean output vector is \(\langle\mathbf{J}\rangle=\mathbf{C}\langle\mathbf{I}\rangle\). However, \(\langle\mathbf{J}\rangle,\sigma_{\mathbf{J}}^{2}\) are an incomplete description of the output statistics; for a full description the cross-correlations between the output levels of the eDAC, the covariance matrix, \(\mathbf{K_{J}}\), must also be specified. The components of \(\mathbf{J}\) are generally statistically correlated whenever the rows of the \(\mathbf{C}\) matrix (the codewords) are non-orthogonal, which is actually the case for the \(\mathbf{C}_{\text{BIP}|\text{UNIP}}\) matrices. The full covariance matrix is readily obtained from the source current variances:  
\[\mathbf{K_{J}}=\mathbf{CK_{i}C}^{T}=\mathbf{C}\ \text{diagMat}\Big{\{}\sigma_{I _{S-1}}^{2},\dots,\sigma_{I_{1}}^{2},\sigma_{I_{0}}^{2}\Big{\}}\mathbf{C}^{T} \tag{120}\]  
In fact, Formula (119) of Theorem 1 provides the diagonal of the covariance matrix, \(\mathbf{K_{J}}\) (120), ignoring the off-diagonal elements of \(\mathbf{K_{J}}\). It is to be noted that, although our statistical models account just for output variances (the diagonal terms in (120)) but not for cross-correlations (off-diagonal terms), the variances nevertheless suffice for modeling symbol-by-symbol optical detection scenarios, not accounting for inter-symbol-interference effects. We now apply Theorem 1 to UNIP | BIP eDACs (be they ThWgt| BiWgt). The output variance vector (119) is generated by linearly transforming the input variance vector through a matrix consisting of the squares of the elements of the code matrix, \(\mathbf{C}\). For UNIP eDACs, the elements of \(\mathbf{C}\) are \(0\,1\), still valued \(0\,1\), invariant when squared.