## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
Here, our main interest is in modulated optical sources for coherent detection, requiring BIP eDACs to drive MZMs, for which we shall use the expression \(\sigma_{\
abla\widetilde{I}_{2^{B-1}}}^{2}=2\sigma_{u}^{2}(2^{B}-1)\) for BIP BiWgt eDAC, per the second line in (163). For the unnormalized worst step variance, multiply both sides of (163) by \(\langle I_{u}\rangle^{2}\):  
\[\text{worst step variance for BIP BiWgt}:\ \ \sigma_{\
abla\widetilde{I}_{2^{B-1}}}^{2}=2(2^{B}-1) \sigma_{u}^{2}\langle I_{u}\rangle^{2} \tag{164}\]  
Consistent with the results obtained here, the DNL and DEV eDAC metrics will be shown in Section 3 to peak at the \"mid-code\" step as well, indicative of the most \"vigorous\" perturbation occurring for that transition. In a coherent communication applications context, with relatively high probability, some eDAC devices in the ASIC fab batch have their constellation mid-code step significantly reduced, e.g., for BIP PAM16 (as used in 256 QAM transmission), the eighth and ninth constellation levels (those with means \(\left\langle I_{8|9}^{\text{BiWgt}}\right\rangle=\mp\frac{1}{2}\langle I_{u}\rangle\)) tend to crowd together the most, i.e., the separation \(|I_{9}^{\text{BiWgt}}-I_{8}^{\text{BiWgt}}|\) tends to be reduced for a larger fraction of the eDAC devices out of the batch (reducing the yield). Therefore, Rx decision errors mistaking the eighth level for the ninth one (or vice-versa) tend to dominate the symbol and bit error rates (as is going to be treated in part II of this paper). The next largest variance contributions are attributed to the transitions at a quarter (fourth to fifth) and three-quarters (twelfth to thirteenth) along the code (the corresponding step variances are \(14\sigma_{u}^{2}\)). Next in line, there are steps contributing 6 units of variance. The least steps perturbation is 2 units of variance. All these perturbations are reflected in the EVM of the constellation (refer to Sections 3.8 and 3.9 for treatment of the EVM of eDACs). This discussion indicates that the second-order statistics of the constellation steps emerges as a significant factor affecting communication link performance, to be explored below. The analysis above for the step variances of BiWgt eDACs is readily adapted to ThWgt eDACs, albeit with normalized source current \(\widetilde{\mathbf{I}}^{(\cdot)\text{ThWgt}}=\underbrace{\{1,1,\ldots,1\}}_ {S=\mathbb{C}-1}=\mathbf{1}_{C-1}^{\forall}\) (79), yielding \(\mathbf{X}^{T}[c]\big{\{}1,1,\ldots,1\big{\}}=\mathbf{X}^{T}[c]\mathbf{1}_{C- 1}^{\forall}=\Sigma\mathbf{X}[c]\). Moreover, the XORs now yield sparse C-1-vectors, each of which with a lone-one element. The sum of bits, \(\Sigma\mathbf{X}[c]\), for each sparse XOR vector with a single one element, is simply unity, \(\Sigma\mathbf{X}[c]=1\), yielding,  
\[\widetilde{\sigma}_{\
abla\widetilde{I}_{c}}^{2}/\sigma_{u}^{2}=\left\{ \begin{array}{ll}1\ \ \text{for UNIP ThWgt}\\ 2\ \ \text{for BIP ThWgt}\end{array} \right. \tag{165}\]  
yielding \(\widetilde{\sigma}_{\
abla\widetilde{I}_{c}^{\text{LNIP}}}^{2}=\sigma_{u}^{2}\), \(\
abla\widetilde{I}_{c}^{\text{BP}}=2\sigma_{u}^{2}\) for UNIP ThWgt eDACs, the minimal possible step variance. It is well-known by eDAC practitioners that nothing surpasses the low DNL of a ThWgt eDAC, if one can afford one. Unfortunately, ThWgt designs incur the energy inefficiency of the power-hungry digital encoder required to map bits to codewords at the full baud rate. For next-gen optical communication, the exorbitant power consumption of the digital encoders of ThWgt eDACs precludes having them adopted as optical modulator drivers. This motivates us to focus on BiWgt eDACs in the following work. Moreover, since our main interest is in coherent optical detection, calling for BIP constellations for the QAM tributaries, we mainly focus on BIP PAM-C constellation generation with BiWgt eDACs.