## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)  
### Matrix-Based Modeling of Current-Steering eDACs: From Current Sources to Output Levels  
The two types of DACs of interest to us (be they eDAC or oDAC) may be described by matrix techniques [13], as partially introduced in [14] for oDACs modeling, ported and adapted here for the first time to eDAC modeling, in preparation of developing an all-analytic approach suitable for statistical modeling of all eDAC types. In this work, we assume current-steering eDACs (most suitable for high-speed communication applications) of the UNIP and BIP type, with more attention on BIP eDACs as used in coherent (optical) communication applications, albeit without neglecting to address UNIP eDACs as well. Note: BIP eDACs, such as those used for generating I or Q tributaries in QAM transmission, generate symmetric constellations, having bipolar antipodal values, around the origin of the output current or voltage domain axis. Fundamentally, random fabrication and material imperfections dictate that the electrical current sources, realized as MOS transistors, in current-steering eDAC ASICs cannot be fabricated to precisely operate at their nominal designed values, but are subject to inevitable statistical variations. The elementary current sources are modeled as independently drawn from a statistical distribution (typically a normal distribution). The relative deviations of the current values of the independent elementary sources relative to a nominal expected common value are referred to as \"current mismatch errors\", or in short \"mismatch\". In fact, mismatch is a fundamental, inevitable eDAC impairment, statistically modeled here using analytic methods. In this paper, all eDAC currents are viewed as random variables perturbed around their nominal values (typically their means) due to the electronic mismatch errors. Other types of eDAC errors such as systematic errors are not addressed in this paper, with the exception of bias (static) errors. Our probabilistic mismatch model then pertains to an ensemble of eDAC chips, exhibiting fundamentally stochastic variations in their physical parameters, chip-to-chip. We aim to develop analytical tools enabling us to assess the quality of the output constellations of electronic or optical transmitters (e.g., based on MZM modulators or on compound oDACs). The communication links of interest use higher-order transmission constellations (e.g., QAM); hence, the Tx-s are to be electronically driven by eDACs. Electronic mismatch errors inherently affecting the eDAC drivers of the (electronic or optical) Tx-s are the key transmitter impairments modeled in this paper. Fortunately, eDACs of the BiWgt and ThWgt type are linear devices, enabling a streamlined matrix-based model developed in the sequel. Consider a stochastic ensemble of nominally identical current-steering eDACs. Each eDAC comprises \(S\) current sources, generating currents \(I_{s}\), for \(s=S-1,S-2,\ldots,3,2,1,0\), with \(S\) the number of current sources. The source currents, \(I_{s}\), are RVs, collected into a random vector \(\mathbf{I}\equiv\{I_{S-1},I_{S-2},\ldots,I_{1},I_{0}\}\). Note the unusual reverse indexing for the elements of \(\mathbf{I}\) from \(S-1\) down to \(0\). Moreover, the source currents means are assumed to be positive means, \(\langle I_{s}\rangle>0\) even for bipolar eDACs, since antipodal negative sources are assumed to be effectively generated by switching circuits, modeled by means of a switching matrix, also known as code matrix, \(\mathbf{C}\), of size \(C\times S\), with elements all \(\pm 1\) for BIP eDACs, \(011\) for UNIP eDACs:  
\[\mathbf{J}=\mathbf{C}\mathbf{I} \tag{1}\]  
where \(\mathbf{J}=\{I_{1},I_{2},\ldots,I_{C}\}\) (a column vector) is the output \(C\)-levels current constellation generated by the eDAC.