## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Modeling of Statistical Properties of the INL \(|\) DNL Metrics for (un)Biased eDACs  
The INL \(|\) DNL vectors are typically inferred for each measured eDAC experimental realization, but it is useful to consider a large ensemble of eDACs subject to the mismatch statistics. For randomly perturbed constellation levels, the INL \(|\) DNL vectors are evidently random; thus, it is useful to evaluate their first- and second-order mismatch statistics. It turns out that the statistics of eDAC constellations may be compactly described by the probability distributions of the **INL** and **DNL** vectors. We first address features common to the statistical descriptions of both INL and DNL metrics. To abbreviate notation, we pair up structurally identical statements pertaining to both INL and DNL, introducing the *NL metric notation to stand for either of INL \(|\) DNL. A concise second-order statistical description of both *NL metrics consists of the mean and variance vectors (the sequences of means and variances of the \(*\mathrm{NL}_{\mathbf{J}}[c]\) samples):  
\[\left\langle*\mathbf{NL}_{\mathbf{J}}\right\rangle=\left\{\left\langle* \mathrm{NL}_{\mathbf{J}}[c]\right\rangle\right\}_{c=1}^{C-1},\quad\sigma_{* \mathbf{NL}_{\mathbf{J}}}^{2}\equiv\left\{\sigma_{*\mathrm{NL}_{\mathbf{J}}[ c]}^{2}\right\}_{c=1}^{C-1}. \tag{311}\]  
Equivalently, the second-order statistics of the *NL metrics is specified by the mean and RMS values of the *NL samples: \(\left\langle*\mathbf{NL}_{\mathbf{J}}\right\rangle,\sqrt{\left\langle\left|* \mathbf{NL}_{\mathbf{J}}\right|^{2}\right\rangle}\). Such alternative description is equivalent to (311), since from elementary probability \(\left\langle\left|*\mathbf{NL}_{\mathbf{J}}\right|^{2}\right\rangle=\left\langle *\mathbf{NL}_{\mathbf{J}}\right\rangle^{2}+\sigma_{*\mathbf{NL}_{\mathbf{J}}} ^{2}\). Generally, INL \(|\) DNL vectors are viewed as random ones; however, they become deterministic whenever the eDAC output constellation is itself deterministic. Examples for which the *NL is deterministic:* Experimentally measured stochastic realizations of the random constellation vector, for one particular eDAC at a time. * Deterministic nominal uniform eDAC constellations driving memoryless nonlinearities such as an MZM. The MZM output constellation is also deterministic but non-uniform. For deterministic constellations, the value of \(\max\{\mathbf{INL}\,|\,\mathbf{DNL}\}\) (most positive sample of \(\star\)NL) and the value of \(\min\{\mathbf{DNL}\}\) (most negative sample) are often stated as concise \(\star\)NL-derived figures of merit. Alternatively, the max of the absolute values of the INL samples is specified, an important specification making it possible to evaluate the \"yield\"--the fraction of the manufactured chips having their max-INL under any specified threshold, evaluation of which calls for the statistical sophistication of Brownian-Bridge methods [8; 9]. The eDAC output constellation deviation sequence, \(\delta_{\mathrm{nom}}\mathbf{J}\), may often be viewed as a superposition of two (non-zero) components: a deterministic deviations sequence referred to as \"deviations; bias and a random mismatch-induced zero-mean perturbation sequence. Examples of constellations with deviations decomposed into bias + zero-mean mismatch:  
* eDAC chips with systematic fabrication structural errors (systematic bias + mismatch). * eDAC feeding an MZM--joint effect of the static nonlinearity of the MZM sine TC and the eDAC electronic-mismatch induced perturbations of the MZM optical output field. For deterministic eDAC constellations, the uniformity of the constellation or lack thereof is the key characteristic. The \"non-uniformity\" of a deterministic constellation is synonymously referred to as the \"bias\" of the constellation. Ideal uniform constellations are referred to as \"unbiased\".