## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Modeling of Statistical Properties of the INL \(|\) DNL Metrics for (un)Biased eDACs
One such concise metric is defined as the root average mean square (RAMS), as defined in Section 1.1, evaluated as the square root of the average of the means of squares of the INL \(|\) DNL samples:  
\[*\text{NL}_{\mathbf{J}}^{\text{RAMS}}\equiv\text{RAMS}\{*\text{NL}_{\mathbf{ J}}\}\equiv\sqrt{\left\langle*\text{NL}_{\mathbf{J}}^{2}\right\rangle}=\sqrt{ \frac{1}{C-1}{\sum_{c=1}^{C-1}\left\langle*\text{NL}_{\mathbf{J}}^{2}[c\right\rangle \right]}} \tag{317}\]  
Another concise scalar metric is the peak of the sequence of RMS values \(\sqrt{\left\langle*\text{NL}_{\mathbf{J}}^{2}[c\right\rangle}\) of the INL \(|\) DNL samples, stated as representative of the entire INL \(|\) DNL random vector:  
\[*\text{NL}_{\mathbf{J}}^{\text{MAX-RMS}}=\max_{c}\left\{\sqrt{\left\langle* \text{NL}_{\mathbf{J}}^{2}[c\right\rangle}\right\} \tag{318}\]  
The scalar metrics \(*\text{NL}_{\mathbf{J}}^{\text{RAMS}}\) and \(*\text{NL}_{\mathbf{J}}^{\text{MAX-RMS}}\) pertain to either biased or bias-free eDACs. For bias-free eDACs, we have \(\left\langle*\text{NL}_{\mathbf{J}}\right\rangle=0\); thus, the mean-square vector \(\left\langle\left|*\text{NL}_{\mathbf{J}}\right|^{2}\right\rangle\) coincides with the variance vector, \(\left\langle\left|*\text{NL}_{\mathbf{J}}\right|^{2}\right\rangle=\sigma_{* \text{NL}_{\mathbf{J}}[c]}^{2}\) reducing (317) to the root average square (RAS) (defined in Section 1.1) of the STD-INL \(|\) DNL vector:  
\[\left\langle*\text{NL}_{\mathbf{J}}\right\rangle=0\Rightarrow* \text{NL}_{\mathbf{J}}^{\text{RAMS}}\equiv\sqrt{\frac{1}{C-1}{\sum_{c=1}^{C- 1}\sigma_{*\text{NL}_{\mathbf{J}}[c]}^{2}}}=\sqrt{\sigma_{*\text{NL}_{ \mathbf{J}}}^{2}} \tag{319}\] \[=\sqrt{\frac{1}{C-1}{\sum_{c=1}^{C-1}\sigma_{*\text{NL}_{ \mathbf{J}}[c]}^{2}}}=\text{RAS}\Big{\{}\sigma_{*\text{NL}_{\mathbf{J}}}\Big{\}}\]  
Therefore, for bias-free eDACs (affected by mismatch but no biases), the concise RAMS scalar metric may be replaced by the RAS of the standard deviations vector, \(\sigma_{*\text{NL}_{\mathbf{J}}}\):  
\[\langle\delta_{\mathrm{nom}}\mathbf{J}\rangle=\langle\mathbf{J}-\mathbf{J}^{ \mathrm{nom}}\rangle=\langle\mathbf{J}\rangle-\langle\mathbf{J}^{\mathrm{nom} }\rangle=0\Leftrightarrow\langle\mathbf{J}\rangle=\langle\mathbf{J}^{\mathrm{nom}}\rangle \tag{312}\]0  
Alternatively, the maximum of all standard deviations of the INL \(|\) DNL samples is stated, as alternative concise scalar metric summarizing the entire INL \(|\) DNL random vector:  
\[\langle\delta_{\mathrm{nom}}\mathbf{J}\rangle=\langle\mathbf{J}-\mathbf{J}^{ \mathrm{nom}}\rangle=\langle\mathbf{J}\rangle-\langle\mathbf{J}^{\mathrm{nom} }\rangle=0\Leftrightarrow\langle\mathbf{J}\rangle=\langle\mathbf{J}^{\mathrm{nom}}\rangle \tag{312}\]1  
A scalar eDAC specification states an upper bound on either of the aforementioned scalar metrics. In the case of the MAX-STD specification, none of standard deviations of the DNL samples should exceed the given DNL\({}_{\mathbf{J}}^{\text{MAX-STD}}\) value. It is useful to analytically derive the STD \(\sigma_{*\text{NL}_{\mathbf{J}}}\) in terms of the Gaussian statistics of the zero-mean current components. Unfortunately, as INL \(|\) DNL is generally non-Gaussian, which implies an intricate statistical endeavor, requiring advanced BrownianBridge concepts as modeled by Radulov et al. [8]. Moreover, the utility of the \(*\)**NLJ** metrics (INL and DNL) for communication performance analysis has never been demonstrated. In the next two subsections, we introduce modified INL and DNL metrics, respectively, referred to as the integral error vector (IEV) and differential error vector (DEV), which are conveniently Gaussian distributed and are amenable to relatively elementary analytic statistical evaluation. Part II of this paper proceeds to establish the direct relevance and utility of the IEV | DEV eDAC descriptions to evaluating transmission metrics of interest in optical or wireless communication, such as the constellation probability of error.