## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Conventional INL Review (in Our Notation)  
In the context of constellations generated by DACs in communication transmitters, the term \"nonlinearity\" refers the deviation of the levels' constellation \(\mathbf{J}\) from an ideal set of corresponding uniformly spaced levels. More precisely, the eDAC \"nonlinearity\" is quantified in terms of the actual levels deviations relative to nominal, also known as reference, constellation \(\mathbf{J}^{\text{nom}}\), which is uniform, i.e., has its levels equispaced, separated by steps \(\
abla\mathbf{J}^{\text{nom}}=\frac{\text{FS}^{\text{nom}}_{\text{C}-1}}{ \mathbf{1}}^{\vee}_{\text{C}-1}\), and has its end levels coinciding with those of the given constellation. **Definition 10**. : _The \"nonimal-referred error vector (\(EV\))\" \(\delta_{\text{nom}}\mathbf{J}\) of the \(C\)-levels constellation \(\mathbf{J}\) at eDAC output is defined as the vector of deviations of the constellation levels \(\{I_{c}\}_{c=1}^{\text{C}}\) with regard to the nominal (reference, intended, ideal) positions \(\{I^{\text{nom}}_{c}\}_{c=1}^{\text{C}}\) of the levels,_  
\[\delta_{\text{nom}}\mathbf{J}\equiv\mathbf{J}-\mathbf{J}^{\text{nom}}=\{ \delta_{\text{nom}}J_{1},\delta_{\text{nom}}J_{2},\ldots,\delta_{\text{nom}}J _{\text{C}}\},\ \ \text{with}\ \delta_{\text{nom}}J_{c}\equiv J_{c}-J^{\text{nom}}_{c},\ \ \text{for}\ c=1,2,\ldots,C-1 \tag{274}\]  
_where the nominal constellation, \(\mathbf{J}^{\text{nom}}\), is a uniform reference constellation with LSL and MSL coinciding with the respective MSL and LSL of \(\mathbf{J}\) (see (28) and the note below it):_  
\[\text{matching the uniform}\ \mathbf{J}^{\text{nom}}\ \text{to}\ \mathbf{J}\ :\ J^{\text{nom}}_{1}=J_{1},J^{\text{nom}}_{c}=J_{c} \tag{275}\]  
The LSL and MSL of the given constellation are adopted as those of the nominal constellation. Thus, \(\mathbf{J}^{\text{nom}}\) is a uniform interpolation between the two end-points of \(\mathbf{J}\). The EV, \(\delta_{\text{nom}}\mathbf{J}\), corresponding to a given constellation, \(\mathbf{J}\), is the deviation of the given constellation with regard to the reference uniform constellation, \(\mathbf{J}^{\text{nom}}\). Note: as \(J_{1|C}\) are generally RVs, so are \(J^{\text{nom}}_{1|C}\) and so are \(J_{c}\) and \(\delta_{\text{nom}}J_{c}\) random for all \(c\) (the underlying randomness stems from the current sources mismatch). Thus, the nominal constellation is a random vector (or finite sequence); nevertheless, all its sample sequences are uniform, though the fixed step in each of the uniform sample constellations is random. The conventional Integral NonLinearity (INL) metric is essentially a normalized version of the \(\mathrm{EV}\), \(\delta_{\mathrm{nom}}\mathbf{J}\equiv\mathbf{J}-\mathbf{J}^{\mathrm{nom}}\) (322). The normalizer in the conventional INL definition is taken as the \"average-step\" or \"average-LSB\", a generally random scalar designated here as \(\overline{\
abla\mathbf{J}}\), \(\overline{\
abla J_{c}}\), or as \(I_{u}\), defined as the arithmetic average (58) of the given constellation steps (increments),  
\[I_{u}\equiv\overline{\
abla\mathbf{J}}\equiv\overline{\
abla J_{c}}\equiv \frac{1}{C-1}\mathrm{FS}=\frac{1}{C-1}(J_{\mathrm{C}}-J_{1}) \tag{276}\]  
Identifying the average step \(\overline{\
abla\mathbf{J}}\) of the constellation as the unary current \(I_{u}\), as per (58), provides new insight into the INL definition, as will be seen presently. The INL samples, \(\mathrm{INL}[c]\) (elements of the INL vector, \(\mathbf{INL}\equiv\{\mathrm{INL}[c]\}_{c=1}^{C-1}\)) are defined below as fractional deviations of the constellation levels: the elements of the vector \(\delta_{\mathrm{nom}}\mathbf{J}\equiv\mathbf{J}-\mathbf{J}^{\mathrm{nom}}\) of the deviations of the levels (relative to the nominal levels) are simply measured in units of \(\overline{\
abla\mathbf{J}}=\overline{\
abla J_{c}}=\frac{\mathrm{FS}}{C-1}=I _{u}\) (referred to as \"average-step\" also known as \"average-LSB\" or average-increment\"). **Definition 11**.