## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Monte Carlo Simulations of Key Properties of the INL, DNL, IEV, DEV eDAC Metrics
In contrast, the pinning at zero of the INL at the two extremities (as in Figure 7(left)) is not a faithful physical representation of eDACs mismatch statistics, since it fails to capture that the noisiest sample (the output current sample likely to have the largest absolute value) of an UNIP eDAC is actually the last one, the MSL (here at \(c=8\))--thus, the pinning the last INL sample to zero suppresses information about the most pronounced noisiness of the MSL sample. In contrast, the advantage of the IEV metric is that its absolute value does provide a faithful indication of the actual noisiness (or biasing distortion) of the sample. The UNIP INL is visibly skew-centrosymmetric, as may be seen by inspecting the five INL cures in (Figure 7(left)).The IEV metric (Figure 8(left)) does not possess (skew-)centrosymmetry properties. In contrast the DEV metric (Figure 8(right)) is seen to retain the centrosymmetry of the DNL metric (Figure 7(right)). Both the DNL an DEV are centrosymmetric, and in fact their respective graphs are nearly identical, with only minute differences between the DEV and DNL for each of the five eDACs (this observation is also apparent in Figure 10). The fact that the DNL and DEV are almost identical, yet the INL and IEV are so different is accounted for by the INL, IEV being accumulations of the DNL, DEV, respectively. Thus, small differences between the DEV and DNL (as apparent in Figure 10) accumulateto a gradual runoff between the IEV and INL (as apparent in Figure 9 as well as in Figure 7(left)). Figure 9 overlays in each of its five frames, the plots of the INL and IEV of five UNIP eDACs. Again, it is visible that the INL (dashed, blue) curves are pinned to zero at their two extremities (at indices 1 and 8), whereas for the IEV curves (solid, orange) are pinned to zero just on their LHS (at \(c=1\)), tending to run off to higher absolute values as the index \(c\) increases. The other visible effect is the lack of skew-centrosymmetry of the IEV--which is a positive aspect as such centrosymmetry is \"unphysical\", inconsistent with the second-order statistics of the output current vectors, \(\mathbf{J}\). Figure 10: Monte Carlo simulations of the DNL (dashed, blue curves) and DEV (solid, orange curves) for each of the 5 BiWgt UNIP PAM8 eDACs (randomly selected out of the eDACs ensemble), pertaining to the DNL curves plotted in Figure 7(right). Figure 10 superposes in each of its five frames the plots of the DNL and DEV of five UNIP eDACs. It is again apparent that there are just small differences (small vertical discrepancies) between the DEV and DNL vectors. Moreover, the centro-symmetry of the DEV is seen to be holding just like that of the DNL in Figure 8(right). We note that since the DEV vector is essentially an affine transformation on the current vector, **J**, and the physical reality is that the levels of **J** do form out a centrosymmetric sequence for BiWgt eDACs, then so should the DEV samples. Figure 11 verifies the second-order statistical properties of the IEV according to Theorem 4 (albeit for unbiased eDACs affected solely by electronic mismatch errors, no systematic errors). The empirical Standard Deviation (STD) of the \(c=1,\,2,\,3,\,\ldots,\,8\) IEV samples is calculated over the entire ensemble of 1500 eDACs for each of the \(C=8\) IEV samples, plotted as 8 red blue dots in the figure. The orange curve, \(\sigma_{u}\sqrt{c-1}=0.1\sqrt{c-1}\) represents the stochastic STD of the IEV samples, plotted in the figure as ideal reference for the empirical STD points. Actually, this function of \(c\) is discrete-valued as it is only defined at discrete indices, \(c=1,\,2,\,3,\ldots,8\), though, for visualization purposes, we had it plotted as continuous-valued graph (as if \(c\) is a real-valued index defined over the closed segment with endpoints \(1,8\)). The expected stochastic STD values are the intersection points between the orange curve and the 8 vertical gridlines at \(c=1,\,2,\,3,\,\ldots,\,8\).