## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### Conventional INL Review (in Our Notation)
: _The (conventional) **INL** vector of a constellation is a random vector obtained as a normalized version of the \(\mathrm{EV}\), \(\delta_{\mathrm{nom}}\mathbf{J}\), essentially the \(\mathrm{EV}\) divided by the age-step, \(\overline{\
abla\mathbf{J}}\) (276):_  
\[ \begin{array}{l}\mathbf{INL}\equiv\frac{\delta_{\mathrm{nom}}\mathbf{J}}{ \overline{\
abla\mathbf{J}}}=\frac{\mathbf{J}-\mathbf{J}^{\mathrm{nom}}}{ \overline{\
abla\mathbf{J}}}\ \mathrm{s.t.}\ J_{1}^{\mathrm{nom}}=J_{1},\ J_{ \mathrm{C}}^{\mathrm{nom}}=J_{\mathrm{C}},\ \ \mathrm{where}\ \overline{\
abla\mathbf{J}}=\overline{\
abla J_{c}}=\frac{ \mathrm{FS}}{C-1}\\ \mathrm{INL}[c]\equiv\mathrm{INL}_{c}\equiv[\mathbf{INL}]_{c}\equiv\frac{ \delta_{\mathrm{nom}}J_{c}}{\overline{\
abla\mathbf{J}}}=\frac{k-\mathbf{J}^ {\mathrm{nom}}}{\overline{\
abla J_{c}}},\ \ \mathrm{for}\ c=1,2,3,\ldots,C\end{array}  \tag{277}\]  
_with the first line in (277) introducing the vector version of the INL, whereas the second line above specifies the elements of the INL vector for \(c=1\), \(2\), \(\ldots\), \(C\) (the \"INL samples\")._  
The equispaced levels of the nominal reference constellation featuring in the INL definition are uniformly spaced at the (generally random) average step, \(\overline{\
abla\mathbf{J}}\), (for \(c=1\), \(2\), \(\ldots\), \(C\)):  
\[J_{c}^{\mathrm{nom}}=J_{1}+(c-1)\ \overline{\
abla\mathbf{J}}=J_{1}+(c-1)\ \frac{\mathrm{FS}}{C-1}=J_{1}+\frac{c-1}{C-1}(J_{c}-J_{1})=\frac{C-c}{C-1} \cdot J_{1}+\frac{c-1}{C-1}\cdot J_{c} \tag{278}\]  
This relation stems from the defined uniformity of the \(\mathbf{J}^{\mathrm{nom}}\) constellation, and from the end-point matching constraints \(J_{1|C}^{\mathrm{nom}}=\ J_{1|C}\) (275), implying that \(\mathbf{J}\) and \(\mathbf{J}^{\mathrm{nom}}\) have the same average step and the same full scales (these quantities generally being \(\mathrm{RV}\)):  
\[\mathrm{FS}\equiv J_{\mathrm{C}}-J_{1}=(C-1)\overline{\
abla\mathbf{J}}=(C-1 )\overline{\
abla\mathbf{J}^{\mathrm{nom}}}=J_{\mathrm{C}}^{\mathrm{nom}}-J_{ 1}^{\mathrm{nom}}=\mathrm{FS}^{\mathrm{nom}} \tag{279}\]  
The average-step \(\overline{\
abla\mathbf{J}}\) is then a fraction \(\frac{1}{C-1}\) of the common FS of \(\mathbf{J},\mathbf{J}^{\mathrm{nom}}\), equal to the constant steps \(\left.\
abla J_{c}^{\mathrm{nom}}\right|_{\forall c\in\{1,2,\ldots,C-1\}}= \frac{\mathrm{FS}}{C-1}\) of the nominal reference constellation, \(\mathbf{J}^{\mathrm{nom}}\):  
\[ \begin{array}{l}\overline{\
abla\mathbf{J}}=\overline{\
abla J_{c}}\ =\overline{\
abla\mathbf{J}^{\mathrm{nom}}}=\ \overline{(\
abla J)_{c}}\Big{|}_{\forall c\in\{0,1,\ldots,C-1\}}=\frac{1}{C-1 }(J_{\mathrm{C}}-J_{1})\\ =\frac{1}{C-1}\mathrm{FS},\ \mathrm{with}\ \mathrm{FS}=J_{\mathrm{C}}-J_{1}\end{array}  \tag{280}\]  
Note: In light of our definition of the unary current, \(I_{u}\), as the arithmetic average (random) step of the constellation (see (58) now repeated as (276) above), the first equation in the first line of the INL conventional definition (277) may be written as  
\[\mathbf{INL}\equiv\frac{\delta_{\mathrm{nom}}\mathbf{J}}{\overline{\
abla \mathbf{J}}}=\frac{\delta_{\mathrm{nom}}\mathbf{J}}{I_{u}}=\frac{\mathbf{J}- \mathbf{J}^{\mathrm{nom}}}{I_{u}}\leftrightarrow\mathrm{INL}_{c}=\frac{ \delta_{\mathrm{nom}}J_{c}}{I_{u}}=\frac{J_{c}-J_{c}^{\mathrm{nom}}}{I_{u}} \tag{281}\]  
Moreover, the nominal (reference) levels (278) may now be written in terms of \(I_{u}\):  
\[J_{c}^{\mathrm{nom}}=J_{1}+(c-1)\ \overline{\
abla\mathbf{J}}=J_{1}+(c-1)I_{u} \tag{282}\]For the \(c\)-th nominal level, just shift the first (nominal or actual) level by \(c-1\) steps of \(I_{u}\).