## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### INL, DNL Relations and Symmetries  
In this subsection we provide, for the first time, to the best of our knowledge, formal derivations of the (skew)-centrosymmetries of the INL and DNL metrics. Those symmetries will be found useful in the sequel. We start by introducing a streamlined notation for uniform constellations:  
**Definition 13**. : _Uniform \(C\)-point constellation with endpoints \(J_{1},J_{C}\):_  
\[\mathbf{J}^{\mathrm{unif}[J_{1},J_{C}]}\equiv\frac{C-c}{C-1}\cdot J_{1}+\frac{ c-1}{C-1}\cdot J_{c}=\left.p_{c}J_{1}+q_{c}J_{C}\right|_{p_{c}:=\frac{C-c}{C-1}J_{c}:=1 -p_{c}=\frac{C-1}{C-1}} \tag{297}\]  
_Note: \(J_{c}^{\mathrm{unif}[J_{1},J_{C}]}\) are convex combinations of the LSL and MSL, \(J_{1},J_{C}\) (\(p_{c}+q_{c}=1\) )._  
As an application of this definition of uniform constellations, the INL definition (277) may be rephrased by leveraging the fact that our earlier introduced nominal constellation is a uniform one:  
\[\mathbf{J}^{\mathrm{nom}}\equiv\mathbf{J}^{\mathrm{unif}[J_{1},J_{C}]} \tag{298}\] \[\therefore\mathbf{INL}\equiv\frac{\mathbf{J}-\mathbf{J}^{\mathrm{nom }}}{\
abla\mathbf{J}}=\frac{\mathbf{J}-\mathbf{J}^{\mathrm{unif}[J_{1},J_{C}] }}{\
abla\mathbf{J}}=\frac{\mathbf{J}-\mathbf{J}^{\mathrm{unif}[J_{1},J_{C}] }}{\frac{1}{C}(J_{C}-J_{1})} \tag{299}\]  
**Definition 14**. : _A general affine transformation is a map \(\mathbf{J}\in\mathbb{R}^{\mathrm{C}}\rightarrow\mathbf{K}\in\mathbb{R}^{ \mathrm{D}}\) given by_  
\[\mathbf{K}=\mathbf{L}\mathbf{J}+\mathbf{t} \tag{300}\]  
_with \(\mathbf{L}\) a \(D\times C\) matrix and \(\mathbf{t}\) a \(D\)-vector. In eDAC theory, we often encounter a particular subclass of affine transformations, having the following form parameterized by a pair scalar coefficients \(a\), \(b\), representing isotropic scaling and translation:_  
\[\mathbf{J}^{\mathrm{aff}[a,b]}\equiv a\mathbf{J}+b\mathbf{1}_{C}^{\forall} \tag{301}\]Note 1: the definition (287) of the **DNL** vector is essentially the image of affine transformation of the form (301) acting on the output constellation steps vector \(\
abla\mathbf{J}\):  
\[ \begin{array}{l}\mathbf{DNL_{J}}\equiv\frac{1}{\
abla\mathbf{J}}\
abla\mathbf{ J}-\mathbf{1}_{\mathrm{C-1}}^{\forall}=\,a\mathbf{J}+b\mathbf{1}_{\mathrm{C}}^{ \forall}\Big{|}\,\,\,\mathbf{J}:=\
abla\mathbf{J}\\ a:=\frac{1}{\
abla\mathbf{J}},b=-1\end{array} \qquad\qquad=(\
abla\mathbf{J})^ {\mathrm{aff}[\frac{1}{\
abla\mathbf{J}},-1]}\\ \end{array} \tag{302}\]  
Note 2: the generation of the **INL** vector from \(\mathbf{J}\) cannot be viewed as an affine transformation of the form (301), but it is rather a more general affine transformation of the form (300), albeit with \(\mathbf{L}\) reduced to a diagonal matrix, \(\mathbf{L}:=\frac{1}{\
abla\mathbf{J}}\mathbf{I}\), and using a non-constant translation vector, \(\mathbf{t}:=\frac{1}{\
abla\mathbf{J}}\mathbf{J}^{\mathrm{unif}[J_{1},J_{ \mathrm{C}}]}\). **Lemma 5**. :  
_(i)_ _An affine transformation of the form (_301_), acting on a uniform constellation, yields a uniform vector, with end-points that are the images of the original endpoints under the same transformation:_  
\[\left(\mathbf{J}^{\mathrm{unif}[J_{1},J_{\mathrm{C}}]}\right)^{\mathrm{aff}[a, b]}=\mathbf{J}^{\mathrm{unif}[J_{1}^{\mathrm{aff}[a,b]},l_{\mathrm{C}}^{\mathrm{aff}[a,b] }]}\Leftrightarrow\left(l_{c}^{\mathrm{unif}[J_{1},J_{\mathrm{C}}]}\right)^{ \mathrm{aff}[a,b]}=J_{c}^{\mathrm{unif}[J_{1}^{\mathrm{aff}[a,b]},l_{\mathrm{C }}^{\mathrm{aff}[a,b]}]} \tag{303}\]  
_(ii)_ _Any uniform constellation (_297_) is generally raised-skew-centrosymmetric._  
**Proof of Lemma 5**. : Relegated to Appendix B.2. \(\square\)  
Thus, affine transformations of the form (301) acting on eDAC constellations preserve constellation uniformity. **Corollary 7**.