## 3 DAC Integral/Differential Nonlinearities--New Normalized Error Vector Metrics
### INL, DNL Relations and Symmetries
:  
_(i)_ _Both the_ **INL** _vector and the_ **DNL** _vector are invariant under any affine transformation of the form (_301_), acting on the current vector, namely_ \(\mathbf{J}^{\mathrm{aff}[a,b]}\equiv a\mathbf{J}+b\mathbf{1}_{\mathrm{C}}^{ \forall}\)_:_  
\[\mathbf{INL_{J}}^{\mathrm{aff}[a,b]}=\mathbf{INL_{J}},\quad\mathbf{DNL_{J}}^{ \mathrm{aff}[a,b]}=\mathbf{DNL_{J}} \tag{304}\]  
_(ii)_ _For any given eDAC topology (or equivalently code matrix), eDACs of the UNIP and BIP polarities have identical_ **INL** _vectors, and also have identical_ **DNL** _vectors, whenever these eDACs are driven by identical source current vectors,_ \(\mathbf{I}\)_. Thus, the_ **INL** _and_ _DNL_ _metrics do not depend on eDAC polarity (_UNIP \(|\) BIP_); however, they generally do depend on eDAC topology or equivalently on the code matrix (BiWgt \(|\) ThWgt). Formally,_  
\[ \begin{array}{l}\mathbf{J}_{\mathrm{UNIP}}=\mathbf{C}_{\mathrm{UNIP}}\mathbf{ I}\\ \mathbf{J}_{\mathrm{BIP}}=\mathbf{C}_{\mathrm{BIP}}\frac{1}{2}\mathbf{I}\end{array}  \bigg{\}}\Rightarrow\mathbf{INL_{J}}_{\mathrm{UNIP}}=\mathbf{INL_{J}}_{ \mathrm{BIP}},\quad\mathbf{DNL_{J}}_{\mathrm{UNIP}}=\mathbf{DNL_{J}}_{ \mathrm{BIP}} \tag{305}\]  
**Proof of Corollary 7**. : The derivation of property (i) is relegated to Appendix B.2. \(\square\)  
Note 1: a more verbose (and informative yet more cumbersome) notation for \(\mathbf{C}_{\mathrm{UNIP}}\), \(\mathbf{C}_{\mathrm{BIP}}\) above would be \(\mathbf{C}_{\mathrm{UNIP}}\equiv\mathbf{C}_{\mathrm{UNIP}}^{\mathrm{BiWgt}[ \mathrm{ThWgt}},\,\,\,\mathbf{C}_{\mathrm{BIP}}\equiv\mathbf{C}_{\mathrm{BIP}}^ {\mathrm{BiWgt}[\mathrm{ThWgt}}.\)  
Note 2: for BiWgt \(|\) ThWgt eDACs (irrespective of polarity), the DNLs are different, whereas the INLs come out the same:  
\[\mathbf{INL}_{\mathbf{C}_{\mathrm{UNIP}}^{\mathrm{BiWgt}}[\mathrm{BIP}]}= \mathbf{INL}_{\mathbf{C}_{\mathrm{UNIP}}^{\mathrm{ThWgt}}}\,\,\,\mathbf{DNL}_{ \mathbf{C}_{\mathrm{UNIP}}^{\mathrm{BiWgt}}[\mathrm{BIP}]}\
eq\mathbf{DNL}_{ \mathbf{C}_{\mathrm{UNIP}}^{\mathrm{ThWgt}}} \tag{306}\]  
Moreover, in light of Corollary 7 (ii), the values of the INL \(|\) DNL quantities in (306) do not change upon transitioning any eDAC (be it BiWgt \(|\) ThWgt) from UNIP to BIP,the transformation of the output constellation of which may be expressed in the affine form (301). Note 3: as a special case of the affine transformation (301), an isotropic scaling transformation \(\mathsf{J}^{\mathsf{aff}[\mathsf{i},\mathsf{0}]}=a\mathsf{J}\) of the current just leaves the INL and DNL invariant. This is easy to make sense of, as the linear scaling of the INL amounts to amplifying the numerator and denominator in the INL and DNL definitions by the factor \(a\), which cancels out. Likewise, the translation \(\mathsf{J}^{\mathsf{aff}[\mathsf{0},\mathsf{j}]}=\mathsf{J}+b\mathsf{I}_{C}^ {\intercal}\) by any constant vector (a vector with equal elements) keeps the INL and DNL invariant. We finally state some usage additional properties of the INL and DNL metrics:  
**Corollary 8**. : _For a UNIP \(|\) BIP BiWgt \(|\) ThWgt eDAC (irrespective of its source currents):_  
_(i)_ _The first and last elements of the INL vector are always zero:_  
\[\mathrm{INL}_{1}=0=\mathrm{INL}_{C}\] (307) _(ii)_ _The sum of the elements of the DNL vector is always zero:_  
\[\sum_{i=1}^{C-1}\mathrm{DNL}_{i}=0\] (308) _(iii)_ _The DNL vector is \"centrosymmetric\":_  
\[\mathrm{DNL}_{\mathrm{c}_{\mathrm{t}\mathrm{C}}}=\mathrm{DNL}_{\mathrm{c}} \Leftrightarrow\mathrm{DNL}_{\mathrm{C}+1-c}=\mathrm{DNL}_{\mathrm{c}} \Leftrightarrow\mathrm{DNL}^{\intercal}=\mathrm{DNL}\] (309) _(iv)_ _The INL vector is \"skew-centrosymmetric\":_  
\[\mathrm{INL}_{\mathrm{c}_{\mathrm{t}\mathrm{C}}}=-\mathrm{INL}_{\mathrm{c}} \Leftrightarrow\mathrm{INL}_{\mathrm{C}+1-c}=-\mathrm{INL}_{\mathrm{c}} \Leftrightarrow\mathrm{INL}^{\intercal}=\mathrm{INL} \tag{310}\]  
**Proof. ** The detailed derivation is relegated to Appendix B. \(\square\)  
The Monte Carlo simulations of INL \(|\) DNL presented in Section 3.11 below verify Corollary 7.