## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Second-Order Statistics of the Steps of the Output Constellation--Matrix-Based Evaluation
\] (154) Component-wise, the corresponding steps variances for BIP BiWgt eDACs are \[\text{BIP BiWgt}: \sigma_{\
abla\widetilde{\gamma}_{c}}^{2}= 2\sigma_{u}^{2}\langle I_{u}\rangle^{2}\mathbf{X}_{\text{BiWgt}}^{ \text{BIP}}[c]\cdot\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\}\] (155) \[\sigma_{\
abla\widetilde{\gamma}_{c}}^{2}= 2\sigma_{u}^{2}\mathbf{X}_{\text{BiWgt}}^{\text{BIP}}[c]\cdot \left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\}\] The steps vector variance is then given by \[\sigma_{\
abla\widetilde{\mathbf{I}}}^{2}= \sigma_{u}^{2}\langle I_{u}\rangle^{2}\left(\mathbf{\
abla C}_{ \text{BiWgt}}^{\text{BIP}}\right)^{\odot 2}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\  
than DD, and it is apparent that the BIP disadvantage of twice mismatch variance for the constellation steps is offset by the large sensitivity advantage enabled by coherent detection of BIP constellations with regard to the direct detection of UNIP constellations.  
The compact key result (152) expresses the doubly normalized variance \(\widetilde{\sigma}_{\
abla_{L}}^{2}/\sigma_{u}^{2}\) for UNIP as an inner product of the XOR vector, \(\mathbf{X}_{\text{BiWgt}}[c]\equiv\mathbf{b}_{\text{BiWgt}}[c]\) XOR \(\mathbf{b}_{\text{BiWgt}}[c+1]\) (151) of successive code-steps with the dyadic sequence \(\left\{2^{B-1},2^{B-2},\ldots,4,2,1\right\}\). Operationally, the inner product scales the bits of the XOR bitstring \(\mathbf{X}^{\text{BiWgt}}[c]\) by the respective dyadic weights \(\left\{2^{B-1},\ldots,4,2,1\right\}\), then sums up the binary weighted bit values to yield the doubly normalized variance of the steps (with a factor of 2 applied in the BIP case relative to the UNIP one).  
The (corresponding) bit positions in \(\mathbf{b}^{T}[c],\mathbf{b}^{T}[c+1]\) contributing to the DNL are those where \(\mathbf{X}[c]\) becomes unity; the heavier contributors are bits where \(\mathbf{X}[c]\) is unity (indicative of corresponding bit flips at those positions, upon transitioning from \(b_{s}[c]\) to \(b_{s}[c+1]\)) with more significant position (having indices \(s\), farther to the left along the rows \(\mathbf{b}^{T}[c],\mathbf{b}^{T}[c+1]\)). In particular, the heaviest contributors are the most significant bit-flip location, \(b_{s}[c]\), \(b_{s}[c+1]\), which are the farthest to the left, as we recall that the index s increases from right to left (but only if the bits at the corresponding locations (same \(s\)) in the two successive codewords are different, \(b_{s}[c]\
eq b_{s}[c+1]\)).  
Let us now formally derive the index of \"worst\" codeword of the BiWgt topology, using (155) of Theorem 2, for the case of a 4-bit 16-level BiWgt eDAC, featuring \(C=16,\ B=\log_{2}C=4,\ \ \widetilde{\mathbf{I}}^{\left(\right)\text{BiWgt}}=\left\{8,4,2,1\right\}\), a behavioral block diagram which is depicted in Figure 1b. For this eDAC example, Table 1 presents the full BiWgt code, the XORs of successive codewords and twice the inner products between the XOR vectors in the third column and the \(\left\{8,\,4,\,2,\,1\right\}\) weights, yielding the (doubly normalized) steps variance \(\sigma_{\
abla\,\widetilde{\mathsf{I}}_{c}}^{2}/\sigma_{u}^{2}\) stated in the fourth column \"Steps VAR\".  
Table 1 pertains to the BIP PAM16 BiWgt eDAC. A similar table for the UNIP PAM16 BiWgt eDAC (not presented) would be obtained by replacing all occurrences of \(-1\) in the BIP code by a 0 bit, and by halving all entries of the \"Step VARs\" column (the variances):  
\[\text{UNIP BiWgt eDACs}:\frac{1}{\sigma_{u}^{2}}\sigma_{\
abla\,\widetilde{ \mathsf{I}}}^{2}=\frac{1}{\sigma_{u}^{2}}\frac{1}{\langle I_{u}\rangle^{2}} \sigma_{\
abla\,\widetilde{\mathsf{I}}}^{2}=\left\{1,3,1,7,1,3,1,15,1,3,1,7,1,3,1\right\}\sigma_{u}^{2} \tag{157}\]  
This result for STEP variances for an UNIP 16-level BiWgt eDAC is generally known (up to different notations).