## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### Matrix-Based Modeling of Current-Steering eDACs: From Current Sources to Output Levels
 \tag{16}\]  
The level-index, \(c\), determines the \(0\,|\,1\) weights of the sources, per the bitstring specified in the \(c\)-th UNIP codeword, \(\mathbf{b}_{\text{UNIP}}[c]\), which embodies a \"counting code\". The codewords (and coding matrix) of BIP BiWgt eDACs are obtained from the UNIP ones by the replacements \(0\to-1\), \(1\to 1\). Thus, BIP BiWgt eDACs, the \(B\) current sources (15) are weighted by \"bipolar bits\", namely \(\pm 1\) values, generating the following linear combination, having the code matrix either add up or subtract the current sources, flipping or not flipping the polarity of each of the contributing current sources in the bipolar linear combination:  
\[ \begin{split}&\left\langle I_{c}^{\text{BiWgt-BIP}}\right\rangle= \pm 2^{B-1}\left\langle I_{0}^{\text{BiWgt-BIP}}\right\rangle\pm 2^{B-2} \left\langle I_{0}^{\text{BiWgt-BIP}}\right\rangle+\\ &\ldots\pm 4\left\langle I_{0}^{\text{BiWgt-BIP}}\right\rangle \pm 2\left\langle I_{0}^{\text{BiWgt-BIP}}\right\rangle\pm\left\langle I_{0}^{ \text{BiWgt-BIP}}\right\rangle\end{split}  \tag{17}\]  
Here, the output level index, \(c\), determines the particular \(\pm\) signs used for weighting each current source, as listed in the \(c\)-th BIP codeword, \(\mathbf{b}_{\text{BIP}}[c]\). For BiWgt UNIP \(|\) BIP eDACs, the total mean source current (applicable to both UNIP \(|\) BIP BiWgt eDACs) is obtained by summing up the elements of dyadic sequence (15), yielding  
\[ \begin{split}&\left\langle I_{\text{tot}}^{\text{BiWgt}}\right\rangle =\left\langle\Sigma^{\text{BiWgt}}\right\rangle=\Sigma\left\langle I^{\text{ BiWgt}}\right\rangle=\sum_{s=0}^{B-1}2^{B-1-s}\left\langle I_{0}\right\rangle= \sum_{s^{\prime}=0}^{B-1}2^{s^{\prime}}\left\langle I_{0}^{\text{BiWgt}} \right\rangle\\ &=(2^{B-1}+2^{B-2}+\ldots+4+2+1)\left\langle I_{0}^{\text{BiWgt} }\right\rangle=(2^{B}-1)\left\langle I_{0}\right\rangle=(C-1)\left\langle I_{0 }^{\text{BiWgt}}\right\rangle\end{split}  \tag{18}\]  
To recapitulate, for BiWgt eDACs, be they UNIP \(|\) BIP,  
\[ \begin{split}&\left\langle\Sigma\mathbf{I}^{\text{BiWgt}}\right\rangle =\Sigma\left\langle\mathbf{I}^{\text{BiWgt}}\right\rangle=\sum_{s=0}^{B-1}2^{B- 1-s}\left\langle I_{0}^{\text{BiWgt}}\right\rangle\\ &=\sum_{s^{\prime}=0}^{B-1}2^{s^{\prime}}\left\langle I_{0}^{ \text{BiWgt}}\right\rangle=(2^{B-1}+2^{B-2}+\ldots+4+2+1)\left\langle I_{0}^{ \text{BiWgt}}\right\rangle\\ &=(2^{B}-1)\left\langle I_{0}^{\text{BiWgt}}\right\rangle=(C-1) \left\langle I_{0}^{\text{BiWgt}}\right\rangle\end{split} \]  
Let us now introduce some as useful eDAC parameters: the eDAC most/least significant level, full scale and full-scale range:  
**Definition 1**. : _The Least-Significant-Level (LSL) and the Most-Significant-Level (MSL), of an eDAC current constellation, \(\mathbf{J}\), are the first and the last element \(J_{1},J_{C}\), respectively. The \"Full-Scale\" (FS) of an eDAC is defined as the difference between the MSL and LSL:_  
\[\mathrm{FS}\equiv J_{C}-J_{1} \tag{19}\]  
_Evidently, the FS is an RV and so are the MSL and LSL. A useful metric is the mean FS,_  
\[\langle\mathrm{FS}\rangle=\langle J_{C}\rangle-\langle J_{1}\rangle \tag{20}\]  
_The segment \(\lfloor\langle J_{1}\rangle\,\langle J_{C}\rangle\rfloor\) of the real axis is referred to as the Full-Scale Range (FSR). Its length equals the mean-FS. The C constellation levels (points) partition the FSR into \(C-1\) constellation \"steps\" also known as \"increments\" also known as nearest-neighbor (NN) intervals._  
_eDAC constellation steps (increments)_  
We now proceed to define and model the steps (increments) of the eDAC constellation levels, i.e., the separations between successive output levels of the constellation. **Definition 2**. : _The \"Steps\", \(\
abla J_{c}\), of a \(1D\) constellation of levels are defined as the lengths of the intervals between pairs of adjacent levels of the constellation:_  
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