## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current
Equivalently, the relative variance in any single particular current source on the chip uniquely determines the second-order statistics of all current sources on the chip. It is this observation that suggests selecting a convenient reference current on the chip, referred to as \"unary current\" (see (58) in Definition 3), then specifying its (relative) variance, subsequently making it possible to evaluate the second-order statistics (means and variances) for any of the current sources and subsequently for all current output levels aboard the eDAC. Moreover, it is useful to normalize all means and STDs of the current sources and current output levels by the mean of the unary current, expressing all currents and their STDs in units of the \"unary current\" as introduced in Section 2.2. The unary-current based normalizations streamline and simplify the all-analytic statistical model of eDACs. _eDAC statistics compactly reformulated using normalizations by the unary current._  
We now show that specifying the mean and relative variance (or relative STD) of the unary current determines the joint statistics of the array of eDAC current sources. We start by noting that the unary current, \(I_{u}\), is an RV, whereas the mean unary current \(\langle I_{u}\rangle\) is a deterministic scalar; as such, \(I_{u}\) is decomposed into mean-plus-perturbation:  
\[\bigg{\langle}\widetilde{I}_{c}^{\text{UNIP BiWgt}[\text{ThWgt}}\bigg{\rangle}= c-1,\ \ \ \bigg{\langle}\widetilde{\mathbf{J}}^{\text{ UNIP BiWgt}[\text{ThWgt}}\bigg{\rangle}=\{0,1,2,\dots,C-1\} \tag{83}\]8  
any given eDAC chip will then be statistically characterized by identifying its unary current distribution (and in particular, the mean unary current and its variance, STD and RSTD metrics). The same distribution of unary current may pertain not only to a particular eDAC chip but to the entire wafer or production lot carrying the eDAC device replicas, provided the \"statistical conditions\" of the material and fabrication be invariant across the wafer and over the production process. It is useful to adopt the relative standard deviation (RSTD)of the unary current RV, \(I_{u}\), as the metric parameterizing the mismatch of current sources (transistors) on the chip, waver or production lot, denoting the relative variance in the unary current, \(I_{u}\), by \(\sigma_{u}^{2}\) (referred to as \"relative mismatch\") as customary in the literature. **Definition 4**. : _\"Relative mismatch\", \(\sigma_{u}\) of an eDAC chip, wafer or production lot (assuming statistical homogeneity, i.e., spatial stationarity) is defined as the relative standard deviation (RSTD) (also known as \"coefficient of variation\") of the selected unary current, \(I_{u}\), for that chip, wafer or production lot:_  
\[\bigg{\langle}\widetilde{I}_{c}^{\text{UNIP BiWgt}[\text{ThWgt}}\bigg{\rangle}= c-1,\ \ \ \bigg{\langle}\widetilde{\mathbf{J}}^{\text{ UNIP BiWgt}[\text{ThWgt}}\bigg{\rangle}=\{0,1,2,\dots,C-1\} \tag{83}\]9  
Squaring the relative mismatch (101) yields the relative variance in the unary current synonymously referred to as \"squared-relative-mismatch\":  
\[\bigg{\langle}\widetilde{\widetilde{I}}_{c}^{\text{ BiWgt}[\text{ ThWgt}}\bigg{\rangle}=\bigg{\{}c-\frac{C+1}{2}\text{ for BIP}\\ c-1\text{ for UNIP} \tag{84}\]0  
Note: The relative mismatch and the relative variance are dimensionless quantities.