## 2 Second-Order Statistical Modeling of Current-Steering Electronic DACs (eDAC)
### All eDAC Currents Are Normalized by the (Mean) Unary Current  
As in most areas of physics, the normalization of various physical quantities by judicious selection of suitable units of measurement is an advantageous approach worth adopting upfront, making it possible to simplify and streamline physical models. eDAC theory is no exception. In this subsection, we introduce \"unary normalizations\" for deterministic and stochastic modeling of eDACs. For a given chip, batch or production lot of eDACs, it is useful to measure all current sources in units of the mean of a conveniently specified reference current, \(I_{u}\), referred to as \"unary current\" (the subscript \(u\) stands for \"unit\" or \"unary\"). The nominal value of the RV \(I_{u}\), i.e., its mean, \(\langle I_{u}\rangle\), is going to be used as the unit of measurement to peg all current sources and all output currents aboard the chip; this normalization makes it possible to most compactly formulate the first and second-order statistical metrics of the array of eDAC current sources. The choice of the current level constituting the mean unary current is in principle arbitrary--smart selection of the unary current is guided by eDAC modeling convenience, such as to yield easiest-to-handle expressions in terms of normalized currents on the eDAC chip. A useful choice of \"unary current\" is to designate the eDAC source current having the least mean as the unary current. This works well for BiWgt! ThWgt eDACs but turns out unsuitable for more general (non-uniform) constellation generation, e.g., as arising in optical transmission. We thus adopt a more versatile unary current specification, applicable to arbitrarily shaped constellations, in particular applicable to non-uniformly spaced constellations, such as predistorted ones, wherein the current sources, \(\mathbf{I}\), are required to depart from the standard ThWgt or BiWgt uniform designs (which, respectively, feature equal mean current sources and dyadic currents sources). To encompass eDACs generating uniform output constellations, as well as extend the model to non-uniform output eDAC constellations, we introduce the following generalized definition for \(\left\langle I_{u}\right\rangle\) (yet reducing to conventional definitions, once the constellation is taken uniform):  
**Definition 3**. : _Unary random current, \(I_{u}\): the unary current is defined as the random current of a fictitious current source, conceptually constructed on the same eDAC chip, equal to the \"average step\" of the output current constellation of the eDAC:_  
\[I_{u}\equiv\overline{\
abla\mathbf{J}}\equiv\overline{\
abla J_{c}}=\frac{1} {C-1}\mathrm{FS}=\frac{1}{C-1}(J_{c}-J_{1})=\frac{1}{C-1}(J_{c}-J_{1}) \tag{58}\]  
The unary current \(I_{u}\) is evidently an RV, since the FS is random; its mean value, namely \(\left\langle I_{u}\right\rangle\), is equal to the \"mean-average step\" of the given (generally nonuniform) constellation,  
\[\left\langle I_{u}\right\rangle\equiv\left\langle\overline{\
abla\mathbf{J}} \right\rangle\equiv\left\langle\overline{\
abla J_{c}}\right\rangle=\frac{1}{ C-1}\langle\mathrm{FS}\rangle=\frac{1}{C-1}\langle J_{C}-J_{1}\rangle= \frac{1}{C-1}[\langle J_{C}\rangle-\langle J_{1}\rangle] \tag{59}\]  
\"mean-average\" meaning here that the stochastic mean is applied to the arithmetic average (58). The \"mean-average\" applied to the sequence of constellation steps yields  
\[\left\langle\overline{\
abla\mathbf{J}}\right\rangle\equiv\left\langle \overline{\
abla J_{c}}\right\rangle=\frac{1}{C-1}\sum\
olimits_{i=1}^{C-1} \langle\
abla J_{i}\rangle=\frac{\left\langle J_{C}\right\rangle-\left\langle L _{1}\right\rangle}{C-1}=\frac{\left\langle\mathrm{FS}\right\rangle}{C-1} \tag{60}\]  
The mean-average step (i.e., \(I_{u}\)) is seen to be \(C-1\) times smaller than the FS of the eDAC. Note 1: the unary current \(I_{u}\) (equal to the \"average step\") was seen to be an RV. In contrast, the mean unary current \(\left\langle I_{u}\right\rangle\) (equal to the \"mean-average step\") is deterministic.