# A Bivariate Volterra Series Model for the Design of Power Amplifier Digital Predistorters
## 2 A Bivariate Volterra Series Perspective
### Bivariate Volterra Models  
PA models implicitly contemplate a nonlinearity dependent on a single variable, the input RF signal. Actually, the FET current source depends on several variables, for example, the gate and drain voltages, \(v_{g}\) and \(v_{d}\). More complete transistor models put forward the presence of an additional mechanism generating an internal variable, such as the aforementioned electrothermal subcircuit.  
In the search of new, possibly beneficial, regressors let us recall the charge-trapping internal mechanism, which is the cause of drain-lag effects in FET devices [17]. From observing the simplified diagram of a one-trap model shown in Figure 1b, it is evident that this subcircuit basically works as an envelope detector. Therefore, the charge-trapping subcircuit generates a new internal variable (the real-valued signal envelope) that additionally controls the current source of the transistor. Another example of an amplifier with a mechanism generating a secondary signal is the high-efficiency envelope tracking (ET) PA [18]. In the ET-PA configuration, shown in Figure 1c, the RF envelope is detected and amplified to provide a dynamically adjusted supply voltage to the basic PA.  
The two examples discussed above fit the schematic representation of Figure 2, where the internally generated envelope acts as the second input to a bivariate block. If this block is a bivariate Volterra system and the two inputs are the RF signal \(\tilde{x}(k)\) and the envelope \(z(k)\), the discrete-time output can be expressed as a double Volterra series [16].  
The output of the bivariate Volterra system is organized into three groups: the first group is a univariate Volterra series dependent on the RF signal \(\tilde{x}(k)\), the second group  
Figure 1: Simplified layouts of a single-FET amplifier (**a**), the associated charge trapping subcircuit (**b**) and the envelope tracking PA (**c**). The following elements are included: gate voltage \(v_{g}\), drain voltage \(v_{d}\), internal variable \(z\), drain current \(l_{d}\), gate-to-source capacitance \(C_{gs}\), frequency-dependent load impedance \(Z_{0}(f)\), capacitance \(C\), and resistances \(R_{fill}\) and \(R_{empty}\).  
is another univariate Volterra series dependent on the envelope \(z(k)\), and the third group contains cross products of \(\tilde{x}(k)\) by \(z(k)\). Therefore, the RF output is given by  
\[\tilde{y}(k)=\hat{H}_{a,0}[\tilde{x}(k)]+\hat{H}_{b,0}[z(k)]+\\ +\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{\mathbf{q}_{m}=\mathbf{0}}^{ \mathbf{Q}_{m}}\sum_{\mathbf{q}_{m}=\mathbf{0}}^{\mathbf{Q}_{m}^{\prime}}h_{n,m }[\mathbf{q}_{n},\mathbf{q}_{m}^{\prime}]\prod_{r=1}^{n}\tilde{x}(k-q_{r})\prod _{s=1}^{m}z(k-q_{s}^{\prime}), \tag{8}\]  
where \(\hat{H}_{a,0}[\cdot]\) and \(\hat{H}_{b,0}[\cdot]\) are univariate Volterra operators. The third group of sums contains the bivariate Volterra kernels \(h_{n,m}[\mathbf{q}_{n},\mathbf{q}_{m}^{\prime}]\) and cross products of the RF input signal \(\tilde{x}(k-q_{r})\) by the envelope \(z(k-q_{s}^{\prime})\). The vector of envelope delays and the vector of maximum delays are denoted as \(\mathbf{q}_{m}^{\prime}\) and \(\mathbf{Q}_{m}^{\prime}\), respectively. Univariate Volterra models are also referred to here as conventional Volterra models.