# A Bivariate Volterra Series Model for the Design of Power Amplifier Digital Predistorters
###### Abstract  
The operation of the power amplifier (PA) in wireless transmitters presents a trade-off between linearity and power efficiency, being more efficient when the device exhibits the highest nonlinearity. Its modeling and linearization performance depend on the quality of the underlying Volterra models that are characterized by the presence of relevant terms amongst the enormous amount of regressors that these models generate. The presence of PA mechanisms that generate an internal state variable motivates the adoption of a bivariate Volterra series perspective with the aim of enhancing modeling capabilities through the inclusion of beneficial terms. In this paper, the conventional Volterra-based models are enhanced by the addition of terms, including cross products of the input signal and the new internal variable. The bivariate versions of the general full Volterra (FV) model and one of its pruned versions, referred to as the circuit-knowledge based Volterra (CKV) model, are derived by considering the signal envelope as the internal variable and applying the proposed methodology to the univariate models. A comparative assessment of the bivariate models versus their conventional counterparts is experimentally performed for the modeling of two PAs driven by a 30 MHz 5G New Radio signal: a class AB PA and a class J PA. The results for the digital predistortion of the class AB PA under a direct learning architecture reveal the benefits in linearization performance produced by the bivariate CKV model structure compared to that of the univariate CKV model. e 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 21 5897 2021 221  
with scalable numerology and augmented bandwidths, which are defined by the fifth generation New Radio (5G-NR) air interface. As the discrete-time models are better suited for the implementation of DPDs, the correct selection of the model structure plays a key role to optimize the linearization performance. Diverse approaches can be adopted for the discrete-time modeling of nonlinear systems, such as multidimensional polynomial filters or artificial neural networks. Volterra series [3; 4] is also a useful mathematical tool for the analysis of nonlinear microwave and wireless systems [5]. It can be considered the extension of Taylor series to nonlinear systems with memory, whose \(n\)-order terms are obtained by means of a nonlinear generalization of the convolution integral that is commonly employed for the study of linear time invariant (LTI) systems. Volterra series produce behavioral models that are linear with respect to the parameters, thus allowing the use of linear regression methods for their identification [6]. With the purpose of modeling wireless nonlinear systems that are driven by communications signals, Volterra series expansions are considered in a discrete, time-sample form. The input complex envelope is sampled at time \(t=kT\), with \(T\) being the sampling period and \(k\in\mathbb{Z}\), thus producing the discrete-time sequence \(x(k)\). Based on the baseband equivalent Volterra model [7], the discrete-time amplifier output at the fundamental frequency zone is described by a linear combination of regressors. These regressors are given by odd-order monomials of the discrete-time input complex envelope \(x(k)\) and its conjugate \(x^{*}(k)\) with different delays that are denoted by \(q,q_{1},q_{2}\), etc. If this univariate Volterra series is truncated to a maximum nonlinear order and memory depth, the resulting structure is denoted as the full Volterra (FV) model. Pioneering publications of baseband DPDs have also proposed the memory polynomial MP [8] and the generalized memory polynomial GMP models [9]. The MP and GMP models involve terms with the cross product of the signal and its exponentiated envelope, e.g., the MP model contains terms  
\[x(k-q)|x(k-q)|^{n-1} \tag{1}\]  
and the GMP model contains terms  
\[x(k-q_{1})|x(k-q_{1}-q_{2})|^{n-1} \tag{2}\]  
for all \(n\), odd and even.