# A Bivariate Volterra Series Model for the Design of Power Amplifier Digital Predistorters
## 4 Application of the Bivariate Model to DPD  
With the aim of highlighting the enhancement in linearization capabilities after the inclusion of bivariate model terms, a direct learning architecture (DLA) [23] was executed in the DPD of a commercial PA. The experimental setup of Figure 3 was employed for that purpose, where the selected operation point for the class AB PA was characterized by an average output power of 31.2 dBm (39.8 dBm of peak output power) and a gain compression of 2.8 dB.  
First, a search of the optimum structure with the DOMP technique was executed over the GMP and the CKV models in the conventional and bivariate forms. The training signals for this experiment were the input and output waveforms of the PA without DPD. In order to have a comparable set of initial regressors, the models were set to seventh order and maximum memory depth of seven taps. The initial numbers of coefficients for the GMP, the CKV and the bi-CKV models were 248, 328 and 812, respectively. The learning curve of the DOMP algorithm with respect to the number of coefficients is shown in Figure 8, where the expectable decreasing error with the number of coefficients shows the best modeling capabilities in the bivariate CKV structure. The enhancement with respect to the conventional version motivates the increase in number of initial coefficients. The bi-CKV model also shows an enhancement with respect to the GMP model, since it represents a richer regressor structure. The BIC was executed over the sweep in number of components to set the optimum number of coefficients of 99, 165 and 210 for the GMP for the CKV and the bi-CKV models, respectively. These selected regressors were fixed and a DPD was designed based on them in a DLA scheme.  
The DPD training was executed for 30 iterations with a step size of \(\mu=1/2\). In this experiment, the linearization performance was measured by means of the adjacent channel power ratio (ACPR) of the output signal. The linearization NMSE exhibits a similar behavior and it is omitted for simplicity. The DPD performance with respect to DLA iterations is shown in Figure 9, where the linearization capabilities of each algorithm in comparison follow a similar trend closely related to the search of the optimum structure.  
Figure 7: NMSE results of the CKV and the bi-CKV models, for a class AB PA operated at 3.6 GHz with an average output power of 27.4 dBm and a class J PA operated at 850 MHz with an average output power of 26 dBm.  
Finally, once the DPD coefficients were trained and fixed, a new validation signal was set as the input to feature the algorithm's linearization performance by their ability to work with signals not used in the model identification stage. The normalized power spectral densities (PSDs) of the output waveform without DPD and with the benchmarked algorithms are presented in Figure 10. This figure reveals the effectiveness of the DPD exhibiting a spectral regrowth reduction in accordance to the attained performance in finding the model structure and training the DPD coefficients.  
Figure 8: Identification NMSE in the search of the optimum DPD structure with respect to the number of selected components for the GMP and the CKV models in its conventional and bivariate versions. A class AB PA at 3.6 GHz was employed with an operation point characterized by an average output power of 31.2 dBm.  
Figure 9: Evolution of the ACPR in the first two adjacent channels with the iterations in a direct learning scheme for class AB PA operated at 3.6 GHz with an average output power of 31.2 dBm.