# A Bivariate Volterra Series Model for the Design of Power Amplifier Digital Predistorters
## 3 PA Modeling Performance  
The experimental setup shown in Figure 3 was employed for the comparative assessment of the proposed approach with bivariate models versus the use of their conventional counterparts. The probing signal, designed according to the 5G-NR standard [22] with a 30 MHz bandwidth, was created by an SMU200A vector signal generator (VSG) from Rohde & Schwarz with built-in arbitrary waveform generator. With a peak-to-average power ratio (PAPR) of 10.5 dB, it contained over 180,000 samples, corresponding to a sampling frequency of 92.16 MSample/s. The VSG was followed by two cascaded Mini-Circuits TVA-4W-422A+ preamplifiers in order to drive the PA under test into a mildly nonlinear operation, while keeping the behavior of the modulator sufficiently linear.  
Two different PAs were employed in the experiments. The first PA under test was the evaluation board of a class AB amplifier based on the CGH40010 GaN HEMT from Cree Inc. (Durham, NC, USA), operated with a drain-to-source current of about 200 mA at 3.6 GHz and an average output power of 27.4 dBm. The second PA under test was a continuous-mode class J amplifier designed over the CGH35015F GaN HEMT from Cree Inc., providing an average output power of 26 dBm at 850 MHz. The output of the PAs was fed to a PXA-N9030A vector signal analyzer (VSA) from Keysight Technologies through a directional coupler and an attenuator in order to avoid introducing undesired distortion from the equipment. The RF output signal was downconverted to baseband and acquired in the VSA, where the measurement dynamic range was optimized by averaging 300 repetitions of the measured signal. Finally, the signals were time-aligned in order to synchronize the input and output data sets.  
First, we compare the different modeling performances of the CKV, FV, bi-CKV and bi-FV models for the class AB PA. An important drawback of the bi-FV model compared to the bi-CKV model is its greater complexity in terms of number of regressors (and coefficients, correspondingly). In order not to go beyond the limits of computing capacity when predicting the PA output, all the models were selected with a ninth nonlinear order and a memory of three samples. Furthermore, the memory is considered only for regressors with nonlinear order below 7 in the case of the FV (univariate and bivariate) models.  
The reduction in the NMSE between the modeled and the measured output of the class AB PA as the DOMP algorithm incorporates new regressors to the active set is shown in Figure 4. For the conventional CKV model (plotted with a dashed line), the best NMSE according to the BIC rule is about \(-48\) dB. An FV model with the same order and memory improves the NMSE (dashed line with circles), an expected result considering its richer regressors set. On the other hand, the improvement is marginal, indicating that the CKV model is quite appropriate despite its sparseness. For comparison purposes, the GMP model curve is also plotted (dotted line), showing the effect of its beneficial terms. The better accuracy of the bi-CKV model (10) (displayed in the figure with a solid line) and the further NMSE improvement obtained with the bi-FV model (solid line with circles) are clear results of the bivariate terms.  
The NMSEs of the conventional CKV and FV models do not reach the level of error of the corresponding bivariate models, even in the case that more regressors are incorporated, evidencing that without bivariate terms, the stock of regressors is insufficient.  
Since the extension of nonlinear order and memory length is impracticable for the bivariate FV model, in the following the focus will be put in the comparison of the conventional and bivariate CKV models with optimized order and memory.  
Figure 3: Photograph of the experimental setup composed of (1) a vector signal generator, (2) two cascaded preamplifiers, (3) a vector signal analyzer, (4) the class AB PA under test, (5) a directional coupler, and (6) an attenuator terminated with a 50 \(\Omega\) load.