## 3 DPD Coefficients Recognition Algorithm
### The JSG-DPD Indirect Learning Structure  
In [28], the indirect learning architecture applied to PA linearization is proposed for the first time. As shown in Figure 5, in the field of satellite communications, the feedback loop for coefficient calculation is almost equivalent to a receiver that would be included in the satellite-transmitter. Therefore, the required cost is unacceptable for satellites; it is also for this reason that dynamic DPD techniques are not widely used in satellite communications. Ref. [29] provides a fifth order static DPD, which means that only the static nonlinearity of the PA can be handled, while the memory effect is ignored. Such a static DPD method would not improve the performance a lot, especially for broadband signals. Figure 4: Data transmission frame structure with noise reduction function. The indirect learning structure of the proposed JSG-DPD technique is illustrated in Figure 6, in which the pre-distorter is not working until the DPD coefficients are identified. At this time, the input signal of the APA \(x(n)=z(n)\), \(z(n)\) is taken as the expected output signal of the DPD coefficients estimation module, and the signal \(\hat{y}(n)\) after noise reduction of the received signal \(\hat{y}(n)\) is taken as the input of the pre-distortion coefficients estimation module, where \(G\) is the gain of the pre-PA and linear APA. The signal \(\hat{y}(n)\) is passed through the DPD coefficients estimation module (i.e., post-distortion) to obtain the output signal \(\hat{z}(n)\), which is subtracted from the desired output signal \(z(n)\) to obtain the error signal \(e(n)\). The adaptive discrimination algorithm obtains the coefficients \(w\) of the optimal DPD coefficients estimation module by iterating the error signal. Finally, the coefficient \(w\) is directly copied as the coefficient of the pre-distorter to make the pre-distorter operate. It is worth pointing out that for the general Volterra system, the \(P\)-order post-inverse is the same as the \(P\)-order pre-inverse [30]. Thus, for our application, the post-distortion response is the same as the pre-distortion response, and that is why the coefficients can be copied directly to the pre-distorter before the APA. In addition, the Volterra series are generalized nonlinear models with memory, and DPD of Volterra models is usually implemented using \(P\)-order inverse techniques [31]. Notably, another communication link is not required because the pilot signal \(z(n)\) is only a fixed segment of the signal known to the ground station for the purpose of calculating the DPD coefficients. Therefore, the proposed JSG-DPD indirect learning structure does not occupy additional space-borne resources. Figure 5: The conventional DPD indirect learning architecture. Figure 6: The indirect learning structure of the proposed JSG-DPD technique. The calculation of the DPD coefficients is very fast after the ground station receives the pilot signal, and this calculation time is even negligible because the computing resources of the ground station are very powerful. Instead, the time to be concerned is the time to upload the DPD coefficients, which depends on the communication rate of the uplink. For this paper, the number of coefficients of the generalized memory polynomial model obtained is several tens, which can be uploaded in a short time (ms level). Therefore, both the DPD coefficient calculation time and the DPD coefficient upload time are very short and do not illegally emit large amounts of spurious emissions to other satellites, which can affect the communication quality. It is only possible that while waiting for the coefficients to be uploaded, the received signal quality of the ground station will degrade because the space-borne DPD coefficients have not been updated yet. After the coefficients are uploaded and the DPD coefficients are updated, the received signal quality at the ground station will be greatly improved.