## 3 DPD Coefficients Recognition Algorithm
### Generalized Memory Polynomial (GMP) Model  
For narrow-band signals, where the memory effect is not obvious, it can usually be implemented using polynomial models, Saleh models [32], etc. For wideband signals, the influence of the memory effect cannot be ignored, and the output signal of the APA is not only related to the current input signal, but also related to the past input signal, so it is necessary to use a nonlinear model with memory for wideband APA modeling. Memory models mainly include Volterra series models [33] and their simplified forms, such as memory polynomial (MP) [28], generalized memory polynomial (GMP) [34], Wiener model [35], Hammerstein model [36], etc. Recently, neural network models [37] have also been investigated for modeling nonlinear systems.  
Usually, the MP model and Volterra series are used to model narrowband memory APAs and wideband memory APAs, respectively. While the coefficients of Volterra series model are too complex and the coefficients of MP model are too simplified. As a compromise, the GMP model can achieve better modeling results than the MP model by considering the generalized memory effect and the number of coefficients can be acceptable for LEO satellite communication.  
The significance of the APA behavior model is that it can inverse model the APA and be used to compensate for the signal distortion caused by the APA nonlinearity. In this broadband OFDM signal and array antennas for the LEO satellite communication application scenario, it is necessary to trade off the computational complexity and algorithm performance. The MP model is a simplified form of Volterra for narrowband signals as well as unit antenna systems with few model coefficients. The MP model with nonlinear order \(K\) and memory depth \(Q\) can be written as  
\[y_{MP}(n)=\sum_{k=0}^{K-1}\sum_{q=0}^{Q-1}c_{kq}x(n-q)|x(n-q)|^{k}, \tag{12}\]  
where \(x(n)\) and \(y_{MP}(n)\) are the input and output signals of the model, and \(c_{kq}\) represents the coefficients of the model. The total number of coefficients is \(O_{MP}=K\cdot Q\).  
Its performance is limited because it contains only the diagonal terms of Volterra. While in some application scenarios, non-diagonal terms can significantly impact the model accuracy. Although the neural network model is very flexible and adaptable, the large number of coefficients requires a large number of space-borne computing resources, so it cannot be deployed on satellites by far. The GMP model performs better for strongly nonlinear dynamical systems because it supplements the consideration of generalized memory effects than the MP model. Therefore, the GMP model is adopted as the model of pre-distorter in this paper  
\[y_{GMP}(n) =\sum_{k=0}^{K_{a}-1}\sum_{l=0}^{L_{a}-1}a_{kl}x(n-l)|x(n-l)|^{k}\] \[+\sum_{k=1}^{K_{b}}\sum_{l=0}^{L_{b}-1}\sum_{m=1}^{M_{b}}b_{klm}x( n-l)|x(n-l-m)|^{k} \tag{13}\] \[+\sum_{k=1}^{K_{c}}\sum_{l=0}^{L_{c}-1}\sum_{m=1}^{M_{c}}c_{klm}x( n-l)|x(n-l+m)|^{k},\]  
where \(x(n)\) and \(y_{GMP}(n)\) are the input and output signals of the model, \(a_{kl}\), \(b_{klm}\), and \(c_{klm}\) represent the model coefficients of the MP sub-model, the lagging term, and the leading term, respectively. \(K_{a}\), \(K_{b}\), and \(K_{c}\) stand for the nonlinear order of each branch term, \(L_{a}\), \(L_{b}\), and \(L_{c}\) denote the memory depth of each branch term, and \(M_{b}\), \(M_{c}\) are the additional degrees of freedom, which determine the maximum selectable range of memory depths for the lagging and leading terms, respectively. The total number of coefficients is \(O_{GMP}=K_{a}L_{a}+K_{b}L_{b}M_{b}+K_{c}L_{c}M_{c}\).