## 2 System Model
### Nonlinear PA Model  
The vector \(\mathbf{s}=[s_{1},s_{2},\ldots,s_{K}]^{T}\in\mathbb{C}^{K\times 1}\) denotes the transmission signal, where \(\mathbb{E}\{\mathbf{s}\}=0\) and \(\mathbb{E}\left\{\mathbf{s}\mathbf{s}^{H}\right\}=\mathbf{I}\) are satisfied. On the \(l\)th satellite, the signal is firstly digitally precoded at baseband by a digital precoding matrix \(\mathbf{F}_{l}^{\mathbf{D}}=\left[\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{ \mathbf{     }}}}}}}}}}}}}} \right]\) is precoded by an analog precoder \(\mathbf{F}_{l}^{\mathbf{A}}=\text{blkdiag}\left[\mathbf{F}_{l}^{\mathbf{A}}, \mathbf{F}_{l}^{\mathbf{A}},\ldots,\mathbf{F}_{l}^{\mathbf{A}}\right]\in \mathbb{C}^{N\times N_{RF}}\), where \(\mathbf{f}_{l,l}^{\mathbf{A}}\) contains non-zero elements corresponding to phase shifters in the network connected to the \(i\)th RF chain of size \(N_{g}\times 1\). \(\mathbf{F}_{l}=\mathbf{F}_{l}^{\mathbf{A}}\) denotes the hybrid precoding matrix on the \(l\)th LEO satellite, where \(\mathbf{f}_{k,l}=\mathbf{F}_{l}^{\mathbf{A}}\mathbf{F}_{k,l}^{\mathbf{D}}\) denotes the precoding of the \(l\)th satellite to the \(k\)th GU. \(\mathbf{F}^{\mathbf{D}}=\left[\mathbf{F}_{1}^{\mathbf{D}},\mathbf{F}_{2}^{ \mathbf{D}},\ldots,\mathbf{F}_{L}^{\mathbf{D}}\right]^{T}\in\mathbb{C}^{LN_{ RF}\times K}\) denotes the digital precoding matrix on \(L\) LEO satellites, \(\mathbf{F}^{\mathbf{D}}=\left[\mathbf{F}_{1}^{\mathbf{D}},\mathbf{F}_{2}^{ \mathbf{D}},\ldots,\mathbf{F}_{L}^{\mathbf{D}}\right]^{T}\in\mathbb{C}^{LN_{ RF}\times K}\) denotes the analog precoding matrix on \(L\) LEO satellites. According to the above hybrid architecture, the expression for the hybrid precoder of the whole system is \(\mathbf{F}=\mathbf{F}^{\mathbf{A}}\mathbf{F}^{\mathbf{D}}\), and \(\mathbf{F}=\left[\mathbf{F}_{1},\mathbf{F}_{2},\ldots,\mathbf{F}_{L}\right]^{T }\in\mathbb{C}^{LN\times K}\). On the \(l\)th satellite, the signal after hybrid precoding is denoted by \(\mathbf{u}_{l}=\mathbf{F}_{l}\mathbf{s}=\mathbf{F}_{l}^{\mathbf{A}}\mathbf{F}_ {l}^{\mathbf{D}}\mathbf{s}\), \(\mathbf{u}_{l}=[u_{l,1},u_{l,2},\ldots,u_{l,N}]\in\mathbb{C}^{N\times 1}\). We can obtain \(\mathbf{U}_{l}=\mathbb{E}\big{\{}\mathbf{u}_{l}\mathbf{u}_{l}^{H}\big{\}}= \mathbf{F}_{l}\mathbf{F}_{l}^{H}\) and \(\mathbf{U}=\mathbf{F}^{H}\). After being precoded, the signal vector \(\mathbf{u}_{l}\) is amplified by the NPAs. Assuming that each NPA has the same input-output relation \(f(\bullet)\), the output signal of the \(n\)th NPA on the \(l\)th satellite is [21]  
\[x_{l,n}=f(u_{l,n})=\sum_{m=0}^{M}\beta_{2m+1}\big{|}u_{l,n}\big{|}^{2m}u_{l,n}, \tag{6}\]  
where \(\beta_{2m+1}\) denotes Complex \((2m+1)\)th-order model coefficient. The instantaneous gain of the \(n\)th NPA is expressed as  
\[\rho_{l,n}=\frac{x_{l,n}}{u_{l,n}}=\sum_{m=0}^{M}\beta_{2m+1}\big{|}u_{l,n} \big{|}^{2m}. \tag{7}\]Each NPA employs a memoryless nonlinear model, which is typically valid within a moderate bandwidth range. As demonstrated in [26], only the odd orders of the polynomial contribute to the in-band distortion. Since the focus is only on the in-band distortion, which can be characterized by odd orders, the nonlinear effects of NPAs are generally smooth. Thus, we can limit our attention to the limited-order polynomial NPA model by setting M = 1 in Equation (7) [27], which is modeled as [23]  
\[\rho_{l,n}=\beta_{1}+\beta_{3}\big{|}u_{l,n}\big{|}^{2}. \tag{8}\]  
On the \(l\)th satellite, the output signal of the NPAs can be expressed as [21]  
\[\mathbf{x}_{l}=\mathbf{G}_{l}\mathbf{u}_{l}=\mathbf{\bar{G}}_{l}\mathbf{u}_{l }+\mathbf{d}_{l}\in\mathbb{C}^{N\times 1}, \tag{9}\]  
where \(\mathbf{G}_{l}=\text{diag}[\rho_{l,1},\rho_{l,2},\cdots,\rho_{l,N}]\in\mathbb{C }^{N\times N}\) denotes the instantaneous gain of the NPAs, and \(\mathbf{\bar{G}}_{l}\in\mathbb{C}^{N\times N}\) denotes the average linear amplification gain.