## Appendix A  
Here, we analyze the complexity of this model, assuming that the signal sampling length is \(S=\)2\(\times S_{symbol}\) and the number of branches in the MRP model is \(N_{step}\). For the linear part of MRP, i.e., the zeroth-order solution term, \(U^{(0)}_{x,y}\), the model needs to perform one fast Fourier transform (FFT) and one inverse fast Fourier transform (IFFT) on the sequence in order to perform dispersion operations without nonlinearity. For the dispersion operations of \(S\)-point radix-2 FFT and sequence length, \(S\), the required complex multiplication and complex addition quantities are as follows:  
\[\begin{split} N^{(complex)}_{\times,fft}(S)=\frac{5}{2}\cdot\log_ {2}S\,\ \ N^{(complex)}_{+,fft}(S)=S\cdot\log_{2}S\\ N^{(complex)}_{\times,D}(S)=S\end{split} \tag{1}\]  
where \(N^{(complex)}_{\times,fft}(S)\) and \(N^{(complex)}_{+,fft}(S)\), respectively, represent the number of complex multiplications and complex additions required for a fast Fourier transform. \(N^{(complex)}_{\times,D}(S)\) represents the complex multiplication required to add dispersion to the signal. Therefore, the complexity of the zeroth-order solution for a sequence of length, \(S\), is \(2\times N^{(complex)}_{\times,fft}(S)+N^{(complex)}_{\times,D}(S)\) complex multiplication and \(2\times N^{(complex)}_{+,fft}(S)\) complex addition. Due to the operation rules of complex numbers, one complex multiplication is equal to four real multiplication and two real addition, and one complex addition is equal to two real addition. Therefore, the computational complexity of the zeroth-order solution is as follows:  
\[N^{(0,real)}_{\times}(S)=4S\cdot(\log_{2}S+1)\ \,\ \ N^{(0,real)}_{+}(S)=6S\cdot \log_{2}S+2S \tag{2}\]  
The number 0 or 1 in the subscript above the letter \(N^{(0,real)}_{\times,fft}\) here represents the order of the solution. For the first-order nonlinear solutions, each branch requires two FFT operations, two IFFT operations, two dispersion operations, and one nonlinear operation. The complexity of single FFT and dispersion operations is the same as Equation (1), while nonlinear operations include 2\(\times\)S complex multiplication and one complex modulo squared operation. The complex modulo squared operation of dual polarization is equivalent to four real multiplication and three real addition operations. Therefore, for the first-order part of an MRP model with \(N_{step}\) branches, the computational complexity is as follows:  
\[\begin{split} N^{(1,real)}_{\times}(S)&=N_{step} \left\{\begin{aligned} & 4N^{(complex)}_{\times,fft}(S)+ \underbrace{2N^{(complex)}_{\times,D}(S)}_{\text{dispersion}}+\underbrace{2S} _{\text{nonlinear}}\end{aligned}\right\}\\ &=N_{step}\cdot[8S\log_{2}S+20S]\end{split} \tag{3}\]
where the last term \(2S\) of \(N_{+}^{(1,\text{real})}\) in Equation (A3) represents the addition operation between the branches of the MRP model. In addition, the reason why there are only two complex multiplication operations for nonlinear operations is that the nonlinear coefficients can be calculated before signal transmission to directly multiply the signal and its modulus square. Therefore, the total computational complexity of the MRP model is show as Equation (13).