## 2 DPS Modeling of LEO-SIN
### DPS Simulation Model  
The above theoretical model of DPS needs to make its envelopes discrete on each frequency to load in the simulations and hardware implementation. Existed models such as MEDS, RMEDS, and MEDS-SP, are all based on SOS, which superpose a limited number of sinusoids and control their amplitudes, frequencies, and phases to approximate Gaussian Colored Noise (GCN). They can simulate different signals fading. In the method of MEDS, the number of sinusoids is limited [24]. The larger the number \(N_{i}\), the better the simulation performances [25]. The Gaussian random process \(\hat{\mu}_{i}(t)\) can be expressed as,  
\[\hat{\mu}_{i}(t)=\sum_{n=1}^{N_{i}}c_{i,n}\cos(2\pi f_{i,n}t+\varphi_{i,n}) \tag{6}\]  
where, \(c_{i,n}\), \(\varphi_{i,n}\), and \(f_{i,n}\) denote the amplitude, phase, and frequency of the \(n\)th sinusoid. \(\varphi_{i,n}\) obeys the uniform distribution range on \((-\pi,\pi]\) and is independent of each other. Based on SOS, \(c_{i,n}\) is set as a constant, \(f_{i,n}\) and \(\theta_{i,n}\) are random variables in MEDS, whose random process is divided into two orthogonal components \(Z_{1}(t)\) and \(Z_{2}(t)\) of I and Q components, and the Gaussian process can be expressed as:  
\[U\left(t\right)=Z_{1}(t)+jZ_{2}(t) \tag{7}\]  
\[Z_{i}(t)=\sqrt{\frac{2}{N_{i}}}\sum_{n=1}^{N_{i}}\cos(2\pi f_{max}\sin(\alpha _{i,n})+\varphi_{i,n}) \tag{8}\]  
\[\alpha_{i,n}=\frac{2\pi n-\pi}{4N_{i}},\;\;i=1,2 \tag{9}\]  
where \(N\) denotes the number of multi-path, and \(f_{max}\) is the maximum DFO. \(\alpha_{i,n}\) denotes the AOA of the nth path determined by the moving direction of the receiver. \(\varphi_{i,n}\) denotes the phase change of the \(n\)th path. With static amplitude-frequency-phase, MEDS cannot describe and simulate the random phase and AOA of multi-path. The reason is that the orthogonality of each superimposed power spectrum of sinusoids would destruct by the introduction of generalized non-stationarity. The method of RMEDS was proposed to improve the orthogonality of each superimposed power spectrum of sinusoids through setting phase and AOA random [25; 26]. Based on MEDS, the AOA \(\alpha_{i,n}\) of the \(n\)th path in RMEDS is,  
\[\alpha_{i,n}=\frac{\pi}{2N_{i}}\bigg{(}n-\frac{1}{2}\bigg{)}+\frac{\theta_{i,n }}{4N_{i}},i=1,2 \tag{10}\]  
where \(\theta_{i,n}\) obeys the uniform distribution of \((-\pi,\pi)\). However, the simulation at one AOA in the traversal only describes the case of that AOA, which cannot enhance the performance for all the cases of AOA, even if we increase the time of simulations in RMEDS. By increasing the number of sinusoids, the complexity of simulations would increase too. The models and performances of MEDS and RMEDS are shown in Figure 4.  
To reduce the complexity of simulations and implementation, MEDS-SP [27] was proposed, with part sinusoids into K sets according to AOA and the average of each set to improve the performances. The model of MEDS-SP is shown in Figure 5.  
Figure 4: (a) The models and performances of MEDS. (b) The models and performances of RMEDS. Figure 5: The models and performance of MEDS-SP. The Gaussian process can be expressed as,  
\[Z_{i}^{(k)}(t)=\sqrt{\frac{2}{N_{i}}}\sum_{n=1}^{N_{i}}\cos\Bigl{(}2\pi f_{max} \sin\Bigl{(}a_{i,n}^{(k)}\Bigr{)}t+\varphi_{i,n}^{(k)}\Bigr{)} \tag{11}\]  
\[U\left(t\right)=Z_{1}(t)+jZ_{2}(t) \tag{7}\]0  
In each set of sinusoids, the traversal of phase and AOA are exploited to a relatively small scale to improve complexity. Meanwhile, the modeling error can also be compensated by the average of each AOA in the set. Results of the Random Method of Exact Doppler Spread Method Set Partitioning (RMEDS-SP) and Correlation Piecewise Convergence  
Traditional DPS modeling methods such as MEDS, RMEDS, and RMEDS-SP cannot provide enough accuracy and efficiency to describe and model the DPS in the simulations for LEO-SIN.