## 2 DPS Modeling of LEO-SIN
### DPS Modeling Accumulated Error Compensations by Converging Piecewise Functions to the ACF  
The above proposed RMEDS-SP model can reduce the modeling error to a certain extent, and we still need to further compensate the accumulated modeling errors from the residual error after the compensations by RMEDS-SP at each time. DPS would change with a certain periodicity by the periodicity of a satellite moving in LEO-SIN. Those periodic changes can be introduced into the simulations model. The accumulated errors can be compensated by a group of converging piecewise functions to the ACF of DPS. First, we analyze the accumulated errors of the DPS simulation model with time. The square error function (SEF) \(E(\tau)\) is observed,  
\[E(\tau)=\left|\vec{R}_{z_{i}z_{i}}(\tau)-R_{z_{i}z_{i}}(\tau)\right|^{2} \tag{19}\]where, \(\widetilde{R}_{z_{2}z_{2}}(\tau)\) denotes the ACF of a simulation model, and \(R_{z_{2}z_{2}}(\tau)\) is the ACF of the reference model. The number of sinusoids is considered permanently. The SEF of RMEDS-SP is shown in Figure 7. As Figure 7 shows, we can calculate that when the relative time is smaller than \(\tau\)\({}_{max}\), the square error of RMEDS-SP is small. Here, \(\tau\)\({}_{max}\) denotes the maximum reliable interval between the observed moment and referenced moment (the relative time). When the relative time exceeds \(\tau\)\({}_{max}\), the square error of ACF will become obviously larger. Figure 7 also shows the performance of the square error of MEDS-SP and RMEDS-SP. The number of sinusoids is set as \(N_{1}=20\), \(N_{2}=21,K=4\), the unit of relative time is set as 1 ms, the maximum Doppler spread caused by multi-path is set as \(f_{max}=100\) HZ, and \(\tau\)\({}_{max}\) is set as 126 ms. Such a parameter setting can provide a better accuracy fit with guaranteed efficiency in simulation and subsequent hardware implementation, and a similar parameter setting also appears in [19; 27]. When relative time is less than 110 ms, the square errors of both models are at a low level of 0.007, and that of RMEDS-SP is lower than that of MEDS-SP. When the relative time is more than 126 ms, the square error would increase obviously, which is more than 0.025. The square error of RMEDS-SP is also lower than that of MEDS-SP. We follow the corresponding relations, \(\tau\)\({}_{max}\) has positive relations with the number of sets \(K\) and the number of sinusoids \(N\) in each set, has negative relation with maximum DPC \(f_{max}\)[27]. Therefore, the \(\tau\)\({}_{max}\) of proposed RMEDS-SP is assumed as \(\tau\)\({}_{max}=\gamma\frac{KN}{f_{max}}\), \(\gamma=\frac{\pi}{20}\). Since the limited number of sinusoids is the essential precondition for improvement, we just exploit the periodic features by the periodicity of a satellite moving to improve the DPS model. There are three features:  
1. The ratio of LOS power to NLOS power is limited. Since existing LOS signals, the ratio \(K_{DPS}\) of LOS power to NLOS power is limited. However, for the randomness power distribution of SOS, some power of multi-path would be larger than that of LOS signals. Therefore, this \(K_{DPS}\) can be exploited to control the accumulated errors of the DPS simulation model. According to Equation (19), when the square errors at this moment are more than the maximum of square range from \(K_{DPS}\), maximum of square error range from \(K_{DPS}\) would be taken to compensate for the modeling error. When the square errors at this moment are less than the maximum square range from \(K_{DPS}\), the square errors of ACF would be taken to compensate. So, we can eliminate some obvious large error of RMEDS-SP by the power controlling, \[Z_{li}^{(k)}(t)=min\Big{\{}Z_{i}^{(k)}(t)S(t),K_{DPS}h_{LOS}(t)\Big{\}}\] (20)  
Figure 7: The error of ACF between the MEDS-SP and RMEDS-SP model. 2. The DPS modeling error is limited for the limited number of AOA \(N_{(a)}\). Since the power of multi-path signals are relatively weak, \(N_{(a)}\) from multi-path can only contribute a little to the whole DPS.