## 2 DPS Modeling of LEO-SIN
### DPS Modeling Accumulated Error Compensations by Converging Piecewise Functions to the ACF
This type of modeling error can be described by the elevation angle. Even if the satellites are moving fast, the serving satellite on the top of UEs still obeys the same track and process. That is, the elevation angle would change from low to 90 degrees, \(N_{(a)}\) obeys the same rule. When the elevation angle is low, \(N_{(a)}\) would increase, while the elevation angle is high, \(N_{(a)}\) would decrease. According to the NTN project, \(N_{(a)}\) can be set by the constraining of different scenarios. 3. The characteristics of DFO and DPS are periodic. The signals from one satellite to the next satellite the UEs receive suffer from the same environment and process. The channel responses and changes are periodic. That is, the DFO and DPS and their modeling error should only have a few differences between this interval (satellite 1 on the top) and the next interval (satellite 2 on the top). Therefore, the former DPS modeling error can be exploited to compensate for the modeling error of DPS at the next interval. So, we model the periodic maximum DFO \(f_{max(a)}\) to improve the DPS modeling error. Since there is an obvious large correlation at the moment of observed starting and the moment of \(\tau\)_max_, we design a Converging Piecewise Function (CPF) for the ACF of RMEDS-SP with many simulations. Note that there is not only one CPF. Here, the CPF is, \[S_{cpf}(t)=\left\{ \begin{array}{l}\frac{1}{\sqrt{1+\left(\frac{t\tau_{max (a)}}{\tau N_{(a)}}\right)^{2}}}\quad,\ 0<t\leqslant\frac{\tau}{2}\\ \frac{1}{\sqrt{1+\left(\frac{\tau-t}{\frac{KN_{(a)}}{\tau}}\right)^{2}}}\,,\ \frac{\tau}{2}<t\leqslant\tau \end{array} \right.\] (21)  
Equation (21) shows the relation between elevation change and modeling error, where \(\tau\) denotes the relative time. The CPF is introduced to the RMEDS-SP model through multiplication by the ACF, the new model is expressed as,  
\[U_{c}^{(k)}\left(t\right)=S_{cpf}(t)\left(Z_{1}^{(k)}(t)+jZ_{2}^{(k)}\left(t \right)\right) \tag{22}\]  
\[Z_{ci}^{(k)}(t)=S_{cpf}(t)\sqrt{\frac{2}{N_{i}}}\sum_{n=1}^{N_{i}}\cos\left(2 \pi f_{max}\sin\left(\alpha_{i,n}^{(k)}\right)t+\varphi_{i,n}^{(k)}\right) \tag{23}\]  
\[R_{czizi}^{(k)}=\int_{-\infty}^{+\infty}S_{cpf}(t)(t)S(t+\tau)z_{i}^{(k)}(t+ \tau)dt \tag{24}\]  
\[R_{czizi}=\frac{1}{K}\sum_{k=1}^{K}R_{zizi}^{(k)} \tag{25}\]  
When \(k\) is finite, since the number of sinusoids and sets of sinusoids is not enough, the large modeling error is inevitable, and it will directly affect the important ACF. After introducing the proposed CPF, the modeling error would be compensated to a certain extent at the divergence part. When \(K\rightarrow\infty\), since there are lots of sinusoids and sets of sinusoids, the effect of the simulation of the DPS model is very close to the theoretical model. At this moment, we do not need CPF to compensate for the accumulated modeling errors. Here, \(\underset{K\rightarrow\infty}{lim}\tau_{max}=\frac{K\tau N}{20/max}=\infty\), that is, the values of \(S_{cpf}(t)\) approaches to 1, and the convergence effect in the time-domain is approximately 0. The ACF of the final model is expressed as follows,  
\[\underset{K\rightarrow\infty}{lim}R_{\mu_{i}\mu_{i}}(\tau)=\frac{2}{\pi}\sum _{n=1}^{N_{i}}\int_{-\frac{\tau}{4}\pi_{i}}^{\frac{\tau}{2N_{i}}}\cos\left\{2 \pi f_{max}\tau\cdot\cos\left[\frac{\pi}{2N_{i}}\left(n-\frac{1}{2}\right)+ \alpha\right]\right\}d\alpha\]\[=\frac{2}{\pi}\sum_{n=1}^{N_{i}}\int_{\frac{\pi(i-1)}{2N_{i}}}^{\frac{ \pi n}{2N_{i}}}\cos(2\pi f_{max}\tau\cos\alpha)d\alpha\] \[=\frac{2}{\pi}\int_{0}^{\pi/2}\cos(2\pi f_{max}\tau\cos\alpha)d\alpha.\] \[=J_{0}(2\pi f_{max}\ \tau) \tag{26}\]  
where \(i=1,2\), \(N_{i}\geq 1\). From Equation (26), we can figure out that when \(K\rightarrow\infty\), the effect of the simulation of the DPS model is very close to the theoretical model. When the number of sinusoids \(K\) and sets of sinusoids \(N\) are small, the CPF can also improve the modeling error. The detailed analysis would be further demonstrated in the next chapter.