## 1 Introduction
To adapt to different scenarios, there are three types of SOS: The SOS with a static amplitude-frequency-phase, the SOS with dynamic amplitude  
Figure 1: Theoretical analysis, simulation model, and hardware implementation model for the channel. frequency-phase, the SOS with static amplitude-frequency, and an assembling dynamic phase. The SOS with static amplitude-frequency-phase, such as Method of Exact Doppler Spread (MEDS), assumes that the amplitude, frequency, and phase of the sinusoids are static. MEDS improved the efficiency of simulations and reduced the complexity of model implementation. However, with static amplitude-frequency-phase, MEDS cannot describe and simulate the random phase and AOA of multi-path, which reduces the accuracy of simulations. The SOS with a dynamic amplitude-frequency-phase such as the Random Method of Exact Doppler Spread method (RMEDS), assumes that the amplitude, frequency, and phase of the sinusoids are random. RMEDS can describe and simulate the random phase and AOA of multi-path by the traversal of AOA. However, the simulation with a fixed AOA in the traversal only describes the case of that AOA, which cannot enhance the performance for all cases of AOA, even if we increase the time of simulations. Increasing the number of sinusoids would increase the complexity of the simulation. The SOS with static amplitude-frequency and assembling dynamic phase, such as Method of Exact Doppler Spread-Set Partitioning (MEDS-SP), assembles different AOA to some pieces. Each AOA can be traversal in each piece, the average value of results can balance the simulation accuracy and complexity. Considering the main path not at the center point of DPS, asymmetrical DPS, and multi-path signals with a random AOA in LEO-SIN, the DPS of LEO-SIN must solve the problem of simulations of random AOA and its variable range. Hence, we combine the advantages of RMEDS and MEDS-SP to design our new method. In the proposed method, SOS would be parted into different sets, then each AOA would be random and operate traversal in the sets. We call our novel method the Random Method of Exact Doppler Spread method Set Partitioning (RMEDS-SP), which has advantages of having a high accuracy of RMEDS and low complexity of RMEDS. In addition, the simulation errors of the simulation model will increase with time. We can compensate for this error by setting a piecewise function to the ACF convergent to the modeling error supposition with time by periodicity, which changes with the satellites moving. Above all, this paper proposes a novel simulation modeling method and hardware implementation for DPS of LEO-SIN based on error compensations by assembling multi-path signals with random AOA and correlation piecewise convergence. The main contributions are as follows:  
1. We first model the DPS of LEO-SIN. Then RMEDS-SP is firstly proposed to improve the accuracy and efficiency of DPS in simulations and hardware implementation;
2. We exploit piecewise functions to the Autocorrelation Function (ACF) of DPS to converge the modeling error supposition with time by periodic changes, which further improve the accuracy of the DPS model in the scenarios of LEO-SIN;
3. We further demonstrate the hardware implementation of this new method. The rest of this paper is organized as follows. In Section 2, the DPS of LEO-SIN is proposed. The Random Method of Exact Doppler Spread method Set Partitioning (RMEDS-SP) is introduced in Section 3. In addition, the piecewise functions to the ACF of DPS are also introduced. In Section 4, the proposed model implemented in the real hardware platform and tests is introduced.