## 5 Results  
In this section, we use Monte Carlo simulations to conduct in-depth research on the evaluation of target estimation and symbol detection performance in different scenarios. We conduct these simulations with an average over \(10^{4}\) OTFS frames. For each OTFS frame, we configure the parameters with \(M_{S}=32\) and \(N_{T}=16\), indicating the presence of 32 subcarriers and 16 time slots within the TF domain. The ISs are four-QAM modulated from the data bits generated randomly. In addition, simulation results show that the proposed hybrid MP detector has a high effectiveness. The effectiveness of the hybrid message-passing algorithm is substantiated, and comprehensive comparisons are made, specifically in relation to conventional detection algorithms encompassing Gaussian-approximation-based MP methods. A uniform maximum iteration limit, denoted as \(I_{\max}\), is uniformly established as 10 for all algorithms under consideration. We investigate two different simulations, one involving four targets and the other involving six targets. Our analysis involves the computation of the ratio between the root mean square error (RMSE) and the corresponding resolution in estimating both R and V. We vary the SNR and determine the RMSEs in velocity estimation and range estimation concerning the SNR. The results of these simulations are visually presented in Figures 5 and 6. As depicted in Figure 5, a noticeable trend emerges when there are four targets: the RMSE associated with the estimated velocity significantly diminishes when the fractional Doppler index is determined through (26). This observation underscores the efficacy of our proposed algorithm. Conversely, when the scenario involves six targets, the RMSE performance deteriorates at identical SNR levels. This degradation is attributed to the heightened interference arising from the overlapping responses of each symbol, rendering the approximation in (25) less precise. Consequently, the accuracy of velocity estimation suffers. Similarly, as illustrated in Figure 6, when the simulation entails four targets, the RMSE in range estimation is reduced, and the results of the target estimation substantially deteriorate only if the integer part of the indices is considered. This phenomenon aligns with the increased interference and inaccuracies associated with the estimation process in scenarios with a greater number of targets. When considering only the integer components, it becomes evident that the RMSE in range estimation remains largely unaffected with respect to varying SNRs. This phenomenon arises due to the limited resolutions of R and V within each grid. The average minimum errors align with half of the resolution values, a correspondence that is consistent with our simulation findings. In the simulations of the proposed hybrid MP detector, we employ a random selection of DD parameters for each path, ensuring the fractional Doppler indices. Note that, for both the Gaussian-approximation-based MP and hybrid MP algorithms, the damping factor is set to 0.7. We initiate our analysis by examining the influence of the parameter \(N_{i}\). Specifically, we apply both the Gaussian-approximation-based MP and the hybrid MP algorithms to various values of \(N_{T_{i}}\) under specific SNR conditions, where the SNR \(=18\) dB and \(P=6\). The acquired findings are depicted graphically in Figure 7.  
Figure 5: The RMSE in velocity estimation concerning the SNR. Figure 6: The RMSE in range estimation concerning the SNR. Remarkably, a substantial improvement in the bit error rate (BER) is discernible as \(N_{T_{i}}\) increases. As \(N_{T_{i}}\) reaches 2 and beyond, it becomes apparent that the performance of the Gaussian-approximation-based MP stabilizes, while the hybrid MP's performance continues to improve. Furthermore, it is worth noting that, across all values of \(N_{i}\), the hybrid MP consistently outperforms the Gaussian-approximation-based MP.