# A Transformer-Based Channel Estimation Method for OTFS Systems
## 4 Performance Evaluation  
### NMSE and BER Performance  
In this section, the simulation results of our proposed algorithm are given to verify the better performance than the threshold method and other deep learning methods. The parameters of our simulation are given in Table 1.  
Figure 4: PCAF results for real parts and imaginary parts. The NMSE of our channel estimation method is computed as  
\[NMSE=\frac{\left\|H^{\prime}-H\right\|^{2}}{\left\|H\right\|^{2}} \tag{20}\]  
where \(H^{\prime}\) denotes the predicted value, and \(H\) is the true value. In Figure 5, we compared the NMSE performance at different \(SNR\) when the \(SNR_{p}\) is fixed at 30 dB, where \(SNR_{p}=E_{pilot}/E_{data}\). The NMSE curves are close to a horizontal line because the transmitter power is constant, which leads to a reduction in the impact of the \(SNR\). Furthermore, we can observe that the performance of DL methods is better than that of the threshold method. Among these neural networks, the performance of RNN and LSTM exceeds that of DNN. This is because RNN and LSTM have a powerful time series processing capability that DNN does not provide. LSTM slightly outperforms RNN due to the short-term dependency bottleneck inherent in RNN. The NMSE performance of our proposed Transformer estimation method exceeds the threshold method by approximately 18 dB. The Transformer estimation method outperforms LSTM due to its utilization of the multi-headed attention mechanism. This mechanism correlates the input sequences of the encoder with those of the decoder, enabling the model to focus on the encoder input sequences that exhibit a higher correlation with the predicted outputs. As a result, it reduces the impact of irrelevant information and enhances the prediction performance. In Figure 6, the \(SNR\) is fixed at 25 dB, and the NMSE performance is compared at different \(SNR_{p}\). It is evident that the channel estimation performance improves as the \(SNR_{p}\) increases. This is because in the case of the lower \(SNR_{p}\), the interference of noise is extremely serious, thus causing the threshold method to fail. We have observed that even at lower \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency & 3.35 GHz \\ No. of subcarriers (M) & 32 \\ No. of OTFS symbols (N) & 32 \\ No. of channel paths & 3 \\ Subcarrier spacing (\(\Delta f\)) & 15 kHz \\ Channel model & Rayleigh Channel \\ Path delay & [0, 0.2, 0.4] \(\mu\)s \\ Path power & [0, \(-10\), \(-10\)] dB \\ Modulation alphabet & 4-QAM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 5: NMSE performance curves between different algorithms. \(SNR_{p}\), our proposed scheme surpasses the performance of the threshold scheme in terms of channel estimation. For example, the NMSE of the threshold scheme is about 12 dB when the \(SNR_{p}\) = 30 dB, while our Transformer estimation model requires a \(SNR_{p}\) of only 10 dB to achieve similar performance, which allows for a reduction in pilot power overhead. Figure 7 illustrates the NMSE capability of the threshold scheme and other neural networks at different speeds. We can observe a decrease in NMSE performance with increasing velocity; this is because the channel we utilize changes at a slower rate in low-speed scenarios, and the channel estimation results are expected to be slightly superior compared to rapidly changing channels in high-speed scenarios. Although the NMSE performance decreases, the variety is not significant, only about 1-2 dB per 100 km/h; this is due to the good robustness of the OTFS system to high Doppler spread. In Figure 8, we show the BER comparison under different \(SNR\), where the case of perfect channel estimation is also taken into account. It is observed that the BER decreases with increasing \(SNR\). In the case of low \(SNR\), the impact of noise can hinder the ability to accurately reflect the difference in channel estimation performance between individual algorithms.