# OTFS: A Potential Waveform for Space-Air-Ground Integrated Networks in 6G and Beyond
## 4 Results and Discussion
### Channels with Doppler Effects  
This section analyses multipath channels under varying levels of mobility, ranging from low speeds (20 km/h) to extreme speeds (2000 km/h). As mobility increases, the Doppler effect becomes more pronounced, leading to time-varying frequency selectivity and ICI. These effects significantly impact the BER performance, making it crucial to evaluate their robustness under different mobility conditions. Figure 7 shows the BER vs. SNR performance of OTFS and OFDM in multipath channels with mobility ranging from 20 to 120 km/h. This is a low-to-moderate mobility scenario, as seen in vehicular communications, drones, and trains in terrestrial networks. It is observed that OFDM performs relatively well in these scenarios, as the effects of ICI caused by Doppler shifts are not significant at these moderate speeds. OFDM is still able to maintain reliable communication with minimal BER degradation, making it suitable for low-to-moderate mobility scenarios. OTFS also demonstrates good performance in these low to moderate mobility scenarios. However, the advantages of OTFS are not significant in low-mobility scenarios because the delay-Doppler domain advantages, which OTFS thrives on, are limited when the Doppler effects are less severe. At lower modulation orders (BPSK and QPSK), OTFS achieves a lower BER than OFDM, highlighting its robustness against interference. However, at higher-order modulations (8-PSK), both OTFS and OFDM exhibit similar BER performance, because higher-order constellations are inherently more sensitive to noise and interference. Figure 8 presents the BER vs. SNR performance of OTFS and OFDM in multipath channels under moderate-to-high mobility conditions (300-500 km/h), typically experienced in high-speed trains and aerial platforms such as high-speed UAVs. Figure 7: BER performance comparison of OTFS and OFDM in a multipath channel with different modulation orders and mobility ranging from 20 to 120 km/h. At these higher mobility levels, Doppler effects become significant, leading to time-varying frequency selectivity and introducing ICI. This results in degradation of the BER performance, particularly at higher SNR values, where the BER curve begins to plateau. OFDM, which is designed for time-invariant or slow-fading channels, fails to track rapid channel variations in high-mobility conditions, making it unsuitable for such scenarios. OTFS, on the other hand, effectively handles Doppler spread by operating in the delay-Doppler domain, maintaining a much lower BER compared to OFDM. Even under high-mobility conditions, OTFS achieves a BER of 10 \({}^{-3}\) at an SNR of approximately 10 dB with BPSK modulation. Unlike OFDM, OTFS maintains consistent performance as mobility increases, demonstrating strong resilience to Doppler shifts and multipath effects, allowing it to sustain lower BER at higher speeds. Figure 9 presents the BER vs. SNR performance of OTFS and OFDM in multipath channels, comparing different modulation orders (BPSK, QPSK, and 8-PSK) under extreme mobility scenarios at speeds between 1000 km/h and 2000 km/h. These conditions are representative of high-speed NTNs such as military jets and supersonic aircrafts. Figure 8: BER performance comparison of OTFS and OFDM in a multipath channel with different modulation orders and high mobility ranging from 300 to 500 km/h. Figure 9: BER performance comparison of OTFS and OFDM in a multipath channel with different modulation orders in extreme mobility scenarios (1000 km/h to 2000 km/h). At these extreme mobility conditions, OFDM experiences severe Doppler effects, leading to significant ICI and poor BER performance, due to time-varying frequency selectivity. As the modulation order increases (from BPSK to 8-PSK), the BER worsens because higher-order modulations are more susceptible to noise and interference. The BER curves for OFDM tend to flatten out, indicating that increasing SNR does not significantly improve performance, making OFDM unsuitable for such extreme mobility conditions.