On the Diversity of Uncoded OTFS Modulation in Doubly-Dispersive Channels (1808.07747v2)

Abstract: Orthogonal time frequency space (OTFS) is a 2-dimensional (2D) modulation technique designed in the delay-Doppler domain. A key premise behind OTFS is the transformation of a time varying multipath channel into an almost non-fading 2D channel in delay-Doppler domain such that all symbols in a transmission frame experience the same channel gain. It has been suggested in the recent literature that OTFS can extract full diversity in the delay-Doppler domain, where full diversity refers to the number of multipath components separable in either the delay or Doppler dimension, but without a formal analysis. In this paper, we present a formal analysis of the diversity achieved by OTFS modulation along with supporting simulations. Specifically, we prove that the asymptotic diversity order of OTFS (as SNR $\rightarrow \infty$) is one. However, in the finite SNR regime, potential for a higher order diversity is witnessed before the diversity one regime takes over. Also, the diversity one regime starts at lower BER values for increased frame sizes. We also propose a phase rotation scheme for OTFS using transcendental numbers. We show that OTFS with this proposed scheme extracts the full diversity in the delay-Doppler domain.

• The paper finds that the asymptotic diversity order of uncoded OTFS in doubly-dispersive channels is one, contrary to prior suggestions of full diversity.
• Uncoded OTFS can achieve higher diversity in the finite SNR regime, particularly with larger frame sizes before the asymptotic diversity one is reached.
• A novel phase rotation scheme using transcendental numbers is proposed to enable uncoded OTFS to achieve full diversity in the delay-Doppler domain.

An Analysis of Diversity in Uncoded OTFS Modulation in Doubly-Dispersive Channels

The paper under review presents a formal analysis of the diversity aspects of Orthogonal Time Frequency Space (OTFS) modulation when applied to communication systems operating in doubly-dispersive channels. The focus of the paper is on assessing the diversity order achievable with OTFS, which is a critical factor for evaluating the robustness and reliability of wireless communication systems.

Key Findings

1. Asymptotic Diversity Order:
   • The paper demonstrates that the asymptotic diversity order of uncoded OTFS, defined as Signal-to-Noise Ratio (SNR) approaches infinity, is one. This finding contrasts with prior suggestions in the literature that OTFS could potentially achieve full diversity in the delay-Doppler domain, which is defined by the number of distinct multipath components that can be separated.
2. Finite SNR Regime:
   • OTFS demonstrates potential for higher diversity orders within the finite SNR regime, particularly observable before the asymptotic regime sets in. This behavior is attributed to the increased frame sizes where the diversity one regime appears at lower Bit Error Rate (BER) values.
3. Phase Rotation Scheme:
   • A novel contribution of the paper is the proposal and analysis of a phase rotation scheme utilizing transcendental numbers. This scheme enables OTFS to achieve full diversity in the delay-Doppler domain. The proposed method effectively manipulates the phases to ensure that, even at high SNR, the diversity order encompasses all separable paths in the channel.
4. Extension to MIMO Systems:
   • The diversity analysis extends to Multiple Input Multiple Output (MIMO) OTFS systems, where it was established that the asymptotic diversity order equals the number of receive antennas. This is aligned with the traditional understanding of MIMO systems where diversity is augmented by spatial information.

Implications and Future Directions

The findings of this paper crucially outline the limits and potential of uncoded OTFS in terms of diversity, emphasizing its advantages in short-term reliability (finite SNR regime) and with strategic enhancements such as phase rotations for long-term diversity gains (asymptotic regime). The proposed phase rotation has immediate practical benefits for increasing robustness in high-mobility environments such as vehicular and high-speed train communications, which are typical use cases for OTFS.

Moreover, these insights provide valuable guidance for the design and implementation of future OTFS systems. Implementing phase rotation could be particularly beneficial for vehicular communication applications like those standardized by IEEE 802.11p, where doubly-dispersive effects are significant due to high mobility and need for reliable communication over rapidly time-varying channels.

The findings open avenues for further research to explore:

• Optimization of phase rotation parameters for enhanced coding gain.
• Integration of coding schemes to complement the diversity gains provided by modulation techniques.
• Investigation into synchronization and adaptation strategies that can work synergistically with the proposed OTFS enhancements.

Conclusion

This paper makes a significant contribution to understanding the theoretical limits and practical enhancements of OTFS modulation in doubly-dispersive channels. It balances between formal mathematical derivations and simulations, providing a comprehensive picture of how phase adjustments can recover full diversity, an essential factor for achieving robust and efficient wireless communications. The insights from this paper can significantly influence the development and optimization of next-generation wireless systems operating under challenging channel conditions.