[Paper Review] On the Diversity of OTFS Modulation in Doubly-Dispersive Channels.

G. D. Surabhi, Rose Mary Augustine+1 more|arXiv (Cornell University)|Aug 23, 2018

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TL;DR

This paper formally analyzes the diversity order of Orthogonal Time Frequency Space (OTFS) modulation in doubly dispersive channels, proving that its asymptotic diversity order is one. However, it demonstrates that finite-SNR gains can yield higher-order diversity, and proposes a phase rotation scheme using transcendental numbers to achieve full diversity in the delay-Doppler domain.

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 $ ightarrow \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.

Motivation & Objective

• To formally analyze the diversity order of OTFS modulation in doubly dispersive channels, addressing a gap in prior literature that assumed full diversity without proof.
• To investigate the behavior of diversity in the finite signal-to-noise ratio (SNR) regime, where higher-order diversity may emerge before the asymptotic regime is reached.
• To propose a novel phase rotation scheme using transcendental numbers to enhance diversity gain in OTFS.
• To demonstrate that the proposed scheme enables OTFS to achieve full diversity in the delay-Doppler domain, overcoming the limitation of asymptotic diversity one.

Proposed method

• Theoretical analysis of the diversity order of OTFS is conducted using stochastic modeling of the delay-Doppler channel, focusing on symbol error rate (SER) behavior as SNR approaches infinity.
• Simulation-based evaluation is used to examine the diversity behavior in the finite SNR regime, particularly the transition from higher-order diversity to asymptotic diversity one.
• A phase rotation scheme is introduced, where pilot or data symbols are rotated using transcendental numbers (e.g., π or e) to decorrelate channel responses across delay-Doppler taps.
• The proposed phase rotation is applied to the OTFS signal constellation to ensure that multipath components in the delay-Doppler domain are distinguishable and contribute independently to diversity gain.
• The performance of the phase-rotated OTFS is evaluated via bit error rate (BER) simulations across various frame sizes and SNR levels.
• Theoretical bounds on diversity order are derived and compared with simulation results to validate the claim of full diversity under the proposed scheme.

Experimental results

Research questions

• RQ1 What is the true asymptotic diversity order of OTFS modulation in doubly dispersive channels, and does it match the commonly assumed full diversity?
• RQ2 How does the diversity behavior of OTFS change in the finite SNR regime, and at what SNR levels does the asymptotic diversity one regime dominate?
• RQ3 Can a phase rotation scheme using transcendental numbers improve the diversity gain in OTFS, and if so, under what conditions?
• RQ4 Does the proposed phase rotation scheme enable OTFS to achieve full diversity in the delay-Doppler domain, as opposed to only asymptotic diversity one?
• RQ5 How does frame size influence the onset of the diversity one regime in OTFS systems?

Key findings

• The asymptotic diversity order of OTFS in doubly dispersive channels is formally proven to be one, contradicting the assumption of full diversity in prior literature.
• In the finite SNR regime, OTFS exhibits a higher-order diversity gain before transitioning into the asymptotic diversity one regime, indicating performance gains at moderate SNR levels.
• The onset of the diversity one regime occurs at lower bit error rate (BER) values for larger frame sizes, suggesting that frame size impacts the SNR threshold for diversity saturation.
• The proposed phase rotation scheme using transcendental numbers successfully enables OTFS to extract full diversity in the delay-Doppler domain, overcoming the limitation of asymptotic diversity one.
• Simulations confirm that the phase-rotated OTFS achieves improved error performance, particularly in the finite-SNR region, validating the theoretical claims.
• The use of transcendental numbers in phase rotation ensures sufficient decorrelation of multipath components, allowing independent fading gains to contribute to diversity.