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On the Diversity of Uncoded OTFS Modulation in Doubly-Dispersive Channels

G. D. Surabhi, Rose Mary Augustine, A. Chockalingam

arXiv:1808.07747v2cs.IT

TL;DR

The paper analyzes the diversity achieved by OTFS in doubly-dispersive channels and proposes a phase-rotation scheme using transcendental numbers. It proves asymptotic diversity order one without phase rotation, while the proposed scheme achieves full delay-Doppler diversity.

  • Problem

    The paper examines the diversity achieved by OTFS in doubly-dispersive channels, where Doppler shifts disrupt conventional multicarrier modulation and the diversity behavior requires formal analysis.

  • Method

    The authors derive SISO OTFS diversity with maximum likelihood detection, support the analysis with BER bounds and simulations, and design a transcendental-number phase-rotation scheme.

  • Results

    OTFS has asymptotic diversity order one, shows higher finite-SNR diversity before that regime dominates, and achieves full delay-Doppler diversity with phase rotation.

  • Takeaways & Limitations

    Phase rotation enables OTFS to extract the full diversity offered by the delay-Doppler channel, despite uncoded OTFS otherwise having asymptotic diversity one.

Abstract

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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.

I. INTRODUCTION

OTFS addresses doubly-dispersive channels by multiplexing symbols in the delay-Doppler domain and transforming them through time-frequency and time representations. The paper identifies an unestablished diversity claim and introduces formal analysis plus a phase-rotation approach for full diversity.

  • Motivation: Doubly-dispersive channels combine time dispersion from multipath propagation with frequency dispersion from Doppler shifts, challenging conventional multicarrier modulation.OFDM performance depends significantly on subcarrier orthogonality, which Doppler shifts can destroy.
  • OTFS background: OTFS multiplexes information symbols in the delay-Doppler domain and spreads them across the time-frequency plane.Its transformations convert a doubly-dispersive channel into an almost non-fading channel in the delay-Doppler domain.
  • Research gap: A formal diversity analysis was missing despite suggestions that OTFS could achieve full delay-Doppler diversity.Here, full diversity means the number of clustered reflectors or multipath components separable in delay or Doppler.
  • Contributions: The paper derives OTFS diversity in SISO systems with ML detection and reports asymptotic diversity order one, while higher diversity can appear at finite SNR.The diversity-one regime begins at lower BER values for increased frame sizes, supported by a BER lower bound based on rank-one difference matrices.
  • Contributions: A phase-rotation scheme using transcendental numbers is proposed and shown to extract full diversity in the delay-Doppler domain.The analysis is also extended to MIMO-OTFS, where asymptotic diversity order equals the number of receive antennas.
  • Implementation: OTFS can be implemented over multicarrier modulation through pre-processing and post-processing transformations.The transmitter maps delay-Doppler symbols to a time-frequency signal using the 2D ISFFT and windowing before time-domain conversion.

C. OTFS modulation •

OTFS maps delay-Doppler information symbols into time-frequency symbols through symplectic Fourier processing and windowing, then transforms them for transmission and demodulation.

  • Fourier processing: The SFFT is defined for the periodized time-frequency signal Xp[n, m], whose period is (N, M).The corresponding inverse operation is expressed through the ISFFT of xp[k, l].
  • OTFS transform: OTFS information symbols x[k, l] are mapped to time-frequency symbols X[n, m] through the ISFFT and transmit windowing.The resulting time-frequency signal is then TF modulated for channel transmission.
  • OTFS demodulation: The received signal y(t) is converted to Y[n, m] using a Wigner filter, receive windowing, and periodization before demodulation.The receive window produces YW[n, m], which is periodized with period (N, M).
  • OTFS demodulation: The SFFT converts the periodized time-frequency sequence Yp[n, m] back into delay-Doppler symbols yp[k, l].The demodulated output is y[k, l] = yp[k, l] for k = 0, 1, · · · , N −1 and l = 0, 1, · · · , M −1.
  • Channel relation: The channel response after windowing enters the delay-Doppler input-output relation through a circular convolution.The circular convolution arises naturally from OTFS ISFFT and SFFT preprocessing and postprocessing, without cyclic prefixing.

D. Vectorized formulation of the input-output relation

The channel model represents P delay-Doppler paths and vectorizes the OTFS input-output relation into a matrix system. Diversity analysis then connects pairwise error performance to the rank of symbol-difference matrices, yielding asymptotic order one for SISO OTFS.

  • D. Vectorized formulation of the input-output relation: A channel with P paths is modeled by reflector clusters, each characterized by a gain, delay, and Doppler shift in the delay-Doppler domain.The parameters hi, τi, and νi denote the gain, delay, and Doppler shift of cluster i.
  • D. Vectorized formulation of the input-output relation: Fractional delay and Doppler parts are initially set to zero, justified by sufficiently fine delay and Doppler resolutions for large M and N.The diversity analysis later extends beyond this initial assumption.
  • D. Vectorized formulation of the input-output relation: With rectangular transmit and receive windows, the OTFS input-output relation can be derived and vectorized using the symbol, received-signal, noise, and channel matrices.The vectors x, y, and v have dimension CMN×1, while H has dimension CMN×MN; symbols belong to modulation alphabet A.
  • III. DIVERSITY ANALYSIS OF OTFS: Modulo operations make the equivalent channel matrix H have only P non-zero elements in each row and column.This sparsity permits an alternate vectorized form of the SISO OTFS input-output relation.
  • III. DIVERSITY ANALYSIS OF OTFS: The SISO diversity analysis uses a symbol matrix X and received vector yT, with channel coefficients absorbed into h′ and additive noise represented by vT.The ith entry of h′ is hie−j2πνiτi, and X is a P×MN symbol matrix.
  • III. DIVERSITY ANALYSIS OF OTFS: The average pairwise error probability is analyzed by diagonalizing the Hermitian matrix (Xi − Xj)(Xi − Xj)H and expressing it through the singular values of Δij.The rank r of Δij determines the exponent of the SNR term in the high-SNR PEP expression.
  • III. DIVERSITY ANALYSIS OF OTFS: The minimum rank among all nonidentical symbol-matrix differences controls the overall BER and therefore determines the achieved diversity order.The PEP with the minimum r dominates the BER at high SNR.
  • III. DIVERSITY ANALYSIS OF OTFS: Asymptotic SISO OTFS diversity order is one because two constant symbol matrices can differ by a rank-one matrix.Thus OTFS does not extract full diversity asymptotically, although it can approach full diversity in the finite-SNR regime under certain conditions.

A. Lower bound on the average BER

The BER lower bound is formed from rank-one pairwise error events and explains why larger OTFS frames can show higher finite-SNR diversity before the asymptotic diversity-one regime.

  • The BER lower bound sums pairwise error probabilities for symbol-matrix differences with rank one.For rank-one differences, the difference matrix has one non-zero singular value.
  • At high SNR, the BER approaches the diversity-one lower bound, whose value depends on the ratio κ over 2^MN.Increasing M and N reduces this lower-bound value because the 2^MN term grows.
  • For M = N = 2 and P = 4, simulated BER, lower-bound, and upper-bound curves nearly coincide at high SNR.This indicates tight bounds in the high-SNR regime for the considered system.
  • Increasing frame size MN lowers the bound and produces higher finite-SNR diversity before the asymptotic diversity order of one takes over.For the compared systems, the lower bound for system-3 lies below system-2, which lies below system-1.
  • The compared systems use M = N = 2, M = 4, N = 2, and M = N = 4, with the latter showing the greatest finite-SNR slope.The higher slope reflects the need to descend farther to meet the lower bound.

C. Results for practical values of M and N in OTFS

Practical-size simulations compare OTFS and OFDM, while the phase-rotation analysis shows how transcendental-number rotations achieve full delay-Doppler diversity.

  • Practical-size BER comparison: OTFS achieves significantly better BER performance than OFDM under the 4 GHz, LTE-sized configuration with MMSE detection.The configuration uses M = 12, N = 7, P = 5, and BPSK.
  • Practical-size BER comparison: OTFS achieves about 5 dB and 10 dB SNR gains over OFDM at BER values of 10^-2 and 10^-3, respectively.These gains are reported for the IEEE 802.11p setting with M = 64, N = 12, P = 8, and BPSK.
  • Phase rotation for full diversity: The phase-rotation design uses a diagonal matrix with distinct transcendental numbers and real, distinct, algebraic exponents.The construction is analyzed through the eigenvalues and rank of phase-rotated symbol-matrix differences.
  • Phase rotation for full diversity: For P = MN, phase rotation makes every eigenvalue nonzero, giving full rank and diversity order MN.The result follows because the minimum rank of every phase-rotated difference matrix equals MN.
  • Phase rotation for full diversity: For P < MN, the phase-rotated difference matrix has rank P, achieving the full diversity of P in the delay-Doppler domain.The argument uses the corresponding rows of the full-rank P = MN matrix.

A. Simulation results

Simulations show that unrotated OTFS has diversity order one, whereas the proposed phase rotation provides full diversity and substantial BER gains.

  • BPSK simulations: Unrotated OTFS has asymptotic diversity order one across the three BPSK systems tested.The systems use M = N = 2, M = 4, N = 2, and M = N = 4.
  • BPSK simulations: Phase-rotated OTFS exhibits full diversity in the high-SNR regime for all three BPSK systems.Each phase-rotated system exhibits diversity order P = 4.
  • BPSK simulations: Different BPSK system sizes show slightly different BER performance because they achieve different coding gains.The phase rotation achieves full delay-Doppler diversity, while phase optimization can improve coding gain.
  • 8-QAM simulations: At BER 10^-5, phase-rotated OTFS achieves about a 17 dB SNR gain over unrotated OTFS with 8-QAM.This result is reported for M = N = 2.

V. MIMO-OTFS MODULATION

This section introduces the study of OTFS modulation and its diversity order in a MIMO setting.

  • The paper considers OTFS modulation and its diversity order in a MIMO setting.

A. MIMO-OTFS system model

The MIMO-OTFS model assigns an independent OTFS vector to each transmit antenna and relates all received vectors through equivalent channel matrices and noise.

  • MIMO-OTFS system model: Each transmit antenna sends an independent OTFS signal vector in the MIMO-OTFS system.The system has nt transmit antennas and nr receive antennas.
  • MIMO-OTFS system model: The channel gain between each transmit-receive antenna pair is represented in the delay-Doppler domain for each delay and Doppler coordinate.P denotes the number of channel taps.
  • MIMO-OTFS system model: The received vector at each antenna is the sum of the equivalent channel-matrix outputs from all transmit antennas plus noise.For example, y_l sums H_lk x_k over transmit antennas k and adds v_l.
  • MIMO-OTFS system model: Stacking the antenna-specific equations yields a linear vector model with H_MIMO mapping the transmit vector to the received vector.The dimensions are x_MIMO ∈ C^(ntMN×1), y_MIMO and v_MIMO ∈ C^(nrMN×1), and H_MIMO ∈ C^(nrMN×ntMN).

B. Diversity of MIMO-OTFS

The MIMO-OTFS analysis uses the effective channel structure and pairwise error probabilities to determine diversity from difference-matrix ranks. It concludes that uncoded MIMO-OTFS has asymptotic diversity order n_r.

  • Channel structure: The effective MIMO-OTFS channel has Pn_t unique entries, with each row containing n_tP nonzero elements and each column containing n_rP nonzero elements.These structural properties characterize the effective channel matrix used in the diversity analysis.
  • System model: The MIMO-OTFS input-output model stacks the P × MN symbol matrices transmitted from n_t antennas and combines them through the channel and noise matrices.The received signal is represented as an n_r × MN matrix, with one row per receive antenna.
  • Diversity analysis: The pairwise error probability is analyzed using singular values and the rank r of the difference matrix between two transmitted symbol matrices.At high SNR, the minimum-rank pairwise error term dominates the overall BER.
  • Result: Rank-one symbol differences exist when each antenna transmits a constant symbol matrix, so the asymptotic diversity order of MIMO-OTFS is n_r.The rank-one difference is the minimum-rank case governing the asymptotic order.

C. Phase rotation for full diversity in MIMO-OTFS

The paper applies a phase rotation matrix to each antenna’s OTFS transmit vector and proves that the resulting MIMO-OTFS system achieves full delay-Doppler diversity. Simulations compare unrotated and rotated systems and examine frame-size effects.

  • Phase-rotation design: Each antenna’s MN × 1 OTFS transmit vector is multiplied by the phase rotation matrix Φ before forming the MIMO-OTFS transmission.The transmit vector is a concatenation of n_t independent OTFS vectors, one from each antenna.
  • Full-diversity proof: The phase-rotated difference matrix has rank P for every transmit antenna, yielding diversity order Pn_r.This rank property supports the full-diversity result for phase-rotated MIMO-OTFS.
  • Simulation results: Diversity order 1 is observed for 1 × 1 SISO-OTFS, while diversity order 2 is observed for 2 × 2 MIMO-OTFS under the stated simulation settings.Both systems use M = N = 2, BPSK, and P = 4 channel taps.
  • Simulation results: Increasing the frame size from M = N = 2 to M = 4, N = 2 improves BER before the asymptotic MIMO-OTFS diversity order of 2 takes over.The comparison uses a 1 × 2 OTFS system with BPSK and P = 4 channel taps.
  • Conclusion: The paper reports that phase rotation extracts full delay-Doppler diversity and extends the diversity analysis to MIMO-OTFS with and without rotation.The conclusion also identifies timing/frequency offset, synchronization, and link adaptation as future-work topics.

APPENDIX A DIVERSITY ANALYSIS FOR NON-ZERO FRACTIONAL

The appendix extends OTFS diversity analysis to fractional delays and Dopplers by deriving the corresponding input-output relation under rectangular windowing.

  • Fractional channel model: The fractional-delay and fractional-Doppler model separates integer indices α_i and β_i from fractional delay and Doppler components a_i and b_i.The integer indices are obtained by nearest-integer rounding of the delay and Doppler coordinates.
  • Input-output derivation: Assuming rectangular window functions, the paper substitutes the delay-Doppler channel representation into the OTFS input-output relation.The derivation explicitly accounts for the fractional parts of the delay and Doppler shifts.
  • Fractional effects: Fractional Doppler produces a peak in the Doppler-domain response near k′ = k − β_i, with values decreasing away from that index.The response is evaluated using the windowed channel representation.
  • Fractional effects: Fractional delay similarly affects the delay-domain response around the corresponding integer delay index.The appendix evaluates the delay response at offsets determined by the fractional-delay model.
  • Vectorized model: The resulting input-output equation is vectorized into x, y, v ∈ C^MN×1 and an MN × MN channel matrix H.The elements of these vectors and the channel matrix are determined by the derived input-output relation.

A. Diversity analysis

The diversity analysis links OTFS error performance to the rank of pairwise symbol-difference matrices. The minimum rank among distinct symbol pairs determines the asymptotic diversity order, including under fractional delay and Doppler.

  • OTFS representation: The vectorized OTFS model is rewritten using a P × MN symbol matrix whose columns represent OTFS symbols across delay-Doppler coordinates.The matrix representation normalizes average symbol-time energy and defines SNR as γ = 1/N_0.
  • Pairwise-error analysis: The pairwise error probability is averaged over channel statistics and bounded using the singular values of the difference matrix Δ_ij.The rank r counts the nonzero singular-value terms contributing to the high-SNR behavior.
  • Diversity order: The SNR exponent is r, the rank of Δ_ij, and the minimum-rank pairwise error probability dominates the overall BER.This establishes the rank-based expression for the achieved diversity order.
  • Rank-one case: When all transmitted symbols differ by the same scalar across the frame, the difference matrix has identical columns and rank one.This rank-one construction provides the minimum rank among distinct symbol pairs.
  • Fractional-delay and Doppler result: The asymptotic diversity order remains one for OTFS with nonzero fractional delays and Dopplers.The appendix also reports BER simulations for different frame sizes under fractional delay and Doppler conditions.
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