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Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation

P. Raviteja, Khoa T. Phan, Yi Hong, Emanuele Viterbo

arXiv:1802.05242v2cs.IT

TL;DR

High-Doppler time-varying channels expose limitations in OFDM, motivating robust OTFS modulation. The paper derives OTFS delay–Doppler input–output relations, analyzes waveform-dependent interference, and proposes a low-complexity message-passing detector. The algorithm compensates for a wide range of Doppler spreads and supports effective interference cancellation and symbol detection.

  • Problem

    OFDM is not robust to time-varying channels with high Doppler spread, motivating modulation techniques that handle channel time variations.

  • Method

    The paper derives uncoded OTFS mod/demod input–output relations and develops a low-complexity message-passing algorithm for joint interference cancellation and symbol detection.

  • Results

    The proposed message-passing algorithm effectively compensates for a wide range of channel Doppler spreads.

  • Takeaways & Limitations

    OTFS can use delay–Doppler channel sparsity for low-complexity joint interference cancellation and detection in large-scale systems.

Abstract

from arXiv · show

The recently proposed orthogonal time frequency space (OTFS) modulation technique was shown to provide significant error performance advantages over orthogonal frequency division multiplexing (OFDM) in Doppler channels. In this paper, we derive the explicit input-output relation describing OTFS modulation and demodulation (mod/demod) for delay-Doppler channels. We analyze the interferences and develop a novel low-complexity yet efficient message passing (MP) algorithm for joint interference cancellation (IC) and symbol detection. The proposed MP algorithm can effectively compensate for a wide range of channel Doppler spreads.

I. INTRODUCTION

The paper motivates OTFS as a modulation technique for time-varying, high-Doppler channels and develops its delay–Doppler input–output relation and message-passing detector. It analyzes waveform-dependent interference and reports robust performance across channel conditions.

  • Motivation: OTFS targets time-varying channels where OFDM is not robust to high Doppler spread, including high-speed railway communications.The motivation is tied to emerging high-data-rate wireless applications and channel time variations.
  • OTFS approach: OTFS spreads each information symbol over a 2D orthogonal basis spanning the frame’s time–frequency domain to combat time-varying multipath channels.The delay–Doppler representation supports this time–frequency signaling structure.
  • Analysis: The paper derives the uncoded OTFS mod/demod input–output relation for delay–Doppler channels using general pulse-shaping waveforms.ISFFT and SFFT are interpreted as pre- and post-processing blocks around time–frequency signaling.
  • Interference analysis: Ideal waveforms produce inter-Doppler interference from fractional Doppler effects, whereas rectangular waveforms additionally produce inter-carrier and inter-symbol interference.The latter effects arise from imperfect time–frequency bi-orthogonality; the rectangular-waveform case assumes no cyclic prefix.
  • Detection: The proposed low-complexity message-passing algorithm jointly performs interference cancellation and detection using delay–Doppler channel sparsity.It uses a sparse factor graph and Gaussian approximation, eliminates ICI and ISI through phase shifting, and mitigates IDI by retaining the largest interference terms.
  • Results: The algorithm compensates for a wide range of Doppler spreads, while practical rectangular-waveform OTFS can match ideal-waveform OTFS performance and outperform OFDM across channel conditions.The reported evidence includes simulation results for uncoded OTFS schemes.

II. SYSTEM MODEL

OTFS maps information symbols between delay–Doppler and time–frequency grids using paired two-dimensional transforms, then transmits the resulting waveform over a time-varying channel.

  • The delay–Doppler plane uses an N×M information grid, while each OTFS frame carries NM symbols over duration T_f = NT and bandwidth B = M∆f.
  • System parameters must satisfy ν_max < 1/T and τ_max < 1/∆f to support the channel’s maximum Doppler and delay.
  • OTFS modulation uses an ISFFT to map delay–Doppler symbols x[k, l] to time–frequency samples X[n, m].
  • The Heisenberg transform converts X[n, m] into the transmitted continuous-time signal s(t).
  • At reception, the Wigner transform maps r(t) to the time–frequency domain, followed by an SFFT for delay–Doppler symbol demodulation.

D. Wireless transmission and reception

The paper models wireless transmission with a sparse delay–Doppler channel and derives explicit OTFS input–output relations for ideal and practical waveforms.

  • The channel impulse response is represented by P propagation paths with gains, delays, and Doppler shifts.
  • Fractional Doppler κ_νi captures the shift from path i’s nearest Doppler tap k_νi, while fractional delays are approximated by the sampling resolution.
  • The receiver applies a matched filter, samples the time–frequency output, and then uses the SFFT to obtain delay–Doppler symbols.
  • Theorem 1 gives the OTFS time–frequency input–output relation, including combined effects of transmit and receive pulses and the channel.
  • Ideal pulses provide a reference case, but cannot be realized in practice; rectangular waveforms introduce interference and serve as a practical analysis case.

A. Time–frequency domain analysis

The delay–Doppler input–output relation reveals localized inter-Doppler interference caused by fractional Doppler, enabling a sparse representation for iterative detection.

  • For ideal pulses, the delay–Doppler input–output relation follows as a special case of the time–frequency-domain theorem.
  • The received symbol y[k, l] is a linear combination of transmitted symbols across Doppler and delay indices.
  • The resulting sparsity is exploited to develop a low-complexity iterative detector based on message passing over a factor graph.
  • Fractional Doppler spreads each path’s contribution across 2N_i + 1 neighboring Doppler symbols centered near k − k_νi.
  • The central shifted symbol contributes most, while neighboring Doppler symbols create inter-Doppler interference.

3) Special channel model cases:

Special channel and waveform cases clarify OTFS behavior: ideal channels reduce to AWGN, integer Doppler yields shifted symbols, and rectangular pulses create delay-dependent interference.

  • Under an ideal channel h(τ, ν) = δ(τ)δ(ν), the received signal behaves as an AWGN channel.
  • With no fractional Doppler, each path circularly shifts the transmitted signal by its delay and Doppler taps and scales it by the path gain.
  • Practical rectangular pulses violate bi-orthogonality and generate interference that degrades system performance.
  • The paper analyzes rectangular-pulse interference and shows that it can be compensated to achieve ideal-pulse performance.
  • For rectangular pulses, the second and third terms correspond to inter-carrier interference and inter-symbol interference, respectively.
  • These interference terms depend on the channel delay τ and arise from samples in the current and previous time slots.

1) ICI analysis:

The analysis characterizes ICI and ISI from rectangular pulses and derives a delay–Doppler input–output relation for OTFS. It shows that interference is localized by Doppler and frequency separation, while the resulting relation remains sparse and differs from ideal pulses mainly through phase shifts.

  • Interference localization: The cross-ambiguity amplitude decreases as interfering subcarriers move farther from the interfered subcarrier, reducing ICI and ISI.The same frequency-separation behavior is reported for both interference types.
  • Interference localization: Increasing Doppler increases the number of neighboring subcarriers that interfere with the present subcarrier.This broadens the effective interference neighborhood in frequency.
  • Delay–Doppler relation: The ICI and ISI summation terms are mutually exclusive, enabling their effects to be distinguished in the delay–Doppler domain.Different sample-index ranges contribute to ICI and ISI.
  • Delay–Doppler relation: Theorem 2 gives the received signal in the delay–Doppler domain for rectangular pulses, with approximation error decreasing as N increases.The relation converts time–frequency ICI and ISI into simple phase shifts in the delay–Doppler domain.
  • Special cases: Rectangular and ideal pulses affect the same number of transmitted signals, but rectangular pulses add a location-dependent delay–Doppler phase.In an ideal channel, rectangular pulses satisfy the bi-orthogonal property and match ideal-pulse behavior.
  • Special cases: With no fractional Doppler, the input–output relation simplifies and IDI does not appear as it does with ideal pulses.This is the stated special-case consequence for κ_νi = 0.

V. MESSAGE PASSING ALGORITHM FOR JOINT INTERFERENCE CANCELLATION AND DETECTION

The paper formulates OTFS detection as inference on a sparse factor graph and develops a message-passing detector for joint interference cancellation and symbol detection. Gaussian interference approximations and sparsity yield linear-in-system-size detection, with practical convergence typically within 20 iterations.

  • Rectangular-pulse detection: Rectangular pulses retain the same number of nonzero elements per channel-matrix row and column as ideal pulses, enabling comparable detector complexity.The channel matrices differ, but their sparsity pattern has the same S-sized support.
  • Detector formulation: The OTFS system is represented by a sparsely connected factor graph with NM variable nodes, NM observation nodes, and S connections per node.Observation and variable nodes connect through the nonzero entries of the channel matrix.
  • Detector formulation: The proposed MP detector solves approximate symbol-by-symbol MAP detection with linear complexity in NM.It isolates one variable from the remaining interference and models that interference as Gaussian noise with computable mean and variance.
  • Message updates: The MP messages comprise Gaussian interference means and variances from observation nodes and symbol-alphabet probability mass functions from variable nodes.These messages are iteratively updated between the two node types.
  • Iteration control: The algorithm updates symbol decisions only when the current convergence indicator gives better estimates than the previous iteration.Stopping criteria include convergence, deterioration relative to the best prior iteration, or reaching the maximum iteration count.
  • Complexity: The overall complexity is O(niterNMSQ), while sparsity keeps S much smaller than NM.The one-iteration computations scale with the number of nonzero channel connections and constellation size.
  • Complexity: The algorithm typically converges within 20 iterations in simulations.The paper attributes complexity reduction to exploiting sparsity and approximating IDI.

B. Application of MP detection algorithm for OFDM over delay–Doppler channels

The paper extends the MP detection framework from OTFS to OFDM over delay–Doppler channels. OFDM detection has a sparse, diagonally dominant frequency-domain channel matrix, allowing the same low-complexity MP approach to compensate Doppler-induced interference.

  • OFDM formulation: The previously developed OTFS MP detector can also be applied to OFDM symbol detection because the OFDM input–output relation has the same form.The OFDM frequency-domain channel matrix is constructed from the time-domain channel matrix using FFT matrices.
  • Channel structure: The OFDM frequency-domain channel matrix is diagonally dominant, with off-diagonal values decaying away from the diagonal.This structure characterizes the frequency-domain interference pattern.
  • Channel structure: The OFDM channel matrix is sparse, enabling the proposed low-complexity MP detector to compensate Doppler effects.The paper explicitly applies the MP algorithm to OFDM for this purpose.

VI. ILLUSTRATIVE RESULTS AND DISCUSSIONS

Simulations evaluate IDI modeling, damping, and OTFS–OFDM performance under delay–Doppler channels. The proposed MP detector benefits from accounting for sufficient interference terms, remains robust across Doppler variations, and allows rectangular-pulse OTFS to approach ideal-pulse performance.

  • Damping-factor selection: ∆ = 0.7 is selected as the optimum damping factor because BER remains stable up to ∆ = 0.7 and convergence then requires the fewest iterations.The evaluation uses 4-QAM, SNR = 18 dB, UE speed 120 Kmph, and Ni = 10.
  • OTFS versus OFDM: OTFS outperforms OFDM by approximately 15 dB at BER 10^-4 with 4-QAM across Doppler frequencies corresponding to 30, 120, and 500 Kmph.The paper attributes this result to OTFS's constant channel gain over transmitted symbols and MP-based IDI cancellation.
  • Pulse-shape comparison: OTFS with rectangular pulses exhibits an error floor without ICI and ISI cancellation, especially at high Doppler.With cancellation, its BER approaches the performance of OTFS with ideal pulses.
  • Pulse-shape comparison: Rectangular-waveform OTFS can achieve ideal-waveform performance across Doppler frequencies when IDI, ICI, and ISI are appropriately canceled.The reported performance remains almost constant despite Doppler variation.
  • OTFS versus OFDM: At high Doppler, OTFS performance is not affected whereas OFDM develops an error floor.This comparison is reported after interference cancellation for OTFS.
  • Higher-order modulation: At 16-QAM and 120 Kmph, OTFS with ICI and ISI cancellation outperforms OFDM by 11 dB at BER = 10^-3.OTFS BER is similar at 30 and 500 Kmph in the additional simulations.

VII. CONCLUSION

The paper derives OTFS input–output relations, characterizes interference in delay–Doppler channels, and proposes a low-complexity message passing algorithm for joint interference cancellation and symbol detection. Simulations show error-performance gains over OFDM and effective compensation across a wide range of Doppler spreads.

  • The study derives OTFS modulation and demodulation input–output relations for delay–Doppler channels.
  • Inter-Doppler, inter-carrier, and inter-symbol interference are characterized using sparse delay–Doppler channel representations.
  • A low-complexity message passing algorithm performs joint interference cancellation and symbol detection for large-scale OTFS systems.
  • The algorithm cancels ISI and ICI through phase shifting and mitigates IDI by accounting for a small number of significant interference terms.
  • The proposed algorithm effectively compensates for a wide range of channel Doppler spreads.
  • Simulations show significant error-performance gains for uncoded OTFS over OFDM under various channel conditions, including practical rectangular waveforms.

APPENDIX A

The appendix derives sampled received-signal expressions through substitutions, variable changes, algebraic simplification, and cross-ambiguity-function representations.

  • The received signal after the Wigner transform is expanded and sampled to obtain Y[n, m].
  • A change of variables and algebraic calculations transform H_n,m[n′, m′] into successive equivalent forms.
  • The final channel expression replaces a bracketed term with the cross-ambiguity function, completing the derivation.
  • Ideal pulses: For ideal pulses, the received signal y[k, l] is expanded using the ISFFT, while h_w[k−k′, l−l′] represents sampled channel response values.

FOR RECTANGULAR PULSES

For rectangular pulses, the appendix decomposes the received signal into interference components and derives approximations using delay–Doppler channel terms and sparse significant contributions.

  • Interference decomposition: The rectangular-pulse received signal y[k, l] is expanded into terms associated with n, m, m′, and p.
  • Interference decomposition: The decomposition separates inter-carrier and inter-symbol interference contributions within y[k, l].
  • Inter-carrier interference: The inter-carrier interference expression is simplified using delay and Doppler taps, channel assumptions, and the cross-ambiguity function.
  • Inter-symbol interference: The inter-symbol interference analysis accounts for the absence of a previous symbol for the first symbol, yielding G_isi(ν_i)=G_ici(ν_i)−1.
  • Sparse approximation: The approximation neglects symbols whose coefficients are very small, approximately 1/N for practical N values such as 64 and 128.
  • Final expression: Combining the derived interference terms produces the final expression for y[k, l].
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