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Cross Domain Iterative Detection for Orthogonal Time Frequency Space Modulation

Shuangyang Li, Weijie Yuan, Zhiqiang Wei, Jinhong Yuan

arXiv:2101.03822v1cs.IT

TL;DR

OTFS detection must balance high-mobility robustness with the complexity of exploiting channel diversity, particularly under fractional Doppler. The paper introduces a cross-domain iterative detector that exchanges extrinsic information between time and delay-Doppler domains, and shows convergence with near-MLSE error performance at much lower complexity. These gains are supported by state-evolution analysis and simulations.

  • Problem

    OTFS detection must achieve strong error performance in high-mobility environments despite the complexity of exploiting its channel diversity, especially with fractional Doppler shifts.

  • Method

    The proposed algorithm applies L-MMSE estimation in the time domain and symbol-by-symbol detection in the delay-Doppler domain, iteratively exchanging extrinsic information through unitary transformations.

  • Results

    The algorithm converges and can approach MLSE error performance even with complex fractional Doppler shifts, while requiring much lower detection complexity.

  • Takeaways & Limitations

    Cross-domain message passing exploits time-domain channel sparsity and delay-Doppler constellation constraints to improve OTFS detection over conventional methods.

Abstract

from arXiv · show

Recently proposed orthogonal time frequency space (OTFS) modulation has been considered as a promising candidate for accommodating various emerging communication and sensing applications in high-mobility environments. In this paper, we propose a novel cross domain iterative detection algorithm to enhance the error performance of OTFS modulation. Different from conventional OTFS detection methods, the proposed algorithm applies basic estimation/detection approaches to both the time domain and delay-Doppler (DD) domain and iteratively updates the extrinsic information from two domains with the unitary transformation. In doing so, the proposed algorithm exploits the time domain channel sparsity and the DD domain symbol constellation constraints. We evaluate the estimation/detection error variance in each domain for each iteration and derive the state evolution to investigate the detection error performance. We show that the performance gain due to iterations comes from the non-Gaussian constellation constraint in the DD domain. More importantly, we prove the proposed algorithm can indeed converge and, in the convergence, the proposed algorithm can achieve almost the same error performance as the maximum-likelihood sequence detection even in the presence of fractional Doppler shifts. Furthermore, the computational complexity associated with the domain transformation is low, thanks to the structure of the discrete Fourier transform (DFT) kernel. Simulation results are consistent with our analysis and demonstrate a significant performance improvement compared to conventional OTFS detection methods.

I. INTRODUCTION

OTFS offers robust high-mobility communication through delay-Doppler representations, but realizing its diversity benefits requires complex detection, especially with fractional Doppler. The paper proposes a cross-domain iterative detector that exchanges extrinsic information between time and delay-Doppler domains, with analysis showing convergence and near-MLSE performance at lower complexity.

  • OTFS motivation: OTFS represents signals in the delay-Doppler domain, where channel responses are relatively sparse and robust, simplifying high-mobility input-output relationships.Each delay-Doppler symbol spreads across the time-frequency domain and experiences the channel's fluctuations over an OTFS frame.
  • OTFS motivation: OTFS can achieve full channel diversity in high-mobility environments by coherently combining energy from different propagation paths.This benefit motivates OTFS despite its more demanding detection requirements.
  • Detection challenge: OTFS requires complex detection algorithms to realize its potential diversity, while reduced-CP structures require advanced equalization for channel-induced interference.Conventional low-complexity approaches exploit delay-Doppler sparsity, but their complexity can rise when fractional Doppler makes the effective channel dense.
  • Detection challenge: Existing OTFS detectors include message passing, variational Bayes, and approximate message passing, but message passing may fail to converge because of short graphical-model cycles.The literature also includes approaches targeting convergence guarantees or reduced-complexity MMSE performance.
  • Proposed approach: The proposed detector exchanges extrinsic information between time-domain and delay-Doppler-domain processing through unitary transformations.It uses L-MMSE equalization in time and symbol-by-symbol detection in delay-Doppler, separating decorrelation and denoising tasks.
  • Analysis and results: State-evolution analysis proves convergence after a few iterations and attributes the performance gain to the non-Gaussian delay-Doppler constellation constraint.In convergence, the method can approach MLSE error performance, including complex fractional-Doppler cases, while requiring much lower detection complexity.
  • Analysis and results: The proposed algorithm has lower complexity than MLSE, and simulations show significant improvement over conventional OTFS detection methods.The paper reports that overall detection complexity does not increase in the presence of fractional Doppler.

II. SYSTEM MODEL

The OTFS system represents symbols in delay-Doppler and transforms them through time-frequency and time domains. Fractional Doppler can densify DD-domain effective channels, whereas the time-domain channel remains sparse, motivating time-domain-assisted detection.

  • Signal representation: OTFS transmits information symbols in the DD domain and maps them to the TF domain through the ISFFT before conventional signal generation.Each DD-domain symbol spreads across the whole TF domain and experiences the TF channel fluctuations.
  • Channel model: The channel is characterized by path gains, delays, Doppler shifts, and fractional Doppler offsets relative to the nearest Doppler grid.Fractional delays are neglected because the typical delay-domain sampling interval is sufficiently small in wide-band systems.
  • Domain models: The system derives equivalent input-output representations for transmitted and received symbols in the DD, TF, and time domains.The time-domain model uses the effective channel matrix and corresponding AWGN sample vector, while DD-domain reception follows SFFT processing.
  • Effective-channel structure: With fractional Doppler shifts, the TF- and DD-domain effective channel matrices can become very dense, but the time-domain matrix remains sparse.At most P entries in each row and column of the time-domain effective channel matrix are nonzero.
  • Detection motivation: This contrast motivates detection based on the effective time-domain channel while retaining DD-domain symbol processing.The paper illustrates the contrasting channel-matrix properties with an example using P = 5 paths.

III. CROSS DOMAIN ITERATIVE DETECTION FOR OTFS MODULATION

The proposed detector combines time-domain estimation with DD-domain symbol detection through iterative cross-domain extrinsic-information exchange. It exploits time-domain channel sparsity and DD-domain constellation constraints, with de-correlation and de-noising performed in the respective domains.

  • Statistical assumptions: The analysis models time-domain symbols as i.i.d. Gaussian variables while exploiting the non-Gaussian DD-domain constellation constraint.The Gaussian model follows the spreading effect of the ISFFT, whereas DD symbols independently take values from A.
  • Cross-domain iteration: The proposed method iteratively exchanges extrinsic information between the time and DD domains through unitary transformations.The transformations connect the time-domain and DD-domain representations while preserving the cross-domain message-passing structure.
  • Domain-specific processing: Time-domain processing performs de-correlation to mitigate fading, multipath, and Doppler, whereas DD-domain processing performs de-noising.The DD-domain detector uses the normalized constellation set A and equal symbol probabilities.
  • Detector architecture: The detector uses two modules: module A estimates the time-domain signal z, while module B performs symbol-by-symbol detection of the DD-domain vector x.Module A passes estimates to module B, and module B uses time-domain estimates for DD-domain detection.

A. Module A: L-MMSE Estimator for Time Domain OTFS Signal

Module A estimates the time-domain OTFS signal with an L-MMSE estimator using received samples, the effective time-domain channel, and a priori information from module B. It returns posterior means and diagonal MSE information for subsequent cross-domain processing.

  • L-MMSE estimation: Module A applies an L-MMSE estimator to the received time-domain vector r using the effective channel and a priori information from module B.The estimator produces the time-domain signal estimate needed by the iterative detector.
  • Initialization: The a priori covariance is diagonal and initialized as I_MN for the first iteration.The diagonal structure follows from the i.i.d. assumption and enables direct computation of the L-MMSE estimation matrix.
  • Estimator outputs: The estimator returns a posteriori means and covariance information for the time-domain signal z.Its implementation computes the estimator matrix, estimation output, and MSE matrix before returning them.
  • Error characterization: Only diagonal entries of the posterior covariance are retained because they represent per-entry MSEs under the i.i.d. assumption.Non-diagonal entries are treated as zero because only diagonal entries are needed for the assumed independent components.

B. Cross Domain Message Passing: from Time Domain to DD Domain

The detector forms DD-domain symbol posteriors from time-domain extrinsic information, then uses symbol-by-symbol detection and transforms the resulting information between domains. Because this DD-domain operation is component-wise, it cannot directly provide extrinsic information; the algorithm instead converts posterior information before continuing message passing.

  • Time-domain extrinsic means and covariance provide the a priori information required for DD-domain symbol detection.
  • The Ungerboeck observation model exploits the unitary transform to obtain linear detection complexity instead of the dense-transform complexity associated with the Forney model.
  • The optimal DD-domain MLSE detection can be expressed symbol by symbol using the transformed estimates and covariance information.
  • The DD-domain symbol-by-symbol operation cannot provide extrinsic information, so posterior DD information is first converted to the time domain before extrinsic information is computed.

D. Cross Domain Message Passing: from DD Domain to Time Domain

The reverse message-passing stage converts DD-domain posterior information into time-domain posterior information and feeds its extrinsic form back to the time-domain estimator. This completes the iterative cross-domain detector.

  • The DD-domain posterior mean and covariance are converted into posterior information for the time-domain OTFS signal.
  • The converted time-domain extrinsic information is fed back as the next iteration’s a priori mean and covariance for module A.
  • Algorithm 3 alternates time-domain L-MMSE estimation and DD-domain symbol-by-symbol detection for up to Lmax iterations.
  • The proposed detector’s error performance and computational complexity are analyzed after the cross-domain message-passing procedure is specified.

IV. PERFORMANCE ANALYSIS

The performance analysis characterizes iterative detection through two recursively updated error-variance states and a state-evolution model. It establishes convergence, explains the role of the DD constellation, and reports MSE behavior for QPSK and 16-QAM.

  • The analysis models each iteration with two module-specific variance states and derives their recursion as the frame size tends to infinity.
  • A fixed point exists when the overall MSE no longer changes with additional iterations, and at convergence time-domain and DD-domain outputs have equal recovery accuracy.
  • For Gaussian DD-domain symbols, iterative extrinsic-information exchange provides no error-performance improvement.
  • The reported improvement for non-Gaussian constellations is attributed to exploiting the DD-domain constellation constraint, unlike the Gaussianized time-domain representation.
  • The performance saturates near 1.3×10^-4 MSE for QPSK and 1.6 × 10^-3 for 16-QAM, while state evolution closely matches the simulated MSE.

B. Analysis of Effective DD Domain SNR

The analysis characterizes the effective DD-domain SNR across iterations and shows that it approaches the maximum receiver SNR, with simulations supporting the state-evolution analysis even when paths share delays.

  • Effective SNR behavior: The effective DD-domain SNR determines BER performance for a given constellation and increases with the number of iterations.The state-evolution SNR closely matches simulation, while the analytical upper bound becomes tighter as iterations increase.
  • Analytical assumptions: Under the distinct-delay assumption, the main diagonal elements of the relevant matrix have the common value ∥h∥2.The analysis assumes distinct delay indices for resolvable paths, although simulations also examine shared-delay paths.
  • Converged performance: With sufficient iterations, the proposed algorithm can theoretically approach the maximum receiver SNR and MLSE error performance.MLSE provides optimal maximum-likelihood error performance but usually has prohibitively high complexity.
  • Analytical validation: The DD-domain effective SNR upper bound becomes tighter as the number of iterations increases and agrees with both simulation and state evolution.The bound is derived under the distinct-delay assumption and is also numerically examined for shared-delay paths.
  • Shared-delay paths: The effective DD-domain SNR analysis remains accurate when different resolvable paths share the same delay index.For this case, QPSK and 16-QAM show similar SNR performance to the distinct-delay case.

C. Analysis of Detection Complexity

The proposed detector has low per-iteration transformation cost and avoids the exponential complexity of MLSE, including when fractional Doppler makes the effective channel dense.

  • Domain transformation: The domain transformation has low computational complexity because the corresponding kernels can exploit their special DFT structure.The component-wise operation in module B has complexity O(MN), while FFT and IFFT operations contribute O(MN log N).
  • Fractional Doppler: The proposed algorithm’s detection complexity does not increase in the presence of fractional Doppler indices.The per-iteration complexity includes matrix-inverse, transformation, and component-wise terms.
  • Comparison with MLSE: MLSE detection has complexity exponential in the number of nonzero elements per channel-matrix row or column, which can become high under fractional Doppler.This gives the proposed method a substantial complexity advantage over optimal MLSE in that setting.

V. NUMERICAL RESULTS

Numerical results show that iterative cross-domain detection improves BER over conventional detectors, with gains that become especially pronounced for dense channels and fractional Doppler.

  • Experimental setup: The numerical results are evaluated against MMSE, SPA, and DD-domain message-passing detectors under integer and fractional Doppler conditions.The SPA comparison is restricted to integer Doppler because its fractional-Doppler complexity is very high.
  • Dense effective channels: For one iteration, the proposed algorithm performs almost the same as DD-domain MMSE detection in the dense-channel setting.The DD-domain message-passing algorithm also has similar error performance at one iteration, but iterative detection subsequently outperforms both methods.
  • P = 10 channels: With five iterations, the SNR gain over MMSE detection increases to around 4.1 dB.The result demonstrates an advantage over conventional detection algorithms under the evaluated complex channel conditions.

VI. CONCLUSION

The paper concludes that cross-domain iterative detection can approach MLSE error performance with much lower complexity, including under complex fractional Doppler shifts.

  • Contribution: The proposed cross-domain iterative detector is analyzed through state evolution, MMSE performance, and effective DD-domain SNR.These analyses are supported by simulation results.
  • Main result: The algorithm can approach MLSE error performance in the presence of complex fractional Doppler shifts while requiring much lower detection complexity.The conclusion identifies this as the central performance-complexity result.
  • Numerical validation: Simulations show significant improvement over conventional detection algorithms for channels with fractional Doppler shifts.The numerical evidence is reported as consistent with the analytical results.
  • Future direction: The authors identify cross-domain signal processing as a possible research direction for OTFS and general multicarrier modulation schemes.The conclusion presents this as a future research direction rather than an established result.
  • Future work: Future work will investigate cross-domain channel-estimation and cross-domain precoding schemes.The conclusion leaves these extensions outside the present work.

APPENDIX A

The appendix develops the delay-Doppler detection formulation and establishes why the proposed symbol-by-symbol procedure remains efficient when diagonal entries differ. It also analyzes extrinsic information, convergence-related bounds, and concavity properties used in the proofs.

  • DD-domain detection: The DD-domain detection uses estimates of the time-domain signal, transforms extrinsic information into the DD domain, and compares it with the DD-domain symbol constellation.The transformed information is used for DD-domain symbol detection after modeling time-domain MMSE estimation inaccuracy as white Gaussian noise.
  • Proof structure: The appendix completes proofs for the stated proposition, lemma, and theorem after deriving the corresponding transformed noise covariance, extrinsic relations, and Jensen-based bounds.These proof steps include the equivalent DD-domain noise covariance and substitutions leading to the theorem’s final expression.
  • Asymptotic analysis: In the asymptotic regime, diagonal entries tend toward a common value by the law of large numbers, enabling straightforward symbol-by-symbol DD detection.The appendix contrasts this asymptotic behavior with the practical case, where specific noise values may produce unequal entries.
  • DD-domain detection: The DD detection formulation bypasses unequal diagonal entries that can make the covariance matrix dense and undermine detection performance.The proposed algorithm can still perform DD detection efficiently in a symbol-by-symbol fashion.
  • Extrinsic information: The appendix derives extrinsic means by excluding each symbol’s own contribution and shows that DD-domain detection cannot provide extrinsic covariance information.The a posteriori probability of x[k] depends only on its corresponding DD-domain mean, while the covariance contribution is zero.
  • Convergence analysis: The convergence analysis applies Jensen’s inequality to a concave function, with bounds tightening as the relevant variance decreases and equality achieved at zero variance.The proof identifies strict positivity conditions for concavity and states that the lower bound becomes tighter as v_a,T decreases.
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