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OTFS Channel Estimation Utilizing Sparse Bayesian Generative Modelling

Louis Anseaume, Benedikt Böck, Franz Weißer, Wolfgang Utschick

arXiv:2609.01074v1eess.SP

TL;DR

Reliable CSI estimation is challenging for OTFS in high-mobility wireless channels, where sparse but complex channel distributions must be recovered. The paper applies a compressive-sensing Gaussian mixture model within a sparse Bayesian generative framework, and reports improved NMSE over the next-best baseline together with arbitrarily fine theoretical distribution approximations.

  • Problem

    High-mobility OTFS requires accurate CSI estimation, while classical sparse estimators may be limited in complex outdoor channel environments.

  • Method

    The paper reformulates OTFS channel estimation as a sparse-grid compressive-sensing problem and applies an EM-trained CSGMM sparse Bayesian generative model.

  • Results

    The CSGMM consistently outperforms OMP, SBL, and M-SBL across the assessed SNR range and benefits from larger sparse grids.

  • Takeaways & Limitations

    OTFS-CSGMM provides a promising framework for high-mobility channel estimation and can, in principle, represent OTFS channel distributions with arbitrarily fine accuracy.

Abstract

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One of the key challenges of future wireless communication systems is ensuring reliability in high-speed mobile scenarios, where accurate recovery of channel state information (CSI) is essential. Many recent studies have concluded that orthogonal time-frequency space (OTFS) modulation is a promising technology for addressing this challenge. Additionally, machine learning (ML)-based methods have the potential to improve channel estimation performance by leveraging ambient information more effectively than classical estimation techniques. This paper particularly addresses channel estimation for OTFS by employing a compressive sensing (CS)-based sparse Bayesian generative model (SBGM), namely the recently introduced compressive sensing Gaussian mixture model (CSGMM). We show that our proposed approach yields significant improvement in normalized mean squared error (NMSE) over the next-best-performing baseline. We additionally provide insights into the theoretical potential of the model to optimally approximate complex channel distributions with arbitrary precision within the Doppler-delay (DD) domain. To summarize, this work establishes the OTFS-CSGMM framework as a promising solution for high mobility wireless channel estimation.

I. INTRODUCTION

High-mobility 6G scenarios expose OFDM’s reliability limits, motivating OTFS and highly accurate CSI estimation. This work combines OTFS channel sparsity with ML-based SBGM learning to improve estimation in complex outdoor environments.

  • 300–500 km/h high-mobility applications motivate alternatives to OFDM, whose high Doppler-induced ICI can degrade signal reconstruction.OTFS operates in the Doppler-delay domain and is presented as a promising candidate for these scenarios.
  • Accurate CSI is required for OTFS to achieve correct and reliable communication.
  • Outdoor high-mobility channels are often sparse because they contain a small number of reflectors, but classical sparse estimators may reach their limits in complex environments.
  • ML-based methods are proposed as promising candidates because they can leverage contextual information inherent to wireless communication systems.The paper applies a CS-based SBGM, specifically CSGMM, to learn channel distributions in high-speed mobile setups.
  • The paper reformulates prior results for CS, applies CSGMM to OTFS channel estimation, and studies both simulated performance and theoretical distribution approximation.It reports improved performance over state-of-the-art baselines and arbitrarily fine approximation under certain channel assumptions.

II. OTFS SYSTEM MODEL

The OTFS system maps DD-domain symbols through TF-domain transforms, transmits them over a time-frequency-selective channel, and recovers DD-domain outputs. A peak pilot with a zero-energy guard isolates channel-induced DD shifts for estimation.

  • The channel is modeled as P multipath components with a time-frequency-dependent impulse response.The formulation includes path-specific ISI coefficients and additive white Gaussian noise.
  • OTFS places symbols on a two-dimensional Doppler-delay modulation grid before transforming them into the time-frequency domain.The ISFFT produces TF symbols, followed by a discrete Heisenberg transform for transmission.
  • At the receiver, Wigner and symplectic Fourier transforms convert the received time-domain signal back into DD-domain output symbols.
  • OTFS channels shift transmitted symbols on the DD grid according to channel parameters, with fractional parameters causing additional energy spread.
  • A central high-energy peak pilot surrounded by zero-energy guard symbols isolates DD-domain shifts from low-energy data symbols.Because interference with other symbols is negligible, the pilot indicates the channel-induced shifts.

III. OTFS CHANNEL ESTIMATION

The channel-estimation framework casts OTFS recovery as a compressive-sensing inverse problem on a sparse DD grid. OTFS-CSGMM combines sparse Bayesian modeling with Gaussian-mixture latent structure and estimates channels using EM-trained posterior means.

  • Sparse Bayesian generative models combine sparse Bayesian learning with a trainable generative model using a latent space.The paper specializes to conditional Gaussian latent models, whose conditioned distributions are Gaussian.
  • Gaussian mixture models approximate complex probability distributions, and incorporating them into the SBGM framework yields CSGMM.The mixture weights satisfy ρk ≥ 0 and sum to one.
  • The OTFS input-output relationship is reformulated with a path dictionary Φ and channel coefficient vector h, then discretized on a sparse DD grid.The sparse representation hsp has only P non-zero entries, while Φdict defines the corresponding dictionary.
  • Under the sparse-grid approximation, the model becomes a compressive-sensing linear inverse problem with noisy observations and a mixture-conditioned sparse prior.The likelihood uses Φdict hsp and Gaussian noise, while each mixture component assigns a diagonal covariance to hsp.
  • Expectation-maximization trains OTFS-CSGMM by estimating posterior statistics in the E-step and maximizing the ELBO over mixture parameters in the M-step.
  • After training, the MSE-optimal conditional mean estimates the sparse channel representation from noisy observations before it is converted to matrix form.

IV. THEORETICAL ANALYSIS

The theoretical analysis addresses distribution learning for OTFS channels under WSSUS-related assumptions, despite the loss of the usual Toeplitz constraint. It shows that CSGMM can approximate realistic channel distributions arbitrarily finely under suitable conditions.

  • For zero-mean WSSUS channels, statistical characteristics are preserved when conditioning on side information, but OTFS lacks the resulting Toeplitz structure constraint.
  • With suitable conditions, CSGMM can optimally learn arbitrarily fine approximations of random OTFS channel distributions.The result provides theoretical motivation for high-resolution DD grids whose learned moments approach those of continuous channel models.

A. Discrete Model Distribution

Under suitable assumptions on side information and path phases, the OTFS model preserves its zero-mean and dictionary-based structure when conditioned on side information. This places the ground-truth distribution within the CSGMM search space in principle.

  • A. Discrete Model Distribution: Theorem 1 states that the OTFS model’s statistical characterization is preserved under conditioning on side information when channel parameters lie on the sparse DD grid.
  • A. Discrete Model Distribution: Under the theorem’s assumptions on side information z and path phases β, the model preserves its zero-mean property and dictionary-based structure.
  • A. Discrete Model Distribution: Because the ground-truth model distribution lies within the CSGMM search space, the model can in principle learn this distribution optimally.

B. Continuous Model Distribution

The continuous-channel result is obtained by refining discrete DD-grid models and showing convergence of their statistical characteristics to the ground-truth continuous channel. This supports arbitrarily accurate CSGMM representations in principle.

  • B. Continuous Model Distribution: A sequence of discrete models on increasingly dense, evenly spaced DD grids is constructed to approximate continuously distributed channel parameters.
  • B. Continuous Model Distribution: The convergence relations for the discrete models and their side-information-dependent forms follow from the dominated convergence theorem.
  • B. Continuous Model Distribution: Theorem 2 establishes convergence of discretely parameterized model statistics to those of the ground-truth continuous channel.
  • B. Continuous Model Distribution: Because the discrete models are learnable by CSGMM, the model can in principle capture arbitrarily accurate representations of realistic OTFS channels.

V. NUMERICAL RESULTS

Experiments evaluate OTFS channel estimation on DeepMIMOv3 using CSGMM against OMP, SBL, and M-SBL. CSGMM consistently outperforms the baselines across SNR values and benefits from larger sparse DD grids.

  • V. NUMERICAL RESULTS: The evaluation uses the DeepMIMOv3 Dynamic Doppler DD1 scenario, with P = 10 paths across 2000 scenes and Ntrain = 12805 and Ntest = 3202.
  • V. NUMERICAL RESULTS: The default setup uses Δf = 30 kHz, T = 33.3 µs, K = 10 components, an 8×8 data grid, and a 32×32 sparse grid at SNR = 20 dB.
  • V. NUMERICAL RESULTS: NMSE is used as the empirical distortion metric for comparing the channel-estimation methods.
  • V. NUMERICAL RESULTS: The baselines are OMP, SBL, and M-SBL, with M-SBL equivalent to a single-component CSGMM model in the stated formulation.
  • V. NUMERICAL RESULTS: CSGMM consistently outperforms all baselines across the tested SNR range and significantly improves over M-SBL, the next-best-performing method.
  • V. NUMERICAL RESULTS: At high SNR, all methods flatten in performance, while fractional Doppler and discrete-grid resolution impose residual generalization bias.
  • V. NUMERICAL RESULTS: For large sparse grids, CSGMM consistently outperforms the baselines; at 64 × 64, it significantly improves over SBL.
  • V. NUMERICAL RESULTS: CSGMM improves monotonically with larger |G|, whereas M-SBL peaks at |G| = 1024 and degrades thereafter, suggesting over-regularization.

VI. CONCLUSION

The paper presents CSGMM as a statistically expressive ML-based SBGM for OTFS channel estimation. It reports improved estimation accuracy and theoretical potential for arbitrarily accurate channel-distribution approximations.

  • VI. CONCLUSION: The paper addresses OTFS channel estimation with a statistically expressive ML-based sparse Bayesian generative model.
  • VI. CONCLUSION: CSGMM accurately captures channel-distribution complexity in high-mobility wireless communication scenarios.
  • VI. CONCLUSION: The approach demonstrates significant estimation-accuracy improvements over state-of-the-art baseline methods.
  • VI. CONCLUSION: The analysis provides theoretical insights into learning arbitrarily accurate approximations of OTFS channel distributions.
  • VI. CONCLUSION: Applying CSGMM and CSVAE to OTFS channel estimation is identified as an opportunity for future research.

VII. APPENDIX

The appendix proof derives Theorem 1 by representing the random vector through a dictionary and sparse coefficients, then applying independence-based changes of variables to obtain the stated moment expressions.

  • VII. APPENDIX: The proof represents r as Ψdict hsp, with each sparse coefficient written as hsp,i = √ηi e−j ˜βi.The representation uses G = |G|, the cardinality of G, and indexes coefficients over i ∈ {1, ..., G}.
  • VII. APPENDIX: Assuming the parameters lie in G, the proof decomposes the entries of hsp into magnitudes ηi and phases ˜βi.
  • VII. APPENDIX: Independence of ˜βi and ηi permits a change of variables in the moments of r, yielding equations (17) and (18).
  • VII. APPENDIX: Under the additional assumption that βp is independent of z, the proof likewise assumes ˜βi is independent of z and obtains equations (19) and (20).
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