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Off-grid Channel Estimation with Sparse Bayesian Learning for OTFS Systems

Zhiqiang Wei, Weijie Yuan, Shuangyang Li, Jinhong Yuan, Derrick Wing Kwan Ng

arXiv:2101.05629v1cs.IT

TL;DR

Fractional delay and Doppler shifts can cause channel spreading that weakens sparsity in the effective DD-domain channel. The paper estimates the original DD-domain response with 1D and lower-complexity 2D off-grid SBL formulations, achieving superior channel-estimation performance, with the 2D method incurring slight degradation relative to 1D.

  • Problem

    Fractional delay and Doppler shifts can cause channel spreading, making the effective DD-domain channel insufficiently sparse for estimation.

  • Method

    The paper proposes off-grid SBL-based channel estimation by reformulating the problem as 1D and 2D off-grid sparse signal recovery problems for the original DD-domain response.

  • Results

    The proposed off-grid schemes achieve superior channel-estimation performance, while the 2D method has much lower computational complexity and only slight performance degradation compared with 1D.

  • Takeaways & Limitations

    Estimating the original DD-domain response avoids spreading from fractional delay and Doppler shifts and effectively exploits DD-domain channel sparsity.

Abstract

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This paper proposes an off-grid channel estimation scheme for orthogonal time-frequency space (OTFS) systems adopting the sparse Bayesian learning (SBL) framework. To avoid channel spreading caused by the fractional delay and Doppler shifts and to fully exploit the channel sparsity in the delay-Doppler (DD) domain, we estimate the original DD domain channel response rather than the effective DD domain channel response as commonly adopted in the literature. OTFS channel estimation is first formulated as a one-dimensional (1D) off-grid sparse signal recovery (SSR) problem based on a virtual sampling grid defined in the DD space, where the on-grid and off-grid components of the delay and Doppler shifts are separated for estimation. In particular, the on-grid components of the delay and Doppler shifts are jointly determined by the entry indices with significant values in the recovered sparse vector. Then, the corresponding off-grid components are modeled as hyper-parameters in the proposed SBL framework, which can be estimated via the expectation-maximization method. To strike a balance between channel estimation performance and computational complexity, we further propose a two-dimensional (2D) off-grid SSR problem via decoupling the delay and Doppler shift estimations. In our developed 1D and 2D off-grid SBL-based channel estimation algorithms, the hyper-parameters are updated alternatively for computing the conditional posterior distribution of channels, which can be exploited to reconstruct the effective DD domain channel. Compared with the 1D method, the proposed 2D method enjoys a much lower computational complexity while only suffers slight performance degradation. Simulation results verify the superior performance of the proposed channel estimation schemes over state-of-the-art schemes.

I. INTRODUCTION

OTFS exploits structured and often sparse delay-Doppler channels, but fractional delay and Doppler shifts spread energy across finite-resolution bins and complicate channel estimation. The paper addresses this with off-grid SBL methods that estimate the original DD channel response and reduce complexity through delay-Doppler decoupling.

  • Motivation: OTFS maps time-varying channels into a quasi-time-invariant delay-Doppler domain whose sparsity can support efficient channel estimation and detection.The DD-domain input-output coupling is simpler than in the time-frequency domain, and DD-domain training enables channel probing.
  • Challenges: Fractional delay and Doppler shifts can spread an originally sparse DD channel across multiple indices, degrading channel estimation.Exact shifts may straddle finite-resolution bins rather than falling into a single bin, making the effective DD channel non-sparse.
  • Prior limitations: Guard spaces mitigate interference from unknown data symbols but incur significant signaling overhead and make existing estimation performance sensitive to guard-space availability.Earlier pilot-based approaches used impulse pilots and sufficient guard spacing in the DD domain.
  • Prior limitations: Existing compressed-sensing approaches face limitations from greedy OMP estimation, integer-Doppler assumptions, coarse grids, and cases where DD sparsity does not hold.The cited SBL scheme improves over OMP but remains essentially on-grid and is valid only in limited settings.
  • Results: The 2D off-grid SBL method has computational complexity proportional to the summed delay and Doppler grid sizes, unlike the 1D method’s corresponding product-scale dependence.Its performance approaches the 1D method with significantly reduced computational complexity, while simulations report superior performance over state-of-the-art schemes.

II. SYSTEM MODEL

OTFS represents information in a two-dimensional delay-Doppler domain, where the original channel response is sparse but fractional shifts spread the sampled effective channel.

  • The effective DD-domain channel is obtained by sampling the original DD-domain channel response through a function incorporating path shifts, pulse shaping, and transceiver windows.
  • The considered input-output model assumes a single-input single-output transceiver, with M subcarriers and N time slots defining the OTFS frame.
  • The system model supports arbitrary pulse shaping filters and arbitrary TF-domain windows, although the paper focuses on ideal pulses and a rectangular window.
  • The original DD-domain response is sparse, but fractional Doppler and delay make the sampling functions nonzero away from the path location, causing effective-channel spreading.
  • The paper therefore estimates the original DD-domain response directly and reconstructs the effective channel for data detection.

III. 1D OFF-GRID COMPRESSED CHANNEL ESTIMATION

The paper formulates OTFS channel estimation as a one-dimensional sparse signal recovery problem.

  • OTFS channel estimation is formulated as a 1D sparse signal recovery problem using the DD-domain input-output relationship.

A. Channel Estimation Problem Formulation

The channel estimation formulation uses pilot observations and a truncated DD-domain signal, then linearizes the shift-dependent measurement model on a virtual grid to separate on-grid and off-grid components.

  • A. Channel Estimation Problem Formulation: The unknown normalized Doppler shifts, normalized delay shifts, and channel coefficients are estimated from received pilot observations through a shift-dependent measurement matrix.
  • A. Channel Estimation Problem Formulation: Unknown path count makes the channel-vector length variable, while unknown shifts enter the measurement matrix and invalidate conventional LS and MMSE estimation.
  • A. Channel Estimation Problem Formulation: Data-symbol effects are treated as independent Gaussian model error, and pilot symbols are used to construct the measurement matrix.
  • A. Channel Estimation Problem Formulation: A single pilot impulse is adopted for simplicity, while the general formulation also covers multiple pilot pulses or pilot sequences.
  • A. Channel Estimation Problem Formulation: Only received samples within the known maximum-Doppler and maximum-delay region are retained, producing a truncated signal that serves as a sufficient statistic and reduces complexity.
  • A. Channel Estimation Problem Formulation: The truncated measurement matrix contains one column per path, with entries determined by the pilot and the sampling function evaluated at each path’s shifts.
  • A. Channel Estimation Problem Formulation: Because Doppler and delay shifts are coupled to the measurement matrix, the estimation model is nonlinear and motivates a first-order linear approximation.
  • B. First-Order Linear Approximation: The approximation uses a virtual DD-domain sampling grid to estimate nearest on-grid shifts and separate their off-grid components using first-order gradients.

C. 1D Off-grid SSR Model

The paper formulates OTFS channel estimation as a 1D off-grid sparse signal recovery problem on a virtual delay-Doppler grid. It separates on-grid path locations from off-grid delay and Doppler components and estimates both using an SBL hierarchy.

  • 1D off-grid SSR formulation: The 1D model recasts the OTFS channel-estimation task as sparse signal recovery on a virtual sampling grid.The linear approximation error is absorbed into the noise term.
  • Off-grid parameterization: The measurement matrix incorporates first-order corrections for off-grid Doppler and delay shifts through κν and ιτ.The resulting matrix is ΦT(κν, ιτ) = ΦT + ΦT,ν diag(κν) + ΦT,τ diag(ιτ).
  • Grid components: Significant entries in the sparse vector determine the jointly estimated on-grid delay and Doppler components.The corresponding off-grid components are estimated separately through the parameters κν and ιτ.
  • Hierarchical SBL model: A hierarchical SBL prior uses a diagonal covariance Λ = diag(α), with α controlling sparse channel coefficients.The hyperprior formulation results in a Laplace prior and sparse estimation.
  • Hierarchical SBL model: The off-grid Doppler and delay variables are modeled as hyper-parameters with uniform distributions within their prescribed bounds.They are updated together with the posterior distribution of the channel vector.

B. Off-grid SBL-based Channel Estimation Algorithm

The proposed 1D estimator alternates conditional channel inference with hyper-parameter updates under an EM-based SBL framework. It converges to a stationary posterior point but can become computationally expensive on large virtual grids.

  • Inference procedure: The algorithm alternates conditional posterior computation with updates of α, κν, ιτ, and β0 until convergence.The hyper-parameters are updated by maximizing the posterior distribution in each iteration.
  • Inference procedure: EM updates the hyper-parameters by forming an expectation objective and maximizing its hyper-parameters alternatively.Closed-form update rules are derived for the hyper-parameters, including the off-grid shift variables.
  • Algorithm output: The converged algorithm returns estimates of channel coefficients, normalized Doppler shifts, and normalized delay shifts.Termination occurs after the iteration limit or when changes in α fall below the convergence tolerance.
  • Convergence: The EM objective is non-decreasing over iterations, guaranteeing convergence to a stationary point of the hyper-parameter posterior.This guarantee applies to the proposed alternating update procedure.
  • Complexity and limitations: Large virtual sampling grids make the 1D estimator computationally expensive because its complexity is dominated by posterior covariance computation.This limitation motivates the paper’s 2D off-grid estimator.
  • Data-aided extension: Unknown data symbols are treated as noise, while an optional data-aided iteration can reuse detected symbols as pilots for further channel updates.The data-aided procedure repeats estimation and detection until the estimated channel stops changing.

V. 2D OFF-GRID SBL-BASED CHANNEL ESTIMATION

To reduce the computational complexity of SBL-based OTFS channel estimation, the paper proposes a 2D off-grid SSR model and corresponding 2D off-grid SBL algorithm. This model differs fundamentally from prior 1D compressed-sensing formulations.

  • 2D off-grid SBL estimation: The paper proposes a 2D off-grid SSR model and a 2D off-grid SBL-based channel-estimation algorithm.The stated motivation is reducing the computational complexity of SBL-based estimation.
  • Relation to prior work: Earlier compressed channel-sensing methods for OTFS use 1D SSR models, whereas the proposed model is fundamentally 2D.The comparison concerns the sparse-recovery formulation rather than merely an implementation detail.

A. 2D Off-grid SSR Model

The 2D formulation exploits separable pilot structure to decouple Doppler and delay estimation on separate virtual grids. It then uses a two-step SBL procedure based on the resulting composite channel representation.

  • 2D off-grid SSR model: When the pilot sequence is separable across Doppler and delay, the 2D model can decouple estimation of normalized Doppler and delay shifts.A single pilot impulse is a special case of this separable structure.
  • Virtual sampling grids: The 2D model defines separate virtual sampling grids for Doppler and delay domains.The Doppler grid spans (−νmax, νmax), while the delay grid spans (0, τmax).
  • 2D off-grid SSR model: The received signals are recast as a 2D SSR problem using the separate grids and a first-order approximation.The measurement model includes approximation error and interference from unknown data symbols in its noise power.
  • Sparse channel representation: The sparse channel matrix contains at most P non-zero entries corresponding to the channel paths.The received-signal matrix collects observations over a compact delay-Doppler region.
  • 2D SBL algorithm: The proposed 2D estimator uses two SBL steps: first estimating a composite matrix D, then estimating the remaining channel representation from D.This procedure is built on the 2D SSR model.

1) First Step:

The first step recovers a row-sparse channel matrix using a multiple-snapshot SBL model, with channel variables and hyper-parameters updated iteratively. Its complexity is dominated by matrix inversions, while the decoupled 2D formulation reduces computational order relative to the 1D formulation.

  • Model: The method models Doppler-domain channel columns as jointly row-sparse and recovers them with a multiple-snapshot SBL algorithm.All columns share the same support, enabling a common row-sparsity structure.
  • Iterative SBL: The unknown channel matrix and hyper-parameters are updated iteratively to obtain the conditional posterior distribution and the recovered channel estimate.The updates include α, κν, and β0, followed by reconstruction of the estimated matrix after convergence.
  • Convergence: The EM-based first step has guaranteed convergence, with each iteration producing a non-decreasing posterior objective.The convergence guarantee applies to the first step of Algorithm 2.
  • Complexity: The 1D method has complexity proportional to the product of delay and Doppler grid sizes, whereas the 2D method scales with their sum.Both methods remain dominated by channel-inversion operations.
  • 2D decomposition: The 2D method decouples Doppler and delay estimation into separate 1D SSR problems that can be solved in parallel.This decoupling is the basis for the lower complexity of Algorithm 2.

VI. SIMULATIONS

The simulations evaluate proposed 1D and 2D off-grid SBL channel estimation against impulse-, OMP-, NOMP-, and on-grid SBL-based schemes. Performance is measured by effective-channel NMSE over averaged OTFS frames under varied sampling resolutions and guard-space settings.

  • Compared methods: The evaluation compares proposed 1D and 2D off-grid SBL estimators with traditional impulse, DC-window, OMP, NOMP, and on-grid SBL methods.The compared methods include both effective-channel and compressed-sensing-based estimators.
  • System parameters: The simulations use N = 32, a 3 GHz carrier frequency, and 15 kHz subcarrier spacing.The maximum relative velocity among transceivers or scatterers is 506.25 km/h.
  • Channel setup: The channel model uses P = 5 paths, maximum normalized Doppler shift kmax = 3, and maximum delay shift lmax = 4.Doppler and delay shifts are generated uniformly within their specified ranges.
  • Sampling and guard space: Virtual Doppler and delay sampling resolutions are varied over rν, rτ ∈ [0.2, 0.9], with guard-space grids reserved around the pilot.The guard space mitigates interference from data symbols.
  • Metric: Effective-channel NMSE is the evaluation metric, and all simulation results are averaged over 1,000 OTFS frames.The reconstructed effective channel is obtained from the estimated original DD-domain response.

A. NMSE versus SNR with Different Virtual Sampling Resolutions

The simulations show that SBL-based estimators generally outperform impulse- and OMP-based alternatives, while the 2D method trades some accuracy for substantially lower complexity. Off-grid estimation is especially useful at low virtual sampling resolution, but its advantage depends on the nonlinear model and interference setting.

  • With guard space: At high virtual sampling resolution rν = rτ = 0.5, the proposed 1D method has the lowest NMSE and gains about 1.5 dB over the on-grid method in the high-SNR regime.The proposed 2D schemes are coarser than the 1D methods because of possible error propagation between their two estimation steps.
  • Resolution effects: The 2D on-grid method can outperform the 2D off-grid method at high virtual sampling resolution because nonlinear hyper-parameter coupling can reduce the off-grid gain.The 2D on-grid model also has fewer unknown parameters.
  • Baseline comparisons: OMP-based methods generally outperform impulse-based methods but remain worse than the proposed SBL methods, reflecting the heuristic nature of OMP.NOMP gains 2 dB over on-grid OMP yet still has substantially higher NMSE than the proposed SBL schemes.
  • Resolution effects: At low virtual sampling resolution, the 2D off-grid method outperforms the 2D on-grid method by estimating off-grid delay and Doppler components.The 1D off-grid SBL method remains the best-performing method in this setting.
  • Without guard space: The proposed 1D off-grid SBL estimator achieves the lowest NMSE at low virtual sampling resolution rν = rτ = 0.8, with about 1 dB gain over 1D on-grid SBL without guard space.This result is reported for channel estimation without guard space.
  • Guard-space impact: Removing guard space causes about 1–2 dB NMSE loss across evaluated schemes because of increased interference from data symbols.Guard-space insertion also incurs signaling overhead of (4kmax + 1)(2lmax + 1) symbols out of MN symbols.

D. Data-Aided Channel Estimation

The paper’s data-aided channel estimation improves NMSE mainly during the first iteration, while the proposed off-grid SBL framework estimates the original DD-domain response and offers a lower-complexity 2D alternative.

  • Data-aided iterations: About 1.5 dB NMSE reduction is achieved in the first data-aided iteration, with only marginal improvement afterward.The first iteration retrieves most unknown data symbols and provides the most significant information for reconstructing the measurement matrix.
  • Data-aided iterations: Only the first data-aided iteration is needed in practice to achieve acceptable channel-estimation performance.This choice reflects the high computational complexity of data detection and channel estimation and the marginal information from later iterations.
  • Proposed estimation framework: The proposed scheme estimates the original DD-domain channel response rather than the effective DD-domain channel.The channel-estimation problem is reformulated as 1D and 2D off-grid SSR problems, separating on-grid and off-grid delay and Doppler components.
  • Proposed estimation framework: The on-grid delay and Doppler components are inferred from the recovered channel vector, while the off-grid components are estimated using an EM algorithm within SBL.The developed algorithms update hyper-parameters alternately to compute the conditional posterior distribution of the channel.
  • Performance and complexity: The proposed off-grid scheme improves channel-estimation accuracy over on-grid methods at low virtual sampling resolution with only slightly higher computational complexity.The 1D method provides superior performance, whereas the 2D method reduces complexity by decoupling delay and Doppler estimation.
  • Performance and complexity: Data-aided channel estimation improves robustness against interference from unknown data symbols and can improve performance without guard space.Extensive simulations demonstrate the proposed channel-estimation performance.
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