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Model-Order-Adaptive Channel Estimation for AFDM Systems with Fractional Delay and Doppler

Ruizhe Wang, Cunhua Pan, Hong Ren, Shaohua Li, Jiangzhou Wang

arXiv:2610.09970v1cs.ITmath.IT

TL;DR

Fractional delay and Doppler spread AFDM path energy, while the unknown number of paths makes channel estimation challenging. The paper analyzes the wrapped response and proposes an adaptive SAGE estimator that initializes paths with dual pilots, uses temporal observations, and adapts model order. Simulations report lower NMSE and BER than the considered benchmarks, including at velocities up to 600 km/h.

  • Problem

    Fractional delay and Doppler spread each path’s energy across DAFT-domain samples, complicating accurate estimation of continuous path parameters needed to exploit AFDM’s channel representation.

    Source: §I · p. 1

  • Method

    An adaptive SAGE estimator combines dual-pilot path initialization, multi-symbol temporal observations, continuous parameter refinement, and statistically controlled path addition and pruning.

    Source: §I · p. 2

  • Results

    The method reports lower NMSE and BER than considered benchmarks and robust performance at terminal velocities up to 600 km/h.

    Source: §I · p. 2, §VIII · p. 12

  • Takeaways & Limitations

    The analyzed response extrema support path initialization and refinement, while the estimator jointly determines model order and continuous path parameters.

    Source: §I · p. 2

Abstract

from arXiv · show

Affine frequency division multiplexing (AFDM) has emerged as a promising waveform for high-mobility communications owing to its ability to exploit multipath diversity. However, channel state information (CSI) acquisition remains challenging in channels with fractional delay and Doppler and an unknown number of propagation paths. In this paper, we investigate the channel estimation for AFDM systems. We first analyze the AFDM response in the presence of fractional-delay-induced frequency wrapping. The analysis reveals that fractional delay may displace the dominant extremum and generate informative secondary extrema. Then, we develop a model-order-adaptive channel estimator within an improved space-alternating generalized expectation-maximization (SAGE) framework. A dual-pilot reference symbol is employed for path and delay initialization, while pilot observations across multiple AFDM symbols provide temporal information for Doppler estimation. New paths are identified through statistically controlled residual tests, whereas unsupported or redundant paths are removed by conditional support pruning. The delay, Doppler frequency, and complex gain of each retained path are subsequently estimated through a coarse-to-fine procedure and refined using the exact AFDM likelihood. Simulation results demonstrate that the proposed method achieves lower normalized mean squared error (NMSE) and bit error rate (BER) than representative SAGE, sparse-recovery, and Bayesian benchmarks. It also provides reliable path detection and model-order estimation and maintains robust performance at terminal velocities of up to 600~km/h.

I. INTRODUCTION

AFDM supports high-mobility communications, but fractional delay and Doppler spread path energy and make continuous channel estimation important for using its delay-Doppler representation. The paper analyzes these response effects and introduces an adaptive SAGE estimator for jointly determining path parameters and model order.

I. INTRODUCTION

  • Fractional delay and Doppler spread path energy across DAFT-domain samples, making accurate continuous delay, Doppler, and gain estimates important for exploiting AFDM’s channel representation.The resulting interference occurs among adjacent path responses.Source: §I · p. 1
  • Fractional delay can displace the dominant response extremum and create informative secondary extrema for parameter initialization and refinement.This analysis concerns the exact AFDM input-output relationship under fractional delay and Doppler.Source: §I · p. 2
  • The estimator combines dual-pilot initialization, multi-symbol temporal observations, and coarse-to-fine refinement with the exact AFDM likelihood.Temporal phase variation supports Doppler estimation, followed by least-squares complex-gain estimation.Source: §I · p. 2
  • Statistically controlled residual tests add paths and conditional support pruning removes unsupported or redundant paths, jointly adapting model order and continuous path parameters without testing prescribed orders.The motivation is that parameter errors can leave structured residuals or split a physical path, while premature stopping or pruning can remove weak paths.Source: §I · p. 2
  • The proposed method reports lower NMSE and BER than considered benchmarks, with reliable path detection and model-order estimation under velocities up to 600 km/h.The introduction also reports improving path-detection F1 with SNR and estimated model order approaching the true average path count.Source: §I · p. 2

A. Single-Antenna AFDM Transmission

  • The single-antenna AFDM setup uses M subcarriers and defines a sampling interval for the transmitted signal.The supplied passage also introduces the effective channel matrix and identifies B(ℓp, ιp, κp) as a path response.Source: A. Single-Antenna AFDM Transmission · p. 3

B. AFDM Input–Output Relationship

This section gives the AFDM input-output relationship and specifies the pilot observations used for estimation. The pilot design places two pilots in the first symbol and adds a tracking-symbol pilot to provide temporal information for Doppler estimation.

B. AFDM Input–Output Relationship

  • The AFDM input-output relationship is presented for fractional delay and Doppler, with a path response matrix and an amplitude response for each propagation path.The supplied passages also state that the estimator can extend to multi-antenna systems by incorporating each path’s spatial response while keeping the delay-Doppler procedure unchanged.Source: B. AFDM Input–Output Relationship · p. 3

C. Pilot-Aided Observation Model

  • The first AFDM symbol contains two pilots, at p1 = B and p2 = M − 1, with B denoting the guard interval between pilot and data.The receiver retains observation windows around the pilots.Source: C. Pilot-Aided Observation Model · p. 3
  • A pilot at M − 1 is also sent in tracking symbols to provide temporal information for improving Doppler estimation.The joint pilot observation contains NJ = (2 + Ntr)W samples.Source: C. Pilot-Aided Observation Model · p. 3

III. EXTREMA AND IDENTIFIABILITY OF THE AFDM ATOM

The section analyzes extrema of the AFDM atom response under fractional delay and examines delay and Doppler identifiability. It explains why the strongest sampled coefficient can localize integer delay coarsely, while secondary extrema contribute information for fractional-delay refinement.

III. EXTREMA AND IDENTIFIABILITY OF THE AFDM ATOM

  • The atom response is analyzed by decomposing its inner summation into geometric components, whose finite intervals yield Dirichlet-type main lobes and sidelobes.The complete amplitude response combines phase-transition terms, and its squared magnitude determines the extrema.Source: §III · p. 4
  • At boundaries where summation membership changes, extrema are determined piecewise using corresponding one-sided response values..Source: §III · p. 4
  • Fractional delay can shift the complete response maximum away from the nominal coherent location, although each geometric term is individually maximized there.The phase-transition components rotate relative to one another, so their weighted sum need not have a stationary point at zero mismatch.Source: §III · p. 4
  • The dominant sampled peak gives a coarse indication of integer delay, but cannot by itself estimate fractional delay exactly.Its continuous location also depends on phase-weighted interactions among the geometric terms.Source: §III · p. 4
  • As fractional delay grows, the sampled magnitude profile becomes asymmetric and spreads across receive indices, with more pronounced secondary extrema and potentially comparable peaks.The dominant and secondary complex coefficients together carry information for off-grid fractional-delay refinement.Source: §III · pp. 4–5, Figure 1 · p. 4

IV. PROPOSED CHANNEL ESTIMATION ALGORITHM

The proposed framework adapts SAGE to jointly estimate AFDM channel parameters and unknown model order. It detects unresolved paths from residuals, estimates their parameters, and sequentially refines accepted paths.

A. Unknown-Order Channel Estimation and the Proposed SAGE Framework

  • The residual-test threshold provides approximate false-alarm calibration under an independent-cell approximation, using K_eff effective search cells despite correlated neighboring atoms.The approximation is stated to apply when neighboring AFDM atoms are correlated.Source: A. Unknown-Order Channel Estimation… · p. 5
  • The model-order objective combines normalized data mismatch with a per-path CFAR evidence penalty, with P_max setting the largest admissible order.The objective is also described as an ℓ0-penalized likelihood.Source: A. Unknown-Order Channel Estimation… · p. 5
  • Residual observations are used to detect unresolved paths, estimate their delay, Doppler, and gain, and incorporate them through a statistically controlled criterion.Accepted path contributions are reconstructed before residual testing and sequential refinement.Source: A. Unknown-Order Channel Estimation… · p. 5, §IV · p. 5
  • For each accepted path, SAGE forms a path-specific observation by removing other paths, estimates parameters using the exact AFDM atom, and refits complex gains.Diagonal loading is used numerically and approaches zero in least-squares gain refits.Source: A. Unknown-Order Channel Estimation… · pp. 5–6

B. Single-Path Delay, Doppler, and Gain Estimation

The single-path procedure uses dual-pilot peak location and temporal phase information to initialize delay and Doppler, then maximizes profile likelihood for continuous parameters and gain. The dominant-extremum score supplies a coarse delay-bin estimate, but interference can shift its peak.

B. Single-Path Delay, Doppler, and Gain Estimation

  • Peak location gives a coarse bin and temporal phase variations give coarse Doppler; these guide candidate integer delays before profile-likelihood estimation of continuous parameters and gain.The complex gain is estimated and jointly refined within the profile-likelihood procedure.Source: B. Single-Path Delay, Doppler, and… · p. 6

1) Dominant-Extremum Delay Initialization:

  • The dual-pilot location score sums squared residual magnitudes at each candidate bin, and its maximizing bin provides the coarse delay estimate.The residuals are the two W-sample observations corresponding to the dual pilots.Source: 1) Dominant-Extremum Delay… · p. 6
  • Because q-sum interference can shift the peak from the true effective-delay location, the coarse bin and Doppler estimates determine candidate integer delays while accepted and tested peaks are excluded.Previously accepted, pruned, and tested locations are excluded using specified neighborhoods.Source: 1) Dominant-Extremum Delay… · p. 6

2) Temporal Doppler Initialization and Integer-Delay Separation:

Temporal pilot observations provide Doppler initialization and help separate integer-delay candidates. The procedure then estimates fractional delay and Doppler from the AFDM response and refines them using bounded updates and a full-window cost.

2) Temporal Doppler Initialization and Integer-Delay Separation:

  • A single-tone temporal response supports maximum-likelihood Doppler estimation, followed by local polynomial interpolation around the largest periodogram sample.A common N_ν-point temporal steering matrix is used for all paths.Source: 2) Temporal Doppler Initialization and… · p. 6
  • The coarse bin and Doppler estimates determine an integer-delay center and neighboring candidates, with further expansion when fractional delay nears ±1/2 or normalized residual remains large.This candidate expansion addresses boundary cases and persistently large residuals.Source: 2) Temporal Doppler Initialization and… · p. 6

3) Joint Fractional-Delay and Doppler Refinement:

  • After cancellation of other paths and the other pilot response, strong peak samples are treated as dominated by the path being updated and normalized using current gain and delay estimates.The coarse fractional delay is obtained by solving a polynomial and selecting an admissible root closest to the unit circle.Source: 3) Joint Fractional-Delay and Doppler… · p. 6
  • Fractional delay and Doppler are refined with alternating updates and bounded Newton steps; a trial is accepted only when the full-window cost decreases.The refinement retains the lowest-cost state and atom.Source: 3) Joint Fractional-Delay and Doppler… · p. 7

4) Fixed-Delay Temporal Doppler and Gain Refinement:

With path delays fixed, tracking observations are used to refine Doppler and estimate time-varying gains. A regularized preliminary least-squares estimate supports innovation and effective-noise assessment before adaptive gain regularization.

4) Fixed-Delay Temporal Doppler and Gain Refinement:

  • A regularized preliminary least-squares estimate is used to assess innovation and effective noise, with ε_g stabilizing the estimate.The resulting variance estimate has a positive floor after preliminary least-squares noise bias is subtracted.Source: 4) Fixed-Delay Temporal Doppler and… · p. 7
  • With delays fixed, path Dopplers are refined using gain-SNR-weighted tracking gains, retaining the geometric Doppler estimate when the temporal estimate is unreliable.The tracking gains use a path-dependent regularization matrix.Source: 4) Fixed-Delay Temporal Doppler and… · p. 7
  • The tracking observations use a pilot transmitted at M −1 across the indexed tracking symbols, providing temporal information for Doppler estimation.The resulting time-varying channel matrix is formed after tracking gains are estimated.Source: 4) Fixed-Delay Temporal Doppler and… · p. 7

C. Adaptive Model-Order Selection

The adaptive model-order procedure detects paths sequentially from the dual-pilot residual, then prunes unsupported or redundant paths. CFAR testing and path-strength thresholds control path retention.

1) Sequential Residual-Domain Path Births:

C. Adaptive Model-Order Selection

  • Conditional pruning removes paths whose normalized strength falls below an adaptive threshold bounded by the CFAR term.The threshold scales with the largest observed path strength and is bounded by zmin and zmax.Source: 1) Sequential Residual-Domain Path… · p. 8

2) SAGE Refinement and Support Pruning:

The refinement stage alternates separated per-path updates with joint gain refitting and support pruning. It then uses all known pilots for missed-path recovery and final Doppler and gain refinement.

2) SAGE Refinement and Support Pruning:

  • Each path is refined against an observation with other-path contributions removed, and a trial is retained only if it reduces cost and preserves separation.The effective location must move by no more than Δmove and remain farther than Δsep from other paths.Source: 2) SAGE Refinement and Support Pruning · p. 8
  • After each path sweep, gains are jointly re-estimated, and paths below the adaptive strength threshold are pruned subject to minimum order Pmin.Pruned locations are recorded to prevent immediate rediscovery.Source: 2) SAGE Refinement and Support Pruning · p. 8
  • The refinement rounds stop when support is unchanged and the convergence condition holds, or when the maximum number of rounds is reached.Source: 2) SAGE Refinement and Support Pruning · p. 8
  • All known pilots are then used for missed-path recovery, joint refinement, and conditional pruning before fixed-support Doppler and gain refinement.Source: 2) SAGE Refinement and Support Pruning · p. 8

V. MONOTONICITY OF THE PENALIZED OBJECTIVE

The proposed estimator outperforms the benchmark schemes in NMSE and BER, while retaining strong path-detection and model-order performance. The section also establishes objective monotonicity under stated conditions and analyzes computational complexity.

V. MONOTONICITY OF THE PENALIZED OBJECTIVE

  • The penalized objective is shown to be monotonic under fixed γord, least-squares gain refitting, the stated birth and stopping rules, cost-decreasing SAGE updates, and bounded pruning.Source: §V · p. 8

VI. COMPUTATIONAL COMPLEXITY

  • The leading computational costs are exact-atom evaluation, observation-domain gain fitting, and temporal Doppler estimation.The analysis expresses these costs in terms of observation dimension, model order, Doppler grid size, and update counts.Source: §VI · p. 9, Table I · p. 10

VII. SIMULATION RESULTS

  • The proposed method achieves the lowest NMSE and BER across the considered SNR range and retains lower NMSE and BER than same-velocity benchmarks at 600 km/h.Its NMSE degradation from 300 to 600 km/h is slight, and its 16-QAM BER degradation is limited.Source: Figure 2 · p. 9, Figure 3 · p. 9, §VII · pp. 9–10
  • Path-support F1 improves from approximately 0.35 at 5 dB to 0.94 at 30 dB, with the proposed method highest or comparable in the moderate-to-high-SNR regime.At high SNR, NOMP-CFAR and SAGE-BIC approach its performance.Source: Figure 4 · p. 10, §VII · p. 10
  • Across considered symbol indices, the proposed method maintains the best NMSE, although accumulated Doppler-estimation phase error causes gradual degradation over time.The degradation is described as limited.Source: §VII · pp. 10–11, Figure 5 · p. 11
  • The estimated path count approaches the true model order as SNR rises, though the method may still omit weak paths because it retains only sufficiently significant contributions.VAMP and SBL tend to underestimate path count, while MDVALSE slightly overestimates the high-SNR average.Source: Figure 6 · p. 11, §VII · p. 11
  • The proposed method has the lowest delay RMSE over the considered SNR range, with estimation error decreasing as SNR increases.The paper attributes the improvement to continuous fractional-delay refinement using the exact AFDM response.Source: §VII · pp. 11–12, Figure 8 · p. 12
  • With true path count increasing from 6 to 10 at 30 dB, NMSE, missed paths, and BER increase, and performance is worse at 600 than 300 km/h.The paper relates this trend to resolving more responses and estimating more gains under a fixed pilot budget.Source: Figure 9 · p. 12, §VII · p. 12, Figure 10 · p. 12

VIII. CONCLUSION

The paper analyzes off-grid AFDM path responses and proposes an adaptive SAGE estimator for channel estimation with fractional delay and Doppler. Simulations show lower NMSE and BER than the considered benchmarks, including under high mobility.

VIII. CONCLUSION

  • The proposed adaptive SAGE estimator combines dual-pilot path initialization, multi-symbol observations, continuous delay–Doppler refinement, and model-order selection.A subsequent temporal refinement reconstructs the time-varying channel over the entire frame.Source: §VIII · p. 12
  • The method achieves lower NMSE and BER than the considered benchmarks over the tested SNR range and maintains its advantage under high-mobility conditions.Source: §VIII · p. 12
  • The paper characterizes the exact off-grid input–output relationship and analyzes extrema and identifiability of the AFDM path response.Source: §VIII · p. 12

APPENDIX A PROOF OF PROPOSITION 1

The proof establishes that path-birth, stopping, refinement, gain-refit, and pruning steps do not increase the objective. It also gives conditions and residual comparisons used in the path-pruning argument.

APPENDIX A PROOF OF PROPOSITION 1

  • When either stopping condition is met, no path is introduced; with fixed support, the order penalty is constant and a retained SAGE update does not increase the global residual.The least-squares gain refit also cannot increase L_P.Source: Appendix A · pp. 12–13
  • The pruning argument compares the normalized residual after deleting a path and refitting retained gains with the residual before deletion.A path is pruned only when G_p < z_th and z_th ≤ γ_ord.Source: Appendix A · p. 13
  • Path births, residual stopping, SAGE refinement, gain refitting, and path pruning are shown not to increase the objective.Source: Appendix A · p. 13
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