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AFDM Channel Parameter Estimation: Exploiting its Shift Properties and Input-Output Relationship

Jialiang Zhu, Zeping Sui, Zilong Liu, Arman Farhang

arXiv:2610.11758v1eess.SP

TL;DR

Existing AFDM channel estimation methods can require large pilot overhead or computationally intensive grid search. This paper proposes an impulse-pilot method that uses AFDM shift properties and SIC for multipath estimation. It reports substantially lower computational cost, less overhead, and improved estimation and BER performance than benchmarks.

  • Problem

    Existing AFDM channel parameter estimation methods either require large pilot overhead or rely on computationally intensive grid search.

    Source: §I · p. 1

  • Method

    The method uses impulse-pilot shift properties to estimate integer delay and Doppler, leakage to estimate fractional Doppler, and successive interference cancellation for multipath channels.

    Source: §I · p. 1

  • Results

    The method requires approximately 300 times and 800 times fewer CMs than peak selection and OMP-based methods, respectively, and improves channel estimation and BER performance while allowing less pilot overhead than benchmarks.

    Source: §I · p. 2, §VI · p. 5

  • Takeaways & Limitations

    At Q = 96, the method achieves approximately 4.8 × 10−3 NMSE, around peak selection's NMSE at the same SNR, with an 11% reduction in guard overhead.

    Source: §VI · p. 6

Abstract

from arXiv · show

Affine frequency division multiplexing (AFDM) has attracted significant attention owing to its excellent backwards compatibility, full diversity achievability, as well as strong resilience to high Doppler and various hardware impairments. Existing AFDM channel parameter estimation methods, however, either require large pilot overhead or rely on computationally intensive grid search methods. This paper proposes a grid-search-independent, low-overhead, impulse-pilot-based channel parameter estimation method for AFDM systems by exploiting its inherent shift properties. Specifically, we show that the effect of a Doppler shift on the impulse pilot is equivalent to a delay shift in the time domain with a phase rotation. Additionally, both delay and Doppler shifts lead to cyclic shifts in the affine domain. By utilizing these properties in both the time and affine domains, two linear equations are formed to estimate the integer delay and Doppler of each path. The fractional part of the Doppler is estimated from the Doppler-induced leakage around the impulse pilot by exploiting the AFDM input-output relationship. For multipath channels, a successive interference cancellation (SIC) framework is developed, in which the strongest propagation path is first estimated, reconstructed, and canceled before the remaining paths are successively estimated. Simulation results demonstrate that the proposed method significantly reduces computational complexity and allows lower pilot overhead than the benchmarks. The proposed method also achieves improved channel parameter estimation and

I. INTRODUCTION

AFDM channel estimation faces trade-offs among pilot overhead and computational complexity, motivating an impulse-pilot method that exploits AFDM shift properties. AFDM uses a DAFT-based affine representation, and the proposed method reports better estimation and BER performance with less overhead than benchmarks.

II. AFFINE FREQUENCY DIVISION MULTIPLEXING

  • AFDM uses a DAFT-based affine channel representation in which paths become shifts, while a chirp-periodic prefix mitigates multipath inter-symbol interference.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

  • Existing AFDM estimation methods face high pilot overhead or computational complexity, including guard-based pilots, high-power pilot designs, doubled guard overhead, and grid-based refinement.Source: §I · p. 1
  • An impulse-pilot method exploits AFDM delay and Doppler shift properties in time and affine domains to estimate integer parameters; Doppler-induced leakage supports fractional Doppler estimation.Source: §I · p. 1
  • The proposed method achieves better channel estimation accuracy and BER performance than peak selection and OMP-based benchmarks, while allowing less pilot overhead.Source: §I · p. 2

III. TIME AND FREQUENCY SHIFT EFFECTS ON AFDM

An affine-domain impulse pilot exhibits complementary shift effects under Doppler and timing offsets. These relationships underpin the proposed channel parameter estimation technique.

III. TIME AND FREQUENCY SHIFT EFFECTS ON AFDM

  • A Doppler shift on an affine-domain impulse pilot appears as a time-domain shift with phase rotation.Source: §III · p. 2
  • A timing offset produces an affine-domain shift with phase rotation.Source: §III · p. 2
  • The two shift properties provide the foundation for the proposed channel parameter estimation technique.Source: §III · p. 2

A. Effect of Doppler shift in time domain

For an impulse pilot, the paper relates Doppler to a circular shift of the pilot in the time domain, accompanied by a constant phase. This equivalence is expressed through a condition relating the pilot index, Doppler, and shift.

A. Effect of Doppler shift in time domain

B. Effect of timing offset in affine domain

A timing offset shifts the impulse pilot in the affine domain while preserving its impulse form and adding a phase factor. The paper relates these shifts to the LCT structure underlying AFDM.

B. Effect of timing offset in affine domain

IV. PROPOSED CHANNEL PARAMETER ESTIMATION

The method estimates paths iteratively, beginning with the strongest path and removing its reconstructed contribution from the received signal. Integer delay and Doppler are jointly identified from time- and affine-domain shifts, with the path count assumed known a priori.

IV. PROPOSED CHANNEL PARAMETER ESTIMATION

  • Each iteration estimates the strongest path’s integer delay and Doppler, then its fractional Doppler and gain, before reconstructing and subtracting that path.The process repeats on the residual until all path parameters are found.Source: §IV · p. 3
  • Assuming a sufficiently high sampling rate, the method models path delays as integer-valued and Doppler shifts as integer plus fractional parts, and assumes the number of propagation paths is known a priori.Source: §IV · p. 3

A. Integer Delay and Doppler Estimation

  • For the strongest path, the received-signal/pilot cross-correlation peak supplies the time-domain index used to relate Doppler-induced shift to path delay.The path Doppler is represented as an integer part plus a fractional part.Source: A. Integer Delay and Doppler Estimation · p. 3
  • Joint time- and affine-domain shift constraints identify integer delay and Doppler even when the affine-domain shift wraps modulo N.Unlike affine-only methods and, the candidate pair must satisfy both relations.Source: A. Integer Delay and Doppler Estimation · p. 3

B. Fractional Doppler Estimation

Fractional Doppler is estimated from the impulse pilot’s leakage using the AFDM input-output relationship. The method estimates its magnitude from the central and stronger neighboring samples, then selects its sign by correlation.

B. Fractional Doppler Estimation

  • The input-output relationship is evaluated for the received signal with an impulse pilot, treating other paths as interference when isolating the strongest path.The simplified derivation omits that interference term, while the simulations retain noise and interference.Source: B. Fractional Doppler Estimation · p. 4
  • The fractional Doppler magnitude is obtained from the central pilot sample and the stronger of its neighboring samples, using a quantity that depends only on the fractional Doppler.The derivation neglects noise and interference for simplicity, although simulations include both.Source: B. Fractional Doppler Estimation · p. 4
  • The sign of fractional Doppler is selected by comparing correlation under the two sign hypotheses using the estimated delay and Doppler parameters.Each hypothesis forms the corresponding effective channel matrix for the path.Source: B. Fractional Doppler Estimation · p. 4

C. Path gain estimation

After estimating a path’s delay and Doppler, the method estimates its complex-valued gain by least squares.

C. Path gain estimation

  • The complex-valued gain of a path is estimated by least squares after its delay and Doppler parameters have been estimated.The received signal is approximated using the estimated effective channel and pilot.Source: C. Path gain estimation · p. 4
  • The path-gain model includes the unknown complex gain and a noise vector in the received-signal approximation.The noise vector is denoted η = [η[0], . . . , η[N −1]]T.Source: C. Path gain estimation · p. 4
  • The method obtains the gain estimate from the resulting least-squares problem.The passage introduces the closed-form solution after specifying the effective-channel model.Source: C. Path gain estimation · p. 4

D. Proposed Multipath Channel Estimation via SIC

The multipath procedure estimates and reconstructs the dominant path, removes its contribution, and repeats on the residual until all paths are estimated. It also uses a preliminary LMMSE data-symbol estimate to reconstruct each path’s contribution.

D. Proposed Multipath Channel Estimation via SIC

  • The SIC procedure estimates the dominant path, reconstructs and removes its pilot and data contributions, then repeats on the residual until all paths are estimated.Removing both components addresses their interference with estimation of remaining paths.Source: D. Proposed Multipath Channel… · pp. 4–5
  • Because the matrices in the stated detector expression are unitary, conventional LMMSE matrix inversion reduces to scalar normalization.Source: D. Proposed Multipath Channel… · p. 4
  • A preliminary LMMSE estimate of the data symbols is used to reconstruct each path’s contribution before that contribution is removed from the received signal.The reconstructed contribution is formed using the path estimate and the pilot-plus-data signal.Source: D. Proposed Multipath Channel… · p. 5

V. COMPUTATIONAL COMPLEXITY ANALYSIS

The proposed estimator has linear per-path complexity, while the dictionary-based and OMP benchmarks require work that scales with their search grids.

V. COMPUTATIONAL COMPLEXITY ANALYSIS

  • The OMP benchmark requires approximately O(PNN_lN_ν) complex multiplications, with N_ν denoting the Doppler-dictionary size.Each iteration correlates the residual with delay-Doppler atoms, and the preliminary estimate adds O(N) operations.Source: §V · p. 5
  • The dictionary-based benchmark has complexity O(NN_lN_αN_a), which increases rapidly with search resolution.N_l, N_α, and N_a are the dictionary sizes for delay, integer Doppler, and fractional Doppler.Source: §V · p. 5
  • Simultaneous correlation-peak detection can let strong or closely spaced paths mask weaker paths or one another.The passage identifies successive OMP estimation and cancellation as a way to avoid this issue.Source: §V · p. 5
  • The proposed method requires O(3PN) complex multiplications for a P-path channel, versus grid-dependent benchmark costs.Its time-domain peak detection and fractional-Doppler sign evaluation each require O(N) complex multiplications per path.Source: §V · p. 5

VI. SIMULATION RESULTS

Simulations compare parameter and channel-matrix estimation, guard overhead, and detection performance. The proposed method improves estimation and BER outcomes against the benchmarks, while requiring fewer complex multiplications.

VI. SIMULATION RESULTS

  • The simulation uses 1024 subcarriers, 4-QAM, three paths, and perfect synchronization in a LOS-dominant Rician TDL channel.The first tap is Rician and the remaining taps are independent Rayleigh fading.Source: §VI · p. 5
  • Under the stated simulation settings, the proposed method requires approximately 300 times fewer CMs than peak selection and 800 times fewer than OMP.The benchmark delay and Doppler grid resolutions are 1 and 0.01, respectively.Source: §VI · p. 5
  • Above 15 dB SNR, the proposed scheme achieves lower delay, Doppler, and path-gain NMSE than the benchmarks.The figures cover Doppler estimation, delay estimation, and path-gain estimation.Source: §VI · p. 5, Figure 3 · p. 6, Figure 2 · p. 6
  • Above 15 dB SNR, the reconstructed channel-matrix NMSE is lower than with OMP and peak selection, while at low SNR it remains comparable.Missed weak paths can raise parameter-estimation NMSE at low SNR, but contribute little to the overall channel.Source: §VI · p. 5, Figure 5 · p. 6
  • At Q=96 and 25 dB SNR, channel-estimation NMSE is approximately 4.8 × 10^-3, comparable to peak selection with 11% less guard overhead.Fig. 6 reports channel-matrix NMSE for different Q.Source: §VI · p. 6, Figure 6 · p. 6
  • The proposed method outperforms OMP and peak selection in BER, has a lower error floor, and is nearly identical to perfect-CSI performance below 18 dB.At higher SNRs, only a small performance gap from perfect CSI is observed.Source: §VI · p. 6

VII. CONCLUSION

The proposed AFDM channel-estimation method uses delay-Doppler shift properties and the input-output relationship, with SIC for multipath channels. Simulations report lower computational complexity and pilot overhead, along with improved channel-estimation and BER performance versus benchmark solutions.

VII. CONCLUSION

  • Integer delay and Doppler are estimated from time- and affine-domain shift properties, while fractional Doppler is estimated from leakage around the impulse pilot.The fractional-Doppler estimate uses the AFDM input-output relationship.Source: §VII · p. 6
  • For multipath channels, an iterative SIC framework successively estimates, reconstructs, and cancels dominant paths.Source: §VII · p. 6
  • The proposed method achieves lower computational complexity and pilot overhead, with improved channel-estimation and BER performance versus benchmark solutions.Fig. 7 presents BER performance.Source: Figure 7 · p. 6, §VII · p. 6
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