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Tensor Decomposition Based Mixed-Field Sensing for XL-MIMO AFDM Systems

Lei Yao, Yin Xu, Aimin Tang, Taohe Chen, Tianyao Ma, Wenjun Zhang

arXiv:2609.11051v1eess.SP

TL;DR

XL-MIMO AFDM sensing remains underexplored, with near-field spherical wavefronts coupling parameters and high mobility producing grid-mismatch challenges. The paper proposes tensor decomposition with decoupled AoA and continuous-domain delay-Doppler estimation. Simulations report orders-of-magnitude higher delay-Doppler accuracy and elimination of angular error floors.

  • Problem

    XL-MIMO AFDM sensing is largely unexplored, while near-field spherical wavefronts cause parameter coupling and high-mobility scenarios pose grid-mismatch challenges.

    Source: §I · p. 1

  • Method

    The paper decomposes received signals as a tensor using Vandermonde structure, then estimates AoA with SD-RDMO and delay-Doppler in the continuous domain with BR-AEG.

    Source: §I · p. 2, 1) Mixed Field Parameters at the… · p. 4, 2) Delay-Doppler Estimation · p. 5

  • Results

    The proposed scheme eliminates angular error floors and achieves orders-of-magnitude higher delay-Doppler accuracy than existing baselines.

    Source: §I · p. 2

  • Takeaways & Limitations

    The framework addresses near-field effects and grid mismatch within a unified sensing approach for XL-MIMO AFDM systems.

    Source: §I · p. 2, §V · p. 6

Abstract

from arXiv · show

Integrated sensing and communications enabled by extremely large-scale MIMO (XL-MIMO) and affine frequency division multiplexing (AFDM) is a highly promising paradigm for vehicular networks. However, the near-field spherical wavefront distortions induce severe non-linear parameter coupling, while the highly dynamic scattering environments exacerbate mismatch errors. To address these critical challenges, this paper proposes a novel tensor-based sensing scheme for XL-MIMO AFDM systems. First, the received signals are reformulated into a tensor, followed by an efficient decomposition approach that exploits the inherent Vandermonde structure of the factor matrices. This allows parameters to be directly estimated from the decomposed matrices, effectively avoiding inter-parameter coupling. Subsequently, a symmetric decoupling and real-domain manifold optimization algorithm is proposed for angle of arrival estimation, circumventing the high-dimensional searches typically induced by near-field effects. Furthermore, a baseband reconstruction and analytical gradient-based algorithm is developed to perform delay-Doppler estimation in the continuous parameter domain, fundamentally eradicating the grid-mismatch errors inherent in high-mobility scenarios. With these decoupled factors, the remaining unknown angle of departure can be readily extracted. Extensive simulation results demonstrate that the proposed scheme achieves orders-of-magnitude improvements in delay-Doppler accuracy and eliminates the error floors in angular estimation that severely bottleneck state-of-the-art baselines.

I. Introduction

The paper addresses the limited study of XL-MIMO AFDM sensing, especially mixed near- and far-field effects and parameter coupling in large apertures. It proposes tensor-based sensing and models a bistatic AFDM system whose modulation uses chirp parameters and a chirp periodic prefix.

I. Introduction

  • XL-MIMO AFDM sensing remains largely unexplored, while near-field spherical wavefronts in large apertures induce severe parameter coupling that motivates a high-precision, low-complexity scheme.Prior mixed-field sensing work is described as needing a broader near-field range and improved robustness to grid mismatch.Source: §I · p. 1
  • The proposed framework decomposes a third-order tensor using Vandermonde structure, then estimates AoA, continuous-domain delay-Doppler, and sequentially AoD.The authors report angular error-floor elimination and orders-of-magnitude higher delay-Doppler accuracy than existing baselines.Source: §I · p. 2

II. System Model

  • The bistatic setup uses an Nt-element transmitter ULA and an Nr-element XL-ULA at the receiving roadside unit, with transmit antennas sending the same DAFT-domain Zadoff-Chu pilot sequentially.Time-division multiplexing is used to extract transmit spatial signatures and facilitate tensor formulation.Source: §II · p. 2

A. AFDM Modulation

  • AFDM maps a time-domain transmit sequence through its modulation process and appends a chirp periodic prefix to support the channel's circular-convolution property and eliminate intercarrier interference.The chirp parameters c1 and c2 are independently tunable; the supplied text gives no further modulation mapping details.Source: A. AFDM Modulation · p. 2

B. Delay-Doppler Representation and AFDM Demodulation

The section defines normalized delay and Doppler in terms of integer and fractional components and specifies chirp-parameter choices. At reception, removing the redundant prefix and mapping the signal to the DAFT domain yields a delay-Doppler manifold representation.

B. Delay-Doppler Representation and AFDM Demodulation

  • Normalized delay and Doppler each comprise an integer part and a fractional offset, denoted by βq = lq + ιq and αq = kq + µq.The fractional delay and Doppler offsets lie in [−0.5, 0.5).Source: B. Delay-Doppler Representation and… · p. 2
  • The chirp parameter c1 is configured using the maximum Doppler and a guard parameter to prevent distinct nonzero entries from overlapping, while c2 is arbitrary irrational or sufficiently small rational.The passage gives c1 = (2(kmax+kv)+1)/2M.Source: B. Delay-Doppler Representation and… · p. 2
  • Receiver processing discards the chirp periodic prefix and maps the received signal back to the DAFT domain before representing each target with a delay-Doppler manifold vector.The delay matrix captures the linear phase shift; ΓCPP,q accounts for the integer-delay periodic-domain effect and equals IM when 2Mc1 is an integer and M is even.Source: B. Delay-Doppler Representation and… · p. 2

C. Manifolds Representation at Transmitter and Receiver

The system uses a planar-wave transmit model and a unified polar-algebraic receiver manifold for targets in the near or far field. The transmitter antennas send sequentially, allowing their echoes to be isolated.

C. Manifolds Representation at Transmitter and Receiver

  • The transmit array uses a far-field planar-wave approximation because its aperture is much smaller than the signal propagation distance.The steering-vector entry is expressed using AoD, wavelength, and transmit-antenna spacing.Source: C. Manifolds Representation at… · pp. 2–3
  • The receiving XL-ULA uses a polar algebraic mapping manifold to represent a mixed-field topology rooted in the spherical wavefront.Its state is represented by ψq = [ϑq, ρq]T.Source: C. Manifolds Representation at… · p. 3
  • When a target is sufficiently distant, ρq approaches zero and the spherical-wave receiver manifold reduces to the classical planar-wave steering vector.The unified expression uses the physical AoA and target distance relative to the reference antenna.Source: C. Manifolds Representation at… · p. 3
  • TDM assigns each transmit antenna a dedicated slot, so its echoes can be perfectly isolated at the receiver.The slot duration is Tsym.Source: C. Manifolds Representation at… · p. 3

A. Tensor Reformulation and Decomposition

The section models the received observations as a tensor to preserve their multidimensional structure, then exploits Vandermonde-structured factor matrices to estimate sensing parameters. Its algebraic decomposition addresses limitations of unconstrained iterative factorization.

A. Tensor Reformulation and Decomposition

  • High-order stacking constructs the observation in a multidimensional algebraic space to avoid structural information loss and parameter coupling from traditional matrix dimensionality reduction.Source: A. Tensor Reformulation and… · p. 3
  • The global observation tensor has dimensions Nr × M × Nt and is expressed through factor matrices, target coefficients, and a noise tensor.Source: A. Tensor Reformulation and… · p. 3
  • The target parameter set includes angle, distance, angle of departure, delay, Doppler, and scattering coefficient, with joint estimation framed as minimizing global Frobenius-norm reconstruction error.Source: A. Tensor Reformulation and… · p. 3
  • Vandermonde structure in the factor matrices offers an alternative to ALS, which is computationally demanding and does not use those constraints, limiting resolvable targets.The proposed estimation approach leverages the factors’ structural properties.Source: A. Tensor Reformulation and… · pp. 3–4
  • An algebraic factor-matrix estimator is proposed to use structural constraints, potentially avoiding iterative processes and improving resolution for more targets.Source: A. Tensor Reformulation and… · p. 4
  • The factor estimation procedure uses SVD signal subspaces and an eigenvalue decomposition to reconstruct the factor matrices.The passages describe recovery of the remaining factors from the resulting decomposition.Source: A. Tensor Reformulation and… · p. 4

B. Sensing Parameters Extraction

After tensor decomposition, the sensing task is recast as extracting parameters from the three factor matrices. The receive spatial manifold estimate is one stated intermediate.

B. Sensing Parameters Extraction

1) Mixed Field Parameters at the Receiver:

The receiver-side method uses array symmetry to separate angle and distance information, then refines the physical parameters through continuous-domain optimization. The paper also introduces a continuous-domain delay-Doppler method to address grid mismatch.

1) Mixed Field Parameters at the Receiver:

2) Delay-Doppler Estimation:

The delay-Doppler procedure starts with a low-resolution grid search and then updates delay and Doppler to overcome discrete-grid accuracy limits. The remaining AoD is extracted after Doppler estimation.

2) Delay-Doppler Estimation:

  • A low-resolution two-dimensional correlation search over delay and Doppler grids provides initial parameter coordinates from the baseband prediction model.Source: 2) Delay-Doppler Estimation · p. 5
  • The delay and Doppler values are subsequently updated to overcome the accuracy limits of discrete grids.Source: 2) Delay-Doppler Estimation · p. 5, 3) AoD · p. 5
  • After Doppler estimation, AoD is extracted from the phase component of the complex variable ẑq using its bijective correspondence with ϕ̂q.Source: 3) AoD · p. 5

C. Complexity Analysis

The scheme’s computational costs span tensor decomposition, AoA estimation, and delay-Doppler estimation, with complexity expressed in operation counts and iteration counts.

C. Complexity Analysis

  • The truncated SVD of YS requires O(QK^3L^3NrM) operations.The signal-subspace pseudo-inverse adds O(K^3NrQ^2) operations, and eigenvalue decomposition takes O(Q^3).Source: C. Complexity Analysis · p. 5
  • SD-RDMO AoA estimation takes O(QNr + QIAoANr) operations, where IAoA is the total iteration count.Source: C. Complexity Analysis · p. 5
  • BR-AEG baseband reconstruction and initial correlation search take O(QM log2 M+QGM), while refinement takes O(QIDDM).G is the number of low-resolution grid points, and IDD is the refinement iteration count required for convergence.Source: C. Complexity Analysis · p. 5

IV. Simulation Results

The simulations evaluate sensing in a high-mobility XL-MIMO AFDM V2I setup, reporting estimation performance across SNR for angle, delay, and Doppler parameters.

IV. Simulation Results

  • The 60 GHz V2I simulation uses three targets at 60, −50, and 80 m/s, including one near-field and two far-field targets.The system uses bandwidth B = 10.24 MHz, M = 256, and 200 receive antennas on each side of the reference antenna; the XL-ULA Rayleigh distance is approximately 100 meters.Source: §IV · pp. 5–6
  • BR-AEG achieves significantly lower delay and Doppler NMSE than CS, AML, and Refined PC, whose high-SNR error floors persist.The comparison uses delay and Doppler NMSE versus SNR.Source: Figure 3 · p. 6, Figure 4 · p. 6, §IV · p. 6
  • SD-RDMO AoA RMSE monotonically decreases as SNR increases, while MUSIC can smear or miss spatial peaks and the subspace method is dictionary-density limited.The figure reports AoA RMSE versus SNR.Source: Figure 2 · p. 6, §IV · p. 6
  • The proposed AoD estimator outperforms MUSIC, while OMP experiences severe performance degradation in the evaluated scenario.The figure reports AoD RMSE versus SNR; OMP relies on predefined discrete spatial-temporal dictionaries.Source: Figure 5 · p. 6, §IV · p. 6

V. Conclusion

The conclusion presents a tensor-based XL-MIMO AFDM sensing framework that decouples parameter estimation and reports improved delay-Doppler and angular performance in simulations.

V. Conclusion

  • The framework reformulates received signals as a third-order tensor and exploits Vandermonde structure to decouple high-dimensional parameters.Source: §V · p. 6
  • SD-RDMO estimates AoA while avoiding high-dimensional search complexity associated with near-field effects.Source: §V · p. 6
  • BR-AEG estimates delay and Doppler in the continuous domain, and the remaining AoD is extracted sequentially; simulations report orders-of-magnitude lower delay-Doppler NMSE and eliminated angular error floors common in grid-dependent methods.Source: §V · p. 6
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