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Covariance-Domain Near-Field Channel Estimation under Hybrid Compression: USW/Fresnel Model, Curvature Learning, and KL Covariance Fitting

Rıfat Volkan Şenyuva

Abstract

Near-field propagation in extremely large aperture arrays requires joint angle-range estimation. In hybrid architectures, only $N_\mathrm{RF}\ll M$ compressed snapshots are available per slot, making the $N_\mathrm{RF}\times N_\mathrm{RF}$ compressed sample covariance the natural sufficient statistic. We propose the Curvature-Learning KL (CL-KL) estimator, which grids only the angle dimension and \emph{learns the per-angle inverse range} directly from the compressed covariance via KL divergence minimisation. CL-KL uses a $Q_θ$-element dictionary instead of the $Q_θQ_r$ atoms of 2-D polar gridding, eliminating the range-dimension dictionary coherence that plagues polar codebooks in the strong near-field regime, and operates entirely on the compressed covariance for full compatibility with hybrid front-ends. At $N_\mathrm{MC}=400$ ($f_c=28$~GHz, $M=64$, $N_\mathrm{RF}=8$, $N=64$, $d=3$, $r\in[0.05,1.0]\,r_\mathrm{RD}$), CL-KL achieves the lowest channel NMSE among all six evaluated methods -- including four full-array baselines using $64\times$ more data -- at $\mathrm{SNR}\in\{-5,0,+5,+10\}$~dB. Running in approximately 70~ms per trial (vs.\ 5~ms for the compressed-domain peer P-SOMP), CL-KL's dominant cost is the $N_\mathrm{RF}{\times}N_\mathrm{RF}$ inversion rather than $M$: measured runtime stays near 70~ms across $M\in\{32,64,128,256\}$, making it aperture-scalable for XL-MIMO deployments. CL-KL is further validated against a derived compressed-domain Cramér-Rao bound and confirmed robust to non-Gaussian (QPSK) source distributions, with a maximum NMSE gap below 0.6~dB.

Covariance-Domain Near-Field Channel Estimation under Hybrid Compression: USW/Fresnel Model, Curvature Learning, and KL Covariance Fitting

Abstract

Near-field propagation in extremely large aperture arrays requires joint angle-range estimation. In hybrid architectures, only compressed snapshots are available per slot, making the compressed sample covariance the natural sufficient statistic. We propose the Curvature-Learning KL (CL-KL) estimator, which grids only the angle dimension and \emph{learns the per-angle inverse range} directly from the compressed covariance via KL divergence minimisation. CL-KL uses a -element dictionary instead of the atoms of 2-D polar gridding, eliminating the range-dimension dictionary coherence that plagues polar codebooks in the strong near-field regime, and operates entirely on the compressed covariance for full compatibility with hybrid front-ends. At (~GHz, , , , , ), CL-KL achieves the lowest channel NMSE among all six evaluated methods -- including four full-array baselines using more data -- at ~dB. Running in approximately 70~ms per trial (vs.\ 5~ms for the compressed-domain peer P-SOMP), CL-KL's dominant cost is the inversion rather than : measured runtime stays near 70~ms across , making it aperture-scalable for XL-MIMO deployments. CL-KL is further validated against a derived compressed-domain Cramér-Rao bound and confirmed robust to non-Gaussian (QPSK) source distributions, with a maximum NMSE gap below 0.6~dB.

Paper Structure

This paper contains 34 sections, 30 equations, 11 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Compressed Covariance
  3. Array, snapshots, and hybrid combining
  4. Exact USW manifold and Fresnel approximation
  5. Snapshot model and compressed covariance
  6. Compressed-Domain Cramér--Rao Bound
  7. Stochastic CRB derivation
  8. Derivative expressions
  9. Error propagation to physical parameters
  10. Baseline: 2-D Polar Gridding
  11. Proposed Method: CL-KL
  12. Angle-only grid with per-angle inverse-range parameter
  13. KL covariance fitting objective with sparsity
  14. CL-KL: power-only main loop with post-loop scan
  15. Phase 1: Power-only main loop
  16. ...and 19 more sections

Figures (11)

  • Figure 2: Channel NMSE (dB) vs. SNR ($N_\mathrm{MC}=400$, $M=64$, $N_\mathrm{RF}=8$, $N=64$, $d=3$). Filled: compressed covariance ($N_\mathrm{RF}^2=64$ values). Open: full snapshot matrix ($M{\times}N=4096$ values). CL-KL leads all six methods at $\mathrm{SNR}\in\{-5,0,+5,+10\}$ dB despite operating on $64\times$ less data.
  • Figure 3: Channel NMSE (dB) vs. $N_\mathrm{RF}\in\{4,8,12,16\}$ ($N_\mathrm{MC}=400$, SNR$=+10$ dB, $M=64$, $N=64$, $d=3$). Filled: compressed-domain. Open: full-array. CL-KL gains $\approx5$ dB per doubling of $N_\mathrm{RF}$; DL-OMP plateaus at $-6.64$ dB due to the two-subarray bottleneck.
  • Figure 4: Channel NMSE (dB) vs. $N\in\{16,32,64,128\}$ ($N_\mathrm{MC}=400$, SNR$=+10$ dB, $M=64$, $N_\mathrm{RF}=8$, $d=3$). Filled: compressed-domain. Open: full-array. CL-KL is nearly flat ($\widehat{\bm{R}}_y$ compresses $N$ snapshots to $N_\mathrm{RF}^2=64$ entries); full-array methods improve more with $N$.
  • Figure 5: Near-to-far transition: RMSE$(r)$ (left), RMSE$(\theta)$ (centre), and NMSE (right) vs. $r_\mathrm{max}/r_\mathrm{RD}$ ($N_\mathrm{MC}=400$, SNR$=+10$ dB, $M=64$, $N_\mathrm{RF}=8$, $d=3$; $r_\mathrm{min}=0.05\,r_\mathrm{RD}$ fixed). Vertical markers: EBRD (dashed purple) and $r_\mathrm{RD}$ (dashed black). Filled: compressed-domain. Open: full-array. CL-KL NMSE varies by less than 1.2 dB over the full sweep, confirming that the compressed covariance budget governs performance.
  • Figure 6: Per-trial runtime (s, left) and channel NMSE (dB, right) vs. $M\in\{32,64,128,256\}$ ($N_\mathrm{MC}=50$, single-core, SNR$=+10$ dB, $N_\mathrm{RF}=8$, $N=64$, $d=3$). CL-KL runtime stays near 70 ms across all tested $M$, confirming that its dominant cost is the $N_\mathrm{RF}{\times}N_\mathrm{RF}$ inversion, not $M$; P-SOMP and BF-SOMP scale more steeply as their dictionaries grow with $M$.
  • ...and 6 more figures