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

Near-field propagation in extremely large aperture arrays requires joint angle-range estimation. In hybrid architectures, only $N_{\mathrm{R}\mathrm{F}}\ll M$ compressed snapshots are available per slot, making the $N_{\mathrm{R}\mathrm{F}}\times N_{\mathrm{R}\mathrm{F}}$ 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 $Q_{\theta }$-element dictionary instead of the $Q_{\theta }{Q}_r$ 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 $N_{\mathrm{M}\mathrm{C}}=400$ ($f_c=28$~GHz, $M=64$, $N_{\mathrm{R}\mathrm{F}}=8$, $N=64$, $d=3$, $r\in [0.05,1.0]r_{\mathrm{R}\mathrm{D}}$), CL-KL achieves the lowest channel NMSE among all six evaluated methods – including four full-array baselines using $64\times$ more data – at $\mathrm{S}\mathrm{N}\mathrm{R}\in \{-5,0,+5,+10\}$~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 $N_{\mathrm{R}\mathrm{F}}\times N_{\mathrm{R}\mathrm{F}}$ 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.