In physical backpropagation, users return loss derivatives over the time-division duplex uplink, giving the base station the downlink gradient without channel reconstruction. Reciprocity calibration conventionally prioritizes base-station chains, as column-normalized zero-forcing reduces a per-user factor to a stream rotation. This priority can reverse. With ideal base-station chains and no noise, per-user amplitude errors make the returned field a weighted sum-rate gradient. With zero-forcing full service at both optima, the loss is independent of array size. Otherwise the selection penalty, serving the wrong users, carries 86 to 94 percent of the amplitude-induced loss in five of six configurations. Per-user phase errors mainly slow the iteration. Base-station chain errors scale the returned field per antenna, and at 3 dB and 25 degrees they cost one user 1.3 dB of signal-to-noise ratio (SNR) that no array size removes. With 80 users, such errors on both sides, and mismatched pilots, oracle per-user and base-station calibration leave 6.2 and 12.3 percent loss at 256 antennas on clustered channels. This ordering holds at 30 dB SNR and reverses for 1 dB, 10 degree user errors. A capped one-echo estimator comes within 2.4 percentage points of oracle per-user calibration but underperforms no calibration at 10 dB SNR.
Over-the-air backpropagation returns a loss gradient through the reciprocal channel, which requires the return port to see the channel of the forward port. If the terminal moves in between, a fluid antenna system can return through the port nearest to the forward position. In ray-traced Chicago streets, with a pedestrian displacement of about half a wavelength and a street-level base station, this rule reverses 20 percent of the gradients at a port spacing of 0.7 wavelengths and 49 percent at 0.9 wavelengths, while one of the four nearest ports stays aligned in all but 2 and 9 percent. We derive a certificate for the sign of the mean correlation from the second moment of a planar angular spectrum. It reduces to a linear program with three variables and infinitely many constraints, is tight, and holds over a region of second moments. In a static Gaussian model, observations collected along a line cannot tell an angular spectrum from its mirror image, so the sign of the cross moment must come from other information. At unsurveyed positions of the same city, a knowledge map that transports surveyed paths through virtual sources and fits them to the causal channel history lowers the reversal rate at 0.9 wavelengths from 49 to 27 percent. Its gain over the nearest port survived every tested error, while its gain over a geometry rule stayed significant at the tested single-error levels of 5 degrees of phase drift per step and 2 mm of position error but not at the next tested levels. Without any survey, across five cities and ten base stations, the nearest port reverses 37 percent of the returns at 0.9 wavelengths and a rule based on the direction and elevation of the base station 20.5 percent. The multilayer perceptrons we tested did not beat these physical rules by the margin fixed in advance in the primary settings.
Accurate acquisition of full channel state information (CSI) is essential for multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) wireless systems, while dense pilot-based estimation incurs substantial overhead. Directly minimizing the prediction error over raw CSI coefficients is fragile in realistic propagation environments, since small-scale fading, mobility, local scattering, and small spatial perturbations can induce large phase-sensitive variations. Nevertheless, wireless channels contain strong spatial, temporal, and frequency-domain correlations that can be more predictably represented in a latent propagation-field domain. Motivated by this observation, we propose a joint-embedding predictive architecture (JEPA)-inspired field world model, defined as a latent predictive model of the wireless propagation field, for sparse-pilot channel reconstruction. The model maps CSI observations into latent time-frequency tokens, predicts masked or future latent field states from historical multi-band context, and fuses the predicted latent prior with sparse noisy current pilots to reconstruct the full MIMO-OFDM channel. Ray-tracing experiments show that the proposed method improves normalized mean square error (NMSE) over pilot-only and direct reconstruction baselines, supporting latent field prediction as an effective prior for sparse-pilot channel acquisition.
A Gaussian random wave field is sampled on a lattice within a bounded aperture. After displacement, re-indexing leaves a residual in the Voronoi cell. We study whether the empirical lag correlation has positive real part throughout the cell. For each realization in a single-output reciprocal link with noiseless over-the-air return, this is equivalent to the negative returned gradient being a descent direction for all residuals and differentiable losses with nonzero error signals. Return noise is treated exactly. For isotropic fields, ball-shaped apertures, unshifted lattice points, and dilations approaching the critical covering radius, every minimizer is a deep hole except with exponentially small probability. A Gaussian vector on the deep holes determines the limiting failure probability, with explicit covariance and lattice-independent limiting normalization. Square, hexagonal, simple cubic, and body-centered cubic lattices have two, three, four, and twelve sign classes. At fixed target failure probabilities below the critical limiting value, this covariance determines the exact leading aperture constant. When the margin and target failure probability vanish jointly, that constant is common to lattice shapes within each dimension. A cubic correction yields vanishing relative tail error in a moderate-deviation regime. A non-asymptotic two-sided aperture law has explicit constants. With a Rayleigh-faded specular path and positive diffuse margin, a failure floor occurs at every positive path power exactly when the cosine of its phase change is negative somewhere in the cell. The floor vanishes for all directions exactly when the covering radius is at most a quarter wavelength.
In over-the-air split learning over an aging single-antenna Rayleigh channel, the mean returned gradient equals the antenna-domain gradient scaled by the temporal correlation. Uncorrected training collapses in negative-correlation lobes. Near a correlation zero, no nonzero scalar calibration improves the ratio of mean power to innovation power for an individual transmission, and, on the tested grid, using sixteen times as many pilots recovers only 7 percent of the training gap. Returning at the first correlation extremum with phase correction, 1.3 ms later at 3.5 GHz and 15 m/s, achieves 90.2 percent accuracy, compared with 91.8 percent with CSI. In the tested line-of-sight setting, the near-zero gap becomes negligible.
Over-the-air split learning returns the receiver's loss gradient through the reciprocal channel, so terminal motion between the forward and backward transmissions corrupts the backward pass. We repair the aged return at a fixed receive array using only second-order statistics. Two numbers taken from the forward-backward cross-covariance, the alignment reachable by the best common phase and by the best unitary correction, bound the main term of every repair. In the 3GPP tapped-delay-line channels TDL-A/B/C, where every antenna ages by the same scalar, the two numbers coincide, and at a correlation zero all tested fixed corrections stay between 58% and 62% accuracy, whereas a return delayed to the next correlation lobe reaches 98.4%. In 3GPP clustered-delay-line channels a statistical unitary correction stays within 0.02 in mean cross-entropy of ideal channel knowledge in all six tested profile and speed combinations, while common-phase correction falls to 69% for CDL-A at 3 km/h and the tested delay loses up to 10 points. In a two-cluster model spatial correction is decisive only where the common-phase alignment nearly vanishes (64% to 95%). With a ResNet-18 on CIFAR-10, only the delayed return trains at a separable correlation zero (86% against at most 16%), and at 30 km/h in a clustered-delay-line channel the unitary correction reaches 80% where the common phase reaches 60%. Given geometry, displacement and angular statistics, a port permutation chosen by linear assignment reaches 99.5-99.7% accuracy for hexagonal and square arrays in oblique motion without pilots or digital feedback during training, where nearest-position matching drops to 89%.