Coherent but Discordant: Rotate, Don't Invert, to Backpropagate Through an Aging Channel

Abstract

A channel can be almost perfectly coherent and still teach a transmitter the wrong direction. In physical ("over-the-air") backpropagation, the receiver sends its loss gradient back through the reciprocal channel; when the terminal moves in between, mobility does not merely add noise to the returned gradient, it rotates it. Under a scalar conditional mean model the rotation is set by the phase of the temporal correlation, so coherence, which measures only its magnitude, is the wrong ruler: in simulation, channels with absolute value of rho greater than 0.9 drive uncorrected training to 51.7% accuracy, while phase correction restores 92.5%. Arrays add a twist. Near zeros of the trace correlation, scalar correction fails even with rho known exactly, yet most of the information is still there: noise has moved it onto neighbouring antennas. Re-indexing the ports by the displacement, with zero digital feedback, recovers the lost performance in simulation, and so does rotating the returned seed with the unitary polar factor of the conditional mean operator, never inverting it; aligning each antenna's phase with its own past makes things worse. An exact array identity explains all three. The lesson fits in one line: correct the rotation, not the stretch.