Optimal Receive Design on the Observation Axis for RSMA Satellite-UAV Downlinks

Abstract

A receiver can change its observation state while the physical signal remains the same. In rate-splitting multiple access (RSMA) downlinks, this distinction separates receive-beam design and phase-reference transport from the motion of the satellite--UAV line of sight. We formulate receive optimization within the physical--observation dual-axis (PODA) structure and obtain three complementary results. For a known line-of-sight response and positive-definite receiver-noise covariance, a noise-weighted matched combiner simultaneously maximizes every decoding SINR and the minimum user rate. Under directional uncertainty, the principal eigenvector of the conditional response covariance globally maximizes mean array gain; for a Gaussian directional model, an equal-amplitude receiver is optimized by steering towards the conditional mean. Mean rate requires a different objective. We prove that the RSMA common-rate water level is jointly concave in the users' receive gains and derive a semidefinite relaxation with a computable global upper bound. A feasible rank-one beam can therefore be compared directly with the optimum of the finite-scenario problem. We also derive a precision-weighted linear predictor that suppresses uncertain physical-velocity estimates after observation-frame transport. Paired dynamic simulations reduce direction-cosine RMSE by 51.29\% relative to unrestricted nine-point extrapolation. In a separate uncertainty experiment, direct rate optimization increases mean minimum spectral efficiency from 0.32425 to 0.41712 bit/s/Hz, an improvement of 28.64\%; the corresponding design objective is within $2.28\times10^{-4}$ bit/s/Hz of a global upper bound. The phase-transport equation fixes a coherent representative of each optimal beam without changing its gain.