How Symmetric Is Uplink–Downlink Duality? Per-User Power Allocation in Multiuser MISO Beamforming

Uplink–downlink (UL–DL) duality is a standard tool in multiuser transmit optimization. With the beamforming directions held fixed, the downlink power minimization problem and its virtual uplink counterpart are feasible for exactly the same signal-to-interference-plus-noise ratio (SINR) targets and attain the same minimum total power, so a downlink power allocation is routinely obtained by solving the uplink problem instead. That guarantee is an aggregate one. In this paper, we ask what it covers for the individual user. Exchanging the two links transposes the interference coupling matrix, and transposition preserves the spectrum but not the solution of the associated linear system. We show that the per-user allocation is preserved for every feasible SINR target if and only if the coupling matrix is symmetric, we introduce a scalar asymmetry measure, and we derive a bound that factorizes the per-user gap into structure, coupling magnitude, and load. Under matched filtering, the symmetry condition reduces to equality of the channel norms, so the asymmetry decays at least as fast as the inverse square root of the number of antennas under homogeneous large-scale fading. Numerically, a 6 dB path-loss spread removes that decay, and regularized zero-forcing suppresses the coupling magnitude rather than the asymmetry.