Rigorous proof of generalization to K>2 users for the spatial upper bound

Prove that, in a multi-user transmission scenario with K >= 3 simultaneously served users whose baseband signals are zero-mean and mutually uncorrelated, and with each transmit branch’s power amplifier modeled as a memoryless third-order nonlinearity with independent additive noise, the directional EIRP in any azimuth direction is upper-bounded by the maximum directional EIRP achieved in the single-user boresight configuration; that is, for all directions φ, EIRP_MU(φ) <= EIRP_SU,max(φ), so the deterministic spatial upper bound determined by the elementary sub-array radiation pattern remains valid and conservative for K >= 3 users. Here, EIRP_MU(φ) denotes the directional EIRP under K-user operation with arbitrary beamforming phase gradients, and EIRP_SU,max(φ) denotes the maximum directional EIRP obtained in the single-user boresight configuration.

Background

The paper derives a deterministic spatial upper bound on radiated power for active antenna arrays, showing that the maximum radiated power occurs at boresight and that the angular envelope is governed by the element or sub-array pattern. This is established analytically for single-user operation and verified experimentally on a 3.5 GHz Massive MIMO antenna.

For multi-user transmission, the authors analyze in detail the two-user case. They show that useful signals and certain intermodulation (IM) components beamform in user directions, while other IM components radiate in additional directions. Due to power sharing and spatial dispersion of IM products, they argue that the directional EIRP in multi-user operation is bounded by the single-user boresight maximum, asserting an inequality that underpins the spatial upper bound’s conservativeness.

They expect the same reasoning to hold for K > 2 users but do not provide a formal proof. Establishing a rigorous proof for arbitrary K would solidify the general applicability of the spatial upper bound framework in practical multi-user Massive MIMO systems and strengthen its use in regulatory coexistence assessments.