Spatial upper bound of radiated power in active antenna systems

The assessment of unwanted radiated emissions from Active Antenna Systems (AAS) has become a critical issue in adjacent-band coexistence scenarios. In this paper, we establish the existence of a deterministic spatial upper bound on the radiated power of active antenna arrays. We show that the maximum radiated power always occurs in the boresight direction, irrespective of frequency or signal nature (useful signal, nonlinear distortion, or noise), or instantaneous beamforming configuration, thereby defining a conservative spatial upper bound whose angular envelope is solely determined by the elementary radiating building block of the antenna architecture, i.e., the element or sub-array radiation pattern. Starting from a two-element array with third-order nonlinearities, we derive the spatial envelope and extend the result to realistic AAS architectures. The theoretical findings are validated by over-the-air (OTA) measurements performed on a 3.5 GHz Massive Multiple-Input Multiple-Output (MIMO) antenna. The proposed approach offers a simple, robust, and measurement-oriented methodology for coexistence assessments involving beamformed radio systems.