Generalized Friis Transmission Formula Using Active Antenna Available Power and Unnamed Power Gain

We use the concept of active antenna available power to derive a generalization of the Friis transmission formula for multiport antenna systems. With beamformer weights chosen, such that the array patterns are the same when transmitting and receiving, the active antenna available power at the receiving antenna divided by the input power at the transmitter is symmetric under link direction reversal in the near field as well as the far field. These results generalize the Friis transmission formula to beamformed multiport antenna systems in an arbitrary reciprocal propagation environment.