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

TL;DR

This work generalizes the Friis transmission formula to beamformed multiport antenna systems in arbitrary reciprocal environments by employing the concept of active antenna available power. It defines the active antenna unnamed power gain as the ratio of receiving-array available power to transmitter-input power, incorporating beamforming networks and near-field coupling via the system impedance matrix. Under pattern-symmetry constraints, the bidirectional gains become equal, establishing a link-direction symmetry condition that extends reciprocity to multiport links beyond the far field. In the far-field limit, the generalized gains factor into conventional antenna gains and path loss, reproducing the classic Friis relation, while near-field and generic-load scenarios are captured through embedded element patterns and network-theory formulations. The results unify noise-based active-power definitions with network-theory gains, provide bounds via generalized eigenvalues, and offer practical implications for MIMO, near-field communications, and metrology of complex antenna arrays.

Abstract

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.

Figures (6)

• Figure 1: Transmitting array 1 fed by ideal current sources and receiving array 2 with ideal voltage sensors having infinite input impedance and a beamforming network (case A). For case B, the direction of transmission and reception is reversed. The dashed line enclosing both antennas represents a multiport network with impedance matrix ${\bf Z}$.
• Figure 2: Arrays 1 and 2 with generic source and load networks and beamformer on array 2. This system diagram represents any array communication channel where signals are transmitted by array 1, received by array 2, amplified, and formed into an output that is a linear combination of the received signals, such as a point-to-point wireless link or an eigenchannel of a MIMO system.
• Figure 3: Active antenna unnamed power gains for a 2 $\times$ 2 multiport antenna system. In the far field, the maximum unnamed power gain converges to the Friis formula with maximum antenna gains combined with the best polarization efficiency. With random weights at both the transmitting and receiving arrays, the antenna gains of the arrays decrease and the active antenna unnamed power gain is lower than the maximum value of the unnamed power gain from ERPEE8. Notice the Friis formula, the maximum gains, and polarization efficiency were evaluated in the far-field region and were then used for all array separations.
• Figure 4: The active antenna unnamed power gain for a 2 $\times$ 2 multiport antenna system with receive beamformer weights optimized for maximum antenna gain and transmit beamformer weights chosen randomly lies between the maximum and minimum values from ERPEE8.
• Figure 5: Model of a perfectly conducting transmitting dipole array consisting of four sections of crossed dipoles backed by a reflector. The receiving array is made of a single section of crossed dipoles. The system is fed and loaded by a diagonal matrix of $50 \, \mathrm{\Omega}$ loads. The system is positioned over an infinite perfectly conducting ground plane at height $H$ equal to four wavelengths. Length of the dipoles $L \approx \lambda / 2$. The distance of the dipoles from the array reflector is $2L/5$. The reflector is a square with side $w = L$. Distance between the array sections is $3L/2$. The figure also shows the directivity patterns of the transmitter (top) and receiver (bottom). The directivity patterns correspond to the maximum directivity in the direction of the transmitter/receiver. The directivity values are $G_1 \approx 15 \, \mathrm{dBi}$, $G_2 \approx 5.2 \, \mathrm{dBi}$.