Exposure-Aware Beamforming for mmWave Systems: From EM Theory to Thermal Compliance

Electromagnetic (EM) exposure compliance has long been recognized as a crucial aspect of communications terminal designs. However, accurately assessing the impact of EM exposure for proper design strategies remains challenging. In this paper, we develop a long-term thermal EM exposure constraint model and propose a novel adaptive exposure-aware beamforming design for an mmWave uplink system. Specifically, we first establish an equivalent channel model based on Maxwell’s radiation equations, which accurately captures the EM physical effects. Then, we derive a closed-form thermal impulse response model from the Pennes bioheat transfer equation (BHTE), characterizing the thermal inertia of tissue. Inspired by this model, we formulate a beamforming optimization problem that translates rigid instantaneous exposure limits into a flexible long-term thermal budget constraint. Furthermore, we develop a low-complexity online beamforming algorithm based on Lyapunov optimization theory, obtaining a closed-form near-optimal solution. Simulation results demonstrate that the proposed algorithm effectively stabilizes tissue temperature near a predefined safety threshold and significantly outperforms the conventional scheme with instantaneous exposure constraints.

💡 Research Summary

The paper addresses a critical gap in millimeter‑wave (mmWave) uplink system design: the lack of a physically accurate, long‑term exposure model that links electromagnetic (EM) radiation to the resulting thermal effects on human tissue. While existing works largely focus on instantaneous exposure metrics such as specific absorption rate (SAR) or power density (PD), regulatory standards for high‑frequency bands actually impose limits on temperature rise over extended periods. To bridge this gap, the authors develop a two‑stage modeling framework and an adaptive beamforming algorithm that respects a long‑term thermal budget rather than a per‑slot SAR/PD ceiling.

Physical EM Radiation Model
Starting from Maxwell’s equations, the authors derive a detailed radiation model for a uniform linear array (ULA) of half‑wave dipoles. They incorporate thin‑wire current distribution, mutual coupling via a rigorously derived impedance matrix, and spherical wavefront effects. The resulting array response vector captures both the normalized gain and phase delay for each transmit element, allowing the electric field at any observation point (including the receive antenna array and the sampled points on a spherical head model) to be expressed as a closed‑form function of the driving voltage vector. This model simultaneously yields the communication channel matrix and the exposure matrix, ensuring that beamforming decisions are grounded in the same physical reality that governs tissue exposure.

Thermal Dynamics via Pennes Bioheat Transfer Equation (BHTE)
The second stage translates the deposited EM power into temperature dynamics using the Pennes bioheat transfer equation, which accounts for thermal conduction, perfusion‑mediated cooling, and metabolic heat generation. By solving the BHTE in the Laplace domain and applying an inverse transform, the authors obtain a closed‑form thermal impulse response (h_T(t)=\frac{1}{\tau}e^{-t/\tau}), where (\tau) is the tissue’s thermal time constant. Convolution of this impulse response with the time‑varying transmit power yields the instantaneous temperature rise at each sampled tissue point.

Long‑Term Thermal Exposure Constraint
Instead of enforcing a hard SAR/PD limit at every time instant, the paper defines a “thermal budget” (B_T) as the maximum allowable cumulative temperature increase over a prescribed window (T_w). Mathematically, the constraint requires the time‑average of the temperature rise to stay below a safety threshold (T_{\max}). This budget is enforced by introducing a virtual queue (Q