This paper introduces a new mathematical framework for dynamic beamforming-based cellular networks, grounded in stochastic geometry. The framework is used to study the electromagnetic field exposure (EMFE) of active and idle users as a function of the distance between them. A novel multi-cosine antenna pattern is introduced, offering more accurate modeling by incorporating both main and side lobes. Results show that the cumulative distribution functions of EMFE and coverage obtained with the multi-cosine pattern align closely with theoretical models, reducing error to less than 2\%, compared to a minimum of 8\% for other models. The marginal distribution of EMFE for each user type is mathematically derived. A unique contribution is the introduction of the SCAIU (\underline{S}patial \underline{C}DF for \underline{A}ctive and \underline{I}dle \underline{U}sers), a metric that ensures coverage for active users while limiting EMFE for idle users. Network performance is analyzed using these metrics across varying distances and antenna elements. The analysis reveals that, for the chosen network parameters, with 64 antenna elements, the impact on idle user EMFE becomes negligible beyond 60~m. However, to maintain active user SINR above 10 dB and idle user EMFE below -50~dBm at 2~m, more than 256 elements are required.

翻译：本文基于随机几何理论，提出了一种新的动态波束成形蜂窝网络数学框架。该框架用于研究活跃用户与空闲用户的电磁场暴露（EMFE）随用户间距离变化的规律。我们引入了一种新颖的多余弦天线方向图模型，通过同时考虑主瓣和旁瓣实现了更精确的建模。结果表明，采用多余弦方向图获得的EMFE与覆盖累积分布函数与理论模型高度吻合，误差降至2%以下，而其他模型的误差至少为8%。我们通过数学推导得出了各类用户EMFE的边缘分布。本研究的独特贡献在于提出了SCAIU（活跃与空闲用户空间累积分布函数）指标，该指标在保障活跃用户覆盖的同时，能有效限制空闲用户的EMFE暴露。我们利用这些指标分析了不同距离和天线单元数量下的网络性能。分析表明：在所选网络参数下，当采用64个天线单元时，对空闲用户EMFE的影响在60米外可忽略不计；然而，若要在2米距离处同时满足活跃用户信干噪比高于10 dB且空闲用户EMFE低于-50 dBm，则需要超过256个天线单元。