We present a propagation model showing that a transmitter randomly positioned in space generates unbounded peaks in the histogram of the resulting power, provided the signal strength is an oscillating or non-monotonic function of distance. Specifically, these peaks are singularities in the empirical probability density that occur at turning point values of the deterministic propagation model. We explain the underlying mechanism of this phenomenon through a concise mathematical argument. This observation has direct implications for estimating random propagation effects such as fading, particularly when reflections off walls are involved. Motivated by understanding intelligent surfaces, we apply this fundamental result to a physical model consisting of a single transmitter between two parallel passive walls. We analyze signal fading due to reflections and observe power oscillations resulting from wall reflections -- a phenomenon long studied in waveguides but relatively unexplored in wireless networks. For the special case where the transmitter is placed halfway between the walls, we present a compact closed-form expression for the received signal involving the Lerch transcendent function. The insights from this work can inform design decisions for intelligent surfaces deployed in cities.

翻译：我们提出了一种传播模型，表明当信号强度是距离的振荡或非单调函数时，空间中随机定位的发射器会在所得功率的直方图中产生无界峰值。具体而言，这些峰值是经验概率密度中的奇点，出现在确定性传播模型的转折点值处。我们通过简洁的数学论证解释了这一现象的内在机制。这一观察结果对估计随机传播效应（如衰落）具有直接意义，特别是在涉及墙壁反射的情况下。受理解智能表面的动机驱动，我们将这一基本结果应用于由两个平行无源墙壁之间的单个发射器组成的物理模型。我们分析了由反射引起的信号衰落，并观察到由墙壁反射产生的功率振荡——这一现象在波导中已被长期研究，但在无线网络中相对未被深入探索。针对发射器恰好位于墙壁中间的特殊情况，我们给出了一个涉及勒奇超越函数的紧凑闭合形式表达式来描述接收信号。这项工作的见解可为城市中部署的智能表面的设计决策提供参考。