Reflected wireless signals under random spatial sampling

TL;DR

This paper introduces a propagation model demonstrating that random transmitter placement can cause unbounded peaks in power distribution histograms due to oscillating signal strength, with implications for wireless fading estimation and intelligent surface design.

Contribution

It provides a mathematical explanation for power peaks caused by reflections and applies this to a model with walls, offering new insights into signal fading and propagation effects.

Findings

Unbounded peaks occur at turning points in power histograms.

Reflections cause power oscillations in wireless signals.

Closed-form expression involving Lerch transcendent for specific transmitter placement.

Abstract

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.