Reflected wireless signals under random spatial sampling

TL;DR

This work shows that random spatial sampling of an oscillatory wireless power function $P(r)$ with turning points yields singularities in the empirical power density, governed by inverse-square-root behavior near $P(t_i)$ where $P'(t_i)=0$. The authors develop a geometrical propagation model for parallel-wall environments via the method of images, deriving both a general framework and a closed-form two-wall result in terms the Lerch transcendent $\Phi(\zeta,s,\gamma)$. They compare two randomness schemes—random transmitter location and random phase—and demonstrate, theoretically and numerically, that turning points produce sharp density peaks under spatial randomization, while random-phase models do not, highlighting the geometric structure underlying fading. The results offer a structural principle for wireless design with intelligent surfaces, enabling geometry-guided prediction of singularities and informing phase-control strategies to mitigate or exploit destructive and constructive interference in urban canyons.

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.

Figures (14)

• Figure 1: A wide class of non-monotonic functions, such as $h(r)=\sin(r)$ on an interval $(r_{\text{L}},r_{\text{U}})$, can be decomposed into a collection of strictly monotonic functions $g_1,\dots,g_{\ell}$ by suitably partitioning the function's domain. $t_i$ are the locations of the turning points where the derivative $h'(t_i) = 0$. Partition: $(r_{\text{L}}, r_{\text{U}}) = A_0\cup A_1 \cup A_2 \cup A_3 \cup A_4$, where the set $A_0$ contains all the endpoints. The partition is such that each function $g_i$, which is defined as $h$ restricted to $A_i$, is strictly monotonic on $A_i$ for $i=1,\dots,4$.
• Figure 2: A turning point in the power function $P(r)$ correspond to a singularity in the probability density. Left: The power function has a minimum at $r = t$ where $P'(t) = 0$. Two distinct locations $r_1$ and $r_2$ produce the same power value (orange points). Right: When the transmitter location is randomized, the resulting probability density $f_V(v)$ has a singularity at $v = P(t)$ where the derivative $P'(r)$ vanishes. The singularity arises from the change-of-variables formula given by expression \ref{['e.densitymonotonic']}.
• Figure 3: Two equivalent ways of representing a signal propagating between a transmitter and receiver, both located next to a reflecting wall. The black solid line is the line-of-sight signal, while the blue dashed line is the non-line-of-sight signal. In both representations the dashed blue lines are of equal total length. Left: The diagram illustrates the signal reflecting off the wall. Right: The diagram uses a virtual node or image on the other side of the wall.
• Figure 4: A receiver $O$ and a transmitter $z$ placed between two infinitely long parallel walls. The black solid line represents the line-of-sight signal path, while the red and blue dashed lines are the respective signals from the first left and right images. The distances between receiver $O$ and the images on the right and left are denoted by $\hat{r}_1$ and $\hat{\ell}_1$ respectively.
• Figure 5: The (non-line-of-sight) signal power $P(x,0)$ for transmitter placement at $a=b=0.5$, attenuation exponent $\beta=4$, power reflection coefficient $\kappa=0.5$, wave number $k=10$, and different values of the horizontal receiver placement $x$.

Theorems & Definitions (10)

• Proposition 4.1
• Remark 4.2
• proof
• Proposition 4.3
• proof
• Proposition 7.1
• proof
• Remark 7.2
• Remark 8.1
• Remark 8.2: Challenges for density estimation