Ch. 2 — Path Loss, Shadowing, and Multipath

Chapter 2 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Abstract

This chapter establishes the mathematical foundations for characterizing wireless signal propagation, distinguishing between large-scale effects like path loss and shadowing and small-scale multipath fading. It introduces deterministic models such as free-space and two-ray propagation, alongside statistical models for shadowing and multipath that account for environmental randomness. These models are critical for link budget analysis, coverage estimation, and the design of robust physical layer strategies in modern communication systems.

Key Concepts

• Large-Scale Propagation Effects Path loss and shadowing represent power variations that occur over distances significantly larger than the signal wavelength, typically ranging from tens to hundreds of meters. Path loss describes the deterministic decay of signal power with transmit-receive distance due to geometry and dissipation, while shadowing models the random attenuation caused by obstacles like buildings or terrain features that block the signal path.

• Free-Space Path Loss This model describes signal propagation in a line-of-sight (LOS) environment devoid of obstructions, where power decays inversely with the square of the distance and the square of the carrier frequency. It serves as the fundamental baseline for link calculations, assuming omnidirectional antennas and no reflections, reflecting the Friis transmission equation.

• Two-Ray Multipath Model This deterministic model approximates propagation environments where a ground-reflected ray interferes constructively or destructively with the direct LOS ray. In the asymptotic limit of large distances, the received power decays with the fourth power of the distance, independent of wavelength, distinguishing it from simple free-space decay.

• Path Loss Exponent A parameter $\gamma$ used in single-slope and multi-slope models to generalize distance-dependent attenuation across diverse environments. The exponent varies by environment type, typically ranging from 2 in free space to values significantly higher in cluttered urban or indoor scenarios, capturing how rapidly signal strength deteriorates with distance.

• Log-Normal Shadowing A statistical model characterizing random fluctuations in received power around the mean path loss value, derived from the central limit theorem applied to signal attenuation through multiple obstacles. The attenuation in decibels is modeled as a Gaussian random variable with a specific standard deviation that depends on the density of scattering objects and frequency.

• Multipath Fading Small-scale variations in the received signal caused by the constructive and destructive interference of multiple signal components arriving with different time delays and phase shifts. Unlike large-scale effects, these variations occur over distances on the order of the signal wavelength and are distinct from the slow variations observed in shadowing.

• Ray Tracing A propagation modeling technique that approximates electromagnetic wave behavior using geometric optics, tracing individual rays for reflection, diffraction, and scattering based on known environment geometry. It allows for detailed site-specific analysis but requires substantial computational resources and precise knowledge of the dielectric properties of the surrounding objects.

• Decorrelation Distance The spatial separation over which shadowing variations become statistically independent, often modeled as an exponential decay of covariance with distance. In outdoor environments, this distance correlates with the size of the blocking objects and typically ranges from 50 to 100 meters for macrocells.

Key Equations and Algorithms

• Friis Transmission Formula $P_r=P_t{G}_t{G}_r{\left(\frac{\lambda }{4\pi d}\right)}^2$ defines the received power $P_r$ for free-space propagation where $P_t$ is transmit power, $G_t$ and $G_r$ are antenna gains, $\lambda$ is the wavelength, and $d$ is the distance. The equation demonstrates that received power is proportional to the square of the wavelength and inversely proportional to the square of the distance.

• Single-Slope Path Loss Model $PL(d)=10{\log }_{10}(K)-10\gamma {\log }_{10}(d/d_r)$ expresses the path loss in decibels as a function of distance $d$, where $\gamma$ is the path loss exponent, $K$ is the path gain at reference distance $d_r$. This model linearizes the attenuation relationship in log-log space to simplify system analysis and link budgeting.

• Log-Normal Shadowing Distribution $f_{\psi dB}({\psi }_{dB})=\frac{1}{\sqrt{2\pi }{\sigma }_{\psi dB}}\exp \left(-\frac{({\psi }_{dB}-{\mu }_{\psi dB}{)}^2}{2{\sigma }_{\psi dB}^2}\right)$ describes the probability density function of the attenuation ${\psi }_{dB}$ in decibels. The mean ${\mu }_{\psi dB}$ is the average path loss and ${\sigma }_{\psi dB}$ is the standard deviation representing shadowing variance in dB.

• Outage Probability Calculation $P_{out}(P_{min},d)=Q\left(\frac{P_{min}-{\bar{P}}_r(d)}{{\sigma }_{\psi dB}}\right)$ calculates the probability that received power $P_r$ falls below a minimum threshold $P_{min}$ at distance $d$. The $Q$-function represents the tail probability of a standard Gaussian variable, indicating the reliability of the link under shadowing.

• Cell Coverage Percentage Formula $C=\frac{1}{2}+\frac{1}{2}\text{erf}\left(\frac{a\ln 10}{\sqrt{8}}\right)+\frac{1}{2}{e}^{2ab+\frac{b^2}{2}}\text{erfc}\left(\frac{a+b\ln 10}{\sqrt{8}}\right)$ determines the fraction of a circular cell area where signal power exceeds $P_{min}$. Here $a$ relates to the margin between boundary power and threshold, and $b$ is related to the path loss exponent and shadowing variance ratio $\gamma /{\sigma }_{\psi dB}$.

• Fresnel-Kirchhoff Diffraction Parameter $v=\alpha \sqrt{\frac{2(d^{\ast }\cdot d^{\star })}{\lambda (d^{\ast }+d^{\star })}}$ characterizes the geometry of a knife-edge obstacle relative to the LOS path, where $d^{\ast }$ and $d^{\star }$ are distances to the obstacle and receiver. The parameter $v$ determines the additional attenuation due to diffraction when a signal bends around an obstruction.

Key Claims and Findings

• Received power in the two-ray model falls off as $d^{-4}$ at large distances, significantly faster than the $d^{-2}$ decay observed in free-space propagation.
• The standard deviation of log-normal shadowing ${\sigma }_{\psi dB}$ typically ranges from 4 dB to 13 dB for outdoor channels, increasing with frequency and distance.
• Cell coverage percentage depends primarily on the ratio of the path loss exponent to shadowing standard deviation $\gamma /{\sigma }_{\psi dB}$ rather than absolute power values alone.
• Multipath fading variations occur over distances on the order of the wavelength, whereas shadowing variations occur over distances proportional to the size of blockage objects.
• General ray tracing techniques are computationally intensive and generally limited to site-specific modeling rather than general theoretical performance analysis.

Terminology

• Path Loss ($PL$): The ratio of transmitted power to received power in decibels, representing the attenuation of signal strength over a communication link.
• Shadowing: Random attenuation of a signal caused by obstacles in the propagation path that block or absorb transmitted energy.
• Multipath: The phenomenon where multiple copies of a signal arrive at the receiver via different paths, resulting in time-delayed and phase-shifted components.
• Line-of-Sight (LOS): A propagation path where a direct, unobstructed ray exists between the transmitter and receiver without intermediate reflections.
• Ray Tracing: A deterministic propagation modeling method that calculates signal paths by simulating reflections and diffractions based on environmental geometry.
• Critical Distance ($d_c$): In the two-ray model, the distance beyond which the received power begins to fall off as the fourth power of the distance.
• Path Loss Exponent ($\gamma$): An integer or real number indicating how rapidly signal power decays with distance in a specific environment.
• Fresnel Zone: A series of concentric elliptical zones between transmitter and receiver where signal propagation experiences phase shifts affecting constructive or destructive interference.