Line-of-sight modeling

In cases where the distinction between LOS (Line-of-Sight) and NLOS (Non-Line-of-Sight) links is made, modeling the LOS probability ${\displaystyle P_{\text{LOS}}}$ becomes critical. The most popular approach to deriving these statistics is based on creating a geometrical model (e.g., Manhattan grid) of the propagation environment.

Simplified 2D model: a popular approach suggested by the International Telecommunication Union (ITU). According to ITU ^[4] , the LOS probability is given by:

${\displaystyle P_{\text{LOS}}=\prod _{n=0}^{m}\left[1-\exp \left(-{\frac {\left[h_{\text{UAV}}-{\frac {(n+{\frac {1}{2}})(h_{\text{UAV}}-h_{\text{G}})}{m+1}}\right]^{2}}{2\Omega ^{2}}}\right)\right],}$

where ${\displaystyle m={\text{floor}}\left(d_{h}{\sqrt {\varsigma \xi }}-1\right)}$, ${\displaystyle d_{h}}$ is the horizontal distance between the UAV and the ground node, ${\displaystyle h_{\text{UAV}}}$ and ${\displaystyle h_{\text{G}}}$ are the terminal heights, ${\displaystyle \varsigma }$ is the ratio of land area covered by buildings compared to the total land area, ${\displaystyle \xi }$ is the mean number of buildings per km², and ${\displaystyle \Omega }$ is the scale parameter of building heights distribution (assumed to follow a Rayleigh distribution). In some cases, it is more convenient to express the LOS probability as a function of incident or elevation angle.^[2]

Note that the expression is independent of the azimuth angle, consequently, the orientation over the city layout is not taken into account resulting in a 2D model even though the terminal heights are used.

The NLOS probability is computed from the LOS probability by the following equation:

${\displaystyle P_{\text{NLOS}}=1-P_{\text{LOS}}.}$

Pathloss modeling

Path loss represents the reduction in power density of an electromagnetic wave as it propagates through space. This attenuation is a critical factor in wireless communication, including A2G channels. A basic path loss model considers a Line-of-Sight (LOS) scenario where the signal travels freely without obstructions between the transmitter and receiver. The formula for calculating the received signal power under these conditions is as follows:^[5]

${\displaystyle P_{\text{R}}=P_{\text{T}}\times G_{\text{T}}\times G_{\text{R}}\times \left({\frac {\lambda }{4\pi d}}\right)^{\eta },}$

Here, ${\displaystyle P_{\text{T}}}$ denotes the power of the transmitted signal, ${\displaystyle G_{\text{T}}}$ and ${\displaystyle G_{\text{R}}}$ are the gains of the transmitting and receiving antennas, respectively, ${\displaystyle \lambda }$ is the wavelength of the carrier signal, and ${\displaystyle d}$ represents the distance between the transmitter and receiver. The Path Loss Exponent (PLE), ${\displaystyle \eta }$, typically has a value of 2 in a free-space environment, indicative of free-space propagation. However, PLE can take other values depending on the propagation environment. Therefore, the general expression for path loss can be represented as:

${\displaystyle \Lambda =\left({\frac {4\pi d}{\lambda }}\right)^{\eta }.}$

However, real-world A2G communication scenarios often differ from ideal free-space conditions. The log-distance path loss model, which considers a reference point for free-space propagation, is frequently utilized to estimate path loss in more complex environments (expressed in decibels):

${\displaystyle \Lambda (d)=\Lambda _{0}+10\eta \log \left({\frac {d}{d_{0}}}\right),}$

where ${\displaystyle \Lambda _{0}}$ is the path loss at a reference distance ${\displaystyle d_{0}}$, which can be calculated or predetermined based on free-space path loss (${\displaystyle \Lambda _{0}=20\log \left[{\frac {4\pi d_{0}}{\lambda }}\right]}$).

Incorporating both LOS and Non-Line-of-Sight (NLOS) conditions, the average path loss can be estimated by combining the path loss values for these two scenarios^[6]

^[7]

${\displaystyle \Lambda =P_{\text{LOS}}\times \Lambda _{\text{LOS}}+(1-P_{\text{LOS}})\times \Lambda _{\text{NLOS}},}$

In this formula, ${\displaystyle \Lambda _{\text{LOS}}}$ and ${\displaystyle \Lambda _{\text{NLOS}}}$ refer to the path loss values for LOS and NLOS conditions, respectively, while ${\displaystyle P_{\text{LOS}}}$ indicates the likelihood of an LOS link between the UAV and the ground station. The respective PLE values for ${\displaystyle \Lambda _{\text{LOS}}}$ and ${\displaystyle \Lambda _{\text{NLOS}}}$ are detailed in various studies.^[2]

Additionally, atmospheric absorption and rain attenuation can also lead to significant power loss for mmWaves and THz frequency bands.

Modeling of shadowing and small-scale fading

Beyond path loss, the presence of large structures like buildings, trees, and vehicles introduces specific, random variations in the power of received signals. These changes, known as Shadow fading, generally evolve at a slower pace, spanning tens to hundreds of wavelengths. Shadow fading at a given distance ${\displaystyle d}$ is typically represented as a normal random variable ${\displaystyle X_{sh}}$ in decibels (dB), with a variance ${\displaystyle \sigma }$. This variance reflects the deviations in received power around the mean path loss.^[2]

On a smaller scale, fading involves rapid changes in received signal strength over shorter distances, typically within the span of a few wavelengths. These fluctuations arise from the interference of Multipath Components (MPCs) that converge at the receiver. To quantify this behavior, statistical models such as the Rayleigh and Rice distributions are frequently used. Both are founded on complex Gaussian statistics. In environments with many MPCs, each with distinct amplitudes and random phases, small-scale fading often adheres to a Rayleigh distribution.^[5] Particularly in Air-to-Air (A2A) and Air-to-Ground (A2G) channels, where Line-of-Sight (LOS) propagation predominates, a Ricean distribution is a more appropriate model.^[5] Additionally, other models like Nakagami, chi-squared (${\displaystyle \chi ^{2}}$), and non-central ${\displaystyle \chi ^{2}}$ distributions are also considered relevant in certain scenarios.^[2] Notably, the ${\displaystyle \chi ^{2}}$ family of distributions encompasses many of these models.

For the modeling of small-scale fading, Geometry-Based Stochastic Channel Models (GBSCM) are among the most widely used methodologies. These models are developed through empirical measurements or geometric analysis and simulation, accommodating the inherently stochastic nature of signal variation. GBSCM is particularly effective in modeling narrow-band channels or the taps of wideband models that employ a tapped delay line approach.^[3]