Standard ITU Manhattan layout

ITU characterizes urban environments through a tuple of built-up parameters $(\alpha, \beta, \gamma)$, where $\alpha$ is the fraction of built area, $\beta$ represents the number of buildings per km $^2$, and $\gamma$ is the Rayleigh scale parameter for building height [Reference Series9]. Based on these parameters, different urban layouts can be generated [Reference Series9] within a Manhattan grid by arranging identical square buildings separated by evenly spaced streets, as shown in Fig. 1(a). The building width $W$ and street width $S$ are derived as $W = 1000\sqrt{\frac{\alpha}{\beta}}, S = 1000\sqrt{\frac{1}{\beta}} - W$, within a $1$ km $^2$ reference area. Building heights follow the Rayleigh probability density function

(1)\begin{equation} f(h) = \frac{h}{\gamma^2}e^{-h^2/(2\gamma^2)}. \end{equation}

Figure 1.

Manhattan and proposed urban layouts for high-rise urban environment ( $\alpha = 0.5, \beta = 300, \gamma = 50$). (a) Manhattan layout, (b) SRU layout, (c) FUU layout, and (d) HEU Layout.

Figure 1 Long description

Four panels compare the simulated urban geometries used in the study for the same high-rise built-up parameters. Panel a shows a regular Manhattan grid with square buildings and structured streets. Panel b shows the Structured Random Urban layout, where the grid structure remains but building sizes and spacing vary. Panel c shows the Fully Unstructured Urban layout with irregular building locations and open spaces. Panel d shows the Highway-Embedded Urban layout, where wide linear highway corridors are inserted through an otherwise irregular urban layout. The figure visually explains how the four layouts keep the same statistical built-up parameters while changing the spatial organization of buildings and open areas.

The simplicity and geometry of the Manhattan layout make it a popular reference for theoretical analysis, such as $P_{\text{LoS}}$ and $PL$. However, it fails to represent the geometric diversity of real cities. These limitations motivate the development of more flexible and irregular urban layouts presented in the following subsections.

Proposed urban layouts

To investigate whether geometric randomness in city structures influences LSF when the underlying built-up parameters are the same, we develop three random urban layouts using identical $(\alpha, \beta, \gamma)$ to ensure consistent building density, coverage, and height distribution [Reference Saboor, Cui, Vinogradov and Pollin19]. The only distinguishing factor among the layouts is the organization of buildings and open spaces in the total area $A_\text{total}$. The total area occupied by buildings is given by $A_\beta = \alpha A_\text{total}$ and $B_{\text{avg}} = \frac{A_\beta}{\beta}$.

Structured random urban (SRU) layout

The SRU layout maintains the general Manhattan alignment but introduces variability in building geometry/size and spacing, as visualized in Fig. 1(b). This controlled irregularity allows investigation of how moderate geometric randomness affects LSF compared to a perfectly regular grid. Each city block is divided into $\beta$ sections, representing the number of buildings per square kilometer. The area of the $i$th building is then randomly varied around the average building area $B_{\text{avg}}$ as

(2)\begin{equation} A_i = B_{\text{avg}}(0.6 + 0.8r_i), \qquad r_i \sim \mathcal{U}(0,1), \end{equation}

ensuring that $A_i \in [0.6B_{\text{avg}}, 1.4B_{\text{avg}}]$. For each building, the width $W_i$ and length $L_i$ are determined as

(3)\begin{equation} W_i = \sqrt{A_i}(0.5 + r_i), \qquad L_i = \frac{A_i}{W_i}, \end{equation}

while the height follows the Rayleigh distribution from equation (1).

Buildings are placed in a grid, creating structured streets with irregular building areas. This hybrid layout represents a more realistic downtown area where city grids exist, yet individual buildings differ in size and shape. As a result, the SRU acts as an intermediate case between deterministic and entirely random city structures, allowing a controlled analysis of how geometric variability affects the LSF.

Fully unstructured urban (FUU) layout

The FUU layout represents the extreme case of geometric randomness, where no predefined grid or symmetry is considered. In FUU, buildings are distributed freely across the simulation area, creating highly irregular streets and non-uniform building clusters, as illustrated in Fig. 1(c). This layout resembles old cities, where building placement and spacing are often highly irregular. Given the total building area $A_\beta$, the individual building areas are assigned using a Dirichlet distribution

(4)\begin{equation} \mathbf{A} = [A_1, A_2, \ldots, A_\beta] \sim \text{Dirichlet}(\mathbf{1}_\beta)\,A_\beta, \end{equation}

which ensures that $\sum_i A_i = A_\beta$. Each building is then assigned a random position $(x_i, y_i)$ within the $A_\text{total}$ region, with an overlap constraint defined as

(5)\begin{equation} G(x,y) = \begin{cases} 1, & \text{if occupied by a building,}\\ 0, & \text{otherwise,} \end{cases} \end{equation}

\begin{equation*} G(x_i:x_i+W_i,\,y_i:y_i+L_i) = 0. \end{equation*}

The resulting layout provides maximum randomness in spatial geometry while maintaining identical statistical density and height characteristics. This allows evaluation of whether LSF behavior depends primarily on environmental density parameters $(\alpha, \beta, \gamma)$ or on the spatial organization of the same built-up area.

Highway-embedded urban (HEU) layout

The HEU layout is an extension of the FUU layout where we embed $n_H$ linear corridors representing highways in the environment, as illustrated in Fig. 1(d). These highways create open spaces within dense areas, mimicking mixed urban environments composed of large highways and compact residential blocks around them. HEU helps in examining how large open corridors affect overall propagation and path-loss behavior.

Let $n_H$ denote the number of highways, each defined by its width $W_H^{(j)}$, length $L_H^{(j)}$, and orientation $\phi_H^{(j)} \in [0, \pi)$. The total area occupied by highways is given by

(6)\begin{equation} A_H = \sum_{j=1}^{n_H} W_H^{(j)} L_H^{(j)}. \end{equation}

This area is subtracted from the available built-up region before placing buildings, resulting in $A'_\beta = A_\beta - A_H$, where $A'_\beta$ is the effective built-up area after reserving highway space. No buildings are placed inside highway regions, enforced by the spatial constraint

(7)\begin{equation} G(x,y) = 0, \qquad \forall (x,y) \in \Omega_H, \end{equation}

where $\Omega_H$ represents the set of grid cells covered by highways.

Buildings are then distributed in the remaining area using the same procedure used in the FUU layout. This combination of dense building clusters and highways helps in examining how the presence of wide corridors within urban areas influences LSF and the resulting PLE.

A2G channel model

The A2G channel between the UAV/ABS and a ground user (GU) can be represented as a combination of deterministic free-space and random losses due to urban obstructions. The average large-scale attenuation $\Lambda(d, h_{\text{ABS}})$ depends on both the 3D distance $d$ and the ABS height $h_{\text{ABS}}$ (or simply on elevation $\theta$), which can be expressed as a weighted sum of LoS and NLoS components

(8)\begin{equation} \begin{aligned} \Lambda(d,h_{\text{ABS}}) =\;& P_{\text{LoS}}(\theta)\,\Lambda_{\text{LoS}}(d,h_{\text{ABS}}) \\ &+ \bigl[1-P_{\text{LoS}}(\theta)\bigr]\Lambda_{\text{NLoS}}(d,h_{\text{ABS}}). \end{aligned} \end{equation}

Here, $P_{\text{LoS}}(\theta)$ depends on the elevation angle $\theta = \tan^{-1}\!\bigl((h_{\text{ABS}} - h_{\text{GU}})/r\bigr)$, which is linked to the 3D distance $d = \sqrt{r^2 + (h_{\text{ABS}} - h_{\text{GU}})^2}$, and $h_{\text{GU}}$ is GU height.

This formulation provides a unified representation of the A2G channel by combining distance-dependent attenuation with the probabilistic influence of the surrounding geometry. The following subsection details $P_{\text{LoS}}$ modelling for different urban configurations.

LoS probability modelling

Ideally, $P_{\text{LoS}}$ depends on both the elevation $\theta$ and azimuth $\varphi$ angles, which can be computed using the geometry-based formulation presented in [Reference Saboor, Vinogradov, Cui, Al-Hourani and Pollin10]. However, since the azimuthal orientation of streets and GU positions tends to vary randomly in realistic city layouts, its effect averages out statistically over multiple realizations. Therefore, the LoS probability is typically modelled as an elevation-only function $P_{\text{LoS}}(\theta)$.

In our recent work [Reference Saboor, Cui, Vinogradov and Pollin19], we demonstrated that the empirical $P_{\text{LoS}}$ can be accurately represented by a four-parameter Sigmoid model as

(9)\begin{equation} P_{\text{LoS}}(\theta) = \frac{1}{1 + \exp(x_1 \theta^3 + x_2 \theta^2 + x_3 \theta + x_4)}, \end{equation}

where $x_1, x_2, x_3, x_4$ are environment-dependent coefficients obtained via nonlinear curve fitting from large-scale RT data.

LSF modelling

Impact of buildings: Buildings are the dominant source of LSF in urban environments, as they directly impact LoS condition. Accordingly, the LSF is modelled as a LoS-dependent log-distance PL with shadow fading

(10)\begin{equation} \Lambda_{x}(d,h_\text{ABS}) = \Lambda_0 + 10\, n_x(h_\text{ABS}) \log_{10}(\frac{d}{d_0}) + \Psi_x, \end{equation}

where $x$ can be LoS or NLoS, $\Lambda_0$ is the reference free-space PL at distance $d_0=1$ m, $n_x(h_\text{ABS})$ are height-dependent PLEs, and $\Psi_x$ is the zero-mean Gaussian shadow fading term with height-dependent standard deviations $\sigma_x(h_\text{ABS})$.

As the ABS height increases, the number of potential obstructions decreases and the elevation angle improves, leading to a gradual transition from NLoS to LoS dominance. We model the height-dependent PLE as

(11)\begin{equation} n(h_\text{ABS}) = n_\infty + (n_0 - n_\infty) \exp(-h_\text{ABS}/h_0^n), \end{equation}

where $n_0$ and $n_\infty$ are limiting PLEs and $h_0$ is a characteristic decay height. The height-dependent standard deviation of shadow fading is similarly modelled as

(12)\begin{equation} \sigma(h_\text{ABS}) = \sigma_\infty + (\sigma_0 - \sigma_\infty) \exp(-h_\text{ABS}/h_0^\sigma), \end{equation}

with $\sigma_0$ and $\sigma_\infty$ serving as limiting values.

Impact of random obstructions: The proposed model focuses on LSF caused by buildings, which dominate attenuation in urban A2G links. In practice, smaller obstacles such as trees or streetlights may introduce additional losses, especially at mmWave frequencies. However, as shown in our earlier work [Reference Saboor, Cui, Vinogradov and Pollin12], the excess loss caused by such objects is typically distance-independent and remains below 3 dB on average, leaving the height-dependent PLE largely unaffected. Therefore, these effects are not explicitly modelled here but can be incorporated as a fixed attenuation offset when finer accuracy is required.

LoS parameterization

Parameter extraction: In the RT simulations, $P_{\text{LoS}}$ or LoS condition is derived directly from geometric visibility between the ABS and GU. For each ABS–GU pair, the RT checks whether the straight line between them intersects any building. If there is no intersection, the link is LoS; else, it is NLoS. For each ABS height, multiple city realizations and user locations are simulated to capture statistical variability. The empirical $P_{\text{LoS}}(\theta)$ is obtained as $P_{\text{LoS}}(\theta) = \frac{N_{\text{LoS}}(\theta)}{N_{\text{Total}}(\theta)}$, where $N_{\text{LoS}}(\theta)$ and $N_{\text{Total}}(\theta)$ are the number of LoS and total links observed for a particular $\theta$, respectively. Therefore, $P_{\text{LoS}}$ is estimated by fitting (9) across urban environments and layouts.

Extracted parameters: The fitting parameters for the four standard urban environments, corresponding to each layout, are given in Table 2. In addition to the four layouts described in “System model” section, we provide the generalized results based on the combined data of all the layouts.

Table 2.

Extracted model parameters for LoS and NLoS links across all layouts and height ranges

Table 2 Long description

The table reports the fitted model parameters for Manhattan, SRU, FUU, HEU, and all layouts combined across suburban, urban, dense urban, and high-rise environments. For LoS probability, it lists the sigmoid coefficients x1, x2, x3, and x4. For the LoS channel model, it reports n and the shadow-fading parameters. For the NLoS channel model, it reports low- and high-altitude path-loss exponent parameters and shadow-fading parameters, including the relevant ABS height ranges. The table shows that LoS PLE remains 2, while NLoS PLE decreases with height and varies moderately across layouts.

LSF parameterization

Parameter extraction: LSF statistics are obtained from the RT dataset by separating LoS/NLoS links and estimating height-dependent PLEs and shadow fading distributions. From the RT dataset, LoS and NLoS channels are separated for each UAV height (all 20 city realizations are used). PLE is obtained via linear regression

(13)\begin{equation} n(h_\text{ABS}) = \frac{\Lambda_\text{RT}(h_\text{ABS}) - \Lambda_0}{10 \log_{10}(d/d_0)}, \end{equation}

and shadow fading as deviations from the fitted PL

(14)\begin{equation} \Psi (h_\text{ABS}) = \Lambda_\text{RT}(h_\text{ABS}) - (\Lambda_0 + 10 n(h_\text{ABS}) \log_{10}(\frac{d}{d_0})). \end{equation}

Height-dependent PLEs and shadow fading standard deviations are fitted with exponential functions (11) and (12).

Extracted channel parameters: The extracted parameters are summarized in Table 2. Characteristic parameters $n_0$, $n_\infty$, $\sigma_0$, $\sigma_\infty$, and $h_0$ provide physical insight into propagation near the ground and asymptotic high-altitude behavior with lower PLE values.

The results of PLE modelling for the combined data illustrated in Fig. 2 reveal clear trends in ABS heights and layouts for NLoS conditions. The results reveal that the height-dependent PLEs decrease toward the 2.5–3 limit at high ABS altitudes, reflecting the gradual transition from heavily obstructed to more favorable propagation. Suburban scenarios display a two-regime behavior, with a breaking point around $100$ m separating a descending PLE regime at low altitudes from an ascending regime at higher altitudes. In contrast, urban and high-rise layouts follow a single descending PLE regime. Note that the LoS PLE equals 2 (free space) in all cases. Shadow fading standard deviations decrease with height, with LoS channels exhibiting lower variability than NLoS links, as expected.

Figure 2.

Combined layouts PLE across ABS altitude and environment under NLoS conditions.

Figure 2 Long description

The plot shows the fitted NLoS path-loss exponent as a function of ABS altitude for the combined-layout case across the considered urban environments. The curves show that NLoS PLE generally decreases as altitude increases, approaching approximately 2.5 to 3 at high ABS heights. The suburban case exhibits a two-regime trend, with a change around 100 m, while urban and high-rise environments mainly follow a descending trend with altitude.

More detailed PLE results are provided in Fig. 3 for all four layouts and the combined data. It can be observed that the layout has a relatively moderate effect ( $\pm$0.2) on the PLE behavior. Since such a difference may still be significant in some cases, Table 2 contains all the details. However, in the further analysis, we mostly focus on the combined data, as it represents a general layout.

Figure 3.

NLoS PLE as a function of ABS deployment altitude for the four urban environments, the combined data, and the fitted height-dependent model for the combined data. (a) Suburban and (b) high-rise urban.

Figure 3 Long description

The two panels compare NLoS path-loss exponent trends versus ABS height for different layouts and the combined model. In the suburban panel, PLE drops sharply at low altitudes and then slowly increases or stabilizes at higher altitudes, showing the two-regime behavior discussed in the text. In the high-rise urban panel, PLE decreases more steadily with altitude. Across both panels, individual layouts differ moderately, while the combined fitted model follows the overall RT-data trend.

The fitted exponential models achieve root mean square errors (RMSEs) of $0.02$– $0.11$ (PLE) and $0.07$– $0.36$ (LoS) and $0.31$– $0.95$ (NLoS) for shadow fading standard deviation, indicating good agreement with the RT data.