Refining the Dual-Slope Path Loss Model with a Distance-Adaptive Exponent and Multi-Breakpoint Calibration

Wireless communication plays a crucial role in modern technology, enabling high-speed data transfer for applications such as 5G, IoT, and smart city infrastructure. One fundamental challenge while designing wireless networks is accurately modeling signal propagation, which directly impacts network planning, interference management, and coverage optimization. Ideally, path loss models are essential in predicting the attenuation of transmitted signals over distance and are widely used in radio wave propagation studies.

Positioning strategies relying on measured signal strength depend greatly on the precision of RF estimations regarding received power [1-5]. These demands have led researchers in RF prediction to re-evaluate the criteria and precision of current breakpoint location and path loss estimation [6, 7].

Moreover, despite the dramatic expansion of wireless cellular communication networks over the past two decades, they continue to face increasing interference, which degrades service quality. This interference arises from suboptimal cellular network design and inadequate optimization, primarily due to the absence of highly accurate propagation models [8]. No RF path loss model can precisely predict signal intensity, as each model has specific validity constraints and is tailored to particular RF scenarios. To enhance their applicability to real-world RF propagation conditions without causing environmental disruptions, it is vital to understand their rationality ranges and apply necessary correction factors [9, 10].

Traditionally, path loss models fall into two categories: single-slope models, like Free-Space Path Loss, and dual-slope models, which adjust the path loss exponent at a defined breakpoint. The Dual-Slope Path Loss Model provides a more realistic representation of signal attenuation by considering two distinct propagation regions. The first region, before the breakpoint, is dominated by free-space propagation, where the path loss exponent is approximately n₁ = 2. Beyond the breakpoint, additional factors such as ground reflection, diffraction, and obstructions contribute to increased signal attenuation, resulting in a higher path loss exponent of n₂ = 4, as noted in reference [11].

In contrast, the traditional Dual-Slope Model suffers from abrupt changes in the path loss exponent at the breakpoint, which can lead to discontinuities in signal prediction. This can introduce significant errors, especially in urban and suburban environments, where signal behavior is influenced by multipath effects, terrain variations, and environmental clutter [8, 12]. Researches like Feuerstein et al. [12] and Elmutasim and Mohd [13] define the breakpoint as the point at which the Fresnel zone starts interfering with the ground, while Perera et al. [14] demonstrated that this model exhibits significant discrepancies when assessed against various measurement campaign findings.

Researchers have developed empirical refinement models, such as the Perera breakpoint, which adjusts the breakpoint distance to match suburban and urban propagation data better [15, 16]. However, this approach still uses fixed exponents before and after the breakpoint rather than responding to environmental variability. Other wireless communication design models include standard models such as 3GPP and WINNER II. Such models use environment-specific parameters and empirical exponents; however, they are rigid and do not allow for smooth changes in exponents. Their breakpoint lengths are frequency-dependent and lack physical [17-19]. Another aspect that recent studies have examined is the use of machine learning (ML) models for path loss prediction, which include neural networks and regression trees. These models frequently outperform traditional equations in site-specific deployments; however, they necessitate large, labelled datasets and function as black boxes, which limits their interpretability and portability [20, 21].

To overcome these limitations, we propose distance-adaptive exponent (DAE) model as an adjustable model, where the path loss exponent n varies continuously with distance rather than switching abruptly at a predefined breakpoint. The key contributions of this study are as follows:

• Proposal of DAE model that dynamically adapts the path loss exponent as a function of distance.

• Integration of multiple breakpoints (Fresnel, Perera, and True breakpoints) to refine transition regions between free-space propagation and multipath-dominated environments.

• Comparison of traditional vs. distance-adaptive exponent (DAE) model, highlighting improvements in accuracy and continuity.

• Validation through MATLAB simulations, demonstrating reduced error in path loss prediction.

Many measurement efforts have confirmed that signal intensity in line-of-sight (LOS) propagation often adheres to the dual-slope attenuation model. The strength of the received signal decreases due to exponential attenuation both prior to and beyond the breakpoint distances, respectively. This dual-slope attenuation model is represented as [22]:

$P L(d)=\left\{\begin{array}{l}P L_1, d \leq d p \\ P L_2, d \geq d p\end{array}\right.$          (1)

where, $P L_1$ and $P L_2$ represent the path loss values before and after the breakpoint location, respectively. d denotes the link separating the transmitter and receiver, while $dp$ is the breakpoint distance that isolate the two slopes.

The conceptual breakpoint distance models rely on the two-ray propagation path loss framework [23, 24], which has been validated through various RF measurement studies. This model describes signal strength, including both direct and reflected waves [25]:

$H(f)=\left|A_t A_r e^{j k d_1}+\Gamma A_t A_r e^{-j k d_2}\right|$          (2)

where, $H(f)$ represents the two-ray path loss model, $A_t$ and $A_r$ are the antenna gains at the transmitter and receiver, respectively, $d_1$ and $d_2$ denote the travelling distances of the direct and reflected beams, respectively, $\Gamma$ is the Fresnel reflection coefficient, while $k$ is the wave factor and equal to 2π/λ.

While according to the study [6], the pathloss could be given by:

$P L=10 \log _{10}\left(\frac{|H(f)|^2}{P_t}\right)$          (3)

where, $P_t$ is transmitted power.

The suggested Perera's breakpoint distance framework, derived by equating the two approximations based on the study [26], is expressed as follows:

$d_{b r k}=8.41 \cdot \frac{h_t h_r}{\lambda}$          (4)

Clearly in Eq. (4) the model predicts $d_{b r k}$ to be more than estimated using the most commonly referenced model introduced in references [8, 12, 13], which can be expressed as $\frac{4 h_t h_r}{\lambda}$. To thoroughly investigate the forecasting precision of Perera's breakpoint distance model, we will compare its effectiveness relative to its underlying framework, the two-ray model. According to studies [24, 26], the communication function of a two-ray radio channel is expressed as follows:

$\begin{aligned} & H_{V, H}=f_t^{V, H}\left(\theta_d^t, \phi_d^t\right) f_r^{V, H}\left(\theta_d^r, \phi_d^r\right) \frac{\lambda}{4 \pi r_d} e^{-j k r_d} \\ & +f_t^{V, H}\left(\theta_g^t, \phi_g^t\right) f_r^{V, H}\left(\theta_g^r, \phi_g^r\right) R_{V, H} \frac{\lambda}{4 \pi r_g} e^{-j k r_g}\end{aligned}$         (5)

where, $f_r^{V, H}(\because)$ and $f_t^{V, H}(\because)$ are receiver and far-field amplitude distributions of the antenna radiation patterns for vertical and horizontal polarizations,, respectively; $\theta_i^{t, r}, \phi_i^{t, r}$ are the transmitter and receiver ray i’s elevation and azimuth angles (direct or reflected), respectively; $r_g$ and $r_d$ denote the propagation distances of the direct and ground (or water) reflected paths, respectively; $k$ is the wave number $\left(k=\frac{2 \pi}{\lambda}\right)$; and $\lambda$ wavelength corresponding to the operating frequency.

Nevertheless, Perera’s model is derivative from the two-ray path loss model and predicts the breakpoint distance $d p$ as:

$d p=\frac{4 h_t h_r}{\lambda}$         (6)

where, $h_t$ and $h_r$ heights of transmitter and receiver antenna, and $\lambda$ is the wavelength of operating frequency.

However, the study [26] introduces a correction term of Perera’s model as:

$Corrected _{d p}=8.41 \cdot \frac{h_t h_r}{\lambda}+C$         (7)

where, $C$ is the correction term given by:

$C=f_1\left(h_r\right)+f_2\left(h_r\right) \frac{h_t}{\lambda}$         (8)

The formulations of $f_1\left(h_r\right)$ and $f_2\left(h_r\right)$ are given in the research [26].

Consequently, our paper’s contribution to improving the dual-slope path loss model by introducing distance-adaptive exponent (DAE) model exponent instead of fixed values. This refinement enables the model to dynamically adjust depending on the link distance between the transmitter and the receiver. Eq. (9) explained the comprehensive concept contribution:

$n(d)=n_1+\gamma \cdot \log _{10}(d)$         (9)

where, $n_1$ is the free-space baseline (typically 2), and $\gamma$ is a tuning parameter reflecting environment dynamics that can clarify $\gamma$ can be empirically related to environmental factors. Thus, the fixed exponent $n_1=2$ which is before breakpoint and $n_2=4$ after the breakpoint, leaded to defined adjustable exponent as:

$n_{D A E}=2+\Upsilon \cdot \log _{10}(d)$         (10)

where, $n_{D A E}$ is the distance-dependent path loss exponent, $\Upsilon$ works as a tunning factor to control the rate of change depends on environment for example in urban = 0.1 up to 0.8 and in rural = 0.05 according to the research [27], whereas $d$ considers the link distance between transmitter and receiver antenna.

Eq. (10) increases the path loss exponent gradually instead of making an abrupt jump from n₁ = 2 to n₂ = 4, which provides a more realistic transition between different propagation conditions. The idea for this contribution arose to avoid the abrupt switch from traditional models while also providing a smooth transition between different distances, more accurately reflecting real-world propagation. In other words, the approach offers customizable advantage via offering different $\Upsilon$ that allows adjustments for urban, suburban, and rural areas.

This work proposes an improved dual-slope path loss model with an adjustable path loss exponent that seamlessly transitions between different propagation settings, particularly above 1 km, where the dual-slope model is severely prone to attenuation. The proposed technique computes path loss in a stepwise manner across Fresnel, Perera, and True breakpoints, resulting in steady and higher accuracy in urban and suburban propagation settings. The results indicate that frequency plays a crucial role, with higher frequencies experiencing greater path loss over distance. This underscores the importance of careful frequency selection. However, while the dual-slope model can be effective at shorter distances not exceeding a few kilometers, the adjustable model provides a more disciplined behavior by dynamically altering the exponent as a function of distance. This eliminates the harsh discontinuities found in typical dual-slope models, especially over kilometer-scale distances. Despite the use of an adjustable path loss exponent enhances accuracy by allowing for a seamless transition in attenuation behaviour. However, it introduces new model parameters, including the tuning factor, which may raise the challenge. But in fact, this difficulty is manageable and does not require large adaptations to existing wireless networks. The adaptability feature could be predefined for various environments (urban, rural, suburban), or dynamically adjusted using simple threshold logic or automated tuning based on terrain classification. Future studies will integrate intelligent reflection with breakpoint placements.