# Path Loss Model for 3.5 GHz and 5.6 GHz Bands in Cascaded Tunnel Environments
###### Abstract
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Methods** & **Models** & **Pros and Cons** & **Ref.** \\ \hline \multirow{8}{*}{Straight tunnel} & Fit measurement results & FI model & Low complexity & [21] \\  & using regression method & & Insufficient accuracy & [22] \\  & Superpose multiple modes in both & & & High accuracy \\  & near and far region & & Limited applicability & [22] \\  & & & High accuracy & \\  & & & Low computational & [23] \\  & Calculate Per-ray cone angle & Improved RT model & & efficiency & \\  & Extract rectangular waveguide & Mixed model based on & Reduced complexity & \\  & model using VPE & waveguide and VPE & Limited Validity & [24; 25] \\ \hline \multirow{8}{*}{Curved tunnel} & Introduce a break point & Improved CI model & High accuracy & \\  & distance into the CI model & & Less stability & [26] \\  & Divide propagation region into & & Realsistic scenario & \\  & LOS and NLOS & & Large deviation & [27] \\  & Define the break point between & Improved FI model & High accuracy & \\  & two waveguiding effects & with break point & Calculations of break & [28] \\  & Estimate the main effects of the & Mixed model based on & point required & \\  & curvature on multimode & waveguide and RT & Insufficient accuracy & \\  & Combine RT method with & & High applicability & [30] \\  & neural network & Improved RT model & High complexity & [31] \\ \hline \multirow{8}{*}{Cascaded tunnel} & Fit measurement results & CI model & Low complexity & [31] \\  & using regression method & & Insufficient accuracy & \\  & & & High accuracy & \\  & Reconstruct a high-precision 3D & & & High-precision 3D & [32] \\  & model of measurement tunnel & & model required & \\  & Divide space into segments to & Improved FDTD & High accuracy & \\  & solve stability constraint & model & high complexity & [33] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: State-of-the-art related works on path-loss models in tunnel environments. propagation neural network is incorporated in the model to predict the path loss in the curved tunnels [30]. For cascaded tunnels, [31] measures five frequency bands (i.e., 2.6 GHz, 3.5 GHz, 5.6 GHz, 10 GHz, and 28 GHz) in a subway tunnel composed of straight and curved sections, and a typical CI model is employed to fit the measurement results. In [32], a high-precision 3D model of a measurement tunnel is reconstructed, and the path loss is predicted using the RT method based on the 3D models. To improve the feasibility of the FDTD method, a segmented FDTD method is proposed to model radio-wave propagation in tunnels, which breaks up the computational space into segments. However, it still needs to solve sets of simultaneous equations [33]. In summary, most research on path loss model for tunnels in the literature mainly deals with a single straight or single curved tunnel, and currently there are only a few works that focus on cascaded tunnels. Meanwhile, it is worth noting that the existing schemes suffer from problems of high computational complexity, extra cost such as high-precision 3D tunnel models, excessive memory, and CPU resources. Regarding this issue, this paper studies the propagation characteristics of radio wave in cascaded tunnels and presents a simplified path-loss model for cascaded tunnels. The extra loss in the far-field region due to the curvature is modeled as a linear function of the propagation distance of the signal in the curved tunnel. Compared with typical RT methods relying on high-precision 3D tunnel models, the proposed model can predict path loss in cascaded tunnels with comparable accuracy but significantly lower complexity. The rest of this paper is summarized as follows. Section 2 proposes the simplified path-loss model for cascaded tunnels based on waveguide and SBR. Section 3 presents the channel measurement at 3.5 GHz and 5.6 GHz bands in a real subway tunnel environment and calibrates the material parameters of RT simulation. In Section 4, the empirical formula between the ELC and the radius of curvature is provided.