# Path Loss Model for 3.5 GHz and 5.6 GHz Bands in Cascaded Tunnel Environments
###### Abstract  
An important and typical scenario of radio propagation in a railway or subway tunnel environment is the cascaded straight and curved tunnel. In this paper, we propose a joint path loss model for cascaded tunnels at 3.5 GHz and 5.6 GHz frequency bands. By combining the waveguide mode theory and the method of shooting and bouncing ray (SBR), it is found that the curvature of tunnels introduces an extra loss in the far-field region, which can be modeled as a linear function of the propagation distance of the signal in the curved tunnel. The channel of the cascaded straight and curved tunnel is thus characterized using the extra loss coefficient (ELC). Based on the ray-tracing (RT) method, an empirical formula between ELC and the radius of the curvature is provided for 3.5 GHz and 5.6 GHz, respectively. Finally, the accuracy of the proposed model is verified by measurement and simulation results. It is shown that the proposed model can predict path loss in cascaded tunnels with desirable accuracy and low complexity. ray tracing; extra loss coefficient; shooting and bouncing ray method; subway tunnel; waveguide effect 2020 April 20220  
arriving at the receiver through line-of-sight (LOS) and reflection paths [18]. The accuracy of the RT methods is reliable when three-dimensional (3D) model of the tunnels and RT parameters are set properly [19], but it requires large computing memory and additional 3D modeling for tunnel environments [20]. As listed in Table 1, considerable efforts have been devoted to improving the path loss models for different tunnel types during the past few decades [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. For straight tunnels, the authors of [21] conducted a measurement campaign in subway tunnels at 900/2400 MHz and established a FI model by fitting the measurement results. While based on waveguide theory, a multimode model is proposed in [22], which gives an expression for the path loss at any frequency and position in tunnels. In [23], the accuracy of the SBR method for RT modeling is improved via per-ray cone angle calculation. In [24; 25], surrogate models of arched tunnels are extracted in the form of rectangular waveguides by combining waveguide models and computationally intensive VPE methods. However, the validity of the surrogate models is affected by transmitter and receiver positions, especially in the regions closer to the boundary walls. For curved tunnels, an improved CI model is proposed by introducing a breakpoint distance based on the measurement data in railway communications, which has been shown to have better accuracy but less stability than the traditional CI model [26]. In [27], channel measurements are conducted in a realistic curved-tunnel environment including the presence and absence of human bodies in terms of LOS and NLOS scenarios. An improved FI path-loss model is proposed to describe the transition between the enhanced and degraded waveguiding mechanisms in curved tunnels with the calculated break-point in [28]. Combined with waveguide theory and the RT method, [29] uses geometrical optics rules to model the main effects of the tunnel curvature and proposes a heuristic model, which can be useful for preliminary path-loss estimation. More recently, the back