Research Article  Open Access
Kai Zhang, Fangqi Zhang, Guoxin Zheng, "SpaceTime Correlation for ThreeDimensional MIMO Channel Model Using Leaky Coaxial Cables in Rectangular Tunnel", International Journal of Antennas and Propagation, vol. 2020, Article ID 3984148, 9 pages, 2020. https://doi.org/10.1155/2020/3984148
SpaceTime Correlation for ThreeDimensional MIMO Channel Model Using Leaky Coaxial Cables in Rectangular Tunnel
Abstract
The rapid development of highspeed train and Metro communications has provided new challenges for the application of MIMO technologies. Therefore, we propose a threedimensional (3D) multipleinput multipleoutput (MIMO) channel model using leaky coaxial cable (LCX) in a rectangular tunnel. The channel model is based on geometrybased singlebounce (GBSB) channel model and the electric field distribution of LCX in the tunnel environment. The theoretical expressions of channel impulse response (CIR) and spacetime correlation function (CF) are also derived and analyzed. The CFs for different model parameters (moving velocity and moving time) and different regions of the tunnel are simulated by Monte Carlo method to verify the theoretical derivation at 1.8 GHz. In the same parametric configuration of nonstationary tunnel scenarios, the time delay of the first minimum value of CFs for LCXMIMO is 1/5 of the time delay of the minimum value of CFs for dipole antennas MIMO when the train moving velocity is 360 km/h. It is shown that, for MIMO system, the performance of using LCXs is better than using dipole antennas.
1. Introduction
Recently, multipleinput multipleoutput (MIMO) technology has been widely used in many scenarios. The nonstationary scenarios of MIMO technologies for highspeed train and Metro communications have drawn attention. Up to present, there are two main cases of nonstationary scenarios: one is the movement of transmitter or receiver and the other is the movement of transmitter and receiver [1–12]. The channel modeling can be based on twodimensional (2D) [1–5] and threedimensional (3D) [6–12]. According to the 3GPP (3GPP TR 25.996) spatial channel model (SCM) [1–6] and WINNER II channel model [7–12], the geometrybased singlebounce (GBSB) [1–5] and geometrybased stochastic model (GBSM) [6–12] are used to describe MIMO propagation channels. The nonstationary scenarios investigated include the high altitude unmanned aerial vehicles (UAV) MIMO airtoground communications [7, 8], the vehicletovehicle communications in tunnel [9, 10] and other scattering environments [11, 12], the basestationtovehicle communications in tunnel [5] and other scattering environments [1, 2], and so on. In these scenarios, the influences of different moving velocities, moving times, moving directions, and moving trajectories on spatial crosscorrelation and autocorrelation are analyzed [6–12]. In [1, 2], the authors have researched the multielement antennas systems in mobile fading channels and have considered that the local scatterers were only around the mobile user (MS) in 2 × 2 MIMO channel, the base station (BS) is fixed, the BS and the MS are both twoelement antennas array, and they also derived a spacetime correlation for MIMO systems in mobile fading channels. It could be called “onering” model in physical MIMO channel models. In [4], the authors analyzed the propagation characteristics of mobiletomobile fading channels for MIMO systems; they derived a great channel model in mobiletomobile fading channels and scatterers around both transmitting antennas and receiving antennas. It could be called “tworing” model in physical MIMO channel models. In [9], the authors studied a nonstationary geometrybased channel model for MIMO vehicletovehicle communications in tunnel environments and derived the theoretical expressions of channel impulse response and spatial correlation functions (SCFs), and they discussed the SCFs for different time delay, different receiving antenna spacing, and different frequency separation. In [11], this paper researched a geometrybased nonstationary MIMO channel model for vehicular communications and discussed the autocorrelation functions (ACFs) for different time separation and different receiving antenna spacing. In [12], this paper proposed a 3D wideband nonstationary multimobility model for vehicletovehicle MIMO channels and derived the channel impulse response (CIR) and correlation functions (CFs); the authors discussed the CFs for different time separations and different times which increased with the increase of time separation and time by simulation.
At the moment, there are some shortcomings in the study of confined area communication such as subway tunnels. Leaky coaxial cable (LCX) is usually used as an antenna (the input signal is entered from single port of every single LCX) and two antennas (the input signal is entered from double ports of every single LCX) to transmit signals in tunnel, due to uniform radiation. Most of the researches on the application of LCX are focused on the measurement and analysis in tunnel and linearcell environments [13, 14]. In [13], the authors present a 2 × 2 MIMO measurement campaign using LCX in tunnel environment at 1.8 GHz, and the results of capacity and condition number indicate that the MIMO performance is not highly dependent on the LCX spacing. In [14], the authors used single LCX as two antennas and researched propagation characteristics of the LCXMIMO systems in linearcell environment at 2.4 GHz and 5 GHz, and the capacity results indicate that the LCXMIMO can achieve more capacity improvement than that of channel using monopole antennas with the simple equal power allocation. A few of them have carried out theoretical description. The channel models of these researches are stationary [15]. In [15], the authors research the theoretical channel model of 2by2 MIMO system using single LCX in free space, and the results show that a 2by2 MIMO system can achieve good performance using single LCX. In [16], the authors proposed a GBSB model for MIMO channel using LCXs in tunnel based on the electric field distribution of the LCX and analyzed the performance of MIMO system in terms of capacity, condition number (CN), and CF by simulation and measurement. But in [16], the authors only studied the stationary scenario. There is little research on nonstationary LCXMIMO communication in tunnel scenarios. With the rapid development of Metro and highspeed railway, there is an urgent need for the study of LCXMIMO in nonstationary tunnel scenarios.
For highspeed railway scenario and Metro scenario, this paper aims to fill these research gaps. We propose a 3D nonstationary LCXMIMO channel model in rectangular tunnel environments. The main contributions and novelties of this paper are summarized as follows: (1) this paper develops a 3D nonstationary LCXMIMO channel model, and the channel impulse response (CIR) and spatial correlation function (CF) are theoretically derived. The proposed model is verified by comparing the spatial CF theory with simulation; (2) this model considers the influence of the receiver’s mobility on the propagation characteristic of LCXMIMO system, in terms of the moving velocity and moving time. The influence of the different positions where the receiver is located in the tunnel on the propagation characteristics is also analyzed; (3) the CFs of LCXMIMO and dipoles MIMO are compared with different moving velocities for different time delays.
The remainder of this paper is organized as follows. In Section 2, we propose a 3D nonstationary LCXMIMO channel model. In Section 3, the theoretical expressions of spacetime CF for our proposed GBSB 3D nonstationary in tunnel scenarios are derived. In Section 4, the spacetime CFs of the proposed model are simulated and validated. Finally, our concluding remarks are given in Section 5.
2. 3D Nonstationary LCXMIMO Channel Model
Figure 1(a) shows the LCX structure; Figure 1(b) shows LCXs (transmitter, Tx) and dipoles (receiver, Rx) installation; and Figure 1(c) shows the 2by2 nonstationary LCXMIMO transmission system in rectangular tunnel environments and the input signals of LCXs are s_{1} and s_{2}. In Figure 1, the O is origin of the coordinate; the parallel direction to axis of tunnel presents the Zaxis of the coordinate system; the cross section of tunnel presents the XOYplane, the vertical direction of the XOYplane presents the Yaxis of the coordinate system, and the horizontal direction of the XOY plane presents the Xaxis of the coordinate system. In Figure 2, for the MIMO system, the parameter names and symbols of LCXMIMO channel model are as shown in Table 1.
(a)
(b)
(c)
(a)
(b)
(c)

In this paper, we only consider LCX with vertical periodic slots, the length of LCX is L, the periodic of slot of LCX is P, and the number of slots of LCX is N. According to the electromagnetic field theory of antenna, the slots of LCX can be seen as a series of magnetic dipoles, and the electric field distribution of a single slot E_{z} can be expressed as [17, 18]where E_{0} denotes the electric field strength of the slot surface, r denotes the distance from a point in the electric field of LCX to a slot, θ is the angle between the direction of r and the axial direction of LCX, k_{0} = 2πf/c denotes the propagation constant in free space, f is the center frequency, and c is velocity of light.
The longitudinal amplitude attenuation of ith slot of LCX α_{i}, phase variation ith slot of LCX β_{i}, and the propagation constant k_{r} in LCX are as follows:where α_{0} is amplitude attenuation of LCX per slot, β_{0} = k_{r}P is phase variation of LCX per slot, and ԑ_{r} is the dielectric constant of LCX.
Because of the rough surface of the tunnel wall, it can be considered that there exists diffuse reflection of electromagnetic wave from the transmitter to the receiver in the tunnel, and the number of effective diffuse reflection locations is M. We assumed that the diffuse reflection locations obey uniform distribution, and all the effective diffuse locations are located on the tunnel side wall opposite to LCX. Therefore, the total electric field strength of a certain point in the tunnel can be considered as the vector superposition of LoS and NLoS components as shown in Figure 2. The LoS and NLoS components of electric field distribution from the pth (p = Tx1, Tx2) LCX to the lth (l = Rx1, Rx2) receiving antenna can be expressed as [16]where E_{i} = E_{0}α_{i}e^{jβi} denotes the electric field strength of the ith slot surface of LCX. E_{0} denotes the original electric field strength of LCX (we assume that E_{0} equals 1 V/m). r_{i,lp} is the distance from the ith slot of pth LCX to the lth receiving antenna. r_{im,p} and r_{im,l} are the distances from the ith slot of pth LCX to mth (m = 1,…,M) effective diffuse point S_{m} and from mth effective diffuse point S_{m} to lth receiving antenna, respectively. θ_{i,lp} and θ_{im,p} are the angles between the direction of r_{i,lp} and the axial direction of LCX and between the direction of r_{im,p} to the axial direction of LCX, respectively.
The LCXMIMO channel model can be described by 2 × 2 complex channel impulse response (CIR) matrix, i.e., H(t) = [h_{lp}(t)]_{2 × 2}, where Ω_{lp} = E[h_{lp}(t)^{2}]. The CIR h_{lp}(t) from pth transmitting antenna (Tx) to lth receiving antenna (Rx) is a vector superposition of the two components CIR of LoS and NLoS and can be expressed as [16]where φ_{im} denotes the phase shift that follows the random variables uniform distribution in the interval [0, 2π) and caused by effective diffuse reflection points in tunnel side wall; K_{lp} denotes Rician factor, i.e., ; and can be calculated by
The particular representation of and in terms of K_{lp}, together with the assumption of and as N ⟶ ∞ and M ⟶ ∞, guarantees that Ω_{lp} = E[h_{lp}(t)^{2}].
3. Spatial Correlation Function
In order to study the performance of the proposed LCXMIMO channel model, this paper assumes that the LoS and NLoS components are independent of each other and the values of amplitude of LoS and NLoS CIR are zeromean processes. According to [1, 2], this paper assumes that the CF of transmitter and the CF of receiver are independent of each other, and the CF of receiver is very small. Therefore, we mainly analyze the spatial CF of LCX at the transmitter, and the spatial CF can be expressed as
According to formulas (4a) and (4b), the spatial CFs of LoS and NLoS can be expressed aswhere ; ; ; ; ; ; ; ; ; ; .
4. Simulation Results and Discussion
According to formulas (5), (7a), and (7b), the spatial CF (LoS + NLoS) is simulated by Monte Carlo method, because the LoS component of CIR is deterministic, and NLoS component of CIR is random process. So, we can multiple iteration NLoS component of CIR by Monte Carlo method. In this paper, we set the number of iterations to 5 × 10^{4}. In this paper, the parameters setup of simulation is mainly a reference to [13]. For the simulation of this paper, we assumed that the LCXs are installed at the tunnel side wall, and the dipole antennas are moving on the axial direction of tunnel ground. K_{11} and K_{12} are set to be of the same values (1 dB, 4 dB, 8 dB, 16 dB). Firstly, the receiver is located in Region 2 of tunnel (i.e., the tunnel’s axial [49 m, 51 m], the horizontal distance from the tunnel’s side wall to receiver is 2.44 m, and the vertical distance from LCX1 to receiver is d_{0} = 1.1 m). The influence of different moving velocities (90 km/h, 180 km/h, and 360 km/h) of the receiver, different relative time delays of the two paths, different LCX spacing, and different moving times on spatial CF is analyzed. Meanwhile, the influence of different regions of receiver in tunnel on spatial CF is analyzed by simulation. For all regions, the horizontal distance from the tunnel’s side wall to receiver is 2.44 m, and the vertical distance from LCX1 to receiver is d_{0} = 1.1 m. The other parameters of simulation are presented in Table 2.

Figure 3(a) describes the simulation and theoretical results of spatial CF as the time delay increases at different moving velocities of receiver. Figure 3(b) describes the change of simulation and theoretical spatial CF as the time delay increases when the receiver is moving at different moving times. The results are consistent with those described in [10]. Figure 3(a) shows that the spatial CF decreases with the increasing of moving velocity of the receiver. Figure 3(b) shows that the spatial CF decreases with the increasing of time delay for various times. Figure 3(c) shows that the spatial CF decreases with the increasing of LCX spacing, and it can also be seen that the spatial CF at different locations in the tunnel has little difference. According to Figures 3(a)–3(c), we can see that the theoretical results are consistent with the simulation results, which proves that the theoretical derivation is reasonable and the proposed channel model is reasonable. The amplitude of CF is decreasing when the time delay increases, in Figures 3(a)–3(b). Meanwhile, the longitudinal power attenuation constant of LCX α is 4.1 dB/100 m and there is uniform radiation along the longitudinal direction of the tunnel, so the difference of amplitude of CF is small in the three regions, in Figure 3(c). Figure 4 shows the correlation function (CF) comparison between dipole MIMO (transmitters are changed to two dipoles and located at the same places of LCXs, and only the Zcoordinate of dipoles of transmitters is changed to 50 m) and LCXMIMO for different velocities with different time delays, and the simulation results indicate that the CFs of LCXMIMO are less than the CFs of dipoles MIMO at the same parameters configuration of nonstationary tunnel scenarios. Based on the results of Figure 3(c), we can see that there is very little difference in the amplitude of CF of the three regions. So, we only simulated in Region 2 for Figure 4. The time delay of the first minimum value of CFs for LCXMIMO is 1/5 of the time delay of the minimum value of CFs for dipoles MIMO when the moving velocity is 360 km/h in Figure 4. In Figure 5, we study the influence of different K factors on CFs with different time delays, and the results show that the CF increases with the increase of the K factor. The multipath component decreases with the increase of the K factor.
(a)
(b)
(c)
5. Conclusion
In this paper, a 3D nonstationary MIMO channel model in rectangular tunnel environment is proposed for Metro and highspeed railway scenarios, the transmitter equipped with two LCXs and the receiver equipped with two dipole antennas, and the receiving antennas array moves along the longitudinal direction of the tunnel. The proposed model and the derived spatial CF expressions are verified by simulation. Comparison of the simulation results and theoretical results indicates that the moving parameters (moving velocity, moving time, and time delay) of the receiver have considerable influences on the LCXMIMO channel characteristics. The simulation results indicate that the CFs of LCXMIMO are less than the CFs of dipoles MIMO in the same parameters configuration of nonstationary tunnel scenarios. In the same parametric configuration of nonstationary tunnel scenarios, the time delay of the first minimum value of CFs for LCXMIMO is 1/5 of the time delay of the minimum value of CFs for dipole antennas MIMO when the train moving velocity is 360 km/h. It is shown that, for MIMO system, the performance of using LCXs is better than using dipole antennas. The simulation results show that the amplitude of CF of the three regions in the tunnel has very little difference.
Data Availability
No data were used to support this study because there are theoretical and simulation results in this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Science Foundation of China (Grant nos. 61571282 and 61871261), National Key R&D Program of China (No. 2016YFB1200602), and Fund of Shanghai Cooperative Innovation Center for Maglev and Rail Transit. The authors particularly appreciate Mr. Saleem Asad for his help during writing of this paper.
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Copyright © 2020 Kai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.