Space-Time Correlation for Three-Dimensional MIMO Channel Model Using Leaky Coaxial Cables in Rectangular Tunnel

+e rapid development of high-speed train and Metro communications has provided new challenges for the application of MIMO technologies. +erefore, we propose a three-dimensional (3D) multiple-input multiple-output (MIMO) channel model using leaky coaxial cable (LCX) in a rectangular tunnel. +e channel model is based on geometry-based single-bounce (GBSB) channel model and the electric field distribution of LCX in the tunnel environment. +e theoretical expressions of channel impulse response (CIR) and space-time correlation function (CF) are also derived and analyzed.+eCFs for different model parameters (moving velocity and moving time) and different regions of the tunnel are simulated byMonte Carlomethod to verify the theoretical derivation at 1.8GHz. 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 MIMOwhen 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.


Introduction
Recently, multiple-input multiple-output (MIMO) technology has been widely used in many scenarios. e nonstationary scenarios of MIMO technologies for high-speed 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][2][3][4][5][6][7][8][9][10][11][12]. e channel modeling can be based on two-dimensional (2D) [1][2][3][4][5] and three-dimensional (3D) [6][7][8][9][10][11][12]. According to the 3GPP (3GPP TR 25.996) spatial channel model (SCM) [1][2][3][4][5][6] and WINNER II channel model [7][8][9][10][11][12], the geometry-based single-bounce (GBSB) [1][2][3][4][5] and geometry-based stochastic model (GBSM) [6][7][8][9][10][11][12] are used to describe MIMO propagation channels. e nonstationary scenarios investigated include the high altitude unmanned aerial vehicles (UAV) MIMO air-to-ground communications [7,8], the vehicle-tovehicle communications in tunnel [9,10] and other scattering environments [11,12], the base-station-to-vehicle 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 cross-correlation and autocorrelation are analyzed [6][7][8][9][10][11][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 two-element antennas array, and they also derived a space-time correlation for MIMO systems in mobile fading channels. It could be called "one-ring" model in physical MIMO channel models. In [4], the authors analyzed the propagation characteristics of mobile-to-mobile fading channels for MIMO systems; they derived a great channel model in mobile-to-mobile fading channels and scatterers around both transmitting antennas and receiving antennas. It could be called "two-ring" model in physical MIMO channel models. In [9], the authors studied a nonstationary geometrybased channel model for MIMO vehicle-to-vehicle 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 geometry-based 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 vehicle-tovehicle 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 linear-cell 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 LCX-MIMO systems in linear-cell environment at 2.4 GHz and 5 GHz, and the capacity results indicate that the LCX-MIMO 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. e channel models of these researches are stationary [15]. In [15], the authors research the theoretical channel model of 2-by-2 MIMO system using single LCX in free space, and the results show that a 2-by-2 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. ere is little research on nonstationary LCX-MIMO communication in tunnel scenarios. With the rapid development of Metro and high-speed railway, there is an urgent need for the study of LCX-MIMO in nonstationary tunnel scenarios.
For high-speed railway scenario and Metro scenario, this paper aims to fill these research gaps. We propose a 3D nonstationary LCX-MIMO channel model in rectangular tunnel environments. e main contributions and novelties of this paper are summarized as follows: (1) this paper develops a 3D nonstationary LCX-MIMO channel model, and the channel impulse response (CIR) and spatial correlation function (CF) are theoretically derived. e 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 LCX-MIMO system, in terms of the moving velocity and moving time. e influence of the different positions where the receiver is located in the tunnel on the propagation characteristics is also analyzed; (3) the CFs of LCX-MIMO and dipoles MIMO are compared with different moving velocities for different time delays. e remainder of this paper is organized as follows. In Section 2, we propose a 3D nonstationary LCX-MIMO channel model. In Section 3, the theoretical expressions of space-time CF for our proposed GBSB 3D nonstationary in tunnel scenarios are derived. In Section 4, the space-time CFs of the proposed model are simulated and validated. Finally, our concluding remarks are given in Section 5.

3D Nonstationary LCX-MIMO 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 2-by-2 nonstationary LCX-MIMO 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 Z-axis of the coordinate system; the cross section of tunnel presents the XOY-plane, the vertical direction of the XOY-plane presents the Y-axis of the coordinate system, and the horizontal direction of the XOY plane presents the X-axis of the coordinate system. In Figure 2, for the MIMO system, the parameter names and symbols of LCX-MIMO channel model are as shown in Table 1.
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. e longitudinal amplitude attenuation of i-th slot of LCX α i , phase variation i-th 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 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 2 International Journal of Antennas and Propagation Jacket Slot P Outer conductor From the inside out  International Journal of Antennas and Propagation 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. erefore, 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. e LoS and NLoS components of electric field distribution from the p-th (p � Tx1, Tx2) LCX to the l-th (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 i-th slot of p-th LCX to the l-th receiving antenna. r im,p and r im,l are the distances from the i-th slot of p-th LCX to m-th (m � 1,. . .,M) effective diffuse point S m and from m-th effective diffuse point S m to l-th 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. e LCX-MIMO channel model can be described by 2 × 2 complex channel impulse response (CIR) matrix, i.e., e CIR h lp (t) from p-th transmitting antenna (Tx) to l-th receiving antenna (Rx) is a vector superposition of the two components CIR of LoS and NLoS and can be expressed as [16] h LoS h NLoS  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 e particular representation of h LoS lp (t) and h NLoS lp (t) in terms of K lp , together with the assumption of

Spatial Correlation Function
In order to study the performance of the proposed LCX-MIMO 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 zero-mean 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. erefore, 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 as International Journal of Antennas and Propagation

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). e 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. e 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. e 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. e 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 Z-coordinate of dipoles of transmitters is changed to 50 m) and LCX-MIMO for different velocities with different time delays, and the simulation results indicate that the CFs of LCX-MIMO 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. e time delay of the first minimum value of CFs for LCX-MIMO 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. e multipath component decreases with the increase of the K factor.

Conclusion
In this paper, a 3D nonstationary MIMO channel model in rectangular tunnel environment is proposed for Metro and high-speed 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. e 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 LCX-MIMO channel characteristics. e simulation results indicate that the CFs of LCX-MIMO 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 LCX-MIMO 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. e 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
e authors declare that they have no conflicts of interest.