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

Authors: Kai Zhang; Fangqi Zhang; Guoxin Zheng
Journal: International Journal of Antennas and Propagation
 (2020)

---

## Abstract

The rapid development of high-speed train and Metro communications has provided new challenges for the application of MIMO technologies. Therefore, we propose a three-dimensional (3D) multiple-input multiple-output (MIMO) channel model using leaky coaxial cable (LCX) in a rectangular tunnel. The channel model is based on geometry-based single-bounce (GBSB) channel model and the electric field distribution of LCX in the tunnel environment. The theoretical expressions of channel impulse response (CIR) and space-time 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 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.


---

## Body

## 1. Introduction

Recently, multiple-input multiple-output (MIMO) technology has been widely used in many scenarios. The 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 [1–12]. The channel modeling can be based on two-dimensional (2D) [1–5] and three-dimensional (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 geometry-based single-bounce (GBSB) [1–5] and geometry-based 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 air-to-ground communications [7, 8], the vehicle-to-vehicle 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 [6–12]. 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 geometry-based 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-to-vehicle 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. The 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. There 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. The 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. 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 LCX-MIMO 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 LCX-MIMO and dipoles MIMO are compared with different moving velocities for different time delays.The remainder of this paper is organized as follows. In Section2, 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.


## 2. 3D Nonstationary LCX-MIMO Channel Model

Figure1(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 s1 and s2. 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.Figure 1
(a) LCX structure; (b) LCXs and dipoles installation; (c) signal propagation for LoS and NLoS path of LCX-MIMO transmission scheme.
(a)(b)(c)Figure 2
Projection view of the LCX-MIMO in a rectangular tunnel: (a) the front view; (b) the side view; (c) the top view.
(a)(b)(c)Table 1
Parameters name and symbol of LCX-MIMO channel model.
NameSymbolTunnel size (length × width × height)L × W × HLength of LCXLLCX spacingDLCX relative permittivityεrReceiving antennas spacinglCoordinate of dipole1 (Rx1)(D, d0, l0)Period of LCX slotPNumber of LCX slotsN = floor (L/P)−1Coordinate ofm-th of scatterer Sm(xm, dm, lm)In this paper, we only consider LCX with vertical periodic slots, the length of LCX isL, 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 Ez can be expressed as [17, 18](1)Ez=E0e−jk0rrsinθ,where E0 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, k0 = 2πf/c denotes the propagation constant in free space, f is the center frequency, and c is velocity of light.The longitudinal amplitude attenuation ofi-th slot of LCX αi, phase variation i-th slot of LCX βi, and the propagation constant kr in LCX are as follows:(2a)αi=α0i,(2b)βi=β0i,(2c)kr=k0εr,where α0 is amplitude attenuation of LCX per slot, β0 = krP is phase variation of LCX per slot, and ԑr 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 isM. 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 p-th (p = Tx1, Tx2) LCX to the l-th (l = Rx1, Rx2) receiving antenna can be expressed as [16](3a)Ez,lpLoS=∑i=1NEie−jk0ri,lpri,lpsinθi,lp,(3b)Ez,lpNLoS=∑i=1N∑m=1MEie−jk0rim,p+rim,lrim,p+rim,lsinθim,p,where Ei = E0αie-jβi denotes the electric field strength of the i-th slot surface of LCX. E0 denotes the original electric field strength of LCX (we assume that E0 equals 1 V/m). ri,lp is the distance from the i-th slot of p-th LCX to the l-th receiving antenna. rim,p and rim,l are the distances from the i-th slot of p-th LCX to m-th (m = 1,…,M) effective diffuse point Sm and from m-th effective diffuse point Sm to l-th receiving antenna, respectively. θi,lp and θim,p are the angles between the direction of ri,lp and the axial direction of LCX and between the direction of rim,p to the axial direction of LCX, respectively.The LCX-MIMO channel model can be described by 2 × 2 complex channel impulse response (CIR) matrix, i.e.,H(t) = [hlp(t)]2 × 2, where Ωlp = E[|hlp(t)|2]. The CIR hlp(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](4a)hlpLoSt=KlpΩlpKlp+1limN⟶∞1N∑i=1NGi,lpLoSte−jk0ri,lpt+βi,(4b)hlpNLoSt=ΩlpKlp+1limN⟶∞M⟶∞1NM∑i=1N∑m=1MGim,lpNLoStejφim−k0rim,p+rim,lt−βi,where φim 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; Klp denotes Rician factor, i.e., Klp=hlpLoSt2/EhlpNLoSt2; Gi,lpLoSt and Gim,lpNLoSt can be calculated by(5)Gi,lpLoSt=αisinθi,lpt/ri,lptN−1∑i=1NEαisinθi,lpt/ri,lpt21/2,Gim,lpNLoSt=αisinθim,p/rim,p+rim,ltNM−1∑i=1N∑m=1MEαisinθim,p/rim,p+rim,lt21/2.The particular representation ofhlpLoSt and hlpNLoSt in terms of Klp, together with the assumption of N−1∑i=1NEGi,lpLoS2⟶1 and NM−1∑i=1N∑m=1MEGim,lpNLoS2⟶1 as N ⟶ ∞ and M ⟶ ∞, guarantees that Ωlp = E[|hlp(t)|2].


## 3. 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. Therefore, we mainly analyze the spatial CF of LCX at the transmitter, and the spatial CF can be expressed as(6)ρRx1Tx1,Rx1Tx2t,τ=EhRx1Tx1thRx1Tx2∗t−τEhRx1Tx1t2EhRx1Tx2t−τ2,=EhRx1Tx1LoSthRx1Tx2LoSt−τ∗+EhRx1Tx1NLoSthRx1Tx2NLoSt−τ∗ΩRx1Tx1ΩRx1Tx2,=ρRx1Tx1,Rx1Tx2LoSt,τ+ρRx1Tx1,Rx1Tx2NLoSt,τ.According to formulas (4a) and (4b), the spatial CFs of LoS and NLoS can be expressed as(7a)ρRx1Tx1,Rx1Tx2LoSt,τ=KRx1Tx1KRx1Tx2KRx1Tx1+1KRx1Tx2+1limN⟶∞∑i=1Nαi2sinθi,Rx1Tx1tsinθi,Rx1Tx2t−τ/ri,Rx1Tx1tri,Rx1Tx2t−τ∑i=1Nαisinθi,Rx1Tx1t/ri,Rx1Tx1t2∑i=1Nαisinθi,Rx1Tx2t−τ/ri,Rx1Tx2t−τ2e−jk0ri,Rx1Tx1t−ri,Rx1Tx2t−τ,(7b)ρRx1Tx1,Rx1Tx2NLoSt,τ=1KRx1Tx1+1KRx1Tx2+1limN⟶∞M⟶∞∑i=1N∑m=1Mαi2sinθim,Tx1sinθim,Tx2/rim,Tx1+rim,Rx1trim,Tx2+rim,Rx1t−τABe−jk0rim,Tx1−rim,Tx2,where A=∑i=1N∑m=1Nαisinθim,Tx1/rim,Tx1+rim,Rx1t2; B=∑i=1N∑m=1Mαisinθim,Tx2/rim,Tx2+rim,Rx1t−τ2; sinθi,Rx1Tx1t=d0+D2/ri,Rx1Tx1t; sinθi,Rx1Tx2t−τ=d0+D2/ri,Rx1Tx2t−τ; ri,Rx1Tx1t=d02+l0−ip+vt2+D21/2; ri,Rx1Tx2t=d0−d2+l0−ip+vt2+D21/2; sinθim,Tx1=xm2+dm2/rim,Tx1; sinθim,Tx2=xm2+dm2/rim,Tx2; rim,Tx1=lm−ip2+dm2+xm2; rim,Tx2=lm−ip2+dm−d2+xm2; rim,Rx1t=l0+vt−lm2+d0−dm2+W/2−xm2.


## 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 × 104. 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. K11 and K12 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 d0 = 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 d0 = 1.1 m. The other parameters of simulation are presented in Table 2.Table 2
Simulations parameters.
NameValueTunnel size (L × W× H)100 m × 5 m × 3 mLength of LCX (Tx)100 mLCX relative permittivity (εr)1.25Receiving antennas spacing (d)0.5 mReceiver (Rx)Dipole antenna (horizontal polarization)x-coordinate of dipole1 (D)2.44 my-coordinate of dipole1 (d0)1.1 mPeriod of LCX slot (P)0.25 mNumber of LCX slots (N)399Longitudinal attenuation constant of LCX (α)4.1 dB/100 mFrequency (f)1.8 GHzLight velocity (c)3 × 108 m/sWave length (λ)c/fRegion 1[0 m, 2 m]Region 2[49 m, 51 m]Region 3[98 m, 100 m]Figure3(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 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. 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 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.Figure 3
Amplitude correlation function (CF: LoS + NLoS): (a) different moving speed and time delay (t = 0 s, d = 0.7λ); (b) different time and time delay (v = 360 km/h, d = 0.7λ); (c) different region Rx1 in tunnel and LCX spacing (τ = 0.5 ms, t = 0.06 s, v = 360 km/h).
(a)(b)(c)Figure 4
Amplitude correlation function (CF: LoS + NLoS) comparison between dipoles MIMO and LCX-MIMO for different velocity with different time delays (t = 0 s, d = 0.7λ, Region 2).Figure 5
Amplitude correlation function (CF: LoS + NLoS) for different K factors with different time delay (t = 0 s, d = 0.7λ, v = 90 km/h).


## 5. 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. 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 LCX-MIMO channel characteristics. The 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. The simulation results show that the amplitude of CF of the three regions in the tunnel has very little difference.



---

*Source: 3984148-2020-08-20.xml*