# Analysis of Nonstationary Characteristics for High-Speed Railway Scenarios

Authors: Tao Zhou; Cheng Tao; Kai Liu
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

This paper presents the analysis of nonstationary characteristics for high-speed railway (HSR) scenarios, according to passive long-term evolution- (LTE-) based channel measurements. The measurement data collected in three typical scenarios, rural, station, and suburban, are processed to obtain the channel impulse responses (CIRs). Based on the CIRs, the nonstationarity of the HSR channel is studied focusing on the stationarity interval, and a four-state Markov chain model is generated to describe the birth-death process of multipath components. The presented results will be useful in dynamic channel modeling for future HSR mobile communication systems.


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## Body

## 1. Introduction

With the rapid development of high-speed railways (HSRs), there appears a growth in demand for new railway communication services, for example, real-time monitoring, train multimedia dispatching, railway emergency communications, railway Internet of Things (IoT), and broadband wireless access for train passengers [1]. To satisfy such ever-increasing requirements, broadband wireless communication systems for HSR have recently attracted much attention in the world. Since 2014, International Union of Railways (UIC) has considered replacing the current global system for mobile communications for railway (GSM-R) with the long-term evolution for railway (LTE-R) [2]. In China, fourth-generation (4G) networks have been deployed along most of HSRs, for a total of 15,000 km by 2014, and as HSRs continue to grow, the dedicated 4G networks that grow with them will exceed 30,000 km in 2020. For future fifth-generation (5G) mobile communication system, it is reported that one of its aims is to provide high-data-rate access under high mobility scenarios [3].Since the radio channel determines the performance of broadband wireless mobile communication systems, detailed knowledge and accurate characterization of its parameters in diverse scenarios are vital. The propagation characteristics are in disparity under various HSR environments. In this case, the characterization of HSR channels should consider the influence of scenarios.So far, a wide variety of the studies have concentrated on the long-term fading behavior, involving path loss and shadowing, in multiple HSR scenarios [4–8]. There are a few research works on the short-term fading behavior, based on wideband channel measurements conducted on HSR. By contrast, the nonstationary behavior, which has been widely studied for vehicle-to-vehicle (V2V) channels [9, 10], is rarely investigated in HSR environments. To fill this research gap, we present the analysis of the nonstationary characteristics for multiple HSR scenarios. Passive channel measurements are conducted for three typical scenarios, rural, station, and suburban, in an HSR LTE network. Considering the interval of stationarity and the dynamic evolution of multipath components (MPCs), the nonstationary behavior is analyzed and compared in different scenarios.The remainder of this paper is outlined as follows. Section2 reviews the related work focusing on short-term fading and nonstationary behaviors. In Section 3, our passive channel measurements based on LTE are introduced. Then, the nonstationary characteristics are analyzed in Section 4, respectively. Finally, conclusions are drawn in Section 5.


## 2. Related Work

Unique HSR scenarios, such as viaduct, cutting, tunnel, station, hilly terrain, rural, and suburban, have a significant impact on propagation characteristics. In our previous work, fading severity and time-frequency-space dispersion of HSR channels in the viaduct and cutting scenarios were deeply characterized based on measurements using Propsound [11–15]. In [15], spatial characteristics, involving angle of arrival (AOA), root-mean-square (RMS) angle spread (AS), and spatial correlation (SC), were analyzed according to a so-called moving virtual antenna array (VAA) scheme. Authors in [16, 17] proposed a statistic model for Ricean K-factor, which investigated the impact of the viaduct height and the cutting width. The WINNER II model [18] and COST 2100 TD [19] provided some measurement results of short-term fading behavior for rural scenarios. There were also a few results of K-factor, RMS delay spread (DS), and Doppler power spectral density (DPSD) measured in the hilly terrain and tunnel scenarios [20–22]. Authors in [23, 24] presented detailed analysis of fading severity and time dispersion in open-type and semiclosed station scenarios. For the nonstationary behavior, stationarity interval (SI) in the viaduct scenario was investigated based on GSM-R measurements, which showed that conventional channel models offered SI much larger than the actual measured ones [25]. We also tried to adopt a RUN test method to obtain the SI for the open-type station scenario in [24]. Additionally, a four-state Markov chain was used to model the birth-death (B-D) process of MPCs in the viaduct scenario [26].Table1 summarizes the existing measurement campaigns about short-term fading and nonstationary behaviors in different HSR environments. It can be found that there are a few results of time-frequency-space characteristics in some HSR scenarios. However, the characterization of the nonstationary behavior is largely neglected in most scenarios. Therefore, this paper aims to investigate the nonstationary behavior in the rural, station, and suburban scenarios.Table 1
Summary of investigation on short-term fading and nonstationary behaviors in HSR environments.
Scenario Short-term fading behavior Nonstationary behavior Fading severity Time dispersion Frequency dispersion Space dispersion Stationarity interval Birth-death process Viaduct [11, 12, 16] [11] [11, 12] [15] [25] [26]  Cutting [13, 14, 17] [13, 14] [13, 14] [15] - -  Rural [18] [18, 19] [19] [18, 19] - -  Hilly terrain [20] [21] [21] - - -  Tunnel [22] [22] [22] - - -  Station [23, 24] [24] -  [24] -  Suburban/urban - - - - - -


## 3. Channel Measurements

### 3.1. Measurement Scenarios

An LTE network deployed on Beijing to Tianjin (BT) HSR in China was chosen in our measurements [27]. It is a hybrid network composed of a dedicated network and a common network. The architecture of the dedicated network is completely different from the common network, which adopts building baseband unit (BBU) plus remote radio unit (RRU) to achieve the special narrow-strip-shaped coverage instead of the cellular coverage. In this architecture, one physical site (PS) deploys two RRUs, which transmit signals by directional antennas in opposite directions along the railway track. The RRUs are connected together via optical fiber and then to a BBU that is in charge of radio frequency (RF) signal processing.The LTE sounder is placed on a high-speed train (HST) to collect the channel data, which experiences multiple scenarios along the BT railway line, such as rural, station, and suburban. In our measurement, the rural and station scenarios are within the coverage of dedicated network, while the suburban scenario is covered by the common network. The detailed measurement parameters for different scenarios in the network are listed in Table2. The carrier frequency is 1.89 GHz for the rural and station scenarios and 2.605 GHz for the suburban scenario. When the HST moves into the suburban area, the speed is decreased from 285 km/h to 185 km/h. The specification of the directional eNB antenna such as gain and beamwidth has slight difference in different frequencies. At Rx side, the LTE sounder is connected to a train-mounted omnidirectional antenna. The average spacing between neighboring PSs is around 1.2 km. The height and distance difference between the PS and the rail track are about 20 m and 30 m, respectively.Table 2
Measurement parameters in the BT HSR LTE network.
Scenario Rural Station Suburban eNB side  Frequency 1.89 GHz 2.605 GHz  Bandwidth 18 MHz 18 MHz  CRS power 12.2 dBm 12.2 dBm  Antenna type Directional Directional  Antenna gain 17.4 dBi 18.6 dBi  Horizontal beamwidth 67 deg 60 deg  Vertical beamwidth 6.6 deg 4.9 deg  Electric tilted angle 3 deg 3 deg  Sounder side  Antenna type Omnidirectional Omnidirectional  Antenna gain 8.5 dBi 8.5 dBi  Speed of HST 285 km/h 185 km/h  Network side  Distance between PSs 1.2 km  Height of PS 30 m  Height of rail track 10 m  Distance between PS and rail track 30 mThe measured scenarios are shown in Figure1. As for the rural scenario, the transmit antenna is much higher than the surroundings, which are light forests and a few buildings with an average height of less than 10 m. The link between the transmitter (Tx) and Rx generally has a strong line-of-sight (LoS) component. However, after a certain distance, the impact of the sparse scatterers will be noticed at the Rx represented by non-LoS (NLoS) components. With regard to the station scenario, in the measurement, the HST runs through the station without stopping. The measured station can be regarded as an open-type station with two awnings that only cover the platform supporting a clear free space over the rail. However, the awnings can still produce lots of NLoS components to complicate the fading behavior. The length of the station is 440 m, the width of the awning is 14 m, and the width of the gap between the two awnings is 9 m. Suburban is a transition zone between the rural and urban. The NLoS components in the suburban environment will be much richer than those in the rural environment. The density of the buildings in the suburban scenario is similar to that in the urban scenario, but the height of the buildings is lower. Since the measured suburban is close to the urban area, some remote high buildings could affect the results.Figure 1
Measurement scenarios. (a) Rural. (b) Station. (c) Suburban.
(a) (b) (c)


### 3.2. Data Processing

Baseband data (BB) collected by the LTE sounder in the multilink regions are used for offline processing. The procedure of data processing is shown in Figure2. Firstly, cell search is implemented to determine the cell identity and obtain synchronized frames for extracting received CRSs and generating local CRSs. Then, frequency-domain correlation is used to estimate channel frequency responses which can be subsequently transformed to the raw CIRs by inverse fast Fourier transform (IFFT) operation [28].Figure 2
The procedure of data processing.


## 3.1. Measurement Scenarios

An LTE network deployed on Beijing to Tianjin (BT) HSR in China was chosen in our measurements [27]. It is a hybrid network composed of a dedicated network and a common network. The architecture of the dedicated network is completely different from the common network, which adopts building baseband unit (BBU) plus remote radio unit (RRU) to achieve the special narrow-strip-shaped coverage instead of the cellular coverage. In this architecture, one physical site (PS) deploys two RRUs, which transmit signals by directional antennas in opposite directions along the railway track. The RRUs are connected together via optical fiber and then to a BBU that is in charge of radio frequency (RF) signal processing.The LTE sounder is placed on a high-speed train (HST) to collect the channel data, which experiences multiple scenarios along the BT railway line, such as rural, station, and suburban. In our measurement, the rural and station scenarios are within the coverage of dedicated network, while the suburban scenario is covered by the common network. The detailed measurement parameters for different scenarios in the network are listed in Table2. The carrier frequency is 1.89 GHz for the rural and station scenarios and 2.605 GHz for the suburban scenario. When the HST moves into the suburban area, the speed is decreased from 285 km/h to 185 km/h. The specification of the directional eNB antenna such as gain and beamwidth has slight difference in different frequencies. At Rx side, the LTE sounder is connected to a train-mounted omnidirectional antenna. The average spacing between neighboring PSs is around 1.2 km. The height and distance difference between the PS and the rail track are about 20 m and 30 m, respectively.Table 2
Measurement parameters in the BT HSR LTE network.
Scenario Rural Station Suburban eNB side  Frequency 1.89 GHz 2.605 GHz  Bandwidth 18 MHz 18 MHz  CRS power 12.2 dBm 12.2 dBm  Antenna type Directional Directional  Antenna gain 17.4 dBi 18.6 dBi  Horizontal beamwidth 67 deg 60 deg  Vertical beamwidth 6.6 deg 4.9 deg  Electric tilted angle 3 deg 3 deg  Sounder side  Antenna type Omnidirectional Omnidirectional  Antenna gain 8.5 dBi 8.5 dBi  Speed of HST 285 km/h 185 km/h  Network side  Distance between PSs 1.2 km  Height of PS 30 m  Height of rail track 10 m  Distance between PS and rail track 30 mThe measured scenarios are shown in Figure1. As for the rural scenario, the transmit antenna is much higher than the surroundings, which are light forests and a few buildings with an average height of less than 10 m. The link between the transmitter (Tx) and Rx generally has a strong line-of-sight (LoS) component. However, after a certain distance, the impact of the sparse scatterers will be noticed at the Rx represented by non-LoS (NLoS) components. With regard to the station scenario, in the measurement, the HST runs through the station without stopping. The measured station can be regarded as an open-type station with two awnings that only cover the platform supporting a clear free space over the rail. However, the awnings can still produce lots of NLoS components to complicate the fading behavior. The length of the station is 440 m, the width of the awning is 14 m, and the width of the gap between the two awnings is 9 m. Suburban is a transition zone between the rural and urban. The NLoS components in the suburban environment will be much richer than those in the rural environment. The density of the buildings in the suburban scenario is similar to that in the urban scenario, but the height of the buildings is lower. Since the measured suburban is close to the urban area, some remote high buildings could affect the results.Figure 1
Measurement scenarios. (a) Rural. (b) Station. (c) Suburban.
(a) (b) (c)


## 3.2. Data Processing

Baseband data (BB) collected by the LTE sounder in the multilink regions are used for offline processing. The procedure of data processing is shown in Figure2. Firstly, cell search is implemented to determine the cell identity and obtain synchronized frames for extracting received CRSs and generating local CRSs. Then, frequency-domain correlation is used to estimate channel frequency responses which can be subsequently transformed to the raw CIRs by inverse fast Fourier transform (IFFT) operation [28].Figure 2
The procedure of data processing.


## 4. Results and Analysis

### 4.1. Stationarity Interval

High mobility leads to the violation of wide sense stationary (WSS) condition for wireless channels under HSR scenarios. The stationarity interval (SI) is defined as the maximum time or distance duration, over which the channel satisfies the WSS condition. It is summarized by [29] that there are several metrics that can be used for measuring the SI, involving local region of stationarity (LRS), correlation matrix distance (CMD), and spectral divergence (SD). Besides, some statistical tests for the WSS of a random process can also be applied to the determination of SI, such as RUN test and reverse arrangement test. The LRS method has been used to estimate the time interval of HSR channels in [25]. The RUN test was applied to the RMS DS data to identify the stationary distance in the station scenario [24]. In this paper, we choose the classical LRS approach to characterize the SI in the measured scenarios.The aim of the LRS method is to find the maximum interval within which the correlation coefficient between two consecutive PDPs exceeds a predefined thresholdcth. The correlation coefficient between PDPs is defined as(1)cxk,Δx=Pxk¯Pxk+Δx¯max⁡Pxk¯2,Pxk+Δx¯2,where (2)Pxk¯=1N∑m=kk+N-1hxm2,where N is the window size and hxm are the samples of channel impulse response.Then, the SI can be estimated as(3)Ik=kmax-kminΔxwhere (4)kmax=argmaxk+1≤m≤L-Ncxk,mΔx<cth(5)kmin=argmin1≤m≤k-1cxk,mΔx<cthwhere L is the length of used data. Here, we consider three typical correlation threshold values: cth=0.7,  cth=0.8, and cth=0.9.Figure3 compares the derived SI results for different cth and scenarios in terms of complementary CDF (CCDF). With the increase of cth, the SI is gradually decreasing. For cth=0.9, only slight difference can be observed in different environments, while there is an obvious deviation for cth=0.7 and cth=0.8. Focusing on the cases of cth=0.7 and cth=0.8, we find that the SI values in the rural scenario are larger than those in the station or suburban scenarios. This is understandable from the completely different scattering environment. Since LoS is dominant in the rural scenario, the MPCs have smaller dynamic changes over time, and thus the stationary distance is longer. When it comes to the station and suburban scenarios, due to the rich reflection and scattering components from the awnings or buildings, the nonstationarity is more serious, and thus the SI decreases. This nonstationarity could be originated from a special physical phenomena, for example, “appearance and disappearance” or “birth and death” of MPCs [26], which will be further investigated in the following subsection.Figure 3
CCDFs of SI in rural, station, and suburban scenarios.The detailed statistical SI results including mean value and 60% and 80% of CCDF for different scenarios are listed in Table3. For cth=0.8, the mean value of SI is 6.46 m in the rural scenario, while those are 3.84 m and 3.74 m in the station and suburban scenarios. In 60% and 80% of cases, the channel could be stationary over a distance of 2.33-4.29 m and 1.25-2.64 m for the measured scenarios, respectively. These values are higher than the results of 1.8 m for 60% and 0.81 m for 80% reported in [25]. From [25], the calculated stationary interval for standard channel models is equal to 3.4 m for 60%, which is shorter than the one of 4.29 m for the rural scenario but is longer than the ones of 2.55 m and 2.33 m for the station and suburban scenarios. It is also observed that the value of SI in 80% of the cases in the measured station scenario is smaller than that of around 4 m in the station scenario reported in [24]. This variance could be due to the use of different calculation methods.Table 3
Analysis results of the SI in different scenarios.
Scenario Method Statistics SI (m) c t h = 0.7 c t h = 0.8 c t h = 0.9 Rural LRS Mean value 14.3 6.46 2.84 60% of CCDF 8.22 4.29 1.65 80% of CCDF 4.74 2.64 0.99  Station LRS Mean value 7.27 3.84 1.87 60% of CCDF 5.02 2.55 1.17 80% of CCDF 3.07 1.25 0.37  Suburban LRS Mean value 7.62 3.74 1.37 60% of CCDF 5.43 2.33 0.86 80% of CCDF 3.54 1.25 0.35  Viaduct [26] LRS 60% of CCDF - 1.8 - 80% of CCDF - 0.81 -  Standard models [26] LRS 60% of CCDF - 3.4 -  Station [25] RUN test 80% of CCDF 4


### 4.2. Birth-Death Process

The nonstationarity of the channel is basically due to the dynamic evolution of MPCs when the Rx is in motion, for example, appearance to disappearance or B-D. To describe this B-D process, a four-state Markov chain model (MCM) is used, where each state is defined as follows [30]:(i)
S0: no “births” or “deaths”(ii)
S1: “births” only(iii)
S2: “deaths” only(iv)
S3: both “births” and “deaths”Note that the state in the MCM only considers the variation of MPCs from the current moment to the next moment. With the motion of the Rx, the states can be transformed to each other. Figure4 illustrates the state transition diagram of the four-state MCM [30]. The probabilistic switching process between states in the MCM is controlled by the state transition probability matrix P given by(6)P=pij=p00p01p02p03p10p11p12p13p20p21p22p23p30p31p32p33where i and j represent the state index, while pij is the transition probability from state Si to state Sj. Note that pij must satisfy the following requirement:(7)0≤pij≤1,i,j=0,1,…,N-1(8)∑j=0N-1pij=1,i=0,1,…,N-1where N is the number of states; that is, N=4 in our case.Figure 4
State transition diagram of the four-state MCM.The steady-state probability can be expressed as(9)PS=PS0PS1PS2PS3which satisfies PS0+PS1+PS2+PS3=1. Each element in PS indicates the overall state occupancy probability.The obtained results of the state transition probability matrix in different scenarios are listed in Table4. Here, the state transition matrix is derived from each CIR. Since the sample rate of CIR is 2000 Hz and the velocity of train is 79 m/s, the reference value for state transition matrix is 0.04 m. It is observed that in the suburban scenario the next state is more likely to transit into the state S3 no matter what the current state is S0, S1, or S2. This means some new MPCs are born and older MPCs die most of the time because of the rich reflection and scattering components in the suburban scenario. For other scenarios, in the case of S0, the next state could be any one of the four states; in the case of S1, S2 or S3 have the maximum probability to be the next state; in the case of S2, the next state is more likely to transit into S1; in the case of S3, the next state is still likely to be S3. The results show the nonsymmetric transition matrix, which means that the transition probability from state A to state B has no relationship with that from state B to state A.Table 4
Analysis results of the B-D process in different scenarios.
Scenario State transition probability matrix Steady-state probability Rural 0.3361 0.4025 0.1203 0.1411 0.0497 0.0315 0.5348 0.3841 0.1658 0.7148 0.0282 0.0912 0.0332 0.0600 0.2505 0.6563 0.10170.25330.25300.3920  Station 0.2791 0.2093 0.1628 0.3488 0.0224 0.0096 0.4936 0.4744 0.0354 0.7235 0.0129 0.2283 0.0081 0.0471 0.0905 0.8544 0.01890.13740.13660.7071  Suburban 0.1870 0.2439 0.1463 0.4228 0.0557 0.0557 0.2256 0.6630 0.0780 0.4150 0.0613 0.4457 0.0151 0.0464 0.0690 0.8695 0.02870.08370.08370.8039  Viaduct [17] 0.2917 0.5208 0.1667 0.0208 0.0200 0.1000 0.5200 0.3600 0.2577 0.4330 0.0928 0.2165 0.0673 0.2115 0.2788 0.4423 0.13710.28000.28570.2971According to the state transition probability matrix, the steady-state probability can be derived, as listed in Table4. It is found that S3 is the most likely state in the station and suburban scenarios. For the rural scenario, either S1, S2, or S3 could be the steady state. It is worth noting that S1 and S2 have approximately similar steady-state probability in any scenario. This confirms that the appearance and disappearance of MPCs are equivalent. The above results in the rural scenario are similar to those in the open viaduct scenario reported in [26]. The obtained results can be applied for ON/OFF tapped delay line models that use the Markov chain to model the ON/OFF process of MPCs [31].


## 4.1. Stationarity Interval

High mobility leads to the violation of wide sense stationary (WSS) condition for wireless channels under HSR scenarios. The stationarity interval (SI) is defined as the maximum time or distance duration, over which the channel satisfies the WSS condition. It is summarized by [29] that there are several metrics that can be used for measuring the SI, involving local region of stationarity (LRS), correlation matrix distance (CMD), and spectral divergence (SD). Besides, some statistical tests for the WSS of a random process can also be applied to the determination of SI, such as RUN test and reverse arrangement test. The LRS method has been used to estimate the time interval of HSR channels in [25]. The RUN test was applied to the RMS DS data to identify the stationary distance in the station scenario [24]. In this paper, we choose the classical LRS approach to characterize the SI in the measured scenarios.The aim of the LRS method is to find the maximum interval within which the correlation coefficient between two consecutive PDPs exceeds a predefined thresholdcth. The correlation coefficient between PDPs is defined as(1)cxk,Δx=Pxk¯Pxk+Δx¯max⁡Pxk¯2,Pxk+Δx¯2,where (2)Pxk¯=1N∑m=kk+N-1hxm2,where N is the window size and hxm are the samples of channel impulse response.Then, the SI can be estimated as(3)Ik=kmax-kminΔxwhere (4)kmax=argmaxk+1≤m≤L-Ncxk,mΔx<cth(5)kmin=argmin1≤m≤k-1cxk,mΔx<cthwhere L is the length of used data. Here, we consider three typical correlation threshold values: cth=0.7,  cth=0.8, and cth=0.9.Figure3 compares the derived SI results for different cth and scenarios in terms of complementary CDF (CCDF). With the increase of cth, the SI is gradually decreasing. For cth=0.9, only slight difference can be observed in different environments, while there is an obvious deviation for cth=0.7 and cth=0.8. Focusing on the cases of cth=0.7 and cth=0.8, we find that the SI values in the rural scenario are larger than those in the station or suburban scenarios. This is understandable from the completely different scattering environment. Since LoS is dominant in the rural scenario, the MPCs have smaller dynamic changes over time, and thus the stationary distance is longer. When it comes to the station and suburban scenarios, due to the rich reflection and scattering components from the awnings or buildings, the nonstationarity is more serious, and thus the SI decreases. This nonstationarity could be originated from a special physical phenomena, for example, “appearance and disappearance” or “birth and death” of MPCs [26], which will be further investigated in the following subsection.Figure 3
CCDFs of SI in rural, station, and suburban scenarios.The detailed statistical SI results including mean value and 60% and 80% of CCDF for different scenarios are listed in Table3. For cth=0.8, the mean value of SI is 6.46 m in the rural scenario, while those are 3.84 m and 3.74 m in the station and suburban scenarios. In 60% and 80% of cases, the channel could be stationary over a distance of 2.33-4.29 m and 1.25-2.64 m for the measured scenarios, respectively. These values are higher than the results of 1.8 m for 60% and 0.81 m for 80% reported in [25]. From [25], the calculated stationary interval for standard channel models is equal to 3.4 m for 60%, which is shorter than the one of 4.29 m for the rural scenario but is longer than the ones of 2.55 m and 2.33 m for the station and suburban scenarios. It is also observed that the value of SI in 80% of the cases in the measured station scenario is smaller than that of around 4 m in the station scenario reported in [24]. This variance could be due to the use of different calculation methods.Table 3
Analysis results of the SI in different scenarios.
Scenario Method Statistics SI (m) c t h = 0.7 c t h = 0.8 c t h = 0.9 Rural LRS Mean value 14.3 6.46 2.84 60% of CCDF 8.22 4.29 1.65 80% of CCDF 4.74 2.64 0.99  Station LRS Mean value 7.27 3.84 1.87 60% of CCDF 5.02 2.55 1.17 80% of CCDF 3.07 1.25 0.37  Suburban LRS Mean value 7.62 3.74 1.37 60% of CCDF 5.43 2.33 0.86 80% of CCDF 3.54 1.25 0.35  Viaduct [26] LRS 60% of CCDF - 1.8 - 80% of CCDF - 0.81 -  Standard models [26] LRS 60% of CCDF - 3.4 -  Station [25] RUN test 80% of CCDF 4


## 4.2. Birth-Death Process

The nonstationarity of the channel is basically due to the dynamic evolution of MPCs when the Rx is in motion, for example, appearance to disappearance or B-D. To describe this B-D process, a four-state Markov chain model (MCM) is used, where each state is defined as follows [30]:(i)
S0: no “births” or “deaths”(ii)
S1: “births” only(iii)
S2: “deaths” only(iv)
S3: both “births” and “deaths”Note that the state in the MCM only considers the variation of MPCs from the current moment to the next moment. With the motion of the Rx, the states can be transformed to each other. Figure4 illustrates the state transition diagram of the four-state MCM [30]. The probabilistic switching process between states in the MCM is controlled by the state transition probability matrix P given by(6)P=pij=p00p01p02p03p10p11p12p13p20p21p22p23p30p31p32p33where i and j represent the state index, while pij is the transition probability from state Si to state Sj. Note that pij must satisfy the following requirement:(7)0≤pij≤1,i,j=0,1,…,N-1(8)∑j=0N-1pij=1,i=0,1,…,N-1where N is the number of states; that is, N=4 in our case.Figure 4
State transition diagram of the four-state MCM.The steady-state probability can be expressed as(9)PS=PS0PS1PS2PS3which satisfies PS0+PS1+PS2+PS3=1. Each element in PS indicates the overall state occupancy probability.The obtained results of the state transition probability matrix in different scenarios are listed in Table4. Here, the state transition matrix is derived from each CIR. Since the sample rate of CIR is 2000 Hz and the velocity of train is 79 m/s, the reference value for state transition matrix is 0.04 m. It is observed that in the suburban scenario the next state is more likely to transit into the state S3 no matter what the current state is S0, S1, or S2. This means some new MPCs are born and older MPCs die most of the time because of the rich reflection and scattering components in the suburban scenario. For other scenarios, in the case of S0, the next state could be any one of the four states; in the case of S1, S2 or S3 have the maximum probability to be the next state; in the case of S2, the next state is more likely to transit into S1; in the case of S3, the next state is still likely to be S3. The results show the nonsymmetric transition matrix, which means that the transition probability from state A to state B has no relationship with that from state B to state A.Table 4
Analysis results of the B-D process in different scenarios.
Scenario State transition probability matrix Steady-state probability Rural 0.3361 0.4025 0.1203 0.1411 0.0497 0.0315 0.5348 0.3841 0.1658 0.7148 0.0282 0.0912 0.0332 0.0600 0.2505 0.6563 0.10170.25330.25300.3920  Station 0.2791 0.2093 0.1628 0.3488 0.0224 0.0096 0.4936 0.4744 0.0354 0.7235 0.0129 0.2283 0.0081 0.0471 0.0905 0.8544 0.01890.13740.13660.7071  Suburban 0.1870 0.2439 0.1463 0.4228 0.0557 0.0557 0.2256 0.6630 0.0780 0.4150 0.0613 0.4457 0.0151 0.0464 0.0690 0.8695 0.02870.08370.08370.8039  Viaduct [17] 0.2917 0.5208 0.1667 0.0208 0.0200 0.1000 0.5200 0.3600 0.2577 0.4330 0.0928 0.2165 0.0673 0.2115 0.2788 0.4423 0.13710.28000.28570.2971According to the state transition probability matrix, the steady-state probability can be derived, as listed in Table4. It is found that S3 is the most likely state in the station and suburban scenarios. For the rural scenario, either S1, S2, or S3 could be the steady state. It is worth noting that S1 and S2 have approximately similar steady-state probability in any scenario. This confirms that the appearance and disappearance of MPCs are equivalent. The above results in the rural scenario are similar to those in the open viaduct scenario reported in [26]. The obtained results can be applied for ON/OFF tapped delay line models that use the Markov chain to model the ON/OFF process of MPCs [31].


## 5. Conclusion

This paper analyzes the nonstationary characteristics in typical HSR scenarios, rural, station, and suburban, depending on the passive LTE-based channel measurements. With regard to the nonstationary characteristics, it is found that the SI is longest in the rural scenario. Additionally, a four-state MCM is established to characterize the B-D process of MPCs, and the corresponding state transition probability matrix and steady-state probability are provided. These results show the realistic channel characteristics in the HSR communication network, which will provide helpful information for nonstationary channel modeling of HSR communication systems.



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*Source: 1729121-2018-06-21.xml*