Construction and Capacity Analysis of High-Rank LoS MIMO Channels in High Speed Railway Scenarios

Abstract

The validity of the maximum capacity criterion applied to realize high-rank line-of-sight (LoS) multiple-input multiple-output (MIMO) channels is investigated for high speed railway scenarios. Performance is evaluated by ergodic capacity. Numerical results demonstrate that by simply adjusting antenna spacing according to the maximum capacity criterion, significant capacity gains are achievable. We find relatively low sensitivity of the system to displacements from the optimal point and angle in relatively short range. Thus, we present two proposals to reconfigure antenna arrays so as to maximize LoS MIMO capacity in the high speed railway scenarios.

1. Introduction

MIMO presents an attractive solution for meeting the requirements of next generation wireless communication system for the high speed railway. Since a high speed data rate is required for efficient voice and data transmission services in the future railways, and the bandwidth resources for the railway are limited, the capacity cannot be improved through increasing bandwidth. So MIMO is considered as an effective technique in long-term evolution for railway (LTE-R) to ensure the efficiency and reliability for data transmissions [1].

The development of MIMO systems over the past decade is based on the assumption of i.i.d Rayleigh fading [2]. Although the assumption simplifies the analysis of MIMO systems, its validity is often violated due to either an specular wavefront or a strong direct component existing; then, the entries of the channel matrix can be more effectively modeled by the Ricean distribution. Conceptually, LoS propagation is viewed to limit MIMO advantages because the channel matrix is normally rank deficient [3, 4]. But some recent investigations have questioned this belief and proposed design methodologies in order to achieve subchannel orthogonally. The orthogonality of subchannels is a key condition for capacity maximization [5–9] and most methodologies are realized by placing the antenna elements sufficiently far apart.

The common characteristics of the above mentioned works are facts about the MIMO performance in a relatively short distance between transmitters and receivers, and meanwhile the low mobility of vehicles in scenarios. Thus, the performance of MIMO optimization design in high speed railways needs to be reassessed under a realistic high speed railway environment. In most scenarios of the high speed railway, the base stations are located less than 30 m away from the tracks, and most of the BS antenna heights are more than 30 meters, hence there are always a strong line-of-sight (LoS) path between the transmitters and the receivers. Moreover, the channels exhibit a sparse multipath structure due to the lack of sufficient scatters in the railway environment [10]. The channel consists of a direct LoS path and sparse multipath, which is known as LoS MIMO channel. The analysis of small-scale fading characteristics in viaduct scenarios is derived from measurements taken along the “Beijing-Tianjin” high-speed railway of China, refer to [11], which indicates that Ricean 𝐾-factor increases when the train gets closer to the base station; meanwhile, the number of resolvable paths and RMS spread for delay firstly increase and then decrease.

In light of these facts, D2a scenario of WINNER II channel model is adopted, which is a realistic high speed railway transmission multipath propagation channel. The primary goal of the present paper is to verify the maximum capacity criterion in the high speed railway and then propose the methodologies for reconfigurable antenna arrays on maximizing the ergodic capacity of MIMO communication links through coverage area of the base station in viaduct scenarios.

The remainder of this paper is organized as follows. Section 2 discusses the simulation scenario and the WINNER II Channel Model. Section 3 develops the maximum capacity criterion. Section 4 concentrates on comparison and analysis of the simulation result and proposes two optimum proposals for reconfiguration antenna array in viaduct scenarios. Finally, Section 5 concludes this paper and summarizes the key findings.

2. High Speed Railway Channel Model

2.1. Scenario Description

According to the environment characteristics of high speed railways, D2a scenario is selected. D2a represents radio propagation in environments where MS is moving, possibly at very high speed, in a rural area. The link between the fixed network and the moving network (train) is typically an LoS type.

Figure 1 shows the setting for simulation in high speed railway scenarios. BS is located 14 m off the track and the height of this transmitter antenna array is 32 m above the ground. The receiver antenna array is installed on the top of the engine and its height is 3 m above the ground. The velocity of the train is set to 360 km/h towards the BS, and the track is parallel to the 𝑋-axis. The center frequency and bandwidth are 2.6 GHz and 100 MHz, respectively. In Figure 1, the geometry parameters are set to 𝐿=1000 m, 𝑑_BS=14 m, thus BS is located at the centre (500,14).

Figure 1 

The simulation scenario.

2.2. WINNER II Channel Model

In WINNER II D2a channel model, each channel realization is generated by summing contributions of eight clusters; each cluster is composed of twenty subpaths, which are associated with different delay, power, angle-of-arrival (AOA), and angle-of-departure (AOD) [12]. As illustrated in Figure 2, consider a single link of a MIMO system with an 𝑆 elements ULA for BS and a 𝑈 elements ULA for MS.

Figure 2 

Single link in WINNER II channel model.

(1) Generate the Delays 𝜏
Delays are drawn randomly from the delay distribution defined in [12]. With exponential delay distribution, calculate 𝜏^′_𝑛=−𝑟_𝜏𝜎_𝜏ln𝑋_𝑛, (1) where 𝑟_𝜏 is the delay distribution proportionality factor (3.8 ns in D2a scenario), and 𝜎_𝜏 is delay spread defined as 40 ns. 𝑋_𝑛∼Uni(0,1) and cluster index 𝑛=1,…,8.
Normalize the delays by subtracting with minimum delay and sort the normalized delays to descending order as follows: 𝜏_𝑛=sort𝜏^′_𝑛−min𝜏^′_𝑛. (2) In the case of LoS condition, additional scaling of delays is required to compensate the effect of LoS peak addition to the delay spread as follows: 𝜏^LoS_𝑛=𝜏_𝑛/𝐷, (3) where 𝐷=0.7705−0.0433𝐾+0.0002𝐾²+0.000017𝐾³;𝐾[dB] is the Ricean 𝐾-factor.

(2) Generate the Cluster Powers 𝑃
With exponential delay distribution the cluster powers are determined by 𝑃^′_𝑛=exp−𝜏_𝑛𝑟_𝜏−1𝑟_𝜏𝜎_𝜏⋅10^{−𝑍𝑛/10}, (4) where 𝑍_𝑛∼𝑁(0,𝜁) is the per cluster shadowing term in [dB]. Average the power so that sum power of all clusters is equal to one as 𝑃_𝑛=𝑃^′_𝑛∑^𝑁_𝑛=1𝑃^′_𝑛. (5)
Assign the power of each ray within a cluster as 𝑃_𝑛/𝑀, where 𝑀=20 is the number of rays per cluster.
For the two strongest clusters, say 𝑛=1 and 2, rays are spread in delay to three subclusters (per cluster), and twenty rays of a cluster are mapped to sub-clusters as presented in Table 1.

Table 1 

Subcluster information for intracluster delay spread clusters.

(3) Generate the Azimuth Arrival Angles 𝜑 and Azimuth Departure Angles 𝜙
The AOA for the 𝑛th cluster is 𝜑_𝑛=𝑋_𝑛𝜑^′_𝑛+𝑌_𝑛−𝑋_𝑛𝜑^′₁+𝑌₁+𝜑_LOS, (6) where 𝜑^′_𝑛=2𝜎_AOA−ln𝑃_𝑛/max𝑃_𝑁𝐶_LOS,𝜎_AOA=𝜎_𝜑/1.4,𝐶_LOS=𝐶×1.1035−0.028𝐾−0.002𝐾²+0.00001𝐾³. (7) In the above equation 𝜎_AOA is the standard deviation of arrival angles (factor 1.4 is the ratio of Gaussian std and corresponding RMS spread). 𝜑_LOS is the LoS direction and component 𝑌_𝑛∼𝑁(0,𝜎_AOA/5). Add the offset angles from Table 2 to cluster angles as 𝜑_𝑛,𝑚=𝜑_𝑛+𝑐_AOA𝛼_𝑚, (8) where 𝜑_𝑛,𝑚 is the AOA for each ray 𝑚 of each cluster 𝑛; 𝑐_AOA is the cluster-wise rms azimuth spread of AOA (𝑐_AOA=3^∘ in D2a scenarios). The corresponding offset angle 𝛼_𝑚 is taken from Table 2. For departure angle 𝜙_𝑛, the procedure is analogous.

Table 2 

Ray offset angles within a cluster, given for 1°rms angle spread.

(4) Draw the Random Initial Phase
{Φ^𝑉𝑉_𝑛,𝑚,Φ^𝑉𝐻_𝑛,𝑚,Φ^𝐻𝑉_𝑛,𝑚,Φ^𝐻𝐻_𝑛,𝑚} are the random initial phases for each ray 𝑚 of each cluster 𝑛 and for four different polarization combinations (𝑣𝑣,𝑣ℎ,ℎ𝑣,ℎℎ). Distribution for the initial phases is uniform, Uni(−𝜋,𝜋). In the LoS case, draw also random initial phases {Φ^𝑉𝑉_LoS,Φ^𝐻𝐻_LoS} for both 𝑉𝑉 and 𝐻𝐻 polarisations.

(5) Generate the Cross-Polarisation Power Ratios (XPR) 𝜅_𝑛,𝑚 for Each Ray 𝑚 of Each Cluster 𝑛
XPR is log-normal distributed. Draw XPR values as 𝜅_𝑛,𝑚=10^𝑋/10, (9) where 𝑋∼𝑁(𝜎,𝜇) is Gaussian distributed with 𝜎 and 𝜇 from [12] for XPR.

(6) Generate the Channel Coefficients for Each Cluster 𝑛 and Each Receiver and Transmitter Element Pair 𝑢,𝑠
For the clusters in D2a, say 𝑛=1,2,…,8, and uniform linear arrays (ULA), the channel coefficients are given by 𝐻_{𝑢,𝑠,𝑛}(𝑡)=1𝐾_𝑅𝐻+1^{′𝑢,𝑠,𝑛}(𝑡)+𝛿(𝑛−1)𝐾_𝑅𝐾_𝑅𝐻+1⁰(𝑡), (10) where 𝛿(⋅) is the Dirac’s delta function and 𝐾_𝑅 is the Ricean 𝐾-factor converted to linear scale. 𝐻₀(𝑡) is the channel coefficient corresponding to the single LoS ray. 𝐻^′_{𝑢,𝑠,𝑛}(𝑡) is the non-LoS channel coefficient component. 𝐻₀(𝑡) and 𝐻^′_{𝑢,𝑠,𝑛}(𝑡) are given by 𝐻₀(𝑡)=𝐹_{𝑡𝑥,𝑠,𝑉}(𝜙_LoS)𝐹_{𝑡𝑥,𝑠,𝐻}(𝜙_LoS)^𝑇⋅exp𝑗Φ^𝑉𝑉_LoS00exp𝑗Φ^𝐻𝐻_LoS𝐹_{𝑟𝑥,𝑢,𝑉}𝜑_LoS𝐹_{𝑟𝑥,𝑢,𝐻}𝜑_LoS⋅exp𝑗𝑑_𝑠2𝜋𝜆₀⁻¹sin𝜙_LoS⋅exp𝑗𝑑_𝑢2𝜋𝜆₀⁻¹sin𝜑_LoS⋅exp𝑗2𝜋𝜈_LoS𝑡,𝐻^′_{𝑢,𝑠,𝑛}(𝑡)=𝑃_𝑛^𝑀_𝑚=1𝐹_{𝑡𝑥,𝑠,𝑉}(𝜙_𝑛,𝑚)𝐹_{𝑡𝑥,𝑠,𝐻}(𝜙_𝑛,𝑚)^𝑇⋅exp𝑗Φ^𝑉𝑉_𝑛,𝑚√𝜅_𝑛,𝑚exp𝑗Φ^𝑉𝐻_𝑛,𝑚√𝜅_𝑛,𝑚exp𝑗Φ^𝐻𝑉_𝑛,𝑚exp𝑗Φ^𝐻𝐻_𝑛,𝑚⋅𝐹_{𝑟𝑥,𝑢,𝑉}𝜑_𝑛,𝑚𝐹_{𝑟𝑥,𝑢,𝐻}𝜑_𝑛,𝑚⋅exp𝑗𝑑_𝑠2𝜋𝜆₀⁻¹sin𝜙_𝑛,𝑚⋅exp𝑗𝑑_𝑢2𝜋𝜆₀⁻¹sin𝜑_𝑛,𝑚⋅exp𝑗2𝜋𝜈_𝑛,𝑚𝑡, (11) where 𝐹_{𝑟𝑥,𝑢,𝑉} and 𝐹_{𝑟𝑥,𝑢,𝐻} are the receiving antenna element 𝑢 field patterns for vertical and horizontal polarizations, respectively, similarly 𝐹_𝑡𝑥 is the transmitting antenna patterns. 𝑑_𝑠 and 𝑑_𝑢 are antenna spacing (m) between transmitter elements and receiver elements, respectively, and 𝜆₀ is the wavelength on carrier frequency. 𝜈_LoS and 𝜈_𝑛,𝑚 are the Doppler frequency shifts for the LoS ray and each ray 𝑚 of each cluster 𝑛, respectively. The Doppler frequency component is calculated from AOA, MS speed 𝜐, and direction of travel 𝜃_𝜐 as follows: 𝜈_LoS=‖𝜐‖cos𝜑_LoS−𝜃_𝜐𝜆₀,𝜈_𝑛,𝑚=‖𝜐‖cos𝜑_𝑛,𝑚−𝜃_𝜐𝜆₀. (12) The CIR for each receiver and transmitter element pair 𝑢,𝑠 is given by 𝐻_(𝑢,𝑠)(𝑡)=^𝑁𝑐_𝑛=1𝐻_{𝑢,𝑠,𝑛}𝑡−𝜏_𝑛. (13)

2.3. Path Loss Model

Path loss model for the WINNER II D2a scenario has been developed based on results of measurements carried out within WINNER, and it is formed as PL=21.5log₁₀(𝐷)+44.2+20log₁₀𝑓_𝑐,5.0 (14) where 𝐷 is the distance between the transmitter and the receiver in (m), 𝑓_𝑐 is the system frequency in (GHz). Path loss factor PL is given as a parameter, which is multiplied to channel matrices.

3. Maximum LoS MIMO Capacity Criterion

Using the simplified maximum capacity criterion in LoS MIMO systems, the problem of reduced capacity in a LoS scenario can be overcome [8]. The validity of the new capacity criterion implementation was investigated by WINNER II channel model in the next section.

3.1. MIMO Channel Capacity

To assess the performance of the stochastic channel in the presence of scatter, the notion of ergodic capacity must be employed. Note that the transmit power is equal to 𝑃_𝑇/𝑆 (at all transmit elements) as UPA (uniform power allocation) is used. The ergodic capacity of a MIMO system to be given by [13] 𝐶=𝐸log₂det𝐼_𝑈+𝜌𝑆𝐻𝐻^𝐻, (15) where 𝜌 corresponds to the average received signal-to-noise ratio (SNR) at the input of the receiver, and [⋅]^𝐻 denotes the conjugate transpose. Extremely, the capacity in (15) is maximized for 𝐻𝐻^𝐻=𝑆𝐼_𝑈, and this response corresponds to a channel with perfectly orthogonal MIMO subchannels.

3.2. Maximum 𝑈×𝑆 Capacity Criterion

It has been demonstrated that the capacity in LoS MIMO channel (ignoring any scatter components at this stage) is maximized, when the following criterion is fulfilled [8]: 𝑑_𝑠𝑑_𝑢1=𝜆𝑆𝐷+𝑝sin𝜔sin𝜃,∀𝑝∈ℤ, (16) where 𝐷 is the Tx-Rx distance, and [8] defines the distance between the first element of each array to equal 𝐷. The 3D geometric configuration is shown in Figure 3. 𝜃 and 𝜔 are the orientations of the arrays for Tx and Rx, respectively. Thus, by knowing the carrier frequency and the Tx-Rx distance, the optimal spacing can be easily calculated from the maximum LoS MIMO capacity criterion. In other words, this criterion for perfectly orthogonal MIMO subchannels architectures defines a number of MIMO architectures with antenna arrays fixed at optimal points.

Figure 3 

Positioning of the elements in a 𝑈×𝑆 MIMO system.

4. Simulation Results

The high speed railway scenarios have its distinctive characteristics, such as a relatively long distance between transmitters and receivers, the high speed of 350 km/h or above, and a number of scatters. But the maximum capacity criterion only considers the LoS component of the channel response. The characteristics of railway scenarios must be accounted for evaluating the criterion. Thus, in this section, we evaluate the criterion (16) in practical high speed railway scenarios described in Figure 1.

We explore a 2×2 MIMO system operating at 2.6 GHz. And the orientations of the antenna arrays for Tx and Rx are perpendicular to the track as 𝜃=90^∘ and 𝜔=90^∘, respectively. Then, the maximum capacity criterion is simplified as 𝑑_𝑠𝑑_𝑢=𝜆𝐷/2,𝑝=0. Assuming interelement spacing 𝑑_𝑠 for reconfigurable antenna arrays equal to 1 m, the 𝑑_𝑢 can be calculated. In the high speed railway scenarios, the separated distance 𝐷 between BS and MS is usually in order of hundreds of meters and is much larger than 𝑑_BS, therefore the approximation as 𝐷=𝑙²+𝑑²_BS≈𝑙 is used for simplification, where 𝑙 is the separated distance between the projection of the BS and the projection of the train on 𝑋-axis. In Figure 4, we have optimal interelement spacings 𝑑_𝑠=1 m, and 𝑑_𝑢=28.85 m for the point 𝑥=0(𝑙=500 m). The 𝐾-factor is set to 9 dB. The ergodic capacity is plotted as a function of SNR for different interelement spacings. The simulated curves are generated according to (15).

Figure 4 

Ergodic capacity as a function of SNR for different antenna spacings (𝐾-factor=9 dB, 𝑙=275 m).

It can be easily seen that the ULA with optimal antenna spacings shows superiority to other geometries over the entire SNR range. When the SNR reaches 20 dB, the capacity of optimal case is about 5 bps/Hz higher than that of ULA with 10𝜆 interelement spacings. In addition, the ULA with 10𝜆 interelement spacings shows superiority to the one with 0.5𝜆 interelement spacings. The reason is that placing the antenna elements far apart results in high resolution and low correlation of channel matrix and then it gives the high capacity.

4.1. Displacement

The criterion for high-rank LoS MIMO channels defines a number of MIMO architectures for systems with ULAs fixed at optimal locations. However, in high speed railway situations, there is a need for high capacity over an area, rather than to a fixed point. To examine the sensitivity of the performance of maximum capacity architectures in high speed railways, the capacity is now evaluated as a function of the displacement from the optimal point. The simulation is explored in the location range from 0 to 500 m, which is the whole trajectory, that is from the midpoint between two BSTs up to the point closest to another BST. And it is equivalent to a 500 m displacement range refering to the optimal point 𝑥=0. The optimal interelement spacings are 𝑑_𝑠=1 m and 𝑑_𝑢=28.85 m. The received power gradually changes.

The variation of capacity with different displacements is shown in Figure 5. From this figure, the performance of optimal case shows superiority to other geometries over the entire trajectory. Above a displacement of 350 m or so, the curve of capacity is escalating faster and faster. The reason is the increased received power and the decreased degree of scattering. When the displacement reaches 500 m, the capacity of optimal case is about 1.5 bps/Hz higher than that of ULA with 0.5𝜆 interelement spacings. In addition, the ULA with 10𝜆 interelement spacings shows superiority to the one with 0.5𝜆 interelement spacings. The reason is the high resolution and the low correlation of channel matrix.

Figure 5 

Ergodic capacity as a function of the displacement from the optimum point.

The sensitivity of the capacity performance for the optimum case is investigated by means of a narrow distance window, see Figure 6. The simulation is explored in the range of 0 m to 100 m, which is equivalent to a 100 m displacement range, refer to the optimal point 𝑥=0 m. From this figure, it is clear that the capacity is relatively insensitive to small displacements from the optimum point. The reason is that the angle changes of 𝜃 and 𝜔 are very small due to short displacements. According to (16), the optimal interelement spacing is local optimum for the premise of sin(𝜃) and sin(𝜔) is basically invariable. So, the same reconfiguration antenna array used in a small regional is feasible.

Figure 6 

Ergodic capacity as a function of the displacement from the optimum point.

4.2. ULA Azimuthal Orientation

There are some disadvantages for antenna array orientation in railway scenarios due to the fast moving of the train and the rapid change of the radio propagation environments. It is difficult to deal with this fast fading. On the other hand, it has some advantages that the route of the train is fixed and the path can be predicted. So, some special antenna types or programs can be chosen. It is clear that there is a dependence of the capacity to the azimuthal orientation of the two arrays (angles 𝜃 and 𝜔) from the maximum capacity criterion of (16). This dependence is now examined using the same channel model as before.

The effect of ULA azimuthal orientation at MS ends on the capacity is shown in Figure 7. The units of angle in MS coordinate axes are degree. The ergodic channel capacity is obtained at 𝜌=20 dB. Each point in Figure 7 stands for different MS azimuthal orientation and fixed BS azimuthal orientation at the optimal point 𝑥=0. At each process of simulation, the antenna orientations of BS and the train are fixed. The results show a little sensitivity (in terms of capacity) to the orientation of the MS arrays. In detail, the channel capacity is seen to vary between the minimum (𝐶min=10.93 bps/Hz) and the maximum (𝐶max=11.25 bps/Hz) values about the curve for the optimal one. The results suggested that this sensitivity needs not focus too much on the design of any practical MIMO system.

Figure 7 

Ergodic capacity as a function of the angle deviation from the optimum angle (𝜌=20 dB).

(1) Maximizing Capacity in Viaduct Scenarios Using Reconfigurable Antenna Arrays
Now theory and methods for exploiting the potential of reconfigurable RF front-ends in high speed railway are not fully developed. The results of simulation offer a reference for reconfigurable antenna arrays on maximizing the capacity of MIMO wireless communication links in high speed railway scenarios. And there is measure data based on Propsound measurements, which is operated in viaduct scenarios [11]. Then, we put forward two proposals through a combination of simulation results and the measurement data.
According to [11], the coverage area of the base station is divided into 4 regions as remote area (RA), toward area (TA), close area (CA), and arrival area (AA). We can design different regions in the area that are involved in different reconfigurable antenna arrays because of the rapidly change of separate distances and different small-scale fading characteristics. The maximum capacity criterion is introduced in the previous section, and then two practical optimum proposals are afforded.

Optimum Proposal 1
Select four midpoints of the four linear regions as the optimal locations and use them to calculate optimal interelement spacings for the different reconfiguration antenna arrays involved in the four regions. The interelement spacings of ULA are adjusted for different regions, when the train travels through the coverage area of base stations.
The maximum capacity criterion defines a number of MIMO architectures for systems with antenna arrays fixed at optimal locations. However, there is a need for high capacity over an area, rather than to a fixed point. In the simulations, to examine the sensitivity of the performance of maximum capacity architectures under high speed railway scenarios, we can determine the advantage of the optimal antenna array basically stable in the small displacements range.

Optimum Proposal 2
Select the edge point of base station coverage as the optimal location and obtain the optimal interelement spacing of the reconfigurable antenna array involved in all regions. The interelement spacing of ULA maintains invariable when the train travels through the linear coverage. This proposal maximizes the capacity of the coverage edge and minimizes the correlation of channel matrix in the coverage.
The corresponding parameters for optimum proposals are taken from Table 3.

Table 3 

The parameters for optimum proposals.

4.3. Scattering

In the maximum LoS MIMO capacity criterion, only LoS component of the channel response is considered. In railway scenarios, some degrees of scatterings are always present in the radio channel, and, hence, their effects must be accounted for the design of the MIMO system.

In Figure 8, the ergodic capacity is plotted against the Ricean 𝐾-factor and compared with two reference geometries. It demonstrates again the superiority of the optimal design obtained by maximum capacity criterion. And meanwhile we see the capacities for ULAs with different antenna spacings decrease when 𝐾-factor increases. The optimal design is not expected in this result because the curve for optimal interelement spacings should increase with the increment of 𝐾-factor, and the reason is high-rank channel for the optimal case [8]. And the unsatisfactory result can be attributed to the alterations of eigenvalue profiles, which are not full rank when 𝐾-factor changes. Meanwhile, UPA used in our investigation becomes quite suboptimal since an amount of power is wasted inevitably on the vanishing or relatively weak spatial dimensions.

Figure 8 

Ergodic capacity as a function of 𝐾-factor for different antenna spacings (𝜌=10 dB, 𝑥=500 m).

5. Conclusion

In this paper, we study the validity of the maximum capacity criterion in real high speed railway environment, which is applied to realize high-rank LoS MIMO channels. D2a scenario of WINNER II channel model is adopted and it is a realistic high speed railway transmission multipath propagation channel. The ergodic capacity is used as the index to discuss the performance of reconfigurable antenna arrays. Numerical results demonstrate that significant capacity gains are achievable by simply adjusting antenna spacing according to the maximum capacity criterion. And we obtain relatively low sensitivity of the optimal antenna arrays to displacement from the optimal point and angle. So, we put forward two proposals for reconfigurable antenna arrays so as to maximize LoS MIMO capacity in the high speed railway scenarios. Then we find that the antenna array geometries obtained from the criterion are suboptimal, since the ergodic capacity decreases with the increment of 𝐾-factor. Thus, it is very necessary to carry on a more thorough research on this optimal methodology for constructing high-rank LoS MIMO channels in high speed railway communication system.

Acknowledgments

The authors would like to express their great thanks to the support from the National Natural Science Foundation of China under Grant 61222105, Beijing Municipal Natural Science Foundation under Grant 4112048, the Fundamental Research Funds for the Central Universities under Grant 2010JBZ008, Program for New Century Excellent Talents in University under Grant NCET-09-0206, and the Key Project of State Key Laboratory under Grant RCS2011ZZ008.