Abstract
This paper presents a suboptimal approach for resource allocation of massive MIMOOFDMA systems for highspeed train (HST) applications. An optimization problem is formulated to alleviate the severe Doppler effect and maximize the energy efficiency (EE) of the system. We propose to decouple the problem between the allocations of antennas, subcarriers, and transmit powers and solve the problem by carrying out the allocations separately and iteratively in an alternating manner. Fast convergence can be achieved for the proposed approach within only several iterations. Simulation results show that the proposed algorithm is superior to existing techniques in terms of system EE and throughput in different system configurations of HST applications.
1. Introduction
Recent development and deployment of highspeed trains (HSTs) have dramatically improved the efficiency and user experience in interstate transportations. However, providing high data rates and good quality of service (QoS) to passengers in the presence of rapidly varying channel conditions and scarce bandwidth availability is a challenging task [1]. Critical challenges have arisen from realtime communications between HSTs and fixed base stations (BS). Existing narrowband railway communication systems, such as GSMR, are not suitable for HSTs due to typically low capacity. 5G technology is currently adopting a socalled network densification approach, which involves the deployment of a large number of base stations (BSs), to increase the network coverage and provide higher throughput to the users [2]. OrthogonalFrequency DivisionMultipleAccess (OFDMA) has been extensively adopted for wideband communications, but severe Doppler shift exists in the communication process because of high mobility, resulting in the difficulties in channel estimation [3] and subsequently destructive intercarrier interference (ICI) [4]. On the other hand, increasing the number of antennas at both transmitters and receivers, also known as MultipleInput MultipleOutput (MIMO), can improve robustness against ICI. Particularly, MIMO with a large number of antennas has been increasingly studied for enhancing quality and reliability of wideband wireless communications. Unfortunately, the benefits do not come for free. Energy consumption would grow substantially, as the number of antennas increases. An energyefficient resource allocation of MIMOOFDMA is expected to balance spectral efficiency and energy efficiency (EE) [5].
There has been a lot of work on wireless resource allocation in static and lowspeed mobile system. In [6], it was revealed that network energy can be saved by assigning nonoverlapping frequency bands to different cells. In [7], a power loading algorithm was proposed to maximize the EE of MIMO. In [8], the authors investigated the energyefficient bandwidth allocation in downlink flat fading OFDMA channels and maximized the numbers of bits transmitted per joule, by using the Lagrangian and timesharing techniques. In [9], the authors proposed a hybrid structure of resource allocation in OFDMA cellular systems, which maximized both the EE and the downlink system capacity. The proposed structure, combined with resource allocation, was shown to improve the EE and the system capacity of OFDMA. In [10], the resource allocation for energyefficient OFDMA systems was formulated as a mixed nonconvex and combinatorial optimization problem and solved by exploiting fractional programming. In [11], the energyefficient configuration of spatial and frequency resources was studied to maximize the EE for downlink MIMOOFDMA systems in the absence of channel state information (CSI) at the BS. However, none of the existing works have taken into account the destructive ICI. For HSTs at a speed of over 500km/h, the fast timevarying channel and the severe Doppler shift have yet to be addressed, and highmobility communication shall be one of the most important and extreme use scenarios in future 5th generation (5G) mobile communication networks [12–16]. The OFDMA resource allocation strategy was designed for fastchanging mobile environments in [17], where a suboptimal allocation policy was developed at a significant cost of computational complexity.
Fog computing, also known as fogging, is an architecture that uses edge devices to carry out a substantial amount of local computation, storage, and communication [18–20]. We use a fog server at the BS to concentrate data, data processing, and applications. The fog server can increase overall computing capability, which helps in efficient resource allocation and utilization.
Fog computing emphasizes proximity to endusers and client objectives, dense geographical distribution and local resource pooling, latency reduction, and backbone bandwidth savings. Therefore, we use this technology to provide practical value for realtime implementation of HST communications.
This paper aims to design an efficient resource allocation strategy to improve the communication performance of HSTs. After analyzing the multiuser MIMOOFDMA downlink system, the influence of mobility on the system is quantified. A mathematical model is put forth to maximize the EE of the system. To tackle the problem, an iterative algorithm with fast convergence is proposed. Specifically, we propose to decouple the problem between the allocations of antennas, subcarriers, and transmit powers and solve the problem by carrying out the allocations separately and iteratively in an alternating manner. Fast convergence can be achieved for the proposed approach within only several iterations. Simulation results demonstrate the gain of the proposed approach in terms of EE and throughput, as compared with existing schemes.
The rest of the paper is organized as follows. We present the system model in Section 2 and formulate and solve the problem of interest in Section 3. In Section 4, the simulation results are provided, followed by conclusions in Section 5.
2. System Model
The system of interest is a multiuser MIMOOFDMA system, as illustrated in Figure 1, where there is a fixed BS equipped with transmit antennas () and user terminals located in a HST. A fog server is employed at the BS to help the resource allocation computation. Each of the user equipment has a single receive antenna. The users share radio resources for down services. Each of the user equipment has a single receive antenna. The users share radio resources for down services. Different users are assigned with different OFDM subcarriers and different antennas, given the large number of transmit antennas. Coherent beamforming is carried out at the BS to produce physical beams towards the users.
The speed of HST can lead to severe Doppler shifts. Let denote the complex channel gain between the BS and user on subcarrier . The total number of subcarriers is . The knowledge on can be inaccurate at the BS, because of the fastchanging HST environment and hence estimation errors. We assumewhere is the estimate of at the BS and is an independent and identically distributed (i.i.d.) measurement error. yields a complex Gaussian distribution due to the use of the Minimum Mean Square Error (MMSE) estimators. andwhere is the subcarrier interval and is the maximum Doppler shift which can be written as . is the speed of light. is the carrier frequency. is the transmit power allocated to user on subcarrier . is the noise power spectral density [10].
We assume that each subcarrier has an equal bandwidth of . Therefore, the total bandwidth of the system is . We also assume that each subcarrier is assigned an equal transmit power; i.e., , where and are the number of subcarriers and the transmit power of the BS allocated to user , respectively.
The Doppler shift can compromise the orthogonality between OFDM subcarriers, resulting in ICI [22]. At a speed of , the power of ICI on a subcarrier can be written as [23]where denotes the duration of an OFDM symbol.
In the case that , the receive signaltonoise ratio (SNR) can be approximated to [17]where is the number of antennas of the BS assigned to user .
The asymptotic rate of the MIMO can be achieved based on the random matrix theory [17]. Specifically, the rate asymptotically converges to the average rate in mean square. The asymptotic rate can be replaced with the average data rate. The total data rate of user converges to
We also consider nonideal circuit power at the BS. We can adopt a linear model [24] at the BS to characterize the circuit power consumption, as given bywhere is the power consumption per active antenna, consisting of the power consumption of filtering, mixing, power amplification, and digitaltoanalog conversion. is the constant part of the power consumption at the BS and is independent of the number of active antennas.
3. Optimization Problem Formulation
The goal of this paper is to maximize the EE of the BS, which can be formulated aswhere, given (5) and (6), the EE of the BS can be written asthe vector collects the subcarrier allocation of all users; collects the antenna allocations for the users; and collects the power allocations of the users. The constraint C1 specifies the total transmit power constraint . C2 specifies the minimum data rate per user. C3 and C4 restrict the total numbers of subcarriers and antennas, respectively.
Clearly, problem (7) is a combinatorial mixed integer programming problem. The objective of (7) also has a fractional form with variables in the denominator of the objective. All this makes (7) a NPhard nonconvex problem with poor tractability. In order to solve the problem efficiently, we develop a suboptimal solution, where the subcarriers allocation, antennas, and transmit powers are optimized separately and sequentially in an alternating manner.
3.1. Subcarrier Allocation
Given and , we first propose to allocate subcarriers to maximize the EE while satisfying the minimum data rates of the users. According to the objective of (8), the subcarrier allocation can be expressed asWe propose to allocate subcarriers based on the criterion of EE. First, we calculate the number of subcarriers allocated to each user according to the minimum data rate of the user. Then, we choose the user with the highest EE and allocate a subcarrier to the user, one user after another, and this repeats until all users are allocated or all subcarriers are assigned. The proposed allocation of subcarriers can be summarized in Algorithm 1.

3.2. Transmit Power and Antenna Allocation
Given the subcarrier allocation developed in Section 3.1, problem (7) can be reformulated to a fractional programing problem with respect to and , as given by [25]This is mixed integer programming. We proceed to relax the integer constraint C4, i.e., to . is the minimum number of antennas to meet the requirements of uninterrupted transmission for all users [10]. As a result, (10) can be further reformulated aswhere is the optimal solution for problem (10).
We can prove that (11) is a concave function by evaluating the Hessian matrix of , as given bywhere , , and . Both the determinant of the Hessian matrix and its th order principal matrix are nonnegative. Thus the Hessian matrix is positive semidefinite. Hence, is strictly convex. As a result, the objective function of problem (11) is jointly concave over while all the constraints are linear. In addition, (11) yields the Slater conditions [26] and therefore holds strong duality. The dual problem of (11) and the primary problem (11) have zero duality gap.
Given , the Lagrangian function can be written aswhere collects the Lagrange multipliers associated with constraint C2 and ; is the Lagrange multiplier associated with constraint C1. The dual problem of (11) is given by
Given , according to the KKT conditions, the optimal power allocation, denoted by , and antenna allocation, denoted by , can be obtained aswhere , , , , and .
The subgradient method can be employed to obtain in an interactive manner, as given bywhere ; is the index for the iterations. and are the step sizes to adjust and , respectively; and and are the subgradients of the Lagrangian function at and , respectively.
The resource allocation policy can be developed based on (15)–(17). Since and can be decoupled in (15), (16), and (17), we can use an improved coordinate ascent (CA) method, where, during each iteration, we first optimize , given and , and then optimize , given , in an alternating fashion until convergence. Given , the proposed allocation of transmit antennas and subcarriers is summarized in Algorithm 2.
Finally, we can use the Dinkelbach method [25] to update . The solution for problem (11) can be summarized in Algorithm 3.

4. Simulation Results
In this section, we simulate the proposed algorithm to verify its effectiveness, where block Rayleigh fading channels are considered. Other simulation parameters are listed in Table 1. We note that the proposed algorithm can be applied under any channel conditions, such as Rician fading channels.
For comparison purpose, the following two resource allocation schemes are also stimulated.
(1) Band allocation based on SNR (BABS) algorithm [27]: it is used for subcarrier allocation. The transmit powers and antennas are allocated in the same way as in the proposed algorithm.
(2) EMMPA algorithm [28]: this algorithm first allocates subcarriers evenly and then allocates the rest of subcarriers to the users with the best channel condition. The scheme developed in [26] is used for the transmit power and antenna allocation.
Figure 2 shows the convergence of the proposed algorithm with different transmit powers, where , , and km/h. It is seen that the EE of the proposed algorithm increases and quickly stabilizes with the growth of iterations. The maximum of the EE can be attained after around only six iterations.
Figure 3 plots the system EE versus the maximum transmit power, where and km/h. We can see that the system EE increases with maximum transmit power. When the transmit power is large enough, the system EE stabilizes. This is because the BS does not need to activate extra antennas or consume extra power when the system maximum EE is reached. The figure also shows that our proposed algorithm performs between the BABS and EMMPA algorithms. The system EE of EMMPA is higher than our proposed algorithm since EMMPA does not have the constraint of and, thus, has a fixed EE value. Additionally, EMMPA is an unconstrained problem to maximize the system EE. The system EE of our proposed algorithm is higher than that of the BABS algorithm because our approach is based on the maximization of EE, while the BABS is based on the minimization of SNR.
Figure 4 presents the system throughput versus the moving speed , where dBm and . We can see that, as increases, the system throughput significantly decreases. This is because ICI power and channel estimation error are increasingly severe and thus increasingly detrimental to communication quality. The system throughput is about 20.7% higher under our proposed algorithm than under BABS algorithm. EMMPA provides the lowest throughput because of its nature of an unconstrained optimization of maximizing EE without constraints. BABS is to minimize the transmit power while allocating subcarriers. The conclusion drawn is that our proposed algorithm can significantly improve throughput.
Figure 5 shows the system EE versus the number of users , where dBm and km/h. It can be seen that the system EE decreases with the number of users. This is because when the number of subcarriers is fixed, each user can be allocated with a less number of subcarriers, resulting in an increase of the transmit powers to satisfy the users data rate requirement. EMMPA has no demand for the data rate, but the subcarriers assigned to the users who have better channel conditions decrease, and thus system EE also decreases.
5. Conclusion
This paper models the resource allocation strategy for multiuser MIMOOFDMA downlink system for HSTs, where subcarriers, transmit power, and antennas are jointly optimized. Specifically, we propose an iterative suboptimal algorithm to optimize the system EE with fast convergence. In terms of the system performance, simulation results show that both EE and throughput are improved. Furthermore, the proposed approach is able to fast stabilize within only several iterations and therefore provides practical value for realtime implementation of HST communications.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request. No additional data are available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project is supported by the National Natural Science Foundation of China (Grant no. 61771002).