Journal of Communications and Information Networks

The standardization process of 5G is being acceleratedcurrently and it is expected to come into commercial use in 2020.5G promises to provide very high data rates(typically of the order of Gbit/s), extremely low latency, a manifold increase in network capacity, and significant improvement in users’ perceived quality-of-service (QoS)^[1]. As energy and environmental issues have become increasingly important, future network functions should not be gained at the expense of increasingly high energy costs; thus, energy conservation will be another feature of 5G^[2]. Massive multiple-input multipleoutput(MIMO)and relaying not only have great advantages in improving spectral efficiency(SE), they also have great potential in energy savings as well. Both are recognized as promising technologies for realizing green 5G^[2].

Massive MIMO, featuring the incorporation of massive antennas and three-dimensional (3D) beamforming, enhances the multiplexing gain among multiple users and significantly improves SE and energy efficiency(EE)without additionally increasing the site density or bandwidth^[3,4]. Relaying, especially the most popular amplify-and-forward(AF)relaying method considered in this paper, which simply amplifies the received signal and forwards it to its destination, can maintain the communication QoS with low power levels at both the transmitter and the receiver and improve the transmission integrity at the same power level. Additionally, it features low cost, high flexibility, and conventional installation, all of which make relaying a green, cost-effective alternative in expanding cell coverage, enhancing indoor communication QoS, and eliminating dead zones. It is known that 5G is to exploit the higher frequency band—millimeter wave, for example. However, there are two fatal flaws in millimeter wave communications: 1)fast signal attenuation; 2)weak penetrating ability^[1]. To overcome these barriers and accomplish the aims of 5G in such frequency bands, AF relay-aided massive MIMO systems—the integration of massive MIMO and AF relaying—are a promising solution.

Remarkably, for relay-aided massive MIMO systems, the circuitry power consumption of a base station(BS)equipped with massive MIMO antennas increases linearly with the number of antennas. In addition, the relay transmission not only makes the network topology further complicated, but also requires additional time and power resources^[5]. How to harvest gains provided by relay-aided massive MIMO systems without increasing resource consumption is a critical issue. In general, power allocation is the focus of resource management in relay-aided massive MIMO systems, and user association, as an important part of resource management, also has a profound impact on system performance; moreover, they influence each other. For these reasons, the design of an energyefficient joint user association and power allocation(JUAPA) scheme is of great interest in relay-aided massive MIMO systems.

The research on AF relay-aided massive MIMO systems has been sparse, let alone energy-efficient resource management in such systems. However, existing works have adopted multiple ways of resource management to enhance the EE of massive MIMO systems^{[6,7,8,9,10,11,12,13,14,15,16]}, which greatly benefit our work. For instance, in Ref. [6], the joint design of a BS dynamic switch and user association to maximize the EE of massive MIMO systems was investigated. The work in Ref. [7]focused on wireless power transfer in massive MIMO systems through exploiting renewable resources to achieve the goal of energy conservation. The SE and EE in massive MIMO systems were analyzed in Refs. [8,9,10]. User association has been treated as a critical method to promote the EE of massive MIMO systems in Refs. [11,12,13]. Energy-efficient JUAPA design in massive MIMO systems was studied in Refs. [14,15].

Note that all the above works assumed a two-tier system where one BS equipped with massive MIMO antennas is underlaid with other types of access points. To the best of our knowledge, the EE of downlink transmissions in AF relayaided massive MIMO systems has not been investigated yet, and an AF relay is very different from other access points in transmission characteristics. Motivated by this, the JUAPA problem in an AF relay-aided massive MIMO system is studied towards maximizing the system EE under the proportional fairness criterion. The main contributions of this paper include the following two aspects: 1)When maximizing the proportionally fair utility of system EE in the AF relay-aided massive MIMO system with JUAPA, practical constraints such as QoS requirements of users and maximum transmit powers of BS and relay stations(RSs)are taken into account; 2)To solve the mixed-integer non-convex joint optimization problem, an iterative algorithm is developed, where the original problem is separated into a master user association problem and a lowerlevel power allocation problem, which are solved by the Lagrangian dual decomposition method and Newton’s method, respectively.

The remainder of this paper is organized as follows. Section Ⅱ introduces the AF relay-aided massive MIMO system model. In section Ⅲ, the energy-efficient JUAPA problem is formulated, and an iterative algorithm is designed to solve the optimization problem. Section Ⅳ presents some simulation results and section V concludes the paper.

Notations: The bold italic case x represents a column vec-tor, the bold upper case $A$ represents a matrix, ${\Vert \cdot \Vert }_p$ represents the p-norm, and E{·}represents the expectation operator.

Consider an AF relay-aided massive MIMO system where a BS equipped with a large-scale array of M antennas together with N single-antenna AF RSs are randomly deployed within the cell range serving K single-antenna mobile stations(MSs, which are also called users in the following)with uniform dis-tribution, as shown in Fig.1. Let B={1, 2, …, N} denote the set of RSs and B₀=0∪B denote the set of BS and RSs, and j∈B₀. Without loss of generality, j=0 refers to the BS. The BS and RSs share the same frequency band. The set of all MSs is denoted by U ={1, 2, …, K}, and i∈U. The transmission power vector of BS and RSs is denoted by P =[p₀, p₁, p₂, …, p_N], which is adjustable, and the upper limit of p _j is $p_j^{\max }$. S denotes the maximum number of down-link data streams that the BS can transmit simultaneously on any given resource block with equal power allocation. In prac-tice, the total number of users associated with the BS is always larger than S, and equal resource sharing among users associ-ated with the BS is assumed. The massive MIMO regime is referred to as the case where $\text{1}\ll \text{S}\ll \text{M}$. The system band-width W is normalized to 1 Hz.

For AF relaying, the time-division duplex operation is assumed with the reciprocity-based channel estimation and perfect channel state information. Because two-hop half-duplex AF relaying is adopted, the downlink transmission process is divided into multiple time slots and each consists of two equal sub-slots. In the first sub-slot, all RSs and a portion of the users associate with the BS for data transfer(the first-hop link); and the rest of MSs (users) associate with RSs in the second sub-slot. It is assumed that each RS can only communicate with one user at a time and the BS can simultaneously communicate with all users. To formulate the user association problem, the association index is denoted by x_{i j}. When user i is associated with the BS or the RS j, x_{i j} =1; otherwise, x_{i j}=0.

In the massive MIMO regime, small-scale fading disappears with the infinitely increasing number of antennas at the BS, and linear zero-forcing beamforming is utilized to further eliminate intra-cell interference. Thus, the achievable downlink data rate r_{i j} of user i associated with the BS can be approximated as^[11]

where γ_{i j} is the corresponding effective signal-to-noise ratio (SNR). Following the signal-to-interference-plus-noise ratio (SINR)derivation in Refs. [11,14]and treating RSs as special MSs in the first sub-slot, we have

where g_{i j} is the large-scale fading channel power gain between the BS and user i, and σ ² is the noise power.

Similarly, the achievable downlink data rate r_{0 j} of relay j associated with the BS is

and the corresponding received SNR of relay j denoted as γ_{0 j} is^[11,14]

where g_{0 j} is the large-scale fading channel power gain between the BS and relay j.

The remaining users associated with RSs receive the amplified signal forwarded by AF relays in the second sub-slot. The signal received at RS j and the signal received at user i associated with it can be respectively expressed as^[17]

where h_{0 j} is the channel coefficient from the BS to RS j, x_i is the signal sent from the BS to user i, and $E\left\{{\Vert x_i\Vert }^2\right\}=1$, n _j is an additive white Gaussian noise (AWGN) with zero mean and variance σ ². β _j is the amplifying gain of relay j. Here, ${\beta }_j=p_j/\left(p_0{\left|h_{oj}\right|}^2+{\sigma }^2\right)=p_j/\left(p_o{g}_{oj}+{\sigma }^2\right)$, where g_{0 j}=|h_{0 j}|², and it satisfies ${\beta }_j\le p_j^{\max }/\left(p_o{g}_{oj}+{\sigma }^2\right)$, h_{i j} is the channel coefficient from RS j to MS i, n_i is an AWGN with variance σ ², and U₁ is the set of users associated with the BS in the first sub-slot. Then, the achievable downlink data rate r_{i j} can be expressed as

where γ_{i j} is the SNR received at user i associated with relay j, which is given by

where g_{i j}=|h_{i j}|² is the large-scale fading channel power gain between RS j and user i.

The power consumption model in Ref. [16]is adopted in this paper, which articulates how the total power consumption of BS scales with the number of antennas M:

where ε₀ is the power amplifier efficiency, c_{m, 0} and c_{m, 1} are coefficients. The details of these coefficients can be found in subsection Ⅱ-B of Ref. [16]. The total power consumption includes the power consumption in transceiver chains, coding and decoding modules, channel estimation and precoding modules, and architectures, as detailed in Ref. [16].

For the power consumption model of RSs, the conventional linear power consumption model in Ref. [18]is adopted, which is composed of the static part $p_j^{sta}$ and the adaptive part $p_j^{adp}$. For AF relays, $p_j^{adp}={\beta }_j/{\epsilon }_{j}E\left\{{\Vert y_j\Vert }^2\right\}=p_j/{\epsilon }_j$. Hence, the power consumption of RS j denoted as $p_j^{total}$ is defined as^[17]

where ε _j is the power amplifier efficiency of RS j. $p_j^{sta}$ is linear to the radiated transmit power of RS j, and $p_j^{sta}$ is related to the power consumption in transceiver chains.

The EE(in bits/Joule)of user i associated with the BS or RS j is defined as $r_{ij}/p_j^{total}$. Note that maximizing the total EE within a system will result in extremely unfair throughput allocation as was shown in Ref. [19]. To avoid this, the EE proportional fairness criterion in Ref. [19]is adopted, where the utility of user i associated with the BS or the RS j is defined as ${\omega }_{ij}=\log \left(r_{ij}/p_j^{total}\right)$. The logarithmic function is concave and hence has diminishing benefits, which encourages allocation fairness and load balancing. The challenge is seeking the optimal user association matrix $X$ and the optimal power allocation vector $P$ with certain realistic constraints that maximize the overall system utility. Thus, the optimization problem is formulated as follows:

where constraints c1 and c2 ensure that each user can only choose one transmission mode: BS-MS or BS-RS-MS; c3 ensures the number of users associated with an RS does not exceed 1 from the relay perspective; c4 indicates the transmit power constraint of the BS or an RS; the SNR constraint is given by c5 and γ^min is the minimum received SNR required by users in the downlink transmission.

Next, we attempt to solve the joint optimization problem P0.However, it is difficult to obtain the globally optimum solution for its mixed-integer and non-convex features. To obtain a sub-optimal solution, the method is to decompose the problem P0 into two sub-problems: the master user association sub-problem P1, and the lower-level power allocation sub-problem P2, and then obtain a sub-optimal solution with an iterative algorithm. The user association matrix $X$ is first derived under fixed transmit powers, and then, power allocation is performed under the given $X$ to further promote the system EE. Accordingly, the two sub-problems are

where $x_{ij}^{\ast }$ is the solution obtained from solving P1.

To solve P1, the Lagrangian dual decomposition method is adopted. When $P$ is fixed, first an auxiliary variable $k_j={\displaystyle {\sum }_{i\in U}{x}_{ij}}$ is introduced to represent the actual number of users associated with the BS. Denote $K=\left[k_0,k_1,\cdots k_N\right]$. The transformed problem P1. 1 is

where

To achieve the optimal solution of P1. 1, the Lagrangian function can be obtained as

where λ _j and µ_i are non-negative Lagrange multipliers. Thus, the corresponding dual function is represented by

where

and

The dual problem with respect to λ and µ can be rewritten as

The maximization of the Lagrangian function has the following analytic solution:

where

This indicates that user i will choose to associate with either the BS or the RS j^∗. Next, by setting∂g_k(λ)/∂k _j=0, k^∗_j can be derived as

To obtain the optimal solution of g(λ, µ), the non-negative Lagrangian multipliers λ and µ can be updated with the subgradient method as

where t represents the tth iteration, and δ(t)is the step size. With the updated λ _j(t)and µ_i(t), x_{i j}(t)and k _j(t)can be updated accordingly via Eqs. (16)-(18). Because the dual problem is always convex, the sub-gradient method is guaranteed to converge to the global optimum of the dual problem P1. 2.

Next, the power allocation sub-problem is solved under the obtained optimal user association matrix $X$. To solve P2, Newton’s method in Ref. [20]is utilized to search for a suboptimal solution.

First, note that the constraint c5 determines the minimum transmit power required at the BS or an RS. Solving the equations for p _j below, one can obtain $p_j^{\min }$:

so that

Before utilizing Newton’s method, an auxiliary variable r_{i j}=log(1+γ_{i j})is introduced. The objective function of P2 can be rewritten as

It is clear that $f_1\left(P\right)$ is a function of p₀, and $f_2\left(P\right)$ is a function of p₀ and p _j. To simplify the process of the first-order and second-order partial derivatives with respect to p₀, it is assumed that the QoS of users associated with RSs is mainly contributed by the transmit power at RS p _j. Therefore, the effect of p₀ is ignored and we have

One can then derive ${\partial }^2{f}_1\left(P\right)/\partial p_0^2$ and ${\partial }^2{f}_2\left(P\right)/\partial p_j^2$, and substitute them into Eq. (24).

Then, Newton’s method is adopted to update the transmit power p _j as

where δ(t)is the step size and∆p _j is in the ascending direction. Here the incremental updating direction is chosen as

It is known that the key to finding the global minimizer of the function p _j(t)+δ(t)∆p _j is to choose an ideal step size δ(t). In this work, the popular back-tracking approach is applied to search for the proper step size δ(t).

Based on the previous analysis, a sub-optimal solution of the joint optimization problem can be obtained by an iterative algorithm for JUAPA with the logarithmic utility of the system EE, which is summarized in Algorithm 1.

Algorithm 1 Joint user association and power allocation

1: Initialize $P\left(0\right)$, $\lambda \left(0\right)$, and $\mu \left(0\right)$, calculate ${\tilde{r}}_{ij}$ for all i and j, set the iteration index t=0;

2: Repeat;

3: Solve sub-problem P1. 1 to obtain optimal $X\left(t\right)$ when $P\left(t\right)$ is fixed;

a)calculate $X\left(t\right)$ from Eq. (16);

b)calculate $K\left(t\right)$ from Eq. (18);

c)update $\lambda \left(t+1\right)$ from Eq. (19);

d)update $\mu \left(t+1\right)$ from Eq. (20);

4: Update $P\left(t+1\right)$ with fixed $X\left(t\right)$ according to Eq. (25)by the backtracking line search for the step size;

5: t=t+1;

6: Until convergence.

In Algorithm 1, the two sub-problems P1 and P2 are solved in an iterative manner until convergence. Specifically, the outer-layer iteration computes $X$ with a given $P$, while the inner-layer iteration computes $P$ with the specific $X$. Note that as long as the goal of JUAPA in each iteration is to maximize the same objective function, Algorithm 1 is guaranteed to converge.

In this section, the theoretical analysis presented in previous sections is verified by MATLAB simulations. As shown in Fig.2, in the simulation scenario, a circular flat area with 500 m radius is depicted, where the BS is located at the center, 20 RSs are evenly located on a circle at a distance of twothirds the radius from the BS, and 100 users are uniformly distributed. The model of large-scale fading channel power gain is borrowed from Ref. [11], where the path loss of the BS-MS link in dB is specified as pl₁=128. 1+37. 6 lg d(km) and the path loss of the RS-MS link is specified as pl₂ =140.7+36. 7 lg d (km). Then, the corresponding large-scale fading channel power gain is derived by $g_{io}=-p{l}_1+z\left({\sigma }_{BS}^2\right)$ and $g_{ij}=-p{l}_2+z\left({\sigma }_{BS}^2\right)$, j> 0.Specifically, $z\left({\sigma }_{BS}^2\right)$ and $z\left({\sigma }_{BS}^2\right)$ represent shadow fading generated from the log-norm Gaussian distribution with standard deviations σ_BS and σ_RS, respectively. Coefficients for power consumption under the linear zero-forcing beamforming scheme are set as c_{0, 0}=4×10⁻³, c_{1, 0}=4. 8×10⁻³, c_{2, 0}=2. 08×10⁻⁸, c_{0, 1}=1×10⁻³, c_{1, 1}=9. 5×10⁻⁸, and c_{2, 1}=6. 25×10^−8[16]. The BS has M antennas and serves a user set with maximum size S=M/10.The remaining simulation parameters are listed in Tab.1.

The performance under the proposed JUAPA algorithm is compared with those of user association-only schemes including the proposed user association, max reference signal received power user association(Max RSRP UA), and Max SNR UA from several different perspectives.

Fig.3(a)shows the geometric mean of EE versus the number of antennas mounted at the BS. In the simulation, the step size is set as δ(t)=1/t. The sub-gradient method and Newton’s method in the proposed JUAPA algorithm achieve fast convergence with less than 50 iterations, and the outermost iterative algorithm converges in less than 5 iterations. Note that the proposed JUAPA algorithm performs best in terms of the EE metric, whereas the proposed UA scheme performs worst and exhibits wide swings, which demonstrates that the JUAPA design for the AF relay-aided massive MIMO system is of great significance. Although the EE under the proposed JUAPA algorithm declines slowly after reaching its peak value, it is still superior to the others, which can be attributed to its capability to maintain low power consumption. The third point to note is that the EE will not continue to increase with an increasing number of antennas because the circuitry power consumption for a large number of antennas is non-negligible at the BS, which restricts the EE from infinitely increasing with the number of antennas.

Fig.3(b) shows the number of users associated with the BS versus the number of antennas at the BS, which can di rectly reflect allocation fairness under the proposed JUAPA algorithm from another viewpoint. Note that under the Max RSRP UA scheme, the number of users associated with the BS continues to increase, and soon, all users associate with the BS, which results in extremely unfair occupation and enormous waste of resources at RSs. The number of users associated with the BS under the Max SNR UA scheme is fixed, whereas the number of users associated with the BS under the proposed JUAPA algorithm is significantly lower, which validates that the proposed JUAPA algorithm can achieve better load balancing than the conventional Max RSRP UA/Max SNR UA schemes because of its proportionally fair utility in the objective function.

In Fig.3(c), the trend of total power consumption with respect to the number of antennas at the BS is shown. It is observed that total power consumption rises steadily with an increasing number of antennas, but the gap between the proposed JUAPA algorithm and the Max SNR UA scheme is large, which further indicates that the joint optimization algorithm can significantly reduce power consumption and elevate the EE owing to efficient power allocation.

Fig.4 shows the transmit powers at the BS and RSs versus the number of antennas at the BS. Obviously, compared with user association-only schemes with p₀=20 W and p _j=1 W, transmit powers at the BS and RSs achieved by the proposed JUAPA algorithm remain at low levels. The average transmit power of an RS is about 0.3 W. Combining with Fig.3(a), note that even if the transmit powers at the BS and RSs are low, the EE obtained by the proposed JUAPA algorithm is high. The reason is that the system SE rises rapidly with an increasing number of antennas at the BS in the early stage, which exceeds the growth rate of power consumption; whereas the situation completely reverses in the late stage with low levels of transmit powers and an increasing number of antennas. Thus, there exist an optimum number of massive MIMO antennas in the AF relay-aided massive MIMO system under the proportional fairness criterion applicable to system EE. From Fig.3(a), the optimum number of massive MIMO antennas is about 150.

Figure4 Transmit powers at BS and RSs versus the number of antennas

The EE of downlink transmissions in an AF relay-aided massive MIMO system has been investigated from the JUAPA perspective. The proportionally fair utility of the system EE was maximized while guaranteeing QoS requirements of users and applying transmit power constraints on the BS and RSs. This optimization problem was solved by decomposing it into a user association sub-problem and a power allocation sub-problem. A low-complexity iterative algorithm was designed to seek a sub-optimal solution of the original problem. Simulation results verify that the proposed algorithm can achieve better load balancing and a higher system EE than user association-only schemes, and keep the BS and RSs operate at low power levels. Moreover, the number of massive MIMO antennas has a significant impact on the system EE and there exist an optimum number of massive MIMO antennas that maximize the proportionally fair utility of the system EE.