Energyefficient communications, namely, green communications, has attracted increasing attention due to energy shortage and greenhouse effect. Motivated by this, we consider the uplink energyefficient resource allocation in multiuser massive multipleinput multipleoutput (MIMO) systems. Specifically, we consider that both the number of antenna arrays at the base station (BS) and the transmit data rate at UE are adjusted adaptively to maximize the energy efficiency. Firstly, we demonstrate the existence of a unique resource allocation solution that is globally optimal by exploiting the properties of objective function. Then we develop an iterative algorithm to solve it. By transforming the originally fractional optimization problem into an equivalent subtractive form using the properties of fractional programming, we develop another efficient iterative resource allocation algorithm. Simulation results have validated the effectiveness of the proposed two algorithms and have shown that both algorithms can fast converge to a nearoptimal solution in a small number of iterations.
The amount of energy consumption for information and communication technology (ICT) increases dramatically with the exponential growth of service requirement [
Recently energyefficient design has emerged as a new trend in wireless communications. The tradeoff between EE (energy efficiency) and SE (spectral efficiency) in downlink multiuser DAS (distributed antenna systems) is addressed in [
It is worth mentioning that all the above works only consider energyefficient resource in singleantenna or fixedbeam OFDM system. There exist few works on the energyefficient design concerning antenna selection for the massive MIMO system. In [
The remainder of this paper is organized as follows. In Section
As shown in Figure
The system model.
As is shown in [
Correspondingly, the sum data rate is lower bounded by
The overall transmit power can be written as
Then, the overall power consumption can be expressed as
The energy efficiency of interest is then given by
Therefore, the resource allocation for the uplink multiuser massive MIMO system can be formulated as the following optimization problem:
The objective function is a ratio of two functions which generally is a nonconvex function. Next, we will propose two approaches to address this issue. The basic idea of the first approach is to first demonstrate the existence of a unique globally optimal point for the data rate and antenna selection by exploiting the properties of the objective function and then to develop an iterative algorithm to directly solve the optimization problem. While, in the second approach, we first transform the originally fractional optimization problem into an equivalent subtractive form by exploiting the properties of fractional programming, we then develop an efficient iterative resource allocation algorithm to obtain this optimum.
In the following, we first develop Lemma
The energy efficiency function
See Appendix
The energy efficiency function
There exists a unique globally optimal number of base station antennas
There exists a unique globally optimal transmit rate vector
Theorem

Conditions 








Conditions 






Theorem
(1)
(2)
adopt
Return
In Algorithm
Obviously, the convergence rate and accuracy of Algorithm
Consider
for
According to Theorem
The problem above is now joint concave with respect to all variables if
Using standard optimization technique, the number of base station antennas is
Therefore, we develop an iterative algorithm to search the optimal
Note that this iterative algorithm guarantees convergence. Please refer to [
(1)
(2) while
do
adopt formula (
adopt formula (
Return
In this section, we provide the simulation results to evaluate the performance of our proposed algorithms varying with the number of users. Moreover, we also provide the simulation results to evaluate the energyefficiency performance varying with the number of iterations and the number of BS antennas. In order to reduce the computational complexity of optimal algorithm, throughout the simulations, we assume that the users are divided into three groups according to their distance to the base station; that is,
Figure
Energy efficiency versus number of users.
Figure
Number of base station antennas versus number of users.
Figure
Overall transmit power versus number of users.
Figure
Spectral efficiency versus number of users.
Figure
Energy efficiency versus the number of iterations.
Figure
Energy efficiency versus number of base station antennas.
Note that we evaluate the energy efficiency via expression (
Energy efficiency versus number of users.
In this paper, we have investigated uplink energyefficient resource allocation in multiuser massive MIMO systems. Our goal is to jointly optimize rate allocation and the number of antenna arrays at BS, in order to make the performance measure in terms of throughput per Joule maximizing, in which the power consumption includes both transmit power and circuit power. We first demonstrate the existence of a unique globally optimal solution by exploiting the properties of objective function and then develop an iterative algorithm to solve the resource allocation problem. Furthermore, we propose a more efficient iterative algorithm by transforming the originally nonconvex optimization problem in fractional form into an equivalent optimization problem in subtractive form. It is proved that the proposed two algorithms guarantee convergence. And the numerical results show that the proposed algorithms converge to a nearoptimal point with a small number of iterations.
(i) For a fixed
According to proposition C.9 of [
The partial derivative of
The derivative of
Because of the
When
When
(ii) The quasiconcavity of
The partial derivative of
The derivative of
When
When
Then, Lemma
Without loss of generality, we define function
The Hessian matrix is given by
As a matter of fact, the function
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Science and Technology Major Project of China under Grant 2012ZX03004005003, National Natural Science Foundation of China under Grants 61201172, 61271018, 61071113, and 61201176, Joint Funds of the National Natural Science Foundation of China under Grant U1404615, Open Funds of State Key Laboratory of Millimeter Waves under Grant K201504, the Research Project of Jiangsu Province under Grants BK20130019 and BE2012167, and the Program for New Century Excellent Talents in University under Grant NCET110088.