1. Introduction
Concerning the vision of ubiquitous wireless intelligence, emerging Internet of Everything applications will require a convergence of communication, sensing, control, and computing functionalities [
1]. Many widely anticipated future services, including eHealth and autonomous vehicles, will be critically dependent on the delivery of high reliability and low latency with high data rates, which consequently calls for a higher area throughput and delivering must-have services due to the imminent data traffic crunch and the rising expectations of service quality. Moreover, the amount of data traffic growing at an exponential pace entails not only a dramatic improvement in spectral efficiency (SE) but also steep power consumption. Consequently, both economic and environmental concerns are in compelling need of consideration in addition to the demand for data traffic [
2]. To this end, energy efficiency emerges as SE is unable to characterize an energy-efficient network, which is highlighted in next-generation network designs. Notably, massive multiple-input multiple-output (MIMO) stands out in various candidate technologies as providing significant improvements in spectral efficiency and energy efficiency, which enables a set of user equipment (UE) to serve over the same time–frequency interval by deploying multiple receive antennas [
3]. Hence, energy efficiency optimization in massive MIMO has received tremendous attention.
Spurred by both economic and environmental concerns, energy efficiency (EE) has been exposed to extensive research to satisfy the energy-efficient performance metric vital to the sixth-generation communication network. Academic researchers have spared no effort to seek seminal contributions to improving energy-efficient performance, such as resource allocation, energy harvesting, and network deployment [
2]. Herein, we cast our attention to power allocation as maximizing the energy efficiency with a given transmit power budget in line with not increasing energy consumption. The study [
4] considered the maximization of the global energy efficiency, as well as of the minimum energy efficiency, which is nonconvex. Exploiting the fractional programming and sequential convex develops a power allocation algorithm that guarantees the convergence to a Karush–Kuhn–Tucker (KKT) point. However, it comes at the cost of high computational complexity and feedback requirements. In the study [
5], EE optimization in the downlink multi-cell massive MIMO was investigated, which took into account different users’ quality-of-service requirements. An iterative optimization algorithm was obtained by alternating optimization about the optimal amount of data rate, the number of antennas, and users. Thus, it was of high complexity incurred from the alternating iterations of multiple parameters. The study [
6] proposed an adaptive power allocation to maximize the SE and EE in a MIMO broadcast channel. Applying the Lagrangian method solved the objective function, which involved a threshold of the effective capacity for each user. However, both [
5,
6] apply to the special case of spatially uncorrelated fading channels with perfect channel state information (CSI). The study [
7] used an energy-efficient low-complexity algorithm (EELCA) to obtain an optimal power allocation solution based on Newton’s methods in the noise-limited scenario, and it exploited the linear power allocation and perfect CSI. In fact, it is challenging to acquire perfect CSI due to the presence of the channel estimation error. The studies [
8,
9] proposed power allocation for an energy-efficient massive MIMO with imperfect CSI, while the power consumption models were linear about antennas and UE. The study [
10] investigated energy efficiency and the spectral efficiency trade-off for a single-cell massive MIMO downlink transmission with statistical channel state information available at the transmitter. A low-complexity suboptimal two-layer water-filling-structured power allocation algorithm was proposed, which reached near-optimal performance. Practical channels were generally spatially correlated, also known as having space-selective fading [
11]. In the study [
12], to address the EE optimization in a multi-cell downlink massive MIMO operating over spatially correlated Rician fading channels with imperfect CSI, the authors transformed the problem into a geometric program and developed an iterative power allocation algorithm under the constraints of a given sum spectral efficiency and a maximum total. Nevertheless, the closed-form solution came at the expense of staggering computational complexity.
From the above analysis, extensive studies have been conducted on EE optimization in massive MIMO with perfect CSI, whereas the case with imperfect CSI in an interference-limited scenario is still open to be studied due to its nonconvexity. In addition, the linear power allocation model taking users and antennas into account has been widely used, which is adopted in studies [
13,
14,
15], for instance. Consequently, we consider spatially correlated Rayleigh fading channels with imperfect CSI and exploit the nonlinear power consumption model, which encompasses dynamic power about throughput and nonlinear power terms from BS’s computation in contrast to the linear power model. The global optimum acquisition comes at the price of unbearable computational complexity as the objective is tough to convert to a convex problem. Herein, we propose a computationally efficient power allocation scheme called Dinkelbach-like power allocation to obtain a suboptimal solution with limited complexity, and the simulation results manifest that the proposed power allocation is a computationally efficient method that jointly increases EE and SE and, meanwhile, performs well in SE fairness compared to several reference schemes.
The remainder of this paper is organized as follows: The spatially correlated channel model is described and spectral efficiency is derived by adopting the minimum mean-squared error (MMSE) estimator and maximum ratio (MR) precoding in
Section 2. In
Section 3, a realistic power consumption model is considered and the EE optimization of multi-cell multi-user massive MIMO is obtained. Subsequently,
Section 4 proposes a sub-optimal power allocation algorithm. In the last section, the simulation results and conclusion are presented.
Notation: In this paper, matrices and column vectors are denoted by upper-case bold letters and lower-case bold letters, respectively; and denote the conjugate transpose and the complex conjugate of matrix A, respectively, and is the transpose; denotes a unit matrix of order M; denotes the Euclidean norm of a scalar, and denotes the mathematical expectation. denotes the (j, k)th element of a matrix .
3. Power Consumption Model and Energy Efficiency
The EE of a cellular network is the number of bits that can be reliably transmitted per unit of energy [
17]. Energy efficiency is the ratio of throughput and energy consumption.
3.1. Power Consumption Model
Tractable but less realistic models may instead reach a misleading conclusion about EE [
20]. Herein, we introduce a power consumption model that quantifies the circuit power incurred by signal processing, backhaul signaling, encoding, and decoding. It can be quantified as
where
and
are the power amplifier efficiencies of UE and BS, respectively, and
is the power for uplink (UL) pilot transmission. For more details, please refer to the monograph [
17], for the sake of space constraints. Notice that the power model is derived concerning the MMSE estimator and MR signal schemes adopted. The parameters are explained in the simulation parameter setting.
3.2. Downlink Energy Efficiency
Global energy efficiency (GEE) as the optimization objective we adopt herein, which sticks to the physical meaning of energy efficiency, is a ratio of throughput and energy consumption. The GEE optimization follows from substantial insight into the above. Subsequently, we put forward the optimization problem, which is mathematically formulated as:
P is the set of all feasible transmit power solutions that satisfy the given constraints on the BS of each cell, where refers to the optimization variable. Before delving into the EE analysis, from intuition, it can be grasped that the numerator is not jointly concave in , nor for the denominator, which means it requires exponential complexity or is even more intractable. For analytical simplicity, we tackle energy-efficient power allocation optimization by keeping the M fixed. As we have observed, no computationally efficient algorithm exists to solve a problem that is not jointly concave in , where . To date, determining the global solutions of energy-efficient power allocation in interference-limited scenarios is still an open problem. There is a performance–complexity tradeoff from the analysis above. It is tricky to obtain a global optimum dealing with a generic optimization.
4. Dinkelbach-Like Power Allocation
In what follows, we propose a suboptimal algorithm based on Dinkelbach’s algorithm called the Dinkelbach-like algorithm to optimize EE in the setup scenario with lower computational complexity. As we can see, the optimization problem belongs to nonconcave programming problems and requires high computational complexity to obtain an optimum. Motivated by Dinkelbach’s algorithm, we transform the EE problem in a fractional form into a subtractive optimization form called the auxiliary sub-problem employing Dinkelbach’s algorithm. To solve globally the auxiliary sub-problem given a parameter in each iteration of Dinkelbach’s algorithm entails unaffordable complexity as it is not a concave problem, which prevents the application of the formal Dinkelbach’s algorithm. To this end, we relax the sub-problem to a concave problem by initializing the interference and omitting the dynamic power term about throughput, subsequently iteratively solving the KKT conditions by bisection search.
4.1. Transformation of EE Optimization Problem
Dinkelbach’s algorithm belongs to the class of parametric algorithms [
21], whose basic idea is to tackle a concave-convex fractional problem (CCFP) by solving a sequence of easier problems that converges to the global solution of the CCFP [
22]. Unfortunately, the objective of the EE optimization problem is a generic fractional problem instead. Similarly, we transform the EE optimization problem via Dinkelbach’s algorithm, expressed as
Then, the global optimum can be obtained using Algorithm 1. Unfortunately, the maximization of the sub-problem is solved globally at the price of high computational complexity, even more impossible to solve. It is critical to tackle the sub-problem properly with affordable complexity.
Algorithm 1. Dinkelbach’s algorithm |
Input |
Output |
1: whiledo |
|
|
|
|
6: end while |
4.2. Solution of Auxiliary Sub-Problem
As we can see, Dinkelbach’s algorithm requires solving globally the auxiliary sub-problem in step 2 given a parameter
value. Herein, we obtain an auxiliary sub-problem expressed as
As aforementioned, the first term of the objective is non-jointly concave in , so the second is also naturally nonconcave due to the presence of , which means it does not enjoy the convexity that enables solving globally with limited complexity. Hence, we propose a low-complexity algorithm at the price of obtaining a sub-optimum of (17) called the primal auxiliary sub-problem. The concave approximation is obtained after specific pretreatment to (17), and then we can iteratively solve the KKT conditions of the former globally. As a consequence, the global optimum guarantees the convergence of the proposed algorithm and is an approximate solution to (17).
4.2.1. Initialization of the Sub-Problem
The difficulty lies in the fact that the first term is not jointly concave, due to the presence of interference and the second term couples the first term. To this end, an approximation more tractable can be obtained through initializing the SINR-like term in equal power allocation, given by
After specific pretreatment,
is
Considering the multi-cell MIMO system is symmetric, we decompose the coupled L cells into L individual cells as each link depends only on the transmit power of its own cell after the initialization operation. In the meanwhile, the power consumption omits the dynamic counterpart related to , which is a valid approximation followed by neglecting small terms as explained in the next section. It can be observed that a concave problem holds provided the nonnegativity of , which is ensured due to the nonnegativity of the numerator and the positivity of the denominator.
4.2.2. Acquisition of KKT Conditions
The global optimum follows from the convexity’s optimality condition, which guarantees the convergence of the Dinkelbach-like algorithm. If and only if each auxiliary sub-problem is solved globally, the generalized Dinkelbach’s algorithm converges to the global solution of the fractional problem [
21]. Herein, (18) could be solved globally utilizing any convex programming algorithm. Denoting the Lagrange multipliers for constraints by
and
, the Lagrangian function of (15) is written as
In addition, the KKT conditions of (18) are expressed as
where (21a) is the Lagrangian stationarity conditions, (21b) is the nonnegativity of the multipliers, (21c) and (21d) are the problem constraints, while (21e) is the complementary slackness condition. Any solution of (21) is also a global solution of (17) given its convexity. To solve the KKT system (21) directly is difficult, so an iterative algorithm can be developed by starting from a feasible transmit power vector and iteratively updating the transmit powers by (22), the updating equation, given by
This can be efficiently solved by Algorithm 2 on the multiplier
. The resulting formal procedure can be stated as follows:
Algorithm 2. Bisection for sub-problem |
Input: Pmax itermax |
Output: |
1: Initialize: compute according to equation (22)
|
2: while Pmax do |
3: compute according to equation (22)
|
4: end while |
5: compute according to equation (22)
|
6: while Pmax + || Pmax + do |
7: if iter > itermax then |
8: break |
9: end if |
10: if Pmax then |
11: |
12: else |
13: |
14: end if |
15:; compute according to equation (22); iter iter + 1
|
16: end while |
17: return |
4.3. Iteration of the Parameter
In the end, we generalize Dinkelbach’s algorithm to obtain a suboptimum of the original EE optimization (15) with affordable complexity after approximating the sub-problem. The Dinkelbach-like algorithm can be stated as in Algorithm 3.
Algorithm 3. Dinkelbach-like algorithm for power allocation |
Input: |
Output: |
1: initialize: |
2: while do |
3: solve the sub-problem according to algorithm 2
|
4: |
5: |
6: |
7: end while |
6. Conclusions
For the multi-cell multi-user massive MIMO downlink energy efficiency optimization in an interference-limited scenario, we propose a sub-optimal Dinkelbach-like algorithm with limited complexity under the constraint of maximum transmit power. To make the scheme more practical, we adopt the more realistic power consumption model and fully consider the impact of dynamic power terms. The simulation results confirm the rationality of the proposed algorithm, which can jointly increase EE and the average sum of SE with lower complexity, and meanwhile obtain satisfactory performance in SE fairness. Hence, it is a candidate as a computationally efficient method adapted for the low-latency requirement of the sixth-generation communication, as the proposed algorithm strikes a good balance between performance and complexity.
As we can see, SE grows with antenna number without bound. However, equipping excessive antennas not only brings extra power consumption but also increases the complexity of system design. Hence, in future work, it is optional to consider the joint optimization of antenna selection and power allocation to promote energy conservation at the marginal loss of SE. In addition, if we want to reach a higher SE and EE in massive MIMO, the simulation result shows that it is wise to search for more energy-efficient transmit chains. Noticeably, power allocation is irreplaceable as a method that improves EE and SE through the rational allocation of resources within a limited resource budget. Therefore, power allocation is of significant consideration in energy-efficient optimization whatever the optimized system design is in future work.