1. Introduction
Non-orthogonal multiple access (NOMA) can be considered to be a promising multiple access technology for fifth-generation networks for accommodating the rapid growth of data traffic [
1,
2,
3,
4]. Unlike orthogonal multiple access, NOMA allows user equipment (UE) to share the same time-frequency resource; hence, the spectral efficiency (SE) and energy efficiency (EE) are improved. The NOMA technique is classified into two main categories: the power domain [
5] and code domain [
6]. Herein, we focus power-domain NOMA, referred to as NOMA for convenience. The key idea of power-domain NOMA is to allocate a higher power to the UE with poorer channel gain. Meanwhile, a UE with a strong channel gain is equipped with successive interference cancellation (SIC) to remove the signal intended to the UE with poorer channel gain before decoding its own signal [
5,
7].
1.1. Related Works and Motivation
Recently, many new approaches have been proposed for UE grouping/clustering [
8,
9,
10,
11]. The number of clusters is selected based on the cell size and channel gain in [
8], whereas a clustering strategy based on channel correlation was investigated in [
11]. In [
9], a tensor model was proposed to manage UE association, in which clusters of UEs were formed by different circular zones. Owing to the channel conditions and computational capacity of UE, SIC is restricted to limited groups/clusters of UE [
8,
9].
To ease implementationin a small-sized network, UE pairing is essential. In [
12], pairing of one near UE and another far UE was shown to be efficient. A cognitive radio inspired NOMA system with a single-antenna base station (BS) was also investigated in [
12], where a cell is divided into inner and outer zones, including cognitive UEs (CUs) with strong channel gain and primary UEs (PUs) with poor channel gain. A two-zone scheme assisted by the multiple-antenna technique was studied in [
13], in which each pair comprised one inner-zone UE and the other outer-zone UE. The authors in [
14] proposed a distance-based UE pairing scheme to improve the sum throughput, in which the far UEs were paired with an available nearest UEs using a distance threshold. Another pairing scheme based on the vector norms of channel responses was developed for a massive MIMO-NOMA system in [
15]. The effect of UE pairing on the performance of NOMA-based systems was studied in [
16], in which a CR-NOMA and an F-NOMA system were considered under fixed power allocation.
To fully exploit the advantages of UE pairing, many algorithms have been developed for pairing two arbitrary UEs without separate zones. In addition to random pairing scheme used in [
17], UE pairing schemes based on channel condition arrangement have been proposed [
17,
18,
19]. The typical pre-processing step is to place all UE into an unpaired set with sorted channel gains (SCG). Under the setting, various strategies have been proposed to pop and pair two UEs until the SCG is empty or only contains one UE. In [
19], two UEs with the best and worst channel gains were popped and paired consecutively. In [
18], two best-channel conditions were selected, whereas the first and middle UEs in the SCG were paired together in [
17]. Unlike the aforementioned work, a more flexible approach was proposed [
20], where any two users in a cell can be dynamically paired together to improve the fairness throughput among all UEs.
Most of the aforementioned works investigated NOMA-based systems to improve SE. In power-domain NOMA, power control is extremely challenging owing to SIC. In addition, the power consumption or EE has been considered to be one of the immediate concerns for the upcoming wireless networks, as indicated by the energy for information and communication technologies, which constitute approximately five percents of the total world’s energy consumption [
21,
22]. Therefore, power consumption has been recently reported in a few studies. The authors in [
10] investigated a joint optimization problem of user assignment and power allocation for a downlink (DL) NOMA network in terms of EE maximization. Another power allocation strategy was proposed in [
23] to evaluate the EE of a broadcast DL NOMA network. These studies investigated the sub-channels assignment for UEs; hence, the UE association is still unknown for exploiting the channel capacity using NOMA. This motivates us to develop algorithms with a joint UE pairing and beamforming design that accounts for both the SE fairness and power consumption.
1.2. Contributions
In this paper, we consider a cellular DL NOMA network, where a multi-objective problem for rate fairness among all UEs and power consumption were investigated under the minimum per-UE rate requirement and signal-to-noise ratio (SNR) constraints for decoding the message. To further improve the channel capacity, the NOMA technique can be applied to two UEs in a cell, instead of UE pairing with two near and far groups. The resulting problem belongs to a mixed-integer non-convex class that is difficult to solve. Therefore, we devise two low-complexity algorithms, based on inner approximation (IA) and graph theory. The main contributions of this paper are described as follows:
First, we investigate a hybrid system comprising NOMA and conventional beamforming to reap their advantages for a unique design. A beamforming design with SIC can be applied to a pair of UE, or conventional beamforming can be used for certain UE depending on the number of UEs and channel conditions. Accordingly, the approach is referred to as dynamic pairing because it enables the pairing of two arbitrary UEs in a cell.
To manage the dynamic pairing, we introduce a new binary variable matrix for UE association. By using the NOMA principle and sorted channel gains, the computational complexity is reduced with the variables forced into an upper triangular matrix. Next, we formulate a problem for joint optimization of UE pairing and beamforming design, thereby enabling the selection of the objective function: the max-min rate (MMR) and power consumption. However, a strong coupling of binary and continuous variables under non-convex constraints results in a hard class of mixed-integer non-convex programs.
To efficiently solve the resulting problem, we propose two approaches based on the IA framework as well as the relaxation method and bipartite graph. The first approach is based on a combination of the relaxation and IA methods; it can transform the complicated original problem into a more tractable form without integer variables. Therefore, an iterative algorithm based on the IA framework is derived to obtain a local optimal solution at the least. To further reduce the time complexity, we develop an algorithm in which UE pairing and the beamforming design are managed using a bipartite graph and the IA method. For UE pairing, this approach uses the channel correlation as the weights of edges on the graph, resulting in a linear bottleneck assignment problem. Hence, we apply the binary search and augmenting path algorithm to obtain the UE pair, and then use the solution of the first approach for the beamforming design based on a specified integer variable value.
1.3. Paper Organization and Notation
The rest of this paper is organized as follows. The system model and problem formulation are described in
Section 2 and
Section 3, respectively. Two proposed solutions based on the IA framework with the relaxation method and graph theory are detailed in
Section 4 and
Section 5, respectively. For the benchmark purpose, the optimal solution based on exhaustive search is also studied in
Section 5.
Section 6 presents numerical results, whereas
Section 7 concludes the paper.
Notation: Lowercase and uppercase letters in bold indicate vectors and matrices, respectively. is the Hermitian transpose of . The set of all complex and real numbers are represented by and , respectively. stands for the Euclidean norm. is the expectation and denotes the real part of a complex number.
2. System Model
We consider a DL NOMA cellular system, where a BS is equipped
N transmit antennas to serve
K single-antenna UEs. As illustrated in
Figure 1, NOMA technique can be applied to a pair of two arbitrary UEs in a set of UEs, denoted by
. For simplicity, we denotes the
k-th UE by
.
2.1. Data Transmission Model
The DL channel from BS to
and the corresponding beamforming vector are denoted as
and
, respectively. Without loss of generality, the channel gains of all UEs are sorted as
. The signal received at
can be written as
where
with
represents the symbol intended for
, and
represents the additive white Gaussian noise (AWGN) at
. To perform UE pairing, we introduce binary variables
, which are expressed as
By defining
, it is clear that
is an upper triangle matrix with the main diagonal of zeros. The signal-to-interference-plus-noise ratio (SINR) at
can be written as
where
. The SINRs for decoding the
’s signals at itself and
are denoted by
and
, respectively, which can be expressed as
where
and
are respectively given as
It is seen that when and are not paired together, i.e., , only is involved in Equation (3), since .
2.2. Power Consumption Model
To measure power-efficiency, we consider the power consumption for transmission with beamforming, which can be expressed as
where
is the efficiency of the power amplifier (PA) at the BS. In other words, the power consumption is the power for data transmission from the BS to all the UE in the form of electromagnetic wave radiation. It is clear that
is a function of beamforming vectors.
Remark 1. In addition to the power for beamforming, the power consumption in a wireless communication system includes the power for signal processing, device activation, and circuit operation [24,25]. However, these values are constant, and hence can be omitted in the objective function without affecting the search for the optimal solution. For convenience, we consider only the power consumption for beamforming in this paper. 3. Problem Formulation
The achievable rate of
in nats/s/Hz can be expressed as
We introduce a parameter
for the objective selection between rate fairness and power consumption optimization. As a result, a multi-objective problem can be formulated as
Objective Equation (8a) can be selected for fairness rate maximization (or power consumption minimization) by setting (or ). Constraints Equation (8c–h) correspond to the NOMA criteria for user pairing. Specifically, Equation (8d) ensures that in a certain pair, UE with the better channel employs SIC to remove the interference from the remaining UE. Constraints Equation (8e–h) guarantee that each UE can be paired with at most one other UE. The signal-to-noise ratio (SNR) threshold in Equation (8i) is used for detecting the signal at the receivers, whereas constraint Equation (8j) provides the minimum bit rate, , such that UEs satisfy the QoS requirement. Owing to a strong coupling of continuous and binary variables under the non-convex objective function and constraints, Equation (8) is a mixed-integer non-convex problem, and thus, it is hard to obtain a globally optimal solution.
Remark 2. Although the proposed dynamic pairing approach can further enhance the system performance, it leads to a large number of possible pairing cases. To overcome this issue and to accelerate the application of NOMA technique, the rest of the paper will devise two algorithms which provide low complexity and fast convergence.
4. Proposed Solution Based on Relaxation Method
To convert Equation (8) to a tractable form, the binary variables in constraint Equation (8c) are relaxed to continuous ones as
. Then, the optimization problem Equation (8) can be rewritten as
where
is a new variable representing the minimum rate among all UEs and
denotes the rate of each UE. Furthermore, Equation (9d) is the QoS constraint, which is equivalent to Equation (8j).
It can be observed that constraints Equations (8i) and (9c) are non-convex and thus difficult to solve directly. To deal with the issue, we first introduce some useful IA functions as follows.
We consider the multiplication function
. Then an upper bound of
is provided in Equation (B.1) in [
26]:
The logarithm function
with
, has a lower bound around a feasible point
[
27]:
A lower bound of the quadratic function
can be expressed as
Let us convexify the QoS constraint Equation (8c). By using Equation (12), Equation (8c) can be approximated as
resulting in an alternative convex constraints:
To convexify constraint Equation (9c), we can split Equation (9c) into two sub-constraints as
Using Equation (11), a lower bound of
around a feasible point
at iteration
is derived as
where
,
, and
are respectively given as
with
,
and
. Clearly,
is still non-convex; hence, we apply Equation (10) to convexify
as
where
is introduced as a new variable satisfying the following second-order cone (SOC) constraint:
Therefore, the convex approximation for constraint Equation (15a) at iteration
can be written as
It can be observed that constraints Equation (15a,b) take the same form. Hence, constraint Equation (15b) at iteration
is approximated by the same steps as Equations (16)–(19), i.e., Equation (15b) is convexified as
where
, and
are respectively defined as
where
,
, and
.
In summary, a successive convex program at iteration
, which provides minorant maximization for Equation (9), can be formulated as
Clearly, when the iterative algorithm terminates, the value of the relax variable
is approximately the binary value, and not a solution of Equation (8). To address this point, the binary values of
for the optimal solution of Equation (8) is recovered by the rounding function Equation (22), i.e.,
where
returns the maximum integer that is not larger than
.
For implementation purposes, it is necessary to find an initial feasible point for Equation (21). This can be achieved by solving the following sub-problem:
where
is a new slack variable. Constraint Equation (23b) guarantees that the initial feasible point can satisfy the QoS requirement. It can be seen that a starting point
is obtained when
is approximately zero. The proposed iterative algorithm based on the IA method to solve the problem Equation (8) is summarized in Algorithm 1. In particular, the solution of continuous variables for the relaxation problem is found in Phase 1, while Phase 2 is devoted to re-compute the solution for the original problem with a fixed value of the integer variable
.
Convergence analysis: Our proposed algorithm based on the IA method has the convergence properties as in [
28]. Let
be the feasible point set of Equation (21) at iteration
i, then the properties of IA method indicate that the optimal solution
obtained at iteration
i satisfies
as well. In addition,
is a connected set [
29]. Therefore, our proposed algorithm can provide at least a local optimal solution that satisfies the Karush-Kuhn-Tucker conditions.
Complexity analysis: The approximated problem Equation (21) includes
SOC/linear constraints and
variables, which complicates the solving of Equation (21) in terms of the big-O expressed as
[
30].
Algorithm 1 Proposed Algorithm for Solving Problem Equation (8) based on IA Method. |
Phase 1:- 1:
Input-1:, , , and . - 2:
Initialization: Set . - 3:
Get an initial point: Execute the sub-problem Equation (23) to obtain an initial point for Equation (21). - 4:
repeat - 5:
Solve the optimization problem Equation (21) to find at the i-th iteration. - 6:
Update . - 7:
Set - 8:
until Convergence - 9:
Use the rounding function Equation (22) to recover . - 10:
Output-1: The optimal solution for the binary variable . Phase 2:- 11:
Input-2:, , , , and . - 12:
Reset and repeat steps 1–7 with binary to obtain the exact beamforming vector . - 13:
Output-2: The optimal solution ().
|
5. Proposed Solutions Based on Integer-Variable Reduction
This section presents the reduction in the computational complexity using integer-variable reduction (IVR). Unlike the relaxation method, this approach first determines the value of for UE pairing; subsequently, power allocation is computed by solving a sub-problem with respect to the given . We investigate two methods to find the UE pairs: (i) a graph-based IVR method to reduce the complexity; and (ii) an IVR using exhaustive search for benchmarking.
5.1. Proposed Solution Using Graph-Based IVR
In this subsection, we propose a low-complexity pairing scheme based on channel vectors. Accordingly, a dynamic pair of UE is determined based on a high channel correlation, to which the SIC technique can be effectively applied. It is observed that when the UEs with high channel correlation are paired to each other, the BS has more degree of freedom for beamforming design to mitigate the inter-pair interference.
To implement this method, we first define a weight matrix
, in which each entry
denoting the channel correlation between the
k-th and
ℓ-th UEs is calculated as
where
denotes the inner product of two vectors
and
. Subsequently, the UE pairs can be determined by solving the following linear bottleneck assignment problem (LBAP):
It is noteworthy that when , no edge exists between the k-th and ℓ-th UEs. To solve problem Equation (25), we use the bipartite graph and binary search methods as follows.
5.1.1. Bipartite Graph for UE Pairing
We employ a balanced bipartite graph
, where
and
are the vertex sets of all UEs with
, and
is the set of edges, i.e.,
, with the weight of an edge
being
. UE pairing can be performed by using the augmenting path algorithm [
31]. Some definitions for developing the pairing algorithm are as follow:
A matching is a subset of without any two edges sharing the same vertex. For convenience, we define an unmatched set . Furthermore, a maximum matching corresponds to a special case, in which is enlarged to the maximum.
A free vertex is defined as a vertex not contained in .
An alternating path is a path in graph , such that two arbitrary successive edges in are from both sets and .
An alternating path becomes an augmenting path when both the starting and ending points of are free vertexes.
We can use the augmenting path algorithm to expand until a maximum matching is obtained. In particular, at each iteration of an iterative algorithm, the augmenting path is determined for a given matching ( at the beginning), and then, is updated as . The algorithm terminates when no augmenting path is found. However, a maximum matching with the augmenting path algorithm merely addresses the maximum pairs of UEs, which is insufficient for obtaining the optimal solution for problem Equation (25) under a balanced bipartite graph in a polynomial-time complexity. Hence, the augmenting path algorithm is integrated into a binary search method as will be described in the following subsection.
5.1.2. Binary Search with Augmenting Path Algorithm
The optimal solution for problem Equation (25) is a maximum matching with channel correlation fairness among UE pairs. In other words, the pairs of UEs are determined such that the minimum channel correlation of UE pairs in the achieved maximum matching is maximized. Moreover, the maximal value for the weight fairness is one of the entries in
. Therefore, the binary search algorithm [
32] is used to find the solution for Equation (25). The search region is given by an auxiliary vector
, of which the components involving all entries of matrix
are sorted in the ascending order, i.e.,
. Let
and
be the lower and upper indexes for the search region on vector
, respectively. Clearly,
and
at the initialization. The search region is denoted by
. At each iteration, the index of the median in the search region is determined as
Subsequently, the augmenting path algorithm is executed on sub-graph
to find the maximum matching
, where
. The search region is updated as
The binary search algorithm terminates when contains only one element, i.e., , which results in the value of . After obtaining the value of , we solve the optimization problem Equation (21) with the given to obtain the solution for Equation (8). The proposed algorithm using graph-based IVR is summarized in Algorithm 2. By applying binary search and augmenting path methods for the bipartite graph, a dynamic pairing solution is computed in Phase 1, at which the binary value of variable is determined to satisfy the constraints of the original problem. In Phase 2, a convex approximation program is solved under the binary value of obtained in Phase 1.
Complexity Analysis: The binary search requires
steps to find the optimal value
, while the augmenting path algorithm takes
[
33]. Therefore, the total complexity for finding
in problem Equation (25) is
. With a given value of
, a sub-problem derived from problem Equation (21) has
variables and
SOC/linear constraints; consequently, the total complexity of Algorithm 2 is
.
Algorithm 2 Proposed Algorithm for Solving Problem Equation (8) using Graph-based IVR. |
Phase 1:- 1:
Input-1:, . - 2:
Compute the weight matrix C as in Equation (24). - 3:
Generate an auxiliary vector v which is sorted in an ascending order. - 4:
Set , . - 5:
repeat - 6:
Compute as in Equation (26). - 7:
Build a sub-graph using the median . - 8:
Apply augmenting path algorithm to find the maximum matching in . - 9:
if then - 10:
Set . - 11:
else - 12:
Set . - 13:
end if - 14:
until
- 15:
Create corresponding to the given . - 16:
Output-1: The pairing solution for the binary variable . Phase 2:- 17:
Input-2:, , , , and . - 18:
Apply Steps 1–7 in Algorithm 1 with the given value of to find the beamforming vector . - 19:
Output-2: The optimal solution ().
|
5.2. Proposed Solution Based on Exhaustive Search
For benchmarking, this section presents the optimal solution using exhaustive search (ES), where all possible cases of are considered while satisfying constraints Equations (8c)–(8h). For each case, a sub-problem is derived from Equation (8) by fixing the value of . Instead of only one subproblem, we obtain many sub-problems as in the graph-based method. Subsequently, we use Equation (21) to find the solutions to the sub-problems. The solution that provides the best performance is selected as the optimal solution. It can be observed that each sub-problem can be solved with a fast convergence since it only needs to handle the continuous variables of the beamforming vectors. However, this approach typical incurs high computational complexity owning to the large number of sub-problems.
Complexity Analysis: The ES method examines sub-problems as all possible cases of . For a given value of , a sub-problem based on Equation (21) contains the same numbers of variables and constraints as in Algorithm 2. Therefore, the total complexity of the ES method is determined as .
7. Conclusions
We studied a hybrid of NOMA and conventional beamforming design, where a dynamic pairing technique was investigated with two arbitrary UEs in a cell that can be paired together. The aim of this paper was to improve the performance in terms of both fairness rate among all UEs and power consumption. To tackle the resulting mixed-integer non-convex problem, we proposed two low-complexity algorithms. The first algorithm was based on a combination of the relaxation and IA methods to jointly optimize UE pairing and beamforming. In the second algorithm, we used the bipartite graph and binary search methods to achieve UE pairing before using the IA method to obtain the beamforming vectors, which can further reduce the computational complexity. Numerical results with realistic parameters demonstrated that the two proposed algorithms outperform the existing schemes in terms of both MMR and the power consumption.
Besides the advantage of novel dynamic UE pairing, some research gaps would be investigated for future works. Even when two proposed algorithms provide good performance and low complexity, an advanced optimization technique for NOMA could be expected to reduce the rate loss. In addition, the channel estimation is not perfect in practice; thus, a robust design under the channel uncertainty would be essential for UE pairing and beamforming in NOMA-based systems. Finally, the effectiveness of dynamic UE pairing in massive device networks is still unknown, which motivates us to develop a new method for the upcoming requirement and environment.