Joint user clustering and resource allocation for device-to-device communication underlaying MU-MIMO cellular networks
- Qiang Wang^{1}Email author,
- Conglin Lai^{1},
- Yue Dong^{1},
- Yuquan Shu^{1} and
- Xiaodong Xu^{1}
https://doi.org/10.1186/s13638-015-0358-6
© Wang et al. 2015
Received: 30 October 2014
Accepted: 11 April 2015
Published: 23 May 2015
Abstract
In this paper, multiple device-to-device (D2D) communication underlaying cellular multiuser multiple inputs multiple outputs (MU-MIMO) systems is investigated. This type of communication can improve spectral efficiency to address future demand, but interference management, user clustering, and resource allocation are three key problems related to resource sharing. Interference alignment (IA) is proposed to better mitigate in-cluster interference compared with a multiplex scheme, and user clustering and resource allocation are jointly investigated using binary-integer programming. In addition to an exhaustive search for a maximum throughput, we propose a two-step suboptimal algorithm by reducing the search space and applying branch-and-bound searching (BBS). To further obtain a good trade-off between performance and complexity, we propose a novel algorithm based on distance-constrained criteria for user clustering. The simulation results show that the IA and multiplex schemes acquiring user clustering gains outperform the orthogonal scheme without user clustering. Besides, the proposed two-step and location-based algorithms achieve little losses compared with the optimal algorithm under low complexities.
Keywords
D2D communication Multiuser MIMO Interference alignment Location-based Resource allocation1 Introduction
In recent years, device-to-device (D2D) communication underlaying cellular infrastructures has attracted considerable attention on both academia and industry. This infrastructure permits peer-to-peer communication without base station (BS) relays but should be under the control of the BS. With the increasing use of local applications, such as short-distance data transmission in social networks, D2D communication which is an effective proximity transmission scheme has seen substantial demand. D2D communication has great potential to improve spectrum efficiency and system performance by reusing cellular resources [1,2].
Multiuser multiple inputs multiple outputs (MU-MIMO), which is regarded as a very important technology, has been applied in numerous systems, including long-term evolution (LTE) uplink and other cooperative networks, to obtain higher multiuser diversity gain [3]. To address future increased demand, the combination of MU-MIMO networks and D2D underlaying communications, as a novel research field, can further improve spectral efficiency and increase the number of access users [4,5]. The relatively sparse literature only focuses on a single D2D pair reusing a cellular user equipment (CUE) resource, and the scenario where multiple D2D pairs reuse resources with CUEs in the MU-MIMO uplinks has not been widely investigated. In this work, we divide CUEs and D2D pairs into several clusters. Each cluster has a certain number of CUEs and D2D pairs. For maximizing the system rate, there are three key issues related to the system optimization problem, i.e., 1) how to mitigate the serious in-cluster interference, 2) how to determine which D2D pairs and CUEs are clustered together, and 3) how to allocate appropriate resources to the clusters.
The investigated communication pattern will cause excessive and serious interference in the network, including D2D links to cellular links and cellular links to D2D links [6]. To obtain a better system or personal performance, the effective methods of interference management in D2D communication underlaying cellular networks include power control [7-10], resource allocation [11-18], interference awareness [19-21], and precoding [22]. Xing and Hakola [8] employ LTE open-loop and closed-loop power control schemes in a D2D scenario. The resource allocation problem is solved by an iterative combinatorial auction method in [14]. By identifying interference, Min et al. [21] propose interference-limited areas of D2D pairs to prohibit them from reusing resources with CUEs if the interference at D2D receivers is greater than a threshold. In MIMO transmission, the interference alignment (IA) technique, which is an effective precoding mechanism that can align the interference at the receiver together to improve receiving signal-to-interference-plus-noise ratio (SINR), has attracted substantial attention in recent years [23-25]. As the earliest research about IA, Cadambe and Jafar [23] provide a linear precoding codebook design in K-user interference channels.
Furthermore, the second issue in clustering and the third issue in resource allocation are coupled with each other. It is thus necessary to jointly study them. Recently, increasingly more works about jointly considering various optimization issues have been discussed [11,17,26-28] and have been shown to further improve system performances. In [27], the authors analyze optimum power control and resource allocation with different sharing modes between CUEs and D2D pairs in a single-cell scenario. Generally, these types of problems are too hard to directly solve, i.e., NP-hard problems [29,30]. An optimal and direct solution to these problems is exhaustive searching, which is used as a benchmark in our paper. In this work, we attempt to design low-complexity suboptimal algorithms for joint user clustering and resource allocation optimization programming. Heuristic greedy algorithms [26], reverse iterative combinatorial auction approaches [14,28], and D2D pair association vector search algorithms [5] are some examples of effective solutions to the NP-hard problem. The inspiration of Xu et al. [28] comes from game theory. The study sets the resources and D2D links as bidders and goods and then conducts a price iteration process. Its result can converge in a finite number of rounds and is better than random allocation, but it exhibits some performance loss.
1.1 Our contribution
- (1)
We consider multiple D2D pairs reusing resources with CUEs in the MU-MIMO uplink system and formulate the joint user clustering of CUEs and D2D pairs and resource allocation optimization programming.
- (2)
We employ a linear IA technique in the scenario to eliminate interference inside clusters, and we then propose evaluation comparisons between the schemes with IA and without IA which is the traditional multiplex scheme.
- (3)
A two-step optimization algorithm is designed to first reduce the search space, thereby decreasing the search difficulty, and then obtain the solution using a branch-and-bound searching (BBS) algorithm. To further reduce the complexity for practical communications, we propose a location-based algorithm that divides a single cell into an inner round employing the IA scheme and an outer circle employing the multiplex scheme.
The simulation results first show that the IA and multiplex schemes acquiring user clustering gains outperform the orthogonal scheme without employing user clustering. The IA scheme outperforms the multiplex scheme in a large range of the D2D user equipment transmit (DUE-TX) power. Second, the two-step algorithm exhibits minimal losses compared to the optimal exhaustive search. The proposed location-based algorithm, whose performance is near optimal under low practical complexity, exhibits a good trade-off between performance and complexity. Third, it is better to keep the distance between D2D users sufficiently small to obtain a better system performance. Finally, the appropriate inner round radius r _{0} of the location-based algorithm monotonously increases with increasing DUE-TX power. With high DUE-TX power, a large r _{0} is more suitable.
The remainder of this paper is organized as follows. Section 2 generalizes the scenario’s system and signal models in the IA scheme and multiplex scheme. We then deduce the SINR expressions. Section 3 proposes the objective and constraint conditions of the joint resource allocation and clustering programming. In Section 4, a two-step algorithm and a location-based algorithm are proposed in detail. Section 5 presents the numerical results, complexities, and analysis. A further discussion is demonstrated in Section 6. Finally, conclusions are shown in Section 7.
1.2 Scenario description and symbol notations
Mathematical notations
Notation | Physical interpretation |
---|---|
K _{ rb } | Denotes the number of all available time-frequency RBs |
K _{ c }, K _{ d } | Denote the number of CUEs or D2D pairs in a cell |
N _{ c }, N _{ d } | Denote the number of CUEs in a CUE subcluster and the number of D2D pairs in a DUE subcluster |
C _{ i } | Denotes the label of CUE in a CUE subcluster, where i varies from 1 to N _{ c } |
D _{ j } | Denotes the label of D2D pairs in a DUE subcluster, where j varies from 1 to N _{ d }, and each D2D pair includes a D _{ j } − T _{ x } and D _{ j } − R _{ x } |
\( {\mathbf{s}}_{C_i}^m,{\mathbf{s}}_{D_j}^m \) | Denote the transmitted signal transmitted at the m-th subcarrier, and the subscript indicates the source of the signal |
\( {\mathbf{E}}_{C_i},{\mathbf{E}}_{D_j} \) | Denote the diagonal matrix of the multiantenna transmit power, where the subscript indicates the source of the signal |
\( {\mathbf{H}}_{i,j}^m \) | Denotes the Rayleigh channel matrix from node j to node i over the m-th subcarrier, with 0 mean and unit variance |
L _{ i,j } | Denotes the diagonal matrix of the path loss from node j to node i |
r _{ i,j } | Denotes the distance between node j and node i |
\( {\mathbf{n}}_{\mathrm{BS}}^m,{\mathbf{n}}_{D_j}^m \) | Denote the additive white Gaussian noise (AWGN) on the m-th subcarrier |
2 System model
In this section, we first introduce a traditional D2D underlaying MU-MIMO uplink scenario employing a multiplex scheme. After deducing a cluster’s received SINRs with minimum mean-square error (MMSE) frequency-domain equalization, we calculate its sum throughput. Then, we formulate the precoding and decoding process of the scenario using IA and give some feasibility conditions and derive the total throughput.
2.1 D2D underlaying MU-MIMO systems using a multiplex scheme
2.2 D2D underlaying MU-MIMO systems using the IA scheme
2.2.1 Signal model through IA
This multiplex scheme is simple and can be easily implemented without channel state information (CSI) feedback, but it has no interference mitigation mechanism and thus results in a degraded performance. We utilize a linear IA mechanism for the clusters. To provide successful receiving, we assume each transmitter only sends a data stream to its corresponding receiver. Thus, each receiver can see N _{ c } + N _{ d } independent data flows. DUE-RX should align N _{ c } + N _{ d } – 1 interfering flows to a certain space to enlarge the space of the target flow. In term of the BS, N _{ c } independent spaces are required to receive N _{ c } target flows.
2.2.2 Limitation and feasibility of the scenario using IA
where b ′ denotes the target of b and a is not the target of b. DOF is the degree of freedom representing the number of data streams in the b-th link. Yetis et al. [25] utilize Bezout’s theorem to determine whether the system is feasible. It presents that if the system is feasible, the number of equations must be less than or equal to the number of variables.
As (36) and (37) show, if N _{ b } = 1, the feasible combinations do not exist. If N _{ b } = 2, then the only two feasible cases are N _{ c } = 1, N _{ d } = 2 and N _{ c } = N _{ d } = 2. In normalization, a relatively lower bound of N _{ b } will be satisfied when N _{ c } = N _{ d } which is without loss of generality. Then, (37) is transformed into N _{ c } = N _{ d } ≤ N _{ b }. That means BS must use at least N _{ d } spaces to align interference. Besides, BS acquires N _{ c } target signals. Combining the target and interference spaces, the total antenna number of BS, N _{ r }, must be at least N _{ c } + N _{ d } for a feasible system.
As (41), the feasible user antenna number increases linearly with D2D user number. As long as there is enough antenna number, the more satisfied users can be accommodated in the system. IA has a great potential to increase the number of users. The feasible implementations for a practical N _{ c } = N _{ d } system of the investigated scenario are N _{ r } = N _{ c } + N _{ d } and N _{ t } ≥ (2N _{ d } + 2)/3. It is worth noting that these two conditions are the sufficient but not the necessity conditions of feasible IA in the investigated scenario.
2.2.3 Precoding and decoding designs
The linear precoding and decoding codebook designs are related to specific scenario parameters, particularly the number of users and antennas. Cadambe and Jafar [23] state that each receiver should use at least one antenna to receive a target signal and at least one antenna to receive the interference. We select a group of the feasible implementation from the last section. As the standard implementation in 3GPP LTE [32,33] presents, the BS is configured with 4 with antennas, i.e., N _{ r } = 4, and each user is configured with 2 antennas, i.e., N _{ t } = 2. As a simple example, we assume N _{ c } = N _{ d } = 2 in a cluster. The parameters submitting formula (40) are general in practical usage.
As (47) expresses, \( {\mathbf{Q}}_{C_1}^m \) can be set as the eigenvector of \( {{\mathbf{G}}_{D_1,{C}_1}^m}^{-1}{\mathbf{G}}_{D_1,{C}_2}^m{{\mathbf{G}}_{D_2,{C}_2}^m}^{-1}{\mathbf{G}}_{D_2,{C}_1}^m \), and then, all precoding matrices \( {\mathbf{Q}}_{C_1}^m,{\mathbf{Q}}_{C_2}^m,{\mathbf{Q}}_{D_1}^m,{\mathbf{Q}}_{D_2}^m \) can be similarly determined.
After deriving precoding matrices, the corresponding ZF decoding matrices also need to be derived to preserve the target spaces. Taking \( {\mathbf{y}}_{D1}^m \) as an example, because \( {\mathbf{G}}_{D1,C1}^m{\mathbf{Q}}_{C1}^m \), \( {\mathbf{G}}_{D1,C2}^m{\mathbf{Q}}_{C2}^m \), and \( {\mathbf{G}}_{D1,D2}^m{\mathbf{Q}}_{D2}^m \) have been aligned in the same space, we select any one of them and then use SVD decomposition, i.e., \( {\mathbf{G}}_{D1,C1}^m{\mathbf{Q}}_{C1}^m=\left[{\mathbf{U}}_1\boldsymbol{\Lambda} \mathbf{V}\right] \). We take the second column of U _{1}, which is denoted as U _{1} ^{(2)}. Then, the ZF matrix at DUE-RX1, D _{1}, is \( {\mathbf{U}}_{D_1}^m={\left({\mathbf{U}}_1^{(2)}\right)}^T \). Because U _{1} is a unitary matrix, \( {\left({\mathbf{U}}_1^{(2)}\right)}^T{\mathbf{G}}_{D1,C1}^m{\mathbf{Q}}_{C1}^m \) can preserve the target signal and eliminate the interference signal. Similar to D _{1}, \( {\mathbf{U}}_{D_2}^m \) can be derived. In terms of the BS receiving the C _{1} signal, after using SVD decomposition to obtain \( {\mathbf{G}}_{\mathrm{BS},C1}^m{\mathbf{Q}}_{C1}^m=\left[{\mathbf{U}}_2\boldsymbol{\Lambda} \mathbf{V}\right] \), we select the second, the third, and the fourth columns as U _{2} ^{(2–4)}, and \( {\mathbf{U}}_{C_1}^m \) is (U _{2} ^{(2–4)})^{ T }. To obtain the C _{2} signal, \( {\mathbf{U}}_{C_1}^m{\mathbf{G}}_{\mathrm{BS},C2}^m{\mathbf{Q}}_{C2}^m=\left[{\mathbf{U}}_3\boldsymbol{\Lambda} \mathbf{V}\right] \), then, we set \( {\mathbf{U}}_{C_2}^m \) is (U _{3} ^{(3–4)})^{ T }.
3 Formulation of joint user clustering and resource allocation problem
- (a)
indicates that each CUE subcluster can share a resource allocation pattern, including an empty resource pattern, with at most one DUE subcluster;
- (b)
indicates that each DUE subcluster can share a resource allocation pattern, including an empty resource pattern, with at most one CUE subcluster;
- (c)
indicates that each resource allocation pattern, except the empty pattern, can only be allocated to at most one CUE subcluster and one DUE subcluster;
- (d)
guarantees that each CUE can only be selected by one CUE subcluster. k1 and k2 represent the indices of the selected CUE subclusters that are allocated resources;
- (e)
guarantees that each D2D pair can only be selected by one DUE subcluster. l1 and l2 represent the indices of selected DUE subclusters that are allocated resources;
- (f)
guarantees that each RB is only allocated to a cluster. n1 and n2 represent the indices of the employed resource pattern.
The joint optimization used to maximize the overall rates of the CUEs and DUEs sharing the same resources is a typical discrete optimization problem and must be non-convex. The usual method to obtain the optimal result of such types of problem is exhaustive searching which is extremely difficult, i.e., NP-hard [29,30]. Because x _{ k,l,n } is 0 or 1, the problem can be transformed as a standard binary-integer programming problem. In this paper, we utilize low-complexity heuristic algorithms to obtain approximately optimal results that do not have much loss of performance compared to the optimal one.
4 Joint user clustering and resource allocation algorithms
In this section, we first present the standard binary-integer programming of the optimization problem and subsequently develop a two-step algorithm to solve it. Finally, a location-based algorithm is provided to obtain a good trade-off between performance and complexity.
4.1 Standard binary-integer programming form
An exhaustive search is a straightforward and basic algorithm used to find the optimal solution of binary-integer programming problems. However, it is overly complex and impractical in real-world scenarios.
4.2 Two-step algorithm
To reduce the complexity, we propose a two-step optimization algorithm. In this algorithm, an exhaustive search algorithm is first used to find the optimal user cluster at each RB. Then, we reserve the elements in (51) that contain the users in RBs’ optimal clusters and abandon the remainder. Based on such a reduced-dimension CUE subcluster subset and DUE subcluster subset, we implement the BBS algorithm [31] to realize the joint optimization.
Let us use an example to illustrate the advantage of reducing the search space by utilizing a two-step algorithm. When K _{ c } = K _{ d } = 12 and K _{ rb } = 6, the extreme case is that 6 CUE and 6 DUE subclusters will be selected to form the reduced-dimension CUE subcluster subset and DUE subcluster subset. Clearly, the dimension of such a CUE subcluster subset is substantially smaller than that of the full CUE subcluster set, i.e., N _{ all. c } = 66. The same effect will occur in the DUE subcluster. Consequently, the search space for the optimization can be reduced, thereby decreasing its complexity.
Low-complexity suboptimal two-step algorithm
Two-step algorithm | |
---|---|
1. | Initialization: S = ∅, K = ∅, L = ∅ |
2. | Clustering Procedure: |
3. | For each resource pattern with only one RB rb ∈ {1,2,…,K _{ rb }} |
4. | Find the CUE sub-cluster and DUE sub-cluster \( \left\{\overline{k},\overline{l}\right\}= \arg \underset{\left\{k,l\right\}}{ \max }{t}_{k,l,rb} \) |
5. | \( K=K{\displaystyle \cup}\kern0.5em \overline{k} \) |
6. | \( L=L\;{\displaystyle \cup}\kern0.5em \overline{l} \) |
7. | End |
8. | Searching Space Reduction Procedure: |
9. | For each \( \widehat{k}\in K;\widehat{l}\in L \) sets |
10. | \( {s}_0=\left\{i\Big|\left(\widehat{k}-1\right)*{N}_{all,d}{N}_{all,rb}+\left(\widehat{l}-1\right)*{N}_{all,rb}+1\le i\le \left(\widehat{k}-1\right)*{N}_{all,d}{N}_{all,rb}+\widehat{l}*{N}_{all,rb}\right\} \) |
12. | S = S ∪ s _{0} |
13. | End |
14. | Joint Solution Procedure: |
15. | Obtain the reduced-dimension optimization problem |
16. | \( {x}_s=\underset{{\mathbf{x}}_S}{ \min}\left\{-{\mathbf{t}}_{\boldsymbol{S}}^{\boldsymbol{T}}{\mathbf{x}}_S\right\} \), s.t. \( {\mathbf{R}}_{:,S}{\mathbf{x}}_S={\mathbf{1}}_{\left({N}_c{N}_d{N}_{rb}+{K}_c+{K}_d\right)\times 1} \) |
17. | Find its solution based on the BBS algorithm |
18. | Return \( {x}_s \) |
19. | End |
4.3 Random clustering scheme based on distance (location-based algorithm)
When there are a large number of CUEs and DUEs in the cell, the computation is quite complex when using exhaustive searching or the two-step algorithm. Therefore, we propose a novel location-based algorithm to randomly select CUEs and D2D pairs based on a distance-constrained criterion to assemble a user cluster.
We assume that area B is divided into six equal sections, denoted by B1 to B6, as illustrated in Figure 2. The main in-cluster interference is between D2D pairs and between CUEs and D2D pairs. Two D2D pairs and two CUEs are randomly located at three non-adjacent sections so that the mutual interference can be reduced to a low level. We select two D2D pairs from two of the three non-adjacent sections and two CUEs from the remaining section. Then, we combine these users into a cluster, which is employed in the multiplex scheme.
In area A, the mutual interference cannot be neglected because the distance between the CUEs and D2D pairs is not sufficiently large. Therefore, the IA scheme is adopted. The area does not need to be divided because the mutual interference is nearly non-existent when the IA scheme is used.
Location-based algorithm
Location-based algorithm | |
---|---|
1. | Initialization: Randomly distribute CUEs and D2D pairs in a cell and calculate their distances to the center of the BS. Determine the value of r _{0} between 0 and the radius of the cell |
2. | Divide users into four groups according to r _{0}, A_{CUE}, A_{D2D}, B_{CUE}, and B_{D2D} |
3. | In area A or B, users randomly constitute clusters (the clustering in area B must satisfy the distance criterion), and each cluster consists of two CUE users and two D2D pairs |
4. | For each RB |
5. | Randomly choose a cluster in the cell and determine whether it is in area A or B |
6. | Calculate the large-scale loss in this cluster |
7. | Case user to BS: 128.1 + 37.6 * log _{10}(d[km]) |
8. | Case user to user: 148 + 40 * log _{10}(d[km]) |
9. | Set the initial transmit power according to area A or B: P _{ C1}, P _{ C2}, P _{ D1}, P _{ D2} |
10. | Do |
11. | For each subcarrier (SC) |
12. | Generate small-scale loss H: multiplex Gaussian with zero mean unit variance |
13. | If cluster in area A |
14. | |
15. | Else cluster in area B |
16. | |
17. | End if |
18. | End SC |
19. | |
20. | |
21. | While SINR does not meet the target, Power Control, then jump 10 |
22. | End each RB |
23. | Calculate sum throughput |
5 Simulation results
Simulation parameters
Carrier frequency: 2 GHz | Macro cell radius R : 500 m |
---|---|
Number of CUEs per cell K _{ c }: 12 | Number of D2D pairs per cell K _{ d }: 12 |
CUE and DUE distribution: uniform | Clustering mode: 2 CUEs + 2 D2D pairs |
IA precoding mode: 1 stream per user | Multiplex mode: 2 streams per user |
K _{ rb }: 6 | Number of subcarriers per RB, M: 12 |
System bandwidth: 1.4 MHz | Subcarrier spacing: 15 kHz |
Multiplex equalizer: MMSE | Small-scale channel: complex Gaussian channel |
User antenna number: 2 | BS antenna number: 4 |
Macro UE path loss: 128.1 + 37.6 log_{10} (d[km]) | D2D path loss: 148 + 40 log_{10} (d[km]) |
CUE target SINR: 5 dB | Noise power density N _{0}: − 174 dBm/Hz |
5.1 Effect of D2D transmit power
Complexity comparison
Schemes | IA optimal | IA two-step | IA random | Location-based |
---|---|---|---|---|
Complexity of the throughput calculation | \( {K}_{rb}{\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d}M \) | \( {K}_{rb}{\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d}M \) | K _{ rb } M | K _{ rb } M |
Complexity of the search | \( \left({\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d}\right)\hat{\mkern6mu} {K}_{rb} \) | \( {K}_{rb}{\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d} \) | K _{ rb } | K _{ rb } |
Schemes | Orthogonal optimal | Orthogonal two-step | Orthogonal random | |
Complexity of the throughput calculation | \( \left({K}_{rb,c}{\mathrm{C}}_{K_c}^{N_c}+{K}_{rb,d}{K}_d\right)M \) | \( \left({K}_{rb,c}{\mathrm{C}}_{K_c}^{N_c}+{K}_{rb,d}{K}_d\right)M \) | K _{ rb } M | |
Complexity of the search | \( {K}_d\hat{\mkern6mu} {K}_{rb,d}+{C}_{K_c}^{N_c}\hat{\mkern6mu} {K}_{rb,c} \) | \( {K}_d{K}_{rb,d}+{\mathrm{C}}_{K_c}^{N_c}{K}_{rb,c} \) | K _{ rb } | |
Schemes | Multiplex optimal | Multiplex two-step | Multiplex random | |
Complexity of the throughput calculation | \( {K}_{rb}{\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d}M \) | \( {K}_{rb}{\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d}M \) | K _{ rb } M | |
Complexity of the search | \( \left({\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d}\right)\hat{\mkern6mu} {K}_{rb} \) | \( {K}_{rb}{\mathrm{C}}_{K_c}^{N_c}{\mathrm{C}}_{K_d}^{N_d} \) | K _{ rb } |
Figure 3 also shows that regardless of whether the scheme is an IA, multiplex, or orthogonal, the two-step algorithm has an average capacity that is similar to the optimal algorithm, and both of them outperform the random algorithm. In summary, the proposed two-step algorithm exhibits not only a higher average capacity but also a lower complexity. Nevertheless, the performance of the location-based algorithm, which employs a random algorithm, is greater than that of any other scheme using a random algorithm, and it is closer to that of the two-step algorithm in the multiplex and IA schemes. The complexity of the location-based algorithm is linear, which is considerably lower than that of the two-step algorithm and is equal to that of the random algorithm. The performance and complexity of the optimal or random algorithm cannot become optimal simultaneously because a conflict always exists between performance and complexity. However, the location-based algorithm can achieve a good trade-off between these two criteria.
5.2 Effect of distance between D2D pairs
5.3 Effect of the inner round radius
6 Discussions
As well as [23,26], our research is based on the assumption of the global CSI known by transmitters or central BS. In terms of the IA scheme, the global CSI acquisition is an important problem and needs significant attention; otherwise, it will cause signals overhead. Jin et al. [34] provided a feedback topology design that can be used to acquire sufficient CSI and reduce signaling overhead. The method helps each transmitter to easily acquire the knowledge of the CSI. However, if the CSI condition cannot be satisfied, the IA scheme will suffer a performance loss. Once it happens, we can only employ statistical CSI as a replacement for instant CSI. The result will most likely be lower than the multiplex performance which does not require CSI knowledge. And this will make the IA scheme no sense.
Additionally, the IA scheme is to a great extent limited by the number of antennas and users. The investigated scenario is regarded as the combination of multiple access channel (MAC) [35] and K-user [23]. But to the best of our knowledge, there is no previous research on joint considering of MAC and K-user scenarios. In this paper, we mainly consider whether a feasible IA can improve the system performance besides a feasibility evaluation. The feasible inequality condition (40) is sufficient but not necessary of IA. The condition is not so tight.
Furthermore, the algorithm convergence is mainly affected by IA and power control in this research. The linear IA [23] is used in this paper which only requires a few calculation steps for precoding and decoding and cannot lead to serious signaling overhead. In order to satisfy the feasibility of linear IA, the number of antennas and users should be implemented appropriately. The iteration method is not our concern which has few differences with the linear method except for IA precoding and decoding. In terms of power control, the powers are iteratively modified by ΔP for the CUE target SINR. The solution will converge more quickly if ΔP is bigger, but the system will be unstable. We employ an abandon mechanism to guarantee convergence. If the number of iterations in an RB is more than a critical value, the corresponding communications are abandoned and reallocated. This mechanism can also prevent the system from signaling overhead.
The CSI acquirement, feasibility, and convergence issues are very important but not completely studied in this paper. In addition to the location-based algorithm utilizing fractional frequency reuse (FFR) and soft frequency reuse (SFR) algorithms in multiple-cell scenarios, they will be considered in our future work.
7 Conclusions
In this paper, we consider a single-cell scenario of multiple D2D communications underlaying MU-MIMO cellular uplink networks. First, we investigate IA and multiplex schemes in user clustering. The IA scheme can eliminate interference, and it obtains higher performance than the multiplex scheme in a large range of DUE-TX power. These two schemes can achieve higher performances compared to the orthogonal scheme, which does not employ user clustering. Second, we generalize a joint optimization problem of user clustering and resource allocation to maximize the overall throughput. To derive the solution, because the global optimal method is an exhaustive search with very high complexity, we propose a two-step algorithm and a location-based algorithm to reduce the complexities with minimal loss of performance. The location-based algorithm, composed of IA and multiplex schemes, has both the low interference advantage of the IA scheme and the easy implementation advantage of the multiplex scheme. The simulation results show that the proposed location-based algorithm produces low levels of mutual interference for multiplex users and that its performance is near optimal under low practical complexity; therefore, it exhibits a good trade-off between performance and complexity. Finally, we evaluate the effect of the distance between D2D pairs and the effect of the appropriate inner round radius r _{0} of the location-based algorithm. The results show that it is better to keep the distance between DUE-TX and DUE-RX sufficiently small to obtain a better performance and that it is more suitable to select a larger r _{0} with a higher DUE-TX power condition.
Declarations
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under Grant 61302082 and in part by the National High-tech Research and Development Program of China under Grant 2014AA01A701.
Authors’ Affiliations
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