Journal of Electrical and Computer Engineering

Volume 2018, Article ID 5676215, 11 pages

https://doi.org/10.1155/2018/5676215

## Linear Processing Design of Amplify-and-Forward Relays for Maximizing the System Throughput

Yulin Normal University, Yulin, China

Correspondence should be addressed to Tiejun Chen; moc.361@0002nujeitnehc

Received 6 August 2017; Revised 6 December 2017; Accepted 11 December 2017; Published 28 January 2018

Academic Editor: Jit S. Mandeep

Copyright © 2018 Qiang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, firstly, we study the linear processing of amplify-and-forward (AF) relays for the multiple relays multiple users scenario. We regard all relays as one special “relay”, and then the subcarrier pairing, relay selection and channel assignment can be seen as a linear processing of the special “relay”. Under fixed power allocation, the linear processing of AF relays can be regarded as a permutation matrix. Employing the partitioned matrix, we propose an optimal linear processing design for AF relays to find the optimal permutation matrix based on the sorting of the received SNR over the subcarriers from BS to relays and from relays to users, respectively. Then, we prove the optimality of the proposed linear processing scheme. Through the proposed linear processing scheme, we can obtain the optimal subcarrier paring, relay selection and channel assignment under given power allocation in polynomial time. Finally, we propose an iterative algorithm based on the proposed linear processing scheme and Lagrange dual domain method to jointly optimize the joint optimization problem involving the subcarrier paring, relay selection, channel assignment and power allocation. Simulation results illustrate that the proposed algorithm can achieve a perfect performance.

#### 1. Introduction

Relay-assisted cooperative communication and orthogonal frequency-division multiplexing (OFDM) are core technologies for the next generation mobile communication system which attract tremendous attentions. Relay-assist cooperative communication can effectively extend communication coverage, and reduce the consumption of transmit power. It can also improve overall system throughput due to its exploiting spatial diversity and combating channel fading [1, 2]. OFDM divides a wideband channel into some orthogonal narrow band subcarriers. Then, the fast data flow can be transformed to a set of slow data flows. Furthermore, inner-symbol interference can be eliminated significantly. OFDM can also increase the flexibility for coding and modulation. Therefore, relay-assisted cooperative communication and OFDM are adopted by a lot of communication standards, such as 3GPP and IEEE 802.16e.

There are two common categories of relay strategies: amplify-and-forward (AF) and decode-and forward (DF). The DF relay can decode and re-encode the received signals, and then retransmit these signals to destination node. Different from DF relay, the AF relay amplifies the received signals linearly and forwards these signals to destination node without decoding. The AF strategy is the most practical relay approach as the result of its low complexity transceiver design [3]. Moreover, AF relays are more transparent to adaptive modulation techniques than DF relays.

Due to the independent fading on each subcarrier, the incoming and outgoing subcarrier should be matched carefully to maximize the overall system throughput. This problem is also called subcarrier paring. Recently, subcarrier paring has gained a lot of attentions, such as [4–6]. Reference [4] proposes an ordered subcarrier pairing (OSP) method, in which it is proved that subcarrier paring according to the channel state is optimality for equal power allocation. The authors in [5] further prove that the OSP is still optimality for optimal power allocation. Reference [6] considers a linear processing design at the relay for amplified-and-forward (AF) relaying communication system without direct path and with direct path under fixed power gain. Different from [4, 5], [6] dose not only consider the cooperative mode without direct path, but also study the the cooperative mode with direct path. In [6], the authors separate the processing structure into two components which are the power amplification matrix and the unitary linear processing matrix, respectively. They show the optimal unitary processing matrix is of permutation structure.

The joint resource allocation for cooperative communication has been also attracted tremendous attentions. For the scenario with multiple relays multiple users, the joint optimization problem involves subcarrier pairing, relay selection, channel assignment and power allocation. Due to the combinational nature of the subcarrier pairing, relay selection and channel assignment, the joint optimization problem is difficult to solve when the number of subcarriers, relays and users are large. Therefore, most previous works only consider a subset of these issues.

In this paper, firstly, we exploit the linear processing scheme for multiple relays multiple users network to maximize the system throughput. We regard all relays as one special “relay”, and the subcarrier pairing, channel assignment and relay selection can be seen as the linear processing of the special “relay” under fixed power allocation. It can be shown that the optimal linear processing matrix of the special “relay” is a promotion of the permutation matrix. To maximize the system throughput under fixed power allocation, the optimal linear processing matrix should be found out. To this end, we propose a linear processing scheme for the AF relays under fixed power allocation which can find out the optimal permutation matrix, and then we prove the optimality of this linear processing design. It is interesting that, through the proposed linear processing scheme, we can obtain the optimal subcarrier paring, relay selection and channel assignment with fixed power allocation in accordance with the ordered SNR over the incoming and outgoing channels in polynomial time. To the best of our knowledge, this paper is the first work to investigate the linear processing of the AF relays for the multiple relays multiple users network. Finally, based on the proposed linear process scheme and Lagrange dual domain method, we propose an iterative algorithm to solve the joint optimization problem involving subcarrier pairing, relay selection, channel assignment as well as power allocation. Simulation results illustrate that the proposed iterative algorithm is effective to find out an* asymptotically* optimal solution in polynomial time.

The rest of the paper is organized as follows. In Section 2, the previous related works are reviewed. We describe the system model in Section 3. The optimal linear processing under fixed power gain is introduced in Section 4. In Section 5, we extend the linear process scheme to the scenario with multiple relays multiple users. In Section 6, we describe the iterative algorithm based on the proposed linear process scheme and Lagrange dual domain method to solve the joint resource allocation problem. In Section 7, simulation results are provided to evaluate the performance of the proposed algorithm. Finally, we conclude this paper in Section 8.

#### 2. Related Work

The resource allocation for the scenario with single BS, multiple relays and multiple users is a joint optimization problem which involves subcarrier paring, relay selection, channel assignment and power allocation. This joint optimization problem can be formulated as a mixed-integer programming due to the combinatorial nature of the subcarrier paring, relay selection and channel assignment. It becomes more and more intractable with the increase of the number of subcarriers, relays and users because the problem of subcarrier pairing, relay selection and channel assignment is NP hard [7]. To decrease the computational complexity, most previous works consider a subset of the joint optimization problems.

References [8–11] consider the scenario with one BS, one relay and one user. Reference [8] jointly optimizes the subcarrier pairing and power allocation, and a mixed-integer programming problem is formulated. Then, the authors in [8] transform the mixed integer programming problem into a convex optimization through continuous relaxation. [9] proposes a hybrid scheme in which the full-duplex and half-duplex relaying modes can be switched opportunistically to obtain a tradeoff between spectral efficiency and self-interference. For DF relay strategy, if the ratio of the time allocation in the first transmission phase over the whole period is not large enough, the DF relay can not decode the received signals accurately. It is necessarily to optimize the the ratio of the time allocation in the first transmission phase over the whole period. Therefore, reference [10] optimizes not only the power allocation but also time allocation for DF relay strategy to minimize the outage probability. The perfect channel state information (CSI) at the source is impractical, thus it is reasonable to study the resource allocation problem with limited feedback [11].

The resource allocation for the scenario with one BS, one relay and multiple users is studied by reference [7]. The authors in [7] propose an optimal algorithm for the joint resource allocation including subcarrier pairing, channel assignment and power allocation through transforming the original problem into some simple linear programming problems. In [12], a combination of transmit-receive weights and Tomlinson-Harashima precoding is used to cancel the interference, and then a low-complexity power allocation is proposed to achieve data rate fairness among different users.

The cooperative communication scenario with one BS, multiple relays and multiple users is studied by [13, 14], and the direct transmission mode is also considered at the same time. The authors in [13] transform the combinational optimization problem of subcarrier paring, channel assignment, relay selection and transmission mode selection into a minimum cost network flow problem, and apply the linear optimal distribution algorithm to solve the minimum cost network flow problem. Then, the Lagrange dual domain method is used to solve the power allocation. Different from [13], reference [14] only considers the relay selection and subcarrier assignment for multiuser cooperative network to decrease the computational complexity. In the first, [14] simplifies the original problem into a new problem through decreasing the number of variables which is easily to handle. Then, branch-and-cut is introduced to optimize the new problem.

There are also a lot of works on the resource allocation for the multiple-input and multiple-output (MIMO) cooperative communication networks, such as [3, 15–21]. Referencee [15] considers the problem of minimizing the total power consumption in a two-hop single-relay MIMO network with QoS requirements. In [15], a nonconvex power allocation problem is approximated with a convex problem, and then the convex problem is computed in closed-form through a multistep procedure. [16] studies the problem of resource allocation in relay-enhanced bidirectional MIMO-OFDM networks to minimize the total power consumption. The authors in [16] propose a green resource allocation scheme to jointly optimize the subchannel assignment, power allocation and phase duration assignment, in which both the separate-downlink (DL)-and-uplink (UL) and mixed-DL-and-UL relaying assignments and the linear block diagonalization (LBD) technique are adopted. Reference [17] study the optimal power allocation structure for the multiple relays network. The authors in [17] show that the power allocation at BS and each relay follow a matching structure. The cooperative communication with virtual MIMO has also attracted a lot of attentions, such as [18]. Reference [18] studies a wireless network where multiple users cooperate with each other. The authors in [18] propose a new auction-based power allocation framework with multiple auctioneers and multiple bidders to maximize the weighted sum-rates of the users.

Recently, energy efficient (EE) resource allocation for MIMO cooperative communication has become attractive, such as [22–27]. Reference [22] considers the energy efficient joint source-relay power allocation problem for MIMO AF relaying system. In [22], the objective of optimization is the number of the bits per second per hertz per Joule with the guarantee of the minimum spectral efficiency, and the objective function and the constraint are not convex. Firstly, [22] transforms the original problem into a pseudo-convex problem by employing the high signal-to-noise ratio (SNR) approximation. Then, the authors in [22] propose a relaxation method according to the Jensen inequality to solve the pseudo-convex problem. [23] studies the energy-spectral efficiency (EE-SE) trade-off of the uplink of a multi-user cellular virtual MIMO system. Finally, a heuristic resource allocation algorithm is proposed to optimize the EE-SE tradeoff in [23]. The energy-efficient resource allocation for OFDMA cellular networks with user cooperation is studied by [24]. In [24], a mixed-integer nonlinear programming problem is formulated, and then an optimal algorithm is proposed to solve the problem.

#### 3. System Model

In this paper, for matrix , we let , and denote the trace, conjugate transpose and determinant of , respectively. denote the identity matrix. represents the block diagonal matrix with on its main diagonal. We assume that the perfect CSI is obtained by the technology of channel estimation, and then the accurate channel gain is known.

In this paper, we intend to exploit the optimal linear processing of relays in OFDMA AF-based relaying network consisting multiple relays multiple users, similar as Figure 1. For easy to exposition, we first study the linear processing scheme for the scenario with multiple relays and single user in this section. Then, we will show that our proposed linear processing scheme can be readily extended to the multiple relays multiple users scenario in the next section. In this work, we assume there are subcarriers and AF relays. The half-duplex relay is considered, that is, each relay can not transmit and receive the signals in the same subcarrier at the same time. The transmission duration is divided into two equal time slot. Let and denote the channel gain matrices between BS and relay and between relay and the user, respectively.