# Performance Analysis of User Pairing Algorithm in Full-Duplex Cellular Networks

Authors: Wonjong Noh; Wonjae Shin; Hyun-Ho Choi
Journal: Mobile Information Systems
 (2017)

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## Abstract

In a full duplexing (FD) wireless cellular network, a base station operates in FD mode, while the downlink (DL) and uplink (UL) users operate in half duplexing (HD) mode. Thus, the downlink and uplink transmissions occur simultaneously so that interuser interference from a UL to a DL user occurs. In an FD network, the main challenge to minimize the interuser interference isuser pairing, which determines a pair of DL and UL users who use the same radio resource simultaneously. We formulate a nonconvex optimization problem for user pairing to maximize the cell throughput. Then, we propose a heuristic user pairing algorithm with low complexity. This algorithm is designed such that the DL user having a better signal quality has higher priority to choose its paired UL user for throughput maximization. Thereafter, we conduct theoretical performance analysis of the FD cellular system based on stochastic geometry and analyze the impact of the user paring algorithm on the performance of the FD cellular system. Results show that the FD system that uses the proposed user pairing algorithm effectively reduces the interuser interference and approaches optimal performance. It also considerably outperforms the FD system using a random user pairing and almost doubles the conventional HD system in terms of cell throughput.


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## Body

## 1. Introduction

In traditional wireless communication, the duplexing mode for transmitting and receiving signals at a node commonly is half duplex (HD), such as frequency division duplexing (FDD) and time division duplexing (TDD). As another duplexing mode, full duplexing (FD) has recently received considerable attention because FD enables a node to transmit and receive signals simultaneously on the same frequency so that spectral efficiency can be improved twice in theory. However, the main challenge in applying the FD to wireless networks is the manner in which one can handle cochannel interference not only between the transmit (Tx) and receive (Rx) antennas at the base station (BS), but also between the uplink (UL) and downlink (DL) communication users. The cochannel interference between the Tx and Rx at the same station is calledself-interference, whereas that between the different stations is calledinteruser interference.To address self-interference, a smart code that divides the transmitting and receiving signals has been invented and a code division duplexing (CDD) system using this coding technique has been studied [1, 2]. Then, the CDD system has been shown to achieve twice the spectral efficiency of the HD system [3]. In addition, advanced antenna cancellation techniques that cancel the self-interference between Tx and Rx on the radio frequency (RF) band have been proposed and real performances by implementation have been demonstrated [4–6]. Unlike these physical layer solutions, scheduling on the medium access control (MAC) layer has been another research issue studied in order to tackle the self-interference problem in an FD system. The self-interference problems at the wireless relay node with the FD have been resolved by dynamic resource allocation schemes [7–9]. In a practical FD cellular system in which multiple Rxs of the BS are distributed at a distance with the Tx of BS, scheduling algorithms that determine both subcarrier allocation and the communication direction between UL and DL users have been designed to maximize cell throughput [10].To address interuser interference in FD systems, various algorithms including resource allocation, scheduling, user pairing, and their optimization have been studied to improve system performance with respect to cell throughput, cell coverage, and outage probability. Some initial studies were conducted to examine various resource allocation-related problems in FD cellular networks [11–13]. Resource block (RB) allocation was considered in [11] for an FD cellular network, and interference between users was reduced as a result. In [12], a two-user FD cellular network was considered and a noncooperative game was proposed for radio resource allocation in such networks. In [13], a power efficient resource allocation algorithm was designed to achieve total transmit power minimization in an FD cellular network.In addition, various scheduling algorithms for an FD system that consider nonconvex and combinatorial optimization problems were studied in [14]. The Janus protocol in [15] also considers a scheduling algorithm for scheduling a transmission mode (HD or FD mode) to maximize in-band FD transmission opportunity, while maintaining fair channel access for all nodes. Recently, a MAC scheduling algorithm that employs cooperative FD relays (FDRs) was proposed in [16]. This study revealed that additional throughput can be achieved by using cooperative FDRs. In [17], joint user scheduling and channel allocation for cellular networks was considered and a suboptimal heuristic algorithm with low computational complexity was proposed. Furthermore, two heuristic user pairing algorithms were designed for throughput maximization and outage minimization, respectively [18]. Simulation results showed that these two algorithms approach optimal performance under the assumption of no residual self-interference at the FD BS. However, up-to-date studies and theoretical performance analysis on user pairing algorithms in FD cellular network environments have not been thoroughly conducted. Compared with [18], our study newly conducts theoretical analysis of the FD cellular system based on stochastic geometry after adopting the user pairing algorithm previously proposed and then investigates the theoretical performance of FD cellular networks according to various system parameters, which have not been considered in [18].In this study, we consider FD wireless cellular networks in which the BS operates in FD mode while DL and UL users operate in HD mode. Because this FD system induces severe interuser interference from UL to DL users, the main concern in maximizing system performance is user pairing, which determines a pair of DL and UL users that use the same radio resource simultaneously. We formulate the optimization problems for user pairing to maximize cell throughput. Solving these optimization problems by means of exhaustive search is highly complex; thus we propose a suboptimal user pairing algorithm with low computation complexity for practical use. In a heuristic manner, the proposed user pairing algorithm considers the signal quality of users and the direct channel between users. Thereafter, we perform theoretical analysis of the FD cellular system based on stochastic geometry and investigate the effects of adopting the user pairing algorithm. Analysis and simulation show that the FD system in conjunction with the proposed user pairing algorithm approaches optimal performance and greatly improves cell throughput compared to the conventional HD system.The rest of this paper is organized as follows. Section2 describes the system model of the considered FD wireless cellular network. In Section 3, user pairing algorithm is proposed for throughput maximization. Section 4 provides stochastic geometry-based analysis of the FD cellular network. In Section 5, analysis and simulation results are given and the performance gain of the proposed pairing algorithm is verified. Conclusion remarks are provided in Section 6.


## 2. System Description

We consider a cellular network composed of multiple cells in which each cell is surrounded byN neighbor cells, as shown in Figure 1. The BS works in FD mode while DL and UL users work in HD mode due to the limited processing ability of user terminals. We assume that a total of 2M users are present in a cell and that both the number of UL users and the number of DL users are M equally. Moreover, one transmission frame is divided into M time slots and each user is allowed to use only one time slot within a frame to transmit or receive data. This time slot allocation corresponds to the manner of round-robin scheduling of radio resources and thus provides some degree of fairness for all users [19].Figure 1
Considered full-duplex cellular networks.Because the DL and UL users transmit signals on the same resource simultaneously, the DL user experiences three different types of interferences from (i) the UL user scheduled in the same slot in the serving cell, (ii) the neighboring BSs, and (iii) the UL users using the same slot in neighboring cells. Therefore, the received signal at DL useri is expressed as(1)yi=P0hd,ixi+Pjφijxj︸interuserintf.+∑n=1NP0ginx^n︸intf.  from  neighbor  BSs+∑n=1NPnφinxˇn︸intf.  from  neighbor  UL  users+ni,where P0 and Pj are the transmission power of the BS and UL user j, respectively. The variables xi, xj, and xn denote the signals sent from the serving BS0, the UL user j using the same time slot, and the nth neighboring cell, respectively. As illustrated in Figure 1, the channel coefficients hd,i and φij correspond to the links to DL user i from its serving BS and the UL user j, respectively. In addition, the channel coefficients gin and φin correspond to the links to DL user i from the BS and UL user in the nth neighboring cell, respectively. Here, all the channel coefficients are assumed to be CN(0,σ2/dijα), where dij is the transmission distance of the corresponding ij link and α denotes the path loss exponent. Finally, ni is the additive white Gaussian noise with zero mean and N0 variance.In addition, the Rx of the BS experiences three kinds of interferences from (i) the Tx of the serving BS, (ii) the neighboring BSs, and (iii) the UL users using the same slot in neighboring cells. Therefore, the signal received from the UL userj at the Rx of the serving BS0 is expressed as(2)yj=Pjhu,jxj+P0g0xi︸self-interference+∑n=1NP0g0nx^n︸intf.  from  neighbor  BSs+∑n=1NPnφ0nxˇn︸intf.  from  neighbor  UL  users+n0,where hu,j, g0, g0n, and φ0n represent the channel coefficients corresponding to the links from the UL user j, the Tx of the serving BS0, the nth neighboring BS, and the nth neighboring UL user, respectively.In the FD system, the Rx of the BS experiences strong interferences from the Txs of both the serving and neighboring BSs (relatively, the interferences from the Txs of UL users in the neighboring cells are weak because of their small transmission power and the long distance to the neighboring BS [10]). However, the BS with FD function can eliminate these kinds of interferences by using the antenna’s cancellation techniques and channel estimation techniques [5, 10]. For the purpose of channel estimation, we assume that the Rx at the BS employs pilot symbols, which are transmitted from the Txs of both the serving BS0 and neighboring BSs. Although the received interference power is very strong compared to the receiving signal strength at the BS, it can be reduced to the same level as the receiver noise floor by applying analog and digital cancellation techniques sequentially [4–6]. Moreover, by adopting a proper channel estimation technique, such as least-square and minimum mean square error, we can estimate the channel coefficients g0 and g0n [20] (details about channel estimation techniques are out of the scope of this study; the channel estimation at the BS is assumed to be perfect for the practicality of the FD). Based on this interference channel estimation and the knowledge of the signals sent from the Txs of the serving BS0 and neighboring BSs through the wired backhaul, the Rx of the serving BS0 can eliminate the corresponding interference parts from the received signal. After interference cancellation, the postprocessed signal of UL user j is expressed as(3)y^j=Pjhu,jxj+P0g0-g^0xi+∑n=1NP0g0n-g^0nx^n+∑n=1NPnφ0nxˇn+n0,where g^0 and g^0n denote the estimated channel coefficients.From (1) and (3), the signal-to-interference-plus-noise ratios (SINRs) of the received signals yi and y^j are, respectively, given by(4)γi=P0hd,i2Pjφij2+∑n=1NP0gin2+∑n=1NPnφin2+N0,(5)γj=Pjhu,j2P0g0-g^02+∑n=1NP0g0n-g^0n2+∑n=1NPnφ0n2+N0.Consequently, the rates (i.e., spectral efficiencies) of DL user i and UL user j are given by(6)Ri=log2⁡1+γi,Rj=log2⁡1+γj.Finally, the cell throughput of FD system is expressed as(7)SFD=∑i=1MRi+∑j=1MRj.In the conventional HD system (e.g., TDD system) in which neither self-interference nor interuser interference exists, the SINRs of DL useri and UL user j are, respectively, expressed as(8)γiHD=P0hd,i2∑n=1NP0gin2+N0,γjHD=Pjhu,j2∑n=1NPnφ0n2+N0.Therefore, the cell throughput of HD system is expressed as(9)SHD=12∑i=1Mlog2⁡1+γiHD+∑j=1Mlog2⁡1+γjHD.


## 3. Proposed User Pairing Algorithm

Based on (4) and (5), the parameters that we can control in the FD system are Pj and φij. The other parameters are given based on the node position or are generally fixed. Here, φij is controlled by the user pairing. Thus, the user pairing influences the DL SINR and cell throughput. Our objective is to design a novel user pairing algorithm to optimize cell throughput.We denote by(i,j) a user pair in which DL user i and UL user j are allocated to the same time slot. Possible user pairs are derived by choosing a number i in {1,2,…,M} and a number j in {1,2,…,M} under the constraint that the chosen number cannot be selected again. Let Π denote the set of all possible user pairs. For example, for M=3, Π becomes(10)Π=1,1,2,2,3,3,1,1,2,3,3,2,1,2,2,1,3,3,1,2,2,3,3,1,1,3,2,1,3,2,1,3,2,2,3,1,where six configurations exist. For a general M, the number of possible configurations for user pairing is given by M!. We denote by πk the kth element of Π, where k=1,2,…,M!.For the purpose of maximizing cell throughput, the optimization problem is formulated as(11)πk=argmax∑i,j∈πkRi+Rjs.t.Pj=Pmaxi,j∈1,2,…,M,where Pmax is the maximum transmission power of the UL user. In this problem, every UL user should set its transmission power Pj to Pmax to maximize its UL throughput. To solve this nonconvex and combinatorial optimization problem, we must search the total M! configurations for the user pairs. This computation complexity is O(M!), which is surprisingly high with a large M (i.e., a large number of users). Therefore, we propose a suboptimal pairing algorithm with low complexity for practical use.To design a user pairing algorithm, we can obtain insight from (4) and (5). As shown in (5), the SINR of UL user γj is not affected by DL users. If we assume that the channel estimation and interference cancellation are perfect and the interference from neighbor UL users is negligible, γj depends on mainly UL channel hu,j. By contrast, the SINR of DL user γi depends not only on the DL channel, hd,i, but also on the direct channel from UL user j to DL user i, φij. Therefore, the user pairing influences only the SINR of DL user γi. To maximize the sum rate of all users, having the user with a better signal quality receive less interference is reasonable [21]. Therefore, we have the DL user with a better signal quality (i.e., a higher value of γiHD) choose first the UL user with a smaller φij as its partner. This approach improves the SINR of the DL user with a better signal quality more by reducing more interference from the UL user, and it eventually increases the sum rate of all users.The proposed user pairing algorithm is given as Algorithm1. First, the transmit power of all UL users is initialized to their available maximum power, Pmax, in order to maximize the UL SINR at the BS. Moreover, the flag indicates whether the UL user is selected for user pairing and is initialized to 0 for all j. Thereafter, the algorithm sorts the DL user in order from the highest to lowest DL SINR value and then runs based on these sorted DL users. For DL user i, the pairing algorithm measures the direct channel φij from the unpaired UL user j to the DL user i. Then, the UL user having the smallest φij value among the unpaired UL users is selected for pairing. Accordingly, the flag of this UL user j is set to 1; thus, the DL user i and selected UL user j are paired with each other. This pairing operation is repeated until all sorted DL users are paired with the remaining UL users. Note that the proposed algorithm depends on only one iteration and one search within each iteration for M DL/UL users so that its computation complexity becomes O(M2).Algorithm 1:Proposed user pairing algorithm.
(1) Initialize:
Pj←Pmax,  j=1,2,…,M
Uj←0,  j=1,2,…,M /∗ flag for selected or not ∗/
(2) Sort the DL users in order of high γiHD value
(3) for  i = [sorted DL users]  do
(4)  Unpaired UL users ← find (Uj = 0)
(5)  Measure φij from unpaired UL user j to DL user i
(6)  Choose the UL user j with the smallest φij among unpaired UL users
(7)  Uj←1
(8)  Complete the user pair (i,j)
(9) end for


## 4. Performance Analysis

### 4.1. Assumptions and Notations

For the stochastic geometry-based analysis, we assume a homogeneous cellular network that consists of macro-cell BSs and users. The BSs are arranged according to a homogeneous Poisson point process (PPP)Φ with intensity λ in the Euclidean plane [22, 23]. We also assume that DL and UL users are distributed according to an independent homogeneous PPP Ω with the same density λ so that the FD BS schedules one UL user and one DL user concurrently during a given time/frequency resource unit [23, 24]. Each user is connected to its closest BS.Figure2 shows the system model where a BS0 simultaneously serves its DL user u0 and UL user u1 through FD. While u0 receives and u1 transmits a signal from and to BS0 using FD, u0 receives interferences from u1, u3, and BS1, and u1 receives interferences from u3, BS0, and BS1. Normal line with (h,·) denotes the signal channel. Also, dotted line with (g,·) and dashed line with (φ,·) denote interference channels by the neighboring BSs and users, respectively. Here, we assume that a tagged BS and tagged user experience Rayleigh fading with unit mean and employ a constant transmit power of 1/μ. In this case, the received power at a typical node at a distance r from its BS is hr-α, where the random variable h follows an exponential distribution with mean 1/μ (i.e., h~exp⁡(μ)). It is denoted as (h,μ). Furthermore, all BSs and users experience a standard power loss propagation with path loss exponent α>2 and additive and constant noise power of σ2. All results are related to a single transmit antenna and single receive antenna, although future extension to multiple antennas is clearly desirable.Figure 2
System model and notations.


### 4.2. Downlink Rate in Full-Duplex

When a typical DL user,u0, has a distance r and fading strength hd~exp⁡(μd) from its serving BS, the user u0 receives a signal with a power of hdr-α. However, the user u0 experiences cumulative interferences Ib and Iu from neighbors, where Ib is the sum of the received powers from all other BSs, each with distance ri and fading strength gi~exp⁡(κbu) to the user u0 such that(12)Ib=∑i∈Φbgiri-α,where Φb is the set of all interfering BSs. In addition, Iu is the sum of the received powers from all other UL users, each with distance ri and fading strength φi~exp⁡(κuu) to the user u0 such that(13)Iu=∑i∈Ωuφiri-α,where Ωu is the set of all interfering users. The SINR of the typical DL user u0 can then be expressed as(14)γDL=hdr-αIb+Iu+σ2.The average rate of the typical DL user u0 is(15)RDLλ,α=EIb,Iu,rln⁡1+γDLIb,Iu,r.Because the probability that the closest BS has a distance r is e-λπr22πλr, (15) is calculated as (16)RDLλ,α=∫r>0e-πλr2EIb,Iuln1+hdr-αIb+Iu+σ2∣r2πλrdr,where (17)EIb,Iuln⁡1+hdr-αIb+Iu+σ2∣r=∫t>0PIb,Iuln⁡1+hdr-αIb+Iu+σ2>t∣rdt=∫t>0PIb,Iuhd>rασ2+Ib+Iuet-1∣rdt=∫t>0EIb,IuPhd>rασ2+Ib+Iuet-1∣r,Ib,Iudtsincehd~exp⁡μd,EIb,Iuln⁡1+hdr-αIb+Iu+σ2∣r=∫t>0EIb,Iue-μdrασ2+Ib+Iuet-1dt=∫t>0EIb,Iue-μdrασ2et-1e-μrαIb+Iuet-1dt=∫t>0e-μdrασ2et-1EIbe-μdrαIbet-1EIue-μdrαIuet-1dt=∫t>0e-μdrασ2et-1LIbμdrαet-1LIuμdrαet-1dt.In (17), the Laplace transforms of Ib and Iu are obtained as the following two lemmas, respectively.Lemma 1.
Laplace transform ofIb(18)LIbμdrαet-1=exp⁡-πλr2βd,buet-12/α∫βd,buet-1-2/α∞11+xα/2dx.Proof.
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### 4.3. Uplink Rate in Full-Duplex

When a typical UL useru1 has a distance r and fading strength hu~exp⁡(μu) to its serving BS, user u1’s signal power at the serving BS is hur-α. Moreover, user u1 experiences cumulative interferences Ib, Iu, and Is. Here, Ib is the sum of the received powers from all other BSs, each with distance ri and fading strength gi~exp⁡(κbb) to the serving BS of user u1, such that(24)Ib=∑i∈Φbgiri-α,where Φb is the set of all interfering BSs. Also, Iu is the sum of the received powers from all other UL users, each with distance ri and fading strength φi~exp⁡(κub) to the serving BS of user u1, such that(25)Iu=∑i∈Ωuφiri-α,where Ωu is the set of all interfering users. Moreover, Is denotes the self-interference by the serving BS of user u1. Therefore, the SINR of typical UL user u1 can be expressed as(26)γUL=hdr-αIb+Iu+Is+σ2.Here, we assume that the self-interference Is amounts to ςσ2 (i.e., ς times noise power σ2). Then, the average rate of a typical UL user u1 is (27)RULλ,α=EIb,Iu,Is,rln⁡1+γULIb,Iu,Is,r=∫r>0e-πλr2∫t>0e-μurασ2+ςσ2et-1LIbμurαet-1LIuμurαet-1dt2πλrdr.Similar to the downlink analysis, we can obtain the Laplace transform of Ib and Iu in this uplink case, as follows:(28)LIbμurαet-1=exp⁡-2πλ∫r∞1-11+μu/κbbet-1r/vαvdvwhere  weputμuκbb=βu,bb,LIbμurαet-1=exp⁡-2πλ∫r∞βu,bbet-1βu,bbet-1+v/rαvdvwhere  weputv2r2=u,LIbμurαet-1=exp⁡-πλr2∫1∞11+βu,bbet-1-2/αuα/2duwhere  weputβu,bbet-1-2/αu=x,LIbμurαet-1=exp-πλr2βu,bbet-12/α∫βu,bbet-1-2/α∞11+xα/2dx,LIuμurαet-1=exp⁡-2πλ∫r∞1-11+μu/κubet-1r/vαvdvwhere  weputμuκub=βu,ub,LIuμurαet-1=exp⁡-2πλ∫r∞βu,ubet-1βu,ubet-1+v/rαvdvwhere  weputv2r2=u,LIuμurαet-1=exp⁡-πλr2∫1∞11+βu,ubet-1-2/αuα/2duwhere  weputβu,ubet-1-2/αu=x,LIuμurαet-1=exp⁡-πλr2βu,ubet-12/α∫βu,ubet-1-2/α∞11+xα/2dx.


### 4.4. Sum Rate in Full-Duplex

The average sum rate,RFD(λ,α), with density λ and path loss exponent α is summarized as(29)RFDλ,α=RDLλ,α+RULλ,α.


## 4.1. Assumptions and Notations

For the stochastic geometry-based analysis, we assume a homogeneous cellular network that consists of macro-cell BSs and users. The BSs are arranged according to a homogeneous Poisson point process (PPP)Φ with intensity λ in the Euclidean plane [22, 23]. We also assume that DL and UL users are distributed according to an independent homogeneous PPP Ω with the same density λ so that the FD BS schedules one UL user and one DL user concurrently during a given time/frequency resource unit [23, 24]. Each user is connected to its closest BS.Figure2 shows the system model where a BS0 simultaneously serves its DL user u0 and UL user u1 through FD. While u0 receives and u1 transmits a signal from and to BS0 using FD, u0 receives interferences from u1, u3, and BS1, and u1 receives interferences from u3, BS0, and BS1. Normal line with (h,·) denotes the signal channel. Also, dotted line with (g,·) and dashed line with (φ,·) denote interference channels by the neighboring BSs and users, respectively. Here, we assume that a tagged BS and tagged user experience Rayleigh fading with unit mean and employ a constant transmit power of 1/μ. In this case, the received power at a typical node at a distance r from its BS is hr-α, where the random variable h follows an exponential distribution with mean 1/μ (i.e., h~exp⁡(μ)). It is denoted as (h,μ). Furthermore, all BSs and users experience a standard power loss propagation with path loss exponent α>2 and additive and constant noise power of σ2. All results are related to a single transmit antenna and single receive antenna, although future extension to multiple antennas is clearly desirable.Figure 2
System model and notations.


## 4.2. Downlink Rate in Full-Duplex

When a typical DL user,u0, has a distance r and fading strength hd~exp⁡(μd) from its serving BS, the user u0 receives a signal with a power of hdr-α. However, the user u0 experiences cumulative interferences Ib and Iu from neighbors, where Ib is the sum of the received powers from all other BSs, each with distance ri and fading strength gi~exp⁡(κbu) to the user u0 such that(12)Ib=∑i∈Φbgiri-α,where Φb is the set of all interfering BSs. In addition, Iu is the sum of the received powers from all other UL users, each with distance ri and fading strength φi~exp⁡(κuu) to the user u0 such that(13)Iu=∑i∈Ωuφiri-α,where Ωu is the set of all interfering users. The SINR of the typical DL user u0 can then be expressed as(14)γDL=hdr-αIb+Iu+σ2.The average rate of the typical DL user u0 is(15)RDLλ,α=EIb,Iu,rln⁡1+γDLIb,Iu,r.Because the probability that the closest BS has a distance r is e-λπr22πλr, (15) is calculated as (16)RDLλ,α=∫r>0e-πλr2EIb,Iuln1+hdr-αIb+Iu+σ2∣r2πλrdr,where (17)EIb,Iuln⁡1+hdr-αIb+Iu+σ2∣r=∫t>0PIb,Iuln⁡1+hdr-αIb+Iu+σ2>t∣rdt=∫t>0PIb,Iuhd>rασ2+Ib+Iuet-1∣rdt=∫t>0EIb,IuPhd>rασ2+Ib+Iuet-1∣r,Ib,Iudtsincehd~exp⁡μd,EIb,Iuln⁡1+hdr-αIb+Iu+σ2∣r=∫t>0EIb,Iue-μdrασ2+Ib+Iuet-1dt=∫t>0EIb,Iue-μdrασ2et-1e-μrαIb+Iuet-1dt=∫t>0e-μdrασ2et-1EIbe-μdrαIbet-1EIue-μdrαIuet-1dt=∫t>0e-μdrασ2et-1LIbμdrαet-1LIuμdrαet-1dt.In (17), the Laplace transforms of Ib and Iu are obtained as the following two lemmas, respectively.Lemma 1.
Laplace transform ofIb(18)LIbμdrαet-1=exp⁡-πλr2βd,buet-12/α∫βd,buet-1-2/α∞11+xα/2dx.Proof.
(19)
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Therefore,(20)LIbμdrαet-1=exp-2πλ∫r∞1-11+μd/κbuet-1r/vαvdvwhere  weputμdκbu=βd,buLIbμdrαet-1=exp-2πλ∫r∞1-11+βd,buet-1r/vαvdv=exp-2πλ∫r∞1+βd,buet-1r/vα-11+βd,buet-1r/vαvdv=exp-2πλ∫r∞βd,buet-1r/vα1+βd,buet-1r/vαvdv=exp-2πλ∫r∞βd,buet-1βd,buet-1+v/rαvdvwhere  weputv2r2=u,LIbμdrαet-1=exp⁡-πλr2∫1∞βd,buet-1βd,buet-1+uα/2du=exp⁡-πλr2∫1∞11+βd,buet-1-2/αuα/2duwhere  weputβd,buet-1-2/αu=x,LIbμdrαet-1=exp-πλr2βd,buet-12/α∫βd,buet-1-2/α∞11+xα/2dx.Lemma 2.
Laplace transform ofIu(21)LIuμdrαet-1=exp⁡-πλr2βd,uuet-12/α∫0∞11+xα/2dx.Proof.
(22)
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Therefore,(23)LIuμdrαet-1=exp-2πλ∫0∞1-11+μd/κuuet-1r/vαvdvwhere  weputμdκuu=βd,uu,LIuμdrαet-1=exp⁡-2πλ∫0∞1-11+βd,uuet-1r/vαvdv=exp⁡-2πλ∫0∞1+βd,uuet-1r/vα-11+βd,uuet-1r/vαvdv=exp⁡-2πλ∫0∞βd,uuet-1r/vα1+βd,uuet-1r/vαvdv=exp⁡-2πλ∫0∞βd,uuet-1βd,uuet-1+v/rαvdvwhere  weputv2r2=u,LIuμdrαet-1=exp-πλr2∫0∞βd,uuet-1βd,uuet-1+uα/2du=exp-πλr2∫0∞11+βd,uuet-1-2/αuα/2duwhere  weputβd,uuet-1-2/αu=x,LIuμdrαet-1=exp⁡-πλr2βd,uuet-12/α∫0∞11+xα/2dx.


## 4.3. Uplink Rate in Full-Duplex

When a typical UL useru1 has a distance r and fading strength hu~exp⁡(μu) to its serving BS, user u1’s signal power at the serving BS is hur-α. Moreover, user u1 experiences cumulative interferences Ib, Iu, and Is. Here, Ib is the sum of the received powers from all other BSs, each with distance ri and fading strength gi~exp⁡(κbb) to the serving BS of user u1, such that(24)Ib=∑i∈Φbgiri-α,where Φb is the set of all interfering BSs. Also, Iu is the sum of the received powers from all other UL users, each with distance ri and fading strength φi~exp⁡(κub) to the serving BS of user u1, such that(25)Iu=∑i∈Ωuφiri-α,where Ωu is the set of all interfering users. Moreover, Is denotes the self-interference by the serving BS of user u1. Therefore, the SINR of typical UL user u1 can be expressed as(26)γUL=hdr-αIb+Iu+Is+σ2.Here, we assume that the self-interference Is amounts to ςσ2 (i.e., ς times noise power σ2). Then, the average rate of a typical UL user u1 is (27)RULλ,α=EIb,Iu,Is,rln⁡1+γULIb,Iu,Is,r=∫r>0e-πλr2∫t>0e-μurασ2+ςσ2et-1LIbμurαet-1LIuμurαet-1dt2πλrdr.Similar to the downlink analysis, we can obtain the Laplace transform of Ib and Iu in this uplink case, as follows:(28)LIbμurαet-1=exp⁡-2πλ∫r∞1-11+μu/κbbet-1r/vαvdvwhere  weputμuκbb=βu,bb,LIbμurαet-1=exp⁡-2πλ∫r∞βu,bbet-1βu,bbet-1+v/rαvdvwhere  weputv2r2=u,LIbμurαet-1=exp⁡-πλr2∫1∞11+βu,bbet-1-2/αuα/2duwhere  weputβu,bbet-1-2/αu=x,LIbμurαet-1=exp-πλr2βu,bbet-12/α∫βu,bbet-1-2/α∞11+xα/2dx,LIuμurαet-1=exp⁡-2πλ∫r∞1-11+μu/κubet-1r/vαvdvwhere  weputμuκub=βu,ub,LIuμurαet-1=exp⁡-2πλ∫r∞βu,ubet-1βu,ubet-1+v/rαvdvwhere  weputv2r2=u,LIuμurαet-1=exp⁡-πλr2∫1∞11+βu,ubet-1-2/αuα/2duwhere  weputβu,ubet-1-2/αu=x,LIuμurαet-1=exp⁡-πλr2βu,ubet-12/α∫βu,ubet-1-2/α∞11+xα/2dx.


## 4.4. Sum Rate in Full-Duplex

The average sum rate,RFD(λ,α), with density λ and path loss exponent α is summarized as(29)RFDλ,α=RDLλ,α+RULλ,α.


## 5. Results and Discussions

For a performance comparison with the proposed algorithms, we consider (i) the conventionalHD mode in which the BS does not use the FD technique, (ii) the FD with therandom pairing that determines the user pair randomly, and (iii) the FD with theoptimal pairing that determines the optimal user pairs through exhaustive search. We observe the system rate by changing the user density λ from 0.1 to 1. For example, the users with λ=1 denote the density of one user per 100 m2 on average [24]. For simulation, we consider the same network environment as the one considered at the analysis [10]. The same number of DL and UL users is randomly distributed according to the density λ in the multicell network areas. The detailed parameters used for evaluation are listed in Table 1. Here, we obtain the channel parameter from neighbor UL user to UL user, κuu, by the simulation after applying each of the considered user pairing algorithms.Table 1
Parameter Setup.
Parameter
Description
Value
λ
User density
0.1~1 (default = 0.5)
σ2
Noise power
1
α
Path loss exponent
4, 3, 2 (default = 4)
ς
Self-interference ratio
0, 10, 100 (default = 0)
μd
Channel parameter from serving BS to DL user
0.1
μu
Channel parameter from UL user to serving BS
1
κbu
Channel parameter from neighbor BS to DL user
0.1
κub
Channel parameter from neighbor UL user to serving BS
1
κbb
Channel parameter from neighbor BS to serving BS
100, 1, 0.1 (default = 100)
κuu
Channel parameter from neighbor UL user to UL user
Variable (obtained by simulation)Figure3 plots the cumulative distribution function (CDF) of the interuser interference power when λ=0.5. This interuser interference level is measured by the simulation when each user pairing algorithm is applied. The distribution of interuser interference is an important measure to verify whether the user pairing algorithm works well. Compared to the random pairing, the proposed pairing remarkably reduces the interuser interference. Moreover, there is no significant difference between the proposed pairing and the optimal pairing.Figure 3
Cumulative density function of interuser interference whenλ=0.5.Figure4 shows the average interuser interference level versus the user density λ. In the case of the random pairing, the interuser interference increases gradually as the user density increases. This is because the random pairing algorithm often generates uncontrolled stronger interferences between the randomly selected DL and UL users as the number of users increases. On the other hand, in the proposed pairing algorithm, the interuser interference decreases gradually as the user density increases because the user diversity increases as the number of users increases so that the proposed algorithm has more opportunity to select a DL and UL user pair with less interuser interference. The optimal pairing shows the similar tendency as the proposed one and has a slightly lower interuser interference level.Figure 4
Average interuser interference versus user densityλ.Figure5 plots the CDF of the uplink rate, downlink rate, and sum rate when λ=0.5. In the case of the UL rate, the FD mode has the same performance regardless of pairing algorithms because the user pairing algorithm affects only the SINR of DL user. However, the UL rate of FD mode is considerably improved compared with that of HD mode. On the other hand, the DL rate is different according to the pairing algorithm. The DL rate is improved in order of the random, proposed, and optimal pairing because their interuser interference level is reduced in this order, as shown in Figure 4. In the case of the sum rate (i.e., sum of UL and DL rates), the similar performance is shown. We can observe that the rate of the proposed pairing algorithm approaches that of the optimal pairing.Figure 5
Cumulative density function of (a) uplink rate, (b) downlink rate, and (c) sum rate whenλ=0.5.
(a)
(b)
(c)Figure6 shows the average rate versus the user density λ. As the user density increases, the sum rate increases because of the effect of user diversity. In each λ, the UL rates of FD mode are equal regardless of the pairing algorithm. However, the DL rate is different according to the pairing algorithm. The proposed pairing algorithm improves the DL rate greater than the random pairing and approaches the optimal pairing. This verifies the proposed pairing algorithm reduces the interuser interference effectively and so contributes to improve the cell throughput. The performance difference between the proposed pairing and the random pairing increases as λ increases because the proposed pairing algorithm has more opportunities to employ the user diversity but the random pairing does not. Compared with the HD mode, the average gains of FD using the random pairing and proposed pairing algorithms are approximately 180 and 195%, respectively, when λ=0.5.Figure 6
Average rate versus user densityλ.Figure7 shows the average sum rate versus the user density λ according to the variations of related parameters: (a) self-interference ratio (ς), (b) channel parameter from neighbor BS to serving BS (κbb), and (c) path loss exponent (α). ς greater than zero means that there remains the self-interference at the BS using FD. Thus, the sum rates of FD mode decrease as ς increases. In addition, the smaller κbb means that there exists more interference from neighboring BSs because of the imperfect channel estimation. This effect also deteriorates the sum rates of FD mode. When we change the path loss exponent α, the sum rate decreases as α decreases and the performance gap between the proposed and random pairing algorithms becomes smaller. This is because the interuser interference increases as the path loss between users weakens. These results show that the FD system depends heavily on the interference cancellation capability at the BS and it is appropriate to apply in the channel environment with high attenuation in order to maximize its performance.Figure 7
Average sum rate versus user densityλ according to the variations of (a) ς, (b) κbb, and (c) α.
(a)
(b)
(c)


## 6. Conclusion

In this study, we investigated the issue of user pairing in FD wireless cellular networks. We proposed a heuristic user pairing algorithm with low complexity to maximize cell throughput. The proposed algorithm was designed such that the DL user having a better signal quality has higher priority to choose its UL user for throughput maximization. Then, we analyzed the performance of the FD cellular network theoretically. The results showed that the proposed user pairing algorithm effectively reduced the interuser interference and achieved near-optimal performance. Thus, the FD system that uses it considerably outperforms the FD system using a random user pairing and almost doubles the conventional HD system in terms of cell throughput. For further study, we will extend the proposed user pairing algorithm to more general user scheduling algorithms that dynamically allocate radio resources and the transmission direction in an FD cellular network.



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*Source: 8182150-2017-03-23.xml*