# Smart Switching Strategy-Based Supervision Rule to Mitigate the Problem of Pilot Contamination in Massive MIMO Systems

Authors: Abdelfettah Belhabib; Mohamed Boulouird; Moha M’Rabet Hassani
Journal: Wireless Communications and Mobile Computing
 (2020)

---

## Abstract

Despite the large benefits that can be fulfilled through the exploitation of Massive Multi-input Multioutput (M-MIMO), this technology still constrained by a well-known constraint, called as pilot contamination problem (PCP), which is the main consequence of, simultaneously, reusing the same set of orthogonal pilot sequences (OPSs) for the users of several cells. Due to the scarcity of the OPS resources, the reuse of the same set of the OPSs for the users of different cells is unavoidable. Hence, this work proposes a novel decontaminating strategy, which is aimed at guaranteeing a trade-off between the use of the OPSs and the mitigation of the PCP. Specifically, to make the use of the available OPSs better, we propose the consolidation of two powerful decontaminating strategies. Under a derived supervision rule (SR), these strategies are the soft pilot reuse-based multicell block diagonalization precoding (SPR-MBDP) and the weighted graph coloring-based pilot assignment (WGC-PA). The SR enables the switching mechanism between the two strategies, which leads to address the PCP with a fewer number of the OPSs compared to the SPR-MBDP, therefore boosting the per-cell achievable rate. Simulation results prove the effectiveness of our proposed strategy.


---

## Body

## 1. Introduction

M-MIMO technology was introduced in 2010 by [1], and it is aimed at serving, simultaneously, tens of the users (K user per cell). This is by employing a massive number (i.e., hundreds) of antennas (i.e., M antennas) at the Base Stations (BSs). This relevant technology has large benefits [2]. Thus, as the number of antennas deployed in the BSs M goes to the infinity, the noisy components and the intracell interferences are made small [1], and the transmit power is reduced through a beamforming technique, which contributes to boosting the energy efficiency [3]. The M-MIMO technology has been matured and adapted to the fifth-generation (5G); this is with a finite number of antennas M=64, and it has been commercialized in the sub-6 GHz bands [4]. However, this is not the end of the M-MIMO, because its large benefits are not yet fully achieved, and still constrained by the PCP [4] (Figure 1).Figure 1
Illustration of the PCP in the case of two hexagonal cells with one user in each cell.The reuse of the same set of the OPSs, across different cells, leads to the problem of intercell interferences. To avoid this problem, each user should be assigned a specific pilot sequence; in other words, the users of all cells must be assigned with a set of OPSs. However, this is not possible due to the scarcity of the available set of OPSs; consequently, the reuse of the same set of OPSs across different cells is unavoidable; hence, the PCP will emerge between the users that employ the same OPSs. Consequently, the channel state information (CSI) will be, inaccurately, estimated and the whole process of communication between the users and the BSs will be dramatically damaged [5].Since the appearance of M-MIMO, a large body of research has been carried out to deal with the PCP. A time-shifted-based strategy has been proposed in [6, 7], which proposes to insert a delay between the phases of communication of the adjacent cells; meanwhile, some of them perform in the data transmission phase; the others are obliged to perform in the pilot transmission phase. However, the estimated CSI at the BSs becomes contaminated by the transmitted data in adjacent cells instead of being contaminated by the pilot sequences. Based on the singular value decomposition (SVD), the null space of interference was exploited to schedule the data signals toward their desired target (i.e., users) [8, 9]. Indeed, extra pilot resources are required to accurately identify the null space of interferences. The space signatures were suggested to spatially identify the desired signals based on their angle of arrivals (AoAs); therefore, the same OPSs are allocated to the users having nonoverlapped AoAs [10, 11]. Nevertheless, a nonoverlapping AoA is assumed which does not necessarily hold in practice.Optimization-based decontamination strategy was suggested to boost the quality-of-services (QoS) for the overall users [12–14]. Specifically, the proposal of [12] is aimed at finding out the optimal pilot assignment scheme, that is, through maximizing the sum rate within cells, while [13] focuses on assigning OPSs to the users of adjacent cells, which are suffering from a severe influence of the PCP; in [14], the users within cells are divided, based on their signal-to-interference-plus-noise ratio (SINR), and those having a low SINR are optimally assigned with OPSs.A weighted graph coloring-based pilot assignment (WGC-PA) decontaminating strategy is proposed in the work of [15], which is aimed at reducing the strength of the PCP upon the users that employ the same OPSs; this is through the employment of the graph coloring theory. The WGC-PA has been coupled with the soft pilot reuse (SPR) strategy in the work of [16, 17], which was, only, focused on the uplink phase. Specifically, [16] proposed to enhance the performance of edge users based on the SPR, while the slight contamination among the center users is addressed based on the WGC-PA. In [17], users within cells are split into three groups, and the problem of the PCP upon the two groups is separately addressed based on the WGC-PA and the SPR strategies, while the PCP upon the group of the center users is neglected.Regarding the fact that the center users are close to the BSs, they benefit from a high QoS compared to the edge users, which are facing a severe degree of the PCP that comes from the edge users of the adjacent cells. To guarantee the fairness between the users of each cell, it is necessary to enhance the QoS of the edge users. Accordingly, a soft pilot reuse-based multicell block diagonalization precoding (SPR-MBDP) was proposed in [9], which suggests selecting the edge users of the overall cells; therefore, each one of the selected users is assigned with a unique OPSs, while the center users of the overall cells are obliged to reuse the same set of the OPSs (i.e., full pilot reuse); therefore, by exploiting the singular value decomposition (SVD) tool, data symbols are scheduled toward the target users. That is through a null subspace of the PCP. However, the problem of pilot overhead can damage the fulfilled performance of the SPR-MBDP [9], due to its requirement for extra OPSs.In the existing decontaminating strategies, less attention has been paid to the problem of pilot overhead, which is—without any doubt—a crucial side that must be respected in any proposed decontaminating strategy; stand from that remark, this paper provides a new decontaminating strategy to boost the performance of the M-MIMO. Specifically, to ensure a trade-off between the use of the OPSs and the mitigation of the PCP, we propose the consolidation of the SPR-MBDP and the WGC-PA. This is under a derived supervision rule (SR) that economizes and controls the use of the precious OPSs while trying to overcome the PCP. The proposed strategy is referred to as a smart switching strategy-based supervision rule (S3SR). In a nutshell, the prime contribution of the present paper can be summarized as follows:(i)
A novel S3SR strategy is proposed to deal with the PCP in M-MIMO(ii)
To ensure a trade-off between the use of OPSs and the mitigation of the PCP, the proposed S3SR is automatically updated based on the SR, which is a new approach that can be exploited to deal with the PCP(iii)
Significant improvements are achieved and proved through simulation resultsTo the best of our knowledge, no prior work has analyzed the S3SR strategy based on the consolidation of the SPR-MBDP and the WGC-PA strategies in the multicell M-MIMO system.This paper is organized as follows: Section2 presents the system model adopted in this work. Section 3 sheds light on the communication schemes and reviews of the influence of PCP upon M-MIMO systems; therefore, Section 4 presents the proposed S3SR strategy. Section 5 provides the simulation results which prove the effectiveness of the proposed S3SR strategy compared to WGC-PA, SPR-MBDP, and the conventional strategy [1]. Finally, Section 6 summarizes the prime contribution of this work.To facilitate the numerical analysis, the following terminologies are considered: the cardinality of a set, the transpose conjugate (i.e., Hermitian), the transpose, the conjugate, and the pseudoinverse are, respectively, denoted ascard., .H, .T, .∗, and .†. The K×K identity matrix is denoted as IK, while j,k refers to the kth user of the jth cell. The Frobenius norm of a matrix is denoted as .F.


## 2. System Model

This section is aimed at presenting the system model, which is adopted in this paper. Hence, a multicell M-MIMO system is considered which is composed ofL hexagonal cells. Each cell is equipped with a centered BS of M antennas, and K<<M single-antenna users [1, 9, 15]. The channel vector that links the kth user of the lth cell (i.e., l,k) to the BS of the jth cell can be expressed as follows:
(1)gl,k,j=hl,k,jβl,k,j,where hl,k,j∈ℂM×1 denotes the small scale fading (SSF) vector. The SSF vectors of the K users are statistically independent and follow a complex Gaussian distribution of zero-mean vector and a covariance matrix IM; therefore, hl,k,j∼Cℵ0,IM. The second coefficient βl,k,j refers to the large-scale fading (LSF) coefficient, which accounts for the path loss and the shadow fading. The LSF in (1) can be expressed [2, 4, 9, 10, 15, 18] as follows:
(2)βl,k,j=zl,k,jrl,k,j/Rα,where rl,k,j denotes the distance between l,k and the BS of the jth cell, zl,k,j refers to the shadow-fading factor, and it has a log-normal distribution (i.e., 10log10zl,k,j) that follows a zero-mean Gaussian distribution, which is characterized by a standard deviation σshadow, while R denotes the cell radius and α represents the path loss exponent. The channel matrix between the K users of the lth cell to the BS of the jth cell can be expressed as
(3)Gj,l=gl,1,j,gl,2,j,⋯,gl,K,j=hl,1,j,hl,2,j,⋯,hl,K,jDj,l1/2,whereDj,l=diagβl,1,j,βl,2,j,⋯,βl,K,j is a diagonal matrix, which includes the LSF coefficients of the K users of the lth cell to the BS of the jth cell.


### 2.1. The Pilot Sequences and Their Utility

In order to shed light on the concept of the OPSs, which is one of the fundamentals of the wireless communication systems, let’s assume that a userk of cell j is assigned with a pilot sequences ϕk=ϕk1,ϕk2,⋯,ϕkτ∈ℂ1×τ of length τ, whereas a user k′ of the same cell is assigned with the pilot sequences ϕk’ of length τ. To estimate the channel vector of the two users at the BS of cell j, the two users uplink their corresponding pilot sequences; therefore, the BS of cell j exploits the received pilot signals and the orthogonality condition to estimate the channel of each user. This is through correlating the received pilot signals with the pilot sequences. The orthogonality condition is expressed as follows:
(4)ϕk’ϕkH=δk,k’,where δk,k’ is the delta function, which is expressed as
(5)δk,k’=1,ifk=k′,0,otherwise.Hence, to avoid the PCP, each user—of the overall cells—should be assigned with an OPS, which would requireKL OPSs to serve the KL users of the considered L cells. Due to the problem of the pilot overhead, using that number of pilots is not allowed.


### 2.2. The Problem of Pilot Overhead

Under the block-fading channel model, based time-division-duplex (TDD) protocol, the communication between the users and their supported BSs can be split into coherence time intervals (i.e., frames) (Figure2), where the channel responses remain constant. Therefore, at the beginning of each frame, the users uplink their pilot sequences of length τ. To avoid the intracell interference, the users that belong to the same cell are allocated with OPSs, which requires K=τ orthogonal pilot sequences.Figure 2
Illustration of a frame, which contains two parts: the first one in the left is dedicated to the pilot transmission, whereas the second part is reserved for data transmission.To reduce the PCP, extra orthogonal pilot sequences can be used at the expense of shrinking the bandwidth left for data transmission, this problem is what we call the problem of pilot overhead, and it is expressed by the ratioμ=τ/T. In other words, increasing the number of OPSs—used for channel training—leads to reduce the subframe of the data transmission (Figure 2).


### 2.3. Asymptotic Orthogonality

Letgl,k,j, gl,k′,j be two M×1 vectors, which are mutually independent, where gl,k,j∼Cℵ0,βl,k,jIM and gl,k′,j∼Cℵ0,βl,k′,jIM. Hence, the asymptotic orthogonality condition can be expressed as in [3, 9, 19] as
(6)limM→∞gl,k,jHgl,k,jM⟶βl,k,j,limM→∞gl,k,jHgl,k′,jM⟶0.The given of (6) is the foothold of the M-MIMO technology, where the channel vectors become mutually orthogonal for M⟶∞. Hence, the LSF can be exploited to describe the channel of each user, instead of using the SSFs, which varies quickly and hard to be exploited.


## 2.1. The Pilot Sequences and Their Utility

In order to shed light on the concept of the OPSs, which is one of the fundamentals of the wireless communication systems, let’s assume that a userk of cell j is assigned with a pilot sequences ϕk=ϕk1,ϕk2,⋯,ϕkτ∈ℂ1×τ of length τ, whereas a user k′ of the same cell is assigned with the pilot sequences ϕk’ of length τ. To estimate the channel vector of the two users at the BS of cell j, the two users uplink their corresponding pilot sequences; therefore, the BS of cell j exploits the received pilot signals and the orthogonality condition to estimate the channel of each user. This is through correlating the received pilot signals with the pilot sequences. The orthogonality condition is expressed as follows:
(4)ϕk’ϕkH=δk,k’,where δk,k’ is the delta function, which is expressed as
(5)δk,k’=1,ifk=k′,0,otherwise.Hence, to avoid the PCP, each user—of the overall cells—should be assigned with an OPS, which would requireKL OPSs to serve the KL users of the considered L cells. Due to the problem of the pilot overhead, using that number of pilots is not allowed.


## 2.2. The Problem of Pilot Overhead

Under the block-fading channel model, based time-division-duplex (TDD) protocol, the communication between the users and their supported BSs can be split into coherence time intervals (i.e., frames) (Figure2), where the channel responses remain constant. Therefore, at the beginning of each frame, the users uplink their pilot sequences of length τ. To avoid the intracell interference, the users that belong to the same cell are allocated with OPSs, which requires K=τ orthogonal pilot sequences.Figure 2
Illustration of a frame, which contains two parts: the first one in the left is dedicated to the pilot transmission, whereas the second part is reserved for data transmission.To reduce the PCP, extra orthogonal pilot sequences can be used at the expense of shrinking the bandwidth left for data transmission, this problem is what we call the problem of pilot overhead, and it is expressed by the ratioμ=τ/T. In other words, increasing the number of OPSs—used for channel training—leads to reduce the subframe of the data transmission (Figure 2).


## 2.3. Asymptotic Orthogonality

Letgl,k,j, gl,k′,j be two M×1 vectors, which are mutually independent, where gl,k,j∼Cℵ0,βl,k,jIM and gl,k′,j∼Cℵ0,βl,k′,jIM. Hence, the asymptotic orthogonality condition can be expressed as in [3, 9, 19] as
(6)limM→∞gl,k,jHgl,k,jM⟶βl,k,j,limM→∞gl,k,jHgl,k′,jM⟶0.The given of (6) is the foothold of the M-MIMO technology, where the channel vectors become mutually orthogonal for M⟶∞. Hence, the LSF can be exploited to describe the channel of each user, instead of using the SSFs, which varies quickly and hard to be exploited.


## 3. Communication Schemes and the Effect of Pilot Contamination

This section is aimed at analyzing the different processes of the wireless communication between the users and their respective BSs, therefore showing the main influence of the PCP on the fulfilled performance of the M-MIMO. Thus, the widely adopted block-fading channel model, based TDD protocol, is considered [4, 9, 15, 20], where the channel responses remain constant. Thus, three specific phases can be distinguished: channel estimation through uplink training, data detection, and data precoding. Therefore, by adopting the system model presented in Section 2, these three phases are provided in the following subsections.


### 3.1. Training Phase and Channel Estimation

During the training phase, the users of each cell, synchronously, uplink their pilot sequences of lengthτ. Thus, the received pilot signal yjp∈ℂM×τ at the BS of the jth cell can be expressed as
(7)yjp=ρp∑l=1L∑k=1Kgl,k,jϕk+njp,where ρp represents the transmitted power, which is used by each user to uplink its pilot sequences; ϕk∈ℂ1×τ is the pilot sequences of the kth user, where the orthogonality condition ϕkϕk′H=δk,k′ is satisfied [21, 22]; njp∈ℂM×τ is an additive white Gaussian noise (AWGN) at the antennas BS of the jth cell. Therefore, by correlating the received pilot signal yjp, with the local pilot sequences ϕk, the estimated channel of j,k at the BS of the jth cell is expressed as
(8)g^j,k,j=yjpϕkHρp=∑l=1Lgl,k,jϕkϕkH+∑l=1L∑k′=1,k′≠kKgl,k′,jϕk′ϕkH+njpϕkHρp=gj,k,j+∑l≠jLgl,k,j+njpϕkHρp.


### 3.2. Data Detection

Users of each cell uplink their data symbols. Accordingly, the userj,k uplink its data symbols xj,ku, where Exj,ku2=1. Therefore, the received data signal yj,ku at the BS of the jth cell can be expressed as
(9)yju=ρu∑l=1L∑k=1Kgl,k,jxl,ku+nju,where nju denotes the AWGN at the antennas BS of the jth cell. Therefore, based on the matched-filter (MF) detector (i.e., maximum ratio combining) [23], the estimated data symbols x^j,ku of the user j,k, at the BS of the jth cell, can be computed as follows:
(10)x^j,ku=g^j,k,jHyju=ρugj,k,jHgj,k,jxj,ku+ρu∑l≠jgl,k,jHgl,k,jxl,ku+εj,ku,where εj,ku arranges the interference and the noisy components, and it is expressed as
(11)εj,ku=ρugj,k,jH∑l≠jgl,k,jHxl,ku+ρu∑l≠jgl,k,jHgj,k,jxj,ku+∑l=1gl,k,jHnju+njpϕkHHρpρu∑l=1L∑k=1Kgl,k,jxl,ku+nju.Therefore, the uplink SINR ofj,k can be expressed as
(12)SINRj,ku=gj,k,jHgj,k,j2∑l≠jLgl,k,jHgl,k,j2+εj,ku2/ρu⟶basedon6βj,k,j2∑l≠jβl,k,j2.Hence, the corresponding average uplink achievable rate is then expressed as(13)Cj,ku=1−μElog21+SINRj,ku,where 0<μ<1 denotes a parameter that evaluates the decrease of the spectral efficiency, which is caused by the problem of pilot overhead. It is well seen from (12) that the SINRj,ku saturates (i.e., tends to constant value however how larger the number of antennas deployed at the BSs) when the number of the antennas BSs tends toward the infinity, which is inconsistent with the main goal of the M-MIMO technology, which is aimed at enhancing the system throughput; this is by using a massive number of antennas at the BSs.


### 3.3. Data Precoding

The data symbols of the users are precoded at the BSs before being transmitted toward the users. Accordingly, based on the MF precoder, the data symbolsxj,kdExj,k2=1 of j,k are precoded at the BS of the jth cell. This is based on the following precoder:
(14)∧j,k=g^j,k,j∗⋏j,k,where ⋏j,k=g^j,k,jTg^j,k,j∗ is a normalization factor. Therefore, the received data signal yj,kd at j,k can be expressed as
(15)yj,kd=ρd∑l=1L∑k=1Kgj,k,lT∧l,kxl,kd+njd=ρd∑l=1Lgj,k,lT∧l,kxl,kd+njd=ρdgj,k,jTg^j,k,j∗⋏j,kxj,kd+ρd∑l≠jLgj,k,lTg^l,k,l∗⋏l,kxl,kd+njd=ρdgj,k,jTgj,k,j∗⋏j,kxj,kd+ρd∑l≠jLgj,k,lTgj,k,l∗⋏l,kxl,kd+ρd⋏j,kgj,k,jTnjpϕkHρp∗xj,kd+ρd∑l≠jgj,k,lT⋏l,knlpϕkHρp∗xl,kd=ρdgj,k,jTgj,k,j∗⋏j,kxj,kd+ρd∑l≠jLgj,k,lTgj,k,l∗⋏l,kxl,kd+εj,kdHere, the transmit power on the downlink phase is denoted asρd, while εj,kd denotes the interference components, which can be expressed as
(16)εj,kd=ρd⋏j,kgj,k,jTnjpϕkHρp∗xj,kd+ρd∑l≠jgj,k,lT⋏l,knlpϕkHρp∗xl,kd.Thus, the downlink SINR ofj,k can be expressed as follows:
(17)SINRj,kd=gj,k,jTgj,k,j∗2/⋏j,k∑l≠jLgj,k,lTgj,k,l∗2/⋏l,k+εj,kd2/ρd⟶basedon6βj,k,j2/⋏j,k∑l≠jβj,k,l2/⋏l,k.Therefore, the average downlink achievable rate can be expressed similarly to (13) as follows:
(18)Cj,kd=1−μElog21+SINRj,kd.It is well seen from (12) and (17) that the fulfilled SINR is saturated. Specifically, by increasing the number of antennas M toward the infinity, the fulfilled SINR remains constant; hence, it is said to be saturated in a fixed value, which is inconsistent with the main target of M-MIMO. Thus, to benefit from the M-MIMO, the increases of M should be accompanied by the increases of the fulfilled SINR. It should be noticed that the saturation of the SINR, in the asymptotic regime (i.e., M⟶∞), is mainly caused by the PCP.


## 3.1. Training Phase and Channel Estimation

During the training phase, the users of each cell, synchronously, uplink their pilot sequences of lengthτ. Thus, the received pilot signal yjp∈ℂM×τ at the BS of the jth cell can be expressed as
(7)yjp=ρp∑l=1L∑k=1Kgl,k,jϕk+njp,where ρp represents the transmitted power, which is used by each user to uplink its pilot sequences; ϕk∈ℂ1×τ is the pilot sequences of the kth user, where the orthogonality condition ϕkϕk′H=δk,k′ is satisfied [21, 22]; njp∈ℂM×τ is an additive white Gaussian noise (AWGN) at the antennas BS of the jth cell. Therefore, by correlating the received pilot signal yjp, with the local pilot sequences ϕk, the estimated channel of j,k at the BS of the jth cell is expressed as
(8)g^j,k,j=yjpϕkHρp=∑l=1Lgl,k,jϕkϕkH+∑l=1L∑k′=1,k′≠kKgl,k′,jϕk′ϕkH+njpϕkHρp=gj,k,j+∑l≠jLgl,k,j+njpϕkHρp.


## 3.2. Data Detection

Users of each cell uplink their data symbols. Accordingly, the userj,k uplink its data symbols xj,ku, where Exj,ku2=1. Therefore, the received data signal yj,ku at the BS of the jth cell can be expressed as
(9)yju=ρu∑l=1L∑k=1Kgl,k,jxl,ku+nju,where nju denotes the AWGN at the antennas BS of the jth cell. Therefore, based on the matched-filter (MF) detector (i.e., maximum ratio combining) [23], the estimated data symbols x^j,ku of the user j,k, at the BS of the jth cell, can be computed as follows:
(10)x^j,ku=g^j,k,jHyju=ρugj,k,jHgj,k,jxj,ku+ρu∑l≠jgl,k,jHgl,k,jxl,ku+εj,ku,where εj,ku arranges the interference and the noisy components, and it is expressed as
(11)εj,ku=ρugj,k,jH∑l≠jgl,k,jHxl,ku+ρu∑l≠jgl,k,jHgj,k,jxj,ku+∑l=1gl,k,jHnju+njpϕkHHρpρu∑l=1L∑k=1Kgl,k,jxl,ku+nju.Therefore, the uplink SINR ofj,k can be expressed as
(12)SINRj,ku=gj,k,jHgj,k,j2∑l≠jLgl,k,jHgl,k,j2+εj,ku2/ρu⟶basedon6βj,k,j2∑l≠jβl,k,j2.Hence, the corresponding average uplink achievable rate is then expressed as(13)Cj,ku=1−μElog21+SINRj,ku,where 0<μ<1 denotes a parameter that evaluates the decrease of the spectral efficiency, which is caused by the problem of pilot overhead. It is well seen from (12) that the SINRj,ku saturates (i.e., tends to constant value however how larger the number of antennas deployed at the BSs) when the number of the antennas BSs tends toward the infinity, which is inconsistent with the main goal of the M-MIMO technology, which is aimed at enhancing the system throughput; this is by using a massive number of antennas at the BSs.


## 3.3. Data Precoding

The data symbols of the users are precoded at the BSs before being transmitted toward the users. Accordingly, based on the MF precoder, the data symbolsxj,kdExj,k2=1 of j,k are precoded at the BS of the jth cell. This is based on the following precoder:
(14)∧j,k=g^j,k,j∗⋏j,k,where ⋏j,k=g^j,k,jTg^j,k,j∗ is a normalization factor. Therefore, the received data signal yj,kd at j,k can be expressed as
(15)yj,kd=ρd∑l=1L∑k=1Kgj,k,lT∧l,kxl,kd+njd=ρd∑l=1Lgj,k,lT∧l,kxl,kd+njd=ρdgj,k,jTg^j,k,j∗⋏j,kxj,kd+ρd∑l≠jLgj,k,lTg^l,k,l∗⋏l,kxl,kd+njd=ρdgj,k,jTgj,k,j∗⋏j,kxj,kd+ρd∑l≠jLgj,k,lTgj,k,l∗⋏l,kxl,kd+ρd⋏j,kgj,k,jTnjpϕkHρp∗xj,kd+ρd∑l≠jgj,k,lT⋏l,knlpϕkHρp∗xl,kd=ρdgj,k,jTgj,k,j∗⋏j,kxj,kd+ρd∑l≠jLgj,k,lTgj,k,l∗⋏l,kxl,kd+εj,kdHere, the transmit power on the downlink phase is denoted asρd, while εj,kd denotes the interference components, which can be expressed as
(16)εj,kd=ρd⋏j,kgj,k,jTnjpϕkHρp∗xj,kd+ρd∑l≠jgj,k,lT⋏l,knlpϕkHρp∗xl,kd.Thus, the downlink SINR ofj,k can be expressed as follows:
(17)SINRj,kd=gj,k,jTgj,k,j∗2/⋏j,k∑l≠jLgj,k,lTgj,k,l∗2/⋏l,k+εj,kd2/ρd⟶basedon6βj,k,j2/⋏j,k∑l≠jβj,k,l2/⋏l,k.Therefore, the average downlink achievable rate can be expressed similarly to (13) as follows:
(18)Cj,kd=1−μElog21+SINRj,kd.It is well seen from (12) and (17) that the fulfilled SINR is saturated. Specifically, by increasing the number of antennas M toward the infinity, the fulfilled SINR remains constant; hence, it is said to be saturated in a fixed value, which is inconsistent with the main target of M-MIMO. Thus, to benefit from the M-MIMO, the increases of M should be accompanied by the increases of the fulfilled SINR. It should be noticed that the saturation of the SINR, in the asymptotic regime (i.e., M⟶∞), is mainly caused by the PCP.


## 4. Proposed S3SR

Since the LSF coefficients are not the same for different users, and the QoS of a user depends on its LSF coefficient (e.g., (12)); the users of each cell benefit from an unfair QoS. Accordingly, the users of each cell are separated into two groups, which are called the center and edge users; therefore, to control the use of the OPSs while mitigating the PCP upon the users of each group, a BS controller is exploited to compute the SR, which is employed to make better use of the OPSs while making the decision on which strategy of the two strategies (i.e., SPR-MBDP or the WGC-PA) is desired to be used to mitigate the PCP upon the users of each group (i.e., center and edge users). This process is described in detail in the following subsections.


### 4.1. User Classification

The users of each cell are randomly distributed and have different LSF coefficients. Thus, by exploiting their respective LSF coefficients, theK users of the jth cell can be classified at the aim of the following threshold:
(19)ρj=λK∑k=1Kβj,k,j2∀j=1,⋯,L,where 0<λ<1 denotes a weighted parameter, which is exploited to adjust ρj regarding the system configuration. Thus, for λ=0, the K users are viewed as center users, while they are considered as edge users for λ=1. Therefore, the user j,k is classified based on its LSF coefficient as follows:
(20)βj,k,j2>ρj⟶Yes,Isacenteruser,No,Isanedgeuser.


### 4.2. Supervision Rule

In order to construct the SR, the number of edge and center users is, separately, computed in each cell. Thus, the number of edgekj,e and center kj,c users of the jth cell is computed based on (19) and (20) as follows:
(21)kj,c=cardk,βj,k,j2>ρj,k=1,2,⋯,K,kj,e=cardk,βj,k,j2≤ρj,k=1,2,⋯,K,and so on for theL cells (i.e., ∀j=1,⋯,L). Therefore, each BS compute and transmit its number of center and edge users to the BS controller, which is in charge of computing the total number of edge users KTot,e and center users KTot,c, which are wandering in the L cells, that is as follows:
(22)KTot,e=∑j=1Lkj,e,KTot,c=∑j=1Lkj,c.Hence, the derived SR at the BS controller can be expressed as follows:(23)KTot,e>KTot,c⟶Yes,WGC‐PAtoedgeusers,whileSPR‐MBDPtocenterusers,No,WGC‐PAtocenterusers,whileSPR‐MBDPtoedgeusers.As the SPR-MBDP strategy requires much OPSs (i.e.,KTot,e+maxkj,c:j=1,2,⋯,L) compared to those required by the WGC-PA strategy (i.e., K). The SR (23) obligates the BSs to apply the SPR-MBDP to the group of users (KTot,e or KTot,c), which requires less number of the OPSs, while the WGC-PA is applied to the other set of users (i.e., unserved users by the SPR-MBDP). Accordingly, the SR is aimed at using the SPR-MBDP and the WGC-PA strategies, flexibly, to achieve two specific goals: reducing the problem of pilot overhead and overcoming the PCP.


### 4.3. The Number of OPSs Required

Referring to (23), the number of OPSs Ksw required by the proposed S3SR can be computed as follows:
(24)KTot,e>KTot,c⟶Yes,Ksw=KTot,c+maxkj,e,j=1,2,⋯,L,No,Ksw=KTot,e+maxkj,c,j=1,2,⋯,L.The number of OPSs required in the conventional SPR-MBDP is computed as follows:(25)KSPR=KTot,e+maxkj,c,j=1,2,⋯,L.To clarify the benefits of the proposed S3SR strategy, let us analyze the example given in Figure3, which depicts a set of three hexagonal cells (i.e., L=3). Each cell is equipped with eight, randomly distributed, users (i.e., K=8). Thus, Figure 3(a) shows the conventional SPR-MBDP, where the edge users are assigned with KTot,e=11 OPSs, while the center users are obliged to reuse maxkj,c,j=1,2,3=5 OPSs. Accordingly, KSPR=16 is the number of the OPSs, which is required by the SPR-MBDP to overcome the PCP. However, from Figure 3(a), it is well recognized that the PCP remains upon the center users. Figure 3(b) depicts the conventional WGC-PA strategy, which is aimed at reducing the PCP upon the users that employ the same OPSs; however, for a large number of users per cell, the WGC-PA tends to perform like a random pilot assignment strategy.Figure 3
(a) and (b) depict, respectively, the SPR-MBDP and the WGC-PA strategies, while (c) and (d) are the two cases of the proposed S3SR strategy.
(a)(b)(c)(d)The two sides of the S3SR are shown in Figures3(c) and 3(d). Specifically, in Figure 3(c)KTot,c=14>KTot,e=10, consequently, the S3SR strategy applies, simultaneously, the SPR-MBDP strategy to the edge users, whereas the WGC-PA is dedicated to the center users. Therefore, the number of the OPSs, which is required in this case based on (24) is Ksw=10+5=15=KSPR. The second side of S3SR is depicted in Figure 3(d), where KTot,e=17>KTot,c=7, and according to (23), the S3SR strategy applies simultaneously the SPR-MBDP strategy to the center users, while the WGC-PA strategy is applied to the edge users. Thus, the number of the required OPSs is only Ksw=7+6=13 OPSs. Still referring to Figure 3(d), the use of the conventional SPR-MBDP will need the use of KSPR=17+3=20OPSs≫Ksw=13. Thus, the S3SR sacrifices with less OPSs compared to the SPR-MBDP.Since the S3SR supports two cases (23), the same analysis can be, symmetrically, done for both cases. The following analysis is restricted to one side of the S3SR, where KTot,e>KTot,c (e.g., Figure 3(d)).


### 4.4. The Allocation of the OPSs

LetΦsw∈ℂKsw×τ be the set of the OPSs (i.e., ΦswΦswH=IKsw), which is required by the S3SR strategy, where Ksw=KTot,c+maxkj,e:j=1,⋯,L. Therefore, Φsw can be expressed as
(26)Φsw=Φsw,cTΦsw,eTT,where Φsw,c∈ℂKTot,c×τ is the set of the OPSs, which is allocated to the KTot,c users, while Φsw,e∈ℂmaxkj,e:j=1,⋯,L×τ denotes the OPSs allocated to the edge users. Since SPR-MBDP is selected to be applied to the center users, the set Φsw,c must be shared by the KTot,c users of the L cells. Therefore, it can be split as follows:
(27)Φsw,c=ϕc,1T,ϕc,2T,⋯,ϕc,LTT,where ϕc,j∈ℂkc,j×τ denotes the subset of the OPSs, which are allocated to the kj,c center users of the jth cell, and it can be expressed as follows:
(28)ϕc,j=pj,1c,pj,2c,⋯,pj,kc,jcT,where pj,kc∈ℂτ×1 denotes the pilot sequences of the kth center user of the jth cell. On the other hand, the set of the OPSs Φsw,e, which is reused by the edge users (i.e., ke,j, ∀j∈1,⋯,L), can be expressed as
(29)Φsw,e=pj,1e,pj,2e,⋯,pj,maxke,jeT,where pj,ke∈ℂτ×1 denotes the pilot sequences assigned to the kth edge user of the jth cell.


### 4.5. The WGC-PA Applied to the Edge Users

A set ofmaxkj,e:j=1,2,⋯,L OPSs is, fully, reused by the edge users of the L cells. Hence, WGC-PA [15] is employed to manage the assignment of the OPSs to the edge users, therefore reducing the strength of the PCP upon these edge users. Specifically, the strength of the PCP between two edge users j,k and l,k′ (i.e., where k≠k′ and j≠l) that employ the same pilot sequence, i.e., pj,ke=pl,k′e, can be expressed as
(30)ζj,kl,k′=βl,k′,j2βj,k,j2+βj,k,l2βl,k′,l2.Notice that asζj,kl,k′ increases, the strength of the PCP between the users j,k and l,k′ increases too and vice versa. Therefore, the WGC-PA is aimed at reusing the same OPSs for the edge users of different cells, where ζj,kl,k′∀k≠k′,j≠l is minimized. To apply the WGC-PA strategy, it is desired first to define an interference graph Gr=E,V, where V denotes the vertexes of Gr, while E denotes the severity of the PCP upon the edge users. Hence, Gr can be expressed similarly to [15] as follows:
(31)Gr=V,E:V=j,k,E=ζj,kl,k′,j≠l,where j,l=1,2,⋯,L, k=1,2,⋯,ke,j, and k′=1,2,⋯,ke,lx.


### 4.6. Channel Estimation

During the training phase, the received pilot signalYjp at the BS of the jth cell can be expressed as
(32)Yjp=ρp∑l=1LGj,lcΦsw,c+∑l=1LGj,leΦsw,e+Njp,where Gj,le∈ℂM×maxkl,e and Gj,lc∈ℂM×KTot,c denote, respectively, the channel matrices of the edge and center users, while Njp∈ℂM×τ is the AWGN at the antennas BS of the jth cell. It should be noticed that, only, the first kl,e rows of Φsw,c are considered when kl,e<maxki,e:i=1,⋯,L.Since the SPR-MBDP strategy serves the center users, their corresponding channels will be, perfectly, estimated. On the other hand, the WGC-PA strategy is dedicated to the edge users of the overall cells. Thus, the strength of the PCP is minimized, upon the edge users. Therefore, the estimated channels at the BS of thejth cell can be expressed as follows:
(33)G^j,jc=1ρpYjpϕc,jH=Gj,jc+Njpρpϕc,jH,G^j,je=1ρpYjpϕeH=Gj,je+∑l≠jLGj,le+NjpρpϕeH.Therefore, based on (4), the noisy components in (33), i.e., Njp/ρpϕc,jH and Njp/ρpϕeH, can be canceled. This is by increasing the number of the antennas at the BSs (i.e., M⟶∞). Notice that only the first kj,e column of ϕc,jH are considered (33) if kj,e<maxki,e,i=1,⋯,L.


### 4.7. Data Detection

Simultaneously, theK users of each cell uplink their data symbols. Thus, the received data signals at the BS of the jth cell can be expressed as follows:
(34)Yju=ρu∑l=1LGj,lcxlu,c+Gj,lexlu,e+Nju,where ρu denotes the transmit power during the uplink phase, while xlu,c=xl,1u,c,xl,2u,c,⋯,xl,kl,cu,cT and xlu,e=xl,1u,e,xl,2u,e,⋯,xl,kl,eu,eT represent, respectively, the data symbols of the kl,c center users and those of the kl,e edge users of the lth cell. The AWGN at the antennas BS of the jth cell is expressed by Nju.For data detection, let us adopt a low-complexity MF detector (i.e.,Dj,jc∈ℂKTot,c×M for the center users and Dj,je∈ℂmaxkj,e,j=1,⋯,L×M for the edge users). Therefore, the detected data symbols can be expressed, separately, for the two groups as follows:


#### 4.7.1. For the Center Users

(35)x^ju,c=Dj,jcYju=Gj,jc+Njpρpϕc,jHHρu∑l=1LGj,lcxlu,c+∑l=1LGj,lexlu,e+Nju=Gj,jcHGj,jcxju,c+Interju,c,where Interju,c denotes the interference terms, and it can be expressed as
(36)Interju,c=Gj,jcHρu∑l≠jGj,lcxlu,c+Nju+ϕc,jNjpHρpρu∑l=1LGj,lcxlu,c+∑l=1LGj,lexlu,e+Nju.Therefore, the detected data symbolsxj,ku,c of the kth center user wandering in the jth cell can be expressed as follows:
(37)x^j,ku,c=ρugj,k,jcHgj,k,jcxj,ku,c+intj,ku,c,where intj,ku,c∈Interju,c refers to the corresponding interference of the kth center user in the jth cell. Therefore, the uplink SINR of the kth center user in the jth cell is
(38)SINRj,ku,c=ρugj,k,jcHgj,k,jc2intj,ku,c2⟶basedon6∞.Based on (6), the denominator of (38) tends toward zero. Consequently, the SINRj,ku,c is significantly boosted, compared to the fulfilled SINR in the conventional (i.e., (12)). Therefore, the average uplink achievable rate, of the kth center user of the jth cell, can be expressed similarly to [9] as follows:
(39)Cj,ku,c=1−KswKμElog21+SINRj,ku,c.


#### 4.7.2. For the Edge Users

(40)x^ju,c=Dj,jeYju=Gj,je+∑l≠jLGj,le+NjpρpϕeHHρu∑l=1LGj,lcxlu,c+Gj,lexlu,e+Nju=Gj,jeHGj,jexju,e+ρu∑l≠jGj,leHGj,lexlu,e+Interju,e.Similar to (37), Interju,e denotes the corresponding interferences. Then, the detected data symbol of the kth edge user wandering in the jth cell is expressed as follows:
(41)x^j,ku,e=ρugj,k,jeHgj,k,jexj,ku,e+ρu∑l≠jgl,k,jeHgl,k,jexj,ku,e+intj,ku,e,where intj,ku,e∈Interju,e denotes the interference term, which corresponds to the kth edge user of the jth cell. Therefore, the uplink SINR of the kth edge user in the jth cell is expressed as follows:
(42)SINRj,ku,e=ρugj,k,jeHgj,k,je2ρu∑l≠jgl,k,jeHgl,k,je2+intj,ku,e2⟶basedon6βj,k,je2∑l≠jβl,k,je2.Even though (42) looks like the conventional strategy (12), they are not similar. Due to the use of the WGC-PA, which is aimed at reducing the interference components in the denominator of (42), hence, the fulfilled SINR is enhanced for the edge users. Therefore, the average uplink achievable rate can be expressed similarly to (39) by replacing SINRj,ku,c by SINRj,ku,e.


### 4.8. Data Precoding

Before transmitting the data symbols toward the target user, each BS precodes its local data symbols. This is based on a constructed precoder (e.g., MF precoder which is considered herein). Thus, the center users will receive their respective data symbols based on SPR-MBDP, while the WGC-PA is used for edge users.


#### 4.8.1.∗For Center Users

From the first line of (33), it is well recognized that the PCP upon the center users is completely canceled. Accordingly, the BSs become able to partially estimate the channel matrices of the center users of the adjacent cells, which are then exploited to find out the subspace spanned by the intercell interference. Therefore, to project data symbols through the null space of interference, this process is explained in more detail in the following.LetA^j∈ℂKTot,c−kj,c×M be the matrix that arranges the estimated channels of intercell interferences, which are generated by the KTot,c−kj,c center users of the L cells to the BS of the jth cell. Thus, the matrix Aj can be expressed as follows:
(43)A^j=G^j,1c,⋯,G^j,j−1c,G^j,j+1c,⋯,G^j,LcT.Therefore, to find out the directions of the null subspace spanned byAj, the SVD tool will be exploited. Thus, the matrix of intercell interferences can be expressed as
(44)A^j=UjΣjVjH,where Uj∈ℂKTot,c−kj,c×KTot,c−kj,c and Vj∈ℂM×M denote, respectively, the left and the right singular vector, while Σj∈ℂKTot,c−kj,c×M refers to the matrix that comprises the singular values and it can be expressed as
(45)Σj=Σ^j,Orj×M−rjOKTot,c−kj,c−rj×rj,OKTot,c−kj,c−rj×M−rj,where rj=rankA^j denotes the rank of A^j, and Σ^j=diagσj,1,σj,2,⋯,σj,rj is an rj×rj diagonal matrix of the singular values, which are arranged as follows:
(46)σj,1≥σj,2≥⋯≥σj,rj>0.SinceM is huge compared to the number of the users, the condition M>rj is necessarily satisfied. Hence, the null subspace of A^j is spanned by the column of a matrix Bj∈ℂM×M−rj, which represents the submatrix of Vj and it can be expressed as
(47)Bj=vj,rj+1,vj,rj+2,⋯,vj,M.Thanks to the SVD tool, the null space of interference is arranged inBj, which can be exploited to construct the projection operator PBj as follows:
(48)PBj=BjBj†,wherePBj=PBjH,where Bj∈NullA^j; therefore, by projecting the conventional MF precoder (14) onto the null subspace of A^j, the new precoder can be expressed as follows:
(49)∧j,jS3SR=PBjG^j,jc∗PBjG^j,jc∗F.Therefore,∀j≠l=1,⋯,L, we have
(50)A^j∧j,jS3SR=0⟶G^j,lcT∧j,jS3SR=0.Hence, based on the enhanced MF precoder (49), the received data signal yjd,c at the kj,c center users wandering in the jth cell can be expressed as follows:
(51)yjd,c=ρd∑l=1LGj,lcT∧j,jS3SRxld,c+Njd,c=ρdGj,jcTPBjG^j,jc∗PBjG^j,jc∗Fxjd,c+Njd,c,where xjd,c refers to the data symbols that will be transmitted to the kj,c users, while Njd,c denotes the AWGN at the antennas of the those kj,c users. Therefore, the received data signal yj,kd,c at the kth center user of the jth cell is expressed as follows:
(52)yj,kd,c=ρdgj,k,jcT.BjBj†gj,k,jc∗PBjG^j,jcFxj,kd,c+ρdgj,k,jcT.BjBj†nj,kd,c∗PBjG^j,jcFxj,kd,c+Nj,kd,c,where xj,kd,c and Nj,kd,c denote, respectively, the respective data symbols of the user j,k and the AWGN at its antenna. Thereafter, the downlink SINR of the kth center user in the jth cell is expressed as
(53)SINRj,kd,c=ρdPBjG∧j,jcF2gj,k,jcT.BjBj†gj,k,jc∗2Θj,kd,c2+Nj,kd,c2,where Θj,kd,c=ρdgj,k,jcT.BjBj†nj,kd,c∗/PBjG^j,jcF arranges the interference components. Notice that Exj,kd,c2=1 still hold. Therefore, based on (6), the denominator of (53) can be forced towards zero, which proves that the SINR is significantly boosted, and the average downlink achievable rate, in this case, is expressed as
(54)Cj,kd,c=1−KswKμElog21+SINRj,kd,c.


#### 4.8.2.∗For the Edge Users

Regarding the fact that the estimated channels of the edge users (i.e., second line of (33)) appear like those derived in the conventional strategy (8), hence, the same analysis given in Subsection 3.3 can be similarly done for the edge users. Thus, the downlink SINR of the kth edge user in the jth cell is then:
(55)SINRj,kd,e=gj,k,jeTgj,k,je∗2/⋏j,k∑l≠jLgj,k,leTgj,k,le∗2/⋏l,k+εj,kd,e2/ρd.Therefore, the average downlink achievable rate for thekth edge user in the jth cell is then expressed as
(56)Cj,kd,e=1−KswKμElog21+SINRj,kd,e,where 0<μ<1 denotes a parameter that evaluates the decrease of the spectral efficiency, which is caused by the problem of pilot overhead.


## 4.1. User Classification

The users of each cell are randomly distributed and have different LSF coefficients. Thus, by exploiting their respective LSF coefficients, theK users of the jth cell can be classified at the aim of the following threshold:
(19)ρj=λK∑k=1Kβj,k,j2∀j=1,⋯,L,where 0<λ<1 denotes a weighted parameter, which is exploited to adjust ρj regarding the system configuration. Thus, for λ=0, the K users are viewed as center users, while they are considered as edge users for λ=1. Therefore, the user j,k is classified based on its LSF coefficient as follows:
(20)βj,k,j2>ρj⟶Yes,Isacenteruser,No,Isanedgeuser.


## 4.2. Supervision Rule

In order to construct the SR, the number of edge and center users is, separately, computed in each cell. Thus, the number of edgekj,e and center kj,c users of the jth cell is computed based on (19) and (20) as follows:
(21)kj,c=cardk,βj,k,j2>ρj,k=1,2,⋯,K,kj,e=cardk,βj,k,j2≤ρj,k=1,2,⋯,K,and so on for theL cells (i.e., ∀j=1,⋯,L). Therefore, each BS compute and transmit its number of center and edge users to the BS controller, which is in charge of computing the total number of edge users KTot,e and center users KTot,c, which are wandering in the L cells, that is as follows:
(22)KTot,e=∑j=1Lkj,e,KTot,c=∑j=1Lkj,c.Hence, the derived SR at the BS controller can be expressed as follows:(23)KTot,e>KTot,c⟶Yes,WGC‐PAtoedgeusers,whileSPR‐MBDPtocenterusers,No,WGC‐PAtocenterusers,whileSPR‐MBDPtoedgeusers.As the SPR-MBDP strategy requires much OPSs (i.e.,KTot,e+maxkj,c:j=1,2,⋯,L) compared to those required by the WGC-PA strategy (i.e., K). The SR (23) obligates the BSs to apply the SPR-MBDP to the group of users (KTot,e or KTot,c), which requires less number of the OPSs, while the WGC-PA is applied to the other set of users (i.e., unserved users by the SPR-MBDP). Accordingly, the SR is aimed at using the SPR-MBDP and the WGC-PA strategies, flexibly, to achieve two specific goals: reducing the problem of pilot overhead and overcoming the PCP.


## 4.3. The Number of OPSs Required

Referring to (23), the number of OPSs Ksw required by the proposed S3SR can be computed as follows:
(24)KTot,e>KTot,c⟶Yes,Ksw=KTot,c+maxkj,e,j=1,2,⋯,L,No,Ksw=KTot,e+maxkj,c,j=1,2,⋯,L.The number of OPSs required in the conventional SPR-MBDP is computed as follows:(25)KSPR=KTot,e+maxkj,c,j=1,2,⋯,L.To clarify the benefits of the proposed S3SR strategy, let us analyze the example given in Figure3, which depicts a set of three hexagonal cells (i.e., L=3). Each cell is equipped with eight, randomly distributed, users (i.e., K=8). Thus, Figure 3(a) shows the conventional SPR-MBDP, where the edge users are assigned with KTot,e=11 OPSs, while the center users are obliged to reuse maxkj,c,j=1,2,3=5 OPSs. Accordingly, KSPR=16 is the number of the OPSs, which is required by the SPR-MBDP to overcome the PCP. However, from Figure 3(a), it is well recognized that the PCP remains upon the center users. Figure 3(b) depicts the conventional WGC-PA strategy, which is aimed at reducing the PCP upon the users that employ the same OPSs; however, for a large number of users per cell, the WGC-PA tends to perform like a random pilot assignment strategy.Figure 3
(a) and (b) depict, respectively, the SPR-MBDP and the WGC-PA strategies, while (c) and (d) are the two cases of the proposed S3SR strategy.
(a)(b)(c)(d)The two sides of the S3SR are shown in Figures3(c) and 3(d). Specifically, in Figure 3(c)KTot,c=14>KTot,e=10, consequently, the S3SR strategy applies, simultaneously, the SPR-MBDP strategy to the edge users, whereas the WGC-PA is dedicated to the center users. Therefore, the number of the OPSs, which is required in this case based on (24) is Ksw=10+5=15=KSPR. The second side of S3SR is depicted in Figure 3(d), where KTot,e=17>KTot,c=7, and according to (23), the S3SR strategy applies simultaneously the SPR-MBDP strategy to the center users, while the WGC-PA strategy is applied to the edge users. Thus, the number of the required OPSs is only Ksw=7+6=13 OPSs. Still referring to Figure 3(d), the use of the conventional SPR-MBDP will need the use of KSPR=17+3=20OPSs≫Ksw=13. Thus, the S3SR sacrifices with less OPSs compared to the SPR-MBDP.Since the S3SR supports two cases (23), the same analysis can be, symmetrically, done for both cases. The following analysis is restricted to one side of the S3SR, where KTot,e>KTot,c (e.g., Figure 3(d)).


## 4.4. The Allocation of the OPSs

LetΦsw∈ℂKsw×τ be the set of the OPSs (i.e., ΦswΦswH=IKsw), which is required by the S3SR strategy, where Ksw=KTot,c+maxkj,e:j=1,⋯,L. Therefore, Φsw can be expressed as
(26)Φsw=Φsw,cTΦsw,eTT,where Φsw,c∈ℂKTot,c×τ is the set of the OPSs, which is allocated to the KTot,c users, while Φsw,e∈ℂmaxkj,e:j=1,⋯,L×τ denotes the OPSs allocated to the edge users. Since SPR-MBDP is selected to be applied to the center users, the set Φsw,c must be shared by the KTot,c users of the L cells. Therefore, it can be split as follows:
(27)Φsw,c=ϕc,1T,ϕc,2T,⋯,ϕc,LTT,where ϕc,j∈ℂkc,j×τ denotes the subset of the OPSs, which are allocated to the kj,c center users of the jth cell, and it can be expressed as follows:
(28)ϕc,j=pj,1c,pj,2c,⋯,pj,kc,jcT,where pj,kc∈ℂτ×1 denotes the pilot sequences of the kth center user of the jth cell. On the other hand, the set of the OPSs Φsw,e, which is reused by the edge users (i.e., ke,j, ∀j∈1,⋯,L), can be expressed as
(29)Φsw,e=pj,1e,pj,2e,⋯,pj,maxke,jeT,where pj,ke∈ℂτ×1 denotes the pilot sequences assigned to the kth edge user of the jth cell.


## 4.5. The WGC-PA Applied to the Edge Users

A set ofmaxkj,e:j=1,2,⋯,L OPSs is, fully, reused by the edge users of the L cells. Hence, WGC-PA [15] is employed to manage the assignment of the OPSs to the edge users, therefore reducing the strength of the PCP upon these edge users. Specifically, the strength of the PCP between two edge users j,k and l,k′ (i.e., where k≠k′ and j≠l) that employ the same pilot sequence, i.e., pj,ke=pl,k′e, can be expressed as
(30)ζj,kl,k′=βl,k′,j2βj,k,j2+βj,k,l2βl,k′,l2.Notice that asζj,kl,k′ increases, the strength of the PCP between the users j,k and l,k′ increases too and vice versa. Therefore, the WGC-PA is aimed at reusing the same OPSs for the edge users of different cells, where ζj,kl,k′∀k≠k′,j≠l is minimized. To apply the WGC-PA strategy, it is desired first to define an interference graph Gr=E,V, where V denotes the vertexes of Gr, while E denotes the severity of the PCP upon the edge users. Hence, Gr can be expressed similarly to [15] as follows:
(31)Gr=V,E:V=j,k,E=ζj,kl,k′,j≠l,where j,l=1,2,⋯,L, k=1,2,⋯,ke,j, and k′=1,2,⋯,ke,lx.


## 4.6. Channel Estimation

During the training phase, the received pilot signalYjp at the BS of the jth cell can be expressed as
(32)Yjp=ρp∑l=1LGj,lcΦsw,c+∑l=1LGj,leΦsw,e+Njp,where Gj,le∈ℂM×maxkl,e and Gj,lc∈ℂM×KTot,c denote, respectively, the channel matrices of the edge and center users, while Njp∈ℂM×τ is the AWGN at the antennas BS of the jth cell. It should be noticed that, only, the first kl,e rows of Φsw,c are considered when kl,e<maxki,e:i=1,⋯,L.Since the SPR-MBDP strategy serves the center users, their corresponding channels will be, perfectly, estimated. On the other hand, the WGC-PA strategy is dedicated to the edge users of the overall cells. Thus, the strength of the PCP is minimized, upon the edge users. Therefore, the estimated channels at the BS of thejth cell can be expressed as follows:
(33)G^j,jc=1ρpYjpϕc,jH=Gj,jc+Njpρpϕc,jH,G^j,je=1ρpYjpϕeH=Gj,je+∑l≠jLGj,le+NjpρpϕeH.Therefore, based on (4), the noisy components in (33), i.e., Njp/ρpϕc,jH and Njp/ρpϕeH, can be canceled. This is by increasing the number of the antennas at the BSs (i.e., M⟶∞). Notice that only the first kj,e column of ϕc,jH are considered (33) if kj,e<maxki,e,i=1,⋯,L.


## 4.7. Data Detection

Simultaneously, theK users of each cell uplink their data symbols. Thus, the received data signals at the BS of the jth cell can be expressed as follows:
(34)Yju=ρu∑l=1LGj,lcxlu,c+Gj,lexlu,e+Nju,where ρu denotes the transmit power during the uplink phase, while xlu,c=xl,1u,c,xl,2u,c,⋯,xl,kl,cu,cT and xlu,e=xl,1u,e,xl,2u,e,⋯,xl,kl,eu,eT represent, respectively, the data symbols of the kl,c center users and those of the kl,e edge users of the lth cell. The AWGN at the antennas BS of the jth cell is expressed by Nju.For data detection, let us adopt a low-complexity MF detector (i.e.,Dj,jc∈ℂKTot,c×M for the center users and Dj,je∈ℂmaxkj,e,j=1,⋯,L×M for the edge users). Therefore, the detected data symbols can be expressed, separately, for the two groups as follows:


### 4.7.1. For the Center Users

(35)x^ju,c=Dj,jcYju=Gj,jc+Njpρpϕc,jHHρu∑l=1LGj,lcxlu,c+∑l=1LGj,lexlu,e+Nju=Gj,jcHGj,jcxju,c+Interju,c,where Interju,c denotes the interference terms, and it can be expressed as
(36)Interju,c=Gj,jcHρu∑l≠jGj,lcxlu,c+Nju+ϕc,jNjpHρpρu∑l=1LGj,lcxlu,c+∑l=1LGj,lexlu,e+Nju.Therefore, the detected data symbolsxj,ku,c of the kth center user wandering in the jth cell can be expressed as follows:
(37)x^j,ku,c=ρugj,k,jcHgj,k,jcxj,ku,c+intj,ku,c,where intj,ku,c∈Interju,c refers to the corresponding interference of the kth center user in the jth cell. Therefore, the uplink SINR of the kth center user in the jth cell is
(38)SINRj,ku,c=ρugj,k,jcHgj,k,jc2intj,ku,c2⟶basedon6∞.Based on (6), the denominator of (38) tends toward zero. Consequently, the SINRj,ku,c is significantly boosted, compared to the fulfilled SINR in the conventional (i.e., (12)). Therefore, the average uplink achievable rate, of the kth center user of the jth cell, can be expressed similarly to [9] as follows:
(39)Cj,ku,c=1−KswKμElog21+SINRj,ku,c.


### 4.7.2. For the Edge Users

(40)x^ju,c=Dj,jeYju=Gj,je+∑l≠jLGj,le+NjpρpϕeHHρu∑l=1LGj,lcxlu,c+Gj,lexlu,e+Nju=Gj,jeHGj,jexju,e+ρu∑l≠jGj,leHGj,lexlu,e+Interju,e.Similar to (37), Interju,e denotes the corresponding interferences. Then, the detected data symbol of the kth edge user wandering in the jth cell is expressed as follows:
(41)x^j,ku,e=ρugj,k,jeHgj,k,jexj,ku,e+ρu∑l≠jgl,k,jeHgl,k,jexj,ku,e+intj,ku,e,where intj,ku,e∈Interju,e denotes the interference term, which corresponds to the kth edge user of the jth cell. Therefore, the uplink SINR of the kth edge user in the jth cell is expressed as follows:
(42)SINRj,ku,e=ρugj,k,jeHgj,k,je2ρu∑l≠jgl,k,jeHgl,k,je2+intj,ku,e2⟶basedon6βj,k,je2∑l≠jβl,k,je2.Even though (42) looks like the conventional strategy (12), they are not similar. Due to the use of the WGC-PA, which is aimed at reducing the interference components in the denominator of (42), hence, the fulfilled SINR is enhanced for the edge users. Therefore, the average uplink achievable rate can be expressed similarly to (39) by replacing SINRj,ku,c by SINRj,ku,e.


## 4.7.1. For the Center Users

(35)x^ju,c=Dj,jcYju=Gj,jc+Njpρpϕc,jHHρu∑l=1LGj,lcxlu,c+∑l=1LGj,lexlu,e+Nju=Gj,jcHGj,jcxju,c+Interju,c,where Interju,c denotes the interference terms, and it can be expressed as
(36)Interju,c=Gj,jcHρu∑l≠jGj,lcxlu,c+Nju+ϕc,jNjpHρpρu∑l=1LGj,lcxlu,c+∑l=1LGj,lexlu,e+Nju.Therefore, the detected data symbolsxj,ku,c of the kth center user wandering in the jth cell can be expressed as follows:
(37)x^j,ku,c=ρugj,k,jcHgj,k,jcxj,ku,c+intj,ku,c,where intj,ku,c∈Interju,c refers to the corresponding interference of the kth center user in the jth cell. Therefore, the uplink SINR of the kth center user in the jth cell is
(38)SINRj,ku,c=ρugj,k,jcHgj,k,jc2intj,ku,c2⟶basedon6∞.Based on (6), the denominator of (38) tends toward zero. Consequently, the SINRj,ku,c is significantly boosted, compared to the fulfilled SINR in the conventional (i.e., (12)). Therefore, the average uplink achievable rate, of the kth center user of the jth cell, can be expressed similarly to [9] as follows:
(39)Cj,ku,c=1−KswKμElog21+SINRj,ku,c.


## 4.7.2. For the Edge Users

(40)x^ju,c=Dj,jeYju=Gj,je+∑l≠jLGj,le+NjpρpϕeHHρu∑l=1LGj,lcxlu,c+Gj,lexlu,e+Nju=Gj,jeHGj,jexju,e+ρu∑l≠jGj,leHGj,lexlu,e+Interju,e.Similar to (37), Interju,e denotes the corresponding interferences. Then, the detected data symbol of the kth edge user wandering in the jth cell is expressed as follows:
(41)x^j,ku,e=ρugj,k,jeHgj,k,jexj,ku,e+ρu∑l≠jgl,k,jeHgl,k,jexj,ku,e+intj,ku,e,where intj,ku,e∈Interju,e denotes the interference term, which corresponds to the kth edge user of the jth cell. Therefore, the uplink SINR of the kth edge user in the jth cell is expressed as follows:
(42)SINRj,ku,e=ρugj,k,jeHgj,k,je2ρu∑l≠jgl,k,jeHgl,k,je2+intj,ku,e2⟶basedon6βj,k,je2∑l≠jβl,k,je2.Even though (42) looks like the conventional strategy (12), they are not similar. Due to the use of the WGC-PA, which is aimed at reducing the interference components in the denominator of (42), hence, the fulfilled SINR is enhanced for the edge users. Therefore, the average uplink achievable rate can be expressed similarly to (39) by replacing SINRj,ku,c by SINRj,ku,e.


## 4.8. Data Precoding

Before transmitting the data symbols toward the target user, each BS precodes its local data symbols. This is based on a constructed precoder (e.g., MF precoder which is considered herein). Thus, the center users will receive their respective data symbols based on SPR-MBDP, while the WGC-PA is used for edge users.


### 4.8.1.∗For Center Users

From the first line of (33), it is well recognized that the PCP upon the center users is completely canceled. Accordingly, the BSs become able to partially estimate the channel matrices of the center users of the adjacent cells, which are then exploited to find out the subspace spanned by the intercell interference. Therefore, to project data symbols through the null space of interference, this process is explained in more detail in the following.LetA^j∈ℂKTot,c−kj,c×M be the matrix that arranges the estimated channels of intercell interferences, which are generated by the KTot,c−kj,c center users of the L cells to the BS of the jth cell. Thus, the matrix Aj can be expressed as follows:
(43)A^j=G^j,1c,⋯,G^j,j−1c,G^j,j+1c,⋯,G^j,LcT.Therefore, to find out the directions of the null subspace spanned byAj, the SVD tool will be exploited. Thus, the matrix of intercell interferences can be expressed as
(44)A^j=UjΣjVjH,where Uj∈ℂKTot,c−kj,c×KTot,c−kj,c and Vj∈ℂM×M denote, respectively, the left and the right singular vector, while Σj∈ℂKTot,c−kj,c×M refers to the matrix that comprises the singular values and it can be expressed as
(45)Σj=Σ^j,Orj×M−rjOKTot,c−kj,c−rj×rj,OKTot,c−kj,c−rj×M−rj,where rj=rankA^j denotes the rank of A^j, and Σ^j=diagσj,1,σj,2,⋯,σj,rj is an rj×rj diagonal matrix of the singular values, which are arranged as follows:
(46)σj,1≥σj,2≥⋯≥σj,rj>0.SinceM is huge compared to the number of the users, the condition M>rj is necessarily satisfied. Hence, the null subspace of A^j is spanned by the column of a matrix Bj∈ℂM×M−rj, which represents the submatrix of Vj and it can be expressed as
(47)Bj=vj,rj+1,vj,rj+2,⋯,vj,M.Thanks to the SVD tool, the null space of interference is arranged inBj, which can be exploited to construct the projection operator PBj as follows:
(48)PBj=BjBj†,wherePBj=PBjH,where Bj∈NullA^j; therefore, by projecting the conventional MF precoder (14) onto the null subspace of A^j, the new precoder can be expressed as follows:
(49)∧j,jS3SR=PBjG^j,jc∗PBjG^j,jc∗F.Therefore,∀j≠l=1,⋯,L, we have
(50)A^j∧j,jS3SR=0⟶G^j,lcT∧j,jS3SR=0.Hence, based on the enhanced MF precoder (49), the received data signal yjd,c at the kj,c center users wandering in the jth cell can be expressed as follows:
(51)yjd,c=ρd∑l=1LGj,lcT∧j,jS3SRxld,c+Njd,c=ρdGj,jcTPBjG^j,jc∗PBjG^j,jc∗Fxjd,c+Njd,c,where xjd,c refers to the data symbols that will be transmitted to the kj,c users, while Njd,c denotes the AWGN at the antennas of the those kj,c users. Therefore, the received data signal yj,kd,c at the kth center user of the jth cell is expressed as follows:
(52)yj,kd,c=ρdgj,k,jcT.BjBj†gj,k,jc∗PBjG^j,jcFxj,kd,c+ρdgj,k,jcT.BjBj†nj,kd,c∗PBjG^j,jcFxj,kd,c+Nj,kd,c,where xj,kd,c and Nj,kd,c denote, respectively, the respective data symbols of the user j,k and the AWGN at its antenna. Thereafter, the downlink SINR of the kth center user in the jth cell is expressed as
(53)SINRj,kd,c=ρdPBjG∧j,jcF2gj,k,jcT.BjBj†gj,k,jc∗2Θj,kd,c2+Nj,kd,c2,where Θj,kd,c=ρdgj,k,jcT.BjBj†nj,kd,c∗/PBjG^j,jcF arranges the interference components. Notice that Exj,kd,c2=1 still hold. Therefore, based on (6), the denominator of (53) can be forced towards zero, which proves that the SINR is significantly boosted, and the average downlink achievable rate, in this case, is expressed as
(54)Cj,kd,c=1−KswKμElog21+SINRj,kd,c.


### 4.8.2.∗For the Edge Users

Regarding the fact that the estimated channels of the edge users (i.e., second line of (33)) appear like those derived in the conventional strategy (8), hence, the same analysis given in Subsection 3.3 can be similarly done for the edge users. Thus, the downlink SINR of the kth edge user in the jth cell is then:
(55)SINRj,kd,e=gj,k,jeTgj,k,je∗2/⋏j,k∑l≠jLgj,k,leTgj,k,le∗2/⋏l,k+εj,kd,e2/ρd.Therefore, the average downlink achievable rate for thekth edge user in the jth cell is then expressed as
(56)Cj,kd,e=1−KswKμElog21+SINRj,kd,e,where 0<μ<1 denotes a parameter that evaluates the decrease of the spectral efficiency, which is caused by the problem of pilot overhead.


## 4.8.1.∗For Center Users

From the first line of (33), it is well recognized that the PCP upon the center users is completely canceled. Accordingly, the BSs become able to partially estimate the channel matrices of the center users of the adjacent cells, which are then exploited to find out the subspace spanned by the intercell interference. Therefore, to project data symbols through the null space of interference, this process is explained in more detail in the following.LetA^j∈ℂKTot,c−kj,c×M be the matrix that arranges the estimated channels of intercell interferences, which are generated by the KTot,c−kj,c center users of the L cells to the BS of the jth cell. Thus, the matrix Aj can be expressed as follows:
(43)A^j=G^j,1c,⋯,G^j,j−1c,G^j,j+1c,⋯,G^j,LcT.Therefore, to find out the directions of the null subspace spanned byAj, the SVD tool will be exploited. Thus, the matrix of intercell interferences can be expressed as
(44)A^j=UjΣjVjH,where Uj∈ℂKTot,c−kj,c×KTot,c−kj,c and Vj∈ℂM×M denote, respectively, the left and the right singular vector, while Σj∈ℂKTot,c−kj,c×M refers to the matrix that comprises the singular values and it can be expressed as
(45)Σj=Σ^j,Orj×M−rjOKTot,c−kj,c−rj×rj,OKTot,c−kj,c−rj×M−rj,where rj=rankA^j denotes the rank of A^j, and Σ^j=diagσj,1,σj,2,⋯,σj,rj is an rj×rj diagonal matrix of the singular values, which are arranged as follows:
(46)σj,1≥σj,2≥⋯≥σj,rj>0.SinceM is huge compared to the number of the users, the condition M>rj is necessarily satisfied. Hence, the null subspace of A^j is spanned by the column of a matrix Bj∈ℂM×M−rj, which represents the submatrix of Vj and it can be expressed as
(47)Bj=vj,rj+1,vj,rj+2,⋯,vj,M.Thanks to the SVD tool, the null space of interference is arranged inBj, which can be exploited to construct the projection operator PBj as follows:
(48)PBj=BjBj†,wherePBj=PBjH,where Bj∈NullA^j; therefore, by projecting the conventional MF precoder (14) onto the null subspace of A^j, the new precoder can be expressed as follows:
(49)∧j,jS3SR=PBjG^j,jc∗PBjG^j,jc∗F.Therefore,∀j≠l=1,⋯,L, we have
(50)A^j∧j,jS3SR=0⟶G^j,lcT∧j,jS3SR=0.Hence, based on the enhanced MF precoder (49), the received data signal yjd,c at the kj,c center users wandering in the jth cell can be expressed as follows:
(51)yjd,c=ρd∑l=1LGj,lcT∧j,jS3SRxld,c+Njd,c=ρdGj,jcTPBjG^j,jc∗PBjG^j,jc∗Fxjd,c+Njd,c,where xjd,c refers to the data symbols that will be transmitted to the kj,c users, while Njd,c denotes the AWGN at the antennas of the those kj,c users. Therefore, the received data signal yj,kd,c at the kth center user of the jth cell is expressed as follows:
(52)yj,kd,c=ρdgj,k,jcT.BjBj†gj,k,jc∗PBjG^j,jcFxj,kd,c+ρdgj,k,jcT.BjBj†nj,kd,c∗PBjG^j,jcFxj,kd,c+Nj,kd,c,where xj,kd,c and Nj,kd,c denote, respectively, the respective data symbols of the user j,k and the AWGN at its antenna. Thereafter, the downlink SINR of the kth center user in the jth cell is expressed as
(53)SINRj,kd,c=ρdPBjG∧j,jcF2gj,k,jcT.BjBj†gj,k,jc∗2Θj,kd,c2+Nj,kd,c2,where Θj,kd,c=ρdgj,k,jcT.BjBj†nj,kd,c∗/PBjG^j,jcF arranges the interference components. Notice that Exj,kd,c2=1 still hold. Therefore, based on (6), the denominator of (53) can be forced towards zero, which proves that the SINR is significantly boosted, and the average downlink achievable rate, in this case, is expressed as
(54)Cj,kd,c=1−KswKμElog21+SINRj,kd,c.


## 4.8.2.∗For the Edge Users

Regarding the fact that the estimated channels of the edge users (i.e., second line of (33)) appear like those derived in the conventional strategy (8), hence, the same analysis given in Subsection 3.3 can be similarly done for the edge users. Thus, the downlink SINR of the kth edge user in the jth cell is then:
(55)SINRj,kd,e=gj,k,jeTgj,k,je∗2/⋏j,k∑l≠jLgj,k,leTgj,k,le∗2/⋏l,k+εj,kd,e2/ρd.Therefore, the average downlink achievable rate for thekth edge user in the jth cell is then expressed as
(56)Cj,kd,e=1−KswKμElog21+SINRj,kd,e,where 0<μ<1 denotes a parameter that evaluates the decrease of the spectral efficiency, which is caused by the problem of pilot overhead.


## 5. Simulation Results

This section is aimed at evaluating the performance of the proposed S3SR. That is through a set of Monte Carlo simulation applied to the typical multicell structure, which is described in Section2, where the K users are randomly distributed within each cell. The specific parameters used in this work are listed in Table 1.Table 1
System settings.
Number of the cellsL=7Number of the antennas BSs32≤M≤512Number of the users per cellK=10Number of OPSs for the WGC-PA [15]S=10Adjustment parameter0≤λ≤1Cell radiusR=500mInner radiusr=30mTransmit powersρp, ρu10 dBmTransmit powerρd12 dBmPath loss exponentα=3log-normal shadowing fadingσshadow=8dBPilot overhead parameter0<μ<1,fixedμ=0.01Figure4 shows the average uplink achievable rate versus the number of antennas deployed in the BSs. As can be recognized from Figure 4, the superiority of the proposed S3SR is well seen compared to the other strategies (i.e., conventional, SPR-MBDP, and WGC-PA), especially for a large number of antennas M≥256, where the gap is increased. On the other hand, the fulfilled performance by the SPR-MBD is too close to those achieved by the WGC-PA, owing to the problem of pilot overhead (Table 2). Furthermore, based on Table 2 where KSPR=Ksw=41, we notice that the S3SR performs better than the SPR-MBDP, even if they employ the same number of the OPSs, that proves the effectiveness of the proposed S3SR over the conventional strategies.Figure 4
Average uplink achievable rate versusM, where λ=0.1, K=10.Table 2
The average number of the OPSs required to serveK×L=70 users.
Number of antennasM3264128256512KSPR based on (25)4645505141Ksw based on (24)4039333541The average downlink achievable rate versusM is depicted in Figure 5, where the superiority of the S3SR compared to the WGC-PA is started from a large number of antennas; specifically, from M≥256, the S3SR performs better compared to the other strategies.Figure 5
Average downlink achievable rate versusM for λ=0.1, K=10.In order to show the robustness of the S3SR versus different distribution of the users within cells, which is controlled by the value ofλ, Figure 6 depicts the mean square error (MSE) [9] of the estimated channels versus λ. The MSE can be expressed as
(57)MSE=E1K∑j=1L∑k=1Kg^j,k,j−gj,k,j22gj,k,j22.Figure 6
Mean square error of the estimated channels based on the CS, SPR-MBDP, and the S3SR strategies, whereλ=0.1, K=10, and M=256.Thus, from Figure6, and for different values of λ, the accuracy of the S3SR is well seen, which is not the case for the SPR-MBDP that loses its accuracy for λ<0.6 and performs better for λ>0.6. Specifically, the accuracy of the SPR-MBDP is enhanced, only, when the users are regarded as edges (i.e., λ⟶1). Indeed, the S3SR keeps approximately the same accuracy for different values of λ, which proves its robustness over different distributions of the users within cells.


### 5.1. Computational Complexity

Unlike the previous decontaminating strategies, where the computational complexity is fixed, the complexity of the S3SR depends on the SR in (23). Hence, the complexity of the S3SR is summarized in Table 3, where kmax,c=maxkj,c,j=1,⋯,L and kmax,e=maxkj,e,j=1,⋯,L denote, respectively, the maximum number of the center and edge users in the L cells.Table 3
Computational complexity of the S3SR, SPR-MBDP [9], and WGC-PA [15].
S3SR if the first line of (24) is consideredOMKTot,c2+Skmax,eL3S3SR if the second line of (24) is consideredOMKTot,e2+Skmax,cL3SPR-MBDPOMKTot,e2+Mkmax,c2WGC-PAOSKL3


## 5.1. Computational Complexity

Unlike the previous decontaminating strategies, where the computational complexity is fixed, the complexity of the S3SR depends on the SR in (23). Hence, the complexity of the S3SR is summarized in Table 3, where kmax,c=maxkj,c,j=1,⋯,L and kmax,e=maxkj,e,j=1,⋯,L denote, respectively, the maximum number of the center and edge users in the L cells.Table 3
Computational complexity of the S3SR, SPR-MBDP [9], and WGC-PA [15].
S3SR if the first line of (24) is consideredOMKTot,c2+Skmax,eL3S3SR if the second line of (24) is consideredOMKTot,e2+Skmax,cL3SPR-MBDPOMKTot,e2+Mkmax,c2WGC-PAOSKL3


## 6. Conclusion

In this paper, we have proposed a novel strategy to mitigate the PCP within multicell M-MIMO systems. Unlike the previous studies, which use a specific decontaminating strategy to resolve the PCP upon the users within cells, our proposed S3SR is continually updated and adapted to a different distribution of the users within cells. This process is done based on a SR, which is aimed at guaranteeing a trade-off between the use of the precious OPSs and mitigating the PCP. In other words, the S3SR is, mainly, based on the consolidation of two powerful decontaminating strategies (i.e., WGC-PA and SPR-MBDP) under a SR. Specifically, the LSF coefficients are exploited, to separate the users of each cell into two groups, which are the center and the edge users; therefore, a derived SR is used by the proposed S3SR to switch between the WGC-PA and the SPR-MBDP. This switching leads to make better use of the OPS resources during the training phase. The superiority and the robustness of the S3SR, compared to the SPR-MBDP, the WGC-PA, and the conventional strategy are proved through simulations.



---

*Source: 8868597-2020-08-17.xml*