LocationAware Pilot Allocation in MultiCell MultiUser Massive MIMO Networks
Abstract
We propose a locationaware pilot allocation algorithm for a massive multipleinput multipleoutput (MIMO) network with highmobility users, where the wireless channels are subject to Rician fading. Pilot allocation in massive MIMO is a hard combinatorial problem and depends on the locations of users. As such, it is highly complex to achieve the optimal pilot allocation in realtime for a network with highmobility users. Against this background, we propose a lowcomplexity pilot allocation algorithm, which exploits the behavior of lineofsight (LOS) interference among the users and allocate the same pilot sequence to the users with small LOS interference. Our examination demonstrates that our proposed algorithm significantly outperforms the existing algorithms, even with localization errors. Specifically, for the system considered in this work, our proposed algorithm provides up to 37.26% improvement in sum spectral efficiency (SE) and improves the sum SE of the worst interferenceaffected users by up to 2.57 bits/sec/Hz, as compared to the existing algorithms.
I Introduction
Massive multipleinput multipleoutput (MIMO) is considered as a key technology for the fifth generation networks, since it offers an increased data rate and improved spectral efficiency (SE). However, challenges such as interference due to pilot contamination prevent us from achieving the full benefit offered by massive MIMO [1, 2]. Pilot contamination affects the performance of massive MIMO even when the number of antennas at the base station (BS) is very large. Consequently, pilot contamination is a widely studied problem in massive MIMO. The existing studies on pilot contamination can be classified into five broad categories: 1) protocol based [1]; 2) precoding based [2]; 3) angleofarrival (AoA) based [4]; 4) blind [5]; and 5) pilot design methods [6].
Recently, an increasing attention has been paid to utilize the location information of users for mitigating pilot contamination. The users’ locations can be estimated or requested from the users directly by the BS and then the location information can be leveraged to allocate pilot sequences in the network such that pilot contamination is minimized (e.g.,[4, 3, 7]). We highlight that the existing locationaware pilot allocation algorithms only consider a simple channel model [3, 8, 7] that may not be generalized enough to depict certain practical channel conditions, such as the channels containing both lineofsight (LOS) and nonlineofsight (NLOS) components. In some cases, a LOS channel component may exist between BSs and users [4]. In order to deal with the pilot allocation problem under LOS conditions or when both LOS and NLOS conditions exist, we consider Rician fading channels in this work. Most existing studies in locationaware pilot allocation algorithms assume that the AoAs of all the users are strictly nonoverlapping [3, 8, 7], which is hard to justify in some practical scenarios. We relax this assumption and demonstrate that the location information is beneficial for pilot allocation, even when the AoAs of interfering users are overlapping. Furthermore, different from existing studies, we assume that the pilot sequences used in a cell are not orthogonal. As such, our system model incorporates both the intercell and the intracell pilot contamination. The work [4] presented the pilot allocation in a singlecell network. Different from [4], we consider a more general multicell network, which encompasses the singlecell network as a special case. Moreover, in this work we derive the multicell LOS interference expression that is valid for an arbitrary number of BS antennas. Furthermore, in this work we perform a thorough comparison between the proposed algorithm and several existing algorithms under varying network conditions. We highlight that the derivation and the comparison were not presented in [4]. Throughout this paper, we define the LOS interference as the interference caused by the LOS components in the channels.
In this correspondence, we propose a lowcomplexity pilot allocation algorithm suitable for a network with highmobility users. Our examination shows that the proposed algorithm improves the sum SE of the network as compared to the existing algorithms, even when the locations of the users suffer from estimation errors.
Ii System Model
We consider an cell massive MIMO network as illustrated in Fig. 1. In each cell, a BS equipped with antennas communicates with singleantenna users. We denote the BS in the ith cell as and the jth user in the ith cell as . Additionally, we denote the location of as , where is the distance from to and is the AoA of at . We represent the smallscale propagation factor between and the as . We assume that is subject to Rician fading. Consequently, consists of a LOS component denoted as and a Rayleigh distributed NLOS component denoted as , where . We assume that the uplink channel between and is affected by largescale propagation effects denoted as . As such, the uplink channel from to is written as
(1) 
where is the Kfactor of at . We assume that the BSs are equipped with uniform linear antenna arrays. As such, depends on the location of and is expressed as
(2) 
where is the distance between two antennas and is the wavelength. We assume that , which is a widely adopted assumption [4]. The uplink channels between all the users in the ith cell and are represented as , where
(3) 
with , , , and .
In this work, we consider the uplink transmission from the users to the BSs, which consists of two phases, i.e., uplink channel estimation and uplink data transmission.
Iia Uplink Channel Estimation
In the uplink channel estimation phase, users from all the cells transmit their preassigned pilot sequences to the BSs. We assume that the length of the pilot sequence is . As such, only orthogonal pilot sequences are available in the massive MIMO network. The pilot sequence assigned to is represented as . Accordingly, the pilot sequences assigned to all the users in the ith cell are represented as . We assume that the pilot sequences are transmitted with unit power. The received matrix at in the uplink pilot transmission phase is given by
(4) 
where is the Gaussian noise matrix at the and the distribution of each independent element in follows . We assume that only knows the estimated location of each user, i.e, . As such, the estimates of the LOS components in (4) are known at . Next, we compute the received matrix corresponding to the NLOS component by subtracting the LOS components from , i.e., , where , , , . We assume that is obtained from (2) using and we obtain and based on the estimated distance . Accordingly, is obtained from (4) as
(5) 
where . We highlight that appears in (IIA) due to localization errors. If the locations of users are precisely known, we have and the received matrix corresponding to the LOS component is completely removed from (4). In other words, without localization error the AoA of each user does not affect the channel estimation.
Remark 1
From (IIA), we note that the channel estimate suffers from intracell pilot contamination when . Additionally, we note that the channel estimates suffer from intercell pilot contamination when the same pilot sequences are repeated throughout the network. Specifically, when the channel estimates suffer from intercell pilot contamination. In realworld telecommunication networks, it is not possible to assign orthogonal pilot sequences to all the users in the network. As such, it is reasonable to assume that and intercell pilot contamination always exists.
IiB Uplink Data Transmission and Spectral Efficiency
During the uplink data transmission, each user in a cell transmits uplink data symbols to the samecell BSs. The multiplies the received vector with the zeroforcing (ZF) matrix to decode the symbols transmitted by the samecell users. Accordingly, using the useandthenforget bound, the signal received from after detection is written as [11]
(7) 
where is the kth column of the ZF detection matrix , is the symbol transmitted by , is the Gaussian noise at , and is the signaltonoise ratio. The uplink SE for is given as , where is the channel uses in a coherence block [11], and is the signaltointerferenceplusnoise ratio (SINR) given by
(8) 
We note that the product of linear detection vector and the channel, i.e., , is important in determining . We next present the proposed pilot allocation algorithm that aims at reducing the interference caused by based on the estimated locations of the users.
Iii LocationAware Pilot Allocation
In this section, we present our lowcomplexity locationaware pilot allocation algorithm. The algorithm requires the knowledge about the largescale fading, AoAs, and factors to perform pilot allocation. Specifically, we first derive the expression for the LOS interference based on the estimated locations of the users. Then, the pilot sequences are allocated to the users sequentially to minimize the LOS interference.
Iiia LOS Interference
We derive the expression for the LOS interference between two users in the network in the following theorem.
Theorem 1
The LOS interference between and based on their estimated locations is given as
(9) 
where
(10) 
Proof:
The proof is provided in Appendix A. \qed
We note that the LOS interference given in (9) consists of two parts. Specifically, part 1 is distancedependent and part 2 is AoAdependent. We note that part 1 is the smallest when the and are the farthest apart from each other. Fig. 2 depicts part 2 at different AoAs for and for . We obtain the following insights regarding AoAdependent part 2 from Fig. 2. The AoAdependent part 2 is

maximum when the AoAs are overlapping, i.e., .

maximum when the AoAs differ by , i.e., .

minimum for certain mutual AoAs for and .

minimum for possible values of or .
We highlight that the observations on part 1 and part 2 can be utilized for pilot allocation. As such, the location information can be used to identify the users with the minimum LOS interference and assign the same pilot sequence to such users to improve the sum SE. We note that the LOS interference (9) is obtained for the LS estimator and the ZF detector. We also clarify that for different combination of estimators and detectors, the LOS interference exists and can be obtained in a similar way to obtaining (9).
Lemma 1
In massive MIMO networks when , we have when and
(11) 
Proof:
Remark 2
According to Lemma 1, when the number of antennas at BSs is large, the LOS interference is high only when the AoAs of the two interfering users are strictly overlapping. Additionally, increasing the number of antennas at BSs increases the possibility of having two users with the minimum LOS interference, because the LOS interference is minimum at values of the mutual AoAs, as depicted in Fig. 2. This observation highlights the benefit of massive MIMO for the proposed locationaware pilot allocation algorithm.
Remark 3
The optimal solution for pilot allocation can be found by performing an exhaustive search to identify the pilot allocation that leads to the highest sum SE. However, this exhaustive search is of a high computational complexity, which makes it infeasible for the network with a large number of highmobility users. For example, for a given and , there are possible pilot allocations to be searched in each cell. This motivates us to propose a lowcomplexity pilot allocation algorithm in the next subsection.
IiiB Pilot Allocation Algorithm
In this subsection, we detail the proposed lowcomplexity pilot sequence allocation algorithm, which results in reduced pilot contamination and improved sum SE. We note that the SINR expression in (8) depends on instantaneous channel realizations, which cannot be accurately obtained when the network suffers from pilot contamination [12]. Due to this limitation, we focus on minimizing the LOS interference given by (9) and allocate the same pilot sequence to the users with low LOS interferences.
We next present the stepbystep algorithm for pilot sequence allocation. We first assign pilot sequences to the center cell and then to the neighboring cells.
Divide the cell in to tiers
The BS divides the cell area into tiers based on the estimated distance as
(12) 
where each of the first tiers consists of users while there are no more than users in the tier .
Assign pilots in tier 1
We next assign the orthogonal pilot sequences to the users in the tier 1, i.e., the tier closest to the BS. The rationale behind assigning orthogonal pilot sequences to the users in tier 1 is to reduce the pilot contamination. Specifically, the interference power from the users that are assigned the same pilot sequence depends on the largescale channel coefficient, i.e, . As such, if two users close to a BS are assigned the same pilot sequence, the interference is large, which results in an increased pilot contamination. This observation can also be validated from the distancedependent part 1 in (9).
Assign pilots in tier 2
We compute using (9) between in the tier 1 and the users in the tier 2. We highlight that we first assign pilot to the user closest to the BS. We assign the same pilot sequence as in tier 1 to the user in tier 2 with minimum . If or are zero, we only used part 2 in (9). We repeat this process for the remaining users in tier 2.
Assign pilots in remaining tiers
We compute the LOS interference between in tier and in tier , where and in tier has been assigned the same pilot sequence. We then compute the average of the LOS interference and assign the same pilot sequence to with the minimum average LOS interference. If the tier has less than users, some pilot sequences are not used in tier .
Assign pilots in remaining cells
After completing pilot allocation for the center cell, we repeat Step 1 to Step 4 for pilot sequence allocation in neighboring cells. In Step 5, we consider all the LOS interference between a tier in the target cell and all the tiers in cells where the pilot sequence allocation has already been carried out, and then compute the average LOS interference and assign pilot sequences accordingly.
The algorithm returns a vector , where the lth element of denotes the pilot allocation for the users in the lth cell.
Iv Numerical Results and Analysis
In this section, we demonstrate the benefits of the proposed pilot allocation algorithm with random pilot allocation [9], the greedy iterative algorithm [10], and sectorbased [13] as benchmarks. Random pilot allocation is the most widely adopted pilot allocation algorithm in massive MIMO [3, 6, 4]. In random pilot allocation, the BS allocates the pilots to all the users in a cell randomly. Greedy iterative algorithm [10] iteratively refines the sum rate by first identifying the user with the lowest rate and then searching a pilot sequence for the user which minimizes the interference. Sectorbased algorithm divides the cellarea in sectors and all users in a sector are assigned the same pilot [13]. The system settings adopted in this section are summarized in Table I. All the results are obtained for an average of 10,000 Monte Carlo simulations.
Fig. 3LABEL:sub@figa depicts the sum SE in the center cell for the proposed pilot allocation, random pilot allocation, and greedy iterative algorithm
Number of cells  2 

Number of users per cell  36 
Cell radius  400 m 
User distance from the BS  
User AoA  
Largescale propagation constant  , where 
Length of pilot sequence  12 
Channel uses in coherence interval  196 
SNR  10 dB 
In Fig. 3LABEL:sub@figb, we compare the performance of the users worst affected by interference. As such, the sum SE of such users is low. We compare the cumulative distribution function (CDF) of five users with minimum sum SE for the three pilot allocation algorithms. We set , , and assume that the locations of users are accurately known to the BSs. Our proposed algorithm outperforms the random pilot allocation and greedy iterative algorithm by and , respectively.
In Fig. 3LABEL:sub@figc, we examine the performance of the proposed algorithm subject to localization errors. The location estimate has an error variance of , where the location error is uniformly distributed. We clarify that certain locationbased schemes may be more sensitive to the errors in AoA than to the errors in distance. However, in this simulation, we consider the errors in both distance and AoA. In the simulations, we use the factor as defined in 3GPP TR25.996 model. Accordingly, we define , where is the estimated distance between and . Furthermore, we assume that the probability of LOS decreases linearly as the distance between users and BSs increases. We highlight that the proposed algorithm still significantly outperforms the other two pilot allocation algorithms when the locations of the users suffer from estimation errors. We note that the sum SE decreases when increases. For example, when increases from to , the sum SE for the proposed algorithm decreases from to , which amounts to a reduction of in the sum SE. However, the proposed pilot allocation algorithm provides an improvement of , , and in the sum SE, respectively, as compared with sectorbased allocation, random allocation, and greedy allocation when .
V Conclusion
We proposed a lowcomplexity locationaware pilot allocation algorithm for a massive MIMO network with highmobility users. The algorithm exploited the behavior of LOS interference between users for pilot sequence allocation. Comparison with existing algorithms demonstrated the advantages of our proposed algorithm in terms of achieving a higher sum SE. In addition, the proposed algorithm is beneficial for the users that are worstaffected by interference and outperforms existing algorithms in the presence of localization errors.
Appendix A
From (8), we note that is important in determining . The vector is based on channel estimates. For example, we have for ZF detection. Consequently, we compute and obtain
(13) 
We highlight that (13) cannot be computed unless the locations of users and channel estimates are known to the BS. However, assuming that the BSs have estimated user locations , we calculate an estimate for the first term, i.e, the pure LOS term in (13), as
(14) 
where we have defined and . We highlight that is a special case of (14) when and . Utilizing this observation we obtain . Normalizing (14) by , we obtain the expression for the LOS interference as
(15) 
where represents the LOS interference between and .
Footnotes
 We note that there are unique pilot allocations in the massive MIMO network, which leads to that identifying the optimal pilot allocation is of a high computational complexity. For a small scale scenario with , , and , we have found that the proposed pilot allocation algorithm achieves between 67.7% to 76.6% of the sum SE achieved by the optimal pilot allocation algorithm, but with a significantly lower computational complexity.
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