Massive multiple-input multiple-output (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; 2) precoding based ; 3) angle-of-arrival (AoA) based ; 4) blind ; and 5) pilot design methods .
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 location-aware 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 line-of-sight (LOS) and non-line-of-sight (NLOS) components. In some cases, a LOS channel component may exist between BSs and users . 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 location-aware pilot allocation algorithms assume that the AoAs of all the users are strictly non-overlapping [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 inter-cell and the intra-cell pilot contamination. The work  presented the pilot allocation in a single-cell network. Different from , we consider a more general multi-cell network, which encompasses the single-cell network as a special case. Moreover, in this work we derive the multi-cell 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 . 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 low-complexity pilot allocation algorithm suitable for a network with high-mobility 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 single-antenna users. We denote the BS in the i-th cell as and the j-th user in the i-th cell as . Additionally, we denote the location of as , where is the distance from to and is the AoA of at . We represent the small-scale 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 large-scale propagation effects denoted as . As such, the uplink channel from to is written as
where is the K-factor 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
where is the distance between two antennas and is the wavelength. We assume that , which is a widely adopted assumption . The uplink channels between all the users in the i-th cell and are represented as , where
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.
Ii-a Uplink Channel Estimation
In the uplink channel estimation phase, users from all the cells transmit their pre-assigned 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 i-th 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
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
where . We highlight that appears in (II-A) 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.
We assume that the obtains the least-square (LS) channel estimates from (II-A) as
where , and .
From (II-A), we note that the channel estimate suffers from intra-cell pilot contamination when . Additionally, we note that the channel estimates suffer from inter-cell pilot contamination when the same pilot sequences are repeated throughout the network. Specifically, when the channel estimates suffer from inter-cell pilot contamination. In real-world 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 inter-cell pilot contamination always exists.
Ii-B Uplink Data Transmission and Spectral Efficiency
During the uplink data transmission, each user in a cell transmits uplink data symbols to the same-cell BSs. The
multiplies the received vectorwith the zero-forcing (ZF) matrix to decode the symbols transmitted by the same-cell users. Accordingly, using the use-and-then-forget bound, the signal received from after detection is written as 
where is the k-th column of the ZF detection matrix , is the symbol transmitted by , is the Gaussian noise at , and is the signal-to-noise ratio. The uplink SE for is given as , where is the channel uses in a coherence block , and is the signal-to-interference-plus-noise ratio (SINR) given by
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 Location-Aware Pilot Allocation
In this section, we present our low-complexity location-aware pilot allocation algorithm. The algorithm requires the knowledge about the large-scale 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.
Iii-a LOS Interference
We derive the expression for the LOS interference between two users in the network in the following theorem.
The LOS interference between and based on their estimated locations is given as
The proof is provided in Appendix A.
We note that the LOS interference given in (9) consists of two parts. Specifically, part 1 is distance-dependent and part 2 is AoA-dependent. 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 AoA-dependent part 2 from Fig. 2. The AoA-dependent 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).
In massive MIMO networks when , we have when and
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 location-aware pilot allocation algorithm.
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 high-mobility users. For example, for a given and , there are possible pilot allocations to be searched in each cell. This motivates us to propose a low-complexity pilot allocation algorithm in the next subsection.
Iii-B Pilot Allocation Algorithm
In this subsection, we detail the proposed low-complexity 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 . 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 step-by-step algorithm for pilot sequence allocation. We first assign pilot sequences to the center cell and then to the neighboring cells.
Iii-B1 Divide the cell in to tiers
The BS divides the cell area into tiers based on the estimated distance as
where each of the first tiers consists of users while there are no more than users in the tier .
Iii-B2 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 large-scale 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 distance-dependent part 1 in (9).
Iii-B3 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.
Iii-B4 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 .
Iii-B5 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 l-th element of denotes the pilot allocation for the users in the l-th cell.
Iv Numerical Results and Analysis
In this section, we demonstrate the benefits of the proposed pilot allocation algorithm with random pilot allocation , the greedy iterative algorithm , and sector-based  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  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. Sector-based algorithm divides the cell-area in sectors and all users in a sector are assigned the same pilot . 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 algorithm111We 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.. We assume that the factors for all the users are the same. The results are obtained using (8) for an average of 10,000 random user locations, where the user locations are accurately known to the BSs. By randomizing user locations we simulate high-mobility scenarios, where the locations of users change for each coherence interval. The advantage of the proposed pilot allocation algorithm is clearly observed from Fig. 3LABEL:sub@figa, where for and the proposed pilot allocation algorithm provides and improvement in sum SE as compared to the random pilot allocation and greedy iterative approach, respectively. Importantly, we observe that the performance improvement increases with , which is due to the fact that the LOS interference dominates the total interference when is large and our proposed algorithm is to minimize the LOS interference. We also observe that the performance improvement increases with , which can be explained by our Lemma 1 and demonstrates the benefit of massive MIMO. We observe that the greedy iterative algorithm achieves a lower sum SE than the random pilot allocation algorithm when is small. This is due to the fact that in high-mobility scenarios, the user with low SINR may have a different location and channel conditions in the next iteration. This makes the greedy iterative algorithm more suitable for low-mobility scenarios.
|Number of cells||2|
|Number of users per cell||36|
|Cell radius||400 m|
|User distance from the BS|
|Large-scale propagation constant||, where|
|Length of pilot sequence||12|
|Channel uses in coherence interval||196|
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 location-based 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 whenincreases. 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 sector-based allocation, random allocation, and greedy allocation when .
We proposed a low-complexity location-aware pilot allocation algorithm for a massive MIMO network with high-mobility 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 worst-affected by interference and outperforms existing algorithms in the presence of localization errors.
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
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
where represents the LOS interference between and .
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