Fractional Pilot Reuse in Massive MIMO Systems
Abstract
Pilot contamination is known to be one of the main impairments for massive MIMO multicell communications. Inspired by the concept of fractional frequency reuse and by recent contributions on pilot reutilization among nonadjacent cells, we propose a new pilot allocation scheme to mitigate this effect. The key idea is to allow users in neighboring cells that are closest to their base stations to reuse the same pilot sequences. Focusing on the uplink, we obtain expressions for the overall spectral efficiency per cell for different linear combining techniques at the base station and use them to obtain both the optimal pilot reuse parameters and the optimal number of scheduled users. Numerical results show a remarkable improvement in terms of spectral efficiency with respect to the existing techniques.
I Introduction
Massive multipleinput multipleoutput (MIMO) refers to a type of cellular network based on multiuser MIMO in which the number of antennas at the base station (BS) is much larger than the number of served user terminals [1, 2, 3]. Under certain propagation conditions, such a system renders quasiorthogonal channels among users and very simple linear processing is shown to be optimal in terms of throughput.
In this context, a problem that appears when obtaining channel state information is that the number of available pilot sequences is finite and limited by the channel behavior, as their duration cannot span larger than the coherence interval of the channel to be estimated. In consequence, mobile users in different cells might have to reuse the same pilot sequences, thus resulting into corrupted channel estimates at each BS: this is known as the pilot contamination effect [4] and represents a major impairment affecting massive MIMO communications.
This problem has been intensively investigated in the literature in the last few years. For instance, [5] proposed a lowrate coordination phase between different BSs; [6] introduced an elaborated scheme for OFDMbased massive MIMO systems, with differentiated downlink and uplink scheduled training phases; [7] proposed a subspace projection to improve the channel estimation accuracy; and [8] also exploited coordination, comparing the amount of training needed with respect to the uncoordinated case and its implications.
More recently, the idea of pilot reuse was introduced in [9]. In this respect, [10, 11] advocated for employing orthogonal pilot subsets in adjacent cells, optimizing the number of required subsets and the number of scheduled users per cell that maximizes the overall spectral efficiency, for both uplink and downlink. This idea is borrowed from the wellestablished concept of frequency reuse in cellular systems [12] and is therein coined as “fractional pilot reuse”: given an integer representing the number of orthogonal pilot subsets (or, equivalently, the ratio between the number of available sequences and the number of scheduled users per cell), for each cell, only a fraction of the interfering cells reuse the same pilot subset.
On the other hand, the concept of fractional frequency reuse (FFR) consists in splitting the available bandwidth into frequency subbands, where the integer specifies the frequency reuse factor among the cell edges and the additional subband is reused in all the cell centers (see [13] and references therein). The difference between traditional frequency reuse and FFR is illustrated in Figure 1.
The idea behind this paper is to adapt the aforementioned concept of FFR to the pilot domain and assign the pilot sequences to the users within the cells in the same way as different frequency subbands are allocated in FFR. Under these premises, we propose a novel channel estimation scheme where a fraction of users within each cell reuses the same pilot subset across the whole system, while the rest are allocated orthogonal subsets depending on a reuse parameter. We also refer to this approach as fractional pilot reuse, but in the sense that a fraction of the users per cell reuse the same pilot subset in all the cells. In this regard, we derive closedform expressions for the spectral efficiency and analyze the performance in the largeantenna regime. Numerical results show that such deployment outperforms the proposal in [10] and offers remarkably good throughput, especially for large numbers of antennas at the BS.
The rest of the paper is structured as follows. In Section II, we introduce the system model on which we build our novel framework based on fractional pilot reuse in Section III. In Section IV, numerical results are reported to corroborate the proposed scheme. Finally, Section V draws some concluding remarks.
Ii System Model
Let us consider a massive MIMO cellular network over a set of cells denoted by . The BS of each cell is equipped with antennas and serves singleantenna users in the uplink; in this context, is fixed while is chosen adaptively. We model the channels as blockfading, where represents the channel response between BS and user in cell , with random coordinates , during its coherence block consisting of channel uses: in the following, we consider that the coherence block of the whole system spans channel uses.^{1}^{1}1A user with high mobility, which would consistently decrease the overall coherence block , can be served in a different timeslot in order not to compromise the channel estimation of the other users. The channel realizations are drawn from a circularly symmetric complex Gaussian distribution, i.e., , where represents the variance of the channel attenuation from BS to user in cell (cf. [11]): this term accounts for propagation impairments such as pathloss and shadowing.
Let us introduce as the vector signal transmitted by user in cell , normalized as , . The received signal at BS in a coherence block, denoted by , is thus given by
(1) 
where is the transmit power allocated to user in cell and is the additive noise at BS , whose elements are distributed independently as . As in [11], we adopt a statisticaware power control,^{2}^{2}2Such powercontrol policy makes the average effective channel gain equal for all users. where the signals from user in cell are allocated the power , with .
We take into account a pilot book of orthogonal pilot sequences defined as
(2) 
with each ; in the following, we will use to denote the index of the pilot sequence assigned to user in cell . Therefore, the vector signal of user in cell can be written as , where represents the information vector. We can thus decompose the received signal in (1) as with
(3)  
(4) 
with . Moreover, we assume that BS applies a linear receive combining vector to the received signal, i.e., , so as to amplify the signal coming from its user while rejecting the interference caused by the others. In this respect, we consider:

maximum ratio combining (MRC), i.e., a traditional passive rejection scheme;

pilotbased zeroforcing combining (PZFC), recently proposed in [11], which actively suppresses both intra and intercell interference.
Iii Fractional Pilot Reuse
Iiia Problem Formulation
Inspired by the concept of FFR, we split the pilot book into subsets, where the integer is referred to as integer reuse factor. The users that, inside each cell , exhibit a low variance of the channel attenuation with respect to their BS (i.e., ) reuse the subsets in nonadjacent cells; on the other hand, the users that, inside each cell , enjoy a high are allocated the same th subset across the whole system.^{3}^{3}3In Section IV, we consider a standard pathloss model in which the users are divided into two groups simply based on the distance from their BS. Hence, we define the new parameter as the fractional reuse factor, which represents the fraction of users per cell that are assigned the additional pilot subset. We refer to this approach as fractional pilot reuse (FPR): an illustrative example is shown in Figure 2. The number of pilot sequences (and, hence, their length) is determined as
(5) 
Note that would imply assigning the same sequences to each cell with no pilot reuse whatsoever (this case is prevented by the definition of ). On the other hand, if , the formulation trivially collapses to as in the baseline [11].
(9)  
(10)  
(11)  
(12) 
Our objective is to find the pilot reuse parameters and , as well as the number of scheduled users per cell , that maximize the overall spectral efficiency within the cell. In the following, we obtain expressions for the spectral efficiency adopting MRC and PZFC combining. Without loss of generality, we will hereafter assume that the user coordinates are uniformly distributed within the cells.
IiiB Achievable Spectral Efficiency
We first give the following preliminary definitions. Let be the set of the users in cell with the highest ; likewise, let be the set of the users in cell with the lowest . Furthermore, we introduce the functions
(6)  
(7) 
for , where the ratio expresses the relative strength of the interference received at BS from user in cell . Lastly, let denote the set of cells using the same subset of pilots as .^{4}^{4}4Observe that occurs only when .
Theorem 1.
The ergodic achievable spectral efficiency in cell , when the users are uniformly distributed within the cell, is given by
(8)  
with defined in (5) and where the expressions of the signaltointerferenceplusnoise ratios (SINRs) are given in (9)–(10) for MRC and in (11)–(12) for PZFC, respectively, at the top of the page.
Proof:
The expression for the ergodic achievable spectral efficiency in cell without FPR is given in [11, Th. 1]:
(13) 
where is the SINR corresponding to user in cell derived in [11, Th. 1, 2] for MRC and PZFC and
(14) 
We first note that, if the coordinates of all users within a cell have the same distribution, then loses its (implicit) dependence on the user index and, as a consequence, we can rewrite the following term as
(15) 
where the last equality derives from the fact that
(16) 
By extending (15) to the case of FPR, we can write
(17)  
(18)  
(19) 
where we have defined
(20) 
and we obtain the SINRs in (9)–(12). Finally, exploiting the fact that , we arrive at the expression of the spectral efficiency in (8). ∎
It is straightforward to conclude that, when , we have and the spectral efficiency in (8) collapses to the expression given in (13).
Remark 1.
The expressions above can be easily extended to the case where the mobile users are not uniformly distributed over the cell at the cost of less tractability. More specifically, when the average relative strength of a user depends on its index, then the terms and cannot be removed from the inner summation. Nonetheless, the resulting formulas can be evaluated by MonteCarlo simulations if required.
Now, let us study the performance of the proposed scheme based on FPR when .
Corollary 1.
Evidently, the asymptotic spectral efficiency derived in Corollary 1 does not depend on the type of combining employed at the BS. Again, we observe that, when , the same expression of the asymptotic spectral efficiency given in [11, Cor. 3] is obtained.
Unfortunately, as in [11], the expressions obtained in (8) and in (1) do not allow for closedform extraction of the optimal ; moreover, a further complication of our scheme is that the terms and intrinsically depend on . Therefore, in the next section, we optimize the system parameters using numerical evaluation for a particular scenario.
Iv Numerical results
In this section, we assess the performance of the proposed scheme based on FPR through numerical evaluation. In doing so, we adopt an equivalent setup to [10] and compare our results against those obtained without using FPR.
Let us consider a symmetric, infinitely large network of hexagonal cells with radius . Furthermore, let us assume a standard pathloss model in which the variance of the channel attenuation is given by , where is the pathloss exponent and specifies the coordinates of BS : in this context, we have (cf. (6)–(7))
(22) 
MRC  PZFC  

For our numerical simulations, the same parameters as in [10] are used: we take into account the interference produced by tiers of surrounding cells, a coherence block of channel uses, a pathloss exponent , an average SNR between any user and any antenna at the corresponding BS of dB, and a number of BS antennas in the range . Lastly, the terms and are computed by MonteCarlo simulations as follows. For each value of , we generate instances of uniformly distributed user coordinates in every cell with a minimum distance of from the corresponding BS; then, for each value of , i) the coordinates that are closest to the BS are assigned to and is computed as in (6), and ii) the other coordinates are assigned to and is computed as in (7).
Remarkably, using FPR always improves the spectral efficiency with respect to [10], as summarized in Table I. Figure 3(a) and Figure 3(b) show the spectral efficiency and optimal number of scheduled users, respectively, when using MRC combining; despite obtaining quite similar optimal values of with respect to the baseline [10], we observe that adopting FPR allows to anticipate the switching point from to , i.e., the channels become quasiorthogonal at lower values of . Likewise, Figure 4 illustrates the same results for the case of PZFC, which produces significantly improved spectral efficiencies with respect to MRC. Note that, with PZFC, choosing yields a higher spectral efficiency with respect to even for low values of , which always allows to schedule more users than without FPR (see Figure 4(b)).
Let us more generally compare MRC and PZFC: in the former, the impact of our approach is more significant as the number of antennas at the BS increases whereas, for the latter, our method exceptionally doubles the spectral efficiency in the whole range of under exam. In addition, by comparing Figure 3(a) and Figure 4(a), it emerges that FPR with MRC always outperforms the baseline [10] with PZFC.
As expected, the spectral efficiency in (8) is concave in for any fixed . The spectral efficiency as a function of is depicted in Figure 5 with BS antennas and using the optimal value of for each setup: in particular, we observe that the optimal changes with both the integer reuse factor and the combining technique. Lastly, the spectral efficiency obtained with PZFC , i.e., the best combination of integer reuse factor and combining technique, is illustrated as a function of in Figure 6 for some values of .
V Conclusions
In this paper, we propose a novel pilot allocation scheme for multicell massive MIMO systems, where users close to their respective base stations get to reuse the same set of pilot sequences across the whole system. We derive expressions for an achievable rate in this scenario and obtain both the pilot reuse parameters and the number of scheduled users per cell that maximize it. Results show a significant gain in terms of spectral efficiency with respect to the existing pilot reuse baseline.
Acknowledgements
The authors would like to thank Emil Björnson and Marios Kountouris for their valuable comments and suggestions.
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