I Introduction
Massive MIMO is an emerging technology to handle the growing demand for wireless data traffic in the next generation cellular networks [1]. A Massive MIMO BS is equipped with hundreds of antennas to spatially multiplex a large number of users on the same time–frequency resource [2]. In a singlecell scenario, there is no need for computationally heavy decoding (or precoding methods) in Massive MIMO as both the thermal noise and mutual interference are effectively suppressed by linear processing, e.g., maximumratio combining (MRC) or regularized zeroforcing (RZF) combining, with a large number of BS antennas [3]. In a multicell scenario, however, pilotbased channel estimation is contaminated by the nonorthogonal transmission in other cells. This results in coherent intercell interference in the data transmission, socalled pilot contamination [4], unless highcomplexity signal processing schemes are used to suppress it [5]. Pilot contamination reduces the benefit of having many antennas and the SEs achieved by lowcomplexity MRC or RZF saturate as the number of antennas grows.
Much research has been dedicated to mitigating the effects of pilot contamination; for example by increasing the length of the pilots [6] or assigning the pilots in a way that reduces the contamination [7]. In practical networks, however, it is not possible to make all pilots orthogonal due to the limited channel coherence block [6]
. Besides, the pilot assignment is a combinatorial problem. Even though heuristic algorithms with relatively low complexity can be developed, this approach still suffers from the asymptotic SE saturation since we only change one contaminating user for a less contaminating user. Instead of combating pilot contamination, one can utilize decoding schemes where the BSs are cooperating
[8]. In the twolayer LSFD (largescale fading decoding) framework, each BS applies an arbitrary local linear decoding method in the first layer. The results are then gathered at a common central station that applies socalled LSFD vectors in a secondlayer to combine the signals from multiple BSs to suppress pilot contamination and other intercell interference. The LSFD vectors are selected only based on the channel statistics (largescale fading) and, therefore, there is no need for the BSs to share their local channel estimates. The new decoding design attains high SE even with a limited number of BS antennas [8]. Previous works on LSFD has either considered uncorrelated Rayleigh fading channels [9, 8] or special correlated Rayleigh fading based on the onering model [10]. The latter paper optimizes the system with respect to networkwide maxmin fairness, which is a criterion that gives all the users the same SE, but usually a very low such SE [11].In this paper, we generalize the LSFD method from [8, 10] to a scenario with arbitrary spatial correlation and also develop a method for joint power control and LSFD vector optimization in the system using the sum SE as the utility. We first quantify the SE in a system with arbitrary processing in the two layers and then derive a closedform expression for the case when MRC is used in the first layer. The LSFD vector that maximizes the SE follows in closed form. Additionally, an uplink sum SE optimization problem with power constraints is formulated. Because it is a hard nonconvex problem, we are not searching for the global optimum but develop an alternating lowcomplexity optimization algorithm that converges to a stationary point. Numerical results demonstrate the effectiveness of the optimized system for Massive MIMO systems with correlated Rayleigh fading.
Notation
: Lower and upper case bold letters are used for vectors and matrices. The expectation of a random variable
is denoted by and is the Euclidean norm. The transpose and Hermitian transpose of a matrix are written as and , respectively. The dimensional diagonal matrix with the diagonal elements is denoted . Finally,is circularly symmetric complex Gaussian distribution.
Ii System Model
We consider a cellular network with cells. Each cell consists of a BS equipped with antennas that serves singleantenna users. The channel vector in the uplink between user in cell and BS is denoted by . We consider the standard blockfading model [11], where the channels are static within a coherence block of size channel uses and take an independent realization in each blocks, according to a stationary ergodic random process. Since practical channels are spatially correlated, we assume that each channel follows a correlated Rayleigh fading model:
(1) 
where is the spatial correlation matrix. The BSs know the channel statistics, but have no prior knowledge of the channel realizations, which need to be estimated in every coherence block.
Iia Channel Estimation
As in conventional Massive MIMO [5], the channels are estimated by letting the users transmit symbol long pilots in a dedicated part of the coherence block, called the pilot phase. All the cells share a common set of mutually orthogonal pilots with . Without loss of generality, we assume that the users in different cells that have the same index use the same pilot and thereby cause pilot contamination to each other [2]. During the pilot phase, at BS the received signal in the pilot phase is denoted and it is given by
(2) 
where is the power of the pilot transmitted by user in cell and is a matrix of independent and identically distributed noise terms, each distributed as . An observation of the channel from user to BS is obtained by using standard MMSE estimation [11]. The channel estimates are used at BS to compute decoding vectors for detecting the signals from the intracell users.
IiB Uplink Data Transmission
In the data phase, it is assumed that user in cell sends a zeromean information symbol with power . The received signal at BS is then
(3) 
where denotes the transmit power of user in cell . Based on the signals in (3), the BSs decode the symbols using the twolayersdecoding technique that is illustrated in Fig. 1. In the first layer, an estimate of the symbol from user in cell is obtained at BS by local linear decoding as
(4) 
where is the linear decoding vector. The symbol estimate contains interference and noise. In particular, the coherent interference caused by pilot contamination from pilotsharing users in other cells is large in Massive MIMO. To mitigate the intercell interference, all the symbol estimates of the pilotsharing users are collected in a vector
(5) 
After the local decoding, a second layer of centralized decoding is performed. The final estimate of the data symbol from user in cell is obtained as
(6) 
where is called the LSFD vector and is the LSFD weight. Unlike previous works, in our framework, arbitrary linear combining methods can be used in the first layer and the LSFD vectors can still be optimized.
In the next section, we use the decoded signals together with the asymptotic channel properties [11, Section 2.5] to derive a closedfrom expression for achievable uplink SE.
Iii Uplink Performance Analysis
This section first derives a SE expression that can be used for any decoding vector and then a closedform expression when using MRC. These expressions are then used to obtain the LSFD vectors that maximize the SE. The useandthenforget capacity bounding technique [3, Chapter 2.3.4], [5, Section 4.3] allows us to compute a lower bound on the uplink ergodic capacity (i.e., an achievable SE) of user in cell as
(7) 
where the effective SINR, denoted by , is
(8) 
where and stand for the desired signal, the pilot contamination, the beamforming gain uncertainty, the noncoherent interference, and the additive noise, respectively, whose expectations are defined as
(9)  
(10)  
(11)  
(12)  
(13) 
We notice that the SE expression in (7) can be applied together with any linear decoding method and any LSFD vector, but the expectations have the evaluated numerically.
Maximizing the SE of user in cell is equivalent to selecting the LSFD vector that maximizes a Rayleigh quotient.
Theorem 1.
If MRC, ZF or RZF is used, for a given set of pilot and data power coefficients, the SE of user in cell is
(14) 
where the matrices are
(15)  
(16)  
(17)  
(18) 
and the vectors are
(19)  
(20) 
In order to attain this SE, the LSFD vector is selected as
(21) 
Proof:
The proof relies on rewriting the SE as a generalized Rayleigh quotient and solving it. The details are available in the journal version of this paper [12]. ∎
We stress that the LSFD vector in (21) is designed to maximize the SE in (14) for every user in the network for a given data and pilot power and a given firstlayer decoding method. This is a nontrivial generalization of the previous works [9, 8, 10], which only considered specific firstlayer decoding methods that could provide closedform expressions.^{1}^{1}1We stress that Theorem 1 also holds in other cases, if we replace as , where . The following theorem states a closedform expression of the SE for the case of MRC in arbitrary spatial correlation, which makes the results more practical than in [10].
Theorem 2.
When MRC is used, the SE in (7) of user in cell is given by
(22) 
where the SINR value is given in (23) on the top of the next page.
(23) 
The values and are given as
(24)  
(25)  
(26) 
where and is defined in the same manner.
Proof.
Theorem 2 describes the exact impact that the spatial correlation has on the system performance through the coefficients and . It is seen that the numerator of (23) grows as the square of the number of antennas, , since the trace in (24) is the sum of terms. This gain comes from the coherent combination of the signals from the antennas. It can also be seen from Theorem 2 that the pilot contamination in (6
) combines coherently, i.e., its variance—the first term in the denominator that contains
—grows as . The other terms in the denominator represent the impact of noncoherent interference and Gaussian noise, respectively. These two terms only grow as . Since the interference terms contain products of correlation matrices of different users, the interference is smaller between users that have different spatial correlation characteristics [11].The following corollary gives the optimal LSFD vector that maximizes the SE of every user for a given set of pilot and data powers.
Corollary 1.
Iv Optimizing the Sum SE
In this section, the sum SE maximization problem is formulated where the optimization variables are the data powers and LSFD vectors. Since this problem is NPhard, an iterative algorithm is proposed to find a stationary point with low computational complexity.
Iva Problem Formulation
We consider sum SE maximization
(31)  
subject to 
Inserting the SE expression (22) into (31), and removing the constant prelog factor, yields the equivalent formulation
(32)  
subject to 
Sum SE maximization with imperfect CSI is known to be a nonconvex and NPhard problem [13] and this applies also to (32), even if the optimal LSFD vectors are given in Corollary 1. Therefore, the global optimum is overly difficult to compute. Nevertheless, solving the ergodic sum SE maximization in (32) for a Massive MIMO system is more practical than maximizing the instantaneous SEs for a given smallscale fading realization, as is normally done in smallscale MIMO systems [14]. Since the sum SE maximization in (32) only depends on the largescale fading coefficients, the solution can be used for as much time as the channel statistics are constant. Another key difference from prior work is that we jointly optimize the data powers and LSFD vectors.
Instead of seeking the global optimum to (32), we will obtain a stationary point to (32) by following the weighted MMSE approach from [14] and adapt it to the problem at hand. To this end, we first formulate the weighted MMSE problem that is equivalent to (32).
Theorem 3.
Proof.
(38) 
(42) 
(43) 
IvB Iterative Algorithm
We will obtain a stationary point to (33) by decomposing it into a sequence of subproblems, each having a closedform solution. To this end, the power variable is substituted with . By alternating between solving the subproblems we obtain the following result.
Theorem 4.
A stationary point to (33) is obtained by iteratively updating . Let the values after iteration . At iteration , the optimization parameters are updated in the following way:

[leftmargin=*]

is updated as
(39) where is given by
(40) 
is updated as in (43) on the top of this page.
If we denote the stationary point to (33) that is attained by the above iterative algorithm as by , , , and , for all , then the solution is also a stationary point to the problem (32).
Proof.
The closedform expression to each optimization variable is obtained by taking the first derivative of the Lagrangian function and equating it to zero, while the same stationary point of problems (32) and (33
) is based on the chain rule. The detail proof is available in the journal version
[12]. ∎If the initial data power values are uniformly distributed over the range
, the initial LSFD vectors can be computed using Corollary 1. The iterative algorithm in Theorem 4 is then used to obtain a stationary point to problem (31). This algorithm is terminated when the variation between two consecutive iterations is sufficiently small.V Numerical Results
We consider a wrappedaround cellular network with four cells. The distance between user in cell and BS is denoted by . The users in each cell are uniformly distributed over the cell area that is at least 35 m away from the BS, i.e., . MonteCarlo simulations are carried out over random sets of user locations. We model the system parameters and largescale fading similar to the 3GPP LTE specifications [15]. The system uses MHz of bandwidth, the noise variance is dBm, and the noise figure is dB. The largescale fading coefficient is computed in decibel scale as where the decibel value of the shadow fading, , has a Gaussian distribution with zero mean and standard derivation . The spatial correlation matrix of the channel from user in cell to BS is described by the exponential correlation model, that models a uniform linear array with the correlation magnitude [16]. The correlation magnitude is multiplied with a unique phaseshift in every correlation matrix, selected as the user’s incidence angle to the array. We assume that the power is fixed to mW for each pilot symbol and it is also the maximum power that each user can allocate to a data symbol, i.e., mW. The following methods are compared in the simulation:

Singlelayer decoding system with fixed data power: Each BS uses MRC for data decoding for the users in the own cell, and all users transmit data symbols with the same power mW.

Singlelayer decoding system with data power control: This benchmark is similar to (i), but the data powers are optimized using a modified version of Theorem 4.

Twolayer decoding system with optimized data power and LSFD vectors: This is the proposed method, where the data powers and LSFD vectors are computed using the weighted MMSE algorithm as in Theorem 4.
Fig. 4 shows the convergence of the proposed method for sum SE optimization in Theorem 4. From the initial random data powers, uniformly distributed in the feasible set, updating the optimization variables gives improved sum SE in every iteration. For the two layer case (iv), the sum SE per cell is about better at the stationary point than at the initial point. At convergence, (iv) gives better sum SE than (ii). The proposed optimization methods need around iterations to converge, but the complexity is low since every iteration in the algorithm consists of evaluating a closedform expression.
Fig. 4 shows the sum SE per cell as a function of the channel correlation magnitude for a multicell Massive MIMO system. First, we observe the substantial gains in sum SE attained by using LSFD. The sum SE increases with up to in the case of equally fixed data powers, while that gain is about for jointly optimized data powers and LSFD vectors. Moreover, this figure shows that the performance is greatly improved when the data powers are optimized. The gain varies from to . The gap becomes larger as the channel correlation magnitude increases. This shows the importance of doing joint data power control and LSFD optimization in Massive MIMO systems with spatially correlated channels.
Fig. 4
compares the cumulative distribution function (CDF) of the sum SE per cell with either MRC or RZF in the first layer, where the latter requires the use of the new general SE expression in Theorem
1. An equal power mW is allocated to each transmitted symbol. Because RZF mitigates noncoherent interference effectively in the first layer, the second layer can increase the average SE by . If MRC is used in the first layer, the SE gain from using LSFD using is only . At the likely point, the two layer decoding system outperforms the single layer counterpart by and when using MRC or RZF, respectively.Vi Conclusion
This paper has investigated the ability of LSFD to mitigate intercell interference in multicell Massive MIMO systems with spatially correlated Rayleigh fading. LSFD is a twolayer decoding method, where a second decoding layer to mitigate intercell interference is applied after the classical decoding. We derived new SE expressions support arbitrary spatial correlation and firstlayer decoding. We used these expressions to optimize the data powers and LSFD vectors, to maximize the sum SE of the network. Even though the sum SE optimization is a nonconvex and NPhard problem, we proposed an iterative approach to obtain a stationary point with low computational complexity. Numerical results demonstrate the effectiveness of LSFD in reducing pilot contamination with the improvement of sum SE for each cell about in the tested scenarios, while the optimized data power control and LFSD design can improve the sum SE with more than . The gains are larger when using RZF in the first layer than when using MRC.
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