I Introduction
Massive multipleinput and multipleoutput (MIMO) has been regarded as a promising technology for future wireless systems owing to its potential of improving both spectral and energy efficiency (EE) with simple signal processing [2]
. This is enabled by the fact that the channel vectors for different users become orthogonal when the number of transmit antennas grows to infinity. However, with massive MIMO arrays, it is generally impractical to equip every antenna of a large array with a radio frequency (RF) chain due to hardware limitations
[3]. Hybrid beamforming techniques, whereby the beamforming process consists of a lowdimensional digital beamforming followed by analog RF beamforming, has emerged as an effective means to address this problem (see, e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]). Both analog and digital components are typically designed separately and locally for a base station (BS) [3].In a cloud radio access network (CRAN) architecture, the baseband signal processing functionalities of multiple BSs are migrated to a baseband processing unit (BBU) in the “cloud”, while RF functionalities are implemented at distributed remote radio heads (RRHs). Therefore, in the CRAN architecture with large antenna arrays at the RRHs, digital precoding across multiple RRHs can be carried out at the BBU, while RF beamforming is performed locally at each RRH. The design problem becomes more challenging by the capacity limitations of the fronthaul links that connect the BBU to the RRHs.
In the downlink of CRAN, the BBU performs joint encoding and precoding of the messages intended for user equipments (UEs), and then the produced baseband signals are quantized and compressed prior to being transferred to the RRH via fronthaul links. The design of precoding and fronthaul compression strategies has been studied in [14, 15, 16]. Specifically, the authors in [14] considered the standard pointtopoint fronthaul compression strategies. In [15, 16], the authors investigated multivariate fronthaul compression.
In this work, we study the application of hybrid beamforming to the CRAN architecture. We tackle the problem of jointly optimizing digital baseband beamforming and fronthaul compression strategies at the BBU along with RF beamforming at the RRHs with the goal of maximizing the weighted downlink sumrate and the network EE. Fronthaul capacity and perRRH transmit power constraints are imposed, as well as constant modulus constraint on the RF beamforming matrices which consist of analog phase shifters [3].
The limited number of RF chains determines the capability of the BBU to acquire channel state information (CSI) through conventional uplink training based on the time division duplex (TDD) operation. In particular, during the uplink training, the received baseband signal depends on a RF beamforming matrix, and hence instantaneous CSI is unavailable when designing the RF beamforming matrices. To address this limitation, the RF beamforming matrices are computed based on the secondorder statistics of the downlink channel vectors, and the digital beamforming and fronthaul compression strategies are adaptive to the estimated effective channel.
Ia Related Work
A hybrid beamforming design has been investigated in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Specifically, in [4], a pointtopoint hybrid precoding and combining algorithm was proposed that uses orthogonal matching pursuit for millimeterwave (mmWave) systems. The authors in [6] provided a lowcomplexity hybrid beamforming scheme to achieve sumrate performance close to that of the zeroforcing (ZF) method for the downlink of multiuser multipleinput singleoutput (MISO) systems. In this case, each RF beamforming vector for a user was determined by projecting the downlink channel onto the feasible RF space with lowdimensional ZF digital beamforming. In addition, for multiuser MIMO mmWave systems, a limited feedback hybrid beamforming scheme was presented in [7]. The work in [8] proved that in hybrid beamforming, the number of RF chains needs to be twice the number of data streams to achieve sumrate performance equal to that of fully digital beamforming. Also, the authors in [8] considered a design of the hybrid beamforming to maximize spectral efficiency for pointtopoint MIMO and multiuser MISO scenarios.
Most of works on hybrid beamforming in [4, 5, 6, 7, 8, 9, 10] have assumed full CSI. However, it is difficult to estimate the channel vectors across all antenna elements, since the estimation operates in the lowdimensional baseband downlink obtained after RF beamforming. To address this issue, in [11] and [12], the RF beamforming matrix was determined by using the longterm CSI, while the digital beamformer was designed based on the lowdimensional effective channel. Recently, a design of hybrid beamforming for CRAN systems has been studied in [17] and [18]. The authors in [17] provided a twostage algorithm that only demands lowdimensional effective CSI. In [18], a RF beamforming was computed based on a weighted sum of secondorder channel statistics and the size of RF and digital beamforming matrices was determined in order to maximize the largescale approximated sumrate with regularized ZF digital beamforming.
Furthermore, the EE maximization problem in CRAN has been studied in [19, 20, 21]. In [19], the authors considered a joint design of beamforming, virtual computing resources, RRH selection, and RRHUE association in a limited fronthaul CRAN, and a global optimization algorithm and a lowcomplexity method were presented. Also, for both singlehop and multihop CRAN scenarios, the problem of EE maximization under both datasharing and compressionbased fronthaul strategies was addressed in [20]. The authors in [21] took into account a realistic power consumption model which is dependent on the data rate and dynamic power amplifier.
IB Main Contributions, Paper Organization and Notation
The main contributions of this paper are as follows:

For downlink CRAN with hybrid analogdigital antenna arrays, we investigate the joint design of fronthauling and hybrid beamforming with the goal of maximizing the weighted sumrate (WSR) and the network EE.

For the case of perfect CSI, we first decompose the problem into two subproblems of the RF beamforming and the digital processing, which turn out to be nonconvex, and then we propose an iterative algorithm based on weighted minimummeansquareerror (WMMSE) approach by relaxing constant modulus constraint. Also, for the imperfect CSI case, we extend the solution with the perfect CSI by applying the sample average approximation (SAA) [22].

Extensive numerical results are provided to validate the effectiveness of the proposed algorithm. In addition, in the presence of channel estimation errors, we show the robustness of the proposed scheme.
The paper is organized as follows: In Sec. II, we present the system model for the downlink of a CRAN with hybrid digital and analog processing and finitecapacity fronthaul links. In Sec. III, for the case of perfect CSI, we describe the problems of WSR maximization and network EE maximization, and an iterative algorithm to tackle the problems is proposed. Sec. IV discusses the problem of CSI estimation in a TDD system. In addition, we introduce an uplink channel training method and provide a RF beamforming matrix design and the digital strategies. Numerical results are illustrated in Sec. V. The paper is closed with the conclusion in Sec. VI.
Variable  Definition 

RF beamforming matrix at the th RRH  
Digital beamforming vector for the th UE  
Quantization noise covariance matrix at the th RRH  
MMSE receiver at the th UE  
Weight variable for the th UE  
Auxiliary matrix at the th RRH  
Auxiliary variable for power consumption  
Additional variable at the th instantaneous channel sample  
Auxiliary variable at the th instantaneous channel sample 
Throughout this paper, boldface uppercase, boldface lowercase and normal letters indicate matrices, vectors and scalars, respectively. The operators , , , and
represent transpose, conjugate transpose, expectation, determinant and trace, respectively. A circularly symmetric complex Gaussian distribution with mean
and covariance matrix is denoted by . The set of all complex matrices is defined as .represents an identity matrix of size
. stands for the Kronecker product. The variables used in this paper are summarized in Table I.Ii System Model
As illustrated in Fig. 1, we consider the downlink of a CRAN system in which a BBU communicates with singleantenna UEs through RRHs, each equipped with transmit antennas. We assume that the th RRH is connected to the BBU via an errorfree digital fronthaul link of capacity bps/Hz [23, 24], and each RRH is equipped with RF chains due to cost limitations. This implies that fully digital beamforming across transmit antennas of each RRH is not enabled [8, 5, 4, 6, 7] and thus hybrid analogdigital solutions are in order. For convenience, we define the sets , , and . We assume that user scheduling is predetermined and hence all the UEs are active. For rate allocation, the priority among active UEs can be controlled by adjusting the weights of the weighted sumrate in Sec. IIIA. The transmitside baseband processing is centralized at the BBU based on CSI reported by the RRHs on the fronthaul links. Assuming a TDD operation, each RRH obtains its local CSI by means of uplink channel training [25, 26].
Iia Channel Model
For the downlink channel from the RRHs to the UEs, we adopt a frequencyflat fading channel model such that the received signal at the th UE is given as
(1) 
where is the transmitted signal of the th RRH subject to the transmit power constraint , equals the channel vector from the th RRH to the th UE, which is distributed as with being the transmitside correlation, represents the signal vector transmitted by all RRHs, indicates the channel vector from all RRHs to the th UE, and denotes the additive noise at the th UE.
IiB Digital Beamforming and Fronthaul Compression
We define the message intended for the th UE as , where stands for the coding block length and is the rate of . The BBU encodes the message into the baseband signal for using a standard random Gaussian channel code. Then, in order to manage interUE interference, the signals are linearly precoded as
(2) 
where is the digital beamforming vector across all the RRHs for the th UE, and represents the th subvector of corresponding to the signal transmitted by the th RRH. Defining the shaping matrices , the th subvector can be expressed as .
Since the BBU communicates with the th RRH via a fronthaul link of finite capacity, the signal is quantized and compressed prior to being transferred to the RRH. Following the approaches in [27, 28], we model the impact of the compression by writing the quantized signal as
(3) 
where the quantization noise is independent of the signal . As for the standard informationtheoretic formulation, the covariance matrix describes the effect of the quantizer. From [29, Ch. 3], the quantized signal can be reliably recovered at the th RRH, if the following condition is satisfied
(4)  
where we define the set of the digital beamforming vectors as .
IiC RF Beamforming
The quantized signal vector decompressed at the th RRH is of dimension , and is input to one of RF chains. In order to fully utilize transmit antennas, the th RRH applies analog RF beamforming to the signal via the beamforming matrix . The RF beamforming obtains signals for the antenna as a combination of output of the RF chains. As a result, the transmitted signal from transmit antennas is given as
(5) 
As summarized in [30], the RF beamforming can be implemented using analog phase shifters and switches. Accordingly, each RF chain is connected to a specific set of transmit antennas through a phase shifter. In this paper, we consider a fully connected phase shifter architecture, whereby each RF chain is connected to all transmit antennas via a separate phase shifter. In the fully connected phase shifter architecture, the th element of the RF beamforming matrix is expressed as for and , where indicates the phase shift between the signals and . Therefore, when designing the RF beamforming matrix , one should satisfy constant modulus constraint for and (see, e.g., [8]).
Iii Design with Perfect CSI
In this section, we discuss the problem of jointly designing the beamforming matrices for RF and digital beamforming, along with the fronthaul quantization noise covariance matrices. As we will see, these problems are interdependent, since the impact of the quantization noise on the receivers’ performance depends on the beamforming matrices. Here, we first consider the case of perfect CSI, while the system with imperfect CSI will be addressed in Sec. IV.
To measure the achievable rate for each UE , we rewrite the signal in (1) under the transmission model (5) as
(6) 
where indicates the effective RF beamforming matrix across all RRHs and stands for the vector of all the quantization noise signals, which is distributed as with .
Assuming that UE decodes the message by treating the interference signals as the additive noise, the achievable rate for the th UE is given as
where , , and we denote the function .
Considering the power consumption in the CRAN, the total power consumption can be modeled as [31]
where the transmission power of the th RRH is obtained as
(9)  
is the circuit power consumed by a UE, and represents the circuit power consumption at each RRH, which is proportional to the number of RF chains.
Iiia Weighted SumRate Maximization
In this work, we tackle the problem of maximizing the WSR of the UEs while satisfying the perRRH transmit power, fronthaul capacity and constant modulus constraints, where is a weight denoting the priority for the th UE. The problem is stated as
(10a)  
s.t.  (10b)  
(10c)  
(10d) 
Problem (10) is nonconvex due to the objective function (10a) and the constraints (10b), (10c) and (10d). In the next subsection, we present an iterative algorithm that computes an efficient solution of the problem. To address problem (10), we propose an iterative algorithm based on block coordinate descent (BCD), whereby the RF beamforming matrices and the digital processing strategies are alternately optimized. We first describe the optimization of the digital part and for a fixed RF beamforming , and then introduce the optimization of the latter.
IiiA1 Optimization of Digital Beamforming and Fronthaul Compression
For a given RF beamforming , problem (10) with respect to the digital beamforming and the fronthaul compression strategies can be written as
(11a)  
s.t.  (11b)  
(11c) 
where we eliminate the constant modulus constraint (10d) which is independent of the digital variables and . Problem (11) is still nonconvex due to the nonconvex objective function (11a) and constraint (11b).
To solve this problem, we extend the WMMSEbased algorithm in [32]. To this end, we introduce two convex lower bounds on (11a) and (11b) by applying a similar approach in [32]. Denoting as the quantity obtained at the th iteration of the BCD, a lower bound on the function in (11a) is written as
(12) 
where we define the function
(13)  
with arbitrary parameters and , and the mean squared error (MSE) function is denoted as
(14)  
Note that the lower bound in (12) is satisfied with equality when the variables and are equal to
(15)  
(16) 
Furthermore, an upper bound of the function in the constraint (11b) is given as
(17) 
for any arbitrary positive definite matrix , where is represented as
(18)  
Here, the matrix that achieves equality in (17) is written as
(19) 
Based on the inequalities (12) and (17), we formulate the problem
(20a)  
(20b)  
(20c) 
where , , and . Although problem (20) is still nonconvex, it is convex with respect to when the variables are fixed and vice versa. As proved in [32], since each variable update yields a nondecreasing objective value in (20a), solving problem (20) alternately over these two sets of variables would yield a solution that is guaranteed to converge to a stationary point. This is detailed in Algorithm 1 below.
IiiA2 Optimization of RF Beamforming
We now discuss the optimization of the RF beamformers in problem (10) for fixed digital variables and . The problem can be stated as
(21a)  
s.t.  (21b)  
(21c) 
The presence of the constant modulus constraint (21c) makes it difficult to solve problem (21). To address this issue, as in [10, Sec. IIIA], we relax the condition (21c) to the convex constraint . Then, the obtained problem can be solved by again applying the WMMSE method in [32]. The procedure for solving problem (21) is summarized in Algorithm 2, where the convex problem of the original problem (21) is stated as
(22a)  
s.t.  (22b)  
(22c) 

: Algorithm for updating 

Set and initialize satisfying the constraints 
(21b)(21c). 
Update for . 
Update for . 
Update as a solution of problem (22) for the 
given . 
Set . 
convergence. 

Since the RF beamforming matrices computed from Algorithm 2, denoted as , may not satisfy the constraint (21c), we propose to obtain a feasible RF beamformer by projecting onto the feasible space [10, Sec. IIIA]. Specifically, we find the RF beamformer such that the distance is minimized. As a result, the beamformer is calculated as for , and [10, Eq. (14)]. In summary, for a joint design of the digital beamforming , the fronthaul compression and the RF beamforming strategies , we run Algorithm 1 and 2 alternately. We note that while both Algorithm 1 and Algorithm 2 are individually convergent in the absence of modulus constraint for the RF beamforming, due to the projection step in the update of RF beamforming, the overall alternating optimization algorithm is not guaranteed to converge. This is also the case for the related algorithms in [33]. Therefore, we will observe the convergence behavior of the proposed algorithm in Sec. V.
IiiB Network Energy Efficiency Maximization
We now consider jointly designing RF and digital beamforming along with fronthaul compression with the aim of maximizing the overall network EE. The network EE is defined as the ratio of the WSR to the corresponding power consumption. Accordingly, the problem is formulated as
(23a)  
s.t.  (23b) 
Problem (23) is also nonconvex due to the objective function (23a) and the constraints (23b). In the following subsection, similar to Sec IIIA, we adopt alternating optimization to tackle problem (23).
IiiB1 Optimization of Digital Beamforming and Fronthaul Compression
For a given RF beamforming , the digital beamforming and the fronthaul compression strategies are optimized by solving the following problem
(24a)  
s.t.  (24b) 
Since problem (24) is nonconvex, we also apply a similar approach proposed in Sec. IIIA1. To make problem (24) more tractable, we first introduce a new objective function as a natural logarithm of the objective function (24a)
(25) 
Then, we consider a convex lower bound of the function (25) as
(26)  
where we define the function
(27)  
with arbitrary parameters , and .
One can show that for fixed , the lower bound in (26) holds with equality when the variables , and are given as
(28)  
(29)  
(30) 
IiiB2 Optimization of RF Beamforming
In this subsection, for fixed digital variables and , we focus on optimizing the RF beamforming by solving the following nonconvex problem
(32a)  
s.t.  (32b) 
To solve problem (32), by using the bound (26) and relaxing the modulus constraint (21c), we express the relaxed problem as
(33a)  
s.t.  (33b) 
Similar in Sec. IIIA2, the sets of variables and are alternately updated until convergence, and then the obtained RF beamforming matrices are projected onto the feasible space to satisfy the modulus constraint (21c). To sum up, the digital beamforming , the fronthaul compression and the RF beamforming are jointly obtained by optimizing alternately and . The effectiveness of the proposed algorithm will be confirmed by numerical results in Sec. V.
Iv Design with Imperfect CSI
In the previous section, we have assumed that the instantaneous channel vectors are perfectly known at the BBU. In this section, we study a more practical case in which lowdimensional effective CSI is acquired by the RRHs via uplink channel training in a TDD operation. The key challenge is that the analog beamforming matrices affect the signal received on the uplink during the training phase. Therefore, the design of the analog beamforming cannot rely on the knowledge of full CSI . Instead, it is assumed that only the covariance matrices of the channel vectors are available at the BBU when designing analog precoding. In practice, this longterm CSI can be estimated by means of time average if the fading channels are stationary for a sufficiently long time [11, 34].
Iva Uplink Channel Training
In the TDD operation, the downlink CSI is obtained based on the uplink training signals by leveraging reciprocity between downlink and uplink channels. The channel matrix between all UEs and the th RRH is estimated at the RRH and forwarded to the BBU. Importantly, since channel estimation is performed based on the lowdimensional output of RF beamforming, the design of the RF beamforming affects the channel estimation as well as the WSR performance. For the rest of this subsection, we describe the relationships between and the channel estimation error.
To elaborate, on the uplink, UE transmits the orthogonal training sequence of symbols with transmit power , where the condition is required in order to ensure the orthogonality of the training sequences. We have for , where denotes the Kronecker delta function. The signal matrix received at the th RRH during uplink training is given as
(34) 
where represents the orthogonal training sequence matrix with is the matrix of training signal powers, and indicates the additive Gaussian noise matrix at the th RRH with for .
To estimate the channel from the received signal , we define the received signal vector of the th RRH as
where denotes the vector obtained by stacking all columns of the matrix on top of each other. Note that the signal (IVA) depends on the RF beamforming matrix .
Minimizing the MSE yields the estimated channel vector as
(36) 
where stands for the th subvector of corresponding to the th UE and
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