# Joint Design of Fronthauling and Hybrid Beamforming for Downlink C-RAN Systems

Hybrid beamforming is known to be a cost-effective and wide-spread solution for a system with large-scale antenna arrays. This work studies the optimization of the analog and digital components of the hybrid beamforming solution for remote radio heads (RRHs) in a downlink cloud radio access network (C-RAN) architecture. Digital processing is carried out at a baseband processing unit (BBU) in the "cloud" and the precoded baseband signals are quantized prior to transmission to the RRHs via finite-capacity fronthaul links. In this system, we consider two different channel state information (CSI) scenarios: 1) ideal CSI at the BBU 2) imperfect effective CSI. Optimization of digital beamforming and fronthaul quantization strategies at the BBU as well as analog radio frequency (RF) beamforming at the RRHs is a coupled problem, since the effect of the quantization noise at the receiver depends on the precoding matrices. The resulting joint optimization problem is examined with the goal of maximizing the weighted downlink sum-rate and the network energy efficiency. Fronthaul capacity and per-RRH power constraints are enforced along with constant modulus constraint on the RF beamforming matrices. For the case of perfect CSI, a block coordinate descent scheme is proposed based on the weighted minimum-mean-square-error approach by relaxing the constant modulus constraint of the analog beamformer. Also, we present the impact of imperfect CSI on the weighted sum-rate and network energy efficiency performance, and the algorithm is extended by applying the sample average approximation. Numerical results confirm the effectiveness of the proposed scheme and show that the proposed algorithm is robust to estimation errors.

## Authors

• 1 publication
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• ### MIMO-Aided Nonlinear Hybrid Transceiver Design for Multiuser mmWave Systems Relying on Tomlinson-Harashima Precoding

Hybrid analog-digital (A/D) transceivers designed for millimeter wave (m...
08/13/2020 ∙ by Kaidi Xu, et al. ∙ 0

• ### Beamforming Design for Large-Scale Antenna Arrays Using Deep Learning

Beamforming (BF) design for large-scale antenna arrays with limited radi...
04/07/2019 ∙ by Tian Lin, et al. ∙ 0

• ### C-RAN with Hybrid RF/FSO Fronthaul Links: Joint Optimization of RF Time Allocation and Fronthaul Compression

This paper considers the uplink of a cloud radio access network (C-RAN) ...
08/15/2018 ∙ by Marzieh Najafi, et al. ∙ 0

• ### Robust Beamforming with Pilot Reuse Scheduling in a Heterogeneous Cloud Radio Access Network

04/23/2018 ∙ by Hao Xu, et al. ∙ 0

• ### Deep Learning Methods for Joint Optimization of Beamforming and Fronthaul Quantization in Cloud Radio Access Networks

Cooperative beamforming across access points (APs) and fronthaul quantiz...
07/06/2021 ∙ by DaeSung Yu, et al. ∙ 0

• ### A Theoretical Performance Bound for Joint Beamformer Design of Wireless Fronthaul and Access Links in Downlink C-RAN

It is known that data rates in standard cellular networks are limited du...
02/13/2021 ∙ by Fehmi Emre Kadan, et al. ∙ 0

• ### Deep unfolding of the weighted MMSE beamforming algorithm

Downlink beamforming is a key technology for cellular networks. However,...
06/15/2020 ∙ by Lissy Pellaco, et al. ∙ 0

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## I Introduction

Massive multiple-input and multiple-output (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 low-dimensional 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 (C-RAN) 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 C-RAN 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 C-RAN, 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 point-to-point fronthaul compression strategies. In [15, 16], the authors investigated multivariate fronthaul compression.

In this work, we study the application of hybrid beamforming to the C-RAN 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 sum-rate and the network EE. Fronthaul capacity and per-RRH 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 second-order statistics of the downlink channel vectors, and the digital beamforming and fronthaul compression strategies are adaptive to the estimated effective channel.

### I-a Related Work

A hybrid beamforming design has been investigated in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Specifically, in [4], a point-to-point hybrid precoding and combining algorithm was proposed that uses orthogonal matching pursuit for millimeter-wave (mmWave) systems. The authors in [6] provided a low-complexity hybrid beamforming scheme to achieve sum-rate performance close to that of the zero-forcing (ZF) method for the downlink of multi-user multiple-input single-output (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 low-dimensional ZF digital beamforming. In addition, for multi-user 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 sum-rate 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 point-to-point MIMO and multi-user 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 low-dimensional baseband downlink obtained after RF beamforming. To address this issue, in [11] and [12], the RF beamforming matrix was determined by using the long-term CSI, while the digital beamformer was designed based on the low-dimensional effective channel. Recently, a design of hybrid beamforming for C-RAN systems has been studied in [17] and [18]. The authors in [17] provided a two-stage algorithm that only demands low-dimensional effective CSI. In [18], a RF beamforming was computed based on a weighted sum of second-order channel statistics and the size of RF and digital beamforming matrices was determined in order to maximize the large-scale approximated sum-rate with regularized ZF digital beamforming.

Furthermore, the EE maximization problem in C-RAN has been studied in [19, 20, 21]. In [19], the authors considered a joint design of beamforming, virtual computing resources, RRH selection, and RRH-UE association in a limited fronthaul C-RAN, and a global optimization algorithm and a low-complexity method were presented. Also, for both single-hop and multi-hop C-RAN scenarios, the problem of EE maximization under both data-sharing and compression-based 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.

### I-B Main Contributions, Paper Organization and Notation

The main contributions of this paper are as follows:

• For downlink C-RAN with hybrid analog-digital antenna arrays, we investigate the joint design of fronthauling and hybrid beamforming with the goal of maximizing the weighted sum-rate (WSR) and the network EE.

• For the case of perfect CSI, we first decompose the problem into two sub-problems of the RF beamforming and the digital processing, which turn out to be non-convex, and then we propose an iterative algorithm based on weighted minimum-mean-square-error (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 C-RAN with hybrid digital and analog processing and finite-capacity 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.

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 C-RAN system in which a BBU communicates with single-antenna UEs through RRHs, each equipped with transmit antennas. We assume that the th RRH is connected to the BBU via an error-free 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 analog-digital 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 sum-rate in Sec. III-A. The transmit-side 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].

### Ii-a Channel Model

For the downlink channel from the RRHs to the UEs, we adopt a frequency-flat fading channel model such that the received signal at the th UE is given as

 yk=∑i∈RhHk,ixi+zk=hHkx+zk, (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 transmit-side 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.

### Ii-B 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 inter-UE interference, the signals are linearly precoded as

 xD=[xHD,1⋯xHD,NR]H=∑k∈KvD,ksk, (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

 ^xD,i=xD,i+qi, (3)

where the quantization noise is independent of the signal . As for the standard information-theoretic 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

 gi(VD,Ωi)≜I(xD,i;^xD,i) (4) =log2det(∑k∈KΞHivD,kvHD,kΞi+Ωi)−log2det(Ωi)≤Ci,

where we define the set of the digital beamforming vectors as .

### Ii-C 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

 xi=VR,i^xD,i=∑k∈KVR,iΞHivD,ksk+VR,iqi. (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

 yk=∑l∈KhHk¯VRvD,lsl+hHk¯VRq+zk, (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

 Rk = fk(VR,VD,Ω)=I(sk;yk) = log2det(|hHk¯VRvD,k|2+ζk(VR,VD,Ω)) −log2det(ζk(VR,VD,Ω)),

where , , and we denote the function .

Considering the power consumption in the C-RAN, the total power consumption can be modeled as [31]

 PT(VR,VD,Ω) ≜ ∑i∈Rpi(VR,i,VD,Ω) +NUPNU+NNRPRF,

where the transmission power of the th RRH is obtained as

 pi(VR,i,VD,Ωi)≜E∥xi∥2 (9) =∑k∈Ktr(VR,iΞHivD,kvHD,kΞiVHR,i)+tr(VR,iΩiVHR,i),

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.

### Iii-a Weighted Sum-Rate Maximization

In this work, we tackle the problem of maximizing the WSR of the UEs while satisfying the per-RRH transmit power, fronthaul capacity and constant modulus constraints, where is a weight denoting the priority for the th UE. The problem is stated as

 % maximize VR,VD,Ω ∑k∈Kwkfk(VR,VD,Ω) (10a) s.t. gi(VD,Ωi)≤Ci, i∈R, (10b) pi(VR,i,VD,Ωi)≤Pi, i∈R, (10c) |VR,i,a,b|2=1, a∈M, b∈N, i∈R. (10d)

Problem (10) is non-convex 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.

#### Iii-A1 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

 maximize VD,Ω ∑k∈Kwkfk(V′R,VD,Ω) (11a) s.t. gi(VD,Ωi)≤Ci, i∈R, (11b) pi(V′R,i,VD,Ωi)≤Pi, i∈R, (11c)

where we eliminate the constant modulus constraint (10d) which is independent of the digital variables and . Problem (11) is still non-convex due to the non-convex objective function (11a) and constraint (11b).

To solve this problem, we extend the WMMSE-based 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

 fk(V′R,VD,Ω)≥1ln2γk(V′R,VD,Ω,u(κ)k,~w(κ)k), (12)

where we define the function

 γk(V′R,VD,Ω,u(κ)k,~w(κ)k) (13) =ln~w(κ)k−~w(κ)kek(V′R,VD,Ω,u(κ)k)+1,

with arbitrary parameters and , and the mean squared error (MSE) function is denoted as

 ek(V′R,VD,Ω,u(κ)k) (14) =|1−(u(κ)k)∗hHk¯V′RvD,k|2+|u(κ)k|2ζk(V′R,VD,Ω).

Note that the lower bound in (12) is satisfied with equality when the variables and are equal to

 u(κ)k=~uk(V′R,VD,Ω) (15) ≜hHk¯V′RvD,k|hHk¯V′RvD,k|2+ζk(V′R,VD,Ω), ~w(κ)k=1ek(V′R,VD,Ω,u(κ)k). (16)

Furthermore, an upper bound of the function in the constraint (11b) is given as

 gi(VD,Ω)≤~gi(VD,Ω,Σ(κ)i), (17)

for any arbitrary positive definite matrix , where is represented as

 ~gi(VD,Ωi,Σ(κ)i)=log2det(Σ(κ)i)−log2det(Ωi) (18) +tr((Σ(κ)i)−1(∑k∈KΞHivD,kvHD,kΞi+Ωi))ln2−Nln2.

Here, the matrix that achieves equality in (17) is written as

 Σ(κ)i=~Σi(VD,Ω)≜∑k∈KΞHivD,kvHD,kΞi+Ωi. (19)

Based on the inequalities (12) and (17), we formulate the problem

 % maximize VD,Ω,u(κ),~w(κ),Σ(κ) ∑k∈Kwkln2γk(V′R,VD,Ω,u(κ)k,~w(κ)k) (20a) s.t.~{} ~gi(VD,Ωi,Σ(κ)i)≤Ci, i∈R, (20b) pi(V′R,i,VD,Ωi)≤Pi, i∈R, (20c)

where , , and . Although problem (20) is still non-convex, it is convex with respect to when the variables are fixed and vice versa. As proved in [32], since each variable update yields a non-decreasing 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.

 Algorithm 1: Algorithm for updating VD and Ω Set κ=1 and initialize V(κ)D and Ω(κ) satisfying the constraints (11b)-(11c). Repeat Update u(κ)k=~uk(V′R,V(κ)D,Ω(κ)) for k∈K. Update ~w(κ)k=1/ek(V′R,V(κ)D,Ω(κ),u(κ)k) for k∈K. Update Σ(κ)i=~Σi(V(κ)D,Ω(κ)) for i∈R. Update {V(κ+1)D,Ω(κ+1)} as a solution of problem (20) for the given {u(κ),~w(κ),Σ(κ)}. Set κ←κ+1. Until convergence.

#### Iii-A2 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

 maximize VR ∑k∈Kwkfk(VR,V′D,Ω′) (21a) s.t. pi(VR,i,V′D,Ω′i)≤Pi, i∈R, (21b) |VR,i,a,b|2=1, a∈M, b∈N, i∈R. (21c)

The presence of the constant modulus constraint (21c) makes it difficult to solve problem (21). To address this issue, as in [10, Sec. III-A], 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

 maximize VR,u(κ),~w(κ) ∑k∈Kwkln2γk(VR,V′D,Ω′,u(κ)k,~w(κ)k) (22a) s.t. pi(VR,i,V′D,Ω′i)≤Pi, i∈R, (22b) |VR,i,a,b|2≤1, a∈M, b∈N, i∈R. (22c)
 Algorithm 2: Algorithm for updating VR Set κ=1 and initialize V(κ)R satisfying the constraints (21b)-(21c). Repeat Update u(κ)k=~uk(V(κ)R,V′D,Ω′) for k∈K. Update ~w(κ)k=1/ek(V(κ)R,V′D,Ω′) for k∈K. Update V(κ)R as a solution of problem (22) for the given {u(κ),~w(κ)}. Set κ←κ+1. Until 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. III-A]. 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.

### Iii-B 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

 % maximize VR,VD,Ω ∑k∈Kwkfk(VR,VD,Ω)PT(VR,VD,Ω) (23a) s.t. (???), (???), (???). (23b)

Problem (23) is also non-convex due to the objective function (23a) and the constraints (23b). In the following subsection, similar to Sec III-A, we adopt alternating optimization to tackle problem (23).

#### Iii-B1 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

 maximize VD,Ω ∑k∈Kwkfk(V′R,VD,Ω)PT(V′R,VD,Ω) (24a) s.t. (???), (???). (24b)

Since problem (24) is non-convex, we also apply a similar approach proposed in Sec. III-A1. To make problem (24) more tractable, we first introduce a new objective function as a natural logarithm of the objective function (24a)

 ln(∑k∈Kwkfk(V′R,VD,Ω))−ln(PT(V′R,VD,Ω)). (25)

Then, we consider a convex lower bound of the function (25) as

 ln(∑k∈Kwkfk(V′R,VD,Ω))−ln(PT(V′R,VD,Ω)) (26) ≥ϵ(V′R,VD,Ω,u(κ),~w(κ),ρ(κ)),

where we define the function

 ϵ(V′R,VD,Ω,u(κ),~w(κ),ρ(κ)) (27) =ln(∑k∈Kwkln2γk(V′R,VD,Ω,u(κ)k,~w(κ)k))−ln(ρ(κ)) −PT(V′R,VD,Ω)ρ(κ)+1,

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

 u(κ)k = ~uk(V′R,VD,Ω), k∈K, (28) ~w(κ)k = 1ek(V′R,VD,Ω,u(κ)k), k∈K, (29) ρ(κ) = PT(V′R,VD,Ω). (30)

Based on the bounds (17) and (26), the problem is formulated as

 maximize VD,Ω,u(κ),~w(κ),Σ(κ),ρ(κ) ϵ(V′R,VD,Ω,u(κ),~w(κ),ρ(κ)) (31a) s.t.~{} (???), (???). (31b)

Similar to Algorithm 1, to obtain a solution , we alternately update the sets of variables and until convergence.

#### Iii-B2 Optimization of RF Beamforming

In this subsection, for fixed digital variables and , we focus on optimizing the RF beamforming by solving the following non-convex problem

 maximize VR ∑k∈Kwkfk(VR,V′D,Ω′)PT(VR,V′D,Ω′) (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

 maximize VR,u(κ),~w(κ),ρ(κ) ϵ(VR,V′D,Ω′,u(κ),~w(κ),ρ(κ)) (33a) s.t. (???), (???). (33b)

Similar in Sec. III-A2, 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 low-dimensional 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 long-term CSI can be estimated by means of time average if the fading channels are stationary for a sufficiently long time [11, 34].

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 low-dimensional 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

 Yi=VHR,iHiΨT+VHR,iNi, (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

 yi = vec(Yi) = (Ψ⊗VHR,i)vec(Hi)+(IL⊗VHR,i)vec(Ni),

where denotes the vector obtained by stacking all columns of the matrix on top of each other. Note that the signal (IV-A) depends on the RF beamforming matrix .

Minimizing the MSE yields the estimated channel vector as

 ^hi=[^hH1,i⋯^hHNU,i]H=Wiyi, (36)

where stands for the th subvector of corresponding to the th UE and