# Joint User Selection and Energy Minimization for Ultra-Dense Multi-channel
C-RAN with Incomplete CSI

Cunhua Pan, Huiling Zhu, Nathan J. Gomes and Jiangzhou Wang, _Fellow_ , _IEEE_
This paper has been accepted by IEEE JSAC with special issue on Deployment
Issues and Performance Challenges for 5G. This work is supported by European
Commission Horizon2020 project iCIRRUS under grant agreement No 644526. C.
Pan, H. Zhu, N. Gomes and J. Wang are with the School of Engineering and
Digital Arts, University of Kent, Canterbury, Kent, CT2 7NZ, U.K. C.Pan is
also with the Queen Mary University of London, London E1 4NS, U.K.
(Email:{C.Pan, H.Zhu, N.J.Gomes, J.Z.Wang}@kent.ac.uk).

###### Abstract

This paper provides a unified framework to deal with the challenges arising in
dense cloud radio access networks (C-RAN), which include huge power
consumption, limited fronthaul capacity, heavy computational complexity,
unavailability of full channel state information (CSI), etc. Specifically, we
aim to jointly optimize the remote radio head (RRH) selection, user equipment
(UE)-RRH associations and beam-vectors to minimize the total network power
consumption (NPC) for dense multi-channel downlink C-RAN with incomplete CSI
subject to per-RRH power constraints, each UE’s total rate requirement, and
fronthaul link capacity constraints. This optimization problem is NP-hard. In
addition, due to the incomplete CSI, the exact expression of UEs’ rate
expression is intractable. We first conservatively replace UEs’ rate
expression with its lower-bound. Then, based on the successive convex
approximation (SCA) technique and the relationship between the data rate and
the mean square error (MSE), we propose a single-layer iterative algorithm to
solve the NPC minimization problem with convergence guarantee. In each
iteration of the algorithm, the Lagrange dual decomposition method is used to
derive the structure of the optimal beam-vectors, which facilitates the
parallel computations at the Baseband unit (BBU) pool. Furthermore, a
bisection UE selection algorithm is proposed to guarantee the feasibility of
the problem. Simulation results show the benefits of the proposed algorithms
and the fact that a limited amount of CSI is sufficient to achieve performance
close to that obtained when perfect CSI is possessed.

##  I Introduction

The fifth-generation (5G) wireless system is expected to offer a thousand
times the throughput [1] of the current fourth-generation (4G) [2, 3, 4] and
provide ubiquitous service access for a large number of user equipments (UEs)
in hot spots such as shopping malls, stadia, etc. To achieve this goal,
heterogeneous and small cell network (HetSNet) is regarded as one of the most
promising techniques by exploiting spatial degrees of freedom through
deploying more and more access points (APs) [5]. However, since all APs reuse
the same frequency, the interference among the APs is a limiting factor [5],
which should be carefully managed. Dense cloud radio access network (C-RAN)
was proposed in [6] as one promising architecture to conquer this issue. In
dense C-RAN, all the base-band processing is performed at the BBU pool through
the recent development of cloud computing techniques [7], while the RRHs are
only responsible for simple radio transmission or reception [8, 9]. Due to
their simple functionality, RRHs can be densely deployed in the network with
low hardware cost. Due to the centralized architecture of dense C-RAN, the
multi-UE interference can be efficiently handled through joint signal
processing techniques such as coordinated multi-point (CoMP), leading to
significant performance gains. Although C-RAN has been introduced in 4G, it is
usually deployed in a large geographical area by connecting macrocell base
stations to the BBU pool through fronthaul links. This conventional C-RAN
incurs large delays on the fronthaul links due to long transmission distance
between RRHs and BBU pool [10], which will violate the stringent latency
requirement in 5G [1], i.e., a roundtrip latency within 1 ms. In contrast,
dense C-RAN studied in this paper is aimed to cover hot spots with much
smaller geographical area. Hence, delays can be significantly reduced.

However, there are many technical and deployment issues associated with dense
C-RAN. First, dense deployment of RRHs will require high power consumption if
all RRHs are activated even when the network traffic load is low. In addition,
if each RRH serves all UEs, significant power will be used on the fronthaul
links. As a result, how to activate the RRHs and select the RRHs for serving
each UE to minimize the total network power consumption (NPC) is a critical
issue. Second, in a dense C-RAN there will be a need for a large number of
fronthaul links, requiring them to be low cost. There may also be a need to
use millimeter wave (mmWave) technology for flexible and low cost deployment.
These cost considerations lead to the likelihood of a capacity constraint on
the fronthaul. Third, in dense C-RAN, the BBU pool will support large number
of RRHs and the number of optimization variables for beam-vectors will become
very large, which will incur high computational complexity and will become
unaffordable. Finally, the dense C-RAN requires more CSI for the facilitation
of CoMP transmission design, which will cause a heavy training overhead. The
amount of training overhead will increase with the number of RRHs and UEs, and
may counteract the cooperative gains provided by CoMP transmission [11]. The
most promising way to deal with this issue is to restrict the number of RRHs
that each UE should measure CSI to. The remaining CSI values can be regarded
as zeros, or only long term channel statistics of the remaining CSI, such as
path loss and shadowing, are considered. How to design transmission strategies
for this incomplete CSI case becomes an imperative task.

Most of current work only deals with parts of the above challenges. For
example, [12, 13, 14, 15] considered the joint RRH selection and beamforming
design to minimize the total NPC subject to UEs’ quality of service (QoS)
targets and per-RRH power constraints. These papers ignored the capacity
constraints on the fronthaul links and assumed that the fronthaul capacity is
unlimited. To address the fronthaul capacity constraints issue, [16]
investigated the problem of minimizing the number of data transfers on the
aggregated fronthaul links with UEs’ QoS constraints and power constraints on
each RRH. However, [16] did not explicitly impose the fronthaul capacity
constraints in the optimization problem. Recently, several papers have
addressed the case when the fronthaul capacity constraints are explicitly
imposed [17, 18, 19]. The case when the optimization problem is infeasible was
not considered. Then, some UEs can be removed to make the optimization problem
feasible again. The UE admission control and total NPC minimization were
jointly optimized in [20], where a single-stage optimization problem was
formulated by introducing a weighting factor in the admission control part.
Recently, [21] extended the work in [20] to multi-channel heterogeneous C-RAN
where the C-RAN is overlaid by a macro-cell. However, for the admission
control designs considered in [20] and [21], one has to carefully choose the
weighting factor associated with the admission control part to ensure that the
selected UEs can satisfy the QoS constraints, which is not easy.

However, the algorithms proposed in [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]
were based on the assumption of full CSI at the BBU pool, which is not
practical as explained. Unfortunately, the algorithms designed for perfect CSI
cannot be directly extended to the case of incomplete CSI. To the best of our
knowledge, only a few papers have considered the incomplete CSI case [22, 23,
24, 25]. [22] proposed a CSI reduction scheme named compressive CSI
acquisition, that can obtain the instantaneous CSIs for a subset of channel
links and the large scale fading gains of the others. Based on the incomplete
CSI, [22] solved a transmit power minimization problem while guaranteeing UEs’
QoS requirements by using a stochastic coordinated beamforming technique.
However, the method needs to solve a high-dimension semi-definite programming
(SDP) problem for each sample, and the number of samples increases with the
size of the network, which incurs an unacceptable complexity for dense C-RAN.
[23] focused on the beamforming algorithm to maximize the sum-rate for
arbitrary UE-centric clustering C-RAN. The “C-cluster method” was introduced
in [23] to reduce channel estimation overhead where only subsets of CSIs for
each UE are measured, and the other unavailable CSIs are regarded as zeros.
Recently, [24] proposed a conservative precoder design with the objective of
maximizing the weighted sum-rate of UEs for arbitrary UE-centric clustering
method with incomplete CSIs, where the long term channel statistic was
incorporated into the optimization. Finally, [25] designed a clustering scheme
maximizing the average net throughput of the dense C-RAN by taking the
training overhead into account. The scheme is based on a hybrid CoMP
transmission mode and operates under a long time duration that may be
performed at the medium access control (MAC) layer since only large-scale CSIs
are required. However, both the beam directions and power allocations were not
optimized in [25]. None of the papers [22, 23, 24, 25] considered the
fronthaul capacity constraints and were mainly focused on sum-rate
maximization problems without incorporating QoS requirements.

The aim of this paper is to provide a complete framework to jointly tackle the
above-mentioned challenges together. Specifically, we investigate the joint
optimization of RRH selection, RRH-UE associations and transmit beamforming to
minimize the NPC for downlink multi-channel C-RAN with incomplete CSI, subject
to fronthaul link capacity constraints, all UEs’ rate requirements and per-RRH
power constraints. The NPC is modeled as the sum of the RRH power consumption
and the fronthaul link power consumption. The low-power sleep mode is
considered in the RRH power consumption model, and the fronthaul link power
consumption is modeled as a linear function of fronthaul traffic. To reduce
the computational complexity, each UE is restricted to be served by its nearby
RRHs since only nearby RRHs contribute significantly to the UE’s signals.
Moreover, to reduce the channel measurement overhead, we introduce the subset
of RRHs that each UE should estimate the CSIs to, while the large-scale fading
(such as path-loss and shadowing) is assumed to be known for the other
unavailable CSI. In general, the candidate set of RRHs for serving UEs and the
CSI estimation set of RRHs for each UE are determined based on UEs’ locations
that may be the task of the upper-layer, which is beyond the scope of this
paper. The NPC minimization problem is an NP-hard mixed-integer non-linear
programming (MINLP) problem due to the indicator functions introduced in both
objective function and fronthaul capacity constraints, whose optimal solution
is intractable. In addition, due to the sum rate constraints and incomplete
CSI, the QoS constraints are non-convex and difficult to handle. Furthermore,
due to the conflicting constraints, the NPC minimization problem may be
infeasible and the initialization solution should be carefully selected. As a
result, the contributions of this paper can be summarized as follows:

  1. 1.

Due to the incomplete CSI, it is intractable to derive the exact closed-form
expression of the data rate for each UE, and thus stringent QoS requirements
for each UE are difficult to be guaranteed. To alleviate this difficulty, we
conservatively replace the data rate of each UE with its lower-bound
expression derived by using the Jensen’s inequality.

  2. 2.

To resolve the feasibility issue, we provide a low-complexity UE selection
algorithm based on bisection search method to maximize the number of admitted
UEs that can achieve their QoS targets, and its complexity only increases
logarithmically with the number of UEs. Simulation results show that this
algorithm can achieve marginal performance loss with respect to (w.r.t.) that
obtained by the exhaustive UE search algorithm with an exponential
computational complexity over the number of admitted UEs.

  3. 3.

Given the feasible set of UEs from the UE selection algorithm, we provide a
low-complexity single-layer iterative algorithm (i.e., Algorithm 1) to solve
the NPC minimization problem. Specifically, the non-smooth indicator function
is approximated as a non-convex function and the successive convex
approximation (SCA) technique [26] is adopted to approximate the non-convex
function as a series of convex functions. To deal with the non-convex QoS
constraints, we translate the technique in [27] that aimed at rate
maximization problem to the NPC minimization problem with rate expressions in
the constraints and incomplete CSI. The convergence of the iterative algorithm
is strictly proved.

  4. 4.

In each iteration of Algorithm 1, there is a subproblem that the beam-vectors
should be optimized. We derive the structure of the optimal beam-vectors by
employing the Lagrange dual decomposition method. Then, each beam-vector can
be obtained in parallel for each sub-channel (SC), which facilitates the
application of the cloud computing technique in BBU pool.

This paper is organized as follows. Section II presents the system model, and
Section III formulates the UE selection problem and NPC minimization problem
along with the complexity analysis. The single-layer iterative algorithm to
solve the NPC algorithm is given in Section IV when the UEs are selected to be
admitted. Then, in Section V, the low-complexity UE selection algorithm is
provided. Simulation results are presented in Section V to evaluate the
performance of the proposed algorithms. Finally, conclusions are drawn in
Section VII.

Notations: For a set ${\cal A}$, $\left|{\cal A}\right|$ denotes the
cardinality of ${\cal A}$, while for a complex number $x$, $\left|x\right|$
denotes the magnitude of $x$. $\bf{1}$ denotes a vector with all elements
equal to ones. ‘s.t.’ is short for ‘subject to’.
${{\mathbb{E}}_{\\{x\\}}}\\{y\\}$ means the expectation of $y$ over $x$. The
complex Gaussian distribution is denoted as ${\cal C}{\cal N}(\cdot,\cdot)$.
We use ${\mathbb{C}}$ to represent the complex set. The lower-case bold
letters denote vectors and upper-case bold letters denote matrices.
${\rm{blkdiag}}(\cdot)$ denotes the block diagonalization operation.

##  II System Model

###  II-A System model

Consider a downlink ultra-dense C-RAN, as shown in Fig. 1, consisting of $I$
RRHs and $K$ UEs111 Due to the simple functionalities of RRHs, the RRHs can be
densely deployed with low hardware cost, wherein the number of RRHs may even
be larger than that of UEs. Hence, the average distance between RRHs and UEs
can be significantly reduced. As a result, the transmission power of the RRHs
can also be reduced due to the decreased path loss. , where each RRH is
equipped with $M$ transmit antennas, and each UE has a single antenna. Denote
the set of RRHs and UEs as ${\cal I}=\left\\{{1,\cdots,I}\right\\}$ and
$\bar{\cal U}=\left\\{{1,\cdots,K}\right\\}$, respectively. Each RRH is
connected to the BBU pool through wireless (e.g. mmWave communication)
fronthaul links. The fronthaul links are represented by dark solid arrows in
Fig. 1. The BBU pool is assumed to have all UEs’ data and distributes each
UE’s data to a carefully selected set of RRHs through the fronthaul links. It
is assumed that all the RRHs send their received data using the Orthogonal
Frequency Division Multiple Access (OFDMA) technique and then cooperatively
transmit to the UEs.

Figure 1: Illustration of a C-RAN with thirteen RRHs and seven UEs. The RRHs
are connected to a BBU pool through wireless fronthaul links. In this
scenario, UE 7 is not selected for serving and the candidate sets of RRHs for
the selected UEs are given by ${{\cal I}_{1}}=\\{1,2,12\\}$, ${{\cal
I}_{2}}=\\{4,7,8\\}$, ${{\cal I}_{3}}=\\{6,9\\}$, ${{\cal
I}_{4}}=\\{8,10,11\\}$, ${{\cal I}_{5}}=\\{8,13\\}$ and ${{\cal
I}_{6}}=\\{10,12\\}$, respectively. The sets of UEs that are potentially
served by the RRHs are given by ${{{\cal U}}_{1}}=\\{1\\}$, ${{{\cal
U}}_{2}}=\\{1\\}$, ${{{\cal U}}_{3}}=\\{\emptyset\\}$, ${{{\cal
U}}_{4}}=\\{2\\}$, ${{{\cal U}}_{5}}=\\{\emptyset\\}$, ${{{\cal
U}}_{6}}=\\{3\\}$, ${{{\cal U}}_{7}}=\\{2\\}$, ${{{\cal U}}_{8}}=\\{2,4,5\\}$,
${{{\cal U}}_{9}}=\\{3\\}$, ${{{\cal U}}_{10}}=\\{4,6\\}$, ${{{\cal
U}}_{11}}=\\{4\\}$, ${{{\cal U}}_{12}}=\\{1,6\\}$ and ${{{\cal
U}}_{13}}=\\{5\\}$, respectively. The sets of RRHs for coordinating the
interference for the selected UEs are given by ${\cal C}{{}_{1}}=\\{10,11\\}$,
${\cal C}{{}_{2}}=\\{9,11\\}$, ${\cal C}{{}_{3}}=\\{7\\}$, ${\cal
C}{{}_{4}}=\\{13\\}$, ${\cal C}{{}_{5}}=\\{9\\}$ and ${\cal
C}{{}_{6}}=\\{13\\}$, respectively. The sets of coordinated UEs by the RRHs
are given by ${{{\cal T}}_{7}}=\\{3\\}$, ${{{\cal T}}_{9}}=\\{2,5\\}$,
${{{\cal T}}_{10}}=\\{1\\}$, ${{{\cal T}}_{11}}=\\{1,2\\}$ and ${{{\cal
T}}_{13}}=\\{4,6\\}$, respectively.

Denote ${{\cal U}}\subseteq\bar{\cal U}$ as the subset of UEs that are
admitted in the C-RAN. To reduce the computational complexity of the large
network, it is assumed that each UE $k\in{\cal U}$ can only be served by its
nearby RRHs since only nearby RRHs contribute significantly to the UE’s signal
quality due to the severe path loss. Denote ${{\cal I}_{k}}\subseteq{\cal I}$
and ${{{\cal U}}_{i}}\subseteq{{\cal U}}$ as the candidate set of RRHs that
potentially serve UE $k$ and the set of UEs that can be potentially served by
RRH $i$, respectively. The transmission links from the RRHs in ${{\cal
I}_{k}}$ to UE $k$ are called the candidate serving links, which are
represented in red solid arrows in Fig. 1. In this paper, it is assumed that
${{\cal I}_{k}}$ and ${{{\cal U}}_{i}}$ are predetermined by some well-known
user-centric cluster methods [28, 29, 25] determined by the MAC layer222In
general, the cluster method is mainly determined based on the large-scale CSI,
which is usually performed in the upper layer such as MAC layer. In some hot
spots such as stadia and shopping malls, the users move slowly. Hence, the
cluster can be kept fixed for a long time compared with the instantaneous CSI.
This paper only focuses on the beam-vectors at the physical layer, and how to
design the optimal cluster method is beyond the scope of this paper.. Please
refer to [30] for a survey on user-centric cluster methods. Note that since no
restrictions are placed on ${{\cal I}_{k}}$, they can overlap with each other,
i.e., there may exist two different UEs $k$ and $k^{\prime}$ that ${{\cal
I}_{k}}\cap{{\cal I}_{k^{\prime}}}\neq\emptyset$, for $\forall
k,k^{\prime}\in{{\cal U}}$. Moreover, the other-cluster interference due to
overlapping coverage can be effectively handled under this user-centric
cluster method. For example, UE 4 and UE 5 have one common serving RRH 8.
Hence, RRH 8 will transmit useful signals to both UE 4 and UE 5, rather than
only interference signals. In addition, the BBU pool has the CSI knowledge
from RRH 3 to UE 4. Thus, the interference from RRH 3 to UE 4 will be
carefully controlled when RRH 3 is serving UE 5. In contrast to the non-
cooperative optimization where each cluster selfishly optimizes its own
performance without considering its impact on the other clusters, in dense
C-RAN all the signal processing operation is performed at the BBU pool, where
the interference among different clusters can be centrally mitigated by
resorting to the powerful cloud computing tool.

Denote the set of available sub-channels (SCs) as ${\cal
N}=\\{1,2,\cdots,N\\}$, where $N$ is the total number of SCs. To maximize the
spectral efficiency, it is assumed that universal frequency reuse is adopted
and the multiuser interference can be efficiently handled by the beamforming
technique. Denoting ${\bf{w}}_{i,k}^{(n)}\in{{\mathbb{C}}^{M\times 1}}$ as the
beam-vector at RRH $i$ for UE $k$ on SC $n$, the transmitted signal of RRH $i$
on SC $n$ is

${\bf{x}}_{i}^{(n)}=\sum\nolimits_{k\in{{\cal
U}_{i}}}{{\bf{w}}_{i,k}^{(n)}s_{k}^{(n)}},$ (1)

where ${s_{k}^{(n)}}$ is the data symbol for UE $k$ on SC $n$. Without loss of
generality, it is assumed that ${{\mathbb{E}}}\\{|s_{k}^{(n)}{|^{2}}\\}=1$ and
${{\mathbb{E}}}\\{s_{{k_{1}}}^{({n_{1}})}s_{{k_{2}}}^{({n_{2}})}\\}=0$ for
$({n_{1}},{k_{1}})\neq({n_{2}},{k_{2}}),\forall{n_{1}},{n_{2}}\in{\cal
N},\forall{k_{1}},{k_{2}}\in{\cal U}$. The baseband received signal at UE $k$
on SC $n$ is given by

$y_{k}^{(n)}=\underbrace{\sum\nolimits_{i\in{{\cal
I}_{k}}}{{\bf{h}}_{i,k}^{(n)}{\bf{w}}_{i,k}^{(n)}s_{k}^{(n)}}}_{{\rm{desired\
signal}}}+\underbrace{\sum\nolimits_{l\neq k,l\in{\cal
U}}{\sum\nolimits_{i\in{{\cal
I}_{l}}}{{\bf{h}}_{i,k}^{(n)}{\bf{w}}_{i,l}^{(n)}s_{l}^{(n)}}}}_{{\rm{interference}}}+z_{k}^{(n)},$
(2)

where ${\bf{h}}_{i,k}^{(n)}\in{{\mathbb{C}}^{1\times M}}$ is the channel
vector from RRH $i$ to UE $k$ on SC $n$, and $z_{k}^{(n)}$ is the additive
complex white Gaussian noise following the distribution of ${\cal C}{\cal
N}(0,\sigma_{k}^{2})$. The channel vector ${\bf{h}}_{i,k}^{(n)}$ can be
written as
${\bf{h}}_{i,k}^{(n)}=\alpha_{i,k}^{(n)}{\bf{\tilde{h}}}_{i,k}^{(n)}$, where
$\alpha_{i,k}^{(n)}$ denotes the large-scale channel gain that includes the
path loss and shadowing, and ${\bf{\tilde{h}}}_{i,k}^{(n)}$ denotes the small-
scale fading vector, where all elements are dependent of each other and each
one has zero mean and unit variance.

For the sake of reduced complexity of decoding at the receivers, we do not
consider the joint decoding of the interfering signals and the multiuser
interference is simply regarded as noise at the receivers. In addition,
coherent joint transmission333This assumption is valid for dense C-RAN. The
reason is that dense C-RAN is usually deployed in hot spots with smaller
coverage area compared with that of the conventional C-RAN that covers
multiple macrocells [10]. Hence, different transmission delays due to
different transmission distances between RRHs and BBU pool can be ignored.
Then, both the synchronization and coherent joint transmission are possible.
is assumed as in most of existing papers [12, 13, 14, 15, 16, 17, 18, 19, 20,
21]. Then, the SINR at UE $k$ on SC $n$ can be obtained from (2) as

$\gamma_{k}^{(n)}({\bf{w}})=\frac{{{{\left|{\sum\nolimits_{i\in{{\cal
I}_{k}}}{{\bf{h}}_{i,k}^{(n)}{\bf{w}}_{i,k}^{(n)}}}\right|}^{2}}}}{{\sum\nolimits_{l\neq
k,l\in{\cal U}}{{{\left|{\sum\nolimits_{i\in{{\cal
I}_{l}}}{{\bf{h}}_{i,k}^{(n)}{\bf{w}}_{i,l}^{(n)}}}\right|}^{2}}}+\sigma_{k}^{2}}}.$
(3)

where ${\bf{w}}$ denotes the collection of all beam-vectors.

As seen in (3), to design the beam-vectors for all UEs, the overall CSI of all
UEs is required. However, it is a formidable task to obtain all CSI for the
dense C-RAN due to the limited training resources. To handle this difficulty,
we introduce the set ${\widetilde{\cal I}_{k}}\supseteq{{\cal I}_{k}}$ for
each UE $k$ that is defined as the set of RRHs that UE $k$ needs to measure
CSI from. Also, we define ${\widetilde{\cal U}_{i}}\supseteq{{\cal U}_{i}}$
for each RRH $i$ as the set of UEs that each RRH $i$ knows the CSI to. In
general, ${\widetilde{\cal I}_{k}}$ are the set of UE $k$’s nearby RRHs and
${\widetilde{\cal U}_{i}}$ are the set of RRH $i$’s nearby UEs. Note that at
least the CSI from all RRHs in ${{\cal I}_{k}}$ is required for cooperative
transmission design. The other CSI from RRHs in ${\cal
C}{{}_{k}}={\widetilde{\cal I}_{k}}\backslash{{\cal I}_{k}}$ to UE $k$ is used
to coordinate the interference, and the links from RRHs in ${\cal C}{{}_{k}}$
are called coordinated interference links, which are shown by blue dashed
arrows in Fig. 1. Also, the UEs in ${\cal T}_{i}={\widetilde{\cal
U}_{i}}\backslash{{\cal U}_{i}}$ are called RRH $i$’s coordinated UEs. For the
CSI from RRHs in ${\cal I}\backslash{\widetilde{\cal I}_{k}}$ to UE $k$, it is
assumed that the BBU pool only knows the large scale gains
$\\{\alpha_{i,k}^{(n)},\forall i\in{\cal I}\backslash{\widetilde{\cal
I}_{k}},k\in{\cal U},n\in\cal N\\}$. This is possible because the large scale
gains change much more slowly than the small-scale fading.

Since the CSI in ${\cal I}\backslash{\widetilde{\cal I}_{k}}$ is unknown, we
consider the following data rate for UE $k$ on SC $n$ (bit/s/Hz) [31]

${\bar{r}}_{k}^{(n)}({\bf{w}})={{\mathbb{E}}_{\left\\{{{\bf{h}}_{i,k}^{(n)},i\in{\cal
I}\backslash{{\widetilde{\cal
I}}_{k}}}\right\\}}}\left\\{{{{\log}_{2}}(1+\gamma_{k}^{(n)}({\bf{w}}))}\right\\}.$
(4)

where the expectation operator is performed over the fast fading of the
unknown CSI in ${\cal I}\backslash{\widetilde{\cal I}_{k}}$. Each UE $k$’s
total data rate should be larger than the minimum rate requirement
${R_{k,\min}}$:

${\rm{C1:}}\ {\bar{r}}_{k,\rm{tot}}({\bf{w}})=\sum\nolimits_{n\in{\cal
N}}{{\bar{r}}_{k}^{(n)}({\bf{w}})}\geq{R_{k,\min}},\forall k\in{\cal U}.$ (5)

In each fronthaul link, the maximum capacity that can be supported is limited.
Hence, the following fronthaul capacity constraint follows:

${\rm{C2}:}\ \sum\nolimits_{k\in{\cal
U}_{i}}{\varepsilon\left(P_{i,k}^{{\rm{tr}}}({\bf{w}})\right){{\bar{r}}_{k,{\rm{tot}}}({\bf{w}})}}\leq{C_{i,\max}},\forall
i\in{\cal I},$ (6)

where $\varepsilon\left(\cdot\right)$ is an indicator function, defined as

$\varepsilon\left(x\right)=\left\\{{\begin{array}[]{*{20}{l}}{1,\;{\rm{if}}\;x\neq
0,}\\\ {0,\;{\rm{otherwise}},}\end{array}}\right.$ (7)

$P_{i,k}^{{\rm{tr}}}({\bf{w}})=\sum\nolimits_{n\in{\cal
N}}{{{\left\|{{\bf{w}}_{i,k}^{(n)}}\right\|}^{2}}}$ denotes the total
transmission power from RRH $i$ to UE $k$, $C_{i,\max}$ is the maximum
capacity that can be supported by the $i$th fronthaul link.

###  II-B Network power consumption model

In this subsection, a practical NPC model is provided that consists of two
parts: power consumption at the RRHs and power consumption on the fronthaul
links.

As in [32], the power consumption of RRH $i$ can be modeled as a piecewise
linear function of the transmit power at RRH $i$:

$P_{i}^{{\rm{rrh}}}({\bf{w}})=\left\\{\begin{array}[]{l}{{{\eta_{i}}}}P_{i}^{{\rm{tr}}}({\bf{w}})+P_{i}^{{\rm{active}}},\
{\rm{if}}\ P_{i}^{{\rm{tr}}}({\bf{w}})>0\\\
P_{i}^{{\rm{sleep}}},\qquad\qquad\quad\ \ {\rm{if}}\
P_{i}^{{\rm{tr}}}({\bf{w}})=0\end{array}\right.$ (8)

where $\eta_{i}>1$ is the constant accounting for the efficiency of the power
amplifier of RRH $i$, $P_{i}^{{\rm{tr}}}({\bf{w}})$ is the total transmit
power at RRH $i$ that should be no larger than ${P_{i,\max}}$, i.e.,

${\rm{C3}}:\ P_{i}^{{\rm{tr}}}({\bf{w}}){\rm{=}}\sum\nolimits_{k\in{{\cal
U}_{i}}}{P_{i,k}^{{\rm{tr}}}({\bf{w}})}\leq{P_{i,\max}},i\in{\cal I},$ (9)

$P_{i}^{{\rm{active}}}$ and $P_{i}^{{\rm{sleep}}}$ represent the circuit power
consumption when RRH $i$ is in active mode and sleep mode, respectively. In
general, $P_{i}^{{\rm{active}}}$ is much larger than $P_{i}^{{\rm{sleep}}}$,
which motivates us strategically to switch off the RRHs to save power in case
of very low traffic.

Fronthaul power consumption model is critical for the optimization of NPC. In
[12] and [13], the fronthaul power consumption was simply modeled as a step
function, with a larger constant value for active mode and smaller one for
sleep mode. In [33], the fronthaul power consumption is modeled to be
proportional to the number of UEs that each one supports. However, these
papers did not take into account the effect of data rate transmitting on each
fronthaul link. Intuitively, to support high fronthaul transmit data rate,
more power should be consumed on the fronthaul links. Compared with [12, 13,
33], we go one step further by modeling the power consumption of each
fronthaul link to be proportional to the total fronthaul transmit data rate as
in [34]:

$P_{i}^{{\rm{fr}}}({\bf{w}})={\rho_{i}}\sum\nolimits_{k\in{\cal
U}_{i}}{\varepsilon\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right){{\bar{r}}_{k,{\rm{tot}}}({\bf{w}})}},$
(10)

where ${\rho_{i}}$ is a constant scaling factor 444In general, this scaling
factor may not be a constant, rather depend on the total transmit data rate on
the fronthaul link. However, how to accurately model this relationship is
still under investigation. To the best of our knowledge, only [34] provided
the detailed study of this model that has been adopted by the existing work
such as [14]..

Based on the above analysis and with some simple manipulations, the NPC is
modeled as

$\displaystyle{P_{{\rm{NPC}}}}({\bf{w}})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum\limits_{i\in{{\cal
I}}}{\left\\{{P_{i}^{{\rm{rrh}}}({\bf{w}})+P_{i}^{{\rm{fr}}}}({\bf{w}})\right\\}}$
(11) $\displaystyle=$ $\displaystyle\\!\\!\\!\sum\limits_{i\in{{\cal
I}}}{\left\\{{{{{{\eta_{i}}}}{P_{i}^{{\rm{tr}}}({\bf{w}})}}\\!+\\!\varepsilon\left(P_{i}^{{\rm{tr}}}({\bf{w}})\right)P_{i}^{\rm{c}}\\!\\!+{\rho_{i}}\sum\limits_{k\in{\cal
U}_{i}}{\varepsilon\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right){{\bar{r}}_{k,{\rm{tot}}}({\bf{w}})}}}\right\\}}\\!+\\!\sum\limits_{i\in{{\cal
I}}}{P_{i}^{{\rm{sleep}}}},$

where ${P_{i}^{{\rm{tr}}}}({\bf{w}})$ is given in (9),
$P_{i}^{\rm{c}}=P_{i}^{{\rm{active}}}-P_{i}^{{\rm{sleep}}},\forall i\in{\cal
I}$.

##  III Problem Formulation and Analysis

Based on the above system model, we formulate the user selection problem and
the NPC minimization problem in a two-stage form. Then, we provide the
complexity analysis for the formulated problems.

###  III-A Problem Formulation

Due to the limited fronthaul capacity constraints C2 in (6) and the power
constraints C3 in (9), the system may not be able to support all UEs with
their rate requirements of C1 in (5). Hence, some UEs may be dropped or
rescheduled in other orthogonal time slots to make the optimization problem
feasible. As a result, we may consider a two-stage optimization problem. In
the first stage, one should find the largest subsets of UEs that can be
supported by the system555Dense C-RAN is usually deployed in hot spots (such
as shopping mall, stadia, et al.) where the number of UEs is huge, and the
amount of available communication resource is limited. Hence, maximizing the
number of admitted users for each time slot should be placed in higher
priority. In some other scenarios, where there are abundant wireless resource,
maximizing the number of admitted UEs in each time slot may not be a good
option in reducing NPC, and dynamically scheduling the UE in different time
slots may further reduce NPC, which will be left for future work. , while in
the second stage, one should optimize the corresponding beam-vectors to
minimize $P_{{\rm{NPC}}}$ with the selected subset of UEs obtained from the
first stage.

As a result, the optimization problem at the first stage is formulated as

$\begin{array}[]{l}{\cal P}_{1}:\ \mathop{\max}\limits_{{\bf{w}},{\cal
U}\subseteq{\overline{\cal U}}}\quad\left|\cal U\right|\\\ \qquad\
{\rm{s}}.{\rm{t}}.\qquad{\kern
1.0pt}{\rm{C1}},{\rm{C2}},{\rm{C3}}.\end{array}$ (12)

Denote ${{\cal U}^{\star}}$ as the solution from Stage I and the corresponding
${{{\cal U}}_{i}}$ becomes ${{{\cal U}}_{i}^{\star}}$. Then, the optimization
problem at the second stage is formulated as

$\displaystyle{\cal P}_{2}:\ \mathop{\min}\limits_{{\bf{w}}}\quad$
$\displaystyle\sum\nolimits_{i\in{{\cal
I}}}{\left\\{{{{{{\eta_{i}}}}{P_{i}^{{\rm{tr}}}({\bf{w}})}}\\!+\\!\varepsilon\left(P_{i}^{{\rm{tr}}}({\bf{w}})\right)P_{i}^{\rm{c}}\\!\\!+{\rho_{i}}\sum\nolimits_{k\in{\cal
U}_{i}^{\star}}{\varepsilon\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right){{\bar{r}}_{k,{\rm{tot}}}({\bf{w}})}}}\right\\}}$
(13a) s.t. $\displaystyle{\rm{C1}},{\rm{C2}},{\rm{C3}}$ (13b)

In the constraints C1, C2, and C3, ${{\cal U}}$ and ${{{\cal U}}_{i}}$ are
replaced by ${{\cal U}^{\star}}$ and ${{{\cal U}}_{i}^{\star}}$, respectively.
Note that the constant term $\sum\nolimits_{i\in{{\cal
I}}}{P_{i}^{{\rm{sleep}}}}$ in (11) has been omitted in the objective function
(13a).

We emphasize that the aim of Stage I is to find the maximum number of admitted
UEs with feasible beam-vectors. These obtained beam-vectors are not guaranteed
to be optimal in terms of NPC. Hence, we need to perform Stage II to optimize
the beam-vectors to reduce the NPC. The beam-vectors obtained from Stage I
will be a feasible initial input that is required by the algorithm developed
in Stage II.

The incomplete CSI at the BBU pool makes the design of beam-vectors very
difficult to solve and the expression for the data rate is difficult to
derive. In the following, we consider its lower-bound and replace the data
rate with its lower-bound, which makes the optimization problem more
tractable.

We first simplify the SINR expression in (3). The beam-vectors for each UE on
each SC $n$ are merged into a single large-dimension vector
${{{\bf{\bar{w}}}}}_{k}^{(n)}={[{{\bf{w}}_{i,k}^{(n){\rm{H}}},\forall
i\in{{\cal I}_{k}}}]^{\rm{H}}}\in{{\mathbb{C}}^{\left|{{{\cal
I}_{k}}}\right|M\times 1}},\forall n\in\cal N$. Then, we define a set of new
channel vectors ${\bf{\bar{h}}}_{l,k}^{(n)}=[{{\bf{h}}_{i,k}^{(n)},\forall
i\in{{\cal I}_{l}}}]\in{{\mathbb{C}}^{1\times\left|{{{\cal
I}_{l}}}\right|M}}$, representing the aggregated CSI from the RRHs in ${\cal
I}_{l}$ to UE $k$ on SC $n$. The SINR expression in (3) can be rewritten as

$\gamma_{k}^{(n)}{({\bf{w}})}=\frac{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}}{{\sum\nolimits_{l\neq
k,l\in{\cal
U}}{{{\left|{{\bf{\bar{h}}}_{l,k}^{(n)}{\bf{\bar{w}}}_{l}^{(n)}}\right|}^{2}}}+\sigma_{k}^{2}}}.$
(14)

Note that ${{\bf{\bar{h}}}_{k,k}^{(n)}}$ is perfectly known in the BBU pool
according to the previous assumption, and only the denominator in (14)
contains the uncertain terms. However, it is difficult to obtain the accurate
rate expression. To deal with this challenge, we consider its lower-bound with
more tractable form. Specifically, since
${\log_{2}}\left({1+{a\mathord{\left/{\vphantom{1x}}\right.\kern-1.2pt}x}}\right)$
is a convex function for any positive $a$, by using Jensen’s inequality [35],
the lower bound of the data rate in (4) can be derived as

$\displaystyle\bar{r}_{k}^{(n)}{({\bf{w}})}$ (15) $\displaystyle\geq$
$\displaystyle{\log_{2}}\left({1+\frac{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}}{{{{\mathbb{E}}_{\left\\{{{\bf{h}}_{i,k}^{(n)},i\in{\cal
I}\backslash{{\tilde{\cal I}}_{k}}}\right\\}}}\left\\{{\sum\nolimits_{l\neq
k,l\in{\cal
U}}{{{\left|{{\bf{\bar{h}}}_{l,k}^{(n)}{\bf{\bar{w}}}_{l}^{(n)}}\right|}^{2}}}}\right\\}+\sigma_{k}^{2}}}}\right)$
(16) $\displaystyle=$
$\displaystyle{\log_{2}}\left({1+\frac{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}}{{\sum\nolimits_{l\neq
k,l\in{\cal
U}}{{\bf{\bar{w}}}_{l}^{(n){\rm{H}}}{\bf{A}}_{l,k}^{(n)}{\bf{\bar{w}}}_{l}^{(n)}}+\sigma_{k}^{2}}}}\right)$
(17) $\displaystyle\buildrel\Delta\over{=}$
$\displaystyle\tilde{r}_{k}^{(n)}{({\bf{w}})}$ (18)

where
${\bf{A}}_{l,k}^{(n)}={{\mathbb{E}}_{\left\\{{{\bf{h}}_{i,k}^{(n)},i\in{\cal
I}\backslash{{\tilde{\cal
I}}_{k}}}\right\\}}}\left\\{{{\bf{\bar{h}}}{{{}_{l,k}^{(n)}}^{\rm{H}}}{\bf{\bar{h}}}_{l,k}^{(n)}}\right\\}\in{\mathbb{C}}^{M{|{\cal
I}_{l}|}\times M{|{\cal I}_{l}|}}$. To obtain the closed-form expression of
${\bf{A}}_{l,k}^{(n)}$, we define the indices of ${{{\cal I}_{l}}}$ as ${{\cal
I}_{l}}=\\{s_{1}^{l},\cdots,s_{|{{\cal I}_{l}}|}^{l}\\}$. Then, we have

${\bf{A}}_{l,k}^{(n)}=\left[{\begin{array}[]{*{20}{c}}{{{\left({{\bf{A}}_{l,k}^{(n)}}\right)}_{1,1}}}&\cdots&{{{\left({{\bf{A}}_{l,k}^{(n)}}\right)}_{1,|{{\cal
I}_{l}}|}}}\\\ \vdots&\ddots&\vdots\\\
{{{\left({{\bf{A}}_{l,k}^{(n)}}\right)}_{|{{\cal
I}_{l}}|,1}}}&\cdots&{{{\left({{\bf{A}}_{l,k}^{(n)}}\right)}_{|{{\cal
I}_{l}}|,|{{\cal I}_{l}}|}}}\end{array}}\right],l\neq k$ (19)

where ${\left({{\bf{A}}_{l,k}^{(n)}}\right)_{i,j}}\in{{\mathbb{C}}^{M\times
M}},i,j\in 1,\cdots,|{{\cal I}_{l}}|$ is the block matrix of
${\bf{A}}_{l,k}^{(n)}$ at the $i$th row and $j$th column, given by

${\left({{\bf{A}}_{l,k}^{(n)}}\right)_{i,j}}=\left\\{\begin{array}[]{l}{\bf{h}}_{s_{i}^{l},k}^{(n){\rm{H}}}{\bf{h}}_{s_{j}^{l},k}^{(n)},\
\ \ \ \ {\rm{if}}\ s_{i}^{l},s_{j}^{l}\in{{\tilde{\cal I}}_{k}},\\\
\left|\alpha_{s_{i}^{l},k}^{(n)}\right|^{2}{{\bf{I}}_{M\times M}},\ {\rm{if}}\
s_{i}^{l},s_{j}^{l}\notin{{\tilde{\cal I}}_{k}},{\rm{and}}\ i=j,\\\
{{\bf{0}}_{M\times M,}}\qquad\ \ \ \ {\rm{otherwise.}}\end{array}\right.$ (20)

It can be easily verified that ${\bf{A}}_{l,k}^{(n)}$ is a positive definite
matrix. Note that the derivations of matrix ${\bf{A}}_{l,k}^{(n)}$ place no
restrictions on the channel distributions and only large-scale channel gains
are required. Hence, the following developed algorithms are applicable for any
channel distributions, such as Rayleigh fading, Ricean channels, Nakagami-$m$
fading channels, et al.

Figure 2: Non-overlapped cluster topology.

Figure 3: Data rate versus the transmit power for the special case.

We now start to check the tightness of this rate lower-bound. It is difficult
to derive the accurate data rate expression for general case. Instead, in
Appendix A, we derive the accurate closed-form expression of data rate for one
special case under three assumptions: 1) The RRH serving cluster is the same
as the CSI cluster for each UE: ${\cal I}_{k}={\widetilde{\cal I}_{k}}$; 2)
The RRH serving cluster for each UE is non-overlapped with each other: ${{\cal
I}_{k}}\cap{{\cal I}_{k^{\prime}}}=\emptyset,\forall k,k^{\prime}\in\cal U$;
3) The small-scale fading vector ${\bf{\tilde{h}}}_{i,k}$ follows the
distribution of ${\cal C}{\cal N}({\bf{0}},{\bf{I}})$ for $\forall i,k$. We
consider one non-overlapped C-RAN scenario deployed within a square area of
coordinates $[-D/2,D/2]\times[-D/2,D/2]$ km as shown in Fig. 3. This network
area is divided into nine $D/3\ {\rm{km}}\times D/3\ {\rm{km}}$ squares. In
each square, one UE is located at the center point and three RRHs are randomly
generated in this square to exclusively serve this UE. For simplicity, only
one SC is considered. The other simulation parameters are the same as in the
simulation Section. It is assumed that each RRH transmits at their maximum
power and the beam direction is chosen to be channel direction. The values of
$D=3$ and $D=1$ are tested, which correspond to sparse and dense scenarios,
respectively. Only UE 5 is considered. Fig. 3 plots three kinds of curves for
comparison: one is the lower bound of data rate derived in (18), one is the
accurate closed-form data rate expression derived in (A.4) in Appendix A, and
the last one is the Monte-Carlo simulations. It is seen from Fig. 3 that the
curve of the closed-form expression coincides with that of the Monte-Carlo
simulations, which verifies the correctness of the derivations. Furthermore,
for the sparse scenarios, when the transmit power is low,
$P_{\rm{max}}<-20{\rm{dB}}$, the lower-bound is quite tight. With the increase
of transmit power, the gap increases and becomes a constant in the high
transmit power regime. Note that only roughly 3% data rate loss will be
incurred when using the lower-bound compared with the accurate data rate,
which is negligible. On the other hand, for the dense scenario, the C-RAN
becomes interference limited and the data rate remains fixed for all ranges of
the transmit power as expected. It is again observed that the gap between the
lower-bound and the exact value is small. Hence, considering the complicated
data rate expression in (A.4) in Appendix A, our derived lower-bound
expression in (18) is much easier to handle and more suitable for algorithm
design.

By replacing the data rate $\bar{r}_{k}^{(n)}$ in Problems ${\cal P}_{1}$ and
${\cal P}_{2}$ with its lower-bound $\tilde{r}_{k}^{(n)}$ given in (18) and
considering the fact that the minimum rate constraints are met with equality
at the optimal point, Problems ${\cal P}_{1}$ and ${\cal P}_{2}$ can be
transformed as

$\displaystyle{\cal P}_{3}:\ \mathop{\max}\limits_{{\bf{w}},{\cal
U}\subseteq{\overline{\cal U}}}\quad$ $\displaystyle\left|\cal U\right|$ (21a)
s.t. $\displaystyle{\rm{C3}},{\rm{C4:}}\ \sum\nolimits_{n\in{\cal
N}}{{\tilde{r}}_{k}^{(n)}{({\bf{w}})}}\geq{R_{k,\min}},\forall k\in{\cal U},$
(21b) $\displaystyle{\rm{C5:}}\ \sum\nolimits_{k\in{\cal
U}_{i}}{\varepsilon\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right){R_{k,\min}}}\leq{C_{i,\max}},\forall
i\in{\cal I},$ (21c)

and

$\displaystyle{\cal P}_{4}:\ \mathop{\min}\limits_{{\bf{w}}}\quad$
$\displaystyle{{\tilde{P}}_{{\rm{tot}}}}({\bf{w}})\triangleq\sum\limits_{i\in{{\cal
I}}}{\left\\{{{{{{\eta_{i}}}}{P_{i}^{{\rm{tr}}}({\bf{w}})}}\\!+\\!\varepsilon\left(P_{i}^{{\rm{tr}}}({\bf{w}})\right)P_{i}^{\rm{c}}\\!\\!+{\rho_{i}}\sum\limits_{k\in{\cal
U}_{i}^{\star}}{\varepsilon\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right)R_{k,\min}}}\right\\}}$
(22a) s.t. $\displaystyle{\rm{C3}},{\rm{C4}},{\rm{C5}},$

respectively.

In the following, we focus on Problems ${\cal P}_{3}$ and ${\cal P}_{4}$.

###  III-B Problem Analysis

By adopting the user-centric clustering method in Section II, the number of
optimization variables in Problems ${\cal P}_{3}$ and ${\cal P}_{4}$ has been
reduced from $NMI\left|{\cal U}\right|$ in fully cooperative transmission
scheme to $NM\sum\nolimits_{k\in{\cal U}}{\left|{{{\cal I}_{k}}}\right|}$
here. By appropriately setting the cluster sizes, the reduced number of
variables $\left({NM\left({I\left|{\cal
U}\right|{\rm{-}}\sum\nolimits_{k\in{\cal U}}{\left|{{{\cal
I}_{k}}}\right|}}\right)}\right)$ may be very large, which significantly
reduces the computational complexity. In addition, some redundant constraints
can be removed, which can additionally reduce the computational complexity.
For example, in Fig. 1, RRH 3 and RRH 5 are not in any UE’s candidate serving
set, and thus the power constraints associated with RRH 3 and RRH 5 in C3 can
be removed. Moreover, if each link supports at most two UEs, then only link 8
(i.e., RRH 8) should be imposed with the fronthaul capacity constraints.
Hence, by employing the user-centric clustering with limited cooperation, the
computational complexity can be reduced significantly.

However, Problems ${\cal P}_{3}$ and ${\cal P}_{4}$ are still difficult to
solve due to the following reasons. Both the objective functions and
constraint C5 contain the non-smooth and non-differential indicator function
or (and) continuous variables, which are usually named as an MINLP problem.
Although the generalized Benders decomposition method [18, 36] is effective in
solving this kind of problems, it is very difficult to directly apply this
method to Problems ${\cal P}_{3}$ and ${\cal P}_{4}$ due to the non-convex sum
data rate constraints over all multiple SCs. An exhaustive search method can
be applied to solve Problems ${\cal P}_{3}$ and ${\cal P}_{4}$. Specifically,
to solve Problem ${\cal P}_{3}$, one should check whether Problem ${\cal
P}_{3}$ is feasible or not for each given user set $\cal U$ and each given set
of UE-RRH associations. This requires $O\left({{2^{\left|{\cal
U}\right|+\left|{\cal U}\right|I}}}\right)$ operations, which will become
prohibitive for large values of $\left|{\cal U}\right|$ and $I$. In addition,
even given the selected UE set ${\cal U}$ and the set of UE-RRH associations,
it is still difficult to check the feasibility since constraint C4 is non-
convex. Moreover, for dense C-RAN, the complexity associated with the
exhaustive search method is unaffordable for BBU pool. Similar difficulties
hold for Problem ${\cal P}_{4}$.

In the next section, we first deal with NPC minimization Problem ${\cal
P}_{4}$ by assuming that the UEs have been selected with feasible beam-
vectors, then one low-complexity UE selection algorithm to deal with Problem
${\cal P}_{3}$ is provided in Section V.

##  IV Low-complexity Algorithm to deal with Problem ${\cal P}_{4}$

In this section, we propose a low-complexity algorithm to solve Problem ${\cal
P}_{4}$ when UEs have been selected by using the UE selection algorithms in
Section V, and denote the selected subset of UEs as $\cal U$. As analyzed in
Section III-B, there are two difficulties to solve Problem ${\cal P}_{4}$: one
is the non-convex sum data rate constraint C4 and the other one is the non-
smooth indicator function.

To deal with the first difficulty, we resort to the relationship between the
data rate and weighted mean square error (MSE). In [27], the authors
considered the sum rate maximization problem by showing that maximizing the
sum rate is equivalent to minimizing the weighted MSE. Unfortunately, there
are two hurdles that preclude the direct application of the technique in [27]:
First, [27] considered the multiple-antenna UEs with perfect CSI. When each UE
has only one antenna with perfect CSI, the rank of channel covariance matrices
will be equal to one, i.e.,
${\rm{rank}}\left({{\bf{\bar{h}}}{{{}_{l,k}^{(n)}}^{\rm{H}}}{\bf{\bar{h}}}_{l,k}^{(n)}}\right)=1,\forall
l,k,n$. However, for the incomplete CSI considered in this paper, the rank of
channel covariance matrix may be larger than 1 according to (19), i.e.,
${\rm{rank}}\left({{\bf{A}}_{l,k}^{(n)}}\right)>1,\forall n,l,k$. Second, in
[27], the rate expression is in the objective function, while the rate
expressions are in the constraints here.

To resolve the first hurdle, we construct an auxiliary signal transmission
model by decomposing each interfering UE into multiple interfering sources.
Specifically, for each UE $k$ on SC $n$, since ${{\bf{A}}_{l,k}^{(n)}},\forall
l\neq k$, are positive definite matrices, they can be decomposed as

${\bf{A}}_{l,k}^{(n)}={\bf{V}}_{l,k}^{(n)}{\bf{V}}_{l,k}^{(n){\rm{H}}},$ (23)

where
${\bf{V}}_{l,k}^{(n)}=\left[{{\bf{v}}_{l,k,1}^{(n)},\cdots,{\bf{v}}_{l,k,d_{l,k}^{(n)}}^{(n)}}\right],\forall
l\neq k$, with $d_{l,k}^{(n)}$ being the rank of ${{\bf{A}}_{l,k}^{(n)}}$.
Then, we construct the following auxiliary signal transmission model for UE
$k$

${\tilde{y}}_{k}^{(n)}={\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}{\tilde{s}}_{k}^{(n)}+\sum\nolimits_{l\in{\cal
U},l\neq
k}{\sum\nolimits_{d=1}^{d_{l,k}^{(n)}}{{\bf{v}}_{l,k,d}^{(n){\rm{H}}}{\bf{\bar{w}}}_{l}^{(n)}{\tilde{s}}_{l,d}^{(n)}}}+z_{k}^{(n)},$
(24)

where $d_{l,k}^{(n)}$ can be regarded as the number of interfering sources
from UE $l$, ${{\bf{v}}_{l,k,d}^{(n){\rm{H}}}}$ can be treated as the CSI from
the $d$th interfering source of UE $l$ to UE $k$, ${\tilde{s}}_{l,d}^{(n)}$ is
the corresponding transmission data. Both ${\tilde{s}}_{l,d}^{(n)}$ and
${\tilde{s}}_{k}^{(n)}$ are assumed to obey the distribution of ${\cal C}{\cal
N}({{0}},{{1}})$. The data from different interfering sources are mutually
independent and independent of ${\tilde{s}}_{k}^{(n)}$. Note that all
interfering sources from the same UE use the same beam-vector. By using the
receive decoding $u_{k}^{(n)}\in\mathbb{C}$ to decode UE $k$’s received signal
on SC $n$ 666The decoding parameters $\\{u_{k}^{(n)}\in\mathbb{C},\forall
n,k\\}$ can be iteratively calculated at the BBU pool by using the following
iterative algorithm. Then BBU pool will send these parameters to the
corresponding UEs for decoding their signals. Note that these parameters
cannot be included in beam-vectors, otherwise they will affect the transmit
power at each RRH. , the estimated signal is given by

${{{\hat{s}}}_{k}^{(n)}}={{u}}_{k}^{(n){\rm{H}}}{{{\tilde{y}}}_{k}^{(n)}},$
(25)

Due to the independence of the transmit data and noise, the mean square error
(MSE) matrix at UE $k$ is given by

$\displaystyle{{\epsilon}_{k}^{(n)}}\left({{\bf{u}},{\bf{w}}}\right)$ (26)
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!{\mathbb{E}_{\\{{{\bf{{\tilde{s}}}},z_{k}^{(n)}}\\}}}\left[{\left({{{{{\hat{s}}}}_{k}^{(n)}}-{{{\tilde{s}}}_{k}^{(n)}}}\right){{\left({{{{{\hat{s}}}}_{k}^{(n)}}-{{{\tilde{s}}}_{k}^{(n)}}}\right)}^{H}}}\right]$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\left(\\!{u_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}\\!-\\!1}\\!\right)\\!\left({u_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}\\!-\\!1}\right)^{\rm{H}}\\!\\!+\\!\\!\sum\limits_{l\in{\cal
U},l\neq
k}\\!{{{\left|{u_{k}^{(n)}}\right|}^{2}}{{\bf{\bar{w}}}_{l}^{(n){\rm{H}}}{\bf{A}}_{l,k}^{(n)}{\bf{\bar{w}}}_{l}^{(n)}}}\\!+\\!\sigma_{k}^{2}{\left|{u_{k}^{(n)}}\right|^{2}},$

where ${\bf{u}}$ and ${\bf{\tilde{s}}}$ are the collections of decoding
variables and data symbols, respectively, and (23) has been used to derive
(26).

To deal with the second hurdle, we successfully find a lower bound of the sum
rate for each UE and this lower bound is tight at certain point. Then, we
replace the sum rate in constraints C4 with its lower bound and iteratively
solve the beam-vectors by using the block coordinate decent method.
Specifically, defining the following functions:

$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)={\log_{2}}e\left({\ln(q_{k}^{(n)})-q_{k}^{(n)}\epsilon_{k}^{(n)}\left({{\bf{u}},{\bf{w}}}\right)+1}\right),\forall
k\in{\cal U},$ (27)

where $q_{k}^{(n)}\geq 0$ is an introduced variable, we have the following
lemma:

Lemma 1: Given the beam-vectors ${\bf{w}}$, function
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ is a lower
bound for $\tilde{r}_{k}^{(n)}({\bf{w}})$. In addition, the optimal
$u_{k}^{(n)}$ and $q_{k}^{(n)}$ for
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ to achieve
$\tilde{r}_{k}^{(n)}({\bf{w}})$ are

$\displaystyle u_{k}^{(n)\star}$ $\displaystyle=$
$\displaystyle{\left({{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}+\sum\nolimits_{l\in{\cal
U},l\neq
k}{{\bf{\bar{w}}}{{{}_{l}^{(n){\rm{H}}}}}{\bf{A}}_{l,k}^{(n)}}{\bf{\bar{w}}}_{l}^{(n)}}+\sigma_{k}^{2}\right)^{-1}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)},$
(28) $\displaystyle q_{k}^{(n)\star}$ $\displaystyle=$
$\displaystyle{\left({\epsilon_{k}^{(n)}}\left({{\bf{u}}^{\star},{\bf{w}}}\right)\right)^{-1}},$
(29)

where $\epsilon_{k}^{(n)}\left({{\bf{u}}^{\star},{\bf{w}}}\right)$ is given by

$\epsilon_{k}^{(n)}\left({{\bf{u}}^{\star},{\bf{w}}}\right)=1-\frac{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}}{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}+\sum\limits_{l\in{\cal
U},l\neq
k}{{\bf{\bar{w}}}{{{}_{l}^{(n)}}^{\rm{H}}}{\bf{A}}_{l,k}^{(n)}}{\bf{\bar{w}}}_{l}^{(n)}+\sigma_{k}^{2}}}.$
(30)

Proof: Please see Appendix B. $\Box$

By replacing $\tilde{r}_{k}^{(n)}(\bf{w})$ in Problem ${\cal P}_{4}$ with its
lower-bound $\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$,
Problem ${\cal P}_{4}$ can be transformed into the following optimization
problem

$\displaystyle{\cal P}_{5}:\
\mathop{\min}\limits_{{\bf{u},\bf{q},\bf{w}}}\quad$
$\displaystyle{{\tilde{P}}_{{\rm{tot}}}}(\bf{w})$ (31a) s.t. $\displaystyle\
{{\rm{C3,C5,}}}$ $\displaystyle\ {\rm{C6}}:{\sum\nolimits_{n\in{\cal
N}}\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)\geq{R_{k,\min}},\forall
k\in{\cal U},}$ (31b)

where ${\bf{u}}$ and ${\bf{q}}$ are the collection of variables
$\left\\{{u_{k}^{(n)},\forall n,k}\right\\}$ and $\left\\{{q_{k}^{(n)},\forall
n,k}\right\\}$, respectively. Note that given ${\bf{u}}$ and ${\bf{q}}$,
constraint C6 is a convex set over beam-vectors, which is more tractable than
Problem ${\cal P}_{4}$, wherein constraint C4 is non-convex. Hence, Problem
${\cal P}_{5}$ can be solved by using the block coordinate decent method:
given ${\bf{w}}$, update ${\bf{u}}$ and ${\bf{q}}$ in (28) and (29),
respectively; update $\\{\alpha_{k}\\}_{k\in{{\cal U}}}$ and ${\bf{w}}$ with
fixed ${\bf{u}}$ and ${\bf{q}}$. We only need to deal with the latter one.
Given ${\bf{u}}$ and ${\bf{q}}$, by inserting the MSE expression in (26) into
C6, Problem ${\cal P}_{5}$ can be transformed as

$\displaystyle{\cal P}_{6}:\ \mathop{\min}\limits_{{\bf{w}}}\ $
$\displaystyle{{\tilde{P}}_{{\rm{tot}}}}(\bf{w})$ (32a) s.t.
$\displaystyle{{\rm{C3,C5,}}}$ $\displaystyle{\rm{C7}}:\sum\limits_{n\in{\cal
N}}{q_{k}^{(n)}\left({{{\left|{u_{k}^{(n)}}\right|}^{2}}{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}-u_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}-u_{k}^{(n)}{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}}}\right)}$
$\displaystyle\ +\sum\limits_{n\in{\cal N}}{\sum\limits_{l\in{\cal U},l\neq
k}{q_{k}^{(n)}{{\left|{u_{k}^{(n)}}\right|}^{2}}{\bf{\bar{w}}}{{{}_{l}^{(n){\rm{H}}}}}{\bf{A}}_{l,k}^{(n)}}{\bf{\bar{w}}}_{l}^{(n)}}\leq{\omega_{k}},\forall
k\in{\cal U},$ (32b)

where ${\omega_{k}}=\sum\nolimits_{n\in{\cal
N}}{\left[{\ln\left({q_{k}^{(n)}}\right)-q_{k}^{(n)}\sigma_{k}^{2}{{\left|{u_{k}^{(n)}}\right|}^{2}}}-q_{k}^{(n)}\right]}+N-{R_{k,\min}}\ln
2$.

Now, we deal with the second difficulty: the non-smooth indicator function
$\varepsilon\left(\cdot\right)$ in (7) in the objective function and C5 in
Problem ${\cal P}_{6}$. The non-smooth indicator function is approximated as a
fractional function ${f_{\theta}}(x)=\frac{x}{{x+\theta}}$, where $\theta$ is
a very small positive value that controls the smoothness of
approximation777Smaller value of $\theta$ will result in more accurate
approximation but leads to less smoothness in the function, while larger value
of $\theta$ leads to high approximation error. From simulations, we find that
$\theta=10^{-5}$ can achieve a good balance between smoothness and
approximation accuracy. In the simulations, when transmit power for each link
is smaller than $10^{-8}$ watt, the transmit power is set to be zero. The
effect on the rate of each user can be negligible. In addition, for practical
analog to digital conversion (ADC) or digital to analog conversion (DAC),
there is a minimum required power to activate it. Hence, when the transmit
power is very small, it can be ignored. . Then,
${{\tilde{P}}_{{\rm{tot}}}}(\bf{w})$ can be approximated as

$\displaystyle{{\tilde{P}}_{{\rm{tot}}}}(\bf{w})$ $\displaystyle\approx$
$\displaystyle\sum\nolimits_{i\in{\cal
I}}{\left\\{{{{{{\eta_{i}}}}{P_{i}^{{\rm{tr}}}(\bf{w})}}+{f_{\theta}}\left({P_{i}^{{\rm{tr}}}(\bf{w})}\right)P_{i}^{\rm{c}}+{\rho_{i}}\sum\nolimits_{k\in{{\cal
U}_{i}}}{{f_{\theta}}\left({P_{i,k}^{{\rm{tr}}}(\bf{w})}\right){R_{k,\min}}}}\right\\}}$
(33) $\displaystyle\buildrel\Delta\over{=}$
$\displaystyle\\!\\!\\!{{\hat{P}}_{{\rm{tot,}}\theta}}(\bf{w}).$ (34)

Note that for any positive $\theta$, the fractional function ${f_{\theta}}(x)$
is strictly smaller than one. Hence, ${{\hat{P}}_{{\rm{tot,}}\theta}}(\bf{w})$
is actually the lower bound of ${{\tilde{P}}_{{\rm{tot}}}}(\bf{w})$. However,
this gap is negligible when $\theta$ is very small and $x$ is comparatively
large. By replacing the indicator function in Problem ${\cal P}_{6}$ with
$f_{\theta}(x)$, we have

$\displaystyle{\cal P}_{7}:\ \mathop{\min}\limits_{{\bf{w}}}\quad$
$\displaystyle{{\hat{P}}_{{\rm{tot,}}\theta}}(\bf{w})$ (35a) s.t.
$\displaystyle\ {{\rm{C3,C7,}}}$ $\displaystyle\
{\rm{C8}}:\sum\nolimits_{k\in{\cal
U}_{i}}{{f_{\theta}}\left({P_{i,k}^{{\rm{tr}}}(\bf{w})}\right){R_{k,\min}}}\leq{C_{i,\max}},\forall
i\in{\cal I}.$ (35b)

Problem ${\cal P}_{7}$ is much more tractable than Problem ${\cal P}_{6}$
since both the objective function and constraints in Problem ${\cal P}_{7}$
are differentiable and continuous. Although Problem ${\cal P}_{7}$ is still
nonconvex due to the concavity of ${f_{\theta}}\left(\cdot\right)$, it is a
well-known difference of convex (d.c.) program, which can be efficiently
solved by the SCA method [37]. The main idea of this method is to approximate
the concave function as its first order Taylor expansion. Specifically, by
using the concavity of ${f_{\theta}}(\cdot)$, one has

${f_{\theta}}\left({P_{i}^{{\rm{tr}}}({\bf{w}})}\right)\leq{f_{\theta}}\left({P_{i}^{{\rm{tr}}}({\bf{w}}(t))}\right)+{{\beta}_{i}}(t)\left({P_{i}^{{\rm{tr}}}({\bf{w}})-P_{i}^{{\rm{tr}}}({\bf{w}}(t))}\right),$
(36)
${f_{\theta}}\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right)\leq{f_{\theta}}\left({P_{i,k}^{{\rm{tr}}}({\bf{w}}(t))}\right)+{{\chi}_{i,k}}(t)\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})-P_{i,k}^{{\rm{tr}}}({\bf{w}}(t))}\right)$
(37)

where ${{\bf{w}}(t)}$ is a collection of beam-vectors at the $t^{\rm{th}}$
iteration, ${{\beta}_{i}}(t)$ and ${{\chi}_{i,k}}(t)$ are given by

${{\beta}_{i}}(t)={{f}_{\theta}^{{}^{\prime}}}\left({P_{i}^{{\rm{tr}}}({\bf{w}}(t))}\right),{{\chi}_{i,k}}(t)={{f}_{\theta}^{{}^{\prime}}}\left({P_{i,k}^{{\rm{tr}}}({\bf{w}}(t))}\right),$
(38)

where ${{f}_{\theta}^{{}^{\prime}}}\left(x\right)$ denotes the first-order
derivative of $x$. By replacing
${f_{\theta}}\left({P_{i}^{{\rm{tr}}}({\bf{w}})}\right)$ and
${f_{\theta}}\left({P_{i,k}^{{\rm{tr}}}({\bf{w}})}\right)$ in Problem ${\cal
P}_{7}$ with the right hand side (RHS) of (36) and (37), respectively, one can
solve the following optimization problem in the $(t+1)^{\rm{th}}$ iteration

$\displaystyle{\cal P}_{8}:\ \mathop{\min}\limits_{{\bf{w}}}\quad$
$\displaystyle\sum\limits_{n\in{\cal N}}{\sum\limits_{k\in{\cal
U}}{{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{G}}_{k}(t){\bf{\bar{w}}}_{k}^{(n)}}}$
(39a) s.t. $\displaystyle\ {{\rm{C3,C7,}}}$ $\displaystyle\
{\rm{C9}}:\sum\limits_{n\in{\cal N}}{\sum\limits_{k\in{{\cal
U}_{i}}}{{\tau_{i,k}}(t){{\left\|{{\bf{w}}_{i,k}^{(n)}}\right\|}^{2}}}}\leq{{\tilde{C}}_{i,\max}}(t),\forall
i\in{\cal I}.$ (39b)

where ${\bf{G}}_{k}(t)$ is given by
${\bf{G}}_{k}(t)={\rm{blkdiag}}\left({{\kappa_{s_{1}^{k}}}(t){{\bf{I}}_{M\times
M}},\cdots,{\kappa_{s_{\left|{{{\cal
I}_{k}}}\right|}^{k}}}(t){{\bf{I}}_{M\times M}}}\right)$ with
${\kappa_{s_{i}^{k}}}(t)=\left({{\eta_{s_{i}^{k}}}+{\beta_{s_{i}^{k}}}(t)P_{s_{i}^{k}}^{\rm{c}}+{\rho_{s_{i}^{k}}}{\chi_{s_{i}^{k},k}}(t){R_{k,\min}}}\right),i=1,\cdots,\left|{{{\cal
I}_{k}}}\right|$, ${\tau_{i,k}}(t)={\chi_{i,k}}(t){R_{k,\min}}$, and
${{\tilde{C}}_{i,\max}}(t)={C_{i,\max}}-\sum\nolimits_{k\in{{\cal
U}_{i}}}{\left({{f_{\theta}}\left({P_{i,k}^{{\rm{tr}}}({\bf{w}}(t))}\right)-{\chi_{i,k}}(t)P_{i}^{{\rm{tr}}}({\bf{w}}(t))}\right){R_{k,\min}}}$.
Note that some constant terms in the RHS of (36) and (37) are omitted in
(39a). Obviously, ${{{\bf{G}}_{k}{(t)}}}$ is a positive definite matrix and
all constraints form a convex set. Then Problem ${\cal P}_{8}$ is a convex
problem. The details to solve it will be given in the next subsection.

Based on the above analysis, an iterative algorithm is given to solve Problem
${\cal P}_{4}$. A straightforward way to solve Problem ${\cal P}_{4}$ would
involve two layers: the inner layer to solve Problem ${\cal P}_{8}$ by using
the SCA method given ${\bf{u}}$ and ${\bf{q}}$; the outer layer to update
${\bf{u}}$ and ${\bf{q}}$ by using (28) and (29) given ${\bf{w}}$. Although
the inner layer is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point
of Problem ${\cal P}_{7}$ as proved in [38], this two-layer algorithm will
incur high computational complexity. Instead, we merge these two layers into
one layer and update $\\{\beta_{i}{(t)},{{\tilde{C}}_{i,\max}}(t),\forall
i\\}$, $\\{\chi_{i,k}{(t)},{\tau_{i,k}}(t),\forall i,k\\}$, ${\bf{u}}(t)$ and
${\bf{q}}(t)$ at the same layer, as given in Algorithm 1. Fortunately,
Algorithm 1 is guaranteed to converge, as proved in Theorem 1.

Algorithm 1 Iterative Algorithm to Solve Problem ${\cal P}_{4}$

1: Initialize the iterative number $t=1$, error tolerance $\delta$. Initialize
${\bf{w}}{(0)}$ with the output from the UE selection algorithm in Section V,
calculate
$\\{\beta_{i}{(0)},\chi_{i,k}{(0)},{\bf{G}}_{k}{(0)},{\tau_{i,k}}(0),{{\tilde{C}}_{i,\max}}(0),\forall
i,k\\}$, calculate ${\bf{u}}{(0)}$ and ${\bf{q}}{(0)}$ by using (28) and (29)
with ${\bf{w}}{(0)}$, calculate the objective value of Problem ${\cal P}_{7}$,
denoted as ${\rm{Obj(}}{{\bf{w}}{(0)}}{\rm{)}}$.

2: Solve Problem ${\cal P}_{8}$ to get ${\bf{w}}{(t)}$ with
$\\{\beta_{i}{(t-1)},\chi_{i,k}{(t-1)},{\bf{G}}_{k}{(t-1)},{\tau_{i,k}}(t-1),{{\tilde{C}}_{i,\max}}(t-1),\forall
i,k\\}$, ${\bf{u}}{(t-1)}$ and ${\bf{q}}{(t-1)}$;

3: Update
$\\{\beta_{i}{(t)},\chi_{i,k}{(t)},{\bf{G}}_{k}{(t)},{\tau_{i,k}}(t),{{\tilde{C}}_{i,\max}}(t),\forall
i,k\\}$ with ${\bf{w}}{(t)}$;

4: Update ${\bf{u}}{(t)}$ and ${\bf{q}}{(t)}$ by using (28) and (29) with
${\bf{w}}{(t)}$;

5: If
${{\left|{{\rm{Obj(}}{{\bf{w}}{(t-1)}}{\rm{)-Obj(}}{{\bf{w}}{(t)}}{\rm{)}}}\right|}\mathord{\left/{\vphantom{{\left|{{\rm{Obj(}}{{\bf{V}}^{(n-1)}}{\rm{)-Obj(}}{{\bf{V}}^{(n)}}{\rm{)}}}\right|}{{\rm{Obj(}}{{\bf{w}}^{(t)}}{\rm{)}}}}}\right.\kern-1.2pt}{{\rm{Obj(}}{{\bf{w}}{(t)}}{\rm{)}}}}<\delta$,
terminate. Otherwise, set $t\leftarrow t+1$ and go to step 2.

Theorem 1: Given the feasible initial input ${\bf{w}}(0)$, Algorithm 1 is
guaranteed to converge both in objective value and variables.

Proof: Please see Appendix C. $\Box$

###  IV-A Lagrange dual decomposition method to solve Problem ${\cal P}_{8}$

In Step 2 of Algorithm 1, Problem ${\cal P}_{8}$ should be solved. Since both
the maximum power limit and fronthaul capacity limit are positive, i.e.,
${C_{i,\max}}>0,{P_{i,\max}}>0,\forall i\in{\cal I}$, the Slater’s condition
of Problem ${\cal P}_{8}$ is satisfied and the duality gap between Problem
${\cal P}_{8}$ and its dual problem is zero [35]. Hence, Problem ${\cal
P}_{8}$ can be equivalently solved by solving its dual problem. In the
following, we derive the optimal form of ${\bf{w}}$ for Problem ${\cal P}_{8}$
by using the Lagrange dual decomposition method. For notation simplicity, we
omit the iteration index $t$ in the following derivations.

Define the following block diagonal matrices

${{\bf{B}}_{i,k}}{\rm{=diag}}\left\\{{\overbrace{{{\bf{0}}_{1\times
M}}}^{s_{1}^{k}},\cdots,\overbrace{{{\bf{1}}_{1\times
M}}}^{s_{j}^{k}},\overbrace{{{\bf{0}}_{1\times
M}}}^{s_{j+1}^{k}},\cdots,\overbrace{{{\bf{0}}_{1\times M}}}^{s_{\left|{{{\cal
I}_{k}}}\right|}^{k}}}\right\\},\ {\rm{if}}\ s_{j}^{k}=i,\forall i\in{\cal
I},k\in{\cal U},$ (40)

then
${\left\|{{\bf{w}}_{i,k}^{(n)}}\right\|^{2}}={\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{{\bf{B}}_{i,k}}{\bf{\bar{w}}}_{k}^{(n)}$.
With some manipulations, the Lagrangian function of Problem ${\cal P}_{8}$ is
given by

$\displaystyle{\cal
L}\left({{\bf{w}},{\bm{\lambda}},{\bm{\mu}},{\bm{\nu}}}\right)$ (41)
$\displaystyle=$ $\displaystyle\sum\limits_{n\in{\cal
N}}{\sum\limits_{k\in{\cal
U}}{{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{J}}_{k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}}-\sum\limits_{n\in{\cal
N}}{\sum\limits_{k\in{\cal
U}}{{\nu_{k}}q_{k}^{(n)}\left({u_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}+u_{k}^{(n)}{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}}}\right)}}$
$\displaystyle+\ln 2\sum\limits_{k\in{\cal
U}}{{\nu_{k}}{R_{k,\min}}}-\sum\limits_{i\in{\cal
I}}{{\lambda_{i}}{P_{i,\max}}}-\sum\limits_{i\in{\cal
I}}{{\mu_{i}}}{{\tilde{C}}_{i,\max}}-\sum\limits_{k\in{\cal
U}}{{\nu_{k}}{\omega_{k}}},$

where ${\bm{\lambda}},{\bm{\mu}},{\bm{\nu}}$ are the collections of non-
negative Lagrangian multipliers corresponding to ${\rm{C3}}$ in (9),
${\rm{C9}}$ in (39b) and $\rm{C7}$ in (32b), respectively,
${\bf{J}}_{k}^{(n)}$ is given by

${\bf{J}}_{k}^{(n)}={{\bf{G}}_{k}}+\sum\limits_{i\in{{\cal
I}_{k}}}{\left({{\lambda_{i}}+{\mu_{i}}{\tau_{i,k}}}\right){{\bf{B}}_{i,k}}}+{\nu_{k}}q_{k}^{(n)}{\left|{u_{k}^{(n)}}\right|^{2}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}+\sum\limits_{l\in{\cal
U},l\neq
k}{{\nu_{l}}q_{l}^{(n)}{{\left|{u_{l}^{(n)}}\right|}^{2}}{\bf{A}}_{k,l}^{(n)}}.$
(42)

Then, the dual function is given by

$\displaystyle g\left({{\bm{\lambda}},{\bm{\mu}},{\bm{\nu}}}\right)$ (43)
$\displaystyle=$ $\displaystyle\\!\\!\\!\mathop{\min}\limits_{{\bf{w}}}{\cal
L}\left({{\bf{w}},{\bm{\lambda}},{\bm{\mu}},{\bm{\nu}}}\right)$ (45)
$\displaystyle=$
$\displaystyle\\!\\!\\!\mathop{\min}\limits_{{\bf{w}}}\sum\limits_{n\in{\cal
N}}{\sum\limits_{k\in{\cal
U}}{{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{J}}_{k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}}-\sum\limits_{n\in{\cal
N}}{\sum\limits_{k\in{\cal
U}}{{\nu_{k}}q_{k}^{(n)}\left({u_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}+u_{k}^{(n)}{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}}}\right)}}$
$\displaystyle+\ln 2\sum\limits_{k\in{\cal
U}}{{\nu_{k}}{R_{k,\min}}}-\sum\limits_{i\in{\cal
I}}{{\lambda_{i}}{P_{i,\max}}}-\sum\limits_{i\in{\cal
I}}{{\mu_{i}}}{{\tilde{C}}_{i,\max}}-\sum\limits_{k\in{\cal
U}}{{\nu_{k}}{\omega_{k}}}.$

Obviously, Problem (45) is a strictly convex problem and the optimal solution
can be easily obtained from its first-order optimality condition as:

${\bf{\bar{w}}}_{k}^{(n)\star}={\nu_{k}}q_{k}^{(n)}u_{k}^{(n)}{\left({{\bf{J}}_{k}^{(n)}}\right)^{-1}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}},\forall
n\in{\cal N},k\in{\cal U}.$ (46)

By inserting the solution of $\\{{\bf{w}}_{k}^{\star},k\in{{\cal U}}\\}$ in
(46) into (45), the dual function can be rewritten as

$\displaystyle g({\bm{\lambda}},{\bm{\mu}},{\bm{\nu}})$ $\displaystyle=$
$\displaystyle-\sum\limits_{n\in{\cal N}}{\sum\limits_{k\in{\cal
U}}{\nu_{k}^{2}q_{k}^{(n)2}{{\left|{u_{k}^{(n)}}\right|}^{2}}{\bf{\bar{h}}}_{k,k}^{(n)}{{\left({{\bf{J}}_{k}^{(n)}}\right)}^{-1}}{\bf{\bar{h}}}_{k,k}^{(n){\rm{H}}}}}$
(47) $\displaystyle+\ln 2\sum\limits_{k\in{\cal
U}}{{\nu_{k}}{R_{k,\min}}}-\sum\limits_{i\in{\cal
I}}{{\lambda_{i}}{P_{i,\max}}}-\sum\limits_{i\in{\cal
I}}{{\mu_{i}}}{{\tilde{C}}_{i,\max}}-\sum\limits_{k\in{\cal
U}}{{\nu_{k}}{\omega_{k}}}.$

Hence, the dual problem of Problem ${\cal P}_{9}$ is given by

$\mathop{\max}\limits_{\left\\{{{\lambda_{i}}\geq 0,{\mu_{i}}\geq
0,{\nu_{k}}\geq 0,\forall
k,i}\right\\}}g({\bm{\lambda}},{\bm{\mu}},{\bm{\nu}}).$ (48)

Fortunately, the objective function of the dual Problem (48) is differentiable
and dual problem is a convex optimization problem as defined in [35]. Hence,
the classic descent methods such as the gradient descent method can be applied
to solve it as detailed in [35].

Remark 1 - Parallel Computations: Note that for given Lagrangian multipliers,
the optimal beam-vectors $\\{{\bf{\bar{w}}}_{k}^{(n)\star},\forall k\\}$ can
be obtained in (46) in closed forms for each SC in parallel. In C-RAN,
multiple-core processors or multiple virtual machines (VMs) are aggregated
together in the BBU pool, which entails C-RAN to be capable of the parallel
computation. Hence, the Lagrange dual decomposition method can run smoothly
under the C-RAN architecture.

##  V A Low-complexity UE Selection Algorithm

In this section, we propose a low-complexity UE selection algorithms to deal
with Problem ${\cal P}_{3}$: the bisection UE selection algorithm, the
complexity of which increases logarithmically with the number of UEs $K$.

Inspired by the UE selection problem formulations (28)-(30) in [39], we first
construct the following alternative optimization problem by introducing a
series of auxiliary variables $\\{\varphi_{k}\\}_{k\in{\bar{\cal U}}}$888The
authors in [39] considered the single-channel case by introducing auxiliary
variables $\\{s_{k},\forall k\\}$ in each UE’s useful signal power. When all
the optimal $\\{s_{k},\forall k\\}$ are no larger than zeros, all UEs can be
admitted. The method in [39] cannot be directly extended to our work since we
consider the multi-channel case. Instead, we introduce the auxiliary variables
$\\{\varphi_{k}\\}_{k\in{\bar{\cal U}}}$ on the right hand side of constraint
C10. When all the optimal $\\{\varphi_{k}\\}_{k\in{\bar{\cal U}}}$ are equal
to one, all UEs can be admitted. Note that [39] optimized the UE selection and
transmit power minimization problem simultaneously by introducing a large $M$.
In our paper, we consider each problem individually. The reason is that we
find from simulations that the big $M$ is difficult to choose and unproperly
chosen $M$ may result in unexpected results.:

$\displaystyle{\cal P}_{9}:\
\mathop{\min}\limits_{\\{\varphi_{k}\\}_{k\in{\bar{\cal U}}},\bf{w}}\quad$
$\displaystyle\sum\nolimits_{k\in{{\cal U}}}{{{({\varphi_{k}}-1)}^{2}}}$ (49a)
s.t. $\displaystyle\ {{\rm{C3,C5,}}}$ $\displaystyle\
{\rm{C10}}:{\sum\nolimits_{n\in{\cal
N}}{\tilde{r}_{k}^{(n)}(\bf{w})}\geq\varphi_{k}^{2}{R_{k,\min}},\forall
k\in\bar{\cal U},}$ (49b)

Obviously, Problem ${\cal P}_{9}$ is always feasible since at least
$\\{\varphi_{k}=0,{{\bf{w}}_{k}^{(n)}}={\bf{0}},\forall k\in{\bar{\cal
U}},n\in{\cal N}\\}$ is a feasible solution. In addition, it is easy to verify
that the optimal $\\{\varphi_{k},k\in{\bar{\cal U}}\\}$ should lie between
zero and one, i.e., $0\leq{\varphi_{k}}\leq 1,\forall k\in\bar{\cal U}$. If UE
$k$ can be admitted, the optimal $\varphi_{k}$ must be equal to one. This can
be easily proved by contradiction. Denote the solution of
$\\{\varphi_{k}\\}_{k\in{\bar{\cal U}}}$ as
${\\{\varphi_{k}^{\star}\\}_{k\in\bar{\cal U}}}$. If
$\varphi_{k}^{\star}=1,\forall k\in{\bar{\cal U}}$, all UEs can be admitted in
the network and output the corresponding optimal beam-vectors for the initial
solution for Algorithm 1 in Section IV. Otherwise, some UEs should be removed.
Intuitively, the UE with a smaller $\varphi_{k}^{\star}$ should have a higher
priority to be removed since it has the largest gap away from its rate
targets. Hence, we sort ${\\{\varphi_{k}^{\star}\\}_{k\in\bar{\cal U}}}$ in
the ascending order:
$\varphi_{{\pi_{1}}}^{\star}\leq\cdots\leq\varphi_{{\pi_{K}}}^{\star}$. Then
admitting the maximum number of UEs is equivalent to finding a minimum
${L_{0}}$ such that all the users in ${\cal
U}=\left\\{{{\pi_{{L_{0}}+1}},\cdots,{\pi_{K}}}\right\\}$ can be supported by
C-RAN with ${L_{0}}=1,\cdots,K-1$. The bisection search procedure can be
adopted to determine the minimum ${L_{0}}$. In each iteration of the bisection
UE search algorithm, we only need to check whether the C-RAN can support all
users in ${\cal U}$ or not. Hence, in each iteration, we need to solve the
following optimization problem

$\displaystyle{\cal P}_{10}:\ \mathop{\min}\limits_{\varphi,\bf{w}}\quad$
$\displaystyle{({\varphi}-1)}^{2}$ (50a) s.t. $\displaystyle\ {{\rm{C3,C5,}}}$
$\displaystyle\ {\rm{C11}}:{\sum\nolimits_{n\in{\cal
N}}{\tilde{r}_{k}^{(n)}(\bf{w})}\geq\varphi^{2}{R_{k,\min}},\forall k\in{\cal
U},}$ (50b)

where $\varphi$ is the introduced optimization variable. Obviously, when the
optimal $\varphi^{\star}$ is equal to one, Problem ${\cal P}_{10}$ is
feasible. Problem ${\cal P}_{10}$ can be similarly solved by using Algorithm
1. Note that all UEs’ rate requirements in ${\cal P}_{10}$ use the same
$\varphi$ and thus Problem ${\cal P}_{10}$ has less variables than Problem
${\cal P}_{9}$.

Finally, the bisection search method is summarized in Algorithm 2. Notice that
Problem ${\cal P}_{10}$ only needs to be solved no more than
$\left\lceil{{{\log}_{2}}(1+K)}\right\rceil$ times.

Algorithm 2 Bisection UE Selection (BUES) Algorithm

1: Solve Problem ${\cal P}_{9}$.

  1. 1.

If $\varphi_{k}^{\star}=1,\forall k\in{\bar{\cal U}}$, terminate and all UEs
can be supported, output the corresponding optimal beam-vectors for the
initial point for Algorithm 1;

  2. 2.

If there exists at least one UE $k$ such that $\varphi_{k}^{\star}<1$, sort
all ${\\{\varphi_{k}^{\star}\\}_{k\in{\cal U}}}$ in the ascending order:
$\varphi_{{\pi_{1}}}^{\star}\leq\cdots\leq\varphi_{{\pi_{K}}}^{\star}$, go to
step 2;

2: Set ${{\cal U}}=\\{{\pi_{K}}\\}$, solve Problem ${\cal P}_{10}$:

  1. 1.

If $\varphi_{\pi_{K}}^{\star}=1$, go to step 3;

  2. 2.

Otherwise, terminate and claim that no UE can be supported;

3: Initialize ${L_{{\rm{low}}}}=0,{L_{{\rm{up}}}}=K$;

4: Repeat

  1. 1.

Set
$l\leftarrow\left\lfloor{\frac{{{L_{{\rm{low}}}}+{L_{{\rm{up}}}}}}{2}}\right\rfloor$;

  2. 2.

Solve Problem ${\cal P}_{10}$ with ${\cal
U}=\left\\{{{\pi_{l+1}},\cdots,{\pi_{K}}}\right\\}$. If $\varphi^{\star}=1$,
set ${L_{{\rm{up}}}}=l$; Otherwise, set ${L_{{\rm{low}}}}=l$;

5: Until ${L_{{\rm{up}}}}-{L_{{\rm{low}}}}=1$. Output the optimal active UE
set ${\cal U}=\left\\{{{\pi_{{L_{\rm{low}}}+1}},\cdots,{\pi_{K}}}\right\\}$
and the corresponding optimal beam-vectors.

##  VI Simulation Results

###  VI-A System parameters

In this section, we present simulation results to evaluate the performance of
the proposed algorithms. The dense C-RAN is within a square area of
coordinates $[-1000,1000]\times[-1000,1000]$ meters. Both the UEs and RRHs are
assumed to be independently and uniformly distributed in this square area. The
channel model consists of three parts: 1) the channel path-loss modeled as
$PL_{i,k}=148.1+37.6{\log_{10}}d_{i,k}\ ({\rm{dB}})$ [40], where $d_{i,k}$ (in
km) is the distance between the $i$th RRH to the $k$th UE; 2) the log-normal
shadowing with zero mean and 8 dB standard derivation; 3) small-scale Rayleigh
fading with zero mean and unit variance. All the UEs are assumed to have the
same rate requirements, i.e., $R_{k,\rm{min}}=R_{\rm{min}},\forall k$. For
ease of exposition, each fronthaul link is assumed to have the same capacity
constraints, i.e., $C_{i,\max}=C_{\max},\forall i$. Then, normalized fronthaul
capacity is considered, i.e.,
${\tilde{C}_{\max}}={{{C_{\max}}}\mathord{\left/{\vphantom{{{C_{\max}}}R}}\right.\kern-1.2pt}R_{\rm{min}}}$,
which represents the maximum number of UEs that can be supported on each
fronthaul link is the same. It is assumed that each UE is potentially served
by its nearest $X$ RRHs, i.e., $\left|{{{\cal I}_{k}}}\right|=X,\forall k$.
Also, each UE is assumed to measure its channel vectors to its nearest $Y$
RRHs, i.e., $\left|{{\widetilde{{\cal I}}_{k}}}\right|=Y,\forall k$. Unless
stated otherwise, the system parameters are set as follows: $M=2$, $K=16$,
$I=20$, $N=3$, system bandwidth $B=10\ \rm{MHz}$, error tolerance
$\delta=10^{-3}$, noise power spectral density is -174 dBm/Hz,
$P_{i}^{{\rm{active}}}=6.8\ {\rm{Watt}},P_{i}^{{\rm{sleep}}}=4.3\
{\rm{Watt}},\eta_{i}=4,\rho_{i}=0.5,P_{i,\rm{max}}=2\ {\rm{Watt}},\forall i$,
$\theta=10^{-5}$, $R_{\rm{min}}=15\ \rm{bit/s/Hz}$, ${\tilde{C}_{\max}}=3$,
$X=3$, $Y=6$.

###  VI-B Numerical Results

Figure 4: The average number of admitted UEs for different algorithms with
$K=8$, $I=12$, $P_{i,\rm{max}}=2\ {\rm{Watt}},\forall i$, $X=3$, and $Y=6$.

Figure 5: Convergence behaviour for the NPC minimization algorithm.

####  VI-B1 Performance of the UE selection algorithm

Fig. 5 shows the average number of admitted UEs for three different
algorithms. Specifically, ‘Joint-BUES-alg.’ denotes the joint beam direction
and power allocation optimization algorithm in Algorithm 2, while ‘MF-BUES-
alg.’ represents that the beam directions are fixed to be the channel
direction, and the power allocation problem is solved by using Algorithm 2.
Note that beam direction is not optimized in ‘MF-BUES-alg.’. This scheme has
lower complexity than ‘Joint-BUES-alg.’, but incurs inferior performance as
seen in the following examples. For comparison, the optimal performance
obtained by exhaustive search 999There are many existing MINLP solvers to
solve the MINLP problems, such as the generalized Benders decomposition method
in [36, 41] and the branch-and-cut (BnC) method in [33]. The main idea of
these two methods is to decompose the original problem into several more
tractable subproblems and iteratively solve the subproblems until convergence.
The condition for convergence is that the globally optimal solution can be
obtained. Unfortunately, we consider the multichannel case, wherein each
subproblem is non-convex and globally optimal solution cannot be obtained as
explained in [42]. Hence, the above two methods are not applicable. Instead,
we adopt the exhaustive search method as the performance benchmark that is
only simulated in a small network. (denoted as ‘Exhaustive-search’) is also
shown, which evaluates every possible subset of UEs and chooses the feasible
subset with the maximum number of UEs. Due to the exponential complexity
associated with the exhaustive search, we only simulate a small network with
$K=8$ and $I=12$. As expected, the number of admitted UEs for all algorithms
decrease with the increase in the rate requirements. The optimal exhaustive
search performs better than the ‘Joint-BUES-alg.’, which comes at the cost of
high computational complexity. However, the performance gap between these two
algorithms is negligible when $R_{\rm{min}}$ is small (e.g., $R_{\rm{min}}<20\
{\rm{bit/s/Hz}}$). By jointly optimizing the beam direction and power
allocation, the ‘Joint-BUES-alg.’ outperforms the ‘MF-BUES-alg.’. However, the
performance gain decreases when $R_{\rm{min}}$ is large. The reason can be
explained as follows. With the increase of $R_{\rm{min}}$, the number of
admitted UEs decreases and these UEs are separated far away from each other.
Hence, interference is not so significant and the channel matching beam
direction approaches the optimal direction.

####  VI-B2 Convergence behaviour of the NPC minimization algorithm

Figure 5 shows the convergence behaviour for the NPC minimization algorithm
(i.e., Algorithm 1), where ‘Joint-NPC-Alg.1’ denotes the joint beam direction
and power allocation optimization performed by Algorithm 1, while ‘MF-NPC-
Alg.1’ denotes that the beam direction is fixed to be channel direction and
the power optimization is carried out by Algorithm 1. The top subplot shows
the NPC trend, the middle and bottom subplots show the numbers of active RRHs
and active links remained in each iteration, respectively. It is seen from
this figure that all these values decrease rapidly and converge within twenty
iterations. Both the convergence speed and the NPC performance for the
considered two algorithms are very similar in this scenario. Moreover, for
‘Joint-NPC-Alg.1’, the NPC, the number of active RRHs and active links
decrease about $65\%$, $45\%$ and $94\%$, respectively, which confirm the
effectiveness of the proposed algorithm in terms of power savings.

In the following, we will evaluate the effects of different system parameters
on both the NPC minimization algorithm (i.e.,Algorithm 1) and UE selection
algorithm (i.e., Algorithm 2). To compare the performance of the NPC
minimization algorithm, the performance of the conventional transmit power
minimization is also considered, where all the RRHs in each UE’s candidate set
are assumed to be active. ‘Joint-Conven’ and ‘MF-Conven’ denote the
conventional method when beam direction and power allocation are jointly
optimized and beam direction is fixed at channel direction, respectively.

####  VI-B3 Effects of the candidate size

Figure 6: Number of admitted UEs versus the candidate size $X$ with $Y=6$.

Figure 7: NPC versus the candidate size $X$ for different algorithms with
$Y=6$.

Figs. 7 and 7 show the numbers of admitted UEs and NPC versus the candidate
size $X$, respectively. The set of UEs that are admitted by the ‘MF-BUES-alg.’
are set as the initialization point for the NPC minimization algorithms, which
is the same in the following simulations. As expected, the larger candidate
size leads to more admitted UEs due to the increasing network degrees of
freedom. However, the number of admitted UEs achieved by ‘Joint-BUES-alg.’ and
‘MF-BUES-alg.’ become flat in the large candidate size regime, which is
consistent with the conclusion in [28]. This means that it is not necessary to
consider the far away RRHs for each UE since they contribute less to their
performance and the candidate size should not be larger than 4 to obtain a
tradeoff between performance and implementation complexity. The similar trend
holds for the ‘Joint-NPC-Alg.1’ and ‘MF-NPC-Alg.1’ in Figure 7. However, in
Figure 7, the conventional transmit power minimization consumes much higher
power than the proposed ‘Joint-NPC-Alg.1’ and ‘MF-NPC-Alg.1’, and the gap
increases with the increase of candidate size. The reason is that with the
increase of candidate size, more RRHs will be in the active mode, which
requires large amount of circuit power consumption. It is surprising to see
from Figure 7 that for the conventional method, ‘MF-Conven’ requires slightly
higher power consumption than ‘Joint-Conven’. On the other hand, ‘MF-NPC-
Alg.1’ requires much higher power than ‘Joint-NPC-Alg.1’. These two facts
confirm that joint beam direction and power allocation optimization is more
important for RRH and link selection.

####  VI-B4 Effects of the amount of CSI

Figure 8: Number of admitted UEs versus the CSI amount $Y$ with $X=3$.

Figure 9: NPC versus the CSI amount $Y$ for different algorithms with $X=3$.

Now, we investigate the effects of limited CSI on the performance of the
proposed algorithms. Figs. 9 and 9 illustrate the number of admitted UEs and
NPC versus the amount of CSI $Y$, respectively. Note that $Y$ denotes the
number of (nearest) RRHs from which CSI is measured. As expected, the number
of admitted UEs increase as the amount of CSI increases, since multi-user
interference can be more accurately suppressed. From Figure 9, it is seen that
the number of admitted UEs increases quickly when $Y<12$ and increases slowly
in the high amount CSI regime. This result indicates that only a moderate
amount of CSI is sufficient for the proposed algorithms to achieve good
performance, which can significantly reduce the channel estimation overhead.
The corresponding NPC increases sightly with the amount of CSI due to more UEs
are admitted. The proposed algorithms are again observed to perform much
better than the conventional transmit power minimization method, highlighting
the importance of joint optimization of transmit power, RRH and link
selection.

####  VI-B5 Effects of fronthaul capacity constraints

Figure 10: Number of admitted UEs versus the normalized fronthaul capacity
$\tilde{C}_{\rm{max}}$.

Figure 11: NPC versus the normalized fronthaul capacity $\tilde{C}_{\rm{max}}$
for different algorithms.

Figs. 11 and 11 show the number of admitted UEs and NPC versus the normalized
fronthaul capacity $C_{\rm{max}}$, respectively. It is seen from Figure 11
that the numbers of admitted UEs for both the ‘Joint-BUES-alg.’ and ‘MF-BUES-
alg.’ increase with $C_{\rm{max}}$ initially due to the fact that more UEs can
be supported by each fronthaul link for large $C_{\rm{max}}$. However, the
number of admitted UEs will be saturated in the large $C_{\rm{max}}$ regime.
It is shown that $C_{\rm{max}}=4$ is enough to achieve a large portion of the
optimal performance, which indicates that the fronthaul link capacity is not
necessary to be very large and the wireless fronthaul link such as mmWave
communication technologies may be applicable in dense C-RAN network. Figure 11
also shows that ‘Joint-BUES-alg.’ outperforms ‘MF-BUES-alg.’ in terms of the
number of admitted UEs and the performance gain increases with $C_{\rm{max}}$.
From Figure 11, it can be seen that the NPC performances of ‘Joint-NPC-Alg.1’
and ‘MF-NPC-Alg.1’ increase with $C_{\rm{max}}$ when $C_{\rm{max}}<=4$, since
more UEs are admitted in this regime as seen from Figure 11. However, when
$C_{\rm{max}}>=4$, the NPC value experiences a slight decrease though the
numbers of UEs are almost the same as seen in Figure 11. This is due to the
fact that more flexibility in the fronthaul link can be exploited to reduce
the numbers of active links and RRHs.

##  VII Conclusions

This paper provided a complete framework to handle the challenges arising in
the dense C-RAN. More specifically, the downlink beam-vectors, RRH selection
and UE-RRH associations were jointly optimized to minimize the total NPC for
dense C-RAN with incomplete CSI subject to fronthaul capacity constraints,
UEs’ QoS targets and per-RRH power constraints. We formulated this problem as
an MINLP problem, which is NP-hard. In addition, the incomplete CSI makes the
QoS constraints difficult to handle. We first replaced the exact expression of
data rate with its lower bound. Then, we developed a low-complexity single-
layer iterative algorithm to solve the NPC minimization problem based on the
successive convex approximation technique and the equivalent relationship
between data rate and MSE. Also, a low-complexity UE selection algorithm was
proposed to guarantee the feasibility of the NPC problem. Simulation results
showed that the proposed UE selection can achieve near-optimal performance
compared to the optimal exhaustive UE search method. Moreover, the proposed
single-layer iterative algorithm can achieve significant power savings in
various setups. Simulation results also showed that only nearest four RRHs are
sufficient to be the candidate set of each UE and limited CSI can contribute
large portion of performance gain from the full CSI case.

The future work lies in the joint optimization of cluster sizes and beam-
vectors when taking into account the cost of computational complexity and
channel training overhead. Also, it is worth studying how to extend the work
to the scenario where each UE is equipped with multiple antennas.

##  Appendix A Accurate closed-form expression of data rate for special case

For the simplicity of notations, the SC index $n$ is omitted in the following
derivations. The SINR for UE $k$ in (14) can be rewritten as

${\gamma_{k}}=\frac{{{{\left|{{X_{k}}}\right|}^{2}}}}{{\sum\nolimits_{l\neq
k,l\in{\cal U}}{{{\left|{{Y_{l,k}}}\right|}^{2}}}+\sigma_{k}^{2}}},$ (A.1)

where ${X_{k}}={{\bf{\bar{h}}}_{k,k}{\bf{\bar{w}}}_{k}}$ and
$Y_{l,k}={{\bf{\bar{h}}}_{l,k}{\bf{\bar{w}}}_{l}}$. Note that
${\bf{\bar{h}}}_{k,k}$ is perfectly known and ${\bf{\bar{w}}}_{l},\forall l$
are deterministic, ${X_{k}}$ is a deterministic value and only
$\\{Y_{l,k},\forall l\in{\cal U},l\neq k\\}$ are random variables. According
to the first two assumptions, all elements in ${\bf{\bar{h}}}_{l,k}$ are
unknown and follow the circular symmetric complex Gaussian distribution.
Specifically, the distribution of ${\bf{\bar{h}}}_{l,k}$ is given by ${\cal
C}{\cal N}({\bf{0}},{\bf{R}}_{l,k})$, where ${\bf{R}}_{l,k}$ is a diagonal
matrix. To obtain the expression of ${\bf{R}}_{l,k}$, we define the indices of
${{{\cal I}_{l}}}$ as ${{\cal I}_{l}}=\\{s_{1}^{l},\cdots,s_{|{{\cal
I}_{l}}|}^{l}\\}$. Then, based on the third assumption, ${\bf{R}}_{l,k}$ can
be easily calculated as
${{\bf{R}}_{l,k}}={\rm{blkdiag}}\left({{{\left|{{\alpha_{s_{1}^{l},k}}}\right|}^{2}}{{\bf{I}}_{M\times
M}},\cdots,{{\left|{{\alpha_{s_{|{{\cal
I}_{l}}|}^{l},k}}}\right|}^{2}}{{\bf{I}}_{M\times M}}}\right)$. Then, given
beam-vector ${\bf{\bar{w}}}_{l}$, $Y_{l,k}$ is a Gaussian random variable with
zero mean and variance given by
$\varpi_{l,k}={\bf{\bar{w}}}_{l}^{H}{{\bf{R}}_{l,k}}{{{\bf{\bar{w}}}}_{l}}$,
i.e., $Y_{l,k}\sim{\cal{CN}}({\bf{0}},{\varpi_{l,k}})$. For convenience,
denote ${Z_{k}}=\sum\nolimits_{l\neq k,l\in{\cal
U}}{{{\left|{{Y_{l}}}\right|}^{2}}}$. Then, $Z_{k}$ follows a generalized chi-
squared distribution, given by [43]

$f({z_{k}})=\sum\nolimits_{l\neq k,l\in{\cal
U}}{{T_{l,k}}{e^{-{{{z_{k}}}\mathord{\left/{\vphantom{{{z_{k}}}{{\varpi_{l,k}}}}}\right.\kern-1.2pt}{{\varpi_{l,k}}}}}}},$
(A.2)

where $T_{l,k}$ is given by

${T_{l,k}}=\frac{1}{{{\varpi_{l,k}}\prod\nolimits_{j\in{\cal U},j\neq
l,k}{\left({1-\frac{{{\varpi_{j,k}}}}{{{\varpi_{l,k}}}}}\right)}}}.$

Then, the data rate is derived as

$\displaystyle\int_{0}^{\infty}{{{\log}_{2}}}\left({1+\frac{{{{\left|{{X_{k}}}\right|}^{2}}}}{{{z_{k}}+\sigma_{k}^{2}}}}\right)f({z_{k}})d{z_{k}}$
$\displaystyle=\sum\limits_{l\neq k,l\in{\cal
U}}{{T_{l,k}}}\int_{0}^{\infty}{{{\log}_{2}}}\left({1+\frac{{{{\left|{{X_{k}}}\right|}^{2}}}}{{{z_{k}}+\sigma_{k}^{2}}}}\right){e^{-\frac{{{z_{k}}}}{{{\varpi_{l,k}}}}}}d{z_{k}}$
$\displaystyle=\sum\limits_{l\neq k,l\in{\cal
U}}{-\frac{{{T_{l,k}}{\varpi_{l,k}}}}{{\ln
2}}}\int_{0}^{\infty}{\left[{\ln\left({{z_{k}}+\sigma_{k}^{2}+{{\left|{{X_{k}}}\right|}^{2}}}\right)-\ln\left({{z_{k}}+\sigma_{k}^{2}}\right)}\right]}d{e^{-\frac{{{z_{k}}}}{{{\varpi_{l,k}}}}}}$
$\displaystyle=\sum\limits_{l\neq k,l\in{\cal
U}}{\frac{{{T_{l,k}}{\varpi_{l,k}}}}{{\ln
2}}\\!\\!\left[{\ln\left({1+\frac{{{{\left|{{X_{k}}}\right|}^{2}}}}{{\sigma_{k}^{2}}}}\right)\\!+\\!\int_{0}^{\infty}{\frac{{{e^{-\frac{{{z_{k}}}}{{{\varpi_{l,k}}}}}}}}{{{z_{k}}+\sigma_{k}^{2}+{{\left|{{X_{k}}}\right|}^{2}}}}d{z_{k}}\\!-\\!\int_{0}^{\infty}{\frac{{{e^{-\frac{{{z_{k}}}}{{{\varpi_{l,k}}}}}}}}{{{z_{k}}+\sigma_{k}^{2}}}d{z_{k}}}}}\right]}$
(A.3) $\displaystyle=\sum\limits_{l\neq k,l\in{\cal
U}}\\!\\!\\!{\frac{{{T_{l,k}}{\varpi_{l,k}}}}{{\ln
2}}\\!\\!\\!\left[{\ln\\!\left(\\!{1+\frac{{{{\left|{{X_{k}}}\right|}^{2}}}}{{\sigma_{k}^{2}}}}\\!\right)\\!-\\!{e^{\frac{{\sigma_{k}^{2}+{{\left|{{X_{k}}}\right|}^{2}}}}{{{\varpi_{l,k}}}}}}{\rm{Ei}}\left(\\!{-\frac{{\sigma_{k}^{2}\\!+\\!{{\left|{{X_{k}}}\right|}^{2}}}}{{{\varpi_{l,k}}}}}\\!\right)+{e^{\frac{{\sigma_{k}^{2}}}{{{\varpi_{l,k}}}}}}{\rm{Ei}}\left(\\!{-\frac{{\sigma_{k}^{2}}}{{{\varpi_{l,k}}}}}\\!\right)}\right]}$
(A.4)

where ${\rm{Ei}}(x)=-\int_{-x}^{\infty}{\left({{e^{-t}}/t}\right)dt}$ is an
exponential integral function, (A.3) is obtained by using integration by
parts, and (A.4) is achieved by invoking [Eq. (3.352.4), [44]].

##  Appendix B Proof of Lemma 1

We prove that
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ is a lower
bound of $\tilde{r}_{k}^{(n)}{({\bf{w}})}$ by showing that given ${{\bf{w}}}$,
the maximum of
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ is equal to
$\tilde{r}_{k}^{(n)}{({\bf{w}})}$.

Obviously, function
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ is
respectively concave over $u_{k}^{(n)}$, $q_{k}^{(n)}$ and ${\bf{w}}$ when the
other two are fixed. As a result, the optimal $u_{k}^{(n)}$ and $q_{k}^{(n)}$
to achieve the maximum value of
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ are obtained
by setting the first order of
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$ to zero,
which are given in (28) and (29), respectively.

By inserting the expression of ${{u}}_{k}^{(n)\star}$ in (28) into (26), the
expression of $\epsilon_{k}^{(n)}\left({{\bf{u}}^{\star},{\bf{w}}}\right)$ can
be obtained by (30). By substituting the optimal ${{u}}_{k}^{(n)\star}$ in
(28) and ${{q}}_{k}^{(n)\star}$ in (29) into function
$\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)},u_{k}^{(n)}}\right)$, we have

$\displaystyle\Psi_{k}^{(n)}\left({{{\bf{w}}},q_{k}^{(n)\star},u_{k}^{(n)\star}}\right)$
$\displaystyle=$
$\displaystyle{\log_{2}}e\ln{\left({1-\frac{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}}{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}+\sum\limits_{l\in{\cal
U},l\neq
k}{{\bf{\bar{w}}}{{{}_{k}^{(n)}}^{\rm{H}}}{\bf{A}}_{l,k}^{(n)}}{\bf{\bar{w}}}_{k}^{(n)}+\sigma_{k}^{2}}}}\right)^{-1}}$
(B.1) $\displaystyle=$
$\displaystyle{\log_{2}}\left({1+\frac{{{{\left|{{\bf{\bar{h}}}_{k,k}^{(n)}{\bf{\bar{w}}}_{k}^{(n)}}\right|}^{2}}}}{{\sum\limits_{l\in{\cal
U},l\neq
k}{{\bf{\bar{w}}}{{{}_{k}^{(n)}}^{\rm{H}}}{\bf{A}}_{l,k}^{(n)}}{\bf{\bar{w}}}_{k}^{(n)}+\sigma_{k}^{2}}}}\right)$
(B.2) $\displaystyle{\rm{=}}$ $\displaystyle\tilde{r}_{k}^{(n)}({\bf{w}}).$
(B.3)

Hence, the proof is complete.

##  Appendix C Proof of Theorem 1

Before proving the theorem, we first construct the following auxiliary problem

$\displaystyle{\cal P}_{X}:\ \mathop{\min}\limits_{{\bf{w}}}\quad$
$\displaystyle{{\hat{P}}_{{\rm{tot,}}\theta}}\left({\bf{w}}\right)$ (C.1a)
s.t. $\displaystyle{\rm{C3}},{\rm{C4}},{\rm{C8}}.$ (C.1b)

Note that Problem ${\cal P}_{X}$ has the same objective function as Problem
${\cal P}_{7}$. In the following, we show that Algorithm 1 actually solves
Problem ${\cal P}_{X}$. In addition, the only difference between the original
Problem ${\cal P}_{4}$ and Problem ${\cal P}_{X}$ is that the indicator
function is replaced by the concave smooth function.

Since $\\{{\bf{w}}{(0)},\forall k\\}$ is initialized by using the output from
the UE selection algorithm, ${\bf{w}}{(0)}$ is a feasible solution of Problem
${\cal P}_{4}$. By using the fact that ${f_{\theta}}(x)<1,\forall x$, we
conclude that ${\bf{w}}{(0)}$ is also feasible for ${\cal P}_{X}$. Note that
${\bf{u}}{(0)}$ and ${\bf{q}}{(0)}$ are calculated by using (28) and (29) with
${\bf{w}}{(0)}$. Then by using Lemma 1, ${\bf{w}}{(0)}$ is a feasible solution
of Problem ${\cal P}_{7}$ with fixed ${\bf{u}}{(0)}$ and ${\bf{q}}{(0)}$. It
is easy to check that ${\bf{w}}{(0)}$ is also a feasible solution of Problem
${\cal P}_{8}$ with
$\\{\beta_{i}{(0)},\chi_{i,k}{(0)},{\bf{G}}_{k}{(0)},{\tau_{i,k}}(0),{{\tilde{C}}_{i,\max}}(0),\forall
i,k\\}$. Now, we consider step 2 of the first iteration (i.e., $t=1$) of
Algorithm 1. Since ${\bf{w}}{(1)}$ is the optimal solution of Problem ${\cal
P}_{8}$, we have

$\begin{array}[]{l}\sum\limits_{n\in{\cal N}}{\sum\limits_{k\in{\cal
U}}{{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}(1){{\bf{G}}_{k}}(0){\bf{\bar{w}}}_{k}^{(n)}(1)}}\leq\sum\limits_{n\in{\cal
N}}{\sum\limits_{k\in{\cal
U}}{{\bf{\bar{w}}}_{k}^{(n){\rm{H}}}(0){{\bf{G}}_{k}}(0){\bf{\bar{w}}}_{k}^{(n)}(0)}}.\end{array}$
(C.2)

For the simplicity of representation, denote
${\psi_{i}}(t)=P_{i}^{{\rm{tr}}}({\bf{w}}(t))$ and
${\xi_{i,k}}(t)=P_{i,k}^{{\rm{tr}}}({\bf{w}}(t))$. Then, we have

$\displaystyle{\rm{Obj}}({{\bf{w}}{(1)}})$
$\displaystyle=\sum\limits_{i\in{{\cal
I}}}{\left({{{{\eta_{i}}}}{\psi_{i}}(1)+{f_{\theta}}\left({\psi_{i}}(1)\right)P_{i}^{\rm{c}}+{\rho_{i}}\sum\limits_{k\in{{\cal
U}_{i}}}{{f_{\theta}}\left({\xi_{i,k}{(1)}}\right){R_{k,\min}}}}\right)}$
$\displaystyle\mathop{\leq}\limits^{(a)}\sum\limits_{i\in{\cal
I}}{\left({{\eta_{i}}{\psi_{i}}(1)+{f_{\theta}}\left({{\psi_{i}}(0)}\right)P_{i}^{\rm{c}}+{\beta_{i}}(0)P_{i}^{\rm{c}}\left({{\psi_{i}}(1)-{\psi_{i}}(0)}\right)}\right)}+$
$\displaystyle\quad\sum\limits_{i\in{\cal I}}{\sum\limits_{k\in{{\cal
U}_{i}}}{{\rho_{i}}{R_{k,\min}}\left({{f_{\theta}}\left({{\xi_{i,k}}(0)}\right)+{\chi_{i,k}}(0)\left({{\xi_{i,k}}(1)-{\xi_{i,k}}(0)}\right)}\right)}}$
$\displaystyle\mathop{\leq}\limits^{(b)}\sum\limits_{i\in{\cal
I}}{\left({{\eta_{i}}{\psi_{i}}(0)+{f_{\theta}}\left({{\psi_{i}}(0)}\right)P_{i}^{\rm{c}}+{\beta_{i}}(0)P_{i}^{\rm{c}}\left({{\psi_{i}}(0)-{\psi_{i}}(0)}\right)}\right)}+$
$\displaystyle\quad\sum\limits_{i\in{\cal I}}{\sum\limits_{k\in{{\cal
U}_{i}}}{{\rho_{i}}{R_{k,\min}}\left({{f_{\theta}}\left({{\xi_{i,k}}(0)}\right)+{\chi_{i,k}}(0)\left({{\xi_{i,k}}(0)-{\xi_{i,k}}(0)}\right)}\right)}}$
$\displaystyle=\sum\limits_{i\in{{\cal
I}}}{\left({{{{\eta_{i}}}}{\psi_{i}}(0)+{f_{\theta}}\left({\psi_{i}}(0)\right)P_{i}^{\rm{c}}+{\rho_{i}}\sum\limits_{k\in{{\cal
U}_{i}}}{{f_{\theta}}\left({\xi_{i,k}{(0)}}\right){R_{k,\min}}}}\right)}$
$\displaystyle={\rm{Obj}}({{\bf{w}}{(0)}})$

where ${\rm{Obj}}({{\bf{w}}{(t)}})$ denotes the objective value of Problem
${\cal P}_{7}$ or ${\cal P}_{X}$, (a) follows by using (36) and (37), (b)
follows due to (C.2).

Next, we show that ${\bf{w}}{(1)}$ is also a feasible solution of Problem
${\cal P}_{X}$. Obviously, ${\bf{w}}{(1)}$ satisfies C3, i.e., the power
constraints. We only need to prove that ${\bf{w}}{(1)}$ satisfies C4 and C8.

Since ${\bf{w}}{(1)}$ is the optimal solution of Problem ${\cal P}_{8}$, we
have

$\sum\nolimits_{n\in{\cal
N}}{\Psi_{k}^{(n)}\left({{{\bf{w}}}(1),q_{k}^{(n)}(0),u_{k}^{(n)}(0)}\right)}\geq{R_{k,\min}}.$
(C.3)

In step 4 of the first iteration of Algorithm 1, ${\bf{u}}{(1)}$ and
${\bf{q}}{(1)}$ are updated by using (28) and (29) with ${\bf{w}}{(1)}$. Then
according to Lemma 1, we have

$\displaystyle\sum\limits_{n\in{\cal N}}{\tilde{r}_{k}^{(n)}}({\bf{w}}(1))$
$\displaystyle=\sum\limits_{n\in{\cal
N}}{\Psi_{k}^{(n)}\left({{\bf{w}}(1),q_{k}^{(n)}(1),u_{k}^{(n)}(1)}\right)}$
(C.4) $\displaystyle\geq\sum\limits_{n\in{\cal
N}}{\Psi_{k}^{(n)}\left({{\bf{w}}(1),q_{k}^{(n)}(0),u_{k}^{(n)}(0)}\right)}$
(C.5) $\displaystyle\geq{R_{k,\min}}.$ (C.6)

Hence, C4 is satisfied.

In addition, we have

$\displaystyle{C_{i,\max}}$ $\displaystyle\geq\sum\limits_{k\in{{\cal
U}_{i}}}{\left({{f_{\theta}}\left({{\xi_{i,k}}(0)}\right)+{\chi_{i,k}}(0)\left({{\xi_{i,k}}(1)-{\xi_{i,k}}(0)}\right)}\right){R_{k,\min}}}$
(C.7) $\displaystyle\geq\sum\limits_{k\in{{{\cal
U}}_{i}}}{{f_{\theta}}\left({{\xi_{i,k}{(1)}}}\right){R_{k,\min}}},$ (C.8)

where (C.7) follows since ${\bf{w}}{(1)}$ is the solution of Problem ${\cal
P}_{8}$ given ${\bf{u}}{(0)}$ and ${\bf{q}}{(0)}$, and (C.8) follows by using
(37). Hence, C8 is satisfied.

As a result, we can conclude that ${\bf{w}}{(1)}$ is also feasible for Problem
${\cal P}_{X}$. By using the similar method, we can obtain

${\rm{Obj}}({{\bf{w}}{(0)}})\geq{\rm{Obj}}({{\bf{w}}{(1)}})\geq{\rm{Obj}}({{\bf{w}}{(2)}})\geq\cdots.$
(C.9)

Obviously, the objective value of Problem ${\cal P}_{7}$ (also ${\cal P}_{X}$)
is lower bounded by zero. Hence, Algorithm 1 is guaranteed to converge in
objective values.

Next, we prove the second part of the theorem: Given the feasible input
${\bf{w}}(0)$, the solution obtained from Algorithm 1 will converge to a
unique point. Obviously, when $\bf{w}$ is given, ${\bf{u}}$ and ${\bf{q}}$ can
be uniquely determined by using (28) and (29), respectively. Since
$\\{{\bf{G}}_{k},\forall k\\}$ are positive definite matrices, the objective
function in Problem ${\cal P}_{8}$ is a strictly convex function of $\bf{w}$.
Furthermore, it can be easily proved that the constraints in Problem ${\cal
P}_{8}$ are convex. As a result, Problem ${\cal P}_{8}$ is a strictly convex
optimization problem. According to [Page 137 in [35]], the globally optimal
solution is unique. Then, by iteratively updating step 2 to step 4 in
Algorithm 1, the algorithm will converge to a unique solution. We emphasize
that since Problem ${\cal P}_{X}$ is a non-convex optimization problem, it may
have multiple locally optimal solutions and its converged unique solution
depends on the initial point. However, once the initial point is given,
Algorithm 1 will converge to a unique solution.

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