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Citation for published version
Pan, Cunhua and Zhu, Huiling and Gomes, Nathan J. and Wang, Jiangzhou (2017) Joint Precoding
and RRH selection for User-centric Green MIMO C-RAN. IEEE Transactions on Wireless Communications,
16 (5). pp. 2891-2906. ISSN 1536-1276.
DOI
https://doi.org/10.1109/TWC.2017.2671358
Link to record in KAR
http://kar.kent.ac.uk/61753/
Document Version
Author's Accepted Manuscript
1
Joint Precoding and RRH selection for User-centric
Green MIMO C-RAN
Cunhua Pan, Huiling Zhu, Nathan J. Gomes, and Jiangzhou Wang, Fellow, IEEE
Abstract—This paper jointly optimizes the precoding matrices
and the set of active remote radio heads (RRHs) to minimize
the network power consumption (NPC) for a user-centric cloud
radio access network (C-RAN), where both the RRHs and
users have multiple antennas and each user is served by its
nearby RRHs. Both users’ rate requirements and per-RRH power
constraints are considered. Due to these conflicting constraints,
this optimization problem may be infeasible. In this paper, we
propose to solve this problem in two stages. In Stage I, a
low-complexity user selection algorithm is proposed to find the
largest subset of feasible users. In Stage II, a low-complexity
algorithm is proposed to solve the optimization problem with the
users selected from Stage I. Specifically, the re-weighted l
1
-norm
minimization method is used to transform the original problem
with non-smooth objective function into a series of weighted
power minimization (WPM) problems, each of which can be
solved by the weighted minimum mean square error (WMMSE)
method. The solution obtained by the WMMSE method is proved
to satisfy the Karush-Kuhn-Tucker (KKT) conditions of the
WPM problem. Moreover, a low-complexity algorithm based on
Newton’s method and the gradient descent method is developed
to update the precoder matrices in each iteration of the WMMSE
method. Simulation results demonstrate the rapid convergence of
the proposed algorithms and the benefits of equipping multiple
antennas at the user side. Moreover, the proposed algorithm is
shown to achieve near-optimal performance in terms of NPC.
Index Terms—Cloud radio access network (C-RAN), User-
centric network, MIMO systems, User selection, Green commu-
nications.
I. INTRODUCTION
Mobile communications has been developing very rapidly
[2]–[4]. In recent years, C-RAN has been proposed as a
promising solution to support the exponential growth of mobile
data traffic [5], [6]. In C-RAN, all the baseband processing is
performed at the baseband unit (BBU) pool with powerful
computation capacity, while the remote radio heads (RRHs)
perform the basic functionalities of signal processing [7], [8].
The RRHs are geographically distributed away from each
other, but connected to the BBU pool through optical fiber
transport links. Under the C-RAN architecture, centralized sig-
nal processing technologies can be realized. Hence, significant
performance gains can be achieved. In addition, the RRHs
Manuscript received March 16, 2016; revised November 27, 2016 and
February 5, 2017; accepted February 10, 2017. Part of this work has
been presented in IEEE Globecom 2016 [1]. This work is supported by
European Commission Horizon2020 project iCIRRUS under grant agreement
No 644526 and NIRVANA Project.
C.Pan was with the School of Engineering and Digital Arts, University of
Kent, Canterbury, Kent, CT2 7NZ, U.K. (Email:{C.Pan}@kent.ac.uk).
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.
(Email:{H.Zhu, N.J.Gomes, J.Z.Wang}@kent.ac.uk).
can be densely deployed in the network with low operation
cost due to their simple functionalities. This will significantly
reduce the average access distance for the users, and thus
lowers the transmission power.
On the other hand, it was reported that the total energy
consumption of wireless communications contributes more
than 3 percent of the worldwide electrical energy consumption
[9], and this portion is expected to grow in the near future
due to the explosive growth of high-data-rate applications
and mobile devices. Hence, energy efficiency has attracted
extensive interest and becomes one of the main performance
metrics in the future fifth generation (5G) systems [10]. When
a large number of RRHs are deployed in the network, the
network power consumption (NPC) of C-RAN will become
considerable due to the increasing circuit power consumption
of the RRHs. Fortunately, it was reported in [11] that the traffic
load varies substantially over both time and space due to user
mobility and varying channel state. Hence, the NPC can be
significantly reduced by putting some RRHs with light load
into sleep mode while maintaining the quality of service (QoS)
requirements of the users, which is the focus of this paper.
Recently, the NPC minimization problem for C-RAN has
been extensively studied in [12]–[22]. These papers formulated
the joint RRH selection and beamforming vector optimization
problem as a mixed-integer non-linear programming (MINLP)
problem, which has a nonconvex discontinuous l
0
-norm in the
objective function or constraints. We summarize the existing
approaches to solve the MINLP problem as follows. The
first approach was proposed in [12], which first reformulated
the problem as an extended mixed integer second-order cone
programming (SOCP) and then applied the branch-and-cut
method to obtain the optimal solution. In the second approach
in [13], [14], the MINLP was first decomposed into a master
problem and a beamforming subproblem. Then, an iterative
algorithm based on the Benders decomposition was derived to
find the optimal solution. Although these two approaches yield
the optimal solution, they have an exponential complexity. The
third approach is the smooth function method, where the l
0
-
norm was approximated as Gaussian-like function in [15], the
exponential function in [16], and arctangent function in [17].
However, the smooth function cannot produce sparse solutions
in general. The last approach was inspired by the compression
sensing, named re-weighted l
1
-norm minimization method
[23]. This method has been widely adopted in the literature
[18]–[22], [24] due to its low computational complexity and
sparsity guarantee, which will also be applied in this paper.
All of the above papers only considered the single-antenna
user (SAU) case. With the increasing development in antenna
2
technology [25], [26], it is possible to equip the wireless
devices with multiple antennas. When both the transmitter
and the receiver are equipped with multiple antennas, multiple
streams can be transmitted simultaneously, rather than only
one stream in the SAU case. Simulation results show that
with the increasing number of receive antennas, more users
can be admitted. Therefore, in this paper, we consider the
multiple-antenna user (MAU) case and jointly optimize the
precoding matrices and the set of active RRHs to minimize the
NPC subject to users’ rate requirements and per-RRH power
constraints.
Unfortunately, the techniques in [12]–[22] dealing with the
SAU case cannot be extended directly to the MAU case.
The reasons are as follows. Firstly, since the rate constraints
and power constraints are conflicting with each other, this
problem may be infeasible. In the SAU networks, the rate
requirements can be equivalently represented as signal-to-
interference-plus-noise ratio (SINR) constraints, which can be
transformed into an SOCP problem. Hence, the feasibility of
the original problem can be easily checked by solving the
SOCP feasibility problem. However, the rate constraints in
the MAU case is non-convex and much more complex due to
the complicated rate expression, which cannot be transformed
into the SOCP formulation as in the SAU case. Hence, new
techniques need to be developed to check the feasibility of the
original problem. Secondly, even though the original problem
is checked to be feasible, how to solve it is still difficult, since
it cannot be transformed into an SOCP problem as in the SAU
case. [27] proposed the weighted minimum mean square error
(WMMSE) method to solve the rate maximization problem
for MIMO interfering broadcast channels, where the rate
expression is in the objective function. Recently, there have
been some work in applying the WMMSE method to solve
the energy efficiency (measured in bit/s/Joule) optimization
problems under rate constraints [28], [29]. However, these
researches have not addressed the feasibility problem due to
the incorporated rate constraints. Only in [29], a heuristic
method was proposed to check the feasibility based on the in-
terference alignment technique, under the assumption that the
transmit power is approaching infinity, which in not practical.
Since the problem considered in this paper imposes power
constraints at each RRH, the heuristic method developed
in [29] is not applicable. More importantly, they have not
revealed the hidden property of applying WMMSE method
to the optimization problem with rate constraints, such as the
convergence property and the optimality of the solutions.
To the best of our knowledge, this paper is the first attempt
to solve the joint RRH and precoding optimization problem
to minimize the NPC in the MAU based user-centric C-RAN,
where each user can be served by an arbitrary subset of
RRHs. Due to the conflicting constraints, this problem may
be infeasible. Some users should be removed or rescheduled
for the next transmission to guarantee the rate requirements
of other users. We provide a comprehensive analysis for this
problem by considering two stages: user selection in Stage I
and algorithm design in Stage II. The main contributions of
this paper are summarized as follows:
1) In Stage I, a low-complexity user selection approach
is proposed to maximize the number of admitted users
that can have their QoS requirements satisfied. Specif-
ically, in each step we solve an alternative problem
by introducing a series of auxiliary variables. This
alternative problem is always feasible. By replacing the
rate expression in the constraints with its lower-bound,
an iterative algorithm is proposed to solve this problem
along with the complexity and convergence analysis of
the algorithm. The alternative problem should be solved
at most K times, where K is the total number of
users. Its complexity is much lower than the optimal
exhaustive user selection method that has an exponential
complexity. Simulation results show that both algorithms
achieve similar performance.
2) In Stage II, a low-complexity algorithm is proposed to
solve the NPC minimization problem with the users
selected from Stage I. Specifically, the re-weighted l
1
-
norm minimization method is adopted to convert the
non-smooth optimization problem into a series of s-
mooth weighted power minimization (WPM) problems.
We again replace the rate expression with its lower-
bound and adapt the WMMSE algorithm originally
designed for a rate maximization problem to solve the
WPM problem. In addition, we strictly prove that when
the WMMSE algorithm is initialized with a feasible
solution, the sequences of precoder matrices generated
in the iterative procedure will finally converge to the
Karush-Kuhn-Tucker (KKT) point of the WPM problem.
3) In each iteration of the WMMSE algorithm, there is
a subproblem for the precoder matrices being updated
with some other fixed variables. Most existing papers
[21], [28]–[31] directly transform it into an SOCP prob-
lem and apply the interior point method [32] to solve
it, which may incur high computational complexity.
In this paper, we go one step further and develop a
low-complexity algorithm to solve this subproblem by
exploiting its special structure. Specifically, we equiva-
lently solve its dual problem because the subproblem is
a convex problem. Fortunately, the objective function
of the dual problem is differentiable, and the block
coordinate descent (BCD) method is adopted to solve
the dual problem. In each iteration of the BCD method,
Newton’s method and the gradient descent method are
applied to update the Lagrangian multipliers. It is strictly
proved that the BCD method can obtain the globally
optimal solution of the subproblem. Complexity analysis
in conjunction with the simulation results show that the
BCD method has a much lower computational complex-
ity than the interior point method.
This paper is organized as follows. In Section II, we
introduce the system model and formulate the optimization
problems. In Section III, a new approach is introduced to select
the maximum number of admitted users. An iterative algorithm
with low complexity is provided in Section IV. Simulation
results are presented in Section V. Conclusions are drawn in
Section VI.
Notations: Uppercase and lowercase boldface denote matri-
3
Fronthaul Links
BBU Pool
BBU
BBU BBU
8VHU
55+
55+8VHU
8VHU
8VHU
8VHU
8VHU
55+
55+
55+
55+
55+
55+55+
55+
55+
55+
´
´
55+
Fig. 1. Illustration of a C-RAN with thirteen RRHs and six users, where user-
centric clustering technique is adopted. In this example, each user is served
by its nearby RRHs within the dotted circle centered at itself. The RRHs that
are not in any users’ candidate set are turned into idle mode, such as RRH3
and RRH 5.
ces and vectors, respectively. For a matrix A, ∥A∥
F
denotes
the Frobenius norm of A and A
H
represents the Hermitian
transpose of A. I
m
denotes a m × m identity matrix. For a
vector a, diag( a) denotes the diagonal matrix with diagonal
elements given by a. blkdiag(·) represent the block-diagonal
matrices. E(·), and Tr(·) represent expectation, trace operators,
respectively. A ≽ B means A − B is a positive semidefinite
matrix. For vector a ∈ C
n×1
, ∥a∥
2
is the Euclidean nor-
m. CN
0, σ
2
I
represents the complex circularly symmetric
Gaussian distribution with zero mean vector and covariance
matrix σ
2
I. For a vector x, ∥x∥
0
is l
0
-norm, means the number
of nonzero entries in a vector.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System model
Consider a downlink C-RAN consisting of I RRHs and
K users
1
, where each RRH is equipped with M transmit
antennas and each user has N receive antennas, as shown in
Fig. 1. Denote the set of RRHs and users as I = {1, ··· , I}
and
¯
U = {1, ··· , K}, respectively. It is assumed that each
RRH is connected to the BBU pool via fronthaul link and the
BBU pool has access to all users’ CSI and data information.
Let U ⊆
¯
U be the set of users that can be admitted to
this networks. To reduce the computational complexity of the
dense network, the user-centric clustering method is adopted,
where each user k ∈ U is assumed to be served by its nearby
RRHs since the distant RRHs contribute less to user’s signal
quality due to the large path loss. The unselected RRHs are
turned into idle mode, such as RRH 3 and RRH 5 in Fig. 1.
1
In dense networks, the number of RRHs may be larger than the number
of users so that the average distance between serving RRHs and users can
be significantly reduced, leading to improved performance. In some extreme
cases, each user may be served by its dedicated RRHs as in [33], [34], where
each RRH serves only one user.
Let I
k
⊆ I and U
i
⊆ U be the candidate set of RRHs for
serving user k and candidate set of users served by RRH i,
respectively. Note that the set of RRHs serving the users may
overlap with each other. For example, in Fig. 1, RRH 12 jointly
serves user 1 and user 6.
Denote V
i,k
∈ C
M×d
as the precoding matrix used by the
ith RRH to transmit data vector s
k
∈ C
d×1
to the kth user,
where d is the number of data streams for each user, and s
k
satisfies E
s
k
s
H
k
= I
d
and E
s
k
s
H
l
= 0, for l = k. Let
¯
V
k
=
V
H
i,k
, ∀i ∈ I
k
H
∈ C
|I
k
|M×d
be the big precoding
matrix for user k from all RRHs in I
k
. In addition, define a set
of new channel matrices
¯
H
j,k
= [H
i,k
, ∀i ∈ I
j
] ∈ C
N×|I
l
|M
,
representing the overall CSI from RRHs in I
j
to user k, where
H
i,k
∈ C
N×M
denotes the channel matrix from the ith RRH
to the kth user. Then, the received signal vector at the kth
user, denoted as y
k
∈ C
N×1
, is given by
y
k
=
¯
H
k,k
¯
V
k
s
k
+
j∈U,j=k
¯
H
j,k
¯
V
j
s
j
+ n
k
,
(1)
where n
k
is the noise vector at the kth user, which satisfies
CN
0, σ
2
k
I
N
. Then, the achievable rate (nat/s/Hz) of the kth
user is given by [35]
R
k
(V) = log
I +
¯
H
k,k
¯
V
k
¯
V
H
k
¯
H
H
k,k
J
−1
k
, (2)
where log(·) is the base of natural logarithm, J
k
=
j∈U,j=k
¯
H
j,k
¯
V
j
¯
V
H
j
¯
H
H
j,k
+ σ
2
k
I is the interference-plus-
noise covariance matrix, and V is the collection of all pre-
coding matrices. Each user’s data rate should be larger than
the minimum requirement:
C1 : R
k
(V) ≥ R
k,min
, ∀k ∈ U. (3)
With densely deployed RRHs, the power consumption on
the RRHs and the corresponding fronthaul links may be
significant. Switching off some RRHs and the corresponding
fronthual links may be a good option to reduce the NPC. To
this end, it is critical to model the NPC.
B. NPC model
The realistic NPC model should consist of three parts:
power consumption at the RRHs, that at the fronthaul links
and that at the BBU pool.
As in [18], the power consumption at RRH i can be modeled
as follows:
P
rrh
i
(V) =
η
i
P
tr
i
(V) + MP
a,rrh
i
, if P
tr
i
(V) > 0
MP
s,rrh
i
, if P
tr
i
(V) = 0
(4)
where η
i
> 1 accounts for the inefficiency of the power
amplifier of RRH i, P
tr
i
(V) is the total transmit power of
RRH i given by P
tr
i
(V)=
k∈U
i
∥V
i,k
∥
2
F
that satisfies the
power constraint:
C2 : P
tr
i
(V) ≤ P
i,max
, ∀i ∈ I, (5)
P
a,rrh
i
and P
s,rrh
i
represent the power consumption for each
antenna (or each RF chain) when RRH i is in active mode
and sleep mode, respectively. In practical systems, P
active
i
is
much higher than P
sleep
i
, which motivates us to switch off
some RRHs.
4
In general, more power consumption will be consumed
on the fronthaul links when they support high data rates. In
[22], this power was modeled to be proportional to the total
fronthaul data rate. We modify the model in [22] to account
for the power when the fronthaul links are in the sleep mode
as follows:
P
fr
i
(V) =
ρ
i
k∈U
i
R
k
(V) + P
a,fr
i
, if P
tr
i
(V) > 0,
P
s,fr
i
, if P
tr
i
(V) = 0.
(6)
where ρ
i
is the proportional factor for fronthaul link i. The
power consumed in the BBU pool mainly depends on the
computational complexity for signal processing. However, how
to accurately model this kind of power consumption is still not
fully understood. As in most papers [12], [18], [19], [22], the
BBU power consumption is modeled as a constant P
BBU
for
simplicity. Let A denote the active RRH set. Then, the NPC
can be modeled as
ˆ
P (A, V) =
i∈I
P
rrh
i
(V) + P
fr
i
(V)
+ P
BBU
(7)
=
i∈A
η
i
P
tr
i
(V) + ρ
i
k∈U
i
R
k
(V) + P
c
i
+
i∈I
P
s
i
+ P
BBU
, (8)
where P
c
i
and P
s
i
are two constants, given by P
c
i
=
M(P
a,rrh
i
−P
s,rrh
i
)+P
a,fr
i
−P
s,fr
i
and P
s
i
= MP
s,rrh
i
+P
s,fr
i
.
C. Problem Formulation
Due to the power constraints C2, the rate requirements
C1 may not be satisfied for all users. Some users should be
removed to make the optimization problem feasible. Hence, we
formulate a two-stage optimization problem. In Stage I, one
should maximize the number of admitted users that can be
supported by the system; in Stage II, one should jointly select
some RRHs and optimize the precoding matrices to minimize
the NPC with the selected users from Stage I.
Specifically, the optimization problem in Stage I can be
formulated as
max
V,U⊆
U
|U|
s.t. C1, C2.
(9)
Then in Stage II, we aim to jointly select the RRHs and
optimize the precoding matrices to minimize the NPC with
the users selected from Stage I, which can be formulated as
2
min
A,V
i∈A
η
i
P
tr
i
(V) + ρ
i
k∈U
⋆
i
R
k
(V) + P
c
i
s.t. C1,
k∈U
⋆
i
∥V
i,k
∥
2
F
≤ P
i,max
, i ∈ A, (10a)
k∈U
⋆
i
∥V
i,k
∥
2
F
= 0, i ∈ I\A, (10b)
where U
⋆
i
is the solution from Stage I. Note that when the
system parameters are given, the last two terms in (8) are
constants, and are omitted in the objective function.
2
In general, the number of transmit antennas should be optimized to
additionally reduce the NPC as seen in the RRH power consumption model
in (4). However, the resulting problem will be much more difficult to solve,
and will be left for future work.
Both the optimization problems in the two stages are
MINLP problems and are difficult to solve. The intuitive
approach to solve this kind of problems is through the ex-
haustive search. For example, to solve the NPC minimization
problem in Stage II, one must solve the precoding matrices
that minimizes the NPC with each given A and obtain the
corresponding objective value. Finally, the A that achieves
the minimum NPC together with the corresponding precoding
matrices is the optimal solution of Problem (10). However,
the exhaustive search has exponentially prohibitive complexity
with respect to the number of RRHs, which is hard to be
implemented in practice in dense C-RANs. The same issue
holds for the user selection problem in Stage I, where the
exhaustive search method has an exponential complexity of
the number of users. Hence, this motivates us to develop low-
complexity algorithms to solve these two Problems.
III. STAGE I: LOW- COMPLEXITY USER SELECTION
ALGORITHM
In this section, we provide a low-complexity user selection
algorithm to guarantee the rate requirements of other users.
Specifically, for an arbitrary given subset of users U, we
construct an alternative problem by introducing a series of
auxiliary variables {α
k
}
k∈U
:
min
{α
k
}
k∈U
,V
k∈U
(α
k
− 1)
2
s.t. C2, R
k
(V) ≥ α
2
k
R
k,min
, ∀k ∈ U,
(11)
Obviously, Problem (11) is always feasible and the optimal
α
k
for each user k should be no larger than one. This can
be easily proved by contradiction. Moreover, user k can be
admitted if and only if the optimal α
k
is equal to one. Hence,
maximizing the number of admitted users is equal to finding
the largest subset of users U, in which all {α
k
}
k∈U
are equal
to one.
Based on the above analysis, we provide a low-complexity
user selection (USC) algorithm to solve Problem (9) in Stage
I. The main idea is to remove each user with the least α
k
< 1
in each iteration. It is intuitive since the user with the least α
k
has the largest gap to its rate target.
Algorithm 1 USC Algorithm
1: Initialize the set of users U = {1, ··· , K};
2: Given U, solve Problem (11) by Algorithm 2 in Subsection
III-A to obtain {α
k
}
k∈U
and V;
3: If α
k
= 1, ∀k ∈ U, output V and U
∗
= U for the
initialization of Stage II and terminate; Otherwise, find
k
∗
= arg min
k∈U
α
k
, remove user k
∗
and update U =
U\k
∗
, go to step 2.
A. Algorithm to solve Problem (11)
In step 2 of Algorithm 1, Problem (11) needs to be solved.
Due to constraints C3 in (11), Problem (11) is a non-convex
problem, which is difficult to solve. To handle this difficulty,
we apply the relationships between WMMSE and the rate
expression.