Coordinated Beamforming for the Multi-Cell
Multi-Antenna Wireless System
Hayssam Dahrouj and Wei Yu
Department of Electrical and Computer Engineering
University of Toronto, Toronto, ON, Canada
Emails: {hayssam,weiyu}@comm.utoronto.ca
Abstract—In a conventional wireless cellular system, signal
processing is performed on a per-cell basis; out-of-cell inter-
ference is treated as background noise. This paper considers
the benefit of coordinating base-stations across multiple cells
in a multi-antenna beamforming system, where multiple base-
stations may jointly optimize their respective beamformers to
improve the overall system performance. This paper focuses
on a downlink scenario where each remote user is equipped
with a single antenna, but where multiple remote users may
be active simultaneously in each cell. The design criterion is
the minimization of the total weighted transmitted power across
the base-stations subject to signal-to-interference-and-noise-ratio
(SINR) constraints at the remote users. The main contribution
is a practical algorithm that is capable of finding the joint
optimal beamformers for all base-stations globally and efficiently.
The proposed algorithm is based on a generalization of uplink-
downlink duality to the multi-cell setting using the Lagrangian
duality theory. The algorithm also naturally leads to a distributed
implementation. Simulation results show that a coordinated
beamforming system can significantly outperform a conventional
system with per-cell signal processing.
I. INTRODUCTION
Conventional wireless systems are designed with a cellular
architecture in which base-stations from different cells com-
municate with their respective remote terminals independently.
Signal processing is performed on a per-cell basis; intercell
interference is treated as background noise. Conventional cel-
lular systems are typically designed to be intercell-interference
limited. Consequently, the performance of conventional sys-
tems can be significantly improved if joint signal processing
is enabled across the different base-stations to minimize or
even to cancel inter-cell interference.
This paper evaluates the benefit of a particular type of
base-station coordination for the multi-cell downlink system.
The focus here is a scenario in which the base-stations are
equipped with multiple antennas and the remote receivers
are equipped with a single antenna each. Within each cell,
multiple remote users may be active simultaneously and are
separated via spatial multiplexing using beamforming. In a
conventional system, the beamforming vectors in each cell
are set indepedently. The main point of this paper is that
significant performance gain is possible if the beamforming
vectors for different base-stations are optimized jointly.
Downlink beamforming for multi-antenna wireless systems
has been studied extensively in the past. A concept known
as uplink-downlink duality has emerged as a main tool for
the beamforming problem. In particular, Rashid-Farrokhi, Liu
and Tassiulas [1] proposed an iterative algorithm to design the
transmit beamforming vectors and power allocations to satisfy
a target SINR for an arbitrary set of transmission links. Their
main contribution is a beamformer-power update algorithm
based on uplink-downlink duality that converges to a feasible
solution to the problem. In the single-cell multi-user downlink
case, the optimality of their algorithm was later proved by
Visotsky and Madhow [2] and Schubert and Boche [3], [4].
Recently, Wiesel, Eldar and Shamai [5] showed that the single-
cell downlink beamforming problem can be formulated as a
second-order cone-programming problem. This crucial insight
allows a new interpretation of duality via Lagrangian theory
in convex optimization [6].
This paper further generalizes the above series of work
by rigorously establishing an uplink-downlink duality for
the multicell multi-user case. It is shown that the multi-cell
downlink problem for minimizing the total weighted trans-
mit power subject to received signal-to-noise-and-interference-
ratio (SINR) constraints can be solved via a dual uplink prob-
lem. A main contribution of this paper is a novel algorithm,
which is capable of efficiently finding the globally optimal
downlink beamforming vector across all base-stations. This
algorithm is a multi-cell generalization of a similar algorithm
proposed in [5] for the single-cell case. A key advantage of the
proposed algorithm as compared to previous solutions based
on beamformer-power update [1] is that the new algorithm
leads naturally to a distributed implementation.
The multi-cell uplink-downlink duality considered in this
paper is related to the concept of network duality proposed by
Song, Cruz and Rao [7], where a general setting with multiple
antennas at both the transmitter and the receivers is considered.
However, the approach in [7] does not allow multiple data
streams per transmitter. Consequently, the network duality of
[7] reduces to a simpler linear programming duality. The
problem setting of this work is also related to the fully
coordinated multi-cell system considered in [8], [9], [10], [11]
in which multiple base-stations are considered as a single
antenna array for transmitting multiple data streams to all
users. The approach proposed in this paper is a first step
toward this vision. As the simulation results of this paper show,
significant performance gain can already be obtained with a
beamformer-level coordination, which is much more practical
to implement than full signal-level coordination.
Fig. 1. A wireless network with seven base-stations and three users per cell.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. Channel Model
This paper considers a multi-cell multiuser spatial multiplex
system with N cells and K users per cell with N
t
antennas
at each base-station and a single antenna at each remote user.
Multiuser transmit beamforming is employed at each base-
station. Let x
i,j
be a complex scalar denoting the information
signal for the jth user in the ith cell, and w
i,j
∈C
N
t
×1
be
its associated beamforming vector. The channel model can be
written down as follows:
y
i,j
=
l
h
H
i,i,j
w
i,l
x
i,l
+
m=i,n
h
H
m,i,j
w
m,n
x
m,n
+ z
i,j
(1)
where y
i,j
∈Cis the received signal at the jth remote user in
the ith cell, h
l,i,j
∈C
N
t
×1
is the vector channel from the base-
station of the lth cell to the jth user in the ith cell, and z
i,j
is the additive white Gaussian complex noise with variance
σ
2
/2 on each of its real and imaginary components. Fig. 1
illustrates the system model for a network with seven cells
and three users per cell.
B. Problem Formulation
The beamformer design problem in this paper consists
of minimizing total transmit power across all base-stations
subject to SINR constraints at the remote users. In practice,
as each base-station has its own power constraint, it is useful
to consider a more general problem of minimizing a weighted
total transmit power, with the power at the ith base-station
weighted by a factor α
i
.
With w
i,j
as the beamforming vectors, the SINR for the jth
user in the ith cell can be expressed as:
Γ
i,j
=
|w
H
i,j
h
i,i,j
|
2
l=j
|w
H
i,l
h
i,i,j
|
2
+
m=i,n
|w
H
m,n
h
m,i,j
|
2
+ σ
2
(2)
Let γ
i,j
be the SINR target for the jth user in the ith cell. The
transmit power minimization problem can then be formulated
as
minimize
i,j
α
i
w
H
i,j
w
i,j
(3)
subject to Γ
i,j
≥ γ
i,j
, ∀i =1···N, j =1···K
where the minimization is over the w
i,j
’s.
The SINR target constraints in (3) may appear to be
nonconvex at a first glance. However, in a study of single-cell
downlink beamforming problem, [5] showed that nonconvex
constraints of this type can be transformed into a second-order-
cone constraint. This crucial observation enables methods for
solving (3) via convex optimization.
C. Conventional Systems
In a conventional wireless cellular system, the multiuser
beamforming problem is solved on a per-cell basis; out-of-
cell interference is regarded as a part of background noise.
In particular, for a fixed base-station
ˆ
i, a conventional system
finds the optimal set of w
ˆ
i,j
, j =1···K, assuming that all
other (N − 1)K beamformers are fixed:
minimize
j
w
H
ˆ
i,j
w
ˆ
i
,j
(4)
subject to Γ
ˆ
i
,j
≥ γ
ˆ
i
,j
, ∀j =1···K
where Γ
ˆ
i,j
is given by (2). This single-cell downlink problem
has a classic solution as given in [1], [2], [3], [5].
Note that in a conventional system, the choice of beam-
formers at each base-station affects the background noise level
at neighboring cells, and hence the setting of beamformers
in neighboring base-stations. Thus, the above per-cell opti-
mization is in practice performed iteratively until the system
converges to a per-cell optimal solution.
D. Motivating Example for Joint Optimization
One of the main points of this paper is that the per-
cell optimization above does not necessarily lead to a joint
optimal solution. Significant performance improvement may
be obtained if base-stations coordinate in jointly optimizing
all of their beamformers at the same time. The following
motivating example illustrates this point.
Consider a multi-cell network but with only a single user per
cell. The per-cell optimization reduces to the optimal transmit
beamforming problem for a multi-input single-output (MISO)
system with a background noise level which includes out-
of-cell interference. Note that regardless of the level of the
background noise, the optimal per-cell transmit beamformer
is a vector that matches the channel. Thus, in this example,
per-cell optimization across the cells converges in one iteration
– every base-station uses a transmit beamformer that matches
the MISO channel.
This channel-matching solution is not necessarily the joint
optimum. For example, when two users belonging to two
different cells are near each other at the cell edge, it may
be advantageous to steer the beamforming vectors for the
two base-stations away from each other so as to minimize
the mutual interference. Such a joint optimal beamforming
solution may lead to higher received SINRs at a fixed transmit
power, or conversely a lower transmit power at fixed SINRs.
One of the first algorithms for solving the multi-cell
joint beamforming optimization problem is given by Rashid-
Farrokhi, Liu and Tassiulas [1]. They showed that the optimal
downlink beamforming problem under SINR constraints can
be solved efficiently by an iterative uplink beamformer and
power update algorithm. It is well known that the uplink
beamforming problem is much easier to solve [12]. Thus, by
transforming the downlink problem into the uplink domain,
the downlink problem may be solved efficiently as well.
The global optimality of the beamformer-power iteration
algorithm has been shown for the single-cell case in [2], [3],
[5]. This paper will first give a rigorous derivation of duality
for the multi-cell case, then propose a new algorithm for
solving the joint multi-cell downlink beamforming problem.
III. U
PLINK-DOWNLINK DUALITY FOR MULTI-CELL
SYSTEMS
Uplink-downlink duality refers to the fact that the minimum
transmit power needed to achieve a certain set of SINR
constraints in a downlink channel is the same as the minimum
total transmit power needed to achieve the same set of SINR
targets in an uplink channel, where the uplink channel is
obtained by reversing the input and the output of the downlink.
This paper establishes uplink-downlink duality for a multi-cell
network. The development here uses a Lagrangian technique,
similar to the approach used in [6].
Theorem 1: The optimal transmit beamforming problem
(3) for the downlink multiuser multi-cellular network can be
solved via a dual uplink channel in which the SINR constraints
remain the same and the noise power is scaled by α
i
. More
precisely, a Lagrangian dual of the optimization problem (3)
is the following minimization problem:
minimize
i,j
λ
i,j
σ
2
(5)
subject to Λ
i,j
≥ γ
i,j
where the minimization is over λ
i,j
, and
Λ
i,j
=max
ˆw
i,j
λ
i,j
| ˆw
H
i,j
h
i,i,j
|
2
(m,l)=(i,j)
λ
m,l
| ˆw
H
i,j
h
i,m,l
|
2
+ α
i
|| ˆw
i,j
||
2
Further, the optimal ˆw
i,j
has the interpretation of being the
receiver beamformer of the dual uplink channel, and is a
scaled version of the optimal w
i,j
. The optimal λ
i,j
has
the interpretation of being the dual uplink power, and it
corresponds to the dual variable associated with the SINR
constraint of (3).
Proof: The proof hinges upon the fact that the SINR
constraints can be reformulated as a second-order cone-
programming problem as shown in [5]. Therefore, strong
duality holds for (3). This allows us to characterize the solution
of (3) via its Lagrangian:
L(w
i,j
,λ
i,j
)=
i,j
α
i
w
H
i,j
w
i,j
−
i,j
λ
i,j
|w
H
i,j
h
i,i,j
|
2
γ
i,j
−
l=j
|w
H
i,l
h
i,i,j
|
2
−
m=i,n
|w
H
m,n
h
m,i,j
|
2
− σ
2
(6)
Rearranging (6), we get:
L(w
i,j
,λ
i,j
)=
i,j
λ
i,j
σ
2
+
i,j
w
H
i,j
α
i
I−
1+
1
γ
i,j
λ
i,j
h
i,i,j
h
H
i,i,j
+
m,n
λ
m,n
h
i,m,n
h
H
i,m,n
w
i,j
(7)
The dual objective is
g(λ
i,j
)=min
w
i,j
L(w
i,j
,λ
i,j
) (8)
It is easy to see that if α
i
I −
1+
1
γ
i,j
λ
i,j
h
i,i,j
h
H
i,i,j
+
m,n
λ
m,n
h
i,m,n
h
H
i,m,n
is not a positive definite matrix, then
there exists a set of w
i,j
that would make g(λ
i,j
)=−∞.
Thus, the Lagrangian dual of (3), which is the maximum of
g(λ
i,j
),is
maximize
i,j
λ
i,j
σ
2
(9)
subject to Σ
i
1+
1
γ
i,j
λ
i,j
h
i,i,j
h
H
i,i,j
where
Σ
i
α
i
I +
m,n
λ
m,n
h
i,m,n
h
H
i,m,n
(10)
Next, we show that the above dual is equivalent to (5). The
problem (5) corresponds to an uplink channel with receive
beamformers ˆw
i,j
, where the noise power of the dual channel
is scaled by α
i
. The optimal receive beamformers ˆw
i,j
that
maximize the SINR are the minimum-mean-squared-error
(MMSE) receivers, which can be expressed as:
ˆw
i,j
=
⎛
⎝
m,l
λ
m,l
σ
2
h
i,m,l
h
H
i,m,l
+ α
i
σ
2
I
⎞
⎠
−1
h
i,i,j
(11)
Plugging back ˆw
i,j
into the SINR constraint of (5), one can
show that the SINR constraint is equivalent to
α
i
I +
m,n
λ
m,n
h
i,m,n
h
H
i,m,n
1+
1
γ
i,j
λ
i,j
h
i,i,j
h
H
i,i,j
Thus, one can rewrite (5) as follows:
minimize
i,j
λ
i,j
σ
2
(12)
subject to Σ
i
1+
1
γ
i,j
λ
i,j
h
i,i,j
h
H
i,i,j
Note that the problems in (9) and (12) are identical except
that the maximization is replaced by minimization and the
inequality constraints are reversed. It can be shown that
the optimal solutions for both problems are such that the
constraints are satisfied with equality. Thus, (9) and (12) give
the same solutions.
In addition, it can be shown that w
i,j
and ˆw
i,j
are scaled
versions of each other. Thus, one would also be able to find
w
i,j
by first finding ˆw
i,j
, then updating it through scalar
multiples named δ
i,j
below:
w
i,j
=
δ
i,j
ˆw
i,j
(13)
It can be shown that these δ
i,j
can be found through a matrix
inversion. Details derivation of this scaling factor can be found
in [6].
IV. OPTIMAL DOWNLINK BEAMFORMING ALGORITHM
The derivation of uplink-downlink duality via Lagrangian
theory forms the basis for numerical algorithms for computing
the optimal downlink beamformers for the multi-cell system.
Our main algorithm is based on an idea of iterative function
evaluation, first proposed for the single-cell case in [5]. This
paper generalizes the algorithm to a multi-cell system.
A. Iterative Function Evaluation Algorithm
The main idea is to solve the downlink beamforming
problem in the dual uplink domain by first finding the optimal
λ
i,j
, then the corresponding ˆw
i,j
. To find the optimal λ
i,j
,we
first take the gradient of the Lagrangian (7) with respect to
w
i,j
and set it to zero:
α
i
I − (1 +
1
γ
i,j
)λ
i,j
h
i,i,j
h
H
i,i,j
+
m,n
λ
m,n
h
i,m,n
h
H
i,m,n
w
i,j
=0. (14)
Thus
Σ
i
w
i,j
=
1+
1
γ
i,j
λ
i,j
h
i,i,j
h
H
i,i,j
w
i,j
(15)
where Σ
i
is as defined in (10).
Now, multiply both sides by h
H
i,i,j
Σ
−1
i
, we get:
h
H
i,i,j
w
i,j
=
1+
1
γ
i,j
λ
i,j
h
H
i,i,j
Σ
−1
i
h
i,i,j
h
H
i,i,j
w
i,j
(16)
Finally, divide both sides of the equation by h
H
i,i,j
w
i,j
,we
obtain a necessary condition for optimal λ
i,j
:
λ
i,j
=
1
1+
1
γ
i,j
h
H
i,i,j
Σ
−1
i
h
i,i,j
(17)
which can be used iteratively to obtain the optimal λ
i,j
.
The algorithm is summarized as follows:
1) Find the optimal uplink power allocation λ
i,j
using the
iterative function evaluation:
λ
i,j
=
1
1+
1
γ
i,j
h
H
i,i,j
Σ
−1
i
h
i,i,j
(18)
where
Σ
i
= α
i
I +
m,n
λ
m,n
h
i,m,n
h
H
i,m,n
(19)
2) Find the optimal uplink receive beamformers based on
the optimal uplink power allocation λ
i,j
:
ˆw
i,j
=
⎛
⎝
m,l
λ
m,l
σ
2
h
i,m,l
h
H
i,m,l
+ σ
2
α
i
I
⎞
⎠
−1
h
i,i,j
(20)
3) Find the optimal transmit downlink beamformers by
scaling ˆw
i,j
:
w
i,j
=
δ
i,j
ˆw
i,j
(21)
The global convergence of this algorithm is guaranteed by
both the duality result discussed in the previous section and
the convergence of the iterative function evaluation which can
be justified by a line of reasoning similar to that in [5]. The
proof is based on the property of standard functions [13]. In
particular, one can stack the dual variables λ
i,j
into one vector
Υ. Then (18) and be rewritten as
λ
(t+1)
i,j
= f
i,j
(Υ
(t)
),i=1···N, j =1···K (22)
The function f satisfies the following properties:
1) If λ
i,j
≥ 0 ∀i, j, then f
i,j
(Υ) > 0.
2) If λ
i,j
≥ λ
i,j
∀i, j, then f
i,j
(Υ) ≥ f
i,j
(Υ
)
3) For ρ>1,wehaveρf
i,j
(Υ) >f
i,j
(ρΥ) ∀i, j.
The proof for these three properties is included in the
Appendix. These properties guarantee that f is a standard
function as defined in [13]. Thus, starting with some initial
Υ
(0)
, the iterative function evaluation algorithm converges to
a unique fixed point, which must be the optimal downlink
power.
B. Comparison with Beamformer-Power Iteration Algorithm
The iterative function evaluation algorithm is based on
finding the optimal λ
i,j
independent of the beamformers. In
[1], [12], Rashid-Farrokhi, Tassiulas and Liu proposed the fol-
lowing beamformers-power iteration algorithm for the uplink,
which, by our previous duality result, must also converge to
the global optimal solution for the downlink:
1) Initialize ˆw
i,j
;
2) Find the λ
i,j
to satisfy the SINR constraints of (5) with
equality;
3) Find the optimal uplink receive beamformers based on
the optimal uplink power allocation λ
i,j
:
ˆw
i,j
=
⎛
⎝
m,l
λ
m,l
σ
2
h
i,m,l
h
H
i,m,l
+ σ
2
α
i
I
⎞
⎠
−1
h
i,i,j
(23)
4) Go to step 2 until convergence;
5) Update the transmit downlink beamformers
w
i,j
=
δ
i,j
ˆw
i,j
(24)
Note that the convergence of the iterations involving steps 2
and 3 was shown in [12].
Both the iterative function evaluation algorithm and the
beamformer-power iteration algorithm provide the optimal
solution for the multi-cell downlink beamforming problem.
However, the iterative function evaluation algorithm has a key
advantage – it can be implemented in a distributed fashion.
Consider the dual uplink channel. The function iteration (18)
for uplink power λ
i,j
involves channel vectors h
i,i,j
within
each cell, which the base-station typically has the knowledge
of, and the matrix Σ
i
. Observe that for the uplink channel,
Σ
i
is precisely the covariance matrix of the received signal
at the base-station i, which includes the intended signal,
the interference, and the background noise. This covariance
matrix may be estimated locally at each base-station. Thus,
the iterative function evaluation for λ
i,j
can be performed
locally, assuming that all other λ
i,j
’s are fixed. Base-station
coordination is achieved via power control (i.e. the update of
λ
i,j
’s, which affect all other Σ
i
’s.)
In fact, these uplink per-cell updates can even be imple-
mented asynchronously with each base-station using possibly
outdated power information. The convergence of such asyn-
chronous update is still guaranteed by the standard function
argument as shown in Theorem 4 of [13].
The interpretation that uplink per-cell updates are exactly
the global optimum is particularly useful for time-division-
duplex (TDD) systems, where uplink and downlink trans-
missions are reciprocals of each other. In such a system,
beamformer and power updates (18) can in fact be done
directly in the uplink on a per-cell basis. These uplink per-
cell iterations always converge. They converge to the global
optimum of the uplink system, which by duality, is also the
global optimum for downlink.
Interestingly, uplink-downlink duality holds for the coordi-
nated multi-cell beamforming problem, but it does not hold for
the per-cell algorithms. The uplink per-cell algorithm provides
the multi-cell optimum; the downlink per-cell algorithm does
not.
V. S
IMULATIONS
This section presents the simulation results for the beam-
forming design problem for a 7-cell network with 3 users per
cell as shown in Fig. 1. Each base-station is equipped with 4
antennas. Standard WiMax parameters are used in simulation:
the noise power spectral density is set to -162 dBm/Hz;
the channel vectors are chosen according to the distance-
dependent path loss L = 128.1+37.6log
10
(d), where d is
the distance in kilometers, with 8dB log-normal shadowing,
and a Rayleigh component. The distance between neighboring
base-stations is set to be 2.8km and the locations of remote
users are chosen at random within each cell. An antenna gain
of 15dBi is assumed. For illustration purposes, the weighting
factors α
i
corresponding to base-station power constraints are
set to be: α
1
= α
2
= ···= α
7
=1.
Fig. 2 shows a plot of the minimum total transmit power
(in dBm) over all base-stations versus the SINR target at the
0 2 4 6 8 10 12 14 16 18 20 2
2
5
10
15
20
25
30
35
40
45
50
55
SINR Target in dB
Total Transmitted Power in dBm
Joint Optimization Algorithm
Per−cell Update Algorithm
Fig. 2. Plot of the total transmitted power versus the SINR targets for both
the joint optimization of beamformers and the per-cell udate algorithm for a
wireless network with seven cells and three users per cell.
remote users. It is observed that while the joint optimization
algorithm has the same performance as the conventional
per-cell update in low SINRs, it offers significantly better
performance at high SINRs. This is due to the fact that at
high SINRs, the multi-cell network becomes predominantly
interference limited. This is the regime in which the joint
optimization approach shows a clear advantage.
To illustrate the convergence behavior of the algorithms,
Fig. 3 plots the norm residue of the uplink transmitted power
(in mW) versus the number of iterations. The norm residue is
defined as:
R
(n)
= σ
2
||Υ
(n)
− Υ
∗
||
2
(25)
where Υ
∗
represents the optimal power vector.
It is observed that while the beamformer-power update
algorithm converges more rapidly than the iterative function
evaluation algorithm at the beginning, the iterative function
evaluation algorithm eventually provides faster convergence.
VI. C
ONCLUSION
This paper provides a solution for the optimal downlink
beamforming design problem for a multi-cell network with
multiple users per cell. Both the uplink and downlink problems
are solved by generalizing uplink-downlink duality to the
multi-cell case using the Lagrangian theory. An iterative func-
tion evaluation algorithm which is capable of finding the global
optimum solution is presented. The algorithm is efficient, and
it can be implemented in a distributed fashion. The distributed
solution outperforms conventional wireless systems with per-
cell signal processing.
A
PPENDIX
This appendix presents the proof of standard function prop-
erties satisfied by f. The proof is similar to the one presented
in [5], and is included here for completeness.