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Title Multigroup Multicast Beamformer Design for MISO-OFDM With Antenna Selection
Authors(s) Venkatraman, Ganesh; Tolli, Antti; Juntti, Markku; Tran, Le-Nam
Publication date 2017-08-21
Publication information IEEE Transactions on Signal Processing, 65 (22): 5832-5847
Publisher IEEE
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1
Multi-Group Multicast Beamformer Design for
MISO-OFDM with Antenna Selection
Ganesh Venkatraman Student Member, IEEE, Antti T
¨
olli Senior Member, IEEE, Markku Juntti Senior
Member, IEEE, and Le-Nam Tran Senior Member, IEEE
Abstract—We study the problem of designing transmit beam-
formers for a multi-group multicasting by considering a multiple-
input single-output (MISO) orthogonal frequency division multi-
plexing (OFDM) framework. The design objective involves either
minimizing the total transmit power for certain guaranteed
quality-of-service (QoS) or maximizing the minimum achievable
rate among the users for a given transmit power budget.
The problem of interest can be formulated as a nonconvex
quadratically constrained quadratic programming (QCQP) for
which the prevailing semidefinite relaxation (SDR) technique
is inefficient for at least two reasons. At first, the relaxed
problem cannot be reformulated as a semidefinite programming.
Secondly, even if the relaxed problem is solved, the so-called
randomization procedure should be used to generate a feasible
solution to the original QCQP, which is difficult to derive for
the considered problem. To overcome these shortcomings, we
adopt successive convex approximation (SCA) framework to find
multicast beamformers directly. The proposed method not only
avoids the need of randomization search but also incurs less
computational complexity compared to an SDR approach. In
addition, we also extend multicasting beamformer design problem
with an additional constraint on the number of active elements,
which is particularly relevant when the number of antennas
is larger than that of radio frequency (RF) chains. Numerical
results are used to demonstrate the superior performance of our
proposed methods over the existing solutions.
I. INTRODUCTION
Physical layer multicasting is gaining significant attention
in the upcoming standards due to services like audio and video
streaming, which simultaneously deliver the same content to
multiple users. Current wireless standards such as the 3rd
Generation Partnership Project (3GPP) Long Term Evolution
(LTE) provide dedicated subframes to deliver multicast con-
tents in addition to regular unicast transmissions due to im-
mense data requirements from on-demand multicast services
[1]–[3]. In order to provide both unicast and multicast services
over cellular networks, evolved Multimedia Broadcast Multi-
cast Service (eMBMS) is specified in the LTE standard. The
challenge is to identify a proper share of wireless resources for
This work has been supported by the Finnish Funding Agency for Inno-
vation (Tekes), Nokia Networks, and the Academy of Finland. This work
has been co-funded by the Irish Government and the European Union under
Irelands EU Structural and Investment Funds Program 2014-2020 through the
SFI Research Centre Program under Grant 13/RC/2077.
G. Venkatraman, A. T
¨
olli and M. Juntti are with Centre for Wireless
Communications (CWC) - Radio Technologies, P.O. Box 4500, University
of Oulu, Finland, FI-90014, (e-mail: firstname.lastname@oulu.fi)
L.-N. Tran was with the Department of Electronic Engineering, Maynooth
University, Maynooth, Co. Kildare, Ireland. He is now with the School of
Electrical and Electronic Engineering, University College Dublin, Ireland, (e-
mail:nam.tran@ucd.ie)
providing the mix of unicast and multicast services. Depending
on the type of service required for each multimedia application
and the number of users requesting for it, the network will
determine the appropriate quality-of-service (QoS).
Physical layer multicasting for both single and multiple
groups has been studied extensively from a signal processing
perspective [4]–[14]. The main challenges that have been ad-
dressed in the context of multicasting problems are the group-
ing of users and the design of transmit beamformers for each
multicast group. In both problems, the knowledge of channel
state information (CSI) is assumed at the base stations (BSs).
While determining the multicast groups, transmit beamformers
are also designed to utilize the available spatial and frequency
dimensions at each transmission instant. The beamformers are
designed with the objective of either minimizing the total
transmit power or maximizing the minimum achievable rate
among the multiplexed multicast groups, where the minimum
rate is defined by the weakest link, i.e., the user with the
minimum rate. However, the design of transmit beamformers
in turn depends on the selection of users for various multicast
groups, which has drawn significant attention in the literature.
A. Multicast Scheduling
In each time slot, the BS involved in multicast transmission
transmits to the user group at a rate determined by the
weakest link. Even though associating users requesting the
same content in a same multicast group is beneficial in terms
of the resource utilization, it may also deteriorate the overall
performance if the users have heterogeneous channel condi-
tions, since the transmission rate is guided by the weakest link.
Thus, user association and resource scheduling for multicast
transmission is not a trivial extension of unicast schedulers.
Multicast scheduling based on a proportional fair metric
has been considered extensively to provide fairness among
multiple multicast groups [15]–[17]. In [15], two variants of
proportional fairness have been proposed based on the achiev-
able rate, namely, inter-group proportional fairness, based on
the sum of all user rates, and multicast proportional fairness,
aims at maximizing the sum log of user rates. Similarly, [16]
designed a scheduler to provide proportional fair utility for
both unicast and multicast users over multiple BSs.
B. Related Work
Upon determining users for various multicast groups, a
scheduler must utilize both spatial and frequency resources
2
provided by multiple-input single-output (MISO) orthogo-
nal frequency division multiplexing (OFDM) framework ef-
ficiently to satisfy certain design criterion. Designing transmit
beamformers for single multicast group with perfect CSI at
the BS was introduced in [4]. Due to the nonconvex nature
of the problem formulation, the beamformers were designed
using semidefinite relaxation (SDR) and the resulting problem
is solved by semidefinite programming (SDP) in [4]. Briefly,
instead of finding a beamformer vector, say, m, the SDR
technique defines a hermitian matrix M = mm
H
and poses
the original problem as an SDP with M as a variable. If the
solution has rank greater than one, then a randomization pro-
cedure proposed in [18] is used to extract a rank-one solution.
An extension to multiple multicast groups was studied in [6].
Alternatively, [19] employed successive convex approximation
(SCA) technique to solve the multicast problem for a single
group. Unlike the SDP based designs, the SCA technique
solves for beamformers directly, thereby avoiding the need
for any randomization procedure. However, for a multi-group
multicasting, the problem becomes difficult as it transforms
into a nonconvex quadratically constrained quadratic program-
ming (QCQP) wherein finding an initial feasible point itself
is difficult. To overcome this, [20] proposed a feasible point
pursuit SCA (FPP-SCA) algorithm by adding slack variables
to nonconvex constraints and a penalty to ensure that they were
all forced to zero, thereby ensuring feasibility of all operating
points (and solutions) throughout the SCA procedure.
A closely related problem of maximizing the minimum
signal-to-interference-plus-noise ratio (SINR) of all users was
also studied extensively in [5], [6], [8], [9], [11], [12], [21]. In
[5], [6], [9], [11] papers, max-min fairness based beamformers
were designed using the SDR approach. An iterative beam-
former design for a single multicast group was proposed in
[12] based on weighted SINR gradients. Alternatively, [8], [22]
proposed beamformer designs based on the SCA technique
for multi-group multicasting. An extension to multiple cells
was considered in [21] based on fractional programming.
Usually, beamformer design for multi-group multicasting with
certain QoS requirement is often not possible when the channel
vectors of users in different groups are collinear. This was
addressed in [23] as a joint beamformer and admission control
design with the objective of maximizing the admitted users.
In addition, various other extensions have been considered
in the literature. A distributed multi-cell beamformer design
for multicasting was proposed in [24] for both min-power and
max-min fairness objectives. An extension to single antenna
multi-group multicasting for relay networks was analyzed in
[25] with a min-power objective. In [26], a robust beamformer
design with imperfect CSI was addressed with a min-power
objective for cloud radio access network (Cloud-RAN). In [27]
and [28], a weighted fair multicast beamforming was proposed
with per antenna power constraints using both the SDR and
the SCA techniques, respectively. A multi-group multicasting
with antenna selection was introduced in [9] based on bisection
search. However, a beamformer design with antenna selection
based on biconvex formulation proposed in [10] was shown to
outperform [9] in terms of total transmit power. The capacity
limits of various multicasting schemes was discussed in [29]
by scaling the number of users and transmit antennas for a
fairness objective. An extension to antenna subset selection
was analyzed in [30] based on the average capacity scaling.
Finally, an extension to multiple-input multiple-output
(MIMO) scenario was considered in [13] wherein a non-
iterative algorithm for designing multicast precoders was pro-
posed to maximize the minimum user rate for a single multi-
cast group. The multiplexing of users over each sub-channel
is based on their channel similarities and the precoders were
evaluated by a weighted sum of the right singular vectors of
the multiplexed users. In [14], a two stage resource allocation
was proposed for multi-group multicasting by performing sub-
carrier assignment followed by a power allocation step over all
sub-carriers to maximize the overall multicasting throughput.
C. Main Contribution
In this paper, we consider the problem of physical layer
resource allocation for multi-group multicasting in a MISO-
OFDM in an isolated cell. In this context, we address the
problem of designing transmit beamformers so as to provide
certain guaranteed QoS in the form of minimum rate. Due
to the presence of multiple sub-channels, the SDR method
proposed in [6], [9]–[11] cannot be used directly as the SINR
requirement for each sub-channel is not fixed. Inspired by the
superior performance of the SCA based solutions in [19], [20],
[22], [28], we adopt the SCA method to solve our problem.
Furthermore, we extend this technique to solve multi-group
multicasting with antenna selection as studied in [9], [10], but
under a MISO-OFDM model. Unlike the approaches such as
the SDR with `
1
/`
∞
norm in [9] and the exact penalty method
in [10], we solve the antenna selection problem by assigning a
binary variable for each element to denote its selection status,
similar to [10]. However, we adopt the SCA based design as in
[19], [20], [22] instead of the SDR based technique proposed
in [10]. Finally, we study the problem of maximizing the
minimum achievable rate by all users in multicast groups for a
given transmit power. Unlike [5], [6], [11], [12], [27], [28], we
consider the fairness problem in a multi-group multiple sub-
channel framework. The performance of proposed schemes are
demonstrated using extensive numerical simulations, including
a uniform linear array (ULA) model for illustrative reasons.
The rest of the paper is organized as follows. Section II
presents both system model and problem formulation, which
is followed by Section III, where the beamformer design for
multicast groups is proposed by employing the SCA technique.
In Section IV, the problem of selecting a subset of antennas
is presented for a power minimization objective. Finally, the
problem of maximizing the minimum achievable rate among
multicast groups for a given transmit power is analyzed in
Section V. The numerical examples are presented in Section
VI together with the complexity figures. Finally, conclusions
are drawn in Section VII. The following notations are used in
this paper. Bold lower and upper case letters denote vectors
and matrices, respectively. (.)
T
, (.)
H
, tr(.), k.k
q
represent the
transpose, Hermitian, the trace operator, and the `
q
norm,
respectively. The ith entry of a vector x is denoted by x
i
.
3
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System Model
We consider a single-cell multi-user MISO system with N
T
transmit antennas transmitting N
G
independent multicast data
streams to K single-antenna receivers over N OFDM sub-
channels (or coherence bands).
1
Each user belongs to one of
the N
G
multicast groups, where the users in each group receive
a common data stream. Let G = {1, 2, . . . , N
G
} denote the
set of all multicast groups present in the system and N =
{1, 2, . . . , N} be the set of all OFDM sub-channels. The set
of all users associated with multicast group g is denoted by
G
g
and we denote the respective group of user k by a positive
integer g
k
. The received symbol y
k,n
on the nth sub-channel
for user k belonging to multicast group g
k
is given by
y
k,n
= h
k,n
m
g
k
,n
d
g
k
,n
+
X
g
0
∈G\{g
k
}
h
k,n
m
g
0
,n
d
g
0
,n
+ e
k,n
(1)
where h
k,n
∈ C
1×N
T
is the channel seen by user k on the
nth sub-channel, and m
g,n
∈ C
N
T
×1
is the beamformer for
multicast group g on sub-channel n. The data symbol d
g,n
,
transmitted for all users in G
g
, is normalized as E[|d
g,n
|
2
] = 1
and e
k,n
is the additive complex white Gaussian noise drawn
from CN(0, N
0
). The SINR seen by user k on the nth sub-
channel, which is represented as Γ
k,n
({m}) be, is given by
Γ
k,n
({m}) =
|h
k,n
m
g
k
,n
|
2
N
0
+
P
g
0
∈G\{g
k
}
|h
k,n
m
g
0
,n
|
2
(2)
where {m} , {m
g,n
}, ∀g ∈ G, ∀n ∈ N denotes the collec-
tion of all transmit beamformers. We remark that Γ
k,n
({m}) is
a function of all transmit beamformers as shown in (2), but for
simplicity, we express it as Γ
k,n
in the following discussions.
B. Problem Formulation
We address three closely related problems on designing
beamformers for a multicast transmission. At first, we study
the problem of minimizing the total transmit power required
to guarantee certain QoS for all users in each multicast group.
Formally, the beamformer design problem is given by
P
1
,
minimize
{m}
P
g∈G
P
N
n=1
km
g,n
k
2
subject to
P
N
n=1
log(1 + Γ
k,n
) ≥ ¯r
g
k
, ∀k
(3a)
(3b)
where ¯r
g
k
is the minimum multicast service rate for all users
belonging to group g
k
∈ G. In unicast transmission, both
joint encoding across all sub-channels and link adaptation by
varying the coding scheme based on the user CSI are optimal
from the information theoretic perspective [31]. Whereas for
multicast transmission, link adaptation is not optimal, since
the code rate is limited by log(1 + min
k∈G
g
{Γ
k,n
}), ∀k ∈ G
g
for each n ∈ N and g ∈ G . Thus, only joint coding across all
the sub-channels with code rate ¯r
g
is optimal for each group
g ∈ G. Despite coding jointly across all the sub-channels,
beamformers are designed specifically for each sub-channel
1
Sub-channel refers to a group of frequency resources for which the channel
is assumed to be relatively constant. Thus, beamformers are designed for a
group of sub-carriers over which multiple data symbols are transmitted.
and multicast group based on the CSI of respective users.
Thus, by varying the SINR on each sub-channel independently,
the overall achievability of joint coding is ensured ∀g ∈ G, ∀k.
As an extension, we consider a design requirement wherein
the number of available radio frequency (RF) chains is smaller
than the number of transmit elements. Such a constraint can be
achieved by forcing certain entries of transmit beamformers to
zero as the power on each antenna is dictated by the respective
beamformer entry. To do so, let us define a vector w as
w = [w
1
, . . . , w
N
T
]
T
, w
t
=
P
g∈G
P
N
n=1
|m
g,n,t
|
2
(4)
where m
g,n,t
is the complex entry corresponding to antenna
index t ∈ {1, 2, . . . , N
T
} of the beamformer vector used to
serve multicast group g on the nth sub-channel and w
t
is the
total transmit power from antenna element t. With the above
notations, the second problem formulation is given as
P
2
,
minimize
{m}
N
T
X
t=1
w
t
subject to kwk
0
≤ N
RF
N
X
n=1
log(1 + Γ
k,n
) ≥ ¯r
g
k
, ∀k
(5a)
(5b)
(5c)
where N
RF
< N
T
is the total number of available RF chains.
Finally, we study above two problems with the objective
of providing fairness, i.e., maximizing the minimum multicast
group rate, for a given power budget. It can be modeled as
P
3
,
maximize
{m}
min
g∈G,k∈G
g
k
n
N
X
n=1
log(1 + Γ
k,n
)
o
subject to kwk
0
≤ N
RF
N
T
X
t=1
w
t
≤ P
tot
(6a)
(6b)
(6c)
where P
tot
is the available transmit power budget. We discuss
above problems and their solutions in subsequent sections.
III. PROPOSED SOLUTION FOR P
1
All the problems outlined in Section II-B are nonconvex due
to SINR expression (2). In order to handle the nonconvexity,
various approaches have been proposed in literature based
on the SDR technique [4]–[6]. However, we resort to the
SCA method as in [20] wherein nonconvex constraints are
relaxed by a sequence of convex ones, which is then solved
iteratively until convergence. Before proceeding further with
the proposed SCA based design, we discuss some drawbacks
in extending the SDR technique to multi-carrier scenario.
A. Limitations of Semidefinite Relaxation
The SDR technique is a powerful signal processing tool that
has been employed widely in wireless communications. For
example, it has been used to demodulate higher order constel-
lations and physical layer beamformers for single and multiple
groups [4], [6], [32]. Unfortunately, the SDR method is not
applicable directly to a multi-carrier multicasting problem. To
4
understand this, let us introduce a positive semidefinite matrix
M
g,n
= m
g,n
m
H
g,n
as an optimization variable along with a
constraint rank(M
g,n
) = 1 so as to extract m
g,n
from M
g,n
.
Now, by using M
g,n
, we can express the SINR Γ
k,n
in (2) as
Γ
k,n
=
tr (H
k,n
M
g,n
)
N
0
+
P
g
0
∈G\{g}
tr (H
k,n
M
g
0
,n
)
, ∀k ∈ G
g
(7)
where H
k,n
= h
H
k,n
h
k,n
is the channel matrix related to h
k,n
.
Using (7), an equivalent formulation for P
1
is written as
minimize
{M}
X
g∈G
N
X
n=1
tr (M
g,n
) (8a)
subject to rank (M
g,n
) = 1, ∀g ∈ G, ∀n ∈ N (8b)
N
X
n=1
log (1 + Γ
k,n
) ≥ ¯r
g
, ∀k ∈ G
g
(8c)
M
g,n
0, ∀g ∈ G, ∀n ∈ N (8d)
where {M} , {M
g,n
}, ∀g ∈ G, ∀n ∈ N is the collection of
all transmit beamformer matrices. The minimum guaranteed
QoS requirement of all users is ensured by the constraint (8c).
The problem (8) is still nonconvex with M
g,n
even when
the rank-one constraint (8b) for all beamformers are omitted.
It follows due to the nonconvex nature of the QoS constraint
(8c). However, for a single sub-channel scenario, i.e., when
N = 1, (8) can be modeled as an SDP problem by discarding
the rank-one constraint (8b) as discussed in [6], [9]. Even
though the rank relaxation of (8) can be solved for a single
sub-channel case, we may still require to extract a rank-one
solution if (8) yields a result with the rank greater than one.
This step is carried out by a randomization procedure in
[18]. The best known randomization algorithm for this case
was proposed in [6], [18] and it requires solving a series of
linear programs. Therefore, as the number of sub-channels and
multicast groups increases, the complexity of (8) scales-up
quickly. Furthermore, designing beamformers for multi-group
multi-carrier multicasting as in (8) by the SDR method is not
a trivial problem, since the QoS constraint in (8c) is defined
over all sub-channels, and therefore cannot be solved for each
sub-channel independently by fixing the SINR arbitrarily by
satisfying (8c). Nonetheless, an iterative solution based on the
SDR and the SCA technique is proposed in [33].
B. Solution based on Successive Convex Approximation
Due to the issues involved with the SDR technique ex-
plained above, we propose an alternative approach to solve P
1
by employing the SCA technique, thereby ensuring a rank-one
solution upon finding the multicast beamformers. In order to
do so, we relax the minimum guaranteed rate constraint in
(3b) with SINR term Γ
k,n
by the following inequalities as
N
X
n=1
log(1 + γ
k,n
) ≥ ¯r
g
, ∀k ∈ G
g
(9a)
|h
k,n
m
g,n
|
2
N
0
+
P
r∈G\{g}
|h
k,n
m
r,n
|
2
≥ γ
k,n
, ∀k ∈ G
g
(9b)
where the newly introduced optimization variable γ
k,n
is an
under-estimator for the actual SINR Γ
k,n
as Γ
k,n
≥ γ
k,n
. By
adding one on both sides of (9b), we rewrite it as
N
0
+
P
r∈G
|h
k,n
m
r,n
|
2
γ
k,n
+ 1
≥
X
r∈G\{g}
|h
k,n
m
r,n
|
2
+ N
0
(10)
where user k belongs to group g ∈ G. Even after replacing
the QoS constraints in (3b) by two inequalities (9a) and (10),
the problem is still nonconvex due to the nonconvexity of the
constraint in (10). Therefore, we adopt the SCA technique in
[34] wherein the nonconvex set (10) is relaxed by a convex
subset around a fixed operating point, which is used in P
1
instead of (10). Upon finding a solution, a new feasible set
is updated by using the current solution as an operating point
for the next iteration and solved for an optimal solution.
In order to find a convex approximation for the nonconvex
constraint (10), we introduce two new stacked channel vectors
¯
h
k,n
and
˜
h
k,n
, which are defined as
¯
h
k,n
, [h
k,n
, h
k,n
, . . . , h
k,n
| {z }
N
G
terms
] ∈ C
1×N
T
·N
G
(11a)
˜
h
k,n
, [h
k,n
, h
k,n
, . . . , h
k,n
| {z }
(N
G
−1) terms
] ∈ C
1×N
T
(N
G
−1)
(11b)
where
¯
h
k,n
and
˜
h
k,n
are the vectors formed by repeating the
channel seen by user k on the nth sub-channel by N
G
and
N
G
−1 times, respectively. In addition, we also introduce two
new stacked vectors m
n
and
˜
m
g,n
such that k ∈ G
g
as
m
n
, [m
T
1,n
, m
T
2,n
, . . . , m
T
N
G
,n
]
T
∈ C
N
T
·N
G
×1
(12a)
˜
m
g,n
, [m
T
1,n
, m
T
2,n
, . . . , m
T
g−1,n
, m
T
g+1,n
,
. . . , m
T
N
G
,n
]
T
∈ C
N
T
(N
G
−1)×1
(12b)
where m
n
is formed by stacking all multicast beamformers for
sub-channel n and the vector
˜
m
g,n
is obtained by stacking
transmit beamformers corresponding to multicast groups in
G\{g} for sub-channel n, i.e., by excluding m
g,n
from m
n
.
The newly defined vectors in (12) are just a rearrangement of
optimization variables {m}.
Now, by using (12), the constraint in (10) becomes
N
0
+ |
¯
h
k,n
m
n
|
2
1 + γ
k,n
≥ N
0
+ |
˜
h
k,n
˜
m
g
k
,n
|
2
(13)
where the l.h.s of (13) is of quadratic-over-linear, i.e., a con-
vex function, and thus can be bounded from below by a linear
first order Taylor approximation L(m
n
, γ
k,n
; m
(i)
n
, γ
(i)
k,n
) as
L(m
n
, γ
k,n
; m
(i)
n
, γ
(i)
k,n
) ,
N
0
+ |
¯
h
k,n
m
(i)
n
|
2
1 + γ
(i)
k,n
+ 2 <
(
m
(i) H
n
¯
h
H
k,n
¯
h
k,n
1 + γ
(i)
k,n
m
n
− m
(i)
n
)
−
N
0
+ |
¯
h
k,n
m
(i)
n
|
2
1 + γ
(i)
k,n
2
γ
k,n
− γ
(i)
k,n
≤ l.h.s (13) (14)
where m
(i)
n
and γ
(i)
k,n
are fixed operating points upon which
the approximation is carried out. Moreover, m
(i)
n
and γ
(i)
k,n
are
the solutions obtained for m
n
and γ
k,n
from the (i − 1)th
SCA iteration, respectively.
Finally, by replacing the l.h.s of (13) with (14), the convex
