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1
Multicast Multigroup Precoding and User
Scheduling for Frame-Based Satellite
Communications
Dimitrios Christopoulos, Member, IEEE, Symeon Chatzinotas, Senior Member, IEEE, and
Bj
¨
orn Ottersten, Fellow, IEEE
Abstract—The present work focuses on the forward link of
a broadband multibeam satellite system that aggressively reuses
the user link frequency resources. Two fundamental practical
challenges, namely the need to frame multiple users per trans-
mission and the per-antenna transmit power limitations, are
addressed. To this end, the so-called frame-based precoding
problem is optimally solved using the principles of physical
layer multicasting to multiple co-channel groups under per-
antenna constraints. In this context, a novel optimization problem
that aims at maximizing the system sum rate under individual
power constraints is proposed. Added to that, the formulation
is further extended to include availability constraints. As a
result, the high gains of the sum rate optimal design are
traded off to satisfy the stringent availability requirements of
satellite systems. Moreover, the throughput maximization with a
granular spectral efficiency versus SINR function, is formulated
and solved. Finally, a multicast-aware user scheduling policy,
based on the channel state information, is developed. Thus,
substantial multiuser diversity gains are gleaned. Numerical
results over a realistic simulation environment exhibit as much
as 30% gains over conventional systems, even for 7 users
per frame, without modifying the framing structure of legacy
communication standards.
Index Terms—Broadband Multibeam Satellite systems, Op-
timal Linear Precoding, Sum Rate Maximization, Multicast
Multigroup beamforming, Per-antenna Constraints
I. I
NTRODUCTION
& R
ELATED
W
ORK
Aggressive frequency reuse schemes have shown to be
the most promising way towards spectrally efficient, high-
throughput wireless communications. In this context, linear
precoding, a transmit signal processing technique that exploits
the offered spatial degrees of freedom of a multi-antenna
transmitter, is brought into play to manage interferences.
Such interference mitigation techniques and subsequently full
frequency reuse configurations, are enabled by the availability
of channel state information (CSI) at the transmitter.
In fixed broadband multibeam satellite communications
(satcoms), the relatively slow channel variations facilitate
the channel acquisition process. Therefore, such scenarios
emerge as the most promising use cases of full frequency
reuse configurations. Nevertheless, the incorporation of linear
precoding techniques is inhibited by the inherent characteris-
tics of the satellite system [1], [2]. The present contribution
The authors are with the SnT, University of Luxembourg. Email:
{dimitrios.christopoulos, symeon.chatzinotas, bjorn.ottersten}@uni.lu. This
work was partially supported by the National Research Fund, Luxembourg
under the projects “CO
2
SAT’ and “SeMIGod :”. Part of this work has been
presented at the IEEE GlobeCom 2014 conference.
focuses on two fundamental constraints stemming from the
practical system implementation. Firstly, the framing structure
of satcom standards, such as the second generation digital
video broadcasting for satellite standard DVB −S2 [3] and its
most recent extensions DVB −S2X [4], inhibit scheduling a
single user per transmission. Secondly, non-flexible on-board
payloads prevent power sharing between beams.
Focusing on the first practical constraint, the physical layer
design of DVB − S2 [3] has been optimized to cope with the
noise limited, with excessive propagation delays and intense
fading phenomena, satellite channel. Therefore, long forward
error correction (FEC) codes and fade mitigation techniques
that rely on an adaptive link layer design (adaptive coding and
modulation – ACM) have been employed. The latest evolution
of DVB − S2X, through its –synchronous over the multiple
beams– superframes (cf. annex E of [4]), allows for the incor-
poration of the aforementioned interference mitigation tech-
niques (cf. annex C of [5]). A small-scale example of the ap-
plication of linear precoding methods within the DVB −S2X
standard is depicted in Fig. 1. Clearly, the underlying framing
structure hinders the calculation of a precoding matrix on a
user-by-user basis. During one transmission period, one frame
per beam accommodates a different number of users, each
with different data requirements. Added to that, the application
of FEC block coding over the entire frame requires that co-
scheduled users decode the entire frame and then extract the
data they need. Also, the unequal data payloads amongst users
simultaneously served in different beams further complicates
the joint processing of the multiple streams. Consequently,
despite the capacity achieving channel based precoding [6],
practical system implementations emanate the consideration
of precoding on a frame-by-frame basis. The notion of frame-
based precoding is presented in more detail in [1], [2].
From a signal processing perspective, physical layer (PHY)
multicasting to multiple co-channel groups [7] can provide the
theoretically optimal precoders when a multi-antenna transmit-
ter conveys independent sets of common data to distinct groups
of users. This scenario is known as PHY multigroup multicast
beamforming (or equivalently precoding). The optimality of
the multicast multigroup precoders for frame-based precod-
ing is intuitively clear, under the following considerations.
In multicasting, the same symbol is transmitted to multiple
receivers. This is the fundamental assumption of frame-based
precoding as well, since the symbols of one frame, regardless
of the information they convey, are addressed to multiple users.
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Fig. 1. Frame-based precoding in DVB − S2X. Function f(∙) denotes the FEC coding operation over the data d
xy
that are uniquely addressed to user
x of beam y, as identified in the right side of the plot. Consequently, the j-th transmitted symbol s
ij
, belonging to the i-th superframe (SF), contains an
encoded bit-stream that needs to be received by all co-scheduled users. In SFs 3 and 4, different number of users are co-scheduled.
Fig. 2. Transmitter functional block diagram, based on DVB-S2 [3], extended
to incorporate advanced interference mitigation techniques.
These users need to receive the entire frame, decode it and then
extract information that is relevant to them. The connection
between PHY multigroup multicast beamforming (precoding)
and frame-based precoding was firstly established in [8].
The second practical constraint tackled in the present work
includes a maximum limit on the per-antenna transmitted
power. Individual per-antenna amplifiers prevent power shar-
ing amongst the antennas of the future full frequency reuse
compatible satellites. On board flexible amplifiers, such as
multi-port amplifiers and flexible traveling wave tube ampli-
fiers [9], come at high costs. Also, power sharing is impossible
in distributed antenna systems (DAS), such as constellations
of cooperative satellite systems (e.g. dual satellite systems [10]
or swarms of nano-satellites).
Enabled by the incorporation of linear precoding in DVB-
S2X, an example of a full frequency reuse transmission chain
is depicted in Fig. 2. The optimal, in a throughput maximizing
sense, precoding matrix, combined with a low complexity user
scheduling algorithm will be presented in the remaining parts
of this work.
A. Related Work
In the PHY multigroup multicast precoding literature, two
fundamental optimization criteria, namely the sum power min-
imization under specific Quality of Service (QoS) constraints
and the maximization of the minimum SINR (max min fair
criterion) have been considered in [7], [11], [12] under a
SPC. Extending these works, a consolidated solution for the
weighted max min fair multigroup multicast beamforming
under PACs has been derived in [13], [14]. To this end,
the well established tools of Semi-Definite Relaxation (SDR)
and Gaussian randomization were combined with bisection to
obtain highly accurate and efficient solutions.
The fundamental attribute of multicasting, that is a single
transmission to be addressed to a group of users, constrains
the system performance according to the worst user. There-
fore, the maximization of the minimum SINR is the most
relevant problem and the fairness criterion is imperative [13].
When advancing to multigroup multicast systems, however,
the service levels between different groups can be adjusted
towards achieving some other optimization goal. The sum rate
maximization (max SR) problem in the multigroup multicast
context was initially considered in [15] under SPC. Therein,
a heuristic iterative algorithm based on the principle of de-
coupling the beamforming design and the power allocation
problem was proposed. In more detail, the SPC max sum rate
problem was solved using a two step optimization algorithm.
The first step was based on the QoS multicast beamforming
problem of [7], as iteratively solved with input QoS targets
defined by the worst user per group in the previous iteration.
The derived precoders push all the users of the group closer
to the worst user thus saving power. The second step of the
algorithm consisted of the gradient based power reallocation
methods of [16]. Hence, a power redistribution takes place
via the sub-gradient method [16] to the end of maximizing
the system sum rate.
In a realistic system design, the need to schedule a large
number users, over subsequent in time transmissions, is of
substantial importance. In the context of multiuser multiple
input multiple output (MU − MIMO) communications, user
scheduling has shown great potential in maximizing the system
throughput performance. In [17], [18], low complexity user
scheduling algorithms allowed for the channel capacity ap-
proaching performance of linear precoding methods when the
number of available users grows large. The enabler for these
algorithms is the exact knowledge of the CSI. Motivated by
these results and acknowledging that the large number of users
served by one satellite can offer significant multiuser diversity
gains, channel based user scheduling over satellite is herein
proposed. Further supporting this claim, the diverse mul-
tiuser satellite environment was exploited towards approaching
the information theoretic channel capacity bounds in [10].
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Therein, user scheduling methods were extended to account
for adjacent transmitters and applied in a multibeam satellite
scenario, exhibiting the importance of scheduling for satcoms.
In the present work, drawing intuitions from the frame-
based design, multicast-aware user scheduling algorithms are
derived. These algorithms, as it will be shown, exploit the
readily available CSI, to glean the multiuser diversity gains of
satellite systems.
Different from the aforementioned works, the sum rate
maximization under PACs has only been considered in [19].
Herein, this principle is used as a stepping stone for the
incremental development of elaborate optimization algorithms
that solve problems inspired by the needs of frame-based
precoding over satellite. The contributions are summarized in
the following points:
•
The max SR multigroup multicast problem under PACs
is formulated and solved.
•
The above max SR problem is extended to account for
minimum rate constraints (MRCs).
•
A novel modulation aware max SR optimization that con-
siders the discretized throughput function of the receive
useful signal power is proposed and heuristically solved.
•
A low complexity, CSI based, user scheduling algorithm
that considers the multigroup multicast nature of the
frame-based precoding system is envisaged.
•
The developed techniques are evaluated over a multi-
beam, full frequency reuse satellite scenario.
The rest of the paper is structured as follows. Section II
models the multigroup multicast system. Based on this model,
the max SR, multigroup multicast optimization problem is
formulated and solved in Sec. III. Extending this optimization,
system dependent problems are tackled in Sec. IV. Further on,
user scheduling is discussed in Sec. V. Finally, in Sec. VI, the
performance of the derived algorithms is evaluated, while Sec.
VII concludes the paper.
Notation: In the remainder of this paper, bold face lower
case and upper case characters denote column vectors and
matrices, respectively. The operators (∙)
T
, (∙)
†
, |∙|, Tr (∙) and
||∙||
2
, correspond to the transpose, the conjugate transpose, the
absolute value, the trace and the Euclidean norm operations,
while [∙]
ij
denotes the i, j-th element of a matrix. An x-
element column vector of ones is denoted as 1
x
. Finally, ∅
denotes an empty set.
II. S
YSTEM
M
ODEL
The focus is on a single broadband multibeam satellite
transmitting to multiple single antenna users. Let N
t
denote
the number of transmitting elements, which for the purposes of
the present work, are considered equal to the number of beams
(one feed per beam assumption) and N
u
the total number of
users simultaneously served. The received signal at the i-th
user will read as y
i
= h
†
i
x + n
i
, where h
†
i
is a 1 × N
t
vector composed of the channel coefficients (i.e. channel gains
and phases) between the i-th user and the N
t
antennas of
the transmitter, x is the N
t
× 1 vector of the transmitted
symbols and n
i
is the complex circular symmetric (c.c.s.)
independent identically distributed (i.i.d) zero mean Additive
White Gaussian Noise (AWGN), measured at the i-th user’s
receiver. Herein, for simplicity, the noise will be normalized
to one and the impact of noise at the receiver side will be
incorporated in the channel coefficients, as will be shown in
the following (Sec. II.A eq. (4) ).
Let us assume that a total of N
t
multicast groups are
realized where I = {G
1
, G
2
, . . . G
N
t
} the collection of index
sets and G
k
the set of users that belong to the k-th multicast
group, k ∈ {1 . . . N
t
}. Each user belongs to only one frame
(i.e. group), thus G
i
∩ G
j
=Ø,∀i, j ∈ {1 ∙∙∙N
t
}, while
ρ = N
u
/N
t
denotes the number of users per group. Let
w
k
∈ C
N
t
×1
denote the precoding weight vector applied to
the transmit antennas to beamform towards the k-th group
of users. By collecting all user channels in one channel
matrix, the general linear signal model in vector form reads as
y = Hx + n = HWs + n, where y and n ∈ C
N
u
, x ∈ C
N
t
and H ∈ C
N
u
×N
t
. Since, the frame-based precoding imposes
a single precoding vector for multiple users, the matrix will
include as many precoding vectors (i.e columns) as the number
of multicast groups. This is the number of transmit antennas,
since one frame per-antenna is assumed. Also, the symbol
vector includes a single equivalent symbol for each frame i.e.
s ∈ C
N
t
, inline with the multicast assumptions. Consequently,
a square precoding matrix is realized, i.e. W ∈ C
N
t
×N
t
.
The assumption of independent information transmitted to
different frames implies that the symbol streams {s
k
}
N
t
k=1
are mutually uncorrelated. Also, the average power of the
transmitted symbols is assumed normalized to one. Therefore,
the total power radiated from the antenna array is equal to
P
tot
=
N
t
X
k=1
w
†
k
w
k
= Trace
WW
†
, (1)
where W = [w
1
, w
2
, . . . w
N
t
]. The power radiated by each
antenna element is a linear combination of all precoders and
reads as [20]
P
n
=
"
N
t
X
k=1
w
k
w
†
k
#
nn
=
h
WW
†
i
nn
, (2)
where n ∈ {1 . . . N
t
} is the antenna index. The fundamental
difference between the SPC of [7] and the proposed PAC is
clear in (2), where instead of one, N
t
constraints are realized,
each one involving all the precoding vectors.
A. Multibeam Satellite Channel
The above general system model is applied over a multi-
beam satellite channel explicitly defined as follows. A 245
beam pattern that covers Europe is employed [22]. For the
purposes of the present work, only a subset of the 245
beams will be considered, as presented in Fig. 3. Such a
consideration is in line with the multiple gate-way (multi-GW)
assumptions of large multibeam systems [21]. However, the
effects of interference from adjacent clusters is left for future
investigations. A complex channel matrix that models the link
budget of each user as well as the phase rotations induced by
the signal propagation is employedin the standards of [22], [9]
and [8]. In more detail, the total channel matrix H ∈ C
N
u
×N
t
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-0.16
-0.15
-0.14
-0.13
-0.12
-0.11
-0.1
-0.09
-0.08
-0.04 -0.02 0 0.02 0.04 0.06 0.08
Fig. 3. Beam pattern covering Europe, provided by [22], with the nine beams
considered herein highlighted.
is generated as
H = ΦB, (3)
and includes the multibeam antenna pattern (matrix B) and
the signal phase due to different propagation paths between
the users (matrix Φ). The real matrix B ∈ R
N
u
×N
t
models
the satellite antenna radiation pattern, the path loss, the receive
antenna gain and the noise power. Its i, j-th entry is given by
[22]:
b
ij
=
p
G
R
G
ij
4π(d
k
∙ λ
−1
)
√
κT
cs
B
u
!
, (4)
with d
k
the distance between the i-th user and the satellite
(slant-range), λ the wavelength, κ the Boltzman constant, T
cs
the clear sky noise temperature of the receiver, B
u
the user
link bandwidth, G
R
the receiver antenna gain and G
ij
the
multibeam antenna gain between the i-th single antenna user
and the j-th on board antenna (= feed). Hence, the beam gain
for each satellite antenna-user pair, depends on the antenna
pattern and on the user position.
An inherent characteristic of the multibeam satellite channel
is the high correlation of signals at the satellite side. Thus a
common assumption in multibeam channel models is that each
user will have the same phase between all transmit antennas
due to the long propagation path [9]. The identical phase
assumption between one user and all transmit feeds is sup-
ported by the relatively small distances between the transmit
antennas and the long propagation distance of all signals to a
specific receiver. Hence, in (3) the diagonal square matrix Φ
is generated as [Φ]
xx
= e
jφ
x
, ∀ x = 1 . . . N
u
where φ
x
is a
uniform random variable in [2π, 0) and [Φ]
xy
= 0, ∀ x 6= y.
B. Average User Throughput
Based on the above link budget considerations, the achiev-
able average user throughput is normalized over the number
of beams, in order to provide a metric comparable with
multibeam systems of any size. Therefore, the average user
throughput, R
avg
as will be hereafter referred to, is given as
R
avg
=
2B
u
1 + α
1
N
t
N
t
X
k=1
f
DVB−S2X
min
i∈G
k
{SINR
i
}, t
, (5)
in [Gbps/beam], where all parameters are defined in Tab. II
of Sec. VI. In (5), the spectral efficiency function f
DVB−S2X
receives as input each users SINR as well as a threshold
vector t. Then, f
DVB−S2X
performs a rounding of the input
SINR to the closest lower floor given by the threshold vector t
and outputs the corresponding spectral efficiency in [bps/Hz].
This operation is denoted as b∙c
t
. The mapping of receive
SINR regions to a spectral efficiency achieved by a respective
modulation and coding (MODCOD) scheme is explicitly
defined in the latest evolution of the satcom standards [4].
It should also be noted, that the conventional four color
frequency reuse calculations are based on the exact same
formula, with the only modifications being the input SINR,
calculated under conventional four color reuse pattern and
with the pre-log factor reduced by four times, equal to the
conventional fractional frequency reuse [22].
III. S
UM
R
ATE
M
AXIMIZATION
For the precoding design, optimal multigroup multicast pre-
coders under per-antenna constraints are proposed to maximize
the throughput of the multibeam satellite system. The design
of throughput maximizing optimal precoders is a complicated
problem without an explicit solution even for the unicasting
case [23]. When advancing to multicasting assumptions, the
structure of the problem becomes even more involved, as
already explained [11]. Consequently, the present work builds
upon the heuristic methods of [15], [16].
Since a multigroup multicasting scenario entails the flexibil-
ity to maximize the total system rate by providing different ser-
vice levels amongst groups, the multigroup multicast max SR
optimization aims at increasing the minimum SINR within
each group while in parallel maximizing the sum of the rates
of all groups. Intuitively, this can be accomplished by reducing
the SINR of users with better conditions than the worst user
of their group. Also, groups that contain compromised users
might need to be turned of, hence driving their users to service
unavailability, in order to save power resources and degrees of
freedom. As a result, power is not consumed for the mitigation
of poor channel conditions. Any remaining power budget is
then reallocated to well conditioned and balanced in terms of
performance groups.
A. Per-antenna Power Constrained Optimization
This section focuses on the per-antenna power constrained
max SR problem, formally defined as
SR : max
{w
k
}
N
t
k=1
N
u
X
i=1
log
2
(1 + γ
i
)
subject to: γ
i
= min
m∈G
k
|w
†
k
h
m
|
2
P
N
t
l6=k
|w
†
l
h
m
|
2
+ σ
2
m
,
∀i ∈ G
k
, k, l ∈ {1 . . . N
t
},
and to:
"
N
t
X
k=1
w
k
w
†
k
#
nn
≤ P
n
,
∀n ∈ {1 . . . N
t
}.
(6)
(7)
Problem SR receives as input the channel matrices as
well as the per-antenna power constraint vector p
ant
=
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[P
1
, P
2
. . . P
N
t
]. Following the notation of [7] for ease of
reference, the optimal objective value of SR will be denoted
as c
∗
= SR(p
ant
) and the associated optimal point as
{w
SR
k
}
N
t
k=1
. The novelty of the SR lies in the PACs, i.e. (7)
instead of the conventional SPC proposed in [15]. Therein,
to solve the elaborate max SR under a SPC problem, the
decoupling of the precoder calculation and the power loading
over these vectors was considered. The first problem was
solved based on the solutions of [7] while the latter on sub-
gradient optimization methods [16]. To the end of solving the
novel SR problem, a heuristic algorithm is proposed herein.
Different than in [15], the new algorithm calculates the per-
antenna power constrained precoders by utilizing recent results
[13]. Also, modified sub-gradient optimization methods are
proposed to take into account the PACs. More specifically,
instead of solving the QoS sum power minimization problem
of [7], the proposed algorithm calculates the PAC precoding
vectors by solving the following problem [13] that reads as
Q : min
r, {w
k
}
N
t
k=1
r
subject to:
|w
†
k
h
i
|
2
P
N
t
l6=k
|w
†
l
h
i
|
2
+ σ
2
i
≥ γ
i
,
∀i ∈ G
k
, k, l ∈ {1 . . . N
t
},
and to:
1
P
n
"
N
t
X
k=1
w
k
w
†
k
#
nn
≤ r,
∀n ∈ {1 . . . N
t
},
(8)
(9)
where r ∈ R
+
. Problem Q receives as input the SINR
target vector g = [γ
1
, γ
2
, . . . γ
N
u
], that is the individual QoS
constraints of each user, as well as the per-antenna power
constraint vector p
ant
. Let the optimal objective value of Q
be denoted as r
∗
= Q(g, p
ant
) and the associated optimal
point as {w
Q
k
}
N
t
k=1
. This problem is solved using the well
established methods of SDR and Gaussian randomization [24].
A more detailed description of the solution of Q can be found
in [13], [14] and is herein omitted for conciseness.
To proceed with the power reallocation step, let us rewrite
the precoding vectors calculated from Q as {w
Q
k
}
N
t
k=1
=
{
√
p
k
v
k
}
N
t
k=1
with ||v
k
||
2
2
= 1 and p = [p
1
. . . p
k
]. By
this normalization, the beamforming problem can be decou-
pled into two problems. The calculation of the beamform-
ing directions, i.e. the normalized {v
k
}
N
t
k=1
, and the power
allocation over the existing groups, i.e. the calculation of
p
k
. Since the exact solution of SR is not straightforwardly
obtained, this decoupling allows for a two step optimization.
Under general unicasting assumptions, the SR maximizing
power allocation with fixed beamforming directions is a
convex optimization problem [16]. Nonetheless, when multi-
group multicasting is considered, the cost function C
SR
=
P
N
t
k=1
log (1 + min
i∈G
k
{SINR
i
}) . is no longer differentiable
due to the min
i∈G
k
operation and one has to adhere to sub-
gradient solutions [15]. What is more, as in detail explained
in [15], the cost function needs to be continuously differen-
tiable, strictly increasing, with a log-convex inverse function.
Nevertheless, this is not the case for SR. Towards providing
a heuristic solution to an involved problem without known
optimal solution, an optimization over the logarithmic power
vector s = {s
k
}
N
t
k=1
= {log p
k
}
N
t
k=1
, will be considered in the
standards of [15]. Therein, the authors employ a function φ
that satisfies the above assumptions to approximate the utility
function of SR. For more information on function φ and
the suggested approximation, the reader is directed to [15]. It
should be noted that the heuristic nature of this solution does
not necessarily guarantee convergence to a global optimum.
Albeit this, and despite being sub-optimal in the max sum
rate sense, the heuristic solutions attain a good performance,
as shown in [15], [16] and in the following. Consequently, in
the present contribution, the power loading is achieved via the
sub-gradient method [16], under specific modifications over
[15] that are hereafter described.
The proposed algorithm, presented in Alg. 1, is an iterative
two step procedure. In each step, the QoS targets g are calcu-
lated as the minimum target per group of the previous iteration,
i.e. γ
i
= min
i∈G
k
{SINR
i
}, ∀i ∈ G
k
, k ∈ {1 . . . N
t
}. There-
fore, the new precoders require equal or less power to achieve
the same system sum rate. Any remaining power is then
redistributed amongst the groups to the end of maximizing
the total system throughput, via the sub-gradient method [16].
Focusing of the later method and using the logarithmic power
vector s = {s
k
}
N
t
k=1
= {log p
k
}
N
t
k=1
, the sub-gradient search
method is given as
s(t + 1) =
Y
P
[s(t) − δ(t) ∙ r(t)] , (10)
where
Q
P
[x] denotes the projection operation of point x ∈
R
N
t
onto the set P ⊂ R
+
N
t
. The parameters δ(t) and r(t)
are the step of the search and the sub-gradient of the SR
cost function at the point s(t), respectively. The number of
iterations this method runs, denoted as t
max
, is predefined.
The projection operation, i.e.
Q
P
[∙], constrains each iteration
of the sub-gradient to the feasibility set of the SR problem.
The analytic calculation of r(t) follows the exact steps of [15],
[16] and is herein omitted for shortness. In order to account
for the more complicated PACs the projection over a per-
antenna power constrained set is considered as follows. The
set of PACs can be defined as
P =
(
p ∈ R
+
N
t
|
"
N
t
X
k=1
p
k
v
k
v
†
k
#
nn
≤ P
n
)
, (11)
where the elements of the power vector p = exp(s) represent
the power allocated to each group. It should be stressed that
this power is inherently different from the power transmitted
by each antenna p
ant
∈ R
+
N
t
. The connection between
p
ant
and p is given by the normalized beamforming vectors
as easily observed in (11). Different from the sum power
constrained solutions of [15], the per-antenna constrained
projection problem is given by
P : min
p
||p − x||
2
2
subject to :
"
N
t
X
k=1
p
k
v
k
v
†
k
#
nn
≤ P
n
,
∀n ∈ {1 . . . N
t
},
(12)
![Fig. 3. Beam pattern covering Europe, provided by [22], with the nine beams considered herein highlighted.](/figures/fig-3-beam-pattern-covering-europe-provided-by-22-with-the-3omaqik2.webp)
