Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 780823, 10 pages
http://dx.doi.org/10.1155/2014/780823
Research Article
Optimization of Power Allocation for Multiusers in
Multi-Spot-Beam Satellite Communication Systems
Heng Wang,1 Aijun Liu,1 Xiaofei Pan,1 and Jianfei Yang1,2
1
2
College of Communications Engineering, PLA University of Science & Technology, Nanjing, Jiangsu 210007, China
Troops 94922 PLA, Jinhua, Zhejiang 321000, China
Correspondence should be addressed to Heng Wang; wangheng0987654321@126.com
Received 13 December 2013; Revised 18 February 2014; Accepted 20 February 2014; Published 24 March 2014
Academic Editor: Changzhi Wu
Copyright © 2014 Heng Wang et al. his is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In recent years, multi-spot-beam satellite communication systems have played a key role in global seamless communication.
However, satellite power resources are scarce and expensive, due to the limitations of satellite platform. herefore, this paper
proposes optimizing the power allocation of each user in order to improve the power utilization eiciency. Initially the capacity
allocated to each user is calculated according to the satellite link budget equations, which can be achieved in the practical satellite
communication systems. he problem of power allocation is then formulated as a convex optimization, taking account of a tradeof between the maximization of the total system capacity and the fairness of power allocation amongst the users. Finally, an
iterative algorithm based on the duality theory is proposed to obtain the optimal solution to the optimization. Compared with
the traditional uniform resource allocation or proportional resource allocation algorithms, the proposed optimal power allocation
algorithm improves the fairness of power allocation amongst the users. Moreover, the computational complexity of the proposed
algorithm is linear with both the numbers of the spot beams and users. As a result, the proposed power allocation algorithm is easy
to be implemented in practice.
1. Introduction
As an important complement of the terrestrial networks,
satellite communication systems provide service to users in
several scenarios where terrestrial networks cannot be used.
In modern satellite communication systems the multi-spotbeam technique has been widely applied, due to its advantage
of concentrating the energy on a small area to provide high
data rate to the users and reusing the same frequency to
increase the total system capacity [1]. However, due to the
limitations of the satellite platform, it is known that the
satellite power resources are scarce and expensive. Moreover,
the real traic demands of each use are also diferent and
time varying. As a result, it is necessary to optimize the power
allocation to each user to satisfy its traic demand.
he problem of power allocation in the multi-spotbeam satellite system has been investigated in [2–9]. In
[2] the problem of power allocation was formulated as
an optimization problem, which is shown to be convex.
hen the Lagrangian multipliers were introduced to solve
the optimization problem. However, the way to ind the
optimal Lagrangian multiplier was not provided in [2]. As a
result, the methods of bisection and subgradient were applied
to search the optimal Lagrangian multipliers in [3, 4]. In
order to improve the total system capacity, a method of
selecting a small number of active beams was proposed in
[5], while keeping the fairness of power allocation amongst
the beams. In [6], a joint power and bandwidth allocation
algorithm was proposed. he algorithm improved both the
total system capacity and the fairness amongst the beams, due
to the dynamic allocation of both the power and bandwidth
resource. he work in [2–6] proposed power allocation
algorithms for the spot beams, without considering the power
allocation to each user in the beams. However, for the users
they only care about the power allocation to them. herefore,
it is signiicant to investigate how to allocate the power
resources to the diferent users in diferent spot beams. In
[7] a power allocation algorithm was proposed to stabilize
the total system capacity even if the channel model and the
speciic arrival rates were unknown, as long as the arrival rate
2
Mathematical Problems in Engineering
vector was inside the capacity region. When the users were
covered by multiple satellites, each of which had multiple
queues for downlink traic, a routing decision was made
to maximize the total system throughput. In [8] an optimal
power allocation algorithm was proposed to maximize the
total system efective capacity in the mobile satellite systems.
he main problem in [2–8] is that the allocated capacity
to each user is calculated through the Shannon capacity
formula. However, the capacity only can be obtained in
theory, which cannot be achieved in the practical satellite
communication system. herefore, the proposed power allocation algorithms in these papers may not be the optimal
algorithm for the practical systems. In order to overcome
this drawback, in [9] a practical capacity formula was applied
in the power allocation, aiming to maximize the number of
users which are satisied with the desired quality of service.
However, only a heuristic algorithm was proposed without
mathematic analysis, and the fairness of power allocation
amongst the users was also ignored.
his paper is aimed to ill these gaps, by optimizing
the power allocation to each user in the multi-spot-beam
satellite communication, according to the practical formula
for calculating the allocated capacity to each user. he
irst step is to calculate the allocated capacity to each user
according to the satellite link budget equations, which can be
achieved in the practical system. It is found that the allocated
capacity to each user is determined by the allocated satellite
power, coding and modulation mode, and channel condition.
At the same time, the allocated capacity is also constrained by
the bandwidth of each user. In order to preciously describe
the impact of these factors on the power allocation, the
problem of power allocation is mathematically formulated as
a nonlinear optimization problem, which is demonstrated as
a convex optimization problem. An iterative algorithm based
on the duality theory is then proposed to obtain the optimal
solution to the optimization. Finally, the impact of the coding
and modulation mode adopted by each user, the bandwidth
of each spot beam, and the channel conditions of each user
on the power allocation results are discussed.
he main contributions of this paper are summarized as
follows:
(1) the mathematical formulation of the problem of
power allocation for multiple users in the multi-spotbeam satellite communication system according to
the practical capacity formula, through a compromise
between the maximization of the total system capacity
and the fairness of the power allocation amongst the
users;
(2) the proposal of an iterative algorithm, which will
obtain the optimal solution to the optimization;
(3) the analysis of the impact of the coding and modulation mode, bandwidth of each spot beam, and channel
conditions of each user on the power allocation
results.
he remainder of this paper is organized as follows. In
Section 2, the model of the multi-spot-beam satellite communication system with multiple users is described, and
the calculation of the capacity allocated to each user according to the satellite link budget equations is also shown. In
Section 3, the problem of power allocation is formulated
as a convex optimization problem. Section 4 proposes the
iterative algorithm to obtain the optimal solution to the
optimization. Section 5 presents the simulation results and
analyzes the impact of the coding and modulation mode,
bandwidth of each spot beam, and channel conditions of each
user on the power allocation result. Section 6 concludes the
paper.
2. A Multi-Spot-Beam Satellite
Communication System Model
Figure 1 shows the coniguration of a multi-spot-beam
satellite communication system, where a regenerative satellite
payload is considered and the single channel per carrier
(SCPC) technique is employed as the access method for the
downlink. In this system uplink signal from user is demodulated and decoded to recover the originally transmitted data
on the satellite. hen the decoded data to user is reencoded
and remodulated using the same or diferent coding and
modulation schemes in the downlink, where diferent users
use diferent signals at diferent frequency and bandwidth.
his paper proposes solving the problem of power allocation
for diferent users in the downlink.
It is assumed that the system consists of � spot beams
�� , � ∈ {1, . . . , �}, and � users �� , � ∈ {1, . . . , �}. he set
of users which are served by the spot beam �� is denoted
by N�� . he traic demand of the �th user is �� , and the
satellite transmitting power allocated to the �th user is �� .
he coding and modulation mode adopted by the �th useris
�� , and the corresponding threshold signal-to-noise ratio per
bit for demodulation is (�� /�0 )�� . It is noted that there are
many schemes of the choice for �� ; however, it is beyond the
scope of this paper. In order to simplify the problem, it is
supposed that each user can only support one kind of coding
and modulation mode. When the user is given, the coding
and modulation mode adopted by the user is determined. It
is meant that �� is only determined by the �th user. hus the
allocated capacity �� of the �th user is calculated according to
the following equations [1]:
�� =
(�/�0 )�
(�� /�0 ) ��
,
(1)
where (�/�0 )� is the downlink carrier power-to-noise power
spectral density ratio of the �th user, which can be calculated
according to the satellite link budget equation [1], given as
follows:
(
� ⋅�
�
�
) = � � ⋅( ),
�0 �
�� ⋅ �
� �
(2)
where � � is the downlink loss of the �th user, which is afected
by the channel condition. It mainly consists of free-space loss,
rain attenuation, and other losses due to catastrophic failure.
(�/�)� is the gain-to-equivalent noise temperature ratio of the
receiving equipment of the �th user. �� is the transmitting
Mathematical Problems in Engineering
3
Satellite
C8
U1
U3
U9
U6
U5
U10
···
U2
U4
B1
U7
U8
UM
···
BK
B2
Figure 1: Coniguration of a multi-spot-beam satellite communication system.
antenna gain of the satellite. It is assumed that the value of
�� is the same for all the users in this paper. � is Boltzmann’s
constant, which is 1.379 × 10−23 W/KHz.
It is noted that the interbeam interference from the
sidelobes of adjacent spot beams will decrease the capacity
of each user. However, the interbeam interference is ignored
here, because the very narrow spot beams over a large number
of spot beams are considered [10].
According to (1) and (2), it is shown that the capacity
allocated to the �th user is determined by the allocated satellite
transmitting power, given as
�� =
�� ⋅ �� ⋅ (�/�)�
.
� � ⋅ (�� /�0 )�� ⋅ �
(3)
It is observed from (3) that the allocated capacity �� of
the �th user is increased as the power allocated to it increases.
However, the total satellite power resources are ixed, so the
capacity of the system is limited. Moreover, the allocated
capacity of each user is also constrained by the bandwidth
resources allocated to it, which are also scarce in the system.
When the coding and modulation mode adopted by the �th
user is given, the bandwidth that needs to be provided to it is
expressed as
� ⋅ [1 + � (�� )]
,
�� = �
� (�� )
Let ��� denote the bandwidth of the �th spot beam. hus
the total bandwidth that can be provided to the users in the �th
spot beam cannot exceed ��� . In other words, the allocated
capacity to the users is also constrained by the bandwidth of
each spot beam.
3. Mathematical Formulation of the
Optimization Problem
In this study, the objective of the power allocation optimization is to minimize the sum of the squared diferences
between the traic demand and the capacity allocated to each
user, taking account of a trade-of between the maximum
total system capacity and the fairness of power allocation
amongst the users. herefore, the optimization problem is
formulated as follows:
min∑(�� − �� )
�
{�� }
subject to
�� =
2
(5)
�=1
�� ⋅ �� ⋅ (�/�)�
≤ ��
� � ⋅ (�� /�0 )�� ⋅ �
∑�� ≤ �total
(6)
�
(4)
where �(�� ) and �(�� ) are the spectral eiciency and roll-of
factor of the coding and modulation mode �� .
(7)
�=1
∑ �� ≤ ��� .
�∈N��
(8)
4
Mathematical Problems in Engineering
he constraint (6) indicates that the allocated capacity to
each user should not exceed the traic demand of it, in order
to avoid the waste of the scarce power resources. Conditions
(7)-(8) imply the constraint for the total power of the satellite
and the total bandwidth of each spot beam, respectively.
It is seen that the problem is a nonlinear optimization
problem with constraints. Moreover, it is obvious that the
objective function in (5) is convex and the functions in
constrains (6)–(8) are linear. As a result, the problem under
consideration is a convex optimization [11].
Due to the nonlinearity of the optimization, it is diicult
to obtain the global optimal solution. In order to make
the above problem tractable, an iterative algorithm based
on the duality theory is proposed in the following section.
It is known that if the optimization problem is a convex
optimization problem, the duality gap between the primal
problem and dual problem is zero, and the optimal value of
the dual problem is equal to the optimal value of the primal
problem. As a result, the dual problem can be irst solved
to obtain the optimal dual solution, and the primal optimal
solution is then computed by solving the primal problem at
the point of the optimal dual solution [11]. Fortunately, it has
been proved that the optimization problem studied here is
a convex optimization problem; thus the power allocation
result obtained by the proposed algorithm is the optimal
power allocation for the users in the multi-spot-beam satellite
communication system.
4. Proposed Power Allocation Algorithm
As mentioned previously, the proposed power allocation
algorithm is based on the duality theory. By introducing
nonnegative dual variables � and � = [�1 , �2 , . . . , �� ] yielded
the Lagrangian, given as
�
2
�=1
�=1
− ∑�� (��� − ∑ �� ) ,
�
�=1
(9)
�∈N��
where P = [�1 , �2 , . . . , �� ].
Maximizing (9) with respect to the nonnegative � and �
brings the following function:
� (P) = max � (P, �, �) .
�≥0,�≥0
(10)
It is seen that if the optimization variables �� are satisied
with the constrains (7)-(8), then �(�total − ∑�
�=1 �� ) ≥ 0 and
∑�
�
)
≥
0.
herefore,
(10)
�
(�
−
∑
will get the
�
��
�∈N�
�=1 �
�
maximal value when �(�total − ∑�
�=1 �� ) = 0 and ∑�=1 �� (��� −
2
∑�∈N� �� ) = 0. As a result, �(P) = ∑�
�=1 (�� − �� ) . To this
�
end, the primal optimization with constraints is changed into
the optimization with no constraints as follows [11]:
�
� = min � (P) = min max � (P, �, �) .
P
P �≥0,�≥0
� (�, �) = min � (P, �, �)
(12)
P
and the dual problem of (11) can be written as
� = max � (�, �) = max min � (P, �, �) .
�≥0,�≥0
�≥0,�≥0 P
(11)
(13)
he work in [12] solved the joint spectrum and power
allocation in cognitive radio networks and proposed a
method to solve the dual problem. Inspired with this paper,
the dual problem (13) is decomposed into the following two
sequentially iterative subproblems.
Subproblem 1: Power Allocation. Given the dual variables �
and �, for any � = [1, . . . , �], maximizing (9) with respect
to �� brings the following equation:
� ⋅ �� ⋅ (�/�)�
2 ⋅ �� ⋅ (�/�)�
(�� − �
)
� � ⋅ (�� /�0 )�� ⋅ �
� � ⋅ (�� /�0 )�� ⋅ �
opt
� ⋅ (�/�)� ⋅ [1 + � (�� )]
= � + �� �
,
� � ⋅ (�� /�0 )�� ⋅ � ⋅ � (�� )
� ∈ N�� .
(14)
he optimized power allocation of the �th user �� can be
easily obtained from (14). It is seen from (14) that nonnegative
dual variables � and � guarantee that �� ≥ �� . As a result, the
constrain (6) is satisied.
opt
Subproblem 2: Dual Variables Update. he optimal dual
variables can be obtained by solving the problem:
(�opt , �opt ) = arg max min [� (Popt , �, �)] .
�,�
� (P, �, �) = ∑(�� − �� ) − � (�total − ∑�� )
�
In addition, the Lagrange dual function can be obtained
from (9) as [11]
(15)
Due to concavity of the dual objective function, here a
subgradient (a generalization of gradient) method is applied
to update the duality variables, shown as [13]
+
��+1 = [�� − Δ�� (�total − ∑�� )] ,
�
opt
(16)
�=1
���+1
=
[��
�
[
−
��
+
(��� − ∑ �� )] ,
�∈N��
]
(17)
where [�]+ = max{0, �},� is the iteration number, and Δ is
the iteration step size of each dual variable.
he subgradient method is very suitable for the situation
that the dual function is not diferentiable. As a result, the
method has been widely applied to solve the optimization
problem [12–18]. It has proven that the above dual variables
update algorithm is guaranteed to converge to the optimal
solution as long as the iteration step size chosen is suiciently
small [13]. A common criterion for choosing the iteration step
size is that the step size must be square summable, but not
absolute summable [13, 18].
Mathematical Problems in Engineering
5
Step 1. Set appropriate initial values for the dual variables.
Step 2. Substitute the values of the dual variables into (14), and then calculate the
optimized power allocation to each user.
Step 3. Substitute the values of the power of each user which is obtained from
step 2, into (16) and (17), and then update the dual variables.
�
�
Step 4. If the conditions of �����+1 (�total − ∑� �� )��� < � and
�� �+1
��
��� (�� − ∑�∈N �� )�� < �, ∀� ∈ {1, . . . , �} are satisied simultaneously, then terminate
�
��
�� �
��
the algorithm. Otherwise, jump to Step 2.
Algorithm 1: he proposed power allocation algorithm.
Table 1: Parameters of the multi-spot-beam satellite communication system.
Parameter
Beam number
User number
User number per spot beam
Traic demand of each user
Total satellite power [�total ]
Satellite transmitting antenna gain [�� ]
Bandwidth of each spot beam
Gain-to-equivalent noise temperature ratio of the receiving equipment [�/�]
Downlink loss [� � ]
Spectral eiciency of the coding and modulation mode [�(�� )]
Roll-of factor of the coding and modulation mode [�(�� )]
hreshold signal-to-noise ratio per bit of the coding and modulation mode [(�� /�0 )�� ]
he whole process of the proposed power allocation
algorithm can be summarized as shown in Algorithm 1.
According to Algorithm 1, it is shown that the computational complexity of step 2 and step 3 is �(�) and �(2�),
respectively. hus the total computational complexity of the
algorithm is �(��+2��), where � is the number of iterations.
It is noted that � is independent of � and �. herefore, the
computational complexity of the proposed algorithm is linear
with both the numbers of the spot beams and users, and the
proposed algorithm is easy to be implemented in practice.
5. Simulation Results and Analysis
For the simulation, a multi-spot-beam satellite communication system model is set up. It is assumed that the values of
downlink loss, gain-to-equivalent noise temperature ratio of
the receiving equipment, and coding and modulation mode
are the same for all the users. he parameters of the system
are shown in Table 1.
5.1. Eiciency of the Proposed Power Allocation Algorithm.
he proposed power allocation algorithm is compared with
the following two traditional allocation algorithms in order
to verify the eiciency of it.
(i) Uniform Resource Allocation Algorithm. he power
allocated to each user is �� = �total /�, � ∈
{1, 2, . . . , �}. he bandwidth allocated to the user in
Value
4
20
5
From 1 Mbps to 20 Mbps by step of 1 Mbps
20 W
20000
100 MHz
20
2�21
1.5
1
2.63
Table 2: Total system capacity of the three algorithms when the
channel conditions of each user are the same.
Algorithms
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
∑ ��
109.1 Mbps
109.1 Mbps
109.1 Mbps
the same spot beam is �� = ��� /|N�� |, � ∈ N�� ,
where |N�� | is the cardinality of the set N�� .
(ii) Proportional Resource Allocation Algorithm. he
power allocated to each user is �� = �� ⋅ �total / ∑�
�=1 �� ,
� ∈ {1, 2, . . . , �}. he bandwidth allocated to the user
in the same spot beam is �� = �� ⋅��� / ∑�∈N� �� , � ∈
�
N�� .
Figure 2 shows the capacity distributions of the users
which are allocated by the three algorithms. Table 2 shows
the total system capacities of the three algorithms. It is
noted that when the channel conditions of each user are
the same, the uniform resource allocation algorithm is a
special case of the water-ill algorithm, which can achieve the
maximal total system capacity [19]. As shown in Figure 2, the
uniform resource allocation algorithm uniformly allocates
the resources to each user, regardless of the traic demand
of each user, even resulting in some users being allocated
more capacity than that is needed. As a result, this uniform
resource allocation algorithm causes a waste of the scarce
6
Mathematical Problems in Engineering
20
×1014
2.5
18
2
14
12
(Ti − Ci )2
Capacity allocated (Mbps)
16
10
8
6
1.5
1
4
0.5
2
0
2
4
6
8
10
12
14
16
18
20
ith user
2
4
6
8
10
12
14
16
18
20
ith user
Traffic demand
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
Figure 2: Comparison of the three algorithms in terms of the
capacity allocated to each user when the channel conditions of each
user are the same.
Table 3: Sum of (�� − �� )2 of the three algorithms when the channel
conditions of each user are the same.
Algorithms
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
0
∑ (�� − �� )2
1.134�15
6.627�14
5.470�14
resources. he proportional resource allocation algorithm
allocates the power resources to each user only according
to its traic demand. he capacity allocated to each user
is linearly increasing, considering the fairness of power
allocation amongst the users to some extent. However, it is
not the optimal solution to the optimization. In order to get a
better fairness, the proposed power algorithms provide more
capacity to the users with higher traic demands and suppress
the capacities of the users with lower traic demands. For
example, the algorithm provides no capacity to the ive lowest
traic demand users. Although the capacities allocated to
each user are diferent, the total system capacities are the same
for the three algorithms, due to the linearity of the capacity
function in terms of the allocated power, and the sameness
of the channel conditions of each user. he conclusion is also
demonstrated by the data in Table 2.
Figure 3 shows the squared diference between the traic
demand and the capacity allocated of each user of the
three algorithms. Table 3 presents the sum of the squared
diferences of the three algorithms. It is shown from Figure 3
that for the uniform and proportional resource allocation
algorithms, although the squared diference between the
traic demand and the capacity allocated to the user with
low traic demand is small, however, the squared diference
increases rapidly when the traic demand increases. On
Figure 3: Comparison of the three algorithms in terms of the
squared diference between the traic demand and the capacity
allocated to each user when the channel conditions of each user are
the same.
the contrast, for the proposed optimal power allocation
algorithm, the squared diference between the traic demand
and the capacity allocated to the users with low traic demand
is larger than that of the former two algorithms. However,
the squared diference is almost the same from user 6 to user
20. As a result, the total squared diference of the proposed
power allocation algorithm is less than that of the former two
algorithms, which is also shown in Table 3. In other words,
the power allocation result of the proposed algorithm is the
best amongst the three algorithms.
5.2. Impact of the Spot Beam Bandwidth on the Power
Allocation Result. As mentioned above, the capacity allocated
to each user is constrained by both the power and bandwidth
allocated to it. Due to the limitation of the bandwidth of each
spot beam, the capacity allocated to the users in the same spot
beam is also constrained. As a result, the power resources
allocated to the users are impacted. In order to show the
impact of the spot beam bandwidth on the power allocation
result, the power allocation results are compared when the
bandwidth of each spot beam is set to be 25 MHz, 50 MHz,
and 100 MHz, and other parameters of the system stay the
same.
From Figure 4, it is obvious that the power allocation
results are diferent when the bandwidth resources of each
spot beam are various. When the spot beam bandwidth is
25 MHz, the capacity allocated to each user is constrained
by the bandwidth. Although the total system power is 20 W,
the total power allocated to all the users is only 13.06 W. As
a result, the power resources in the system are wasted and
the total system capacity is decreased. When the bandwidth
is 50 MHz, the capacities allocated to the users in the last
Mathematical Problems in Engineering
7
20
×1014
2.5
18
2
14
12
(Ti − Ci )2
Capacity allocated (Mbps)
16
10
8
6
1.5
1
4
0.5
2
0
2
4
6
8
10
12
14
16
18
20
ith user
0
2
4
6
8
10
12
14
16
18
20
ith user
Traffic demand
Spot beam bandwidth is 25 MHz
Spot beam bandwidth is 50 MHz
Spot beam bandwidth is 100 MHz
Spot beam bandwidth is 25 MHz
Spot beam bandwidth is 50 MHz
Spot beam bandwidth is 100 MHz
Figure 4: Comparison of the three diferent spot beams bandwidths
in terms of the capacity allocated to each user.
Figure 5: Comparison of the three diferent spot beams bandwidths
in terms of the squared diference between the traic demand and
the capacity allocated to each user.
Table 4: Total system capacity of three diferent spot beams
bandwidths.
Table 5: Sum of (�� − �� )2 of the three diferent spot beams
bandwidths.
Bandwidth of each spot beam
25 MHz
50 MHz
100 MHz
Bandwidth of each spot beam
25 MHz
50 MHz
100 MHz
∑ ��
71.25 Mbps
109.1 Mbps
109.1 Mbps
two spot beams are constrained by the bandwidth, due to
the high traic demand of users. hus the power resources
will be provided to the users with low traic demands in
the former two spot beams. When the bandwidth of each
spot beam is 100 MHz, the system has more than enough
bandwidth to be allocated to each user, thus the capacity
allocated to each user is limited by the total system power
resources. In order to improve the fairness of power allocation
amongst the users, the power resources are rarely or never
provided to the users with low traic demand. Although the
power resources allocated to each user are diferent when the
spot beam bandwidth is 50 MHz and 100 MHz, the power
resources are suiciently utilized. As a result, the total system
capacity is the same, which is also seen from Table 4.
As mentioned in Figure 5 and Table 5, when the bandwidth of each spot beam is lower, more power resources will
be provided to the users with low traic demand. herefore,
it is seen from Figure 5 that the squared diference between
traic demand and allocated capacity to users with low traic
demand is smaller. However, the squared diference is lager
for the users with high traic demand. As a result, the total
squared diference is larger when the bandwidth of each spot
beam is lower. his conclusion can be also observed from
Table 5.
∑ (�� − �� )2
15.34�14
7.476�14
5.470�14
5.3. Impact of the Coding and Modulation Mode of Each
User on the Power Allocation Result. It is known that the
power eiciency and spectral eiciency of a given coding
and modulation mode are usually contradictory to each
other. In other word, a higher spectral eiciency coding and
modulation code can support more capacity in the limited
bandwidth. However, more power must be provided to it to
support the coding and modulation mode, due to a higher
value of �� /�0 , resulting in lower power eiciency, and vice
versa. It is seen from the analysis in Section 5.2 that when
the bandwidth of each spot beam is 25 MHz, the capacity
allocated to each other is limited by the bandwidth and the
power resources are wasted. In order to solve the problem,
a higher bandwidth eiciency coding and modulation mode
can be adopted by each user. he capacity allocation results
are compared when each user adopts the three diferent
coding and modulation modes, as shown in Table 6.
It is known that when mode 1 is adopted by each user, the
power resources are wasted, due to the low spectral eiciency.
When mode 2 is adopted by each user, it is seen from Figure 6
that more capacity will be allocated to the users in spot beam
2 to spot beam 4, due to the higher spectral eiciency of
the mode and suicient utilization of the power resource.
As a result, the total system capacity is increased. When
8
Mathematical Problems in Engineering
20
×1014
2.5
18
2
14
12
(Ti − Ci )2
Capacity allocated (Mbps)
16
10
8
6
1.5
1
4
0.5
2
0
2
4
6
8
10
12
14
16
18
20
0
ith user
Table 6: hreshold signal-to-noise ratio per bit and spectral
eiciency of the three coding and modulation modes.
Mode 1
Mode 2
Mode 3
hreshold signal-to-noise
ratio per bit
Spectral
eiciency
2.63
3.63
4.47
1.5
1.75
2.15
Table 7: Total system capacity of three diferent coding and
modulation modes.
Adopted coding and modulation mode of each user
Mode 1
Mode 2
Mode 3
6
8
10
12
14
16
18
20
Mode 1
Mode 2
Mode 3
Figure 6: Comparison of the three diferent coding and modulation
modes in terms of the capacity allocated to each user.
Coding and
modulation mode
4
ith user
Mode 2
Mode 3
Traffic demand
Mode 1
2
∑ ��
71.25 Mbps
79.02 Mbps
64.24 Mbps
mode 3 is adopted, although the spectral eiciency is further
improved, the power eiciency is further reduced. herefore,
the capacity allocated to each user is limited by the power
resources allocated to it. Due to the low power eiciency, the
total system capacity is increased, which is shown in Table 7.
When the spectral eiciency of the coding and modulation mode is higher, the users with high traic demand
in the last several spot beams are provided more capacity
due to the higher spectral eiciency, resulting in a lower
squared diference as shown in Figure 7. herefore, the
total system squared diference between traic demand and
capacity allocated to the users is smaller, especially for mode
3. his conclusion is obviously seen from Table 8.
5.4. Impact of the Channel Condition of Each User on the Power
Allocation Result. It is known that the channel conditions of
each user are afected by many kinds of factor, causing that
the downlink losses of each user are not the same. In order to
Figure 7: Comparison of the three diferent spot beams bandwidths
in terms of the squared diference between the traic demand and
the capacity allocated to each user.
Table 8: Sum of (�� − �� )2 of the three diferent spot beams
bandwidths.
Adopted coding and
modulation mode of each user
Mode 1
Mode 2
Mode 3
∑ (�� − �� )2
1.533�15
1.366�15
1.298�15
Table 9: Total system capacity of the three algorithms when the
channel conditions of each user are not the same.
Algorithms
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
∑ ��
63.91 Mbps
60.73 Mbps
55.10 Mbps
show the impact of channel condition on the power allocation
result, the channel conditions of the users in the same spot
beam are set to be 2�21 , 3�21 , 4�21 , 5�21 , and 6�21 . Moreover,
the traic demands of the users in the same spot beam are
set the same, and the traic demands of the users in the four
diferent spot beams are set to be 3 Mbps, 8 Mbps, 13 Mbps,
and 18 Mbps. he simulation results are shown in Figure 8
and Table 9.
It is seen from Figure 8 that the proposed power allocation
algorithm provides more capacity to the users with higher
traic demand, in order to minimize the total system squared
diference between the traic demand and capacity allocated
to each user. he proposed algorithm allocates the same
capacities to the users in spot beam 3 or 4, which implied
that more power resource will be allocated to the users with
Mathematical Problems in Engineering
9
20
×1014
2.5
18
2
14
12
(Ti − Ci )2
Capacity allocated (Mbps)
16
10
8
6
1.5
1
4
0.5
2
0
2
4
6
8
10
12
14
16
18
20
ith user
2
4
6
8
10
12
14
16
18
20
ith user
Traffic demand
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
Figure 8: Comparison of the three algorithms in terms of the
capacity allocated to each user when the channel conditions of each
user are not the same.
Table 10: Sum of (�� − �� )2 of the three algorithms when the channel
conditions of each user are not the same.
Algorithms
Uniform resource allocation
Proportional resource allocation
Proposed optimal power allocation
0
∑ (�� − �� )2
1.719�15
1.414�15
1.363�15
worse channel conditions in these two spot beams. As a result,
compared with the other two resource allocation algorithms,
the total system capacity of the proposed power allocation
algorithm is decreased, as clearly shown in Table 9.
As mentioned in Figure 9 and Table 10, the proposed
power allocation algorithm provides more capacity to the
users with higher traic demand. herefore, the squared
diferences between the traic demand and capacity allocated
to these users are lower. Compared with the other two
algorithms, although the squared diferences of the users with
lower traic demand are higher, the total squared diference
of the proposed power allocation algorithm is lower, as shown
in Table 10. As a result, it is observed that the proposed
algorithm improves the fairness of power allocation amongst
the user at cost of the total system capacity.
6. Conclusion
In the multi-spot-beam satellite system it is crucial for us to
improve the power resources utilization eiciency, due to the
scarceness of the satellite power resources. To this end, the
problem of power allocation was mathematically formulated
as a convex optimization problem and an optimal power
Figure 9: Comparison of the three algorithms in terms of the
squared diference between the traic demand and the capacity
allocated to each user when the channel conditions of each user are
not the same.
allocation algorithm was proposed to solve the problem.
In the optimization, the capacity allocated to each user
was calculated according to satellite link budget equations
rather than the Shannon capacity formula. As a result, the
capacity allocated to each user can be achieved and the power
allocation result is more suitable for the practical multispot-beam satellite communication system. Moreover, the
computational complexity of proposed algorithm is linear
with both the numbers of the spot beams and users. As a
result, it can be implemented in the practical system.
It is shown from the simulation results that, compared
with the traditional power allocation algorithms, the proposed algorithm improved the fairness of the power allocation amongst the users. Both the coding and modulation
mode adopted by each user and the bandwidth of each spot
beam have a signiicant impact on the power allocation result.
When the bandwidth of each spot beam is suicient, more
power resources will be provided to the users with higher
traic demand to improve the fairness of power allocation
amongst the users. On the contrast, when the bandwidth
of each spot beam is limited, more power will be provided
to the users with lower traic demand. Even the satellite
power resources are wasted, due to the further reduction of
bandwidth of each spot beam. he impact of the coding and
modulation mode on the power allocation result is similar
to that of the bandwidth of each spot beam. Moreover,
the channel conditions of each user also afect the power
allocation result. he proposed algorithm provides more
resource to the users with the high traic demand. As a result,
if the channel conditions of these high traic demand users
are worse, the total system capacity will be decreased.
10
Conflict of Interests
he authors declare that they do not have any commercial
or associative interest that represents a conlict of interests in
connection with the work submitted.
Acknowledgment
he authors would like to thank the project support by the
National High-Tech Research & Development Program of
China under Grant 2012AA01A508.
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