Research ArticleOptimization of Power Allocation for Multiusers inMulti-Spot-Beam Satellite Communication Systems

Heng Wang1 Aijun Liu1 Xiaofei Pan1 and Jianfei Yang12

1 College of Communications Engineering PLA University of Science amp Technology Nanjing Jiangsu 210007 China2 Troops 94922 PLA Jinhua Zhejiang 321000 China

Correspondence should be addressed to Heng Wang wangheng0987654321126com

Received 13 December 2013 Revised 18 February 2014 Accepted 20 February 2014 Published 24 March 2014

Academic Editor Changzhi Wu

Copyright copy 2014 Heng Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 communicationHowever satellite power resources are scarce and expensive due to the limitations of satellite platform Therefore this paperproposes optimizing the power allocation of each user in order to improve the power utilization efficiency Initially the capacityallocated to each user is calculated according to the satellite link budget equations which can be achieved in the practical satellitecommunication systems The problem of power allocation is then formulated as a convex optimization taking account of a trade-off between the maximization of the total system capacity and the fairness of power allocation amongst the users Finally aniterative algorithm based on the duality theory is proposed to obtain the optimal solution to the optimization Compared withthe traditional uniform resource allocation or proportional resource allocation algorithms the proposed optimal power allocationalgorithm improves the fairness of power allocation amongst the users Moreover the computational complexity of the proposedalgorithm is linear with both the numbers of the spot beams and users As a result the proposed power allocation algorithm is easyto be implemented in practice

1 Introduction

As an important complement of the terrestrial networkssatellite communication systems provide service to users inseveral scenarios where terrestrial networks cannot be usedIn modern satellite communication systems the multi-spot-beam technique has been widely applied due to its advantageof concentrating the energy on a small area to provide highdata rate to the users and reusing the same frequency toincrease the total system capacity [1] However due to thelimitations of the satellite platform it is known that thesatellite power resources are scarce and expensive Moreoverthe real traffic demands of each use are also different andtime varying As a result it is necessary to optimize the powerallocation to each user to satisfy its traffic demand

The problem of power allocation in the multi-spot-beam satellite system has been investigated in [2ndash9] In[2] the problem of power allocation was formulated asan optimization problem which is shown to be convexThen the Lagrangian multipliers were introduced to solve

the optimization problem However the way to find theoptimal Lagrangian multiplier was not provided in [2] As aresult themethods of bisection and subgradient were appliedto search the optimal Lagrangian multipliers in [3 4] Inorder to improve the total system capacity a method ofselecting a small number of active beams was proposed in[5] while keeping the fairness of power allocation amongstthe beams In [6] a joint power and bandwidth allocationalgorithm was proposed The algorithm improved both thetotal system capacity and the fairness amongst the beams dueto the dynamic allocation of both the power and bandwidthresource The work in [2ndash6] proposed power allocationalgorithms for the spot beams without considering the powerallocation to each user in the beams However for the usersthey only care about the power allocation to themThereforeit is significant to investigate how to allocate the powerresources to the different users in different spot beams In[7] a power allocation algorithm was proposed to stabilizethe total system capacity even if the channel model and thespecific arrival rates were unknown as long as the arrival rate

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 780823 10 pageshttpdxdoiorg1011552014780823

2 Mathematical Problems in Engineering

vector was inside the capacity region When the users werecovered by multiple satellites each of which had multiplequeues for downlink traffic a routing decision was madeto maximize the total system throughput In [8] an optimalpower allocation algorithm was proposed to maximize thetotal system effective capacity in the mobile satellite systemsThe main problem in [2ndash8] is that the allocated capacityto each user is calculated through the Shannon capacityformula However the capacity only can be obtained intheory which cannot be achieved in the practical satellitecommunication system Therefore the proposed power allo-cation algorithms in these papers may not be the optimalalgorithm for the practical systems In order to overcomethis drawback in [9] a practical capacity formula was appliedin the power allocation aiming to maximize the number ofusers which are satisfied with the desired quality of serviceHowever only a heuristic algorithm was proposed withoutmathematic analysis and the fairness of power allocationamongst the users was also ignored

This paper is aimed to fill these gaps by optimizingthe power allocation to each user in the multi-spot-beamsatellite communication according to the practical formulafor calculating the allocated capacity to each user Thefirst step is to calculate the allocated capacity to each useraccording to the satellite link budget equations which can beachieved in the practical system It is found that the allocatedcapacity to each user is determined by the allocated satellitepower coding andmodulationmode and channel conditionAt the same time the allocated capacity is also constrained bythe bandwidth of each user In order to preciously describethe impact of these factors on the power allocation theproblem of power allocation is mathematically formulated asa nonlinear optimization problem which is demonstrated asa convex optimization problem An iterative algorithm basedon the duality theory is then proposed to obtain the optimalsolution to the optimization Finally the impact of the codingand modulation mode adopted by each user the bandwidthof each spot beam and the channel conditions of each useron the power allocation results are discussed

The main contributions of this paper are summarized asfollows

(1) the mathematical formulation of the problem ofpower allocation for multiple users in the multi-spot-beam satellite communication system according tothe practical capacity formula through a compromisebetween themaximization of the total system capacityand the fairness of the power allocation amongst theusers

(2) the proposal of an iterative algorithm which willobtain the optimal solution to the optimization

(3) the analysis of the impact of the coding and modula-tionmode bandwidth of each spot beam and channelconditions of each user on the power allocationresults

The remainder of this paper is organized as follows InSection 2 themodel of themulti-spot-beam satellite commu-nication system with multiple users is described and

the calculation of the capacity allocated to each user accord-ing to the satellite link budget equations is also shown InSection 3 the problem of power allocation is formulatedas a convex optimization problem Section 4 proposes theiterative algorithm to obtain the optimal solution to theoptimization Section 5 presents the simulation results andanalyzes the impact of the coding and modulation modebandwidth of each spot beam and channel conditions of eachuser on the power allocation result Section 6 concludes thepaper

2 A Multi-Spot-Beam SatelliteCommunication System Model

Figure 1 shows the configuration of a multi-spot-beamsatellite communication system where a regenerative satellitepayload is considered and the single channel per carrier(SCPC) technique is employed as the access method for thedownlink In this system uplink signal from user is demodu-lated and decoded to recover the originally transmitted dataon the satellite Then the decoded data to user is reencodedand remodulated using the same or different coding andmodulation schemes in the downlink where different usersuse different signals at different frequency and bandwidthThis paper proposes solving the problem of power allocationfor different users in the downlink

It is assumed that the system consists of 119870 spot beams119861119894 119894 isin 1 119870 and 119872 users 119880

119894 119894 isin 1 119872 The set

of users which are served by the spot beam 119861119894is denoted

by N119861119894 The traffic demand of the 119894th user is 119879

119894 and the

satellite transmitting power allocated to the 119894th user is 119875119894

The coding and modulation mode adopted by the 119894th useris120572119894 and the corresponding threshold signal-to-noise ratio per

bit for demodulation is (1198641198871198730)120572119894 It is noted that there are

many schemes of the choice for 120572119894 however it is beyond the

scope of this paper In order to simplify the problem it issupposed that each user can only support one kind of codingand modulation mode When the user is given the codingand modulation mode adopted by the user is determined Itis meant that 120572

119894is only determined by the 119894th user Thus the

allocated capacity119863119894of the 119894th user is calculated according to

the following equations [1]

119863119894=(1198621198730)119894

(1198641198871198730)120572119894

(1)

where (1198621198730)119894is the downlink carrier power-to-noise power

spectral density ratio of the 119894th user which can be calculatedaccording to the satellite link budget equation [1] given asfollows

(119862

1198730

)

119894

=119875119894sdot 119866119878

119871119894sdot 119896

sdot (119866

119879)119894

(2)

where 119871119894is the downlink loss of the 119894th user which is affected

by the channel condition It mainly consists of free-space lossrain attenuation and other losses due to catastrophic failure(119866119879)

119894is the gain-to-equivalent noise temperature ratio of the

receiving equipment of the 119894th user 119866119878is the transmitting

Mathematical Problems in Engineering 3

Satellite

C8

U1

U2

U3

U4

B1

U5U6

U7U8

B2

U9

U10

UM

BK

middot middot middot

middot middot middot

Figure 1 Configuration of a multi-spot-beam satellite communication system

antenna gain of the satellite It is assumed that the value of119866119878is the same for all the users in this paper 119896 is Boltzmannrsquos

constant which is 1379 times 10minus23WKHzIt is noted that the interbeam interference from the

sidelobes of adjacent spot beams will decrease the capacityof each user However the interbeam interference is ignoredhere because the very narrow spot beams over a large numberof spot beams are considered [10]

According to (1) and (2) it is shown that the capacityallocated to the 119894th user is determined by the allocated satellitetransmitting power given as

119863119894=

119875119894sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198730)120572119894sdot 119896 (3)

It is observed from (3) that the allocated capacity 119863119894of

the 119894th user is increased as the power allocated to it increasesHowever the total satellite power resources are fixed so thecapacity of the system is limited Moreover the allocatedcapacity of each user is also constrained by the bandwidthresources allocated to it which are also scarce in the systemWhen the coding and modulation mode adopted by the 119894thuser is given the bandwidth that needs to be provided to it isexpressed as

119882119894=119863119894sdot [1 + 120588 (120572

119894)]

120578 (120572119894)

(4)

where 120578(120572119894) and 120588(120572

119894) are the spectral efficiency and roll-off

factor of the coding and modulation mode 120572119894

Let119882119861119894denote the bandwidth of the 119894th spot beam Thus

the total bandwidth that can be provided to the users in the 119894thspot beam cannot exceed 119882

119861119894 In other words the allocated

capacity to the users is also constrained by the bandwidth ofeach spot beam

3 Mathematical Formulation of theOptimization Problem

In this study the objective of the power allocation opti-mization is to minimize the sum of the squared differencesbetween the traffic demand and the capacity allocated to eachuser taking account of a trade-off between the maximumtotal system capacity and the fairness of power allocationamongst the users Therefore the optimization problem isformulated as follows

min119875119894

119872

sum

119894=1

(119879119894minus 119863119894)2 (5)

subject to

119863119894=

119875119894sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198730)120572119894sdot 119896

le 119879119894 (6)

119872

sum

119894=1

119875119894le 119875total (7)

sum

119894isinN119861119895

119882119894le 119882119861119895 (8)

4 Mathematical Problems in Engineering

The constraint (6) indicates that the allocated capacity toeach user should not exceed the traffic demand of it in orderto avoid the waste of the scarce power resources Conditions(7)-(8) imply the constraint for the total power of the satelliteand the total bandwidth of each spot beam respectively

It is seen that the problem is a nonlinear optimizationproblem with constraints Moreover it is obvious that theobjective function in (5) is convex and the functions inconstrains (6)ndash(8) are linear As a result the problem underconsideration is a convex optimization [11]

Due to the nonlinearity of the optimization it is difficultto obtain the global optimal solution In order to makethe above problem tractable an iterative algorithm basedon the duality theory is proposed in the following sectionIt is known that if the optimization problem is a convexoptimization problem the duality gap between the primalproblem and dual problem is zero and the optimal value ofthe dual problem is equal to the optimal value of the primalproblem As a result the dual problem can be first solvedto obtain the optimal dual solution and the primal optimalsolution is then computed by solving the primal problem atthe point of the optimal dual solution [11] Fortunately it hasbeen proved that the optimization problem studied here isa convex optimization problem thus the power allocationresult obtained by the proposed algorithm is the optimalpower allocation for the users in themulti-spot-beam satellitecommunication system

4 Proposed Power Allocation Algorithm

As mentioned previously the proposed power allocationalgorithm is based on the duality theory By introducingnonnegative dual variables 120582 and 120590 = [120590

1 1205902 120590

119870] yielded

the Lagrangian given as

119871 (P120590 120582) =119872

sum

119894=1

(119879119894minus 119863119894)2minus 120582(119875total minus

119872

sum

119894=1

119875119894)

minus

119870

sum

119894=1

120590119894(119882119861119894minus sum

119895isinN119861119894

119882119895)

(9)

where P = [1198751 1198752 119875

119899]

Maximizing (9) with respect to the nonnegative 120582 and 120590brings the following function

119911 (P) = max120582ge0120590ge0

119871 (P120590 120582) (10)

It is seen that if the optimization variables 119875119894are satisfied

with the constrains (7)-(8) then 120582(119875total minus sum119872

119894=1119875119894) ge 0 and

sum119870

119894=1120590119894(119882119861119894minus sum119895isinN119861119894

119882119895) ge 0 Therefore (10) will get the

maximal value when 120582(119875total minussum119872

119894=1119875119894) = 0 andsum119870

119894=1120590119894(119882119861119894minus

sum119895isinN119861119894

119882119895) = 0 As a result 119911(P) = sum

119872

119894=1(119879119894minus 119863119894)2 To this

end the primal optimization with constraints is changed intothe optimization with no constraints as follows [11]

119901 = minP

119911 (P) = minP

max120582ge0120590ge0

119871 (P120590 120582) (11)

In addition the Lagrange dual function can be obtainedfrom (9) as [11]

119863 (120590 120582) = minP

119871 (P120590 120582) (12)

and the dual problem of (11) can be written as

119889 = max120582ge0120590ge0

119863 (120590 120582) = max120582ge0120590ge0

minP

119871 (P120590 120582) (13)

The work in [12] solved the joint spectrum and powerallocation in cognitive radio networks and proposed amethod to solve the dual problem Inspired with this paperthe dual problem (13) is decomposed into the following twosequentially iterative subproblems

Subproblem 1 Power Allocation Given the dual variables 120582and 120590 for any 119894 = [1 119872] maximizing (9) with respectto 119875119894brings the following equation

2 sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

(119879119894minus119875opt119894

sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

)

= 120582 + 120590119895

119866119878sdot (119866119879)119894 sdot [1 + 120588 (120572119894)]

119871119894sdot (1198641198871198990)120572119894sdot 119896 sdot 120578 (120572

119894) 119894 isin N

119861119895

(14)

The optimized power allocation of the 119894th user119875opt119894

can beeasily obtained from (14) It is seen from (14) that nonnegativedual variables 120582 and 120590 guarantee that 119879

119894ge 119863119894 As a result the

constrain (6) is satisfied

Subproblem 2 Dual Variables Update The optimal dualvariables can be obtained by solving the problem

(120590opt 120582

opt) = argmax

120590120582

min [119871 (Popt120590 120582)] (15)

Due to concavity of the dual objective function here asubgradient (a generalization of gradient) method is appliedto update the duality variables shown as [13]

120582119899+1

= [120582119899minus Δ119899

120582(119875total minus

119872

sum

119894=1

119875opt119894)]

+

(16)

120590119899+1

119894= [

[

120590119899

119894minus Δ119899

120590(119882119861119894minus sum

119895isinN119861119894

119882119895)]

]

+

(17)

where [119909]+ = max0 119909119899 is the iteration number and Δ isthe iteration step size of each dual variable

The subgradient method is very suitable for the situationthat the dual function is not differentiable As a result themethod has been widely applied to solve the optimizationproblem [12ndash18] It has proven that the above dual variablesupdate algorithm is guaranteed to converge to the optimalsolution as long as the iteration step size chosen is sufficientlysmall [13] A common criterion for choosing the iteration stepsize is that the step size must be square summable but notabsolute summable [13 18]

Mathematical Problems in Engineering 5

Step 1 Set appropriate initial values for the dual variablesStep 2 Substitute the values of the dual variables into (14) and then calculate the

optimized power allocation to each userStep 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 variablesStep 4 If the conditions of 1003816100381610038161003816120582

119899+1(119875total minus sum119894 119875119894)

1003816100381610038161003816 lt 120576 and100381610038161003816100381610038161003816120590119899+1

119894(119882119861119894minus sum119895isinN119861119894

119882119895)100381610038161003816100381610038161003816lt 120576 forall119894 isin 1 119870 are satisfied simultaneously then terminate

the algorithm Otherwise jump to Step 2

Algorithm 1 The proposed power allocation algorithm

Table 1 Parameters of the multi-spot-beam satellite communication system

Parameter ValueBeam number 4User number 20User number per spot beam 5Traffic demand of each user From 1Mbps to 20Mbps by step of 1MbpsTotal satellite power [119875total] 20WSatellite transmitting antenna gain [119866

119878] 20000

Bandwidth of each spot beam 100MHzGain-to-equivalent noise temperature ratio of the receiving equipment [119866119879] 20Downlink loss [119871

119894] 211989021

Spectral efficiency of the coding and modulation mode [120578(120572119894)] 15

Roll-off factor of the coding and modulation mode [120588(120572119894)] 1

Threshold signal-to-noise ratio per bit of the coding and modulation mode [(1198641198871198730)120572119894] 263

The whole process of the proposed power allocationalgorithm can be summarized as shown in Algorithm 1

According to Algorithm 1 it is shown that the computa-tional complexity of step 2 and step 3 is 119874(119872) and 119874(2119870)respectively Thus the total computational complexity of thealgorithm is119874(119878119872+2119878119870) where 119878 is the number of iterationsIt is noted that 119878 is independent of 119870 and 119872 Therefore thecomputational complexity of the proposed algorithm is linearwith both the numbers of the spot beams and users and theproposed algorithm is easy to be implemented in practice

5 Simulation Results and Analysis

For the simulation a multi-spot-beam satellite communica-tion system model is set up It is assumed that the values ofdownlink loss gain-to-equivalent noise temperature ratio ofthe receiving equipment and coding and modulation modeare the same for all the users The parameters of the systemare shown in Table 1

51 Efficiency of the Proposed Power Allocation AlgorithmThe proposed power allocation algorithm is compared withthe following two traditional allocation algorithms in orderto verify the efficiency of it

(i) Uniform Resource Allocation Algorithm The powerallocated to each user is 119875

119894= 119875total119872 119894 isin

1 2 119872 The bandwidth allocated to the user in

Table 2 Total system capacity of the three algorithms when thechannel conditions of each user are the same

Algorithms sum119862119894

Uniform resource allocation 1091MbpsProportional resource allocation 1091MbpsProposed optimal power allocation 1091Mbps

the same spot beam is 119882119895= 119882119861119894|N119861119894| 119895 isin N

119861119894

where |N119861119894| is the cardinality of the setN

119861119894

(ii) Proportional Resource Allocation Algorithm Thepower allocated to each user is 119875

119894= 119879119894sdot 119875totalsum

119872

119894=1119879119894

119894 isin 1 2 119872 The bandwidth allocated to the userin the same spot beam is119882

119895= 119879119895sdot119882119861119894sum119896isinN119861119894

119879119896 119895 isin

N119861119894

Figure 2 shows the capacity distributions of the userswhich are allocated by the three algorithms Table 2 showsthe total system capacities of the three algorithms It isnoted that when the channel conditions of each user arethe same the uniform resource allocation algorithm is aspecial case of the water-fill algorithm which can achieve themaximal total system capacity [19] As shown in Figure 2 theuniform resource allocation algorithm uniformly allocatesthe resources to each user regardless of the traffic demandof each user even resulting in some users being allocatedmore capacity than that is needed As a result this uniformresource allocation algorithm causes a waste of the scarce

6 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 2 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are the same

Table 3 Sum of (119879119894minus119862119894)2 of the three algorithms when the channel

conditions of each user are the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 113411986415

Proportional resource allocation 662711986414

Proposed optimal power allocation 547011986414

resources The proportional resource allocation algorithmallocates the power resources to each user only accordingto its traffic demand The capacity allocated to each useris linearly increasing considering the fairness of powerallocation amongst the users to some extent However it isnot the optimal solution to the optimization In order to get abetter fairness the proposed power algorithms provide morecapacity to the userswith higher traffic demands and suppressthe capacities of the users with lower traffic demands Forexample the algorithm provides no capacity to the five lowesttraffic demand users Although the capacities allocated toeach user are different the total system capacities are the samefor the three algorithms due to the linearity of the capacityfunction in terms of the allocated power and the samenessof the channel conditions of each user The conclusion is alsodemonstrated by the data in Table 2

Figure 3 shows the squared difference between the trafficdemand and the capacity allocated of each user of thethree algorithms Table 3 presents the sum of the squareddifferences of the three algorithms It is shown from Figure 3that for the uniform and proportional resource allocationalgorithms although the squared difference between thetraffic demand and the capacity allocated to the user withlow traffic demand is small however the squared differenceincreases rapidly when the traffic demand increases On

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 3 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arethe same

the contrast for the proposed optimal power allocationalgorithm the squared difference between the traffic demandand the capacity allocated to the userswith low trafficdemandis larger than that of the former two algorithms Howeverthe squared difference is almost the same from user 6 to user20 As a result the total squared difference of the proposedpower allocation algorithm is less than that of the former twoalgorithms which is also shown in Table 3 In other wordsthe power allocation result of the proposed algorithm is thebest amongst the three algorithms

52 Impact of the Spot Beam Bandwidth on the PowerAllocationResult Asmentioned above the capacity allocatedto each user is constrained by both the power and bandwidthallocated to it Due to the limitation of the bandwidth of eachspot beam the capacity allocated to the users in the same spotbeam is also constrained As a result the power resourcesallocated to the users are impacted In order to show theimpact of the spot beam bandwidth on the power allocationresult the power allocation results are compared when thebandwidth of each spot beam is set to be 25MHz 50MHzand 100MHz and other parameters of the system stay thesame

From Figure 4 it is obvious that the power allocationresults are different when the bandwidth resources of eachspot beam are various When the spot beam bandwidth is25MHz the capacity allocated to each user is constrainedby the bandwidth Although the total system power is 20Wthe total power allocated to all the users is only 1306W Asa result the power resources in the system are wasted andthe total system capacity is decreased When the bandwidthis 50MHz the capacities allocated to the users in the last

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

vector was inside the capacity region When the users werecovered by multiple satellites each of which had multiplequeues for downlink traffic a routing decision was madeto maximize the total system throughput In [8] an optimalpower allocation algorithm was proposed to maximize thetotal system effective capacity in the mobile satellite systemsThe main problem in [2ndash8] is that the allocated capacityto each user is calculated through the Shannon capacityformula However the capacity only can be obtained intheory which cannot be achieved in the practical satellitecommunication system Therefore the proposed power allo-cation algorithms in these papers may not be the optimalalgorithm for the practical systems In order to overcomethis drawback in [9] a practical capacity formula was appliedin the power allocation aiming to maximize the number ofusers which are satisfied with the desired quality of serviceHowever only a heuristic algorithm was proposed withoutmathematic analysis and the fairness of power allocationamongst the users was also ignored

This paper is aimed to fill these gaps by optimizingthe power allocation to each user in the multi-spot-beamsatellite communication according to the practical formulafor calculating the allocated capacity to each user Thefirst step is to calculate the allocated capacity to each useraccording to the satellite link budget equations which can beachieved in the practical system It is found that the allocatedcapacity to each user is determined by the allocated satellitepower coding andmodulationmode and channel conditionAt the same time the allocated capacity is also constrained bythe bandwidth of each user In order to preciously describethe impact of these factors on the power allocation theproblem of power allocation is mathematically formulated asa nonlinear optimization problem which is demonstrated asa convex optimization problem An iterative algorithm basedon the duality theory is then proposed to obtain the optimalsolution to the optimization Finally the impact of the codingand modulation mode adopted by each user the bandwidthof each spot beam and the channel conditions of each useron the power allocation results are discussed

The main contributions of this paper are summarized asfollows

(1) the mathematical formulation of the problem ofpower allocation for multiple users in the multi-spot-beam satellite communication system according tothe practical capacity formula through a compromisebetween themaximization of the total system capacityand the fairness of the power allocation amongst theusers

(2) the proposal of an iterative algorithm which willobtain the optimal solution to the optimization

(3) the analysis of the impact of the coding and modula-tionmode bandwidth of each spot beam and channelconditions of each user on the power allocationresults

The remainder of this paper is organized as follows InSection 2 themodel of themulti-spot-beam satellite commu-nication system with multiple users is described and

the calculation of the capacity allocated to each user accord-ing to the satellite link budget equations is also shown InSection 3 the problem of power allocation is formulatedas a convex optimization problem Section 4 proposes theiterative algorithm to obtain the optimal solution to theoptimization Section 5 presents the simulation results andanalyzes the impact of the coding and modulation modebandwidth of each spot beam and channel conditions of eachuser on the power allocation result Section 6 concludes thepaper

2 A Multi-Spot-Beam SatelliteCommunication System Model

Figure 1 shows the configuration of a multi-spot-beamsatellite communication system where a regenerative satellitepayload is considered and the single channel per carrier(SCPC) technique is employed as the access method for thedownlink In this system uplink signal from user is demodu-lated and decoded to recover the originally transmitted dataon the satellite Then the decoded data to user is reencodedand remodulated using the same or different coding andmodulation schemes in the downlink where different usersuse different signals at different frequency and bandwidthThis paper proposes solving the problem of power allocationfor different users in the downlink

It is assumed that the system consists of 119870 spot beams119861119894 119894 isin 1 119870 and 119872 users 119880

119894 119894 isin 1 119872 The set

of users which are served by the spot beam 119861119894is denoted

by N119861119894 The traffic demand of the 119894th user is 119879

119894 and the

satellite transmitting power allocated to the 119894th user is 119875119894

The coding and modulation mode adopted by the 119894th useris120572119894 and the corresponding threshold signal-to-noise ratio per

bit for demodulation is (1198641198871198730)120572119894 It is noted that there are

many schemes of the choice for 120572119894 however it is beyond the

scope of this paper In order to simplify the problem it issupposed that each user can only support one kind of codingand modulation mode When the user is given the codingand modulation mode adopted by the user is determined Itis meant that 120572

119894is only determined by the 119894th user Thus the

allocated capacity119863119894of the 119894th user is calculated according to

the following equations [1]

119863119894=(1198621198730)119894

(1198641198871198730)120572119894

(1)

where (1198621198730)119894is the downlink carrier power-to-noise power

spectral density ratio of the 119894th user which can be calculatedaccording to the satellite link budget equation [1] given asfollows

(119862

1198730

)

119894

=119875119894sdot 119866119878

119871119894sdot 119896

sdot (119866

119879)119894

(2)

where 119871119894is the downlink loss of the 119894th user which is affected

by the channel condition It mainly consists of free-space lossrain attenuation and other losses due to catastrophic failure(119866119879)

119894is the gain-to-equivalent noise temperature ratio of the

receiving equipment of the 119894th user 119866119878is the transmitting

Mathematical Problems in Engineering 3

Satellite

C8

U1

U2

U3

U4

B1

U5U6

U7U8

B2

U9

U10

UM

BK

middot middot middot

middot middot middot

Figure 1 Configuration of a multi-spot-beam satellite communication system

antenna gain of the satellite It is assumed that the value of119866119878is the same for all the users in this paper 119896 is Boltzmannrsquos

constant which is 1379 times 10minus23WKHzIt is noted that the interbeam interference from the

sidelobes of adjacent spot beams will decrease the capacityof each user However the interbeam interference is ignoredhere because the very narrow spot beams over a large numberof spot beams are considered [10]

According to (1) and (2) it is shown that the capacityallocated to the 119894th user is determined by the allocated satellitetransmitting power given as

119863119894=

119875119894sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198730)120572119894sdot 119896 (3)

It is observed from (3) that the allocated capacity 119863119894of

the 119894th user is increased as the power allocated to it increasesHowever the total satellite power resources are fixed so thecapacity of the system is limited Moreover the allocatedcapacity of each user is also constrained by the bandwidthresources allocated to it which are also scarce in the systemWhen the coding and modulation mode adopted by the 119894thuser is given the bandwidth that needs to be provided to it isexpressed as

119882119894=119863119894sdot [1 + 120588 (120572

119894)]

120578 (120572119894)

(4)

where 120578(120572119894) and 120588(120572

119894) are the spectral efficiency and roll-off

factor of the coding and modulation mode 120572119894

Let119882119861119894denote the bandwidth of the 119894th spot beam Thus

the total bandwidth that can be provided to the users in the 119894thspot beam cannot exceed 119882

119861119894 In other words the allocated

capacity to the users is also constrained by the bandwidth ofeach spot beam

3 Mathematical Formulation of theOptimization Problem

In this study the objective of the power allocation opti-mization is to minimize the sum of the squared differencesbetween the traffic demand and the capacity allocated to eachuser taking account of a trade-off between the maximumtotal system capacity and the fairness of power allocationamongst the users Therefore the optimization problem isformulated as follows

min119875119894

119872

sum

119894=1

(119879119894minus 119863119894)2 (5)

subject to

119863119894=

119875119894sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198730)120572119894sdot 119896

le 119879119894 (6)

119872

sum

119894=1

119875119894le 119875total (7)

sum

119894isinN119861119895

119882119894le 119882119861119895 (8)

4 Mathematical Problems in Engineering

The constraint (6) indicates that the allocated capacity toeach user should not exceed the traffic demand of it in orderto avoid the waste of the scarce power resources Conditions(7)-(8) imply the constraint for the total power of the satelliteand the total bandwidth of each spot beam respectively

It is seen that the problem is a nonlinear optimizationproblem with constraints Moreover it is obvious that theobjective function in (5) is convex and the functions inconstrains (6)ndash(8) are linear As a result the problem underconsideration is a convex optimization [11]

Due to the nonlinearity of the optimization it is difficultto obtain the global optimal solution In order to makethe above problem tractable an iterative algorithm basedon the duality theory is proposed in the following sectionIt is known that if the optimization problem is a convexoptimization problem the duality gap between the primalproblem and dual problem is zero and the optimal value ofthe dual problem is equal to the optimal value of the primalproblem As a result the dual problem can be first solvedto obtain the optimal dual solution and the primal optimalsolution is then computed by solving the primal problem atthe point of the optimal dual solution [11] Fortunately it hasbeen proved that the optimization problem studied here isa convex optimization problem thus the power allocationresult obtained by the proposed algorithm is the optimalpower allocation for the users in themulti-spot-beam satellitecommunication system

4 Proposed Power Allocation Algorithm

As mentioned previously the proposed power allocationalgorithm is based on the duality theory By introducingnonnegative dual variables 120582 and 120590 = [120590

1 1205902 120590

119870] yielded

the Lagrangian given as

119871 (P120590 120582) =119872

sum

119894=1

(119879119894minus 119863119894)2minus 120582(119875total minus

119872

sum

119894=1

119875119894)

minus

119870

sum

119894=1

120590119894(119882119861119894minus sum

119895isinN119861119894

119882119895)

(9)

where P = [1198751 1198752 119875

119899]

Maximizing (9) with respect to the nonnegative 120582 and 120590brings the following function

119911 (P) = max120582ge0120590ge0

119871 (P120590 120582) (10)

It is seen that if the optimization variables 119875119894are satisfied

with the constrains (7)-(8) then 120582(119875total minus sum119872

119894=1119875119894) ge 0 and

sum119870

119894=1120590119894(119882119861119894minus sum119895isinN119861119894

119882119895) ge 0 Therefore (10) will get the

maximal value when 120582(119875total minussum119872

119894=1119875119894) = 0 andsum119870

119894=1120590119894(119882119861119894minus

sum119895isinN119861119894

119882119895) = 0 As a result 119911(P) = sum

119872

119894=1(119879119894minus 119863119894)2 To this

end the primal optimization with constraints is changed intothe optimization with no constraints as follows [11]

119901 = minP

119911 (P) = minP

max120582ge0120590ge0

119871 (P120590 120582) (11)

In addition the Lagrange dual function can be obtainedfrom (9) as [11]

119863 (120590 120582) = minP

119871 (P120590 120582) (12)

and the dual problem of (11) can be written as

119889 = max120582ge0120590ge0

119863 (120590 120582) = max120582ge0120590ge0

minP

119871 (P120590 120582) (13)

The work in [12] solved the joint spectrum and powerallocation in cognitive radio networks and proposed amethod to solve the dual problem Inspired with this paperthe dual problem (13) is decomposed into the following twosequentially iterative subproblems

Subproblem 1 Power Allocation Given the dual variables 120582and 120590 for any 119894 = [1 119872] maximizing (9) with respectto 119875119894brings the following equation

2 sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

(119879119894minus119875opt119894

sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

)

= 120582 + 120590119895

119866119878sdot (119866119879)119894 sdot [1 + 120588 (120572119894)]

119871119894sdot (1198641198871198990)120572119894sdot 119896 sdot 120578 (120572

119894) 119894 isin N

119861119895

(14)

The optimized power allocation of the 119894th user119875opt119894

can beeasily obtained from (14) It is seen from (14) that nonnegativedual variables 120582 and 120590 guarantee that 119879

119894ge 119863119894 As a result the

constrain (6) is satisfied

Subproblem 2 Dual Variables Update The optimal dualvariables can be obtained by solving the problem

(120590opt 120582

opt) = argmax

120590120582

min [119871 (Popt120590 120582)] (15)

Due to concavity of the dual objective function here asubgradient (a generalization of gradient) method is appliedto update the duality variables shown as [13]

120582119899+1

= [120582119899minus Δ119899

120582(119875total minus

119872

sum

119894=1

119875opt119894)]

+

(16)

120590119899+1

119894= [

[

120590119899

119894minus Δ119899

120590(119882119861119894minus sum

119895isinN119861119894

119882119895)]

]

+

(17)

where [119909]+ = max0 119909119899 is the iteration number and Δ isthe iteration step size of each dual variable

The subgradient method is very suitable for the situationthat the dual function is not differentiable As a result themethod has been widely applied to solve the optimizationproblem [12ndash18] It has proven that the above dual variablesupdate algorithm is guaranteed to converge to the optimalsolution as long as the iteration step size chosen is sufficientlysmall [13] A common criterion for choosing the iteration stepsize is that the step size must be square summable but notabsolute summable [13 18]

Mathematical Problems in Engineering 5

Step 1 Set appropriate initial values for the dual variablesStep 2 Substitute the values of the dual variables into (14) and then calculate the

optimized power allocation to each userStep 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 variablesStep 4 If the conditions of 1003816100381610038161003816120582

119899+1(119875total minus sum119894 119875119894)

1003816100381610038161003816 lt 120576 and100381610038161003816100381610038161003816120590119899+1

119894(119882119861119894minus sum119895isinN119861119894

119882119895)100381610038161003816100381610038161003816lt 120576 forall119894 isin 1 119870 are satisfied simultaneously then terminate

the algorithm Otherwise jump to Step 2

Algorithm 1 The proposed power allocation algorithm

Table 1 Parameters of the multi-spot-beam satellite communication system

Parameter ValueBeam number 4User number 20User number per spot beam 5Traffic demand of each user From 1Mbps to 20Mbps by step of 1MbpsTotal satellite power [119875total] 20WSatellite transmitting antenna gain [119866

119878] 20000

Bandwidth of each spot beam 100MHzGain-to-equivalent noise temperature ratio of the receiving equipment [119866119879] 20Downlink loss [119871

119894] 211989021

Spectral efficiency of the coding and modulation mode [120578(120572119894)] 15

Roll-off factor of the coding and modulation mode [120588(120572119894)] 1

Threshold signal-to-noise ratio per bit of the coding and modulation mode [(1198641198871198730)120572119894] 263

The whole process of the proposed power allocationalgorithm can be summarized as shown in Algorithm 1

According to Algorithm 1 it is shown that the computa-tional complexity of step 2 and step 3 is 119874(119872) and 119874(2119870)respectively Thus the total computational complexity of thealgorithm is119874(119878119872+2119878119870) where 119878 is the number of iterationsIt is noted that 119878 is independent of 119870 and 119872 Therefore thecomputational complexity of the proposed algorithm is linearwith both the numbers of the spot beams and users and theproposed algorithm is easy to be implemented in practice

5 Simulation Results and Analysis

For the simulation a multi-spot-beam satellite communica-tion system model is set up It is assumed that the values ofdownlink loss gain-to-equivalent noise temperature ratio ofthe receiving equipment and coding and modulation modeare the same for all the users The parameters of the systemare shown in Table 1

51 Efficiency of the Proposed Power Allocation AlgorithmThe proposed power allocation algorithm is compared withthe following two traditional allocation algorithms in orderto verify the efficiency of it

(i) Uniform Resource Allocation Algorithm The powerallocated to each user is 119875

119894= 119875total119872 119894 isin

1 2 119872 The bandwidth allocated to the user in

Table 2 Total system capacity of the three algorithms when thechannel conditions of each user are the same

Algorithms sum119862119894

Uniform resource allocation 1091MbpsProportional resource allocation 1091MbpsProposed optimal power allocation 1091Mbps

the same spot beam is 119882119895= 119882119861119894|N119861119894| 119895 isin N

119861119894

where |N119861119894| is the cardinality of the setN

119861119894

(ii) Proportional Resource Allocation Algorithm Thepower allocated to each user is 119875

119894= 119879119894sdot 119875totalsum

119872

119894=1119879119894

119894 isin 1 2 119872 The bandwidth allocated to the userin the same spot beam is119882

119895= 119879119895sdot119882119861119894sum119896isinN119861119894

119879119896 119895 isin

N119861119894

Figure 2 shows the capacity distributions of the userswhich are allocated by the three algorithms Table 2 showsthe total system capacities of the three algorithms It isnoted that when the channel conditions of each user arethe same the uniform resource allocation algorithm is aspecial case of the water-fill algorithm which can achieve themaximal total system capacity [19] As shown in Figure 2 theuniform resource allocation algorithm uniformly allocatesthe resources to each user regardless of the traffic demandof each user even resulting in some users being allocatedmore capacity than that is needed As a result this uniformresource allocation algorithm causes a waste of the scarce

6 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 2 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are the same

Table 3 Sum of (119879119894minus119862119894)2 of the three algorithms when the channel

conditions of each user are the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 113411986415

Proportional resource allocation 662711986414

Proposed optimal power allocation 547011986414

resources The proportional resource allocation algorithmallocates the power resources to each user only accordingto its traffic demand The capacity allocated to each useris linearly increasing considering the fairness of powerallocation amongst the users to some extent However it isnot the optimal solution to the optimization In order to get abetter fairness the proposed power algorithms provide morecapacity to the userswith higher traffic demands and suppressthe capacities of the users with lower traffic demands Forexample the algorithm provides no capacity to the five lowesttraffic demand users Although the capacities allocated toeach user are different the total system capacities are the samefor the three algorithms due to the linearity of the capacityfunction in terms of the allocated power and the samenessof the channel conditions of each user The conclusion is alsodemonstrated by the data in Table 2

Figure 3 shows the squared difference between the trafficdemand and the capacity allocated of each user of thethree algorithms Table 3 presents the sum of the squareddifferences of the three algorithms It is shown from Figure 3that for the uniform and proportional resource allocationalgorithms although the squared difference between thetraffic demand and the capacity allocated to the user withlow traffic demand is small however the squared differenceincreases rapidly when the traffic demand increases On

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 3 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arethe same

the contrast for the proposed optimal power allocationalgorithm the squared difference between the traffic demandand the capacity allocated to the userswith low trafficdemandis larger than that of the former two algorithms Howeverthe squared difference is almost the same from user 6 to user20 As a result the total squared difference of the proposedpower allocation algorithm is less than that of the former twoalgorithms which is also shown in Table 3 In other wordsthe power allocation result of the proposed algorithm is thebest amongst the three algorithms

52 Impact of the Spot Beam Bandwidth on the PowerAllocationResult Asmentioned above the capacity allocatedto each user is constrained by both the power and bandwidthallocated to it Due to the limitation of the bandwidth of eachspot beam the capacity allocated to the users in the same spotbeam is also constrained As a result the power resourcesallocated to the users are impacted In order to show theimpact of the spot beam bandwidth on the power allocationresult the power allocation results are compared when thebandwidth of each spot beam is set to be 25MHz 50MHzand 100MHz and other parameters of the system stay thesame

From Figure 4 it is obvious that the power allocationresults are different when the bandwidth resources of eachspot beam are various When the spot beam bandwidth is25MHz the capacity allocated to each user is constrainedby the bandwidth Although the total system power is 20Wthe total power allocated to all the users is only 1306W Asa result the power resources in the system are wasted andthe total system capacity is decreased When the bandwidthis 50MHz the capacities allocated to the users in the last

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Satellite

C8

U1

U2

U3

U4

B1

U5U6

U7U8

B2

U9

U10

UM

BK

middot middot middot

middot middot middot

Figure 1 Configuration of a multi-spot-beam satellite communication system

antenna gain of the satellite It is assumed that the value of119866119878is the same for all the users in this paper 119896 is Boltzmannrsquos

constant which is 1379 times 10minus23WKHzIt is noted that the interbeam interference from the

sidelobes of adjacent spot beams will decrease the capacityof each user However the interbeam interference is ignoredhere because the very narrow spot beams over a large numberof spot beams are considered [10]

According to (1) and (2) it is shown that the capacityallocated to the 119894th user is determined by the allocated satellitetransmitting power given as

119863119894=

119875119894sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198730)120572119894sdot 119896 (3)

It is observed from (3) that the allocated capacity 119863119894of

the 119894th user is increased as the power allocated to it increasesHowever the total satellite power resources are fixed so thecapacity of the system is limited Moreover the allocatedcapacity of each user is also constrained by the bandwidthresources allocated to it which are also scarce in the systemWhen the coding and modulation mode adopted by the 119894thuser is given the bandwidth that needs to be provided to it isexpressed as

119882119894=119863119894sdot [1 + 120588 (120572

119894)]

120578 (120572119894)

(4)

where 120578(120572119894) and 120588(120572

119894) are the spectral efficiency and roll-off

factor of the coding and modulation mode 120572119894

Let119882119861119894denote the bandwidth of the 119894th spot beam Thus

the total bandwidth that can be provided to the users in the 119894thspot beam cannot exceed 119882

119861119894 In other words the allocated

capacity to the users is also constrained by the bandwidth ofeach spot beam

3 Mathematical Formulation of theOptimization Problem

In this study the objective of the power allocation opti-mization is to minimize the sum of the squared differencesbetween the traffic demand and the capacity allocated to eachuser taking account of a trade-off between the maximumtotal system capacity and the fairness of power allocationamongst the users Therefore the optimization problem isformulated as follows

min119875119894

119872

sum

119894=1

(119879119894minus 119863119894)2 (5)

subject to

119863119894=

119875119894sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198730)120572119894sdot 119896

le 119879119894 (6)

119872

sum

119894=1

119875119894le 119875total (7)

sum

119894isinN119861119895

119882119894le 119882119861119895 (8)

4 Mathematical Problems in Engineering

The constraint (6) indicates that the allocated capacity toeach user should not exceed the traffic demand of it in orderto avoid the waste of the scarce power resources Conditions(7)-(8) imply the constraint for the total power of the satelliteand the total bandwidth of each spot beam respectively

It is seen that the problem is a nonlinear optimizationproblem with constraints Moreover it is obvious that theobjective function in (5) is convex and the functions inconstrains (6)ndash(8) are linear As a result the problem underconsideration is a convex optimization [11]

Due to the nonlinearity of the optimization it is difficultto obtain the global optimal solution In order to makethe above problem tractable an iterative algorithm basedon the duality theory is proposed in the following sectionIt is known that if the optimization problem is a convexoptimization problem the duality gap between the primalproblem and dual problem is zero and the optimal value ofthe dual problem is equal to the optimal value of the primalproblem As a result the dual problem can be first solvedto obtain the optimal dual solution and the primal optimalsolution is then computed by solving the primal problem atthe point of the optimal dual solution [11] Fortunately it hasbeen proved that the optimization problem studied here isa convex optimization problem thus the power allocationresult obtained by the proposed algorithm is the optimalpower allocation for the users in themulti-spot-beam satellitecommunication system

4 Proposed Power Allocation Algorithm

As mentioned previously the proposed power allocationalgorithm is based on the duality theory By introducingnonnegative dual variables 120582 and 120590 = [120590

1 1205902 120590

119870] yielded

the Lagrangian given as

119871 (P120590 120582) =119872

sum

119894=1

(119879119894minus 119863119894)2minus 120582(119875total minus

119872

sum

119894=1

119875119894)

minus

119870

sum

119894=1

120590119894(119882119861119894minus sum

119895isinN119861119894

119882119895)

(9)

where P = [1198751 1198752 119875

119899]

Maximizing (9) with respect to the nonnegative 120582 and 120590brings the following function

119911 (P) = max120582ge0120590ge0

119871 (P120590 120582) (10)

It is seen that if the optimization variables 119875119894are satisfied

with the constrains (7)-(8) then 120582(119875total minus sum119872

119894=1119875119894) ge 0 and

sum119870

119894=1120590119894(119882119861119894minus sum119895isinN119861119894

119882119895) ge 0 Therefore (10) will get the

maximal value when 120582(119875total minussum119872

119894=1119875119894) = 0 andsum119870

119894=1120590119894(119882119861119894minus

sum119895isinN119861119894

119882119895) = 0 As a result 119911(P) = sum

119872

119894=1(119879119894minus 119863119894)2 To this

end the primal optimization with constraints is changed intothe optimization with no constraints as follows [11]

119901 = minP

119911 (P) = minP

max120582ge0120590ge0

119871 (P120590 120582) (11)

In addition the Lagrange dual function can be obtainedfrom (9) as [11]

119863 (120590 120582) = minP

119871 (P120590 120582) (12)

and the dual problem of (11) can be written as

119889 = max120582ge0120590ge0

119863 (120590 120582) = max120582ge0120590ge0

minP

119871 (P120590 120582) (13)

The work in [12] solved the joint spectrum and powerallocation in cognitive radio networks and proposed amethod to solve the dual problem Inspired with this paperthe dual problem (13) is decomposed into the following twosequentially iterative subproblems

Subproblem 1 Power Allocation Given the dual variables 120582and 120590 for any 119894 = [1 119872] maximizing (9) with respectto 119875119894brings the following equation

2 sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

(119879119894minus119875opt119894

sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

)

= 120582 + 120590119895

119866119878sdot (119866119879)119894 sdot [1 + 120588 (120572119894)]

119871119894sdot (1198641198871198990)120572119894sdot 119896 sdot 120578 (120572

119894) 119894 isin N

119861119895

(14)

The optimized power allocation of the 119894th user119875opt119894

can beeasily obtained from (14) It is seen from (14) that nonnegativedual variables 120582 and 120590 guarantee that 119879

119894ge 119863119894 As a result the

constrain (6) is satisfied

Subproblem 2 Dual Variables Update The optimal dualvariables can be obtained by solving the problem

(120590opt 120582

opt) = argmax

120590120582

min [119871 (Popt120590 120582)] (15)

Due to concavity of the dual objective function here asubgradient (a generalization of gradient) method is appliedto update the duality variables shown as [13]

120582119899+1

= [120582119899minus Δ119899

120582(119875total minus

119872

sum

119894=1

119875opt119894)]

+

(16)

120590119899+1

119894= [

[

120590119899

119894minus Δ119899

120590(119882119861119894minus sum

119895isinN119861119894

119882119895)]

]

+

(17)

where [119909]+ = max0 119909119899 is the iteration number and Δ isthe iteration step size of each dual variable

The subgradient method is very suitable for the situationthat the dual function is not differentiable As a result themethod has been widely applied to solve the optimizationproblem [12ndash18] It has proven that the above dual variablesupdate algorithm is guaranteed to converge to the optimalsolution as long as the iteration step size chosen is sufficientlysmall [13] A common criterion for choosing the iteration stepsize is that the step size must be square summable but notabsolute summable [13 18]

Mathematical Problems in Engineering 5

Step 1 Set appropriate initial values for the dual variablesStep 2 Substitute the values of the dual variables into (14) and then calculate the

optimized power allocation to each userStep 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 variablesStep 4 If the conditions of 1003816100381610038161003816120582

119899+1(119875total minus sum119894 119875119894)

1003816100381610038161003816 lt 120576 and100381610038161003816100381610038161003816120590119899+1

119894(119882119861119894minus sum119895isinN119861119894

119882119895)100381610038161003816100381610038161003816lt 120576 forall119894 isin 1 119870 are satisfied simultaneously then terminate

the algorithm Otherwise jump to Step 2

Algorithm 1 The proposed power allocation algorithm

Table 1 Parameters of the multi-spot-beam satellite communication system

Parameter ValueBeam number 4User number 20User number per spot beam 5Traffic demand of each user From 1Mbps to 20Mbps by step of 1MbpsTotal satellite power [119875total] 20WSatellite transmitting antenna gain [119866

119878] 20000

Bandwidth of each spot beam 100MHzGain-to-equivalent noise temperature ratio of the receiving equipment [119866119879] 20Downlink loss [119871

119894] 211989021

Spectral efficiency of the coding and modulation mode [120578(120572119894)] 15

Roll-off factor of the coding and modulation mode [120588(120572119894)] 1

Threshold signal-to-noise ratio per bit of the coding and modulation mode [(1198641198871198730)120572119894] 263

The whole process of the proposed power allocationalgorithm can be summarized as shown in Algorithm 1

According to Algorithm 1 it is shown that the computa-tional complexity of step 2 and step 3 is 119874(119872) and 119874(2119870)respectively Thus the total computational complexity of thealgorithm is119874(119878119872+2119878119870) where 119878 is the number of iterationsIt is noted that 119878 is independent of 119870 and 119872 Therefore thecomputational complexity of the proposed algorithm is linearwith both the numbers of the spot beams and users and theproposed algorithm is easy to be implemented in practice

5 Simulation Results and Analysis

For the simulation a multi-spot-beam satellite communica-tion system model is set up It is assumed that the values ofdownlink loss gain-to-equivalent noise temperature ratio ofthe receiving equipment and coding and modulation modeare the same for all the users The parameters of the systemare shown in Table 1

51 Efficiency of the Proposed Power Allocation AlgorithmThe proposed power allocation algorithm is compared withthe following two traditional allocation algorithms in orderto verify the efficiency of it

(i) Uniform Resource Allocation Algorithm The powerallocated to each user is 119875

119894= 119875total119872 119894 isin

1 2 119872 The bandwidth allocated to the user in

Table 2 Total system capacity of the three algorithms when thechannel conditions of each user are the same

Algorithms sum119862119894

Uniform resource allocation 1091MbpsProportional resource allocation 1091MbpsProposed optimal power allocation 1091Mbps

the same spot beam is 119882119895= 119882119861119894|N119861119894| 119895 isin N

119861119894

where |N119861119894| is the cardinality of the setN

119861119894

(ii) Proportional Resource Allocation Algorithm Thepower allocated to each user is 119875

119894= 119879119894sdot 119875totalsum

119872

119894=1119879119894

119894 isin 1 2 119872 The bandwidth allocated to the userin the same spot beam is119882

119895= 119879119895sdot119882119861119894sum119896isinN119861119894

119879119896 119895 isin

N119861119894

Figure 2 shows the capacity distributions of the userswhich are allocated by the three algorithms Table 2 showsthe total system capacities of the three algorithms It isnoted that when the channel conditions of each user arethe same the uniform resource allocation algorithm is aspecial case of the water-fill algorithm which can achieve themaximal total system capacity [19] As shown in Figure 2 theuniform resource allocation algorithm uniformly allocatesthe resources to each user regardless of the traffic demandof each user even resulting in some users being allocatedmore capacity than that is needed As a result this uniformresource allocation algorithm causes a waste of the scarce

6 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 2 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are the same

Table 3 Sum of (119879119894minus119862119894)2 of the three algorithms when the channel

conditions of each user are the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 113411986415

Proportional resource allocation 662711986414

Proposed optimal power allocation 547011986414

resources The proportional resource allocation algorithmallocates the power resources to each user only accordingto its traffic demand The capacity allocated to each useris linearly increasing considering the fairness of powerallocation amongst the users to some extent However it isnot the optimal solution to the optimization In order to get abetter fairness the proposed power algorithms provide morecapacity to the userswith higher traffic demands and suppressthe capacities of the users with lower traffic demands Forexample the algorithm provides no capacity to the five lowesttraffic demand users Although the capacities allocated toeach user are different the total system capacities are the samefor the three algorithms due to the linearity of the capacityfunction in terms of the allocated power and the samenessof the channel conditions of each user The conclusion is alsodemonstrated by the data in Table 2

Figure 3 shows the squared difference between the trafficdemand and the capacity allocated of each user of thethree algorithms Table 3 presents the sum of the squareddifferences of the three algorithms It is shown from Figure 3that for the uniform and proportional resource allocationalgorithms although the squared difference between thetraffic demand and the capacity allocated to the user withlow traffic demand is small however the squared differenceincreases rapidly when the traffic demand increases On

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 3 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arethe same

the contrast for the proposed optimal power allocationalgorithm the squared difference between the traffic demandand the capacity allocated to the userswith low trafficdemandis larger than that of the former two algorithms Howeverthe squared difference is almost the same from user 6 to user20 As a result the total squared difference of the proposedpower allocation algorithm is less than that of the former twoalgorithms which is also shown in Table 3 In other wordsthe power allocation result of the proposed algorithm is thebest amongst the three algorithms

52 Impact of the Spot Beam Bandwidth on the PowerAllocationResult Asmentioned above the capacity allocatedto each user is constrained by both the power and bandwidthallocated to it Due to the limitation of the bandwidth of eachspot beam the capacity allocated to the users in the same spotbeam is also constrained As a result the power resourcesallocated to the users are impacted In order to show theimpact of the spot beam bandwidth on the power allocationresult the power allocation results are compared when thebandwidth of each spot beam is set to be 25MHz 50MHzand 100MHz and other parameters of the system stay thesame

From Figure 4 it is obvious that the power allocationresults are different when the bandwidth resources of eachspot beam are various When the spot beam bandwidth is25MHz the capacity allocated to each user is constrainedby the bandwidth Although the total system power is 20Wthe total power allocated to all the users is only 1306W Asa result the power resources in the system are wasted andthe total system capacity is decreased When the bandwidthis 50MHz the capacities allocated to the users in the last

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

The constraint (6) indicates that the allocated capacity toeach user should not exceed the traffic demand of it in orderto avoid the waste of the scarce power resources Conditions(7)-(8) imply the constraint for the total power of the satelliteand the total bandwidth of each spot beam respectively

It is seen that the problem is a nonlinear optimizationproblem with constraints Moreover it is obvious that theobjective function in (5) is convex and the functions inconstrains (6)ndash(8) are linear As a result the problem underconsideration is a convex optimization [11]

Due to the nonlinearity of the optimization it is difficultto obtain the global optimal solution In order to makethe above problem tractable an iterative algorithm basedon the duality theory is proposed in the following sectionIt is known that if the optimization problem is a convexoptimization problem the duality gap between the primalproblem and dual problem is zero and the optimal value ofthe dual problem is equal to the optimal value of the primalproblem As a result the dual problem can be first solvedto obtain the optimal dual solution and the primal optimalsolution is then computed by solving the primal problem atthe point of the optimal dual solution [11] Fortunately it hasbeen proved that the optimization problem studied here isa convex optimization problem thus the power allocationresult obtained by the proposed algorithm is the optimalpower allocation for the users in themulti-spot-beam satellitecommunication system

4 Proposed Power Allocation Algorithm

As mentioned previously the proposed power allocationalgorithm is based on the duality theory By introducingnonnegative dual variables 120582 and 120590 = [120590

1 1205902 120590

119870] yielded

the Lagrangian given as

119871 (P120590 120582) =119872

sum

119894=1

(119879119894minus 119863119894)2minus 120582(119875total minus

119872

sum

119894=1

119875119894)

minus

119870

sum

119894=1

120590119894(119882119861119894minus sum

119895isinN119861119894

119882119895)

(9)

where P = [1198751 1198752 119875

119899]

Maximizing (9) with respect to the nonnegative 120582 and 120590brings the following function

119911 (P) = max120582ge0120590ge0

119871 (P120590 120582) (10)

It is seen that if the optimization variables 119875119894are satisfied

with the constrains (7)-(8) then 120582(119875total minus sum119872

119894=1119875119894) ge 0 and

sum119870

119894=1120590119894(119882119861119894minus sum119895isinN119861119894

119882119895) ge 0 Therefore (10) will get the

maximal value when 120582(119875total minussum119872

119894=1119875119894) = 0 andsum119870

119894=1120590119894(119882119861119894minus

sum119895isinN119861119894

119882119895) = 0 As a result 119911(P) = sum

119872

119894=1(119879119894minus 119863119894)2 To this

end the primal optimization with constraints is changed intothe optimization with no constraints as follows [11]

119901 = minP

119911 (P) = minP

max120582ge0120590ge0

119871 (P120590 120582) (11)

In addition the Lagrange dual function can be obtainedfrom (9) as [11]

119863 (120590 120582) = minP

119871 (P120590 120582) (12)

and the dual problem of (11) can be written as

119889 = max120582ge0120590ge0

119863 (120590 120582) = max120582ge0120590ge0

minP

119871 (P120590 120582) (13)

The work in [12] solved the joint spectrum and powerallocation in cognitive radio networks and proposed amethod to solve the dual problem Inspired with this paperthe dual problem (13) is decomposed into the following twosequentially iterative subproblems

Subproblem 1 Power Allocation Given the dual variables 120582and 120590 for any 119894 = [1 119872] maximizing (9) with respectto 119875119894brings the following equation

2 sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

(119879119894minus119875opt119894

sdot 119866119878sdot (119866119879)119894

119871119894sdot (1198641198871198990)120572119894sdot 119896

)

= 120582 + 120590119895

119866119878sdot (119866119879)119894 sdot [1 + 120588 (120572119894)]

119871119894sdot (1198641198871198990)120572119894sdot 119896 sdot 120578 (120572

119894) 119894 isin N

119861119895

(14)

The optimized power allocation of the 119894th user119875opt119894

can beeasily obtained from (14) It is seen from (14) that nonnegativedual variables 120582 and 120590 guarantee that 119879

119894ge 119863119894 As a result the

constrain (6) is satisfied

Subproblem 2 Dual Variables Update The optimal dualvariables can be obtained by solving the problem

(120590opt 120582

opt) = argmax

120590120582

min [119871 (Popt120590 120582)] (15)

Due to concavity of the dual objective function here asubgradient (a generalization of gradient) method is appliedto update the duality variables shown as [13]

120582119899+1

= [120582119899minus Δ119899

120582(119875total minus

119872

sum

119894=1

119875opt119894)]

+

(16)

120590119899+1

119894= [

[

120590119899

119894minus Δ119899

120590(119882119861119894minus sum

119895isinN119861119894

119882119895)]

]

+

(17)

where [119909]+ = max0 119909119899 is the iteration number and Δ isthe iteration step size of each dual variable

The subgradient method is very suitable for the situationthat the dual function is not differentiable As a result themethod has been widely applied to solve the optimizationproblem [12ndash18] It has proven that the above dual variablesupdate algorithm is guaranteed to converge to the optimalsolution as long as the iteration step size chosen is sufficientlysmall [13] A common criterion for choosing the iteration stepsize is that the step size must be square summable but notabsolute summable [13 18]

Mathematical Problems in Engineering 5

Step 1 Set appropriate initial values for the dual variablesStep 2 Substitute the values of the dual variables into (14) and then calculate the

optimized power allocation to each userStep 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 variablesStep 4 If the conditions of 1003816100381610038161003816120582

119899+1(119875total minus sum119894 119875119894)

1003816100381610038161003816 lt 120576 and100381610038161003816100381610038161003816120590119899+1

119894(119882119861119894minus sum119895isinN119861119894

119882119895)100381610038161003816100381610038161003816lt 120576 forall119894 isin 1 119870 are satisfied simultaneously then terminate

the algorithm Otherwise jump to Step 2

Algorithm 1 The proposed power allocation algorithm

Table 1 Parameters of the multi-spot-beam satellite communication system

Parameter ValueBeam number 4User number 20User number per spot beam 5Traffic demand of each user From 1Mbps to 20Mbps by step of 1MbpsTotal satellite power [119875total] 20WSatellite transmitting antenna gain [119866

119878] 20000

Bandwidth of each spot beam 100MHzGain-to-equivalent noise temperature ratio of the receiving equipment [119866119879] 20Downlink loss [119871

119894] 211989021

Spectral efficiency of the coding and modulation mode [120578(120572119894)] 15

Roll-off factor of the coding and modulation mode [120588(120572119894)] 1

Threshold signal-to-noise ratio per bit of the coding and modulation mode [(1198641198871198730)120572119894] 263

The whole process of the proposed power allocationalgorithm can be summarized as shown in Algorithm 1

According to Algorithm 1 it is shown that the computa-tional complexity of step 2 and step 3 is 119874(119872) and 119874(2119870)respectively Thus the total computational complexity of thealgorithm is119874(119878119872+2119878119870) where 119878 is the number of iterationsIt is noted that 119878 is independent of 119870 and 119872 Therefore thecomputational complexity of the proposed algorithm is linearwith both the numbers of the spot beams and users and theproposed algorithm is easy to be implemented in practice

5 Simulation Results and Analysis

For the simulation a multi-spot-beam satellite communica-tion system model is set up It is assumed that the values ofdownlink loss gain-to-equivalent noise temperature ratio ofthe receiving equipment and coding and modulation modeare the same for all the users The parameters of the systemare shown in Table 1

51 Efficiency of the Proposed Power Allocation AlgorithmThe proposed power allocation algorithm is compared withthe following two traditional allocation algorithms in orderto verify the efficiency of it

(i) Uniform Resource Allocation Algorithm The powerallocated to each user is 119875

119894= 119875total119872 119894 isin

1 2 119872 The bandwidth allocated to the user in

Table 2 Total system capacity of the three algorithms when thechannel conditions of each user are the same

Algorithms sum119862119894

Uniform resource allocation 1091MbpsProportional resource allocation 1091MbpsProposed optimal power allocation 1091Mbps

the same spot beam is 119882119895= 119882119861119894|N119861119894| 119895 isin N

119861119894

where |N119861119894| is the cardinality of the setN

119861119894

(ii) Proportional Resource Allocation Algorithm Thepower allocated to each user is 119875

119894= 119879119894sdot 119875totalsum

119872

119894=1119879119894

119894 isin 1 2 119872 The bandwidth allocated to the userin the same spot beam is119882

119895= 119879119895sdot119882119861119894sum119896isinN119861119894

119879119896 119895 isin

N119861119894

Figure 2 shows the capacity distributions of the userswhich are allocated by the three algorithms Table 2 showsthe total system capacities of the three algorithms It isnoted that when the channel conditions of each user arethe same the uniform resource allocation algorithm is aspecial case of the water-fill algorithm which can achieve themaximal total system capacity [19] As shown in Figure 2 theuniform resource allocation algorithm uniformly allocatesthe resources to each user regardless of the traffic demandof each user even resulting in some users being allocatedmore capacity than that is needed As a result this uniformresource allocation algorithm causes a waste of the scarce

6 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 2 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are the same

Table 3 Sum of (119879119894minus119862119894)2 of the three algorithms when the channel

conditions of each user are the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 113411986415

Proportional resource allocation 662711986414

Proposed optimal power allocation 547011986414

resources The proportional resource allocation algorithmallocates the power resources to each user only accordingto its traffic demand The capacity allocated to each useris linearly increasing considering the fairness of powerallocation amongst the users to some extent However it isnot the optimal solution to the optimization In order to get abetter fairness the proposed power algorithms provide morecapacity to the userswith higher traffic demands and suppressthe capacities of the users with lower traffic demands Forexample the algorithm provides no capacity to the five lowesttraffic demand users Although the capacities allocated toeach user are different the total system capacities are the samefor the three algorithms due to the linearity of the capacityfunction in terms of the allocated power and the samenessof the channel conditions of each user The conclusion is alsodemonstrated by the data in Table 2

Figure 3 shows the squared difference between the trafficdemand and the capacity allocated of each user of thethree algorithms Table 3 presents the sum of the squareddifferences of the three algorithms It is shown from Figure 3that for the uniform and proportional resource allocationalgorithms although the squared difference between thetraffic demand and the capacity allocated to the user withlow traffic demand is small however the squared differenceincreases rapidly when the traffic demand increases On

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 3 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arethe same

the contrast for the proposed optimal power allocationalgorithm the squared difference between the traffic demandand the capacity allocated to the userswith low trafficdemandis larger than that of the former two algorithms Howeverthe squared difference is almost the same from user 6 to user20 As a result the total squared difference of the proposedpower allocation algorithm is less than that of the former twoalgorithms which is also shown in Table 3 In other wordsthe power allocation result of the proposed algorithm is thebest amongst the three algorithms

52 Impact of the Spot Beam Bandwidth on the PowerAllocationResult Asmentioned above the capacity allocatedto each user is constrained by both the power and bandwidthallocated to it Due to the limitation of the bandwidth of eachspot beam the capacity allocated to the users in the same spotbeam is also constrained As a result the power resourcesallocated to the users are impacted In order to show theimpact of the spot beam bandwidth on the power allocationresult the power allocation results are compared when thebandwidth of each spot beam is set to be 25MHz 50MHzand 100MHz and other parameters of the system stay thesame

From Figure 4 it is obvious that the power allocationresults are different when the bandwidth resources of eachspot beam are various When the spot beam bandwidth is25MHz the capacity allocated to each user is constrainedby the bandwidth Although the total system power is 20Wthe total power allocated to all the users is only 1306W Asa result the power resources in the system are wasted andthe total system capacity is decreased When the bandwidthis 50MHz the capacities allocated to the users in the last

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Step 1 Set appropriate initial values for the dual variablesStep 2 Substitute the values of the dual variables into (14) and then calculate the

optimized power allocation to each userStep 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 variablesStep 4 If the conditions of 1003816100381610038161003816120582

119899+1(119875total minus sum119894 119875119894)

1003816100381610038161003816 lt 120576 and100381610038161003816100381610038161003816120590119899+1

119894(119882119861119894minus sum119895isinN119861119894

119882119895)100381610038161003816100381610038161003816lt 120576 forall119894 isin 1 119870 are satisfied simultaneously then terminate

the algorithm Otherwise jump to Step 2

Algorithm 1 The proposed power allocation algorithm

Table 1 Parameters of the multi-spot-beam satellite communication system

Parameter ValueBeam number 4User number 20User number per spot beam 5Traffic demand of each user From 1Mbps to 20Mbps by step of 1MbpsTotal satellite power [119875total] 20WSatellite transmitting antenna gain [119866

119878] 20000

Bandwidth of each spot beam 100MHzGain-to-equivalent noise temperature ratio of the receiving equipment [119866119879] 20Downlink loss [119871

119894] 211989021

Spectral efficiency of the coding and modulation mode [120578(120572119894)] 15

Roll-off factor of the coding and modulation mode [120588(120572119894)] 1

Threshold signal-to-noise ratio per bit of the coding and modulation mode [(1198641198871198730)120572119894] 263

The whole process of the proposed power allocationalgorithm can be summarized as shown in Algorithm 1

According to Algorithm 1 it is shown that the computa-tional complexity of step 2 and step 3 is 119874(119872) and 119874(2119870)respectively Thus the total computational complexity of thealgorithm is119874(119878119872+2119878119870) where 119878 is the number of iterationsIt is noted that 119878 is independent of 119870 and 119872 Therefore thecomputational complexity of the proposed algorithm is linearwith both the numbers of the spot beams and users and theproposed algorithm is easy to be implemented in practice

5 Simulation Results and Analysis

For the simulation a multi-spot-beam satellite communica-tion system model is set up It is assumed that the values ofdownlink loss gain-to-equivalent noise temperature ratio ofthe receiving equipment and coding and modulation modeare the same for all the users The parameters of the systemare shown in Table 1

51 Efficiency of the Proposed Power Allocation AlgorithmThe proposed power allocation algorithm is compared withthe following two traditional allocation algorithms in orderto verify the efficiency of it

(i) Uniform Resource Allocation Algorithm The powerallocated to each user is 119875

119894= 119875total119872 119894 isin

1 2 119872 The bandwidth allocated to the user in

Table 2 Total system capacity of the three algorithms when thechannel conditions of each user are the same

Algorithms sum119862119894

Uniform resource allocation 1091MbpsProportional resource allocation 1091MbpsProposed optimal power allocation 1091Mbps

the same spot beam is 119882119895= 119882119861119894|N119861119894| 119895 isin N

119861119894

where |N119861119894| is the cardinality of the setN

119861119894

(ii) Proportional Resource Allocation Algorithm Thepower allocated to each user is 119875

119894= 119879119894sdot 119875totalsum

119872

119894=1119879119894

119894 isin 1 2 119872 The bandwidth allocated to the userin the same spot beam is119882

119895= 119879119895sdot119882119861119894sum119896isinN119861119894

119879119896 119895 isin

N119861119894

Figure 2 shows the capacity distributions of the userswhich are allocated by the three algorithms Table 2 showsthe total system capacities of the three algorithms It isnoted that when the channel conditions of each user arethe same the uniform resource allocation algorithm is aspecial case of the water-fill algorithm which can achieve themaximal total system capacity [19] As shown in Figure 2 theuniform resource allocation algorithm uniformly allocatesthe resources to each user regardless of the traffic demandof each user even resulting in some users being allocatedmore capacity than that is needed As a result this uniformresource allocation algorithm causes a waste of the scarce

6 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 2 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are the same

Table 3 Sum of (119879119894minus119862119894)2 of the three algorithms when the channel

conditions of each user are the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 113411986415

Proportional resource allocation 662711986414

Proposed optimal power allocation 547011986414

resources The proportional resource allocation algorithmallocates the power resources to each user only accordingto its traffic demand The capacity allocated to each useris linearly increasing considering the fairness of powerallocation amongst the users to some extent However it isnot the optimal solution to the optimization In order to get abetter fairness the proposed power algorithms provide morecapacity to the userswith higher traffic demands and suppressthe capacities of the users with lower traffic demands Forexample the algorithm provides no capacity to the five lowesttraffic demand users Although the capacities allocated toeach user are different the total system capacities are the samefor the three algorithms due to the linearity of the capacityfunction in terms of the allocated power and the samenessof the channel conditions of each user The conclusion is alsodemonstrated by the data in Table 2

Figure 3 shows the squared difference between the trafficdemand and the capacity allocated of each user of thethree algorithms Table 3 presents the sum of the squareddifferences of the three algorithms It is shown from Figure 3that for the uniform and proportional resource allocationalgorithms although the squared difference between thetraffic demand and the capacity allocated to the user withlow traffic demand is small however the squared differenceincreases rapidly when the traffic demand increases On

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 3 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arethe same

the contrast for the proposed optimal power allocationalgorithm the squared difference between the traffic demandand the capacity allocated to the userswith low trafficdemandis larger than that of the former two algorithms Howeverthe squared difference is almost the same from user 6 to user20 As a result the total squared difference of the proposedpower allocation algorithm is less than that of the former twoalgorithms which is also shown in Table 3 In other wordsthe power allocation result of the proposed algorithm is thebest amongst the three algorithms

52 Impact of the Spot Beam Bandwidth on the PowerAllocationResult Asmentioned above the capacity allocatedto each user is constrained by both the power and bandwidthallocated to it Due to the limitation of the bandwidth of eachspot beam the capacity allocated to the users in the same spotbeam is also constrained As a result the power resourcesallocated to the users are impacted In order to show theimpact of the spot beam bandwidth on the power allocationresult the power allocation results are compared when thebandwidth of each spot beam is set to be 25MHz 50MHzand 100MHz and other parameters of the system stay thesame

From Figure 4 it is obvious that the power allocationresults are different when the bandwidth resources of eachspot beam are various When the spot beam bandwidth is25MHz the capacity allocated to each user is constrainedby the bandwidth Although the total system power is 20Wthe total power allocated to all the users is only 1306W Asa result the power resources in the system are wasted andthe total system capacity is decreased When the bandwidthis 50MHz the capacities allocated to the users in the last

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 2 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are the same

Table 3 Sum of (119879119894minus119862119894)2 of the three algorithms when the channel

conditions of each user are the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 113411986415

Proportional resource allocation 662711986414

Proposed optimal power allocation 547011986414

resources The proportional resource allocation algorithmallocates the power resources to each user only accordingto its traffic demand The capacity allocated to each useris linearly increasing considering the fairness of powerallocation amongst the users to some extent However it isnot the optimal solution to the optimization In order to get abetter fairness the proposed power algorithms provide morecapacity to the userswith higher traffic demands and suppressthe capacities of the users with lower traffic demands Forexample the algorithm provides no capacity to the five lowesttraffic demand users Although the capacities allocated toeach user are different the total system capacities are the samefor the three algorithms due to the linearity of the capacityfunction in terms of the allocated power and the samenessof the channel conditions of each user The conclusion is alsodemonstrated by the data in Table 2

Figure 3 shows the squared difference between the trafficdemand and the capacity allocated of each user of thethree algorithms Table 3 presents the sum of the squareddifferences of the three algorithms It is shown from Figure 3that for the uniform and proportional resource allocationalgorithms although the squared difference between thetraffic demand and the capacity allocated to the user withlow traffic demand is small however the squared differenceincreases rapidly when the traffic demand increases On

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 3 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arethe same

the contrast for the proposed optimal power allocationalgorithm the squared difference between the traffic demandand the capacity allocated to the userswith low trafficdemandis larger than that of the former two algorithms Howeverthe squared difference is almost the same from user 6 to user20 As a result the total squared difference of the proposedpower allocation algorithm is less than that of the former twoalgorithms which is also shown in Table 3 In other wordsthe power allocation result of the proposed algorithm is thebest amongst the three algorithms

52 Impact of the Spot Beam Bandwidth on the PowerAllocationResult Asmentioned above the capacity allocatedto each user is constrained by both the power and bandwidthallocated to it Due to the limitation of the bandwidth of eachspot beam the capacity allocated to the users in the same spotbeam is also constrained As a result the power resourcesallocated to the users are impacted In order to show theimpact of the spot beam bandwidth on the power allocationresult the power allocation results are compared when thebandwidth of each spot beam is set to be 25MHz 50MHzand 100MHz and other parameters of the system stay thesame

From Figure 4 it is obvious that the power allocationresults are different when the bandwidth resources of eachspot beam are various When the spot beam bandwidth is25MHz the capacity allocated to each user is constrainedby the bandwidth Although the total system power is 20Wthe total power allocated to all the users is only 1306W Asa result the power resources in the system are wasted andthe total system capacity is decreased When the bandwidthis 50MHz the capacities allocated to the users in the last

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandSpot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

ith user

Figure 4 Comparison of the three different spot beams bandwidthsin terms of the capacity allocated to each user

Table 4 Total system capacity of three different spot beamsbandwidths

Bandwidth of each spot beam sum119862119894

25MHz 7125Mbps50MHz 1091Mbps100MHz 1091Mbps

two spot beams are constrained by the bandwidth due tothe high traffic demand of users Thus the power resourceswill be provided to the users with low traffic demands inthe former two spot beams When the bandwidth of eachspot beam is 100MHz the system has more than enoughbandwidth to be allocated to each user thus the capacityallocated to each user is limited by the total system powerresources In order to improve the fairness of power allocationamongst the users the power resources are rarely or neverprovided to the users with low traffic demand Although thepower resources allocated to each user are different when thespot beam bandwidth is 50MHz and 100MHz the powerresources are sufficiently utilized As a result the total systemcapacity is the same which is also seen from Table 4

As mentioned in Figure 5 and Table 5 when the band-width of each spot beam is lower more power resources willbe provided to the users with low traffic demand Thereforeit is seen from Figure 5 that the squared difference betweentraffic demand and allocated capacity to users with low trafficdemand is smaller However the squared difference is lagerfor the users with high traffic demand As a result the totalsquared difference is larger when the bandwidth of each spotbeam is lower This conclusion can be also observed fromTable 5

2 4 6 8 10 12 14 16 18 200

05

15

25

1

2

Spot beam bandwidth is 25MHzSpot beam bandwidth is 50MHzSpot beam bandwidth is 100MHz

times1014

(TiminusCi)2

ith user

Figure 5 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 5 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Bandwidth of each spot beam sum (119879119894minus 119862119894)2

25MHz 153411986414

50MHz 747611986414

100MHz 547011986414

53 Impact of the Coding and Modulation Mode of EachUser on the Power Allocation Result It is known that thepower efficiency and spectral efficiency of a given codingand modulation mode are usually contradictory to eachother In other word a higher spectral efficiency coding andmodulation code can support more capacity in the limitedbandwidth However more power must be provided to it tosupport the coding and modulation mode due to a highervalue of 119864

1198871198730 resulting in lower power efficiency and vice

versa It is seen from the analysis in Section 52 that whenthe bandwidth of each spot beam is 25MHz the capacityallocated to each other is limited by the bandwidth and thepower resources are wasted In order to solve the problema higher bandwidth efficiency coding and modulation modecan be adopted by each user The capacity allocation resultsare compared when each user adopts the three differentcoding and modulation modes as shown in Table 6

It is known that when mode 1 is adopted by each user thepower resources are wasted due to the low spectral efficiencyWhenmode 2 is adopted by each user it is seen from Figure 6that more capacity will be allocated to the users in spot beam2 to spot beam 4 due to the higher spectral efficiency ofthe mode and sufficient utilization of the power resourceAs a result the total system capacity is increased When

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandMode 1

Mode 2Mode 3

ith user

Figure 6 Comparison of the three different coding andmodulationmodes in terms of the capacity allocated to each user

Table 6 Threshold signal-to-noise ratio per bit and spectralefficiency of the three coding and modulation modes

Coding andmodulation mode

Threshold signal-to-noiseratio per bit

Spectralefficiency

Mode 1 263 15Mode 2 363 175Mode 3 447 215

Table 7 Total system capacity of three different coding andmodulation modes

Adopted coding and modulation mode of each user sum119862119894

Mode 1 7125MbpsMode 2 7902MbpsMode 3 6424Mbps

mode 3 is adopted although the spectral efficiency is furtherimproved the power efficiency is further reduced Thereforethe capacity allocated to each user is limited by the powerresources allocated to it Due to the low power efficiency thetotal system capacity is increased which is shown in Table 7

When the spectral efficiency of the coding and modu-lation mode is higher the users with high traffic demandin the last several spot beams are provided more capacitydue to the higher spectral efficiency resulting in a lowersquared difference as shown in Figure 7 Therefore thetotal system squared difference between traffic demand andcapacity allocated to the users is smaller especially for mode3 This conclusion is obviously seen from Table 8

54 Impact of the Channel Condition of EachUser on the PowerAllocation Result It is known that the channel conditions ofeach user are affected by many kinds of factor causing thatthe downlink losses of each user are not the same In order to

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Mode 1Mode 2Mode 3

times1014

(TiminusCi)2

ith user

Figure 7 Comparison of the three different spot beams bandwidthsin terms of the squared difference between the traffic demand andthe capacity allocated to each user

Table 8 Sum of (119879119894minus 119862119894)2 of the three different spot beams

bandwidths

Adopted coding andmodulation mode of each user sum (119879

119894minus 119862119894)2

Mode 1 153311986415

Mode 2 136611986415

Mode 3 129811986415

Table 9 Total system capacity of the three algorithms when thechannel conditions of each user are not the same

Algorithms sum119862119894

Uniform resource allocation 6391MbpsProportional resource allocation 6073MbpsProposed optimal power allocation 5510Mbps

show the impact of channel condition on the power allocationresult the channel conditions of the users in the same spotbeam are set to be 211989021 311989021 411989021 511989021 and 611989021 Moreoverthe traffic demands of the users in the same spot beam areset the same and the traffic demands of the users in the fourdifferent spot beams are set to be 3Mbps 8Mbps 13Mbpsand 18Mbps The simulation results are shown in Figure 8and Table 9

It is seen fromFigure 8 that the proposed power allocationalgorithm provides more capacity to the users with highertraffic demand in order tominimize the total system squareddifference between the traffic demand and capacity allocatedto each user The proposed algorithm allocates the samecapacities to the users in spot beam 3 or 4 which impliedthat more power resource will be allocated to the users with

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform resource allocationProportional resource allocationProposed optimal power allocation

ith user

Figure 8 Comparison of the three algorithms in terms of thecapacity allocated to each user when the channel conditions of eachuser are not the same

Table 10 Sumof (119879119894minus 119862119894)2 of the three algorithmswhen the channel

conditions of each user are not the same

Algorithms sum (119879119894minus 119862119894)2

Uniform resource allocation 171911986415

Proportional resource allocation 141411986415

Proposed optimal power allocation 136311986415

worse channel conditions in these two spot beams As a resultcompared with the other two resource allocation algorithmsthe total system capacity of the proposed power allocationalgorithm is decreased as clearly shown in Table 9

As mentioned in Figure 9 and Table 10 the proposedpower allocation algorithm provides more capacity to theusers with higher traffic demand Therefore the squareddifferences between the traffic demand and capacity allocatedto these users are lower Compared with the other twoalgorithms although the squared differences of the users withlower traffic demand are higher the total squared differenceof the proposed power allocation algorithm is lower as shownin Table 10 As a result it is observed that the proposedalgorithm improves the fairness of power allocation amongstthe user at cost of the total system capacity

6 Conclusion

In the multi-spot-beam satellite system it is crucial for us toimprove the power resources utilization efficiency due to thescarceness of the satellite power resources To this end theproblem of power allocation was mathematically formulatedas a convex optimization problem and an optimal power

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

Uniform resource allocationProportional resource allocationProposed optimal power allocation

times1014

(TiminusCi)2

ith user

Figure 9 Comparison of the three algorithms in terms of thesquared difference between the traffic demand and the capacityallocated to each user when the channel conditions of each user arenot the same

allocation algorithm was proposed to solve the problemIn the optimization the capacity allocated to each userwas calculated according to satellite link budget equationsrather than the Shannon capacity formula As a result thecapacity allocated to each user can be achieved and the powerallocation result is more suitable for the practical multi-spot-beam satellite communication system Moreover thecomputational complexity of proposed algorithm is linearwith both the numbers of the spot beams and users As aresult it can be implemented in the practical system

It is shown from the simulation results that comparedwith the traditional power allocation algorithms the pro-posed algorithm improved the fairness of the power allo-cation amongst the users Both the coding and modulationmode adopted by each user and the bandwidth of each spotbeamhave a significant impact on the power allocation resultWhen the bandwidth of each spot beam is sufficient morepower resources will be provided to the users with highertraffic demand to improve the fairness of power allocationamongst the users On the contrast when the bandwidthof each spot beam is limited more power will be providedto the users with lower traffic demand Even the satellitepower resources are wasted due to the further reduction ofbandwidth of each spot beam The impact of the coding andmodulation mode on the power allocation result is similarto that of the bandwidth of each spot beam Moreoverthe channel conditions of each user also affect the powerallocation result The proposed algorithm provides moreresource to the users with the high traffic demand As a resultif the channel conditions of these high traffic demand usersare worse the total system capacity will be decreased

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

The authors would like to thank the project support by theNational High-Tech Research amp Development Program ofChina under Grant 2012AA01A508

References

[1] D Roddy Satellite Communication McGraw-Hill New YorkNY USA 2001

[2] J P Choi and V W S Chan ldquoOptimum power and beam allo-cation based on traffic demands and channel conditions oversatellite downlinksrdquo IEEE Transactions on Wireless Communi-cations vol 4 no 6 pp 2983ndash2993 2005

[3] YHongA Srinivasan B Cheng LHartman andPAndreadisldquoOptimal power allocation for multiple beam satellite systemsrdquoin Proceedings of the IEEE Radio andWireless Symposium (RWSrsquo08) pp 823ndash826 2008

[4] F Qi L Guangxia F Shaodong and G Qian ldquoOptimumpower allocation based on traffic demand for multi-beamsatellite communication systemsrdquo in Proceedings of the IEEE13th International Conference on Communication Technology(ICCT rsquo11) pp 873ndash876 2011

[5] U Park H W Kim D S Oh and B-J Ku ldquoOptimum selectivebeam allocation scheme for satellite network with multi-spotbeamsrdquo in Proceedings of the 4th International Conference onAdvances in Satellite and Space Communications (SPACOMMrsquo12) pp 78ndash81 2012

[6] H Wang A Liu and X Pan ldquoOptimization of joint power andbandwidth allocation in multi-spot-beam satellite communica-tion systemsrdquoMathematical Problems in Engineering vol 2014Article ID 683604 9 pages 2014

[7] M J Neely E Modiano and C E Rohrs ldquoPower allocation androuting in multibeam satellites with time-varying channelsrdquoIEEEACM Transactions on Networking vol 11 no 1 pp 138ndash152 2003

[8] S Vassaki A D Panagopoulos and P Constantinou ldquoEffectivecapacity and optimal power allocation for mobile satellitesystems and servicesrdquo IEEE Communications Letters vol 16 no1 pp 60ndash63 2012

[9] A Destounis and A D Panagopoulos ldquoDynamic powerallocation for broadband multi-beam satellite communicationnetworksrdquo IEEE Communications Letters vol 15 no 4 pp 380ndash382 2011

[10] J Guo S Ren Y Si and J Wu ldquoAnalysis of other spot-beaminterference in TD-SCDMA compatible satellite systemrdquo inProceedings of the International Conference onWireless Commu-nications and Signal Processing (WCSP rsquo11) pp 1ndash4 2011

[11] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[12] G Ding Q Wu and J Wang ldquoSensing confidence level-based joint spectrum and power allocation in cognitive radionetworksrdquoWireless Personal Communications vol 72 no 1 pp283ndash298 2013

[13] W Yu and L Raymond ldquoDual methods for nonconvex spec-trum optimization of multicarrier systemsrdquo IEEE Transactionson Communications vol 54 no 7 pp 1310ndash1322 2006

[14] R Wang V K N Lau L Lv and B Chen ldquoJoint cross-layerscheduling and spectrum sensing for OFDMA cognitive radiosystemsrdquo IEEE Transactions onWireless Communications vol 8no 5 pp 2410ndash2416 2009

[15] G M Antonio X Wang and G B Giannakis ldquoDynamicresource management for cognitive radios using limited-ratefeedbackrdquo IEEE Transactions on Signal Processing vol 57 no9 pp 3651ndash3666 2009

[16] U B Filik and M Kurban ldquoFeasible modified subgradientmethod for solving the thermal unit commitment problem asa new approachrdquo Mathematical Problems in Engineering vol2010 Article ID 159429 11 pages 2010

[17] U Basaran Filik and M Kurban ldquoSolving unit commitmentproblem using modified subgradient method combined withsimulated annealing algorithmrdquo Mathematical Problems inEngineering vol 2010 Article ID 295645 15 pages 2010

[18] D Bertsekas Nonlinear Programming Athena Scientific Bel-mont Mass USA 1999

[19] T M Cover and J A Thomas Elements of Information TheoryJohn Wiley amp Sons New York NY USA 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of