research article optimization of power allocation for a...

Research ArticleOptimization of Power Allocation for a Multibeam SatelliteCommunication System with Interbeam Interference

Heng Wang Aijun Liu Xiaofei Pan and Jiong Li

College of Communications Engineering PLA University of Science and Technology No 2 Yudao Street Nanjing 210007 China

Correspondence should be addressed to Heng Wang wangheng0987654321126com

Received 22 October 2013 Accepted 24 December 2013 Published 16 January 2014

Academic Editor Hanan Luss

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 multibeam satellite communication systems it is important to improve the utilization efficiency of the power resources due tothe scarcity of satellite power resourcesThe interbeam interference between the beamsmust be considered in the power allocationtherefore it is important to optimize the power allocated to each beam in order to improve the total system performance Initiallythe power allocation problem is formulated as a nonlinear optimization considering a compromise between the maximization oftotal system capacity and the fairness of the power allocation amongst the beams A dynamic power allocation algorithm based onduality theory is then proposed to obtain a locally optimal solution for the optimization problem Compared with traditional powerallocation algorithms this proposed dynamic power allocation algorithm improves the fairness of the power allocation amongstthe beams and in addition the proposed algorithm also increases the total system capacity in certain scenarios

1 Introduction

In satellite communication systems a satellite may providecoverage of the entire region of the earth visible from thesatellite by using a single beam In this case the gain ofthe satellite antenna will be limited by the beamwidth asimposed by the coverage For instance for a geostationarysatellite global coverage implies a 3 dB beamwidth of 175and consequently an antenna gain of no more than 20 dB [1]Therefore each user must be equipped with a large apertureantenna to support the high traffic rate which results in greatinconvenience In order to solve this problem the multibeamtechnique has been widely applied in modern satellite com-munication systems In multibeam satellite communicationsystems the satellite provides coverage of only part of theearth by means of a narrow beam The benefit of a highersatellite antenna gain is obtained due to a reduction in theaperture angle of the antenna beam [1] As a result a userwith a small aperture antenna can support a high traffic rateMoreover the multibeam technique supports the reuse offrequencies for different beams in order to increase the totalsystem capacityWhen two beams utilize the same frequencyinterbeam interference is introduced to the two beams due tothe nonzero gain of the antenna side lobe It has been noted

that when there is interbeam interference between the beamsthe capacity allocated to each beam is determined not onlyby the power allocated to the beam but also by the powerallocated to the other beams

Due to the limitations of satellite platform it is knownthat satellite power resources are scarce and expensive It isthus important to optimize the utilization efficiency of thepower resources Moreover the traffic demands of each beamare different with varying times due to the different coverageareas and the interbeam interference between the differentbeams is also different As a result it is critical to optimizethe power allocation to each beam to meet the specific trafficdemands

Power allocation algorithms were proposed in earlierworks [2ndash7] The mathematical formulation and analyticsolutions of the optimum power allocation problem havebeen presented [2] however the mathematical algorithmto solve the optimization problem was not provided As aresult bisection and subgradient methodologies have beenutilized to solve the optimization problem [3 4] In orderto improve the total system capacity a method to select asmall number of active beams has been proposed [5] whichmaintained the fairness of the power allocation amongst the

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 469437 8 pageshttpdxdoiorg1011552014469437

2 Journal of Applied Mathematics

beamsThemain problem in [2ndash5] was that the authors failedto consider the interbeam interference between the beamswhich cannot be ignored in determining power allocationsA novel resource allocation scheme for multibeam satellitecommunication systems has been described offered maxi-mum communication capacity [6] The scheme optimizedthe frequency bandwidth the satellite transmission powerthe modulation level and the coding rate to each beamin order to manage the ever-changing user distributionsand the interbeam interference conditions However thescheme ignored the fairness of the power allocations amongstthe beams A joint optimization allocation algorithm forthe power and the carrier was proposed [7] in order tobest match the asymmetric traffic requests The algorithmattempted to support the greatest degree of fairness in thepower allocation to each beam regardless of the total systemcapacity

This paperrsquos research is aimed at resolving this deficiencyby optimizing the power allocations for a multibeam satellitecommunication system with full consideration of the impactof interbeam interference The first step is to mathematicallyformulate the problem of power allocation as a non-linearoptimization compromising between the maximization oftotal system capacity and the fairness of the power allocationsto each beam It is found that in the optimization process theoptimal variables are coupled with each other As a result itis difficult to determine whether the optimization is convexor not and to obtain the globally optimal solution for theoptimization To this end a dynamic power allocation algo-rithm based on duality theory is proposed to obtain a locallyoptimal solution for the optimization Finally the simulationresults show the efficiency of the proposed dynamic powerallocation algorithm

The main contributions of this research are summarizedas follows

(1) themathematical formulation of the power allocationproblem formultibeam satellite communication withconsideration of interbeam interference through acompromise between the maximization of the totalsystem capacity and the fairness of the power alloca-tion amongst the beams

(2) the proposal of an algorithm based on duality theorywhich will obtain a locally optimal solution for theoptimization problem

(3) a demonstration of the effects of the interbeaminterference and the channel conditions of each beamon the power allocation results

The remainder of this paper is organized as follows InSection 2 the model of a multibeam satellite communicationsystemwith interbeam interference is described In Section 3a mathematical formulation of the optimization problemof the power allocation is presented Section 4 presents theproposal of a dynamic power allocation algorithm designedto obtain a locally optimal solution to the optimizationSection 5 presents the simulation results and analyzes theeffects of the interbeam interference and channel conditions

InterferedOther

Ci

TiI

totali

beams beams

Figure 1 Configuration of the multibeam satellite communicationsystem

of each beam on the power allocation results Section 6presents the conclusion of the paper

2 A Multibeam Satellite CommunicationSystem Model

Figure 1 shows the system configuration of the multibeamsatellite communication system that is studied here where

119873 is the quantity of the beams119879119894is the traffic demand of the 119894th beam

119875119894is the power allocated to the 119894th beam

119868total119894

is the total interference on the 119894th beam from theother beams120574 is the signal attenuation factor of the 119894th beam andit is noted that 120574 mainly consisted of the effects ofweather conditions free space loss and antenna gainand119875total is the total satellite power resources within thesystem

To precisely describe the interbeam interference withinthe system the interbeam interference matrix H is intro-duced which is defined as follows

119867 =

[[[[

[

0 ℎ12

sdot sdot sdot ℎ1119899

ℎ21

0 sdot sdot sdot ℎ2119899

d

ℎ1198991

ℎ1198992

sdot sdot sdot 0

]]]]

]

(1)

where the element ℎ119894119895denotes the interference coefficient

from the 119895th beamon the 119894th beam It is noted that the elementℎ119894119894is zero because the interference from the same beam is

ignored It is obvious from (1) that the total interference on the

Journal of Applied Mathematics 3

119894th beam from other beams 119868total119894

issum119873119896=1119896 = 119894

119875119896ℎ119894119896 As a result

using time sharing for Gaussian broadcast channels [8] theShannon bounded capacity Ci for the 119894th beam is given as

119862119894= 119882 log

2(1 +

119875119894

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896

) (2)

where 1198730is the noise power density of each beam and W

is the bandwidth of each beam It is shown in (2) that thecapacity119862

119894of the 119894th beam is increased as the power allocated

to the beam increases However the capacity is decreasedas the power allocated to other beams increases due to theinterbeam interference As a result the capacity of each beamis determined not only by the power allocated to it but alsoby the power allocated to the other beams

3 Mathematical Formulation ofthe Power Allocation

In this paper the metric to evaluate the power alloca-tion results minimizes the sum of the squared differencesbetween the traffic demand and the capacity allocated toeach beam As a result the metric will ensure a relativelygreater capacity allocation to the beamswhen there are highertraffic demands which will achieve greater fairness of thepower allocations amongst the beams At the same time themetric will also work to maximize the total system capacityTherefore the metric considers a compromise between themaximization of total system capacity and the fairness ofthe power allocations amongst the beams As a result theoptimization is formulated as follows

min119875119894

119873

sum

119894=1

(119879119894minus 119862119894)2

st 119862119894= 119882 log

2(1 +

119875119894

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896

)

119873

sum

119894=1

119875119894le 119875total

(3)

When there is no interbeam interference between thebeams each of the elements in the interbeam interferencematrix is equal to zero As a result the optimization is convex[2] and the globally optimal solution can be obtained by theoptimizationHowever when interbeam interference actuallyexists it is seen that the optimal variables 119875

119894are coupled

with each otherTherefore it is difficult to determine whetherthe optimization is convex or not and to obtain the globallyoptimal power solution for the optimization To this endan algorithm based on duality theory is proposed to obtaina locally optimal solution for the optimization [9ndash12] aspresented in the following section

4 Proposed Dynamic PowerAllocation Algorithm

Asmentioned above the proposed dynamic power allocationalgorithm is based on duality theory [13] By introducing thenonnegative dual variable 120582 the Lagrange function is givenby

119871 (P 120582) =119873

sum

119894=1

(119879119894minus 119862119894)2minus 120582(119875total minus

119873

sum

119894=1

119875119894) (4)

where P = [1198751 1198752 119875

119873]

From (4) the Lagrange dual function can be obtained by

119863 (120582) = minP

119871 (P 120582) (5)

and the dual problem can be written as

119889lowast= max120582ge0120590119894ge0

119863 (120582) (6)

The dual problem in (6) can be further decomposed intothe following two sequentially iterative subproblems [9]Subproblem 1 Power Allocation Given the dual variable 120582 forany 119894 isin 1 2 119873 differentiating (4) with respect to 119875

119894

results in the equation below

120597119863 (120590 120582)

120597119875119894

= 2

119873

sum

119895=1

(119879119895minus 119862119895)

120597119862119895

120597119875119894

minus 120582 = 0 (7)

The optimized power allocation of the 119894th beam 119875119894can

be obtained in (7) by numerical calculation methods forexample the golden section Moreover if the optimized 119875

119894

is less than zero then 119875119894is set to be zero The detailed

expressions in (7) are shown in the appendixSubproblem 2 Dual Variable Update The optimal dual vari-able can be obtained by solving the dual problem

120582opt

= argmax120582

min [119871 (Popt 120582)] (8)

Because the dual function is always convex a subgradientmethod (a generalization of the gradient) can be used here toupdate the dual variable as shown below [9]

120582119899+1

= [120582119899minus Δ119899

120582(119875total minus

119873

sum

119894=1

119875opt119894

)]

+

(9)

where [119909]+ = max0 119909 119899 is the iteration number and Δ isthe iteration step size

It has been proven that the above dual variable updatingalgorithm is guaranteed to converge to the optimal solutionas long as the iteration step chosen is sufficiently small [9]

The whole process of the proposed dynamic powerallocation algorithm is summarized as follows

Step 1 Set appropriate values to 120582 and 119875119894 119894 isin 1 2 119873

Step 2 Calculate the value of 119875119894from (7)


Step 3 Substitute the power values of each beam as obtainedfrom Step 2 into (9) and then update the dual variable

Step 4 If the condition of |120582119899+1(119875total minus sum119873

119894=1119875119894)| lt 120576 is

satisfied then terminate the algorithm otherwise jump toStep 2

Utilizing the above process the allocated power to eachbeam is obtained

5 Simulation Results and Analysis

For the simulation a multibeam satellite communicationsystem model is set up The system has 10 beams For eachbeam the bandwidth resource is 50MHz and the normalizednoisepower spectral density parameter 120574119873

0is 02119890minus6 Total

satellite power is 200W The traffic demand of each beam isincreased from 80Mbps to 170Mbps by steps of 10Mbps

51The Efficiency of the Proposed Power Allocation AlgorithmIn order to show the efficiency of the proposed dynamicpower allocation algorithm it is compared with the followingtwo traditional algorithms

(1) Uniform power allocation algorithm 119875119894= 119875total119873

(2) Proportional power allocation algorithm 119875119894

=

119875total119879119894119879total where 119879total is the total traffic demand ofall the beams

Moreover comparisons are made of the power allocationresults for the three algorithms in the following two scenarioswith different interbeam interference matrixes

Scenario 1 In this system each beam interferes with the threeadjacent beams As a result the element in the interbeaminterference matrix is set as follows

ℎ119894119895=

03 if 1003816100381610038161003816119895 minus 1198941003816100381610038161003816 = 1 or 1003816100381610038161003816119895 minus 119894 plusmn 10

1003816100381610038161003816 = 1


1003816100381610038161003816 = 2


1003816100381610038161003816 = 3

0 esle

(10)

Figure 2 shows the capacity allocated to each beam forthe three power allocation algorithms in Scenario 1 Table 1presents the total system capacities of the three powerallocation algorithms in Scenario 1 As shown in Figure 2the uniform power allocation algorithm uniformly allocatespower to each beam regardless of the traffic demand ofthe beams or the fairness of the power allocations amongstthe beams Moreover the total interference from the otherbeams is the same for each beam and as a result thecapacity allocated to each beam is the sameThe proportionalpower allocation algorithm allocates the power resources toeach beam solely according to the traffic demand of eachbeam regardless of the interbeam interference Thereforethe capacity allocated to a beam with high traffic demandis higher than that allocated to a beam with low trafficdemand Compared with the proportional power allocationalgorithm the proposed dynamic power allocation algorithm

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

ith beam

Capa

city

allo

cate

d (M

bps)

Traffic demandUniform power allocation

Proportional power allocationProposed dynamic powerallocation

Figure 2 Distribution of the capacity allocated to each beam forthe three algorithms in Scenario 1

Table 1 Total system capacity of the three algorithms in Scenario 1

Algorithm sum119862119894

Uniform power allocation 69283 (Mbps)Proportional power allocation 68520 (Mbps)Proposed dynamic power allocation 67503 (Mbps)

Table 2 Systemrsquos total squared difference for the three algorithmsfrom Scenario 1

Algorithm sum (119879119894minus 119862119894)2

Uniform power allocation 39311986416

Proportional power allocation 35411986416

Proposed dynamic power allocation 34811986416

allocatesmore power resources to beams having higher trafficdemands in order to minimize the systemrsquos total squareddifference between the traffic demand and the capacityallocated to each beam However due to the concavity ofthe capacity function in terms of allocated power the totalsystem capacity is decreased which is also shown by the datain Table 1

Figure 3 shows the squared difference between the traf-fic demand and allocated capacity of each beam for thethree algorithms from Scenario 1 Table 2 presents the totalsquared difference for the three algorithms in Scenario 1 Asmentioned above the proposed dynamic power allocationalgorithm provides more power resources to the beams withhigher traffic demands As a result when the results of theproposed dynamic power allocation algorithm are comparedto the results of the other two algorithms the squareddifference for the beams with high traffic demands is lowerand the squared difference for the beams with low trafficdemand is higher Moreover the total squared difference forthe proposed dynamic power algorithm is the lowest of the


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

ith beam

(TiminusCi)2

Uniform power allocationProportional power allocationProposed dynamic power allocation

times1015

Figure 3 Distribution of the squared difference between the trafficdemand and the allocated capacity of each beam for each of thethree algorithms from Scenario 1

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam



Figure 4 Distribution of the allocated capacity to each beam forthe three algorithms in Scenario 2

three algorithmsThis conclusion is also demonstrated by thedata shown in Table 2

Scenario 2 In this scenario it is assumed that there is ahostile interference source in Beam 1 Thus we consider thatonly Beam 1 interferes with the other beams The interbeaminterference matrix is set as follows

ℎ119894119895=

03 119894 = 2 10 119895 = 1

0 else(11)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

(TiminusCi)2

times1015

ith beam


Figure 5 Distribution of the squared difference between the trafficdemand and the capacity allocated to each beam for the threealgorithms in Scenario 2

Figure 4 shows the allocated capacity for each beamfor the three power allocation algorithms in Scenario 2Table 3 presents the total system capacities of the threepower allocation algorithms from Scenario 2 It is knownthat only Beam 1 interferes with the other beams It willseem reasonable that by allocating less power to Beam 1the interference on the other beams will decrease and thetotal system capacity will increase To this end the proposeddynamic power allocation algorithm allocates no power toBeam 1 to decrease its interference with the other beamsHowever both the uniform and the proportional powerallocation algorithms allocate power to each beam regardlessof the interbeam interference matrix in the system Thus thepower allocated to each beam in Scenario 2 is the same asthat in Scenario 1 and the power allocated to Beam 1 is notdecreased As a result the total system capacity obtained bythe two algorithms is less than that obtained by the proposeddynamic power allocation algorithm as shown in Table 3

Figure 5 shows the squared difference between the trafficdemand and the capacity allocated to each beam for the threealgorithms in Scenario 2 Table 4 presents the total squareddifference for the three algorithms in Scenario 2 Figure 5shows that the squared difference of Beam 1 obtained bythe proposed algorithm is higher than that obtained by theother two algorithms However the squared differences fromBeams 2 to 10 are lower This is because when comparedwith the other two algorithms the proposed dynamic powerallocation algorithm provides no power resources to Beam 1and provides more power to Beams 2 through 10 Moreoverthe total squared difference of the proposed dynamic powerallocation algorithm is less than that of the other two powerallocation algorithms Taken together with the conclusion


Table 3 Total system capacity for the three algorithms in Scenario2



Table 4 Systemrsquos total squared difference for the three algorithmsin Scenario 2








about the total system capacity it is clear that the proposeddynamic power allocation algorithm improves both thesystem capacity and the fairness of the power allocationsamongst the beams in this scenario

It is noted that the traffic demand and the channelconditions of each beam are the same in the two scenariosand only the interbeam interferencematrix is different How-ever the power allocation result obtained by the proposedalgorithm shows a great difference in the two scenarios Inother words the interbeam interference between the beamshas a significant impact on the power allocation results Inaddition the proposed algorithm dynamically allocates thepower resource to each beam taking into account the impactof the interbeam interference between the beams making thebest effort in removing the adverse impacts of the interbeaminterference

52 The Effects of the Channel Condition of Each Beam on thePower Allocation Results It is known that signal attenuationfactor 120574 is affected by channel conditions To show theimpact of the channel conditions of each beam on the powerallocation results the following scenario is set up

Scenario 3 The normalized noise power spectral densityparameters 120574119873

0from Beams 3 through 5 are set to be 02119890minus6

12119890minus6 and 22119890

minus6 The traffic demand of the three beams isset to be the same as 100Mbps The interbeam interferencematrix is set to be the same as that in Scenario 2 and otherparameters in the system remained the same

Figure 6 shows the capacity allocated to each beam forthe three power allocation algorithms when the channel con-ditions of each beam are different Table 5 presents the totalsystem capacity for the three power allocation algorithms

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam









in Scenario 3 It is noted that the traffic demand and totalinterference from the other beams for Beams 3 through 5are the same and only the channel conditions of the threebeams are different As shown in Figure 6 the proposeddynamic power allocation algorithm provides more powerresources to the beams that have better channel conditionsand rarely or never provides power to beams with worsechannel conditions Beam 5 for example with the worstchannel condition is provided with no power resourcesTherefore the proposed dynamic power allocation algorithmnot only considers the fairness of the power allocationsamongst the beams but also tries tomaximize the throughputof the system and achieves a good system performance aspredicted The proportional or uniform power allocationalgorithms cannot dynamically allocate the power resourcesto each beam according to their channel conditions thustheir total system capacities are less than that of the proposeddynamic power allocation algorithm as clearly demonstratedby the data shown in Table 5

Figure 7 shows the squared difference between the trafficdemand and the capacity allocated to each beam for the threealgorithms in Scenario 3 Table 6 presents the total squareddifference of the three algorithms in Scenario 3Asmentionedabove the proposed dynamic power allocation algorithmprovides more power resources to the beams having betterchannel conditions As a result the squared difference of the


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015

Figure 7 Distribution of squared difference between the trafficdemand and the capacities allocated to each beam for the threealgorithms in Scenario 3

traffic demand and the capacity allocated to the beams withbetter channel conditions is smaller than that of the othertwo algorithms as shown in Figure 7 Moreover the totalsquared difference of the proposed dynamic power allocationalgorithm is also the smallest of the three algorithms Thusthe fairness of the power allocations amongst the beamsthat is obtained by the proposed dynamic power allocationalgorithm provides the optimal optimization

6 Conclusions

In multibeam satellite communication systems due to thereusing of frequencies there exists interbeam interferencebetween the beams which cannot be ignored in determiningpower allocations To precisely describe the impact of theinterbeam interference the problem of power allocation asa non-linear optimization with constraints was formulatedincluding a compromise between the maximization of totalsystem capacity and the fairness of the power allocationsto the beams A dynamic power allocation algorithm wasthen proposed to obtain a locally optimal solution to theoptimization

It was shown that compared with the traditional uniformor proportional power allocation algorithms the proposeddynamic power allocation algorithm improved the fairnessof the power allocations to the beams and also increased thetotal system capacity in certain scenarios such as Scenarios2 and 3 as presented in Section 5 In addition the interbeaminterference between both the beams and the channel con-ditions of each beam had a significant impact on the powerallocation results The proposed dynamic power allocationalgorithm functioned to remove the adverse impacts of these

factors for example the algorithm allocated less power to thebeams which had greater interference on the other beams orwhich had worse channel conditions

Appendix

Equation (7) is expressed in detailWhen 119894 = 119895 120597119862

119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)

When 119894 = 119895 120597119862119895120597119875119894is expressed as

120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)

Substituting (A1) and (A2) into (7) the following equa-tion is obtained

(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)

According to (A3) the optimized power allocation ofthe 119894th beam 119875

119894could be obtained by numerical calculation

methods

Conflict of Interests

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

Acknowledgment

The authors would like to thank the project support providedby the National High-Tech Research amp Development Pro-gram of China under Grant 2012AA01A508


References

[1] G Maral M Bousquet and Z Sun Communications SystemsSystems Techniques and Technology JohnWiley and Sons NewYork NY USA 5th edition 2009

[2] J P Choi and V W S Chan ldquoOptimum power and beamallocation based on traffic demands and channel conditionsover satellite downlinksrdquo IEEE Transactions on Wireless Com-munications vol 4 no 6 pp 2983ndash2992 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 January 2008

[4] Q Feng G Li S Feng andQ Gao ldquoOptimumpower allocationbased on traffic demand for multi-beam satellite communica-tion systemsrdquo in Proceedings of the 13th International Conferenceon Communication Technology (ICCT rsquo11) pp 873ndash876 Septem-ber 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] K Nakahira K Kobayashi andM Ueba ldquoCapacity and qualityenhancement using an adaptive resource allocation for multi-beam mobile satellite communication systemsrdquo in Proceedingsof the IEEE Wireless Communications and Networking Confer-ence (WCNC rsquo06) pp 153ndash158 April 2006

[7] J Lei and M A Vazquez-Castro ldquoJoint power and carrierallocation for the multibeam satellite downlink with individualSINR constraintsrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo10) May 2010

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

[9] W Yu and R Lui ldquoDual methods for nonconvex spectrumoptimization of multicarrier systemsrdquo IEEE Transactions onCommunications vol 54 no 7 pp 1310ndash1322 2006

[10] 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

[11] 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

[12] A G Marques 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

[13] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


beamsThemain problem in [2ndash5] was that the authors failedto consider the interbeam interference between the beamswhich cannot be ignored in determining power allocationsA novel resource allocation scheme for multibeam satellitecommunication systems has been described offered maxi-mum communication capacity [6] The scheme optimizedthe frequency bandwidth the satellite transmission powerthe modulation level and the coding rate to each beamin order to manage the ever-changing user distributionsand the interbeam interference conditions However thescheme ignored the fairness of the power allocations amongstthe beams A joint optimization allocation algorithm forthe power and the carrier was proposed [7] in order tobest match the asymmetric traffic requests The algorithmattempted to support the greatest degree of fairness in thepower allocation to each beam regardless of the total systemcapacity

This paperrsquos research is aimed at resolving this deficiencyby optimizing the power allocations for a multibeam satellitecommunication system with full consideration of the impactof interbeam interference The first step is to mathematicallyformulate the problem of power allocation as a non-linearoptimization compromising between the maximization oftotal system capacity and the fairness of the power allocationsto each beam It is found that in the optimization process theoptimal variables are coupled with each other As a result itis difficult to determine whether the optimization is convexor not and to obtain the globally optimal solution for theoptimization To this end a dynamic power allocation algo-rithm based on duality theory is proposed to obtain a locallyoptimal solution for the optimization Finally the simulationresults show the efficiency of the proposed dynamic powerallocation algorithm

The main contributions of this research are summarizedas follows

(1) themathematical formulation of the power allocationproblem formultibeam satellite communication withconsideration of interbeam interference through acompromise between the maximization of the totalsystem capacity and the fairness of the power alloca-tion amongst the beams

(2) the proposal of an algorithm based on duality theorywhich will obtain a locally optimal solution for theoptimization problem

(3) a demonstration of the effects of the interbeaminterference and the channel conditions of each beamon the power allocation results

The remainder of this paper is organized as follows InSection 2 the model of a multibeam satellite communicationsystemwith interbeam interference is described In Section 3a mathematical formulation of the optimization problemof the power allocation is presented Section 4 presents theproposal of a dynamic power allocation algorithm designedto obtain a locally optimal solution to the optimizationSection 5 presents the simulation results and analyzes theeffects of the interbeam interference and channel conditions

InterferedOther

Ci

TiI

totali

beams beams

Figure 1 Configuration of the multibeam satellite communicationsystem

of each beam on the power allocation results Section 6presents the conclusion of the paper

2 A Multibeam Satellite CommunicationSystem Model

Figure 1 shows the system configuration of the multibeamsatellite communication system that is studied here where

119873 is the quantity of the beams119879119894is the traffic demand of the 119894th beam

119875119894is the power allocated to the 119894th beam

119868total119894

is the total interference on the 119894th beam from theother beams120574 is the signal attenuation factor of the 119894th beam andit is noted that 120574 mainly consisted of the effects ofweather conditions free space loss and antenna gainand119875total is the total satellite power resources within thesystem

To precisely describe the interbeam interference withinthe system the interbeam interference matrix H is intro-duced which is defined as follows

119867 =

[[[[

[

0 ℎ12

sdot sdot sdot ℎ1119899

ℎ21

0 sdot sdot sdot ℎ2119899

d

ℎ1198991

ℎ1198992

sdot sdot sdot 0

]]]]

]

(1)

where the element ℎ119894119895denotes the interference coefficient

from the 119895th beamon the 119894th beam It is noted that the elementℎ119894119894is zero because the interference from the same beam is

ignored It is obvious from (1) that the total interference on the



issum119873119896=1119896 = 119894

119875119896ℎ119894119896 As a result


119862119894= 119882 log

2(1 +

119875119894

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896

) (2)







min119875119894

119873

sum

119894=1

(119879119894minus 119862119894)2

st 119862119894= 119882 log

2(1 +

119875119894

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896

)

119873

sum

119894=1

119875119894le 119875total

(3)


119894are coupled




119871 (P 120582) =119873

sum

119894=1


119873

sum

119894=1

119875119894) (4)

where P = [1198751 1198752 119875

119873]


119863 (120582) = minP

119871 (P 120582) (5)


119889lowast= max120582ge0120590119894ge0

119863 (120582) (6)


119894


120597119863 (120590 120582)

120597119875119894

= 2

119873

sum

119895=1

(119879119895minus 119862119895)

120597119862119895

120597119875119894

minus 120582 = 0 (7)



119894



120582opt

= argmax120582

min [119871 (Popt 120582)] (8)


120582119899+1

= [120582119899minus Δ119899


119873

sum

119894=1

119875opt119894

)]

+

(9)









119894=1119875119894)| lt 120576 is










=




ℎ119894119895=


1003816100381610038161003816 = 1


1003816100381610038161003816 = 2


1003816100381610038161003816 = 3

0 esle

(10)


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

ith beam

Capa

city

allo

cate

d (M

bps)















1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

ith beam

(TiminusCi)2


times1015


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam






ℎ119894119895=

03 119894 = 2 10 119895 = 1

0 else(11)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

(TiminusCi)2

times1015

ith beam






















12119890minus6 and 22119890



1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam












1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015



6 Conclusions




Appendix


119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)


120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)


(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)



methods



Acknowledgment



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











issum119873119896=1119896 = 119894

119875119896ℎ119894119896 As a result


119862119894= 119882 log

2(1 +

119875119894

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896

) (2)







min119875119894

119873

sum

119894=1

(119879119894minus 119862119894)2

st 119862119894= 119882 log

2(1 +

119875119894

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896

)

119873

sum

119894=1

119875119894le 119875total

(3)


119894are coupled




119871 (P 120582) =119873

sum

119894=1


119873

sum

119894=1

119875119894) (4)

where P = [1198751 1198752 119875

119873]


119863 (120582) = minP

119871 (P 120582) (5)


119889lowast= max120582ge0120590119894ge0

119863 (120582) (6)


119894


120597119863 (120590 120582)

120597119875119894

= 2

119873

sum

119895=1

(119879119895minus 119862119895)

120597119862119895

120597119875119894

minus 120582 = 0 (7)



119894



120582opt

= argmax120582

min [119871 (Popt 120582)] (8)


120582119899+1

= [120582119899minus Δ119899


119873

sum

119894=1

119875opt119894

)]

+

(9)









119894=1119875119894)| lt 120576 is










=




ℎ119894119895=


1003816100381610038161003816 = 1


1003816100381610038161003816 = 2


1003816100381610038161003816 = 3

0 esle

(10)


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

ith beam

Capa

city

allo

cate

d (M

bps)















1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

ith beam

(TiminusCi)2


times1015


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam






ℎ119894119895=

03 119894 = 2 10 119895 = 1

0 else(11)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

(TiminusCi)2

times1015

ith beam






















12119890minus6 and 22119890



1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam












1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015



6 Conclusions




Appendix


119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)


120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)


(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)



methods



Acknowledgment



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894=1119875119894)| lt 120576 is










=




ℎ119894119895=


1003816100381610038161003816 = 1


1003816100381610038161003816 = 2


1003816100381610038161003816 = 3

0 esle

(10)


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

ith beam

Capa

city

allo

cate

d (M

bps)















1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

ith beam

(TiminusCi)2


times1015


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam






ℎ119894119895=

03 119894 = 2 10 119895 = 1

0 else(11)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

(TiminusCi)2

times1015

ith beam






















12119890minus6 and 22119890



1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam












1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015



6 Conclusions




Appendix


119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)


120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)


(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)



methods



Acknowledgment



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

ith beam

(TiminusCi)2


times1015


1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam






ℎ119894119895=

03 119894 = 2 10 119895 = 1

0 else(11)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

(TiminusCi)2

times1015

ith beam






















12119890minus6 and 22119890



1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam












1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015



6 Conclusions




Appendix


119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)


120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)


(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)



methods



Acknowledgment



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























12119890minus6 and 22119890



1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Capa

city

allo

cate

d (M

bps)

ith beam












1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015



6 Conclusions




Appendix


119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)


120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)


(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)



methods



Acknowledgment



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

ith beam

(TiminusCi)2


times1015



6 Conclusions




Appendix


119895120597119875119894is given as

120597119862119894

120597119875119894

=119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

(A1)


120597119862119895

120597119875119894

=minus119882

ln 2

sdot 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A2)


(119879119894minus 119862119894)2119882

ln 2sdot

1

1205741198821198730+ sum119873

119896=1119896 = 119894119875119896ℎ119894119896+ 119875119894

minus 120582

=

119873

sum

119895=1119895 = 119894

119882

ln 2(2119879119895minus 2119862119895) sdot 119872119895119894

(A3)

where

119872119895119894= 119875119895ℎ119895119894times ((120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896)

2

+119875119895(120574119882119873

0+

119873

sum

119896=1119896 = 119895

119875119896ℎ119895119896))

minus1

(A4)



methods



Acknowledgment



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimization of power allocation for a...

Documents