Optimal channel switching in multiuser systems under average capacity constraints

(1)

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Optimal

channel

switching

in

multiuser

systems

under

average

capacity

constraints

✩

Ahmet

Dundar Sezer,

Sinan Gezici

∗

DepartmentofElectricalandElectronicsEngineering,BilkentUniversity,Bilkent,Ankara06800,Turkey

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline20January2017 Keywords: Channelswitching Capacity Multiuser Timesharing Powerallocation

Inthispaper, the optimalchannel switchingproblemis studiedfor averagecapacitymaximizationin the presenceofmultiple receiversin thecommunicationsystem. First, theoptimal channelswitching problemisproposedforaveragecapacitymaximizationofthecommunicationbetweenthetransmitter and thesecondaryreceiverwhilefulfilling theminimumaveragecapacityrequirementoftheprimary receiver and considering the average and peak power constraints. Then, an alternative equivalent optimizationproblemisprovidedanditisshownthatthesolutionofthisoptimizationproblemsatisfies the constraints with equality. Based on the alternative optimizationproblem, it is obtainedthat the optimal channel switching strategy employs at most three communication links in the presence of multiple availablechannelsinthesystem. Inaddition,theoptimal strategiesarespecifiedintermsof thenumberofchannelsemployedbythetransmittertocommunicatewiththeprimaryandsecondary receivers.Finally,numericalexamplesareprovidedinordertoverifythetheoreticalinvestigations.

1. Introduction

Optimal power allocation has critical importance for enhanc-ingperformanceofcommunicationsystems.Forexample,infading environments,performance ofcommunicationbetweentwousers canbeimprovedbyemployinganefficientpowerallocation strat-egy(e.g.,water-fillingalgorithm[1])comparedtotheconventional uniformpower allocation approach. In the literature, the studies related to power allocation have mostly focused on the perfor-mancemetricssuchaschannelcapacity(e.g.,[1–3]),biterrorrate (BER)(e.g.,[4–8]),andoutageprobability(e.g.,[9–11]) ingeneral. In[1],theoptimalpowerallocationstrategyisderivedforcapacity maximizationoverafadingadditivewhiteGaussiannoise(AWGN) channelinthepresenceofperfectchannelstateinformation(CSI) at both the transmitter and the receiver. It is obtained that the optimalstrategythatmaximizesthechannelcapacityisthe water-fillingsolutioninwhichmorepowerisallocatedtobetterchannel statesifthesignal-to-noiseratio(SNR)isaboveacertainthreshold andnopoweristransmittedotherwise.Viaoptimalpower

alloca-✩ _This_research_was_supported_in_part_by_the_{Distinguished}_Young_Scientist_Award ofTurkishAcademyofSciences(TUBA-GEBIP2013).A.D.Sezerisalsosupportedby ASELSANGraduateScholarshipforTurkishAcademicians.

*

Correspondingauthor.Fax:+903122664192.

E-mailaddresses:adsezer@ee.bilkent.edu.tr(A.D. Sezer),gezici@ee.bilkent.edu.tr (S. Gezici).

tion, the ergodic capacity and the outage capacity is maximized in[2]forsecondaryusersina cognitiveradio network.Ina sim-ilarcontext,theoptimalpowerallocationschemesareconsidered in[4]forcognitiveradionetworksinordertominimizethe aver-ageBERofsecondaryusers.In[9],theoptimalpowerallocationis studied inorderto reducethe outageprobability in fading chan-nels.

Inadditiontothepowerallocationapproach,timesharing(i.e., randomization) is another method forimproving performance of communication systems. The mechanism behind the benefits of the time sharing (randomization) method is related to a phe-nomenoncalledstochasticresonance(SR).Thecounterintuitive ef-fectsofSRprovidesperformance benefitsinthecontextof statis-ticalaverageforasysteminwhichnonlinearities andsuboptimal parametersareobserved[12,13].Intheliterature,thetimesharing approach hasbeen studied inthe context ofnoise enhanced de-tectionandestimation(e.g.,[14–18]),errorperformance improve-ment (e.g., [16,19–24]), and jamming performance enhancement (e.g.,[25–27]).Althoughanincreaseinthenoisedegradesthe sys-temperformanceingeneral, additionofnoisetoasystemin con-junctionwithtimesharingamongacertainnumberofsignallevels can provide performance benefits [14–18]. In a similar context, stochasticsignaling,i.e.,timesharingamongmultiplesignalvalues foreachinformationsymbol,isperformedforaveragepower con-strainednon-Gaussianchannelstoimprovetheerrorperformance ofthe system[19,20]. In[19],it ispresented that randomization

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(timesharing)isrequiredamongnomorethanthreedifferent sig-nal values in order to achieve the optimal error performance in thepresenceofsecondandfourthmomentconstraints.Also,time sharingamongmultipledetectors (i.e.,detectorrandomization)is employed over additive time-invariant noise channels [16,21]. In

[16], itis obtainedthat time sharingbetweentwo antipodal sig-nalpairsandthecorrespondingmaximuma-posterioriprobability (MAP)detectorscansigniﬁcantlyenhancethesystemperformance inthepresenceofsymmetricGaussianmixturenoise.Inasimilar manner,thestudyin[21]investigatesbothdetectorrandomization andstochastic signalingapproachesforan M-ary communication system in which an additive noise channel is considered with a known distribution. In the context of jamming performance en-hancement, a jammer can employ time sharing among multiple powerlevelsinordertoreducethedetectionperformanceofa re-ceiver or to degrade the error performance of a communication system[25–27].

Inthe presenceofmultiplechannelsin acommunication sys-tem, time sharing (i.e., channel switching) can be employed to enhance certain performance metrics such asaverage probability oferror,averagenumberofcorrectlyreceivedsymbols, and chan-nel capacity [28–31]. The channel switching problem is studied

in[28] for M-ary communicationsystems inwhicha transmitter

communicates with a receiver by employing a stochastic signal-ingapproachinordertominimizetheaverageprobabilityoferror underan averagepower constraint. It isshown that the optimal strategycorrespondstoeitheroneofthefollowingstrategies: de-terministicsignalingover a single channel,time sharing between two different signal constellations over a single channel, or time sharing between two channels with deterministic signaling over each channel. The channel switching problem is also studied in

[29]formaximizingtheaveragenumberofcorrectlyreceived sym-bolsbetweena transmitteranda receiverin thepresence of av-erage power and cost constraints. It is proved that the optimal strategy corresponds to channel switching eitheramong at most threedifferent channelswithfull channel utilization(i.e.,no idle periods), orbetween atmosttwo different channelswithpartial channelutilization. Unlikethestudies in[28] and[29],the chan-nelswitchingstrategyisemployedtogether withpowerallocation in order to enhance the capacity of a communication systemin

[30,31].In[30],theoptimalchannelswitchingstrategiesare inves-tigatedforacommunicationsysteminwhichasingle transmitter communicateswithasinglereceiverinthepresenceoftheaverage andpeak powerconstraints.Itisobtainedthattheoptimal chan-nelswitchingstrategycorrespondstotheexclusiveuseofasingle channel or to channel switching betweentwo channels. In [31], thestudyin[30] isextendedfora communicationsystemwhere thechannel switchingdelays(costs) are considered dueto hard-ware limitations.Itis shownthat anychannel switchingstrategy consistingofmorethantwodifferentchannelscannotbeoptimal. Althoughthe channelswitching problemhasbeenstudied for communicationbetweenasingle transmitterandasinglereceiver inthepresenceofaverageandpeak powerconstraintsandinthe considerationofchannel switchingdelays,no studiesinthe liter-aturehaveconsideredthechannelswitchingprobleminthe pres-ence of multiple receivers in the communication system. In this study,atransmittercommunicateswithtworeceivers(classiﬁedas primaryandsecondary)byemployingachannelswitchingstrategy amongavailable multiplechannelsin thesystem. The aimofthe transmitter is to enhance the average capacity of the secondary receiver while satisfying the minimum average capacity require-mentfortheprimaryreceiverinthepresenceofaverageandpeak

power constraints.1 _Also, _due _to _hardware _limitations, _the

trans-mittercanestablishonlyonecommunicationlinkwithoneofthe receivers at a given time by employing one of the communica-tion channelsavailable inthe system. It isobtained that ifmore thanone channelisavailable,thentheoptimalchannelswitching strategy which maximizesthe average capacity of the secondary receiverconsistsofnomorethan3 communicationlinks.(Itis im-portant to note that each channel inthe system constitutestwo communicationlinks;thatis,oneforthecommunicationbetween thetransmitterandtheprimaryreceiverandoneforthe commu-nication between thetransmitter andthe secondary receiver.) In addition,withregard tothenumberofchannelsemployed inthe optimalchannelswitchingstrategy,itisconcludedthatthe trans-mittereithercommunicateswiththeprimaryreceiveroveratmost two channelsandemploys asingle channelforthesecondary re-ceiver, or communicates withthe primary receiver over a single channel andemploysatmosttwochannelsforthesecondary re-ceiver.Inadditiontothecommunicationsystemwithasingle pri-maryreceiver,thechannelswitchingprobleminthisstudyisalso extended forcommunicationsystems inwhich there exist multi-ple primary receivers, each having a separate minimum average capacityrequirementforthecommunicationwiththetransmitter. Lastly,numericalexamplesareprovidedtoexemplifythe theoreti-calresults.

Compared tothis manuscript, the studies in[30] and[31] do notconsiderthemulti-userscenarioandconsequentlytheoptimal channelstrategiesobtainedinthosestudiesarenotapplicablefora communicationsysteminwhichmultipleuserscommunicatewith each other.Eventhough thestudies in[30] and[31]do not pro-vide anyapproaches formulti-user communicationsystems,they constitute a fundamental aspect for the optimal channel switch-ing strategiesobtainedinthismanuscript.Therefore,themethods and approaches employed in this study bear a certain level of resemblance to those in [30] and [31]. On the other hand, it is important to note that the contributions ofthisstudy tothe lit-erature aresigniﬁcantly differentfromthe onesin[30] and[31]. Moreprecisely,theconstraintrelatedtotheminimumaverage ca-pacityrequirementoftheprimary receiverinthecommunication system modeled in this study alters the analysis of the optimal channelswitchingstrategyandrequiresnewproofapproachesthat aremostlydifferentfromtheonesemployedin[30]and[31].

Themaincontributionsofthispapercanbesummarizedas fol-lows:

•

Fortheﬁrsttimeintheliterature,thechannelswitching prob-lemisstudiedforaveragecapacitymaximization inthe pres-enceofmultiple receiversina communicationsystemwhere thetransmittercommunicateswiththeprimaryandsecondary receiversinordertoimprovetheaveragecapacityofthe sec-ondaryreceiverundertheaverageandpeakpowerconstraints and the minimum average capacity requirement forthe pri-maryreceiver.

•

It isobtainedthattheoptimalchannel switchingstrategy in-cludes no morethan 3 communication links inthe presence ofmultipleavailablecommunicationchannelsinthesystem.

•

Itisshownthattheoptimalchannelswitchingstrategy

corre-spondstooneofthefollowingstrategies:

– The transmitterperformscommunicationwiththeprimary receiver over at most two channels and employs a single channelforthesecondaryreceiver.

1 _In_this_study,_the_channel_switching_delays_are_omitted_in_order_to_simplify_the systemmodel.However,themaincontributionsofthemanuscriptarevalidinthe presenceofswitchingdelays,aswell.

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Fig. 1. Blockdiagramofacommunicationsysteminwhichtransmitter communi-cateswithprimaryandsecondaryreceiversviachannelswitchingamongK chan-nels(frequencybands).Itisnotedthatthe channelcoeﬃcientscanbedifferentfor thesamechannels.

– The transmitter communicates with the primary receiver over a single channel and atmosttwo channels are occu-piedforthecommunicationtothesecondaryreceiver.

•

A low-complexity solutionto the channel switching problem

isprovided,whichrequiresthecomparisonoftheaverage ca-pacities obtained by two optimizationproblems,each having signiﬁcantlylowercomputationalcomplexitythantheoriginal channelswitchingproblem.

•

As an extension, the channel switching problem is reformu-lated in the consideration of multiple primary receivers and theircorrespondingminimumaveragecapacityrequirements.

2. Systemmodelandproblemformulation

Consider acommunication systeminwhich K different chan-nels (frequencybands) are available foratransmitter to commu-nicate with two receivers classiﬁed as primary and secondary.2 It is assumed that, due to hardware constraints, the transmitter can establish only one communication link with one of the re-ceivers at a given time by performing communication over one of the channels [30,31]. The reason for this assumption is that thetransmitterandthereceiversareassumedtohaveasingle RF chaineachduetocomplexityandcostconsiderations.The restric-tioncausedbythisassumption simpliﬁesthecircuit andantenna designattransmittersandreceiverswhile reducingthe hardware costsby allowing toemploya single RFchainto transmit/receive data.Thetransmittercanswitch(timeshare)amongthese

K

chan-nels to improve the average capacity of the secondary receiver whilesatisfyingtheminimumaveragecapacityrequirementforthe primary receiver. The channelsare modeled asstatistically inde-pendentﬂat-fadingadditiveGaussiannoisechannelswithconstant powerspectral densitylevels over the channel bandwidths. Also, thechannel state information(CSI) is assumedtobe available at both the transmitter and the associated receiver, and the chan-nels canhavedifferent bandwidths andconstantspectral density levels ingeneral. Fig. 1illustrates the system modelwith K

dif-ferentchannels(frequencybands),wherethetransmitter commu-nicateswithone primaryandone secondaryreceivervia channel switching(i.e.,time sharing). Inpractice,the transmittercan ini-tiatecommunicationwiththeprimary receiverandcommunicate overone channel fora certain fractionoftime. Then,it switches toanother channel and communicateswiththe primary receiver over that channel for another fraction of time. The similar

pro-2 _Extensions_to_multiple _receivers_are_presented_in_Section₄_. _Also,_the _terms, primaryandsecondary,usedinthestudyhavedifferentmeaningsfromtheones usedinthecognitiveradioliteraturewhereprimaryusersarelicensedusersand secondaryusersareunlicensedusersthatareallowedtoaccessthespectrumwhen primaryusersarenotactive.

cess continues forthe remaining channels. Later, the transmitter establishescommunicationwiththesecondaryreceiverandit ap-plies the same procedure asemployed for the primary receiver; that is, fora certain fraction of time, it communicates with the secondarytransmitteroverone channelanditswitchestothe re-mainingchannelsinorderandcommunicatesoverthosechannels forcertain fractionsoftime.Itisimportanttoemphasizethatthe receiversareclassifiedasprimaryandsecondaryinthestudysince the transmitterprimarily satisfies the minimumaverage capacity requirementfortheprimaryreceiverandthenperforms communi-cationwiththesecondaryreceivertoenhancetheaveragecapacity of thecommunication withthe secondary receiver. Thisscenario is applicable to wireless sensor networks in which child nodes can employ the channel switching strategy in order to improve theiraveragecapacitywhilefulfillingtheminimumaverage capac-ity constraintof theparent node. Also, it can be statedthat the channel switching strategy may improvethe energyefficiency of thecommunicationsystembyrequiringaloweraveragepowerto achieve the same average channel capacityachieved by the con-ventionalmethods[32,33].

Let Bi and Ni

/

2 denote, respectively, the bandwidth and the constant power spectral density level of the additive Gaussian noiseforchannel

i,

where

i

∈ {

1

,

. . . ,

K

}

,andlet

h

k

i representthe complex channel gain forchannel i between the transmitterand receiver

k,

where

k

∈ {

p

,

s

}

denotesthelabelforeithertheprimary orthesecondaryreceiver.Then,thecapacityofchannel

i between

thetransmitterandreceiver

k is

expressedas

C

k_i

(

P

)

=

B

i

log

2

1 +

h

k i

2

P

N

i

B

i

bits/sec

(1)

where P represents theaveragetransmitpower[34].

The main objective of this study is to determine the opti-mal channel switching strategy that maximizes the average ca-pacity of the communication between the transmitter and the secondary receiver while ensuring the minimum average capac-ity constraint forthe primary receiver with the consideration of average and peak power constraints. To provide a mathematical formulation, time-sharing (channel switching) factors are deﬁned as

λ

p₁

,

. . . ,

λ

p_K

,

λ

s₁

,

. . . ,

λ

s_K,where

λ

p_i and

λ

s_i denotethefractionsof time whenchannel

i is

utilized by thetransmitterfor communi-cation with the primary receiver andthe secondary receiver, re-spectively.Then,thefollowingoptimalchannelswitchingproblem isproposedforaveragecapacitymaximizationofthelinkbetween thetransmitterandthesecondaryreceiverunderaminimum av-eragecapacityconstraintoftheprimaryreceiver:

max {λp i,λsi,P p i,Pis}Ki=1 K

i=1

λ

s_iC_is

(

Ps_i

)

(2a) subject to K

i=1

λ

p_iCp_i

(

Pp_i

)

≥

Creq (2b) K

i=1

(λ

p_i Pp_i

+ λ

_isPs_i

)

≤

Pav

,

Pp_i

,

P_is

∈ [

0

,

Ppk

] , ∀

i

∈ {

1

, . . . ,

K

}

(2c) K

i=1

(λ

p_i

+ λ

s_i

)

=

1

,

λ

p_i

, λ

s_i

∈ [

0

,

1

] , ∀

i

∈ {

1

, . . . ,

K

}

(2d)

where Ck_i

(

Pi

)

fork

∈ {

p

,

s

}

is asin (1), Pp_i and P_is represent the average transmit powers allocated tochannel i in orderto com-municatewith theprimary andsecondary receivers, respectively,

(4)

Creqistheminimumaveragecapacityrequirementfortheprimary receiver,

P

pk denotesthepeakpowerlimit,andPavrepresentsthe average powerlimit for thetransmitter. The averagepower limit canbeassociatedwiththepowerconsumptionand/orthebattery life at the transmitter. On the other hand, the peak power con-straintrefers to themaximum powerlevel that canbe produced by the transmittercircuitry (i.e.,a hardware constraint).It is as-sumed that Pav

<

Ppk and Creq

>

0. It is also important to note thatthereexistsatotalof2K communicationlinksinthesystem since each of the K channels (frequency bands) can be used for communicatingwiththeprimaryreceiverorsecondaryreceiver.

3. Optimalchannelswitchingforcommunicationbetweenthe transmitterandthesecondaryreceiver

Since the optimization problem in (2) is not convex and re-quiresasearchovera4K dimensionalspaceingeneral,itishard to obtainthe solutionof theproblemin its currentform. There-fore,theaimistoconverttheoptimizationproblemin(2)intoa tractableequivalentoptimizationproblem,thesolutionofwhichis thesameasthatof(2).Thefollowingoptimizationproblem repre-sentssuchanalternativeoptimizationproblem.

Proposition1.

The following optimization problem results in the same

maximum average capacity for the secondary receiver as the original op-timization problem in (2): max {λp i,λsi,P p i,Psi}Ki=1 K

i=1

λ

s_iCs_max

(

P_is

)

(3a) subject to K

i=1

λ

p_iC_maxp

(

Pp_i

)

≥

Creq (3b) K

i=1

(λ

p_i Pp_i

+ λ

s_iPs_i

)

≤

Pav

,

Pp_i

,

P_is

∈ [

0

,

Ppk

] , ∀

i

∈ {

1

, . . . ,

K

}

(3c) K

i=1

(λ

p_i

+ λ

s_i

)

=

1

,

λ

p_i

, λ

s_i

∈ [

0

,

1

] , ∀

i

∈ {

1

, . . . ,

K

}

(3d) where Ck max

(

P

)

is deﬁned as

C

kmax

(

P

)

max

{

C

1k

(

P

), . . . ,

C

k K

(

P

)

}

(4)

for k

∈ {

p

,

s

}

.

Proof. Let

{˜λ

p_i

,

˜λ

s_i

,

P

˜

p_i

,

P

˜

s_i

}

K_i₌₁ denotethesolutionofthe optimiza-tion problemin (2) and C∗ denote the corresponding maximum average capacity. Then, the achieved maximum average capacity forthecommunicationbetweenthetransmitterandthesecondary receiver can be written asC∗

=

K_i₌₁

˜λ

s

iCis

( ˜

Psi

)

. From the deﬁni-tionof

C

k

maxin(4),thefollowingrelationisobtained:

C

∗

=

K

i=1

˜λ

s i

C

is

( ˜

P

si

)

≤

K

i=1

˜λ

s i

C

smax

( ˜

P

is

).

(5)

It is notedthat

{˜λ

p_i

,

˜λ

s_i

,

P

˜

_ip

,

P

˜

s

i

}

Ki=1 satisﬁesthe constraints in (3). Therefore, itis deduced that the problem in(3) can achieve the maximumaveragecapacityobtainedbytheproblemin(2);thatis,

C∗

≤

C,where

C

_denotes_the_maximum_average_capacity

accord-ingto(3).Next,considerthesolutionoftheoptimizationproblem in (3). The maximum average capacity obtained by (3) can be

expressed as

C

₌

K i=1

¯λ

siCmaxs

( ¯

Psi

)

,where

{¯λ

p i

,

¯λ

s i

,

P

¯

p i

,

P

¯

s i

}

iK=1 de-notesthesolutionof(3).Now,deﬁnefunctions

g

(k)

₍

_i

₎

_for

_k

_{∈ {}

_p

_,

_s

_}

andsets

S

(mk) for

k

∈ {

p

,

s

}

asfollows3:

g(k)

(

i

)

arg max l∈{1,...,K} Cl

( ¯

P k i

) ,

∀

i

∈ {

1

, . . . ,

K

}

(6) and Sm(k)

{

i

∈ {

1

, . . . ,

K

} |

g(k)

(

i

)

=

m

} , ∀

m

∈ {

1

, . . . ,

K

}.

(7) Then,thefollowingrelationscanbeobtainedfor

k

∈ {

p

,

s

}

:

K

i=1

¯λ

k iCkmax

( ¯

Pki

)

=

K

i=1

¯λ

k iCkg(k)₍_i₎

( ¯

P k i

)

(8)

=

K

i=1

n∈S(_ik)

¯λ

k nCki

( ¯

Pnk

)

(9)

≤

K

i=1

n∈S(_ik)

¯λ

k n

Ck_i

n∈S(_ik)

¯λ

k nP

¯

nk

n∈S(_ik)

¯λ

k n

(10)

=

K

i=1

ˆλ

k iC k i

( ˆ

P k i

)

(11) where

ˆλ

k

i andP

ˆ

ki aredeﬁnedas

ˆλ

k i

n∈S(k)_i

¯λ

k n

and

P

ˆ

ki

n∈S(k)_i

¯λ

k n

P

¯

kn

n∈S(k)_i

¯λ

k n

(12)

for i

∈ {

1

,

. . . ,

K

}

. The equalities in (8) and (9) are obtained fromthe deﬁnitionsin (6)and(7),respectively, andthe inequal-ity in (10) follows from Jensen’s inequality due to the concav-ity of the capacity function [34,35]. Based on the inequality in

(8)–(11),itisobtainedthat

ˆλ

p_i’sandP

ˆ

p_i’ssatisfytheminimum av-erage capacityrequirement in(2); that is,

K_i₌₁

ˆλ

p_iCp_i

( ˆ

P_ip

)

≥

Creq since

_iK₌₁

ˆλ

p_iC_ip

( ˆ

P_ip

)

≥

K_i₌₁

¯λ

p_iCpmax

( ¯

P p i

)

and

K i=1

¯λ

p iC p max

( ¯

P p i

)

≥

Creq. Also, it is notedfrom (12), based on (6) and (7), that

ˆλ

ki’s and P

ˆ

k

i’s for k

∈ {

p

,

s

}

satisfy the other constraints in (2); that is,

K_i₌₁

(ˆλ

p_i P

ˆ

_ip

+ ˆλ

_isP

ˆ

_is

)

≤

Pav, P

ˆ

p_i

,

P

ˆ

s_i

∈ [

0

,

P

pk

]

,

∀

i

∈ {

1

,

. . . ,

K

}

,

K i=1

(ˆλ

p i

+ ˆλ

s i

)

=

1,and

ˆλ

p i

,

ˆλ

s i

≥

0,

∀

i

∈ {

1

,

. . . ,

K

}

. Therefore,the inequality in (8)–(11), namely, C

_≤

K i=1

ˆλ

iCi

( ˆ

Pi

)

, implies that the optimal solution of (3) cannot achieve a higher average ca-pacity than that achieved by (2); that is, C

_≤

_C∗_. _Hence, _it _is concludedthat

C

₌

_C∗_since

_C

_≥

_C∗_must_also_hold_as_mentioned

atthebeginningoftheproof.

2

BasedonProposition 1,thesolutionoftheoriginalproblemin

(2)canbeobtainedfromtheoptimizationproblemin(3),whichis moretractablethantheonein(2),asinvestigatedinthefollowing.

Proposition 1alsoimpliesthat anoptimalstrategyalways utilizes thebestchannelforagivenpowerlevel,asnotedfrom(3a),(3b), and(4),whichisintuitive duetothemonotoneincreasing nature ofthecapacityexpression.

Asaﬁrststeptowardscharacterizingthesolutionof(3),the fol-lowingpropositionprovidesausefulstatementthattheconstraints in(3b)and(3c)alwaysholdwithequality.

3 _In_the_case_of_multiple_maximizers_in₍₆₎_,_any_maximizing_index_can_be_chosen forg(k)₍_i₎_.

(5)

Proposition2.

The solution of the optimization problem in

(3)satisﬁes the constraints in (3b)and (3c)with equality; that is,

iK=1

¯λ

p iC p max

( ¯

P p i

)

=

Creqand

iK=1

¯λ

p i P

¯

p i

+ ¯λ

s iP

¯

s i

=

Pav, where

{¯λ

p i

,

¯λ

s i

,

P

¯

p i

,

P

¯

s i

}

K i=1

de-notes the solution of (3).

Proof. Assumethat

{¯λ

p_i

,

¯λ

s_i

,

P

¯

p_i

,

P

¯

s

i

}

Ki=1 isthe solution of(3) such that

_iK₌₁

(¯λ

p_i P

¯

_ip

+ ¯λ

s

iP

¯

si

)

<

Pav.Then,thefollowingcasesare con-sidered4_:

•

If

¯λ

s_i

=

0

,

∀

i

∈ {

1

,

. . . ,

K

}

, then there exists at least one P

¯

p_i

such that P

¯

_ip

<

Ppk since

K i=1

λ

p i P p i

≤

Pav and

K i=1

λ

p i

=

1 due to the constraints in (3c) and (3d), respectively, and

Pav

<

Ppk by the assumption for (2). Let P

¯

pl denote one of them.Then,consideranalternativesolution

{ˆλ

p_i

,

ˆλ

s_i

,

P

ˆ

p_i

,

P

ˆ

_is

}

_iK₌₁, where

ˆ

P

_lp

=

min

P

pk

, ¯

P

_lp

+

P

av

−

K

i=1

¯λ

p i

P

¯

p i

/¯λ

_lp

,

(13)

ˆλ

p l

=

¯λ

p l

C

p max

( ¯

P

p l

)

C

pmax

( ˆ

P

p_l

)

,

(14)

ˆλ

p i

= ¯λ

p i

,

∀

i

∈ {

1 , . . . ,

K

} \ {

l

},

(15)

ˆ

P

_ip

= ¯

P

_ip

,

∀

i

∈ {

1 , . . . ,

K

} \ {

l

},

(16)

ˆλ

s 1

= ¯λ

p l

− ˆλ

p l

,

(17)

ˆ

P

₁s

= ˆ

P

_lp

,

(18)

ˆλ

s i

= ¯λ

si

,

∀

i

∈ {

2 , . . . ,

K

},

(19)

ˆ

P

_is

= ¯

P

s_i

,

∀

i

∈ {

2 , . . . ,

K

}.

(20)

Thesolution

{¯λ

p_i

,

¯λ

s_i

,

P

¯

p_i

,

P

¯

s_i

}

K

i=1achievesanaveragecapacityof

¯

Cs

=

0 dueto

¯λ

s_i

=

0

,

∀

i

∈ {

1

,

. . . ,

K

}

.Ontheother hand,the alternativesolutionsatisﬁestheconstraintsin(3)andachieves a larger capacity;that is C

ˆ

s

_{= ˆλ}

s

1Csmax

( ˆ

P1s

)

>

0 since

ˆλ

s1

>

0 and P

ˆ

s₁

>

0.Therefore,

{¯λ

p_i

,

¯λ

s_i

,

P

¯

p_i

,

P

¯

s_i

}

K_i₌₁ cannotbeoptimalif

¯λ

s

i

=

0

,

∀

i

∈ {

1

,

. . . ,

K

}

,whichcontradictswiththeassumption atthebeginningoftheproof.

•

Forthecasethat

¯λ

s_i

>

0

,

∃

i

∈ {

1

,

. . . ,

K

}

,deﬁneasetas

M

{

i

∈ {

1 , . . . ,

K

} | ¯λ

s_i

>

0 } .

(21)

Next,considerthefollowingcases:

– IfP

¯

s_k

=

Ppk

,

∀

k

∈

M, thenthereexistsatleastone P

¯

pi that satisﬁes P

¯

p_i

<

Ppk since the constraints in (3c) and (3d) hold. Let P

¯

p_l representone of them andconsideran alter-native solution

{ˆλ

p_i

,

ˆλ

s_i

,

P

ˆ

_ip

,

P

ˆ

s_i

}

K_i₌₁,where P

ˆ

_lp,

ˆλ

p_l,

ˆλ

p_i forall

i

∈ {

1

,

. . . ,

K

}

\ {

l

}

, P

ˆ

p_i forall

i

∈ {

1

,

. . . ,

K

}

\ {

l

}

,

ˆλ

s₁,and P

ˆ

₁s

areasin(13)–(18)andtheremainingtermsareasfollows:

ˆλ

s 2

=

k_∈M

¯λ

s k

,

(22)

ˆ

P

₂s

=

P

pk

,

(23)

ˆλ

s i

=

0 ,

∀

i

∈ {

3 , . . . ,

K

},

(24)

ˆ

P

_is

=

0 ,

∀

i

∈ {

3 , . . . ,

K

}.

(25)

4 _In_this_case,_it_is_assumed_that_multiple_channels_are_available_for communica-tion;thatis,K>1.Inthecaseofasinglechannelavailableforcommunication(i.e., K=1),asimilarapproachcanbeemployed.

Theachievedaveragecapacityby

{¯λ

p_i

,

¯λ

s_i

,

P

¯

p_i

,

P

¯

_is

}

K

i=1 isC

¯

s

=

K

i=1

¯λ

siCmaxs

( ¯

Psi

)

,whichislowerthanthatachievedbythe alternativesolutionduetothefollowingrelation:

¯

Cs

=

K

i=1

¯λ

s iCsmax

( ¯

Psi

)

=

k∈M

¯λ

s kCsmax

(

Ppk

)

(26)

<

k∈M

¯λ

s kCsmax

(

Ppk

)

+ ˆλ

s 1Cmaxs

( ˆ

Ps1

)

(27)

=

K

i=1

ˆλ

s iCmaxs

( ˆ

Psi

)

(28)

= ˆ

Cs (29)

where(26)followsfromthecondition that P

¯

s_k

=

Ppk

,

∀

k

∈

M, the inequality in (27) is due to

ˆλ

s₁

>

0 and P

ˆ

s₁

>

0,

(28)is obtained basedon (13)–(18) and(22)–(25), and ﬁ-nally C

ˆ

s in (28)denotes the achievedaverage capacityby the alternative solution. Based on (26)–(29), it is obtained that C

¯

s

< ˆ

Cs. Therefore,

{¯λ

p_i

,

¯λ

s_i

,

P

¯

p_i

,

P

¯

s_i

}

K_i₌₁ cannot be opti-mal andconsequently the assumption atthe beginning of theproofmustbefalseifP

¯

s

k

=

Ppk,

∀

k

∈

M for thecasethat

¯λ

s

i

>

0

,

∃

i

∈ {

1

,

. . . ,

K

}

.

– If P

¯

_ks

<

Ppk,

∃

k

∈

M, then based on a similar approach to thatinLemma 1of[30],analternativesolution

{ˆλ

p_i

, ˆλ

s_i

, ˆ

Pp_i

,

ˆ

Ps_i

}

K_i₌₁canbeexpressedas

ˆλ

p i

= ¯λ

p i

,

∀

i

∈ {

1 , . . . ,

K

},

(30)

ˆ

P

p_i

= ¯

P

p_i

,

∀

i

∈ {

1 , . . . ,

K

},

(31)

ˆ

P

_ls

=

min

P

pk

, ¯

P

ls

+

P

av

−

K

i=1

¯λ

s i

P

¯

s i

/¯λ

s_l

,

(32)

ˆ

P

s_i

= ¯

P

s_i

,

∀

i

∈ {

1 , . . . ,

K

} \ {

l

},

(33)

ˆλ

s i

= ¯λ

si

,

∀

i

∈ {

1 , . . . ,

K

}

(34)

where P

¯

_lsisoneofthepowerlevelsthatsatisﬁes P

¯

_ls

<

Ppk. Since P

ˆ

s_l

> ¯

P_ls and C_maxs

(

P

)

in (4) is a monotone increas-ingfunctionof

P ,

itisobtainedthatthealternativesolution achieves a larger average capacity than

{¯λ

p_i

,

¯λ

_is

,

P

¯

_ip

,

P

¯

s_i

}

K

i=1 does. Therefore, the assumption at the beginning of the proofmustnotbetrue.

Basedonthecasesspeciﬁedabove,itisconcludedbycontradiction that the solutionof theoptimization problemin(3) satisﬁes the constraintin(3c)withequality;thatis,

K_i₌₁

¯λ

p_i P

¯

p_i

+ ¯λ

s_iP

¯

s_i

=

Pav.

In the second part of the proof, the aim is to prove that the solution of (3) satisﬁes the constraint in (3b) with equality. Assume that

{¯λ

p_i

,

¯λ

s_i

,

P

¯

p_i

,

P

¯

s_i

}

_iK₌₁ is the solution of (3) such that

K i=1

¯λ

p iC p max

( ¯

P p

i

)

>

Creq.Since

C

req

>

0 byassumption, there ex-ists at least one

{¯λ

p_i

,

P

¯

p_i

}

pair such that

¯λ

p_i

>

0 and P

¯

p_i

>

0. Let

{¯λ

p

l

,

P

¯

p

l

}

denote one of them. Then, there exists a non-negative

ˆ

Pp_l

< ¯

P_lp suchthat

_iK₌₁

ˆλ

p_iCmaxp

( ˆ

Ppi

)

≥

Creq,where

ˆλ

pi

= ¯λ

p i forall

i

∈ {1

,

. . . ,

K

}

andP

ˆ

p_i

= ¯

P_ip forall

i

∈ {1

,

. . . ,

K

}

\ {

l

}

since

C

pmax

(

P

)

isamonotoneincreasingandcontinuousfunctionof P .

•

If

¯λ

s_i

=

0

,

∀

i

∈ {

1

,

. . . ,

K

}

,thenconsideranalternativesolution

{ˆλ

p i

,

ˆλ

si

,

P

ˆ

p i

,

P

ˆ

si

}

iK=1,where

ˆλ

s 1

= ¯λ

p l

,

(35)

(6)

ˆ

P

s₁

= ¯

P

_lp

− ˆ

P

_lp

,

(36)

ˆλ

s i

= ¯λ

s i

,

∀

i

∈ {

2 , . . . ,

K

},

(37)

ˆ

P

s_i

= ¯

P

s_i

,

∀

i

∈ {

2 , . . . ,

K

}.

(38)

•

Forthecasethat

¯λ

s_i

>

0

,

∃

i

∈ {

1

,

. . . ,

K

}

,deﬁneasetas

M

{

i

∈ {

1 , . . . ,

K

} | ¯λ

s_i

>

0 }.

(39)

Next,considerthefollowingcases: – If P

¯

s

k

=

Ppk

,

∀

k

∈

M, then consider an alternativesolution

{ˆλ

p

i

,

ˆλ

si

,

P

ˆ

p

i

,

P

ˆ

si

}

Ki=1,where

ˆλ

s1and P

ˆ

1s areasin(36)and(37), respectively,and

ˆλ

s 2

=

k∈M

¯λ

s k

,

(40)

ˆ

P

₂s

=

P

pk

,

(41)

ˆλ

s i

=

0 ,

∀

i

∈ {

3 , . . . ,

K

},

(42)

ˆ

P

s_i

=

0 ,

∀

i

∈ {

3 , . . . ,

K

}.

(43)

– If P

¯

_ks

<

Ppk,

∃

k

∈

M, then based on a similar approach to thatinLemma 1of[30],analternativesolution

{ˆλ

p_i

,

ˆλ

s_i

, ˆ

Pp_i

,

ˆ

P_is

}

K_i₌₁ canbeexpressedas

ˆ

P

_ls

=

min

{

P

pk

, ¯

P

ls

+ ¯λ

p l

( ¯

P

p l

− ˆ

P

p l

)/¯λ

s l

},

(44)

ˆ

P

s_i

= ¯

P

s_i

,

∀

i

∈ {

1 , . . . ,

K

} \ {

l

},

(45)

ˆλ

s i

= ¯λ

si

,

∀

i

∈ {

1 , . . . ,

K

}

(46)

whereP

¯

s_l isoneofthepowerlevelsthatsatisﬁesP

¯

s_l

<

Ppk. Similartotheﬁrstpartoftheproof,allalternativesolutions spec-iﬁed for the cases above achieve a larger average capacity than

{¯λ

p i

,

¯λ

s i

,

P

¯

p i

,

P

¯

s i

}

K

i=1 does. Therefore, it is proved by contradiction thatthesolutionsatisﬁestheconstraintin(3b)withequality;that is,

_iK₌₁

¯λ

p_iCmaxp

( ¯

Ppi

)

=

Creq.

2

Even though Proposition 2 states that the constraints in (3b)

and(3c) are satisfiedwithequality,itisstill difficulttosolvethe optimizationproblem in(3). Therefore,the following proposition is presented in order to provide a further simplification for the optimizationproblemin(3).

Proposition3.

The optimal channel switching strategy

based on the op-timization problem in (3)employs at most min

{

3

,

2K

}

communication links.

Proof. If K

≤

1, then the assertion in Proposition 3 holds obvi-ously. Otherwise,(if K

>

1),then considerthefollowing transfor-mations:

ν

i

=

λ

p_i

,

if i

≤

K

λ

s_m

,

if i

>

K

,

∀

i

∈ {

1

, . . . ,

2K

}

(47) Pi

=

Pp_i

,

if i

≤

K Ps_m

,

if i

>

K

,

∀

i

∈ {

1

, . . . ,

2K

}

(48)

where

m

i

−

K . Also,deﬁnethefollowingfunctions:

C_maxp _,_i

(

P

)

=

Cmaxp

(

P

),

if i

≤

K 0

,

if i

>

K

,

∀

i

∈ {

1

, . . . ,

2K

}

(49) C_maxs _,_i

(

P

)

=

0

,

if i

≤

K Cs max

(

P

),

if i

>

K

,

∀

i

∈ {

1

, . . . ,

2K

}

(50)

forall P

∈ [

0

,

P

pk

]

.Basedonthetransformationsin(47)and(48) andthefunctionsin(49)and(50),theoptimizationproblemin(3)

canbewritteninthefollowingform:

max {νi,Pi}2Ki=1 2K

i=1

ν

iCsmax,i

(

Pi

)

(51a) subject to 2K

i=1

ν

iCpmax,i

(

Pi

)

≥

Creq (51b) 2K

i=1

ν

iPi

≤

Pav

,

Pi

∈ [

0

,

Ppk

] , ∀

i

∈ {

1

, . . . ,

2K

}

(51c) 2K

i=1

ν

i

=

1

,

ν

i

∈ [

0

,

1

] , ∀

i

∈ {

1

, . . . ,

2K

}

(51d) Next,deﬁnethefollowingsets:

V =

_2K

i=1

ν

i

C

maxp ,i

(

P

i

),

2K

i=1

ν

i

C

smax,i

(

P

i

),

2K

i=1

ν

i

P

i

∈ R

3

2K

i₌1

ν

i

=

1 ,

ν

i

∈ [

0 ,

1 ] ,

P

i

∈ [

0 ,

P

pk

] , ∀

i

∈ {

1 , . . . ,

2K

}

(52)

W =

w

= {

u

1

, . . . ,

u

2K

}

u

i

∈

U

i

,

∀

i

∈ {

1 , . . . ,

2K

}

(53)

where

U

i

=

P

,

Cp_max_,_i

(

P

),

Cs_max_,_i

(

P

)

∈ R

3

_P

_{∈ [}

₀

_,

_P pk

]

,

∀

i

∈ {

1

, . . . ,

2K

} .

(54)

It is noted that set

V

includes the solution of the optimization problem in (51) by deﬁnition.Let

W

i represent the ith element of set

W

,which is alsoa set. Then, set

V

is equalto the union of the convex hulls of set

W

i

,

∀

i

∈ {

1

,

. . . ,

|

W|}

; that is,

V =

|W|

i=1conv

(W

i

)

.Therefore,

i|W|=1conv

(W

i

)

alsoincludesthe solu-tion ofthe optimizationproblemin(51). Thedeﬁnition ofunion implies that the solution of (51) is an element of conv

(

W

i

)

for some i

∈ {

1

,

. . . ,

|

W|}

.Without loss of generality, let l be one of them. Since the optimization problemin (51) is a maximization problem, thesolutionof(51)residesontheboundaryofthe con-vex hullofset

W

l.Then, byCarathéodory’s theorem[36,37],any pointontheboundaryoftheconvexhullofset

W

l canbe repre-sented by a convex combination of at mostd points in set

W

l, where d is the dimension of space in which

W

l resides. Since

W

l

⊂ R

3 andtheoptimalsolution of(51)corresponds toa point ontheboundaryofconv

(

W

l

)

,theoptimalchannelswitching strat-egyemploysatmost3 communicationlinks.

2

BasedonProposition 3andthestudyin[30],theoptimal chan-nel switching strategy can be investigated as follows: Let C

¯

req denote the achieved maximumaverage capacity forthe commu-nication betweenthe transmitterandthe primary receiver when there is no secondary receiver in the system. Then, C

¯

req can be calculatedasfollows:

¯

Creq

=

max {λp i,P p i}iK=1 K

i=1

λ

p_i Cp_max

(

P_ip

)

(55a) subject to K

i=1

λ

p_i Pp_i

=

Pav

,

(7)

P_ip

∈ [

0

,

Ppk

] , ∀

i

∈ {

1

, . . . ,

K

}

(55b) K

i=1

λ

p_i

=

1

, λ

p_i

∈ [

0

,

1

], ∀

i

∈ {

1

, . . . ,

K

}

(55c)

If the maximum average capacity achieved by the optimization problemin (55) is strictly lower than the minimum average ca-pacity requirement for the primary receiver (i.e., C

¯

req

<

Creq), thenthereisnopossiblechannelswitchingstrategyforthe prob-lem in(2) since the optimizationproblem in (3) is infeasible.If

¯

Creq

=

Creq,theoptimalchannelswitchingstrategycorrespondsto switching betweenat most two channels between the transmit-terandthe primary receiverbased on the optimization problem in(2) andProposition 4in [30]. In that case, nocommunication isperformedbetweenthetransmitterandthesecondaryreceiver. Therefore,the achievedmaximumaverage capacityis C

₌

_0.

Fi-nally, ifC

¯

req

>

Creq,then the optimal channel switching strategy correspondstooneofthefollowingtwostrategies:

•

Strategy-1(Communicatewiththeprimaryreceiveroverat most twochannelsandemploysingle channelforthe sec-ondaryreceiver): Inthisstrategy,thetransmitteremploysone or twochannelsto satisfythe minimumaveragecapacity re-quirementoftheprimary receiverandusesonlyone channel inordertomaximize theaveragecapacityofthe communica-tiontothesecondaryreceiver.Theachievedmaximumaverage capacityforthecommunicationtothesecondaryreceiver, de-notedby

C

str,1,canbecalculatedasfollows:

Cstr,1

=

max

λ1,λ2,λ3,P1,P2,P3

λ

1Cmaxs

(

P1

)

(56a)

subject to

λ

2Cpmax

(

P2

)

+ λ

3Cpmax

(

P3

)

=

Creq (56b)

λ

1P1

+ λ

2P2

+ λ

3P3

=

Pav

,

P1

,

P2

,

P3

∈ [

0

,

Ppk

] ,

(56c)

λ

1

+ λ

2

+ λ

3

=

1

, λ

1

, λ

2

, λ

3

∈ [

0

,

1

]

(56d)

•

Strategy-2(Communicatewiththesecondaryreceiveroverat most twochannelsandemploysingle channelforthe pri-mary receiver): In this case, the transmitter maximizes the average capacityof the communication to the secondary re-ceiver by employing one or two channelswhile meeting the minimum average capacity requirement for the primary re-ceiver via communicationover asingle channel. In thiscase, theachievedaveragecapacitycanbeexpressedas

Cstr,2

=

max

λ1,λ2,λ3,P1,P2,P3

λ

1Cmaxs

(

P1

)

+ λ

2Csmax

(

P2

)

(57a)

subject to

λ

3Cpmax

(

P3

)

=

Creq (57b)

λ

1P1

+ λ

2P2

+ λ

3P3

=

Pav

,

P1

,

P2

,

P3

∈ [

0

,

Ppk

] ,

(57c)

λ

1

+ λ

2

+ λ

3

=

1

, λ

1

, λ

2

, λ

3

∈ [

0

,

1

]

(57d)

BasedonStrategy 1andStrategy 2,themaximumaverage ca-pacity for the communication between the transmitter and the secondaryreceiver,whichisthesolutionof(2),canbecalculated as

C

=

max

{

Cstr,1

,

Cstr,2

}

(58)

where

C

str,1 and

C

str,2areasin(56)and(57),respectively. Itisimportant tonote that theoptimizationproblemsin(56)

and(57)havesigniﬁcantly lower computational complexity com-pared to the original optimization problem in (2) since each of

(56)and(57)requiresasearch onlyover a3 dimensional space5

whereasasearch overa4K dimensionalspaceisrequiredforthe problemin(2),where K

>

1.

4. Optimalchannelswitchinginthepresenceofmultiple primaryreceivers

In thepresence ofmultiple primary receivers, each havingan individual minimum average capacity requirement, the optimiza-tionproblemin(2)canbeextendedasfollows:

max λs_i,Ps_i,λp j_i ,Pp j_i N_j₌₁K_i₌₁ K

i=1

λ

s_iC_is

(

Ps_i

)

(59a) subject to K

i=1

λ

p_ijC_ipj

(

P_ipj

)

≥

Creqj

,

∀

j

∈ {

1

, . . . ,

N

},

(59b) K

i=1

⎛

⎝λ

s iPsi

+

N

j=1

λ

p_ijPpj i

⎞

⎠ ≤

Pav

,

(59c) P_is

,

Pp_ij

∈ [

0

,

Ppk

], ∀

i

∈ {

1

, . . . ,

K

}, ∀

j

∈ {

1

, . . . ,

N

},

(59d) K

i=1

⎛

⎝λ

s i

+

N

j=1

λ

p_ij

⎞

⎠ =

1

,

(59e)

λ

s_i

, λ

p_ij

∈ [

0

,

1

], ∀

i

∈ {

1

, . . . ,

K

}, ∀

j

∈ {

1

, . . . ,

N

},

(59f)

where

λ

p_ij and P_ipj denote, respectively, the time-sharing factor andtheaveragetransmitpowerallocatedtochannel

i for

the com-municationbetweenthetransmitterandthe jth primary receiver,

N is thenumberofprimaryreceiversinthesystem,

C

pj

i

(

P

)

as de-ﬁnedin(1),

C

reqj istheminimumaveragecapacityrequirementfor the jth primaryreceiver,andtheotherparametersareasin(2).

It is noted that the optimization problem in (2) is a special case of (59) when there exists only one primary receiver; that is,when N

=

1. Therefore,it isin generalmorediﬃcult to solve the optimization problem in(59) since it requires a search over a 2K

(

N

+

1

)

dimensional space, which ishigher than 4K , corre-spondingto (2),for N

>

1. However, theresultsobtainedfor the problemin(2) canbe extendedfor (59),asexplained inthe fol-lowingremark.

Remark1. Based on a similar approach to that in Proposition 1, it can be shown that an alternative optimizationproblem to the problem in (59)can be obtained. Also, the approach in Proposi-tion 2alsoholdsfortheoptimizationproblemin(59)and conse-quentlythe solutionof(59) satisﬁestheconstraints in(59b)and

(59c)withequality.Moreover,similartotheproofinProposition 3, itcanbestatedfortheoptimizationproblemin(59)thatthe op-timal channel switching strategy based on (59)employs atmost min

{

N

+

2

,

K

(

N

+

1

)

}

communication linksinthe system, where

K

(

N

+

1

)

linksareavailableintotal.Speciﬁcally,theoptimal chan-nelswitchingstrategycanberealizedbyswitchingamongatmost

(

N

+

2

)

communicationlinksinthepresenceofmultipleavailable channelsinthesystem;thatis,

K

>

1.

It is concluded from Remark 1 that the solution of (59) can becalculatedbysolving atotalof

(

N

+

1

)

optimizationproblems, 5 _Note_that_the_search_space _dimensions_of_the_optimization _problems_in₍₅₆₎ and(57)areobtainedbysubstitutingtheequalityconstraintsin(56b)–(56d)and (57b)–(57d)intotheobjectivefunctionsin(56a)and(57a),respectively.

(8)

Fig. 2. Capacityofeachlinkversuspowerforthecommunicationbetweenthe trans-mitterand theprimaryreceiver,where B1=1 MHz,B2=3 MHz, B3=4 MHz, B4=5 MHz,B5=10 MHz, N1=N2=N3=N4=N5=10−12 W/Hz,|hp1|2=1, |hp₂|2₌₀ .1,|hp₃|2₌₀ .1,|hp₄|2₌₀ .1,and|hp₅|2₌₀ .05.

each requiring a search over a 2

(

N

+

2

)

dimensional space, and thenchoosingthe maximumamongtheobtainedaverage capaci-ties. Hence, theoptimal channel switchingstrategy based onthe optimizationproblemin(59)canbeobtainedwithlow computa-tionalcomplexity.

For complexity comparisons, assume that there exist ﬁnitely many possible values of

λ

k_i and Pk

i for each k

∈ {

p

,

s

}

and i

∈

{

1

,

. . . ,

K

}

,where

λ

k_i

∈ [

0

,

1

]

and Pk_i

∈ [

0

,

P

pk

]

forall

k

∈ {

p

,

s

}

and

i

∈ {

1

,

. . . ,

K

}

.Let

N

λ denotethenumberofdifferent

λ

valuesfor

λ

∈ [

0

,

1

]

andNP representthenumberofdifferent P values for

P

∈ [

0

,

P

pk

]

.Then, theoriginal optimizationproblemin(2)hasa computational complexity of

O(

N2K

λ

×

N2KP

)

. Onthe other hand, the complexity ofeach optimizationproblemin (56) and(57) is in the order of

O(

N3_λ

×

N3_P

)

. Therefore, in the presence of mul-tiple available channels, instead ofsolving theoriginal optimiza-tion problem in (2) with a complexity of

O(

N2K_λ

×

N2K_P

)

where

K

>

1, thesolution of(2) can be obtainedwith alower compu-tational complexityby solving twooptimizationproblems in(56)

and(57),eachhavingacomputationalcomplexityof

O(

N_λ3

×

N3_P

)

. In the presence of N primary receivers in the communication system, the complexity of the optimization problem in (59) is

O(

N_λK(N+1)

×

N_PK(N+1)

)

.However,thesolutionof(59)canbe calcu-latedwithalowercomplexitybysolving

N

+

1 optimization prob-lems,eachhavingacomputationalcomplexityof

O(

N_λN+2

×

NN_P+2

)

.

5. Numericalresults

In this section, several numerical examples are presented to investigatethe performance of theproposed strategiesandto il-lustrate the optimal strategy for various values of the average power limit andthe minimum average capacity requirement for theprimaryreceiver.Tothataim,acommunicationsystemis con-sideredwith

K

=

5 channels,thebandwidthsandthenoiselevels ofwhicharegivenby

B

1

=

1 MHz,

B

2

=

3 MHz,

B

3

=

4 MHz,

B

4

=

5 MHz,

B

5

=

10 MHz,and

N

1

=

N2

=

N3

=

N4

=

N5

=

10−12W

/

Hz (cf. (1)). It is assumedthat all the channels are available forthe transmitterand can be used to communicate with both the pri-maryandsecondaryreceivers.Also,forthesechannels,thechannel powergainsfromthetransmittertotheprimaryandsecondary re-ceiversaregivenby

|

hp₁

|

2

₌

_1,

_|

_hp

2

|

2

=

0

.

1,

|

h p 3

|

2

=

0

.

1,

|

h p 4

|

2

=

0

.

1,

|

hp₅

|

2

₌

₀

_.

_05,

_|

_hs 1

|

2

=

1,

|

h s 2

|

2

=

0

.

1,

|

h s 3

|

2

=

0

.

1,

|

h s 4

|

2

=

0

.

1, and

Fig. 3. Capacityofeachlinkversuspowerforthecommunicationbetweenthe trans-mitterandthesecondaryreceiver,whereB1=1 MHz,B2=3 MHz,B3=4 MHz, B4=5 MHz,B5=10 MHz,N1=N2=N3=N4=N5=10−12 W/Hz, |hs1|2=1, |hs

2|2=0.1,|hs3|2=0.1,|hs4|2=0.1,and|hs5|2=0.1.

Fig. 4. AveragecapacityversusaveragepowerlimitforStrategy1,Strategy2,and theoptimalchannelswitchingstrategyforthescenarioinFig. 2andFig. 3,where Creq=5 Mbps.

|

hs₅

|

2

=

0

.

1.Inthisscenario,thepeakpowerconstraintin(2)isset to Ppk

=

0

.

1 mW.Thecapacityofeachlinkavailableforthe trans-mittertocommunicate withtheprimaryandsecondary receivers isplottedasafunctionofpowerinFig. 2andFig. 3,respectively.

Inordertoinvestigatetheeffectoftheaveragepowerlimiton the performance of theoptimal channel switchingstrategies, the minimum average capacityconstraintfor theprimary receiver in

(2) is set to Creq

=

5 Mbps ﬁrst. Then, by considering the chan-nel linksin Fig. 2 and Fig. 3,the optimalaverage capacities are obtained fordifferentaverage power limits( Pav) basedon Strat-egy 1 in (56)andStrategy 2 in(57),andtheachievedmaximum average capacities are presented in Fig. 4. From Fig. 4, it is ob-served that C

=

0 for Pav

<

0

.

031 mW sincethere isnofeasible solutionoftheoptimizationproblemin(2)for

C

req

=

5 Mbps and

Pav

<

0

.

031 mW.Ontheotherhand,for

P

av

≥

0

.

031 mW,the op-timal channel switching strategy can be obtainedbased on (56)

and(57),anditcorrespondstoStrategy 1 forall Pav

≥

0

.

031 mW since Strategy 1 outperforms Strategy 2 in terms of the