Maximization of average number of correctly received symbols over multiple channels in the presence of idle periods

(1)

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Maximization

of

average

number

of

correctly

received

symbols

over

multiple

channels

in

the

presence

of

idle

periods

✩

Musa

Furkan Keskin

a

,

1

,

Mehmet

Necip Kurt

a

,

1

,

Mehmet

Emin Tutay

b

,

Sinan Gezici

a

,

∗

,

Orhan Arikan

a

,

1

a_Department_{of Electrical}_{and Electronics}_Engineering,_Bilkent_University,_Ankara_06800,_Turkey b_{Department of}_{Electrical and}_{Electronics Engineering,}_Dicle_University,_Diyarbakir_{21280, Turkey}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Article history:

Availableonline7April2016 Keywords:

Channelswitching Gaussianchannel Fading

Probabilityofcorrectdecision Partialtransmission Logarithmiccost

Inthisstudy,optimalchannelswitching(timesharing)strategiesareinvestigatedunderaveragepower and cost constraints for maximizing the average number of correctly received symbols between a transmitter andareceiverthatare connectedviamultipleﬂat-fadingchannelswithadditiveGaussian noise.The optimal strategyisshownto correspondtochannel switchingeitheramong atmostthree differentchannelswithfullchannelutilization(i.e., noidleperiods),orbetweenatmosttwodifferent channelswithpartialchannelutilization.Also,itisstatedthattheoptimalsolutionmustoperateatthe maximumaveragepowerand themaximumaveragecost,whichfacilitateslow-complexityapproaches forobtaining theoptimal strategy.Fortwo-channelstrategies,anupper boundisderived,intermsof theparametersoftheemployedchannels,ontheratiobetweentheoptimal powerlevels.Inaddition, theoreticalresultsarederivedforcharacterizingtheoptimalsolutionforchannelswitchingbetweentwo channels,andforcomparingperformanceofsinglechannelstrategies.Suﬃcientconditionsthatdepend solelyonthesystemsparametersareobtainedforspecifyingwhenpartialchannelutilizationcannotbe optimal.Furthermore, theproposed optimalchannel switchingproblemis investigatedforlogarithmic cost functions,and various theoretical resultsare obtained relatedtothe optimal strategy.Numerical examplesarepresentedtoillustratethevalidityofthetheoreticalresults.

1. Introduction

Timesharing(randomization)hasattractedasigniﬁcantdealof interest in the literature due to its capability to provide perfor-manceimprovementsforcommunicationsystems[1–11].In[2],it isdemonstratedthattheaverageprobability oferroroveradditive noise channels witharbitrary noise probability densityfunctions (PDFs)canbereducedviaoptimalstochasticsignaling,which per-forms time sharing among at most three different signal levels foreach informationsymbol.Thestudyin[5]investigates perfor-mancegainsthat canbe achievedbydetectorrandomizationand stochastic signaling, and proves that the optimal receiver design

✩ _This_research_was_supported_in_part_by_the_{Distinguished}_Young_Scientist_Award ofTurkishAcademyofSciences(TUBA-GEBIP2013).Partofthisworkwaspresented atIEEEWirelessCommunicationsandNetworkingConference(WCNC)2016.

*

Correspondingauthor.Fax:+90-312-266-4192. E-mail addresses:keskin@ee.bilkent.edu.tr(M.F. Keskin),

mnkurt@ee.bilkent.edu.tr(M.N. Kurt),

emin.tutay@dicle.edu.tr

(M.E. Tutay), gezici@ee.bilkent.edu.tr(S. Gezici),

oarikan@ee.bilkent.edu.tr

(O. Arikan).

1 _Fax:_{+90-312-266-4192.}

isrealizedby time sharing(randomization)betweenatmosttwo maximum a-posterioriprobability (MAP)detectors corresponding totwodeterministicsignalvectors.Forthedownlinkofamultiuser communicationssystem, [8]performs jointoptimizationofsignal amplitudes, detectors, and detector randomization factors in or-dertoreducetheworst-caseaverageprobabilityoferror.Similarly, jamming performanceofaverage powerconstrainedjammerscan be enhanced by performing time sharing among different power levels [3,6,7].In[3],theoptimaltime sharing strategyfora jam-merthatoperatesoverachannelwithasymmetricunimodalnoise densityisshowntocorrespondtoon-offjammingwhenthe aver-age power constraintis belowa certain threshold.The optimum jamming strategy that minimizes the probability of detection in the Neyman–Pearson framework is considered in[7],where it is stated that power randomization between atmost two different powerlevels canresultinthehighestjamming performanceover anadditivenoisechannelwithagenericPDF.

Performance enhancements via time sharing can also be re-alized in communicationsystems where the transmitter and the receiver are connected through multiple channels [3,10–13]. In such a scenario,channelswitching canbe performedby

(2)

tingovera channelduringacertainperiodoftimeandswitching to another channel during the next period. In [3], the optimal channel switchingstrategy is studied forminimizing the average probabilityoferroroverasetofchannelswithadditive unimodal noiseunderanaveragepowerconstraint,anditisprovedthatthe optimumperformance can be achievedvia timesharing between at mosttwo channels and power levels. An average power con-strainedM-arycommunicationsysteminthepresenceofmultiple additivenoisechannelswithgeneric noisePDFsisstudiedin[11]

in the context of minimizing the average probability oferror by joint optimizationof channel switching, stochastic signaling, and detection strategies.It is demonstrated that the optimal strategy is to employ deterministic signaling or time sharing between at most two signal constellations over a single channel, or to per-formchannelswitchingbetweentwochannelswithdeterministic signaling. The beneﬁts of channel switching are investigated in

[13]foradditiveGaussian noisechannelsunderaverageandpeak power constraints, where the objective is to maximize average channel capacity.It is proved that the optimalsolution performs channel switching between at most two different channels. The studyin[10] formulatesthechannelswitchingproblemby incor-poratingchannelcosts associatedwiththe usageofeach channel fortransmissionandimposinganaveragecostconstraint.The opti-malchannelswitchingstrategyoverasetofGaussianchannelsand underaveragepowerandcostconstraintsisshowntocorrespond totimesharingamongatmostthreedifferentchannels[10].

Thepreviousstudiesonthechannelswitchingproblemmainly employ the average probability of error [3,10,11] or the average channelcapacity[13] astheobjectivefunctions, andassume that the channelsare fully utilized; i.e., there always exists transmis-sion over one of the channels and there are no idle periods. In this manuscript, the channel switching problem is investigated for maximizing the average number of correctly received sym-bolsinthe absenceofthe full transmission/utilizationconstraint. Morespeciﬁcally,theoptimalchannelswitchingstrategiesare de-signedoverasetofﬂat-fading channelsunderaveragepowerand cost constraints for the maximization of the average number of correctlyreceived symbols. Ratherthan forcing full utilization of channels(i.e.,noidleperiods)asin[10],anovelandmoregeneral formulationisdevelopedforchannel switching,where communi-cation may not occur during a certain period oftime, which,in somescenarios,isshowntoattainahigheraveragenumberof cor-rectly received symbols than full channel utilization. In addition, unlikethenofading assumptionin[10],Rayleighfadingchannels are also considered in designing the optimal channel switching strategies. Furthermore, the proposed optimal channel switching problem is studied for logarithmic cost functions, where a log-arithmic relation is employed between the signal-to-noise ratio (SNR)ofeachchannelanditsutilizationcost,whichisin compli-ancewiththecostfunctionsintheliterature[10,14–16].Themain contributionsandnovelty ofthe studyinthismanuscriptcan be summarizedasfollows:

•

The optimalchannel switchingproblem isformulated inthe presenceofpartialdatatransmissionfortheﬁrsttime.

•

It isshown that the optimumsolution is achievedvia

chan-nel switchingeither amongatmost threechannels withfull transmission or between at most two channels with partial transmission(Proposition 1).

•

Theoretical results are obtained for characterizing the opti-malsolutionforchannelswitchingbetweentwochannels,and for comparing the performance of single channel strategies (Propositions 2 and 3).

•

Suﬃcient conditionsfor theoptimality of full data transmis-sion are derived in terms of channel costs, standard devia-tionsofchannel noise,andchannel fadingstatistics ( Proposi-tion 4).Undertheseconditions,partialtransmission strategies

Fig. 1. Channelswitchingamong

K channels,

where

C

i denotesthecostofusing

channel

i.

areguaranteedtobenotoptimal,whichfacilitatesalow com-plexitysolutionfortheoptimalchannelswitchingproblem.

•

For logarithmic cost functions, it is shown that the partial

transmission strategies are not optimalif the average power limitishigherthanacertainthreshold,whichdependsonthe parametersoftheworstandbestchannels,aswellasthe pa-rameters of the cost function and the probability of correct decision(Proposition 5).

Inaddition,numericalexamplesarepresentedforthe demonstra-tionofthetheoreticalresults.

An important practical application of the channel switching problem considered in this manuscript is a cognitive radio (CR) system,inwhichprimaryusersareregardedasownersofthe fre-quency spectrum, and secondary users can utilize the frequency bands of primary users under certain conditions [10,17]. In the spectrum tradingframework proposed in[18],primary users can sellcertain partoftheirspectrumto secondaryusersfortheaim of revenue maximization. In that case, there can exist multiple available frequency bands(channels) with different costs for the useofsecondaryusers,andasecondaryusercanperformchannel switchingamongdifferentavailable channelstoimproveits com-munication performance [10]. Similar to the effort of secondary usersinCRnetworksforobtaining thebestperformance overthe availablebands,theaimofthisstudyistodesignoptimalchannel switching strategiesfor an arbitrarysignal constellation to maxi-mize the average numberof correctly received symbolsbetween the transmitter and the receiver under average cost and power constraints over multiple fading channels corrupted by additive whiteGaussiannoise.

The remainder of themanuscriptis organized asfollows: The systemmodelandtheproblemformulationare presentedin Sec-tion 2.The solutionof theoptimal channelswitching problemis characterizedandtheoreticalresultsareobtainedinSection3.The optimalchannelswitchingproblemisstudiedforlogarithmic cost functionsin Section4. InSection 5,numericalexamples are pro-vided,whichisfollowedbytheconcludingremarksinSection6.

2. Systemmodelandproblemformulation

The channel switching problem is formulated for an M-ary

communication system with K channels between the transmit-ter and the receiver, as shown in Fig. 1. Channel switching is performedovera communicationintervalthat consistsofa suﬃ-ciently largenumberofsymbols. Dependingonfadingconditions, thefollowingtwocasesareconsidered:

•

Case1: In this case, it is assumed that slow fading occurs

andthe channel coeﬃcient ofeach channel isﬁxed over the whole communicationinterval during whichchannel switch-ingisperformed.Also,thetransmitterisassumedtohavethe channelstateinformation(CSI)forallthechannels,whichcan beprovidedinpracticeviafeedbackfromthereceiver.

•

Case2: In this case, block fading is considered, where each

block consists of a number of symbols and the block dura-tion is signiﬁcantly shorter than the communicationinterval

(3)

during which channel switching is performed. It is assumed that the channel coefficientschange fromblock to block (in-dependently)butthestatistics ofthechannel coefficientsare fixedforeachchannelinthecommunicationinterval.Also,the transmitterhasthechanneldistributioninformation(CDI)for allthechannelsbutitdoesnothaveCSI.

Inbothcases,channelsareassumedtobefrequencynon-selective (i.e.,ﬂatfading)toeliminateinter-symbolinterference.

To enhance system performance, the transmitter performs channel switching among K channels over time in perfect syn-chronization withthereceiver, that is,time sharingis performed amongdifferentchannels byusing onlyone channel in a certain fraction of time [3,11].2 Fraction of time when transmission is performed over channel i is denoted by

λ

i, which is called the channelswitchingfactor for channel i. The channel switching fac-torssatisfy

_iK₌₁

λ

i

≤

1 and

λ

i

≥

0,

∀

i

∈ {

1

. . .

K

}

.Thus,unlikethe

previous studies such as[3,10,11], it ispossible to haveidle pe-riodsof communicationswhere symbol transmission/reception is notperformed(inthecaseof

_iK₌₁

λ

i

<

1),whichcanprovide

per-formanceimprovementsincertain conditionsascompared tofull utilizationofchannels(seeProposition 1andSection5).

Remark1.Fortheimplementationofchannelswitching,the trans-mitterand the receiver are assumed to be synchronized so that thereceiverknowswhichchanneliscurrentlyinuseorifitisthe idleperiod.Then,thereceiveremploysadecisionruleforthe cor-respondingchannelordoesnotperformanydecisionfortheidle period.Inpractice,thisassumptioncanbe realizedby employing acommunicationsprotocolthatallocatestheﬁrst Ns,1 symbolsin thepayloadforchannel 1,thenextNs,2 symbolsinthepayloadfor channel 2, . . . ,andthelast Ns,K+1symbolsfortheidleperiod.The informationon thenumberofsymbolsfordifferentchannelsand fortheidleperiodcanbeincludedintheheaderofa communica-tionspacket.

Generic M-ary modulation with an arbitrary one-dimensional or two-dimensional signal constellation3 is considered for com-munication over each channel. The complex received signal cor-respondingtochanneli canbeexpressedas

y

=

P

i

α

i

s

(ij)

+

n

i

(1)

for j

∈ {

0

,

1

,

. . . ,

M

−

1

}

and i

∈ {

1

,

. . . ,

K

}

, where s(_i0)

,

s(_i1)

,

. . . ,

s(_iM−1) denotethesetof(complex)transmittedsignals(withunit averageenergy)employedforM-arycommunicationsoverchannel

i, Pi determines the average powerof the transmittedsignal for

channeli,

αi

isthecomplexfading coeﬃcient oftheithchannel, andni is circularly-symmetric complex Gaussian noise for

chan-nel i with mean zero and variance 2

σ

2

i. It is assumed that the

noise componentsare independent across the channelsand they arealsoindependentofthefadingcoeﬃcientsandthetransmitted signals. Inaddition,equallylikely symbolsareconsidered; hence, thepriorprobabilityofeachsymbols(_ij)for j

∈ {

0

,

1

,

. . . ,

M

−

1

}

is equalto1

/

M.Itisassumedthat

αi

’sareperfectlyestimatedatthe receiver.Thesignal-to-noiseratio(SNR)persymbolisdeﬁnedas

γ

i

=

P

i

|

α

i

|

2

2 σ

2

i

· (2)

Foroptimum coherentdemodulation, a generic expression for the probability of symbol error corresponding to the SNR in (2)

2

ItisassumedthatthetransmitterandthereceiverhavesingleRFunitssothat theycanuseonlyonechannelatagiventime

[3,11]

.

3 _{One-dimensional}_and _{two-dimensional} _signal _{constellations} _(e.g., _PAM, _PSK, QAM)areemployedinalmostallpracticaldigitalcommunicationssystems.

overGaussianchannelscanbestatedexactlyorapproximately (de-pendingonthemodulationtypeandorder)as[19]

P

s

(

γ

i

)

=

η

Q

κ

√

γ

i

(3)

where Q

(

·)

denotes the Q -function,

γi

isasin(2),and

η

and

κ

areconstant parametersthatdependon themodulationtype and order. It should be notedthat the expression in (3) is exact for severaltypesofmodulationssuchasBPSK,BFSKandM-PAM,and itholdsapproximatelyforothertypesofmodulationathighSNRs

[19].

In Case 2, the fading coeﬃcient

αi

for channel i is modeled (overthefadingblocks)asazero-mean,circularly-symmetric com-plex Gaussian random variable with variance

ς

2

i, which

corre-spondstoRayleighfading.Then,

γi

in(2)becomesanexponential randomvariable,andthe averageprobability ofsymbolerrorcan beobtainedbycalculatingtheexpectedvalueof(3)overthat ex-ponentialdistribution,whichyields[19]

g

i

(

P

)

= ˜

η

1 −

˜

κ

P

˜

κ

P

+

σ

2 i

/

ς

i2

(4)

where gi

(

P

)

represents the average probability of symbol error

over channel i fora power levelof P ,

η

˜

η

/

2 and

κ

˜

κ

2

_/

_2._It isnotedthat gi

(

P

)

isaconvexandmonotonedecreasingfunction

of P forP

≥

0.

InCase 1,thefadingcoeﬃcientsofthechannelsareﬁxed dur-ing the channel switchingoperation and they are known by the transmitterandthereceiver.Sincetheprobabilityofsymbolerror dependson

αi

and

σi

only through the

|

α

i

|

2

/

σ

i2 term(see (2)),

|

α

i

|

=

√

2 canbeemployedfori

=

1

,

. . . ,

K withoutlossof gener-ality,andthedifferencesbetweenthechannel coeﬃcientscanbe reﬂectedto the

σ

2

i termsaccordingly.Then,basedon(2),(3)can

beexpressedforCase 1as

g

i

(

P

)

=

η

Q

κ

P

σ

2 i

(5)

whichholds exactlyforBPSK,BFSK andM-PAMmodulations and approximately for other types of modulation at high SNRs [19]. For rectangular M-QAM andQPSK constellations, the exact error probabilityofsymbolerrorforCase 1canbestatedas

g

i

(

P

)

=

1 −

η

Q

κ

P

σ

2 i

2

(6)

where

η

and

κ

aredeterminedbythemodulationtype.

Theanalysisinthisstudyisgenerictoacertainextentsinceit employs (6)forQPSK andM-QAMmodulations inCase 1, (5)for other typesofmodulation inCase 1, and(4)inCase 2. Although

[10] considers (5) in Case 1 for scenarios with full channel uti-lization,thereexistnostudiesintheliteraturethatinvestigatethe channel switchingproblembased on(6) inCase 1 (i.e.,forQPSK and M-QAMmodulations with slow fading)and based on (4) in Case 2(i.e.,forRayleighfading).Inaddition,thescenariowith par-tial channel utilizationis proposed forchannel switchingfor the ﬁrsttimeinthisstudy.

IntheconsideredsystemmodelinFig. 1,thereexists K

chan-nels fortransmission,andeachchannel hasa costvalue,denoted by Ci for i

∈ {

1

,

. . . ,

K

}

, which represents the cost of utilizinga

channelperunittime[10,18,20].Costvaluesarenonnegative,and therelationbetweencostsofdifferentchannelsisgivenbyCi

>

Cj

if

σ

2 i

/

ς

2 i

<

σ

2 j

/

ς

2 j inCase 2andif

σ

2 i

<

σ

2 j inCase 1,

∀

j

=

i.This

ismotivatedbythefactthatachannelwithahigher

ς

2

i

/

σ

i2value,

ora lower

σ

2

i value (equivalently,higher SNR)yieldsa lower

(4)

Fig. 2. Acommunicationintervalof

T seconds

duringwhichchannelswitchingisperformedbetweenthetransmitterandthereceiver.Communicationoccursduringthe transmissionperiodandstopsduringtheidleperiod.λi showsthepercentageoftimechannel i isemployedfortransmissionwith

i

∈ {1,. . . ,K},andλK+1 denotesthe percentageoftheidleperiodintheinterval.Thesymbolrateofthecommunicationlinkisassumedtobe

R in

symbolspersecond.

requiressuch a channel tohave a highercost [20,21]. In the re-mainder ofthemanuscript,

β

i isemployedto denotethechannel parameter oftheithchannelforbothCase 1andCase 2,whichis deﬁnedas

β

i

σ

2 i

,

Case 1

σ

_i2

/

ς

_i2

,

Case 2

(7)

Thechannelparameterssatisfy

β

i

< β

jforCi

>

Cj.

Inthisstudy,severalassumptions/propertiesare stated regard-ingtheprobabilityoferrorfunction gi

(

·)

inordertoderivegeneric

theoreticalresultswhicharevalidforvarioustypesofmodulations. The following assumptions state the convexity andmonotonicity propertiesoftheerrorfunction.

Assumption1. gi

(

P

)

isaconvexfunctionofP for P

>

0.

Assumption2. gi

(

P

)

is a monotone decreasingfunction of P

/β

i,

that is, gi

(

P

)

= ˜

g

(

P

/β

i

)

where g is

˜

amonotone decreasing

func-tion.

Theconvexityassumptionissatisﬁedforallthreetypesoferror functionsin(4)–(6).Similarly, Assumption 2is validforall types oferrorprobabilityfunctions, whichareactuallyfunctionsofSNR ratherthanpower.

Theaim istoperform jointoptimizationofchannelswitching factors andsignal powers in order tomaximize theaverage (ex-pected) number of correctly received symbols per unit of time between a transmitter and a receiver under average power and costconstraints.Itisassumedthatthetransmitterknowsthe fad-ingcoeﬃcientsofthechannels,

αi

’s,inCase 1.Ontheotherhand, the transmitter has the knowledge of the

ς

2

i

/

σ

i2 term for each

channelbutdoesnotknowthefadingcoeﬃcientforeachsymbol inCase 2.Acommunicationintervalforthechannelswitching op-eration is shownin Fig. 2, where T denotes theduration ofthe interval in seconds and R denotes the symbol rateover a given communicationlinkinsymbolspersecond,whichisthesamefor allthe K channels.AccordingtoFig. 2,forthecommunication in-terval of T seconds, the average (expected) number of correctly receivedsymbolsovertheithchannelcanbeexpressedas

N

c,i

= λ

i

T R P

c,i

(8)

where

λ

i is the channel switching factor for channel i and Pc,i

istheaverage probabilityof correctdecisionover channel i fora powerlevelofPi,whichcanbecalculatedas

P

c,i

=

1 −

g

i

(

P

i

)

(9)

with gi

(

Pi

)

denoting the average probability of symbol error as

computedin(4)–(6)fordifferenttypesofmodulationsandcases. Theexpressionin(8)correspondstotheaveragenumberof sym-bolsthatarecorrectlyreceivedduringthecommunicationinterval of length

λ

iT . Extending (8) to all the channels yields the

aver-agenumberofcorrectlyreceivedsymbolsduring aninterval ofT

secondsoverallchannels:

N

c

=

T R

K

i=1

λ

i

P

c,i

.

(10)

Foracommunicationintervalof T seconds,theobjectivefunction tomaximizeisgivenbytheexpressionin(10).SinceT and R can

be assumedto be constant design/system parameters, the maxi-mizationof(10)isequivalenttomaximizing

¯

P

c

=

K

i=1

λ

i

P

c,i

.

(11)

Deﬁninghi

(

P

)

1

−

gi

(

P

)

asthecorrectdecisionprobabilityover

channel i fora powerlevelof P ,(11)canbe expressedbasedon

(9)as

¯

P

c

=

K

i₌1

λ

i

h

i

(

P

i

) .

(12)

Remark2.Theweighted sumofthecorrectdecisionprobabilities in(12)representstheaverageprobabilityofcorrectdecisionifthe sumofthechannelswitchingfactorsequalsto1;otherwise,it cor-respondstothe“normalized”averagenumberofcorrectlyreceived symbols,normalizedbyT R,whichisthenumberofsymbols trans-mittedduringa communicationinterval,assumingaﬁxedsymbol duration. In the restof themanuscript, theobjective function in

(12)willbereferredtoasthe“averageprobabilityofcorrect deci-sion”,regardlessofwhetherthechannelswitchingfactorssumto 1 ornot.

The reasoning behind thechoice of theaverage probability of correctdecisioninstead oftheaverageprobability oferrorasthe optimizationcriterioncanbeexplainedasfollows:Whenthe prob-ability of error metric,i.e.,

K_i₌₁

λ

igi

(

Pi

)

, is employed in partial

utilizationofchannels,thetransmittermaychoosenottosendany symbols, that is,

λ

i

=

0,

∀

i (which isnot possiblein full

utiliza-tionofchannels),thusleadingtozeroaverageprobabilityoferror, which isthe minimumthat canbe achieved. On theother hand, fortheprobabilityofcorrectdecisionmetricinpartialutilization, ifnosymbol transmissionoccursduringa certain periodoftime, the average probability of correct decision,

_iK₌₁

λ

ihi

(

Pi

)

, turns

outtobezeroduringthatperiod,whichisundesirable.Hence,the average probabilityofcorrectdecision, asopposedto theaverage probabilityoferror,astheoptimizationcriterion,forcesthe trans-mittertoexploitthecommunicationchannelsaseﬃcientlyas pos-sible.(Inthiscontext,thetermgoodput canbeusedtoreplacethe average probability of correctdecision (with appropriate scaling) whenitreferstotheratioofthetotalnumberofcorrectlyreceived symbolsto thetotal transmissiontime ata systemlevel without takingintoaccountencoding/decoding,thepacket-by-packet trans-missionschemeandthelayeredconceptofnetworking.)

Itisnotedthatforanytwochannels,theonewithahighercost always results in a higher probability of correct decisionfor the samepowerlevel;thatis,ifCi

>

Cj (whichimplies

β

i

< β

j),then hi

(

P

)

>

hj

(

P

)

forall P

>

0 (cf.(4)–(7)). Severalconstraints must

be imposed while maximizing the average probability of correct decisioninorderforthechannelswitchingstrategiestobe appli-cablein practicalsettings. Namely,there exists anaverage power constraint,whichcanbestatedas

K_i₌₁

λ

iPi

≤

Ap,where Ap rep-resentstheaveragepowerlimit.Also,anaveragetransmissioncost

(5)

constraintcanbeexpressedas

K_i₌₁

λ

iCi

≤

Ac,where Ac denotes theaveragecostlimit[10].Then,thefollowingoptimization prob-lemisproposed: max {λi,Pi}iK=1 K

i=1

λ

ihi

(

Pi

)

subject to K

i=1

λ

iPi

≤

Ap

,

K

i=1

λ

iCi

≤

Ac

,

(13) K

i=1

λ

i

≤

1

,

λ

i

≥

0

,

∀

i

∈ {

1

. . .

K

} .

Theoptimizationproblemin(13)searchesoverbothfull transmis-sion strategies(i.e.,

_iK₌₁

λ

i

=

1) andpartialtransmission strategies

(i.e.,

K_i₌₁

λ

i

<

1) in order to achieve the maximum probability

ofcorrectsymbol decision overavailable channels underaverage power and cost constraints. As investigated in the remainder of thestudy,thepartialtransmissionstrategymayyieldhigher aver-ageprobabilities ofcorrectdecisionin certainscenarios than the fulltransmissionstrategyandcanbethesolutionofthe optimiza-tion problem in (13). In such scenarios, no transmission during a certain period of time facilitates a more eﬃcient usage of the costbudget.Hence,bygeneralizingtheconceptofchannel switch-ing to scenarios with possible idle periods, the average number ofcorrectly received symbolscan be improved ina communica-tionsystemthat issubjecttoaveragecost andpowerconstraints. Infact,thisimprovementcanbeachievedwithoutanysigniﬁcant complexityincreasecomparedtothechannelswitchingsystemsin theliterature[10].Inaddition,theaveragenumberofcorrectly re-ceived symbolscan be considered as an important parameter in practicalsystems.

Forthetheoreticalanalyses,itisassumedwithoutlossof gen-erality that the channel parameter

β

i is distinct for each

chan-nel.This isbased on the fact that ifthere are multiplechannels withthesamechannelparameter,channelswitchingbetweensuch channelscanneverincreasetheaverageprobabilityofcorrect deci-sioncomparedtoemployingonlyoneofthematthesameaverage powerforthe totaldurationoftime,whichisduetothe concav-ityof the correct decision probability expressions, hi

(

·)

. For this

reason,theproblemformulationthatconsiders onlythechannels withdistinctchannelparametersissuﬃcienttoobtaintheoverall optimalsolution.

3. Optimalchannelswitching–generalanalysis

Inthissection,theoptimalchannelswitchingproblemin(13)is examinedindetail.Inparticular,theproblemin(13)isreducedto asimplerequivalent formandtheoptimalstrategiesareobtained basedonlow-complexitycalculations.Theassumptionmadeabout theorderingofchannelcostswithoutlossofgeneralityisthatthe costvaluessatisfyC1

>

C2

>

· · · >

CK,thusthechannelparameters

areorderedas

β

1

< β

2

<

· · · < β

K.Inthiscase, theprobabilityof

correctdecisionfunctionssatisfy h1

(

P

)

>

h2

(

P

)

>

· · · >

hK

(

P

)

for

all P

≥

0.

Based on the ordering ofthe channel costs, it is clearthat if

Ac

≥

C1,theoptimalsolutionof(13)istotransmitoverchannel 1 exclusivelywithpower Ap.Inotherwords,sincetransmissionover channel 1 results in the highest probability of correct decision amongallthechannels,theoptimalapproachbecomes theuseof thebestchannel(channel 1)allthetime atthemaximumpower limitwhenthecostbudgetallowsit.

Since(13)caneasilybesolvedforAc

≥

C1,thecaseofAc

<

C1 isconsidered in theremainder of thestudy. Itis straightforward toshow that thesolution ofthe problemin(13)always satisﬁes

theaveragepowerconstraintwithequalitysincehi

(

P

)

isa

mono-toneincreasingfunctionof P foralli

∈ {

1

,

. . . ,

K

}

.Mathematically speaking, if

{λ

∗_i

,

P_i∗

}

_iK₌₁ denotes the solution of the optimization problemin (13), then

K_i₌₁

λ

_i∗P_i∗

=

Ap.Furthermore, based ona similar approachtothe proofof Proposition 1in[10] withslight modiﬁcationstoconsiderthepartialtransmissionstrategies,itcan beinferredthattheoptimalchannelswitchingsolutionoperatesat theaveragecostlimit,thatis,

K_i₌₁

λ

∗_iC∗_i

=

Ac.Hence,anoptimal channel switching strategy must utilize all the available average power and average cost for Ac

<

C1. Therefore, the optimization problemin(13)can besolvedby consideringequalityconstraints (insteadofinequalityconstraints)fortheaveragepowerand aver-agecost,whichprovidesanimportantreductionincomputational complexity.

The followingremarkispresented toreveal thereasoning be-hindtheuseofpartialtransmission.

Remark3.Partialdatatransmissioncouldnotbeanoptimal strat-egy ifthe average cost constraintdid not exist in the optimiza-tionproblemin(13);thatis,theoptimalsolutionof(13)satisﬁes

K

i=1

λ

i

=

1 intheabsenceofthecostconstraint.

Proof. Theproofcanbeobtainedbycontradiction.Let

{λ

∗_i

,

P_i∗

}

K i=1 denotethesolution oftheoptimizationproblemin(13).Suppose that the average cost constraint does not exist and the optimal solution satisﬁes

_iK₌₁

λ

∗_i

<

1. Deﬁne the idle period as

λ

∗_K₊₁

1

−

_iK₌₁

λ

∗_i.Then,followingrelationscanbeestablished:

K

i=1

λ

∗_ihi

(

P∗i

) <

K

i=1

λ

∗_ih1

(

Pi∗

)

(14)

<

K

i=1

λ

∗_ih1

(

Pi∗

)

+ λ

∗K+1h1

(0)

(15)

<

h1

_K

i=1

λ

∗_iP∗_i

+ λ

∗_K₊₁0

(16)

=

h1

_K

i=1

λ

∗_iP∗_i

(17) wheretheﬁrstinequalityfollowsfromthefactthath1

(

P

)

>

hi

(

P

)

,

∀

P (since C1

>

C2

>

· · · >

CK), the second inequality uses the

factsthat

λ

∗_K₊₁

>

0 (duetopartialtransmission)andh1

(

0

)

=

1

/

M with M denoting the modulation order, and the third inequal-ity is obtained from the strict concavity of h1. The inequality in (14)–(17), namely,

_iK₌₁

λ

∗_ihi

(

Pi∗

)

<

h1

iK=1

λ

∗iP∗i

, indicates that the optimal solution yields a lower average probability of correctdecisionthanthesolutionwhichutilizeschannel 1 exclu-sively(i.e.withchannelswitchingfactor1)withthesameaverage power

K_i₌₁

λ

∗_iP_i∗, butoperates atan average cost C1 that satis-ﬁesC1

>

Ki=1

λ

∗i

C1

>

Ki=1

λ

∗iCi sinceC1

>

Ci fori

∈ {

2

. . .

K

}

.

Hence,intheabsenceofthecostconstraint,therealwaysexistsa fulltransmissionstrategythatachievesahigheraverage probabil-ityofcorrectdecisionthan apartial transmissionstrategy,which implies that

_iK₌₁

λ

∗_i

=

1 must hold, leading to a contradiction. Therefore, partial transmission cannot be optimal in theabsence oftheaveragecostconstraint.

2

Remark 3pointstoanimportantfactabouttheconditions un-der whichpartial transmissionstrategy canbe appliedinstead of full utilizationofchannels. Remark 3 statesthatpartial transmis-sioncan be reasonable onlyifthebudget islimited. The optimal

(6)

strategyforanunlimitedbudgetistoutilizeexclusivelythe chan-nelwiththehighestcost (i.e.,the highestcorrectdecision proba-bility)atthemaximumaveragepower.Thus,partialdata transmis-sionmaybeoptimalwhenthebudgetshouldbeusedeﬃcientlyto maximizetheprobabilityofcorrectdecision(e.g.,insteadofusing alow-costchannelexclusively,itmaybebettertouseahigh-cost channelpartially).

Inthefollowingproposition,itisstatedthattheoptimal chan-nel switchingstrategy, which isobtained asthe solutionof (13), correspondstochannelswitchingeitheramongatmostmin

{

K

,

3

}

channels with full transmission or between at most min

{

K

,

2

}

channelswithpartialtransmission.

Proposition1.AssumethatthepowerlevelssatisfyPi

∈ [

0

,

Pmax]for some ﬁnite Pmax.Then, theoptimalchannelswitchingstrategy isto

switcheitheramongatmostmin

{

K

,

3

}

channelswithfulltransmission, orbetweenatmostmin

{

K

,

2

}

channelswithpartialtransmission.

Proof. The proof is based on Carathéodory’s theorem [22], and

similar arguments to those in [10,23] can be employed. Assume that K

≥

3 since the statement in the proposition alreadyholds otherwise.First,sets

U

and

W

aredeﬁnedas

U

=

(

P

,

hi

(

P

),

Ci

) ,

∀

i

∈ {

1, . . . ,K

} ,

∀

P

∈ [

0,Pmax

]

∪ {(

0,0,0)

}

(18)

W

=

_K

i=1

λ

iPi

,

K

i=1

λ

ihi

(

Pi

) ,

K

i=1

λ

iCi

,

∀λi

≥

0

,

K

i=1

λ

i

≤

1

,

∀

Pi

∈ [

0,Pmax

]

.

(19)

From(19),itisobservedthatthesolutionof(13)mustbean ele-mentof

W

.Also, it can be concludedthat

W

isa subset ofthe convex hull of

U

according to the deﬁnitions in (18) and (19). Inaddition, duetothe aimofmaximization in(13),the solution canbeshowntocorrespondtoan elementof

W

that liesonthe boundaryofthe convexhullof

U

basedonsimilar argumentsto thosein[10,23].Then,Carathéodory’stheoremstatesthat any el-ementofontheboundary ofthe convexhullof

U

(includingthe solution of(13)) can be expressed asthe convexcombination of atmostdim

(

U)

=

3 elementsin

U

[22].Therefore,thesolutionof

(13)correspondstochannelswitchingeither(i) amongatmost3 channelswith full transmission if

(

0

,

0

,

0

)

isnot one ofthe ele-mentsof

U

employedintheconvexcombinationforthesolution, or(ii) betweenatmost2 channelswithpartialtransmission

oth-erwise.

2

BasedonProposition 1,anoptimalchannelswitchingsolution corresponds to one of the following strategies: Partial/full trans-mission over a single channel, partial/full transmission over two channels,andfull transmissionoverthreechannels.Thefollowing sectionsexplorethedetailsofthosestrategies.

3.1. Singlechannelstrategies

The optimal solutions forfull andpartial transmission over a singlechannelareinvestigatedinthissection.

Strategy1P–partialtransmissionoverasinglechannel: Inthis case, one ofthe channels is employed partially; that is,a single channelisusedduringthebusyperiod,andan idleperiodexists, aswell.Apartialtransmissionstrategythatemploysasingle chan-nelwithacostsmallerthanAccannotbeoptimal(i.e.,thesolution of(13))sincetheoptimalsolutionmustoperateattheaveragecost limit Ac,asdiscussedabove(thethirdparagraphofSection3).

In some cases, the optimal channel switching strategy corre-sponds to Strategy 1P. In those scenarios, the optimal solution mustbe searchedamongthechannelswithcostshigherthan Ac. Let

S

g

{

l

∈ {

1

,

. . . ,

K

}

: Cl

>

Ac}. Assume that channel i

∈

S

g is

employed withchannel switching factor

λ

i and power Pi. Then,

λ

iPi

=

Ap and

λ

iCi

=

Ac.Therefore,theoptimalsolutionfor chan-nel i is obtained as

λ

∗_i

=

Ac

/

Ci and P∗i

=

ApCi

/

Ac. Hence, the averageprobabilityofcorrectdecisionisgivenby

λ

∗_ihi

(

P∗i

)

=

Ac Ci hi

Ap Ci Ac

(20) andthechannelthatyieldstheoptimalsolutionunderStrategy 1P isobtainedas i∗

=

arg max i∈Sg Ac Ci hi

Ap Ci Ac

.

(21)

Strategy1F–fulltransmissionoverasinglechannel: In this case, one of thechannels isemployed all thetime. This strategy may be the optimal channel switching strategy if there exists a channel withcost Ac since otherwisethe averagecost cannot be equalto Ac.

3.2. Two-channelstrategies

There exist two strategiesfor channel switchingbetweentwo channels: Partial transmission over two channels and full trans-missionovertwochannels.

Strategy2P–channelswitchingbetweentwochannelswith

partial transmission: In this strategy, channel switching is per-formed betweentwo differentchannels and the sum ofchannel switchingfactorsissmallerthan1,i.e.,thereexistsan idleperiod withnodatatransmission.Letchanneli andchannel j denotethe channelsemployedinthisstrategy.Then,theproblemin(13)can beformulatedunderStrategy 2Pas

max

λi, λj,Pi,Pj

λ

i

h

i

(

P

i

)

+ λ

j

h

j

(

P

j

)

subject to

λ

i

P

i

+ λ

j

P

j

=

A

p

,

λ

i

C

i

+ λ

j

C

j

=

A

c

,

λ

i

+ λ

j

<

1 , λ

i

, λ

j

∈ [

0 ,

1 ) .

(22)

It is notedthat Strategy 1P is covered asa specialcase of Strat-egy 2P.Itisobservedfromtheaveragecostconstraintin(22)that, fortheoptimalchannel switchingbetweentwo channels,atleast oneofthechannelsshouldhaveacostgreaterthan Ac.Therefore, in ordertoobtain theoptimalsolution forStrategy 2P,the prob-lem in(22)should besolved for Kg

(

K

−

1

)

channel pairs,where

Kg isthenumberofchannelsthecosts ofwhichare greaterthan

Ac andK isthetotalnumberofchannels.

Based on the argument in the previous paragraph, assume, without loss ofgenerality, that Ci

>

Cj for the problemin (22).

From theaverage powerandcostconstraintsin(22),theoptimal value of

λ

j and Pj canbeexpressedintermsoftheoptimal

val-uesof

λ

i and Pi as

λ

j

= (

Ac

− λ

iCi

)/

Cj and Pj

= (

Ap

− λ

iPi

)/λ

j.

Therefore,theoptimizationproblemin(22)canbesimpliﬁed sig-niﬁcantlybyoptimizingovertwovariablesinsteadoffourvariables basedonthetwoequalityconstraints.Theoptimizationproblemin

(22)canthenbeexpressedasfollows: max λi∈[0,Ac/Ci) Pi∈[0,Ap/λi]

λ

ihi

(

Pi

)

+ λ

jhj

Ap

− λi

Pi

λ

j

(23)

where

λ

j

= (

Ac

− λ

iCi

)/

Cj andthe constraintsfor

λ

i and Pi are

obtainedfrom therelations

λ

iCi

+ λ

jCj

=

Ac and

λ

iPi

+ λ

jPj

=

Ap.From (23), it isobserved that the optimalsolution for Strat-egy 2P requires a search over a two-dimensional spaceonly (for

(7)

eachpossiblechannelpair).Thistwo-dimensionalsearch mustbe executedby ﬁrstdetermining avalue for

λ

i andthen ﬁndingthe

optimal Pi for the current

λ

i value since the search interval for Pi dependsonthevalueof

λ

i.Finally,themaximumforallthose

(λ

i

,

Pi

)

pairs iscalculatedandthe pairthat yields themaximum

valueoftheobjectivefunctionisdeterminedtobeoptimal.

Strategy2F–channelswitchingbetweentwochannelswith

fulltransmission: Inthisstrategy,channelswitchingisperformed betweentwodifferentchannelsandthesumofchannelswitching factorsisequalto1.Theformulationoftheproblemin(13)under Strategy 2Fisasfollows:

max

λi, λj,Pi,Pj

λ

i

h

i

(

P

i

)

+ λ

j

h

j

(

P

j

)

subject to

λ

i

P

i

+ λ

j

P

j

=

A

p

,

λ

i

C

i

+ λ

j

C

j

=

A

c

,

λ

i

+ λ

j

=

1 , λ

i

, λ

j

∈ [

0 ,

1 ] .

(24)

In this case, the optimization can be performed over a single variablesince thesum ofchannel switchingfactors forms anew equality.Strategy 2F reducestoStrategy 1F ifone ofthe channel switchingfactorsisequalto1.

Thefollowingassumptionsaremadeabouttheerrorfunctions

gi

(

·)

toprovideabasisforthenextproposition.

Deﬁnition1.AssumeCi

>

Cjandlet Pi jbedeﬁnedasthesolution

toequation g_i

(

x

)

−

g_j

(

x

)

=

0,thatis,g_i

(

Pi j

)

=

gj

(

Pi j

)

.

Assumption3.

(i) g_i

(

P

)

>

g_j

(

P

)

ifP

>

Pi j. (ii) g_i

(

Pi j

)

=

g_j

(

Pi j

)

.

(iii) g_i

(

P

)

<

g_j

(

P

)

if0

<

P

<

Pi j. (iv) g_i

(

0

)

= −∞ ∀

i

∈ {

1

,

. . . ,

K

}

.

Assumption4.Ifg_i

(

Pi

)

=

gj

(

Pj

)

issatisﬁed,then (i) Pj Pi

<

βj βi for Pj

>

Pi

>

Pi j (ii) Pi Pj

<

βj βi for Pj

<

Pi

<

Pi j

Assumption 3isvalidforall typesofmodulationswhoseerror probability expressions are givenby (4)–(6). On the other hand,

Assumption 4issatisﬁedfortheerrorfunctionsin(4)and(5),but notsatisﬁedfor(6).

Thefollowingpropositionderivesupperandlower boundsfor theoptimalpowerlevelsobtainedforStrategy 2PandStrategy 2F andtheirratios.

Proposition2.SupposethatAssumptions 1,3,and4hold.Letthe solu-tionoftheoptimizationproblemin(13)underthetwo-channel strate-giesbedenotedby

{λ

∗_i

,

P_i∗

,

λ

∗_j

,

P∗_j

}

andsupposethat

λ

∗_i

>

0,

λ

∗_j

>

0,

andCi

>

Cj.Then,theoptimalpowerlevelsandthechannelswitching factorssatisfythefollowingrelationsdependingontheaveragepower limit:

(i) IfAp

=

Pi j

(λ

∗_i

+ λ

∗_j

)

,thenP∗_i

=

P∗_j

=

Pi j. (ii) IfAp

>

Pi j

(λ

∗i

+ λ

∗j

)

,thenP∗j

>

P∗i

>

Pi j. (iii) IfAp

<

Pi j

(λ

∗_i

+ λ

∗_j

)

,thenP∗_j

<

P∗_i

<

Pi j.

wherePi jisasinDeﬁnition 1.Inaddition,theratiobetweentheoptimal powerlevelscannotexceed

β

j

/β

i;thatis,

max

P

∗_j

P

∗_i

,

P

_i∗

P

∗_j

<

β

j

β

i

.

(25)

Proof. Theoptimizationproblems in(22)and(24)can besolved together by re-writing the constrainton the sum ofthe channel switching factors as

λ

i

+ λ

j

≤

1. In this case, the optimization

problem againbecomes the one in (23).Consider the ﬁrst-order derivativeoftheobjectivefunctionin(23)withrespecttoPi:

λ

i

h

_i

(

P

i

)

−

h

j

A

p

− λ

i

P

i

λ

j

(26)

or,equivalently,

λ

i

(

h

i

(

P

i

)

−

h

j

(

P

j

)) .

(27)

Due to Assumption 1, g_i

(

P

)

is an increasing function of P and h_i

(

P

)

= −

g_i

(

P

)

isadecreasingfunctionofP for P

>

0.Hence,the expressionin(26)isadecreasingfunctionof Pi for Pi

>

0,

start-ingfrom

∞

at Pi

=

0 anddecreasingmonotonicallytowards

−∞

atPi

=

Ap

/λ

i(duetoAssumption 3).Therefore,giventhevalueof

λ

i, thereisa unique maximizer P∗i for theoptimizationproblem

in (23), which corresponds to the point at which the ﬁrst-order derivativeiszero.Equatingtheﬁrst-orderderivativein(27)tozero andsettingh_i

(

P

)

= −

g_i

(

P

)

yieldsthefollowingnecessaryand suf-ﬁcientconditionfortheoptimalsolutionof(23):

g

_i

(

P

i

)

=

g

_j

(

P

j

) .

(28)

If Ap

=

Pi j

(λ

i

+ λ

j

)

, it is obvious, by using the deﬁnitionof Pi j

inDeﬁnition 1 andAssumption 3,that Pi

=

Pj

=

Pi j satisﬁes the

conditionin(28).Sincethesolutionof(28)isunique,theoptimal solutionof(23)isobtainedas P∗_i

=

P∗_j

=

Pi j,asstatedintheﬁrst

partofProposition 2.

Inorderto provethesecond partoftheproposition,it isﬁrst observedthat theﬁrst-orderderivative in(26)is amonotone de-creasingfunctionof Ap andamonotoneincreasingfunctionofPi.

Therefore, thevalue of Pi atwhich the ﬁrst-orderderivative

be-comeszerogetslargerasApincreases.Sincetheﬁrst-order deriva-tive becomes zeroat Pi

=

Pi j when Ap

=

Pi j

(λ

i

+ λ

j

)

(asproved

intheﬁrstpart),theﬁrst-orderderivativebecomeszeroatavalue larger than Pi j when Ap

>

Pi j

(λ

i

+ λ

j

)

.Hence, the optimal

solu-tionof(23)satisﬁes Pi

>

Pi j for Ap

>

Pi j

(λ

i

+ λ

j

)

.In addition,it

isconcluded from(28)that as Pi increases, the optimalvalue of Pj shouldalsoincreasesince gi

(

P

)

isanincreasing functionof P

for P

>

0,asstatedabove. In other words, P_i∗

>

Pi j alsoimplies P∗_j

>

Pi j basedontherelationin(28).Next,theorderingbetween P∗_i and P∗_j should be determined.Due to Assumption 3and the condition(28),g_j

(

P∗_j

)

=

g_i

(

P∗_i

)

>

g_j

(

P_i∗

)

since P∗_i

>

Pi j,which

re-quires P∗_j

>

P_i∗duetothemonotoneincreasingpropertyof g_j

(

P

)

. Therefore,forAp

>

Pi j

(λ

i

+λ

j

)

,theinequality P∗j

>

P∗i

>

Pi jis

ob-tained.Similarly,forAp

<

Pi j

(λ

i

+λ

j

)

,theinequality P∗_j

<

P_i∗

<

Pi j

canbeobtained.

Sincetheoptimalpowerlevelssatisfytheconditionin(28),the ﬁnal statement in the proposition can be reached by using As-sumption 4andthe resultsobtainedinthe previous partsofthe proposition related to the ordering of the optimal power levels. For Ap

=

Pi j

(λ

i

+ λ

j

)

, Pi∗

/

P∗j

=

1 asstatedintheﬁrstpartofthe

proposition. Overall, the ratio between the optimal power levels is upperbounded by

β

j

/β

i forany value of Ap,as statedin the proposition.

2

ThesearchspaceoftheoptimizationforStrategy 2Pcanbe re-ducedbasedonProposition 2asfollows.Foreach

λ

i

∈

0

,

Ac

/

Ci

,

λ

jiscalculatedas

λ

j

= (

Ac

− λ

iCi

)/

Cjandtheoptimalpower

(8)

•

IfAp

=

Ai j

(λ

i

+ λ

j

)

,theoptimalsolutionisgivenby

P

∗_i

=

P

∗_j

=

P

i j

.

(29)

•

IfAp

>

Ai j

(λ

i

+ λ

j

)

,theoptimizationproblemin(23)issolved

for

P

i

∈

max

P

i j

, β

i

A

p

/β

j

,

min

A

p

/λ

i

, β

j

A

p

/(β

i

λ

j

)

}

(30)

which is obtained from (25) and the relation in the second partoftheproposition.

•

IfAp

<

Ai j

(λ

i

+ λ

j

)

,theproblemin(23)issolvedfor

P

i

∈

A

p

,

min

P

i j

,

A

p

/λ

i

, β

j

A

p

/(β

i

λ

j

)

(31)

whichisobtainedfrom(25)andtherelationinthethirdpart oftheproposition.

Applyingtheprocedureabove,theoptimalvalueof Pi andthe

correspondingvalueoftheobjectivefunctionin(23)canbe deter-minedforagiven

λ

i

∈

0

,

Ac

/

Ci

.Theoptimalvalueof

λ

i,denoted

by

λ

∗_i,isthevaluethatyieldsthemaximumoftheobjective func-tionin(23) andthe corresponding valueof P_i∗ givesthe optimal value of Pi.Oncetheoptimalpair

{λ

∗i

,

P∗i

}

is obtained,the

opti-malvaluesof

λ

jandPj arecalculatedas

λ

∗_j

= (

Ac

− λ

∗_iCi

)/

Cjand P∗_j

=

Ap

− λ

∗_iP∗_i

/λ

∗_j,respectively.

3.3. Three-channelstrategies

BasedonProposition 1,thereexistsonlyonestrategyfor chan-nelswitchingamongthreechannels.

Strategy3–channelswitchingamongthreechannels: Inthis strategy,transmissionisperformedbyswitchingamongthree dif-ferentchannelsandthechannelswitchingfactorsaddupto1 (i.e., fulltransmission).Theformulationoftheoptimizationproblemin

(13)underStrategy 3isgivenasfollows:

max

λi, λj, λk,Pi,Pj,Pk

λ

i

h

i

(

P

i

)

+ λ

j

h

j

(

P

j

)

+ λ

k

h

k

(

P

k

)

subject to

λ

i

P

i

+ λ

j

P

j

+ λ

k

P

k

=

A

p

,

λ

i

C

i

+ λ

j

C

j

+ λ

k

C

k

=

A

c

,

λ

i

+ λ

j

+ λ

k

=

1 , λ

i

, λ

j

, λ

k

∈ [

0 ,

1 ]

(32)

where i, j and k are the employedchannels. Dueto theaverage costconstraintin(32),atleastoneofthechannelsshouldhavea costgreater than Ac andatleastoneofthemshould haveacost smallerthan Ac.Thus, theoptimization problemin (32)mustbe solvedfor KgKs

(

K

−

2

)

channel triples andthetriplethat yields the highestaverage probability of correct decisionis determined to be optimal. Here, Kg and Ks denote, respectively, the num-ber ofchannels withcosts greater than Ac and smallerthan Ac. It is observedfrom (32)that the optimization can be performed overthreevariablesinsteadofsixvariablesbyimposingthethree equalityconstraints.

ItshouldbenotedthatStrategy 1FandStrategy 2Farecovered as special cases of Strategy 3. Therefore, in order to obtain the optimalsolutionincaseoffulldatatransmission,theoptimization problemin(32)canbesolved ﬁrst,whichrevealsthetype ofthe transmissionstrategytobeapplied.

3.4. Comparisonofchannelswitchingstrategies

In thissection, theoretical results are obtained forcomparing theperformance ofthedifferentstrategies.First,thesingle chan-nelstrategies areexamined intermsofthe probability ofcorrect decision to put forward a suboptimal solution when only a sin-glechannel isemployed.Conditionsare investigatedunderwhich

full or partial transmission over a single channel (Strategy 1P or Strategy 1F) is optimal. Strategy 1F can be optimalonly if there exists a channel withcost Ac; otherwise,the cost budget would be usedpartially and thesolution would not be optimal. Hence, thecomparisonofStrategy 1FversusStrategy 1Pascandidatesfor theoveralloptimalsolutioncanbemadeby

hi∗

(

Ap

)

max j∈Sg Ac Cj hj

Ap Cj Ac

(33) where

S

g

{

l

∈ {

1

, . . . ,

K

} :

Cl

>

Ac},i∗istheindexofthechannel

satisfyingCi∗

=

Ac,andtheleft-hand-sideandtheright-hand-side of(33)representtheaverageprobabilitiesofcorrectdecision cor-respondingtoStrategy 1FandStrategy 1P,respectively.Forpartial transmissionoverchannel j with j

∈

/

S

g(i.e.,Cj

<

Ac),theaverage probabilityofcorrectdecisioncanbeexpressedas

λ

jhj

(

Pj

) < λ

jhj

(

Pj

)

+ (

1

− λ

j

)

hj

(0)

(34)

<

hj

λ

jPj

+ (

1

− λ

j

)

0

(35)

≤

hj

(

Ap

)

(36)

<

hi∗

(

Ap

)

(37)

wherethe secondinequality resultsfromtheconcavityofhj,the

third inequality is due to the average power constraint, and the last inequality isobtainedfromCj

<

Ac

=

Ci∗.The inequalitiesin (34)–(37),namely,

λ

jhj

(

Pj

)

<

hi∗

(

Ap

)

,demonstratewhychannels notin

S

gneednotbeincludedin(33).

Thefollowingpropositionpresentsasuﬃcientconditionfor de-ciding between two channels in terms of optimality under the singlechannelstrategies,Strategy 1PorStrategy 1F.

Proposition3.SupposethatAssumptions 1 and2hold.Considera chan-nelpair

(

i

,

j

)

suchthatCi

>

Cj

≥

Ac.Ifthecondition

β

j

β

i

≤

1

−

Cj Ci gi

(0)

+

Cj Ci (38)

is satisﬁed, then partial/full transmission over channel j achieves a higher probabilityof correct decisionthan partial transmission over channeli.

Proof. Assumethattheinequalityin(38)issatisﬁed.First,the in-equalityin(38)isstatedas