THEJOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.X,NO.X,XXX.200X 1 Spectrum Sharing-based Multi-hop Decode-and-Forward Relay Networks under Interference Constraints: Performance Analysis and Relay Position Optimization Vo Nguyen Quoc Bao, Tran Thien Thanh, Tuan Duc Nguyen, Thanh Dinh Vu 3 Abstract: Theexactclosed-formexpressionsforoutageprobability band is detected vacant [8–10]. For the latter case, sec- 1 0 and bit error rate of spectrum sharing-based multi-hop decode- ondaryusers(SUs)consistingofthesourceandrelaysmay 2 and-forward(DF)relaynetworksinnon-identicalRayleighfading take advantage of a PU’s frequency band by opportunis- channelsarederived.Wealsoprovidetheapproximateclosed-form tically transmitting with high power as long as the PU’s n expressionforthesystemergodiccapacity.Utilizingthesetractable a transmission is strictly protected [11,12]. In spite of the J analytical formulas, we can study theimpact of key network pa- burdenonPUs, the SS approachcanimprovespectraleffi- rametersontheperformanceofcognitivemulti-hoprelaynetworks 3 ciency more aggressivelythan the OSA approach. underinterferenceconstraints. Usingalinearnetworkmodel,we ] derive an optimum relay position scheme by numerically solving Recently, the performance analysis for SS-based cogni- T anoptimizationproblemofbalancingaveragesignal-to-noiseratio tive relay networks has gained great attention, (see, e.g. I (SNR) of each hop. Thenumerical resultsshow that theoptimal [13–18]), due to its adaptability to extend the coverage . s scheme leads to SNR performance gains of more than 1 dB. All for wireless networks. This situation may arise in prac- c theanalyticalexpressionsareverifiedbyMonte-Carlosimulations tice when data is sent from a cognitive source to a given [ confirmingtheadvantageofmultihopDFrelayingnetworksincog- cognitive destination on a hop-by-hopbasis via various in- 1 nitiveenvironments. termediatecognitiverelaynodes. Inadditiontotheadvan- v tageofextendingthecoveragewithoutusinglargepowerat 4 Index Terms: Ergodic Capacity, Rayleigh fading channels, Cog- thetransmitter,cognitiverelaynetworksareabletoreduce 8 nitive radio, Underlay relay networks, Amplify-and-Forward, 3 interference causing to PUs. Decode-and-Forward. 0 Inparticular,Guoet. al. in[13]derivedtheupper-bound . 1 for outage probability of underlay selective DF relay net- I. Introduction 0 worksoperatingwithintheconstraintimposedonthepeak 3 Cognitive radio (CR) has drawn considerable attention power receivedat the primary receiver. In [14], only based 1 in the academic and industrial communities in the past on the partial channel state information (CSI) of primary : v few years and has been considered as one of the main fea- user link and partial CSI between the relays and the pri- i ture for future wireless networks [1,2]. As an evolution of maryuserreceiver,Liproposedtworelayselectionschemes X software-defined radio (SDR), CR with ability of learning for cognitive relay networks. The system performance in r a from its surroundings and adapting its transmitting con- termsofoutageprobabilitywasalsoprovidedoverRayleigh figurations allows secondary (unlicensed) users to oppor- fading channels. For amplify-and-forward (AF) relaying, tunistically transmits data in bands licensed to primary [15] and [16] studies the dual hop relaying networks with users [3–5]. As a result, it can alleviate the problem of and without considering combining technique at the sec- spectrum congestionand thus allowing for a more efficient ondary destination, respectively. Equipped multi antenna spectrum utilization [6]. for secondary nodes, the work in [17] investigated the er- Recently, CRhas alsobeenconsideredasthe radioplat- godic capacity for secondary underlay multi-input multi- form for relaying networks. Previous works on cognitive output (MIMO) networks, where transmit antenna selec- networks have assumed two types of cognitive operational tion (TAS) and maximal ratio combining (MRC) are used modesincluding opportunisticspectrumaccess(OSA)and at the transmitter and receiver, respectively. In parallel, spectrum sharing (SS) [6,7]. In the OSA approach, CR DongLi[18]analyzedtheeffectofmaximum-ratiocombin- usersareallowedtotransmitoverthesamefrequencyband ing diversity on the performance of underlay single-input licensed to primary users (PUs) only when the frequency multi-output (SIMO) systems where the transmit power and the interference power constraint are taken into ac- V. N. Q. Bao is with Posts and Telecommunications Insti- counts. All of the theoretical performance analyses of cog- tute of Technology (PTIT), Ho Chi Minh City, Vietnam, email: nitive relaynetworksmentioned abovehave just only been [email protected]. T.T. Thanhand T.D.VuarewithHoChi MinhCityUniversity solvedfortheparticularcaseoftwoconsecutivehopsexcept ofTechnology(HCMUT),VietnamNationalUniversityHoChiMinh for the paper [19]. In [19], the authors studied the perfor- City(VNUHCM). mance of cognitive underlay multihop decode-and-forward T. D. Nguyen is with International University (IU), Vietnam Na- tionalUniversityHoChiMinhCity(VNUHCM). relayingnetworksoverRayleighfadingchannels. However, 1229-2370/03/$10.00 (cid:13)c 2003 KICS 2 THEJOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.X,NO.X,XXX.200X the results in this paper is limited while it only provided to concurrently transmit with PU over the same licensed theclosed-formexpressionofthesystemoutageprobability band while adhering to the interference constraint on the under interference constraints. PU-Rx, i.e., any data transmission of SUs resulting in a Therefore,inthispaper,wedevelopaperformanceanal- higher interference level than an interference temperature ysis framework for SS-based multi-hop CR networks. In at the PU-Rx is prohibited. To represent the maximum particular,wederiveexactclosed-formexpressionsforout- allowable interference power level at the PU-Rx, the in- ageprobability(OP)andbiterrorrate(BER)aswellasthe terference temperature ( ), is used [12–14]. Let h p D,k I approximateclosed-formexpressionforergodiccapacityfor and h be the channel coefficients of the link from the I,k the considered system over independent but non-identical k-th SU-Tx to the next SU-Rx and to the PU-Rx, re- distributed (i.n.d.) Rayleigh fading channels. To gain in- spectively. Under Rayleigh fading, h 2 and h 2 are D,k I,k | | | | sights, the asymptotic approximation for outage probabil- exponential distributed with their corresponding parame- ity,biterrorrateathighSNRregime,isalsoprovided. We ters λ = E h 2 and λ = E h 2 , where E . D,k D,k I,k I,k {| | } {| | } {} have shown that the system diversity order is always one standsfortheexpectationoperation. Perfectchannelstate regardlessthenumberofhopsandthecodinggainincreases information (CSI) of the CR PU’s Rx link is assumed k → accordingtotheincreaseofthenumberofhops. Forapre- at the CR 1. We further assume that the additive white k determined position of a primary receiver, the problem of Gaussiannoise(AWGN)associatedwitheachhopisazero- relay position optimization is also considered and solved mean Gaussian random variable with variance . 0 N by using the numerical approach. Finally, simulation re- III. Performance Analysis sults are provided to validate the analytical performance assessments. Consider the k-th hop of CR multihop networks,the in- The remainder of this paper is organized as follows. In stantaneous signal-to-noiseratio is written as Section 1, the system model for underlay cognitive multi- h 2 hop network is described. In Section III, the system per- γ =P | D,k| . (1) k k formance metrics in terms of outage probability, bit error 0 N rate and ergodic capacity are derived for Rayleigh fading To ensure that the interference power at the PU-Rx is al- channels. In Sect. IV, we are concerned with the problem ways below the interference temperature, p, the transmit I of relay position optimization. Numerical results are given power is upper bounded by in Section V, where the advantage of cognitive underlay P /h 2. (2) k p I,k multhop systems is investigated. Finally, conclusions are ≤I | | drawn in Section VI. Aiming to enhance the system performance, we adopt P = /h 2, (3) II. System Model k p I,k I | | yieFldoringRaγykle=igNhIp0f|a|hhdDIi,,nkk.|g|22.channels, the probability density Data link PU PU function (PDF) of γ = h 2 with Z D,I is of Interference Tx Rx Z,k |γZ,k| ∈ { } link the form fγZ,k(γ) = γ¯Z1,ke−γ¯Z,k, where γ¯Z,k = λZ,k. The PDFofthe receivedSNRathopk, γ ,is derivedas[22,p. k h h h 187, eq. 6-60] I,1 I,2 I,3 ∞x 0 xγ 0 CR1 hD,1 CR2 hD,2 CR3 hD,3 CR4 fγk(γ) = Z0 INp fγD,k(cid:18) INp (cid:19)fγI,k(x)dx α k = , (4) (γ+α )2 k Fig.1. Systemmodelofamultihopspectrum sharingcommunication where α = γ¯D,k / . From (4), the corresponding cu- system. k γ¯I,kIp N0 mulative distribution function (CDF) is given by γ We considera multi-hopSSsystemwith the coexistence F (γ) = f (x)dx γk γk of PUs, i.e., licensed users, and SU, i.e., unlicensed users, Z0 γ as shown in Fig. 1. Assume that PUs and SUs share the = . (5) γ+α same narrow-band frequency with bandwidth B, which is k licensed to PUs. In the primary network, the PU trans- Having the PDF and CDF of each γ in hands, we are k mitter(PU-Tx)transmitsitsdatatowardsthePUreceiver now in a position to derive the performance metrics of the (PU-Rx). Inthesecondarynetwork,theSU-Tx(CR1)indi- system including outage probability, bit error probability, rectly transmits the message towards the SU-Rx (CRK+1) and ergodic capacity. with the help of K-1 cognitive decode-and-forward(regen- 1Itcanberealizedbymanymechanisms,e.g.,directfeedbackfrom erative) relays in between denoted by CR ,...,CR . Fol- 2 K primaryreceivers,indirectfeedbackfrombandmanager[20]orusing lowing the SS-based (underlay) approach,SUs are allowed CSIofthePU-CRk linkwithchannel reciprocityproperty[21]. VONGUYENQUOCBAOetal.: SPECTRUMSHARING-BASEDMULTI-HOPDECODE-AND-FORWARDRELAYNETWORKS 3 A. Outage Probability B. Bit Error Rate In an interference-limited multi-hop regenerative relay Inthissection,westudythemostgeneralizedscenarioin system, an outage event is declared whenever any of the a multihop network where all the single-hops in the route instantaneous forwarded SNRs of the K hops falls below havethedifferentstatisticalbehavior2,i.e.,allthelinksare a given threshold, γth. In other words, the overall system i.n.d. withdifferentaveragechannelpower,γ¯p6=γ¯q. Taking outage is dominated by the weakest hop. Thus, the end- into account the fact that a wrong bit transmission from to-end (e2e) outage probability is written as [23] nodep tonodeq (q >p)isequivalentto anoddnumberof wrongsingle-hopbittransmissionbetweenbothnodesand OP=Pr[min(γ ,...,γ )<γ ]. (6) employing the recursive error relation, we have the exact 1 K th e2e BER of the system as [25] Since γ withk =1,...,K areassumedto be independent k K K of each other, (6) is rewriten as BER = BER 1 2BER , (11) e2e p q − p=1 q=p+1 K X Y (cid:0) (cid:1) OP=1− [1−Fγk(γth)]. (7) where BERp denotes the average BER for square M- k=1 ary quadrature amplitude (M-QAM) modulation (M = Y 4m,m=1,2, ) in hop p and is given by Substituting (5) into (7), we have ··· ∞ K αk BERp = BERAWGNfγp(γ)dγ. (12) OP=1 . (8) Z − γ +α 0 th k k=1 Y In the above equation, BER is the instantaneous AWGN BER of hop p, namely [26] Theorem 1: At high SNR regime, the system OP can be approximated as log2√M υj φjnerfc √ωnγ OP γth K λI,k . (9) BERAWGN = jP=1 √nP=M0log √M(cid:0) (cid:1), (13) 2 → Ip λD,k Proof: We start the pNr0ooXk=f 1by using the fact that the where υj=(1−2−j)√M−1, ωn= (2n+12)M23l2og2M, and φjn= ncreogslse-ctteerdmcso,mFpγakr(eγdtht)oFγFlγ(kγ(tγht)h)wfiothrvkalu6=eslofinint(e7r)e,stc.anAsbae (−1)jn√2jM−1k 2j−1− n√2Mj−1+12 . Here, ⌊.⌋− and erfc(x) = result, (7) can be approximatedas 2 ∞e t2d(cid:16)t are djefined ask(cid:17)the floor and complementary √π − x K erroRr function, respectively. Substituting (4) and (13) into OP≈ Fγk(γth). (10) (12) and swapping integration and summation order, we k=1 have3 X Plugging (5) into (10) and making use x/(1+x) x for log2√M υj φj ≈ BER = n , (14) smallx,wearriveatthedesiredresult. Thiscompletesthe p p √Mlog √MJ proof. ✷ Xj=1 nX=0 2 From (9), it is worth noting that the outage probability where is defined as follows: p J at high SNRs is determined by the channel gain ratios be- tween the data and interference channels rather than the ∞ α p averagechannelpowers. Togainfurther insights,we prove Jp = erfc √ωpγ (γ+α )2dγ. (15) the following theorem. Z0 (cid:0) (cid:1) p Theorem 2: The system diversity order and coding Using integration by parts, it is shown that is of the gain are G =1 and G = γ K λI,k −1. form Jp Proof:d Observincg ((cid:18)9),thiktP=1isλDo,bkv(cid:19)ious to see that γerfc √ωpγ ∞ ωp ∞√γe−ωpγ = + dγ. (16) p the system diver1sity gain is one and the coding gain is J γ+(cid:0) αp (cid:1)(cid:12)(cid:12)γ=0 π Z0 γ+αp (cid:18)γthkK=1λλDI,,kk(cid:19)− according to OP → (GcI/N0)−Gd [24✷]. J1 (cid:12)(cid:12)(cid:12) J2 P 2Other sc|enarios in{czluding in}depende|nt and{izdentica}l distributed From Theorem 2, it is worth noting that similar to con- (i.i.d.) caseareaspecialcaseofthenetworkunderconsideration. ventional multihop networks, the system diversity order is 3It should be noted that in this paper we only consider square MQAM;howevertheemployedapproachcouldbeeasilyextendedfor always one regardless of number of hops and the increase other modulation schemes such as MPSK, MPAM and rectangular of hops results in an increase of the system coding gain. MQAM. 4 THEJOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.X,NO.X,XXX.200X To determine Jp, we need to compute J1 and J2. For J1, where a = √M√lMog−√1M and b = 23(lMog2M1). Employing the making use the l’Hopital rule, we have 2 − same steps as for (20) and then making use the infinite se- ries representation for the error function, i.e., erfc(√x) ω γ 1 = erfc √ωpγ p e−ωpγ e−x 1 1 for large x [29], we can have the desired r≈e- J (cid:20) −r π (cid:21)γ= √πx − 2x (cid:0) (cid:1) ∞ sults as in (22). = 0. (17) (cid:0) (cid:1) For i.i.d. case, it follows immediately from the result of ForJ2,introducingachangeofvariables,namely,u=√γ, the i.n.d. case by letting α={αp}Kp=1. ✷ we can rewrite as 2 J C. Ergodic Capacity J2 = 2 ∞u2u+2 be−ωpu2du paBcietysidisesanouotthagereiamnpdobrittanertrporerpforormbaabniclietym, tehaesuerreg,oddeificncead- Z 0 astheexpectedvalueoftheinstantaneousmutualinforma- = 2 ∞e−ωpu2du− ∞u2b+be−ωpu2du. (18) tniaotniobne.twTeheenetrhgeodcoicgnciatpivaecistoyurce(iannbdittsh/eseccoognndi)tivpeerdeusntiit- Z0 Z0 bandwidth can be expressed asC   Together with the identities [27, eq. (3.23.3)] and [28, eq. 1 ∞ (7.4.11)], it is shown that = log (1+γ)f (γ)dγ, (25) C K 2 γe2e Z0 p =1 √ωnαkeωnαk√πerfc(√ωnαk). (19) J − whereγ is the equivalentinstantaneouse2eSNR forthe e2e Plugging (19) into (14), we achieve the closed-form ex- multi-hop CR network. To derive the ergodic capacity, we pression for BERp as firstneedanexpressionforthePDFofγe2e. However,with regenerative relaying, an exact closed-form expression for logj2=√1Mnυ=j0φjn 1−√ωnαpeωnαp√πerfc √ωnαp tthraectPaDbiFlitiys, nwoetumseatthheemaaptpicraolxlyimvaitaibonle.apFporroamcha.thAemccaotridcs- BER = .(20) p P P (cid:2) √Mlog √M (cid:0) (cid:1)(cid:3) ing to [30], regardlessof the modulation scheme used, γe2e 2 can be tightly approximated as Substituting (20) into (11) yields the average e2e BER. For i.i.d. fading channels,i.e., α K =α, (11) simpli- γ γ˜ = min γ . (26) { p}p=1 e2e ≈ e2e k=1,...,K k fies as (21) shown at the top of the next page. Theorem 3: AthighSNR regime,the end-to-endBER Having been widely adopted in the performance studies ofcognitiveunderlaymultihopDF relayingnetworksoper- of DF relay networks (see, e.g., [31,32]), the advantage of ating in Rayleigh fading channels is approximated as this analytical approach is able to provide a mathematics tractable form for the CDF and PDF of the end-to-end K a 1 , i.n.d.channels SNR. Consequently, the PDF of γ˜ is given by BERe2e → 2bp=1αp , (22) e2e  2PKbαa, i.i.d.channels f (γ) = dFγ˜e2e(γ) γ˜e2e dγ where a= √M√lMog−2√1M and b= 23(lMog−2M1). K K Proof: Observing (11) and using the fact that at = f (γ) (1 F (γ)), (27) K γk − γn high SNR regime the product term, 1 2BERq , Xk=1 n=Y1,n6=k − q=p+1 approaches to one. We are able to apprQoxim(cid:0)ate the end(cid:1)- K whereF (γ)=1 [1 F (γ)]. Substituting(4)and to-end BER as γ˜e2e − − γk k=1 K K (5) into (27), we get Q BER = BER 1 2BER e2e p q − K K Xp=1 q=Yp+1(cid:0) →1 (cid:1) fγ˜e2e(γ)=Xk=1(γ+αkQ) n=Kn1=α1n(γ+αn). (28) K | {z } Q BER . (23) With the current form of (28), it is very difficult to ob- p ≈ p=1 tain the closed-form expression for the end-to-end capac- X ity. To facilitate the analysis, we sort and renumber α k Byneglectingsomeofthehigherordertermsin(13),BER p in the ascending order as α = = α = β < is expressed as [26, eq. (18)] 1 ··· r1 1 < α = = α = β and ···N r1+r2+···+rN−1+1 ··· r1+r2+···+rN N r =K with r being a positive integer. Stated an- ∞ α n=1 n k p BER = aerfc bγ dγ, (24) other way, β , ,β are distinct elements of α , ,α . p (γ+α )2 P 1 ··· N 1 ··· K Z0 (cid:16)p (cid:17) p Using the partial-fractionexpansion,(28)canbe rewritten VONGUYENQUOCBAOetal.: SPECTRUMSHARING-BASEDMULTI-HOPDECODE-AND-FORWARDRELAYNETWORKS 5 K 1 2 log2√M υj BERe2e = 21−1− √Mlog √M φjn 1−√ωnαeωnα√πerfc(√ωnα)   (21) 2 j=1 n=0  X X (cid:0) (cid:1)      as For the special case of β =1, (36) simplifies to n fγ˜e2e(γ)= Kk=1αk nN=1lr=n1 (γ+Aβnn,l)l+1!, (29) Il(βn) = ll1n2Z0∞ (γ+dγ1)l+1, Y XX 1 = . (37) where An,l is the coefficient of the partial-fraction expan- l2ln2 sion determined as [33,34]4 For i.i.d. case, from (27), we have An,l = (rn1−l)!(cid:26)∂∂s((rrnn−−ll))[(γ+βn)rnfγ˜e2e(γ)](cid:27)(cid:12)(cid:12)(cid:12)γ=−βn.(31) fγ˜e2e(γ)=K[1K−αFKγk(γ)]K−1fγk(γ) Plugging (29) in (25), we get (cid:12) = . (38) (γ+α)K+1 K α N rn k=1 k A I (β ), (32) Combining(38)and(25),wehavethe e2eergodiccapacity n,l l n C ≈ K for this case as Q n=1l=1 XX whereIl(βn)withl 1is the auxiliaryfunctiondefinedas ∞log (1+γ) ≥ =αK 2 C (γ+α)K+1 I (β )= ∞ log2(1+γ)dγ. (33) Z0 l n Z0 (βn+γ)l+1 =αKIK(α). (39) Using integration by parts, we have Plugging (36) (or (37)) into (32) (or (39)), we have the closed-form,integral-free,expressionfor the system capac- I (β )= log2(1+γ) ∞ ity. Itisworthnotingthatoursuggestedmethodisprecise l n "− l(γ+βn)l #γ=0 and tractable with the determination of the appropriate parameters being done straightforwardly. Additionally, as 0 is givenina closed-formfashion,its evaluationis instan- → |+ 1 {z∞ }dγ . (34) Ctaneous regardless of the number of hops and the value of lln2Z0 (1+γ)(γ+βn)l the maximum interference temperature. Itisofinteresttocomparetheergodiccapacityofunder- Whenlisaninteger,afterusingpartialfractionexpansion, layDFandAFmultihopnetworks. Thefollowingtheorem we have (35) as shown at the top of the next page. Note is provided to answer such the question. that the integral in (35) is not converging due to the first Theorem 4: For the same network and channel set- term. Todealwiththeproblem,byappropriaterearrange- tings of underlay multihop networks,DF relayingprovides ments andthenperformingthe integrations,weobtainthe slightly better ergodic capacity than its AF counterpart. final closed-form expression for I (β ) as (36). l n Proof: Denote and be the ergodic capacity DF AF C C for the underlay DF and AF multihop network, respec- 4Forconvenience,thecoefficientsAn,l canbeobtainedmoreeasily by solving the system of K equations, which is established by ran- tively. According to the min-cut max-flow theorem [35], domlychoosing K distinctvalues of γ but not equal to any βn [34]. namely the end-to-end system capacity cannot be larger LetusdenoteK chosenvaluesofγ asBu withu=1,···,K,wecan than the capacity of each hop, the end-to-end ergodic ca- obtainalinearsystemofequationsas pacity for underlay DF multihop networks can be written N rn An,l K 1 as nX=1Xl=1(γ+βn)l+1 =kX=1(γ+αk)QKn=1(γ+αn) (30) min(c ,c ,...,c ), (40) DF 1 2 K C ≤ where A = [ A1,1 ··· An,l ··· AN,rn ]T is obtained by A = C−1D, where [.]T denotes the transpose operator; C is whereck withk =1,...,K istheShannoncapacityofhop a K × K matrix, whose entries are Cu,v = (Bu+β1p)q+1 with k, given by p−1 v = q+mP=1rm; D = [ D1 ··· Du ··· DK]T with Du = 1 ∞ K 1 K 1 andu,v=1,···,K. ck = K log2(1+γk)fγk(γ)dγ. (41) Qn=1 (Bu+αn)kP=1(Bu+αk) Z0 6 THEJOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.X,NO.X,XXX.200X 1 ∞ 1 l 1 I (β )= dγ (35) l n lln2Z0 "(βn−1)l(γ+1) −Xk=1(βn−1)k(γ+βn)l+1−k# 1 1 ∞ 1 1 l−1 1 ∞ dγ I (β )= dγ l n lln2(βn−1)l Z0 (cid:18)γ+1 − γ+βn(cid:19) −kX=1(βn−1)k Z0 (γ+βn)l+1−k  l 1  1 logβn − 1 = (36) lln2"(βn−1)l −Xk=1(βn−1)k(l−k)βnl−k# Making use the Jensen’s inequality, we easily see that 1 PU(x ,y ) > E min[log (1+γ ),...,log (1+γ )] . P P CDF K γ1,...,γK{ 2 1 2 K } (42) d d d Since the binary logarithm is strictly concave, DF can be I,1 I,2 I,3 C rewritten as CR(0,0) CR (1,0) 1 4 > 1 E log [1+min(γ ,...,γ )] . (43) d d CDF K γ1,...,γK{ 2 1 K } D,1 d D,3 D,2 Based on the results reported in [36], i.e. 1 1 < E log [1+min(γ ,...,γ )] , (44) CAF K γ1,...,γK{ 2 1 K } we can complete the proof after making a comparison be- Fig.2. Cognitiveunderlay3-hopDFrelaynetwork inastraightline. tween (43) and (44). ✷ Before moving on to the next section, here we would like stress that although DF performs slightly better er- communication between road-side units placed along the godic capacity, it needs more complicated implementation road [39,40]. as compared to AF [23]. We furtherassumethatthe overalldistancebetweenthe source and the destination is normalized to one, i.e., IV. Relay Position Optimization d + +d =1, (45) Inthissection,wefocusontheproblemofrelayposition D,1 ··· D,K optimization. In particular, for given underlay DF multi- whered denotesthephysicaldistanceofdatahopk. Un- D,k hop network parameters including the coordinates of the dera predeterminedpositionofthe primaryreceiveranda primary receiver,the secondarysource,secondarydestina- fixed number of hops K, the problem of finding the opti- tion,andthenumberofhops,ourproblemistofindoptimal malpositionoftherelaysthatminimizesthesystemoutage positions for relays, which makes the system performance probability can be mathematically stated as follows: (in terms of outage probability or system error probabil- K ity) minimize. We first provide the solution for the case of min γth λI,k outage probability. Ip k=1λD,k For simplicity, we consider the network scenario illus- N0 P K (46) d =1 trated in Fig. 2, where all secondary nodes are ordered in subject to D,k .  k=1 thesequenceandpositionedalongthestraightlineconnect-  dD,k >P0, k =1,...,K ing the secondary source and the secondary destination. Suchamodelismathematicallytractableandwell-adopted Based on a single-slope distance-dependent path loss model for the average channel powers [41], we can write in the literature in studying multihop networks [37,38]. In addition,itisreadilyextendedtothemoregeneralizedcase λI,k dI,k−η dD,k η oftwodimension(2-D)networks. Interestinglyenough,al- = = , (47) though the linear multihop network model is slightly sim- λD,k dD,k−η (cid:18)dI,k (cid:19) plified model of the real world, we can find it in practical, whereη 2denotesthepathlossexponent. Recallingthat ≥ e.g. the communication between cars on a highway, or the η typically has value of 2 in free-space environments and VONGUYENQUOCBAOetal.: SPECTRUMSHARING-BASEDMULTI-HOPDECODE-AND-FORWARDRELAYNETWORKS 7 upto5and6inshadowedareasandobstructedin-building Todetermined andd ,observingFig.2andmaking ∗D,k ∗I,k scenarios,respectively[42,Table4.2]. Combining(46)and use the Pythagoreantheorem, we have (47) and noting that γth and Ip are the givenconstraints, 2 the optimization problem in (N460) can be simplified as d 2 =y 2+ x K−1d (55) I,k P P D,k − k=1 (cid:18) (cid:19) K η X dD,k for k = 2,...,K. It is obvious for the case of k = 1 that min k=1(cid:18)dI,k (cid:19) dI,1 = xP2+yP2. Using(53),(55)isrewrittenasfollows: X K pk 1 2 2 subjectto dD,k =1, . (48) x − d +y 2 = x 2+y 2 dD,k . (56)  kdP=D1,k >0, k =1,...,K P−Xp=1 D,k! P (cid:0) P P (cid:1)(cid:18)dD,1(cid:19) Combining (45) and (56), a system of K equations for  Theorem 5: For a given coordinate of primary re- d ,...,d is formulated as follows: D,1 D,K ceiver (x ,y ), the secondary network outage probability P P achieves its minimum at dD,1+ +dD,K 1=0 ··· − 2 OPmin = γNItp0hK kYK=1dd∗D∗I,,kk!Kη , (49)  y 2y+P2+x(xPK−−d1Dd,1)2−2(cid:0)x...P2x+2y+P2y(cid:1)(cid:16)2ddDD,,d12D(cid:17),K=20=0 . wofhehroepdk∗D,,kbewinitghtkhe=ro1o,t.s..o,fKthearneotnhlieneoaprtismyastledmistoafnKce  P (cid:18) P− kP=1 D,k(cid:19) −(cid:0) P P (cid:1)(cid:16)dD,1(cid:17) (57) equations as follows: Withthecurrentformof (57),itseemsimpossibletoobtain the closed-form expression for d . Consequently, in this d + +d 1=0 D,k D,1 D,K ··· − 2 case,theonlypossibilityistosolve(57)numerically. Using  x(xPK−−d1Dd,1)2−2(cid:0)xP2x+2...y+P2y(cid:1)(cid:16)2ddDD,,d12D(cid:17),K+2y+P2y=20=0 . dNtthh∗Deee,wkrpt(ero0con)uo’=rsfs.mioK1net.fhoTordhael[l4ik3n.]i,tFi(ar5ol7mv)ac(lau5ne3)bfoaerndddeD(t5e,kr7m)c,aiwnneedbceabnysecmleocemtaepndlseato✷esf  (cid:18) P− kP=1 D,k(cid:19) −(cid:0) P P (cid:1)(cid:16)dD,1(cid:17) P (50) Wthee naveexrtapgeressyensttemoptbimitaelrrroelrayrapteo.sitDioiffnse,rewnhtifcrhommionuimtaigzee probability, which serves as a lower bound to the frame And d∗I,k with k = 2,...,K are determined by using the errorrateforblockfadingenvironmentandprovidesanin- relationship sight into the theoretic-informationperformance limit, the averagebiterrorrateshowstheactualsystemperformance d d d ∗D,1 = = ∗D,k = = ∗D,K. (51) foradesiredtargetspectralefficiency,i.e. modulationlevel. d∗I,1 ··· d∗I,k ··· d∗I,K As such, the following theorem is of importance in this re- Proof: To solve the above optimization problem, we gard. canuse Cauchytheorem5. Inparticular,from(48),we can Theorem 6: For a predetermined coordinate of pri- have mary receiver (x ,y ), linear DF multihop networks un- P P der interference constraints provide the best performance K d η K d η in terms of bit error probability if and only if D,k K K D,k . (52) kX=1(cid:18)dI,k (cid:19) ≥ vuukY=1(cid:18)dI,k (cid:19) dD,1 = = dD,k = = dD,K. (58) t d ··· d ··· d I,1 I,k I,K Equality holds if and only if And the corresponding system bit error probability under dD,1 dD,k dD,K optimal relay positions is = = = = . (53) dI,1 ··· dI,k ··· dI,K a K d∗D,k Kη BER = K . (59) Deqeunaoltitinygocdc∗Du,rk,athnedmd∗Iin,kimasizethdeOoPptiismal values making the Proof: Thepreo2oefis2obmit tekYd=h1edre∗I,kdu!etothesimilarity oftheformbetweenOPandBERatthehighSNRregime. η γth K d∗D,k K Then the Theorem 6 is easily inferred from Theorem 5 ✷ OP = K (54) min NIp0 kY=1 d∗I,k ! opFtirmomalTrehlaeyorpeomsit5ioannsd(6b,oitthisinwtoerrtmhysotofopuotiangteopurtotbhaabtiltihtye andbiterrorrate)donotdependonthepathlossexponent 5In some mathematics books, this theorem also is named as the inequalityofarithmeticandgeometricmeans. . 8 THEJOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.X,NO.X,XXX.200X V. Numerical Results and Discussion The purpose of this section is twofolds. We first provide 100 16−QAM numerical results to confirm the derived analytical expres- sions and then show the network performance advantage K decreasing 5 → 1 offered by relay position optimization. 10−1 For illustrative purpose, we consider a linear multi-hop e network in a 2-D plane, where all SUs are co-linearly lo- Rat cated and the distance between the cognitive source and Error 10−2 the cognitive destination is normalized to one. Further- Bit more,thecognitivesourceandthecognitivedestinationare located at points with coordinates (0,0) and (1,0), respec- 10−3 tively. Each cognitive relay node is equidistant from each Analysis (Exact) Analysis (Approximation) other, i.e. d =1/K. The average channel power CRk,CRk+1 Simulation forthetransmissionbetweennodeAandnodeBismodeled 10−4 as λA,B = dA,B−η where η denotes the path loss exponent −10 0 10 20 30 40 withA CR ,...,CR andB PU,CR ,...,CR . Ip/ N0 [dB] 1 K 1 2 K ∈{ − } ∈{ } In all examples, we locate the PU-Rx at coordinate (0.35, Fig.4. BiterrorprobabilityversusaverageIp/N0. 0.35) and set η =4. 2 100 AAnnaallyyssiiss ((EAxpapcrot)ximation) 1.8 ASnimaulylasitsio (nApproximation) Simulation 1.6 1.4 y Probability10−1 dic Capacit 1.12 Outage Ergor 00..68 K increasing, 1 → 5 10−2 0.4 K = {1,2,3,4,5} 0.2 0 −10 −5 0 5 10 15 20 10−3 I/ N [dB] −10 0 10 20 30 40 p 0 I/ N [dB] p 0 Fig.5. ErgodiccapacityversusaverageIp/N0. Fig.3. OutageprobabilityversusaverageIp/N0. the number of hops is equivalent to reducing the effective InFig.3andFig.4,werespectivelyillustratetheoutage transmission bandwidth of each hop. In addition, the nu- probability and BER of the multi-hop cognitive networks mericalresultsshowthatthe analyticalresultsareingood asafunctionofinterferencetemperaturefordifferentnum- agreement with the simulation results. ber of hops. As can be observed from the two figures, the performance is enhanced as the number of cognitive relay Fig. 6 compares the capacity performance of multi-hop hops K increases. It is important to note that the dimin- cognitive relay networks for different positions of the PU- ishing gain returns as the number of hops increases. The Rx given the same number of hops. Observing the results numerically evaluated results demonstrate the correctness in the figure, we can see that the system performance im- of the presented analysis. proveswhentheprimarynodeislocatedfartherawayfrom Figure 5 displays the capacity performance for cogni- the secondary relay transmitters, as expected. tivemulti-hoptransmissionbyvaryingthenumberofhops, Up to this point, we have not studied the effect of the K = 1,2,...,5. It is worth noting that for interference- proposed relay position optimization. In doing so, we con- limitedregime(low / ),thesystemwithlargeK offers siderer three relay position profiles: randomization, equal- p 0 I N improved performance. For a high interference tempera- ization and optimization, denoted as Profile A, Profile B ture level, multi-hop transmission with small hops is more and Profile C, respectively. In Profile A, all secondary re- favorable. It can be explained by using the fact that with lays are chosen randomly from a uniform distribution. In the channel model and time-sharing schedule used, at low ProfileB,the distance betweenany twonodes is the same. / transmission over shorter distance corresponds to And in Profile C, secondary relays are set using the rule, p 0 I N increased effective SNRs while at high / increasing proposed in Sect. IV. Table 1 demonstrates the results for p 0 I N VONGUYENQUOCBAOetal.: SPECTRUMSHARING-BASEDMULTI-HOPDECODE-AND-FORWARDRELAYNETWORKS 9 4 1.8 Profile A Profile B 3.5 PU(x,y)=(0.3,0.3) 1.6 Profile C PU(x,y)=(0.6,0.6) 3 PU(x,y)=(0.9,0.9) 1.4 η = 6 Simulation Ergordic Capacity12..255 Ergordic Capacity01..182 η = 4 η = 2 1 0.6 0.4 0.5 0.2 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 −10 −5 0 5 10 15 20 K I/ N [dB] p 0 Fig.8. Effectofpathlossexponentonergodiccapacity. Fig.6. EffectofPU-Rx’slocationonergodiccapacity. K =2 and 4. that makes the system ergodic capacity maximize. It can be explained by using the fact that with small η, the ben- Profile A Profile B Profile C efit of path loss gain is not enough to compensate the loss d =0.1767 d =0.5 d =0.4192 due to the use of multi orthogonal time slots for multihop K =2 D,1 D,1 D,1 d =0.8333 d =0.5 d =0.5808 communications. D,2 D,2 D,2 d =0.20 d =0.25 d =0.1915 D,1 D,1 D,1 VI. Conclusion d =0.28 d =0.25 d =0.1900 K =4 D,2 D,2 D,2 dD,3 =0.36 dD,3 =0.25 dD,3 =0.2492 We have investigated the performance of cognitive re- dD,4 =0.16 dD,4 =0.25 dD,4 =0.3693 generativemulti-hoprelaynetworksusingtheunderlayap- Table1. Comparisionofthreerelaypositionprofiles proach. We have derived the closed-form expressions for the outage probability, BER, and ergodic capacity over i.n.d. Rayleigh fading channels. High analysis for outage probability and bit error rate have also made to provide 100 insights into the system behaviors. The numerical results Profile A show that under the interference constraints inflicted by Profile B Profile C the primary network, the multi-hop transmission still of- fersaconsiderablegainascomparedtodirecttransmission and thus makes it an attractive proposition for cognitive bility networks. a b e Pro10−1 K = 2 ACKNOWLEDGMENTS g a ut O This research was supported by the Vietnam Na- tionalFoundationforScienceandTechnologyDevelopment (NAFOSTED) (No. 102.01-2011.22). K = 4 10−2 REFERENCES 0 5 10 15 20 25 30 I/ N [dB] [1] J.Wang,M.Ghosh,andK.Challapali,“Emergingcognitivera- p 0 dioapplications: Asurvey,”IEEECommun.Mag.,vol.49,no.3, pp.74–81,Mar.2011. Fig.7. Comparisonofthreerelaypositionprofiles. [2] S.Filin,H.Harada,H.Murakami,andK.Ishizu,“International standardization of cognitive radio systems,” IEEE Commun. Mag.,vol.49,no.3,pp.82–89, Mar.2011. In Fig. 7, we can see that Profile C outperforms Profile [3] I. Mitola, J. and J. Maguire, G. 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(ICASSP ’03)., vol. 4, pp. IV–189– Vo Nguyen Quoc Bao was born in Nha 92vol.4. Trang,KhanhHoaProvince,Vietnam,in1979. [24] Z. Wang and G. B. Giannakis,“A simple and general param- He received the B.E. and M.Eng. degree in eterization quantifying performance in fading channels,”IEEE electrical engineering from Ho Chi Minh City Trans. Commun.,vol.51,no.8,pp.1389–1398, Aug.2003. UniversityofTechnology(HCMUT),Vietnam, [25] E.Morgado,I.Mora-Jimenez,J.J.Vinagre,J.Ramos,andA.J. in 2002 and 2005, respectively, and Ph.D. de- Caamano,“End-to-end average BER in multihop wireless net- gree in electrical engineering from University works over fading channels,”vol. 9, no. 8, pp. 2478–2487, Sep. of Ulsan, South Korea, in 2010. In 2002, he 2010. joined the Department of Electrical Engineer- [26] K.ChoandD.Yoon,“OnthegeneralBERexpressionofone-and ing,PostsandTelecommunicationsInstituteof two-dimensional amplitude modulations,” IEEE Trans. Com- Technology(PTIT),asalecturer. SinceFebru- mun.,vol.50,no.7,pp.1074–1080, Jul.2002. ary 2010, he has been with the Department of Telecommunications, [27] I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, PTIT, where he is currently an Assistant Professor. He is the au- Table of integrals, series and products, 7th ed. Amsterdam ; thor or coauthor of more than 60 technical papers in the area of Boston: Elsevier,2007. wireless and mobile communications. His major research interests [28] M. Abramowitz and I. A. Stegun, Handbook of mathemati- are modulation and coding techniques, MIMO systems, combining cal functions with formulas, graphs, and mathematical tables. techniques, cooperative communications, and cognitive radio. Dr. Washington: U.S.Govt.Print.Off.,1972. BaoisamemberofKoreaInformationandCommunicationsSociety [29] C. Tellambura and A. Annamalai, “Efficient computation of (KICS), The Institute of Electronics, Information and Communica- erfc(x) for large arguments,” IEEE Trans. Commun., vol. 48, tionEngineers(IEICE)andtheInstituteofElectricalandElectronics no.4,pp.529–532, Apr.2000. Engineers(IEEE).