Performance Analysis of Cognitive Relay Networks …folk.uio.no/yanzhang/IEEEJSACZhangMay2015.pdfIEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.33,NO.5,MAY2015 865 Performance

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015 865

Performance Analysis of Cognitive Relay NetworksOver Nakagami-m Fading Channels

Xing Zhang, Senior Member, IEEE, Yan Zhang, Senior Member, IEEE,Zhi Yan, Jia Xing, and Wenbo Wang, Member, IEEE

Abstract—In this paper, we present performance analysis forunderlay cognitive decode-and-forward relay networks with theN th best relay selection scheme over Nakagami-m fading chan-nels. Both the maximum tolerated interference power constraintand the maximum transmit power limit are considered. Specif-ically, exact and asymptotic closed-form expressions are derivedfor the outage probability of the secondary system with the N thbest relay selection scheme. The selection probability of the N thbest relay under limited feedback is discussed. In addition, wealso obtain the closed-form expression for the ergodic capacityof the secondary system with a single relay. These expressionsfacilitate in effectively evaluating the network performance in keyoperation parameters and in optimizing system parameters. Thetheoretical derivations are extensively validated through MonteCarlo simulations. Both theoretical and simulation results showthat the fading severity of the secondary transmission links hasmore impact on the outage performance and the capacity thanthat of the interference links does. Through asymptotic analysis,we show that the diversity order for the N th best relay selectionscheme is min(m1,m3) × (M − N) + m3, where M denotesthe number of cognitive relays, and m1 and m3 represent thefading severity parameters of the first-hop transmission link andthe second-hop transmission link, respectively.

Index Terms—Cognitive relay networks, N th best relay selec-tion, Nakagami-m fading, outage probability, ergodic capacity.

I. INTRODUCTION

RADIO spectrum is an important and scarce resourcewhich is increasingly demanded by many kinds of users.

Cognitive radio is an efficient technology to improve the spec-trum resources utilization and has gained much attention in re-cent years [1]. There are three main cognitive radio paradigms:underlay, overlay and interweave [2]. The underlay paradigmallows cognitive (secondary) users to utilize the licensed spec-

Manuscript received January 4, 2014; revised May 8, 2014 and July 15, 2014;accepted August 23, 2014. Date of publication September 30, 2014; date ofcurrent version April 21, 2015. This work was supported in part by the NationalNatural Science Foundation of China (NSFC) under Grant 61372114, by theNational 973 Program of China under Grant 2012CB316005, by the Joint Fundsof NSFC-Guangdong under Grant U1035001, and by Beijing Higher EducationYoung Elite Teacher Project under Grant YETP0434.

X. Zhang, J. Xing, and W. Wang are with the Wireless Signal Processingand Network Laboratory, Key Laboratory of Universal Wireless Communica-tion, Ministry of Education, Beijing University of Posts and Telecommunica-tions, Beijing 100876, China (e-mail: [email protected]; [email protected];[email protected]).

Y. Zhang is with Simula Research Laboratory, Fornebu 1364, Norway(e-mail: [email protected]).

Z. Yan is with the School of Electrical and Information Engineering, HunanUniversity, Changsha 410082, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2014.2361081

trum if the interference caused to primary users is below agiven interference threshold. In overlay systems, both primaryand secondary users can utilize the licensed spectrum simul-taneously through sophisticated signal processing and coding.In interweave systems, secondary users opportunistically uti-lize spectrum holes to communicate without interfering thetransmission of primary users. In this paper, we focus on theunderlay systems.

On the other hand, relay communication has emerged asa powerful spatial diversity technology for effectively com-bating channel fading and greatly improving the transmissionperformance of wireless communication systems [3]. Thereare two kinds of classical relay communication protocols, i.e.,amplify-and-forward (AF) and decode-and-forward (DF) [4]. Inmultiple-relay systems, the relay-selection-based transmissionprotocol can get higher spectrum efficiency compared with thetraditional relay communication protocol. Best relay selection[5], [6] is an ideal protocol to achieve the best performance. Inpractice, however, the best relay might not be available due tosome scheduling, load balancing or channel side information(CSI) imperfect feedback conditions. In this case, the secondbest relay or more generally the N th best relay might beselected. Therefore, the study of the N th best relay selectionis of great need.

Inspired by cognitive radio and relay communication, thecognitive relay network which combines these two techniquesis proposed. In cognitive relay networks, both the spectrumefficiency and the transmission performance can be improved.Recently, the research of cognitive relay networks has attractedmuch attention, especially on the underlay paradigm [7]–[19].In [7], the outage probability of a cognitive dual-hop networkwith a single AF relay under the interference power constraintwas derived. The outage probability of a cognitive DF relaynetwork without a direct transmission link and with best relayselection was evaluated in [8]. In [9], a rough upper boundon outage probability for cognitive DF relay networks witha direct transmission link and with best relay selection overindependent and identically distributed (i.i.d) Rayleigh fadingchannels was obtained. However, these works can be improvedby considering the dependence among the received signal-to-noise ratios (SNRs) in the first hop. Considering this kind ofdependence, the accurate upper and lower bound on outageprobability for such systems were respectively obtained in [10]and [11].

In [12], the exact outage probability expression for cognitiveDF relay network was derived. The authors in [13] investigatedthe diversity performance of cognitive relay networks with

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866 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 5, MAY 2015

three different relaying protocols, i.e., selective AF, selectiveDF, and AF with partial relay selection. In [14], the authorsstudied the outage performance of a cognitive network adoptingincremental DF protocol. In [15], the outage probability, sym-bol error probability and ergodic capacity were derived for cog-nitive AF relay networks with best relay selection. References[7]–[15] focused on Rayleigh fading channels while references[16]–[19] studied the more general fading environment, i.e.,Nakagami-m fading channels. In [16]–[18], the authors studiedthe outage performance of cognitive relay networks with asingle relay over Nakagami-m fading channels. In [19], theoutage performance of cognitive DF relay network with bestrelay selection over independent and non-identically distributedNakagami-m fading channels was studied.

The above mentioned references all focused on the bestrelay selection scheme. Only a few studies involved the N thbest relay selection scheme. As we have mentioned, the N threlay selection is more of practical significance. In [20], theperformance for conventional AF and DF relay networks withthe N th best relay selection over Rayleigh fading channels wasstudied. The asymptotic symbol error rate for a conventionalAF relay network with the N th best relay selection overNakagami-m fading channels was derived in [21]. In [22],the authors investigated the outage behavior of a conventionaldual-hop N th-best DF relay system in the presence of co-channel interference over Rayleigh fading channels. Theoutage performance for cognitive relay networks with the N thbest relay selection over Rayleigh fading channels were studiedin [23]. In summary, it is observed that there have been no priorworks on the performance of cognitive relay network withthe N th best relay selection scheme over Nakagami-m fadingchannels. While in practical networks, the channels will notalways be simply Rayleigh-distributed. Thus, a comprehensivestudy of cognitive relay network with N th best relay selectionover the general Nakagami-m fading channel will be beneficialfor the design in practical cognitive relay systems.

In this paper, we investigate the performance of an underlaycognitive DF relay network over Nakagami-m fading channels.Our main contributions are as follows:

• The exact outage probability of the secondary system withthe N th best relay selection is derived over Nakagami-mfading channels, which build the relationship between theoutage performance and the related system parameters. Inaddition, the selection of the N th best relay in the limitedfeedback scenario is discussed.

• An asymptotic analysis is carried out to get the asymptoticoutage probability of the secondary system with the N thbest relay selection. The diversity order is also obtained.

• The closed-form expression for the ergodic capacity ofthe secondary system with single relay is derived overNakagami-m fading channels.

• The results show that the fading severity of the secondarytransmission links has more impact on the outage perfor-mance and the ergodic capacity than the fading severity ofthe interference links.

The rest of this paper is organized as follows. Section IIdescribes the system model. Sections III and IV present the de-

Fig. 1. System model.

tailed analysis of exact and asymptotic outage performance ofthe secondary system with the N th best relay selection scheme,respectively. The selection probability of the N th best relayunder limited feedback is discussed in Section V. In Section VI,the exact ergodic capacity is derived and analyzed. Numericalresults are shown in Section VII. Finally, conclusions are givenin Section VIII.

Notation: Ckn represents the binomial coefficient and n!

represents the factorial of n. Γ(α) =∫∞0 tα−1e−tdt, Γ(α, x) =∫∞

x tα−1e−tdt and γ(α, x) =∫ x

0 tα−1e−tdt denote the gammafunction [24, eq. (8.310.1)], the upper incomplete gamma func-tion [24, eq. (8.350.2)] and the lower incomplete gammafunction [24, eq. (8.350.1)], respectively. Ei(x) = −

∫∞−x

e−t

t dtrepresents the exponential integral function [24, eq. (8.211.1)].The cumulative distributed function (CDF) and the probabilitydensity function (PDF) of random variable X are expressed asFX(·) and fX(·), respectively.

II. SYSTEM MODEL

We consider an underlay cognitive DF relay network, asillustrated in Fig. 1. It involves one primary user receiver (PU)and a secondary system. The secondary system is a dual-hoprelay communication system which consists of one secondarysource (SS), one secondary destination (SD) and M sec-ondary relays (SRi, i = 1, . . . ,M). All nodes are equippedwith a single antenna and operate in half-duplex mode. Theinterference from the primary transmitter is assumed to beneglected as in [7]–[19]. This can be possible if the primarytransmitter is located far away from the secondary users, or theinterference is modeled as the noise term [8]. Like [13], [15],[16], [18], etc., we assume that there is no direct link betweenSS and SD due to the severe shadowing and path loss. Weemploy the CSI-assisted DF relaying protocol and the N th bestrelay selection scheme. A whole transmission process of thesecondary system consists of two phases. In the first phase,the SS broadcasts messages to M relays under a transmitpower constraint which guarantees that the interference on theprimary user receiver does not exceed a threshold. In the secondphase, the N th best relay that is selected from the successfuldecoding relay set based on the channel quality of the second-hop links forwards source messages to the SD. Finally, the SD

ZHANG et al.: PERFORMANCE OF COGNITIVE RELAY NETWORKS OVER NAKAGAMI-m FADING CHANNELS 867

TABLE IPARAMETERS NOTATIONS

decodes source messages. The transmitters of the secondarysystem are with the maximum transmit power constraint Pmax,and the maximum interference power constraint of the primaryuser receiver is Q. All links are assumed to be independentNakagami-m flat fading channels with integer values of thefading severity parameters and unit average power. The channelgain of SS-SRi, SS-PU , SRi-SD, and SRi-PU are denotedas gsi, gsp, gid, and gip, whose fading severity parameters arem1, m2, m3, and m4, respectively. The thermal noise at eachreceiver is modeled as additive white Gaussian noise (AWGN)with variance σ2. More details of the parameters used in thispaper are given in Table I.

III. EXACT OUTAGE PERFORMANCE ANALYSIS

In this part, we derive the exact outage probability expressionfor the previously described underlay cognitive relay networkwith the N th best relay selection, which can be used to evaluatethe impact of the related parameters on the outage performance,which include the maximum interference power constraint Q,the maximum transmit power constraint Pmax, the fading sever-ity parameters mi (i = 1, 2, 3, 4), the number of relays M andthe order of the selected relay N .

Considering the maximum transmit power constraint Pmax ofthe secondary transmitters and the interference power constraintQ of the primary user, the transmit power of the SS andthe ith relay SRi should be no more than min(Pmax, Q/gsp)and min(Pmax, Q/gip), respectively. In order to maximizethe transmission performance of the secondary system, thesecondary transmitters transmit signals with the maximumallowable transmit power. Hence, the transmit power at SScan be written as PS = min(Pmax, Q/gsp), where gsp denotesthe channel coefficient of the link between SS and PU . Simi-larly, the transmit power at SRi is given by PRi

= min(Pmax,Q/gip).

In the first-hop transmission, the SS broadcasts messages torelays. As a result, the received SNR at the ith relay SRi is

written as

γsi=min(Pmax, Q/gsp)gsi

σ2=

{Pmaxgsi

σ2 , for Pmax<Qgsp

Qgsiσ2gsp

, for Pmax≥ Qgsp

.

(1)

From the above equation, it is worth noting that γsi for eachi is related with gsp while Pmax > Q/gsp. Hence, the receivedSNRs at relays are correlated in the first-hop transmission whilePmax > Q/gsp, but they are independent while Pmax ≤ Q/gsp.We denote the target transmission rate of secondary system asR. Then the received SNR at the ith relay should meet thefollowing inequality if the ith relay can successfully decodesource messages.

R ≤ 1

2log2(1 + γsi). (2)

We define γth = 22R − 1, so the successful decoding constraintof the ith relay can be simplified to γsi ≥ γth. We denotethe successful decoding relay set as R(s) in the first-hoptransmission.

Lemma 1: The probability of the successful decoding relayset R(s) is given by the expression (3), shown at the bottom ofthe next page, where n = |R(s)| denotes the number of relays

in R(s), γPmax= Pmax/σ

2, γQ = Q/σ2, and H =m1−1∑j=0

jwj .

Proof: See Appendix A. �In the second-hop transmission, the N th best relay selected

from the successful decoding relay set R(s) forwards thesource messages to the SD. N should be less than or equalto |R(s)|. It is worth noting that the N th best relay selectionmakes sense only when R(s) is not empty, since N ≥ 1.

With the N th best relay selection scheme, the received SNRat the SD is

γrd = N th maxi∈R(s)

(γid) = N th maxγsi≥γth

(γid), (4)


where N thmax(·) denotes the N th maximum item, γid =min(Pmax,Q/gip)gid

σ2 denotes the received SNR at the SD if theith relay of R(s) forwards source messages to the SD in thesecond phase.

Lemma 2: The CDF of γrd is given as (5), shown at thebottom of the page.

Proof: See Appendix B. �Finally, the SD decodes source messages. Hence, the equiv-

alent end-to-end received SNR is γrd in the transmission proce-dure, and the mutual information of secondary system is given as

C =1

2log2(1 + γrd). (6)

According to the Shannon information theory, the outage oc-curs when C < R. We denote the event that the N th best relayis selected as SN . Therefore, the conditional outage probabilityof the secondary system given SN and R(s) (|R(s)| ≥ N) iscalculated as

Pr (outage|SN ,R(s)) = Fγrd(γth). (7)

Considering all the possibilities of R(s), the outage proba-bility of the secondary system with the N th best relay selectionscheme can be written as the following expression according tothe law of total probability.

Pr(outage|SN ) =M∑

n=N

CnM Pr (R(s))Fγrd

(γth). (8)

Substituting (3) and (5) into (8), the exact outage probability forthe N th best relay selection can be obtained.

IV. ASYMPTOTIC OUTAGE PERFORMANCE ANALYSIS

In this part, we derive the asymptotic outage probabilityexpression in high SNR regions to reveal the diversity per-formance of the secondary system with the N th best relayselection scheme.

For a relay selection diversity communication system, thediversity order is an important performance metric. It is definedas d = − limγ→∞(logPout(γ)/ log(γ)), where γ denotes theSNR of systems. The diversity order essentially indicates thenumber of received independent fading signals at the receiver.To derive the diversity order of the secondary system withthe N th best relay selection scheme, the asymptotic outageprobability in high SNR regions should be obtained firstly. Forthe simplicity of analysis, as [13] and [25], we set γ = 1/σ2 torepresent the SNR of the secondary system in the subsequentdiscussions. Hence, the high SNR region arises while σ2 → 0.According to the asymptotic behavior of γ(n, x), we have

limx→0

γ(n, x) =xn

n. (9)

Note that the n in (9) is not necessarily an integer. Therefore,the asymptotic outage performance analysis in this section isapplicable to the cases of arbitrary Nakagami fading parame-ters, including the non-integer ones.

Lemma 3: The probability of the successful decoding relayset R(s) can be asymptotically approximated as (10), shown atthe bottom of the next page,where∝ represents “proportional to.”

Proof: See Appendix C. �Lemma 4: The asymptotic expression for the CDF of γrd is

written as (11), shown at the bottom of the next page.Proof: See Appendix D. �

By utilizing Lemma 3 and Lemma 4, and substituting (10)and (11) into (8), the asymptotic outage probability of thesecondary system with the N th best relay selection scheme canbe obtained as (12), shown at the bottom of the next page.

Meanwhile, we can see from (12) that

Pr(outage|SN ) ∝M∑

n=N

(1

γ

)m1(M−n)+m3(n−N+1)

. (13)

In high SNR regions, the higher order terms of 1/γ can beomitted. From (13), the n = N term (i.e., (1/γ)m1(M−N)+m3)

Pr (R(s))=

⎡⎣1− γ

(m1,

m1γth

γPmax

)Γ(m1)

⎤⎦n ⎡⎣γ

(m1,

m1γth

γPmax

)Γ(m1)

⎤⎦M−n

γ(m2,

m2QPmax

)Γ(m2)

+n∑

i=0

M−n+i∑l=0

∑w0+...+wm1−1=l

{Ci

nClM−n+i(−1)i+l

×l!mm2

2 Γ(m2 +H,

m2γQ+m1γthlγPmax

)Γ(m2)

(m1γthγQ

)H (γQ

m2γQ +m1γthl

)m2+H m1−1∏j=0

[1

wj !

(1

j!

)wj]}

(3)

Fγrd(x) =

N∑k=1

k−1∑u=0

Ck−1n Cu

k−1(−1)k−1−u

⎧⎨⎩

γ(m3,

m3xγPmax

)Γ(m3)

γ(m4,

m4QPmax

)Γ(m4)

+Γ(m4,

m4QPmax

)Γ(m4)

− mm44

Γ(m4)

m3−1∑i=0

⎡⎣Γ

(m4 + i,

xm3+m4γQ

γPmax

)i!

(xm3

γQ

)i (xm3

γQ+m4

)−(m4+i)⎤⎦⎫⎬⎭

n−u

(5)


is left when m1 ≤ m3, and the n = M term (i.e.,(1/γ)m3(M−N+1)) is left when m1 > m3. To sum up,the asymptotic outage probability of secondary system inhigh SNR regions is proportional to (1/γ)m1(M−N)+m3 whenm1 ≤ m3, but it is proportional to (1/γ)m3(M−N+1) whenm1 > m3. Hence, the diversity order of the secondary systemis min(m1,m3)× (M −N) +m3. It is indicated that thediversity performance of the secondary system with the N thbest relay selection scheme is affected by the channel fadingseverity parameters of the transmission links, as well as thedifference between the number of relays M and the order ofthe selected relay N . The channel fading severity parametersof the interference links have no impact on the diversity order.

V. SELECTION PROBABILITY OF THE NTH BEST RELAY

UNDER LIMITED FEEDBACK

The relay selection process is mainly based on the obtainedchannel knowledge. In practice, due to imperfect CSI feedback,the chosen relay may not be the best one. Limited feedback isoften used to perform relay selection [26]. In this section, wediscuss the impact of limited feedback on the relay selectionprocess.

The relay is selected according to the SNR of SRi-SD γid(i ∈ R(s)). Considering channel reciprocity, we assume relaySRi can acquire the channel fading coefficient of SRi-SD.Therefore, γid is available at SRi. In order to select a relay,SRi should feedback its SNR γid to the decision-making nodewho performs relay selection.

We assume that L bits are used to feedback the SNRs, sothere are q = 2L quantization intervals. Given the quantizedcodebook {γ̂1, γ̂2, · · · , γ̂q}, the quantized value of γid (i ∈R(s)) is determined by

γ̂id = arg minγ̂∈{γ̂1,γ̂2,···,γ̂q}

|γid − γ̂|. (14)

Then each relay transmits the index of its quantized value tothe decision-making node. We assume that γid falls into eachquantization interval with equal probability p = 1/q throughsome non-uniform quantizer.

With limited feedback, the SNRs of different relays may fallinto the same quantization interval. The “best” relay selectiondepends on the best quantization interval, i.e., the quantizationinterval with the largest quantized value that contains at leastone relay. We denote the set of relays whose SNR falls intothe best quantization interval as R(b). R(b) is a subset of thesuccessful decoding relay set R(s). For a given R(s) withn = |R(s)|, when n = 0, i.e., R(s) is empty, no relay would beselected. When n > 0, i.e., R(s) is not empty, the probabilitythat there are nb relays in the best quantization interval for thecase of nb < n can be calculated as

Pr (|R(b)| = nb|R(s)) = Cnbn pnb

q−1∑a=1

(1− ap)n−nb . (15)

For the case of nb = n, i.e., all relays are in the same quantiza-tion interval, the probability that there are nb relays in R(b) for

Pr (R(s))γ→∞≈

[1

Γ(m1 + 1)

(m1γth

γ

)m1]M−n

1

Γ(m2)

[(1

Pmax

)m1(M−n)

γ

(m2,

m2Q

Pmax

)

+

(1

m2Q

)m1(M−n)

Γ

(m1(M − n) +m2,

m2Q

Pmax

)](10)

Fγrd(x)

γ→∞≈

N∑k=1

{(m3x

γ

)m3(n−k+1)

Ck−1n

(1

Γ(m3 + 1)Γ(m4)

)n−k+1

×[(

1

Pmax

)m3

γ

(m4,

m4Q

Pmax

)+

(1

m4Q

)m3

Γ

(m3 +m4,

m4Q

Pmax

)]n−k+1}

(11)

Pr(outage|SN )γ→∞≈

M∑n=N

N∑k=1

(γthγ

)m1(M−n)+m3(n−k+1)Cn

MCk−1n

Γ(m2)

(mm1

1

Γ(m1 + 1)

)M−n (mm3

3

Γ(m3 + 1)Γ(m4)

)n−k+1

×[(

1

Pmax

)m3

γ

(m4,

m4Q

Pmax

)+

(1

m4Q

)m3

Γ

(m3 +m4,

m4Q

Pmax

)]n−k+1

×[(

1

Pmax

)m1(M−n)

γ

(m2,

m2Q

Pmax

)+

(1

m2Q

)m1(M−n)

Γ

(m1(M − n) +m2,

m2Q

Pmax

)](12)


the given R(s) can be expressed as

Pr (|R(b)| = nb|R(s)) = qpn. (16)

The “best” relay is selected from R(b) randomly, i.e., each relayin R(b) is chosen with probability 1

nb. We denote the event

that the N th best relay is chosen as SN (N = 1, 2, · · · , n).Therefore, the conditional selection probability of the N th bestrelay given R(s) is expressed as

Pr (SN |R(s)) =n∑

nb=N

1

nbPr (|R(b)| = nb|R(s)) . (17)

According to the law of total probability, the selection prob-ability of the N th best relay under limited feedback is given by

Pr(SN ) =

M∑n=N

CnM Pr (R(s)) Pr (SN |R(s)) , (18)

where Pr(R(s)) can be calculated as (3).Combining this with the results in Section III, we can obtain

the outage probability of the secondary system with limitedfeedback as

Pout=Pr (R(s) = ∅)+M∑

N=1

Pr(SN ) Pr(outage|SN ). (19)

VI. EXACT ERGODIC CAPACITY ANALYSIS

In this section, we derive the exact ergodic capacity of thecognitive DF relay network over Nakagami-m fading channels.Specifically, we study the special case where there is only onesingle relay in the secondary system, i.e., M = 1 and N = 1.Additionally, we consider symmetric channel conditions wherethe channels associated with the first hop and the second hophave the same parameters. The fading severity parameters of thesecondary transmission links are denoted as m1 = m3 = ms

while those of the interference links are denoted asm2=m4=mp.For the case of M = 1 and N = 1, the end-to-end SNR of

the secondary system is given by

γe2e = min(γsr, γrd). (20)

Thus the CDF of γe2e can be written as

Fγe2e(x) = Pr {min(γsr, γrd) ≤ x}

=1− Pr{γsr > x}Pr{γrd > x}=1− (1− Fγsr

(x)) (1− Fγrd(x)) . (21)

Since there is only one relay node, the CDF of γrd is thesame as the CDF of γid in (42). Due to the symmetric channelconditions, γsr has the same distribution as γrd. We rewrite it as

Fγsr(x)=Fγrd

(x)=γ(ms,

msxγPmax

)Γ(ms)

γ(mp,

mpQPmax

)Γ(mp)

+Γ(mp,

mpQPmax

)Γ(mp)

− mmpp

Γ(mp)

ms−1∑i=0

[1

i!

(msx

γQ

)i (msx

γQ+mp

)−(mp+i)

× Γ

(mp + i,

msx+mpγQγPmax

)]. (22)

According to the definition, the ergodic capacity of thesecondary system can be expressed as

C̄ =1

2

∫ ∞

0

log2(1 + x)fγe2e(x) dx. (23)

By using the same method as [27], we can rewrite the expres-sion for the ergodic capacity as

C̄ =1

2 ln 2

∫ ∞

0

1

1 + x(1− Fγe2e

(x)) dx

=1

2 ln 2

∫ ∞

0

1

1 + x(1− Fγsr

(x)) (1− Fγrd(x)) dx.

(24)

By substituting (22) into (24) and utilizing the series repre-sentation of the incomplete Gamma function [24, eq. (8.352.6)],we can get (25), shown at the bottom of the page, where A andB(v) are defined as

A =

∫ ∞

0

xi+j

1 + xe− 2msx

γPmax dx (26)

and

B(v) =

∫ ∞

0

xi+j

(1 + x)(msx+mpγQ)ve− 2msx

γPmax dx, (27)

respectively.

C̄ =1

2 ln 2

1

(Γ(mp))2

ms−1∑i=0

ms−1∑j=0

mi+js

i!j!

{(γ

(mp,

mpQ

Pmax

))21

γi+jPmax

A

+ γ

(mp,

mpQ

Pmax

)1

γiPmax

(mpγQ)mpΓ(mp + j)e−

mpQ

Pmax

m+j−1∑k=0

1

k!

1

γkPmax

B(mp + j − k)

+ γ

(mp,

mpQ

Pmax

)1

γjPmax

(mpγQ)mpΓ(mp + i)e−

mpQ

Pmax

mp+i−1∑l=0

1

l!

1

γlPmax

B(mp + i− l)

+[(mpγQ)

mpe−mpQ

Pmax

]2Γ(mp + i)Γ(mp + j)

mp+i−1∑l=0

mp+j−1∑k=0

1

l!k!

1

γl+kPmax

B(2mp + i+ j − l − k)

}(25)


By using [24, eq. (3.353.5)], we obtain

A = (−1)i+j−1e2ms

γPmax Ei

(− 2ms

γPmax

)

+

i+j∑h=1

(h− 1)!(−1)i+j−h

(γPmax

2ms

)h

. (28)

To calculate the functionB(v), we should consider two cases.For the case of ms = mpγQ, by splitting the term 1/(1 +

x)(msx+mpγQ)v , we get

B(v) =1

(mpγQ −ms)v

∫ ∞

0

xi+j

1 + xe− 2msx

γPmax dx︸︷︷︸Φ1

−v∑

a=1

ms

(mpγQ−ms)a

∫ ∞

0

xi+j

(msx+mpγQ)v+1−ae− 2msx

γPmax dx︸︷︷︸Φ2

.

(29)

The integral term Φ1 is the same as A. The integral term Φ2

can be calculated with the help of the Meijer’s G-function [28].Specifically, the term 1/(msx+mpγQ)

v+1−a in Φ2 can beexpressed as

1

(msx+mpγQ)v+1−a

=1

(mpγQ)v+1−a

1

Γ(v + 1− a)G1,1

1,1

(msx

mpγQ

∣∣∣∣a− v

0

).

(30)

By substituting (30) into Φ2 and using [24, eq. (7.813.1)], wehave

Φ2 =1

(mpγQ)v+1−a

1

Γ(v + 1− a)

(γPmax

2ms

)i+j+1

× G1,22,1

(Pmax

2mpQ

∣∣∣∣−i− j, a− v

0

). (31)

Thus B(v) is obtained for the case of ms = mpγQ.For the case of ms = mpγQ, B(v) can be written as

B(v) =1

mvs

∫ ∞

0

xi+j

(1 + x)v+1e− 2msx

γPmax dx. (32)

By the same way we get Φ2, we can obtain the closed-formexpression for (32).

In conclusion, we obtain B(v) as (33), shown at the bottomof the page.

By substituting A and B(v) which are given by (28) and (33),respectively, into (25), we can get the closed-form expressionfor the system ergodic capacity.

Fig. 2. Outage probability versus interference power constraint to noiseratio for different channel fading severity parameters with M = 6, N = 2,Pmax/σ2 = 10 dB and R = 1 bit/s/Hz.

VII. NUMERICAL RESULTS AND DISCUSSIONS

In this section, we present numerical results to validate ourtheoretical analysis in Sections III–VI. A detailed investigationis given on the impact of the interference power constraint,the maximum transmit power constraint, the fading severityparameters, the number of relays and the relay selection schemeon the outage and diversity performance of the secondarysystem. The effect of limited feedback on the relay selectionprobability is also studied.

Fig. 2 illustrates the exact outage probability of the secondarysystem versus the interference power constraint to noise ratioQ/σ2 for various channel fading severity parameters. Thenumber of relays M and the order of the selected relay N are setto 6 and 2, respectively. The solid lines represent our analyticalresults, and the square symbols represent the Monte Carlosimulation results. From Fig. 2, we can see that our analyticalresults match well with the simulation results, which validatesour theoretical analysis. The outage performance improves withthe increase of the fading severity parameters. It is worth notingthat the outage probability has a decline in the low Q region.This is because when Q gets smaller, there are fewer relaysavailable in the successful decoding relay set, so the chancethat the N th best relay makes sense (N ≤ |R(s)|) gets smaller.Besides, the outage performance improves with the increase ofQ in the median Q region, but will reach saturation when Q islarge enough due to the existence of Pmax.

Fig. 3 plots the outage probability of the secondary systemversus the maximum transmit power constraint to noise ratioPmax/σ

2. For the similar reason of the case in Fig. 2, the outage

B(v)=

⎧⎪⎨⎪⎩

1(mpγQ−ms)v

A−v∑

a=1

ms

(mpγQ−ms)a1

(mpγQ)v+1−a1

Γ(v+1−a)

(γPmax

2ms

)i+j+1

G1,22,1

(Pmax

2mpQ

∣∣∣−i−j,a−v0

), for ms = mpγQ

1mv

s

1Γ(v+1)

(γPmax

2ms

)i+j+1

G1,22,1

(γPmax

2ms

∣∣∣−i−j,−v0

), for ms = mpγQ

(33)


Fig. 3. Outage probability versus maximum transmit power constraint tonoise ratio for different channel fading severity parameters with M = 6, N =2, Q/σ2 = 10 dB and R = 1 bit/s/Hz.

Fig. 4. Outage probability versus interference power constraint to noise ratiofor different numbers of relays (M) with N = 2, m1 = m2 = m3 = m4 =3, Pmax/σ2 = 10 dB and R = 1 bit/s/Hz.

probability has a decline in the low Pmax region and reachessaturation in the high Pmax region.

Fig. 4 illustrates the impact of the number of relays M onthe outage performance of the secondary system with the N thbest relay selection (N = 2). In the median and high Q regions,more relays can provide better outage performance. It is alsoobserved that the turning point between the low Q region andthe median Q region shifts left with the increase of M .

Fig. 5 gives the outage performance of the secondary systemfor different relay selection schemes. The number of relays Mis set to 6. It shows that in the median and high Q regions, theoutage performance decreases with the increase of the order ofthe selected relay (i.e., N ) since the performance of the secondhop is worsened. In addition, we can observe that the turningpoint between the low Q region and the median Q region shiftsright with the increase of N .

The impact of the fading severity of the transmission linksand the interference links on the outage performance is illus-trated in Fig. 6. It is observed that the fading severity of the

Fig. 5. Outage probability versus interference power constraint to noise ratiofor different orders of selected relays (N) with M = 6, m1 = m2 = m3 =m4 = 2, Pmax/σ2 = 10 dB and R = 1 bit/s/Hz.

Fig. 6. Outage probability versus interference power constraint to noise ratiofor different fading severity of the transmission links and the interference linkswith M = 6, N = 2, Pmax/σ2 = 10 dB and R = 1 bit/s/Hz.

transmission links has great impact on the outage performance,but the fading severity of the interference links has little impacton the outage performance. This is in compliance with theresults in [29] which investigates the imperfect CSI of thetransmission links and the interference links.

Fig. 7 plots the impact of the fading severity of the first-hop links and the second-hop links on the outage performance.From this figure, we can see that when the number of relaysM is relatively small, the fading severity of the second-hoplinks has more influence on the outage performance than thatof the first-hop links. But when M is relatively large, these twohops have nearly equal impact on the outage performance of thesecondary system.

Figs. 8 and 9 present the exact and asymptotic outage proba-bility curves according to Sections III and IV. From these twofigures, we can observe that the asymptotic outage probabilityis very close to the exact one in high SNR regions. It isindicated that our asymptotic outage probability expressioncan be used to effectively evaluate the outage performance


Fig. 7. Outage probability versus interference power constraint to noise ratiofor different fading severity of first-hop links and second-hop links with N = 2,Pmax/σ2 = 10 dB and R = 1 bit/s/Hz.

Fig. 8. The exact and asymptotic outage probability versus system SNR (γ =1/σ2) for different fading severity of the transmission links with M = 4, N =2 and m2 = m4 = 2, Q = 10 dB, Pmax = 10 dB and R = 1 bit/s/Hz.

of the secondary system in high SNR regions, even for non-integer fading severity parameters. According to our analysisin Section IV, the diversity order of the secondary systemis min(m1,m3)× (M −N) +m3, which coincides with theslope of the curves in these figures.

In Fig. 10, the selection probability of the N th best relaywith limited feedback is illustrated. From this figure, we canobserve that the selection probability of the relay decreaseswith the increase of N . This means the probability that abetter relay is selected is larger than the probability that aworse relay is selected in the limited feedback scenario. Forthe best relay, as the number of feedback bits L increases, itsselection probability gets larger. For the worse relays (N ≥ 3),the selection probability decreases with the increase of L.

Fig. 11 depicts the ergodic capacity of the secondary systemwith one single relay. It can be observed that the ergodiccapacity improves with the increase of the interference powerconstraint Q and will reach saturation when Q is large enough

Fig. 9. The exact and asymptotic outage probability versus system SNR (γ =1/σ2) for different relay selection scheme with m1 = m2 = m3 = m4 = 2,Q = 10 dB, Pmax = 10 dB, σ2 = 1 and R = 1 bit/s/Hz.

Fig. 10. Selection probability of the N th best relay versus the number of feed-back bits (L) with M = 10, m1 = m2 = 3, Q/σ2 = 10 dB, Pmax/σ2 =10 dB and R = 1 bit/s/Hz.

Fig. 11. Ergodic capacity versus interference power constraint to noise ratiofor different fading severity of the transmission links and the interference linkswith M = 1, N = 1 and Pmax/σ2 = 10 dB.


due to the maximum transmit power constraint Pmax. It canalso be seen that the fading severity of the transmission linkshas much greater impact on the ergodic capacity than that ofthe interference links. As the fading of the transmission linksgets severer, the capacity of the secondary system will decreasegreatly. However, as the fading of the interference links getsseverer, the capacity will increase when Q is small and willdecrease when Q is large.

VIII. CONCLUSION AND FUTURE WORKS

In this paper, we study the performance of an underlaycognitive DF relay network with the N th best relay selec-tion over Nakagami-m fading channels, considering both themaximum transmit power limit and the interference powerconstraint. The exact and asymptotic outage probability ex-pressions for such system are derived. Through asymptoticanalysis, we obtain the diversity order of the secondary system,which is min(m1,m3)× (M −N) +m3, where M representsthe number of relays and m1, m3 denote the fading severityparameters of the first-hop transmission link and the second-hop transmission link. It is indicated that the fading severity ofchannels, the number of relays and the relay selection schemehave great impact on the outage performance of the secondarysystem. The selection probability of the N th best relay isgiven in the limited feedback scenario. Besides, we obtain theexact ergodic capacity for the special case where there is onesingle relay in the secondary system. The theoretical analysisis validated by simulations. The results show that the fadingseverity of the transmission links has more impact on the outageperformance as well as the ergodic capacity than that of theinterference links. In this paper the direct link for the secondarysystem is not considered. We believe that considering the directlink can be an interesting topic in our future works.

APPENDIX APROOF OF LEMMA 1

According to the definition of R(s), the probability of the setR(s) can be written as

Pr(R(s))=Pr

⎡⎣ ⋂i∈R(s)

(γsi≥γth),⋂

i∈R(s)

(γsi<γth)

⎤⎦

= Pr

⎡⎣ ⋂i∈R(s)

(Pmaxgsi

σ2≥γth

),⋂

i∈R(s)

(Pmaxgsi

σ2<γth

), Pmax<

Q

gsp

⎤⎦

︸︷︷︸I1

+ Pr

⎡⎣ ⋂i∈R(s)

(Qgsiσ2gsp

≥γth

),

⋂i∈R(s)

(Qgsiσ2gsp

<γth

), Pmax≥

Q

gsp

⎤⎦

︸︷︷︸I2

,

(34)

where the first summand denotes that all relays can successfullydecode in the set R(s), and the second summand denotes thatthe other relays decode failed.

The first summand in (34) can be calculated as

I1=

[Pr

(Pmaxgsi

σ2≥γth

)]n[Pr

(Pmaxgsi

σ2< γth

)]M−n

× Pr

(Pmax <

Q

gsp

)

=

[1−Fgsi

(γthσ

2

Pmax

)]n [Fgsi

(γthσ

2

Pmax

)]M−n

Fgsp

(Q

Pmax

)

=

⎡⎣1− γ

(m1,

m1γth

γPmax

)Γ(m1)

⎤⎦n ⎡⎣γ

(m1,

m1γth

γPmax

)Γ(m1)

⎤⎦M−n

×γ(m2,

m2QPmax

)Γ(m2)

. (35)

In the second summand, it is found that all parts are corre-lated with the variable gsp. Hence, the second summand can bewritten as

I2=

∫ ∞

QPmax

[Pr

(Qgsiσ2t

≥γth

)]n[Pr

(Qgsiσ2t

<γth

)]M−n

fgsp(t)dt

=

∫ ∞

QPmax

⎡⎣1− γ

(m1,

m1γthtγQ

)Γ(m1)

⎤⎦n ⎡⎣γ

(m1,

m1γthtγQ

)Γ(m1)

⎤⎦M−n

× mm22 tm2−1e−m2t

Γ(m2)dt

=

n∑i=0

{Ci

n(−1)i∫ ∞

QPmax

⎡⎣γ

(m1,

m1γthtγQ

)Γ(m1)

⎤⎦M−n+i


Γ(m2)dt

}(36)

Then, utilizing the following expansion for an incompletegamma function [24, eq. (8.352.6)]:

γ(n, x) = Γ(n)

(1− e−x

n−1∑i=0

xi

i!

). (37)

I2 can be transformed into

I2=

n∑i=0

{Ci

n(−1)i∫ ∞

QPmax

⎡⎣γ

(m1,

m1γthtγQ

)Γ(m1)

⎤⎦M−n+i


Γ(m2)dt

}

=

n∑i=0

{Ci

n(−1)i∫ ∞

QPmax

⎡⎢⎣1−e

−m1γtht

γQ

m1−1∑j=0

(m1γtht

γQ

)j

j!

⎤⎥⎦M−n+i


Γ(m2)dt

}


=

n∑i=0

M−n+i∑l=0

{Ci

nClM−n+i(−1)i+l mm2

2

Γ(m2)

∫ ∞

QPmax

tm2−1

× e−m1γthl+m2γQ

γQt

⎡⎣m1−1∑

j=0

1

j!

(m1γtht

γQ

)j⎤⎦l

dt

}(38)

According to the multinomial theorem, the term[m1−1∑j=0

1j!

(m1γtht

γQ

)j]l

in (38) can be expanded as

⎡⎣m1−1∑

j=0

1

j!

(m1γtht

γQ

)j⎤⎦l

= l!∑

w0+w1+···+wm1−1=l

m1−1∏j=0

{1

wj !

[1

j!

(m1γtht

γQ

)j]wj

}

= l!∑

w0+w1+···+wm1−1=l

⎧⎨⎩(m1γtht

γQ

)Hm1−1∏j=0

[1

wj !

(1

j!

)wj]⎫⎬⎭,

(39)

where H =m1−1∑j=0

jwj . So I2 can be rewritten as

I2=

n∑i=0

M−n+i∑l=0

∑w0+w1+...+wm1−1=l

{Ci

nClM−n+i(−1)i+l m

m22 l!

Γ(m2)

×(m1γthγQ

)H m1−1∏j=0

[1

wj !

(1

j!

)wj]

×∫ ∞

QPmax

tm2−1+He−m1γthl+m2γQ

γQtdt

}

=

n∑i=0

M−n+i∑l=0

∑w0+w1+...+wm1−1=l

{Ci

nClM−n+i(−1)i+l m

m22 l!

Γ(m2)

×(m1γthγQ

)H (γQ

m2γQ +m1γthl

)m2+H

× Γ

(m2+H,

m2γQ +m1γthl

γPmax

)×

m1−1∏j=0

[1

wj !

(1

j!

)wj]}

.

(40)

Substituting (35) and (40) into (34), we can obtain (3).

APPENDIX BPROOF OF LEMMA 2

According to the expression of γrd (4), the CDF of γrd canbe written as

Fγrd(x)= Pr

(N th max

i∈R(s)γid ≤ x

)

=

N∑k=1

Ck−1n [Pr(γid ≤ x)]n−k+1 [Pr(γid < x)]k−1

=N∑

k=1

Ck−1n [Fγid

(x)]n−k+1[1−Fγid(x)]k−1, (41)

where Fγid(x) is expressed as

Fγid(x) = Pr(γid ≤ x) = Pr

(min(Pmax, Q/gip)gid

σ2≤ x

)=Pr

(Pmaxgid

σ2≤x, Pmax<

Q

gip

)+Pr

(Qgidσ2gip

≤x, Pmax≥Q

gip

)

= Pr

(Pmaxgid

σ2≤ x

)Pr

(Pmax <

Q

gip

)︸︷︷︸

J1

+

∫ ∞

QPmax

Pr

(Qgidσ2t

≤ x

)fgip(t)dt︸︷︷︸

J2

. (42)

Utilizing the CDF of gip and gid,J1 can be easily calculated as

J1 =γ(m3,

m3xγPmax

)Γ(m3)

γ(m4,

m4QPmax

)Γ(m4)

. (43)

For J2, the integral can be calculated as following by usingthe expansion expression (37) as follows:

J2 =

∫ ∞

QPmax

γ(m3,

m3xtγQ

)Γ(m3)

mm44 tm4−1e−m4t

Γ(m4)dt

=mm4

4

Γ(m4)

∫ ∞

QPmax

tm4−1e−m4t

[1−e

−m3x

γQtm3−1∑i=0

1

i!

(m3x

γQt

)i]dt

=Γ(m4,

m4QPmax

)Γ(m4)

− mm44

Γ(m4)

m3−1∑i=0

[1

i!

(xm3

γQ

)i

×(xm3

γQ+m4

)−(m4+i)

Γ

(m4 + i,

xm3 +m4γQγPmax

)].

(44)

Substituting (43) and (44) into (42), we can get Fγid(x).

Then from (41), we can obtain Fγrd(x) as (5).

APPENDIX CPROOF OF LEMMA 3

According to (9), I1 can be written as the following expres-sion in high SNR regions:

I1γ→∞≈

⎡⎣1−

(m1γth

Pmaxγ

)m1

Γ(m1+1)

⎤⎦n⎡⎣(

m1γth

Pmaxγ

)m1

Γ(m1+1)

⎤⎦M−n

γ(m2,

m2QPmax

)Γ(m2)

≈[

1

Γ(m1 + 1)

(m1γthPmaxγ

)m1]M−n γ

(m2,

m2QPmax

)Γ(m2)

. (45)


For I2, it can be approximated written as

I2γ→∞≈

∫ ∞

QPmax

⎡⎣1−

(m1γtht

Qγ

)m1

Γ(m1 + 1)

⎤⎦n ⎡⎣(

m1γthtQγ

)m1

Γ(m1 + 1)

⎤⎦M−n


Γ(m2)dt

≈

⎡⎣(m1γth

Qγ

)m1

Γ(m1+1)

⎤⎦M−n(

1m2

)m1(M−n)

Γ(m2)Γ

(m1(M−n)+m2,

m2Q

Pmax

).

(46)

Substituting (45) and (46) into (34), we can obtain (10).

APPENDIX DPROOF OF LEMMA 4

According to (9), J1 can be written as the following expres-sion in high SNR regions:

J1γ→∞≈ 1

Γ(m3 + 1)

(m3x

Pmaxγ

)m3 γ(m4,

m4QPmax

)Γ(m4)

. (47)

For J2, it can be written as

J2γ→∞≈

∫ ∞

QPmax

1

Γ(m3 + 1)

(m3xt

Qγ

)m3 mm44 tm4−1e−m4t

Γ(m4)dt

=

(m3x

m4Qγ

)m3

⎛⎝Γ

(m3 +m4,

m4QPmax

)Γ(m3 + 1)Γ(m4)

⎞⎠ . (48)

Substituting (47) and (48) into (42), we can obtain theasymptotic expression for the CDF of γid as

Fγid(x)

γ→∞≈

(m3x

γ

)m3 1

Γ(m3 + 1)Γ(m4)

[(1

Pmax

)m3

× γ

(m4,

m4Q

Pmax

)+

(1

m4Q

)m3

Γ

(m3+m4,

m4Q

Pmax

)]. (49)

From (49), we can easily obtain that

1− Fγid(x)

γ→∞≈ 1. (50)

Substituting (49) and (50) into (41), we can obtain theasymptotic expression for the CDF of γrd as (11).

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Xing Zhang (M’10–SM’14) received the Ph.D. de-gree from Beijing University of Posts and Telecom-munications (BUPT), Beijing, China, in 2007. SinceJuly 2007, he has been with the School of Infor-mation and Communications Engineering, BUPT,where he is currently an Associate Professor. He isthe author/coauthor of two technical books and morethan 50 papers in top journals and international con-ferences and filed more than 30 patents (12 granted).His research interests are mainly in wireless com-munications and networks, green communications

and energy-efficient design, cognitive radio and cooperative communications,traffic modeling, and network optimization.

Prof. Zhang has served on the editorial boards of several international jour-nals, including KSII Transactions on Internet and Information Systems and theInternational Journal of Distributed Sensor Networks, and as a TPC Cochair/TPC member for a number of major international conferences, including Mobi-Quitous 2012, IEEE ICC/GLOBECOM/WCNC, CROWNCOM, Chinacom,etc. He received the Best Paper Award in the 9th International Conferenceon Communications and Networking in China (Chinacom 2014) and the 17thInternational Symposium on Wireless Personal Multimedia Communications(WPMC 2014).

Yan Zhang (SM’10) received the Ph.D. degree fromNanyang Technological University, Singapore. SinceAugust 2006, he has been with Simula ResearchLaboratory, Fornebu, Norway, where he is currentlythe Head of the Department of Networks. His recentresearch interests include wireless networks,machine-to-machine communications, and smartgrid communications. He is a Regional Editor or anAssociate Editor on the editorial board and a GuestEditor of a number of international journals.

Zhi Yan received the B.Sc. degree in mechani-cal engineering and automation and the Ph.D. de-gree in communication and information system fromBeijing University of Posts and Telecommunica-tions (BUPT), Beijing, China, in 2007 and 2012,respectively. From August 2012 to March 2014, hewas a Researcher with the Network Technology Re-search Center, China Unicom Research Institute. Heis currently an Assistant Professor with the Schoolof Electrical and Information Engineering, HunanUniversity, Changsha, China. His current research

interests are in the cognitive radio, cooperative communication, and cellularnetwork traffic analysis and modeling.

Jia Xing received the B.S. degree in communica-tion engineering in 2012 from Beijing Universityof Posts and Telecommunications, Beijing, China,in 2012 where she is currently working towardthe M.S. degree in the Key Laboratory of Univer-sal Wireless Communications, School of Informa-tion and Communication Engineering. Her researchinterests include cognitive radio and cooperativecommunication.

Wenbo Wang received the B.S., M.S., and Ph.D. de-grees from Beijing University of Posts and Telecom-munications (BUPT), Beijing, China, in 1986, 1989,and 1992, respectively. He is currently a Professorwith and the Executive Vice Dean of the GraduateSchool, BUPT. He is also the Deputy Director of theKey Laboratory of Universal Wireless Communica-tion, Ministry of Education. He has published morethan 200 journal and international conference papersand 6 books. His current research interests includeradio transmission technology, wireless network the-

ory, and software radio technology.

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Performance Analysis of Cognitive Relay Networks …folk.uio.no/yanzhang/IEEEJSACZhangMay2015.pdfIEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.33,NO.5,MAY2015 865 Performance