Performance Analysis of Incremental Amplify-and-Forward Relaying Protocols with Nth Best Partial Relay Selection Under Interference Constraint

Wireless Pers Commun (2013) 71:2741–2757DOI 10.1007/s11277-012-0968-9

Performance Analysis of IncrementalAmplify-and-Forward Relaying Protocols with Nth BestPartial Relay Selection Under Interference Constraint

Tran Trung Duy · Hyung-Yun Kong

Published online: 20 December 2012© Springer Science+Business Media New York 2012

Abstract In this paper, we investigate two incremental amplify-and-forward relaying pro-tocols in cognitive underlay networks. In the proposed protocols, whenever the secondarydestination cannot receive the secondary source’s signal successfully, it requests a retrans-mission from one of M secondary relays. In the first protocol, we assume that a secondaryrelay with the Nth best channel gain to the secondary source is used to forward the receivedsignal to the secondary destination. In the second protocol, relying on the quality of channelsbetween the secondary relay and secondary destination and between the secondary relay andprimary user, the Nth best relay is chosen for the retransmission. We derive exact closed-formexpressions of the outage probability and average number of time slots for both protocolsover Rayleigh fading channel. Finally, these mathematical expressions are then verified byMonte Carlo simulations.

Keywords Cooperative relaying · Cognitive underlay network · Amplify-and-forward ·Outage probability · Relay selection

1 Introduction

Cooperative relaying transmission protocols [1,2] have been used to enhance the performanceof wireless communication systems under fading environments. In cooperative communica-tion with multiple relays, the systems expect to choose the best relay, which will cooperatewith the source to transmit signals to the destination. In [3,4], by relying on the channel stateinformation (CSI) across two hops of each relay, the best relay is selected for the cooperation.In [5,6], the best partial relay selection method, in which only the CSI of the source-relaylink is used, has been studied. However, in some cases, the system cannot choose the best

T. T. Duy · H.-Y. Kong (B)University of Ulsan, Ulsan, South Koreae-mail: [email protected]

H.-Y. Konge-mail: [email protected]

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2742 T. T. Duy, H.-Y. Kong

relay due to scheduling or load balancing issues. In [7,8], the authors considered cooperativeschemes in which the Nth best relay would be used to forward the source’s information tothe destination.

Recently, spectrum sharing protocols with interference constraints [9] (or cognitive under-lay protocols) have gained much attention in cognitive radio network research. In these mod-els, the secondary transmitters must adapt their transmit power so that the interference createdat the primary user is below the maximum allowable interference. Due to the constraint onthe transmit power; the performance of cognitive underlay protocols is severely degraded infading environments. To enhance the performance of the secondary network, some recentlypublished works proposed the use of cooperative communication. In [10,11], the authorsproposed a cognitive decode-and-forward relaying protocol with partial relay selection andfull relay selection, respectively. In [12,13], the authors investigated relaying schemes usingthe best partial and full amplify-and-forward relay. In [14], the cooperative spectrum shar-ing network with amplify-and-forward and selection diversity has been proposed. However,the protocols in [10–14] use two time slots to transmit the signal over orthogonal channels,which results in ineffective spectrum usage. To solve the spectral efficiency problem in suchschemes, incremental cooperative transmission protocols [4,15] can be used. In incrementalrelaying schemes, the destination will feed back a message to indicate when a relay is allowedto relay the source’s information to the destination.

In this paper, we investigate the Nth best partial relay selection scheme in cognitive under-lay networks. In particular, we propose two incremental cooperative relaying protocols inwhich the relay uses amplify-and-forward mode to forward the signal received from thesecondary source to the secondary destination. As discussed above, the incremental relay-ing scheme is used to reduce the number of time slots used, as compared to conventionalcooperative protocols [1,2]. In addition, we propose to use the partial relay selection methodbecause this technique is simpler and easier to implement than the full relay selection method.In both proposed protocols, the transmission occurs as follows. First, the secondary sourcebroadcasts the signal to the secondary destination during the first time slot. If the secondarydestination can receive the signal successfully, then transmission of the signal ends. Other-wise, the secondary destination will request a retransmission from a chosen relay. In the firstproposed protocol, we assume that the system can choose the relay with the Nth best channelgain of the source-relay link for the retransmission. In the second one, the Nth best relay isselected, relying on the quality of the channels between the secondary relay and secondarydestination and between the secondary relay and primary user.

The main contribution of this paper is that we derive exact closed-form expressions ofthe average outage probability and number of time slots under Rayleigh fading channel.We can observe that the derivations in [12,13] did not include the direct link in the per-formance analysis. When considering the direct link, due to the common link between thesecondary source and primary user, the instantaneous signal-to-noise (SNR) of the directlink and that of the link between the secondary source and secondary relays are correlatedrandom variables (RVs). Next, to determine the diversity order of the proposed protocols,we derive the approximate expressions for the average outage probability. Finally, we per-form Monte-Carlo simulations to verify the derivations. Results show that the simulation andtheoretical results are in good agreement and the diversity gain of the proposed protocols istwo.

The rest of the paper is organized as follows. The system model and the proposedschemes are described in Sect. 2. In Sect. 3, a performance evaluation of the protocols isdescribed. The simulation results are presented in Sect. 4. Finally, the paper is concluded inSect. 5.

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Performance Analysis of Incremental Amplify-and-Forward Relaying Protocols 2743

Fig. 1 System model of thecooperative relaying schemein a cognitive underlay network

SS SD

SR j

()

11, j

dh

()2

2,

j

dh( )4 4, jd h

( )0 0,d h

1SR SR M

PU

( )3 3,d h

2 System Model

In Fig. 1, we present the system model of the cooperative relaying scheme in the cognitiveunderlay network. In this model, the secondary source SS attempts to transmit its data to thesecondary destination SD with the help of M secondary relays (SRs), i.e., SR1,SR2, . . .,SRM .In the underlay approaches [9–14], the secondary source and secondary relay must adapttheir transmit power so that the interference caused at the primary user (PU) is below anallowable level IP . We assume that distances from the secondary source (or from the sec-ondary destination) to the secondary relays are same. As presented in Fig. 1, we denoted0, d1, d2, d3, and d4 as the distance of links SS − SD, SS − S R, S R − SD, SS − PU , andS R − PU , respectively. Also, let us denote h0, h1 j , h2 j , h3 and h4 j as the channel coeffi-cient of links SS − SD, SS − S R, S R − SD, SS − PU , and S R − PU , respectively, wherej ∈ {1, 2, . . . ,M}. It is assumed that all of the channels are modeled by independent and flatRayleigh fading channels. If we set γ0 = |h0|2, γ3 = |h3|2 and γi j = |hi j |2, i ∈ {1, 2, 4};then γ0, γ3 and γi j are exponential RVs with parameters [4]: λ0 = dβ0 , λ3 = dβ3 and λi = dβi ,respectively, where β is path-loss exponent that varies from 2 to 6.

Before transmitting the signal, we assume that the system can choose the Nth best sec-ondary relay for the cooperation (1 ≤ N ≤ M). If N = 1, then the best secondary relay isused to help the secondary source. When N = M , the system must use the worst secondaryrelay to cooperate with the direct transmission between the secondary source and secondarydestination.

In the first proposed protocol, named IAF1, the relay selection strategy is expressed as

Rb : γ1b = Nth maxj=1,2,...,M

(γ1 j). (1)

In the second proposed protocol, named IAF2, the system can choose a relay, relying on thefollowing strategy:

Rc : γ2c

γ4c= Nth max

j=1,2,...,M

(γ2 j

γ4 j

). (2)

In (1) and (2), we denote Rb and Rc as the chosen relay of the IAF1 and IAF2 protocols,respectively. Now, the data transmission of both protocols occurs as describe below.

At the first time slot, the secondary source transmits its data to the secondary destination,which can be received by the chosen secondary relay. Then, the destination attempts to decodethe received data. If the secondary destination can decode the data successfully, then it feedsback an ACK message to the secondary source and chosen secondary relay. In this case, thedata transmission is successful and the chosen secondary relay does nothing. Otherwise, thedestination sends a NACK message to request a retransmission from the chosen secondaryrelay. Then, this relay forwards the received data to the destination using amplify-and-forwardmode.

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3 Performance Analysis

3.1 Exact Outage Probability

In this work, we assume that all nodes are equipped with a single antenna and operate inhalf-duplex mode. Hence, the data transmission occurs on orthogonal channels, which usetime division multiple access technique (TDMA). First, we consider the direct transmissionprotocol (DT) in which the data is directly transmitted from the secondary source to thesecondary destination without the help of relays. In this protocol, the source transmits thedata at the allowed maximum power. Similar to [16, (1)], the instantaneous SNR received atthe destination is expressed given as

ψ0 = IP |h0|2N0|h3|2 = Qγ0

γ3, (3)

where Q = IP/N0 and N0 is variance of Gaussian noise, which is assumed to be the sameat all of the receivers such as the secondary relays and secondary destination.

Since achievable rate of the data transmission between the secondary source and secondarydestination is RDT

1 = log2 (1 + ψ0), the average outage probability of the DT protocol isgiven similarly to [16, (3)] as

PoutDT = Pr

[RDT

1 < R]

= Pr[ψ0 < 2R − 1

]= λ0

(2R − 1

)

λ0(2R − 1

)+ λ3 Q, (4)

where R is the target rate of the system.Considering the IAF1 protocol; from (1), the cumulative density function (CDF) of the

channel gain γ1b is expressed similarly to [7, (22)] as

Fγ1b (x) =N∑

n=1

Cn−1M (1 − exp (−λ1x))M−n+1 exp (− (n − 1) λ1x)

= 1 +N∑

n=1

M−n+1∑

m=0,n+m>1

(−1)mCn−1M Cm

M−n+1 exp (− (n + m − 1) λ1x) (5)

where Cxy = y!

x !(y−x)! with x and y are integers and y ≥ x .In addition, the instantaneous SNR of the relay link in the IAF1 protocol is given as

[13, (5)]:

γ AF1 = ψ1ψ2

ψ1 + ψ2 + 1, (6)

where ψ1 = Q γ1bγ3

and ψ2 = Q γ2bγ4b

.

Since the achievable rate of the direct and relay links is 12 log2 (1 + ψ0) and 1

2 log2(1 + γ AF

1

), respectively, the average outage probability of the IAF1 protocol is formulated

as follows:

PoutIAF1 = Pr

[1

2log2 (1 + ψ0) < R,

1

2log2

(1 + γ AF

1

)< R

]

= Pr[ψ0 < γth, γ

AF1 < γth

](7)

where γth = 4R − 1.

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Since the RV γ3 appears in both ψ0 and γ AF1 , so ψ0 and γ AF

1 are not independent, i.e.,Pout

IAF1 �= Pr[ψ0 < γth

]Pr[γ AF

1 < γth]. Hence, we use “Appendix 1” to calculate Pout

IAF1 asfollows:

PoutIAF1 = λ0ρ

λ3 + λ0ρ+

N∑

n=1

M−n+1∑

m=0,n+m>1

(−1)m Cn−1M Cm

M−n+1λ4

λ2ρ + λ4

×

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

λ3

λ3 +�(λ2γth + λ4 Q

) − λ3

λ3 + λ0ρ +�(λ2γth + λ4 Q

)

− �λ2λ3(1 + γth

)

(� (λ4 Q − λ2)+ λ3)2

[

ln

(�(λ2γth + λ4 Q

)+ λ3

�λ2(1 + γth

)

)

− �(λ4 Q − λ2)+ λ3

�(λ2γth + λ4 Q

)+ λ3

]

+ �λ2λ3(1 + γth

)

(� (λ4 Q − λ2)+ λ0ρ + λ3)2

[

ln

(�(λ2γth + λ4 Q

)+ λ0ρ + λ3

�λ2(1 + γth

)

)

− �(λ4 Q − λ2)+ λ0ρ + λ3(γth + κ2 Q

)�+ λ0ρ + λ3

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

(8)

where ρ = γth/Q and � = (n + m − 1) λ1ρ

λ2γth + λ4 Q.

Considering the IAF2 protocol; the instantaneous SNR of the relay link is expressed as

γ AF2 = ψ3ψ4

ψ3 + ψ4 + 1, (9)

where ψ3 = Q γ1cγ3

and ψ4 = Q γ2cγ4c

.

Since Pr[

Qγ2 jγ4 j

< x]

= λ2xλ2x+λ4 Q , j ∈ {1, 2, . . . ,M}, similarly to [8, (30)], we can

present the probability density function (PDF) of the channel gain ψ4 = Q γ2cγ4c

as follows:

fγ1c (x) = M !(N − 1)! (M − N )!

(λ2x

λ2x + λ4 Q

)M−N (λ4 Q

λ2x + λ4 Q

)N−1λ2λ4 Q

(λ2x + λ4 Q)2

= M !(N − 1)! (M − N )! (κQ)N x M−N

(x + κQ)M+1 (10)

where κ = λ4/λ2.Also, the average outage probability of the IAF2 protocol can be formulated as

PoutIAF2 = Pr

[ψ0 < γth, γ

AF2 < γth

]. (11)

Now, we use “Appendix 2” to obtain PoutIAF2 as follows:

PoutIAF2 = λ0ρ

λ3 + λ0ρ− M !(N − 1)! (M − N )!

(κQ)N λ3

γ M−Nth (γth + κQ)M

M−N∑

i=0

N−1∑

j=0

(−1)N−1− j CiM−N C j

N−1− j

(κ

ρ

)M−N−i

×

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(−1)M−i− j ((1+γth))M−i− j

M−i− j+1

⎡

⎢⎢⎣

2 F1

(1,M−i− j+1;M−i− j+2,

(1+γth )+λ3(γth+κQ)+λ3

)

((γth+κQ)+λ3)M−i− j+1

− 2 F1

(1,M−i− j+1;M−i− j+2,

(1+γth)+λ3+λ0ρ(γth+κQ)+λ3+λ0ρ

)

((γth+κQ)+λ3+λ0ρ)M−i− j+1

⎤

⎥⎥⎦

+M−i− j−1∑

u=0

(−1)uu!∏uz=0 (M−i− j−z)

[((1+γth))

u

((γth+κQ)+λ3)u+1 − ((1+γth ))

u

((γth+κQ)+λ3+λ0ρ)u+1

]

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

(12)

where = λ3ργth+κQ .

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3.2 Asymptotic Outage Probability

Using [13, (21)], for (6), the average outage probability of the IAF1 protocol can be approx-imated as

PIAF1out ≈ Pr

[ψ0 < γth,min (ψ1, ψ2) < γth

]

︸︷︷︸P1

out

. (13)

First, we rewrite the probability P1out in (13) under the following form as

P1out = Pr

[ψ0 < γth

]− Pr[ψ0 < γth,min (ψ1, ψ2) ≥ γth

]

= Pr[ψ0 < γth

]− Pr[ψ2 ≥ γth

]Pr[ψ0 < γth, ψ1 ≥ γth

]

= Pr[ψ0 < γth


]Pr[γ1b ≥ ργ3 > γ0

]

= λ0ρ

λ0ρ + λ3− λ4

λ2ρ + λ4

+∞∫

0

fγ0 (x) dx

+∞∫

xρ

fγ3 (y) dy

+∞∫

ρy

fγ1b (z) dz (14)

where fγ1b (z) is the PDF of γ1b which can easily obtain by differentiating Fγ1b (z) presentedin (5) with respect to z.

Therefore, substituting the PDFs fγ1b (z) , fγ3 (y) and fγ0 (x) into (14) and after somesimple manipulation, we obtain

PIAF1out ≈ P1

out = λ0ρ

λ0ρ + λ3

− λ4

λ2ρ + λ4

⎡

⎣N∑

n=1

M−n+1∑

m=0,n+m>1

(−1)m+1 Cn−1M Cm

M−n+1λ0λ3ρ

((n + m − 1) λ1ρ + λ3) (λ0ρ + (n + m − 1) λ1ρ + λ3)

⎤

⎦

(15)

From (15), when Q is large, i.e., Q → +∞, (or ρ is very small, i.e., ρ → 0), by using thesecond-order derivative, we can approximate PIAF1

out as follows:

PIAF1out ≈ P1

out (ρ = 0)+(∂P1

out

∂ρ(ρ = 0)

)ρ +

(∂2P1

out

∂ρ2 (ρ = 0)

)ρ2

2

≈⎛

⎝λ0λ2

λ3λ4+

N∑

n=1

M−n+1∑

m=0,n+m>1

2 (−1)m Cn−1M Cm

M−n+1 (n + m − 1) λ0λ1

λ23

⎞

⎠ ρ2,

(16)

In (16), it is noted that P1out (ρ = 0) = 0 and ∂P1

out∂ρ

(ρ = 0) = 0.From (16), we can easily calculate the diversity order of the IAF1 protocol as

dIAF1 = − limQ→+∞

log(P IAF1

out

)

log (Q)

= − limρ→0

log

((λ0λ2λ3λ4

+∑Nn=1

∑M−n+1m=0,n+m>1

2(−1)m Cn−1M Cm

M−n+1(m+n−1)λ0λ1

λ23

)ρ2)

log (ϕth)− log (ρ)(17)

= 2

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Considering the PR2 protocol, similar to (13), we can approximate the average outage prob-ability PIAF2

out as

PIAF2out ≈ Pr

[ψ0 < γth,min (ψ3, ψ4) < γth

]

≈ Pr[ψ0 < γth


]Pr[ψ0 < γth, ψ3 ≥ γth

]

≈ λ0ρ

λ0ρ + λ3− Pr

[ψ4 ≥ γth

]+∞∫

0

fγ0 (x) dx

+∞∫

xρ

fγ3 (y) dy

+∞∫

ρy

fγ1c (z) dz.

(18)

First, we have

+∞∫

0

fγ0 (x) dx

+∞∫

xρ

fγ3 (y) dy

+∞∫

ρy

fγ1c (z) dz = λ0λ3ρ

(λ1ρ + λ3) (λ0ρ + λ1ρ + λ3), (19)

Second, with the help of [7, (22)], we can obtain Pr[ψ4 ≥ γth

]in (18) as

Pr[ψ4 ≥ γth

] = 1−Pr

[γ2c

γ4c< ρ

]=1−

N∑

m=1

Cm−1M

(λ2ρ

λ2ρ + λ4

)M−m+1 (λ4

λ2ρ + λ4

)m−1

= 1 −N∑

m=1

Cm−1M

λM−m+12 λm−1

4 ρM−m+1

(λ2ρ + λ4)M

(20)

Substituting (19), (20) into (18), we obtain

PIAF2out ≈ λ0ρ

λ0ρ + λ3− λ0λ3ρ

(λ1ρ + λ3) (λ0ρ + λ1ρ + λ3)[

1 −N∑

m=1

Cm−1M

λM−m+12 λm−1

4 ρM−m+1

(λ2ρ + λ4)M

]

(21)

Similar to (16), by using the second-order derivative for (21), we can obtain an approximateexpression for PIAF2

out at high value of Q as follows:

PIAF2out ≈

⎧⎪⎨

⎪⎩

2 λ0λ1λ2

3ρ2; If M > N

(2 λ0λ1λ2

3+ M λ0λ2

λ3λ4

)ρ2; If M = N

. (22)

Relying on (22), it is easy to obtain the diversity gain of this protocol as

dIAF2 = − limQ→+∞

log(P IAF2

out

)

log (Q)= 2 (23)

3.3 Average Number of Time Slots

In the proposed protocols, if the secondary destination can successfully decode the signalreceived from the secondary source, the system only uses one time slot. Otherwise, two time

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Fig. 2 The outage probabilityas a function of Q in dB whenxR = xP = 0.5 and M = 2

slots are used to transmit the signal. Therefore, the average number of time slots used in bothprotocols can be calculated as follows:

T S = 1 × Pr[ψ0 < γth

]+ 2 × Pr[ψ0 ≥ γth

] = 2 − Pr[ψ0 < γth

]

= λ0ρ + 2λ3

λ0ρ + λ3(24)

4 Simulation Results

In this section, we present Monte Carlo simulations to verify the derived expressions above.In a two-dimensional plane, we assume that the coordinates of the secondary source SS, thesecondary destination SD, the secondary relay SR and the primary receiver PR are (0,0),(1,0), (xR , 0) and (xP , yP ), respectively, where 0 < xR, xP < 1. Therefore, d0 = 1, d1 =xR, d2 = 1 − d1, d3 =

√x2

P + y2P and d4 =

√(xR − xP )

2 + y2P . In all simulations, we

assume the target rate of the system R equals 1 (bps), the path-loss exponent equals to 3 andthe co-ordinate yP equals to 0.5.

In Fig. 2, we present the outage probability as a function of Q in dB. In this figure, weconsider a symmetric network with xR = xP = 0.5 and M = 2. Because the proposedprotocols can use more than one time slot to transmit the data, the transmission rate of theproposed protocols can be lower than that of the DT protocol. Indeed, at low Q value, whenthe transmit power of the secondary source and relays is strictly limited, we can see that theDT protocol outperforms the proposed protocols. However, at high values of Q, the outageperformance of the IAF1 and IAF2 protocols is better than that of the DT protocol. This isdue to the fact that the proposed protocols use cooperative transmission, which improvesthe diversity gain. Indeed, as we can see, the proposed protocols obtain the higher slope ofthe outage probability, which means that their diversity order is higher than that of the DT

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Fig. 3 The outage probability as a function of Q in dB when xR = 0.6, xP = 0.4 and M = 3

protocol. Next, it can be observed that when the best relay (N = 1) can be chosen for thecooperation, the outage performance of the proposed protocols is better. In addition, the IAF2protocol outperforms the IAF1 protocol when the best relay is chosen. However, when thesecond best relay (N = 2) is selected, the performance of both protocols is quite similar.

Figure 3 shows the outage performance of the protocols in an asymmetric network withxR = 0.6, xP = 0.4 and M = 3. Similar to Fig. 2, the proposed protocol outperforms theDT protocol at high Q region. We can also see that the outage performances of the proposedprotocols tend to parallel with each other, which means that they obtain same diversity gain.Different from Fig. 2, in this figure, the outage performance of the IAF1 protocol is betterthan that of the IAF2 protocol when N = 1 and worse than that of the IAF2 protocol whenN = 3. From the results shown Figs. 2 and 3, we can see that the performance of the proposedprotocols depends on the position of the secondary relays and primary user. Hence, for eachspecific position of these terminals, the cognitive underlay network can choose the IAF1protocol or the IAF2 protocol for obtaining the better performance. Finally, it is also seenthat the simulation and theoretical results are in good agreement.

To determine the diversity order of the protocols, in Figs. 4 and 5, we present the outageperformance at high value of Q. As presented in these figures, the diversity order of theproposed protocols equals to 2 while that of the DT protocol equals to 1. Moreover, we alsocan see that at high value of Q, the approximate results match with exact results very well.Therefore, the results presented in Figs. 4 and 5 can validate the exactness of Eqs. (15), (16),(17), (21), (22) and (23).

In Fig. 6, we present the impact of the distance d1 on the outage performance of theproposed protocols. In this simulation, we place the PU at the position (0.5, 0.5). In addition,we assume that the system can select the best relay for the cooperation (N = 1) and the valueof Q is 5 dB. First, it is evident that the outage performance of the proposed protocol increaseswith the number of relays M. Second, the outage probability of the IAF2 protocol increases

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Fig. 4 The outage probability as a function of Q in dB when xP = 0.5, xP = 0.6,M = 2 and N = 1

Fig. 5 The outage probability as a function of Q in dB when xP = 0.5, xP = 0.7 and M = 2

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Fig. 6 The outage probability as a function of distance d1 when xP = 0.5, Q = 5 dB and N = 1

Fig. 7 The outage probability as a function of xP when xR = 0.5, Q = 7.5 dB and M = 4

with increasing d1 and that of the IAF1 protocol reaches an optimal value when the distancebetween the SS and SR is about 0.75. In addition, from this figure, we can determine theposition of relays where the proposed protocols outperform the DT protocol. For example,the performance of the IAF2 protocol is better than that of the DT protocol when d1 is lowerthan 0.75.

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Fig. 8 The average number oftime slots used in two proposedprotocols as a function of xP

In Fig. 7, we present the outage probability as a function of xP when xR = 0.5, Q = 7.5dB and M = 4. As presented in this figure, the outage probability of the protocols decreasesas xP increases. This is due to the fact that as xP increases, the distances from the SS andSR to the PU also increase so the SS and SR can use a higher transmit power for sending thesignal.

Figure 8 presents the average number of time slots used in two proposed protocols as afunction of xP . We can see that the average number of time slots reduce with increasing xP

or Q. Again, the simulation and theoretical results are in good agreement which verify ourderivation in (24).

5 Conclusions

In this paper, we investigated the outage performance of two incremental amplify-and-forward protocols with Nth best relay selection in the cognitive underlay network. Wederived the exact closed-form expressions of the outage probability and then verified ouranalysis by Monte Carlo simulations. The results showed that the proposed protocol has ahigher diversity gain compared with the direct transmission protocol. Moreover, the perfor-mance of the proposed protocols is significantly affected by the relay selection, the numberof relays, the positions of the relays and the primary user.

Appendix A: Derivation of (8)

Relying on [16], the CDF and PDF, respectively, of the RVs ψ2 are given as

Fψ2 (x) = λ2x

λ2x + λ4 Q, (25)

fψ2 (x) = λ2λ4 Q

(λ2x + λ4 Q)2(26)

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Setting w = γ3; the outage probability PoutIAF1 conditioned on γ3 is calculated as

PoutIAF1|γ3

(w) = Pr[ψ0 < γth, γ

AF1 < γth |γ3

]

= Pr[

Qγ0

w< γth

]Pr

[ψ1ψ2

ψ1 + ψ2 + 1< γth |γ3

]

= Pr[γ0 < ρw

]⎡

⎣Fψ2 (γth)++∞∫

γth

Fψ1|γ3

(γth x + γth

x − γth

)fψ2 (x) dx

⎤

⎦

(27)

where ρ = γth/Q and Fψ1|γ3 (.) is the CDF of ψ1 conditioned on γ3, which can be given byusing (5) as

Fψ1|γ3 (x) = Pr[

Qγ1b

w< x

]= Fγ1b

(wx

Q

)

= 1 +N∑

n=1

M−n+1∑

m=0,n+m>1

(−1)mCn−1M Cm

M−n+1 exp

(− (n + m − 1) λ1w

Qx

)

(28)

Substituting (25), (26) and (28) into (27), we obtain (29) as

PoutIAF1|γ3

(w) = (1 − exp (−λ0ρw))

×

⎡

⎢⎢⎢⎢⎢⎢⎣

1 +N∑

n=1

M−n+1∑

m=0,n+m>1

(−1)mCn−1M Cm

M−n+1λ2λ4 Q

+∞∫

γth

exp

(− (n + m − 1) λ1ρw

x + 1

x − γth

)

(λ2x + λ4 Q)2dx

︸︷︷︸I1

⎤

⎥⎥⎥⎥⎥⎥⎦

(29)

Now, we consider the integral I1 in (29). First, by setting y = 1 + λ2γth+λ4 Qλ2(x−γth )

, we canrespectively get

λ2x + λ4 Q = (λ2γth + λ4 Q) y

y − 1, (30)

x + 1

x − γth= λ2 (1 + γth) y + λ4 Q − λ2

λ4 Q + λ2γth, (31)

dy = − λ2γth + λ4 Q

λ2 (x − γth)2 dx = − λ2 (y − 1)2

λ2γth + λ4 Qdx (32)

Therefore, by using the change of variable y = 1 + λ2γth+λ4 Qλ2(x−γth )

and results given in (30–32),we can express the integral I1 under the following form:

I1 =0∫

+∞

(y − 1)2

(λ2γth + λ4 Q)2 y2exp

(− (n + m − 1) λ1ρw

λ2 (1 + γth) y + λ4 Q − λ2

λ4 Q + λ2γth

)(−λ2γth + λ4 Q

λ2 (y − 1)2dy

)

= 1

λ2 (λ2γth + λ4 Q)exp

(− (n + m − 1) λ1ρw

λ4 Q − λ2

λ4 Q + λ2γth

)

×+∞∫

0

1

y2 exp

(− (n + m − 1) λ1ρw

λ2 (1 + γth) y

λ4 Q + λ2γth

)dy (33)

123


Substituting (33) into (29), we can rewrite PoutIAF1|γ3

(w) as follows:

PoutIAF1|γ3

(w) = (1 − exp (−λ0ρw))⎡

⎢⎣

1 +∑Nn=1

∑M−n+1m=0,n+m>1 (−1)mCn−1

M CmM−n+1

λ4 Q exp (−�(λ4 Q − λ2)w)

λ2γth + λ4 Q

× ∫ +∞1

exp (−�λ2 (1 + γth)wy)

y2 dy

⎤

⎥⎦ , (34)

where � = (n+m−1)λ1ρλ2γth+λ4 Q .

Because

+∞∫

1

exp(−�λ2(1+γth)wy)y2 dy =exp (−�λ2 (1+γth)w)−�λ2 (1+γth) wE1 (�λ2 (1+γth) w),

where E1 (.) is exponential integral, we obtain (35) from (34) as follows:

PoutIAF1|γ3

(w) = 1 − exp (−λ0ρw)+N∑

n=1

M−n+1∑

m=0,n+m>1

(−1)mCn−1M Cm

M−n+1λ4 Q

λ2γth + λ4 Q

×⎡

⎣exp (−�(λ2γth + λ4 Q) w)− exp (− (� (λ2γth + λ4 Q)+ λ0ρ)w)

−�λ2 (1 + γth) w exp (−�(λ4 Q − λ2)w) E1 (�λ2 (1 + γth)w)

+�λ2 (1 + γth) w exp (− (� (λ4 Q − λ2)+ λ0ρ)w) E1 (�λ2 (1 + γth)w)

⎤

⎦

(35)

Similar to [14], the probability PoutIAF1 can be formulated as

PoutIAF1 =

+∞∫

0

PoutIAF1|γ3

(w) fγ3 (w) dw =+∞∫

0

PoutIAF1|γ3

(w) λ3 exp (−λ3w) dw (36)

In addition, with a, b and c as positive constants, we have (33) as

+∞∫

0

(ax) b exp (−cx) E1 (ax) dx = ab

c2

[ln

(a + c

a

)− c

a + c

](37)

Now, we substitute (35) into (36), then we use (37) to calculate the corresponding integral.After some careful manipulation, we obtain (8).

Appendix B: Derivation of (12)

Similar to (27), by settingw = γ3, the outage probability PoutIAF2 conditioned onγ3 is calculated

as

PoutIAF2|γ3

(w) = Pr[γ0 < wρ

]⎡

⎣Fψ3 (γth)++∞∫

γth

Fψ3|γ3

(γth x + γth

x − γth

)fψ4 (x) dx

⎤

⎦ .

(38)

123


By using Fψ3|γ3 (x) = 1 − exp(− λ3wx

Q

)and (10) for (38), we obtain

PoutIAF2|γ3

(w) = (1 − exp (−λ0ρw))

×⎡

⎣1− M !(N − 1)! (M − N )! (κQ)N

+∞∫

γth

x M−N

(x + κQ)M+1 exp

(−λ3ρw

x + 1

x − γth

)dx

⎤

⎦.

(39)

Now, by applying the change of variable y = x+κQx−γth

, we rewrite (39) as

PoutIAF2|γ3

(w)= (1 − exp (−λ0ρw))

⎡

⎢⎢⎣

1 − M !(N−1)!(M−N )!

γ M−Nth (κQ)N

(γth+κQ)M exp (−(κQ − 1) w)

×+∞∫

1

(y+ κ

ρ

)M−N(y−1)N−1

yM+1 exp (−(1 + γth) wy) dy

⎤

⎥⎥⎦

(40)

where = λ3ργth+κQ .

Using binomial expansion for(

y + κρ

)M−Nand (y − 1)N−1 in (40), we have

PoutIAF2|γ3

(w) = (1 − exp (−λ0ρw))

×

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 − M !(N − 1)! (M − N )!

γ M−Nth (κQ)N

(γth + κQ)Mexp (−(κQ − 1) w)

×M−N∑

i=0

N−1∑

j=0(−1)N−1− j Ci

M−N C jN−1− j

(κ

ρ

)M−N−i +∞∫

1

exp (−(1 + γth) wy)

yM−i− j+1 dy

︸︷︷︸J1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

.

(41)

Applying [17, 3.351.4], for the integral J1 in (42) and E1 (x) = −Ei (−x), we can obtain(42) as

PoutIAF2|γ3

(w) = 1 − exp (−λ0ρw)

− M !(N − 1)! (M − N )!

γ M−Nth (κQ)N

(γth + κQ)M

M−N∑

i=0

N−1∑

j=0

(−1)N−1− j CiM−N C j

N−1− j

(κ

ρ

)M−N−i

×

⎧⎪⎪⎨

⎪⎪⎩

(−1)M−i− j ((1+γth ))M−i− j

(M−i− j)![wM−i− j exp (−(κQ − 1) w) E1 ( (1 + γth) w)

−wM−i− j exp (− ( (κQ − 1)+ λ0ρ)w) E1 ( (1 + γth) w)

]

+M−i− j−1∑

u=0

(−1)u ((1+γth ))u

∏uz=0 (M−i− j−z)

[wu exp (−(γth + κQ) w)−wu exp (− ( (γth + κQ)+ λ0ρ)w)

]

⎫⎪⎪⎬

⎪⎪⎭

(42)

Also, the probability PoutIAF2 can be formulated as

PoutIAF2 =

+∞∫

0

PoutIAF2|γ3

(w) fγ3 (w) dw =+∞∫

0

PoutIAF2|γ3

(w) λ3 exp (−λ3w) dw (43)

Now, we substitute (42) into (43) and use [17, 6.228.2] with E1 (x) = −Ei (−x) for thecorresponding integrals to obtain (12).

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Author Biographies

Tran Trung Duy received the B.E. degree in Electronics andTelecommunications Engineering from Ho Chi Minh City Universityof Technology, Vietnam, in 2007. He is currently working toward theMaster degree in the Department of Electrical Engineering, Universityof Ulsan, Korea. His major research interests are mobile ad-hoc net-works, wireless sensor networks, cooperative communications, cooper-ative routing, cognitive radio, combining techniques.

Hyung-Yun Kong received the M.E. and Ph.D. degrees in electricalengineering from Polytechnic University, Brooklyn, New York, USA,in 1991 and 1996, respectively, He received a B.E. in electrical engi-neering from New York Institute of Technology, New York, in 1989.Since 1996, he has been with LG electronics Co., Ltd., in the multi-media research lab developing PCS mobile phone systems, and from1997 the LG chairman’s office planning future satellite communica-tion systems. Currently he is a Professor in electrical engineering at theUniversity of Ulsan, Korea. His research area includes channel coding,detection and estimation, cooperative communications, cognitive radioand sensor networks. He is a member of IEEK, KICS, KIPS, IEEE, andIEICE.

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Performance Analysis of Incremental Amplify-and-Forward Relaying Protocols with Nth Best Partial Relay Selection Under Interference Constraint

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