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Proactive Relay Selection with Joint Impact of Hardware Impairmentand Co-channel Interference
Duy, T. T., Duong, T. Q., da Costa, D. B., Bao, V. N. Q., & Elkashlan, M. (2015). Proactive Relay Selection withJoint Impact of Hardware Impairment and Co-channel Interference. IEEE Transactions on Communications,63(5), 1594-1606. https://doi.org/10.1109/TCOMM.2015.2396517
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Download date:31. Mar. 2020
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. X, NO. X, X 1
Proactive Relay Selection with Joint Impact ofHardware Impairment and Co-channel InterferenceTran Trung Duy, Trung Q. Duong, Senior Member, IEEE, Daniel Benevides da Costa, Senior Member, IEEE,
Vo Nguyen Quoc Bao, Member, IEEE, and Maged Elkashlan, Member, IEEE
Abstract—In this paper, we investigate the end-to-end per-formance of dual-hop proactive decode-and-forward relayingnetworks with N th best relay selection in the presence of twopractical deleterious effects: i) hardware impairment and ii) co-channel interference. In particular, we derive new exact andasymptotic closed-form expressions for the outage probability andaverage channel capacity of N th best partial and opportunisticrelay selection schemes over Rayleigh fading channels. Insightfuldiscussions are provided. It is shown that, when the system cannotselect the best relay for cooperation, the partial relay selectionscheme outperforms the opportunistic method under the impactof the same co-channel interference (CCI). In addition, withoutCCI but under the effect of hardware impairment, it is shownthat both selection strategies have the same asymptotic channelcapacity. Monte Carlo simulations are presented to corroborateour analysis.
Index Terms—Hardware impairment, decode-and-forward re-laying, partial relay selection, opportunistic relay selection, out-age probability, channel capacity.
I. INTRODUCTION
Along the last decade, the concept of cooperative diversity[1] has been well exploited as an efficient means to enhancethe performance of wireless communications. The basic ideais to allow single-antenna terminals to share their antennasin order to mimic a physical multiple-antenna array so thatspatial diversity can be explored. However, the use of multiplerelays may invoke a spectral efficiency loss and relay selectionschemes arise as a promising solution for alleviating thisproblem. Two proactive relay selection strategies1 that havebeen widely investigated in the literature are opportunisticrelay selection (ORS) [2]–[11] and partial relay selection(PRS) [12]–[19]. In ORS the best relay is chosen relyingon the channel state information (CSI) of both source-relayand relay-destination links. The pioneering idea of ORS wasproposed in [2], while [3] presented an asymptotic analysis ofthe symbol error rate (SER) of a selection amplify-and-forward
Manuscript received April 20, 2014; revised August 14, 2014 and December15, 2014. The associate editor coordinating the review of this paper andapproving it for publication was Prof. Ali Ghrayeb.
T. T. Duy and V. N. Q. Bao are with Posts and Telecommunica-tions Institute of Technology, Vietnam (e-mail: [email protected],[email protected]).
T. Q. Duong is with Queen’s University Belfast, UK (email:[email protected]).
D. B. da Costa is with Federal University of Ceara, CE, Brazil (email:[email protected]). The work is supported by the CNPq (Grant No.302106/2011-1)
M. Elkashlan is with Queen Mary University of London, UK (email:[email protected]).
1In proactive relay selection, the relay is chosen before the source trans-mission.
(AF) network. In [4], it was shown that optimal transmission ofa single relay among a set of multiple AF relays minimize theoutage probability (OP) and outperform any other strategiesthat involve simultaneous transmissions from more than oneAF relay under an aggregate power constraint. In [5], the OPof a cooperative network with multiple potential decode-and-forward (DF) relays and multiple simultaneous transmissionswas investigated, in which a selection cooperation scheme wasproposed. In [6], closed-form expressions for the OP and thebit error rate (BER) of uncoded threshold-based ORS werederived assuming arbitrary signal-to-noise ratio (SNR) levels,arbitrary number of available DF relays, and arbitrary source-destination channel conditions. In [7], with independent non-identically distributed (i.n.i.d.) Rician fading channels, approx-imate formulas for the SER of ORS were derived. Consideringi.n.i.d. Nakagami-m fading and a selection combining (SC)receiver at the destination, the outage performance of ORS wasexamined in [8], while [9] derived closed-form expressions forthe SER. In [10], exact closed-form expressions for the OPand ergodic capacity (EC) of selection cooperative relayingwere derived assuming a maximal-ratio combiner (MRC) atthe destination. In [11], an incremental DF ORS scheme wasproposed in which the selected relay chooses to cooperate onlyif the source-destination channel is of an unacceptable quality.A closed-form expression for the OP was derived.
A common feature of all the aforementioned papers is thatfull diversity gain can be attained. On the other hand, in theseworks there is the need for continuous channel feedback fromall the links, which results in a high power consumption andlarge overhead, a non-desirable feature for ad-hoc and sensornetworks. To alleviate this problem, PRS was proposed in[12], where only CSI of the source-relay link is used to selectthe best relay. Thus, by monitoring the connectivity of onlyone-hop rather than two-hop, the lifetime of the network canbe prolonged. In [13], tight closed-form approximations forthe EC of dual-hop AF relaying networks with PRS werederived. Relying on the channel quality of the second-hop forselecting the best relay, the work in [14] examined the outageperformance of DF relaying networks subject to Nakagami-m and employing a MRC receiver at the destination. In [15],a comprehensive performance analysis of dual-hop relayingnetworks with fixed-gain semi-blind relays was carried out.In particular, closed-form expressions for the OP, probabilitydensity function (PDF), moment generating functions (MGFs),and generalized moments of the end-to-end SNR were derived.In addition, the second-order statistics of the end-to-end en-velope was studied and the corresponding level crossing rate
and average fade duration were obtained in an exact manner. In[16], assuming the presence of the direct link between sourceand destination, an exact performance analysis of DF dual-hopnetworks with relay selection and subject to i.n.i.d Nakagami-m fading was presented. The diversity and coding gains ofPRS schemes subject to Nakagami-m fading were attained in[17], while the impact of feedback delay was analyzed in [18].Finally, in [19], three novel PRS schemes were proposed.
Common to all these works dealing with ORS and PRS isthe assumption of perfect transceiver hardware (i.e., ideal hard-ware) of the terminals. However, in practice, the transceiverhardware is imperfect due to phase noise, I/Q imbalanceand amplifier nonlinearities [20]–[22]. Very few works haveinvestigated the effect of hardware impairments in dual-hopcooperative networks and they are briefly discussed next. In[23], the authors quantified the impact of hardware impair-ments on dual-hop AF and DF relaying networks subjectto Nakagami-m fading. Expressing the OP as a functionof the effective end-to-end signal-to-noise-and-distortion ratio(SNDR), exact closed-form and asymptotic formulas for theOP were derived considering hardware impairments at thesource, relay, and destination. Upper bounds for the EC werederived as well. In that work, fundamental design guidelinesfor selecting hardware that satisfies the requirements of apractical relaying system were pointed out. In [24], the authorsanalyzed the impact of hardware impairments at the relay onthe OP and the SER in two-way AF relaying.
Another channel impairment that may be taken into ac-count in practical systems is co-channel interference (CCI).Differently from hardware impairments, the study of CCI incooperative networks has already been extensively investigatedalong the last years. In the sequel, three representative workswill be discussed. In [25], the outage behavior of dual-hopDF ORS schemes was investigated with CCI at both therelays and the destination. It was shown that the co-channelinterferers do not affect the diversity gain. However, suchinterferers degrade the outage performance by affecting thecoding gain of the system. In [26], assuming a multiuser relaynetwork composed by a single source, a single AF relay, andmultiple destinations, the outage performance of opportunisticscheduling was examined in which the relay and the multipledestinations undergo CCI. Exact expressions and closed-formlower bounds for the OP were derived. In addition, the impactof CSI feedback delay when CCI is considered only at therelay was studied. Finally, in [27], the impact of CCI in two-way AF relaying systems was analyzed.
In this paper, we investigate the end-to-end performance ofdual-hop DF relaying networks in the presence of two practicaldeleterious effects: i) hardware impairment and ii) CCI. BothORS and PRS schemes are considered. To the best of theauthors’ knowledge, this is the first attempt to analyze the jointimpact of hardware impairment and CCI in a dual-hop relayingnetwork. Assuming Rayleigh fading, new exact and asymptoticclosed-form expressions for the OP and the average channelcapacity are derived. Insightful discussions are provided. Itis shown that, when the system cannot select the best relayfor cooperation, PRS scheme outperforms the opportunisticmethod under the impact of the same CCI. In addition, without
S
Ds
1 1
Rb
s
2
2
1I v
vs
1v
2I t ts
2t
1R
RM
1bh 2bh
vg
tl
Fig. 1. Dual-hop relay networks in presence of hardware impairments andco-channel interference.
CCI but under the effect of hardware impairment, it is shownthat both selection strategies have the same asymptotic channelcapacity. Monte Carlo simulations are presented to corroborateour analysis.
The rest of this paper is organized as follows. The systemmodel is described in Section II. In Section III, closed-formexpressions for the OP and average channel capacity arederived. Simulation results are presented in Section IV alongwith representative numerical results. Finally, this paper isconcluded in Section V. Appendices A-F present the proofsof the Lemmas and the Theorems.
II. SYSTEM/CHANNEL MODELS AND PRELIMINARYRESULTS
A. System and Channel Models
Consider a dual-hop proactive relay network in which asource S attempts to transmit its data to the destination Dthrough the help of M available relays Rm, m = 1, 2, · · · ,M ,as shown in Fig. 1. Each terminal is equipped with a singleantenna and operates in a half-duplex mode. Assuming that thedirect link between S and D experiences deep shadowing, thecommunication is realized into two time-slots. In our analysis,depending on the available CSI, we consider two well-knownproactive relay selection methods: partial relay selection [12]and opportunistic relay selection [28]. In this case, only onerelay Rb satisfying a predefined criterion is selected for helpingto forward the source message.
In the first time slot, the source transmits its signal s tothe chosen relay Rb. Assume that there are K1 interferencesources I1v , v = 1, 2, . . . ,K1, which are currently using thesame channel, and hence creating interferences to the relay Rb.In the second time slot, the relay Rb forwards the source signalto the destination by using a DF protocol. Also, we assumethat there are K2 interference sources I2t, t = 1, 2, . . . ,K2.In the presence of the hardware impairments and co-channelinterference, the received signal at Rb and D can be expressed,
respectively, as
yRb = h1b (s+ η1) +
K1∑v=1
gv (sv + η1v) + µ1 + nRb , (1)
yD = h2b (s+ η2) +
K2∑t=1
lt (st + η2t) + µ2 + nD, (2)
where nRb and nD are, respectively, the additive white Gaus-sian noise (AWGN) terms at R and D, with zero mean andvariance N0, sv and st are the signals transmitted by theinterference sources I1v and I2t, respectively, h1b, h2b, gv , andlt are the channel coefficients of the links S → Rb, Rb → D,I1v → Rb, and Rb → D, respectively. In addition, η1, η1v , η2
and η2t denote the noises caused by the hardware impairmentsat the transmitters S, I1v, Rb, and I2t, respectively, while µ1
and µ2 are the noises generated by the hardware impairmentsat the receivers Rb and D, respectively.
Assume that all the channels follow a Rayleigh distribution.Thus, the corresponding channel gains ϕSRm
= |h1m|2,ϕRmD = |h2m|2 for m = 1, 2, . . . ,M , |gv|2, and |lt|2 areexponential random variables (RVs) with parameters λSRm ,λRmD, λRI1v , and λDI2t , respectively.
Remark 1: Similar to [23], [24], we can model the dis-tortion noises η1, η1v , η2, η2t, µ1 and µ2 as circularly-symmetric complex Gaussian distribution with zero-mean andvariance σ2
1PS, σ23vPI, σ2
2PS, σ24tPI, σ2
3
(|h1b|2PS + |gv|2PI
),
and σ24
(|h2b|2PS + |lt|2PI
), respectively. In this case, PS and
PI denote the transmit powers of the source (and relays) andthe interference sources, respectively, while σ1, σ3v , σ2, σ4t,σ3 and σ4 present the level of the hardware impairments atthe corresponding transmitters and receivers. Without loss ofgenerality, it is also assumed that all of the nodes have thesame structure so that the impairment levels are the same, i.e.,σ1 = σ3v = σ4t = σa, and σ3 = σ4 = σb [23], [24]. 2
From (1) and (2), the received signal-to-interference-plus-noise ratio (SINR) at Rb and D can be written, respectively,as
ψSRb=
PSϕSRb
(σ21 + σ2
3)PSϕSRb+
K1∑v=1
(1 + σ21v + σ2
3)PI|gv|2 +N0
=γSRb
κγSRb+ Z1 + 1
, (3)
ψRbD =PSϕRbD
(σ22 + σ2
4)PSϕRbD +K2∑t=1
(1 + σ22t + σ2
4)PI|lt|2 +N0
=γRbD
κγRbD + Z2 + 1, (4)
2In case of different levels of hardware impairment, our results can beapplied to derive the upper-bound and/or lower-bound of the outage probabil-ity and average channel capacity. Moreover, in practice, with knowledge ofimpairment transceiver levels, we should select the transceivers with similarimpairment levels, in order to optimize the system performance (see [23,Corollary 3].
where
γSRb=PSϕSRb
N0, γRbD =
PSϕRbD
N0, Z1 =
K1∑v=1
(1 + κ)PI|gv|2
N0,
Z2 =
K2∑t=1
(1 + κ)PI|lt|2
N0, κ = σ2
a + σ2b .
B. Preliminary Results
In partial relay selection method, the N th best relay Rb isselected by the following strategy:
Rb = N th argmaxm=1,2,...,M
(ϕSRm) . (5)
On the other hand, in the opportunistic relay selection strategy,the N th best relay Rb is chosen according to
Rb = N th argmaxm=1,2,...M
min (ϕSRm, ϕRmD) . (6)
We can observe from (5) and (6) that the relay selectionprocess in the ORS protocol requires each relay to obtain thechannel state information (CSI) of the S → R and R → Dlinks, while that in the PRS only needs the CSI of the firstlink. Hence, the implementation of the ORS protocol is morecomplex than that of the PRS protocol. Moreover, we notethat the relay selection operation in the ORS protocol can berealized by a distributed manner as presented in [2].Remark 2: Throughout this paper, we assume clustering relaynetworks where data links are independent and identically dis-tributed (i.i.d.), i.e., λSRm = λSR and λRmD = λRD for all m.In addition, since the interferers can originate from differentcells, the interference links are presumed to be independentnon-identically distributed (i.n.i.d.), i.e., λRI1m 6= λRI1n ifm 6= n, and λDI2m 6= λDI2n if m 6= n.3
The PDF of Za, a ∈ 1, 2, can be expressed as
fZa (za) =
Ka∑u=1
αXIau exp (−ΩXIau), (7)
where X ≡ R if a = 1, X ≡ D if a = 2,
ΩXIau=
ΩXIau
γ, ΩXIau
=λXIau
(1 + κ) rP,
γ =PI
N0=PS
N0=PN0
, αXIau=αXIau
γ rP,
αXIau = ΩXIau
Ka∏w=1,w 6=u
ΩXIaw
ΩXIaw − ΩXIau
.
Since the DF relaying protocol is employed, the end-to-endSINR is given by
ψYe2e = min (ψSRb , ψRbD) , (8)
where Y ∈ (ORS,PRS).
3Our derivation can be easily extended to i.n.i.d. data links and/or i.i.d.interference links.
III. PERFORMANCE ANALYSIS
A. Outage Probability
In this subsection, exact closed-form expressions for the OPof both PRS and ORS schemes will be derived. By definition,the OP is the probability that the end-to-end received SINR islower than a pre-determined threshold γth.
1) Partial Relay Selection (PRS): The outage probabilityof the PRS protocol can be formulated as
P outPRS = Pr(ψPRSe2e < γth
)= FψPRS
e2e(γth) , (9)
where FψPRSe2e
(.) denotes the CDF of ψPRSe2e .
Theorem 1: If x ≥ κ−1, then FψPRSe2e
(x) = 1, and if x <κ−1, it follows that
FψPRSe2e
(x) = 1−N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
∆1∆2
× (1− κx)2
(Φ1 + x) (Φ2 + x)exp
(− (Θ1 + Ω2)x
1− κx
),
(10)
where Ω1 = N0λSR/P , Cab = b!a!(b−a)! , Θ1 =
(n+m− 1) Ω1, ∆1 = Cm−1M CnM−m+1αRI1v
/(Θ1−κΩRI1v),
Φ1 = ΩRI1v/ (Θ1 − κΩRI1v
), Ω2 = λRDN0/P , ∆2 =αDI2t
/ (Ω2 − ΩDI2tκ) and Φ2 = ΩDI2t
/ (Ω2 − κΩDI2t).
Proof 1: The proof is presented in Appendix A.Lemma 1: Without interference sources, i.e., by settingPI → 0 or rP → 0, and x < κ−1, the CDF FψPRS
e2e(·) can
be expressed as
FψPRSe2e
(x) = 1−N∑m=1
M−m+1∑n=0,m+n 6=1
(−1)n+1
Cm−1M CnM−m+1
× exp
(− (Θ1 + Ω2)x
1− κx
). (11)
Proof 2: The proof is given in Appendix B.Theorem 2: At high transmit SNR and assuming x < κ−1,
the CDF FψPRSe2e
(·) can be approximated by
FψPRSe2e
(x)γ→+∞≈ 1−
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×∆1∆2(1− κx)
2
(Φ1 + x) (Φ2 + x). (12)
Proof 3: For high values of γ, (3) and (4) can be approxi-mated by
ψSRb
γ→+∞≈ γSRb
κγSRb+ Z1
,
ψRbD
γ→+∞≈ γRbD
κγRbD + Z2. (13)
From (13), with the same manner with Appendix A, we canobtain
FψSRb(x)
γ→+∞≈ 1−
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
(−1)n+1
∆11− κxΦ1 + x
,
FψRbD(x)
γ→+∞≈ 1−
K2∑t=1
∆21− κxΦ2 + x
.
By substituting the results above into (A.1), (12) can beattained.
Then, similar to Appendix A, (12) can be attained.Lemma 2: Without interference sources, i.e., by setting
PI → 0 or rP → 0, and x < κ−1, the CDF FψPRSe2e
(·) athigh transmit SNR can be expressed as
FψPRSe2e
(x)γ→+∞≈
Ω2x/ (1− κx) ; if N < M(MΩ1 + Ω2)x/ (1− κx) ; if N = M
. (14)
Proof 4: The proof is given in Appendix C.From Lemma 2, one can observe that when x < κ−1, thediversity order equals 1.
2) Opportunistic Relay Selection (ORS): The OP of theORS scheme can be formulated as
P outORS = Pr(ψORSe2e < γth
)= FψORS
e2e(γth) , (15)
Theorem 3: If x ≥ κ−1, then FψORSe2e
(x) = 1, and if x <κ−1, it follows that
FψORSe2e
(x) = 1−N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×[
∆3
Φ3 + x+
∆4
Φ4 + x+
∆5
Φ6 + x+
∆6
Φ7 + x
]× (1− κx)
2
(Φ5 + x)exp
(− Θ2x
1− κx
), (16)
where Ω = Ω1 + Ω2, Θ2 = (n+m− 1) Ω,Φ3 = ΩRI1v
/ (Ω1 − κΩRI1v), Φ4 = ΩRI1v
/ (Θ2 − κΩRI1v),
Φ5 = (ΩRI1v + ΩDI2t) / (Θ2 − κ (ΩRI1v + ΩDI2t)),Φ6 = ΩDI2t/ (Ω2 − κΩDI2t), Φ7 = ΩDI2t/ (Θ2 − κΩDI2t),
∆3 = (n+m− 1)Cm−1M CnM−m+1
Ω2αRI1vαDI2t
Ω2 + (n+m− 2) Ω
× 1
(Θ2 − κ (ΩRI1v + ΩDI2t)) (Ω1 − κΩRI1v),
∆4 = (n+m− 2)Cm−1M CnM−m+1
Ω1αRI1vαDI2t
Ω2 + (n+m− 2) Ω
× 1
(Θ2 − κΩRI1v) (Θ2 − κ (ΩRI1v
+ ΩDI2t)),
∆5 = (n+m− 1)Cm−1M CnM−m+1
Ω1αRI1vαDI2t
Ω1 + (n+m− 2) Ω
× 1
(Θ2 − κ (ΩRI1v + ΩDI2t)) (Ω2 − κΩDI2t),
∆6 = (n+m− 2)Cm−1M CnM−m+1
Ω2αRI1vαDI2t
Ω1 + (n+m− 2) Ω
× 1
(Θ2 − κΩDI2t) (Θ2 − κ (ΩRI1v
+ ΩDI2t)).
Proof 5: The proof is presented in Appendix D.Lemma 3: Without interference sources, i.e., by setting
PI → 0 or rP → 0, and x < κ−1, the CDF FψORSe2e
(·) can beexpressed as
FψORSe2e
(x) = 1−N∑m=1
M−m+1∑n=0,n+m>1
(−1)n+1
Cm−1M CnM−m+1
× exp
(− (n+m− 1)
Ωx
1− κx
). (17)
Proof 6: From (D.3), (17) can be obtained.Theorem 4: At high transmit SNR and assuming x < κ−1,
the CDF FψPRSe2e
(·) can be approximated by
FψORSe2e
(x)γ→+∞≈ 1−
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×[
∆3
Φ3 + x+
∆4
Φ4 + x+
∆5
Φ6 + x+
∆6
Φ7 + x
]× (1− κx)
2
Φ5 + x. (18)
Proof 7: Note that, for high γ values, (3) and (4) can beapproximated by (13). Hence, similar as obtained (13) from(12), (18) can be attained by omitting the term exp
(− Θ2x
1−κx
)from (16).
Lemma 4: Without interference sources and considering x <κ−1, the CDF FψORS
e2e(·) at high transmit SNR can be written
as
Fγ1b(x)
γ→+∞≈ CN−1
M
(Ωx
1− κx
)M−N+1
. (19)
Proof 8: The proof is similar to that of Lemma 2.From (19), one can attest that the diversity order of the ORSstrategy equals to M −N + 1.
B. Average Channel Capacity
The average channel capacity can be mathematically definedas
CYavg =
1
2E
log2
(1 + ψY
e2e
)=
1
2 ln 2
∫ κ−1
0
ln (1 + x) fψYe2e
(x) dx, (20)
where Y ∈ PRS,ORS, E . symbolizes expectation, andfψY
e2e(·) denotes the PDF of ψY
e2e.From (10) and (16), (20) can be rewritten as
CYavg =
1
2 ln 2
∫ κ−1
0
1− FψYe2e
(x)
1 + xdx. (21)
Proposition 1: In the presence of hardware impairments,i.e., κ > 0, the average channel capacity of both PRS andORS methods is bounded by
CAavg ≤
1
2 ln 2ln
(1 +
1
κ
). (22)
Proof 9: From (3) and (4), it is easy to see that ψ1b ≤κ−1 and ψ2b ≤ κ−1, which implies in ψY
e2e ≤ κ−1. Thus,combining with (20), (22) can be readily obtained.Before calculating the average capacity of the PRS and ORSstrategies, the following integral will be introduced.
J (κ,Ω,Φ) =
∫ κ−1
0
1
Φ + xexp
(− Ωx
1− κx
)dx
= exp
(ΩΦ
Φκ+ 1
)E1
(ΩΦ
Φκ+ 1
)− exp
(Ω
κ
)E1
(Ω
κ
), (23)
where E1(.) denotes the exponential integral function [29].Proof 10: By interchanging the variable t = 1/(1 − κx),
J (κ,Ω,Φ) can be rewritten as
J (κ,Ω,Φ) =exp (Ω/κ)
κΦ + 1
∫ +∞
1
exp (−tΩ/κ)
t (t− 1/ (κΦ + 1))dt
= exp
(ΩΦ
κΦ + 1
)∫ +∞
κΦκΦ+1
exp (−tΩ/κ)
tdt
− exp
(Ω
κ
)∫ +∞
1
exp (−tΩ/κ)
tdt.
Then, by using the definition of the exponential integralfunction E1 (x) =
∫ +∞x
exp(−t)t dt, we can easily obtain (23).
1) Partial Relay Selection (PRS):Theorem 5: The average channel capacity of the PRS
method can be expressed as (24), shown at the top ofnext page, with δ1 = (1 + κΦ1)
2/ (Φ2 − Φ1) and δ2 =
(1 + κΦ2)2/ (Φ1 − Φ2).
Proof 11: The proof is presented in Appendix E.Lemma 5: Without interference sources, the average channel
capacity of the PRS method is given by
CPRSavg =
1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×∆1∆2J (κ,Θ1 + Ω2, 1) . (25)
Proof 12: Relying on (1), (21), and (23), Lemma 5 can beeasily proved.
Theorem 6: At high transmit SNR γ, the asymptotic averagechannel capacity of the PRS method can be derived as (26),shown at the top of next page.
Proof 13: (26) can be attained from (24) by performingthe appropriate substitutions, i.e., replacing J (κ,Θ1 + Ω2, 1),J (κ,Θ1 + Ω2,Φ1), and J (κ,Θ1 + Ω2,Φ2) by J (κ, 0, 1),J (κ, 0,Φ1) and J (κ, 0,Φ1), respectively. In addition, notethat J (κ, 0,Ω) = ln ((1 + κΦ) /κΦ).Finally, one can see that without interference sources, theasymptotic average capacity of the PRS method is given asin (22), i.e.,
CPRSavg
γ→+∞≈ 1
2 ln 2ln
(1 +
1
κ
). (27)
2) Opportunistic Relay Selection (ORS):Theorem 7: The average channel capacity of the ORS
method can be given as (28), shown at the top ofnext page, with δ3 = (1 + κΦ3)
2/ (Φ5 − Φ3), δ4 =
(1 + κΦ5)2/ (Φ3 − Φ5), δ5 = (1 + κΦ4)
2/ (Φ5 − Φ4), δ6 =
(1 + κΦ5)2/ (Φ4 − Φ5), δ7 = (1 + κΦ6)
2/ (Φ5 − Φ6), δ8 =
(1 + κΦ5)2/ (Φ6 − Φ5), δ9 = (1 + κΦ7)
2/ (Φ5 − Φ7), and
δ10 = (1 + κΦ5)2/ (Φ7 − Φ5).
Proof 14: The proof is presented in Appendix F .Lemma 6: Without interference sources, the average channel
capacity can be rewritten as
CORSavg =
N∑m=1
M−m+1∑n=0, n+m>1
(−1)nCm−1M CnM−m+1
× J (κ, (n+m− 1) Ωx, 1) . (29)
CPRSavg =
1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
∆1∆2
×[(
δ1Φ1 − 1
+δ2
Φ2 − 1+ κ2
)J (κ,Θ1 + Ω2, 1)− δ1
Φ1 − 1J (κ,Θ1 + Ω2,Φ1)− δ2
Φ2 − 1J (κ,Θ1 + Ω2,Φ2)
]. (24)
CPRSavg
γ→+∞≈ 1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
∆1∆2
×[(
δ1Φ1 − 1
+δ2
Φ2 − 1+ κ2
)ln
(1 + κ
κ
)− δ1
Φ1 − 1ln
(1 + κΦ1
κΦ1
)− δ2
Φ2 − 1ln
(1 + κΦ2
κΦ2
)]. (26)
CORSavg =
1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×(
∆3δ3Φ3 − 1
+∆4δ5
Φ4 − 1+
∆5δ7Φ6 − 1
+∆6δ9
Φ7 − 1+
∆3δ4 + ∆4δ6 + ∆5δ8 + ∆6δ10
Φ5 − 1+ (∆3 + ∆4)κ2
)J (κ,Θ2, 1)
− ∆3δ3Φ3 − 1
J (κ,Θ2,Φ3)− ∆4δ5Φ4 − 1
J (κ,Θ2,Φ4)− ∆5δ7Φ6 − 1
J (κ,Θ2,Φ6)− ∆6δ9Φ7 − 1
J (κ,Θ2,Φ7)
−(
∆3δ4 + ∆4δ6 + ∆5δ8 + ∆6δ10
Φ5 − 1
)J (κ,Θ2,Φ5) . (28)
CORSavg
γ→+∞≈ 1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×(
∆3δ3Φ3 − 1
+∆4δ5
Φ4 − 1+
∆5δ7Φ6 − 1
+∆6δ9
Φ7 − 1+
∆3δ4 + ∆4δ6 + ∆5δ8 + ∆6δ10
Φ5 − 1+ (∆3 + ∆4)κ2
)ln
(1 + κ
κ
)− ∆3δ3
Φ3 − 1ln
(1 + κΦ3
κΦ3
)− ∆4δ5
Φ4 − 1ln
(1 + κΦ4
κΦ4
)− ∆5δ7
Φ6 − 1ln
(1 + κΦ6
κΦ6
)− ∆6δ9
Φ7 − 1ln
(1 + κΦ7
κΦ7
)−(
∆3δ4 + ∆4δ6 + ∆5δ8 + ∆6δ10
Φ5 − 1
)ln
(1 + κΦ5
κΦ5
). (30)
Proof 15: Based on (17), (21), and (23), Lemma 6 can bereadily proved.
Theorem 8: At high transmit SNR γ, the asymptotic averagechannel capacity of the PRS method can be derived as (30),shown at the top of next page.
Proof 16: The proof of Theorem 8 is similar to that ofTheorem 6.One can easily prove that the asymptotic average capacity ofthe PRS method at high γ is given as in (27), i.e.,
CORSavg
γ→+∞≈ 1
2 ln 2ln
(1 +
1
κ
). (31)
From (27) and (31), note that under the impact of hardwareimpairment and without co-channel interference, the PRS andORS schemes have the same average channel capacity at hightransmit SNR.
IV. NUMERICAL RESULTS AND SIMULATIONS
In this Section, representative numerical results are pre-sented to illustrate the performance of the two proposed relayselection schemes in the presence of hardware impairmentand CCI. Monte Carlo simulation results are also shownto corroborate the proposed analysis. Without any loss ofgenerality, we set γth < κ−1.
In Fig. 2, the outage probability is plotted as a functionof transmit SNR γ. The following parameters are employed:M = 4, K1 = K2 = 2, rP = 1, γth = 1, κ = 0.075, λSR =0.3, λRD = 0.5, λRI1v ∈ 1, 2, and λDI2t ∈ 1.5, 2.5. Itcan be observed that the outage performance of the ORS andPRS schemes is better if the system can select the best relayfor the cooperation (N = 1). In addition, when N = 1, theoutage probability of the ORS scheme is lower than that ofthe PRS one. However, such a metric of the PRS is higherthan that of the ORS when N = 2. It is because that whenthe best relay cannot be selected, the end-to-end SINR of the
-5 0 5 10 15 20 250.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
SNR (dB)
PRS-Sim (N=2) ORS-Sim (N=2) PRS-Sim (N=1) ORS-Sim (N=1) Theory-Exact Theory-Asym
Fig. 2. Outage probability as a function of the transmit SNR γ when M = 4,K1 = K2 = 2, rP = 1, γth = 1, κ = 0.075, λSR = 0.3, λRD = 0.5,λRI1v ∈ 1, 2, and λDI2t ∈ 1.5, 2.5.
0 5 10 15 201E-5
1E-4
1E-3
0.01
0.1
1
10
SNR (dB)
PRS-Sim (N=1) ORS-Sim (N=1) PRS-Sim (N=2) ORS-Sim (N=2) Theory-Exact Theory-Asym
Out
age
Prob
abili
ty
Fig. 3. Outage probability as a function of the transmit SNR γ when M = 3,rP = 0, γth = 1.5, κ = 0.08, λSR = 1.1, and λRD = 1.1.
ORS protocol is no longer maximum. Hence, PRS can providea higher end-to-end SINR than ORS. Finally, it can be seenthat the outage probability decreases when the transmit SNRincreases. However, the outage performance of both protocolsconverges to positive constant at high SNR regime. Therefore,we can conclude that the system obtains the zero-diversityorder when there are the interference sources in the network.
In Fig. 3, the outage performance is depicted as a functionof transmit SNR γ when there is no interference source andby setting M = 3, rP = 0, γth = 1.5, κ = 0.08 and λSR =λRD = 1.1. Note that the ORS scheme outperforms the PRSone for both N = 1, 2, with the performance gap being higher
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0
0.1
0.2
0.3
0.4
0.5
rP
PRS-Sim ORS-Sim Theory-Exact PRS-Theory (r
P=0)
ORS-Theory (rP=0)
Fig. 4. Outage probability as a function of the ratio rP when γ = 10dB,M = 6, N = 2, K1 = K2 = 1, γth = 0.5, κ = 0.08, λSR = 1,λRD = 0.5, λRI11
= 1.5, and λDI21= 2.
0 5 10 15 20 25 300.05
0.10
0.15
0.20
0.25
0.30
SNR (dB)
PRS-Sim (N=1) ORS-Sim (N=1) PRS-Sim (N=2) ORS-Sim (N=2) Theory-Exact Theory-Asym
Fig. 5. Average channel capacity as a function of the transmit SNR γ whenM = 2, K1 = K2 = 1, rP = 1, κ = 0.075, λSR = 1.1, λRD = 1.3, andλRI11
= λDI21= 0.7.
for the case N = 1. The reason is that the diversity order4 ofthe ORS scheme equals to 3 for N = 1, while it is 2 forN = 2. Indeed, for N = 2, the performance of both schemesis almost the same at low and medium SNRs, and only athigh SNR region a practical difference in performance can bedetected.
In Fig. 4, we investigate the impact of the ratio rP (PI/PS)on the outage performance of the proposed protocols. Forthe illustrative purpose, we fix the parameters γ, M , N , K1,K2, γth, κ, λSR, λRD, λRI11 and λDI21 by 10 dB, 6, 2, 1,
4The diversity order of the PRS scheme is always 1, regardless of the valueof N .
0 5 10 15 20 25 301.0
1.5
2.0
2.5
3.0
3.5
SNR (dB)
PRS-Sim (\κ=0.05) ORS-Sim (\κ=0.05) ORS-Sim (\κ=0.01) ORS-Sim (\κ=0.01) Theory-Exact Theory-Asym
Fig. 6. Average channel capacity as a function of the transmit SNR γ whenM = 4, N = 1, rP = 0, and λSR = λRD = 0.1.
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.100.1
0.2
0.3
0.4
0.5
κ
PRS-Sim (N=1) ORS-Sim (N=1) PRS-Sim (N=3) ORS-Sim (N=3) Theory-Exact
Ave
rage
Cha
nnel
Cap
acity
Fig. 7. Average channel capacity as a function of κ when γ = 10dB,M = 5, K1 = K2 = 1, rP = 1, λSR = λRD = 1, λRI11
= 0.5, andlambdaRI11
= 0.75.
1, 0.5, 0.08, 1, 0.5, 1.5 and 2, respectively. It can be seenfrom this figure that the outage performance of both protocolsdecreases when the ratio rP increases. Different with theresults presented in Fig. 2, although the system can only selectthe second-best relay for the cooperation, the ORS protocolobtains better performance as compared with the PRS protocol.
Fig. 5 presents the average channel capacity of the PRSand ORS protocols as a function of the transmit SNR γ.In this figure, we fix the parameters as follows: M = 2,K1 = K2 = 1, rP = 1, κ = 0.075 and λSR = 1.1,λRD = 1.3, and λRI11 = λDI21 = 0.7. Similar to Fig. 2,the ORS scheme achieves higher channel capacity than PRSone when the system can select the best relay (i.e., N = 1) for
cooperation. Otherwise, for N = 2, i.e., the system selects onethe second best relay for cooperation, the PRS strategy attainsbetter performance. Finally, note that the channel capacity ofboth schemes converges to the asymptotic values at high SNRregion.
In Fig. 6, the effect of the hardware impairment level κ onthe average channel capacity is investigated. It is assumed thatthere is no CCI, i.e., rP = 0. The remaining parameters aredesigned as follows: M = 4, N = 1, and λSR = λRD =0.1. One can notice that the PRS and ORS schemes have thesame asymptotic channel capacity. In addition, it is shown thatboth strategies obtain better performance as the value of κdecreases, with ORS presenting better performance than PRS.
Fig. 7 presents the average channel capacity as a functionof κ when γ = 10dB, M = 5, K1 = K2 = 1, rP = 1,λSR = λRD = 1, λRI11 = 0.5, and λRI11 = 0.75. It can beobserved from this figure that the channel capacity of the PRSand ORS protocols decreases with the increasing of κ. Again,we can obverse that the performance of the ORS scheme isbetter than that of the PRS scheme when the best relay canbe selected for the cooperation.
V. CONCLUSIONS
In this paper, analyzing the impact of hardware impairmentand CCI, the end-to-end performance of dual-hop proactiveDF relaying networks with N th PRS and N th ORS is inves-tigated. Exact and asymptotic closed-form expressions for theoutage probability and average channel capacity of both relayselection schemes were derived. Insightful discussions wereprovided. For instance, it was shown that, when the systemcannot select the best relay for cooperation, the partial relayselection scheme outperforms the opportunistic method underthe impact of the same co-channel interference.
ACKNOWLEDGMENTS
This research is funded by Vietnam National Foundation forScience and Technology Development (NAFOSTED) undergrant number 102.01-2014.33.
APPENDIX A: PROOF OF THEOREM 1
Firstly, we rewrite FψPRSe2e
(x) as follows
FψPRSe2e
(x) = 1−(
1− FψSRb(x))(
1− FψRbD (x)). (A.1)
Thus, in order to attain (A.1), the CDFs FψSRb(·) and FψRbD
(·)are required. Considering first the CDF of ψSRb
, we have that
FψSRb(x) = Pr (ψSRb
< x)
=
1; if x ≥ κ−1
γSRb< x+xZ1
1−κx ; if x < κ−1 (A.2)
For x < κ−1, (A.2) can be formulated as
FψSRb(x) =
∫ +∞
0
FγSRb
(x+ xz1
1− κx
)fZ1
(z1) dz1. (A.3)
Now, using the N -best order statistics [30], the CDF of γSRb
can be written as
FγSRb(y) =
N∑m=1
Cm−1M (1− exp (−Ω1y))
M−m+1
× exp (− (m− 1) Ω1y)
= 1−N∑m=1
M−m+1∑n=0,n+m>1
(−1)n+1
Cm−1M CnM−m+1
× exp (− (n+m− 1) Ω1y) , (A.4)
where Ω1 = N0λSR/P and Cab = b!a!(b−a)! , with a and b being
integers and b > a.Combining (7), (A.3) and (A.4), and after some algebraic
manipulation, it follows that
FψSRb(x) = 1−
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
(−1)n+1
∆1
× 1− κxΦ1 + x
exp
(− Θ1x
1− κx
), (A.5)
where Θ1 = (n+m− 1) Ω1, ∆1 =Cm−1M CnM−m+1αRI1v/(Θ1 − κΩRI1v), and Φ1 =
ΩRI1v/ (Θ1 − κΩRI1v). For simplicity, we assume thatΘ1 − κΩRI1v
6= 0.Similarly, one can see that, if x ≥ κ−1, FψRbD
(x) = 1,while if x < κ−1, then
FψRbD(x) = 1−
K2∑t=1
∆21− κxΦ2 + x
exp
(− Ω2x
1− κx
), (A.6)
where Ω2 = λRDN0/P , ∆2 = αDI2t/ (Ω2 − ΩDI2t
κ), Φ2 =ΩDI2t
/ (Ω2 − κΩDI2t), and Ω2 − κΩDI2t
6= 0.Finally, by substituting (A.5) and (A.6) into (A.1), (10) is
attained, which completes the proof.
APPENDIX B: PROOF OF LEMMA 1
Without interference sources, (3) and (4) can be rewrittenas
ψSRb=
γSRb
κγSRb+ 1
,
ψRbD =γRbD
κγRbD + 1. (B.1)
Similar to (A.2)-(A.5), the CDFs FψSRb(·) and FψRbD
(·) canbe obtained as
FψSRb(x) = 1−
N∑m=1
M−m+1∑n=0,m+n 6=1
(−1)n+1
× Cm−1M CnM−m+1 exp
(− Θ1x
1− κx
),
FψRbD(x) = 1− exp
(− Ω2x
1− κx
). (B.2)
Then, combining the above results with (A.1), the proof ofLemma 1 is concluded.
APPENDIX C: PROOF OF LEMMA 2From (A.1) in Appendix A, we can approximate F
ψPRSe2e
(x)
at high SNR region by
FψPRSe2e
(x)γ→+∞≈ FψSRb
(x) + FψRbD (x) , (C.1)
where ψSRband ψRbD are given as (B.1) in Appendix B.
In addition, since 1 − exp (−t)t→0≈ t and exp (−t)
t→0≈ 1,
asymptotic expressions for (B.2) can be written as
Fγ1b(x)
γ→+∞≈
N∑m=1
Cm−1M
(Ω1x
1− κx
)M−m+1
γ→+∞≈ CN−1
M
(Ω1x
1− κx
)M−N+1
,
Fγ2b(x)
γ→+∞≈ Ω2x
1− κx. (C.2)
Combining (C.1) and (C.2), (14) is attained, which completesthe proof.
APPENDIX D: PROOF OF THEOREM 3Firstly, it is easy to see that FψORS
e2e (x) = 1 when x ≥ κ−1.Thus, considering the case when x < κ−1, (16) can berewritten as
FψORSe2e
(x) = 1− Pr (ψSRb≥ x, ψRbD ≥ x)
= 1− Pr
(γSRb
≥ x+ xZ1
1− κx, γRbD ≥
x+ xZ2
1− κx
). (D.1)
Since γSRband γRbD are not independent, the method pro-
posed in [11] will be employed to calculate (D.1). Initially,we will derive the probability Pr (γSRb
≥ u1, γRbD ≥ u2). Tothis end, similar to [11], this probability can be formulated as
Pr (γSRb≥ u1, γRbD ≥ u2) =
∫ +∞
0
∂G (z)
∂z
fTmax(z)
fTi (z)dz.
(D.2)
In (D.2), Tmax = Nth maxm=1,2,...,M
min (γSRm , γRmD), in
which its CDF can be expressed similarly to (A.4) as
FTmax(z) = 1−
N∑m=1
M−m+1∑n=0,n+m>1
(−1)n+1
Cm−1M CnM−m+1
× exp (− (n+m− 1) Ωz) , (D.3)
where Ω = Ω1 + Ω2. Thus, the PDF of Tmax can be derivedas
fTmax(z) =
N∑m=1
M−m+1∑n=0,m+n>1
(−1)nCm−1M CnM−m+1
× (m+ n− 1) Ω exp (− (n+m− 1) Ωz) . (D.4)
By its turn, in (D.2), Ti = min (γSRi, γRiD) , i = 1, 2, . . . ,M ,
such that its PDF can be expressed as
fTi (z) = Ω exp (−Ωz) . (D.5)
Finally, the term G(z) in (D.2) can be formulated as
G (z) = Pr (γSRi≥ u1, γRiD ≥ u2,min (γSRi
, γRiD) < z) .(D.6)
In order to calculate G(z), two cases will be considered:• Case 1: u1 ≥ u2
In this case, G(z) can be obtained as
G (z) =
0; if z ≤ u2
exp (−Ω1u1 − Ω2u2)− exp (−Ω1u1 − Ω2z) ; if u2 ≤ z < u1
exp (−Ω1u1 − Ω2u2)− exp (−Ωz) ; if z ≥ u1
(D.7)
• Case 2: u1 < u2
In this case, it follows that
G (z) =
0; if z ≤ u1
exp (−Ω1u1 − Ω2u2)− exp (−Ω2u2 − Ω1z) ; if u2 ≤ z < u1
exp (−Ω1u1 − Ω2u2)− exp (−Ωz) ; if z ≥ u1
(D.8)
Combining (D.4), (D.5), (D.7) and (D.8), and after some al-gebraic manipulations, Pr (γSRb
≥ u1, γRbD ≥ u2) is derivedfor Case 1 and Case 2 in (D.9) and (D.10), respectively, shownat the top of next page.
Now, replacing u1 = (x+ xZ1) / (1− κx) and u2 =(x+ xZ2) / (1− κx) in (D.9) and (D.10), the outage prob-ability FψORS
e2e(x) can be calculated as5
FψORSe2e
(x) = 1− S1 − S2, (D.11)
where
S1 =
∫ +∞
0
∫ z1
0
Pr
(γSRb
≥ x+ xz1
1− κx, γRbD ≥
x+ xz2
1− κx
)fZ1 (z1) fZ2 (z2) dz2dz1, (D.12)
S2 =
∫ +∞
0
∫ z2
0
Pr
(γSRb
≥ x+ xz1
1− κx, γRbD ≥
x+ xz2
1− κx
)fZ1
(z1) fZ2(z2) dz1dz2. (D.13)
By substituting (7) and (D.9) into (D.12), and after somealgebraic manipulations, it follows that
S1 =N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×
[∆3 (1− κx)
2
(Φ3 + x) (Φ5 + x)+
∆4 (1− κx)2
(Φ4 + x) (Φ5 + x)
]
× exp
(− Θ2x
1− κx
), (D.14)
where Θ2 = (n+m− 1) Ω, Φ3 =ΩRI1v
/ (Ω1 − κΩRI1v), Φ4 = ΩRI1v
/ (Θ2 − κΩRI1v),
Φ5 = (ΩRI1v+ ΩDI2t
) / (Θ2 − κ (ΩRI1v+ ΩDI2t
)), and
∆3 = (n+m− 1)Cm−1M CnM−m+1
Ω2αRI1vαDI2t
Ω2 + (n+m− 2) Ω
× 1
(Θ2 − κ (ΩRI1v + ΩDI2t)) (Ω1 − κΩRI1v),
∆4 = (n+m− 2)Cm−1M CnM−m+1
Ω1αRI1vαDI2t
Ω2 + (n+m− 2) Ω
× 1
(Θ2 − κΩRI1v) (Θ2 − κ (ΩRI1v
+ ΩDI2t)).
5u1 ≥ u2 is equivalent to Z1 ≥ Z2, and vice versa.
Similarly, from (7), (D.10) and (D.13), S2 can be obtained as
S2 =
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×
[∆5 (1− κx)
2
(Φ6 + x) (Φ5 + x)+
∆6 (1− κx)2
(Φ7 + x) (Φ5 + x)
]
× exp
(− Θ2x
1− κx
), (D.15)
where Φ6 = ΩDI2t/ (Ω2 − κΩDI2t), Φ7 =ΩDI2t/ (Θ2 − κΩDI2t), and
∆5 = (n+m− 1)Cm−1M CnM−m+1
Ω1αRI1vαDI2t
Ω1 + (n+m− 2) Ω
× 1
(Θ2 − κ (ΩRI1v+ ΩDI2t
)) (Ω2 − κΩDI2t),
∆6 = (n+m− 2)Cm−1M CnM−m+1
Ω2αRI1vαDI2t
Ω1 + (n+m− 2) Ω
× 1
(Θ2 − κΩDI2t) (Θ2 − κ (ΩRI1v + ΩDI2t)).
Finally, combining (D.11), (D.14) and (D.15), the proof isconcluded.
APPENDIX E: PROOF OF THEOREM 5Firstly, we rewrite (10) as
FψPRSe2e
(x) = 1−N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
∆1∆2
×(
δ1Φ1 + x
+δ2
Φ2 + x+ κ2
)× exp
(− (Θ1 + Ω2)x
1− κx
), (E.1)
where δ1 = (1 + κΦ1)2/ (Φ2 − Φ1) and δ2 =
(1 + κΦ2)2/ (Φ1 − Φ2). Now, by substituting (E.1) into
(21), we have
CPRSavg =
1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
∆1∆2
×∫ κ−1
0
(δ1
(1 + x) (Φ1 + x)+
δ2(1 + x) (Φ2 + x)
+κ2
1 + x
)× exp
(− (Θ1 + Ω2)x
1− κx
)dx. (E.2)
Next, rewriting (E.2) as
CPRSavg =
1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
∆1∆2
×∫ κ−1
0
L1 (x) exp
(− (Θ1 + Ω2)x
1− κx
)dx, (E.3)
in which
L1 (x) =δ1
Φ1 − 1
(1
1 + x− 1
Φ1 + x
)+
δ2Φ2 − 1
(1
1 + x− 1
Φ2 + x
)+
κ2
1 + x.
Pr (γSRb≥ u1, γRbD ≥ u2)=
N∑m=1
M−m+1∑n=0,m+n>1
(−1)nCm−1M CnM−m+1
×[
(n+m− 1) Ω2
Ω2 + (n+m− 2) Ωexp (−Ω1u1 − (Ω2 + (n+m− 2) Ω)u2) +
(n+m− 2) Ω1
Ω2 + (n+m− 2) Ωexp (− (n+m− 1) Ωu1)
](D.9)
Pr (γSRb≥ u1, γRbD ≥ u2)=
N∑m=1
M−m+1∑n=0,m+n>1
(−1)nCm−1M CnM−m+1
×[
(n+m− 1) Ω1
Ω1 + (n+m− 2) Ωexp (−Ω2u2 − (Ω1 + (n+m− 2) Ω)u1)+
(n+m− 2) Ω2
Ω1 + (n+m− 2) Ωexp (− (n+m− 1) Ωu2)
](D.10)
Finally, applying (23) for the corresponding integral in (E.3),we finish the proof of Theorem 5.
APPENDIX F: PROOF OF THEOREM 7Firstly, we rewrite (16) as (F.1), shown at the top of
this page, where δ3 = (1 + κΦ3)2/ (Φ5 − Φ3), δ4 =
(1 + κΦ5)2/ (Φ3 − Φ5), δ5 = (1 + κΦ4)
2/ (Φ5 − Φ4), δ6 =
(1 + κΦ5)2/ (Φ4 − Φ5), δ7 = (1 + κΦ6)
2/ (Φ5 − Φ6), δ8 =
(1 + κΦ5)2/ (Φ6 − Φ5), δ9 = (1 + κΦ7)
2/ (Φ5 − Φ7), and
δ10 = (1 + κΦ5)2/ (Φ7 − Φ5). Now, by substituting (F.1) into
(21), and after some algebraic manipulations, it follows that
CORSavg =
1
2 ln 2
N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
×∫ κ−1
0
L2 (x) exp
(− Θ2x
1− κx
)dx, (F.2)
where L2(x) is a function of x, which is given in (F.3),shown at the top of next page. Next, applying (23) for thecorresponding integral in (F.2), the proof is concluded.
REFERENCES
[1] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” vol. 50,no. 12, pp. 3062–3080, Dec. 2004.
[2] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperativediversity method based on network path selection,” vol. 24, no. 3, pp.559–572, Mar. 2006.
[3] Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-and-forward relay systems,” IEEE Commun. Lett., vol. 10, no. 11, pp.757–759, Nov. 2006.
[4] A. Bletsas, H. Shin, and M. Z. Win, “Outage optimality of opportunisticamplify-and-forward relaying,” IEEE Commun. Lett., vol. 11, no. 3, pp.261–263, Mar. 2007.
[5] E. Beres and R. Adve, “Selection cooperation in multi-source coopera-tive networks,” vol. 7, no. 1, pp. 118–127, Jan. 2008.
[6] D. S. Michalopoulos and G. K. Karagiannidis, “Performance analysis ofsingle relay selection in Rayleigh fading,” vol. 7, no. 10, pp. 3718–3724,Oct. 2008.
[7] B. Maham and A. Hjorungnes, “Performance analysis of amplify-and-forward opportunistic relaying in Rician fading,” vol. 16, no. 8, pp.643–646, Aug. 2009.
[8] F. Xu, F. C. M. Lau, Q. F. Zhou, and D.-W. Yue, “Outage performanceof cooperative communication systems using opportunistic relaying andselection combining receiver,” vol. 16, pp. 237–240, Apr. 2009.
[9] T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “On the performance ofselection decode-and-forward relay networks over Nakagami-m fadingchannels,” vol. 13, no. 8, pp. 172–174, Mar. 2009.
[10] A. Adinoyi, Y. Fan, H. Yanikomeroglu, H. V. Poor, and F. Al-Shaalan,“Performance of selection relaying and cooperative diversity,” vol. 8,no. 12, pp. 5790–5795, Dec. 2009.
[11] K. Tourki, H.-C. Yang, and M.-S. Alouini, “Accurate outage analysis ofincremental decode-and-forward opportunistic relaying,” vol. 10, no. 4,pp. 1021–1025, Apr. 2011.
[12] I. Krikidis, J. Thompson, S. McLaughlin, and N. Goertz, “Amplify-and-forward with partial relay selection,” IEEE Commun. Lett., vol. 12, no. 4,pp. 235–237, Apr. 2008.
[13] D. B. da Costa and S. Aıssa, “Capacity analysis of cooperative systemswith relay selection in Nakagami-m fading,” vol. 13, no. 9, pp. 637–639,Sep. 2009.
[14] K. Yan, J. Jiang, Y. G. Wang, and H. T. Liu, “Outage probabilityof selection cooperation with MRC in Nakagami-m fading channels,”vol. 16, no. 12, pp. 1031–1034, Dec. 2009.
[15] D. B. da Costa and S. Aissa, “End-to-end performance of dual-hopsemi-blind relaying systems with partial relay selection,” vol. 8, no. 8,pp. 4306–4315, Aug. 2009.
[16] G. C. Alexandropoulos, A. Papadogiannis, and K. Berberidis, “Per-formance analysis of cooperative networks with relay selection overNakagami-m fading channels,” vol. 17, no. 5, pp. 1–4, May 2010.
[17] H. Ding, J. Ge, D. B. da Costa, and Z. Jiang, “Diversity and codinggains of fixed-gain amplify-and-forward with partial relay selection inNakagami-m fading,” vol. 14, no. 8, pp. 734–736, Aug. 2010.
[18] H. Suraweera, M. Soysa, C. Tellambura, and H. Garg, “Performanceanalysis of partial relay selection with feedback delay,” vol. 17, no. 6,pp. 531–534, June 2010.
[19] Y. Chen, C.-X. Wang, H. Xiao, and D. Yuan, “Novel partial selectionschemes for AF relaying in Nakagami-m fading channels,” vol. 60,no. 7, pp. 3497–3503, Sept. 2011.
[20] E. Costa and S. Pupolin, “m-QAM-OFDM system performance inthe presence of a nonlinear amplifier and phase noise,” IEEE Trans.Commun., vol. 50, no. 3, pp. 462–472, Mar. 2002.
[21] M. Mokhtar, A. Gomaa, and N. Al-Dhahir, “OFDM AF relaying underI/Q imbalance: Performance analysis and baseband compensation,”vol. 61, no. 4, pp. 1304–1313, Apr. 2013.
[22] M. Mokhtar, A.-A. A. Boulogeorgos, G. K. Karagiannidis, and N. Al-Dhahir, “OFDM opportunistic relaying under joint transmit/receive I/Qimbalance,” vol. 62, no. 5, pp. 1458–1468, May 2014.
[23] E. Bjornson, M. Matthaiou, and M. Debbah, “A new look at dual-hop relaying: Performance limits with hardware impairments,” vol. 61,no. 11, pp. 4512–4525, Nov. 2013.
[24] M. Matthaiou and A. Papadogiannis, “Two-way relaying under thepresence of relay transceiver hardware impairments,” vol. 17, no. 6,pp. 1136 – 1139, Jun. 2013.
[25] A. Salhab, F. Al-Qahtani, S. Zummo, and H. Alnuweiri, “Exact outageprobability of opportunistic DF relay systems with interference at boththe relay and the destination over Nakagami-m fading channels,” vol. 62,no. 2, pp. 920 – 927, Feb. 2013.
[26] K. T. Hemachandra and N. C. Beaulieu, “Outage analysis of opportunis-tic scheduling in dual-hop multiuser relay networks in the presence ofinterference,” vol. 61, no. 5, pp. 1786 – 1796, May 2013.
[27] E. Soleimani-Nasab, M. Matthaiou, M. Ardebilipour, and G. Karagianni-dis, “Two-way AF relaying in the presence of co-channel interference,”vol. 61, no. 8, pp. 3156 – 3169, Aug. 2013.
FψORSe2e
= 1−N∑m=1
M−m+1∑n=0,m+n>1
K1∑v=1
K2∑t=1
(−1)n+1
exp
(− Θ2x
1− κx
)×∆3
(δ3
Φ3 + x+
δ4Φ5 + x
+ κ2
)+ ∆4
(δ5
Φ4 + x+
δ6Φ5 + x
+ κ2
)+ ∆5
(δ7
Φ6 + x+
δ8Φ5 + x
+ κ2
)+ ∆6
(δ9
Φ7 + x+
δ10
Φ5 + x+ κ2
). (F.1)
L2 (x) = ∆3
(δ3
Φ3 − 1
(1
1 + x− 1
Φ3 + x
)+
δ4Φ5 − 1
(1
1 + x− 1
Φ5 + x
)+
κ2
1 + x
)+ ∆4
(δ5
Φ4 − 1
(1
1 + x− 1
Φ4 + x
)+
δ6Φ5 − 1
(1
1 + x− 1
Φ5 + x
)+
κ2
1 + x
)+ ∆5
(δ7
Φ6 − 1
(1
1 + x− 1
Φ6 + x
)+
δ8Φ5 − 1
(1
1 + x− 1
Φ5 + x
)+
κ2
1 + x
)+ ∆6
(δ9
Φ7 − 1
(1
1 + x− 1
Φ7 + x
)+
δ10
Φ5 − 1
(1
1 + x− 1
Φ5 + x
)+
κ2
1 + x
). (F.3)
[28] A. Bletsas, H. Shin, and M. Z. Win, “Cooperative communications withoutage-optimal opportunistic relaying,” vol. 6, no. 9, pp. 3450–3460,Sept. 2007.
[29] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts, 6th ed. San Diego, CA: Academic, 2000.
[30] S. Ikki and M. Ahmed, “On the performance of adaptive decode-and-forward cooperative diversity with the nth best-relay selection scheme,”in Proc. Global Telecommunications Conference, Honolulu, HI, Nov.-Dec. 2009.
Tran Trung Duy was born in Nha Trang city,Vietnam, in 1984. He received the B.E. degreein Electronics and Telecommunications Engineeringfrom the French-Vietnamese training program forexcellent engineers (PFIEV), Ho Chi Minh CityUniversity of Technology, Vietnam in 2007. In 2013,he received the Ph.D degree in electrical engineeringfrom University of Ulsan, South Korea. In 2013, hejoined the Department of Telecommunications, Postsand Telecommunications Institute of Technology(PTIT), as a lecturer. His major research interests
are cooperative communications, cognitive radio, and physical layer security.
Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Swedenin 2012, and then continued working at BTH as aproject manager. Since 2013, he has joined Queen’sUniversity Belfast, UK as a Lecturer (AssistantProfessor). He held a visiting position at PolytechnicInstitute of New York University and SingaporeUniversity of Technology and Design in 2009 and2011, respectively. His current research interestsinclude cooperative communications, cognitive radio
networks, physical layer security, massive MIMO, cross-layer design, mm-waves communications, and localization for radios and networks.
Dr. Duong has served as an Editor for the IEEE COMMUNICATIONSLETTERS, IET COMMUNICATIONS, WILEY TRANSACTIONS ON EMERGINGTELECOMMUNICATIONS TECHNOLOGIES. He has also served as the LeadGuest Editor of the special issue on “Location Awareness for Radios andNetworks” of the IEEE JOURNAL IN SELECTED AREAS ON COMMUNI-CATIONS, the Lead Guest Editor of the special issue on “Secure PhysicalLayer Communications” of the IET COMMUNICATIONS, Guest Editor ofthe special issue on “Green Media: The Future of Wireless MultimediaNetworks” of the IEEE WIRELESS COMMUNICATIONS MAGAZINE, GuestEditor of the special issue on “Millimeter Wave Communications for 5G”and “Energy Harvesting Communications” of the IEEE COMMUNICATIONSMAGAZINE, Guest Editor of the special issue on “Cooperative CognitiveNetworks” of the EURASIP JOURNAL ON WIRELESS COMMUNICATIONSAND NETWORKING, Guest Editor of special issue on “Security Challengesand Issues in Cognitive Radio Networks” of the EURASIP JOURNAL ONADVANCES SIGNAL PROCESSING. He was awarded the Best Paper Awardat the IEEE Vehicular Technology Conference (VTC-Spring) in 2013, IEEEInternational Conference on Communications (ICC) 2014, and the ExemplaryReviewer Certificate of the IEEE Communications Letters in 2012.
Daniel Benevides da Costa was born in Fortaleza,Ceara, Brazil, in 1981. He received the B.Sc. degreein telecommunications from the Military Institute ofEngineering, Rio de Janeiro, Brazil, in 2003 andthe M.Sc. and Ph.D. degrees in telecommunicationsfrom the University of Campinas, Campinas, Brazil,in 2006 and 2008, respectively. His Ph.D. disserta-tion was awarded the Best Ph.D. Thesis in ElectricalEngineering by the Brazilian Ministry of Education(CAPES) at the 2009 CAPES Thesis Contest. From2008 to 2009, he was a Postdoctoral Research Fel-
low with INRS-EMT, University of Quebec, Montreal, QC, Canada. At thattime, he was the recipient of two scholarships: 1) the Merit ScholarshipProgram for Foreign Students in Quebec and 2) the Natural Sciences andEngineering Research Council of Canada Postdoctoral Scholarship. Since2010, he has been with the Federal University of Ceara, Brazil, where heis currently an Assistant Professor.
Prof. da Costa has authored or coauthored more than 60 papers in IEEE/IETjournals and more than 45 papers in international conferences. His researchinterests lie in the area of wireless communications and include channelmodeling and characterization, relaying/multihop/mesh networks, cooperativesystems, cognitive radio networks, tensor modeling, physical layer security,and performance analysis/design of multiple-input multiple-output systems.He is currently an Editor of the IEEE COMMUNICATIONS LETTERS, theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, the EURASIPJOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING,and the KSII TRANSACTIONS ON INTERNET AND INFORMATIONSYSTEMS. He has also served as Associate Technical Editor for the IEEECOMMUNICATIONS MAGAZINE. He served as the Lead Guest Editor forEURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NET-WORKING in the Special Issue on “Cooperative Cognitive Networks”, LeadGuest Editor for KSII TRANSACTIONS ON INTERNET AND INFORMA-TION SYSTEMS in the Special Issue on “Cognitive Radio Networks: Survey,Tutorial, and New Introduction”, and a Guest Editor for IET COMMUNICA-TIONS in the Special Issue on “Secure Physical Layer Communications”.Also, he was a Workshop Chair of the 2nd International Conference onComputing, Management and Telecommunications (ComManTel 2014) and heserved as a TPC chair for the IEEE GLOBECOM 2013, Workshop on TrustedCommunications with Physical Layer Security. He also acts as a reviewer formajor international journals of the IEEE and IET, and he has been Memberof the Technical Program Committee of several international conferences,such as ICC, WCNC, GLOBECOM, PIRMC, and VTC. He is currently aScientific Consultant of the National Council of Scientific and TechnologicalDevelopment (CNPq), Brazil, and of the Brazilian Ministry of Education(CAPES). He is also a Productivity Research Fellow of CNPq. From 2010to 2012, he was a Productivity Research Fellow of the Ceara Council ofScientific and Technological Development (FUNCAP). Currently, he is amember of the Advisory Board of FUNCAP, Area: Telecommunications. Heis also the recipient of three conference paper awards: one at the 2009 IEEEInternational Symposium on Computers and Communications, one at the 13thInternational Symposium on Wireless Personal Multimedia Communicationsin 2010, and another at the XXIX Brazilian Telecommunications Symposiumin 2011. In 2013, he received the Exemplary Reviewer Certificate of theIEEE Wireless Communications Letters and the Certificate of Appreciationof Top Associate Editor for outstanding contributions to IEEE Transactionson Vehicular Technology. He is Senior Member of IEEE, Member of IEEECommunications Society, IEEE Vehicular Technology Society, and BrazilianTelecommunications Society.
Vo Nguyen Quoc Bao was born in Khanh Hoa,Vietnam, in 1979. He received the B.E. and M.Eng.degree in electrical engineering from Ho Chi MinhCity University of Technology (HCMUT), Vietnam,in 2002 and 2005, respectively, and Ph.D. degreein electrical engineering from University of Ul-san, South Korea, in 2010. In 2002, he joined theDepartment of Electrical Engineering, Posts andTelecommunications Institute of Technology (PTIT),as a lecturer. Since February 2010, he has beenwith the Department of Telecommunications, PTIT,
where he is currently an Assistant Professor. His major research interestsare modulation and coding techniques, MIMO system, combining technique,cooperative communications and cognitive radio. Dr. Bao is a Member ofKorea Information and Communications Society (KICS), The Institute ofElectronics, Information and Communication Engineers (IEICE) and theInstitute of Electrical and Electronics Engineers (IEEE).
Maged Elkashlan (M’06) received the Ph.D. de-gree in Electrical Engineering from the Universityof British Columbia, Canada, 2006. From 2006 to2007, he was with the Laboratory for AdvancedNetworking at University of British Columbia. From2007 to 2011, he was with the Wireless andNetworking Technologies Laboratory at Common-wealth Scientific and Industrial Research Organiza-tion (CSIRO), Australia. During this time, he heldan adjunct appointment at University of TechnologySydney, Australia. In 2011, he joined the School
of Electronic Engineering and Computer Science at Queen Mary Universityof London, UK, as an Assistant Professor. He also holds visiting facultyappointments at the University of New South Wales, Australia, and BeijingUniversity of Posts and Telecommunications, China. His research interests fallinto the broad areas of communication theory, wireless communications, andstatistical signal processing for distributed data processing, millimeter wavecommunications, heterogeneous networks, and security.
Dr. Elkashlan currently serves as an Editor of IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON VEHICULARTECHNOLOGY, and IEEE COMMUNICATIONS LETTERS. He also serves asLead Guest Editor for the special issue on “Green Media: The Future ofWireless Multimedia Networks” of the IEEE WIRELESS COMMUNICATIONSMAGAZINE, Lead Guest Editor for the special issue on “Millimeter WaveCommunications for 5G” of the IEEE COMMUNICATIONS MAGAZINE, GuestEditor for the special issue on “Energy Harvesting Communications” of theIEEE COMMUNICATIONS MAGAZINE, and Guest Editor for the special issueon “Location Awareness for Radios and Networks” of the IEEE JOURNAL ONSELECTED AREAS IN COMMUNICATIONS. He received the Best Paper Awardat the IEEE International Conference on Communications (ICC) in 2014,the International Conference on Communications and Networking in China(CHINACOM) in 2014, and the IEEE Vehicular Technology Conference(VTC-Spring) in 2013. He received the Exemplary Reviewer Certificate ofthe IEEE Communications Letters in 2012.