Outage Probability of Dual-Hop Multiple Antenna AF Relaying Systems with Interference

108 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 1, JANUARY 2013

Outage Probability of Dual-Hop Multiple AntennaAF Relaying Systems with Interference

Caijun Zhong, Member, IEEE, Himal A. Suraweera, Member, IEEE, Aiping Huang, Senior Member, IEEE,Zhaoyang Zhang, Member, IEEE, and Chau Yuen, Member, IEEE

Abstract—This paper presents an analytical investigation onthe outage performance of dual-hop multiple antenna amplify-and-forward relaying systems in the presence of interference.For both the fixed-gain and variable-gain relaying schemes, exactanalytical expressions for the outage probability of the systemsare derived. Moreover, simple outage probability approximationsat the high signal-to-noise-ratio regime are provided, and thediversity order achieved by the systems are characterized. Ourresults suggest that variable-gain relaying systems always outper-form the corresponding fixed-gain relaying systems. In addition,the fixed-gain relaying schemes only achieve diversity order ofone, while the achievable diversity order of the variable-gainrelaying scheme depends on the location of the multiple antennas.

Index Terms—Amplify-and-forward relaying, dual-hop sys-tems, interference, multiple antenna system, outage probability.

I. INTRODUCTION

DUE to the ability of significantly improving the through-put, coverage, and energy consumption of the commu-

nications systems, dual-hop relaying technique has attractedenormous attention from both the industry [1] and academia[2, 3]. Among various relaying schemes proposed in the lit-erature, amplify-and-forward (AF) relaying scheme, whichsimply amplifies the received signal and re-transmits it to thedestination, is of particular interest because of its simplicityand low implementation cost.

The AF relaying scheme generally falls into two categories,i.e., fixed-gain relaying [2] and variable-gain relaying [3]. Bothschemes have received great attention and a large body ofliteratures has investigated the performance of the two relay-ing schemes in various propagation environments (see [4, 5]and references therein). While these works have significantlyimproved our understanding on the performance of dual-hop

Paper approved by M.-S. Alouini, the Editor for Modulation and DiversitySystems of the IEEE Communications Society. Manuscript received October11, 2011; revised March 5 and May 29, 2012.

This work was supported in part by the National Key Basic ResearchProgram of China (No. 2012CB316104), the National Natural ScienceFoundation of China (61201229, 60972058 and 60972057), the ZhejiangProvincial Natural Science Foundation of China (No. LQ12F01006 and No.LR12F01002), the Fundamental Research Funds for Central Universities(2012QNA5011), and the Supporting Program for New Century ExcellentTalents in University (NCET-09-0701). The work of H. Suraweera and C.Yuen were partly supported by the Singapore University Technology andDesign (grant no. SUTD-ZJU/RES/02/2011).

C. Zhong, A. Huang, and Z. Zhang are with the Institute of Information andCommunication Engineering, Zhejiang University, Hangzhou, 310027, China(e-mail: {caijunzhong, aiping.huang, ning ming}@zju.edu.cn).

H. A. Suraweera and C. Yuen are with the Engineering Product Develop-ment Department, Singapore University of Technology and Design, Singapore(e-mail: {himalsuraweera, yuenchau}@sutd.edu.sg).

Digital Object Identifier 10.1109/TCOMM.2012.102512.110679

AF relaying systems, a key feature of wireless communicationsystems, namely, co-channel interference (CCI), is neglected.

This important observation has recently promoted a surgeof research interest in understanding the impact of CCI on theperformance of dual-hop AF relaying systems. In [6], the out-age performance of dual-hop fixed-gain AF relaying systemswith interference-limited destination was investigated, while[7] addressed case with variable-gain relaying scheme andinterference-limited relay, and later [9] extended the analysisof [7] to the more general Nakagami-m fading channels, while[8] studied the performance of fixed-gain dual-hop systemswith a Rician interferer. Meanwhile, the more general casewith interference at both the relay and destination nodes hasbeen investigated in [10–15]. In [10], the outage performanceof dual-hop fixed-gain AF relaying scheme was examined,and the case with variable-gain relaying scheme was dealtwith in [11, 12]. [13] presented an approximated error analysisof the system employing variable-gain relaying scheme, and[14] studied the outage performance of both fixed-gain andvariable-gain schemes assuming a single dominant interfererat both the relay and destination, while [15, 16] addressedthe case with Nakagami-m fading. Most recently, resourceallocation problems in the AF relaying systems have beenstudied in [17, 18].

It is worth pointing out that most of the prior works assumethe interference-limited scenario, hence, the impact of thejoint effect of CCI and noise on the outage performance ofdual-hop AF relaying system has not been well-understood.In addition, all the prior works consider the single antennasystems, therefore, the effect of employing multiple antennasin the presence of CCI in the dual-hop context remains un-known. In light of these two key observations, we investigatethe outage performance of dual-hop multiple antenna AFrelaying systems in the presence of CCI as well as noise.For mathematical tractability, we limit the analysis to thecase where only one of the nodes is equipped with multipleantennas, hence, three scenarios are of interest: (1) multipleantenna source, single antenna relay and destination (N-1-1);(2) multiple antenna destination, single antenna source andrelay (1-1-N); (3) multiple antenna relay, single antenna sourceand destination (1-N-1). We also assume that the relay nodeis subjected to a single dominant interferer and noise whilethe destination is corrupted by the noise only1. Although thesystem model is less general, it enables us to gain key design

1Our model assumes that the relay and destination nodes experience differ-ent patterns of interference, which is particularly suitable for the frequencydivision duplex system, where the source-relay link and the relay-destinationlink operate over different frequency [19, 20].

0090-6778/13$31.00 c© 2013 IEEE

ZHONG et al.: OUTAGE PROBABILITY OF DUAL-HOP MULTIPLE ANTENNA AF RELAYING SYSTEMS WITH INTERFERENCE 109

insights on the joint impact of CCI and noise, as well as thebenefit of implementing multiple antennas.

The main contributions of the paper are summarised asfollows:

• For fixed-gain relaying systems, we derive exact closed-form expressions for the outage probability of all threesystems.

• For variable-gain relaying systems, we present analyticalexpressions involving a single integral for the outageprobability of all three systems. In addition, we proposesimple and tight closed-form lower bound of the outageprobability of the system.

• For both fixed-gain and variable-gain relaying systems,we give simple and informative high signal-to-noise-ratio(SNR) approximations for the outage probability for allthree systems.

• These analytical expressions not only provide fast andefficient means for the evaluation of the outage perfor-mance of the systems, they also enable us to gain valuableinsights on the impact of key system parameters on theoutage performance of the system.

The remainder of the paper is organized as follows: SectionII introduces the system model. Section III presents theexact as well as asymptotical analytical expressions for theoutage probability of the systems, and numerical results anddiscussions are provided in Section IV. Finally, Section Vconcludes the paper and summarizes the findings.

Notations: We use bold upper case letters to denote ma-trices, bold lower case letters to denote vectors and lowercase letters to denote scalers. ||h||2 denotes the Frobeniusnorm, E{x} stands for the expectation of random variable x, ∗denotes the conjugate operator, while † denotes the conjugatetranspose operator. n! denotes the factorial of integer n andΓ(x) is the gamma function.

II. SYSTEM MODEL

Let us consider a dual-hop multiple antenna AF relayingsystem as illustrated in Figure 1, where the relay node issubjected to a single dominate interferer2 and additive whiteGaussian noise (AWGN), while the destination node is cor-rupted by AWGN only. During the first phase, the sourcetransmits signal symbol to the relay node, and the signalreceived at the relay node can be expressed as

yr = H1x+ hIsI + n1, (1)

where H1 denotes the channel for the source-relay link, and itsentries follow identically and independently distributed (i.i.d.)complex Gaussian distribution with zero-mean and varianceσ21 , x is the source symbol vector with E{x†x} = P , hI

is the channel for the interference-relay link, and its entriesare i.i.d. complex Gaussian random variables with zero-meanand variance σ2

I , sI is the interference symbol satisfying

2We assume a single interferer at the relay node for the mathematicaltractability. Although less general, the single interferer model is still ofpractical interest and importance. For instance, in a well planned cellularnetwork, it is very likely that the system will be subjected to a single dominantinterferer. Hence, it has been adopted in a number of previous works, i.e.,[21, 22].

S R D

I

H1 H2

hIf1

f1 f2

Fig. 1. System model: S, R, D and I denote source, relay,destination and interferer node, respectively. The source-relayand relay-destination link operate over different frequency f1and f2.

E{sIs∗I} = PI , and n1 is the AWGN noise at the relay nodewith E{n†

1n1} = N0I.At the second phase, the relay node transmits a transformed

version of the received signal to the destination, and the signalat the destination can be expressed as

yd = H†2Wyr + n2, (2)

where H2 denotes the channel for the relay-destination link,and its entries are i.i.d. complex Gaussian random variableswith zero-mean and variance σ2

2 . W is the transformationmatrix with E{||Wyr||2F } = Pr, and n2 is the AWGN withE{n2n

†2} = N0I. We assume that H1, H2, and hI are

mutually independent.Note, for the sake of a concise presentation, we have, in

the above, provided a fairly general dual-hop multiple antennaAF relaying system model on purpose. However, we do notspecify the size of the matrices H1, H2 or vectors hI here.Instead, they will be defined explicitly whenever appropriatein the following sections.

For notational convenience, we define ρ1 � Pσ21

N0, ρ2 �

Prσ22

N0, and ρI � PIσ

2I

N0.

III. THE N-1-1 SYSTEM

This section considers the case where the source node isequipped with N antennas, while the relay and destinationnodes only have a single antenna. For such system, we assumethat beamforming scheme is adopted at the source node, i.e.,x = wts, where wt is the transmit beamforming vector with|wt|2 = 1 and s is the transmit symbol with E{ss∗} = P .To conform to the notation convention, we will use vectorh1 ∼ CN 1×N to denote the source-relay link instead H1.Similarly hI and h2, will be used to denote channel for theinterference-relay link and relay-destination link, respectively.

The end-to-end signal-to-interference-and-noise ratio(SINR) of the system can be expressed as:

γ =|w|2|h2|2|h1wt|2P

|w|2|h2|2|hI |2PI + |w|2|h2|2N0 +N0. (3)

It is easy to observe that the optimal beamforming vector is

to match the first hop channel h1, i.e., wt =h†

1

|h1| . Therefore,the end-to-end SINR is given by

γ =|w|2|h2|2|h1|2P

|w|2|h2|2|hI |2PI + |w|2|h2|2N0 +N0. (4)


In the following, we provide a separate treatment for the fixed-gain and variable-gain relaying schemes.

A. Fixed-Gain Relaying

For fixed-gain relaying scheme, the relaying gain is givenby w2 = Pr

NPσ21+PIσ2

I+N0, and we have the following key

result.Theorem 1: The outage probability of the N-1-1 dual-hop

fixed-gain AF relaying systems is given by

Pout(γth) = 1− e− γth

ρ1

N−1∑m=0

(γthρ1

)m2

m!

m∑j=0

(m

j

) j∑k=0

(j

k

)

Γ(k + 1)ρkIρk+11

(ρIγth + ρ1)k+1

(γthρ1

) k−j+12(Nρ1 + ρI + 1

ρ2

) j−k+12

×Kk−j+1

(2

√(Nρ1 + ρI + 1)γth

ρ1ρ2

), (5)

where Kv(x) is the v-th order modified Bessel function of thesecond kind [23, Eq. (8.407.1)].

Proof: See Appendix B-A.Theorem 1 only involves standard functions, and hence

offers an efficient means to evaluate the outage probabilityof the N-1-1 dual-hop fixed-gain AF relaying systems. Forthe special case N = 1, the above expression reduces toprior results presented in [8, Eq. (12)] and [16, Eq. (10)].To gain more insights, we look into the high SNR regime,where simple expressions can be obtained.

Theorem 2: At the high SNR regime, i.e., ρ2 = μρ1, ρ1 →∞, the outage probability of the N-1-1 dual-hop fixed-gainAF relaying systems can be approximated as

P lowout (γth) ≈ (6){ (

1μ

(ln μρ1

γth+ ψ(1) + ψ(2)

)+ ρI + 1

)γthρ1, N = 1,

Nμ(N−1)

γthρ1, N ≥ 2,

where ψ(x) is the digamma function [23, Eq. (8.360.1)].Proof: See Appendix B-B.

Theorem 2 indicates that fixed-gain relaying schemes onlyachieve diversity order one regardless of the number of an-tennas N . However, increasing N helps improve the outageperformance by providing extra array gain. Moreover, Theo-rem 2 suggests that the impact of CCI vanishes at the highSNR regime when N ≥ 2.

B. Variable-Gain Relaying

For variable-gain relaying scheme, the relaying gain is givenby w2 = Pr

|h1|2P+|hI |2PI+N0, and we have the following key

result.Theorem 3: The outage probability of the N -1-1 variable-

gain relaying systems is given by

Pout(γth) = 1− 2e− γth

ρ1− γth

ρ2

σ2I

N1−1∑m=0

(γthρ1

)m1

m!

m∑j=0

(m

j

)

(1

ρ2

)m−j j∑k=0

(j

k

)(γthρ2

)j−k ((γth + 1)γth

ρ1ρ2

) k−m+12

I1(γth)

where I1(γth) is defined as

I1(γth) =∫ ∞

0

e−(

PIγthN0ρ1

+ 1

σ2I

)y3(y3PIN0

+ 1

) k+m+12

Kk−m+1

(2

√(γth + 1)γth

ρ1ρ2

(y3PIN0

+ 1

))dy3. (7)

Proof: See Appendix B-C.To the best of the authors’ knowledge, the integral I1 does

not admit a closed-form expression. However, this single inte-gral expression can be efficiently evaluated numerically, whichstill provides computational advantage over the Monte Carlosimulation method. Alternatively, we can use the followingclosed-form lower bound on the outage probability, which istight across the entire SNR range, and becomes exact at thehigh SNR regime.

Corollary 1: The outage probability of the N-1-1 variable-gain relaying systems is lower bounded by

P lowout (γth) = 1− (8)

e−γth

ρ1− γth

ρ2

N−1∑m=0

(γthρ1

)m1

m!

m∑j=0

(m

j

)Γ(j + 1)ρjIρ

j+11

(ρ1 + ρIγth)j+1.

Proof: We first notice that the end-to-end SINR can beupper bounded by

γ =

|h1|2PN0

|h2|2Pr

N0(|h3|2PI

N0+ 1)(

|h2|2Pr

N0+ 1)+ |h1|2P

N0

≤ min

⎛⎝ |h1|2P

N0

|h3|2PI

N0+ 1

,|h2|2PrN0

⎞⎠ . (9)

Hence, due to the independence of h1, h2 and h3, the outageprobability of the system can be lower bounded by

P lowout (γth) = 1−

Pr

⎛⎝ |h1|2P

N0

|h3|2PI

N0+ 1

≥ γth

⎞⎠Pr

( |h2|2PrN0

≥ γth

). (10)

To this end, the desired result can be computed after somesimple algebraic manipulations with the help of Lemma 2presented in Appendix A.

Now, we look into the high SNR regime, and investigatethe diversity order achieved by the system.

Theorem 4: At the high SNR regime, i.e., ρ2 = μρ1, ρ1 →∞, the outage probability of the N-1-1 variable-gain relayingsystems can be approximated as

P lowout (γth) ≈

{ (1μ + ρI + 1

)γthρ1, N = 1,

1μγthρ1, N ≥ 2.

(11)

Proof: See Appendix B-D.Clearly, the variable-gain relaying system also achieves di-

versity order of one. Now, comparing Theorem 4 and Theorem2, it is evident that the variable-gain relaying scheme outper-forms the fixed-gain relaying scheme at the high SNR regime.Moreover, the performance gain is much more pronounced forsmall N , and gradually diminishes when N becomes large.Similarly, we see that the impact of CCI disappears when


N ≥ 2, which suggests that implementing multiple antenna atthe source can effectively help combat the CCI at the relay.

IV. THE 1-1-N SYSTEM

This section considers the case where the destination nodeis equipped with N antennas, while the source and relaynodes only have a single antenna. Similarly, to conform tothe notation convention, we will use scaler h1, hI and vectorh2 ∼ CNN×1 to denote the source-relay, interference-relayand relay-destination links, respectively. After applying themaximum ratio combining at the destination node, the end-to-end SINR can be expressed as

γ =|w|2|h2|2|h1|2P

|w|2|h2|2|hI |2PI + |w|2|h2|2N0 +N0. (12)

For notational convenience, we define y1 � |h1|2, y2 � |h2|2,y3 � |hI |2.


For fixed-gain relaying scheme, the relaying gain is givenby w2 = Pr

Pσ21+PIσ2

I+N0, and the outage probability of the

system is given in the following theorem.Theorem 5: The outage probability of the 1-1-N fixed-gain

relaying systems is given by

Pout(γth) = 1− 2ρ1e− γth

ρ1

Γ(N)(ρIγth + ρ1)

((ρ1 + ρI + 1)γth

ρ1ρ2

)N2

×KN

(2

√(ρ1 + ρI + 1)γth

ρ1ρ2

). (13)

Proof: From the definition, the outage probability is givenby

Pout(γth) = Pr

(y1 ≤ γth

P

(y3PI +N0 +

N0

|w|2y2

)).

Conditioned on y2 and y3, the outage probability can be shownas

Pout(γth) = 1− e−γthN0

Pσ21 e

− γthN0

Pσ21|w|2y2 e

−γthy3PIPσ2

1 . (14)

Averaging over y2 and y3, the unconditional outage probabilitycan be obtained as

Pout(γth) = 1− e− γthN0

Pσ21

2

Γ(N)σ2N2

(γthN0σ

22

Pσ21 |w|2

)N2

KN

(2

√γthN0

Pσ21σ

22 |w|2

)1

PIσ2Iγth

Pσ21

+ 1. (15)

To this end, substituting w into Eq. (15), the desired result canbe obtained after some simple algebraic manipulations.

Having obtained the exact outage probability expression, wenow establish the asymptotical outage probability approxima-tion at the high SNR regime.

Theorem 6: At the high SNR regime, i.e., ρ2 = μρ1, ρ1 →∞, the outage probability of the 1-1-N dual-hop fixed-gainAF relaying systems can be approximated by

Pout(γth) ≈(ρI + 1 +

1

(N − 1)μ

)γthρ1, N ≥ 2. (16)

Proof: Utilizing the asymptotic expansion (36), thedesired result can be obtained after some basic algebraicmanipulations.

Theorem 6 indicates that the 1-1-N system achieves diver-sity order one. Also, it suggests that a large N and relaytransmit power helps to reduces the outage probability byproviding a larger array gain. Moreover, it shows that the CCIalways degrades the outage performance of the system.


For variable-gain relaying scheme, the relaying gain is givenby w2 = Pr

y1P+y3PI+N0, and the outage probability of the

system is given in the following theorem.Theorem 7: The outage probability of the 1-1-N dual-hop

variable-gain AF relaying systems can be expressed as

Pout(γth) = 1− 2e−γth

ρ1− γth

ρ2

σ2IΓ(N)

(17)

×N−1∑k=0

(N − 1

k

)ρN−k−12

((γth + 1)γth

ρ1ρ2

) k+12

I2(γth),

where

I2(γth) =∫ ∞

0

e−γthPIy3

ρ1N0− y3

σ2I

(PIy3N0

+ 1

)k+12

×Kk+1

(2

√(γth + 1)γth

ρ1ρ2

(PIy3N0

+ 1

))dy3. (18)

Proof: The result can be obtained by following similarlines as in the proof of Theorem 3, along with some simplealgebraic manipulations.

Corollary 2: The outage probability of the 1-1-N dual-hopvariable-gain AF relaying systems is lower bounded by

P lowout (γth) = 1− ρ1e

− γthρ1

ρ1 + ρIγth

(1− 1

Γ(N)γ

(N,

γthρ2

)),

(19)

where γ(n, x) is the lower incomplete gamma function [23,Eq. (8.350.1)].

Proof: The result can be obtained by following similarlines as in the proof of Corollary 1, along with some simplealgebraic manipulations.

Now, we look into the high SNR regime, and investigatethe diversity order achieved by the system.

Theorem 8: At the high SNR regime, i.e., ρ2 = μρ1, ρ1 →∞, the outage probability of the system can be approximatedas

P lowout (γth) ≈ (1 + ρI)

γthρ1, N ≥ 2. (20)

Proof: Utilizing the asymptotical expansion of the incom-plete gamma function [23, Eq. (8.354.1)], the desired resultcan be obtained after some simple algebraic manipulations.

Not surprisingly, we see that the 1-1-N system withvariable-gain relaying also achieves diversity order one. Com-pared with Theorem 6, we see that the variable-gain relay-ing scheme outperforms the fixed-gain relaying scheme byachieving a higher array gain. Also, Theorem 8 suggests a


rather interesting result that increasing N beyond two doesnot produce any advantage at the high SNR regime.

V. THE 1-N-1 SYSTEM

This section considers the case where the relay node isequipped with N antennas, while the source and destinationnodes only have a single antenna. Similarly, to conform to thenotation convention, we will use h1 ∼ CNN×1, hI ∼ CNN×1

and h2 ∼ CN 1×N to denote the source-relay, interference-relay and relay-destination links, respectively. Then it is easyto show that the end-to-end SINR can be expressed as

γ =|h†

2Wh1|2P|h2WhI |2PI + |h†

2W|2N0 +N0

. (21)


With fixed-gain relaying scheme, the relay transformationmatrix is simply a scaled identity matrix, i.e., W = wI, withw2 = Pr

NPσ21+NPIσ2

I+N0. Hence, the end-to-end SINR reduces

to

γ =w2|h†

2h1|2Pw2|h†

2hI |2PI + w2|h†2|2N0 +N0

. (22)

Theorem 9: The outage probability of the 1-N-1 dual-hopfixed-gain AF relaying systems is given by

Pout(γth) = 1− 2ρ1e− γth

ρ1

Γ(N)(ρ1 + ρIγth)

((Nρ1 +NρI + 1)γth

ρ1ρ2

)N2

×KN

(2

√(Nρ1 +NρI + 1)γth

ρ1ρ2

). (23)

Proof: See Appendix C-A.Theorem 10: At the high SNR regime, ρ2 = μρ1, ρ1 → ∞,

the outage probability of the 1-N-1 fixed-gain relaying systemscan be approximated as

Pout(γth) ≈(1 + ρI +

N

μ(N − 1)

)γthρ1, N ≥ 2. (24)

Proof: Utilizing the asymptotic expansion (36), thedesired result can be obtained after some basic algebraicmanipulations.

Theorem 10 indicates that the 1-N-1 system with fixed-gainAF relaying only achieves diversity order one. Moreover, itsuggests that the CCI degrades the outage performance whileincreasing N helps improve the outage performance.


When the channel state information (CSI) is available atthe relay node, the optimal relay transformation matrix Wcould be obtained by solving Eq. (21). However, due to thenon-convex nature of the problem, finding the optimal W inanalytical form does not seem to be tractable. Therefore, wehereafter propose a heuristic W and investigate its perfor-mance.

With CSI at the relay node, it is natural to apply themaximal ratio combining/transmitting principle. Hence, the

relay transformation matrix is given by W = wh2∗h†

1

|h2||h1| .Depending on the availability of the interference channelinformation (ICI) at the relay node, we consider two separatecases.

1) Without ICI: In this case, to meet the power constraintat the relay node, we have

w2 =Pr

NPσ21 + PIσ2

I +N0. (25)

Hence, the end-to-end SINR can be expressed as

γ =|h2|2|h1|2P

|h2|2 |h1hI |2|h1|2 PI + |h†

2|2N0 +N0/w2. (26)

Theorem 11: The outage probability of the 1-N-1 dual-hop variable-gain AF relaying systems without ICI can beexpressed as

Pout(γth) = 1− 2e−γthρ1

Γ(N)

N−1∑m=0

γmthρm1 m!

m∑i=0

(m

i

) i∑j=0

(i

j

)

ρjIΓ(j + 1)(γthρIρ1

+ 1)j+1

(γthρ1

)N+j−i2

(Nρ1 + ρI + 1

ρ2

)N+i−j2

×KN+j−i

(2

√(Nρ1 + ρI + 1)γth

ρ1ρ2

). (27)

Proof: See Appendix C-B.

Theorem 12: At the high SNR regime, ρ2 = μρ1, ρ1 → ∞,the outage probability of the 1-N-1 dual-hop variable-gain AFrelaying systems without ICI can be approximated as

Pout(γth) ≈ (28)N−1∑i=0

(−1)N−i(ln Nγthμρ1

− ψ(1)− ψ(N − i+ 1))

Γ(N)Γ(i+ 1)Γ(N − i+ 1)

(Nγthμρ1

)N.

Proof: The result can be obtained by following similarlines as in the proof of Theorem 1 with the help of Lemma1.

2) With ICI: When the ICI is available at the relay node,we have

w2 =Pr

h†1h1P + |h†

1hI |2PI/|h1|2 +N0

. (29)

Hence, the end-to-end SINR can be expressed as

γ = (30)

|h2|2|h1|2P|h2|2 |h1hI |2

|h1|2 PI + |h†2|2N0 +

N0

Pr

(|h1|2P +

|h†1hI |2|h1|2 PI +N0

) .Theorem 13: The outage probability of the 1-N-1 dual-hop

variable-gain AF relaying systems with ICI can be expressedas

Pout(γth) = 1− 2e−γthρ1

− γthρ2

σ2IΓ(N)

N−1∑m=0

(γthρ1

)m1

m!

m∑j=0

(m

j

)(

1

ρ2

)m−j N+j−1∑k=0

(N + j − 1

k

)(γthρ2

)N+j−1−k

((γth + 1)γth

ρ1ρ2

) k−m+12

I3(γth), (31)


where

I3(γth) =∫ ∞

0

Kk−m+1

(2

√(γth + 1)γth

ρ1ρ2

(PIy3N0

+ 1

))(PIy3N0

+ 1

) k+m+12

e−(

PIγthρ1N0

+ 1

σ2I

)y3dy3. (32)

Proof: The result can be obtained by following similarlines as in the proof of Theorem 3, along with some simplealgebraic manipulations.

Corollary 3: The outage probability of the 1-N-1 dual-hopvariable-gain AF relaying systems with ICI can be lowerbounded by

P lowout (γth) = 1−

⎛⎝1−

γ(N, γthρ2

)Γ(N)

⎞⎠ (33)

⎛⎝e− γth

ρ1

N−1∑m=0

(γthρ1

)m1

m!

m∑j=0

(m

j

)Γ(j + 1)ρjIρ

j+11

(ρ1 + ρIγth)j+1

⎞⎠ .

Proof: The result can be obtained by following similarlines as in the proof of Corollary 1, along with some simplealgebraic manipulations.

Theorem 14: At the high SNR regime, the outage proba-bility of the system when N ≥ 2 can be approximated

P lowout (γth) ≈ (34)

1

Γ(N + 1)

(ρINe

1ρI Γ

(N + 1,

1

ρI

)+

1

μN

)(γthρ1

)N,

where Γ(n, x) is the upper incomplete gamma function [23,Eq. (8.350.2)].

Proof: See Appendix C-C.While Theorem 10 implies that the fixed-gain relaying

scheme only achieves diversity order of one, both Theorem12 and 14 reveal that diversity order of N is achieved by1-N-1 systems with variable-gain relaying scheme.

VI. NUMERICAL RESULTS AND DISCUSSION

In this section, Monte Carlo simulation results are providedto validate the analytical expressions presented in the previoussections. Note, the integral expressions presented in Theorem3 and Theorem 7 are evaluated numerically with the build-in functions in Matlab, i.e., the “quad” command, and wechoose the default absolute error tolerance value 1.0 ∗ 10−6

to control the accuracy of the numerical integration. Forall the simulations, we set γth = 0 dB, and ρI = 0 dB.Also, all the simulation results are obtained by 107 runs.In general, deploying multiple antenna helps to combat theimpact of CCI at the relay node, and the variable-gain relayingscheme outperforms the fixed-gain relaying scheme at the highSNR regime, see Table I for a summary of the performancecomparison between the two relaying schemes.

Figure 2 plots the outage probability of the N-1-1 dual-hop AF relaying systems for both fixed-gain and variable-gain relaying schemes when μ = 1. First of all, we cansee that the analytical results are in exact agreement withthe Monte Carlo simulation results, and the outage lowerbound for the variable-gain relaying system is sufficiently tight

0 10 20 30 40 5010

−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Out

age

Pro

babi

lity

Monte Carlo simulationExact analyticalAsymptotic approximationAnalytical lower bound

N=1

N=2

variable−gain

fixed−gain

Fig. 2. Outage probability of the N-1-1 dual-hop relayingsystems: fixed-gain vs. variable-gain.

0 5 10 15 20 25 3010

−3

10−2

10−1

100

SNR (dB)

Out

age

Pro

babi

lity


fixed−gain, N=2, 10

variable−gain, N=2, 10

Fig. 3. Outage probability of the 1-1-N dual-hop relayingsystems with different N : fixed-gain vs. variable-gain.

across the entire SNR range of interest, while the high SNRapproximations works quite well even at moderate SNRs (i.e.,ρ1 = 18 dB). It can also be observed that, for both N = 1and N = 2, the same diversity order of one is achievedby both the fixed-gain and variable-gain relaying schemes,which implies that increasing N does not provide additionaldiversity gain for the N-1-1 system. However, it does improvethe outage performance of the system by offering extra codinggain. Moreover, the variable-gain relaying schemes in generaloutperforms the fixed-gain relaying schemes.

Figure 3 examines the outage probability of the 1-1-N dual-hop AF relaying systems for both fixed-gain and variable-gain relaying schemes when μ = 1. Similar to the N-1-1dual-hop AF relaying systems, we observe that only diversityorder of one is achieved for both the fixed-gain and variable-gain relaying systems regardless of N , and variable-gainrelaying systems achieve superior outage performance than thefixed-gain relaying systems. However, such performance gaindiminishes gradually N becomes larger. As illustrated in thefigure, the outage gap when N = 10 is much narrower whencompared with N = 2. This rather interesting phenomenon ismainly due to the fact that the outage performance improve-


TABLE I. High SNR Outage Performance ComparisonN-1-1 1-1-N 1-N-1

Fixed-gainKey results Theorem 2 Theorem 6 Theorem 10

Diversity order 1 1 1Impact of N N provides array gain, not

diversity gain.N provides array gain, notdiversity gain.

N provides array gain, notdiversity gain.

Variable-gainKey results Theorem 4 Theorem 8 Theorem 12 & 14

Diversity order 1 1 NImpact of N N provides no array gain

and diversity gainN provides no array gainand diversity gain

N provides both array gainand diversity order

0 5 10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Out

age

Pro

babi

lity


variable−gain without IC

fixed−gain

variable−gain with IC

Fig. 4. Outage probability of the 1-N-1 dual-hop relayingsystems with N = 2: fixed-gain vs. variable-gain.

ment of the variable-gain relaying schemes due to increasingN is almost intangible at the high SNR regime, as manifestedin Theorem 8.

Figure 4 illustrates the outage probability of the 1-N-1 dual-hop AF relaying systems for both fixed-gain and variable-gain relaying schemes when N = 2 and μ = 1. As shownin the figure, the diversity order achieved by the fixed-gainrelaying scheme is one, while the diversity order achievedby the variable-gain relaying schemes is two. Moreover, wesee that additional improvement of the outage performance isachieved when the ICI is available at the relay node. Bothobservations suggest the critical importance of have CSI atthe relay node for the 1-N-1 AF dual-hop systems.

Figure 5 provides an outage performance comparison be-tween the N-1-1, 1-1-N and 1-N-1 dual-hop AF relayingsystems with fixed-gain relaying scheme. Let us first lookback at Theorem 1, 5 and 9, we see that the coefficients ofthe high SNR approximations for the N-1-1, 1-1-N and 1-N-1are given by aN11 = N

μ(N−1) , a11N = ρI + 1 + 1μ(N−1) , and

a1N1 = 1 + ρI +N

μ(N−1) , respectively. It is easy to observethat a1N1 ≥ {aN11, a11N}. Now the difference of aN11 anda11N can be computed as aN11−a11N = 1

μ − (1+ρI), whichsuggests the N-1-1 system outperforms the 1-1-N system onlyif 1

μ ≤ 1 + ρI . In Figure 4, it can be observed that the 1-N-1 system always has the worst outage performance, whilewhether the outage performance of N-1-1 systems is superiorthan that of the 1-1-N systems depends on μ, which confirmsthe above analysis.

0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

SNR

Out

age

Pro

babi

lity

Monte Carlo simulationExact analyticalAsymptotic approximation

N−1−1

1−1−N

1−N−1

μ=2

μ=0.2

Fig. 5. Comparison of the outage probability of N-1-1, 1-1-Nand 1-N-1 dual-hop systems with fixed-gain relaying.

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Out

age

Pro

babi

lity

Monte Carlo simulationExact analyticalAsymptotic approximation

1−N−1

N−1−1

1−1−N

Fig. 6. Comparison of the outage probability of N-1-1, 1-1-Nand 1-N-1 dual-hop systems with variable-gain relaying.

Figure 6 provides an outage performance comparison be-tween the N-1-1, 1-1-N and 1-N-1 dual-hop AF systems withvariable-gain relaying scheme for μ = 0.2, 2. Recall Theorem12 and 14 that the 1-N-1 system achieves diversity orderof N , hence, it definitely outperforms the 1-1-N and 1-N-1systems which only achieve diversity order of one. While aclose observation at Theorem 4 and 8 shows that whether the1-1-N system is superior to the N-1-1 system depends on therelationship between 1

μ and 1 + ρI . In Figure 5, we see thatthe 1-N-1 system always has the best outage performance,


while the outage performance of the N-1-1 system is betterthan 1-1-N system when μ = 2, and worse than the 1-1-Nwhen μ = 0.2.

VII. CONCLUSION

This paper has investigated the outage performance of dual-hop multiple antenna AF relaying systems with both fixed-gain and variable-gain relaying schemes. Exact analyticalexpressions for the outage probability of the systems underconsideration were presented, which provide fast and efficientmeans to evaluate the outage probability of the systems. Inaddition, simple and informative high SNR approximationswere derived, while shed lights on how key parameters, suchas CCI, antenna number N , and relay power μ, affect theoutage performance of the systems.

The findings suggest that, for all three scenarios, thevariable-gain relaying scheme outperforms the fixed-gain re-laying scheme. Moreover, for the N-1-1 and 1-1-N systems,the performance advantage of variable-gain relaying schemediminishes as N increases. On the contrary, for the 1-N-1system, the performance advantage of variable-gain relayingis substantially increased when N becomes large. Moreover,it is demonstrated that whether the N-1-1 system outperformsthe 1-1-N system depends on the interference power and therelay power.

It is explicitly proven that, for all three scenarios, the fixed-gain relaying scheme only achieves diversity order of one.On the other hand, the variable-gain relaying scheme achievesdiversity order of one for the N-1-1 and 1-1-N systems, butprovides diversity order of N for the 1-N-1 system, whichsuggests that it is beneficial to put the multiple antennas at therelay node when variable-gain relaying scheme is adopted.

APPENDIX ARELATED LEMMAS

In this section, we present three Lemmas which will beused in the proof of the main results. Specifically, Lemma 1will be used in the derivation of the asymptotical high SNRapproximation for the N-1-1 system employing fixed-gainrelaying scheme, Lemma 2 will be used in the derivation ofthe lower bound of the outage probability of the N-1-1 systememploying variable-gain relaying scheme, while Lemma 3 willbe used in the derivation of the exact outage probability of theN-1-1 system employing variable-gain relaying scheme.

Lemma 1: Let U = y1min(

1y3+1 ,

y2C

), where y1, y2 are

independent random variables with probability density func-tion (p.d.f.) fyi(x) =

xNi−1

Γ(Ni)e−x, i = 1, 2. y3 is independently

distributed exponential random variable with p.d.f. fy3(x) =1λ3e− x

λ3 . C is a positive constant. Then the asymptoticalexpansion of the cumulative distribution function (c.d.f.) of Unear zero can be expressed as FU (x) = Cx

N1−1 when N2 = 1,

and FU (x) =∑N−1i=0

(−1)N−i(lnCx−ψ(1)−ψ(N−i+1))Γ(N)Γ(i+1)Γ(N−i+1) (Cx)N

when N1 = N2 = N .Proof: Define v �

(1

y3+1 ,y2C

), we first take a look at

the asymptotic behavior of v near zero. To do so, we start by

deriving the c.d.f. of v as follows

Fv(x) = 1− Pr

(1

y3 + 1≥ x

)Pr(y2C

≥ x)

= 1− Pr

(y3 ≤ 1

x− 1

)(1− Pr (y2 ≤ Cx))

= 1−(1− exp

(− 1

λ3

(1

x− 1

)))(1− Fy2(Cx)).

Hence, conditioned on y1, the c.d.f. of U can be expressed as

FU (x) =

1−(1− exp

(− y1λ3

(1

x− 1

)))(1− Fy2

(Cx

y1

)).

Now, we observe that when x → 0, exp(− 1λ3

(y1x − 1

)) →0, hence, the conditional c.d.f. of U can be well approximatedby FU (x) ≈ Fy2

(Cxy1

). For the N2 = 1 case, the c.d.f. of y2

near zero can be approximated as Fy2(Cxy1

)≈ Cx

y1. To this

end, average over y1 yields the desired result.

The N1 = N2 = N case is a bit more tricky since we cannot approximate the c.d.f. of y2 near zero because of the factthat the resulting integration does not converge. Therefore, weadopt an alternative method. We first explicitly approximatethe c.d.f. of U near zero as

FU (x) ≈ 1− 2

Γ(N)

N−1∑i=0

(Cx)i

Γ(i + 1)(Cx)

N−i2 KN−i

(2√Cx).

(35)

Invoking the asymptotic expansion of the Kv(x) [23, Eq.(8.446)], we have

(Cx)N−i

2 KN−i(2√Cx) =

1

2

N−i−1∑k=0

Γ(N − i− k)

Γ(k + 1)(−Cx)k

− (−Cx)N−i

2

∞∑k=0

ln x− ψ(k + 1)− ψ(N − i+ k + 1)

Γ(k + 1)Γ(N − i+ k + 1)(Cx)k.

The next key observation is that

N−1∑i=0

xi

Γ(i+ 1)

N−i−1∑k=0

Γ(N − i− k)

Γ(k + 1)(−x)k = Γ(N). (36)

Hence, the first non-zero term in Eq. (35) after the asymptoticexpansion is given by

FU (x) ≈ (Cx)N

Γ(N)

N−1∑i=0

(−1)N−i(lnCx− ψ(1)− ψ(N − i+ 1))

Γ(i+ 1)Γ(N − i+ 1),

which completes the proof.

Lemma 2: Let y1 and y2 be independent random variableswith p.d.f. fy1(x) = xN1−1

λN11 Γ(N1)

e−xλ1 , fy2(x) = 1

λ2e−

xλ2 , and

a, b are positive constant, then the c.d.f. of random variableU � ay1

by2+1 is given by

FU (x) =

1− e− x

aλ1

N1−1∑m=0

(x

aλ1

)m1

m!

m∑j=0

(m

j

)Γ(j + 1)bj

λ2

(bxaλ1

+ 1λ2

)j+1 .


Proof: Starting from the definition, the c.d.f. of randomvariable U can be computed as

FU (x) = Pr(U ≤ x) = Pr(y1 ≤ x

a(by2 + 1)

)=

∫ ∞

0

Fy1

(xa(bt+ 1)

)fy2(t)dt. (37)

To this end, plunging the corresponding c.d.f. of y1 andp.d.f. of y2, the desired result follows after some algebraicmanipulations.

Lemma 3: Let y1 and y2 be independent random variableswith p.d.f. fyi(x) =

xNi−1

λNii Γ(Ni)

e− x

λi , i = 1, 2, and a, b are pos-

itive constant, then the c.d.f. of random variable U � y1y2−aby2+a

is given by

FU (x) = 1− e− x

λ1− ab

λ2

λN2

2 Γ(N2)

N1−1∑m=0

(x

λ1

)m1

m!

m∑j=0

(m

j

)am−j

N2+j−1∑k=0

(N2 + j − 1

k

)(ab)

N2+j−1−k

2

(a(b+ 1)λ2x

λ1

) k−m+12

Kk−m+1

⎛⎝2

√a(b+ 1)x

λ1λ2

⎞⎠ .

Proof: Starting from the definition, the c.d.f. of randomvariable U can be computed as

FU (x) = Pr(U ≤ x) = Pr

(y1y2 − ab

y2 + a≤ x

)

=

∫ ∞

ab

Fy1

((t+ a)x

t− ab

)fy2(t)dt+

∫ ab

0

fy2(t)dt. (38)

Utilizing the c.d.f. of random variable y1, the c.d.f. of U canbe expressed as

FU (x) = 1− 1

λN22 Γ(N2)

N1−1∑m=0

(x

λ1

)m1

m!∫ ∞

ab

e− (t+a)x

(t−ab)λ1

(t+ a

t− ab

)mtN2−1e

− tλ2 dt (39)

Making a change of variable s = t− ab, and simplifying, wehave

FU (x) = 1− e− x

λ1− ab

λ2

λN22 Γ(N2)

N1−1∑m=0

(x

λ1

)m1

m!∫ ∞

0

e−a(b+1)x

sλ1− s

λ2

(s+ ab+ a

s

)m(s+ ab)

N2−1ds. (40)

Applying the binomial expansion, we arrive at

FU (x) = 1− e−xλ1

− abλ2

λN22 Γ(N2)

N1−1∑m=0

(x

λ1

)m1

m!

m∑j=0

(m

j

)am−j

N2+j−1∑k=0

(N2 + j − 1

k

)(ab)N2+j−1−k

∫ ∞

0

e− a(b+1)x

sλ1− s

λ2 sk−mds. (41)

To this end, the desired result can be obtained with the helpof [23, Eq. (3.471.9)].

APPENDIX BPROOF FOR THE N -1-1 SYSTEMS

A. Proof of Theorem 1

Starting from the definition, the outage probability can beexpressed as

Pout(γth) = Pr

(y1P

y3PI +N0 +N0/(|w|2y2) ≤ γth

), (42)

where y1 � |h1|2, y2 � |h2|2, y3 � |hI |2.Conditioned on y2 and y3, the outage probability can be

evaluated as

Pout(γth) = 1− e−y3PIγth

σ21P e

−N0γthσ21P e

− N0γthσ21P |w|2y2

N−1∑m=0

(γthN0

Pσ21

)m

1

m!

m∑j=0

(m

j

)j∑

k=0

(j

k

)(y3PI

N0

)k (1

|w|2y2

)j−k

. (43)

Hence, averaging over y2 and y3, the unconditional outageprobability can be computed as

Pout(γth) = 1− e−N0γth

σ21P

N−1∑m=0

(γthN0

Pσ21

)m1

m!

m∑j=0

(m

j

)j∑

k=0

(j

k

)(PI

N0

)k (PIγthσ21P

+1

σ2I

)−k−1Γ(k + 1)

σ2I

(1

|w|2)j−k

2

σ22(

N0γthσ22

σ21P |w|2

) k−j+12

Kk−j+1

(2

√N0γth

P |w|2σ21σ

22

). (44)

To this end, substituting w into Eq. (44), the desired resultcan be obtained after some simple algebraic manipulations.

B. Proof of Theorem 2

We find it convenient to give a separate treatment for theN = 1 and N ≥ 2 cases. When N = 1, the outage probabilityreduce to

Pout(γth) = 1− 2e− γth

ρ1

√γthμρ1

(1 +

ρIγthρ1

)−1

K1

(2

√γthμρ1

).

(45)

Applying the asymptotical expansion of Kv(x) according toEq. (36), we have

Pout(γth) ≈ 1−(1− γth

ρ1

)(1− ρIγth

ρ1

)

×(1 +

γthμρ1

(lnγthμρ1

− ψ(1)− ψ(2)

)). (46)

To this end, the desired result can be obtained after somesimple algebraic manipulations.

When N ≥ 2, due to the complex multi-summationof Kv(x) in the outage expression, directly utilizing theasymptotic expansion Eq. (36) does not seem to be tractable.Therefore, we adopt the following alternative approach. Wenote that the end-to-end SINR is statistically equivalent to

γ =y1y2ρ1ρ2

(y3ρI + 1)y2ρ2 + (1 +Nρ1 + ρI), (47)

where yi has the p.d.f. fyi(x) = xNi−1

Γ(Ni)e−x, and N1 = N ,

N2 = N3 = 1. Hence, the outage probability of the systemcan be alternatively computed as


Pout(γth) = Pr

(y1y2

(y3ρI + 1)y2 + (1 +Nρ1 + ρI)/ρ2≤ γthρ1

).

(48)

At the high SNR regime, the outage probability can be tightlylower bounded by

Pout(γth) ≥ Pr

(min

(y1y2N/μ

,y1

y3ρI + 1

)≤ γthρ1

). (49)

To this end, involving Lemma 1 yields the desired result.

C. Proof of Theorem 3

The outage probability of the system can be expressed as

Pout(γth) = Pr

(y1y2PPr

(y2Pr +N0)(y3PI +N0) + y1PN0≤ γth

)

= Pr

(y1y2 − N0γth

Pr

y2 +N0Pr

≤ γthP

(y3PI +N0)

). (50)

Invoking Lemma 3, we obtain the following outage probabilityexpression conditioned on y3

Pout(γth) = 1− e− γth

Pσ21(y3PI+N0)−N0γth

Prσ22

σ22

N1−1∑m=0

(γthN0

Pσ21

)m

1

m!

m∑j=0

(m

j

)(N0

Pr

)m−j j∑k=0

(j

k

)(N0γthPr

)j−k

2

(N2

0 (γth + 1)γthσ22

PrPσ21

) k−m+12

(y3PIN0

+ 1

)k+m+12

Kk−m+1

(2

√N2

0 (γth + 1)γthPPrσ2

1σ22

(y3PIN0

+ 1

)).

To this end, the desired result can be obtained by furtheraveraging over y3, along with some simple basic algebraicmanipulations.

D. Proof of Theorem 4

Starting from Eq. (10), conditioned on y3, the outage lowerbound can be expressed as

P lowout (γth) = Fy1

(γthN0

P

(y3PIN0

+ 1

))+ Fy2

(γthN0

Pr

)

− Fy2

(γthN0

Pr

)Fy1

(γthN0

P

(y3PIN0

+ 1

)). (51)

When ρ1 becomes large, utilizing the asymptotic expansionof lower incomplete gamma function [23, Eq. (8.354.1)], it iseasy to show that

Fy1

(γthN0

P

(y3PIN0

+ 1

))≈(y3PI

N0+ 1)N

Γ(N + 1)

(γthρ1

)N,

(52)

and

Fy2

(γthN0

Pr

)≈ γthμρ1

. (53)

Hence, quick observation reveals that the outage probabilityis dominated by the second term in Eq. (51) when N ≥ 2.

On the other hand, when N = 1, the outage probability canbe approximated by

P lowout (γth) ≈

(E{y3}PIN0

+1

μ+ 1

)γthρ1. (54)

Thus, the desired result follows after explicitly computing thefirst moment of y3.

APPENDIX CPROOF FOR THE 1-N -1 SYSTEMS

A. Proof of Theorem 9

We start the proof by expressing the end-to-end SINR as

γ =Py1

PIy2 +N0 +N0

a2y3

, (55)

where y1 � |h†2h1|2|h2|2 , y2 � |h†

2hI |2|h2|2 and y3 � |h2|2.

From [24], we know that y1 and y2 follows exponentialdistribution with parameter σ2

1 and σ2I , respectively. Moreover,

y1, y2, and y3 are mutually independent. Hence, conditionedon y2 and y3, the outage probability of the system can becomputed as


Pσ21 e

−PIγthy2Pσ2

1 e− N0γth

a2Py3σ21 . (56)

To this end, averaging over y2 and y3, we have


Pσ21

∫ ∞

0

e−PIγthy2

Pσ21

1

σ2I

e− y2

σ2I dy2

×∫ ∞

0

e− N0x

a2Pσ21y3

1

σ2N2 Γ(N)

yN−13 e

− y3σ22 dy3

= 1− 2

σ2N2 Γ(N)

e−N0γth

Pσ21

1 +PIγthσ2

I

Pσ21

(N0σ

22γth

a2Pσ21

)N2

×KN

(2

√N0γth

a2Pσ21σ

22

). (57)

Finally, substituting w into Eq. (57), the desired result followsafter some simple algebraic manipulations.

B. Proof of Theorem 11

The end-to-end SINR can be alternatively expressed as

γ =y1y2P

y2y3PI + y2N0 +N0

w2

, (58)

where y1 � |h1|2 and y2 � |h2|2, and y3 � |h†1h||2

|h1|2 . Noticingthat y3 is exponentially distributed with parameter σ2

I , and y3is independent of y1, the outage probability conditioned on y2and y3 can be computed as


Pσ21 e

−γthPIPσ2

1y3e− γthN0

Pw2σ21y2

N−1∑m=0

γmthNm0

σ2m1 m!Pm

(PIN0

y3 +1

w2y2+ 1

)m. (59)


Now, applying the binomial expansion, we have


Pσ21 e

−γthPIPσ2

1y3e− γthN0

Pw2σ21y2

N−1∑m=0

γmthNm0

σ2m1 m!Pm

m∑i=0

(m

i

) i∑j=0

(i

j

)(PIy3N0

)j (1

w2y2

)i−j.

Averaging over y3 and y2, we have

Pout(γth) = 1− 2e−γthN0

Pσ21

σ2N2 Γ(N)

N−1∑m=0

γmthNm0

σ2m1 m!Pm

m∑i=0

(m

i

)i∑

j=0

(i

j

)(PIN0

)j (1

w2

)i−j1

σ2I

Γ(j + 1)(γthPI

Pσ21+ 1

σ2I

)j+1

(γthN0σ

22

Pw2σ21

)N+j−i2

KN+j−i

(2

√γthN0

Pw2σ21σ

22

).

(60)

Finally, substituting w into (60), the desired result followsafter some simple algebraic manipulations.

C. Proof of Theorem 14

Due to the double summation involved in Corollary 3, itis difficult to obtain the asymptotic expansion directly. Hence,we adopt a different approach. Following similar lines as in theproof of Corollary 1, the outage lower bound can be expressedas

P lowout (γth) = 1− Pr

(y1PN0

y3PI

N0+ 1

≥ γth

)Pr

(y2PrN0

≥ γth

),

(61)

where y1 � |h1|2 and y2 � |h2|2, and y3 � |h†1h||2

|h1|2 .Conditioned on y3, the outage lower bound can be expressed

as

P lowout (γth) =

1−⎛⎝1−

γ(N, γthρ2

)Γ(N)

⎞⎠⎛⎝1−

γ(N,(y3PI

N0+ 1)γthρ1

)Γ(N)

⎞⎠ .

Then, utilizing the asymptotic expansion of incomplete gammafunction [23, Eq. (8.354.1)], the outage lower bound can beapproximated as

P lowout (γth) ≈

1

Γ(N + 1)

(1

μN+

(y3PIN0

+ 1

)N)(γthρ1

)N.

Finally, averaging over y3 yields the desired result.

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Caijun Zhong (S’07-M’10) received the B.S. de-gree in Information Engineering from the Xi’anJiaotong University, Xi’an, China, in 2004, and theM.S. degree in Information Security in 2006, Ph.D.degree in Telecommunications in 2010, both fromUniversity College London, London, United King-dom. From September 2009 to September 2011, hewas a research fellow at the Institute for Electron-ics, Communications and Information Technologies(ECIT), Queen’s University Belfast, Belfast, UK.Since September 2011, he has been with Zhejiang

University, Hangzhou, China, where he is currently an assistant professor. Hisresearch interests include multivariate statistical theory, MIMO communica-tions systems, cooperative communications and cognitive radio systems.

Himal A. Suraweera (S’04, M’07) received theB.Sc. degree (first class honors) in electrical andelectronics engineering Peradeniya University, Per-adeniya, Sri Lanka, in 2001 and the Ph.D. degreefrom Monash University, Melbourne, Australia, in2007. He was also awarded the 2007 Mollie Hol-man Doctoral and 2007 Kenneth Hunt Medals forhis doctoral thesis upon graduating from MonashUniversity.

From 2001 to 2002, he was with the Depart-ment of Electrical and Electronics Engineering, Per-

adeniya University, as an Instructor. From October 2006 to January 2007 hewas at Monash University as a Research Associate. From February 2007 toJune 2009, he was at the center for Telecommunications and Microelectronics,Victoria University, Melbourne, Australia as a Research Fellow. From July2009 to January 2011, he was with the Department of Electrical and ComputerEngineering, National University of Singapore, Singapore as a Research Fel-low. He is now a Research Fellow at the Singapore University of Technologyand Design, Singapore. His main research interests include relay networks,cognitive radio, MIMO, wireless security and Green communications.

Dr. Suraweera is an associate Editor of the IEEE COMMUNICATIONSLETTERS. He received an IEEE COMMUNICATIONS LETTERS exemplaryreviewer certificate in 2009 and an IEEE Communications Society Asia-Pacific Outstanding Young Researcher Award in 2011. He was a recipientof an International Postgraduate Research Scholarship from the AustralianCommonwealth during 2003-2006.

Aiping Huang graduated from Nanjing Institute ofPost and Telecommunications, China in 1977, re-ceived MS degree in Communication and ElectronicSystem from Southeast University, China in 1982,and received Licentiate of Technology degree fromHelsinki University of Technology, Finland in 1997.

She worked from 1977 to 1980 as an engineer atDesign and Research Institute of Post and Telecom-munication Ministry, China. From 1982 to 1994, shewas with Zhejiang University, China, as an assistantprofessor and then an associate professor in the

Department of Scientific Instrumentation. She was a research scientist atHelsinki University of Technology, Finland from 1994 to 1998. She is afull professor and director of Institute of Information and CommunicationEngineering at Zhejiang University, China.

She has published a book and more than 100 papers in refereed journalsand conferences on signal processing, communications and networks. Hercurrent research interests include cognitive wireless networks and planning/optimization of cellular mobile communication systems. She currently servesas vice chair of IEEE Communications Society Nanjing Chapter, councilmember of China Institute of Communications, and member of StandingCommittee of Zhejiang Provincial Institute of Communications. She is anadjunct professor of Nanjing University of Posts and Telecommunications,China.

Zhaoyang Zhang (M’02) received his B.Sc. de-gree in radio technology and Ph.D degree in com-munication and information system from ZhejiangUniversity, Hangzhou, China, in 1994 and 1998,respectively. Since 1998, he has been with theDepartment of Information Science and ElectronicEngineering, Zhejiang University, where he is nowa full professor. His current research interests aremainly focused on information theory and codingtheory, cooperative relay and mobile relay, cognitiveradio and cognitive radio networks, communication

and network signal processing, etc. He is currently serving as an associateeditor for the International Journal of Communication Systems.

Chau Yuen received the B. Eng and PhD degreefrom Nanyang Technological University, Singaporein 2000 and 2004 respectively. He was a Post DocFellow in Lucent Technologies Bell Labs (MurrayHill) during 2005, and a Visiting Assistant Pro-fessor of Hong Kong Polytechnic University on2008. During the period of 2006-2010, he workedat Institute for Infocomm Research (Singapore) as aSenior Research Engineer. He has published over100 research papers at international journals orconferences. His present research interests include

green communications, cognitive network, cooperative transmissions, networkcoding, and distributed storage. He joined Singapore University of Technologyand Design as an assistant professor from June 2010. He also serves as anAssociate Editor for IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.

Outage Probability of Dual-Hop Multiple Antenna AF Relaying Systems with Interference

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