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3156 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

Two-Way AF Relaying in the Presence ofCo-Channel Interference

Ehsan Soleimani-Nasab, Student Member, IEEE, Michail Matthaiou, Member, IEEE,Mehrdad Ardebilipour, and George K. Karagiannidis, Senior Member, IEEE

Abstract—In this paper, we investigate the performance of two-way interference-limited amplify-and-forward relaying systemsover independent, non-identically distributed Nakagami-m fad-ing channels. Our analysis generalizes several previous results,since it accounts for interference affecting all network nodes.In particular, tight lower bounds on the end-to-end outage andsymbol error probability are derived in closed-form, while auseful expression is presented for the asymptotically low outageregime. Some special cases of practical interest (e.g., no interfer-ence power and interference-limited case) are also studied. Usingthe derived lower bounds as a starting point and for the caseof Rayleigh fading, we formulate and solve analytically threepractical optimization problems, namely, power allocation underfixed location for the relay, optimal relay position with fixedpower allocation, and joint optimization of power allocation andrelay position under a transmit power constraint. The numericalresults provide important physical insights into the implicationsof model parameters on the system performance; for instance, itis demonstrated that relay position optimization offers significantperformance enhancement over the non-optimized case for anasymmetric interference power profile, whilst the optimizationgains are marginal for a symmetric one.

Index Terms—Amplify-and-forward, interference limited sys-tems, outage probability, two-way relaying.

I. INTRODUCTION

THE increasing data rate demands of concurrent and futurewireless applications have fostered the development of

cooperative diversity schemes. Many cooperative strategieshave been proposed in the literature based on different relayingtechniques, such as amplify-and-forward (AF) and decode-and-forward (DF). More specifically, in AF relaying schemes,the relay simply amplifies the received signal from the sourcebefore retransmitting it to the destination (without performingany demodulating and decoding of the received signal) [1, 2].On this basis, AF relaying systems have low implementation

Manuscript received November 3, 2012; revised February 1, March 20, andMay 14, 2013. The editor coordinating the review of this paper and approvingit for publication was Y. Chen.

E. Soleimani-Nasab and M. Ardebilipour are with the Facultyof Electrical and Computer Engineering, K. N. Toosi University ofTechnology, P.O. Box 16315-1355, Tehran 1431714191, Iran (e-mail:[email protected], [email protected]).

M. Matthaiou is with the Department of Signals and Systems,Chalmers University of Technology, 412 96, Gothenburg, Sweden (e-mail:[email protected]).

G. K. Karagiannidis is with the Department of Electrical and ComputerEngineering, Aristotle University of Thessaloniki, 54 124, Thessaloniki,Greece (e-mail: [email protected]).

Part of this paper has been submitted for potential publication to the IEEEGlobal Communications Conference (GLOBECOM), Dec. 2013.

Digital Object Identifier 10.1109/TCOMM.2013.053013.120840

complexity and are anticipated to be deployed in futurevehicular and sensor networks among others. For this reason,we henceforth assess the performance of AF relaying schemes.

As wireless networks evolve towards high load deploymentswith aggressive cellular frequency reuse, a dominant factoris inter-cell co-channel interference (CCI). For this reason,CCI has been recently investigated in the context of wirelessrelaying (see e.g., [3–7] and references therein). In [3–7],however, the main aim was to study the effect of CCI inunidirectional relay links. For example, in [3], the outageprobability (OP) was derived in closed-form in the presence ofCCI at the destination. Considering CCI at the relay, the OPwas obtained in [4] for Rayleigh fading channels. An extensionof [4] to Nakagami-m fading channels was presented in [5].More recently, the impact of feedback delay with beamformingand CCI at the relay was investigated in [6]. In [7], the OP andbit error rate (BER), with a single interferer impairing boththe relay and the destination, were analyzed. In [8], the OP fordual-hop interference-limited relaying systems was deduced,where both the relay and destination nodes are subject tointerference.

The spectral efficiency (SE) in unidirectional relaying sys-tems is inherently low since the communication occupiestwo time slots. Therefore, the coded bidirectional or two-way relaying techniques have recently received significantattention, since they can improve the SE [9]. In two-wayrelaying systems, in the first time slot, two nodes transmitsimultaneously to the relay, and the relay will transmit datato the designated destinations in the second time slot. Theauthors in [10] obtained the OP and symbol error probability(SEP) of interference-limited systems over Rayleigh fadingchannels, where they worked with the upper bound of theharmonic mean. However, they introduced an assumption onindependency of two dependent random variables (RVs) toderive their closed-form results for the OP and SEP. In [11],the authors derived the OP and SEP of two multiple-inputmultiple-output two-way relaying schemes over Nakagami-mfading channels, in an interference-free network. The authorsin [12] investigated the SEP of two-way relaying systemsusing network coding schemes, though they did not considerany interference in the network. In [13], the authors derivedan approximation for the end-to-end OP of a fixed gainAF two-way relay network, suffering from CCI and overRayleigh fading channels. In [14], the authors examined theoutage performance of dual-hop AF relaying systems withCCI over independent, non-identically distributed Nakagami-

0090-6778/13$31.00 c© 2013 IEEE

SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3157

m fading channels. They extended their work to two-wayrelaying systems in [15], where they considered the impact ofinterference only at the relay; in addition, they approximatedthe probability distribution function (PDF) of the sum ofinterferers’ powers by a gamma RV. In [16], the authorsexamined the OP of two-way AF relaying systems with CCIover Nakagami-m fading channels, where the relay was notsubject to interference. While all previous works have im-proved our knowledge on the performance characterization oftwo-way interference-limited AF relaying, the most importantdifferences between our work and [15, 16] are: 1) In [16],interference affects only the source nodes, whilst the relayis subject to noise only; moreover, all analytical results arelimited to the OP and closed-from results were derived onlyfor Rayleigh fading, 2) In [15], interference affects only therelay, while the relay gain does not contain the interferers’effect.

Motivated by the above mentioned limitations of [15] and[16], we herein pursue a detailed and generalized performanceanalysis of dual-hop two-way AF relaying systems, whereCCI is considered at all nodes in the network (i.e. both thesource nodes and relay). In this light, we derive tight lowerbounds on the OP and SEP of two-way interference limitedAF relaying networks over Nakagami-m fading channels, atarbitrary signal to interference plus noise ratios (SINRs), alongwith asymptotic expressions in the low outage regime. Thecontributions of this paper can be summarized as follows:

• We consider a two-way dual-hop configuration, wherethe single-antenna source nodes and relay are affected bymultiple interferers. This is a practical but complicatedsetup which has scarcely appeared in the literature. Wefocus on the Nakagami-m fading model, that has beenextensively used for the performance analysis of wirelesscommunication systems [17]. In particular, tight closed-form lower bounds on the OP and SEP are derived thatextend and complement several previous results in theliterature (e.g., those in [10, 15, 16]).

• Based on the derived lower bounds and for the case ofRayleigh fading, we formulate three interesting optimiza-tion problems which seek to minimize the OP. In partic-ular, we consider power allocation under fixed locationfor the relay, optimal relay position with fixed powerallocation, and joint optimization of power allocation andrelay position under a transmit power constraint.

• In order to get some additional insights into the impact ofsystem parameters, such as fading parameters and numberof interferers, we consider the asymptotically low outageregime and obtain the diversity order and coding gain.Finally, we particularize our results to the cases of nointerference power and interference-limited case.

The rest of the paper is organized as: Section II introducesthe system model. In Section III, we derive closed-formlower bounds for the OP and SEP. The asymptotic analysisand optimization results are given in Section IV and V,respectively. Section VI particularizes the results of Section IIIto some special cases of interest. Finally, Section VII presentsour numerical results, while Section VIII concludes the paper.

Notation: We use fh(.) and Fh(.) to denote the PDF andcumulative distribution function (CDF) of a RV h, respec-

. . .. . . . . .

gf1S 2SR

1,1Sh1 , sS Nh ,1Rh , rR Nh

2 ,1Sh2 , tS Nh

Fig. 1. Schematic illustration of the cooperative system under consideration.

tively. We consider that g (a, b) demonstrates the Gammadistribution, with a and b being its shape and scale parameters.Recall that the PDF and CDF of a Gamma RV are respectively

fγ (γ) =γa−1

baΓ (a)e−

γb and Fγ (γ) = 1− Γ

(a, γ

b

)Γ (a)

(1)

where Γ(n) =∫∞0

e−ttn−1 dt is the gamma function [18,Eq. (8.310.1)], and Γ(b, x) =

∫∞x

e−ttb−1 dt is the upperincomplete gamma function [18, Eq. (8.350.2)]. The operatorE[.] stands for expectation.

II. SYSTEM MODEL AND FADING STATISTICS

We consider a cooperative relaying system with two single-antenna source nodes (S1 and S2), which exchange informa-tion via the relay R (see Fig. 1). Moreover, R, S1 and S2 areimpaired by Nr, Ns and Nt sources of CCI from other users inthe network. Additionally, f is the channel coefficient betweenS1 and R and viceversa (i.e., the S1 → R and R → S1 links)and g is the channel coefficient between S2 and R which isreciprocal (i.e., the S2 → R and R → S2 links). Also, hR,i,hS1,j and hS2,k are the channel coefficients between R, S1

and S2 and the i-th (i = 1, ..., Nr), j-th (j = 1, ..., Ns) andk-th (k = 1, ..., Nt) interferer at R, S1 and S2, respectively.Additionally, PR, PS1 and PS2 are the transmitted powersof R, S1 and S2, respectively. Furthermore, PRi, PS1j andPS2k is the power of the i-th, j-th and k-th CCI signalimpairing R, S1 and S2, respectively and σ2 denotes the noisevariance at all nodes. Hence, the instantaneous SNRs for theS1 → R, S2 → R, R → S1, and R → S2 links are given by

γ1 =PS1 |f |2

σ2 , γ2 =PS2 |g|2

σ2 , γ3 = PR|f |2σ2 and γ4 = PR|g|2

σ2 ,respectively. Also, the instantaneous interference-to-noise ratiofor the i-th CCI at R, j-th CCI at S1 and k-th CCI atS2 is given by γR,i =

PRi|hR,i|2σ2 , γS1,j =

PS1j |hS1,j |2σ2 and

γS2,k =PS2k|hS2,k|2

σ2 , respectively.As was previously mentioned, we assume that the amplitude

of all links follows the Nakagami-m distribution, where m ≥0.5 represents the fading severity parameter [19]. As such,the distribution of the corresponding SNRs are Gamma RVs,where the shape parameter is m and the scale parameter isΩ/m, where Ω is the mean value of Gamma RVs. As such,the distributions of |f |2, |g|2, γS1,j , γS2,k and γR,i can beexpressed, via the corresponding parameters, as follows

|f |2 d∼ g (m1, 1/a) , |g|2 d∼ g (m2, 1/b) ,

γS1,jd∼ g (ms, 1/α) , γS2,k

d∼ g (mt, 1/η) ,

γR,id∼ g (mr, 1/β) (2)

where the symbold∼ denotes ”distributed as”. Moreover, a �

m1

Ω1, b � m2

Ω2, α � msσ

2

ΩsPS1j, β � mrσ

2

ΩrPRiand η � mtσ

2

ΩtPS2k. The

3158 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

signal received at the relay is as follows

yR =√PS1fxS1 +

√PS2gxS2 +

Nr∑i=1

√PRihR,ixR,i + nR

(3)

where xS1 , xS2 and xR,i are the signals generated from S1, S2

and the i-th interferer affecting the relay, respectively, whilenR is the additive white Gaussian noise (AWGN) at the relay.After amplification at the relay by a variable gain factor G,the signal received at S1 can be written as

yS1 =√PrGfyR +

Ns∑j=1

√PS1jhS1,jxS1,j + nS1 (4)

where xS1,j is the signal generated from the j-th interfereraffecting S1 and nS1 is the AWGN noise at S1, while

G−1 �

√√√√PS1 |f |2 + PS2 |g|2 +Nr∑i=1

PRi|hR,i|2 + σ2. (5)

Since S1 knows its transmitted signal, it can eliminate the self-interference term.1 Then, the signal received at S1 becomes

yS1 =√PRPS2GfgxS2 +Gf

√PR

Nr∑i=1

√PRihR,ixR,i

+

Ns∑j=1

√PS1jhS1,jxS1,j +

√PRfGnR + nS1 . (6)

The received SINR at S1 can then be expressed as

γS1 =PRPS2G

2|f |2|g|2[G2PR|f |2

Nr∑i=1

PRi|hR,i|2 +Ns∑j=1

PS1j |hS1,j|2

+σ2(G2PR|f |2 + 1

)].

(7)

After some algebraic manipulations, the SINR at S1 becomes

γS1 =

PRPS2 |f |2|g|2[Ns∑j=1

PS1j|hS1,j|2+σ2

][Nr∑i=1

PRi|hR,i|2+σ2

]

PS1 |f |2+PS2 |g|2[Nr∑i=1

PRi|hR,i|2+σ2

] + PR|f |2[Ns∑j=1

PS1j|hS1,j|2+σ2

] + 1. (8)

By assuming PS1 = PS2 = PS [20, 21], without significantloss of generality, the SINR at S1 further simplifies to

γS1 =

�γ1γ2[Ns∑j=1

γS1,j+1

][Nr∑i=1

γR,i+1

]

γ1+γ2[Nr∑i=1

γR,i+1

] + �γ1[Ns∑j=1

γS1,j+1

] + 1(9)

where � � PR

PS. Likewise, the received SINR at S2 is

γS2 =

�γ1γ2[Nt∑k=1

γS2,k+1

][Nr∑i=1

γR,i+1

]

γ1+γ2[Nr∑i=1

γR,i+1

] + �γ2[Nt∑k=1

γS2,k+1

] + 1. (10)

Moreover, the instantaneous SNR for S1 → R and S2 → R

1We have implicitly assumed that the channel coefficient between S1 andR is known at S1. Likewise, the channel coefficient between S2 and R isknown at S2. For a detailed discussion about this assumption, see [9].

are given by γ1 = γ|f |2 and γ2 = γ|g|2, respectively, whereγ = PS

σ2 is the average SNR per symbol. Note that in thefollowing section, we analytically investigate the OP and SEP,starting from the expressions in (9) and (10).

III. PERFORMANCE ANALYSIS

By setting γS �Ns∑j=1

γS1,j + 1 and γR �Nr∑i=1

γR,i + 1, the

received SINR at S1 can be tightly upper bounded in theinterference-limited regime, according to

γS1 ≤�γ1γ2

γSγR

γ1+γ2

γR+ �γ1

γS

=�γ1γ2

γ1 (γS + �γR) + γ2γS

=� γ1γ2

(γS+�γR)

γ1 +γ2

(γS+�γR)γS=

�γ1γ2

(γS+�γR)γS

γ1

γS+ γ2

(γS+�γR)

= �XY

X + Y

(11)

where X � γ1

γS, Y � γ2

γS+�γR. Note that a similar ex-

pression can be derived for the received SINR at S2 where

γT �Nt∑k=1

γS2,k +1. It is well known that the min(X,Y ) is a

tight upper bound of XYX+Y ; in fact, as X and Y go to infinity

the bound becomes exact. Hence, we use this bound for allderivations henceforth.2 As such, the upper bounded SINR atS1 and S2 can be expressed as

γupS1

= �min

(γ1γS

,γ2

γS + �γR

)(12)

γupS2

= �min

(γ2γT

,γ1

γT + �γR

). (13)

Finally, the end-to-end SINR of this system can be written as

γe2e = min (γS1 , γS2) ≤ min(γupS1, γup

S2

)� γup

e2e. (14)

Note that some works, such as [10], derived analytical expres-sions based on γS1 , which is not the true end-to-end SINR oftwo-way relaying systems. To compute the OP of the end-to-end SINR, we first need to derive the CDFs of the intermediatevariables X and Y . In general, it is known that the sum of Li.i.d. Gamma RVs with shape parameter k and scale parameterθ, is also a Gamma RV with parameters kL and θ. By defining

γs �Ns∑j=1

γS1,j , γt �Nt∑k=1

γS2,k and γr �Nr∑i=1

γR,i, the distri-

butions of γs, γt and γr are respectively as g (Nsms, 1/α),g (Ntmt, 1/η) and g (Nrmr, 1/β). The CDFs of X and Y aregiven by the following proposition:

Proposition 1: The CDFs of X and Y are respectively

FX (z) = 1− αNsms

Γ (Nsms)e−

azγ

m1−1∑i=0

i∑j=0

(ij

)(az)

i

γii!

× Γ (j +Nsms)(azγ + α

)j+Nsms(15)

2Note that this is a standard methodology in the performance analysisof unidirectional and bidirectional relaying systems affected by interference,which facilitates the, otherwise tedious, mathematical manipulations [10, 15].More importantly, it guarantees that our asymptotic results are exact.

SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3159

FY (z)=1− αNsms

Γ (Nsms)

βNrmr

Γ (Nrmr)

m2−1∑i=0

i∑j=0

i−j∑k=0

(ij

)(i− jk

)�k(az)

i

γii!

(�+ 1)i−j−k

e2bzγ

Γ (j +Nsms)(bzγ + α

)j+Nsms

Γ (k +Nrmr)(�bzγ + β

)k+Nrmr.

(16)

Proof: See Appendix I.From (15) and (16), it is clear that m1 and m2 should beintegers. After computing the CDFs of X and Y , we nowproceed to derive the CDFs of γup

S1and γup

S2:

Proposition 2: The CDFs of γupS1

and γupS2

are given by

FγupS1

(z)=1− αmsNs

Γ (msNs)

βmrNr

Γ (mrNr)e−((�+1)b+a) z

�γ

m1−1∑i=0

m2−1∑j=0

j∑k=0

i∑l=0

j−k∑t=0

(�+ 1)j−k−t

i!j!

(jk

)(j − kt

)(il

)(az)

i(bz)

j

�i+j−tγi+j

Γ (msNs + l + k)(az�γ + bz

�γ + α)msNs+l+k

Γ (mrNr + t)(β + bz

γ

)mrNr+t.

(17)

FγupS2

(z) = 1− ηmtNt

Γ (mtNt)

βmrNr

Γ (mrNr)e−((�+1)a+b) z

�γ

m2−1∑i=0

m1−1∑j=0

j∑k=0

i∑l=0

j−k∑t=0

(�+ 1)j−k−t

i!j!

(jk

)(j − kt

)(il

)(az)

j(bz)

i

�i+j−tγi+j

Γ (mtNt + l + k)(az�γ + bz

�γ + η)mtNt+l+k

Γ (mrNr + t)(β + az

γ

)mrNr+t.

(18)

Proof: See Appendix II.As before, from (17) and (18), it is clear that m1 and m2

should be integers. With these results in our hands, we cannow evaluate the CDF of the upper bounded end-to-end SINR.

Proposition 3: The CDF of the upper bounded end-to-endSINR, γup

e2e, is given by

Fγupe2e

(z)=1−(P11 (z) + P12 (z)

)(P21 (z) + P22 (z)

)(19)

where P11 (z), P12 (z), P21 (z) and P22 (z) are defined in(20)-(23) at the top of next page.

Proof: See Appendix III.Note that m1, m2, msNs and mtNt should have integervalues. Hereafter, we investigate the most important perfor-mance metrics i.e. the lower bounded OP and SEP for two-way interference-limited systems based on (19).

A. Outage Probability

The OP is the probability that either the S1-to-relay linkSINR or the S2-to-relay link SINR falls bellow a certainthreshold, γth = γ0

γ . By using (19), we can now obtain thefollowing lower bound on the exact OP of the system

Pout(γth) ≥ P lbout (γth) = 1− P1 (γ0)P2 (γ0) (24)

where P1 (γ0) � P11 (γ0) + P12 (γ0), P2 (γ0) � P21 (γ0) +P22 (γ0). Note that (24) can be efficiently evaluated, as it

includes finite summations of elementary functions.

B. Symbol Error Probability

We now turn our attention to the SEP, which for most digitalcommunication modulations can be expressed as [17]

P lbe =

c

2

√d

π

∞∫0

e−dγ

√γFγup

e2e(γ) (γ)dγ (25)

where the constants c and d depend on the type of themodulation. For example, we have

T =

(aγ

)msNs+mrNr

(α+ az

γ

)msNs(α+ az

γ + β)mrNr

=

msNs∑i1=1

τi1(z + αγ

a

)i1 +

mrNr∑i2=1

λi2(z + (α+β)γ

a

)i2 (26)

where

τi1� limz→−αγ

a

∂msNs−i1

(msNs − i1)!∂zmsNs−i1

(z +

αγ

a

)msNs

T

λi2� limz→− (α+β)γ

a

∂mrNr−i2

(mrNr − i2)!∂zmrNr−i2

(z+

(α+β)γ

a

)mrNr

T.

We also invoke the following identity [22, Eq. (2.1.3.1)]∞∫0

xμe−sx

(x+ z)v dx = Γ (μ+ 1) zμ−v+1Ψ(μ+ 1, μ− v + 2; sz)

(27)

where Ψ(·; ·; ·) is the Tricomi confluent hypergeometric func-tion [18, Eq. (9.210.2)]. By substituting (19) into (25), usingpartial fractions expansions as in (26) and utilizing (27),a lower bound on the SEP of two-way interference-limitedrelaying systems can be derived in closed-from; however, thederived expression is omitted due to space limitations. TheSEP for some special cases is investigated in Section IV.

IV. ASYMPTOTIC OUTAGE ANALYSIS

Since the exact results of the previous section providelimited physical insights, we now focus on the low outageregime. The threshold value of the OP is defined as γth = γ0

γ ,where γ0 is a finite threshold value. As SNR γ increases, γthtends to zero and we can approximate the PDF distribution ofthe end-to-end SINR around the origin via a Taylor’s series.The following approximations will be useful in our analysis

(1 + γth)−n =

K∑i=0

( −ni

)γith + o

(γKth

)(28)

eγth =K∑i=0

γith

i!+ o

(γKth

), as γth → 0 (29)

where n and K are positive integers. Recall that, in the lowoutage regime the proposed upper bound on the end-to-endSINR becomes exact and we can precisely predict the diversityorder and coding gain, with the aid of (29):

Proposition 4: The asymptotic OP of the end-to-end SINRis given by

P∞out (γth) = Gcγ

Gd

th + o(γGd

th

)(30)

3160 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

P11 (z) �βmrNr

Γ (mrNr)

ηmtNt

Γ (mtNt)

αmsNs

Γ (msNs)e−(

azγ ( 1

�+1)+�α)m1−1∑l=0

l∑r=0

msNs+r−1∑i=0

i∑j=0

i−j∑k=0

(lr

)(ij

)(i− jk

)alzl

�l+k−iγll!i!

× Γ (msNs + r)(α+ az

�γ

)msNs+r−i

Γ (mrNr + j)(�α+ az

γ + β)mrNr+j

Γ (mtNt + k)(α+ az

�γ + η)mtNt+k

(20)

P12 (z)�βmrNr

Γ (mrNr)

ηmtNt

Γ (mtNt)e−

azγ ( 1

�+1)m1−1∑l=0

l∑j=0

l−j∑k=0

(lj

)(l− jk

)alzl(�+ 1)

l−j−k

�l−j γll!

Γ (mrNr + j)(azγ + β

)mrNr+j

Γ (mtNt + k)(az�γ + η

)mtNt+k

− ηmtNt

Γ (mtNt)

βmrNr

Γ (mrNr)e−( az

γ ( 1�+1)+�α)

m1−1∑l=0

l∑r=0

msNs−1∑i=0

i∑j=0

l−r∑w=0

i−j∑k=0

(ij

)(lr

)(l − rw

)(i− jk

)

×alzlαi(�+ 1)l−r−w

�l+k−i−r γll!i!

Γ (mrNr + j + r)(azγ + �α+ β

)mrNr+j+r

Γ (mtNt + k + w)(az�γ + α+ η

)mtNt+k+w(21)

P21 (z)�βmrNr

Γ (mrNr)

αmsNs

Γ (msNs)e−

2bzγ ( 1

�+1)m2−1∑l=0

l∑j=0

l−j∑k=0

(lj

)(l − jk

)blzl(�+ 1)

l−j−k

�l−j γll!

Γ (mrNr + j)(bzγ + β

)mrNr+j

Γ (msNs + k)(bz�γ + α

)msNs+k

− βmrNr

Γ (mrNr)

αmsNs

Γ (msNs)e−( bz

γ (1�+1)+�η)

m2−1∑l=0

l∑r=0

mtNt−1∑i=0

i∑j=0

l−r∑w=0

i−j∑k=0

(ij

)(lr

)(l− rw

)(i− jk

)

×blzlηi(�+ 1)l−r−w

�l+k−i−r γll!i!

Γ (mrNr + j + r)(bzγ + �η + β

)mrNr+j+r

Γ (msNs + k + w)(bz�γ + α+ η

)msNs+k+w(22)

P22 (z) �βmrNr

Γ (mrNr)

ηmtNt

Γ (mtNt)

αmsNs

Γ (msNs)e−(

bzγ (

1�+1)+�η)

m2−1∑l=0

l∑r=0

mtNt+r−1∑i=0

i∑j=0

i−j∑k=0

(lr

)(ij

)(i − jk

)blzl

�l+k−iγll!i!

× Γ (mtNt + r)(η + bz

�γ

)mtNt+r−i

Γ (mrNr + j)(�η + bz

γ + β)mrNr+j

Γ (msNs + k)(α+ bz

�γ + η)msNs+k

. (23)

where the diversity order and coding gain are respectively

Gd =min (m1,m2)

Gc =

⎧⎨⎩Gc1 m2 > m1

Gc2 m1 > m2

Gc1 +Gc2 m1 = m2

(31)

while Gc1 and Gc2 are given at the top of next page.Proof: At high SNRs, where γ → ∞, we have 1 −

Γ(m, azγ )

Γ(m) → ( azγ )

m

Γ(m+1) [22, Eq. (06.06.06.0001.02)]. Followingthe proof of Proposition 3, we can get (30).

The above result implies that, when all interferers’ powers(i.e. PRi, PS1j and PS2k) are kept constant, interference doesnot affect the diversity order. However, when the interferencepower is growing large while the ratio of transmit powers ofboth sources versus the interferers’ powers is kept constant (ascenario corresponding to the special case b in Section VI),the performance of the system cannot be improved due to theinterference becoming dominant; as such, the diversity orderin this case is equal to 0. These results are consistent with[10] and [15]. For the sake of simplicity, we now elaborateon the case of Rayleigh fading, by setting all m parameters to

one, while the interference power is assumed constant; then,the asymptotic OP becomes equal to

P∞out (γth) =

((a+ b)

(�+ 1

�+

Nr

β

)+

aNt

�η+

bNs

�α

+βNr

Γ (Nr)

ηNt

Γ (Nt)e−�α

Ns−1∑i=0

i∑j=0

i−j∑k=0

aαi−1 (Ns − i)

�k−i+1i!

(ij

)

×(

i− jk

)Γ (Nr + j)

(�α+ β)Nr+j

Γ (Nt + k)

(η + α)Nt+k

+βNrαNs

Γ (Nr) Γ (Ns)

× e−�ηNt−1∑i=0

i∑j=0

i−j∑k=0

bηi−1 (Nt − i)

�k−i+1i!

(ij

)(i− jk

)× Γ (Nr + j)

(�η + β)Nr+j

Γ (Ns + k)

(α+ η)Ns+k

)γth + o(γth). (34)

Note that (34) is different from [10, Eq. (12)] since, aspreviously mentioned, the authors therein worked on the OP ofγS1 and assumed that X and Y are independent. In this paper,however, this assumption has been relaxed. In the asymptoticregime, by substituting (34) into (25), the SEP for Rayleigh

SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3161

Gc1 =βmrNr

Γ (mrNr)

ηmtNt

Γ (mtNt)

am1

Γ (m1 + 1)

⎡⎣msNs−1∑i=0

i∑j=0

m1∑l=0

i−j∑k=0

m1−l∑w=0

(ij

)(m1

l

)(i− jk

)

×(

m1 − lw

)αi(�+ 1)

m1−l−w

�m1+k−i−li!e−α� Γ (mrNr + l + j)

(α�+ β)mrNr+l+j

Γ (mtNt + k + w)

(α+ η)mtNt+k+w−

m1∑l=0

m1−l∑w=0

(m1 − l

w

)

× (�+ 1)m1−l−w

�m1−l

Γ (mrNr + l)

βmrNr+l

Γ (mtNt + w)

ηmtNt+w− αmsNs

Γ (msNs)

m1∑r=0

msNs+r−1∑i=0

i∑j=0

i−j∑k=0

(ij

)

×(

m1

r

)(i− jk

)αi

�m1+k−ii!e−α�Γ (msNs + r)

αmsNs+r

Γ (mrNr + j)

(α�+ β)mrNr+j

Γ (mtNt + k)

(α+ η)mtNt+k

](32)

Gc2 =βmrNr

Γ (mrNr)

αmsNs

Γ (msNs)

bm2

Γ (m2 + 1)

⎡⎣mtNt−1∑i=0

i∑j=0

m2∑l=0

i−j∑k=0

m2−l∑w=0

(ij

)(m2

l

)(i − jk

)

×(

m2 − lw

)ηi(�+ 1)

m2−l−w

�m2+k−i−li!e−η� Γ (mrNr + l + j)

(η�+ β)mrNr+l+j

Γ (msNs + k + w)

(α+ η)msNs+k+w−

m2∑l=0

m2−l∑w=0

(m2 − l

w

)

× (�+ 1)m2−l−w

�m2−l

Γ (mrNr + l)

βmrNr+l

Γ (msNs + w)

αmsNs+w− ηmtNt

Γ (mtNt)

m2∑r=0

mtNt+r−1∑i=0

i∑j=0

i−j∑k=0

(ij

)

×(

m2

r

)(i− jk

)ηi

�m2+k−ii!e−η�Γ (mtNt + r)

ηmtNt+r

Γ (mrNr + j)

(η�+ β)mrNr+j

Γ (msNs + k)

(α+ η)msNs+k

]. (33)

fading channels can be obtained as

P∞e =

c

4dγ

((a+ b)

(�+ 1

�+

Nr

β

)+

aNt

�η+

bNs

�α

+βNr

Γ (Nr)

ηNt

Γ (Nt)e−�α

Ns−1∑i=0

i∑j=0

i−j∑k=0

aαi−1 (Ns − i)

�k−i+1i!

(ij

)

×(

i− jk

)Γ (Nr + j)

(�α+ β)Nr+j

Γ (Nt + k)

(η + α)Nt+k

+βNrαNs

Γ (Nr) Γ (Ns)

× e−�ηNt−1∑i=0

i∑j=0

i−j∑k=0

bηi−1 (Nt − i)

�k−i+1i!

(ij

)(i− jk

)× Γ (Nr + j)

(�η + β)Nr+j

Γ (Ns + k)

(α+ η)Ns+k

)+ o

(γ−1

). (35)

As anticipated, the diversity order is equal to 1 for the par-ticular Rayleigh case under consideration. When interferenceexists only at the relay (i.e. Ns = Nt = 0), (34) and (35)reduce respectively to

P∞out (γth) = (a+ b)

(�+ 1

�+

Nr

β

)γth + o(γth)

P∞e =

c(a+ b)(

�+1� + Nr

β

)4dγ

+ o(γ−1

). (36)

The above expressions reveal straightforwardly the impactof the model parameters on the system performance. Morespecifically, we can see that by increasing Nr and the nodes’power or by reducing the interference power, the OP and SEPwill reduce. For the interference-free system (i.e. Ns = Nr =

Nt = 0), (34) and (35) reduce respectively to

P∞out =

�+ 1

�(a+ b)γth + o(γth) (37)

P∞e =

c�+1� (a+ b)

4dγ+ o

(γ−1

). (38)

Note that, in this case γupe2e = �min

(γ1

�+1 ,γ2

�+1

), while the

diversity order is 1, which is in agreement with min(m1,m2),predicted by Proposition 4.

V. RELAY POSITION OPTIMIZATION

The optimal placement of relays and/or the optimal powerallocation for improving the system performance (e.g., min-imizing the OP) has been a very hot area of research. Forexample, recently the authors in [23] investigated an opti-mization problem based on the error probability for one-wayrelaying networks over Rayleigh fading channels. In [24],they extended their work to Nakagami-m fading channels,where they worked with the high SNR approximation of theOP. In the context of two-way relaying, the authors in [25]minimized the individual OPs of the source nodes at highSNRs. The authors in [26] derived the optimal achievable end-to-end rate of two-way interference-free systems. In [27], theauthors presented optimal power allocation results in order tomaximize the sum of the achievable rates of both hops, wheninterference is present in the network. The authors in [28]worked out some optimal power allocation results for the OPof DF two-way relay networks. Note that, to the best of ourknowledge, the work presented here on two-way interference-limited relaying networks, where CCI affect all nodes, is new.In the following, we consider some optimization problems

3162 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

which seek to optimally allocate power to the network nodesand also find the optimal relay position, in order to minimizethe OP for constant interference power. Note that an arbitrarySINR analysis is intractable and for this reason we elaborateon the asymptotically low outage regime and assume the caseof Rayleigh fading for the S1 → R and R → S2 links (i.e.m1 = m2 = 1).3

A. Optimization of OP at S1 and S2

In our first optimization formulation we seek to minimizethe OP at S1.4 Referring back to (17), the lower bound on theOP at S1, P lb

outS1, should be minimized according to

mindS1,R, dS2,R

P lboutS1

(γth)

subject to 0 < dS1,R, 0 < dS2,R, 2PS + PR = Ptot (39)

where dS1,R and dS2,R are the distances between S1 andthe relay and S2 and the relay, respectively, while Ptot isthe total power of the system. We now make the standardassumption that the average SNR, Ω, decays exponentiallywith the distance such that a = m1

PSd−vS1,R

, b = m2

PSd−vS2,R

. Note

that v is the path-loss exponent, whose value is normally inthe range of 2 to 6. By setting m1 = m2 = 1 in (17), we havei = j = k = l = t = 0. By doing so, the lower bound on theOP at node S1 can be simplified as

P lboutS1

(γth) = 1− αmsNsβmrNre−(b(1�+1)+ a

� )γth(a+b� γth + α

)msNs

(bγth + β)mrNr

.

(40)

In the low outage regime, a+bα and b

β tend to zero. Thus, using(29), and by keeping only the dominant term, the lower boundin (40) becomes exact and equal to

P∞outS1

(γth) � L(γth) = (�+ 1)b+ a

�γth

+a+ b

�αmsNsγth +

b

βmrNrγth + o(γth). (41)

Hereafter, our objective function is L(γth) which, fortunately,lends itself to algebraic manipulations;5

1) Power Allocation Optimization under Fixed Relay Loca-tion: By substituting the path-loss based definition of a andb and assuming a fixed location for the relay, (41) can berewritten as

L(γth) = A1

dvS1,R

PR+A1

dvS2,R

PR+B1

dvS2,R

PS(42)

where A1 �(1 + msNs

α

)γth and B1 �

(1 + mrNr

β

)γth. The

second derivative of (42) with respect to PS is equal to

∂2L(γth)∂P 2

S

= 8A1

dvS1,R

P 3R

+ 8A1

dvS2,R

P 3R

+ 2B1

dvS2,R

P 3S

> 0 (43)

3We emphasize the fact that in order to analytically find the optimal valuesfor the case of arbitrary m1 and m2, the optimization problem can be solvedonly numerically.

4We should note that several previous works have considered deriving theOP at S1 as well [10, 12, 29, 30, 31].

5Note that a similar optimization problem can be defined based on SEPby using (25), where the optimization results are the same with the onlydifference pertaining to the replacement of A1, B1 with A1 = c

4dA1, B1 =

c4dB1.

which is, of course, strictly positive. As such, the optimizationproblem is convex. By introducing the Lagrangian multipliersand applying the Karush-Kuhn-Tucker (KKT) conditions, wecan obtain the optimal powers as

∂L(γth)∂PS

= 2A1

dvS1,R

P 2R

+ 2A1

dvS2,R

P 2R

−B1

dvS2,R

P 2S

= 0

⇒ P ∗S =

Ptot

2 +

√2A1

(dvS1,R

+dvS2,R

)B1dv

S2,R

. (44)

It can be seen that, if the relay is closer to S2, S1 needs morepower to achieve the same performance. Interestingly, the op-timal power allocation is independent of γth. For i.i.d. fadingand symmetric interference channels (i.e., ms = mr = mt,Ωs = Ωr = Ωt and Ns = Nr = Nt), where the optimumrelay position is in the middle of the two source nodes’distance [23], the optimal power allocation solution becomesP ∗R = 2P ∗

S = Ptot

2 . By substituting the optimal values from(44) into (42) and setting dS1,R = dS2,R = 0.5, the minimumvalue of OP at S1, can be written as

L∗sc1(γth) = A1

dvS1,R

P ∗R

+A1

dvS2,R

P ∗R

+B1

dvS2,R

P ∗S

=

√A1

(√A1 +

√B1

)2v−1Ptot

+

√B1

(√A1 +

√B1

)2v−1Ptot

. (45)

The minimum value of OP at S1, for this first scenario, is

L∗sc1 (γth) =

A1 + 2√A1B1 +B1

2v−1Ptot

. (46)

It is obvious that by decreasing ms, mr, Ns, Nr and γth andalso by increasing α, β, v and Ptot, L∗

sc1(γth) will decrease.2) Relay Position Optimization under Fixed Power Alloca-

tion: To minimize the effects of path-loss, it is obvious that therelay should be aligned between S1 and S2. This is reasonablesince the direct line link between any two nodes minimizesthe loss of transmitted power [24]. By setting dS1,R = d anddS2,R = 1− d, the OP in (41) can be simplified as

L(γth) = A1dv + (1− d)

v

PR+B1

(1− d)v

PS+ o(γth). (47)

The second derivative of (47) with respect to d is equal to

∂2L(γth)∂d2

=A1v (v − 1) dv−2

PR+A1

v (v − 1) (1− d)v−2

PR

+B1v (v − 1) (1− d)v−2

PS> 0 (48)

which is, of course, strictly positive. By introducing theLagrangian multipliers and applying the KKT conditions, weobtain the optimal relay position as

∂L(γth)∂d

=A1vdv−1

PR−A1

v(1− d)v−1

PR−B1

v(1− d)v−1

PS= 0

⇒ d∗ =1

1 +

(A1PR

A1PR

+B1PS

) 1v−1

. (49)

By observing (49), it is obvious that if the allocated power toS1 and S2 decreases and also if mr or Nr increase, the relayshould be closer to S2; this is reasonable since the qualityof the R → S2 link needs to be improved to avoid beingin outage. An interesting result appears for i.i.d. fading andsymmetric interference channels (i.e., ms = mr = mt, Ωs =

SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3163

Ωr = Ωt and Ns = Nr = Nt), for which A1 = B1; this statesthat the optimal relay location is near to in the S2, which is inagreement with [23]. This is rather intuitive: referring back to(40), we can see that the OP at S1 includes exponential termsdepending on the S1 → R and S2 → R links. Given that theformer term scales twice as fast as the latter, the relay nodeshould be close to S2. By substituting the optimal value from(49) into (47), and setting PR = 2PS = Ptot

2 , the minimumvalue of OP at S1 can be rewritten as

L∗sc2(γth) = A1

d∗v + (1− d∗)v

PR+B1

(1− d∗)v

PS

=2A1

Ptot

[1 +

(A1

A1+2B1

) 1v−1

]v +2(

A1

A1+2B1

) vv−1

(A1 + 2B1)

Ptot

[1 +

(A1

A1+2B1

) 1v−1

]v .

(50)

Hence, the minimum value of OP at S1 can be obtained as

L∗sc2 (γth) =

2Av

v−1

1 (A1 + 2B1) + 2A1(A1 + 2B1)v

v−1

Ptot

[A

1v−1

1 + (A1 + 2B1)1

v−1

]v .

(51)

We can see that by decreasing ms, mr, Ns, Nr and γth andalso by increasing α, β, v and Ptot, the minimal value of OPin (51) will be reduced.

3) Joint Power Allocation and Relay Position Optimization:The joint optimization problem can be expressed as

minPS , PR, d

L (γth) = A1dv

PR+A1

(1− d)v

PR+B1

(1− d)v

PS

subject to 2PS + PR = Ptot, and 0 < PS , PR, 0 < d < 1.(52)

Proposition 5: The optimization problem in (52), when v ≥2, d ∈ [0, 1] and PS ∈ [0, Ptot], is convex.

Proof: By differentiating the objective function in (52)twice with respect to PS and d, we can show that the Hessianmatrix is positive semi-definite when v ≥ 2, d ∈ [0, 1] andPS ∈ [0, Ptot].

By differentiating (41) with respect to PS and d and settingboth derivatives equal to zero, we end up with

dv−1

(1− d)v−1 = 1 +

B1PR

A1PS,

B1P2R

2A1P 2S

− 1 =dv

(1− d)v. (53)

We can now obtain the optimal power allocation and relayposition by numerically solving the following equations[

dv−1

(1− d)v−1 − 1

]2=

2B1

A1

[dv

(1− d)v + 1

],

P ∗S =

Ptot

A1

B1

[dv−1

(1−d)v−1 − 1]+ 2

. (54)

Our numerical results show that joint optimization and relayposition optimization yield almost the same performance.

B. Optimization of OP at S1 with constrained OP at S2

The previous optimization problem focuses on the OP at S1

by ignoring the OP at S2. Alternatively, we can define a newproblem which seeks to minimize the OP at S1 by consideringthe OP at S2. The main advantage of this approach is that we

can adaptively select this threshold to meet some stipulatedOP tolerance. By assuming � = 1, we now have

mind

P∞outS1

(γth) =A1

PSdv +

C1

PS(1− d)v

subject to P∞outS2

(γth) =C2

PSdv +

A2

PS(1− d)

v ≤ OPS2 ,

and, 0 < d < 1 (55)

where A2 �(1 + mtNt

η

)γth, C1 � A1+B1 and C2 � A2+

B1, while OPS2 is the maximum tolerable OP at S2. Note thatP∞outS2

(γth) is derived by using (18) and in a similar way as(41). Using the Lagrange multiplier definition, we can provethat the optimization problem in (55) is convex.

L(γth) = A1

PSdv +

C1

PS(1− d)

v

+ λ

(C2

PSdv +

A2

PS(1− d)

v −OPS2

)(56)

∂2L(γth)∂d2

= v (v − 1)

[(A1

PS+

λC2

PS

)dv−2

+

(C1

PS+

λA2

PS

)(1− d)

v−2

]> 0. (57)

Hence, by using the Lagrange multiplier, the KKT conditionsare as follows:

(A1 + λC2) dv−1 = (C1 + λA2) (1− d)

v−1 (58)

λ ≥ 0, 0 < d < 1 (59)

λ

(C2

PSdv +

A2

PS(1− d)

v −OPS2

)= 0 (60)

where λ is the Lagrangian coefficient. Here, we have twocases:

1. λ = 0 ⇒ d∗ = 1

1+(

A1C1

) 1v−1

.

2. λ = 0 ⇒ d∗ = argd

(C2

PSdv + A2

PS(1− d)

v −OPS2 = 0)

.

Assuming v = 4, the optimal value is the solution to thefollowing 4-th order polynomial

d4− 4A2

A2 + C2d3 +

6A2

A2 + C2d2 − 4A2

A2 + C2d

+A2 − PSOPS2

A2 + C2= 0. (61)

Using [32, Sec. (5)], we can solve this polynomial analytically,though the results are omitted due to space limitations.

We can now take a closer look into the actual impact of relayposition optimization. We begin with the optimization schemein (47). For i.i.d. fading and symmetric interference channels(i.e. ms = mr = mt, Ns = Nr = Nt and Ωs = Ωr = Ωt)for which A1 = B1, we have

Lnon−opt (γth) = L∗sc1 (γth) =

A1

Ptot2v−3

v=4=

0.5A1

Ptot

L∗sc2 (γth) =

6A1

Ptot

(1 + 3

1v−1

)v−1

v=4≈ 0.41A1

Ptot

(62)

where Lnon−opt(γth) indicates the OP at S1 where no opti-mization is being performed; in this non-optimized case, weset PR = 2PS = Ptot

2 and dS1,R = dS2,R = 0.5 in (47).Note that (62) implies that in the case of identical fadingconditions, the system performance cannot improve via power

3164 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

optimization.We can now compare the cases of symmetric and asymmet-

ric interference power with the same total interference powerconstraint. In the case of symmetric interference power profile,where Ns = Nr,Ωs = Ωr, we have A1 = B1 and as suchthere is no need for optimization. On the other hand, in theasymmetric interference power profile, if we consider that theinterference at the relay is stronger than at S1 (i.e., we increasePRi while PS1j is kept fixed, and hence B1 is increasing whileA1 is kept fixed), (47) and (51) simplify respectively to

Lnon−opt (γth) ≈ B1

Ptot2v−2, L∗

sc1 (γth) ≈B1

2v−1Ptot

L∗sc2 (γth) ≈

2A1

Ptot

. (63)

Note that for the non-optimized case we have assumedd = 0.5 and 2PS2 = PR = 2PS1 = Ptot

2 . As can beobserved from (63), significant performance enhancement canbe attained via the use of relay position optimization. Thisverifies the importance of optimization for the asymmetricinterference profile case. In fact, by increasing B1, while A1

is kept fixed (i.e. by increasing the SNR), the OPs in (63) areproportional to B1, B1 and B0

1 , respectively.

VI. PRACTICAL CASES OF INTEREST

In this section, we particularize the previously reportedresults to some practical cases of interest.

a) Interference-free (Ns = Nr = Nt = 0)When we set (PS1j = PS2k = PRi = PI = 0), (24)

simplifies to

P lbout (γth)

= 1− e−(�+1)(a+b)γth

m1−1∑i=0

m2−1∑j=0

(�+ 1)i+j

aibjγi+jth

i!j!(64)

where, in this case, γupS1

= �min(γ1,

γ2

�+1

), which is a tight

upper bound for �γ1γ2

(�+1)γ1+γ2while γup

S2= �min

(γ2,

γ1

�+1

).

Note that the derived CDF in (17), when Ns = Nr = 0, is atight lower bound for [29, Eq. (4)].

b) Interference-limited case: For simplicity, we assume thata = b, α = β = η,m1 = m2,msNs = mtNt. By settingPS1j = PS2k = PRi = PI , PS = PR and PS

PI= ρ ≥ 1 where

PS , PI → ∞, (24) after some manipulations simplifies to

P lbout (γth) = 1−

(P3 (γth) + P4 (γth)

)2

(65)

where

P3 (γth) �1

Γ (mrNr)

1

[Γ (msNs)]2

m1−1∑l=0

msNs+l−1∑i=0

i∑j=0(

ij

)( m1Ωs

ρmsΩf

)lγlth

l!i!

Γ (msNs + l)(1 + m1Ωs

ρmsΩfγth

)msNs+l−j−k

× Γ (mrNr + j)(2 + m1Ωs

ρmsΩfγth

)mrNr+j

Γ (msNs + i− j)(2 + m1Ωs

ρmsΩfγth

)msNs+i−j

0 5 10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

100

SNR (dB)

End

−to

−E

nd O

utag

e Pr

obab

ility

SimulationAnalytical lower bound (24)Low outage approximation (30)

m = 1, N = 2, PI = 5dB

N = 2, PI = 15dB

N = 2, PI = 5dB N = 10, PI = 15dB

Fig. 2. Outage probability at S1 (PI is a constant) for m1 = mt = 2,m2 =ms = 3, mr = 2.5,Ω1 = Ω2 = 1,Ωs = Ωr = Ωt = 0.01, γ0 = 3.

P4 (γth) �1

Γ (mrNr)

1

Γ (msNs)

(m1−1∑l=0

l∑j=0

(lj

)γlth

l!

× Γ (mrNr + j)(m1Ωs

ρmsΩfγth + 1

)mrNr+j

Γ (msNs + l− j)(m1Ωs

ρmsΩfγth + 1

)msNs+l−j

×(

m1Ωs

ρmsΩf

)l

−m1−1∑l=0

l∑r=0

msNs−1∑i=0

i∑j=0

(ij

)(lr

)

×(

m1Ωs

ρmsΩf

)lγlth

l!i!

Γ (mrNr + j + r)(m1Ωs

ρmsΩfγth + 2

)mrNr+j+r

× Γ (msNs + l − r + i− j)(m1Ωs

ρmsΩfγth + 2

)msNs+l−r+i−j

)

where Ωs � E[|hS1,j|2] and Ωf � E[|f |2]. The diversity orderis equal to 0 which means that the OP will saturate when theratio of signal to interference power is constant.

VII. SIMULATION RESULTS

In this section, the presented theoretical results are validatedby a set of Monte-Carlo simulations, where we assume thatNt = Nr = Ns = N . Note that all curves are plotted as afunction of the average SNR γ.

Figure 2 demonstrates the analytical lower bound for the OPin (24) along with the low outage approximation in (30), wherePI is a constant (i.e. a scenario corresponding to high SNR aswell). For the sake of completeness, we also consider the caseof Rayleigh fading, where all m parameters are equal to 1while � = 1. As expected, the diversity order for Nakagami-m and Rayleigh fading channels is respectively equal to 2and 1 (i.e. minimum of m1 and m2). As can be seen, theOP increases as the power of interference, PI , increases. Theproposed analytical lower bound yields excellent tightnessacross the entire SNR range and becomes exact at highSNRs. Likewise, the asymptotic outage approximation canvery efficiently predict the exact OP.

Figure 3 illustrates the analytical lower bound for the OP,where PS/PI is kept constant, while � = 1. We observe thatby increasing the number of interferers, the OP increases too.As the SNR increases, the OP reaches an error floor since

SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3165

0 5 10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

100

SNR (dB)

EE

nd−

to−

End

Out

age

Prob

abili

ty

SimulationAnalytical lower bound (24)

N = 10, P s/PI = 20dB

N = 0, P s/PI = 20dB

m = 1, N = 10, P s/PI = 20dB

N = 10, P s/PI = 15dB

Fig. 3. Outage probability at S1 (PS/PI is a constant) for m1 = mt =2, m2 = ms = 3, mr = 2.5,Ω1 = Ω2 = 1,Ωs = Ωr = Ωt = 0.01, γ0 =3.

0 5 10 15 20 25 30 35 40

10−6

10−4

10−2

100

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

SimulationAnalytical lower bound (25)

m1 = 2

m1 = 4

m1 = 1

Fig. 4. Symbol error probability at S1 for m2 = 2,ms = mr = mt =0.5,Ω1 = Ω2 = 1,Ωs = Ωr = Ωt = 0.01, N = 2, PS/PI = 20dB.

the effect of interference becomes dominant, while for theinterference-free case the error floor does not occur. Note thatthe diversity order in the former case is equal to 0.

Figure 4 shows the analytical lower bound for the SEP forBPSK modulation (c = d = 1) and different values of the m1

parameter where we have assumed that � = 1. As observed, byincreasing m1 the SEP reduces systematically. Also, it can beseen that the relative distance between the curves reduces forhigher values of the m1 factor. This implies that the impact ofm1 becomes increasingly less pronounced. This is consistentwith the results of [33]. Similar observations can be madewhen m2 increases and m1 is kept fixed. For example, m2 =2m1 = 2 has the same SEP value as m1 = 2m2 = 2.

Figure 5 elaborates on the optimization scenarios outlinedin Section V, where PS

PIis a constant. Note that, in this

case, all curves are plotted against Ptot. Since the objectivefunction is the OP at S1, we have that 2PS1 = 2PS2 =2PS = PR = Ptot/2 which means � = 2. In the first casePS − PS1j = 30 dB, PS − PRi = 0 dB, PS − PS2k = 0 dB,which represents an asymmetric interference power profile,while in the other case PS − PS1j = 15 dB, PS − PRi =15 dB, PS − PS2k = 15 dB, which represents a symmetricinterference power profile. In the asymmetric case, the OP ofrelay position optimization is decreased by approximately 10orders of magnitude compared to the non-optimized case.

0 5 10 15 20 25 30 35 40 45 5010

−4

10−3

10−2

10−1

100

Ptot

End

−to

−E

nd O

utag

e Pr

obab

ility

Non−optimizedPower optimizedDistance optimizedJoint optimization

14 16 18 2010

−1.9

10−1.4

Asymmetric

Symmetric

Fig. 5. Outage probability optimization at S1 for m1 = m2 = ms =mr = 1, Ns = Nr = 6, γ0 = 3, υ = 4.

0 5 10 15 20 25 30 3510

−4

10−3

10−2

10−1

100

End

−to

−E

nd O

utag

e Pr

obab

ility

Ptot

Non−optimizedDistance optimized: OP at S

1

Distance optimized: OP at S2

24 26 28

10−2.4

10−2.1

Asymmetric

Symmetric

Asymmetric

Fig. 6. Outage probability optimization at S1 for m1 = m2 = ms =mr = mt = 1, Ns = Nr = 1, γ0 = 3, υ = 4.

Our results show that the asymmetric power case outper-forms the symmetric case according to (62), Also, in thesymmetric case, the power allocation optimization and non-optimized case (d = 0.5, PR = 2PS = Ptot/2) have the samevalues. This means that performing optimization, leaves theOP unaffected. As a result, in the low SNR regime, if wewant to optimize the OP at S1, the best choice of powers andrelay position are d = 0.5, PR = 2PS = Ptot/2.

Figure 6 presents the outage optimization results at S1,where the OP at S2 is kept under a fixed threshold OPS2 =103

γ . In this case, PS1 = PS2 = PS = PR = Ptot

3 (� = 1). Thegraph also depicts the following curves: the OP at S2, whichis obtained by substituting the optimal value of d into theconstraint in (55), and the non-optimized OP at S1 by settingd = 0.5 in the objective function in (55). In the first case PS−PS1j = 20 dB, PS −PRi = 10 dB, PS −PS2k = 0 dB, whichrepresents an asymmetric interference power profile, while inthe other case PS −PS1j = PS −PRi = PS −PS2k = 10 dB,which represents a symmetric interference power profile. Inthe former case, the OP at S1 is smaller than in the symmetriccase. For the symmetric profile, the optimized OP at S1 hasa marginal performance gain against the non-optimized one.

VIII. CONCLUSION

We assessed the performance of a dual-hop two-way AFrelaying system over Nakagami-m fading channels, where all

3166 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

nodes are impaired by CCI. More specifically, we have derivednew tight lower bounds for the OP and SEP of the system atarbitrary SINRs. Moreover, simplified asymptotic results in thelow outage regime were deduced. For the case of Rayleighfading, we examined three practical optimization problems,where we observed that relay position optimization yieldssubstantial performance improvement. Finally, the distributionof the interferers’ power can significantly affect the OP, sincethe asymmetric interference profile case yields lower OP thanthe symmetric profile case in the low outage regime. Note thatall the presented expressions herein can be easily evaluatedand efficiently programmed. We finally point out that thepresented results complement and extend several previousresults reported in the literature over the past years.

APPENDIX I: PROOF OF PROPOSITION 1

To begin with, the CDFs of X and Y are equal to

FX (z) = Pr (X ≤ z) = Pr (γ1 ≤ zγS |γS ) = Fγ1 (zγS |γS )FY (z) = Pr (Y ≤ z)

= Pr (γ2 ≤ z (γS + �γR) | γS , γR)= Fγ2 (z (γS + �γR) | γS , γR) .

The CDFs of X and Y can be written in integral form as

FX (z) =

∞∫0

Fγ1 (zγS |γS ) fγS (γS)dγS (66)

FY (z) =

∞∫0

∞∫0

Fγ2 (z (γS +�γR) |γSγR )

× fγS (γS)fγR (γR)dγSdγR. (67)

Defining γS = γs+1 and γR = γr +1, substituting the PDFsof γs and γr and the CDF of γ1 from (2) into (66), we get

FX (z)=αNsms

Γ (Nsms)

∞∫0

⎡⎣1−Γ(m1,

azγ (x+ 1)

)Γ (m1)

⎤⎦xNsms−1

eαxdx.

(68)

Using [18, Eq. (8.352.2)] for integer m1, (68) becomes

FX (z) = 1− αNsms

Γ (Nsms)

m1−1∑i=0

(az)i

γii!

×∞∫0

e−azγ (x+1)(x+ 1)

ixNsms−1e−αxdx. (69)

Using the definition of binomial coefficients, (69) becomes

FX (z) = 1− αNsms

Γ (Nsms)

m1−1∑i=0

i∑j=0

(az)i

γii!

(ij

)e−

azγ

×∞∫0

e−(azγ +α)xxj+Nsms−1dx. (70)

Using [18, Eq. (17.13.3)], (70) can be written as in (15).Defining γT = γt + 1, (67) can be expanded, for integer m2,as

FY (z) = 1− αNsms

Γ (Nsms)

βNrmr

Γ (Nrmr)

m2−1∑i=0

i∑j=0

(bz)i

γii!

(ij

)

× Γ (j +Nsms)(bzγ + α

)j+Nsms

∞∫0

(�y + �+ 1)i−j

ebzγ (�y+�+1)

yNrmr−1

eβydy. (71)

Once more, by using the definition of binomial coefficientsand [18, Eq. (17.13.3)], (71) simplifies to (16).

APPENDIX II: PROOF OF PROPOSITION 2

The CDF of S1, by considering r = γ1

γs+1 and w =γ2

γs+�γr+�+1 , can be mathematically expressed as

FγupS1

(z) = Pr (�min(r, w) < z)

= 1− Pr (�min(r, w) > z) = 1− Pr (�r > z, �w > z) .

The above probability can be alternatively evaluated as

FγupS1

(z) = 1− Eγs,γr

[Pr

(�γ1 > z (γs + 1)

∣∣∣∣∣γs, γr)

× Pr

(�γ2 > z (γs + �γr + �+ 1)

∣∣∣∣∣γs, γr)]

= 1−(1− Eγs,γr

[Fγ1

(�−1z (γs + 1)

) ])

×(1− Eγs,γr

[Fγ2

(�−1z (γs + �γr + �+ 1)

) ]). (72)

By assuming integer values for m1 and m2, we get

FγupS1

(z) = 1− Eγs,γr

[e−

az�γ (γs+1)− bz

�γ (γs+�γr+�+1)

m1−1∑i=0

m2−1∑j=0

(az)i(bz)

j

(�γ)i+j

i!j!(γs + 1)

i(γs + �γr + �+ 1)

j

]. (73)

Using the definition of binomial coefficients, (73) becomes

FγupS1

(z) = 1− Eγs,γr

[e−

az�γ (γs+1)− bz

�γ (γs+�γr+�+1)

m1−1∑i=0

m2−1∑j=0

i∑l=0

j∑k=0

j−k∑t=0

(il

)(jk

)(j − kt

)

× (az)i(bz)

j

(�γ)i+j

i!j!γl+ks �tγt

r(�+ 1)j−k−t

]. (74)

By integrating (74) over γs and γr, (17) is derived. The prooffor Fγup

S2(z) follows a similar line of reasoning.

SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3167

APPENDIX III: PROOF OF PROPOSITION 3

Since X and Y are dependent to each other, we utilize thefollowing methodology,

Fγupe2e

(z) = 1− Pr

(min

(γupS1, γup

S2

)≥ z

)

= 1− Pr

(γupS1

≥ z, γupS2

≥ z

)

= 1− Pr

(min

( �γ1γs + 1

,�γ2

γs + �γr + �+ 1

)≥ z,

min

(�γ2

γt + 1,

�γ1γt + �γr + �+ 1

)≥z

)

= 1− Pr

(�γ1

γs + 1≥ z,

�γ2γs + �γr + �+ 1

≥ z,

�γ2γt + 1

≥ z,�γ1

γt + �γr + �+ 1≥ z

)

= 1− Eγs,γt,γr

[Pr

(�γ1 ≥ z (γs + 1) ,

�γ1 ≥ z (γt + �γr + �+ 1)

∣∣∣∣∣γs, γt, γr)]

× Eγs,γt,γr

[Pr

(�γ2 ≥ z (γs + �γr + �+ 1) ,

�γ2 ≥ z (γt + 1)

∣∣∣∣∣γs, γt, γr)]

= 1− P1 (z)P2 (z) (75)

where P1 (z) and P2 (z) can be expressed as

P1 (z) = Eγs,γt,γr

[Pr

(�γ1 ≥ z (γs + 1) ,

γs + 1 ≥ γt + �γr + �+ 1

∣∣∣∣∣γs, γt, γr)]

+ Eγs,γt,γr

[Pr

(�γ1 ≥ z (γt + �γr + �+ 1) ,

γt + γr + �+ 1 ≥ γs + 1

∣∣∣∣∣γs, γt, γr)]

P2 (z) = Eγs,γt,γr

[Pr

(�γ2 ≥ z (γs + �γr + �+ 1) ,

γs+ �γr+�+ 1 ≥ γt + 1

∣∣∣∣∣γs, γt, γr)]

+ Eγs,γt,γr

[Pr

(�γ2 ≥ z (γt + 1) ,

γt + 1 ≥ γs + �γr + �+ 1

∣∣∣∣∣γs, γt, γr)]

.

Now, P1 (z) can be written in integral form according to

P1 (z) = Eγt,γr

[ ∞∫γt+�γr+�

∞∫�−1z(y+1)

fγ1 (x)fγs (y) dxdy

]

+ Eγt,γr

[γt+�γr+�∫0

∞∫�−1z(γt+�γr+�+1)

fγ1 (x)fγs (y) dxdy

].

(76)

Utilizing [18, Eq. (2.33.10)] and [18, Eq. (8.350.4)], the firstterm in (76) can be obtained as

P11 (z) = Eγt,γr

[αmsNs

Γ (msNs)

am1

γm1Γ (m1)

∞∫γt+�γr+�

∞∫�−1z(y+1)

xm1−1e−axγ dxymsNs−1e−αydy

]

= Eγt,γr

[αmsNs

Γ (msNs)

∞∫γt+�γr+�

Γ(m1,

az�γ (y + 1)

)Γ (m1)

ymsNs−1e−αydy

].

For integer values of m1, the above expression becomes

P11 (z) = Eγt,γr

[αmsNs

Γ (msNs)

m1−1∑l=0

alzl

�lγll!

∞∫γt+�γr+�

(y + 1)le−

az�γ (y+1)ymsNs−1e−αydy

]

= Eγt,γr

[αmsNse−

az�γ

Γ (msNs)

m1−1∑l=0

l∑r=0

(lr

)alzl

�lγll!

∞∫γt+γr+1

e−(az�γ +α)yymsNs+r−1dy

](77)

where the second equality is obtained by using the definition ofbinomial coefficients. By solving the integral in (77) utilizing[18, Eq. (2.33.10)], and assuming integer values for msNs

and applying [18, Eq. (8.352.2)]), we have

P11 (z) = Eγt,γr

[αmsNse−

az�γ

Γ (msNs)

m1−1∑l=0

l∑r=0

(lr

)alzl

�lγll!

Γ(msNs + r,

(α+ az

�γ

)(γt + �γr + �)

)(α+ az

�γ

)msNs+r

]

= Eγt,γr

[m1−1∑l=0

l∑r=0

msNs+r−1∑i=0

1

i!

(lr

)alzl

�lγll!

αmsNse−az�γ

Γ (msNs)

(γt + �γr + �)iΓ (msNs + r)(

α+ az�γ

)msNs+r−ie−(α+

az�γ )(γt+�γr+�)

]. (78)

3168 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 8, AUGUST 2013

By using [34, Eq. (2.1.3.2)], (78) can be written as

P11 (z) = Eγt

[βmrNr

Γ (mrNr)

m1−1∑l=0

l∑r=0

msNs+r−1∑i=0

i∑j=0

(ij

)(

lr

)alzl

�l−j γll!i!

αmsNse−az�γ −(α+ az

�γ )(γt+�)

Γ (msNs)Γ (msNs + r)

× (γt + �)i−j(

α+ az�γ

)msNs+r−i

∞∫0

e−(α�+azγ +β)xxmrNr+j−1dx

].

The above expression admits the following manipulations

P11 (z)=Eγt

[βmrNr

Γ (mrNr)

m1−1∑l=0

l∑r=0

msNs+r−1∑i=0

i∑j=0

(ij

)(lr

)

× alzl

�l−j γll!i!

αmsNse−az�γ −(α+ az

�γ )(γt+�)

Γ (msNs)

(γt + �)i−j(

α+ az�γ

)msNs+r−i

× Γ (mrNr + j)(�α+ az

γ + β)mrNr+j

]=

βmrNr

Γ (mrNr)

ηmtNt

Γ (mtNt)

αmsNs

Γ (msNs)

× e−az�γ −(�α+ az

γ )m1−1∑l=0

l∑r=0

msNs+r−1∑i=0

i∑j=0

i−j∑k=0

(lr

)(ij

)×(

i− jk

)Γ (msNs + r)(

α+ az�γ

)msNs+r−i

Γ (mrNr + j)(�α+ az

γ + β)mrNr+j

× alzl

�l+k−iγll!i!

∞∫0

e−(α+az�γ +η)yymtNt+k−1dy. (79)

By integrating over γt in (79) using [18, Eq. (3.351.3)],we arrive to P11 (z) in (19). Likewise, P12 (z), P21 (z) andP22 (z) can be derived for integer m2 and mtNt.

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SOLEIMANI-NASAB et al.: TWO-WAY AF RELAYING IN THE PRESENCE OF CO-CHANNEL INTERFERENCE 3169

Ehsan Soleimani-Nasab (S’11) was born in Ker-man, Iran in 1984. He obtained the B.Sc. in Electri-cal Engineering from the Iran University of Scienceand Technology, Iran in 2006. He then received theM.Sc. in Communication Systems from the K. N.Toosi University of Technology, Iran. He is currentlypursuing his Ph.D. degree at the the K. N. ToosiUniversity of Technology, Iran. From April throughOctober 2012, he was with the Department of Sig-nals and Systems, Chalmers University of Tech-nology, Sweden working as a Visiting Researcher.

His research interests span signal processing for wireless communications,cooperative communications, MIMO systems, and cognitive radio networks.

Michail Matthaiou (S’05–M’08) was born in Thes-saloniki, Greece in 1981. He obtained the Diplomadegree (5 years) in Electrical and Computer En-gineering from the Aristotle University of Thessa-loniki, Greece in 2004. He then received the M.Sc.(with distinction) in Communication Systems andSignal Processing from the University of Bristol,U.K. and Ph.D. degrees from the University ofEdinburgh, U.K. in 2005 and 2008, respectively.From September 2008 through May 2010, he waswith the Institute for Circuit Theory and Signal

Processing, Munich University of Technology (TUM), Germany working as aPostdoctoral Research Associate. In June 2010, he joined Chalmers Universityof Technology, Sweden as an Assistant Professor and in 2011 he was awardedthe Docent title. His research interests span signal processing for wirelesscommunications, random matrix theory and multivariate statistics for MIMOsystems, and performance analysis of fading channels.

Dr. Matthaiou is the recipient of the 2011 IEEE ComSoc Young ResearcherAward for the Europe, Middle East and Africa Region and a co-recipient ofthe 2006 IEEE Communications Chapter Project Prize for the best M.Sc.dissertation in the area of communications. He was an Exemplary Reviewerfor IEEE COMMUNICATIONS LETTERS for 2010. He has been a memberof Technical Program Committees for several IEEE conferences such asICC, GLOBECOM, etc. He currently serves as an Associate Editor for theIEEE TRANSACTIONS ON COMMUNICATIONS, IEEE COMMUNICATIONS

LETTERS and was the Lead Guest Editor of the special issue on “Large-scalemultiple antenna wireless systems” of the IEEE JOURNAL ON SELECTED

AREAS IN COMMUNICATIONS. He is an associate member of the IEEE SignalProcessing Society SPCOM and SAM technical committees.

Mehrdad Ardebilipour was born in Iran, in Febru-ary 1954. He received B.Sc. and M.Sc. Degrees inelectrical engineering from K. N. Toosi Universityof Technology, Tehran, Iran, in 1977 and TarbiatModarres University, Tehran, Iran, in 1989, respec-tively. He has also been awarded the degree of Ph.D.by the University of Surrey, Guildford, England, in2001. Since 2001, he has been an assistant professorat K. N. Toosi University of Technology and wasdirecting the Communications Engineering Depart-ment for six years. Currently, he is the director of

the Spread Spectrum and Wireless Communications research laboratory. Hiscurrent research interests are cognitive radio, cooperative communication, ad-hoc and sensor networks, MIMO communication, OFDM, game theory andcross-layer design for wireless communications.

George K. Karagiannidis (SM’2003) was bornin Pithagorion, Samos Island, Greece. He receivedthe University Diploma (5 years) and Ph.D degree,both in electrical and computer engineering from theUniversity of Patras, in 1987 and 1999, respectively.From 2000 to 2004, he was a Senior Researcherat the Institute for Space Applications and RemoteSensing, National Observatory of Athens, Greece. InJune 2004, he joined the faculty of Aristotle Univer-sity of Thessaloniki, Greece where he is currentlyProfessor and Director of Digital Telecommunica-

tions Systems and Networks Laboratory. His research interests are in the broadarea of digital communications systems with emphasis on communicationstheory, energy efficient MIMO and cooperative communications, cognitiveradio, smart grid and optical wireless communications. He is the author or co-author of more than 240 technical papers published in scientific journals andpresented at international conferences. He is also author of the Greek editionof a book on “Telecommunications Systems” and co-author of the bookAdvanced Wireless Communications Systems (Cambridge Publications, 2012).He is co-recipient of the Best Paper Award of the Wireless CommunicationsSymposium (WCS) in the IEEE International Conference on Communications(ICC’07), Glasgow, U.K., June 2007.

Dr. Karagiannidis has been a member of Technical Program Committees forseveral IEEE conferences such as ICC, GLOBECOM, VTC, etc. In the pasthe was Editor for Fading Channels and Diversity of the IEEE TRANSAC-TIONS ON COMMUNICATIONS, Senior Editor of IEEE COMMUNICATIONSLETTERS and Editor of the EURASIP Journal of Wireless Communicationsand Networks. He was Lead Guest Editor of the special issue on “OpticalWireless Communications” of the IEEE JOURNAL ON SELECTED AREAS INCOMMUNICATIONS and Guest Editor of the special issue on “Large-scalemultiple antenna wireless systems”. Since January 2012, he is the Editor-inChief of the IEEE COMMUNICATIONS LETTERS.

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