Performance of Antenna Selection for Two-WayRelay Networks with Physical Network Coding

Kang Song∗, Baofeng Ji∗†, Yongming Huang∗‡ and Luxi Yang∗∗ School of Information Science & Engineering, Southeast University, Nanjing, China 210096

† Electronic & Information Engineering College, Henan University of Science and Technology, Luoyang, China 471003‡ Key Laboratory of System Control & Information Processing, Ministry of Education, Shanghai Jiao Tong University,

Shanghai, China 200240Email: {sk, baofengji, huangym, lxyang}@seu.edu.cn

Abstract—In this paper, the performance of a two-way amplifyand forward (AF) multi-antenna relay network is presented,where the worse receive signal-to-noise ratio (SNR) of the twoend users is maximized. We first derive the exact probabilitydensity function (PDF) of receive SNR for two users. Thenan approximate expression of SNR in the high-SNR regimeis obtained. A closed-form average sum bit error rate (BER)approximation is also derived to quantify the performance of theantenna selection system at high SNR. It is shown by simulationthat our analytical result is efficient in estimating the networkreliability, especially in the high transmit SNR region.

I. INTRODUCTION

Physical Network Coding (PNC) has recently receivedgreat attention in academical research field of wireless commu-nication [1]–[3], while the original concept of network codinghas been proposed for a decade [4]. In relay assisted system,network coding technology provides a significant enhancementto the throughput performance of multi-hop wireless networks.Meanwhile, by taking advantage of additive nature of simul-taneously arriving electromagnetic waves, the complexity ofPNC based relay is relatively low [1].

It has been proved that multiple-input multiple-out (MI-MO) technology promises significant improvements in termsof spectral efficiency and link reliability [5]–[8]. PNC withMIMO relaying has also been studied in recent years [9].However, as the number of antennas grows, the complexityand cost of system increase. In consideration of performance,cost and complexity, antenna selection has been proposed asa compromise.

Most paper focus on antenna selection in one-way re-lay network. In [10], optimal SNR-based transmit selectionscheme is derived and the conclusion that the proposed an-tenna selection scheme could achieve the full diversity orderis drawn. Two suboptimal low-complexity transmit antennaselection scheme which could also achieve full diversity isproposed in [11]. In [12], [13], an end-to-end (e2e) bestantenna selection scheme is investigated.

To the best knowledge of the authors, there is very littleliterature on the performance of antenna selection for two-wayrelay networks with PNC in Nakagami-m fading channel. Inthis paper, a two-way AF MIMO relay network with Max-Min-Max antenna selection is considered. As discussed in [10]–[13], in spite of worse performance than transmit beamformingor maximal-ratio combining (MRC) system, the complexity

�� � ��

������ ��� �����

Fig. 1. Illustration of system model

and power consumption in the circuit of antenna selectionsystem is relatively low.

This paper is organized as follows. First, in Section II, wedescribe our system model and assumptions and propose ourMax-Min-Max antenna selection scheme. Then we derive theperformance of antenna selection for two-way relay networkswith PNC in Nakagami-m fading channel in Section III, and inSection IV simulations are performed to validate our derivationin section III.

II. SYSTEM MODEL

As illustrated in Fig. 1, we consider a two-way AF relaysystem where S1, Relay and S2 are equipped with N1, NR andN2 antennas respectively. It is assumed that all nodes operatein half-duplex mode, and there is no direct link between S1

and S2. The channel between each pair of transmitter andreceiver antennas is modeled by flat Nakagami-m fading. His a NR ×N1 matrix, denoting the channel from S1 to Relay,while G is a NR×N2 matrix, denoting the channel from S2 toRelay. We assume all elements of H and G are independentand identically distributed (i.i.d.). Based on known channelstate information (CSI), all nodes select one of their antennasseparately before transmission.

The transmission between S1 and S2 is carried out intwo time slots. During the first slot, S1 and S2 transmit theirmessage simultaneously. When the ith, jth and kth antenna ofS1, Relay and S2 are selected, the received signal at Relay isgiven by

yR =√P1hijx1 +

√P2gkjx2 + nR (1)

where hij is the element at the ith row and jth column of H,gkj is the element at the kth row and jth column of G, andnR is the Additive White Gaussian Noise (AWGN) at Relay,satisfying nR ∼ CN(0, σ2

R).

978-1-4799-0308-5/13/$31.00 © 2013 IEEE

Due to the power constraint at Relay, the receive signal yRis multiplied by a gain α, where α =

√PR

P1‖hij‖2+P2‖gkj‖2+σ2R

.During the second slot, Relay transmit the multiplied signal.With assumption of channel reciprocity, the signal after can-celing the self-interference at S1 and S2 is given by

y1 =√

P2αh∗ijgkjx2 + αh∗

ijnR + n1 (2)

y2 =√

P1αg∗kjhijx1 + αg∗kjnR + n2 (3)

where (·)∗ denotes complex conjugate, n1 and n2 is theAWGN at S1 and S2, satisfying n1 ∼ CN(0, σ2

1) andn2 ∼ CN(0, σ2

2).

Without loss of generality, we assume σ2R = σ2

1 = σ22 = 1.

The SNR at the two end nodes could be obtained from (2) and(3)

γ1 =P2PR‖hij‖2‖gkj‖2

(PR + P1)‖hij‖2 + P2‖gkj‖2 + 1(4)

γ2 =P1PR‖hij‖2‖gkj‖2

P1‖hij‖2 + (PR + P2)‖gij‖2 + 1(5)

In [14], it is proved that the best-worse selection in AFrelay system is optimal in the sense of maximizing the worsereceive SNR of the two-way communications. For antennaselection in two way AF networks, similar idea is employed.We first find the best antenna index of S1 and S2 for eachantenna of relay. Then we use best-worse to select the antennaof relay. Once the antenna of relay is fixed, the best antennaof S1 and S2 is also fixed. This Max-Min-Max criterion canbe expressed mathematically by the following formula

j = argmaxj

min

{max

i{‖hij‖2},max

k{‖gkj‖2}

}(6)

i = argmaxi

‖hij‖2 (7)

k = argmaxk

‖gkj‖2 (8)

III. PERFORMANCE ANALYSIS

In this section, we derive the analytical and high SNRapproximate Probability Density Function (PDF) of e2e SNR,and analyze the system’s e2e asymptotic BER. We assumehij ∼ Nakagami(m1,Ω1), gij ∼ Nakagami(m2,Ω2), wherem1 and m2 are integer no less than 1.

A. Probability Density Function of e2e SNR

Theorem 1: If X = max1≤i≤N1

{Xi}, Xi ∼ Γ(m1, θ1) and

Y = max1≤j≤N2

{Yj}, Yj ∼ Γ(m2, θ2) are independently dis-

tributed, the PDF of Z = XYaX+bY+1 is

pZ(z) = A∑Φ1

l1+m1−1∑i=0

l2+m2−1∑j=0

B

× zl2+m2+l1+m1−2−i(abz + 1)l2+m2−1−j

× e−k1+1θ1

bz− k2+1θ2

azI(z)

(9)

where

∑Φ

�∑

(k1,l1,k2,l2)∈Φ

A =N1N2(m1 − 1)!(m2 − 1)!

Γ(m1)N1Γ(m2)N2θm11 θm2

2

B =

(N1 − 1

k1

)(N2 − 1

k2

)(l1 +m1 − 1

i

)×(l2 +m2 − 1

j

)(−1)k1+k2al1al2

× al2+m2−1−jbl1+m1−1−i

a0 = 1, an =1

n

n∑m=1

mk − n+m

m!an−m

I(z) = aI1(z) + (2abz + 1)I0(z) + (abz + 1)bzI−1(z)

In(z) = 2

[(k2 + 1)θ1(k1 + 1)θ2

(abz + 1)z

] i−j+n2

×Ki−j+n

⎛⎝2

√(k1 + 1)(k2 + 1)

θ1θ2(abz + 1)z

⎞⎠Φ1 =

{(k1, l1, k2, l2)|0 ≤ k1 ≤ N1 − 1, 0 ≤ l1 ≤ (m1 − 1)k1,

0 ≤ k2 ≤ N2 − 1, 0 ≤ l2 ≤ (m2 − 1)k2}

Proof: Since Xi ∼ Γ(m1, θ1), Yj ∼ Γ(m2, θ2), the PDFof Xi and Yi is given by

fXi(x) =1

Γ(m1)θm11

xm1−1e−xθ1 (10)

fYi(y) =1

Γ(m2)θm22

ym2−1e−yθ2 (11)

According to [15], the PDF and CDF of X and Y could bewritten as

fX(x) =N1

Γ(m1)N1θm11

[γ(m1,

x

θ1)

]N1−1

xm1−1e−xθ1 (12)

fY (y) =N2

Γ(m2)N2θm22

[γ(m2,

y

θ2)

]N2−1

ym2−1e−yθ2 (13)

FX(x) =1

Γ(m1)N1

[γ(m1,

x

θ1)

]N1

(14)

FY (y) =1

Γ(m2)N2

[γ(m2,

y

θ2)

]N2

(15)

As X and Y are independently distributed, the joint probabilitydistribution of X and Y is

pXY (x, y) =N1N2

Γ(m1)N1Γ(m2)N2θm11 θm2

2

×[γ(m1,

x

θ1)

]N1−1 [γ(m2,

y

θ2)

]N2−1

× xm1−1ym2−1e−xθ1

− yθ2

(16)

Assuming z = xyax+by+1 , w = ax2+x

ax+by+1 , we could get x =

w + bz, y = az + (abz+1)zw . The Jacobian matrix is

JF (z, w) =

[∂x∂z

∂x∂w

∂y∂z

∂y∂w

](17)

The Jacobian determinant is det (JF (z, w)) = a + 2abz+1w +

(abz+1)bzw2 . The joint probability distribution of Z and W is

pZW (z, w) =det (JF (z, w))N1N2

Γ(m1)N1Γ(m2)N2θm11 θm2

2

[γ(m1,

w + bz

θ1)

]N1−1

×[γ(m2,

az

θ2+

(abz + 1)z

wθ2

)]N2−1

(w + bz)m1−1

×[az +

(abz + 1)z

w

]m2−1

e−w+bzθ1

− 1θ2[az+ (abz+1)z

w ]

(18)

The probability distribution of Z could be derived as pZ(z) =∫∞0

pZW (z, w)dw, which could be rewritten as

pZ(z) =N1N2

Γ(m1)N1Γ(m2)N2θm11 θm2

2

×∫ ∞

0

det (JF (z, w))

[γ(m1,

w + bz

θ1)

]N1−1

×[γ(m2,

az

θ2+

(abz + 1)z

wθ2

)]N2−1

(w + bz)m1−1

×[az +

(abz + 1)z

w

]m2−1

e−w+bzθ1

− 1θ2[az+ (abz+1)z

w ]dw

(19)

From [16],

γ(m,x) = (m− 1)!

[1− e−x

m−1∑l=0

xl

l!

](20)

[γ(m,x)]N−1

[(m− 1)!]N−1=

[1− e−x

m−1∑m=0

xm

m!

]N−1

=N−1∑k=0

(N − 1

k

)(−1)ke−kx

[m−1∑l=0

xl

l!

]k

=N−1∑k=0

(N − 1

k

)(−1)ke−kx

(m−1)k∑l=0

alxl

(21)

where(··)

is binomial coefficient, al is coefficient defined as0.314 in [16].

With the help of 0.316 in [16]

(w + bz)n1

[(a+

abz + 1

w)z

]n2

=zn2

[n1∑i=0

(n1

i

)wi(bz)n1−i

][n2∑j=0

(n2

j

)an2−jw−j(abz + 1)j

]

=zn2

n1∑i=0

n2∑j=0

(n1

i

)(n2

j

)(bz)n1−ian2−jwi−j(abz + 1)j

(22)

Substituting (21) and (22) into (19),

pZ(z) = A

∫ ∞

0

det(JF (z, w))∑Φ1

l1+m1−1∑i=0

×l2+m2−1∑

j=0

Bzl2+m2+l1+m1−2−i(abz + 1)j

× wi−je−k1+1θ1

(w+bz)− k2+1θ2

(a+ abz+1w )zdw

(23)

Using 2.3.16.1 in [17]∫ ∞

0

xa−1e−px−q/xdx = 2

(q

p

)a/2

Ka (2√pq) (24)

Eq. (23) could be rewritten as (9).

Corollary 1: The probability density function of e2e SNRis given by

pγ1(z) = A

∑Φ1

l1+m1−1∑i=0

l2+m2−1∑j=0

B

×(zc

)l2+m2+l1+m1−2−i(abz

c+ 1

)l2+m2−1−j

× e−k1+1θ1c bz− k2+1

θ2c azI(zc

)where a = PR + P1, b = P2, c = P2PR.

pγ2(z) = A∑Φ1

l1+m1−1∑i=0

l2+m2−1∑j=0

×B(zc

)l2+m2+l1+m1−2−i(abz

c+ 1

)l2+m2−1−j

× e−k1+1θ1c bz− k2+1

θ2c azI(zc

)where a = P1, b = PR + P2, c = P1PR.

Proof: Note that since hij ∼ Nakagami(m1,Ω1), gij ∼Nakagami(m2,Ω2) are independent, ‖hij‖2 ∼ Γ(m1, θ1),‖gkj‖2 ∼ Γ(m2, θ2), where θ1 = Ω1

m1, θ2 = Ω2

m2. The

requirement of Theorem 1 is satisfied, and the result couldbe easily derived.

Corollary 2: In large SNR regime, the probability densityfunction of e2e SNR could be approximated as

pγ(z) = A∑Φ1

l1+m1−1∑i=0

l2+m2−1∑j=0

C(zc

)2l2+2m2+l1+m1−3−i−j

× e−k1+1θ1c bz− k2+1

θ2c az I(zc

)where

C = B (ab)l2+m2−1−j

I(z) = aI1(z) + (2abz + 1)I0(z) + (abz + 1)bzI−1(z)

In(z) = 2

[(k2 + 1)θ1(k1 + 1)θ2

ab

] i−j+n2

zi−j+n

×Ki−j+n

⎛⎝2z

√(k1 + 1)(k2 + 1)

θ1θ2ab

⎞⎠

Proof: When SNR is high, The SNR at the two end nodesis given by

γ1 ≈ P2PR‖hij‖2‖gkj‖2(PR + P1)‖hij‖2 + P2‖gkj‖2 (25)

γ2 ≈ P1PR‖hij‖2‖gkj‖2P1‖hij‖2 + (PR + P2)‖gij‖2 (26)

The remainder of proof is similar to that of Theorem 1.

B. Average sum Bit Error Ratio (BER)

Theorem 2: The average sum BER of the best-worse an-tenna selection scheme is given by

Pe ≈na∑l=1

NR∑m=0

m∑n=0

[∑Φ2

C1Γ(m1 + p)

(q + 1/θ1)m1+p+∑Φ3

C2Γ(m2 + p)

(q + 1/θ2)m2+p

](27)

where p = l1 + l2 + l−12 , q = k1/θ1 + k2/θ2 + 1, n1 =

N1(NR −m+ n), n2 = N2(NR − n), and

C1 = cl(√2)l−1

(NR

m

)(m

n

)(n1 − 1

k1

)×(n2

k2

)(−1)NR−m+k1+k2

n1al1al2θm1+l11 θl22

× [(m1 − 1)!]n1−1[(m2 − 1)!]n2

Γ(m1)n1Γ(m2)n2

C2 = cl(√2)l−1

(NR

m

)(m

n

)(n1

k1

)×

(n2 − 1

k2

)(−1)NR−m+k1+k2

n2al1al2θl11 θm2+l2

2

× [(m1 − 1)!]n1 [(m2 − 1)!]n2−1

Γ(m1)n1Γ(m2)n2

Φ2 ={(k1, l1, k2, l2)|0 ≤ k1 ≤ n1 − 1, 0 ≤ l1 ≤ (m1 − 1)k1,

0 ≤ k2 ≤ n2, 0 ≤ l2 ≤ (m2 − 1)k2}

Φ3 ={(k1, l1, k2, l2)|0 ≤ k1 ≤ n1, 0 ≤ l1 ≤ (m1 − 1)k1,

0 ≤ k2 ≤ n2 − 1, 0 ≤ l2 ≤ (m2 − 1)k2}

Proof: As derived in [18], the CDF of equivalent SNRafter best-worse antenna selection is

Pγmax−min(z) = [1− (1− FX(z)) (1− FY (z))]NR (28)

Using the binomial theorem, we rewrite the expression as

Pγmax−min(z) =

NR∑m=0

m∑n=0

(NR

m

)(m

n

)(−1)NR−m

× FNR−m+nX (z)FNR−n

Y (z)

(29)

The PDF of equivalent SNR after best-worse antenna selectionis

pγmax−min(z) =

NR∑m=0

m∑n=0

(NR

m

)(m

n

)(−1)NR−m

× FNR−m+n−1X (z)FNR−n−1

Y (z)[(NR −m+ n)

× fX(z)FY (z) + (NR − n)FX(z)fY (z)]

(30)

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

SNR/dB

SimulationApproxmationAnalytical result

P1=P2=PR=1

P1=P2=PR=5

Fig. 2. The PDF of e2e SNR against transmit SNR

For BPSK, the average sum BER at destination is mathe-matically formulated as [19]

Pe = E[Q(√2z)] ≈

∫ +∞

0

Q(√2z)pγmax−min(z)dz (31)

where Q(x) = 1√2π

∫∞x

exp(−u2

2

)du is the tail probability

of the standard normal distribution.

As derived in [20],

Q(x) ≈ e−x2

2

na∑l=1

clxl−1 (32)

where cl = (−1)l+1Al

B√π(

√2)l+1l!

. A = 1.98 and B = 1.135 arediscussed in [21]. na is the terms of the Taylor series.

Substituting (32) into (31),

Pe ≈∫ +∞

0

e−zna∑l=1

cl(√2)l−1z

l−12 pγmax−min(z)dz (33)

Substituting (12) (13) (14) (15) (21) (29) into (33), andafter some algebraic manipulation, Eq. (27) is easily obtained.

IV. SIMULATION

In this section, numerical results are provided to validatethe analysis result derived above.

Fig. 2 shows the PDF of e2e SNR from S1 to S2, whenN1 = 4, N2 = NR = 1, m1 = 3, m2 = 2, Ω1 = Ω2 = 2.Observe that result obtained using the closed-form expressionderived in this paper and Monte Carlo simulation results arein excellent agreement. Approximate results from Corollary 2,as we expected, are more precise when transmit SNR is high.

In Fig. 3, the system outage probability based on (28)is plotted when m1 = 3, m2 = 2, Ω1 = 4, Ω2 = 2 andthreshold rate R0 = 1 bit/s/Hz. As can be seen from this figure,the approximation in [18] is also effective in our antennaselection system where all of 3 nodes select their antennasbefore transmission. Although channel quality of S1 to S2 isbetter than that of S2 to S1, the overall outage probability isdominated by the worse link.

0 5 10 15 20

10−4

10−3

10−2

10−1

100

SNR/dB

outa

ge p

roba

bilit

y

analytical result from (28)simulationS1−>Relay−>S2 (simulation)S2−>Relay−>S1 (simulation)

N1=N2=1, NR=2

N1=N2=NR=2

Fig. 3. The overall outage probability against transmit SNR

0 2 4 6 8 10 12 1410−6

10−5

10−4

10−3

10−2

10−1

100

SNR/dB

aver

age

sum

BE

R

SimulationAnalytical result

N1=N2=1,NR=2

N1=1,N2=NR=2

N1=N2=NR=2

Fig. 4. The average sum BER against transmit SNR.

Fig. 4 compares the approximate average sum BER withMonte Carlo simulation when m1 = 3, m2 = 2, Ω1 = Ω2 = 2,na = 9. It can be seen that the approximate result is an upperbound of actual performance. As the transmit BER decreases,the gap between approximation and actual performance be-come narrower. In high SNR regime, the closed-form resultprovides a good approximation to the simulation results.

V. CONCLUSION

In this work we derived the performance for the Max-Min-Max antenna selection two-way relay system. The analyticaland asymptotic PDF of e2e SNR was obtained. Furthermore,the asymptotic BER in high SNR regime was also derived.The accuracy of our derivations was finally confirmed bysimulation results.

ACKNOWLEDGMENT

This work was supported by National Science and Tech-nology Major Project of China under Grant 2012ZX03004005-003, National Natural Science Foundation of China underGrants 61271018, 61071113, 61201172 and 61201176, Re-search Project of Jiangsu Province under Grants BK20130019,BK2011597, BK2012021 and BE2012167, and Open ResearchFund of Key Laboratory of System Control and InformationProcessing, Ministry of Education.

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