Performance Analysis of Relay Selection for Two-Way Cooperative Relay Networks

Baofeng Ji, Kang Song, Yongming Huang, Luxi Yang School of Information Science and Engineering, Southeast University, Nanjing 210096, China {fengbaoji@126.com, songhealth@gmail.com, huangym@seu.edu.cn, lxyang@seu.edu.cn}

Abstract—In this paper, we investigate relay selection for a two-way cooperative relaying network. In our scheme, the best-worse criterion is used to select a preferred relay each time to perform amplify-and-forward relaying. The closed-form expression of probability density function (PDF) and cumulative density function(CDF) of the two-way SNRs for our scheme are derived. It is shown that the best-worse scheme achieves full diversity gain. The asymptotic packet error rate (PER) performance of our scheme is also examined through rigorous analysis. Simulation results show that our derived results for the best-worse relay selection scheme is consistent with the numerical results.

Keywords- Cooperative relay; Two-way relay; Relay selection

I. INTRODUCTION

Cooperative relaying [1] has emerged as a promising technology for future development in wireless communications. A combination of cooperative relay and precoding technology [2-4] is able to further improve performance. In particular, different nodes in a cooperative network can act as relay nodes and collaborate with each other to establish a communication link between a transmitter and a receiver. This relaying strategy promises to enhance communication reliability, expand the coverage, and improve the spectrum efficiency without increasing power at the transmitter. Motivated by this, a number of cooperative schemes and corresponding performance analysis have been given in previous literatures such as [5-11].

The existing relaying strategy can be divided into two basic classes, i.e., one-way relay and two-way relay. Our focus is on the two-way relaying strategy, which can realize a data exchange between two users only through two or three time slots. In general, there are two different two-way cooperative relay modes: one is the analog mode, i.e., non-regenerative cooperative relay; the other is the digital mode, i.e., regenerative cooperative relay. In the analog mode, the signals received by the cooperative relays are forwarded after linear processing, such as multiplying with the amplification factor. In this way the signal and the noise are both amplified. In the digital mode, the received signals at the relays are first demodulated and detected through non-linear processing. Thus, the signal is extracted from noise and interference, and is re-encoded/re-modulated through physical network coding (such as XOR or other mapping) before forwarding. One of the key ideas on two-way relay networks is that each end node can cancel the self-interference from its received signal to help decode the information from the other node[13-14].

When a relay network consists of multiple relay stations, a proper relay selection (RS) scheme will play a key role in the

system performance. By far, the relay selection optimization for two-way cooperative relay networks has not been extensively studied as for one-way relay networks. In this paper we are concerned with the RS optimization for the two-way cooperative relay network where each relay station is equipped with a single antenna and employs non-regenerative relaying strategy. Specifically, we concentrate on the RS design with the criterion of maximizing the worse receive SNR of two terminal nodes and aim to analyze its performance. In particular, the closed-form probability density function (PDF) and the asymptotic packet error ratio (PER) for the max-min-SNR RS scheme are derived and validated via simulation results.

The remainder of the paper is organized as follows. Section II introduces the system model of a two-way cooperative relay. Section III gives the closed-form PDF and CDF expressions for maximizing the worse link SNR. Section IV provides the simulation results to validate our analysis results; Section V draws the conclusions. Involved proofs are illustrated in the appendix.

II. SYSTEM MODEL

Consider a two-way cooperative relay system as illustrated in Fig.1, where 1S and 2S represent two source nodes which exchange information between each other through the relaying of N two-way cooperative relay nodes. Each node is equipped with a single antenna. It is assumed that the cooperative relay nodes have full channel state information. We define that ih is a random channel vector between the source 1S and the cooperative relay nodes, and ig is a random channel vector between the source 2S and the cooperative relay nodes. The entries of ih and ig are independent identical distributed according to (0,1)CN , where the index i denotes the index of the cooperative relay nodes. Due to the channel reciprocity, the channel of cooperative relay

1S and 2S cooperative relay links are H

ih and Hig . The power constraint at

the two source nodes 1S , 2S and cooperative relay i are denoted as 1P , 2P and iQ , respectively. denotes the Frobenius norm throughout this paper. H denotes the conjugate transpose of the entries.

The two source nodes 1S and 2S exchange information through two time slots. In the first time slot, 1S and 2S send source information to the cooperative relay nodes respectively. The signal received by the ith cooperative relay node can be expressed as

This work was supported by the National Natural Science Foundation ofChina under Grants 60902012 and 61071113, the National Science andTechnology Major Project of China under Grants 2011ZX03003-001-02 and 2011ZX03003-003-03, the Ph.D. Programs Foundation of Ministry ofEducation of China under Grants 20090092120013 and 20100092110010, the Natural Science Foundation of Jiangsu Province (BK2011598, BK2011019)

978-1-4577-1010-0/11/$26.00 ©2011 IEEE

1 1 2 2r i i i iy P h s P g s n (1)

where in denotes the noise at the relay i.

Figure 1 Two-way cooperative relay system

III. RELAY SELECTION SCHEME

In the second timeslot, one out of the N cooperative relays is selected to forward the information symbols from both two sources. Without loss of generality, assuming the cooperative relay j is selected, relay j amplifies its received signal r iy and forwards it to both end nodes. The transmit signal is written as

j j r jT y

where 2 2 21 2 1

jj

j j j

Q

P h P g

, 2

1 j denotes that the noise

variance in the first timeslot at relay j.

Thus the signals received by 1S and 2S can be expressed as

1, 1 1 2 2 1

int

H H H Hj j j j j j j j j j j j

self erference

x h T P h h s P h g s h n w

2, 1 1 2 2 2

int

H H H Hj j j j j j j j j j j j

self erference

x g T P g h s P g g s g n w (2)

where uw is the noise at node u for 1, 2u . Assume that all noises are i.i.d complex Gaussian random variables following

20,CN . Two source nodes can obtain the other node’s information by canceling the self-interference. Specifically, the two nodes can obtain 1, jx and 2, jx written as follows:

1, 2 2 1H H

j j j j j j jx P h g s h n w

2, 1 1 2H H

j j j j j j jx P g h s g n w (3) The two source nodes then can use the Maximum-

Likelihood (ML) decoding rules to obtain the estimatte of its information symbol as [12]:

1 2, 11ˆ arg min H

j j j js Ss x Pg h s

2 1, 22ˆ arg min H

j j j js Ss x P h g s

(4)

From (3) we can obtain the signal to noise (SNR) at the two end nodes as

2

2

1, 2 22 2 2 2 21 1,2 1 2 1,2 1 1,2( )

Hj j j

j

w j j j w j j w

P Q h g

P Q h P g

2

1

2, 2 22 2 2 2 22 1 1 12,2 2,2 2,2

( )

Hj j j

j

j j j j jw w w

PQ g h

P Q g P h

(5)

where 1,2

2

w denotes the noise variance of 1w in the second

timeslot, 2,2

2

w denotes the noise variance of 2w in the second

time slot.

In [12], it is proved that the best-worse RS is optimal in the sense of maximizing the worse receive SNR of the two-way communications. Thus we can select the cooperative relay i as follows

1, 2,ˆ arg max min{ , }j j

ji . (6)

This RS scheme maximizes the worse SNR of the two-way communications, which is equivalent to minimizing the higher error probability of the two nodes in the network.

It is easy to know that 2

hj jh and 2

gj jg . The

eigenvectors correspond to nonzero eigenvalue hj and gj are

/j jh h and /j jg g , respectively. Other eigenvectors can be selected arbitrarily as long as the orthogonality between them is satsified.

If the two source nodes have the same transmission power constraint, we can obtain the relation formula as follows, which is similar to that in [12]:

2 2

1, 2,j j j jh g (7)

If the transmission powers of the two source nodes are different, we cannot obtain the formula (7). Then the formula (6) can be equivalent to

2 2ˆ arg max min{ , }j j

ji h g (8)

IV. PERFORMANCE ANALYSIS

In this section, we analyze the diversity order and packet error ratio of this RS scheme by deriving the PDF and CDF of the best-worse SNR ˆ 1, 2,max min{ , }j ji j

.

Theorem 1: Based on the result that 2

jh and 2

jg follows

central chi-square distribution, with their PDF expressed as

212 2 21( ) 0jN

h

j j jp h N h e h

, the PDF of 1, j and

2, j can be obtained as follows

1 1( )( ) 1

2 221, 0 0

2

2( ) ( ) ( )

[ ( )] ( * )

2( ) ( 1) (2 )

*

j i i jN N i Ni jc d bN N iNj i j

i j

j i

dp e C C

N c d b c b

b bKb c d

1 1( )( ) 1

2 222, 0 0

2

2

2( ) ( ) ( )

[ ( )] ( * )

( ) ( 1) (2 )

*

j i i jN N i Ni jc d bN N iNj i j

i j

j i

dp e C C

N c d b c b

b bKb c d

(9)

where 1,2

1,2

2 21 1

2 21

w

w

j j

j

P Qc

,

221 j

Pd

, 1,2

1,2

21

2 21 1

2 21

w

w

j j

j j

j

Qb

P Q

121 j

Pc

, 2,2

2,2

2 22 1

2 21

w

w

j j

j

P Qd

, 2,2

2,2

21

2 22 1

2 21

w

w

j j

j j

j

Qb

P Q

.

Theorem 2: When the powers of the two end nodes and the cooperative relay nodes are high compared to the power of noise, we can simplify (8) to (12) as follows

1,2

1,2

1,2

2 222 21

1, 2 22 21 1 2

2 2 21 1

( )w

w

w

jj j

j

jj j

j jj j

P Qh g

P Q Ph g

2,2

2,2

2,2

2 212 21

2, 2 22 22 1 1

2 2 21 1

( )w

w

w

jj j

j

jj j

j jj j

PQh g

P Q Pg h

(10)

Then we have the PDF of 1, j and 2, j , i.e.,

1 1( )( )

21,

22 1 2

20

2( )

[ ( )] ( * )

2 ( ) ( ) ( )

*

c d bNj

i NNN i

N i Ni

p eN c d b

dC K

b c bc d

1 1( )( )

22,

22 1 2

20

2( )

[ ( )] ( * )

2 ( ) ( ) ( )

*

c d bNj

i NNN i

N i Ni

p eN c d b

dC K

b c bc d

(11)

where c , d ,b and c , d andb are the same as the formula(9).

Theorem 3：It is derived that the CDF of the 1, j and 2, j as follows.

1

0 0

1

0

2/11

2

1

!

!121

21

,1

k

k

l mk

lmk

ml

k

keP

j

221

12/)1(

2 2 aa

K mllm

(12)

2, j can be obtained in the same way. The proof can be seen appendix A.

Therefore we can obtain the CDF of 1, 2,min{ , }j j j as

1, 2,( ) 1 [1 ( )][1 ( )]j j jj j

F (13)

Then the CDF of ˆ 1, 2,max min{ , }j ji j can be expressed as

1ˆ ˆ ˆ1, 2,( ) {1 [1 ( )][1 ( )]}R

ji i ij jF (14)

Theorem 4: When all the nodes use the same power level, 1, j and 2, j can be expressed as (20) in large SNR regime.

2 2 2 2

0

1, 2 2 2 22 2

j j j j

j

j j j j

p h g h g

h g h g

2 2 2 2

0

2, 2 2 2 22 2

j j j j

j

j j j j

p h g h g

h g h g

(15)

Then the outage probability of the best-worse selection can be given by:

022

02

2( , )

1( )

RR

out

NP

N

(16)

The asymptotic expression is

0 0

12 1 2 1

0 0

2 2

( ) ( )

NRRNR NR N

NoutP

N N N N

(17)

The diversity gains is NR , the coding gains is 0

12 12

( )

NR N

N N

.

The proof is given in appendix B.

Theorem 5 If the interference is ignorable, the average sum BER of the best-worse selection scheme is given by

1

2 20

20 0 0

1( )

20

2 21( 1)

( )4

1 (2 )

2

nqpRp q n

BER pp q n

q n

R p bp

p q

q q n

(18)

The proof is given in appendix C.

Theorem 6 The amount of fading (AF) can be shown to be:

22 2

2 2 1x x

Ax

Fx

x

(19)

where the 2x and 2x are respectively as follows:

21 2 12 0

2 20 0 0

2

1 2 1 2 ( 1) ( )=

( )

1 ( 2)

v u nR vn

vv u n

u n

R v R bx

v u

u u n

11 2 10

2 20 0 0

1

1 2 1 2 ( 1) ( )

( )

1 ( 1)

v u nR vn

vv u n

u n

R v R bx

v u

u u n

The proof is given in appendix D.

V. SIMULATION

In this section, we simulate the analytical results to compare with the numerical results. We assume that the modulation adopt QPSK, and the relay numbers K can be 1,2 and 3. Figure 2 and 3 show respectively the outage probability versus 0 and the average symbol error probability versus 0 . Result shows that the derived analytical results are consistent with the numerical results.

0 5 10 15 20 25 30 35 4010

-6

10-5

10-4

10-3

10-2

10-1

100

0,dB

out

ag

e p

rob

abi

lity

analytical resultsimulationS1->b->S2 (simulation)

S2->b->S1 (simulation)

Figure 2 Outage probability against 0

0 5 10 15 20 25 30 35 4010

-6

10-5

10-4

10-3

10-2

10-1

100

0,dB

Ave

rage

sym

bol

err

or

pro

bab

ility

S

1->b->S

2 (simulation)

S2->b->S

1 (simulation)

S2->b->S

1 (simulation)

S2->b->S

1 (simulation)

analytical results R=1analytical results R=2

Figure 3 Average symbol error probability against 0

VI. CONCLUSION

In this paper, we derived some statistical results for the best-worst RS scheme in a two-way relaying network. Through

rigorous derivation, we have obtained the closed-form PDF and CDF of the SNR, as well as the asymptotic ASEP. Based on these, the achievable diversity order on two-way cooperative relay selection is analyzed. The accuracy of our derivations was confirmed by simulation results.

Appendix A

The formula (10) can be rewritten as 2 2 2 2

1, 2 2 2 2

1 2

j j j j

j

j j j j

h g h g

db h cb g a h a g

2 2 2 2

2, 2 2 2 2

1 2

j j j j

j

j j j j

h g h g

c b g d b h a g a h

(20)

We firstly derived the CDF of 1, j .Due to the 2

jh and2

jg are following the gamma distribution, so we can

derive the general expression as formula (20), the general PDF expression of the gamma distribution is given by

1

( ) ( )( )

x

X

x ep x U x

(21)

Then the CDF can be given as( , )

( ) 1 ( )( )X

x

P x U x

.

We can substitute the2

jh and2

jg for 1x and 2x for

simplicity, then we can obtain the formula

1 21,

1 1 2 2j

x x

a x a x

(22)

Then the CDF of 1, j can be expressed as

2

2

,1 11121

112

0

11121

112 |Pr|Pr

a

a

dxxpxax

xaxdxxpx

ax

xaxP

j

(23)

Then the first item can be expressed as

2

2

1 12 1 1 10

1 2

( , )Pr | ( ) 1

( )

a

aa x

x x p x dxx a

(24)

Then the second item can be expressed as

1221

1111

2

2

2

1

,1

,

dxaxxa

exa

a

x

(25)

then using [15,8.352.2, 3.471.9], the CDF of 1, j is given by

1

0 0

1

0

2/11

2

1

!

!121

21

,1

k

k

l mk

lmk

ml

k

keP

j

221

12/)1(

2 2 aa

K mllm

(26)

Similar to the 1, j , the CDF of 2, j can be obtained as

1

0 0

1

0

2/1'1

2

1

!

!121

'2

'1

,2

k

k

l mk

lmk

ml

k

keP

j

2

'2

'1

1

2/)1('2 2

aaK ml

lm

(27)

When the parameter 1a and 2a are 2 and 1 respectively, the results are the same to the [18]. The results above is the general form to the gamma distribution.

Appendix B

0 01

0 0

Pr{min( , ) } Pr{min( , ) }

1 Pr{ , }

R R

out ARB BRA ARB BRAi

R

ARB BRA

P R R R R R R

R R R R

Through slightly approximation as [18], we can obtain the outage probability as follows

2 0

02

2( , )

1( )

R

out

WN

PN

(28)

Due to the fact that the gamma function 0

0

2( , )

WN

can be

rewritten as 0

0

2( , )

Wr N

, the 0

0

2( , )

Wr N

is the lower incomplete

gamma function. The asymptotic outage probability can be expressed as (29) in large SNR.

0 0

12 1 2 1

0 0

2 2

( ) ( )

NRRNR NR N

NoutP

N N N N

(29)

Therefore the diversity gains is NR, the coding gains

0

12 12

( )

NR N

N N

.

Appendix C

The gamma distribution PDF is given by (49), we can

consider the PDF of 2

0 jh , then we can obtain the formula

about the 0 , the (21) can be rewritten it as

01

0

( ) ( )( ) ( )

x

X

x ep x U x

(30)

Then the CDF of 2

0 jh can be given by

0

( , )( ) 1 ( )

( )X

x

P x U x

The PDF of max min can be calculated as

0

1max min min min

12 1

0 02 2

0

( ) ( ( )) ( )

( , ) ( , ) 2 1

( ) ( ) ( )

R

xR

p x R P x p x

x xx e

R

(31)

Due to the CDF of min ( )p x is given

by2

0min 2

( , )( ) 1 ( )

( )

x

P x U x

.So the PDF about min ( )p x can be

calculated as

010min 2

0

2 ( , )( )

( ) ( )

xx

p x x e

the CDF of max min ( )p x is given by

2

0max 2

( , )( ) 1

( )

Rx

P x

max min

max min0

2

020

0

2

020

0

1( ( 2 ))

2

1 ( )

4

( , )1

( 1)( )4

( , )1

= ( 1)( )4

BER

x

p

x Rp

p

pxR

pp

p

p Q

eP x dx

x

xRe

dxpx

xR e

dxp x

Then the BERp can be equivalent to the integral. Using [15,0.314] we obtain the results.

1

2 20

20 0 0

1( )

20

2 21( 1)

( )4

1 (2 )

2

nqpRp q n

BER pp q n

q n

R p bp

p q

q q n

Where the nb is given by :

00qb a ,

10

1( ) 1, is a natural number

m

m n m nn

b nq m n a c for m qma

where the 1n

is Pochhammer symbol defined in [15].

Appendix D

Using the CDF of max ( )p x , we can calculate the expectation and the variance of the SNR as follows

11 2 10

2 20 0 0

1

1 2 1 2 ( 1) ( )

( )

1 ( 1)

v u nR vn

vv u n

u n

R v R bx

v u

u u n

0

0

12 1

2 2 0 02 20

0

2 111

2 2000 0

0

( , ) ( , )2 1

( ) ( ) ( )

12 = ( 1) 1 ( , )

( ) ( ) ( )

1 2 1 =

xR

xv

Rv

vv

n

x xx e

x x R dx

RR x e xr dx

v

R v

v u

21 2 10

2 20 0

2

2 ( 1) ( )

( )

1 ( 2)

v u nR vn

vv u

u n

R b

u u n

So the amount of fading (AF) is calculated as:

22 2

2 2 1x x

Ax

Fx

x

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