Performance Analysis of Two-Way AF Cooperative Relay Networks Over Weibull Fading Channels

Journal of Communications Vol. 8, No. 6, June 2013

Performance Analysis of Two-Way AF Cooperative Relay Networks over Weibull Fading Channels

Xinjie Wang1,2, Hao Zhang1,3, T. Aaron Gulliver3, Wei Shi1 and Hongjiao Zhang1 1 Department of Electrical Engineering, Ocean University of China, Qingdao, China 2 College of Communication and Electronics, Qingdao Technological University, China 3Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada Email: [email protected]; [email protected]; [email protected]; [email protected] [email protected]

Abstract—The performance of two-way amplify-and-forward and symbol error probability (SEP) for two-way AF (AF) cooperative relay networks over independent but not relaying were analyzed with perfect channel state necessarily identically distributed Weibull fading channels is information (CSI) over Rayleigh fading channels in [11], studied. Tight closed form approximations for the overall [12]. In [13], two-way relaying in mixed Rayleigh and outage probability (OOP) and the average symbol error Rician fading channels was considered. The exact probability (ASEP) are derived. The analysis is verified by expressions for OP, SEP and average sum-rate were simulation, and the accuracy is shown to be excellent, especially derived over Nakagami-m fading channels in [14]. at high signal noise ratio (SNR). According to the analysis, increasing the fading parameter, the power scaling parameter, or However, the above results only consider the performance in terms of single user outage and SER. the number of relays, can improve the system performance when the best relay is selected. Since the quality of service (QoS) requirements for bidirectional links are the same as for two-way relaying, Index Terms—Two-way relay network; Weibull fading channel; the overall outage probability (OOP) and average SEP Outage probability; Symbol error probability; Amplify and performance can be used to evaluate the performance of forward both links. In [15], the OOP and average symbol error

probability (ASEP) performance of two-way AF based I. INTRODUCTION relaying protocols over Rayleigh fading channels was investigated. The corresponding performance over Cooperative relaying techniques have recently gained attention as an efficient way to mitigate fading in wireless Nakagami-m fading channels was considered in [16], [17]. The performance with a cascaded fading networks. Most conventional cooperative relay networks are half-duplex and employ a relaying mode such as distribution such as the cascaded generalized-K amplify-and-forward (AF), decode-and-forward (DF), or distribution [18] has also been investigated. The Weibull incremental relaying [1]. The performance of cooperative distribution has been recommended for mobile radio diversity networks was investigated in [2]. To enhance systems as well as the Nakagami-m distribution [19]. The performance, selective DF and opportunistic relaying performance of a conventional cooperative relay network networks have been proposed, and the outage over a Weibull fading channel was analyzed in [20]-[23]. performance and symbol error rate (SER) analyzed in [3]. However, the overall performance of two-way relaying Further, the average symbol error probability of a multi- has not been investigated over these channels. branch dual-hop cooperative network in Rayleigh and In this paper, two-way AF relaying with multiple Nakagami-m fading channels was analyzed in [4], [5]. relays over independent but not necessarily identically

The strategy of selecting the best relay and Nth best relay distributed Weibull fading channels is examined. Tight from amongst several relays has been proposed to closed-form approximations for the OOP and ASEP are enhance the spectral efficiency [6]-[9]. The performance derived. These results are validated by simulation, and of conventional one-way cooperative systems has been the effect of the channel parameters on performance is investigated extensively. examined. To further improve the spectral efficiency, two-way The remainder of this paper is organized as follows. AF and DF cooperative relaying protocols have been Section II presents the system model, and the overall introduced [10]. In this case, only two time steps are outage probability is derived in Section III. The overall required to transmit a symbol. The outage probability (OP) error probability is derived in Section IV. Performance results are given in Section V based on the analysis in the previous sections. Finally, some conclusions are given in Manuscript received April 20, 2013; revised June 15, 2013, accepted Section VI. June 18, 2013 This work was supported by the International S&T Cooperation Program of Qingdao, China under Grant No. 12-1-4-137-hz. II. SYSTEM MODEL Corresponding author email: [email protected] doi:10.12720/jcm.8.6.372-377

The two-way relay model is shown in Fig. 1. Two respectively. The signal-to-noise ratios (SNRs) of the source nodes S1 and S2 exchange information via the links S1  R j  S2 and S2  R j  S1 are given by best relay Rk . Thus, there is no direct connection 12jj between S1 and S2 . Rk is selected from the set of relay 12j  (6) 12jj21 nodes R j , j=1, 2, ..., N. All nodes operate in half-duplex 12jj fashion and are equipped with a single antenna. hij is the  21j  (7) 21 12jj channel coefficient between source node Si (i=1, 2) and 2 R j . The channel coefficients hij are independent but not respectively, where  ij P S| h ij | / N0 , i=1, 2. When PN , (6) and (7) can be expressed as necessarily identically distributed (i.n.d.) random S 0 variables that follow the Weibull distribution. 12jj 12j  (8) 12jj 2 and    12jj (9) 21j 212jj

In the second time slot, the best relay Rk is selected to relay the signal, and is selected as follows [15]

k  arg max min1jj 2 , 2 1 (10) jN{1.. }  

Figure. 1 The two-way AF relay cooperative network. The channel coefficient hij is a Weibull random variable, so the probability density function (PDF) of h According to the protocol of two-way relay networks ij [10], in the first time slot the signal received at the jth can be written as [23] relay is given by 2ij 2ij 21  x ij  f||h  x  x exp  (11) y P h x  P h x  n (1) ij    ij ij j s1 j 1 s 2 j 2 j   ij ij where Ps is the transmitted signal energy from Si , xi is where Eh(2 )/  1 1/  ,  ()x is the Gamma the normalized signal with unit energy, and n is additive ij ij ij  j function,  is the power scaling parameter for the white Gaussian noise (AWGN) at relay R j with zero ij channel between S and R , and  is the fading mean and variance N0 . i j ij severity parameter for the channel between S and R . In the second time slot, if relay R j is selected to relay i j The PDF of the instantaneous SNR  ij can be expressed the signal, it scales the received signal by G j , and as broadcasts the resulting signal to S1 and S2 . Then the ij ij  1 x signal received at Si from relay R j is given by ij  f  x  x exp  (12) ij    ij  ij  ij ij ySi G j h ij y j n i (2)

where  PN/ . With perfect channel state information at R , we have ij ij S 0 j

P III. OVERALL OUTAGE PROBABILITY G  R (3) j P| h |22 P | h | N s1 j s 2 j 0 For a two-way AF relaying system employing the best relay R , an overall outage event occurs when at least k where PR is the transmitted signal energy from R j . For one of 12k and  21k fall below the threshold  th . Thus convenience, let P = P . s R the overall outage probability can be defined as After the self-interference is removed, the received signals at S1 and S2 are given by Pout = Pr min(1k 2 ,  2 k 1 )   th 

y G h() P h x  n  n (4) = Pr arg max min(1j 2 ,  2 j 1 )   th S1 j 1 j s 2 j 2 j 1  jN{1.. }  and N = Pr min(1j 2 ,  2 j 1 )   th  (13) yS2 G j h 2 j() P s h 1 j x 1  n j  n 2 (5) j1

We have 1  where Q x  exp t2 2 dt , and u and v 2 x Pr min(1j 2 ,  2 j 1 )   th  depend on the modulation employed, e.g. for BPSK u =1 =1 Pr min( ,  )    1j 2 2 j 1 th  and v =2 [24]. We then have

     1j 2 j 1 j 2 j Pe  = P( e | ) f (  ) d  =1 Pr , (14) e e  k 22 th   th 0 1j 2 j 1 j 2 j 2 2 t  ut = F() e2 dt Defining X 1  and X 1  , from (12) the  0  k 11jj 22jj 2 v corresponding PDF and CDF are  N 2212jj u  22tt     1 exp       0       2 j1 vv12jj ij 1 1     fxX    1 exp   (15) ij ij ij ij  x x 2 ij  ij  t expdt (22) and 2   1 Asymptotically, 1 exp(x )  x as x  0 , so at high FxX   exp (16) ij ij xij  SNRs (  large), (21) can be expressed as  ij respectively. Then 12jj2 N 22t u  22tt     Pr min( ,  )   Pe     e 2 dt  1j 2 2 j 1 th  e   0      2 j1 vv12jj C     1 x  =1 FXX f x dx   ijj  21jj 2 2 2 N  22 th u 2 0   L   D   2 i121 i  1 iN  1  j1 ijv  j 1  FXX 2 x f x dx (17)  21jj 1 N th -    N C 2 ijj 1  2 j1   (23) ij 2 j1 j where C 1/3 th and D 1/ 2 th . Substituting (17) into (13) provides the exact overall outage probability. where  is the fading severity parameter for the channel However, this expression cannot easily be evaluated. ijj

between S and R i =1, 2, and  is the power scaling Therefore an approach similar to that in [15], [18] is used i j j , j ijj to obtain the following tight closed-form lower bound on parameter for the channel between S and R . (17) . i j j

V. NUMERICAL RESULTS Pr min(1j 2 ,  2 j 1 )   th  1 1  1 F F (18) X1 j X 2 j  2 th 2 th Substituting (18) in (13), the overall outage probability is given by

N 11    Pout = 1 FFXX    (19) 12jj j1 22th   th 

In a similar manner, the CDF of the instantaneous SNR of the best relay Rk can be expressed as

N 11    Figure. 2 Overall outage probability for different fading severity Fx= 1 FF (20) parameters.  k   XX12jj    j122xx    In this section, we examine the overall outage performance and average symbol error probability for

IV. AVERAGE SYMBOL ERROR PROBABILITY binary phase shift keying (BPSK) modulation. Fig. 2

The ASEP is another important metric used to evaluate presents the overall outage probability versus PNS / 0 with performance. The instantaneous SEP can be expressed as 2 2  th = 1, N0 = 1, Eh()1 j = Eh()2 j = 1, fading severity

Pee  |   = uQ v  (21) parameters 1 j = v1 and 2 j = v2 , and N=1 relay. This

shows that the closed-form expression for the overall performance characteristics are similar for BPSK and outage probability is in good agreement with the exact QPSK. Fig. 5 shows that in the high SNR region, results expression, and this is confirmed by Monte Carlo using the closed form expression (23) match well with simulation. In addition, increasing one of v1 and v2 those obtained using (22). Only for large N and small provides only a small improvement in the overall outage SNR is the difference between (22) and (23) significant. probability, but at high SNRs, increasing both results in a The ASEP is very large in this case, and so has no significant improvement. This is because the overall practical relevance. The ASEP is reduced by increasing outage probability is dominated by the smaller of v and v1 and v2 or the number of relays N. The ASEP can be 1 v , as observed from (14) . reduced significantly by increasing the number of relays 2 N, This also holds for the overall outage probability. As an example, The ASEP is reduced from 10-6 to less than

10-10 when the number of relays is changed from 1 to 3

with v1 =2 and v2 =2 when SNR is 25 dB.

Figure. 3 Overall outage probability for different numbers of relays.

Fig. 3 presents the overall outage probability versus PN/ for different numbers of relays and fading severity S 0 values. The other parameters are the same as for Fig. 2. Figure. 5 Average symbol error probability for different power scaling This shows that increasing the number of relays improves parameters with BPSK modulation. the performance, as expected, particularly for larger

fading severity parameters v1 and v2 . For example, the -3 -9 outage probability is reduced from 10 to less than 10 when the number of relays is changed from 1 to 3 with

v1 =2 and v2 =2 when SNR is 20 dB.

Figure. 6 Average symbol error probability for different power scaling parameters with QPSK modulation.

VI. CONCLUSION Figure. 4 Overall outage probability for different power scaling parameters. An analysis of two-way AF relay cooperative network s with best relaying over independent but not necessarily 2 Fig. 4 shows that increasing Eh()ij improves the identically distributed (i.n.d.) Weibull fading channels overall outage performance. This figure also indicates was presented. Closed form approximations for the that for sufficiently high SNRs, the change in overall outage probability (OOP) and the average symbol 2 error probability (ASEP) were derived. These expressions performance is a constant as Eh() is varied. ij were shown to be tight, especially at high SNRs. Fig. 5 and Fig. 6 present the average symbol error Performance results presented show that the overall probability for BPSK and quadrature phase shift keying outage probability and the average symbol error (QPSK) modulation, respectively. From these figures, we probability can be improved by increasing the fading find that the ASEP for QPSK is slightly hi gh er than severity and power scaling parameters as well as the number of relays. BPSK for the same parameters. However, the

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Electron. Lett. vol. 47 no. 2, pp. 150–152, Jan 2011. University of Victoria, Victoria, BC, Canada in 1989. From 1989 to 1991 he was employed as a [17] P. K. Upadhyay and S. Prakriya, ―Performance of two way Defence Scientist at Defence Research opportunistic relaying with analog network coding over Establishment Ottawa, Ottawa, ON, Canada. He

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channels,‖ in Proc. Nat. Conf. on Commun., New Delhi, India, Department of Electrical and Computer Engineering. In 2002 he became a Fellow of the Engineering Institute of Canada, and in 2012 a Fellow of Feb 2013, pp. 1-5. the Canadian Academy of Engineering. His research interests include [19] IEEE Vehicular Technology Society Committee on Radio information theory and communication theory, algebraic coding theory, Propogation, ―Coverage prediction for mobile radio systems cryptography, smart grid and ultra wideband communications.

Wei Shi received the B.S. degree in 2009 from Hongjiao Zhang is pursuing the Ph.D. degree at Ludong University, Yantai, China. Currently, he is Ocean University of China. He received the B.S. pursuing a Ph.D. degree in the Department of degree in Changchun University of Science and Electrical Engineering, Ocean University of China. Technology in 2006 and the Master degree in

His research interests include OFDM, UWB, 60 Communication and information System from GHz wireless communications and cooperative Ocean University of China in 2010. His research communication networks. interests include 60GHz wireless communication and ultra-wideband radio systems.