Performance evaluation of half-duplex relay-based opportunistic cooperation diversity

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1 . RESEARCH PAPERS. SCIENCE CHINA Information Sciences January 200 Vol. 53 No. 2: 0 doi: 0.007/s Performance evaluation of half-duplex relay-based opportunistic cooperation diversity ZOU YuLong, ZHENG BaoYu & ZHU Jia Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunications, Nanjing 20003, China Received April 7, 2008; accepted September 6, 2009 Abstract Cooperative diversity is proposed as a means to improve the wireless transmission performance, which can effectively combat the wireless fading through sharing the antennas between source and relay nodes. In this paper, considering the practicability, we investigate the use of half-duplex relay, rather than full-duplex relay, for the opportunistic cooperation diversity. Two combining methods, namely selection diversity combining (SDC) and maximum ratio combining (MRC), are utilized for the implementation of the half-duplex relay-based opportunistic cooperation. Closed-form expressions of outage probability are derived for the proposed opportunistic cooperation as well as the known deterministic cooperation over Rayleigh fading channels. Numerical results show that the opportunistic cooperation is superior to the traditional deterministic cooperation in terms of outage probability and, moreover, the MRC-based case outperforms the SDC-based case no matter which duplex mode relay (i.e., half-duplex and full-duplex) is used for the opportunistic cooperation. Keywords half-duplex relay, cooperative diversity, SDC, MRC, outage probability Citation Zou Y L, Zheng B Y, Zhu J. Performance evaluation of half-duplex relay-based opportunistic cooperation diversity. Sci China Inf Sci, 200, 53: 0, doi: 0.007/s Introduction It is, in general, a difficult task to improve the channel capacity due to the fading effect of wireless channels. Multiple-input and multiple-output (MIMO) technology [, 2] proposed in recent years makes full use of space resources by employing multiple antennas at both the transmitter and receiver, thus largely increasing the channel capacity. However, in many wireless applications, wireless transreceivers may not be able to support multiple antennas due to the limitation in physical size and power consumption. As an alternative spatial diversity technique, user cooperation has been proposed as a means to provide robustness against channel fluctuations by sharing different users antennas [3, 4]. Recent years have witnessed an increased interest in this topic. And cooperative diversity systems have been extensively studied in terms of different techniques and different performance measures [3 9]. The term opportunistic was first used by Viswanath et al. [0]. In their work, the base station always selects the best user for transmission in a fading environment. It has also been used in the context of efficient flooding of signals in multi-hop networks to increase the communication range []. Recently, an opportunistic decode-and-forward (ODF) scheme has been proposed in [9], where the relay terminal Corresponding author ( zouyulong9842@26.com; yulong.zou@stevens.edu) c Science China Press and Springer-Verlag Berlin Heidelberg 200 info.scichina.com

2 2 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 may be utilized depending on the overall network state with dynamic power and time allocation. The ODF scheme is only adaptive for the occasion where the cooperative relay has the ability to decode the received signal from source node. In [7, 8], we have proposed an opportunistic cooperation scheme for nonregenerative relay systems. It has been shown that the opportunistic cooperation diversity outperforms the traditional cooperation diversity in terms of outage probability and bit error rate (BER). However, the relay used in [7, 8] are based on full-duplex mode that is difficult to implement in practice due to the high power level difference between the transmit and receive signals. Considering the practicability, we would utilize a half-duplex relay to investigate the performance of opportunistic cooperation. This paper differs from the previous research in the following aspects. First, a half-duplex relay is employed for the implementation of opportunistic cooperation in a non-regenerative relay system where the relay node is incapable of decoding. Second, two combining methods, i.e., SDC and MRC, are used for the half-duplex relay-based opportunistic cooperation along with the outage probability analysis over Rayleigh fading channels. Third, we also derive a closed-form expression of outage probability for the known deterministic cooperation, with numerical results verifying the merits of the proposed opportunistic cooperation diversity. The remainder of this paper is organized as follows. After a brief description of the system model in section 2, we present a half-duplex relay-based opportunistic cooperation framework in section 3, where both the SDC and the MRC are considered. Next, in section 4, closed-form outage probability expressions are derived for the half-duplex relay-based opportunistic cooperation with the two combining methods. Besides, the outage probability analysis for both the non-cooperation and the deterministic cooperation are also presented in this section. Numerical results are provided in section 5, showing the merits of the opportunistic cooperation diversity. Finally, in section 6, we make some concluding remarks. 2 System model We consider a wireless communication network with a source (s), a destination (d) andacooperative relay (r) as shown in Figure. Assume that each transmission link between any two nodes is modeled as a Rayleigh block fading channel, i.e., the fading coefficients of the time-varying channel are regarded as constant within two consecutive symbol periods. There is also independent, zero-mean additive white Gaussian noise (AWGN) with one sided power spectral density N 0 at each receiver. Without loss of generality, let s l [t (k )T ] be the equivalent lowpass form of the transmitted signal with the power P T at the source node. Thus, with the coherent reception, the signals received at the cooperative relay and the destination can be expressed as and r sr [t (k )T ]= α sr (k) 2 P T s l [t (k )T ]+α sr(k)n sr [t (k )T ], () r sd [t (k )T ]= α sd (k) 2 P T s l [t (k )T ]+α sd (k)n sd[t (k )T ], (2) where the subscripts sr and sd denote the transmission from the source to the cooperative relay and that from the source to the destination, respectively. For non-regenerative systems, the cooperative relay amplifies and forwards the received signal from the source node without any sort of decoding, which can be referred to as analogy relaying in contrast to digital relaying that is employed in regenerative systems. Note that all the wireless channels are modeled as Rayleigh block fading channels, meaning α rd (k) = α rd (k + ). Hence, through the analog relaying with gain G, the signal received at the destination can be given by r srd (t kt) = α rd (k) 2 G r sr [t (k )T ]+α rd(k)n rd (t kt), (3) where r sr [t (k )T ] is given in eq. (2). The random variables (RVs), α sr (k), α sd (k) andα rd (k), are the fading coefficients of the channel from the source to cooperative relay, from the source to the destination,

3 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 3 Figure System model of cooperative diversity. and from cooperative relay to the destination, respectively. Notice that the variances of RVs α sr (k), α sd (k) andα rd (k) areσ sr (k), σ sd (k) andσ rd (k), respectively. Throughout this paper, we make the following three assumptions: ) the relay used in the system model is half-duplex out of consideration for practicability; 2) all the channels are independent of each other in space; 3) the receivers can accurately estimate the fading coefficients in their received signals. 3 Proposed half-duplex relay-based opportunistic cooperation diversity A basic deterministic cooperation scheme has already been presented in [4], where the original signal from source is coherently detected by cooperative relay at the kth symbol period, and then at the next symbol period, the relay amplifies the received signal and forwards it to the destination regardless of its quality. In [7, 8], we proposed an opportunistic cooperation diversity scheme to avoid forwarding the severely deteriorated signal over the source-relay channel. However, the relay used in [7, 8] was full-duplex, which would be difficult to implement in practical systems. In this paper, we will consider the use of half-duplex relay for the opportunistic cooperation diversity. In half-duplex relay systems, there is a loss in spectral efficiency due to the pre-log factor /2 in corresponding channel capacity, which can be observed in the following equation formulations. To avoid much repetition, we do not present here the framework of the opportunistic cooperation scheme that has been illustrated already in Figure 2 of [7] (for details see [7]). In what follows, we focus on the formulations of signal model for the half-duplex relay-based opportunistic cooperation. From eq. (), the signal-to-noise-ratio (SNR) of the received signal at the cooperative relay can be computed as SNR r = α sr (k) 2 γ T, (4) where γ T = P T /N 0. According to Shannon coding theorem, outage event is deemed to occur when the channel capacity falls below a predetermined threshold R. Thus, we can describe the source-relay outage as 2 log 2 ( + SNR r ) <R, (5) where the factor /2 in the front of log function is due to the fact that two orthogonal channels are needed for each half-duplex relay transmission. When the source-relay outage occurs, the source would repeat its signal transmission with the power P T. Thus, during the next symbol period, the received signal at the destination can be expressed as r sd (t kt) = α sd (k) 2 P T s l [t (k )T ]+α sd (k)n sd(t kt). (6) Now, the destination combines the two copies of the received signal, namely r sd [t (k )T ]andr sd (t kt) as given in eqs. (2) and (6), respectively. In particular, if the SDC method is adopted, only the signal copy with a higher SNR will be used for maximum-likelihood (ML) decision; else the MRC method is

4 4 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 adopted, both the two signal copies are added together to achieve an enhanced signal version that will be utilized for ML decision. Therefore, we can obtain ˆr sd (t kt) = α sd (k) 2 P T s l [t (k )T ]+α sd(k)n sd (t kt), (7) for the half-duplex relay-based opportunistic cooperation with SDC (also referred to as half-duplex and SDC-based opportunistic cooperation) and ˆr sd (t kt) =2 α sd (k) 2 P T s l [t (k )T ]+α sd(k)n sd [t (k )T ]+α sd(k)n sd (t kt), (8) for the half-duplex relay-based opportunistic cooperation with MRC, called half-duplex and MRC-based opportunistic cooperation. When no outage event occurs in the source-relay channel, we have 2 log 2 ( + SNR r ) >R. (9) Then, the cooperative relay will amplify the received signal and forward it to the destination with a relaying gain G. Throughout this paper, we consider the gain G =/ α sr (k) 2. Thus, the final signals combined at the destination are given by eqs. (0) and () for the SDC-based and the MRC-based opportunistic cooperation diversity schemes, respectively. α sd (k) 2 P T s l [t (k )T ]+α sd (k)n sd[t (k )T ]; α sd (k) 2 > α sr(k) 2 α rd (k) 2 ˆr d (t kt) = α sr (k) 2 + α rd (k) 2, (0) α rd (k) 2 P T s l [t (k )T ]+ α rd(k) 2 α sr (k) n sr[t (k )T ] +α rd (k)n rd(t kt); others, ˆr d (t kt) =[ α sd (k) 2 + α rd (k) 2 ] P T s l [t (k )T ]+ α rd(k) 2 α sr (k) n sr[t (k )T ] + α sd(k)n sd [t (k )T ]+α rd(k)n rd (t kt). Based on the combined signal ˆr d (t kt), the destination would make its decision result of the transmitted signal s l [t (k )T ] in accordance with the ML criterion. Now, we have finished the formulation of signal models for the half-duplex relay-based opportunistic cooperation. It is worth mentioning that by detecting the SNR of received signal only, the cooperative relay can determine whether or not the source-relay channel is in outage, and thus this opportunistic cooperation method is straightforward in implementation. () 4 Outage probability analysis of the half-duplex relay-based opportunistic cooperation diversity First of all, let us consider the non-cooperative transmission from the source to the destination. From eq. (2), we can easily calculate the outage probability for the non-cooperative transmission scheme as Pout non =Pr [ log 2 ( + α sd (k) 2 γ T ) <R ]. (2) Notice that RV α sd (k) 2 follows the exponential distribution with parameter /σsd 2. Hence, the probability integral as given in eq. (2) is derived as ( ) Pout non = exp 2R, (3) where = σ 2 sd γ T is regarded as the mean source-destination SNR.

5 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No Half-duplex and SDC-based opportunistic cooperation As can be seen from section 3, there are two possible cases for the data transmission depending on whether the source-relay channel is in outage. For notational convenience, let the case θ = represent the source-relay channel being in outage and let the case θ = 2 represent the other case. Case θ = : This case corresponds to source-relay channel being in outage (for example, due to the deep fading). Following eq. (5), we can calculate the occurrence probability of case θ = as { } Pr SDC Opp (θ =)=Pr 2 log 2( + α sr (k) 2 γ T ) <R. (4) Notice that RV α sr (k) 2 follows an exponential distribution with the parameter /σsr. 2 Thus, we can compute eq. (4) as ( ) Pr SDC Opp (θ =)= exp 22R, (5) γ sr where γ sr = σsr 2 γ T is the mean source-relay channel SNR. From eq. (7), the corresponding outage probability of the half-duplex and SDC-based opportunistic cooperation for case θ = can be given by { } ( ) Pout SDC Opp (θ =)=Pr 2 log 2( + α sd (k) 2 γ T ) <R = exp 22R, (6) where = σsd 2 γ T is the source-destination channel SNR. Case θ = 2: This case corresponds to a good condition of the source-relay channel, which indicates that no outage event occurs as described by eq. (9). Hence, the occurrence probability of case θ = 2is calculated as ( ) Pr SDC Opp (θ =2)=exp 22R. (7) γ sr Following eq. (0) and noting that all the channels are independent of each other in space, we can obtain the outage probability of the half-duplex and SDC-based opportunistic cooperation diversity in case θ =2 as Pout SDC Opp (θ = 2) = 8(I) 8(II), (8) where the terms 8(I) and 8(II) are given by { } ( ) 8(I) = Pr α sd (k) 2 < 22R = exp 22R, (9) γ T and 8(II) = Pr { Γ < 2 2R }, ( ) Γ= α sr (k) 2 + γ T α rd (k) 2. γ T (20) Obviously, RVs α sr (k) 2 and α rd (k) 2 follow exponential distribution with the parameters, /σsr 2 and /σrd 2, respectively. Hence, we can easily obtain α sr(k) 2 γ T ε(/γ sr )and α rd (k) 2 γ T ε(/γ rd ), where γ sr = σsr 2 γ T and γ rd = σrd 2 γ T. Using the results of Appendix A of [7], we obtain the CDF of RV Γ, P Γ (γ), as follows: ( ) ( 2γ 2γ P Γ (γ) = K exp γ γ ). (2) γsr γ rd γsr γ rd γ sr γ rd Thus, combining eq. (20) and eq. (2) gives ( 8(II) = 22R R+ ) ( ) 2 K exp 22R 22R. (22) γsr γ rd γsr γ rd γ sr γ rd

6 6 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 Besides, if we ignore the received noise at cooperative relay, the corresponding SNR received at the destination for the half-duplex and SDC-based opportunistic cooperation in case θ = 2 can be given by SNR SDC Opp (θ =2)=max( α sd (k) 2 γ T, α rd (k) 2 γ T ), from which a tight lower bound on the outage probability is calculated as [ ( )] [ ( )] SDC Opp(θ =2)= exp 22R exp 22R. (23) γ rd Using eqs. (5) (8), we can obtain an exact outage probability expression for the half-duplex and SDC-based opportunistic cooperation as Pout SDC Opp =Pr SDC Opp (θ =)Pout SDC Opp (θ =) +Pr SDC Opp (θ =2)Pout SDC Opp (θ =2). (24) Hence, a lower bound on outage probability for the half-duplex and SDC-based opportunistic cooperation is given by SDC Opp =Pr SDC Opp(θ =)Pout SDC Opp (θ =) +Pr SDC Opp (θ =2) SDC Opp(θ =2), (25) where SDC Opp (θ = 2) is given in eq. (23). 4.2 Half-duplex and MRC-based opportunistic cooperation Also, let case θ = represent an outage event occurs in the source-relay channel; otherwise, θ = 2. In what follows, we derive a closed-form expression of the outage analysis for the MRC-based opportunistic cooperation. Case θ = : In a similar way, the occurrence probability of this case is given by ( ) Pr MRC Opp (θ =)= exp 22R. (26) γ sr Then, from eq. (8), we can obtain the outage probability of the half-duplex and MRC-based opportunistic cooperation in case θ =as { } ( ) Pout MRC Opp (θ =)=Pr 2 log 2( + 2 α sd (k) 2 γ T ) <R = exp 22R, (27) 2 where = σsd 2 γ T is viewed as the mean source-destination SNR. Case θ = 2: The occurrence probability of case θ =2isgivenby ( ) Pr MRC Opp (θ =2)=exp 22R. (28) γ sr For the convenience of theoretical analysis, we ignore the noise at the cooperative relay. This is reasonable since the case θ = 2 occurs only when the instantaneous SNR of source-relay channel is comparatively high. Thus, eq. () can be further simplified into ˆr d (t kt) =[ α sd (k) 2 + α rd (k) 2 ] P T s l [t (k )T ] + α sd (k)n sd[t (k )T ]+α rd (k)n rd(t kt), from which the corresponding SNR is calculated by SNR MRC Opp (θ =2)= α sd (k) 2 γ T + α rd (k) 2 γ T. (29)

7 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 7 Following eq. (29), a tight lower bound on the outage probability for the half-duplex and MRC-based opportunistic cooperation in case θ =2isgivenby { } MRC Opp(θ =2)=Pr 2 log 2( + [ α sd (k) 2 + α rd (k) 2 ]γ T ) <R. (30) Note that RVs α sd (k) 2 and α rd (k) 2 follow exponential distribution with parameters /σsd 2 and /σ2 rd, respectively, and are independent of each other. Therefore, we can obtain ( MRC Opp (θ =2)= exp x y ) dxdy, (3) Θ σsd 2 σ2 rd where the parameter Θ is defined as Θ = {(x, y) x + y<(2 2R )/γ T }. Performing the double integration, we can further obtain ( ) ( ) + 22R exp 22R ; = γ rd, MRC Opp (θ =2)= γ rd exp γ rd σ 2 sd ( 22R exp γ rd ) σ 2 rd ( 22R γ rd ) ; γ rd. Combining eqs. (26) (32), we can easily obtain a tight lower bound on the outage probability for the half-duplex and MRC-based opportunistic cooperation as follows: MRC Opp =Pr MRC Opp(θ =) Pout MRC Opp (θ =) +Pr MRC Opp (θ =2) MRC Opp (θ =2). (33) Then, we present the closed-form expressions of outage probability for the half-duplex relay-based deterministic cooperation diversity, in which, as the name implies, the cooperative relay must forward the received signal to the destination node regardless of its quality. For a fair comparison with the opportunistic cooperation, we also consider two combining methods (namely SDC and MRC) for the deterministic cooperation, and the corresponding schemes are referred to as the half-duplex and SDCbased as well as the half-duplex and MRC-based deterministic cooperation, respectively. The closed-form outage probability expressions for the two deterministic schemes can be easily given by (32) Pout SDC Deter =Pout SDC Opp (θ =2), (34) and MRC Deter = MRC Opp(θ =2), (35) where Pout SDC Opp (θ = 2) and MRC Opp (θ = 2) are given in eqs. (8) and (32), respectively. So far, we have conducted the performance analysis of the non-cooperation, the half-duplex relay-based opportunistic cooperation, as well as the half-duplex relay-based deterministic cooperation, leading to closed-form expressions of outage probability as shown in eqs. (3), (24), (33), (34) and (35), which will be used in the next section to conduct numerical evaluations. 5 Numerical results and analysis In this section, we first present the numerical results of outage probability for the half-duplex and the fullduplex relay-based opportunistic cooperation. Note that closed-form expressions of the outage probability of the full-duplex relay-based opportunistic cooperation with the two combining methods, SDC and MRC, are given by eqs. (28) and (37) in [7], respectively. Next, we conduct the performance comparison between the opportunistic cooperation and the traditional deterministic cooperation, verifying the advantage

8 8 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 of our schemes. Finally, the SDC-based opportunistic cooperation is compared with the MRC-based opportunistic cooperation, with numerical results showing the merits of the MRC-based case. Figure 2 shows the plots of eqs. (24) and (33) as well as eqs. (28) and (37) of [7] as a function of the mean source-destination SNR. One can see from Figure 2 that the full-duplex relay-based opportunistic cooperation always outperforms the half-duplex relay-based opportunistic cooperation no matter which combining method (i.e., SDC and MRC) is adopted. This is because there is a spectrum efficiency loss for the half-duplex relay transmissions since two orthogonal channels are needed to complete each one relay transmission. However, the full-duplex relay is very difficult to implement in practice due to the high power level difference between the transmit and receive signals. Besides, it is shown from Figure 2 that the MRC-based case performs better than the SDC-based case no matter which duplex mode is used in the opportunistic cooperation system. Figure 2 Outage probability versus the mean source-destination SNR of the half-duplex relay-based and the full-duplex relay-based opportunistic cooperation with R=2 b/s/hz, α =0.5 andγ sr = = γ rd +0 db. Figure 3 Outage probability versus the data rate R of the deterministic cooperation and the opportunistic cooperation with α =0.5, γ sr = =5 db and γ rd =5 db.

9 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 9 Figure 4 Outage probability comparison between the SDC-based and the MRC-based opportunistic cooperation for different values of η = γ sr with R=2 b/s/hz and γ rd =. Figure 5 The SDC-based versus the MRC-based opportunistic cooperation for different values of μ = γ rd with R=2 b/s/hz and =. Figure 3 illustrates the outage probability comparison between the deterministic cooperation and the opportunistic cooperation, where both the cases of half-duplex and full-duplex are considered. As can be observed, the outage probabilities of the half-duplex and the full-duplex relay-based opportunistic cooperation are smaller than that of the corresponding deterministic cooperation, showing the advantage of the opportunistic cooperation. In what follows, we focus on the performance comparison between the half-duplex and SDC-based opportunistic cooperation as well as the corresponding MRC-based case under different relay channel conditions. Figure 4 depicts the outage probability versus the mean source-destination SNR,, of the SDC-based and the MRC-based opportunistic cooperation with the use of half-duplex relay, where the performance curves are the plots of eqs. (24) and (33) for the different values of η = γ sr, respectively. As shown in Figure 4, the outage probabilities of the SDC-based opportunistic cooperation are larger than that of the corresponding MRC-based opportunistic cooperation when η = 0 db, η = 5 dbandη=0 db. In Figure 5, we show a comparison of the SDC-based opportunistic cooperation with the MRC-base

10 0 ZOU YuLong, et al. Sci China Inf Sci January 200 Vol. 53 No. 2 case under the different channel conditions from the relay to the destination. As is evident in Figure 5, the outage probability performances of MRC-based opportunistic cooperation scheme corresponding to η = 0, 5 and 0 db are superior to the SDC-based opportunistic cooperation, respectively. Therefore, the MRC-based opportunistic cooperation outperforms the SDC-based case in terms of outage probability. 6 Conclusions In this paper, we have investigated the use of half-duplex relay to the opportunistic cooperation diversity system, where two combining methods (i.e., SDC and MRC) are utilized for the implementation of the half-duplex relay-based opportunistic cooperation. We have derived the closed-form expressions of outage probability for the proposed opportunistic cooperation diversity over Raleigh fading channels. Also, the outage probabilities of the traditional non-cooperation and deterministic cooperation have been developed for the purpose of performance comparison. It has been shown that the proposed opportunistic cooperation performs better than the known deterministic cooperation and moreover, the MRC-based case is superior to the SDC-based case in terms of outage probability. Acknowledgements This work was supported by the Postgraduate Innovation Program of Scientific Research of Jiangsu Province (Grant Nos. CX08B 080Z, CX09B 50Z), the National Natural Science Foundation of China (Grant No ), the Key Project of Natural Science Funding of Jiangsu Province (Grant No. BK ), the National High-Tech Research & Development Program of China (Grant No. 2009AA0Z24), and the Major Development Program of Jiangsu Educational Committee (Grant No. 06KJA500). References Goldsmith A, Jafar S A, Jindal N, et al. Capacity limits of MIMO channels. IEEE J Select Area Commun, 2003, 2: Zheng J, Rao B D. LDPC-coded MIMO systems with unknown block fading channels: soft MIMO detector design, channel estimation, and code optimization. IEEE Trans Signal Process, 2006, 54: Sendonaris A, Erkip E, Aazhang B. User cooperation diversity part I: System description. IEEE Trans Commun, 2003, 5: Laneman J N, Tse D N C, Wornell G W. Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans Inf Theory, 2004, 50: Hunter T E, Sanayei S, Nosratinia A. Outage analysis of coded cooperation. IEEE Trans Inf Theory, 2006, 52: Zou Y L, Zheng B, Zhu X. A new cooperative diversity scheme for next generation wireless network. In: The Fifth IEEE Consumer Communications and Networking Conference, Las Vegas, USA, Zou Y L, Zheng B Y, Zhu J. Outage analysis of opportunistic cooperation over Rayleigh fading channels. IEEE Trans Wirel Commun, 2009, 8: Zou Y L, Zheng B Y, Zhu W P. An opportunistic cooperation scheme and its BER analysis. IEEE Trans Wirel Commun, 2009, 8: Gunduz D, Erkip E. Opportunistic cooperation by dynamic resource allocation. IEEE Trans Wirel Commun, 2007, 6: Viswanath P, Tse D N C, Laroia R. Opportunistic beam-forming using dumb antennas. IEEE Trans Inf Theory, 2002, 48: Scaglione A, Hong Y W. Opportunistic large arrays: Cooperative transmission in wireless multihop ad hoc networks to reach far distances. IEEE Trans Signal Process, 2003, 5: Gradshteyn I S, Ryzhik I M. Table of Integrals, Series, and Products. 5th ed. San Diego, CA: Academic Press, Abramowitz M, Stegun I A. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. 9th ed. New York: Dover Publications, Oberhettinger F, Badii L. Tables of Laplace Transforms. New York: Springer-Verlag, 973