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LETTER

Distributed Switch and Stay Combining with Partial Relay Selection over Rayleigh Fading Channels Vo Nguyen Quoc BAO†a) , Student Member and Hyung Yun KONG††b) , Member

SUMMARY Switch and stay combining (SSC) is an attractive diversity technique due to its low complexity and compatibility to resourceconstrained wireless networks. This letter proposes a distributed SSC for partial relay selection networks in order to achieve spatial diversity as well as to improve spectral efficiency. Simulation results show that the performance loss (in terms of bit error probability) of the proposed networks relative to partial relay selection networks with selection combining is not substantial. key words: switch and stay combining, partial relay selection, selection combining, decode-and-forward (DF), Rayleigh fading

1.

Introduction

To achieve spatial diversity as well as to improve the system’s overall capacity and coverage, a new communication paradigm named as cooperative communications is recently proposed [1], [2]. Different from conventional pointto-point communications, it allows different users or nodes in a wireless network to share resources and transmit jointly. Various cooperation protocols have been proposed for wireless networks. Among them, the opportunistic cooperative protocol is considered as a promising one in terms of spectral efficiency and achievable diversity [2]. However, its need for having perfect time synchronization and centralized processing approach is a crucial issue in some resource constrained wireless networks (especially, ad-hoc or wireless sensor networks) [3]. To address this concern, partial relay selection schemes for amplify-and-forward (AF) in which the best relay is selected based on instantaneous signal-tonoise ratios (SNRs) of the first hop instead of instantaneous SNRs composed of the SNRs across the two-hops is proposed [3], [4]. However, these models may not be the case in practical applications where relaying link is proposed with an aim to improve the quality of service over an existing link being under deep fades. Furthermore, without the help of direct link between the source and the destination, the dual hop relaying with partial relay selection (DRPRS) does not offer any diversity gain except 3 dB coding gain relative Manuscript received November 5, 2009. Manuscript revised May 3, 2010. † The author is with the Department of Telecommunications, Posts and Telecommunications Institute of Technology (PTIT), Ho Chi Minh City, Vietnam. †† The author is with the Department of Electrical Engineering, University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, 680-749 Korea. a) E-mail: [email protected] b) E-mail: [email protected] (Corresponding author) DOI: 10.1587/transcom.E93.B.2795

to direct communication [3]. To improve the performance of DRPRS, in this paper, we propose a new class of partial relay selection networks, which not only exploits the direct source-destination link to provide spatial diversity but also increases the system spectral efficiency. To that effect, the concept of switch and stay combining is used at the destination. Furthermore, by utilizing appropriate feedback information, the destination is able to switch cyclically between the direct link and the relaying link. The switching occurs when the instantaneous SNR of the currently selected link falls below the given threshold regardless the current instantaneous SNR of the alternative link. Thus far, the literature on distributed SSC with relaying concept is still relatively sparse. In particular, in [5], a distributed version of SSC is proposed for one relay equipped with decode-and-forward. It is then extended for two relays including direct transmission offering a simpler alternative of cooperative relaying [6]. Different from the schemes in [5], [6], in our proposed scheme many relays are involved in the cooperative transmission; however, with the use of partial relay selection, the system can be viewed as a virtual dual-branch switch-and-stay combining. In particular, since relaying link is being used, only the best relay whose link in the first hop has highest SNR is selected to forward the source information to the destination. Whereas in [6], the number of relays in cooperative transmission is limited by two and the system can be thought of as a virtual triplebranch switch-and-stay combining system since the direct transmission is taken into account. Furthermore, to facilitate the derivation, in our paper the relationship at high SNRs of the two hop relaying is employed instead of the exact approach as proposed in [5], [6]. The contributions of this paper are as follows. First, we propose the spectral-efficient, low-complexity combination of distributed SSC and partial relay selection and then derive the closed form statistics for each link SNR over Rayleigh fading channels, namely cumulative distribution function (CDF) and probability density function (PDF). Second, based on the PDF and CDF, the closed-form expressions of outage probability, bit error probability for square M-QAM and achievable spectral efficiency for the proposed system are presented. The theoretical analysis is confirmed through a comparison with Monte Carlo simulations. Our study shows that diversity order of two can be achieved for partial relay selection networks in conjunction with distributed SSC.

c 2010 The Institute of Electronics, Information and Communication Engineers Copyright 

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 2.

Fβ1 (γ) =

System Model

Distributed Switch-and-Stay Combining (DSSC)

Assuming that the relaying link is being used, i.e., B = R where B denotes the active branch. For partial relay selection, the best relay with highest instantaneous SNR in the first hop is selected to forward the source information to the destination. Let β1 be the instantaneous SNR of the link from the source to the best relay, we have β1 = max γ1,i i=1,...,N

fβ1 (γ)dγ =

0

Consider a wireless network with N + 2 nodes where the source (S) communicates with the destination (D) with help of the N relaying nodes (Ri , i = 1, . . . , N). Due to the use of distributed SSC along with the appropriate feedback signal sent to the source and all cooperative relays, the source information reaches to the destination either directly from the source or indirectly through the relaying link, denoted by D and R, respectively. In each transmission slot, the destination compares the received SNR with the predetermined switching threshold, T . The switching occurs since the SNR on the active link falls below T and becomes effective during the following transmission. The fading is further assumed to be slow enough that the fading envelope of each signal is constant over two consecutive transmission slots. Whenever the relaying link is connected to the destination, the communication is divided into two sub-slots. With partial relay selection, only the best relay providing highest SNR of the links from the source in the first subslot will serve as the forwarder in the second subslot. Due to Rayleigh fading, the channel powers, denoted by α0 = |hS D |2 , α1,i = |hS Ri |2 and α2,i = |hRi D |2 , are independent and exponential random variables whose means are λ0 , λ1,i and λ2,i , respectively. The average transmit power for the source and the relays is denoted by P s . Let us define the instantaneous SNRs for the source to the destination, the source to relay i, and relay i to the destination links as γ0 = P s α0 , γ1,i = P s α1,i and γ2,i = P s α2,i , respectively. In this paper, we deal with a network in which relays are grouped into a cluster due to their proximity where the chosen criterion is based on average SNRs, therefore this system model ensures that all channels from the source to the relays as well as from the relays to the destination have the same average channel power, i.e., γ¯ 1,i = E{γ1,i } = P s λ1,i = γ¯ 1 and γ¯ 2,i = E{γ2,i } = P s λ2,i = γ¯ 2 for all i where E{.} denotes the expectation operator. 3.

γ

(1)

Suppose that γ1,i is independent and identically distributed according to the exponential distribution, the PDF of γ1,i is −γ given by fγ1,i (γ) = γ¯11 e γ¯ 1 . If the branches from the source are independently faded, thanks to [7, p.246], the PDF and CDF of β1 is respectively written by   N N−1  N i − γ¯iγ N − γ¯γ  − γ¯γ i−1 e 1 (2) fβ1 (γ) = e 1 1− e 1 = (−1) i γ¯ 1 γ¯ 1 i=1

N  i=1

i−1

(−1)

   N − iγ 1 − e γ¯ 1 (3) i

Further, let β2 represents the instantaneous SNR of the link from the best relay to the destination, in a flat Rayleigh fading channel, the PDF and CDF of β2 can be expressed as fβ2 (γ) =

1 − γ¯γ −γ e 2 , Fβ2 (γ) = 1 − e γ¯ 2 γ¯ 2

(4)

Assuming un-coded modulation is used, the fixed decodeand-forward relay cannot detect any erroneous detection, and it may forward incorrectly decoded signals to the destination. Hence, similarly as in [8, Property 1], the equivalent instantaneous SNR of the DF dual hop relaying at the destination can be tightly approximated in the high SNR regime as follows: γR ≈ min{β1 , β2 }

(5)

With the help of [7, p.194, Eq. (6-81)] together with (2), (3) and (4), the joint PDF of γR is given after some manipulations as   N  N 1 − μγ fγR (γ) = (−1)i−1 e i (6) i μi i=1 where μi = (i/¯γ1 +1/¯γ2 )−1 . The CDF relative to γR is   γ N  i−1 N FγR (γ) = fγR (γ)dγ = (−1) 1 − e−γ/μi i i=1

(7)

0

In the another case, i.e., the direct link is in active (B = D), the PDF and CDF of γD in Rayleigh fading channel are fγD (γ) =

1 − γ¯γ −γ e 0 , FγD (γ) = 1 − e γ¯ 0 γ¯ 0

(8)

where γ¯ 0 = P s λ0 . According the operation mode of distributed SSC, the end-to-end performance of distributed SSC schemes with partial relay selection depends on the steady state selection probability for each branch, defined as the fraction of time that branch will serve as an active branch. Mathematically speaking, we have

 (9a) pD = Pr(B = D) = FγR (T ) FγD (T ) + FγR (T )

 pR = Pr(B = R) = FγD (T ) FγD (T ) + FγR (T ) (9b) where FγR (T ) and FγD (T ) can be readily obtained by evaluating FγR (γ) and FγD (γ) at γ = T , respectively. 3.1 Outage Probability The end-to-end outage probability for distributed SSC schemes with partial relay selection can be derived by using the law of total probability as 

Pr(γD ≤ T ) Pr {O|(B = R} + Pr(O) = pD Pr(γD > T ) Pr {O|(B = D, γD > T )}

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+ pR

Pr(γR ≤ T ) Pr {O|(B = D)} + Pr(γR > T ) Pr {O|(B = R, γR > T )}

 I(a, b, c, z) =

(10)

D R = 2r − 1 and γth = 22r − 1 with r denotes the target where γth spectral efficiency in case of direct transmission.

3.2 Bit Error Probability Similarly, the bit error probability of distributed SSC networks with partial relay selection is given by 

Pr(γD ≤ T ) Pr {E|(B = R} + Pr(E) = pD Pr(γD > T ) Pr {E|(B = D, γD > T )} 

Pr(γR ≤ T ) Pr {E|(B = D)} + + pR (12) Pr(γR > T ) Pr {E|(B = R, γR > T )} Recalling that Pr {E|(B = Z)} = Pr {E|(B = Z, γZ ≥ 0)}, the conditional average bit error probability over Rayleigh fading channels for M-ary square quadrature amplitude (MQAM) modulation (M = 4m , m = 1, 2, . . .) with Gray mapping can be expressible in general form as

j=1

(13)

k=0

 √  √ j−1 where υ j = (1 − 2− j ) M − 1, ϕkj = (−1) k.2 / M (2 j−1 −  √  √ √ k.2 j−1 / M + 1/2 )/( M log2 M), ςk = (2k + 1)2 3 log2 M/(2M − 2). Furthermore, we define . and erfc(.) as the floor and complementary error function, respectively. The conditional PDF of γZ |(γZ ≥ z) can be obtained by using conditional probability as follows [7]:  0, γ
Substituting (8) and (14) (with Z = D) into (13) and then taking the integral with respect to γ, we achieve the conditional bit error probability for the direct link as follows: √

υj ∞ log 2 M 

Pr{E|(B = D, γD > z)} =

j=1

z

k=0 √

z

= e γ¯ 0

√ ϕkj erfc( ςk γ) fγD (γ)dγ 1 − FγD (z)

υj log 2 M j=1

where I(.) is defined as follows:

k=0

I(ϕkj , ςk , γ¯ 0 , z)

(15)

 γ bγ 1c e− c dγ

   √  z z bc (1 = a e− c erfc bz − 1+bc erfc + bc) c

(16)

Similarly, from (6), (7) and (14), Pr {E|(B = R, γR > z)} can be written, after some manipulation, in the following closed form as √ υj N log2 M  

Pr{E|(B = R, γR ≥ z)} =

th

0

a erfc

z

where Pr(γZ ≤ T ) = 1 − Pr(γZ > T ) = FγZ (T ) with Z ∈ {R, D}. By introducing z ∈ {0, T } and noting that Pr {O|(B = Z)} = Pr {O|(B = Z, γZ ≥ 0)}, we get the compact general formula for Pr {O|(B = Z, γZ > z)} as ⎧ Z Fγ (γth )−FγZ (z) ⎪ Z ⎪ ⎪ , z ≤ γth ⎨ Z1−F γZ (z) (11) Pr {O|(B = Z, γZ > z)} = ⎪ ⎪ ⎪ ⎩ 0, z > γZ

Pr {E|(B = Z, γZ > z)} √ υj √ ∞ log 2 M  = ϕkj erfc ςk γ fγZ |(γZ ≥z) (γ)dγ

∞

j=1

k=0 i=1

(−1)i−1 (Ni )I(ϕkj ,ςk ,μi ,z) 1−FγR (z)

(17)

It is of interest to determine the optimal value of T , defined as the value that makes Pr(E) minimal. With the current form of Pr(E) in (12), this optimal value T ∗ can be found by numerically solving the following equation  d Pr(E)   =0 (18) dT T =T ∗ 3.3 Achievable Spectral Efficiency Same as the distributed SSC networks with one relay [5], the achievable spectral efficiency of the proposed scheme can be expressed as R¯ = pD r + pR r/2 4.

(19)

Distributed Selection Combining (DSC)

Selection combining (SC) can be viewed as an optimal implementation of the switch-and-stay diversity system [9]. In here, we study the performance of the distributed selection combining with partial relay selection for comparison purposes. In particular, since SC technique is employed at the destination, the combined instantaneous SNR at the output of the selection combiner is written by γΣ = max{γR , γD }

(20)

From the elementary theory of order statistics, we can obtain the joint PDF for γΣ as [7, p.194, Eq. (6.79)] fγΣ (γ) = FγR (γ) fγD (γ) + FγR (γ) fγD (γ) ⎡ −γ ⎤ −γ  ⎢⎢ μ1i e μi + γ10 e γ0 − ⎥⎥⎥ N  ⎢ ⎥   i−1 N ⎢ ⎢⎢ = (−1) −γ 1 + 1 ⎥⎥⎥⎦ i ⎢ ⎣ 1 1 μi γ0 i=1 μi + γ0 e

(21)

From (21), the outage probability and average bit error probability of partial relay selection networks with SC can be written respectively as follows: γ ⎤ γ   ⎡⎢ − th − th N  ⎢⎢ 1 − e  μi − e γ0 ⎥⎥⎥⎥ i−1 N ⎢ ⎢⎢ ⎥⎥⎦ (−1) (22) Pr(O) = 1 1 i ⎣ +e−γth μi + γ0 i=1 ⎤ ⎡ j √   ⎢⎢⎢⎢I(ϕk , ςk , μi , 0)+ ⎥⎥⎥⎥ υj  log N 2 M ⎥ N ⎢⎢⎢ j ⎢⎢⎢I(ϕk , ςk , γ¯ 0 , 0)− ⎥⎥⎥⎥⎥ (−1)i−1 Pr(E) = ⎥⎥⎦ i ⎢⎢⎣ j j=1 k=0 i=1 i γ0 , 0) I(ϕk , ςk , μμi +γ 0 (23)

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Fig. 3 Spectral efficiency for partial relay selection networks with DSSC and DSC (r = 1).

Fig. 1 Outage probability for partial relay selection networks with DSSC (r = 1).

Fig. 2, we use the same modulation scheme, i.e. 4-QAM, for both the direct and relaying links, whereas in Fig. 1 the same rate, i.e. r = 1 bps/Hz, is used for both the links. Therefore, D = 2r − 1 for the outage thresholds are calculated by, i.e. γth R 2r the direct link and γth = 2 − 1 for the relaying link, respectively. The difference in the outage thresholds is due to the fact that data transmission from the source to the destination over the direct link and relaying link occurs in one and two timeslot(s), respectively. The achievable spectral efficiency for the systems is plotted in Fig. 3. Compared with DSC systems, the benefit of the DSSC systems is the improvement of spectral efficiency with the cost of a limited feedback from the destination. 6.

Fig. 2 Bit error probability for partial relay selection networks with DSSC (4-QAM).

where γth = 22r − 1. 5.

Numerical Results

Numerical results are presented for N = 1, 3 and 5 with N N = 2 and {λ2,i }i=1 = 3. In Fig. 1, the outλ0 = 1, {λ1,i }i=1 age probability of partial relay selection networks with distributed SSC is compared to that with selection combining as well as with DRPRS. We found that the outage probability is minimal since the switching threshold is equal to the R , as expected. Furthermore, outage threshold, i.e., T = γth for the same target spectral efficiency r, the distributed SSC outperforms the DSC and DRPRS over the whole average SNR range. In Fig. 2, the bit error probability of three schemes (DSSC† , DSC and DRPRS) is investigated. It is interesting to show that due to the use of partial relay selection, increasing number of relays does not result in any advantage except 3 dB gain relative to the case of N = 1. Furthermore, the bit error probability curves confirm that, under same channel conditions, the performance of a system employing SC receiver is always better as compared to an equivalent system using distributed SSC by around 4 dB at high SNRs. It is worth noting that in here we obtain the reverse results relative to Fig. 1. It can be explained by using the fact that in †

The optimum switching threshold is used.

Conclusion

We have introduced an effective combination of SSC and partial relay selection. The system performance metrics of outage probability, bit error probability, and spectral efficiency on Rayleigh fading channels have been presented. The numerical results confirm the advantage of the proposed scheme over the existing ones in [2]–[5]. Specifically, with [2], it can solve the problem of low complexity requirement where the relay selection protocol demanding the instantaneous SNR of the two hops is not applicable. With [3], [4], it provides better performance in terms of bit error probability and spectral efficiency. And with [5], it offers better bit error probability around 3 dB with the same average SNR. On the other hand, the bit error probability of the proposed system is close to that of distributed SC systems but at a lower complexity, i.e., no actual diversity combiner is utilized at the destination. Analysis results are in excellent agreement with simulation results. Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0073895). References [1] J.N. Laneman, D.N.C. Tse, and G.W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol.50, no.12, pp.3062–3080, 2004.

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[2] A. Bletsas, H. Shin, and M.Z. Win, “Cooperative communications with outage-optimal opportunistic relaying,” IEEE Trans. Wirel. Commun., vol.6, no.9, pp.3450–3460, 2007. [3] I. Krikidis, J. Thompson, S. McLaughlin, and N. goertz, “Amplifyand-forward with partial relay selection,” IEEE Commun. Lett., vol.12, no.4, pp.235–237, 2008. [4] H.A. Suraweera, D.S. Michalopoulos, and G.K. Karagiannidis, “Semi-blind amplify-and-forward with partial relay selection,” Electron. Lett., vol.45, no.6, pp.317–319, 2009. [5] D.S. Michalopoulos and G.K. Karagiannidis, “Distributed switch and stay combining (dssc) with a single decode and forward relay,” IEEE

Commun. Lett., vol.11, no.5, pp.408–410, 2007. [6] D.S. Michalopoulos and G.K. Karagiannidis, “Two-relay distributed switch and stay combining,” IEEE Trans. Commun., vol.56, no.11, pp.1790–1794, 2008. [7] A. Papoulis and S.U. Pillai, Probability, random variables, and stochastic processes, 4th ed., McGraw-Hill, Boston, 2002. [8] T. Wang, A. Cano, G.B. Giannakis, and J.N. Laneman, “Highperformance cooperative demodulation with decode-and-forward relays,” IEEE Trans. Commun., vol.55, no.7, pp.1427–1438, 2007. [9] M.K. Simon and M.S. Alouini, Digital Communication over Fading Channels, 2nd ed., John Wiley & Sons, Hoboken, N.J., 2005.