www.ietdl.org Published in IET Communications Received on 23rd June 2010 Revised on 15th November 2010 doi: 10.1049/iet-com.2010.0517
ISSN 1751-8628
Switch-and-stay partial relay selection over Rayleigh fading channels A. Gharanjik K. Mohamed-pour Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran, Iran E-mail:
[email protected]
Abstract: In this study, a switch-and-stay partial relay selection (SS-PRS) scheme is analysed, where only a single relay from two cooperating open-form decode-and-forward (DF) relays is active in each transmission time slot. The performance of the proposed relay selection scheme is presented in terms of the outage probability and the average bit error rate (ABER). In this study, closedform expressions are derived for the outage probability and the ABER of the system. In addition, an optimum threshold for the relay switching is found in a minimum ABER sense for the binary phase-shift keying modulation. The numerical and analytical results show that the SS-PRS scheme and the conventional PRS scheme give almost the same performance, whereas the SS-PRS has the advantage of notably lower complexity.
1
Introduction
Cooperative relaying techniques have attracted much research interest in the recent years among the industry and the academia [1, 2]. In the cooperative relaying schemes one or more relay nodes are selected according to some system parameters. Various relay selection schemes have already been introduced in the literature [3 –11]. In [4], Bletsas et al. proposed a relay selection scheme, where the selected criterion is the best instantaneous signal-to-noise ratio (SNR) composed of the SNR across both source-relay and relay-destination hops. However, this scheme is not suitable to implement in the resource-constrained wireless systems such as ad hoc and sensor networks because monitoring the connectivity among the different links globally can limit the network lifetime. Also, it requires that the nodes be carefully synchronised which is difficult to implement. A partial relay selection (PRS) scheme has recently been proposed in [5], where the source just needs partial information [the first hop channel state information (CSI)] for relay selection. The performance of the PRS scheme has been analysed for a CSI-assisted amplify-and-forward (AF) relaying scheme in [5 – 9] and for a semi-blind AF relaying system in [10 – 12]. In the PRS scheme, the source monitors the connectivity among the nodes locally (one-hop) rather than globally (two-hop). Based on the local feedbacks received from the relays, the source selects the best link (source-relay) for data transmission. However, the PRS scheme has to find the best relay node at every transmission time slot. In this regard, a certain level of complexity and overhead still arises since the CSI of all participating links is needed. Also, when the PRS scheme is implemented in real time, owing to the feedback delay the selected relay may not be the best among the relay set, and thus the system performance is adversely affected [8]. IET Commun., 2011, Vol. 5, Iss. 9, pp. 1199–1203 doi: 10.1049/iet-com.2010.0517
Motivated by these challenges, this paper proposes the switch-and-stay partial relay selection (SS-PRS) scheme where there is no need for the source node to continuously monitor all the channels in the first hop. In this scheme, only a single relay is active in every transmission time slot remaining active as long as the first hop SNR is sufficiently high. Note that in this scheme only one relay is active at each transmission time slot, and the active relay feedbacks to the source only when the relay switching is required. This relay selection scheme decreases the complexity and power consumption of the system. In this paper, the performance of the proposed SS-PRS scheme is evaluated with respect to the outage probability and average bit error rate (ABER). A closed-form expression is provided for the outage probability and then another closed-form expression is derived for ABER concerning the M-ary phase-shift keying (MPSK) and M-ary quadrature amplitude modulation (M-QAM). Also, for binary PSK (BPSK) modulation, an expression for the optimum switching threshold in a minimum average error rate sense is found. The analytical results are verified using computer simulations. Both the analytical and numerical results show that the proposed scheme using an appropriate switching threshold (T ) and the one proposed in [5] have relatively similar performance especially at high average SNRs, meanwhile the SS-PRS scheme has the advantage of lower complexity. The remainder of this paper is organised as follows. Section 2 introduces the System and Channels Model for SS-PRS scheme. In Section 3, closed-form expression for the Outage Probability and ABER of the system are derived. Also, the Optimum Switching Thresholds in the sense of minimum outage and minimum ABER are analysed. Numerical Results are presented in Section 4. Concluding remarks are provided in Section 5. 1199
& The Institution of Engineering and Technology 2011
www.ietdl.org 2
System and channels model
A dual-hop transmission system consisting of one source (S), one destination (D) and two relays (R1 and R1 ) is considered. The source terminal has no direct link with the destination, and the transmission is performed only through relays. These relays are willing to assist the transmission by decoding (demodulating and then remodulating) the received signal and forwarding it to D. Also, the relays are supposed to operate in the half-duplex mode. Therefore each transmission occurs in two time subslots. In the first subslot the source transmits its data to the active relay and in the second subslot the active relay retransmits the processed data (decoded signal) to the destination. The proposed scheme activates only one of the two relays in a switch-and-stay manner [13, 14]. In the first subslot, the SNR of the received signal is compared with a switching threshold T by the active relay. If the SNR is lower than T, then relay switching occurs. This is implemented by an appropriate feedback sent to the source by the active relay. This feedback conveys the message that the channels condition between the source and the active relay is not appropriate, and so by the next transmission time slot the active relay will switch to the idle mode. In this case, the source urges the the idle relay to switch to the active mode. The active relay (selected relay) is denoted by R, R [ {R1 , R2 }. The instantaneous SNRs of the S − Ri , i ¼ 1, 2 and R − D channels are represented by gSRi and gRD , respectively. Also, it is assumed that these channels experience independent, flat and slow Rayleigh fading, with average SRi and g RD . Therefore the PDF of gSRi SNRs denoted by g and gRD can be written as fgSR (g) = 1/ gSRi exp(− g/ gSRi ) i and fgRD (g) = 1/ gRD exp (−g/ gRD ).
3 3.1
Performance analysis Outage probability
In this subsection, we derive a closed-form expression for the outage probability of the SS-PRS scheme. For this purpose, first we must find the probability density function (PDF) and cumulative density function (CDF) of the received instantaneous SNR at the destination. We define a random variable g representing the received instantaneous SNR at the destination. Then, in the dual-hop DF relaying system, we can write the PDF of g as [15] 1 −g/gRD fG (g) = Ad(g) + (1 − A) e (1) RD g where d(g) is the delta function and A = Pr[gSR , gth ]. A can be considered as the outage probability of the first hop, Pout−hop1 (gth ), which is derived by utilising the outage analysis of the switch-and-stay combining (SSC) systems [13, 14] as A = Pout−hop1 (gth ) ⎧ FgSR (T )FgSR (T )(FgSR (gth ) + FgSR (gth )) ⎪ ⎪ 1 2 1 2 ⎪ , gth , T ⎪ ⎪ FgSR (T ) + FgSR (T ) ⎪ ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ ⎨ FgSR (T )FgSR (T )(FgSR (gth ) + FgSR (gth ) − 2) 1 2 1 2 = ⎪ ⎪ FgSR (T ) + FgSR (T ) ⎪ ⎪ 1 2 ⎪ ⎪ ⎪ FgSR (gth )FgSR (T ) + FgSR (T )FgSR (gth ) ⎪ ⎪ 1 2 1 2 ⎪ , gth ≥ T ⎪ ⎩ + FgSR (T ) + FgSR (T ) 1
2
(2) 1200 & The Institution of Engineering and Technology 2011
where FgSR (x) = 1 − exp (−x/ gSRi ). The setting of the i predetermined switching threshold (T ) is an important system design issue and significantly affects the outage probability of the first hop and therefore affects the end-to-end outage performance of the system. For a given gth , the optimal switching threshold in the minimum outage probability sense is given by T = gth (see [14, ch. 9.8.1.3]). In this condition, (2) results in A = FgSR (gth )FgSR (gth ) = (1 − e−gth /gSR1 ) 1
2
× (1 − e =
2
−gth / gSR2
)
−gth / gSRj
[1 − e
(3) ]
j=1
The end-to-end outage probability of the SS-PRS scheme can be defined as Pout (gth ) = Pr[g , gth ] = FG (gth ), where FG (g) is the CDF of g. Then, FG (gth ) can be evaluated by integrating the (1) and replacing g with gth as FG (gth ) = AU(gth ) + (1 − A)(1 − e−gth /gthRD )
(4)
where U(x) is unit function. Finally, by substituting (3) into (4), we obtain a closed-form expression for the end-to-end outage probability of the system as Pout (gth = 1 − e−g/gRD 1 −
2
−gth / gSRj
[1 − e
]
(5)
j=1
This is the optimal outage probability of the SS PRS scheme which equals that of the scheme (proposed in [5] with N ¼ 2, where N is the number of the participating relays) that selects the best relay based on the first hop SNRs (gSR1 and gSR2 ) for each transmission slot. 3.2
Average bit error rate
Another important performance criterion is the ABER that is E−hop1 and P E−hop2 as studied in this subsection. Denoting P the ABERs of the links S R and R D, respectively, the end-to-end ABER of the dual-hop decode and forward relaying system can be expressed as [16] E = P E−hop1 + P E−hop2 − 2P E−hop1 P E−hop2 P
(6)
The straightforward approach to obtain the ABER of each E−hopi , is to average the conditional BER, PE (E|g), hop, P over the PDF fhop i (g) as 1 E−hopi = P
PE (E|g)fhopi (g)dg,
i = 1, 2
(7)
0
In the cases of the MPSK and the M-QAM, PE (E|g) can be √
expressed in the form of aQ cg , where a and c are constants determined by the modulation scheme and √
is Q(x) the Gaussian Q-function defined as Q(x) = 1/ 2p
1 exp(−t 2 /2)dt. Therefore for MPSK and M-QAM
1 x E−hopi = a Q modulations the ABER of each hop, P 0 √
( cg)fhopi (g)dg, can be calculated using [17] as E−hopi = √a
P 2p
1 0
2 2 t t exp − dt Fhopi c c
(8)
IET Commun., 2011, Vol. 5, Iss. 9, pp. 1199–1203 doi: 10.1049/iet-com.2010.0517
www.ietdl.org
c gRD c gSR 2 × 1− +a c gRD + 2 c gSR + 2
c gRD T(c gSR + 2) × 1− Q SR c gRD + 2 g
Assuming the i.i.d channels in the first hop SR2 = g SR ), CDF of gSR1 and gSR2 are expressed ( gSR1 = g as FgSR (g) = FgSR (g) = 1 − exp(− g/ gSR ). Therefore with 1 2 the help of (2), CDF of the first hop can be written as 1 − (e−T /gSR + e−g/gSR ), g,T (9) Fhop1 (g) = 1 + e−(T +g)/gSR − 2e−g/gSR , g ≥ T Also, CDF of the second hop is expressed as Fhop2 (g) = 1 − e−g/gRD . By substituting the CDFs in (8), we can calculate the ABER of each hop as follows E−hop1 = √a
P 2p × e−t
1
(1 − e 0 2
/2
a − √
2p
dt −
1
√
cT
ae−T /cgSR √
2p
−(T / gSR +t2 /c gSR )
+e
√
cT
e−t
2
/2
) (10)
dt
0
2 e−t (cgSR +2/2cgSR ) dt
and E−hop2 = √a
P 2p
1
(1 − e−t
2
/c gRD
)e−t
2
/2
dt
(11)
0
Integrals in (10) and (11) can be evaluated using [18, eq. 3.321.3] and after some manipulations the ABERs can be expressed as
a c gSR = (1 − eT /gSR ) 1 − c gSR + 2 2 √
+ ae−T /gSR Q cT
c gSR T (c gSR + 2) Q −a SR c gSR + 2 g
E−hop1 P
As we can see in (14), the ABER of the system depends on the switching threshold (T ) and by selecting an appropriate T, we can minimise the ABER of the proposed system. 3.3
−t 2 /c gSR
In this subsection, we find the optimum switching threshold in the minimum ABER sense for the MPSK modulation. It is clear from (12) and (13) that only ABER of the first hop E−hop1 ) depends on T. Therefore for minimising the (P ABER (6) we have to find the optimum T that minimises E−hop1 . The straightforward approach for finding the the P optimal T is to take the derivative of (14) with respect to T and set it to zero in order to solve the resultant equation. In the case of (14), however, this approach renders too difficult to be used. Thus, an alternative expression for the ABER of the first hop when MPSK modulation is used can be written as [14] E−hop1 = 1 P p
a c gSR T / g E = (1 − e SR ) 1 − P 2 c gSR + 2
√
a c gRD −T / gSR 1− Q cT + + ae c gRD + 2 2
c gSR T (c gSR + 2) Q −a SR c gSR + 2 g
√
c gRD − a2 e−T /gSR 1 − Q cT c gRD + 2
a2 c gSR T / gSR ) 1− − (1 − e 2 c gSR + 2 IET Commun., 2011, Vol. 5, Iss. 9, pp. 1199–1203 doi: 10.1049/iet-com.2010.0517
Mhop1
0
gPSK df sin2 f
(15)
1 where Mhop1 (s) = 0 fhop1 (g)e−sg dg is the moment generating function (MGF) of the fhop1 (g) and gPSK = sin2 (p/M ). Taking the derivative of (9) and using the definition of MGF, Mhop1 (s) can be expressed as 1
fg (g)e−sg dg
(16)
T
gSR )e−g/gSR and where Fg (g) = 1 − e−g/gSR , fg (g) = (1/ −1 Mg (s) = (1 − s gSR ) . Substituting (16) in (15) leads to
(13)
E−hop1 = 1 Fg (T ) P p
(M −1)p/M
1 + Fg (T ) p
Substituting (12) and (13) into (6) yields an end-to-end ABER expression for the SS-PRS which is given by
(M −1)p/M
Mhop1 (s) = Fg (T )Mg (s) +
E−hop2 P
Optimum switching threshold
(12)
and
a c gRD 1− = c gRD + 2 2
(14)
Mg
0
(M −1)p/M 1
gPSK df sin2 f fg (g)e
0
−(ggPSK / sin2 f)
dg df
T
(17) The optimum threshold value, Topt , is obtained by solving the following equation E−hop1 dP dT
=0
(18)
T =Topt
Substituting (17) in (18) and after simplification, we obtain (M −1)p/M 0
Mg
(M −1)p/M Topt gPSK − gPSK sin2 f df = 0 e df − 2 sin f 0 (19)
To the best of our knowledge, there is no closed-form solution for (19) in general case of MPSK modulation. But for the case 1201
& The Institution of Engineering and Technology 2011
www.ietdl.org of BPSK modulation (M ¼ 2), (19) reduces to
p/2 Mg 0
p/2 1 − Topt / sin2 f e df = 0 df − sin2 f 0
(20)
Using the same approach as the one used in [14], we obtain the optimum switching threshold in the minimum ABER sense for BPSK as
Topt
4
2 SR 1 g −1 1 Q 1− = SR + 1 g 2 2
(21)
Numerical results
In this section, we present the numerical results of the analytical outage probability and the ABER for the BPSK modulation and then a discussion of computational complexity of the proposed scheme is provided. In all simulations, channels are assumed to experience independent and identically distributed (i.i.d) SR2 = g RD = g ). We plotted the Rayleigh fading ( gSR1 = g outage and error rate performance curves against the average SNR of the channels ( g). Fig. 1 illustrates the outage probability of the SS-PRS scheme for different values of T and gth = 1. Note that we could plot the outage probability for different values of gth and analyse the effect of selecting different values of T. It is clearly observed that using the optimum relay switching threshold, in the sense of minimum outage probability (T = gth ), the outage probability of the system will be successfully minimised. It is clear again that, if the switching threshold is chosen as too high, the source is continuously switching between the two relays, which results in poor outage performance for the low SNRs. On the other hand, if the threshold level is chosen as too low, the source is almost locked to one of the relays and again poor performance is achieved for the medium and high SNRs. Fig. 2 compares the error rate performance of the SS-PRS scheme using the optimal switching threshold in the sense of minimum error rate (Topt ) with that of PRS scheme proposed in [5] when N ¼ 2. Note that these schemes have relatively similar performance especially at high average SNRs. We also plot the simulation result of the SS-PRS scheme which verifies the analytical results.
Fig. 1 Outage probability of SS-PRS scheme for different values of T and gth 1202 & The Institution of Engineering and Technology 2011
Fig. 2 ABER of the SS-PRS scheme using (Topt) compared with that of PRS scheme proposed in [5] when number of the relays is two (N ¼ 2)
Fig. 3 plots the ABER for the BPSK modulation against the switching threshold T for different average SNRs of the channels ( g). As expected, there exists an optimal switching threshold that minimises the ABER. These optimal switching thresholds could be calculated using (21) and as one can see in Fig. 3, they vary by average SNR. As mentioned in the introduction section, in the PRS scheme all channels in the first hop should be monitored and estimated in every transmission slot that entails a certain level of complexity. This complexity grows as the number of cooperating relays increases. For example, if the number of cooperating relays is N, in every transmission slot N channels should be estimated. In Comparison with with the previous scheme, in the proposed scheme only a single relay is active in each transmission slot. As a result, only one channel should be estimated in each transmission slot. It is clear that the proposed scheme has complexity N times less than the PRS scheme. It is worth mentioning, that two optimal relay switching thresholds have been found for the SS-PRS scheme: one for minimising the outage probability and the other for minimising the ABER. Owing to the performance demands of the system one of these optimal thresholds can be used.
Fig. 3 ABER of the SS-PRS against the switching threshold T for different average SNRs ( g¯ ) IET Commun., 2011, Vol. 5, Iss. 9, pp. 1199–1203 doi: 10.1049/iet-com.2010.0517
www.ietdl.org Also, the simplicity of the SS-PRS system stems from the fact that only one relay is active in each transmission time slot with no need for all relay to be active simultaneously. Besides, the active relay sends feedback to the source just in cases where relay switching occurs. As a result, the system’s power consumption is reduced and its complexity decreases. It should be also noted that the proposed scheme in this paper can be generalised for the cases where more than two relays are used which is hoped to be further studied by the next research.
5
Conclusion
The performance of the SS-PRS is analysed in terms of the outage probability and the ABER, then the respective closed-form expressions are derived. The computer simulations validate the analytical results and show that the performance of the SS-PRS scheme is approximately similar to the conventional PRS scheme, and meanwhile it does have the advantage of more simplicity.
6
Acknowledgment
This work is partially supported by Iran Telecommunication Research Center (ITRC) under Grant no. 500/8982.
7
References
1 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, pp. 3062–3080 2 Gharanjik, A., Mohamedpour, K.: ‘Performance study of cooperative diversity with selection combining’. Proc. 21st IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications (PIMRC’10), Istanbul, Turkey, September 2010, pp. 2121– 2124 3 Jing, Y., Jafarkhani, H.: ‘Single and multiple relay selection schemes and their achievable diversity orders’, IEEE Trans. Wirel. Commun., 2009, 8, pp. 1414–1423
IET Commun., 2011, Vol. 5, Iss. 9, pp. 1199–1203 doi: 10.1049/iet-com.2010.0517
4 Bletsas, A., Khisti, A., Reed, D.P.: ‘A simple cooperative diversity method based on network path selection’, IEEE J. Sel. Areas Commun., 2006, 24, (3), pp. 659 –672 5 Krikidis, I., Thompson, J., McLaughlin, S.: ‘Amplify-and-forward with partial relay selection’, IEEE Commun. Lett., 2008, 12, (4), pp. 235–237 6 Kim, J.B., Kim, D.: ‘Comparison of tightly power-constrained performances for opportunistic amplify-and-forward relaying with partial or full channel information’, IEEE Commun. Lett., 2009, 13, pp. 100–102 7 Sun, L., Zhang, T., Niu, H., Wang, J.: ‘Effect of multiple antenna at the destination on the diversity performance of amplify-and-forward system with partial relay selection’, IEEE Signal Process. Lett., 2010, 17, pp. 631–634 8 Suraweera, H.A., Soya, M., Tellambura, C., Garg, H.K.: ‘Performance analysis of partial relay selection with feedback delay’, IEEE Signal Process. Lett., 2010, 6, pp. 531– 534 9 Ding, H., Ge, J., Jiang, Z.: ‘Asymptotic performance analysis of amplify-and-fowrard with partial relay selection in Rician fading’, Electron. Lett., 2010, 46, pp. 248–250 10 Suraweera, H.A., Michalopoulos, D.S., Karagiannidis, G.K.: ‘Semiblind amplify-and-forward with partial relay selection’, Electron. Lett., 2009, 45, pp. 317–319 11 da Costa, D.B., Aissa, S.: ‘End-to-end performance of dual-hop semiblind relaying with partial relay selection’, IEEE Trans. Wirel. Commun., 2009, 8, pp. 4306– 4315 12 da Costa, D.B., Aissa, S.: ‘Capacity analysis of cooperative systems with relay selection in nakagami-m fading’, IEEE Commun. Lett., 2009, 13, pp. 637–639 13 Michalopoulos, D.S., Karagiannidis, G.K.: ‘Two-relay distributed switch and stay combining’, IEEE Trans. Wirel. Commun., 2008, 56, pp. 1790– 1794 14 Simon, M.K., Alouini, M.S.: ‘Digital communication over fading channels’ (Wiley, New York, 2005, 2nd edn.) 15 Beaulieu, N.C., Hu, J.: ‘A closed-form expression for the outage probability of decode-and-forward relaying in dissimilar rayleigh fading channels’, IEEE Commun. Lett., 2006, 10, pp. 813–815 16 Hasna, M.O., Alouini, M.S.: ‘End-to-end performance of transmission systems with relays over rayleigh fading channels’, IEEE Trans. Wirel. Commun., 2003, 2, pp. 1226– 1131 17 Zhao, Y., Adve, R., Lim, T.J.: ‘Symbol error rate of selection amplifyand-forward relay systems’, IEEE Commun. Lett., 2006, 10, pp. 757–759 18 Gradshteyn, I.S., Ryzhik, I.M.: ‘Table of integrals, series, and products’ (Acdemic Press, San Diego, 2000, 6th edn.)
1203
& The Institution of Engineering and Technology 2011
