IEICE TRANS. COMMUN., VOL.E92–B, NO.4 APRIL 2009

1369

LETTER

Symbol Error Rate Expression for Decode-and-Forward Relaying Using Generalized Selection Combining over Rayleigh Fading Channels Bao Quoc VO-NGUYEN†a) , Nonmember and Hyung Yun KONG† , Member

SUMMARY Cooperative transmission is an efficient approach to improve the performance of wireless communications over fading channels without the need for physical co-located antenna arrays. In this paper, we propose a novel cooperative protocol with selective decode-and-forward relays and generalized selection combining (GSC) technique at destination. The advantage of this scheme is that it not only allows us to optimize the structure of destination but also to fully exploit the diversity offered by the channels with an appropriate number of chosen strongest paths. For an arbitrary number of relays, an exact and closed-form expression of the Symbol Error Rate (SER) is derived for M-ary PSK in independent but not identically distributed Rayleigh fading channels. Various simulations are performed and their results exactly match the results of analyses. key words: generalized selection combining (GSC), Symbol Error Rate (SER), decode-and-forward (DF)

1.

Introduction

Diversity is an effective technique used in wireless communication systems to combat the performance degradation caused by fading. Recently, communication systems in which spatial diversity is achieved by multiple communication nodes collaborating together to form a virtual antenna array have been proposed [1]. Furthermore, [2] presents several viable protocols to achieve the benefits of these distributed spatial diversity systems. In this letter, we focus on the decode-and-forward (DF) relaying protocol [2]. Diversity gain is obtained by combining the relayed signals as well as the signal from the source using a variety of diversity combining technique [3], [4]. A crucial issue in these diversity systems is how to combine the available diversity paths in the destination in order to achieve optimum performance within acceptable complexity. There are three conventional combining techniques: selective combining (SC) which selects the signal from that diversity path with the largest instantaneous SNR; equal-gain combining (EGC) which coherently combines all available paths weighting each with equal gain; and maximal-ratio combining (MRC) which also coherently combines all available paths but weighs each with the respective gain of the path. Among them, SC gives the most inferior performance, MRC gives the best and the optimum performance, and EGC has a performance quality in between the others. SC and MRC are the two exManuscript received December 10, 2007. Manuscript revised July 11, 2008. † The authors are with University of Ulsan, Korea. a) E-mail: [email protected] DOI: 10.1587/transcom.E92.B.1369

tremes of complexity quality tradeoff. The SC receiver is much simpler than the alternatives. However, it can not exploit all diversity gain offered by the channels because the contribution from the other paths are wasted, irrespective of their strength. MRC combines all the outcome of all paths resulting in the best possible combining performance gain. The cost for this performance is the heavy processing complexity and extremely complicated circuitry required for phase coherence and amplitude estimation on each path. It is known as the optimal in the BER performance sense. However, when both the performance and complexity should be considered, as are the cases in mobile system or wireless sensor networks, then a scheme that satisfies good balance point between performance and complexity is preferred. In addition, wireless agents (e.g. mobile units or wireless sensors) using high-order receiver diversity can not be installed MRC because of power or processing limitation. In addressing this problem, [5] proposed a suboptimal scheme that retains most of the advantages of the MRC scheme and has been widely studied [5], [6], [8]. It is called generalized selection combining (GSC), which adaptively combines the strongest (on the basis of instantaneous SNR) among available ones and then combines them according to the maximal-ratio combining rule [6]. It means that the destination chooses only strongest paths it received from source and relays to combine using MRC technique. The advantage of this technique is that it not only allows us to optimize the structure of destination but also to fully exploit the diversity offered by the channels. In addition, this scheme provides the best tradeoff between the receiver complexity and performance of the system. In this paper, we also derived an exact closed-form for the average Symbol Error Rate expression of selective DF using GSC at destination in independent and not identically distributed Rayleigh fading channels with an arbitrary number of relays. The rest of this paper is organized as follows. In Sect. 2, a system model is described. In Sect. 3, the performance of the proposed protocol is evaluated in terms of end-to-end Symbol Error Rate. In Sect. 4, the results obtained from numerical analysis and Monte-Carlo simulation are compared. Finally, conclusions are drawn in Sect. 5. 2.

System Model

We consider the wireless network illustrated in Fig. 1. It

c 2009 The Institute of Electronics, Information and Communication Engineers Copyright 

IEICE TRANS. COMMUN., VOL.E92–B, NO.4 APRIL 2009

1370

Fig. 1

Selective decode-and-forward relaying model with N relays.

By using the theorem on total probability [7], the average SER for the system can be derived as a weighted sum of the average SER for the GSC corresponding to set C D . In order to calculate P¯ S , we have to consider all possible cases at relays:  ¯ PR (C D )P¯ D (LC , C D ) P¯ S = |C D |=0  ¯ + PR (C D )P¯ D (LC , C D ) + · · · |C D |=1 P¯ R (C D )P¯ D (LC , C D ) + · · · + (1) |C D |=r  ¯ PR (C D )P¯ D (LC , C D ) + |C D |=N−1  ¯ + PR (C D )P¯ D (LC , C D ) |C D |=N

is assumed that every channel between the nodes experiences slow, flat, Rayleigh fading. Due to Rayleigh fading,  2 the channel powers, denoted by α0 = |hS D |2 , α1,i = hS Ri   2 and α2,i = hRi D  where i = 1, . . . , N are independent and exponential random variables whose means are λ0 , λ1,i and λ2,i , respectively. The average transmit signal-to-noise ratio (SNRs) for the source and the relays are denoted by ρS and ρRi with i = 1, · · · , N. A time-division channel allocation scheme with N + 1 time slots is occupied in order to realize orthogonal channelization. In the first time slot, the source broadcasts its data to destination and N relays. At the end of the first time, relays will demodulate and check whether their received data are right or wrong. We define a decoding set C D , whose members are relays which decode successfully. So it is obvious that C D is a subset of C = {R1 , R2 , . . . , RN }. During the following N time slots, members of the decoding set C D forward the source information to the destination in their respective time slots. And to simplify notation, we define a new set C D , which represents all nodes that transmit or relay the source information to the destination. That is, C D = {S } ∪ C D . A maximum-likehood receiver with GSC installed in the destination will coherently combine the LC diversity paths with the largest instantaneous SNRs by weighting them by the complex conjugate of their respective fading gains. It is assumed that the receivers at the destination and relays have perfect channel state information but no transmitter channel state information is available at the source and relays. 3.

SER Analysis

In this section, we use the mathematical probability model to derive an end-to-end closed-form SER of the proposed protocol. In this paper, we just consider the general case of independent and not identically distributed channels with M-ary PSK modulation. With GSC, the number of diversity paths to be combined in the destination depends not only on number of relays of set C D but also on LC . The value of LC is almost predetermined because the complexity of the system is fixed.

where |C D | denotes the cardinality of C D . P¯ R (C D ) is the probability of each decoding set C D . P¯ R (C D ) can be expressed as follows: P¯ R (C D ) =

N 

S¯ i(1−ai )

i=1

N   ai 1 − S¯ i

(2)

i=1

where ai = 1 represents the status that the i-th relay node belongs to the decoding set C D and vice versa, that is:  1, Ri ∈ C D ai = , i = 1, 2, · · · , N (3) 0, Ri  C D S¯ i denotes the average symbol error rate at the i-th relay and can be obtained by applying [8, (8.113) on p.256]:

  gPS K λ1,i ρS M S¯ i = M−1 1 − M 1+gPS K λ1,i ρS (M−1)π  (4)    gPS K λ1,i ρS π × π2 + tan−1 1+gPS K λ1,i ρS cot M i

where gPS K = sin2 (π/M).   For each r, there are Nr possible subsets of size r. By applying [9, (3.1.6) on p.11], we totally have 2N of subsets of all size r from 0 to N. So we can rewrite P¯ S as follows: ⎧⎡ N ⎫ ⎤ N  B ⎪⎢ ⎪   aki⎥⎥ ⎪ ⎪   ⎢ (1−aki ) ⎨ ⎬ ⎥ ⎢  ⎢⎢⎣ S¯ P¯ S = 1 − S¯ i ⎥⎥⎦ P¯ D LC , C D ⎪ (5) ⎪ ⎪ i ⎪ ⎩ ⎭ k=0

i=1

i=1

where B = 2N − 1 and [ ak1

ak2

· · · akN ] = de2bi(k)

(6)

where de2bi(.) is a function which converts decimal numbers to binary numbers with N column output [10]. P¯ D (LC , C D ) denotes average SER for the combined signals obtained by using GSC technique after the destination receives M-ary PSK modulated signals from the set C D where LC is the number of strongest paths which are chosen. In most cases, the destination has a fixed processing complexity, so the number of largest SNR paths (LC ) is predetermined. In addition, the number of diversity paths to be combined at destination depends on number of relays belonging to set C D . In case it is less than LC , the number of the actual paths to be combined at destination is just |C D |.

LETTER

1371

  So letting Lc = min(C D  , LC ), L = |C D | and applying [8, (9.485) on p.496], we obtain:  ! L L γ¯ i−1  l l −1 P¯ D (LC , C D ) = γ¯ i1 ,i2 ,...,iL =1 l=1

i1 i ⎛ ⎞2 ···iL ⎜⎜⎜ L ⎟⎟  L  ⎜⎜ ⎟⎟⎟ M−1  cl ⎟ ⎜ × ⎜⎜ ⎟⎠ M c −c l k⎟ l=1 ⎝ k=1 kl ⎧   cl ⎪ M ⎪ 1 − 1+c ⎪ ⎪ (M−1)π l ⎨  ×⎪ ⎪ ⎪ ⎪ ⎩ × π + tan−1 2

k=1

ik

(7)

cl 1+cl

⎫ ⎪ ⎪ ⎪ ⎪

 ⎬ ⎪ ⎪ ⎪ ⎭ cot Mπ ⎪

where γ¯ j denotes the average SNR of each path received by the destination from the set C D with γ¯ 1 = λ0 ρS and γ¯ j+1 = γ¯ i if Ri ∈ C D

(8)

where γ¯ i = λ2,i ρRi with j = 1, 2, · · · , L − 1; i = 1, 2, · · · , N and: ⎞−1 ⎛ l ⎟⎟⎟ ⎜⎜  ⎜ −1 cl = gPS K min(l, Lc ) ⎜⎜⎝⎜ γ¯ ik ⎟⎟⎟⎠ (9)

SER of the system of the case (0 → 2) is better than the other because GSC technique at the destination only combines the strongest diversity paths and diminishes severely faded paths which harm on system performance. Figure 3 illustrates the average SER of the proposed with 6 relays when we change value of LC . The cases of SC and MRC correspond to LC = 1 and LC = 7, respectively. Under the assumed channel conditions, the MRC case outperforms the SC case by around 2 dB at SER 10−2 with QPSK modulation. It is easy to recognize that there are not much difference between the performance of the system when LC = 4 and LC = 7. As expected, we can exploit fully diversity order offered by channels without using MRC at the destination. In Fig. 4, we study the effect of number of relays on the performance when we fix the number of strongest paths we choose (LC ). By increasing the number of relays from 3 to 6 and always choosing 4 strongest paths to combine at the destination, the performance of the protocol is nearly

k=1

4.

Numerical Results and Discussion

In this section, performance evaluation of the proposed protocol for the case of independent but not identically distributed Rayleigh fading channels is conducted. Results computed using our theoretical analysis and Monte Carlo simulations are compared. For ease of analysis, it is assumed that the average transmit SNRs for all transmit nodes are equal, and λ0 , λ1,i and λ2,i are uniformly distributed between 0 and 1 (Figs. 3, 4). In Fig. 2, we study the impact of varying parameters λ0 , λ1,i and λ2,i on the system performance by letting them be uniformly distributed between 0 and 1 (indicated as 0 → 1) and between 0 and 2 (indicated as 0 → 2). As expected,

Fig. 2

SER of the selective DF relaying for QPSK with N = 6.

Fig. 3

Fig. 4

SER of the selective DF relaying for QPSK & 8-PSK with N = 6.

BER of the selective DF relaying for BPSK with N = 3, 4, 5, 6.

IEICE TRANS. COMMUN., VOL.E92–B, NO.4 APRIL 2009

1372

same when SNRs are lower than 12 dB. With higher SNRs, it does not strongly affect on the number of relays. It is obvious that we can get the same performance compared with MRC technique without using all forward channels from N relays. In addition, the results obtained by using the exact closed-form expression derived in this paper and by simulation results are in excellent agreement. 5.

Conclusion

The performance of DF relaying with GSC in independent but not identically distributed Rayleigh fading channels was examined. The advantage of this protocol is that it not only allows us to optimize the structure of the destination but also fully exploits diversity order offered by the channels. Compared to the destination-sited MRC, the destination installed GSC can achieve nearly same performance and can save power resources. An exact closed-form SER expression for selective decode-and-forward cooperative communication protocol with arbitrary number of relays was established. Its validity was demonstrated by a variety of MonteCarlo simulations. One of advantages of the SER expression provided in this paper is that it can support MRC and SC systems simply by setting LC = 1 and LC = N + 1, respectively. Acknowledgments This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea

government (MOST) (No. R01-2007-000-20400-0). References [1] A. Nosratinia, A. Hedayat, and T.E. Hunter, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol.42, no.10, pp.74–80, Oct. 2004. [2] 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, Dec. 2004. [3] I.-H. Lee and D. Kim, “BER analysis for decode-and-forward relaying in dissimilar Rayleigh fading channels,” IEEE Commun. Lett., vol.11, no.1, pp.52–54, Jan. 2007. [4] N.C. Beaulieu and J. Hu, “A closed-form expression for the outage probability of decode-and-forward relaying in dissimilar Rayleigh fading channels,” IEEE Commun. Lett., vol.10, no.12, pp.813–815, Dec. 2006. [5] M.-S. Alouini and M.K. Simon, “An MGF-based performance analysis of generalized selection combining over Rayleigh fading channels,” IEEE Trans. Commun., vol.48, no.3, pp.401–415, March 2000. [6] M.K. Simon and M.-S. Alouini, “A compact performance analysis of generalized selection combining with independent but nonidentically distributed Rayleigh fading paths,” IEEE Trans. Commun., vol.50, no.9, pp.1409–1412, Nov. 2002. [7] A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, Fourth ed., McGraw-Hill, 2002. [8] M.K. Simon, Digital Communication over Fading channels, Second ed., John Wiley & Sons, Hoboken, New Jersey, 2005. [9] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., Dover Press, New York, NY, 1972. [10] www.mathworks.com/access/helpdesk/help/toolbox/comm/ug/de2bi. html