IEICE TRANS. COMMUN., VOL.E91–B, NO.11 NOVEMBER 2008

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BER Performance of Decode-and-Forward Relaying Using Equal-Gain Combining over Rayleigh Fading Channels Bao Quoc VO NGUYEN†a) , Nonmember and Hyung Yun KONG† , Member

SUMMARY This paper provides a closed form expression for calculating the bit error rate of the decode-and-forward relay protocol that uses equal-gain combining (EGC) at the destination with an arbitrary number of relays. We have shown that EGC technique for decode-and-forward relay scheme offers remarkable diversity advantage over direct transmission. In addition, we also study the impact of combining techniques on the performance of the system by comparing a system that uses EGC to one that uses maximum ratio combining (MRC) & selection combining (SC). Simulations are performed to confirm our theoretical analysis. key words: cooperative communication, decode-and-forward, equal-gain combining

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]. Various protocols have been proposed to achieve the benefit offered by cooperative communication such as: amplify-and-forward, decodeand-forward, coded cooperation, etc. . .. This paper focuses on regenerative relaying (also called hybrid decode-andforward [1] or selection relaying [2]). It is one of the simple cooperative communications protocols where the relay must make an independent decision on whether or not to decode and forward the source information [3]–[6]. Therefore, it avoids the noise enhancement in fixed amplify-and-forward relaying and remedies the decoding error retransmission in fixed decode-and-forward relaying [2] (both drawbacks induced by the relay). At the destination, the receiver can employ a variety of diversity combining techniques to obtain diversity from the multiple signal replicas available from the relays and the source such as: MRC, SC and EGC. Among them, MRC has the highest complexity of all combining techniques since it requires knowledge of fading amplitude in each signal branch and gives the best performance compared with the others [7]. Although optimum performance is highly desirable, practical systems often sacrifice some performance in order to reduce their complexity. Instead of using Maximal Ratio Combining which requires exact knowledge of the channel state information, EGC is often used in practice because of its reduced complexity relative Manuscript received April 10, 2008. Manuscript revised July 16, 2008. † The authors are with University of Ulsan, Korea. a) E-mail: [email protected] DOI: 10.1093/ietcom/e91–b.11.3760

to the optimum MRC scheme. Indeed EGC equally weights each branch before combining, and therefore does not require estimation of the branch fading amplitudes. Thus, another benefit of using EGC as opposed to MRC is reduced power consumption at the receiver. In the past, relatively few contributions concerning evaluating performance of the DF relaying protocol with multi relays have been published [3]–[6]. Some previous analyses always assumed that the channels between the source, relay(s) and destination are independent and identically distributed (i.i.d.) Rayleigh. However, in real scenarios, the condition of i.i.d. between channels is hard to obtain and considering independent but not identically distributed (i.n.d.) channels is more generalized and appropriate. Under this condition, closed form expressions for outage probability and bit error rate of DF relaying systems that use MRC and SC at the destination are presented in [4]–[6]. In this paper, we present a closedform expression for BER of the DF relaying system with an arbitrary number of relays that uses EGC at the destination for both cases of i.n.d. and i.i.d. channels. These derivations were done for the system with BPSK modulation. In addition, we also study the impact of combining techniques on the performance of the system by comparing a system that uses EGC to one that uses MRC & SC. The rest of this paper is organized as follows. In Sect. 2, we introduce the model under study. Section 3 shows the formulas allowing for evaluation of the average BER. Section 4, we contrast the simulations and the results yielded by theory. Finally, the paper is closed in Sect. 5. 2.

System Model

We consider the wireless network illustrated in Fig. 1. It 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. To guarantee orthogonal transmissions, we consider a Time Division Multiple Access (TDMA) arrangement with N + 1 time slots. However, the basic idea and operation of our proposed protocol does not depend on the specifics of the channel access protocol. In the first time slot, the

c 2008 The Institute of Electronics, Information and Communication Engineers Copyright 

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+

i1 ,i2 ,···,in =1 i1
+··· +

Fig. 1

Decode-and-forward relaying model with N relays.

source broadcasts its data to the destination and N relays. At the end of the first time slot, 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. In real scenario, the decoding set is determined after receiving one frame from the source by employing cyclic-redundancy-check (CRC). However, in this paper, we assumed that the decoding set can be decided by symbol by symbol for mathematical tractability of BER calculation [3], [4]. 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. It is assumed that the receivers at the relays and destination have exact channel state information so that equal gain combiner can be employed; however, no transmitter channel state information is available at transmitters of the source or relays. 3.

Performance Analysis

We first consider the general case of independent and not identically distributed channels and then provide a compact solution for the case when the channels are assumed to be independent and identically distributed. Using the theorem on total probability, the average BER at the destination can be derived as a weighed sum of the BER for EGC at the destination corresponding to set of decoding relays [4]. Because C D is a random set, the number of relays in the decoding set C D is  a random variable n, n = 0, 1, . . . , N. For each n, there are Nn possible subsets of size n. Thus, the average BER at the destination can be written as Pb = Pr(C D = {∅})BD (C D = {∅}) N      + Pr C D = {Ri1 } BD C D = {Ri1 } i1 =1    N  Pr CD = {Ri1 , Ri2 }  + i1 ,i2 =1 ×BD C D = {Ri1 , Ri2 }

   Pr CD = {Ri1 , Ri2 , . . . , Rin }  ×BD C D = {Ri1 , Ri2 , . . . , Rin }

N 

N 

i1 ,i2 ,···,iN =1 i1
   Pr CD = {Ri1 , Ri2 , . . . , RiN }  ×BD C D = {Ri1 , Ri2 , . . . , RiN }

(1)

  where Pr C D = {Ri1 , Ri2 , . . . , Rin } denotes the probability for decoding set C D whose cardinality equals to n, BD C D = {Ri1 , Ri2 , . . . , Rin } denotes average BER for the combined signal obtained by using EGC after the destination received forwarded signals from the decoding set C D as well as from the source (S). The probability for decoding set C D can be obtained by: ⎤⎡ ⎤ ⎡ ⎢⎢⎢  ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎥ ⎢ ⎢ ¯ ¯ Pr(C D ) = ⎢⎢⎣ 1 − S i ⎥⎥⎦ ⎢⎢⎣ S i ⎥⎥⎥⎦ (2) Ri ∈C D

Ri C D

where S¯ i denotes the average symbol error rate of BPSK modulated symbols transmitted from the source to the i-th relay. The average symbol error rate S¯ i for BPSK in slow and flat Rayleigh fading channels can be derived by averaging the symbol error rate for the AWGN channel over the pdf of the SNR in Rayleigh fading of each path [7, p.124, Eqs. (5.1)–(5.6)].    1  sin2 θ dθ S¯ i = E Q 2ρS α1,i = 2 π sin θ + ρS λ1,i 0  ⎛ ⎞ ⎜ ⎟ ρS λ1,i ⎟⎟⎟ 1 ⎜⎜ ⎟⎟ = ⎜⎜⎜⎝1 − 2 1 + ρS λ1,i ⎠ π/2

where Q(x) =

1 π

π/2  0

(3)

  x2 exp − 2 sin dθ is defined in [7, p.85, 2 θ

Eq. (4.2)]. With EGC at the destination, the signals forwarded from the decoding set C D as well as from the source (S) are cophased, summed, and coherently demodulated [7], [8]. To simply notation, we define a new set C D , which represents all nodes that transmit or relay the source   information to the destination, i.e., C D = {S } ∪ C D and C D  = n + 1. Let us define γ1 , γ2 , . . . , γn+1 as the instantaneous SNR of each path received by the destination from the set C D with their expected values γ¯ 1 = λ0 ρS , γ¯ 2 = λ2,i1 ρRi1 , . . . ,¯γk = λ2,ik−1 ρRik−1 , . . . , γ¯ n+1 = λ2,in ρRin , respectively. To obtain the average BER of BPSK with EGC on i.n.d. Rayleigh fading channels, we proceed analogous to [8, Eq. (11)] and use Hermite integration: ∞  1 1  BD C D = {Ri1 , Ri2 , . . . , Rin } = − G(z) exp(−z2 )dz 2 π 0

1 1 − wmG(zm ) (4) 2 π m=1 M

≈

i1
+···

where

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  ⎧ ⎡ ⎪ ⎢⎢⎢ 1 F1 − 21 ; 12 ; xm (k) ⎪ ⎪ n+1 ⎪  ⎪ ⎨ ⎢⎢⎢⎢ πγ¯ k G(zm ) = Im ⎪ ⎢⎢⎢ + jzm ⎪ n+1 ⎪  ⎢ ⎪ ⎪ n+1+ γ¯ l ⎩ k=1 ⎣ l=1

γ¯ k z2m n+1 

xm (k) =

n+1+

l=1

⎤⎫ ⎥⎥⎥⎪ ⎪ ⎪ ⎥⎥⎥⎪ ⎪ −1 ⎥⎥⎥⎬ .zm ⎪ ⎥⎥⎦⎪ ⎪ ⎪ ⎪ ⎭

(5)

(6) γ¯ l

and zm & wm can be obtained in [9, p.924]; 1 F1 (a, b, z) denotes the confluent hyper-geometric function [9, ch.13]. M is set to 10 to ensure good accuracy with N ≤ 5. Note that the integral is defined over the positive half axis, therefore, only abscissas of positive values are used in the calculation [8]. For the case of i.i.d., the BER of the system is obtained by simplifying (1) which can be expressed under binomial distribution. Letting λ0 = λ1,i = λ2,i = λ, ρS = ρRi = ρ for i = 1, . . . , N, hence γ¯ k = γ¯ for k = 1, . . . , n + 1, it is straightforward to arrive at  n  N−n (7) Pr(C D ) = 1 − S¯ S¯

Fig. 2 N.

BER for DF with EGC at the destination for difference values of

where S¯ i = S¯ for i = 1, . . . , N. In addition, G(zm ) can be reduced to ⎧⎡ % &⎤n+1 ⎫ ⎪ ¯ 2m ⎪ ⎪ ⎥⎥ ⎪ ⎢⎢⎢1 F1 − 1 ; 1 ; γz ⎪ ⎪ ⎪ 2 2 (n+1)(1+γ) ¯ ⎥ ⎬ −1 ⎨⎢⎢ ⎥⎥⎥ ⎪ ' ⎢ G(zm ) = Im ⎪ (8) .zm ⎪ ⎥ ⎢ ⎪ ⎪ ⎥ ⎢ π γ ¯ ⎪ ⎪ ⎦ ⎣ ⎪ ⎪ + jz ⎭ ⎩ m (n+1)(1+γ) ¯

Substituting (4), (7) & (8) into (1), we can obtain the end-to-end average bit error rate for DF relaying system for an arbitrary number of relays with EGC at the destination over i.i.d. Rayleigh channels as ⎡( ) ⎤ ⎢⎢⎢ N 1 − S¯ n S¯ N−n ⎥⎥⎥ N  ⎢⎢⎢ n ⎥⎥ ⎢⎢⎢  ⎥⎥⎥⎥ Pb = (9) M ⎢⎢⎢ 1 1  ⎥⎥⎥ ⎦ n=0 ⎣ × − w G(z ) m m 2 π m=1

4.

Numerical Results and Discussion

Fig. 3

Using the analysis presented in Sect. 3, various number of performance evaluation will be presented here and will be compared with simulation results. For fair of comparison, we assumed that the total average transmit signal-to-noise ratio is fixed as: ρS +

N 

ρRi = ρDT

(10)

i=1

where ρDT is the average transmit signal to noise ratio of S in case of direct transmission. For simplicity, it is assumed that the average transmit signal to noise ratio (SNRs) for all transmit nodes (the source and the relays) are equal. That is: ρS = ρRi =

ρDT = ρ, i = 1, 2, . . . , N N+1

(11)

BER of DF relaying with EGC over i.i.d. and i.n.d. channels.

In addition, for ease of analysis, it is assumed that λ0 , λ1,i and λ2,i with i = 1, . . . , N in Fig. 2 & Fig. 4 are uniformly distributed between 0 and 1. Figure 2 shows the average BER for the DF relaying system with EGC at the destination for different number of cooperative nodes as well as for direct transmission schemes. As shown in the figure, the improvement of the average BER will be proportional to the number of relays. In addition, the proposed protocol always outperforms the direct transmissions schemes unless the SNR is less than 4 dB. In Fig. 3, the BER of DF relaying with 1, 2 and 3 relays in both i.i.d. and i.n.d. channels were examined. The results are based on the assumption that the total channel power for i.i.d. channels is equal to that for i.n.d. channels, i.e. λ0 =

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in excellent agreement with the derived results. A closedform BER expression for BPSK modulation was also established to facilitate in evaluating the performance without time-consuming computer simulations. In addition, it is straightforward to obtain average BER of the DF relaying protocol using EGC for coherent BFSK by simply replacing all λ0 , λ1,i and λ2,i in the above expressions with λ0 /2, λ1,i /2 and λ2,i /2. 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

Fig. 4 Comparing DF relaying with EGC, MRC and SC at the destination.

λ1,1 = λ1,2 = λ1,3 = λ2,1 = λ2,2 = λ2,3 = 1 for i.i.d. channels and λ0 = 1, λ1,1 = 0.9, λ1,2 = 1.5, λ1,3 = 0.6, λ2,1 = 1.1, λ2,2 = 0.5, λ2,3 = 1.4 for i.n.d. channels. It is seen that our analytical results and the simulation results are in excellent agreement. In Fig. 4, the performances of the DF relaying systems with difference diversity combining techniques at the destination are illustrated. Among them, SC gives the worst performance, MRC [4] gives the best performance and EGC has a performance quality in between the others. In addition, BER curves confirm that, under same channel conditions, the performance of a system employing MRC receiver is always better as compared to an equivalent system using EGC by around 1 dB. 5.

Conclusion

The performance of DF relaying protocol using EGC technique at the destination under both i.i.d. and i.n.d. Rayleigh channels was examined in this paper. Simulation results are

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