2008 International Conference on Advanced Technologies for Communications

BER Performance of Decode-and-Forward Relaying Using Selection Combining over Rayleigh Fading Channels Bao Vo Nguyen Quoc∗ , Hyung Yun Kong∗ , Chien Hoang Dinh† , Thuong Le Tien† ∗ School

of Electrical Engineering University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749 Email: {baovnq,hkong}mail.ulsan.ac.kr † Telecommunications Department, Hochiminh University of Technology, Vietnam Email: {hdchien,thuongle}@hcmut.edu.vn

Abstract—This paper offers performance analysis of decodeand-forward protocol employing selection combining (SC) at the destination. For an arbitrary number of relays, BER for M PSK of the system is investigated in both independent identically distributed (i.i.d) and independent but not identically distributed (i.n.d.) Rayleigh fading channels. A variety of simulations are performed and show that they match exactly with analytic ones. In addition, our results also show that the optimum number of relays depend not only on channel conditions (operating SNR) but also on modulation scheme which we exploit.

I. I NTRODUCTION Signal fading is a serious problem in wireless communications and spatial diversity owing to deploying multiple antennas at both transmitter and receiver is an efficient solution to mitigate this effect [1]. However, when wireless users may not be able to support multiple antennas due to size and power limitations or other constraints, this diversity technique is not exploited. To overcome such a restriction, cooperative communications was mentioned to allow single-antenna users to gain some benefits of the spatial diversity [2]. The philosophy is that a relay can assist information transmission of a source to a destination. Therefore, the destination will receive transmitted information more reliably since from statistical viewpoint, the probability that all channels to the destination are deeply faded is significantly reduced. Various protocols have been proposed to achieve the benefit offered by cooperative communication such as: amplify-andforward, decode-and-forward, coded cooperation. This paper focuses on regenerative relaying (or hybrid decode-andforward [2] or selection relaying [3]). 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. Therefore, it avoids the noise enhancement in fixed amplify-and-forward relaying and remedies the decoding error retransmission in fixed decodeand-forward relaying [3] (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

978-1-4244-2681-2/08/$25.00 ©2008 IEEE

source. Although optimum performance is highly desirable, practical systems often sacrifice some performance in order to reduce their complexity. Instead of using MRC which requires exact knowledge of the channel state information, a system may use SC which is the simplest combining method. It only selects the best signal out of all replicas for further processing and neglects all the remaining. This reduces the computational costs and may even lead to a better performance than MRC, because channels with very low SNR can not accurately estimated and contribute much noise [4]. In addition, another benefit of using SC as opposed to MRC is reduced hardware complexity at the receiver. It is appropriate for sensor networks which require fixed processing complexity at each node and reduce more cost in implementation. In the past, relatively few contributions concerning evaluating performance of the DF relaying protocols with multi relays have been published [5][9]. The performance is often evaluated by outage probability and bit error rate. Some previous analyses always assumed that the channels between the source, relays and destination are 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 channels is more generalized and appropriate. Under this condition, a closed form expression for outage probability and bit error rate of DF relaying systems that exploited MRC at the destination are presented in [5]- [8]. In [9], outage probability for the relaying system that use SC at the destination is provided also for both i.n.d. and i.i.d. channels. In this paper, we present exact and closed-form expressions for BER of the DF relaying protocol with an arbitrary number of relays that uses SC at the destination for both cases of i.n.d. and i.i.d. channels. These derivations were done for the system with M -PSK modulation. In addition, we also study the impact of combining techniques on the performance of the system by comparing a system that uses SC to one that uses MRC. II. S YSTEM M ODEL We consider the wireless network illustrated in Fig. 1. It is assumed that every channel between the nodes experiences

301

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  n = 0, 1, · · · , N . For each n, there are N n possible subsets of size n. Thus, the average BER at the destination can be written as ¯D (CD = {∅}) P¯b = Pr (CD = {∅}) B N  ¯D (CD = {Ri }) Pr (CD = {Ri1 }) B + 1 i1 =1   N Pr (CD = {Ri1 , Ri2 })  + ¯D (CD = {Ri , Ri }) ×B 1 2 i1 ,i2 =1

RN

hRN D

hSRN Ri

hSRi

hSR1

hRi D R1

hR1D

hSD S

S: Source ; Ri: i-th Relay ; D: Destination

D

i1
+··· Fig. 1.

+

Selective Decode-and-Forward Relaying model with N relays



N  i1 ,i2 ,··· ,in =1 i1
+··· slow, flat, Rayleigh fading. Due to Rayleigh fading, the 2 2 channel powers, denoted by α0 = |hSD | , α1,i = |hSRi | 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 . For medium access, frequency division multiple access (FDMA), code division multiple access (CDMA) and time division multiple access (TDMA) can be used. However, for convenience, a time-division channel allocation scheme with (N + 1) time slots is occupied in order to realize orthogonal channelization, thus no inter-relay interference is considered in the signal model. In the first time slot, the source broadcasts its data to destination and N relays. At the end of first time slot, relays will demodulate and check whether their received data are right or wrong. We define a decoding set CD , 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 [5]. It is obvious that CD is a subset of C = {R1 , · · · , Ri , · · · , RN }. During the following N time slots, members of the decoding set CD forward the source information to the destination in their respective time slots. 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. III. P ERFORMANCE A NALYSIS We first consider the general case of independent and not identically distributed (i.n.d.) channels and then provide a compact solution for the case when the channels are assumed to be independent and identically distributed (i.i.d.). In this paper, we assumed that the bit-symbol mappings follow a Gray code. Using the theorem on total probability, the average BER at the destination can be derived as a weighed sum of the BER for SC at the destination corresponding to each set of decoding relays CD . Because CD is a random set, the number of relays in the decoding set CD is a random variable n, i.e. |CD | = n,

+



N  i1 ,i2 ,··· ,iN =1 i1
Pr (CD = {Ri1 , Ri2 , . . . , Rin }) ¯D (CD = {Ri , Ri , . . . , Ri }) ×B 1 2 n Pr (CD = {Ri1 , Ri2 , . . . , RiN }) ¯D (CD = {Ri , Ri , . . . , Ri }) ×B 1 2 N



(1)



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

Ri ∈C / D

where S¯i denotes the average symbol error rate of MPSKmodulated symbols transmitted from the source to the ith relay. For the case of coherently detected M -PSK, to evaluate the average SER from the source to the ith relay on Rayleigh fading, we merely replace γS with log2 (M )ρS α1,i in [10, eq. (5)] and use MGF-based approach, namely 

gM P SK 1 (M −1)π/M dθ (3) M γi − S¯i = π 0 sin2 θ where g = sin2 (π/M ) and Mγi (s) = 1/(1 − s log2 (M )ρS λ1,i ) for Rayleigh fading [11]. Finally, calculating (3) gives the desired result [11, eq. (5A.14)]:  (M −1)π/M sin2 θ dθ S¯i = π1 0 g log2 (M )ρS λ1,i +sin2 θ    ⎡ ⎤ g log (M )ρS λ1,n ×  M −1  1 − 1+g log2 2 (M )ρS λ1,n (MM −1)π   ⎦ = M ⎣  g log2 (M )ρS λ1,n π π −1 2 + tan 1+g log (M )ρS λ1,n cot M 2

With SC at the destination, the signal with the largest received SNR is always selected. To simply notation, we define  , which represents all nodes that transmit or relay a new set CD  = {S}∪CD the source information to the destination, i.e., CD  and |CD | = n + 1. Let us define γ1 , γ2 , . . . , γn+1 denotes the instantaneous SNR of each path received by the destination  with their expected values γ¯1 = λ0 ρS , from the set CD γ¯2 = λ2,i1 ρRi1 , . . . , γ¯n+1 = λ2,in ρRin , respectively. So the

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(4)

instantaneous SNR at the output of the selection combiner can then be expressed as

0

10

−1

10

−2

γ = max(γ1 , γ2 , . . . , γK )

10

Average Bit Error Rate

(5)

where K = n + 1. If the branches are independently faded then order statistics gives the cumulative distribution function (CDF). Fγ (γ) = P [γ1 ≤ γ, . . . , γK ≤ γ] =

K 

−3

10

−4

10

QPSK − N=3 − Analysis QPSK − N=3 − Simulation 8−PSK − N=3 − Analysis 8−PSK − N=3 − Simulation QPSK − N=4 − Analysis QPSK − N=4 − Simulation 8−PSK − N=4 − Analysis 8−PSK − N=4 − Simulation QPSK − N=5 − Analysis QPSK − N=5 − Simulation 8−PSK − N=5 − Analysis 8−PSK − N=5 − Simulation

−5

10

−6

10

−7

10

Fγj (γ)

(6)

K K K   ∂  Fγj (γ) = fγj (γ) P (γl ≤ γ) fγ (γ) = ∂γ j=1 j=1

(7)

l=1 l=j

where, for the Rayleigh fading channel case: fγj (γj ) =

1 −γj /¯γj e , P (γl ≤ γ) = 1 − e−γ/¯γl γ¯j

Substituting (8) into (7), we obtain [12]: ⎡ fγ (γ) =

where Γj =

K  ⎢ ⎣(−1)j−1

j=1 j  l=1

K  i1 ,i2 ,...,ij =1 i1
(8)

0

5

(9)

signal at the destination for M -PSK with SC on i.n.d. Rayleigh fading channels, we proceed analogous to [13]. M  1 eu Pr {θ ∈ Θu } log2 M u=1

(10)

u u where Θu = [θL , θU ] = [(2u − 3)π/M, (2u − 1)π/M ] for u = 1, . . . , M and eu is the number of bit errors in the decision region Θu . With no loss of generality, it is assumed that φ = 0 , the probability Pr {θ ∈ Θu } is u

θU  ∞ fθ (θ |φ , γ)fγ (γ)dγdθ Pr {θ ∈ Θu } = u 0 θL ⎡

10 15 Average SNR per Bit [dB]

20

25

Average Bit Error Rate for M -PSK

where

  u u 1 I θU , θL , Γj =

  tan−1 (αu U) + 12 βUu 12 + π   tan−1 (αu L) − 12 βLu 12 + π   u u ), μuL = log2 (M )/Γj sin(θL ) μuU = log2 (M )/Γj sin(θU   u u log2 (M )/Γj cos(θU log2 (M )/Γj cos(θL ) u ) u  u  u αU = , αL = (μU )2 + 1 (μL )2 + 1 βUu = 

γ¯i−1 . To obtain the BER of the combined l

¯D = B

Fig. 2.

⎤

⎥ Γj e−Γj γ ⎦

QPSK

−8

10

j=1

and Fγj (γ) = P (γj ≤ γ) is the corresponding CDF of γj . We know that when the strongest diversity branch is selected from a total K available i.n.d. diversity branches, the joint pdf of γ for K-branch SC is given by differentiating (6):

8−PSK

u u θU −θL 2π

μuU μu , βLu =  u L u (μU )2 + 1 (μL )2 + 1

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¯j = γ¯ for j = 1, . . . , K, it is straightforward to arrive at  n  N −n S¯ (13) Pr(CD ) = 1 − S¯ where S¯i = S¯ for i = 1, . . . , N . From (12), we can rewrite ¯D under simplified forms as B ⎧ ⎡   ⎤⎫ j−1 K M ⎨ K ⎬ (−1)   j × 1  ⎦ ¯D = ⎣  B eu (14) γ ¯ u u ⎭ log2 M u=1 ⎩ j=1 I θU , θL ,j Substituting (13)-(14) into (1), we can obtain the end-to-end average bit error rate for DF relaying system for an arbitrary number of relays with SC at the destination over i.i.d. Rayleigh channels.

⎤

(11) IV. N UMERICAL R ESULTS AND D ISCUSSION Using the analysis presented in Section III, various number = u of performance evaluation will be presented here and will be j=1 i1 ,i2 ,...,ij =1 θL 0 i1
K K   ⎥ ⎢ 1 1, . . . , N in Fig. 2 are uniformly distributed between 0 and u u ⎥ (12) ⎢(−1)j−1 I θU , θL , Pr {θ ∈ Θu } = ⎦ ⎣ 1 and the average transmit SNRs for all transmit nodes are Γ j j=1 i1 ,i2 ,...,ij =1 equal, i.e., ρS = ρR1 = · · · = ρRN = ρ = ρDT /(N + 1). i1
⎢ ⎣(−1)j−1

K 

u θU ∞

⎥ fθ (θ |φ , γ)Γj e−Γj γ dγdθ⎦

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1-3 dB.

0

10

V. C ONCLUSION

−1

10

The performance of DF relaying system with SC diversity receiver operating over i.n.d. and i.i.d. Rayleigh fading channels has been analyzed. Simulation results are in excellent agreement with the derived expression. The derived BER expression is general and offers a convenient way to evaluate the DF relaying system which exploits SC technique at the destination. In addition, results also show that the loss in performance of system employed SC technique is not much when compared to DF system that uses more complex MRC technique. Moreover, our analysis reveals an interesting result for this relaying protocol: the optimal number of cooperative relays under average BER viewpoint is a complex function of two variables: operating SNR and modulation scheme.

−2

Average Bit Error Rate

10

−3

10

−4

10

QPSK − i.n.d. − Analysis QPSK − i.n.d. − Simulation 16−PSK − i.n.d. − Analysis 16−PSK − i.n.d. − Simulation QPSK − i.i.d. − Analysis QPSK − i.i.d. − Simulation 16−PSK − i.i.d. − Analysis 16−PSK − i.i.d. − Simulation

−5

10

−6

10

−7

10

Fig. 3.

0

5

10 Average SNR per Bit [dB]

15

20

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

ACKNOWLEDGMENT

0

10

N = 1 − MRC N = 1 − SC N = 2 − MRC N = 2 − SC N = 3 − MRC N = 3 − SC

−1

10

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).

−2

Average Bit Error Rate

10

−3

10

R EFERENCES

−4

10

−5

10

−6

10

−7

10

Fig. 4.

0

5

10 15 20 Average SNR per Bit [dB]

25

30

BER of DF relaying with SC and MRC (16-PSK, N = 1, 2, 3)

Fig. 2 shows the average BER of the DF relaying system with different numbers of cooperative nodes. As shown in the figure, in high SNR regime, the improvement of the average BER is proportional to the number of relays. However with low SNR regime, using more relays could make the system performance worsen. For example, in Fig. 2, we can see that the worst performance is obtained with 5 relays for QPSK when SNR is lower than 15 dB. It means that the optimal number of relays in DF relaying system depends not only on SNR but also on modulation scheme. In Fig. 3, the BER of DF relaying with 3 relays in both i.i.d and i.n.d. channels was examined. The results are based on the assumption that the total average SNR for i.i.d. channels is equal to that for i.n.d channels, i.e., λ0 = λ1,1 = λ1,2 = λ1,3 = λ2,1 = λ2,2 = λ2,3 = 2 for i.i.d. channels and λ0 = 2, λ1,1 = 2.8, λ1,2 = 2.7, λ1,3 = 3.5, λ2,1 = 1.2, λ2,2 = 1.3, λ2,3 = 0.5 for i.n.d. channels. It is seen that the performance of DF relaying system under i.i.d. channels is a little bit better than that under i.n.d. channels. In addition, our analytical results and the simulation results are in excellent agreement. In Fig. 4, BER curses confirm that, under same channel conditions, the performance of system employing MRC receiver [5] is always better as compared to an equivalent system using SC by around

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