Exact Bit Error Probability of Multi-hop Decode-and-Forward Relaying with Selection Combining Bao Quoc Vo-Nguyen and Hyung Yun Kong Department of Electrical Engineering, University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, 680-749 Korea {baovnq,hkong}@mail.ulsan.ac.kr http://wcomm.ulsan.ac.kr

Abstract. In this paper, an exact closed-form bit error rate expression for M -PSK is presented for multi-hop Decode-and-Forward Relaying (MDFR) scheme, in which selection combining technique is employed at each node. We have shown that the proposed protocol offers remarkable diversity advantage over direct transmission as well as the conventional decode-and-forward relaying (CDFR) scheme. Simulations are performed to confirm our theoretical analysis. Keywords: Bit Error Rate (BER), Decode-and-Forward Relaying, Rayleigh fading, Selection Combining, M -PSK, cooperative communication.

1

Introduction

Recently, relaying dual-hop transmission has gained more attention under forms of cooperative communications and it is treated as one of the candidates to overcome the channel impairment like fading, shadowing and path loss [1]. The main idea is that in a multi-user network, two or more users share their information and transmit jointly as a virtual antenna array. This enables them to obtain higher diversity than they could have individually [1-9]. In the past, relatively few contributions concerning evaluating performance of the DF relaying protocol with multi relays and maximal ratio combining (MRC) or selection combining (SC) have been published [2-9]. In particular, in [2], Jeremiah Hu and Norman C. Beaulieu derived a closed-form expression for outage probability of the CDFR networks with SC when the statistics of the channels between the source, relays, and destination are assumed to be independent and identically distributed (i.i.d.) and independent but not identically distributed (i.n.d.). In [4, 5], the performance of CDFR with maximal ratio combining at the destination in terms of outage probability and bit error probability over independent but not identically distributed channels was also examined. In [3, 6-9], a class of multi-hop cooperative scheme employing decode-and-forward relaying with MRC, called multi-hop Decode-and-Forward Relaying (MDFR) scheme, was proposed, and various performance metrics were also provided. D.-S. Huang et al. (Eds.): ICIC 2009, LNAI 5755, pp. 718–727, 2009. c Springer-Verlag Berlin Heidelberg 2009 

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However, to the best of the authors’ knowledge, there is no publication concerning the exact expression for bit error rate of the MDFR with selection combining in both i.i.d. and i.n.d. Rayleigh fading channels. In this paper, we focus on selective decode-and-forward relaying where the relay must make an independent decision on whether or not to decode and forward source information [1]. In addition, a concept of cooperative diversity protocols for multi-hop wireless networks, which allows relay nodes to exploit all information they overhear from their previous nodes along the route to the destination to increase the change of cooperation, is applied. To that effect, the receiver at each node can employ a variety of diversity combining techniques to obtain diversity from the multiple signal replicas available from its preceding relaying nodes and the source. 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, a system may use selection combining which is the simplest combining method. It only selects the best signal out of all replicas for further processing and neglects all the remaining ones. The benefit of using SC as opposed to MRC is reduced hardware complexity at each node in the network. In addition, it also reduces the computational costs and may even lead to a better performance than MRC, because in practice channels with very low SNR can not accurately estimated and contribute much noise. The contributions of this paper are as follows. We derive an exact closedform expression bit error rate for M -PSK of the MDFR scheme. In addition, the comparison between the performance of MDFR and that of CDFR [2] is performed and it confirms that the proposed protocol outperforms CDFR in all range of operating SNRs. The rest of this paper is organized as follows. In Sect. 2, we introduce the model under study and describe the proposed protocol. Section 3 shows the formulas allowing for evaluation of average BER of the system. In Sect. 4, we contrast the simulations and the results yielded by theory. Finally, the paper is closed in Sect. 5.

2

System Model

We consider a wireless relay network consisting of one source, K relays and one destination operating over slow, flat, Rayleigh fading channels as illustrated in Fig. 1. The source terminal (T0 ) communicates with the destination (TK+1 ) via K relay nodes denoted as T1 , · · · , Tk , · · · , TK . Due to Rayleigh fading, the channel powers, denoted by αTi ,Tj = |hTi ,Tj |2 are independent and exponential random variables where hTi ,Tj is the fading coefficient from node Ti to node Tj with i = 0, · · · , K, j = 1, · · · , K + 1 and i < j. We define λTi ,Tj as the expected value of αTi ,Tj . The average transmit powers for the source and the relays are denoted by ρTi with i = 0, · · · , K, respectively. We further define γTi ,Tj = ρTi αTi ,Tj as the instantaneous SNR per bit for the link Ti → Tj .

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T2 T1

T0

T3

T4

Fig. 1. A MDFR system with 3 relays (K = 3)

For medium access, a time-division channel allocation scheme with K +1 time slots is occupied in order to realize orthogonal channelization, thus no inter-relay interference is considered in the signal model. According to the selective DF relaying protocol [1], the relay decides to cooperate or not with the source in its own time slot, based on the quality of its received signals. Since selection combining technique is used, the relay adaptively chooses the strongest signal (on the basic of instantaneous SNR) among available ones to demodulate and then check whether its received data are right or wrong. If they are right, that relay will cooperate with the source in its transmission time slot, otherwise, it will keep silent. We define a decoding set D(Tk ) for node Tk , k = 1, 2, · · · , K + 1, whose members are its preceding relays which decode successfully. So it is obvious that D(Tk ) is a subset of C = {T1 , T2 , · · · , TK }. In real scenario, the decoding set is determined after receiving one frame 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 [4]. We further assume that the receivers at the destination and relays have perfect channel state information (CSI) but no transmitter CSI is available at the source and relays.

3

BER Analysis

Similarly as in [2-7], namely applying the theorem of total probability, the bit error rate of the multi-hop decode-and-forward relaying can be derived as a weighted sum of the bit error rate for SC at the destination, BD [D(TK+1 )], corresponding to each set of decoding relay D(TK+1 ). Thus the end-to-end bit error rate for M -PSK of the system Pb can be written as  Pb = Pr [D(TK+1 )] BD [D(TK+1 )] (1) D(TK+1 )∈2C

where 2C denotes the power set of C that is the set of all subsets of C.

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Since selection combining is exploited at each relay and the destination, the signal with largest SNR is always selected from the signals received from its k decoding set as well as from the source. Let us define {γi }ni=1 as the instantaneous SNR per bit of each path received by the node Tk from the set D∗ (Tk ) with their k expected values {¯ γi }ni=1 , respectively, where D∗ (Tk ) = D(Tk ) ∪ {T0 } and nk is the cardinality of the set D∗ (Tk ), i.e., nk = |D∗ (Tk )|. Under the assumption that all links are subject to independent fading, the cumulative distribution function (CDF) of βk = max ρTi αTi ,Tk = max γi ∗ i=1,...,nk

Ti ∈D (Tk )

can be determined by [10] Fβk (γ) = Pr[γ1 < γ, . . . , γi < γ, . . . , γnk < γ] =

nk  

1 − e−γ/¯γi

 (2)

i=1

Hence, the joint pdf of βk is given by differentiating (2) with respect to γ [11]. ⎡ ⎤ nk nk   ∂ ⎢ ⎥ i−1 Fβ (γ) = fβk (γ) = (3) ωi e−ωi γ ⎦ ⎣(−1) ∂γ k m ,...,m =1 i=1 1

i

m1 <···
i −1 where ωi = l=1 γ¯m . It is noted that the joint pdf of βk is expressed under a l mathematic tractable form, which offers a convenient way to derive the average bit error probability of the system. Due to the fact that the decoding set is determined on symbol-by-symbol basics, the conditional probability that relay node Tk is involved in the cooperative transmission is obtained as follows: Pr[Tk ∈ D(TK+1 )|D∗ (Tk )] = 1 − STk

(4)

where STk denotes the average symbol error rate (SER) of M -PSK modulated symbols transmitted from D∗ (Tk ). For the case of coherently detected M -PSK, to evaluate STk , the MGF-based approach is used [12], namely  

1 (M−1)π/M log (M )gMP SK STk = M βk − 2 dθ (5) π 0 sin2 θ where gMP SK = sin2 (π/M ) and Mβk (s) is defined as follows [12, eq. (1.2)]: ⎡ ⎤

∞ nk nk   1 ⎢ ⎥ i−1 Mβk (s) = fβk (γ)esγ dγ = (6) ⎣(−1) −1 ⎦ 1 − sω i m ,...,m =1 i=1 1

0

i

m1 <···
Finally, substituting (6) into (5) and after some manipulations [12, eq. (5.79)] give us the desired result as eq. (7) as follows: ⎡ ⎤ (M−1)π/M n n k k  sin2 θ 1 ⎣(−1)i−1 STk = dθ ⎦ π sin2 θ+gM P SK ωi −1 log M i=1

m1 ,...,mi =1

2

0

m⎡ 1 <···
2

1+gM P SK ωi

log2 M

⎤⎤⎫ (7) ⎬  ⎦⎦ π ⎭

M

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By using the relation of joint probability of mass function (pmf) and sequence of conditional pmf [13], the decoding set probability can be written as follows: Pr[D(TK+1 )] =   ∗ Pr [Tk ∈ D(TK+1 )|D (Tk )] ×  Tk ∈D(TK+1 )   (1 − Pr [Tl ∈ / D(TK+1 )|D∗ (Tl )])

(8)

Tl ∈C\D(TK+1 )

The closed-form expression for conditional bit error rate at the destination can be obtained by proceeding analogous to [14]. M  1 em Pr {θ ∈ Θm } BD [D(TK+1 )] = log2 M m=1

(9)

m m where Θm = [θL , θU ] = [(2m − 3)π/M, (2m − 1)π/M ] for m = 1, · · · , M and em is the number of bit errors in the decision region Θm . Without loss of generality, it is assumed that φ = 0, the probability Pr{θ ∈ Θm } is ⎤ ⎡ m

θU ∞ nK+1 nK+1 ⎢   ⎥ m m Pr{θ ∈ Θm }= fθ (θ |φ , γ)fβK+1(γ)dγdθ= ⎣(−1)i−1 I θU , θL ; ωi−1 ⎦(10) m0 θL

i=1

m1 ,...,mi =1 m1 <···
where fθ (θ|φ, γ) is defined by [14, eq. (9b)]. Furthermore, using the analysis in m m , θL ; ωi−1 ) can be derived as follows: [14], I(θU      m m  m m −1  θU 1 m 1 tan−1 (αm tan−1 (αm −θL m 1 U) L) + ψU + + , θL ; ω i = I θU −ψL (11) 2π 2 2 π 2 π with

  −1 m m m = log2 (M )ωi sin(θU ), μL = log2 (M )ωi−1 sin(θL )   m m log2 (M )ωi−1 cos(θU ) log2 (M )ωi−1 cos(θL ) m ! ! αm , α = = U L 2 2 (μm (μm U) +1 L) +1 m m μ μ m m ψU = ! mU = ! mL , ψL 2 (μU ) + 1 (μL )2 + 1 μm U

(12a) (12b) (12c)

Substituting (8)-(9) into (1), we can obtain the exact closed-form expression for bit error rate of the system.

4

Numerical Results

In this section, we provide some simulation results of the proposed protocol and verify these results with our derived formula. We consider a linear network

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consisting of multiple nodes. The average channel power due to transmission between node Ti and node Tj is modeled as λTi ,Tj = κ0 d−η where dTi ,Tj is the Ti ,Tj distance from node Ti to node Tj , η is the path loss exponent and κ0 captures the effects due to antenna gain, shadowing, etc [15]. However, for a fair of comparison to direct transmission, the overall distance of all hops is normalized to be one, i.e., K k=0 dTk ,Tk+1 = 1, and the uniform power allocation is employed in order to keep the total power constraint. Without loss of generality, we assume κ0 = 1, η = 3 and each node is equidistant from each other, i.e., dTi ,Tj = (j − i)/(K + 1) for all results except those in Fig. 4.

0

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DT − Analysis DT − Simulation K=1 Analysis K=1 Simulation K=2 Analysis K=2 Simulation Increasing K K=3 Analysis K=3 Simulation K=4 Analysis K=4 Simulation

−10

10

−15

10

0

5

10 15 20 Average SNR per Bit [dB]

25

30

Fig. 2. Effect of increasing the number of relays on the average bit error rate of the multi-hop decode-and-forward cooperative networks with BPSK modulation (M = 2)

Fig. 2 shows the bit error rate of MDFR for BPSK with different numbers of cooperative nodes. As can clearly be seen in high SNR regime, the improvement of bit error rate is proportional to the number of relays. In Fig. 3, we study the average BER performance for different levels of M PSK. Note that with Gray code used for bit-symbol mapping, average BER for QPSK is same with that of BPSK. Furthermore, as expected, the results from theory and from simulation are in excellent agreement. In Fig. 4, the performance of MDFR in both independent and identically distributed (i.i.d.) and independent but not identically distributed (i.n.d.) channels is examined. The results are based on the assumption that λTi ,Tj is set to be one and to be uniformly distributed between 0 and 1 for the i.i.d and i.n.d. case, respectively. It can be seen that the performance of MDFR under i.i.d. channels is better than that under i.n.d. channels. In Fig. 5, the performance of CDFR and MDFR are compared and illustrated. As clearly shown, MDFR always outperforms direct transmission as well as CDFR with transmit power gain of about 2 dB and 4 dB for K = 2 and K = 4, respectively, for any value of operating SNRs.

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10

Increasing M

−2

Bit Error Rate

10

−4

10

BPSK Analysis BPSK Simulation QPSK Analysis QPSK Simulator 8−PSK Analysis 8−PSK Simulation 16−PSK Analysis 16−PSK Simulation 32−PSK Analysis 32−PSK Simulation 64−PSK Analysis 64−PSK Simulation 128−PSK Analysis 128−PSK Simulation

−6

10

−8

10

−10

10

0

5

BPSK & QPSK

10 15 20 Average SNR per Bit [dB]

25

30

Fig. 3. Effect of modulation levels on the average bit error rate of the multi-hop decodeand-forward cooperative networks with K = 2

0

10

i.n.d.

−2

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10

K=1 Analysis −i.i.d. K=1 Simulation − i.i.d. K=1 Analysis −i.n.d. K=1 Simulation − i.n.d. K=3 Analysis −i.i.d. K=3 Simulation − i.i.d. K=3 Analysis −i.n.d. K=3 Simulation − i.n.d. K=5 Analysis −i.i.d. K=5 Simulation − i.i.d. K=5 Analysis −i.n.d. K=5 Simulation − i.n.d.

−6

10

−8

10

−10

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i.i.d.

−12

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0

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Fig. 4. BEP for 4-PSK (M = 4) of the multi-hop decode-and-forward cooperative networks under i.i.d. channels (λTi Tj = 1) and i.n.d. channels (λTi Tj are uniformly distributed between 0 and 1) with i = 0, · · · , K, j = 1, · · · , K + 1 and i < j

In Fig. 6, we investigate the bit error rate of the multi-hop decode-and-forward cooperative networks in different communication environments, η. More specifically, η typically varies between 2 (free-space path loss) and 5 to 6 (shadowed areas and obstructed in-building scenarios) [16]. It can be seen that under same conditions, the MDFR offers more benefit in poor communication environments.

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0

10

Direct Transmission

−5

Bit Error Rate

10

DT − Analysis DT − Simulation K=2 CDFR Analysis K=2 CDFR Simulation K=2 MDFR Analysis K=2 MDFR Simulation K=4 CDFR Analysis K=4 CDFR Simulation K=4 MDFR Analysis K=4 MDFR Simulation

−10

10

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10

0

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25

30

Fig. 5. Performance comparison between MDFR and CDFR with QPSK modulation (M = 4) 0

10

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10

−3

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−4

10

E /N =0 dB b

0

E /N =5 dB b

−5

0

E /N =10 dB

10

b

0

Eb/N0=15 dB −6

10

2

3

4 η

5

6

Fig. 6. Effect of pass loss exponents on the average bit error rate of the multi-hop decode-and-forward cooperative networks with 16-PSK modulation (M = 16)

Finally, in Fig. 7, we study the effect of diversity technique on the average BER of the multi-hop decode-and-forward cooperative networks. The BER curses confirm that, under same channel conditions, the performance of the system employing MRC receiver is always better as compared to an equivalent system using SC. Furthermore, the performance-loss gap between two systems tends to increase to be proportional to the number of relays. For example, the performance loss for the cases K = 1, 2 and 3 are 1,2 and 3 dB, respectively. However, it is reasonable because the advantage of MRC over SC comes at the price of complexity of the receiver at each node in the networks.

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K=1 − SC K=1 − MRC K=2 − SC K=2 − MRC K=3 − SC K=3 − MRC

−8

10

−10

10

−12

10

0

10 20 30 Average SNR per Bit [dB]

40

Fig. 7. Effect of combining technique on the average bit error rate of the multi-hop decode-and-forward cooperative networks with 8-PSK modulation (M = 8)

5

Conclusion

We have presented the exact closed-form expression for bit error rate of multihop DF relaying over Rayleigh fading channels. Its validity was demonstrated by a variety of Monte-Carlo simulations. The expression is general and offers a convenient way to evaluate MDFR system which exploits SC technique. In addition, the results were shown that employing the MDFR significantly enhances the system performance compared to that of CDFR.

Acknowledgment This research was financially supported by the Ministry of Commerce, Industry and Energy (MOCIE) and Korea Industrial Technology Foundation (KOTEF) through the Human Resource Training Project for Regional Innovation.

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