IEICE TRANS. COMMUN., VOL.E92–B, NO.11 NOVEMBER 2009

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A Simple Performance Approximation for Multi-Hop Decode-and-Forward Relaying over Rayleigh Fading Channels Bao Quoc VO-NGUYEN†a) , Student Member and Hyung Yun KONG†b) , Member

SUMMARY This letter provides a study on the end-to-end performance of multi-hop wireless communication systems equipped with re-generative (decode-and-forward) relays over Rayleigh fading channels. More specifically, the probability density function (pdf) of the tightly approximated end-to-end signal-to-noise ratio (SNR) of the systems is derived. Using this approximation allows us to avoid considering all possible combinations of correct and erroneous decisions at the relays for which the endto-end transmission is error-free. The proposed analysis offers a simple and unifying approach as well as reduces computation burden in evaluating important multi-hop system’s performance metrics. Simulations are performed to verify the accuracy and to show the tightness of the theoretical analysis. key words: multi-hop relaying, rayleigh fading, decode-and-forward relaying, bit error probability, symbol error probability, spectral efficiency, probability of SNR gain

1.

Introduction

While wireless communication will be the core technique, a direct communication between a transmitter and a receiver is faced with many limitations. In particular, communication over long distances is only possible using prohibitively high transmission power. Recently, multi-hop transmission is introduced as a promising technique to achieve broader coverage and to mitigate wireless channel impairment as well as to solve the high transmit-power problem [1]. The main idea is that the channel from a source node to a destination node is split into multiple possibly shorter links by using some intermediate nodes in between. The use of intermediate nodes as relays can reduce the total transmit power and will be a necessary ingredient in some specific networks, e.g., wireless sensor networks. The design and performance analysis of multi-hop DF relaying with single-input single-output (SISO) in each hop has been well studied, i.e. see [2]–[8]. In particular, Hasna et al studied outage probability of multi-hop systems over Rayleigh [2], [5], [6] and Nakagami-m [4] fading channels. In [3], an exact symbol error probability (SEP) of M-PSK for multi-hop systems was provided. In [7], an upper bound for error probability of multi-hop systems was investigated and it also introduced a concept of multi-hop diversity. Although these expressions enable numerical evaluation of the system performance, however, in general they do not offer insights as to what parameters determine system perManuscript received November 4, 2008. Manuscript revised June 25, 2009. † The authors are with University of Ulsan, Korea. a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1587/transcom.E92.B.3524

formance in the presence of fading channels. Furthermore, even when the such exact expressions exist, they may be cumbersome to work with in some certain cases, e.g., using them as criteria for optimizing system design. In this letter, we aim at filling this gap. In particular, the compact form of pdf of the tightly approximated end-to-end SNR is derived, it is then used to derive the closed-form expressions for symbol error probability and bit error probability (BEP) for M-PSK. Furthermore, the spectral efficiency of the system is also considered. The approach employed in this letter offers a convenient way to evaluate the system’s performance where there is no need for a brute-force summation over all possible correct and erroneous decision at immediate relays in the expressions. The benefit achieved by using multi-hop relaying schemes instead of single-hop communications is further investigated via the probability of SNR gain where SNR gain is defined as an average ratio of the end-to-end SNR of the multi-hop system to the SNR of the direct transmission. 2.

System Model

We consider a wireless relay network consisting of one source, K − 1 relays and one destination operating over Rayleigh fading channels. The source terminal (T 0 ) communicates with the destination (T K ) via K − 1 relay nodes denoted as T 1 , · · · , T k , · · · , T K−1 . Intermediate terminals always fully decode the received signal, and then forward the re-encoded version to their respective successor node. We assume that every relay processes only the signals received from its preceding nodes, what allows for reducing the computational costs and hardware complexity at the receiver of each node. Furthermore, it is appropriate for wireless sensor networks, which require fixed processing complexity at each node. It is assumed that every channel between the nodes experiences slow, flat, Rayleigh fading. Due to Rayleigh fading, the channel power of each hop, denoted by |hk |2 , is an independent and exponential random variable whose mean is λk where hk is the fading coefficient from the (k−1)th node to the kth node where k = 1, · · · , K. The average transmit powers for the source and the relays are denoted by ρk with k = 1, · · · , K. Let us define γk = ρk |hk |2 as the instantaneous SNR for each hop. We further define the received SNR for the single-hop system as γ0 = ρ0 |h0 |2 where ρ0 and h0 are the transmit power of the source in case of direct transmission and the fading coefficient of the channel between the

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

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source and the destination, respectively. 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. For medium access, a time-division channel allocation scheme with K time slots is occupied in order to realize orthogonal channelization, thus no inter-relay interference is considered in the signal model. 3.

Performance Analysis

3.1 pdf of End-to-End SNR Consider the kth hop and yk as the received signal at node k, then we have yk = hk s + nk

(1)

where hk is a zero-mean complex Gaussian random variable with a Rayleigh-distributed amplitude and a uniformly distributed phase angle, s is a complex baseband transmitted signal and nk is a zero-mean complex Gaussian random variable representing the AWGN with variance N0 which is the one-sided power spectral. The probability density function and probability distribution function (CDF) of γk can be defined as follows: 1 −γ −γ (2) fγk (γ) = e γ¯ k , Fγk (γ) = 1 − e γ¯ k γ¯ k where γ¯ k = E{γk } = ρk λk where E{.} denotes the expectation operator. Note that, owing to the imperfect detection at the relay, it may forward incorrectly decoded signals to the destination. Hence, similarly as in [9], the multi-hop DF channel can be modeled as an equivalent single hop whose output SNR γeq can be tightly approximated as follows: γeq = min γk

(3)

k=1,...,K

Under the assumption that the hops are subject to independent but not necessarily identically distributed Rayleigh fading, order statistics give the CDF of γeq as   Fγeq (γ) = 1 − Pr γ1 > γ, . . . , γK > γ (4) = 1−

K  

1 − Fγk (γ)



k=1

Hence, the joint pdf of γeq for K hops is given by differentiating (4). fγeq (γ) =

K  k=1

fγk (γ)

K  

 1 − Fγ j (γ)

(5)

j=1 jk

Substituting (2) into (5) and after some manipulations, the pdf of γeq can be determined as follows: K K K  1 − γ¯γ  − γ¯γj  1 −γ Kj=1 γ¯ −1 j fγeq (γ) = e k e = e γ ¯ γ ¯ k k j=1 k=1 k=1 jk

= χe−γχ K −1 γ¯ k . where χ = k=1

(6)

3.2 Symbol Error Probability Using moment generating function (MGF) approach [10], we derive the closed-form expression of SEP for the multihop DF relaying system. In particular, the SEP of the multi-hop DF relaying system for M-PSK modulation can be given by 

g MPS K log2 M 1 π−π/M dθ (7) Mγeq − Ps = π 0 sin2 θ where g MPS K = sin2 (π/M) and Mγeq is the MGF of γeq defined as Mγeq (s) = Eγeq {e sγeq } [10, Eq. (1.2)]. Taking the expectation with respect to the random variable γeq whose pdf is given in (6) yields the result of Mγeq (s) = (1 − sχ−1 )−1 and substituting it into (7), we obtain the closed-form expression for SEP as follows [10, p.142, Eq. (5.79)]: π−π/M

sin2 θ dθ sin2 θ + ai 0  ⎡   ai

 ⎢⎢ M 1 − M − 1 ⎢⎢⎢  1+ai (M−1)π  ⎢⎢ =   ai π M ⎢⎣ × π + tan−1 cot 2 1+ai M

1 Ps = π

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎦ (8)

where ai = χ−1 g MPS K log2 M. In addition, for BPSK, letting M = 2 in (8) and simplifying gives  ⎞ ⎛ 1 χ−1 ⎟⎟⎟⎟⎟ 1  1 ⎜⎜⎜⎜⎜ P s = ⎜⎜⎝1 − (9) 1 − (1 + χ)− 2 ⎟⎟⎠ = −1 2 2 1+χ At high SNRs (χ → 0) and applying the asymptotic exponential approximation, i.e., (1 + x)n ≈ 1 + nx for small x, we obtain [11, Eq. (14.3-13)] ! 



χ→0 1 χ 1 1 1 1 Ps ≈ +· · ·+ (10) 1− 1− χ = = 2 2 4 4 γ¯ 1 γ¯ K 3.3 Bit Error Probability The closed form expression bit error probability for M-PSK, M = 2m with m = 1, · · · , M, can be obtained by proceeding analogous to [12]. Pb =

M 1  em Pr {θ ∈ Θm } log2 M m=1

(11)

where Θm = [θmL , θUm ] = [(2m − 3)π/M, (2m − 1)π/M] and em is the number of bit errors in the decision region Θm . With no loss of generality, it is assumed that ϕ = 0, the probability Pr {θ ∈ Θm } is θU ∞ m

fθ (θ |ϕ , γ) fγeq (γ)dγdθ

Pr {θ ∈ Θm } = θm L

0

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=

⎛ ⎛ ⎞ −1 m ⎞ θUm −θmL 1 m ⎜⎜⎜1 tan−1 (αm ⎟⎟⎟ 1 m ⎜⎜⎜⎜1 tan (αL )⎟⎟⎟⎟ U )⎟ + βU ⎜⎝ + β + − ⎝ ⎠ ⎠ 2π 2 2 π 2 L 2 π (12)

where fθ (θ|ϕ, γ) is defined by [12, Eq. (9b)] and   −1 sin(θm ), μm = = log Mχ log2 Mχ−1 sin(θmL ) μm 2 U U L " " log2 Mχ−1 cos(θUm ) m log2 Mχ−1 cos(θmL ) m , αL = αU =   2+1 (μm ) (μmL )2 + 1 U   m m m m 2 m 2 βm U = μU / (μU ) + 1, βL = μL / (μL ) + 1 3.4 Spectral Efficiency Spectral efficiency is one of the important information theoretic measures of the system. Here, the achievable spectral efficiency of the system is considered and can be obtained by averaging the instantaneous spectral efficiency over the fading distribution as follows: ! 1 1 χ C = Eγeq log2 (1 + γeq ) = e Γ(0, χ) (14) K K ln 2 #∞ where Γ(a, x) = x ta−1 e−t dt is an incomplete Gamma function [13, p.260, Eq. (6.5.3)].

∞ Ω=1 −

(

1 − e−γμχ

0

4.

) 1 −γ/γ¯ 0 1 e dγ = γ¯ 0 1 + γ¯ 0 μχ

Numerical Results and Discussion

We consider a linear network consisting of multiple nodes. The average channel power due to transmission between node T i and node T j is modeled as λTi ,T j = κ0 dT−η where i ,T j dTi ,T j is the distance from node T i to node T j , η is the path loss exponent and κ0 captures the effects due to antenna gain, shadowing, etc. [11]. More specifically, η typically varies between 2 (free-space path loss) and 5 to 6 (shadowed areas and obstructed in-building scenarios) [15]. For a fair comparison to direct transmission,  the overall distance of all K−1 dTk ,Tk+1 = 1 and the hops is normalized to be one, i.e., k=0 uniform power allocation is employed in order to keep the K = ρ0 /K. Withtotal transmit-power constraint, i.e., {ρi }i=1 out loss of generality, we assume κ0 = 1 and each node is equidistant from each other, i.e., dTi ,T j = ( j − i)/K for all results in this section. In Figs. 1, 2, we study the average SEP and average BEP of the system as functions of average signal-to-noise ratio (SNR) per bit. We also compare the performance of

3.5 Probability of SNR Gain The purpose of this subsection is to study the probability of SNR gain of multi-hop DF systems over a single-hop system. Such a SNR gain offers us an explicit view about the advantage achieved by multi-hop systems. Mathematically speaking, the probability of SNR gain achieved by multihop DF systems over direct transmission is defined as follows [14]: $ % $ % γeq γeq Δ > μ = 1 − Pr ≤μ Ω = Pr γ0 γ0 ∞ & ' = 1 − Pr γeq ≤ μγ0 |γ0 = γ fγ0 (γ)dγ

Fig. 1

SEP for multi-hop systems.

Fig. 2

BEP for multi-hop systems.

0

∞ = 1−

Fγeq (μγ) fγ0 (γ)dγ

(15)

0

where μ is a pre-determined SNR gain we wish to obtain and Fγeq (γ) is the corresponding CDF of γeq obtained by integration of (6) between 0 and γ as γ Fγeq (γ) =

fγeq (γ)dγ = 1 − e−γχ

(16)

0

Substituting (16) into (15), we can obtain the probability of SNR gain for the system as

(17)

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and probability of SNR gain of the system are also investigated. The derived expressions are compact and reduce a lot of computation burden in evaluating important multihop system’s performance metrics. In addition, results were shown that the MDF is superior to DT only in poor communication environments. Acknowledgments

Fig. 3

Spectral Efficiency for multi-hop systems.

This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (No. R01-2007-000-20400-0). References

Fig. 4

Probability for SNR gain for multi-hop systems.

the multi-hop DF system (denoted as MDF) with that of the direct transmission (DT). It is can be clearly observed from Fig. 1 that the performance of MDF (K > 1) is better than that of DF (K = 1) for all values of SNRs. Furthermore, the performance of the multi-hop system with the improvement of the average SEP or average BEP is proportional to the number of hops (K). For example, at the target average SEP 10−1 for 16-PSK (Fig. 1), MDF with five hops (K = 5) outperforms DT with transmit power gain of about 7 dB. The spectral efficiency for the system is also shown in Fig. 3. We can see that the more number of hops the system uses, the lower spectral efficiency of the system we can achieve. Consequently, MDF can provide a good trade-off between system performance and spectral efficiency. In Fig. 4, we show the probability of the SNR gain achievable by MDF in different communication environments, η. In comparing the probabilities of SNR gain for different values of η, it can be seen that we only benefit from increasing the number of hops in poor communication environments, i.e., η > 2. In addition, it is obvious that our analytical results and the simulation results are in excellent agreement. 5.

Conclusion

We have presented closed-form expressions for the average SEP and BEP of the multi-hop DF relaying system over Rayleigh fading channels. Moreover, the spectral efficiency

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