Error Probability Performance for Multi-hop Decode-and-Forward Relaying over Rayleigh Fading Channels Vo Nguyen Quoc Bao∗ , Hyung Yun Kong∗ ∗ School

of Electrical Engineering University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749 Email: [email protected], [email protected] Abstract—This paper provides a complete study on the end-toend performance of multi-hop wireless communication systems equipped with re-generative (decode-and-forward) relays over Rayleigh fading channels assuming single-antenna terminals. More specifically, the probability density function (pdf) of the tightly approximated end-to-end 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 and the destination for which the end-to-end transmission is error-free, thus reducing 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.

for M -QAM. In addition, spectral efficiency of the system is also considered. The approach employed in this letter offers a convenient and compact way to evaluate the system’s performance where there is no need 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 communications instead of the 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.

I. I NTRODUCTION 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 the outage and the error probability of multi-hop systems over Rayleigh [2, 5, 6] and Nakagami-m [4] fading channels. In [3], an exact symbol error probability of M -PSK for multi-hop systems was provided. In [7], an upper bound for error probability of the multi-hop systems was investigated and it also introduced a concept of multi-hop diversity. In this letter, we present a complete performance analysis of the multi-hop DF relaying system over Rayleigh fading channels. The main contribution of this letter is the derivation of the compact form of pdf of the tightly approximated endto-end SNR, which is then used to derive the closed-form expressions for symbol error probability, bit error probability

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II. S YSTEM M ODEL We consider a wireless relay network consisting of one source, K − 1 relays and one destination operating over Rayleigh fading channels. The source terminal (T0 ) communicates with the destination (TK ) via K − 1 relay nodes denoted as T1 , · · · , Tk , · · · , TK−1 . Intermediate terminals always perform hard decisions on the received symbols before forwarding them 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 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, respectively. Let us define γk = ρk |hk |2 as the instantaneous SNR for each hop. We further define the received SNR for the signal-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 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.

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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. III. P ERFORMANCE A NALYSIS A. 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

where γ¯k = E{γk } = ρk λk . 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: min

k=1,...,K

γk

(3)

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 − P [γ1 > γ, . . . , γK > γ]

=

1−

K 

The outage probability Po is defined as the probability that the output SNR γeq of the equivalent single hop falls below a certain predetermined threshold SNR γth . Hence, the outage probability of the system can be obtained by integrating the pdf of γth as γth fγeq (γ)dγ Po = Pr{γeq < γth } = =

(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 − γ e k , Fγk (γ) = 1 − e γ¯k (2) fγk (γ) = γ¯k

γeq =

B. Outage Probability

(4)

[1 − Fγk (γ)]

k=1

fγeq (γ) =

fγk (γ)

K 

  1 − Fγj (γ)

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

=

where χ =

K

=

j=k

  K   1 −γ K γ ¯k−1 j=1 e γ¯k

k=1 −γχ

χe

¯k−1 . k=1 γ

ISBN 978-89-5519-139-4

(7)

which is in agreement with the previous known results [5, eq. (28)] or [4], as expected. C. Symbol Error Rate In this paper, we consider only SER and BER for M -ary square quadrature amplitude (M -QAM) modulation (M = 4m , m = 1, 2, · · · ) due to limit of space. However, the closedform expressions of those for other modulation schemes, e.g. M PSK, M PAM, can be determined in the same manner [1011]. In particular, the SER for square M -QAM in the AWGN channel is given in [11, p. 278, eq. (5.279)] as √ √ (9) Ps (ε |γ) = 2pQ ( qγ) − p2 Q2 ( qγ) √ where p = 2(1 − 1/ M ), q = 3 log2 M/(M − 1),

 π/2 Q(x) = π1 0 exp −x2 2 sin2 θ dθ is the Gaussian Qfunction defined in [12, p. 85, eq. (4.2)] and Q2 (x) =

 1 π/4 exp −x2 2 sin2 θ dθ is also defined in [12, p.88, eq. π 0 (4.9)]. The error rate for square M -QAM in slow and flat Rayleigh fading channels can be derived by averaging the error rate for the AWGN channel over the pdf of the SNR in Rayleigh fading. ∞ Ps (ε) =

Ps (ε |γ )fγeq (γ)dγ

(10)

0

(5)

j=1 j=k

k=1

0

As a check, consider the conventional dual-hop transmission where K = 2. Then, from (7), we have:

1 1 −γ + (8) Po = 1 − e th γ¯1 γ¯2

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

1 − e−γth χ

(6)

where Ps (ε |γ ) is the symbol error rate conditioned on γ. By substituting (6) and (9) into (10) and using moment generating function (MGF) approach [12], the closed-form expression of SER for the system can be evaluated with the help of [12, p. 151, eq. (5A.11) and eq. (5A.12)] as ⎡ ⎤ π/2 2p sin2 θ dθ ⎥ ⎢ π sin2 θ+qχ−1 /2 ⎢ ⎥ 0 Ps = ⎢ ⎥ ⎣ p2 π/4 ⎦ 2 sin θ dθ −π sin2 θ+qχ−1 /2 0  ⎤ ⎡ qχ−1 1 − p4 − 2+qχ −1 ×   ⎦ (11) = p⎣  −1 1 − πp tan−1 2+qχ −1 qχ

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integration of (6) between 0 and γ as

D. Bit Error Rate Similarly as SER, the BER of the system over Rayleigh fading channels for M -ary square QAM with Gray mapping can be given as [13] ⎤ ⎡ √ υj log M  2 ∞ 1 j √ φn ⎥ ⎢ √ (12) Pb = ⎣ M log2 M ⎦dγ

√ j=1 n=0 × erfc ω γ f (γ) n γeq 0

Fγeq (γ) =

∞ Ω

In addition, for 4-QAM (M = 4), at high SNRs and by using the asymptotic exponential approximation, i.e., (1 + x)n ≈ 1 + nx for small x, (13) reduces to   1 1 χ 1 (14) Pb = = + ··· + 4 4 γ¯1 γ¯K E. 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 log2 (1 + γeq ) = eχ Γ(0, χ) C = Eγeq K K ln 2 ∞ where Γ(a, x) = x ta−1 e−t dt is an incomplete Gamma function [14, p. 260, eq. (6.5.3)].

=

0

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

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1 − e−γμχ

 1 −γ/¯γ0 e dγ γ¯0

1 1 + γ¯0 μχ

(17)

. IV. N UMERICAL RESULTS AND DISCUSSION We consider a linear network consisting of multiple nodes. The average channel power due to transmission between node where Ti and node Tj is modeled as λTi ,Tj = κ0 d−η Ti ,Tj dTi ,Tj is the distance from node Ti to node Tj , η is the path loss exponent and κ0 captures the effects due to antenna gain, shadowing, etc [11,16]. More specifically, η typically varies between 2 (free-space path loss) and 5 to 6 (shadowed areas and obstructed in-building scenarios) [16]. For a fair comparison to direct transmission, the overall distance of all K−1 hops is normalized to be one, i.e., k=0 dTk ,Tk+1 = 1 and the uniform power allocation is employed in order to keep the K total power constraint, i.e., {ρi }i=1 = ρ0 /K. Without loss of generality, we assume κ0 = 1 and each node is equidistant from each other, i.e., dTi ,Tj = (j − i)/K for all results this section. 0

10

γth=20

−1

10

F. Probability of SNR gain The purpose of this subsection is to study the probability of SNR gain of multi-hop DF systems over the 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 multi-hop DF systems over direct transmission is defined as follows [15]: ! ! Δ γ γ Ω = Pr γeq0 > μ = 1 − Pr γeq0 ≤ μ ∞ = 1 − Pr {γeq ≤ μγ0 |γ0 = γ} fγ0 (γ)dγ (15) 0 ∞ = 1 − Fγeq (μγ)fγ0 (γ)dγ

= 1− 0

=

Furthermore, we define . and erfc(.) as the floor and complementary error function [12, p. 121, eq. (4A.6)], respectively. Substituting (6) into (12) and taking the integral with respect to γ, we achieve the closed-form expression for BER as follows: √    log2 M υj   1 ωn χ−1 j √ (13) φn 1 − Pb = √ 1 + ωn χ−1 M log2 M j=1 n=0

(16)

Substituting (17) into (16), we can obtain the probability of SNR gain for the system as

=

(2n+1)2 3 log2 M . 2M −2

fγeq (γ)dγ = 1 − e−γχ

0

Outage Probability

√ (1 − 2−j ) M − 1, φjn where υj  =    j−1 n2 j−1 √ M √ (−1) 2j−1 − n2 and ωn + 12 M

γ

−2

10

K=1−Simulation (DT) K=1−Analysis (DT) K=3−Simulation K=3−Analysis K=5−Simulation K=5−Analysis

−3

10

γ =2 th

−4

10

0

5

Fig. 1.

10 15 20 Average SNR per bit [dB]

25

30

Outage Probability for multi-hop systems.

Fig. 1 illustrates the outage probability of the multi-hop system since the number of hops increases. Herein, under the assumption that there is no latency in processing in each node, i.e., decoding and then forwarding the received signal to the

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next node, we can see that increasing the number of hops K reduces the outage probability of the system. −1

10

64−QAM

0

−1

64−QAM

Symbol Error Probability

10

−2

10

K=1 4−QAM Simulation (DT) K=1 4−QAM Analysis (DT) K=3 4−QAM Simulation K=3 4−QAM Analysis K=5 4−QAM Simulation K=5 4−QAM Analysis K=1 64−QAM Simulation (DT) K=1 64−QAM Analysis (DT) K=3 64−QAM Simulation K=3 64−QAM Analysis K=5 64−QAM Simulation K=5 64−QAM Analysis

−3

10

−4

10

−5

10

Bit Error Probability

10

0

5

−2

10

K=2 16−QAM Simulation K=2 16−QAM Analysis K=4 16−QAM Simulation K=4 16−QAM Analysis K=6 16−QAM Simulation K=6 16−QAM Analysis K=2 64−QAM Simulation K=2 64−QAM Analysis K=4 64−QAM Simulation K=4 64−QAM Analysis K=6 64−QAM Simulation K=6 64−QAM Analysis

−3

10

−4

10

−5

10

0

4−QAM

10 15 20 Average SNR per bit [dB]

5

Fig. 4. 25

16−QAM

10 15 20 Average SNR per bit [dB]

25

30

BEP for multi-hop systems with different K.

30 0

10

Fig. 2.

SEP for multi-hop systems with different K.

Increasing M −1

Bit Error Probability

10

0

10

Increasing M

−1

Symbol Error Probability

10

4−QAM Simulation 4−QAM Analysis 16−QAM Simulation 16−QAM Analysis 64−QAM Simulation 64−QAM Analysis 256−QAM Simulation 256−QAM Analysis 1024−QAM Simulation 1024−QAM Analysis 4096 64−QAM Simulation 4096−QAM Analysis

−2

10

−3

10

0

5

Fig. 3.

10 15 20 Average SNR per bit [dB]

−3

10

−4

−5

10

0

5

Fig. 5. 25

10 15 20 Average SNR per bit [dB]

25

30

BEP for multi-hop systems with different M .

30

SEP for multi-hop systems with different M .

In Figs. 2-5, we study the average SER and average BER of the system as functions of average signal to noise ratio (SNR) per bit. In addition, we compare the performance of the multi-hop DF system (denoted as MDF) with that of the direct transmission (DT), i.e., K = 1. For all values of SNRs, the performance of our proposed system is better than that of the other. It is obvious that the performance of the multi-hop system with the improvement of the average SER or average BER is proportional to the number of hops (K). For example, at the target average SER 10−1 for 64-QAM (Fig. 2), MDF with five hops (K = 5) outperforms DT with transmit power

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4−QAM Simulation 4−QAM Analysis 16−QAM Simulation 16−QAM Analysis 64−QAM Simulation 64−QAM Analysis 256−QAM Simulation 256−QAM Analysis 1024−QAM Simulation 1024−QAM Analysis 4096 64−QAM Simulation 4096−QAM Analysis

10

−4

10

−2

10

gain of about 8 dB. Moreover, we can see that there are a small gap between simulation and analysis results for high constellations at low SNRs, and this gap reaches to zero in high SNRs. It is due to the fact that the condition for the approximation of the equivalent single channel whose SNR is expressed by the minimum SNR of all hops is not satisfied at low SNRs. The spectral efficiency for the system is also shown in Fig. 6. We can see that the more number of hops the systems use, the lower spectral efficiency of the system we can achieve. Then, the advantages of the multi-hop system come at the price of spectral efficiency. Consequently, MDF can provide a good trade-off between system performance and spectral efficiency. In Fig. 7-8, we show the probability of the SNR gain

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14

10 8 6 4

0.9 0.8 Probability of SNR Gain ( Ω)

Spectral Efficiency

12

2 0 −10

0

K=2 Simulation K=2 Analysis K=3 Simulation K=3 Analysis K=4 Simulation K=4 Analysis K=5 Simulation K=5 Analysis K=6 Simulation K=6 Analysis K=7 Simulation K=7 Analysis

1

K=1−Simulation (DT) K=1−Analysis (DT) K=2−Simulation K=2−Analysis K=3−Simulation K=3−Analysis K=4−Simulation K=4−Analysis K=5−Simulation K=5−Analysis

10 20 Average SNR per bit [dB]

30

40

0.7 0.6 0.5 0.4 0.3

Increasing K

0.2 0.1

Fig. 6.

Spectral Efficiency for multi-hop systems

Probability of SNR Gain ( Ω)

1

}

K=3 K=5 K=7

0.8

X

0

2

3

4 η [dB]

5

6

Simulation

Fig. 8.

Analysis

Probability for SNR gain for multi-hop systems.

η=2

0.6

η=3 η=4

0.4 0.2 0 −10

Fig. 7.

0

10

20 μ [dB]

30

40

50

Probability for SNR gain for multi-hop systems with different K.

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. V. C ONCLUSION We have presented the closed-form expressions for the average SER and BER of the multi-hop DF relaying system over Rayleigh fading channels. Moreover, the spectral efficiency 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 multi-hop system’s performance metrics. In addition, the results were shown that the MDF only has advantage over DT in poor communication environments. ACKNOWLEDGMENTS 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 R EFERENCES [1] O. Oyman, J. N. Laneman, and S. Sandhu, ”Multihop Relaying for Broadband Wireless Mesh Networks: From Theory to Practice,” IEEE Communications Magazine, November 2007.

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[2] M. O. Hasna, ”Optimal power allocation for relayed transmissions over Rayleigh-fading channels,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1999-2004, 2004. [3] A. Muller and J. Speidel, ”Exact symbol error probability of m-psk for multihop transmission with regenerative relays,” Communications Letters, IEEE, vol. 11, pp. 952-954, 2007. [4] M. O. Hasna and M.-S. Alouini, ”Outage Probability of Multihop Transmission Over Nakagami Fading Channels,” IEEE Communications Letters, vol. 7, pp. 216-218, May 2003. [5] M. O. Hasna and M.-S. Alouini, ”End-to-End Performance of Transmission System with Relays over Rayleigh-Fading Channels,” IEEE Transactions on Wireless Communications, vol. 2, pp. 1126-1131, November 2003. [6] M. O. Hasna and M. S. Alouini, ”Harmonic mean and end-to-end performance of transmission systems with relays,” Communications, IEEE Transactions on, vol. 52, pp. 130-135, 2004. [7] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, ”Multihop Diversity in Wireless Relaying Channels,” IEEE Transactions on Communications, vol. 52, pp. 1820-1830, October 2004. [8] Y. Lin, M. O. Hasna, and M. S. Alouini, ”Average outage duration of multihop communication systems with regenerative relays,” Wireless Communications, IEEE Transactions on, vol. 4, pp. 1366-1371, 2005. [9] T. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, ”HighPerformance Cooperative Demodulation With Decode-and-Forward Relays,” IEEE Transactions on Communications, vol. 55, pp. 1427-1438, July 2007. [10] S. Chennakeshu and J. B. Anderson, ”Error Rates for Rayleigh Fading Multichannel Reception of MPSK,” IEEE Transactions on Communications, vol. 43, pp. 338-346, February/March/April 1995. [11] J. G. Proakis, Digital communications, 4th ed. Boston: McGraw-Hill, 2001. [12] M. K. Simon and M.-S. Alouini, Digital communication over fading channels, 2nd ed. Hoboken, N.J.: John Wiley & Sons, 2005. [13] K. Cho and D. Yoon, ”On the General BER Expression of Oneand Two-Dimensional Amplitude Modulations,” IEEE Transactions on Communications, vol. 50, pp. 1074-1080, July 2002. [14] M. Abramowitz, I. A. Stegun, and Knovel (Firm), ”Handbook of mathematical functions with formulas, graphs, and mathematical tables,” 10th printing, with corrections. ed Washington, D.C.: U.S. Dept. of Commerce : U.S. G.P.O., 1972, pp. xiv, 1046 p. [15] L. In-Ho and K. Dongwoo, ”Probability of SNR Gain by Dual-Hop Relaying over Single-Hop Transmission in SISO Rayleigh Fading Channels,” Communications Letters, IEEE, vol. 12, pp. 734-736, 2008. [16] H. Karl and A. Willig, Protocols and architectures for wireless sensor networks. Hoboken, NJ: Wiley, 2005.

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