IEICE TRANS. COMMUN., VOL.E93–B, NO.5 MAY 2010

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LETTER

Incremental Relaying with Partial Relay Selection∗ Vo Nguyen Quoc BAO†a) , Student Member and Hyung Yun KONG†b) , Member

SUMMARY We propose an incremental relaying protocol in conjunction with partial relay selection with the aim of making efficient use of the degrees of freedom of the channels as well as improving the performance of dual hop relaying with partial relay selection (DRPRS). Specifically, whenever the direct link from the source to the destination is not favorable to decoding, the destination will request the help of the opportunistic relay providing highest SNR of the links from the source. Theoretical analyses, as well as simulation results, verify that our scheme outperforms the DRPRS scheme in terms of bit error probability. key words: amplify-and-forward, incremental relaying, partial relay selection

1.

Introduction

Recently, A. Bletsas et al. have proposed a spectral-efficient, full spatial-diversity achievable relaying strategy in which only the best relay possessing the highest instantaneous channel gain to the destination (e.g. using distributed timers) is involved in the cooperative transmission [1]. However, this cooperative strategy requires the full channel state information composed of instantaneous signal-to-noise (SNR) ratios of the first hop and the second hop for the best relay selection procedure resulting in implementation difficulties in certain practical scenarios. To address this concern, [2] proposed opportunistic amplify-and-forward (AF) relaying where the best relay is selected solely based on partial channel state information (i.e., only neighboring (1 hop) channel information is available to the nodes), which can prolong the network lifetime and also reduce the need for perfect time synchronization and the centralized processing approach. More recently, this scheme has been further studied and extended in [3]–[5]. Most of the above-mentioned protocols, however, suffer from one disadvantage — a loss in spectral efficiency as multiple orthogonal channel allocations (including one broadcasting phase and one or more retransmission phases) are required to transmit one data packet. However, it can be observed that when the signal-to-noise ratio (SNR) is sufficiently large, the destination may be able to decode the Manuscript received October 17, 2009. Manuscript revised January 13, 2010. † The authors are with the Department of Electrical Engineering, University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, 680-749 Korea. ∗ This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (No. R01-2007-000-20400-0). a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1587/transcom.E93.B.1317

message based only on the signal from the source. In such cases, a limited feedback from the destination can avoid the retransmission phase, thus improving the spectral efficiency as in [6]. In this letter, we propose and analyze the performance of an incremental relaying employing partial relay selection (IRPRS), which offers a better performance and also achieves higher spectral efficiency than dual hop relaying with partial relay selection [2]. The contributions of this paper are as follows. First, we derive tight closed-form expressions for bit error probability (BEP) of the two cases: with and without using diversity combiner at the destination and then provide an asymptotic form of BEP for both schemes. Second, based upon these statistical expressions, we perform numerical investigation and a performance comparison of two these schemes showing that: • The analysis results are in excellent agreement with the simulated one in all range of operating SNRs. The IRPRS with diversity combiner performs slightly better than that without MRC at low and medium SNR regimes; however, they both converge to the S-D channel since the switching threshold is small in comparison with average SNRs. • Our schemes not only provide better performance but also achieve higher spectral efficiency than the conventional dual hop relaying with partial relay selection. 2.

System Model

We consider a wireless relay network consisting of one source (S), N relays Ri with i = 1, 2, . . . , N and one destination (D). Each node is equipped with single antenna and operates in half-duplex mode. The system uses orthogonal channels for transmission to eliminate mutual interference. To facilitate the protocol description, we assume that a timedivision protocol is used. In the first time slot, the source broadcasts the information to N relays as well as to the destination. Under assumption that the receiver at the destination employs symbol-bysymbol detection, at the end of the first time slot, the destination checks the quality of the direct link between the source and the destination. If it is greater than a predetermined threshold (γth ), the “success” message (i.e., one-bit feedback signal may be used [6]) will be broadcast, then the source will start transmitting with a new signal while the relays still keep silent in the following time slot. Otherwise,

c 2010 The Institute of Electronics, Information and Communication Engineers Copyright 

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the “failure” message will be sent to demand the help from the best relay. Let hS D , hS Ri and hRi D be the link coefficients between the source to the destination, the source to relay i and relay i to the destination, respectively. Due to Rayleigh fading, the channel powers, denoted by α0 = |hS D |2 , α1,i = |hS Ri |2 and α2,i = |hRi D |2 , are independent and exponential random variables with parameters λ0 , λ1,i and λ2,i , respectively. The average transmit powers for the source and the relays in two time slots are denoted by ρ1 = ζρ0 and ρ2 = (1 − ζ)ρ0 , respectively where ρ0 denotes the transmit power for the source in case of direct transmission and ζ is a power allocation ratio (0 ≤ ζ ≤ 1). Let us define the effective instantaneous SNRs for the S → D, S → Ri and Ri → D links as γ0 = ρ1 α0 , γ1,i = ρ1 α1,i and γ2,i = ρ2 α2,i , respectively. Assuming that all relay nodes are selected and clustered together by long-term routing process based on average channel powers, it therefore ensures that all channels from S → Ri and Ri → D have the same average channel power, i.e., γ¯ 1,i = ρ1 λ1,i = γ¯ 1 and γ¯ 2,i = ρ2 λ2,i = γ¯ 2 , respectively. We further assume 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. We begin by considering the case when the destination requires the help from the relays. Due to the use of partial relay selection, only the relay providing highest SNR of the links from the source will serve as the forwarder in the second phase. Let β1 denotes the instantaneous SNR of the link from the source to the best relay, we have β1 = max γ1,i

(1)

i=1,...,N

Suppose that γ1,i are independent and identically distributed according to the exponential distribution, the probability density function (PDF) of γ1,i is given by fγ1,i (γ) = exp(−γ/¯γ1 )/¯γ1 . If the links from the source are independently faded, thanks to [7], the PDF of β1 is written by   N N−1 N i − γ¯iγ N − γ¯γ  − γ¯γ i−1 fβ1 (γ) = e 1 1−e 1 = (−1) e 1 i γ¯ 1 γ¯ 1 i=1

relaying link at high SNR regime is dominated by the weakest link [8], then it can be modeled at high SNRs as a direct communication link whose equivalent SNR is written as follows: β = β1 β2 /(β1 + β2 + 1) ≈ min{β1 , β2 } From (5), the joint PDF of β is given by [7, Eq. (6-81)] fβ (γ) = fβ1 (γ)+ fβ2 (γ)−Fβ1 (γ) fβ2 (γ)−Fβ2 (γ) fβ1 (γ) Substituting (2)-(4) into (6) yields the PDF of β as   N  N 1 − μγ fβ (γ) = (−1)i−1 e i i μi i=1

(6)

(7)

where μi = (i/¯γ1 + 1/¯γ2 )−1 . 3.

Performance Analysis

3.1 Bit Error Probability In incremental relaying, the destination will request the assistance from the best relay if the SNR of the direct link is less than the pre-determined threshold, i.e., γ0 ≤ γth . Otherwise, the destination will decode the message only based on the direct signal sent by the source. Therefore, by using the rule of total probability, the end-to-end BEP of the system is derived as follows: (2) Pb = Pr(γ0 > γth )P(1) D + Pr(γ0 ≤ γth )PD

= exp(−γth /¯γ0 )P(1) γ0 )]P(2) D + [1 − exp(−γth /¯ D

(8)

where P(1) D is the conditional average BEP of the S-D link given that γ0 > γth and P(2) D denotes the conditional average BEP that an error occurs in the cooperative transmission. The conditional average BEP of the S-D link over Rayleigh fading channel for M-ary square quadrature amplitude (MQAM) modulation (M = 4m , m = 1, 2, . . .) with a Gray code mapping of bits to symbols can be given as [9] √

(2) P(1) D =

The corresponding cumulative distribution function (CDF) of β1 can be readily obtained by integrating fβ1 (γ) between 0 and γ.

(5)

∞ log υj 2 M 0

j=1

ϕkj erfc

√

ςk γ fγ0 |(γ0 ≥γth ) (γ)dγ

(9)

k=0

(4)

√ (2k+1)2 3 log2 M where υ j = (1 − 2− j ) M − 1, ςk = and (2M−2)



 j−1  

k2 √ √ √ j−1 M log2 M . Furtherϕkj = (1) M 2 j−1 − k2√ M + 12 more, we define . and erfc(.) as the floor and complementary error function, respectively. In a flat Rayleigh fading channel, the conditional PDF of γ0 |(γ0 ≥ γth ) can be obtained by using conditional probability [7] as follows:  0, γ < γth fγ0 |(γ0 ≥γth ) (γ) = eγth /γ¯ 0 −γ/γ¯ 0 (10) e , γ ≥ γth γ¯ 0

Under the assumption that two hops are subject to independent fading and making use the fact that the dual hop AF

where γ¯ 0 = E[γ0 ] = ρλ0 and E[.] denotes the expectation operator. Substituting (10) into (9) and taking the integral with respect to γ, we achieve the conditional BEP P1D as

 Fβ1 (γ) =

γ 0

fβ1 (γ)dγ =

  N   N − iγ (−1)i−1 1 − e γ¯ 1 i i=1

(3)

Further, let β2 represents the instantaneous SNR of the link from the best relay to the destination. In a flat Rayleigh fading channel, the PDF and CDF of β2 can be expressed as fβ2 (γ) =

1 − γ¯γ −γ e 2 , Fβ2 (γ) = 1 − e γ¯ 2 γ¯ 2

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Similarly, based on the PDF of γΣ given in (14), the second term on the right hand side of (8) for the case of using combining technique can be derived as

follows: P(1) D =

√ ∞ log υj 2 M γth γth

= e γ¯ 0

j=1

ϕkj erfc

√

ςk γ

k=0

√ υj log 2 M j=1

eγth /γ¯ 0 − γ¯γ e 0 dγ γ¯ 0

 I1 ϕkj , ςk , γ¯ 0 , γth

√

P(2) D = (11)

k=0

  ∞ γ I1 (a, b, c, γth ) = a erfc bγ 1c e− c dγ γ   γ 

th   th γth bc (1 + bc) erfc = a e− c erfc bγth − 1+bc c

Whenever the S-D channel is not sufficiently high, i.e., γ0 < γth , the destination sends a request-to-forward (“failure”) message to the relays to demand the help from the best relay. It is emphasized that the destination does not utilize a maximal ratio combiner (MRC) in this case, then, from (7), it is straightforward to derive the second term, P2D , in (8) as follows: √ ∞ log υj 2 M √ = ϕkj erfc ςk γ fβ (γ)dγ P(2) D

=

j=1 k=0

k=0

ϕkj

    N  ς k μi i−1 N (−1) 1− (12) i 1+ςk μi i=1

3.1.2 With Diversity Combiner The destination will combine two received signals in two time phases as per the rule of MRC, the combined instantaneous SNR is given by γΣ = γ0 |(γ0 < γth ) + β

(13)

Under the assumption of independence of γ0 and β and without getting into the details of the derivation, the joint PDF of γΣ can be obtained by ⎧ N ⎪ (−1)i−1 (Ni ) fL (γ) ⎪ ⎪ ⎪ i=1 ⎪ ⎪ , γ < γth ⎨ 1−e−γth /γ¯ 0 fγΣ (γ) = ⎪ (14) ⎪ N  i−1 N ⎪ (γ) (−1) f ⎪ ( ) U i ⎪ ⎪ ⎩ i=1 , γ ≥ γth 1−e−γth /γ¯ 0

where fL (γ) and fU (γ) are defined as follows: ⎧ γ¯ 0  1 − γ μi  1 − γ γ¯ 0 ⎪ ⎪ + μi−¯γ0 μi e μi , γ¯ 0  μi ⎪ ⎨ γ¯ 0−μi γ¯ 0 e fL (γ) = ⎪ γ ⎪ γ − γ¯ ⎪ ⎩ e 0, γ¯ 0 = μi γ¯ 02 ⎧  −γ  1 − 1   γ ⎪ ⎪ 1−e th γ¯ 0 μi 1 − μi ⎪ ⎪ , γ¯ 0  μi ⎪ 1−γ¯ 0 /μi μi e ⎨ fU (γ) = ⎪ ⎪ γ ⎪ ⎪ γth − γ¯ ⎪ ⎩ e 0, γ¯ 0 = μi γ¯ 2 0

ςk γ fγΣ (γ)dγ

k=0

√ υj  log N 2 M 

3.1.1 Without Diversity Combiner

υj log 2 M

√

⎧ γth

√   j ⎪ ⎪ ⎪ ϕk erfc ςk γ fγΣ (γ)dγ+ υj ⎪ log ⎪ 2 M ⎪ ⎪ ⎨ 0 = ⎪

√  ∞ j ⎪ ⎪ ⎪ j=1 k=0 ⎪ ϕ erfc ςk γ fγΣ (γ)dγ ⎪ ⎪ k ⎩γ

=

j=1

j=1

0

ϕkj erfc

√

where I1 (.) is defined as

0 √

∞ log υj 2 M

j=1

th

(−1)i−1

N 

k=0 i=1



1 − e−γth /γ¯ 0

i

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

K (¯γ0 , μi )



(15)

where K(¯γ0 , μi ) is defined as ⎧⎡ ⎤   ⎪ ⎥⎥⎥ ⎢⎢⎢ γ¯ 0 I2 ϕ j , ςk , γ¯ 0 , γth ⎪ ⎪ ⎪ γ ¯ −μ k ⎥⎥⎥ ⎢ 0 i ⎪ ⎢⎢⎢ ⎪ ⎪ ⎥⎥⎥



 ⎪ ⎢⎢⎢ ⎪ j μ ⎪ ⎥⎥⎥ ⎪ ⎢⎢⎢+ μ −iγ¯ I2 ϕk , ςk , μi , γth ⎪ ⎪ ⎥⎥⎥, γ¯ 0  μi ⎪ ⎢⎢⎢  i 0  ⎪  ⎪ ⎪

j ⎥⎥⎥⎥ ⎨⎢⎢⎢ 1−e−γth γ¯10 − μ1i ⎢⎣+ K(¯γ0 , μi ) = ⎪ I1 ϕk , ςk , μi , γth ⎥⎦ ⎪ ⎪ 1−γ¯ 0 /μi ⎪ ⎪ ⎪ ⎡ ⎤ ⎪  ⎪ ⎪ ⎢⎢I3 ϕ j , ςk , γ¯ 0 , γth ⎪ ⎢ ⎥⎥⎥⎥ ⎪ ⎪ k ⎢ ⎪ ⎢⎢⎢ ⎥⎥, ⎪ γ¯ 0 = μi

 ⎪ ⎪ ⎪ ⎩⎣+¯γth /γ¯ 0 I1 ϕ j , ςk , γ¯ 0 , γth ⎥⎦ k and I2 (.) and I3 (.) can be solved in closed-form as

  γth γ I2 (a, b, c, γth ) = a erfc bγ 1c e− c dγ 0   

   γth γth bc (1 = a 1−e− c erfc bγth − 1+bc erf + bc) c 0

  γth γ I3 (a, b, c, γth ) = a erfc bγ cγ2 e− c dγ 0 ⎤ ⎡



  bγ (bc+1)γ th − c th ⎥ ⎢⎢⎢1−e− γcth 1+ γth erfc bγth + ⎥⎥⎥ e 2 ⎢ c ⎥⎥⎥   π(bc+1) = a⎢⎢⎢⎢⎢  2 ⎥⎥⎦ (bc+1)γth ⎣ − (bc+3/2) 3bc erf c (bc+1) where erf(.) is the error function. 3.1.3 Asymptotic Analysis of BEP Although (8) provides an evaluation of the average BEP for square M-QAM over the SNR region of interest for both cases, the form of this expression does not lend any insight into the asymptotic behavior of IRPRS. To get a better picture of this behavior, we investigate the system performance at high SNR regime. In particular, it should be noted that the first term in (8) becomes the dominant term since γ0 approaches to ∞. Mathematically speaking, since the switching threshold γth is very small in comparison with the average SNR, the second term in (8) vanishes due to the fact that Pr(γ0 > γth ) = 1 − Pr(γ0 ≤ γth ) ≈ 1 leading to the approximation form of Pb at high SNRs as follows: &

'

√ υj log 2 M 

Pb ≈ 1−Pr (γ0 < γth ) P1D =

j=1 k=0

I1 (ϕkj , ςk , γ¯ 0 , γth) (16)

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which is purely dependent on the quality of the sourcedestination transmission, modulation scheme and the predetermined threshold, γth . 3.2 Spectral Efficiency Same as the incremental protocol with one relay [11], the proposed protocol offers a spectral efficiency, which is less than R but larger than R/2 where R denotes the spectral efficiency of direct transmission. In particular, the spectral ef¯ can be obtained ficiency of IRPRS schemes, denoted by R, as R¯ = R Pr(γ0 ≥ γth ) + R/2 Pr(γ0 < γth ) & ' = R/2 1 + exp(−γth /¯γ0 )

(17)

From (17), it is straightforward to arrive that R/2 ≤ R¯ ≤ R 4.

(18)

Numerical Results

In order to validate our analysis given in the previous section, numerical results are provided in the sequel. Figure 1 plots the BEPs for 4-QAM versus average SNR per bit for the systems with 1, 3 and 5 relays. For a fair comparison to direct transmission (DT), the uniform power allocation is employed in order to keep the total transmit-power constraint, i.e., ζ = 0.5. As can be clearly seen, IRPRS outperforms DRPRS in all ranges of operating SNRs. The bound on the BEP is loose at low SNR regime; however, it gets tighter at high SNR regime. As predicted by the analysis, the average BEP of IRPRS converges to the performance of direct transmission given that γ0 > γth . We can see that the gap between the respective BEP for IRPRS and DT merely depends on the underlying modulation scheme and the predetermined threshold. For example, with γth = 5 and 4-QAM, the curve of the approximated BEP versus average SNR per bit in decibels for IRPRS is parallel to those of the equivalent

curves for DRPRS and DT and shifted around 30 dB to the left. The effect of using diversity combiner on the performance of IRPRS schemes is further investigated in Fig. 1. It can be observed that the IRPRS scheme with combining technique gains advantage over its counterpart without combining technique at low SNR values and both their performance approach the bound totally determined by the quality of the first time slot communication at high SNR values. This behavior can be explained by the following arguments. For high SNR values, the IRPRS rarely requests the help from the best relay and therefore the end-to-end BEP is dominated by the source-destination communication. On the other hand, for channels with low SNR values, the communication in the first time slot signal is not sufficient for error-less transmission and therefore leading to the high demand of cooperative transmission. Figure 2 illustrates the BER performance of the proposed scheme as a function of γth . We can see that the optimum value of γth is a complex function of average SNRs. For example, for a fixed average SNR of 20 dB, the BER performance will be better if γth increases from 0 to 6. However, if γth is varying outside of this range, e.g., 6 ≤ γth < ∞, no improvement on the BER performance can be observed. The advantage of SSCSR schemes can be further ascertained by referring to Fig. 3 where the spectral efficiency of the IRPRS system is shown. Compared with DRPRS schemes, the advantage of IRPRS schemes is the improve-

Fig. 2 Effect of γth on the system performance (4-QAM, γth = 5, λ0 = 1, λ1,i = 1, λ2,i = 1, ζ = 0.5).

Fig. 1

BEP for IRPRS (4-QAM, γth = 5, λ0 = 1, λ1,i = 2, λ2,i = 1).

Fig. 3 Achievable spectral efficiency for IRPRS (γth = 5, λ0 = 1, λ1,i = 2, λ2,i = 1).

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average SNRs results in different optimal values for ζ. The results exacted from the figures provide some insight information on how much power should be allocated to improve performance. In particular, with a fixed value of threshold, as average SNR increases, more power should be allocated at the relays. On the other hand, as clearly seen in Fig. 5, due to the use of partial relay selection, number of cooperative relays in the networks has no influence on the optimum value of the power allocation ratio. 5.

Fig. 4 Varying power allocation (γth = 5, λ0 = 1, λ1,i = 1, λ2,i = 1, N = 3).

Conclusion

We developed and analyzed a novel combination of incremental relaying and partial relay selection. Compared to the DRPRS, IRPRS is more spectrally efficient, better performing and readily applicable to wireless sensor/ad-hoc networks. Our analytical derivations have been verified by simulations results and it is shown that the derived BEPs are remarkable close to the simulated results. References

Fig. 5 Varying power allocation (γth = 5, λ0 = 1, λ1,i = 1, λ2,i = 1, Eb /N0 = 20 dB).

ment of spectral efficiency with the cost of a limited feedback from the destination. At high SNR values, the spectral efficiency of IRPRS reaches to that of direct transmission. This point can be explained by considering the fact that at such SNRs the destination can successfully decode based only on the signal sent by the source. Up to this point, we did not consider the effect of the power allocation ratio, ζ, on the system performance. It is obvious to see that although equal power allocation, i.e., ζ = 0.5, is a natural and reasonable choice in practice; it is not optimal for IRPRS networks. In Figs. 4, 5, we show the performance of incremental relaying protocol as a function of power allocation ζ. In general, the different value of

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