Joint Relay Selection and Power Allocation for ... - Semantic Scholar
IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013
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Joint Relay Selection and Power Allocation for Two-Way Relaying with Physical Layer Network Coding Thinh Phu Do, Jin Soo Wang, Iickho Song, Fellow, IEEE, and Yun Hee Kim, Senior Member, IEEE Abstract—We consider a two-way relay (TWR) network with multiple relays, where the relays adopt physical layer network coding based on decode-and-forward. A joint relay selection and power allocation (RS-PA) scheme is proposed to improve the symbol error probability (SEP) of the TWR network using BPSK modulation under the total power constraint. For the proposed scheme and the benchmark scheme performing RS without PA, the average SEP of the TWR network is derived analytically and confirmed with simulation results. It is shown that the proposed scheme provides an asymptotic SNR gain of about 1.76 dB over the benchmark scheme at the same diversity order. Index Terms—Physical layer network coding, power allocation, relay selection, symbol error probability, two-way relay.
I. I NTRODUCTION WO-WAY relay (TWR) has drawn much attention from wireless researchers due to its capability of compensating the spectral loss incurred by conventional one-way relay (OWR) [1], [2]. Recent TWR protocols complete the information exchange between two source nodes in two or three transmission phases (TPs) by employing the network coding designed for multicast [3], [4] while the conventional OWR completes the task in four TPs. The TWR protocols adopt either analog network coding (ANC) based on amplifyand-forward [1] or physical layer network coding (PLNC) based on decode-and-forward [2]. The ANC makes the relay operation simple but necessitates self-interference cancelation (SIC) at the source nodes, whereas the PLNC eliminates the SIC operation at the cost of increased complexity of the relay nodes which decode the XOR bits of the source nodes. With the advantages of the TWR networks, many efforts have been devoted to the performance improvement. On one hand, power allocation (PA) among the nodes has been addressed in the single relay TWR network [5], [6], where the PA is performed once with an average channel state information (CSI) at the transmitting nodes for the improvement of average performance. On the other hand, when multiple relays are available, relay selection (RS) has been studied for the ANC TWR [7] and PLNC TWR [8] networks to reduce the symbol error probability (SEP): Recently, the two approaches are combined into joint relay selection and power allocation (RS-PA) to improve the performance further under different objective
T
Manuscript received September 23, 2012. The associate editor coordinating the review of this letter and approving it for publication was M. Xiao. This work was supported by the National Research Foundation of Korea, with funding from the Ministry of Education, Science, and Technology, under Grants 2012-0001867, 2012R1A1A2040091 and 2012-0005622. T. P. Do, J. S. Wang, and Y. H. Kim (corresponding author) are with the Department of Electronics and Radio Engineering, Kyung Hee University, Yongin, Korea (e-mail: [email protected], {delta310, yheekim}@khu.ac.kr). I. Song is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2013.122013.122134
functions and conditions [9]–[11]. Specifically, the schemes in [9] and [10] aim at maximizing the minimum signal-to-noise ratio (SNR) at the source nodes to reduce the SEP and the scheme in [11] aims at minimizing the total transmit power under the rate constraint. Unlike the PA schemes in [5] and [6] exploiting the average CSI to determine the best solution once, the joint RS-PA schemes in [9]–[11] leverage the performance by exploiting an instantaneous CSI in the determination of the best solution for each information exchange. It is noteworthy that RS-PA schemes have so far been considered only for the ANC, but not for the PLNC. In this letter, a novel joint RS-PA scheme is addressed in the PLNC TWR network with multiple relays for the improvement of the SEP. The metric for the RS with the proposed PA exploiting an instantaneous CSI is derived in a closed form. The average SEP and its asymptotic behavior is analyzed, and verified with simulation results, for the proposed scheme and its benchmark scheme performing RS only. II. S YSTEM M ODEL Consider a TWR network in which two source nodes S1 and S2 exchange BPSK modulated information via M relay nodes {Rm }M m=1 in two TPs. Every node, equipped with a single antenna, operates in half-duplex. For any i = 1, 2 and m = 1, 2, . . . , M , the channel between Si and Rm is reciprocal, Rayleigh fading, and time-invariant over two TPs; the channel is described by him ∼ CN (0, Ωim ), where ∼ stands for ‘distributed as’ and CN (μ, σ 2 ) denotes the circularly symmetric complex Gaussian distribution with mean μ and variance σ 2 . We denote the information bit of Si and its BPSK symbol by bi ∈ {0, 1} and xi = 1−2bi ∈ {−1, 1}, respectively. In the first TP called the multiple access, S1 and S2 transmit their symbols x1 and x2 simultaneously to the relay nodes. Then, relay node Rm receives the signal (1) yRm = P1 h1m x1 + P2 h2m x2 + nRm , where Pi is the transmit power of Si and nRm ∼ CN (0, σ 2 ) is the additive noise at Rm . Assume that Rmo is the only relay node chosen to transmit the signal. The relay node Rmo detects b = b1 ⊕ b2 , where ⊕ denotes the bitwise XOR operation. Equivalently, Rmo detects the BPSK symbol z = 1 − 2b = x1 x2 by employing the maximum likelihood criterion [2] as zˆ=1 e−Dmo (x1 ,x2 ) ≷ e−Dmo (x1 ,x2 ) , (2) {(x1 ,x2 )|x1 x2 =1}
zˆ=−1
{(x1 ,x2 )|x1 x2 =−1}
2 2 √ 1 Pi him xi and where Dm (x1 , x2 ) = σ2 yRm − i=1 (x1 , x2 ) ∈ {(−1, −1), (−1, 1), (1, −1), (1, 1)}. In the second
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TP called the broadcast, Rmo forwards the detected symbol zˆ to S1 and S2 . The signal received at Si is then given by yi = PR himo zˆ + ni , (3)
and
where PR is the transmit power of a relay and ni ∼ CN (0, σ 2 ) is the additive noise at Si . Finally, Si retrieves the bit transmitted from Sj as xˆj = z˜i xi , where z˜i = 1 if h∗imo yi ≥ 0 and z˜i = −1 if h∗imo yi < 0, and (i, j) ∈ Ω with Ω = {(1, 2), (2, 1)}.
γm is maximized with the maximum value
III. J OINT R ELAY S ELECTION AND P OWER A LLOCATION Let us now derive the criterion for joint RS-PA using instantaneous CSI for the PLNC TWR network employing BPSK modulation. When Rm transmits zˆ, the end-to-end (e2e) SEP at Si , with hm = [h1m h2m ] given, can be expressed as Pe2e,i|hm = Pmac|hm 1 − Pbc,i|hm + 1 − Pmac|hm Pbc,i|hm , (4) where Pmac|hm = Pr {ˆ z = z|hm } and Pbc,i|hm = Pr {˜ zi = zˆ|hm }. The SEP of the network, defined by 2 Pe2e|hm = 12 Pe2e,i|hm , is clearly upper bounded as i=1
2
Pe2e|hm ≤ Pmac|hm
1 + Pbc,i|hm . 2 i=1
(5)
2 √ Since Pmac|hm ≤ Q 2Pi αim and Pbc,i|hm = i=1 √ Q 2PR αim as derived in [2], we have 2
1
2Pi αim + Q 2PR αim Q Pe2e|hm ≤ , (6) 2 i=1 2
∞ 1 √ exp − u2 du and αim = where Q(x) = x 2π 2 2 |him | /σ . Now, for positive constants c1 and c2 of similar order, we have c1 Q(u1 )+c2 Q(u2 ) ≈ cko Q(min(u1 , u2 )) if u1 = u2 and c1 Q(u1 ) + c2 Q(u2 ) = (c1 + c2 )Q(min(u1 , u2 )) if u1 = u2 , where ko = arg min (uk ). Hence, based on (6), it is not k∈{1,2}
unreasonable to approximate the SEP of the network as
2γm , Pe2e|hm κQ Q
(7)
where γm = min (P1 α1m , P2 α2m , PR min(α1m , α2m )) and κQ =
2 i=1
ei +
1 2
2 i=1
(8)
fi with ei = 1 if Pi αim = γm and
ei = 0 otherwise and fi = 1 if PR αim = γm and fi = 0 otherwise. The proposed RS-PA scheme selects (P1 , P2 , PR ) and the relay Rmo in such a way that γm is maximized under the total power constraint. Specifically, we first determine the PA as max min(P1 α1m , P2 α2m , PR min(α1m , α2m ))
(9)
subject to P1 + P2 + PR ≤ PT , Pi ≥ 0, and PR ≥ 0. It can be shown after some manipulations that, when P1 α1m = P2 α2m = PR min(α1m , α2m ), or equivalently, when PT αjm , (i, j) ∈ Ω (10) Pi = α1m + α2m + max(α1m , α2m )
PR =
γm =
PT max(α1m , α2m ) , α1m + α2m + max(α1m , α2m )
PT α1m α2m . α1m + α2m + max(α1m , α2m )
(11)
(12)
Once the PA is completed as described above, the RS is performed to maximize the SNR as mo = arg max γm , after 1≤m≤M
which Si and Rmo set their power as specified in (10) and (11) with m = mo for the TWR as described in Section II. The proposed RS-PA scheme can be implemented practically by extending the RS scheme in [12]. Firstly, each source node broadcasts a beacon signal containing a preamble to the relay nodes so that relay Rm can estimate α1m and α2m and then compute γm . Each relay then constructs a feedback signal containing a preamble and the information on γm , and starts a timer Tm , the duration of which is inversely proportional to γm . The relay node Rmo , with its timer being expired first, transmits the feedback signal: Sensing the feedback signal of Rmo , the other relays reserve their transmission. Subsequently, source node Si retrieves γmo from the feedback signal sent by Rmo and estimates αimo , with which Pi = γmo /αimo is computed for data exchange. IV. P ERFORMANCE A NALYSIS By noting that κQ = 52 from (10) and (11), the average SEP of the TWR network employing the proposed RS-PA scheme can be approximated as (z) 5 5 ∞ e−z Fγmax ¯ √ √ dz (13) 2γmax Pe2e ≈ E Q = 2 4 0 π z from (7), where γmax =
max γm and FX (·) denotes the
1≤m≤M
cumulative distribution function (cdf) of a random variable (rv) X. The equality in (13) is obtained easily from integration M (z), we need to derive by parts. Since Fγmax (z) = m=1 Fγm (z) for the evaluation of (13). Fγm Theorem 1: For two independent exponential rvs X and Y with parameters λ1 and λ2 , respectively, we have XY ∼ T (λ1 , λ2 ), X + Y + max(X, Y )
(14)
where the cdf of T (λ1 , λ2 ) is FT (λ1 ,λ2 ) (z) = 1 + e−3(λ1 +λ2 )z − e−(λ1 +λ2 )z · λi e−λj z H 2λj z 2 , λi , 2z (i,j)∈Ω
for z > 0 with H(a, b, c)
∞ c
(15)
exp − ua + bu du.
Proof: See Appendix A. Lemma 1: An upper bound on FT (λ1 ,λ2 ) (z) is given by FTU(λ1 ,λ2 ) (z) = 1 + e−3(λ1 +λ2 )z − e−(λ1 +λ2 )z · λi e−λj z H L 2λj z 2 , λi , 2z , (i,j)∈Ω
(16)
DO et al.: JOINT RELAY SELECTION AND POWER ALLOCATION FOR TWO-WAY RELAYING WITH PHYSICAL LAYER NETWORK CODING
where
303
L
H (a, b, c) =
√
4a 1 K1 1 − e−bc 4ab − bc a
b a (17) · ce− c + aEi − c
with Kν (x) the ν-th order modified Bessel function of the ∞ −t second kind and Ei(x) = − −x e t dt for x < 0 [13, 8.211.1]. Proof: See Appendix B. (12) and Theorem 1, it is clear that γm ∼ From −1 −1 T ρ1m , ρ2m , where ρim = E {PT αim } = ρΩim with ρ = appr1 PT /σ 2 . Thus, from (13) and (15), we have P¯e2e ≈ P¯e2e , where ∞ M e−z 5 appr1 P¯e2e = FT (ρ−1 ,ρ−1 ) (z) √ √ dz. (18) 1m 2m 4 0 π z m=1
Fig. 1. Average SEP of the PLNC TWR employing the RS-PA and RS schemes.
Replacing FT (ρ−1 ,ρ−1 ) (z) with FTU ρ−1 ,ρ−1 (z) in (18), we ( 1m 2m ) 1m 2m appr2 will get P¯e2e ≈ P¯e2e , where ∞ M 5 e−z appr2 U P¯e2e = FT (ρ−1 ,ρ−1 ) (z) √ √ dz. (19) 1m 2m 4 0 π z m=1 Let us now investigate the behavior of P¯e2e at high SNR using (19). As ρim → ∞, or equivalently as λi → 0 in (16), we have H L 2λj z 2 , λi , 2z ≈ λ1i − 2z since K1 (x) ≈ x1 , e−x ≈ 1 − x, and Ei(x) ≈ ln |x| as x → 0. We then have FTU(λ1 ,λ2 ) (z) ≈ 2(λ1 + λ2 )z for λi → 0, which leads to 1 1 + FTU(ρ−1 ,ρ−1 ) (z) ≈ 2 z. (20) 1m 2m ρ1m ρ2m Employing (20) in (19) and ρim = Ωim ρ, we obtain M 1 5 2M Γ M + 12 1 √ M P¯e2e (ρ) ≈ + , (21) 4 πρ Ω1m Ω2m m=1 ∞ where Γ(x) = 0 e−t tx−1 dt is the Gamma function and the argument ρ in P¯e2e (ρ) is shown explicitly for convenience. In Appendix C, when P1 = P2 = PR = PT /3, we have derived the approximate average SEP M e−z 3 ∞ −3 ρ 1 + ρ 1 z eq ¯ 1m 2m √ √ dz (22) 1−e Pe2e ≈ 4 0 m=1 π z of the TWR network employing the RS without PA. In the high SNR regime (ρim → ∞), (22) becomes M 1 3 3M Γ M + 12 1 eq ¯ √ M Pe2e (ρ) ≈ + (23) 4 πρ Ω1m Ω2m m=1
−3 ρ 1 + ρ 1 z 1 1 1m 2m z when ρim → ≈ 3 ρ1m + ρ2m since 1 − e ∞. Clearly, it is observed from (21) and (23) that the proposed RS-PA has a diversity order M , identical to that of the RS without PA, and that M 5 2 P¯e2e (ρ) ≈ . (24) eq 3 3 P¯e2e (ρ)
Fig. 2.
Comparison of PLNC and ANC with the RS-PA and RS schemes.
eq It is interesting to note from (24) that P¯e2e (ρ) ≈ P¯e2e (βρ) with 1/M β = 1.5(0.6) : An important implication of this observation is that the proposed scheme can provide an asymptotic SNR gain of about 1.76 dB over the RS without PA when M → ∞.
V. P ERFORMANCE E VALUATION Assuming that Ωim = 1, or equivalently ρim = PT /σ 2 , for m = 1, 2, . . . , M , let us now evaluate the average SEP of the PLNC TWR network employing the proposed RS-PA scheme. In Fig. 1, the average SEP of the PLNC TWR is shown for the proposed scheme (denoted by ‘RS-PA’) and the RS without PA (denoted by ‘RS’), where ‘sim’ denotes results from simulation; ‘appr1’, ‘appr2’, and ‘asymp’ in RSPA represent (18), (19), and (21), respectively; and ‘appr’ and ‘asymp’ in RS represent (22) and (23), respectively. Fig. 1(a) shows that the approximate SEPs agree with simulation results quite well when the SEP is less than 10−2 . The figure also shows that RS-PA outperforms RS by about 0.86 dB when M = 8 at the SEP of 10−6 . Fig. 1(b) shows that the asymptotic SEPs agree with the approximate SEPs in the high SNR regime, especially for the proposed RS-PA. It is observed/confirmed that (i) the asymptotic SNR gain β of the RS-PA over the RA increases with M and (ii) the diversity order is identically M for both the RS-PA and RS.
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In Fig. 2, the performance of PLNC and ANC is compared via simulation when RS-PA and RS are employed. It is observed that PLNC outperforms ANC and that the performance gain increases with M . The gain results from the facts that (i) the relay in PLNC transmits one modulation symbol while the relay in ANC transmits a linear combination of two modulation symbols and (ii) no noise amplification occurs in the relayed signal of PLNC: Thus, although the gain may vary with the SNR and system parameters due to the nonlinear nature of (i) and (ii), PLNC in general outperforms ANC. VI. C ONCLUSION We have proposed a joint RS-PA scheme for the PLNC TWR network with BPSK modulation in multiple-relay environments. The proposed scheme selects the relay with an instantaneous CSI to reduce the SEP, taking also into account PA among the nodes under the total power constraint. We have derived the average SEP and its asymptotic behavior of the TWR network employing the proposed scheme and the benchmark scheme employing the RS without PA. The results show that the proposed scheme provides the same diversity order as, and an asymptotic SNR gain of about 1.76 dB when compared to, the RS without PA. It is also observed that, with RS-PA, the PLNC outperforms the ANC significantly. A PPENDIX A. Proof of Theorem 1 The cdf of Z =
XY X+Y +max(X,Y )
can be expressed as
FZ (z) = Pr{Z ≤ z, X ≤ Y } + Pr{Z ≤ z, X > Y }. (25) I1
I2
With the probability density functions (pdfs) fX (x) = λ1 e−λ1 x and fY (y) = λ2 e−λ2 y for x, y ≥ 0, we have XY ≤ z, X ≤ Y I1 = Pr X + 2Y ∞ = Pr{(y − z)X ≤ 2yz, X ≤ y}fY (y)dy 0 ∞ 3z 2yz = Pr{X ≤ y}fY (y)dy + Pr X ≤ fY (y)dy y−z 0 3z 3z ∞ 2λ1 yz = 1 − λ2 e−λ1 y e−λ2 y dy − λ2 e− y−z e−λ2 y dy 0 ∞ 3z 2λ1 (u+z)z λ1 + λ2 e−3(λ1 +λ2 )z −λ2 (u+z) u = − λ2 e− du. λ1 + λ2 2z (26) Due to the symmetry, it is straightforward to obtain I2 from I1 by exchanging λ1 and λ2 . Then, the cdf FT (λ1 ,λ2 ) (z) = I1 +I2 of Z ∼ T (λ1 , λ2 ) is expressed as (15). B. Proof of Lemma 1 The upper bound FTU(λ1 ,λ2 ) (z) is obtained by deriving the lower bound H L (a, b, c) on ∞ c a a H(a, b, c) = e−( u +bu) du − e−( u +bu) du, (27) 0 0 √ (g)√ 4a ¯ H(a,b,c) = K 4ab ( ) 1 b
where (g) is from [13, 3.324.1]. First, c 1 c −a ¯ u e du e−bu du H(a, b, c) ≤ c 0 0
(28)
from the Chebyshev’s inequality for integrals [13, 12.314]. c a c Next, noting that 0 e−bu du = 1b (1−e−bc ) and that 0 e− u du a 1 −a = ce c + aEi − c from a change of variable t = u and −bc a ¯ integration by parts, we have H(a, b, c) ≤ 1−ebc ce− c a +aEi − c , eventually leading to H(a, b, c) ≥ H L (a, b, c). eq C. Derivation of P¯e2e
Let us derive the average SEP of the TWR network employing RS without PA when P1 = P2 = PR = 13 PT . We eq have κQ = 32 and γm = γm P3T min (α1m , α2m ) in (7). eq is selected, the average SEP When the RS that maximizes γm is derived with an approach similar to that for (13) as eq (z) 3 ∞ e−z Fγmax eq √ √ P¯e2e dz, (29) ≈ 4 0 π z eq = where γmax
eq eq max γm . Since the cdf of γm is easily
1≤m≤M
obtained to be F
eq γm
(z) = 1 − e
−3 ρ 1 + ρ 1 z 1m
2m
, we have (22).
R EFERENCES [1] B. Rankov and A. Wittneben, “Spectral efficient protocols for half-duplex fading relay channels,” IEEE J. Sel. Areas Commun., vol. 25, no. 2, pp. 379–389, Feb. 2007. [2] M. Ju and I. M. Kim, “Error performance analysis of BPSK modulation in physical layer network coded bidirectional relay channels,” IEEE Trans. Commun., vol. 58, no. 10, pp. 2770–2775, Oct. 2010. [3] M. Xiao and T. Aulin, “On the bit error probability of noisy channel networks with intermediate node encoding,” IEEE Trans. Inf. Theory., vol. 54, no. 11, pp. 5188–5198, Nov. 2008. [4] M. Xiao and T. Aulin, “Optimal decoding and performance analysis of a noisy channel network with network coding,” IEEE Trans. Commun., vol. 57, no. 5, pp. 5188–5198, May 2009. [5] Y. Zhang, Y. Ma, and R. Tafazolli, “Power allocation for bidirectional AF relaying over Rayleigh fading channels,” IEEE Commun. Lett., vol. 14, no. 2, pp. 145–147, Feb. 2010. [6] H. Bagheri, M. Ardakani, and C. Tellambura, “Resource allocation for two-way AF relaying with receive channel knowledge,” IEEE Trans. Wireless Commun., vol. 11, no. 6, pp. 2002–2007, June 2012. [7] L. Song, G. Hong, B. Jiao, and M. Debbah, “Joint relay selection and analog network coding using differential modulation in two-way relay channels,” IEEE Trans. Veh. Technol., vol. 59, no. 6, pp. 2932–2939, July 2010. [8] Y. Li, R. H. Y. Louie, and B. Vucetic, “Relay selection with network coding in two-way relay channels,” IEEE Trans. Veh. Technol., vol. 59, no. 9, pp. 4489–4499, Nov. 2010. [9] S. Talwar, Y. Jing, and S. Shahbazpanahi, “Joint relay selection and power allocation for two-way relay networks,” IEEE Signal Process. Lett., vol. 18, no. 2, pp. 91–94, Feb. 2011. [10] Y. Yang, J. Ge, and Y. Gao, “Power allocation for two-way opportunistic amplify-and-forward relaying over Nakagami-m channels,” IEEE Trans. Wireless Commun., vol. 10, no. 7, pp. 2063–2068, July 2011. [11] M. Zhou, Q. Cui, R. Jantti, and X. Tao, “Energy-efficient relay selection and power allocation for two-way relay channel with analog network coding,” IEEE Commun. Lett., vol. 16, no. 6, pp. 816–819, June 2012. [12] A. Beltsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006. [13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edition. Academic, 2007.
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Joint Relay Selection and Power Allocation for Two-Way Relaying with Physical Layer Network Coding Thinh Phu Do, Jin Soo Wang, Iickho Song, Fellow, IEEE, and Yun Hee Kim, Senior Member, IEEE Abstract—We consider a two-way relay (TWR) network with multiple relays, where the relays adopt physical layer network coding based on decode-and-forward. A joint relay selection and power allocation (RS-PA) scheme is proposed to improve the symbol error probability (SEP) of the TWR network using BPSK modulation under the total power constraint. For the proposed scheme and the benchmark scheme performing RS without PA, the average SEP of the TWR network is derived analytically and confirmed with simulation results. It is shown that the proposed scheme provides an asymptotic SNR gain of about 1.76 dB over the benchmark scheme at the same diversity order. Index Terms—Physical layer network coding, power allocation, relay selection, symbol error probability, two-way relay.
I. I NTRODUCTION WO-WAY relay (TWR) has drawn much attention from wireless researchers due to its capability of compensating the spectral loss incurred by conventional one-way relay (OWR) [1], [2]. Recent TWR protocols complete the information exchange between two source nodes in two or three transmission phases (TPs) by employing the network coding designed for multicast [3], [4] while the conventional OWR completes the task in four TPs. The TWR protocols adopt either analog network coding (ANC) based on amplifyand-forward [1] or physical layer network coding (PLNC) based on decode-and-forward [2]. The ANC makes the relay operation simple but necessitates self-interference cancelation (SIC) at the source nodes, whereas the PLNC eliminates the SIC operation at the cost of increased complexity of the relay nodes which decode the XOR bits of the source nodes. With the advantages of the TWR networks, many efforts have been devoted to the performance improvement. On one hand, power allocation (PA) among the nodes has been addressed in the single relay TWR network [5], [6], where the PA is performed once with an average channel state information (CSI) at the transmitting nodes for the improvement of average performance. On the other hand, when multiple relays are available, relay selection (RS) has been studied for the ANC TWR [7] and PLNC TWR [8] networks to reduce the symbol error probability (SEP): Recently, the two approaches are combined into joint relay selection and power allocation (RS-PA) to improve the performance further under different objective
T
Manuscript received September 23, 2012. The associate editor coordinating the review of this letter and approving it for publication was M. Xiao. This work was supported by the National Research Foundation of Korea, with funding from the Ministry of Education, Science, and Technology, under Grants 2012-0001867, 2012R1A1A2040091 and 2012-0005622. T. P. Do, J. S. Wang, and Y. H. Kim (corresponding author) are with the Department of Electronics and Radio Engineering, Kyung Hee University, Yongin, Korea (e-mail: [email protected], {delta310, yheekim}@khu.ac.kr). I. Song is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2013.122013.122134
functions and conditions [9]–[11]. Specifically, the schemes in [9] and [10] aim at maximizing the minimum signal-to-noise ratio (SNR) at the source nodes to reduce the SEP and the scheme in [11] aims at minimizing the total transmit power under the rate constraint. Unlike the PA schemes in [5] and [6] exploiting the average CSI to determine the best solution once, the joint RS-PA schemes in [9]–[11] leverage the performance by exploiting an instantaneous CSI in the determination of the best solution for each information exchange. It is noteworthy that RS-PA schemes have so far been considered only for the ANC, but not for the PLNC. In this letter, a novel joint RS-PA scheme is addressed in the PLNC TWR network with multiple relays for the improvement of the SEP. The metric for the RS with the proposed PA exploiting an instantaneous CSI is derived in a closed form. The average SEP and its asymptotic behavior is analyzed, and verified with simulation results, for the proposed scheme and its benchmark scheme performing RS only. II. S YSTEM M ODEL Consider a TWR network in which two source nodes S1 and S2 exchange BPSK modulated information via M relay nodes {Rm }M m=1 in two TPs. Every node, equipped with a single antenna, operates in half-duplex. For any i = 1, 2 and m = 1, 2, . . . , M , the channel between Si and Rm is reciprocal, Rayleigh fading, and time-invariant over two TPs; the channel is described by him ∼ CN (0, Ωim ), where ∼ stands for ‘distributed as’ and CN (μ, σ 2 ) denotes the circularly symmetric complex Gaussian distribution with mean μ and variance σ 2 . We denote the information bit of Si and its BPSK symbol by bi ∈ {0, 1} and xi = 1−2bi ∈ {−1, 1}, respectively. In the first TP called the multiple access, S1 and S2 transmit their symbols x1 and x2 simultaneously to the relay nodes. Then, relay node Rm receives the signal (1) yRm = P1 h1m x1 + P2 h2m x2 + nRm , where Pi is the transmit power of Si and nRm ∼ CN (0, σ 2 ) is the additive noise at Rm . Assume that Rmo is the only relay node chosen to transmit the signal. The relay node Rmo detects b = b1 ⊕ b2 , where ⊕ denotes the bitwise XOR operation. Equivalently, Rmo detects the BPSK symbol z = 1 − 2b = x1 x2 by employing the maximum likelihood criterion [2] as zˆ=1 e−Dmo (x1 ,x2 ) ≷ e−Dmo (x1 ,x2 ) , (2) {(x1 ,x2 )|x1 x2 =1}
zˆ=−1
{(x1 ,x2 )|x1 x2 =−1}
2 2 √ 1 Pi him xi and where Dm (x1 , x2 ) = σ2 yRm − i=1 (x1 , x2 ) ∈ {(−1, −1), (−1, 1), (1, −1), (1, 1)}. In the second
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TP called the broadcast, Rmo forwards the detected symbol zˆ to S1 and S2 . The signal received at Si is then given by yi = PR himo zˆ + ni , (3)
and
where PR is the transmit power of a relay and ni ∼ CN (0, σ 2 ) is the additive noise at Si . Finally, Si retrieves the bit transmitted from Sj as xˆj = z˜i xi , where z˜i = 1 if h∗imo yi ≥ 0 and z˜i = −1 if h∗imo yi < 0, and (i, j) ∈ Ω with Ω = {(1, 2), (2, 1)}.
γm is maximized with the maximum value
III. J OINT R ELAY S ELECTION AND P OWER A LLOCATION Let us now derive the criterion for joint RS-PA using instantaneous CSI for the PLNC TWR network employing BPSK modulation. When Rm transmits zˆ, the end-to-end (e2e) SEP at Si , with hm = [h1m h2m ] given, can be expressed as Pe2e,i|hm = Pmac|hm 1 − Pbc,i|hm + 1 − Pmac|hm Pbc,i|hm , (4) where Pmac|hm = Pr {ˆ z = z|hm } and Pbc,i|hm = Pr {˜ zi = zˆ|hm }. The SEP of the network, defined by 2 Pe2e|hm = 12 Pe2e,i|hm , is clearly upper bounded as i=1
2
Pe2e|hm ≤ Pmac|hm
1 + Pbc,i|hm . 2 i=1
(5)
2 √ Since Pmac|hm ≤ Q 2Pi αim and Pbc,i|hm = i=1 √ Q 2PR αim as derived in [2], we have 2
1
2Pi αim + Q 2PR αim Q Pe2e|hm ≤ , (6) 2 i=1 2
∞ 1 √ exp − u2 du and αim = where Q(x) = x 2π 2 2 |him | /σ . Now, for positive constants c1 and c2 of similar order, we have c1 Q(u1 )+c2 Q(u2 ) ≈ cko Q(min(u1 , u2 )) if u1 = u2 and c1 Q(u1 ) + c2 Q(u2 ) = (c1 + c2 )Q(min(u1 , u2 )) if u1 = u2 , where ko = arg min (uk ). Hence, based on (6), it is not k∈{1,2}
unreasonable to approximate the SEP of the network as
2γm , Pe2e|hm κQ Q
(7)
where γm = min (P1 α1m , P2 α2m , PR min(α1m , α2m )) and κQ =
2 i=1
ei +
1 2
2 i=1
(8)
fi with ei = 1 if Pi αim = γm and
ei = 0 otherwise and fi = 1 if PR αim = γm and fi = 0 otherwise. The proposed RS-PA scheme selects (P1 , P2 , PR ) and the relay Rmo in such a way that γm is maximized under the total power constraint. Specifically, we first determine the PA as max min(P1 α1m , P2 α2m , PR min(α1m , α2m ))
(9)
subject to P1 + P2 + PR ≤ PT , Pi ≥ 0, and PR ≥ 0. It can be shown after some manipulations that, when P1 α1m = P2 α2m = PR min(α1m , α2m ), or equivalently, when PT αjm , (i, j) ∈ Ω (10) Pi = α1m + α2m + max(α1m , α2m )
PR =
γm =
PT max(α1m , α2m ) , α1m + α2m + max(α1m , α2m )
PT α1m α2m . α1m + α2m + max(α1m , α2m )
(11)
(12)
Once the PA is completed as described above, the RS is performed to maximize the SNR as mo = arg max γm , after 1≤m≤M
which Si and Rmo set their power as specified in (10) and (11) with m = mo for the TWR as described in Section II. The proposed RS-PA scheme can be implemented practically by extending the RS scheme in [12]. Firstly, each source node broadcasts a beacon signal containing a preamble to the relay nodes so that relay Rm can estimate α1m and α2m and then compute γm . Each relay then constructs a feedback signal containing a preamble and the information on γm , and starts a timer Tm , the duration of which is inversely proportional to γm . The relay node Rmo , with its timer being expired first, transmits the feedback signal: Sensing the feedback signal of Rmo , the other relays reserve their transmission. Subsequently, source node Si retrieves γmo from the feedback signal sent by Rmo and estimates αimo , with which Pi = γmo /αimo is computed for data exchange. IV. P ERFORMANCE A NALYSIS By noting that κQ = 52 from (10) and (11), the average SEP of the TWR network employing the proposed RS-PA scheme can be approximated as (z) 5 5 ∞ e−z Fγmax ¯ √ √ dz (13) 2γmax Pe2e ≈ E Q = 2 4 0 π z from (7), where γmax =
max γm and FX (·) denotes the
1≤m≤M
cumulative distribution function (cdf) of a random variable (rv) X. The equality in (13) is obtained easily from integration M (z), we need to derive by parts. Since Fγmax (z) = m=1 Fγm (z) for the evaluation of (13). Fγm Theorem 1: For two independent exponential rvs X and Y with parameters λ1 and λ2 , respectively, we have XY ∼ T (λ1 , λ2 ), X + Y + max(X, Y )
(14)
where the cdf of T (λ1 , λ2 ) is FT (λ1 ,λ2 ) (z) = 1 + e−3(λ1 +λ2 )z − e−(λ1 +λ2 )z · λi e−λj z H 2λj z 2 , λi , 2z (i,j)∈Ω
for z > 0 with H(a, b, c)
∞ c
(15)
exp − ua + bu du.
Proof: See Appendix A. Lemma 1: An upper bound on FT (λ1 ,λ2 ) (z) is given by FTU(λ1 ,λ2 ) (z) = 1 + e−3(λ1 +λ2 )z − e−(λ1 +λ2 )z · λi e−λj z H L 2λj z 2 , λi , 2z , (i,j)∈Ω
(16)
DO et al.: JOINT RELAY SELECTION AND POWER ALLOCATION FOR TWO-WAY RELAYING WITH PHYSICAL LAYER NETWORK CODING
where
303
L
H (a, b, c) =
√
4a 1 K1 1 − e−bc 4ab − bc a
b a (17) · ce− c + aEi − c
with Kν (x) the ν-th order modified Bessel function of the ∞ −t second kind and Ei(x) = − −x e t dt for x < 0 [13, 8.211.1]. Proof: See Appendix B. (12) and Theorem 1, it is clear that γm ∼ From −1 −1 T ρ1m , ρ2m , where ρim = E {PT αim } = ρΩim with ρ = appr1 PT /σ 2 . Thus, from (13) and (15), we have P¯e2e ≈ P¯e2e , where ∞ M e−z 5 appr1 P¯e2e = FT (ρ−1 ,ρ−1 ) (z) √ √ dz. (18) 1m 2m 4 0 π z m=1
Fig. 1. Average SEP of the PLNC TWR employing the RS-PA and RS schemes.
Replacing FT (ρ−1 ,ρ−1 ) (z) with FTU ρ−1 ,ρ−1 (z) in (18), we ( 1m 2m ) 1m 2m appr2 will get P¯e2e ≈ P¯e2e , where ∞ M 5 e−z appr2 U P¯e2e = FT (ρ−1 ,ρ−1 ) (z) √ √ dz. (19) 1m 2m 4 0 π z m=1 Let us now investigate the behavior of P¯e2e at high SNR using (19). As ρim → ∞, or equivalently as λi → 0 in (16), we have H L 2λj z 2 , λi , 2z ≈ λ1i − 2z since K1 (x) ≈ x1 , e−x ≈ 1 − x, and Ei(x) ≈ ln |x| as x → 0. We then have FTU(λ1 ,λ2 ) (z) ≈ 2(λ1 + λ2 )z for λi → 0, which leads to 1 1 + FTU(ρ−1 ,ρ−1 ) (z) ≈ 2 z. (20) 1m 2m ρ1m ρ2m Employing (20) in (19) and ρim = Ωim ρ, we obtain M 1 5 2M Γ M + 12 1 √ M P¯e2e (ρ) ≈ + , (21) 4 πρ Ω1m Ω2m m=1 ∞ where Γ(x) = 0 e−t tx−1 dt is the Gamma function and the argument ρ in P¯e2e (ρ) is shown explicitly for convenience. In Appendix C, when P1 = P2 = PR = PT /3, we have derived the approximate average SEP M e−z 3 ∞ −3 ρ 1 + ρ 1 z eq ¯ 1m 2m √ √ dz (22) 1−e Pe2e ≈ 4 0 m=1 π z of the TWR network employing the RS without PA. In the high SNR regime (ρim → ∞), (22) becomes M 1 3 3M Γ M + 12 1 eq ¯ √ M Pe2e (ρ) ≈ + (23) 4 πρ Ω1m Ω2m m=1
−3 ρ 1 + ρ 1 z 1 1 1m 2m z when ρim → ≈ 3 ρ1m + ρ2m since 1 − e ∞. Clearly, it is observed from (21) and (23) that the proposed RS-PA has a diversity order M , identical to that of the RS without PA, and that M 5 2 P¯e2e (ρ) ≈ . (24) eq 3 3 P¯e2e (ρ)
Fig. 2.
Comparison of PLNC and ANC with the RS-PA and RS schemes.
eq It is interesting to note from (24) that P¯e2e (ρ) ≈ P¯e2e (βρ) with 1/M β = 1.5(0.6) : An important implication of this observation is that the proposed scheme can provide an asymptotic SNR gain of about 1.76 dB over the RS without PA when M → ∞.
V. P ERFORMANCE E VALUATION Assuming that Ωim = 1, or equivalently ρim = PT /σ 2 , for m = 1, 2, . . . , M , let us now evaluate the average SEP of the PLNC TWR network employing the proposed RS-PA scheme. In Fig. 1, the average SEP of the PLNC TWR is shown for the proposed scheme (denoted by ‘RS-PA’) and the RS without PA (denoted by ‘RS’), where ‘sim’ denotes results from simulation; ‘appr1’, ‘appr2’, and ‘asymp’ in RSPA represent (18), (19), and (21), respectively; and ‘appr’ and ‘asymp’ in RS represent (22) and (23), respectively. Fig. 1(a) shows that the approximate SEPs agree with simulation results quite well when the SEP is less than 10−2 . The figure also shows that RS-PA outperforms RS by about 0.86 dB when M = 8 at the SEP of 10−6 . Fig. 1(b) shows that the asymptotic SEPs agree with the approximate SEPs in the high SNR regime, especially for the proposed RS-PA. It is observed/confirmed that (i) the asymptotic SNR gain β of the RS-PA over the RA increases with M and (ii) the diversity order is identically M for both the RS-PA and RS.
304
IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013
In Fig. 2, the performance of PLNC and ANC is compared via simulation when RS-PA and RS are employed. It is observed that PLNC outperforms ANC and that the performance gain increases with M . The gain results from the facts that (i) the relay in PLNC transmits one modulation symbol while the relay in ANC transmits a linear combination of two modulation symbols and (ii) no noise amplification occurs in the relayed signal of PLNC: Thus, although the gain may vary with the SNR and system parameters due to the nonlinear nature of (i) and (ii), PLNC in general outperforms ANC. VI. C ONCLUSION We have proposed a joint RS-PA scheme for the PLNC TWR network with BPSK modulation in multiple-relay environments. The proposed scheme selects the relay with an instantaneous CSI to reduce the SEP, taking also into account PA among the nodes under the total power constraint. We have derived the average SEP and its asymptotic behavior of the TWR network employing the proposed scheme and the benchmark scheme employing the RS without PA. The results show that the proposed scheme provides the same diversity order as, and an asymptotic SNR gain of about 1.76 dB when compared to, the RS without PA. It is also observed that, with RS-PA, the PLNC outperforms the ANC significantly. A PPENDIX A. Proof of Theorem 1 The cdf of Z =
XY X+Y +max(X,Y )
can be expressed as
FZ (z) = Pr{Z ≤ z, X ≤ Y } + Pr{Z ≤ z, X > Y }. (25) I1
I2
With the probability density functions (pdfs) fX (x) = λ1 e−λ1 x and fY (y) = λ2 e−λ2 y for x, y ≥ 0, we have XY ≤ z, X ≤ Y I1 = Pr X + 2Y ∞ = Pr{(y − z)X ≤ 2yz, X ≤ y}fY (y)dy 0 ∞ 3z 2yz = Pr{X ≤ y}fY (y)dy + Pr X ≤ fY (y)dy y−z 0 3z 3z ∞ 2λ1 yz = 1 − λ2 e−λ1 y e−λ2 y dy − λ2 e− y−z e−λ2 y dy 0 ∞ 3z 2λ1 (u+z)z λ1 + λ2 e−3(λ1 +λ2 )z −λ2 (u+z) u = − λ2 e− du. λ1 + λ2 2z (26) Due to the symmetry, it is straightforward to obtain I2 from I1 by exchanging λ1 and λ2 . Then, the cdf FT (λ1 ,λ2 ) (z) = I1 +I2 of Z ∼ T (λ1 , λ2 ) is expressed as (15). B. Proof of Lemma 1 The upper bound FTU(λ1 ,λ2 ) (z) is obtained by deriving the lower bound H L (a, b, c) on ∞ c a a H(a, b, c) = e−( u +bu) du − e−( u +bu) du, (27) 0 0 √ (g)√ 4a ¯ H(a,b,c) = K 4ab ( ) 1 b
where (g) is from [13, 3.324.1]. First, c 1 c −a ¯ u e du e−bu du H(a, b, c) ≤ c 0 0
(28)
from the Chebyshev’s inequality for integrals [13, 12.314]. c a c Next, noting that 0 e−bu du = 1b (1−e−bc ) and that 0 e− u du a 1 −a = ce c + aEi − c from a change of variable t = u and −bc a ¯ integration by parts, we have H(a, b, c) ≤ 1−ebc ce− c a +aEi − c , eventually leading to H(a, b, c) ≥ H L (a, b, c). eq C. Derivation of P¯e2e
Let us derive the average SEP of the TWR network employing RS without PA when P1 = P2 = PR = 13 PT . We eq have κQ = 32 and γm = γm P3T min (α1m , α2m ) in (7). eq is selected, the average SEP When the RS that maximizes γm is derived with an approach similar to that for (13) as eq (z) 3 ∞ e−z Fγmax eq √ √ P¯e2e dz, (29) ≈ 4 0 π z eq = where γmax
eq eq max γm . Since the cdf of γm is easily
1≤m≤M
obtained to be F
eq γm
(z) = 1 − e
−3 ρ 1 + ρ 1 z 1m
2m
, we have (22).
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