DSpace at VNU: Pairwise Error Probability of Distributed Space-Time Coding Employing Alamouti Scheme in Wireless Relays...
Wireless Pers Commun (2009) 51:231–244 DOI 10.1007/s11277-008-9640-9 Pairwise Error Probability of Distributed Space–Time Coding Employing Alamouti Scheme in Wireless Relays Networks Trung Q Duong · Ngoc-Tien Nguyen · Trang Hoang · Viet-Kinh Nguyen Published online: 13 November 2008 © Springer Science+Business Media, LLC 2008 Abstract In this paper, we analyze the pairwise error probability (PEP) of distributed space–time codes, in which the source and the relay generate Alamouti space–time code in a distributed fashion We restrict our attention to the space–time code construction for Protocol III in Nabar et al (IEEE Journal on Selected Areas Communications 22(6): 1099–1109, 2004) In particular, we derive two closed-form approximations for PEP when the relay is either close to the destination or source and an upper bound for any position of the relay Using the alternative definition of Q-function, we can express these PEPs in terms of finite integral whose integrand is composed of trigonometric functions We further show that with only one relay assisted source-destination link, system still achieves diversity order of two, assuming single-antenna terminals We also perform Monte-Carlo simulations to verify the analysis Keywords (PEP) Distributed space–time codes · Relay channels · Pairwise error probability This paper was presented in part at the 7th International Symposium on Communications and Information Technologies, Sydney, Australia, Oct., 2007 T Q Duong (B) Blekinge Institute of Technology, Ronneby 372 25, Sweden e-mail: quang.trung.duong@bth.se N.-T Nguyen Ministry of Posts and Telematics (MPT), Hanoi, Vietnam e-mail: nntien@mpt.gov.vn T Hoang Heterogeneous Silicon Integration Department, CEA/LETI, Grenoble, France e-mail: trang.hoang@cea.fr V.-K Nguyen University of Technology-Hanoi National University, Hanoi, Vietnam e-mail: kinhnv@vnu.edu.vn 123 232 T Q Duong et al Introduction It is well-known that the multiple-input multiple-output (MIMO) systems can remarkably improve the capacity and reliability of wireless communications over fading channels using multiplexing scheme and space–time coding [2–5] However, the demand for low-cost and small-size portable devices has prohibited practical implementation of MIMO system Recently, with the increasing interests in ad-hoc networks, cooperative diversity has been proposed to exploit MIMO benefits in a distributed fashion [6–8] Distributed space–time coding (DSTC) has been considered to achieve cooperative diversity in wireless relay networks, in which different relays work as co-located transmit antennas and construct a space–time code to realize spatial diversity gain [1,9–11] Recently, a closed form expression of bit-error-rate (BER) has been presented for DSTC in [11] Also, a simple DSTC for two relays has been investigated in [12,13] More specifically, Alamouti scheme has been applied into relay systems where two single-antenna1 relays simultaneously receive a noisy signal from the source and generate Alamouti space–time codes in a distributed fashion before relaying the signals to the destination This model has been demonstrated to obtain a full diversity order, i.e., the degree of diversity is order of two [14] In practice, such code design is difficult due to distributed and ad-hoc nature, as opposed to codes designed for co-located (MIMO) systems Furthermore, coherent reception of multiple relays transmission obliges synchronization (at the symbol and carrier level) among multiple transmit-receive pairs, which enhances the receiver complexity The problem we are interested in is whether we still achieve a full diversity order with only one relay The considered relay network model is similar to the Protocol III in [1] Unlike [1], in which the channel of relay-destination link was assumed static, in this paper we analyze the PEP taking into account random fading channels for all links The source communicates with the destination during the first hop In the second hop, both relay and source communicate with the destination We also assume the channel information is only available at the destination A key feature of our work is that the relay simply amplifies the signal and transmits to the destination without any sort of signal regeneration, called non-regenerative or amplifyand-forward protocol This simplification of relay operations avoid imposing bottlenecks on the rate of the relays Our contribution in this work can be briefly described as follows We analyze the PEP of DSTC, in which the relay and the source construct a distributed-Alamouti space-time code (each terminal transmits one row of Alamouti code [15] to the destination) We then successfully derive two closed-form expressions for PEP as the relay approaches both ends and an upper bound for any position of relay Assessing two tight approximations of PEP in the high signal-to-noise ratio (SNR) regime, we further show that the considered distributed-Alamouti system can achieve a full diversity gain with only one relay assisted the direct communication We also perform Monte-Carlo simulations to validate our analysis The rest of this paper is organized as follows In Sect 2, we briefly review the cooperative system of one non-regenerative relay based on Alamouti scheme Two closed-form approximations for PEP when the relay is either near to the destination or source and an upper bound for any position of relay are derived in Sect Using the asymptotic (high SNR) PEP formulas we show that the distributed system employing Alamouti codes achieves a diversity gain of order two in Sect Numerical results are given in Sect Finally, Sect concludes the paper Notation: Throughout the paper, we shall use the following notations Vector is written as bold lower case letter and matrix is written as bold upper case letter The superscripts ∗ and † In this paper, we only consider the single-antenna terminal 123 Pairwise Error Probability of Distributed Space–Time Coding 233 Fig Schematic of relay channel stand for the complex conjugate and transpose conjugate, respectively I n represents the n ×n identity matrix A F denotes Frobenius norm of the matrix A and |x| indicates the envelope of x Ex {.} is the expectation operator over the random variable x A complex Gaussian dism, ) tribution with mean µ and variance σ is denoted by CN (µ, σ ) Let us denote N˜ m (m as a complex Gaussian random vector with mean vector m and covariance matrix log is the natural logarithm (a, x) is the incomplete gamma function defined as (a, x) = ∞ a−1 −t e dt and Kn (.) is the nth-order modified Bessel function of the second kind x t System Models We consider a wireless relay network with three terminals as shown in Fig Every terminal has a single antenna, which can not transmit and receive simultaneously The relay terminal assists in communication by simply amplifying and forwarding received signals We also assume all terminals are synchronized in the symbol level and channel remains constant for a coherence time (at least two symbol-intervals) and changes independently to a new value for each coherence time The source transmits the first row of Alamouti code to the destination during the first hop with average transmit power per symbol Ps For the first hop transmission, the receive signal at the relay is given by y R = h SRs + n R (1) where h SR ∼ CN (0, SR ) is Rayleigh-fading channel coefficient for the source-relay link, s = [ s1 s2 ] is the first row of Alamouti code, si (i = 1, 2) is selected from signal constellation S , and n R ∼ CN (0, N0 ) is complex additive white Gaussian noise (AWGN) at the relay In the second hop, the source sends the second row of Alamouti code to the destination, whereas the relay retransmits a scaled version of y R to the destination with the same power constraint as in the first hop The relaying gain is determined only to satisfy the average power constraint with statistical channel state information (CSI) on h SR For the second hop transmission, the receive signal at the destination is readily written as y D = h RD ωyy R + h SDs + n D (2) where ω is the scalar relaying gain, h RD ∼ CN (0, RD ) and h SD ∼ CN (0, SD ) are Rayleigh-fading channel coefficients for the relay-destination and source-destination links, 123 234 T Q Duong et al respectively, s = [ −s2∗ s1∗ ] is the second row of Alamouti code, and n D ∼ CN (0, N0 ) is the AWGN at the destination Note that all the random quantities hA and nB , A ∈ {SD, SR, RD}, B ∈ {R, D}, are statistically independent and the variations in A capture the effect of distance-related path loss in each link To constrain transmit power at the relay, we have E ωyy R F =E si F (3) yielding ω2 = SR + SNR −1 (4) s where SNR = P N0 is the common SNR of each link without fading [8] The receive signal at the destination in (2) is now formed as follows yD = hS + n where h = (5) s1 s2 is Alamouti space–time code, and n = −s2∗ s1∗ 2 0, N0 + |h RD | ω I h RD h SR ω h SD , S = h RD ωnn R + n D ∼ N˜ Pairwise Error Probability As mentioned above, assuming the destination knows channel information for all links, it is easy to see that y D |hA is a Gaussian random vector with mean vector h S and covariance matrix N0 + |h RD |2 ω2 I Hence, the maximum-likelihood (ML) decoding of the system can be readily seen to be Sˆ = arg y D − h S F S (6) where the minimization is performed over all possible codeword matrices S With the ML decoding in (6), the PEP, given the channel coefficients hA , of mistaking S by E is obtained as P(SS → E | hA ) = Q h (SS − E ) 2F 2N0 + |h RD |2 ω2 (7) It is important to note that S and E (SS = E ) are the two possible codewords of Alamouti space–time code, hence, we have (SS − E ) (SS − E )† = Ps d E2 I (8) d E2 = |s1 − e1 |2 + |s2 − e2 |2 > (9) where π/2 x dθ Applying the alternative definition of Q-function [16]: Q(x) = π1 exp − sin 2θ into (7) and integrating over all channel realizations, we obtain the unconditional PEP as follows: P(SS → E ) = Eγ {P(SS → E |γ )} = π 123 π/2 φγ SNRd E2 sin2 θ dθ (10) Pairwise Error Probability of Distributed Space–Time Coding 235 where γ = |h RD |2 |h SR |2 ω2 + |h SD |2 h 2F = , + |h RD |2 ω2 + |h RD |2 ω2 (11) and φγ (ν) Eγ {exp (−νγ )} is the moment-generating function (MGF) of the random variable γ To evaluate the integration in (10), next we discuss two specific cases: (i) when the relay is close to the destination (ii) when the relay is close to the source 3.1 The Relay is Close to the Destination If the relays are much closer to the destination than the source, then we may have + |h RD |2 ω2 high SNR ≈ |h RD |2 ω2 In this case (11) can be approximated as γ ≈ |h SR |2 + |h SD |2 (12) |h RD |2 ω2 Recalling that hA , A ∈ {SD, SR, RD}, are assumed to be statistically independent, the MGF of γ , φγ (ν), can be determined by φγ (ν) = φ|h SR |2 (ν) φ y where y = |h SD |2 |h RD |2 Since hA ∼ CN (0, A distribution with hazard rate 1/ written as A ), ν ω2 (13) it is obvious that |hA |2 obeys an exponential The probability density function (p.d.f.) of |hA |2 can be p|hA |2 (x) = exp(−x/ A A ), (14) yielding φ|h SR |2 (ν) = E|h SR |2 exp − |h SR |2 ν = (1 + ν −1 SR ) (15) Following Lemma in the Appendix 6, p.d.f of y is given by pY (y) = SD RD ( SD + −2 , RD y) (16) yielding φy ν ω2 ∞ = exp − = ν ω2 SD νy ω2 exp RD SD ν ω2 RD ( SD RD SD + −1, −2 dy RD y) ν ω2 SD where (17) follows immediately from the change of variable u = Eq (3.381.1)] Substituting (13), (15), and (17) into (10), we obtain P (SS → E ) = π (17) RD SD + RD y and [17, π/2 1+ξ SR −1 ξˆ exp ξˆ −1, ξˆ dθ (18) 123 236 T Q Duong et al where ξ= SNRd E2 (19) sin2 θ and ξˆ = SD RD ω ξ (20) 3.2 The Relay is Close to the Source On the other hand, if the relays are much close to the source, the following approximation may be hold + |h RD |2 ω2 high SNR ≈ In this case, (11) can be readily written as γ ≈ |h RD |2 |h SR |2 ω2 + |h SD |2 (21) φγ (ν) = φz ω2 ν φ|h SD |2 (ν) (22) yielding with φ|h SD |2 (ν) = E|h SD |2 exp −ν |h SD |2 = (1 + SD ν) −1 (23) and φz ω2 ν , where z = |h RD |2 |h SR |2 , is given by φz ω2 ν = Ez exp −ω2 νz ∞ = exp −ω2 νz SR RD K0 z SR dz (24) RD = λ exp(λ) (0, λ) (25) −1 where λ = ω2 SR RD ν , (24) follows immediately from [14, Theorem 3], and (25) can be obtained from the change of variable t = ω2 · ν · z along with [17, Eq (8.352.4)] Combining (22), (23), (25), and (10), the PEP becomes P (SS → E ) = π π/2 1+ξ SD −1 ζ exp (ζ ) (0, ζ ) dθ (26) −1 where ξ is given in (19) and ζ = ω2 SR RD ξ We can clearly see that the PEPs are given in closed-form expressions when the relay is either close to the destination or source, shown in (18) and (26), respectively These results can be readily calculated by common mathematical software packages such as MATHEMATICA or MAPLE 3.3 An Upper Bound of Pairwise Error Probability So far, the closed-form approximations of PEP have been considered for the case when the relay is close to both ends In this subsection, we will derive the upper bound of PEP for any position of relay From (11), the value of γ can be shown as 123 Pairwise Error Probability of Distributed Space–Time Coding γ > |h RD |2 |h SR |2 + |h RD |2 ω2 237 = γ0 (27) Applying (27) into (10), we can upper bound the PEP as P (SS → E ) < π = π π/2 SNRd E2 φγ0 sin2 θ π/2 1+ξ SR dθ + ρξ SR exp (ρ) (0, ρ) dθ (28) where (28) follows immediately from Lemma in the Appendix, ξ is given in (19), and ρ = ω2 RD 1+ξ −1 SR Diversity Order Noting that, in [1] the diversity order was obtained from the Chernoff bound (not a tight bound) with an assumption that the relay-to-destination link is static (non-fading) and deduced from the upper bound of PEP (assuming fading relay-destination link) when the relay is close to the source In this section, we quantify the effect of our relay protocol on the PEP curve in the high-SNR regime when the relay approaches both ends Using the two tight approximations of PEP derived Sects 3.1 and 3.2, we can assess the diversity order of distributed-Alamouti systems by the following theorem Theorem (Achievable Diversity Order) The non-regenerative cooperation of our scheme provides maximum diversity order, i.e., D = 2, when the relay is near both ends Proof The diversity has been defined as the absolute values of the slopes of the error probability (e.g., PEP) curve plotted on a log–log scale in high SNR regime [16], i.e., D − log P (SS → E ) log (SNR) SNR→∞ lim (29) As can be seen from (10), the PEP is expressed in a form of finite integral whose integrand is the MGF of random variable γ Therefore, the asymptotic behavior of the MGF φγ (ν) at large SNR reveals a high-SNR slope of the PEP curve, we have D= − log φγ (ν) |θ =π/2 log (SNR) SNR→∞ lim (30) • The relay is close to the destination Substituting (15) and (17) into (13), the diversity order D in (30) is now shown as D = lim x→∞ log(1+αx) log x − lim x→∞ log(βx exp(βx) (−1,βx)) log x = − (−1) = d E2 (31) d E2 SD 4ω2 RD , x = SNR, and (31) follows immediately by applying where α = SR , β = l’Hospital rule • The relay is close to the source Substituting (23) and (25) into (22), the diversity order D is now given by 123 238 T Q Duong et al Fig Collinear topology with an exponential-decay path loss model where and RD = (1 − ε)−α SD with α = D = lim x→∞ log(1+ηx) log x − lim x→∞ −α , SD ∝ d log( ϕx exp( ϕx ) log x SD 2 where η = , ϕ = dE ω l’Hospital rule This completes the proof.2 SR RD −1 SD , (0, ϕx )) = − (−1) = d E2 −α SR = ε (32) , and (32) follows immediately from Numerical Results In this section, we validate our analysis by comparing with simulation In the following numerical examples, we consider the non-regenerative relay protocol employing Alamouti code as in Sect We assume collinear geometry for locations of three communicating terminals, as shown in Fig The path loss of each link follows an exponential-decay model: if the distance between the source and destination is equal to d, then SD ∝ d −α where the exponent α = corresponding to a typical non line-of-sight propagation Then, SR = −α SD and −α RD = (1 − ) SD For Alamouti code transmission, the source send symbols selected from binary phase-shift keying (BPSK) The reason we use BPSK modulation is to simplify the calculation of PEP as described in the following For the normalization, the BPSK constellation points are −1 and When only one of the corresponding symbols in S and E are different, for example s1 = e1 and s2 = e2 , we have d E2 = On the other hand, when both corresponding symbols in the transmit and receive codewords are different from each other, it holds d E2 = Figures and draw the PEP versus SNR when the relay approaches the destination for = 0.6, 0.7, 0.8 with d E2 = and the source for = 0.2, 0.3, 0.4 with d E2 = As can be clearly seen from both figures, analytical and simulated PEP curves match exactly when the relay is located near by both ends Observe that the PEP slops for = 0.2, = 0.3, and = 0.4 are identical at the high SNR regime, as speculated in Theorem Similar observation can be made for = 0.6, = 0.7, and = 0.8 However when the relay moves far away from both ends, analytical and simulated curves not completely agree together, for example = 0.4 and = 0.6, since the approximations in (12) and (21) only satisfy when the relay is closely located to the destination and source, respectively In Fig 5, the PEP at SNR=20 dB with d E2 = is depicted as a function of the fraction It is clear to see that the performance is decreased when the relay approaches both end It has been confirmed by using mathematical software package MATHEMATICA 123 Pairwise Error Probability of Distributed Space–Time Coding 239 Fig Pairwise error probability of BPSK versus SNR in non-regenerative relay channels employing Alamouti scheme when = 0.6 , 0.7 , 0.8 with = (the relay is close to the dE destination) SD = 16 Fig Pairwise error probability of BPSK versus SNR in non-regenerative relay channels employing = (the relay is close to the source) Alamouti scheme when = 0.2 , 0.3 , 0.4 with d E SD = 16 and the symmetric geometry ( = 0.5) shows an optimal performance For comparison, we also plot the upper bound of PEP The upper bound closely matches the simulated curve when the relay approaches the destination It can be explained in the following: At the relatively high SNR regime, we have ω2 ≈ 1SR which makes the numerator of (11) becomes |h RD |2 |h SR |2 + |h SD |2 In addition, when the relay is close to the destination, small leading to γ ≈ γ0 , given in (27) SR SR is relatively 123 240 T Q Duong et al = and SNR = 15 dB in Fig Pairwise error probability of BPSK as a function of the fraction with d E non-regenerative relay channels employing Alamouti scheme We also plot the upper bound of PEP Conclusion In this paper, we have analyzed the PEP of cooperative system, in which the source and the relay generate Alamouti space–time code in a distributed fashion to exploit the benefit of MIMO system in relay fading channels Specifically, two tight approximations of PEP as the relay approaches both ends and an upper bound of PEP for any position of relay have been derived in closed-form expressions We have quantified the effect of PEP in the high SNR regime and shown that the full diversity order can be achieved Appendix 1: Auxiliary Results The following lemmas will be helpful in the paper Lemma Let X and Y be statistically independent and not necessarily identically distributed (i.n.i.d.) exponential random variables with hazard rate x and y , respectively Suppose that the ratio of Z takes the form X Y Then, we obtain the p.d.f of random variable Z as Z= p Z (z) = x y x + (33) yz −2 (34) Proof Note that ∞ p Z (z) = y · p X Y (yz, y) dy 123 (35) Pairwise Error Probability of Distributed Space–Time Coding ∞ y = x = x y y z exp − + x x 241 + y dy (36) y −2 yz (37) where (36) follows immediately from (14) and the statistical independence of X and Y This completes the proof Lemma Let X and Y be statistically independent and not necessarily identically distributed (i.n.i.d.) exponential random variables with hazard rate x and y , respectively Suppose that the ratio of Z takes the form XY , a+Y Z= a > (38) Then, we obtain the MGF of random variable Z as φ Z (ν) = 1+ν + τν x x exp (τ ) (0, τ ) (39) with a + ν (1 y τ= (40) x) Proof The cumulative distribution function (c.d.f.) of Z is given by XY ≤ z} a+Y = EY FZ |Y (z) FZ (z) = Pr{ = EY − exp − = 1− y z (a + Y ) xY ∞ exp − z (a + y) y dy − xy y (41) The p.d.f of Z is obtained by differentiating (41) with respect to z as follows ∞ p Z (z) = x y = x y x x + exp − ∞ y = ∞ + y y exp − z (a + y) y − dy xy y z (a + y) y − dy y x y z (a + y) y a exp − − dy y y x y exp − 2a x a+y y az z exp − x y x z x K0 az K1 x az x (42) y y (43) 123 242 T Q Duong et al where (43) follows immediately from [17, eq (3.471.9)] The MGF of Z can be expressed by φ Z (ν) = E Z {exp (−νz)} = x + ∞ a y x = 1+ν z 1/2 exp − ∞ 2a x y y x exp − x + τν x x + ν z K0 exp (τ ) az + 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on Selected Areas Communications, 16(8), 1451–1458 16 Simon, M K & Alouini, M.-S (2000) Digital communication over fading channels: A unified approach to performance analysis New York: Wiley 123 Pairwise Error Probability of Distributed Space–Time Coding 243 17 Gradshteyn, I S & Ryzhik, I M (2000) Table of integrals, series, and products, 6th ed San Diego: Academic Author Biographies Trung Q Duong was born in Hoi An Town, Quang Nam Province, Vietnam, in 1979 He received the B.S degree in electrical-electronics engineering from Ho Chi Minh City University of Technology, Vietnam, in 2002, and the M.S degree in computer engineering from Kyung Hee University, South Korea, in 2005 In April 2004, he joined the Faculty of Electrical Engineering, Faculty of Ho Chi Minh City University of Transport, Vietnam He was a recipient of the Korean Government IT Scholarship Program for International Graduate Students In December 2006, he was awarded the Best Paper Award of IEEE Student Paper Contest—IEEE Seoul Section He finished a two-year Ph.D course in radio communications engineering from Kyung Hee University, South Korea, in 2007 In December 2007, he joined the staff of the School of Engineering, Blekinge Institute of Technology (BTH), Sweden His current research interests include wireless and mobile communications Ngoc-Tien Nguyen was born in Hanoi, Vietnam, on February 26, 1964 He received the B.S degree in electronics and communications from Military Technical Academy, Vinh Yen, Vietnam, in 1985, and the M.S degree in electronics and telecommunications from Hanoi National University Currently, he is a Ph.D candidate at the Vietnam Posts and Telecommunications Institute of Technology (PTIT) Since 1993, he has been working at the Ministry of Posts and Telematics (MPT) of Vietnam His research interests include multi-user detection for DS-CDMA systems, OFDM, MC-CDMA, and MIMO Trang Hoang received the B.S and M.S degree in electronicstelecommunications engnieering from Ho Chi Minh City University of Technology, Viet Nam in 2002 and 2004, respectively Currently, he is a research staff in CEA/LETI-Minatec, Heterogeneous Integartion on Silicon Division, Grenoble working towards his Ph.D degree in the field of MEMS at University Joseph Fourier, France His research interest is in the domain of integration of passive components, SAW devices, IC design, SoC and MEMS fabrication 123 244 T Q Duong et al Viet-Kinh Nguyen received the Ph.D degree in electrical engineering from Poland Academy of Sciences, in 1977 Nowadays, he is a professor teaching at University of Technology, Hanoi National University, Vietnam His current research interests are CDMA, MC-CDMA, OFDM and MIMO 123 ... MATHEMATICA 123 Pairwise Error Probability of Distributed Space–Time Coding 239 Fig Pairwise error probability of BPSK versus SNR in non-regenerative relay channels employing Alamouti scheme when = 0.6... channel of relay-destination link was assumed static, in this paper we analyze the PEP taking into account random fading channels for all links The source communicates with the destination during... version of y R to the destination with the same power constraint as in the first hop The relaying gain is determined only to satisfy the average power constraint with statistical channel state information