40120140501016

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40120140501016

  1. 1. International Journal of ElectronicsJOURNAL OF ELECTRONICS AND INTERNATIONAL and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET) ISSN 0976 – 6464(Print) ISSN 0976 – 6472(Online) Volume 5, Issue 1, January (2014), pp. 138-147 © IAEME: www.iaeme.com/ijecet.asp Journal Impact Factor (2013): 5.8896 (Calculated by GISI) www.jifactor.com IJECET ©IAEME OPTIMAL DECODING FOR WIRELESS RELAY NETWORKS WITH DECODE-AND-FORWARD COOPERATION PROTOCOL Taha Abdelshafy Abdelhakim Khalaf Department of Electrical Engineering, Assiut University, Assiut, Egypt, 71516 ABSTRACT In this paper, we derive an exact upper bound (UB) on bit error rate (BER) of the decode-and-forward (DF) cooperation protocol. The destination uses the maximum a posterior (MAP) decoder to estimate the data sent from the source. The MAP decoder is optimal in the sense that it minimizes the the end-to-end error probability. The complexity of derivation comes from the fact that however the source generates data with equal probability, the data received at the destination does not have the same a priori probability. That is because of the error that occurs in the source-to-relay link. Hence, the MAP decoding rule can not be simplified to the maximum likelihood (ML) decoding rule. The results show that the upper bound is very tight. Therefore, the closed form expression of the upper bound can be used to fully study and understand the diversity performance of the DF cooperation protocol. Keywords: Relay Network, Diversity, Decode-and-Forward, MAP Decoder. I. INTRODUCTION Cooperative relaying is gaining a significant attention in that multiple intermediate relay nodes can collaborate with each other to enhance the overall network efficiency. It exploits the physical-layer broadcast property offered by the wireless medium that transmitted signals can be received and processed by any node in the neighborhood of a transmitter. The cooperative relaying approach has a great potential to provide substantial benefits in terms of reliability (diversity gain) [1]-[4] and rate (bandwidth or spectral efficiency)[5]-[7]. These benefits can extend the coverage, reduce network energy consumption, and promote uniform energy drainage by exploiting neighbors’ resources. They can be of great value in many applications, including ad-hoc networks, mesh networks, and next generation wireless local area networks and cellular networks. Through cooperative diversity, relay nodes forward the signal received from the source to propagate redundant signals over multiple paths in the network. This redundancy allows the ultimate receiver to essentially average channel variations resulting from fading, shadowing, and other forms of interference [2]. Several cooperation protocols 138
  2. 2. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME have been proposed in the literature such as amplify-and-forward (AF), decode-and-forward (DF), and amplify forward compress-and-forward (CF) [10, 11]. CF) The error performance of the cooperative communications with decode-anddecode -forward protocol with different combining techniques was investigated in the literature. The end-to-end performance of end end wireless communication systems with relays over Rayleigh-fading channels was studied in [12] when fading the direct link between the source and the destination does not exist. A general framework for ML detection of both coherent and noncoherent uncoded cooperative diversity was presented in [13] where the authors dervived a high SNR approximations based on the closed-form BER expressions. The hors closed form performance of the maximum likelihood decoding for network coded cooperative communication is studiied in [14]. In this paper, we drive a closed from expression for the upper bound on the bit error probability of the decode-and-forward cooperation protocol with MAP decoder. Unlike all previous work, the forward receiver does not use any combining techniques. However, we consider the data received from the consider source together with the data received from the relay node as a codeword. The decoding rule is to find the codeword that maximizes the a posterior probability. The MAP decoding rule is optimal in the sense that is minimizes the end- end error probability. Since the error probability in the -to-end source-to-relay link is not necessarily 0.5 , the codewords received at the destination are not relay equaprobable. Therefore, the MAP decoding rule can not be simplified to the maximum likelihood (ML) decoding rule. We used the closed form expression we derived to find an approximate upper bound at high SNR regime. The approximate upper bound can be used to clearly understand the effect of the SNR of the source-to-relay link on the system diversity. relay The remainder part of this paper is organized as follows. The system model is described in Section II. The MAP decoding scheme is described in Section III. The derivation of the exact upper . . bound on the bit error probability is provided in Section IV. Section V presents numerical results and discussions. Finally, the conclusions are drawn in Section VI. II. SYSTEM MODEL Figure 1: System Model We consider a relay network composed of one source (S), one relay (R), and one destination (D) as shown in Fig. 1. The data transmission occurs over two phases. In the first phase, the source sends its data to the destination. Due to the broadcast nature of the wireless medium, the relay may also of receive the source’s data (possibly with some errors). In the second phase, the relay node decodes the data received from the source and then forwards the decoded data to the destination. We assume that all data are sent using binary phase shift keying (BPSK) modulation scheme and the source generates e its bits with equal probability, i.e., p(bs = 0) = p (bs = 1) = 0.5 where bs is the source’s bit. We assume that all channel are Rayleigh fading channels with additive white Gaussian noise (AWGN). We also assume that each node is equipped with single antenna. 139
  3. 3. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME The signals received at the relay node and the destination in both phases are given by − yr = hsr d srm xs + nr , (1) − y1 = hsd d sdm xs + n1 , (2) − y2 = hrd d rdm xr + n2 , (3) where • yr is the signal received at the relay node in the first phase from the source. • y1 is the signal received at the destination in the first phase from the source. • y2 is the signal received at the destination in the second phase from the relay. • xs ∈{+ Es ,− Es } is the BPSK modulated signal of the source’s bit, bs ∈ {0,1} , where Es is the transmit energy per the source’s bit. • xr ∈ {+ Er ,− Er } is the BPSK modulated signal of the relay’s bit, br ∈ {0,1} , where Er is the transmit energy per the relay’s bit. • hsr , hsd , and hrd are the channel fading gains of the S-R, S-D, and R-D links, respectively. • d sr , d sd , and d rd are the lengths of the S-R, S-D, and R-D links, respectively. • m is the path loss exponent. • nr is the AWGN noise at the relay node which has zero mean and variance N 0 /2 . r • n1 and n2 are independent AWGN noise components at the destination which have zero mean and variance N 0 /2 . The bit br represents an estimate of the source’s bit bs at the relay node and it is given by 0 br =  1 yr ≥ 0 yr < 0 (4) Let the random variable e ∈{0,1} captures the error events on the source-to-relay channel, i.e, br = bs ⊕ e , where “ ⊕ ” denotes the binary “xor” operation. Hence, e = 1 means br ≠ bs , i.e. the source’s bit is received in error at the relay node. In the case of BPSK modulation over flat fading channel with AWGN, the probability of error in the source-relay link, i.e. P(e = 1) , is given by [9] γ sr  1 1 −  2 1 + γ sr  1 = [1 − Γsr ] 2 Pe = sr (5) − where γ sr = d srm Es /N 0 is the average received SNR of the link between the source and the relay and r Γsr = γ sr /(1 + γ sr ) . The received signals at the destination can be written in the matrix form as follows 140
  4. 4. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME y = Hx + n (6) where the received vector y = [ y1 y2 ]T , the transmitted vector x = [ xs xr ]T , the noise vector n = [n1 n2 ]T , and the channel matrix H is given by h d − m H =  sd sd 0     − d rdm   0 hrd (7) III. MAP DECODING In this section, we present the MAP decoding scheme for cooperative communication networks with decode-and-forward cooperation protocol. The MAP decoding scheme is optimal in the sense that it minimizes the error probability. Let b = [bs br ]T denote the transmitted vector where br = bs ⊕ e . This vector takes one of four possible values b1 = [0 0]T , b2 = [11]T , b3 = [0 1]T , and b4 = [1 0]T based on the values bs and e . Since P (e = 1) is not necessarily equal to P(e = 0) , These four vectors are not equaprobable. The probability of transmitting a specific vector bi is given by 1 − Pesr   2 P(bi ) =   Pe  sr  2 i = 1,2 (8) i = 3,4 The MAP decoder finds the transmitted vector bi that maximizes the a posterior probability P(bi | y ) . An estimate of the transmitted vector is calculated as follows b = arg maxb P (b | y ) i i (9) and the maximization process is over the 4 possible vectors. Applying Bayes theorem to (9) yields b = arg maxb i P( y | bi ) P(bi ) P( y ) = arg maxb P( y | bi ) P (bi ) i (10) where P ( y | bi ) = 1 −|| y − Hxi || 2 /N 0 e (πN 0 ) (11) and xi is the modulated vector that corresponds to the vector bi , e.g., x3 = [+ Es − Er ]T . Substituting from (11) into (10) yields 141
  5. 5. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME b = arg maxb i 1 −|| y − Hxi || 2 /N 0 e P (bi ) (πN 0 ) = arg minb (|| y − Hxi || 2 − N 0 log ( P (bi )) ) i (12) IV. PROBABILITY OF BIT ERROR In this section, we derive the upper bound on the bit error probability for the MAP decoding scheme. For mathematical traceability, we assume that the source and the relay node adjust their transmit power such that all signals are received at the destination with the same SNR γ = E/N 0 . This m m can be achieved by choosing E s = d sr E and Er = d rd E . The bit error probability is given by PE ≤ (P(b1 → b2 ) + P(b1 → b4 ) )P(b1 ) + (P (b2 → b1 ) + P (b2 → b3 ) )P(b2 ) (13) + (P(b3 → b2 ) + P(b3 → b4 ) )P(b3 ) + (P(b4 → b1 ) + P(b4 → b3 ) )P(b4 ) where P (bi → b j ) is the pairwise error probability of confusing bi with b j when bi is transmitted and when these are the only two hypothesis. When the vector bi is transmitted, the received vector would be yi = Hxi + n (14) and the probability P (bi → b j ) is given by P(bi → b j ) = P || yi − Hx j || 2 − N 0 log( P(b j )) < || yi − Hxi || 2 − N 0 log( P (bi ))     || H ( xi − x j ) || 2 N 0 P(bi )    = P (< n, H ( xi − x j ) >) > + log (15) 2 2 P(b j )     ( When h = [ hsd hrd ]T is given, < n, H ( xi − x j ) > would be a Gaussian random variable with zero mean and variance N0 || H ( xi − x j ) ||2 . Accordingly, 2  || H ( xi − x j ) || N 0 /2 log(α ij )   P (bi → b j | h) = Q +  || H ( xi − x j ) ||  2N0   (16) where α ij = P (bi )/P (b j ) . From (16), we notice that the probability P (bi → b j ) depends on: 1- The Hamming distance between bi and b j , wij , (i.e., the number of positions at which bi and b j are different); 2- Whether bi and b j have the same probability. Let P (α ij ) = P (bi → b j ) when wij = 1 and P2 = P (bi → b j ) when wij = 2 . From (16), we notice 1 that P2 doesn’t depend on α ij . Hence, substitute from (8) into (13), yield 142
  6. 6. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME ( PE ≤ 2 P2 + [P (α 41 ) + P (α 32 )]Pe + [P (α14 ) + P (α 23 )] 1 − Pe 1 1 1 1 sr sr ) (17) In order to derive P (α ij ) , we consider the case when i = 1 and j = 4 . Substitution in (16) yields 1  log(α14 )  2  P (b1 → b4 | hsr ) = Q 2hsr γ + 2  2 2hsr γ    (18) 2 where γ = E/N 0 . In order to find P (α14 ) , we average (18) over the distribution of hsr . Since all 1 channels have the same distribution, {∀i, j : wij = 1} , we have ∞  log (α ij )   f X ( x)dx P (α ij ) = ∫ Q 2 x + 1  0 2 2x    (19) where f X ( x) = 1 γ e − x/γ (20) Substituting from (20) into (19) with considering Q (u ) = 0.5erfc(u/ 2 ) yields 1 2γ P (α ij ) = 1 ∞  ∫ erfc  0 x+ b  − x/γ e dx x (21) where b = log (α ij )/4 . Integrating (21) using integration by parts where w = erfc( x + b/ x ) , dv = e − x/γ dx , dw = − 1  b  − ( x +b ) 2 /x dx , and v = −γe − x/γ , yields 1 −  e πx  x   b  − x/γ γ  e − − γerfc x + x π   b≥0  I = 1 1 + I1 b < 0 1 P (α ij ) = 1 2γ ∞ ∫ 1  b  − x/γ −( x + b )2 /x  dx  1 − e x  x 0 (22) where I1 = −1 2 π ∫ ∞ 0 1  b  − x/γ − ( x + b ) 2 /x dx 1 − e x  x (23) After some mathematical manipulations, we can write (23) as follows 2 (u + d )  γ d −  − e u du (24) ∫0  1 + γ u    where r = 2b(1 − 1 + 1/γ ) , d = b 1 + 1/γ , and u = x(1 + 1/γ ) . We can write (24) as follows − e −r I1 = 2 π ∞ 1 u 143
  7. 7. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME − e −r I1 = 2 π ∫ ∞ 0 1 u  I =  −r 2 − e + I 2 2 (u + d )  γ d −  −1 + + 1 − e u du  1+ γ u   b≥0 b<0 (25) where  I 2 = 1 −    e −r  1+ γ  2 π  γ ∫ 1 − (u e u ∞ 0 + d )2 u du (26) By changing variables, let y = u and dy = dy/(2 u ) . After substitution in (26) we have  I 2 = 1 −   γ  e −r  1+ γ  π   γ = 1 −  1+ γ  ∫ ∞ 0 e  e − (2 d + r )   π   y2 +d −  y  ∫ ∞ 0     2 − y2 − e dy d2 y2 dy (27) Using integration tables to find the integral in (27) yields [8] 1  γ  −4 d −r e  1 −  1+ γ  2   I2 =    1 1 − γ e − r   2 1+ γ     b≥0 (28) b<0 After substituting for the d, r and b , we have 1 1  γ  − 2 (1+  1 − α ij 1+ γ  2  I2 =  1 γ  − 2 (1− 1  1− α ij  2  1+ γ    1+ 1+ 1 γ 1 γ ) α ij ≥ 1 (29) ) α ij < 1 Substituting from (29) into (25) then into (22) yields  1 − Γ − (1+1/ Γ )/2 α ij if α ij ≥ 1  2  P1 (α ij ) =   1 + Γ − (1−1/ Γ )/2 if α ij < 1 1 − 2 α ij  γ (30) In order to derive P2 , we consider the case when i = 1 and j = 2 . Substituting 1+ γ α ij = 1 into (16) yields where Γ = 144
  8. 8. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME ( 2 2 P(b1 → b2 | h) = Q 2(hsr + hrd )γ ) (31) 2 2 Averaging (31) over the joint distribution of hsr and hrd yields [9] 2  1 − Γ  1  m + 1 1 + Γ   P2 =   ∑     2  m = 0 m  2  1 1  2 = (1 − Γ ) 1 + Γ  2  2  m (32) It is clear from (5) that if Es ≥ 0 or equivalently 0 ≤ Γsr ≤ 1 then the error probability in the S-R link would be 0 ≤ Pe ≤ 0.5 . Therefore, substitution from (30) and (32) into the right hand side of (17) sr results in the following upper bound    1 − Γsr 1 1  1 2 PUB = (1 − Γ ) 1 + Γ  + (1 − Γsr )1 − Γ 1+ Γ 2  2  2 sr    1− Γ   2Γ        (33) At higher receive SNR’s γ sr and γ , the upper bound of (33) can be approximated as follows PUB ≈ γ 1  1 + 2  4γ  γ sr     (34) V. NUMERICAL RESULTS AND DISCUSSIONS In this section, we present numerical results for both analysis and simulations. 0 10 -1 10 -2 Bit Error Rate BER 10 -3 10 -4 Upper Bound γSR = 0 dB 10 Simulations γSR = 0 dB -5 Upper Bound γSR = 15 dB 10 Simulations γSR = 15 dB -6 10 Upper Bound γSR = 50 dB Simulations γSR = 50 dB -7 10 0 5 10 15 20 Receive SNR at the destination, γ 25 30 Figure 2: Bit error probability versus γ = E/N 0 based on simulations and upper bound 145
  9. 9. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME Fig. 2 compares the bit error probability obtained from simulations with the exact upper bound given by (33). The BER is plotted against the receive SNR γ = E/N 0 of S-D and R-D links at different values of the the receive SNR of the S-R link γ sr or equivalently different values of the distance d sr . We found that the derived upper bound is very tight. Therefore, the closed form expression of the upper bound can be used to fully study and understand the diversity performance of the DF cooperation protocol. We also found that the SNR of the S-R link highly affects the end-to-end error rate of the system. At low values of γ sr the error rate of the S-R link Pe would be high and sr accordingly the system does not provide the expected diversity gain. We notice that the diversity gain increases as the value of γ sr increases. 2 1.9 1.8 Diversity Order (L) 1.7 1.6 1.5 1.4 1.3 γ1 = 20 dB 1.2 γ1 = 30 dB γ1 = 40 dB 1.1 γ1 = 50 dB 1 0 10 20 30 40 50 60 70 Receive SNR at the relay, γsr 80 90 100 Figure 3: Diversity gain versus γ sr at different values of γ 1 Since the diversity order is related to the slope of the BER curve with respect to γ and in order to the study the effect of the γ sr on the diversity, let’s define the diversity order L as follows L= log10 PUB (γ 2 ) − log10 PUB (γ 1 ) (γ 2 (dB) − γ 1 (dB))/10 (35) where PUB (γ i ) is the upper bound at γ i and i = 1,2 . The definition of (35) measures the change of exponent of the BER with respect to one decade of the SNR in dB. Fig. 3 shows the diversity order defined in (35) against the receive SNR of the S-R link γ sr at different values of γ 1 where γ 2 = γ 1 + 10 dB. We found that full diversity of 2 is achieved at higher values of γ sr . We also found that at fixed value of γ sr , higher diversity order is achieved at lower values γ , e.g., at γ sr = 40 dB, the diversity order is ≈ 1.7 when γ 1 = 20 dB, and the diversity order is ≈ 1 when γ 1 = 50 dB. This result is consistent with the approximation of (34) where the diversity order is close to 1 when γ > γ sr and close to 2 when γ < γ sr and both of them are high. The intuitive explanation for this phenomenon is that the enhancement of BER performance is dominated by the link with the lower SNR. In other words, the link with the lower SNR is considered as the bottleneck of the BER improvement. Therefore, when γ is small it will be the bottleneck and the BER is proportional to 1/γ 2 . 146
  10. 10. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME VI. CONCLUSIONS In this paper, we presented the MAP decoding scheme for the wireless relay network that adopts the decode-and-forward as a cooperation protocol. We derived a closed form expression for the upper bound on the bit error probability. The numerical results show that the upper bound is very tight. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] S. Alam and S. H. Gupta, “Performance analysis of cooperative communication wireless network,” International Journal of Electronics and Communication Engineering and Technology IJECET, pp. 301–309, vol. 3, no. 2, July–September 2012. J. N. Laneman, D.N.C.Tse, G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Transactions on Information Theory, pp. 3062–3080, Dec. 2004. A. Sendonaris, E. Erkip, B.Azhang, “User cooperation diversity – Part I: System description,” IEEE Transactions on Communications, pp. 1927–1938, Nov. 2003. A. Nosratinia, et al., “Cooperative communication in wireless networks,” IEEE Communications Magazine, pp. 74–80, Oct. 2004. S. W. Kim, “Cooperative spatial multiplexing in mobile ad hoc networks,” Proc. of IEEE International Conference on Mobile Ad hoc and Sensor Systems Conference (MASS,) 2005. Y. Zhang, G. Wang, M. G. Amin, “Cooperative spatial multiplexing in multi-hop wireless networks,” Proc. of IEEE ICASSP06, pp. IV821–IV824, May 2006. A. Host-Madsen and A. Nosratinia, “The multiplexing gain of wireless networks,” Proc. of IEEE International Symposium on Information Theory, 2005. I.S.Gradshteyn and I.M.Ryzhik, Tables of Integrals, Series, and Products, pp.341, Eq. 3.472, Academic Press, 1980. Andrea Goldsmith, Wireless communications, Cambridge University Press , 2005. T. M. Cover, A. A. El Gamal, “Capacity theorems for the relay channel”, IEEE Transactions on Information Theory, vol. IT-25, No 5, pp 572-584, Sept. 1979. A.D. Wyner, “The rate-distortion function for source coding with side information at the decoder –II: General Sources”, Information and Control, Vol 38, pp 60-80, 1978. Mazen O. Hasna and Mohamed S. Alouini, “End-to-end performance of transmission systems with relays over Rayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1126–1131, Nov. 2003. D. Chen and J. N. Laneman, “Modulation and demodulation for cooperative diversity in wireless systems,” IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 1785–1794, Jul. 2006. Taha Khalaf, “Performance of Maximum Likelihood Decoder in Network Coded Cooperative Communications,” IEEE Wireless Days Conference, Valencia, Spain, Nov. 2013. 147

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