DSpace at VNU: Noncont decode-and-forward cooperative systems with maximum energy selection

The 2013 International Conference on Advanced Technologies for Communications (ATC'13) Noncoherent Decode-and-Forward Cooperative Systems with Maximum Energy Selection Ha X Nguyen1, Cuu V Ho2 , Chan Dai Truyen Thai3 , Danh T Nguyen4 ha.nguyen@ttu.edu.vn, School of Engineering, Tan Tao University Tan Duc Ecity, Duc Hoa, Long An Province, Vietnam cuu.hv@cb.sgu.edu.vn, Department of Electronics and Telecommunications, Saigon University 273 An Duong Vuong, District 5, Ho Chi Minh City, Vietnam chan.thai@ifsttar.fr, Univ Lille Nord de France F-59000, Lille, IFSTTAR, LEOST, F-59650, Villeneuve d’Ascq, France nthanhdanh0410@gmail.com, Faculty of Electronics and Telecommunications, University of Science Vietnam National University, Ho Chi Minh City, Vietnam Abstract—This paper investigates the performance of a maximum energy selection receiver of an adaptive decode-andforward (DF) relaying scheme for a cooperative wireless system In particular, a close-form expression for the bit-error-rate (BER) is analytically derived when the system is deployed with binary frequency-shift keying (BFSK) modulation The thresholds used at the relays to address the issue of error propagation are optimized to minimize the BER While finding the optimal thresholds requires information on the average signal-to-noise ratios (SNRs) of all the transmission links in the system, the approximate threshold at each relay that requires only information on the average SNR of the source-corresponding relay is investigated It is also shown that the system achieves a full diversity order with the approximate thresholds Both analytical and simulation results are provided to validate our theoretical analysis receiver, i.e., selecting the maximum output from the squarelaw detectors of all branches to perform a detection, for a threshold-based (i.e., adaptive) DF cooperative system While the destination in [7] relies on the maximum magnitude of the energy difference, the destination in this paper employs the maximum energy from the square-law detectors to detect the transmitted signal The approximate thresholds that achieve full diversity are provided in this paper Note that the direct link between the source and destination is considered in this work while the work in [7] assumes that there is no such a link II S YSTEM M ODEL 5HOD\ I I NTRODUCTION Frequency shift keying (FSK) is a popular modulation scheme in noncoherent communications in which the receiver does not require any channel state information (CSI) to decode the transmitted signals [1] Consequently, using FSK signals in cooperative systems has been focused recently since there is a complexity advantage in decoding [2]–[7] It is due to the fact that there are many wireless fading channels involved in the systems [8], [9], which makes the task of channel estimation more difficult With the decode-and-forward (DF) protocol employing FSK in cooperative systems, reference [3] proposed maximum likelihood (ML) and suboptimal piecewise linear (PL) schemes to decode the signals at the destination However, it was shown that the system could not achieve a full diversity order due to the error forwarding at the relays References [6], [7] proposed to use a threshold at the relays to address the issue of error propagation for binary frequencyshift keying (BFSK) modulation While the destination in [6] combines all the signals from the retransmitting relays, the destination in [7] selects only one signal with the largest magnitude of the energy difference to decode Unfortunately, designing the optimal thresholds to minimize the average biterror-rate (BER) of the system relies on the MATLAB Optimization Toolbox and a theoretical analysis of the diversity order is not available This paper studies the maximum energy selection (MES) 978-1-4799-1089-2/13/$31.00 ©2013 IEEE θ  > θ 'HVWLQDWLRQ WK U 5HPDLQ 6LOHQW < 'HFRGHDQG )RUZDUG 0D[LPXP (QHUJ\ 6HOHFWLRQ 6RXUFH 5HOD\ θ  > θ UWK 5HPDLQ 6LOHQW < 'HFRGHDQG )RUZDUG Fig System description of the proposed scheme Fig illustrates the signal transmission from the source (node 0) to destination (node K + 1) with the assistance of K half-duplex relays (node i, i = 1, , K) The relays retransmit signals to the destination in orthogonal channels In this paper, we assume that the fading channel coefficient between transmit node i and receive node j, denoted by hi,j , and the noise component at receive node j, denoted by ni,j , are modeled as zero-mean complex Gaussian random variables and N0 , respectively The instantaneous with variances σi,j signal-to-noise ratio (SNR) of the channel between node i and node j, which is denoted by γi,j , is given as γi,j = 136 The 2013 International Conference on Advanced Technologies for Communications (ATC'13) Ei |hi,j |2 /N0 where Ei is the average transmitted energy of /N0 node i The corresponding average SNR is γ i,j = Ei σi,j In the first phase, the source broadcasts the signal xm and the received signals at node i, i = 1, , K + 1, are written as y0,i = E0 h0,i xm + n0,i , i = 1, 2, , K + (1) A BER Computations The law of total probability is employed to compute the average BER of the system First, denote Ω1 , Ω2 , and Ω3 as the sets of the relays that forward a correct bit, an incorrect bit, and remain silent, respectively It is clear that K = |Ω1 | + |Ω2 | + |Ω3 | where |Ω| denotes the cardinality of set Ω The probability of occurrence for the specific set {Ω1 , Ω2 , Ω3 } is [7]: where xm is the mth symbol of an BFSK constellation I1 (θrth , γ 0,i ) − I1 (θrth , γ 0,i ) Without loss of generality, assume that the first symbol P (Ω1 , Ω2 , Ω3 ) = i∈Ω3 i∈(Ω1 ∪Ω2 ) from the signal constellation is transmitted The outputs of th the square-law detector for the first and second symbols at × I2 (θrth , γ 0,i ) (11) − I2 (θr , γ 0,i ) node i, i = 1, , K + are written, respectively, as i∈Ω1 i∈Ω2 y0,i,1 y0,i,2 = = E0 h0,i + n0,i,1 , (2) n0,i,2 , (3) As in [6], the difference of the outputs of the squarelaw detector, namely θ0,i = y0,i,1 − y0,i,2 , is considered as a reliability measure of the detection at node i Therefore, node i decodes and retransmits a BFSK signal only if θ0,i > θrth When node i transmits a correct bit in the second phase, the outputs of the square-law detector for the first and second symbols at the destination are yi,K+1,1 yi,K+1,2 = = Ei h0,i + ni,K+1,1 , ni,K+1,2 (4) (5) Meanwhile, the outputs of the square-law detector for the first and second symbols at the destination can be written as follows if node i transmits an incorrect bit: yi,K+1,1 yi,K+1,2 = = ni,K+1,1 , (6) Ei h0,i + ni,K+1,2 , (7) where A ∪ B denotes the union of sets A and B The function I1 (θrth , γ 0,i ) is the probability of the event θ0,i < θrth and is computed as [7]: + γ 0,i th − e−θr /(1+γ 0,i ) I1 (θrth , γ 0,i ) = + γ 0,i th 1 − e−θr (12) + + γ 0,i On the other hand, I2 (θrth , γ 0,i ) is the probability of error at node i, i = 1, , K, given the event θ0,i > θrth and is determined by [7] th 1 e−θr (13) I2 (θrth , γ 0,i ) = + γ 0,i − I1 (θrth , γ 0,i ) Now let Ww,m (w ∈ {Ω1 ∪ {0}}), Vv,m (v ∈ Ω2 ) and Rr,m (r ∈ Ω3 ) denote the outputs of the square-law detector for the mth symbol, m = 1, 2, measured at the destination With the assumption that the first symbol from the signal constellation is transmitted, the probability density functions (pdfs) of Ww,m , Vv,m and Rr,m are given, respectively, by If θ0,i < θrth , node i remains silent in the second phase and the outputs of the square-law detector for the first and second symbols at the destination are fWw,m (x) = fw,1 (x), m = fw,2 (x), m = (14) = fv,2 (x), m = fv,1 (x), m = (15) = yi,K+1,1 = ni,K+1,1 | , (8) fVv,m (x) yi,K+1,2 = ni,K+1,2 |2 (9) fRr,m (x) Finally, the destination compares and chooses the maximum output from all the outputs of the square-law detectors, i.e., employs the maximum energy selection, to detect the transmitted information In other words, the decision rule is of the following form: ˆi, m ˆ = arg max yi,K+1,m i=0, ,K m=1,2 (10) fr,2 (x), m = or m = −x/(N0 (1+γ k,K+1 )) , N0 (1+γ k,K+1 ) e −x/N0 e , x ≥ N0 where fk,1 (x) = (16) x ≥ and fk,2 (x) = An error occurs at the destination if among the 2(K + 1) statistics Ww,m , Vv,m and Rr,m , w ∈ {Ω1 ∪{0}}, n ∈ Ω2 , r ∈ Ω3 , m = 1, 2, the one with the largest value is 1) Case (Θ = 1) one of Ww,1 , 2) Case (Θ = 2) one of Vv,1 , and 3) Case (Θ = 3) one of Rr,1 Thus, given the set {Ω1 , Ω2 , Ω3 }, the BER can be computed as III BER C OMPUTATIONS AND T HRESHOLDS In this section, the BER analysis for MES scheme is first carried out for a network with arbitrary qualities of sourcerelay and relay-destination links Then, the optimal thresholds are chosen to minimize the average BER are discussed Finally, the approximate thresholds are proposed to achieve a full diversity order 137 PΩ1 ,Ω2 ,Ω3 (ε, Θ = i) PΩ1 ,Ω2 ,Ω3 (ε) = i=1 P W w,2 − Ww,2 < + = P V v,2 − Vv,2 < v∈Ω2 w∈Ω1 ∪{0} P Rr,2 − Rr,2 < + r∈Ω3 (17) The 2013 International Conference on Advanced Technologies for Communications (ATC'13) where W w,2 V v,2 = Rr,2 = max = max i=w m=1,2 max i=v m=1,2 i=r m=1,2 (Wi,m , Ww,1 , Vv,m , Rr,m ), (Ww,m , Vi,m , Vv,1 , Rr,m ), and (Ww,m , Vv,m , Ri,m , Rr,1 ) The conditional BER PΩ1 ,Ω2 ,Ω3 (ε, Θ = i), i = 1, 2, 3, can be computed1 as (18), (19), and (20) on the top of this page, where (G1 ∪ G2 ) = Ω means that G1 and G2 are two disjoint subsets of Ω and the union of those disjoint subsets is Ω Obviously, the average BER with a given threshold θrth can be expressed as On the other hand, the conditional BER PΩ1 ,Ω2 ,Ω3 (ε) = i=1 PΩ1 ,Ω2 ,Ω3 (ε, Θ = i) can be evaluated from the large SNR behavior by considering the value of the first non-zero order derivative of the PDF at the origin [10] According to [11], one can verify that PΩ1 ,Ω2 ,Ω3 (ε, Θ = 1) = PΩ1 ,Ω2 ,Ω3 (ε, Θ = 2) = Ω1 ∈P(S) Ω2 ∈P(S\Ω1 ) PΩ1 ,Ω2 ,Ω3 (ε, Θ = i)P (Ω1 , Ω2 , Ω3 ) (21) i=1 where P(Ω) denotes the power set of Ω The set S = {1, , K} = arg BER(θrth ) θrth lim γ →∞ 1− cγ Q 1+γ log(γ)/γ =Q γ0 = Qσ02 , γ (23) it follows from (11) that3 P (Ω1 , Ω2 , Ω3 ) ≤ I1 (θrth , γ 0,i ) |Ω3 | I2 (θrth , γ 0,i ) |Ω3 | (log(γ)/γ) The |Ω2 | 1/γ Q+1 |Ω2 | (24) pdfs of W w,2 , V v,2 and Rr,2 are given in Appendix A using the approximate thresholds, besides the information collection, the system does not need to find the optimal thresholds centrally and send to the relays, hence, reducing the complexity and implementation costs of the system With two positive real functions f (x) and g(x), we say f (x) g(x) if (x) = d where d < ∞ is a positive constant lim supx→∞ fg(x) By PΩ1 ,Ω2 ,Ω3 (ε, Θ = 3) = |Ω1 |+|Ω2 |+1 ∞ ∞ , (25) fV v,1 (x)e−x/N0 (1+γ ) dx if |Ω1 | + |Ω2 | = if |Ω1 | + |Ω2 | > (26) fRr,1 (x)e−x/N0 dx |Ω1 |+|Ω2 |+1 (1/γ ) (27) Thus, one has PΩ1 ,Ω2 ,Ω3 (ε) ⎧ K+1 ⎪ , if |Ω2 | = ⎨(1/γ ) Q+1+|Ω3 | (1/γ ) , if |Ω1 | = and |Ω2 | = ⎪ ⎩ K+|Ω2 |(Q+1) , if |Ω1 | > and |Ω2 | > (1/γ ) (22) It is clear from (21) that the system need to collect information on the average SNRs of all the transmission links to find the optimal thresholds Unfortunately, an close-form solution for optimal threshold values is very difficult, if not impossible, to find Therefore, to further reduce the complexity of the system2 , in what follows, we propose approximate thresholds and prove that by using those thresholds, the system can achieve the maximum diversity order Lemma 1: If the relays use the threshold θrth = Q log cγ where γ = E0 /N0 , the system achieves a full diversity order of K + for any Q ≥ K and a positive constant c Proof: To simplify our derivation, we consider the i.i.d case, i.e., γ 0,i = γ i,K+1 = γ 0,K+1 = γ , i = 1, , K where γ = E0 σ02 /N0 and N0 = Since θrth = Q log cγ and fW w,1 (x)e−x/N0 dx 1, |Ω |+|Ω2 | (1/γ ) , θrth (1/γ ) BER θrth = B Optimal and Approximate Thresholds Given the closed-form expression of the average BER in (21), one can choose the threshold θrth to minimize the average BER of the system by using the MATLAB Optimization Toolbox The optimization problem can be set up as follows: ∞ (28) Therefore, for sufficiently large values of SNR and θrth = Q log cγ where Q ≥ K, it follows from (21) that BER θrth K+1 (1/γ ) (29) So Lemma is proved IV S IMULATION R ESULTS This section presents analytical and simulation results for the BER performance of different noncoherent DF cooperative systems In conducting the simulations, it is assumed that the noise components at the receivers, i.e., relays and destination are modeled as i.i.d CN (0, 1) random variables Fig plots the average BERs of the proposed scheme, PL scheme and the scheme in [6] in a two-relay system when the variances of 2 = 0.1σi,K+1 = Rayleigh fading channels are set to be 2σ0,i = 1, i = 1, From the figure, both the analytical 5σ0,K+1 (shown as marker symbols) and simulation (shown in line with marker symbols) results are identical, hence verifying our analysis in Section III The figure also shows that the BER of the proposed scheme is significantly better than the BER of the PL scheme It is institutively clear since the PL scheme suffers from the error propagation The scheme in [6] outperforms the other two schemes due to the fact that the destination in [6] combines all the signals from the retransmitting relays besides dealing with the problem of error propagation However, the proposed scheme does not require any statistical information of the fading channels to perform a 138 The 2013 International Conference on Advanced Technologies for Communications (ATC'13) ⎡ K+|Ω3 | K + |Ω3 | ⎣ l l=0 i∈(Ω1 ∪Ω2 ∪{0}) (G1 ∪G2 )=((Ω1 ∪Ω2 ∪{0})\{i}) ⎞⎤ ⎛ 1 ⎠⎦ ⎝ (−1)K+|Ω3 |+|G2 |−l K+|Ω3 |−l+1 1 N0 + γ i,K+1 t∈G2 N0 (1+γ t,K+1 ) + N0 (1+γ i,K+1 ) + N0 ⎡ K+|Ω3 |−1 (|Ω1 | + 1) (K + |Ω3 |) K + |Ω3 | − ⎣ + l N0 l=0 (G1 ∪G2 )=(Ω1 ∪Ω2 ∪{0}) ⎞⎤ ⎛ ⎠⎦ (−1)K+|Ω3 |+|G2 |−l−1 ⎝ K+|Ω3 |−l+1 + t∈G2 N0 (1+γ ) N0 PΩ1 ,Ω2 ,Ω3 (ε, Θ = 1) = (|Ω1 | + 1) (18) t,K+1 K+|Ω3 |+1 PΩ1 ,Ω2 ,Ω3 (ε, Θ = 2) = l=0 (−1)K+|G2 |+|Ω3 |−l+1 ⎡ K + |Ω3 | + l ⎛ N0 + γ i,K+1 + N0 K+|Ω3 | l=0 ⎣ v∈Ω2 i∈(Ω1 ∪Ω2 ∪{0})\{v} (G1 ∪G2 )=((Ω1 ∪Ω2 ∪{0})\{i,v}) ⎝ t∈G2 N0 (1+γ t,K+1 ) K + |Ω3 | l ⎡ ⎛ PΩ1 ,Ω2 ,Ω3 (ε, Θ = 3) = |Ω3 | l=0 (−1)K+|G2 |+|Ω3 |−l + N0 1 + γ i,K+1 |Ω3 | (K + |Ω3 |) N0 + N0 (1+γ v,K+1 ) + K+|Ω3 |−l+1 N0 (G1 ∪G2 )=((Ω1 ∪Ω2 ∪{0})\{v}) N0 (1+γ t,K+1 ) + (N0 (1+γ v,K+1 ) + K+|Ω3 |−l+1 N0 i∈((Ω1 ∪Ω2 ∪{0})) (G1 ∪G2 )=((Ω1 ∪Ω2 ∪{0})\{i}) ⎝ N0 (1+γ t,K+1 ) + ⎡ t∈G2 l=0 N0 (1+γ i,K+1 ) + K+|Ω3 |−l+1 N0 ⎞⎤ t∈G2 (19) ⎠⎦ K + |Ω3 | − ⎣ l (G1 ∪G2 )=(Ω1 ∪Ω2 ) ⎛ (−1)K+|G2 |+|Ω3 |−l−1 ⎝ ⎠⎦ ⎞⎤ ⎛ K+|Ω3 |−1 ⎠⎦ ⎞⎤ t∈G2 ⎡ K + |Ω3 | ⎣ l + N0 (1+γ i,K+1 ) ⎣ v∈Ω2 (−1)K+|G2 |+|Ω3 |−l ⎝ K+|Ω3 | ⎞⎤ N0 (1+γ t,K+1 ) + K+|Ω3 |−l+1 N0 ⎠⎦ (20) detection Such the information is required for the PL scheme and the scheme in [6] Fig presents the average BERs obtained by simulation and analysis for two different schemes in a three-relay cooperative 2 = σi,K+1 = σ0,K+1 = 1, i = 1, 2, The system Here σ0,i figure again confirms the analysis performed in Section III At sufficient large values of SNR, the proposed scheme yields a superior performance compared to the PL scheme BFSK signals A closed-form BER expression is obtained and used to choose the optimal thresholds to minimize the average BER Approximate thresholds are proposed and the diversity order is verified Performance comparison reveals that the proposed scheme outperforms the other two schemes with a lower complexity V C ONCLUSION This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.33 This paper studies the maximum energy selection receiver for an adaptive decode-and-forward (DF) relaying system with VI ACKNOWLEDGEMENT 139 The 2013 International Conference on Advanced Technologies for Communications (ATC'13) 10 10 10 fV v,2 (x) = −1 d P (V v,2 < x) = dx fi,1 (x)× i∈(Ω1 ∪Ω2 ∪{0})\{v} Fj,1 (x) (F1,2 (x))K+|Ω3 |+1 + −2 BER j∈((Ω1 ∪Ω2 ∪{0})\{i,v}) 10 × e−x/N0 (F1,2 (x))K+|Ω3 | −3 (K + |Ω3 | + 1) N0 Fj,1 (x) (31) j∈(Ω1 ∪Ω2 ∪{0})\{v} 10 10 10 −4 PL Two−threshold [6] MES (Opt threshold − Simulation) MES (Opt threshold − Analysis) MES (Approx threshold − Simulation) MES (Approx threshold − Analysis) −5 −6 10 15 fRr,2 (x) = d P (Rr,2 < x) = dx fi,1 (x)× i∈(Ω1 ∪Ω2 ∪{0}) Fj,1 (x) (F1,2 (x)) K+|Ω3 | + j∈((Ω1 ∪Ω2 ∪{0})\{i}) 20 25 × e−x/N0 (F1,2 (x)) 30 Average Power per Node (dB) K+|Ω3 |−1 (K + |Ω3 |) N0 Fj,1 (x) (32) j∈(Ω1 ∪Ω2 ∪{0}) Fig BERs of a two-relay network with different schemes when = 0.1σ 2 2σ0,i i,K+1 = 5σ0,K+1 = 10 BER 10 10 10 10 where Fk,1 (x) = − e−x/(N0 (1+γ k,K+1 )) and Fk,2 (x) = − e−x/N0 R EFERENCES −2 −4 PL MES (Opt threshold − Simulation) MES (Opt threshold − Analysis) MES (Approx threshold − Simulation) MES (Approx threshold − Analysis) −6 −8 10 15 20 25 30 Average Power per Node (dB) Fig BERs of a three-relay network with different schemes when = σ2 σ0,i i,K+1 = σ0,K+1 = A PPENDIX A P DFS OF W w,1 , V v,1 , AND Rr,1 RANDOM VARIABLES The pdf of W w,1 , V v,1 , and Rr,1 can be found, respectively, as follows: fW w,2 (x) = d P (W w,2 < x) = dx fi,1 (x)× i∈(Ω1 ∪Ω2 ∪{0}) Fj,1 (x) (F1,2 (x)) K+|Ω3 | + j∈((Ω1 ∪Ω2 ∪{0})\{i}) e−x/N0 (F1,2 (x)) [1] M K Simon and M.-S Alouini, Digital Communication over Fading Channels Wiley, 2005 [2] R Annavajjala, P Cosman, and L Milstein, “On the performance of optimum noncoherent amplify-and-forward reception for cooperative diversity,” Proc IEEE Military Commun Conf., pp 3280–3288, October 2005 [3] D Chen and J Laneman, “Modulation and demodulation for cooperative diversity in wireless systems,” IEEE Trans Wireless Commun., vol 5, pp 1785–1794, July 2006 [4] M R Souryal, “Non-coherent amplify-and-forward generalized likelihood ratio test receiver,” IEEE Trans Wireless Commun., vol 9, pp 2320–2327, July 2010 [5] G Farhadi and N Beaulieu, “A low complexity receiver for noncoherent amplify-and-forward cooperative systems,” IEEE Trans Commun., vol 58, pp 2499–2504, September 2010 [6] H X Nguyen and H H Nguyen, “Adaptive relaying in noncoherent cooperative networks,” IEEE Trans Signal Process., vol 58, pp 3938– 3945, July 2010 [7] H X Nguyen and H H Nguyen, “Selection combining for noncoherent decode-and-forward relay networks,” EURASIP Journal on Wireless Communications and Networking, vol 2011, no 1, p 106, 2011 [8] J Laneman and G Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans Inform Theory, vol 49, pp 2415–2425, October 2003 [9] J Laneman, D Tse, and G Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans Inform Theory, vol 50, pp 3062–3080, December 2004 [10] Z Wang and G Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans Commun., vol 51, pp 1389–1398, August 2003 [11] Y Zhao, R Adve, and T J Lim, “Symbol error rate of selection amplifyand-forward relay systems,” IEEE Commun Letters, vol 10, pp 757– 759, November 2006 K+|Ω3 |−1 K + |Ω3 | × N0 Fj,1 (x) (30) j∈(Ω1 ∪Ω2 ∪{0}) 140 ... IMULATION R ESULTS This section presents analytical and simulation results for the BER performance of different noncoherent DF cooperative systems In conducting the simulations, it is assumed that... the destination compares and chooses the maximum output from all the outputs of the square-law detectors, i.e., employs the maximum energy selection, to detect the transmitted information In other... Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.33 This paper studies the maximum energy selection receiver for an adaptive decode-and-forward