Exact BEP of Cooperative MC-CDMA Systems using Selective Threshold Digital Relaying

2 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications Exact BEP of Cooperative MC-CDMA Systems using Selective Threshold Digital Relaying Hela Hakim, Hatem Boujemaa and Wessam Ajib 2 Carthage University, Higher School of Communication of Tunis, TECHTRA Research Unit, Tunisia Emails: hela.hakim@gmail.com and boujemaa.hatem@supcom.rnu.tn 2 Department of Computer Sciences, Université dequébec à Montréal, Canada Email: ajib.wessam@uqam.ca Abstract In this paper, we derive the end-to-end (e2e Bit Error Probability (BEP of cooperative Multi Carrier Code Division Multiple Access (MC-CDMA systems using selective threshold digital relaying (STDR. In STDR, a set of potential relays whose received Signal-to-Noise Ratio (SNR exceeds a threshold value γ t, called reliable relays, is formed. Then, only the best relay among reliable relays is allowed to retransmit the received signal. We activate the relay with the largest SNR in relay-destination link. The derived BEP results are valid for any multipath intensity profile of the channel. I. INTRODUCTION Cooperative diversity exploits broadcast nature of wireless transmission to create spatial diversity by introducing relay node(s to assist the communication between a transmitter and a receiver ] 3]. Relaying protocols can be classified into two categories: (i analog relaying when the relay amplifies the noisy received signal without detection and then forwards the amplified signal, and (ii digital relaying when the relay decodes and then forwards the remodulated signal. This paper focuses on digital relaying. If the relay correctly detects the received signal, the symbol error probability at the destination is significantly decreased by combining signal copies coming from two branches: sourcedestination and relay-destination. However, if the relay forwards an erroneous detected signal, a symbol error at the destination is strongly probable. This event is called error propagation, and can significantly deteriorate e2e performance of the relaying protocol. An obvious way to mitigate error propagation is to use coding schemes such as Cyclic redundancy check (CRC to detect errors at relays. This requires high computational capacity. Moreover, the error detection causes additional time delay. In heterogeneous networks including devices with a wide range of computational capacities, it is interesting to use relaying schemes that are transparent to coding. An alternate approach to coding schemes is the use of the instantaneous SNR of the source-relay link as an indication of the reliability of the relay detection. If the SNR of the source-relay link is above a predetermined threshold value, the relay is likely able to correctly decode the received This work was supported by a National Priorities Research Program grant from the Qatar National Research Fund under the reference (NPRP8-577-2-24 and NSERC Discovery grant. signal. Otherwise, a symbol error at the relay is strongly probable. These kind of relaying schemes are called threshold digital relaying. They have been proposed and their asymptotic e2e BEP have been derived in 4] 5]. Namely in 5], Onat et. al have proposed a threshold digital relaying cooperation protocol (TDRC only relays whose received SNR are above a threshold value, called reliable relays, are authorized to retransmit. The selection of a single relay among reliable relays is proved to achieve full diversity order and increase spectral efficiency 6]-8]. In this work, only the best relay among reliable relays cooperates, we call this kind of protocol selective threshold digital relaying (STDR. The best relay is the one with the largest SNR in relay-destination link. In 9]- ], the performance of STDR in terms of capacity outage and BEP have been derived for Rayleigh fading environments in the presence of a single path channel. To the best of our knowledge, the exact e2e BEP of STDR protocol with best relay selection has not been derived for MC CDMA cooperative systems in the presence of multipath propagation. All the previous works dealing with STDR protocol considered a single path channel. In this paper, we derive the exact e2e BEP of STDR protocol in the presence of multipath propagation for cooperative MC CDMA systems. We consider a cooperative wireless network using MC CDMA a source user communicates with a single destination D, that can be a Base-Station (BS or an Access Point (AP and the remaining N- users serve as potential relays. The system use the STDR and the activated relay offers the largest SNR in the relay-destination link. The derived results are valid for any multipath intensity profile of the channel. The remainder of this paper is organized as follows. In section II, we present the system model. In section III, we derive exact e2e BEP of cooperative MC CDMA systems using STDR. In section IV, we present simulation and theoretical results. Section V draws concluding remarks and a summary of our findings. II. SYSTEM MODE A. Relaying protocol We consider a wireless network with a circular cell. We assume that N uniformly distributed users within the cell communicate with a destination (D, that can be a BS/AP, 978--4577-348-4//$26. 2 IEEE 4

located at the center of the cell. Each user transmits a signal to D while the remaining N- users serve as potential relays. Nodes are assumed to communicate in half-duplex mode, i.e., they can not transmit and receive simultaneously. A two-phase relaying protocol is considered. In the first phase, during the first time slot, a source user (S transmits a signal to D while the N- remaining users listen, as shown in Fig.. Users which receive the source signal with an SNR above the threshold value γ t, send training symbols so that D can determine the SNR of the relay-destination link. Based on the collected SNR information, D activates the relay which offers the largest SNR in relay-destination link. In the second phase of the considered relaying protocol, during the second time slot, the activated relay re-transmits the source signal to D, as shown in Fig. 2. Fig. 2. Phase 2: A selected relay among reliable relays retransmits the source signal to D Fig.. listen Phase : S transmits a signal to D while the remaining N users B. Models of transmitted and received signals An MC-CDMA transmitter spreads the original signal using a spreading code in the frequency domain ]-2]. Without loss of generality, the number of subcarriers,, is assumed to be equal to the spreading factor. The equivalent base-band transmitted signal can be written as e(t = s k g(t kt s c X,k+m e j2πfmt, ( k m= s k is the k-th transmitted symbol, E X is the transmitted energy per symbol by X, T s is the symbol period, {c X,k+m } m= is a unit modulus spreading sequence used by X, g(t is a rectangular pulse response with unit useful energy and duration T s = Ts u + η, Ts u is the useful symbol period and η is the guard interval, f m = f + mδf is the m-th subcarrier frequency, f is the frequency of the first subcarrier and Δf =/Ts u is the subcarrier separation. If the channel delay spread is lower than the guard interval η, then the restriction of the received signal to the interval kt s + η, (k +T s ] can be written as r(t = s k c X,k+m e j2πfmt F (f m ; t Ts η m= +n (t, (2 F (f m ; t = f (τ; te j2πfmτ dτ, (3 f (τ; t is the impulse response of the Rayleigh multipath fading channel of the X-Y link at the time instant t, and n (t is an AWGN with two-sided power spectral density N. An MC-CDMA receiver uses a Discrete Fourier Transformation (DFT to recover the transmitted signal over the different subcarriers ]-2]. In the following, a perfect synchronization on the different subcarrier frequencies is assumed. After removing the received signal during the guard interval and compensating the modulation due to the spreading sequence, the DFT outputs for symbol s k can be written as z k = (z k,...,z, k T = s kf + n, (4 F = (F (f ; kt s,...,f (f ; kt s T and n =(n k,,...,nk, T is a vector of AWGNs with covariance matrix N I, I is the identity matrix with size, and (. T is the transpose operator. Assuming perfect channel estimation, the optimal soft output of the MC-CDMA receiver is given by Λ k = F zk N, (5 5

(. denotes the Hermitian operator. Hence the instantaneous SNR at the output of the receiver is given by γ = E X F F N. (6 III. E2E BEP ANAYSISOFTHESYSTEM In this section, we derive the e2e BEP at D for binary phase shift keying (BPSK modulation. The e2e BEP at D can be written as P e,d = N m= P ( Θ = mp (e Θ = m, (7 Θ denotes the set of users which receive the source signal with an SNR above γ t and Θ denotes the cardinality of Θ. Following the same methodology presented in 3], the PDF of the SNR in (6 is given by p γ (x = = exp( x, if x, (8 = E X λ(j (9 N k k j λ (j λ (j, ( λ(k λ (j is the j-th eigenvalue of F correlation matrix Q =E(F F. The probability that γ <γ t is given by P (γ <γ t = exp( γ t ]. ( The probability that γ γ t is given by P (γ γ t = exp( γ t. (2 Hence, for m>, wehave ( N m P ( Θ = m = exp( γ t n= k U(n i U(n ( exp( γ t, (3 U(n is the n-th possible combination of m reliable relays among the set of N- potential relays. P (e Θ = m = ( N m n= P (e Θ = m, Θ=U(n P (Θ = U(n Θ = m. (4 The last term of the above equation is given by P (Θ = U(n Θ = m = exp( γ t i U(n k U(n P ( Θ = m exp( γ t ]. (5 et I = { Θ = m, Θ=U(n}. Hence, the conditional bit error probability P (e I can be written as P (e I = q Θ P (e I,R SelΘ = qp (R SelΘ = q, (6 R SelΘ is the activated relay in Θ. We have γ RSelΘ D = max R i Θ {γ R i D}. Hence, the probability P (R SelΘ = q is given by P (R SelΘ = q = P (γ RqD >γ Rk D. (7 k Θ k q The obtained expression of P (R SelΘ = q is given by P (R SelΘ = q = k Θ k q R qd l= π (l R k D β (l R k D β (l R k D β(j R qd β(l R k D R q D + β(l R k D. (8 If R SelΘ decodes incorrectly the received signal, it retransmits an erroneous signal to D leading to error propagation event. The bit error probability at D due to error propagation is denoted by P eprop,d. The bit error probability at D given that R SelΘ has retransmitted a correctly decoded signal is denoted by P ecoop,d. Hence, the probability P (e I,R SelΘ = q can be written as P (e I,R SelΘ = q = P e,rselθ P eprop,d +( P e,rselθ P ecoop,d, (9 P e,rselθ is the bit error probability at R SelΘ. The bit error probability at D due to error propagation P eprop,d can be bounded by the worst case value i.e., P eprop,d 2 4], 5]. The bit error probability P e,rselθ can be written as P e,rselθ = γ t Q( 2xp γsrselθ γ SRSelΘ γ t (xdx, (2 Q(x = e t2 /2. The conditional PDF x p γsrselθ γ SRSelΘ γ t (x is given by 2π + p γsrselθ (x p γsrselθ γ SRSelΘ γ t (x = Υ SRSelΘ (γ t if x γ t (2 o.w, 6

p γrselθd (x= i Θ k= π (k β (k m = π (m R l(θ,i,d... m Θ = π (m 2 Θ Θ R l(θ,i, Θ D n= ( ξ(n x exp(. (22 α nikm...m Θ P ecoop,d = SD i Θ k= π (k R i D β (k +Ψ(α nikm...m Θ m = π (m R l(θ,i, D... α 2 nikm...m Θ α nikm...m Θ SD π (m 2 Θ Θ R l(θ,i, Θ D ( ξ(n Ψ( SD β(j SD α nikm...m Θ n= SD α nikm...m Θ ]. (23 m Θ = Υ (γ t =P(γ γ t. Using integration by parts and an adequate variable substitution we obtain SR P e,rselθ = SelΘ Q( γ t β 2γ t e (j Υ SRSelΘ (γ t ( ] β(j Q 2γ + t ( +. (24 The bit error probability P ecoop,d is given by P ecoop,d = Q( 2(x + up γsd (xp γrselθd (udxdu. To determine the pdf of γ RSelΘ D, we use the following result 6] p γrselθd (γ = i Θ p γrid (γ l Θ l i (25 P γrld (γ, (26 P X (γ is the cumulative Distribution Function (CDF of X ] P γ (γ = π (k exp( γ (27 k= β (k et {l(θ,i,p} Θ p= be the set of relays indices which belong to Θ and different from i. The obtained expression of the pdf of γ RSelΘ D is given in (22 at the top of this page (see 3] for details, ɛ n, Θ =(ɛ n, Θ (,...ɛ n, Θ ( Θ is the binary representation of n 2 Θ, and ξ(n = Θ p= α nikm...m Θ = β (k + ɛ n, Θ (p, (28 Θ p= ɛ n, Θ (p. (29 β (m p R l(θ,i,pd To determine the expression of P ecoop,d in (25, we use the following result which can be obtained using integration by parts Q( x exp( a 2(x + u exp( u b dxdu = a b a Ψ(a a b +Ψ(b b b a, (3 Ψ(x = ] x (3 2 x + Using the pdf of γ RSelΘ D in (22 and the equations above, we obtain the expression of P ecoop,d given in (23. IV. NUMERICA AND SIMUATION RESUTS This section provides some numerical and simulations results of the considered cooperative MC-CDMA system. Subcarriers separation was set to Δf =5kHz corresponding to T s =25μs and η =5μs. The number of subcarriers was set to =6. Simulations results were performed for ITU Pedestrian B channels. The Multipath Intensity Profile (MIP of the ITU channels is as follows Φ (τ = P i= E( f i 2 δ(τ τ i, (32 P, f i and τ i are respectively, the number of paths, the complex gain and the delay of the i-th path of the X-Y link, δ(. is the Dirac function and E(. is the expectation operator. The average power of the delay of the i-th path depends of the distance d eff between X and Y as follows E( f i 2 = p iς d ϱ, (33 d = d eff /d is the normalized distance between X and Y, d is the arbitrary reference distance, ς is the path loss at the reference distance, < p i is the relative average power of the i-th path so that P i= p i = and ϱ is the path loss exponent. ς and ϱ were set to and 3, respectively. The reference distance d is chosen to be d SD. The cell radius was set to m and simulations were carried out for different random topologies. We have allocated the same power to source and relay i.e., E X = E b /2. Fig. 3 shows the BEP of the considered cooperative MC CDMA system using STDR for ITU pedestrian B channel and 7

BEP 2 Simu: Direct Transmission Simu: STDR, γ t =6 db Simu: SDTR, γ t =2 db Theo: Direct Transmission Theo: STDR, γ t =6 db Theo: SDTR, γ t =2 db 3 2 4 6 8 Eb/N (db Fig. 3. BEP of cooperative MC CDMA systems using STDR for ITU Pedestrian B channels, N=3. BEP 2 Simu: Direct Transmission Simu: STDR, N=3 Simu: STDR, N=6 Theo: Direct Transmission Theo: STDR, N=3 Theo: STDR, N=6 3 2 4 6 8 E b /N (db Fig. 4. Effect of users numbers on BEP of cooperative MC CDMA systems using STDR for ITU Pedestrian B channels, γ t =2dB. number of users equal 3. We used two different SNR threshold values. At low SNR, the BEP of STDR for γ t =6dB is close to that of direct transmission. This is because at low SNR, received SNRs at relays will rarely exceed the threshold value and hence no cooperation will be performed. We observe that there is a match between theoretical and simulations curves which validates our derived BEP expressions. Fig. 4 shows the BEP for different number of users in the cell. We observe that the BEP performance improves as the number of relays increases. This is because the destination will be more lucky to select a better suitable relay. Theoretical and simulations results are in perfect match. V. CONCUSION In this paper, we have considered a cooperative wireless network using MC CDMA N users communicate with a single destination D that can be a BS/AP. We have derived exact e 2e BEP at D of the considered cooperative MC- CDMA systems using Selective Threshold Digital Relaying (STDR with best relay selection in the presence of multipath propagation. The derived results are valid for any multipath intensity profile of the channel. REFERENCES ] A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity Part I: system description. IEEE Trans. Commun., vol. 5, no., pp. 927 938, Nov. 23. 2] S. Yang and J.-C. Beliore, Towards the optimal amplify-and-forward cooperative diversity scheme. IEEE Trans. Inf. Theory, vol. 53, no. 9, pp. 34 326, Sept. 27. 3] J. N. aneman, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans. Inf. Theory, vol. 5, no. 2, pp. 362 38, Dec. 24. 4] F. A. Onat, Y. Fan, H. Yanikomeroglu, and J. Thompson, Asymptotic BER analysis of threshold digital relaying in cooperative wireless systems. IEEE Trans. Wireless Commun., vol. 7, no. 2, pp. 4938 4947, Dec. 28. 5] Furuzan Atay Onat, Yijia Fan, Halim Yanikomeroglu, and H. Vincent Poor, A threshold based relay selection in cooperative wireless networks. in Proc. Global Communication Conference, 28, pp. 5. 6] P. A. Anghel and M. Kaveh, Exact symbol error probability of a cooperative network in a Rayleigh-fading environment, IEEE Trans. Wireless Commun., vol. 3, pp. 46 42, Sept. 24 7] A. Ribeiro, X. Cai, and G. Giannakis, Symbol error probabilities for general cooperative links, IEEE Trans. Wireless Commun., vol. 4, pp. 264 273, May 25. 8] Y. Zhao, R. Adve, and T. J. im, Symbol error rate of selection amplifyand-forward relay systems, IEEE Commun. ett., vol., pp. 757 759, Nov. 26. 9] E. Beres and R. Adve, Selection cooperation in multi-source cooperative networks. IEEE Trans. Wireless Commun., vol. 7, no., pp. 8 27, Jan. 28. ] D. Michalopoulos and G. Karagiannidis, Performance analysis of single relay selection in Rayleigh fading. IEEE Trans. Wireless Commun. vol. 7, no. pp. 378 3724, Oct. 28. ] S. Hara and R. Prasad, Overveiw of multicarrier CDMA. IEEE Commun. Mag., vol. 35, no. 2, pp. 26 33, Dec. 997. 2] S. Hara and R. Prasad, Design and performance of multicarrier CDMA system in frequency-selective Rayleigh fading channels. IEEE Trans. Veh. Technol., vol. 48, no. 5, pp. 584 595, Sep. 999. 3] H. Boujemaa, Exact and Asymptotic BEP of Cooperative DS-CDMA Systems using Decode and Forward Relaying in the Presence of Multipath Propagation. IEEE Trans. Wireless Commun., vol. 8, no. 9, pp. 4464 4469, Sep. 29. 4] P. Herhold, E. Zimmermann, and G. Fettweis, A simple cooperative extension to wireless relaying. in Proc. International Zurich Seminar on Communications, 24, pp. 36 39. 5] A. Adinoyi and H. Yanikomeroglu Cooperative relaying in multiantenna fixed relay networks. IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 533 544, Feb. 27. 6] N. Kong, T. Eng and. B. Milstein A selection combining scheme for rake receivers. in Proc. ICUPC, 995 pp. 426 429. 8