Analysis of Outage and Throughput for Opportunistic Cooperative HARQ Systems over Time Correlated Fading Channels

Transcription

1 Analysis of Outae and Throuhput for Opportunistic Cooperative HARQ Systems over Time Correlated Fadin Channels Xuanxuan Yan, Haichuan Din,3, Zhen Shi, Shaodan Ma, and Su Pan Department of Electrical and Computer Enineerin, University of Macau, Macau Nanjin University of Posts and Telecommunications, China 3 School of Information and Electronics, Beijin Institute of Technoloy, China Abstract In this paper, an opportunistic cooperative HARQ system is analyzed. Different from prior analyses, time correlated fadin channels are considered. Based on moment eneration function and Laplace transform, the outae probability and throuhput in terms of lon-term averae transmission rate (LATR of this opportunistic cooperative HARQ system are derived in closed-forms. The accuracy of the analytical results is verified by computer simulations. From the analytical results, the impacts of the time correlation and other system parameters on the performance are investiated and the optimal packet rate selection to maximize the LATR is discussed. I. INTRODUCTION As an effective error control technique to provide reliable communications, hybrid automatic repeat request (HARQ has been widely adopted in various communication systems []. There are a number of HARQ schemes proposed in the literature. One representative HARQ scheme is HARQ with chase combinin (HARQ-CC. In this scheme, the same packets are retransmitted in every HARQ round and the receiver combines the retransmitted packets with the previously failed packets throuh maximum ratio combinin (MRC for better detection. It has superior performance and relative low complexity and has attracted considerable research interest in recent years [] [7]. A lot of studies have been conducted on HARQ-CC operatin over independent and identically distributed (i.i.d. fadin channels, but none of them obtain the eneral results for the correlated fadin channels. Some but not all of them are mentioned here. In [], two suboptimal rate selection schemes are proposed for HARQ-CC in i.i.d. Rayleih fadin channels to reduce the computational load. Optimal power allocation for HARQ-CC is investiated in [3]. By usin an approximate outae probability and formulatin the problem as eometric prorammin problem, a low complexity power allocation scheme is proposed. In [4], the HARQ-CC is applied in opportunistic cooperative systems and the throuhput in terms of lon-term averae transmission rate (LATR is derived, based on which the optimal rate selection is enabled to maximize the LATR. As pointed out in [7], time correlated fadin channel is more realistic in low to medium mobility scenarios. The time correlation in the channels is expected to decrease the diversity and lead to certain performance deradation. How the time correlation derades the performance and the exact performance of HARQ-CC operatin over time correlated channels are of reat importance for system desin. In [7], the delay limited throuhput of HARQ-CC in multiple input sinle output (MISO systems over time correlated channels is analyzed under specific condition. Due to the variety in the system topoloies and the difficulty to find the distribution of the sum of correlated and non-identically fadin channels, there is no eneral analytical approach for the analysis of HARQ-CC over time correlated channels and the analytical result in [7] is not applicable to other systems. In this paper, we analyze the performance of HARQ-CC in opportunistic cooperative systems over time correlated Rayleih fadin channels. Due to the involvement of a cooperative channel and the presence of the time correlation, the analysis is by no means straihtforward. Based on moment eneratin function and Laplace transform, the outae and throuhput in terms of LATR are derived in closed-forms. The impacts of the time correlation and other system parameters are investiated. In addition, the optimal packet rate to maximize the LATR is discussed. The accuracy of the analytical results is also verified by simulations. The rest of this paper is oranized as follows. In Section II, the system model is introduced. In Section III, the outae probability and the LATR are derived and the impacts of various parameters on system performance are studied. Finally, conclusions are drawn in Section IV. II. SYSTEM MODEL A cooperative network with a source, a relay and a destination is considered here as shown in Fi.. Similarly to [4] [6], HARQ-CC is adopted and the communication protocol of this cooperative HARQ network is as follows. A. Communication Protocol The relay opportunistically helps the source node transmit messae to the destination throuh decode-and-forward(df

2 protocol. Specifically, the communication process is partitioned into two phases. In the first phase, the source broadcasts a packet at a selected modulation and codin rate (MCS R to the relay and the destination with transmit power P S,. The relay and the destination receive and decode the packet. If the destination receives the packet correctly, it feeds back an acknowledement (ACK messae. Otherwise, a nonacknowledement (NACK messae is fed back to the source node and the failed packet is stored. Upon receivin an ACK messae, the source node beins to send a new packet. While a NACK messae is received, the source node retransmits the packet and the destination will combine the received packet with previous failed packets throuh MRC. The relay node adopts a similar process as the destination, except that it sends the feedback messae to the source node only when it can successfully decode the packet. If the relay cannot successfully decode the packet, the second phase transmission will never occur and the transmission process just follows the traditional HARQ-CC protocol between the source and the destination. When the relay node decodes the packet successfully and the destination cannot, the second phase transmission beins and the relay cooperates with the source to transmit the packet to the destination throuh maximum-ratio-transmission (MRT in the subsequent HARQ retransmissions. The transmission powers of the source and the relay nodes in the second phase are P S, and P R respectively. This process will continue until the destination decodes the packet successfully or the maximum number of retransmissions has been reached. B. Sinal Model In the first phase, the sinals received at the relay and the destination in the lth HARQ round are y l, R = hl x + z l R, y l, D = hl x + z l D, ( where x is the transmitted sinal, zr l and zl D are zero mean complex circular symmetry Gaussian noises at the relay and the destination respectively, i.e., zr l C(0, N R and zd l C(0, N D, and h l and hl are the channel coefficients between the source and the relay/destination respectively. Different from previous works on cooperative HARQ systems [4] [6], we consider time-correlated fadin channels. Followin the Gauss-Markov time correlation model [7], [8], the channel coefficient h l can be formulated as h l = ρ l h 0 + ρ l w l, ( where h 0 denotes the channel coefficient between the source and the destination at the first HARQ round which follows a complex Gaussian distribution with zero mean and variance σ, i.e., h0 CN (0, σ, ρ denotes the time correlation factor, and w l denotes the independent fadin term which is independent with h 0 and follows an identical and independent complex Gaussian distribution with zero mean The first HARQ round is the initial transmission and the lth (l > HARQ round is the l th retransmission. Fi.. A cooperative relay network and variance σ, that is, wl CN (0, σ. The channel coefficient between the source and the relay h l can be formulated similarly. From (, when operatin in the first phase, the received SNRs at the relay and the destination in the lth HARQ round can be written as γ l, = h l (P S, /N R, γ l, = h l (P S, /N D. (3 In the second phase, the source and the relay cooperatively transmit the packet to the destination throuh MRT. Thus the received SNR at the destination in the lth HARQ round is γ l, = h l RD (P R /N R + h l (P S, /N D, (4 where h l RD is the time correlated channel between the relay and the destination and is modeled similarly to (. III. THROUGHPUT ANALYSIS One widely adopted metric to characterize the throuhput performance of HARQ-CC systems is lon-term averae transmission rate (LATR. It is enerally defined as [0] T = R N, (5 where R is the packet rate and N is the averae number of transmissions. In the followin, the LATR of the considered opportunistic cooperative HARQ system will be analyzed. Definin (k as the outae probability after the kth HARQ round, i.e. the probability that the packet still can not be successfully decoded after the kth HARQ round, we have N = + Nmax k= (k []. Clearly, to analyze the LATR T, the outae probability (k is needed and will be derived as follows. A. Outae Probability In the cooperative HARQ system, the communication is conducted in two phases and thus the analysis is more challenin. Noticin that the second phase transmission will occur only when the relay node decodes the messae successfully,

3 the outae probability of this cooperative HARQ system can be formulated as (k = k r= + ( k = P suc (rpout (k r (k Pout(k k < k N max, (6 where P suc (r is the probability that the relay successfully decodes the packet for the first time in the rth HARQ round, (r is the probability that the relay still can not decode the packet after the rth HARQ round, (k r is the probability that the destination cannot decode the messae after the kth HARQ round iven that the relay successfully decodes the packet in the rth HARQ round, and (k k is the probability that the destination cannot decode the packet after the kth HARQ round iven that the relay fails to decode the packet after the k th HARQ round. By definition, the followin equations hold: P suc (r = (k k = (k r k, (7 Clearly from (6-8, proceed, ( r = (r Pout (r < r N max (k r and Pout (k r will be derived first. Accordin to its definition,. (8 (r are necessary. To (k r can be formulated as (k r = P ( ( ( lo + γ R,, lo + γ k R, (9 where γ i is the effective SNR after packet combinin at the destination. Since the HARQ-CC technique is adopted, the effective SNR is iven as i γ l, γ i i r = r γ l, + i. (0 γ l, i > r l=r+ The outae probability (9 can then be rewritten as ( r k (k r = P γ l, + γ l, R. ( From (3 and (4, ( follows as (k r = P r + k l=r+ + k l=r+ l=r+ h l (P S, /N D h l (P S, /N D h l RD (P R /N D < R. ( With the Gauss-Markov time correlation model in ( and iven h 0 and ρ, the squared channel coefficient between the source and the destination h l follows a non-central chi-squared distribution with two derees of freedom. The probability density function (p.d.f. of h l iven h 0 and ρ is f (x = (λ,l / i e λ,l/ h l h f 0 i (x, x > 0,,ρ i! i=0 (3 where f i (x is the p.d.f. of a chi-squared distributed random variable with freedom of + i, i.e., χ ( + i, λ,l = ρ l h 0 / σ,l and σ,l = σ ( ρl. The conditional p.d.f. of the squared channel coefficient between the relay and the destination h l RD can be formulated in a similar way as (3 with λ RD,l = ρ l h 0 / RD σrd,l and σrd,l = σ RD( ρ l. Clearly, iven h 0, h 0 RD and ρ, the squared coefficients h l and h l RD are independent noncentral chi-squared random variables. Given h 0, h 0 RD and ρ, the left-hand side of the inequality in ( is a sum of independent non-identical distributed chi-squared random variables whose distribution is hard to find. To tackle this problem, we turn to the moment eneratin function (m..f. method. By settin a = P S, /N D, b = P S, /N D, c = P R /N D, the m..f. of a h l iven h 0 and ρ can be derived as M a h l h 0,ρ (s = ( aσ,ls exp asλ,l σ,l aσ,l s, ( aσ,l > s (4 Given h 0, h 0 RD, and ρ, the m..f. of Z = r hl a + k l=r+ hl b + k l=r+ hl RD c can then be derived as M Z h 0, h 0 RD,ρ (s = Πr M a h l h 0,ρ (s Π k l=r+m b h l h 0,ρ (s Πk l=r+m c h l RD h 0 RD,ρ (s = Π r ( aσ,ls Π k l=r+( bσ,ls r [ λ,l Π k l=r+( cσrd,ls σ,l exp as ] aσ,l s [ k λ,l σ,l + bs ] [ k λrd,l σrd,l bσ,l s + cs ] cσrd,l s. l=r+ l=r+ (5 Settin = R, with (5 and the numerical inversion of Laplace transform [], the cumulative distribution function of Z conditioned on h 0, h 0 RD and ρ can be derived as F Z h 0, h 0 RD,ρ( = L [ s M Z h 0, h ( 0 RD,ρ( s] = Q e A/ Q M+q q=0 q m=0 ( m β m Re s M Z h 0, h 0 RD,ρ( s + E(A, M, Q, (6

4 where β m = if m = 0, if m =,,..., M, s = (A + πjm/, and j =, Re represents the real part, and E(A, M, Q denotes the error term which includes the discretization error and the truncation error. Specifically, A is the parameter due to the adoption of the trapezoidal rule and it is shown in [] that the discretization error can be e bounded as A. Thus we can control the discretization e A error to be smaller than 0 0 by settin A = 0ln0 = On the other hand, M and Q are the parameters due to the truncation of the infinite sum and the adoption of Euler summation. By properly choosin M and Q, e.. M =, Q = 5 as suested in [], we can make the error term E(A, M, Q neliible compared to the outae probability. Let u = h 0 and v = h 0 RD. Their p.d.f.s follow f(u = exp( u and f(v = exp( v for u > 0 σ σ σrd σrd and v > 0. By interatin out the random variables u and v, the outae probability (k r is obtained as: (k r = = Q e A/ 0 0 σ exp F (f(uf(vdudv Z h 0 0 0, h 0 RD,ρ ( M+q Q ( m q β q=0 m=0 m Re s M Z ( s u,v,ρ ( ( u v dudv σ σrd exp = Q e A/ ( M+q Q q q=0 m=0 Re sw w w 3 σ σ RD σ RD ( m β m ( t + t + σ ( t 3 + σ RD (7 where w = Π r (+aσ,l s, w = Π k l=r+ (+bσ,l s, w 3 = Π k l=r+ ( + cσ RD,l s, t = r [ ρ l a s +aσ,l s], t = k l=r+ [ ρ l b s +bσ,l s], and t 3 = k l=r+ [ ρ l c s +cσ RD,l s]. On the other hand, (r can be written as (r = P ( lo ( + γ R,, lo ( + γ r R, (8 where γ i = i γ l, due to the adoption of HARQ-CC. Thus (r can be reformulated as ( r (r = P γ l, R. (9 Fi.. Outae probability v.s. packet rate. Noticin the similarity between ( and (9, (r can be derived followin a similar approach as (r = Q e A/ Re q=0 ( M+q Q ( m q β m=0 m sw 4 σ (t 4 + σ, (0 r where w 4 = ( + aσ,l s and t 4 = r [ ρ l a s +aσ Substitutin (7 and (0 into (6,,l s]. the outae probability of the cooperative HARQ system can be finally obtained in closed-form. The outae probability is investiated and shown in Fi., by takin a cooperative HARQ system with the followin settin as an example: the maximum number of HARQ rounds N max = 4, the variances σ = σ = σ RD = and N D = N R =, and the transmission powers as P S, = 0, P S, = 0, P R = 5. It is clear from Fi. that our analytical results match with the simulation results well, which, validates the correctness of our analysis. Meanwhile, the outae probability unsurprisinly increases with the packet rate increasin. Furthermore, the outae probability increases with the time correlation factor ρ, since lare ρ means that the channel conditions are more similar durin different HARQ rounds and thus results in less time diversity. Next the considered opportunistic relayin HARQ scheme is compared with the traditional noncooperative scheme by investiatin the cooperative ain which is defined as the ratio of the outae probability of the traditional HARQ scheme to that of the opportunistic relayin HARQ scheme. For fairness, the transmission powers are set as P S, = P S, + P R, i.e., P S, = 5, P S, = 0, P R = 5, while the other parameters are the same as those in Fi.. The cooperative ain is shown in Fi. 3. It is shown that the cooperative ain keeps nearly constant in low time correlation reion however it drops sinificantly in the hih time correlation reion. Intuitively, the relayin channel can contribute not only spatial diversity but also time diversity when ρ is small. However, when ρ is hih, the time diversity in the relayin channel sinificantly

5 has verified the accuracy of our analysis and theoretically Fi. 3. Cooperative ain v.s. time correlation factor ρ. Fi. 4. LATR v.s. packet rate. decreases, which results in the decrease of the cooperative ain. B. Throuhput and Optimal Packet Rate With the results in the above section, the LATR can be analytically derived by substitutin (6, (7 and (0 into (5. Clearly the LATR depends on the packet rate. The analytical result of the LATR would enable the desin of the packet rate. Specifically, the packet rate can be optimally chosen by solvin the followin optimization problem which maximizes the LATR subject to a certain outae constraint as R = ar max R R N, s.t. (N max ϵ, ( where ϵ is the maximum tolerable outae in the system. Alternatively, the optimal rate can be directly found from the LATR plot shown in Fi. 4, takin the same system as Fi. with outae constraint ε = 0.0 as an example. It is clear from Fi. 4 that when the time correlation factors are ρ =, 0.9, 0, the optimal rate R should be chosen as.04,.79, 3. respectively. Notice that with the result of outae probability, the delay limit throuhput of this opportunistic cooperative HARQ system can easily be derived with DLT = N max R k [Pout k= it is not discussed here. (k Pout (k]. Due to the limit of space, IV. CONCLUSIONS In this paper, the outae and throuhput (i.e., LART of an opportunistic cooperative HARQ system were analyzed based on moment eneratin function and Laplace transform. Unlike prior analyses, time correlated fadin channels were considered and the distribution of the sum of correlated and non-identically fadin channels was obtained. With the derived analytical results, the impact of the channel time correlation factor on the outae and cooperative ain was investiated. Furthermore, the optimal rate to maximize the LATR under certain outae constraint was obtained numerically. The results demonstrated the advantaes of cooperation and HARQ-CC techniques. ACKNOWLEDGEMENT This work was supported in part by the National Natural Science Foundation of China under contract No 6735 and in part by the Macau Science and Technoloy Development Foundation under Grant 067/03/A. REFERENCES [] E. Dahlman, S. Parkvall, J. Sköld, and P. Bemin, 3G Evolution: HSPA and LTE for Mobile Broadband, nd edition. Elsevier, 00. [] S. H. Kim, S. J. Lee and D. K. Sun, Low-complexity rate selection of HARQ with Chase combinin in Rayleih block-fadin channels, IEEE Trans. Veh. Technol., vol. 6, no. 6, pp , Jul. 03. [3] T. V. K. Chaitanya and E. G. Larsson, Optimal power allocation for hybrid ARQ with Chase combinin in i.i.d. Rayleih fadin channels, IEEE Trans. Commun., vol. 6, no. 5, pp , May 03. [4] S. H. Kim, S. J. Lee, and D. K. Sun, Rate-adaptation-based cooperative hybrid ARQ relayin scheme in Rayleih block-fadin channels, IEEE Trans. Veh. Technol., vol. 60, no. 9, pp , Nov. 0. [5] S. H. Kim, S. J. Leey, D. K. Sun, H. Nishiyamaz and N. Kato, Optimal rate selection scheme in a two-hop relay network adoptin Chase combinin HARQ in Rayleih block-fadin channels, in Proc. 0 IEEE Wireless Communications and Networkin Conference (WCNC 0. [6] M. Maaz, P. Mary and M. Hélard, Enery efficiency analysis in relay assisted hybrid-arq communications, in Proc. 0 IEEE 3rd International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 0. [7] S. H. Rastear, S. Vakilinia, and B. H. Khalaj, Rate adaptation and power allocation for time-correlated MISO Rayleih fadin channel with delay-limited HARQ, in Proc. IEEE International Conference on Communications (ICC 0. [8] S. M. Kim, W. Choi, T. W. Ban and D. K. Sun, Optimal rate adaptation for hybrid ARQ in time-correlated Rayleih fadin channels, IEEE Trans. Wireless Commun., vol. 0, no. 3, pp , Mar. 0. [9] Y. C. Ko, M. S. Alouini and M. K. Simon, Outae probability of diversity systems over eneralized fadin channels, IEEE Trans. Commun., vol. 48, no., pp , Nov [0] P. Wu, N. Jindal, Performance of hybrid-arq in block-fadin channels: a fixed outae probability analysis, IEEE Trans. Commun., vol. 58, no. 4, pp. 9 4, Apr. 00. [] H. Boujemâa, Delay analysis of cooperative truncated HARQ with opportunistic relayin, IEEE Trans. Veh. Technol., vol. 58, no. 9, pp , Nov [] J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distribution, ORSA J. Computin, vol. 7, no., pp , 995.