Capacity Performance Analysis for Decode-and-Forward OFDMDual-Hop System

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In this paper, we propose an exact analytical technique to evaluate the average capacity of a dual-hop OFDM relay system with decode-and-forward protocol in an independent and identical distribution (i.i.d.) Rayleigh fading channel. Four schemes, (no) matching “and” or “or” (no) power allocation, will be considered. First, the probability density function (pdf) for the end-to-end power channel gain for each scheme is described. Then, based on these pdf functions, we will give the expressions of the average capacity. Monte Carlo simulation results will be shown to confirm the analytical results for both the pdf functions and average capacities.

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Capacity Performance Analysis for Decode-and-Forward OFDMDual-Hop System

  1. 1. IEICE TRANS. COMMUN., VOL.E93–B, NO.9 SEPTEMBER 2010 2477 LETTER Capacity Performance Analysis for Decode-and-Forward OFDM Dual-Hop System Ha-Nguyen VU†a) , Le Thanh TAN† , Nonmembers, and Hyung Yun KONG†∗ , Member SUMMARY In this paper, we propose an exact analytical technique to evaluate the average capacity of a dual-hop OFDM relay system with decode-and-forward protocol in an independent and identical distribution (i.i.d.) Rayleigh fading channel. Four schemes, (no) matching “and” or “or” (no) power allocation, will be considered. First, the probability den- sity function (pdf) for the end-to-end power channel gain for each scheme is described. Then, based on these pdf functions, we will give the expres- sions of the average capacity. Monte Carlo simulation results will be shown to confirm the analytical results for both the pdf functions and average ca- pacities. key words: OFDM transmission, decode and forward, multi-hop, ergodic capacity, probability density function 1. Introduction Multi-hop relay networks have attracted much attention in the recent years among the wireless communications and ad hoc network researchers [1], [2]. Multi-hop links provide a convenient solution to the range problem encountered in wireless networks. Thus, relaying the signal through one or several relay nodes gains a power efficient means of achiev- ing the enviable distance of the communication link. Two main relaying strategies have been identified to be usable in such scenarios: amplify-and-forward (AF) and decode- and-forward (DF). AF means that the received signal is am- plified to achieve an expected power and then retransmit- ted by the relay without performing any decode process. In the DF strategy, the signal is decoded at the relay and re- encoded for retransmission. Orthogonal frequency division multiplexing (OFDM) is a mature technique to mitigate the problems of frequency of selectivity and inter-symbol inter- ference. Therefore, for the wide bandwidth multi-hop sys- tem, the combination of multi-hop system and OFDM mod- ulation is an even more promising way to increase capacity and coverage [3]–[6]. In [3], multi relays (R) are chosen to help the transmission between the source (S) and the desti- nation (D) where each relay is assigned for each subchannel. On the other hand, just only one R with OFDM modulation communication is occurred in [4]–[6]. However, as the fad- ing gains of different channels are mutually independent, the subcarriers which experience deep fading over S-R channel may not be in deep fading over R-D channel. Thus, the sub- Manuscript received January 15, 2010. Manuscript revised May 23, 2010. † The authors are with the School of Electrical Engineering, University of Ulsan, Korea. ∗ Corresponding author. a) E-mail: hanguyenvu@mail.ulsan.ac.kr DOI: 10.1587/transcom.E93.B.2477 carrier matching is considered to utilize the independence of different subcarriers and links so that improving the capac- ity of both AF [4] and DF [5], [6] relaying. In the optimal matching scheme, the kth worst channel from S to R will be matched with the kth worst one from R to D. Moreover, the exact capacity performance analysis of the AF scheme is given by Suraweera et al. in [4]. However, to our best knowl- edge, the analytical expressions for the DF scheme have not been investigated yet. Hence, this paper provides an exact technique to eval- uate the dual-hop OFDM relay system capacity with DF protocol in the i.i.d. Rayleigh fading channels. We con- sider four schemes to make a comparison for the wide band transmission such as: (1) normal scheme with no match- ing and no power allocation (NOR scheme), (2) power al- location scheme — considering the power allocation with- out matching subcarriers [5] (PA scheme), (3) the scheme with matching subcarriers but no power allocation (MA scheme), (4) optimal scheme — applying both matching problem and power allocation (MA-PA scheme). To ana- lyze these scheme, we first derive closed-form expressions for end-to-end channel power gain probability density func- tion (pdf) for all schemes with different subcarrier pairs. Next, based on these pdf the average capacities of the sub- carrier pairs are obtained using numerical integration with the constraint equaling power transmission for each subcar- rier pair. Then, the total capacity of the OFDM relay system is obtained by aggregating all the capacities of the individ- ual subcarrier pairs. Finally, the simulation results are also shown to confirm the accuracy of the proposed analytical procedure; the demonstrate the potential gains of subcarrier matching schemes. 2. System Model and the System Capacity 2.1 System Model We consider an OFDM dual-hop system including S, D and R. The relay strategy is DF and every node only uses one receive antenna and one transmit antenna. Assume that D can receive signal from R but not from S. The transmission protocol takes place in two equal time slots. In the first time slot, S transmits an OFDM symbol with N subcarriers to R. Then, R receives, decodes and arranges the subcarriers to forward the data to D in the second time slot also over N subcarriers. The destination decodes the signal based on the received signal from R. When the data on the ith subcarrier Copyright c 2010 The Institute of Electronics, Information and Communication Engineers
  2. 2. 2478 IEICE TRANS. COMMUN., VOL.E93–B, NO.9 SEPTEMBER 2010 of the S-R channel is forwarded to destination through jth subcarrier of the R-D channel, we call that subcarrier i is matched with subcarrier j (j = α(i)). We assume that the power consumption for each matched subcarrier pair is fixed as P = PT /N where PT is the total power transmission. In this paper, we will consider the capacity performances of four OFDM transmission strategies: 1. NOR scheme — no subcarrier matching and no power allocation — the bits transmitted on subcarrier i at S will be retransmitted on subcarrier i at R; the power consumption is equally allocated at S and R (equal P/2). 2. PA scheme — power allocation but no subcarrier matching — the bits transmitted on subcarrier i at S will be retransmitted on subcarrier i at R; the power at S and R is allocated to achieve the maximum capacity for each subcarrier pair [6]. 3. MA scheme — subcarrier matching but no power allo- cation — the bits transmitted on the ith worst subcarrier at S will be retransmitted on the ith worst one at R; the power consumption is equally allocated at S and R. 4. MA-PA scheme — subcarrier matching and power allo- cation — the bits transmitted on the ith worst subcarrier at S will be retransmitted on the ith worst one at R; the power at S and R is allocated to maximize capacity for each subcarrier pair. 2.2 The Capacity Formulation The capacity of two-hop DF relaying scheme in the ith matched subcarrier pair can be given as Ci =(1/2) min log2 1+pS,iγS,i , log2 1+pR,α(i)γR,α(i) (1) where pS,i, pR,α(i) repectively denote the transmitted power for the subcarrier pair ith at S and R, with the constraint pS,i + pR,α(i) = P. γS,i and γR,α(i) are the power gain over noise of the channel from S to R and R to D over the sub- carrier pair ith. In this paper, we assume that the channels experience the i.i.d. Rayleigh fading with the unit variance. The power spectral densities of noise are equal at S and R with zero mean and unit variance . Hence, γS,i and γR,α(i) are the exponential distributed random variable (r.v.) with the unit variance. • NOR scheme This is scheme, we have α(i) = i and pS,i = pR,α(i) = P/2. Thus, the capacity of the ith pair is given as CNOR i = (1/2) log2 1 + P min γS,i, γR,i 2 (2) • PA scheme Similar to the previous scheme, α(i) = i; however, the power allocation is occurred. From (1), the maximal capacity is obtained when [6] (Eq. (5)) pS,iγS,i = pR,iγR,i = PγS,iγR,i γS,i + γR,i (3) Hence, the capacity in this case can be given as CPA i = (1/2) log2 1 + PγS,iγR,i γS,i + γR,i (4) • MA scheme It has been shown in [5], [6] that the optimal subcarrier matching for a maximum end-to-end overall capacity should be that the kth worth subcarrier in Hop1 (S-R) is matched to the kth worth subcarrier in Hop2 (R-D). Denote γ(k) S and γ(k) R as the kth smallest values of the sets γS,i i = 1, ..., N and γR,i i = 1, ..., N , respectively. Hence, the optimal match- ing algorithm is γ(k) S ∼ γ(k) R . However, the power allocation is not achieved in this scheme. Therefore, the capacity of this matched subcarrier pair is CMA (k) = (1/2) log2 1 + P min γ(k) S , γ(k) R 2 (5) • MA-PA scheme In this scheme, we combine both results of the two previous schemes where the optimal matching and power allocation are considered together. Thus, the optimal capacity for the matched subcarrier pair γ(k) S ∼ γ(k) R can be described as CMA−AP (k) = (1/2) log2 1 + Pγ(k) S γ(k) R γ(k) S + γ(k) R (6) 3. Exact Valuation of System Capacity 3.1 Order Statistics of Rayleigh Distribution Let X1, X2, ..., XN be i.i.d. exponential distributed r.v.s with the unit variance; hence, the pdf f(x) and the cumulative distribution function (CDF) F(x) of these r.v.s can be given as f (x) = e−x ; F (x) = 1 − e−x (7) We define the r.v. X(k) as the kth smallest value from the observed r.v. set X1, X2, ..., XN. Based on [7], the pdf and CDF of X(k) can be described as f(k) (x) = NCk−1 N−1 1 − e−x k−1 e−(N−k+1)x = k−1 i=0 uie−vi x F(k) (x) = x 0 f(k) (z) dz = k−1 i=0 ui 1 − e−vi x vi (8) where Cn m is the choose function of n and m, ui = NCk−1 N−1Ci k−1 (−1)i and vi = i + N − k + 1. 3.2 Performance Analysis of Relay System Capacity • NOR scheme According to the exponential distribution, the pdfs of two i.i.d. r.v.s γS,i, γR,i can be given as (7). Hence, the r.v. z (z = min(γS,i, γR,i)) becomes [7] fz (z) = 2e−z 1 − 1 − e−z = 2e−2z (9)
  3. 3. LETTER 2479 Using this result, the capacity of the ith subcarrier pair can be calculated as CNOR i = (1/2) ∞ 0 log2 (1 + Pz/2)fz (z) dz = −e−4/P Ei (4/P) (2 ln 2) (10) where Ei (.) is the exponential integral function [8] (Eq. 8.211.1). • PA scheme Let a = 1 γS,i, b = 1 γR,i and c = 1/(a + b) . According to [7], the pdfs of a, b can be defined as fa (x) = fb (x) = 1 x2 e−1/x (11) With the help of [8] (Eq. 3.471.9), the moment gener- ating function (MGF) of a and b is obtained as Φa (s) = Φb (s) = 2 √ sK1 2 √ s where Kυ (z) is the Kυ (z) order mod- ified Bessel function of the second kind [8] (Eq. 8.432.6). Then, the MGF of (a + b) can be given by multiplying both MGFs of a and b. Φ(a+b) (s) = Φa (s) Φb (s) = 4s K1 2 √ s 2 (12) From (12), based on Eq. 13.2.20 of [9], we obtain the closed-form expression of CDF and pdf of r.v. c as follows: Fc(c) = 1 − F(a+b) (1/c) =1− L−1 Φ(a+b)(s) s 1/c =1−2ce−2c K1(2c) fc (c) = ∂Fc (c)/∂c = 4ce−2c [K0 (2c) + K1 (2c)] (13) where L−1 {.} denotes the inverse Laplace transform. Then, the capacity of the ith subcarrier pair can be obtained as CPA i = (1/2) ∞ 0 log2 (1 + Pc) fc (c) dc (14) • MA scheme Let q(k) = min γ(k) S , γ(k) R . Now, we define the pdf of q(k) by applying the results of Sect. 3.1. According to the determi- nation of γ(k) S and γ(k) R , the CDF and pdf of them can be given as in (8). Hence, the pdf of the r.v. q(k) becomes fq(k) (q) = fγ(k) S (q) 1−Fγ(k) R (q) + fγ(k) R (q) 1−Fγ(k) S (q) = 2 k−1 i=0 uie−vi x × ⎡ ⎢⎢⎢⎢⎢⎢⎣1 − k−1 j=0 uj vj 1 − e−vj x ⎤ ⎥⎥⎥⎥⎥⎥⎦ = 2 k−1 i=0 uie−vi x − 2 k−1 i=0 k−1 j=0 uiuj vj e−(vi+vj)x (15) Similar to (10), the average capacity of the matched subcar- rier pair γ(k) S ∼ γ(k) R is given as CMA (k) = (1/2) ∞ 0 log2 (1 + Pq/2)fq(k) (q) dq (16) • MA-PA scheme Let a(k) = 1 γ(k) S , b(k) = 1 γ(k) R and c(k) = 1 a(k) + b(k) . By analyzing similarly to PA scheme, the pdf and MGF of a(k) and b(k) are given as fa(k) (x) = fb(k) (x) = 1 x2 f(k) 1 x = k−1 i=0 ui e−vi/x x2 Φa(k) (s) = Φb(k) (s) = k−1 i=0 ui2 √ visK1 2 √ vis (17) Hence, the MGF of a(k) + b(k) is achieved as Φa(k)+b(k) (s)= k−1 i,j=0 uiuj2 √ vivjsK1 2 √ vis K1 2 √ vjs (18) Due to applying the inverse Laplace transform, and the note that c(k) = 1 a(k) + b(k) , the pdf of c(k) can be obtained by the same way with (13) as fc(k) (c) = k−1 i,j=0 2uiujc √ vivj e−(vi+vj)c × 2 √ vivjK0 2 √ vivj + vi + vj K1 2 √ vivj (19) Finally, the capacity for the optimal matched subcarrier pair γ(k) S ∼ γ(k) R with the power allocation between S and R is given by CMA−PA (k) =(1/2) ∞ 0 log2 1+Pc(k) fc(k) c(k) dc(k) (20) Then, the total capacity for each scheme can be achieved by taking the sum of all the subcarrier pair’s ca- pacities. Hence, the total capacity of the network is CX total = N i=1 CX i (21) where X presents for NOR, PA, MA, MA-PA. 4. Numerical Results Figure 1 shows the numerical results of the probability den- sity functions of the kth smallest matched subcarrier pair in a dual-hop OFDM relay system with and without power allocation fc(k) (x) and fq(k) (x), respectively. An OFDM sys- tem with N = 16 subcarriers is considered and the figure shows the value of the density functions for k = 6, 10, 12. It can be observed that both fc(k) (x) and fq(k) (x) for large k have the larger mean and the variance when comparing to that of them for the smaller k. To validate the accuracy of the theoretical probability density function of the subcarrier matching systems, the derived simulation results were also obtained. As can be seen, the theoretical and simulation re- sults for the density functions match perfectly. Figure 2 shows the capacity versus the i.i.d Rayleigh
  4. 4. 2480 IEICE TRANS. COMMUN., VOL.E93–B, NO.9 SEPTEMBER 2010 Fig. 1 Plots of fc(k) (x) and fq(k) (x) for k = 6, 10, 12 versus the value of random variable. Fig. 2 Channel capacity against the total power PT (N = 16). Fig. 3 Per channel capacity distribution for different total power transmission value (SNR = 5 and 10 dB). fading channel with 16 subcarriers (N = 16). The figure il- lustrates that if there is no subcarrier matching, power allo- cation increases the system capacity, which can be obtained by comparing the capacity of NOR scheme to that of PA scheme. Similarly, the subcarrier matching can improve the capacity by comparing the capacity of NOR scheme with that of MA scheme. However, by comparing PA and MA schemes, the subcarrier matching method is preferred. It means that MA scheme solves the “bottleneck problem” be- tween two hops; hence the higher capacity can be attained without the power allocation. Finally, when both these methods are achieved we can get the highest capacity; it can be observed as the capacities of the MA-PA scheme are higher than those of the other schemes. In Fig. 3, we provide comparisons of capacity of each subcarrier pair (CMA (k) and CMA−PA (k) ) which is evaluated using the accurate analytical or- der statistics based method and simulations, for a dual-hop OFDM relay systems with subcarrier matching. In simula- tions we selected N = 16 and SNR = 5, 10 dB. The capacity per channel without subcarrier matching is also shown as a reference. As can be seen, the analytical results based on order statistics exactly match with those of simulations. 5. Conclusion In this letter, we proposed an exact analytical technique of evaluating the average capacity of a dual-hop decode- and-forward OFDM relay system with subcarrier mapping and power allocation for each subcarrier pair and the other non-optimal schemes. Closed form density function expres- sions were derived for the end-to-end channel power gains of all schemes. Monte Carlo simulation results and compar- isons with the analytical results were presented to confirm the accuracy of the proposed analytical technique, and also demonstrate the outperformance of the MA-PA scheme. Acknowledgement This research was supported by Basic Science Research Pro- gram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0073895). References [1] M.O. Hasna and M.S. Alouini, “A performance study of dual-hop transmissions with fixed gain relays,” IEEE Trans. Wireless Com- mun., vol.3, no.6, pp.1963–1968, 2004. [2] B.Q. Vo-Nguyen and H.Y. Kong, “A simple performance approxima- tion for multi-hop decode-and-forward relaying over Rayleigh fading channels,” IEICE Trans. Commun., vol.E92-B, no.11, pp.3524–3527, Nov. 2009. [3] H.V. Khuong and H.Y. Kong, “Energy saving in OFDM systems through cooperative relaying,” ETRI J., vol.29, no.1, pp.27–35, Feb. 2007. [4] H.A. Suraweera and J. Armstrong, “Performance of OFDM-based dual-hop amplify-and-forward relaying,” IEEE Commun. Lett., vol.11, no.11, pp.726–728, 2007. [5] W. Wenyi and W. Renbiao, “Capacity maximization for OFDM two- hop relay system with separate power constraints,” IEEE Trans. Veh. Technol., vol.58, no.9, pp.4943–4954, 2009. [6] Y. Wang, X.-C. Qu, T. Wu, and B.-L. Liu, “Power allocation and sub- carrier pairing algorithm for regenerative OFDM relay system,” IEEE 65th, Vehicular Technology Conference, VTC2007-Spring, pp.2727– 2731, 2007. [7] A. Papoulis and S.U. Pillai, Probability, random variables, and stochastic processes, 4th ed., McGraw-Hill, Boston, 2002. [8] I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of integrals, series and products, 7th ed., Amsterdam, Boston, Elsevier, 2007. [9] H.K.G.E. Roberts, Table of Laplace Transforms, W.B. Saunders, 1966.

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