IEICE TRANS. COMMUN., VOL.E93–B, NO.9 SEPTEMBER 20102477

LETTER

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

Ha-Nguyen VU†a), Le Thanh TAN†, Nonmembers, and Hyung Yun KONG†∗, Member

SUMMARY In this paper, we propose an exact analytical techniqueto evaluate the average capacity of a dual-hop OFDM relay system withdecode-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 schemeis described. Then, based on these pdf functions, we will give the expres-sions of the average capacity. Monte Carlo simulation results will be shownto confirm the analytical results for both the pdf functions and average ca-pacities.key words: OFDM transmission, decode and forward, multi-hop, ergodiccapacity, probability density function

1. Introduction

Multi-hop relay networks have attracted much attention inthe recent years among the wireless communications and adhoc network researchers [1], [2]. Multi-hop links providea convenient solution to the range problem encountered inwireless networks. Thus, relaying the signal through one orseveral relay nodes gains a power efficient means of achiev-ing the enviable distance of the communication link. Twomain relaying strategies have been identified to be usablein 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. Inthe DF strategy, the signal is decoded at the relay and re-encoded for retransmission. Orthogonal frequency divisionmultiplexing (OFDM) is a mature technique to mitigate theproblems 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 capacityand coverage [3]–[6]. In [3], multi relays (R) are chosen tohelp 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 modulationcommunication is occurred in [4]–[6]. However, as the fad-ing gains of different channels are mutually independent, thesubcarriers which experience deep fading over S-R channelmay 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.krDOI: 10.1587/transcom.E93.B.2477

carrier matching is considered to utilize the independence ofdifferent subcarriers and links so that improving the capac-ity of both AF [4] and DF [5], [6] relaying. In the optimalmatching scheme, the kth worst channel from S to R willbe matched with the kth worst one from R to D. Moreover,the exact capacity performance analysis of the AF scheme isgiven by Suraweera et al. in [4]. However, to our best knowl-edge, the analytical expressions for the DF scheme have notbeen investigated yet.

Hence, this paper provides an exact technique to eval-uate the dual-hop OFDM relay system capacity with DFprotocol in the i.i.d. Rayleigh fading channels. We con-sider four schemes to make a comparison for the wide bandtransmission 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 schemewith matching subcarriers but no power allocation (MAscheme), (4) optimal scheme — applying both matchingproblem and power allocation (MA-PA scheme). To ana-lyze these scheme, we first derive closed-form expressionsfor 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 withthe constraint equaling power transmission for each subcar-rier pair. Then, the total capacity of the OFDM relay systemis obtained by aggregating all the capacities of the individ-ual subcarrier pairs. Finally, the simulation results are alsoshown to confirm the accuracy of the proposed analyticalprocedure; the demonstrate the potential gains of subcarriermatching schemes.

2. System Model and the System Capacity

2.1 System Model

We consider an OFDM dual-hop system including S, D andR. The relay strategy is DF and every node only uses onereceive antenna and one transmit antenna. Assume that Dcan receive signal from R but not from S. The transmissionprotocol takes place in two equal time slots. In the first timeslot, S transmits an OFDM symbol with N subcarriers to R.Then, R receives, decodes and arranges the subcarriers toforward the data to D in the second time slot also over Nsubcarriers. The destination decodes the signal based on thereceived signal from R. When the data on the ith subcarrier

Copyright c© 2010 The Institute of Electronics, Information and Communication Engineers

2478IEICE TRANS. COMMUN., VOL.E93–B, NO.9 SEPTEMBER 2010

of the S-R channel is forwarded to destination through jthsubcarrier of the R-D channel, we call that subcarrier i ismatched with subcarrier j ( j = α(i)). We assume that thepower consumption for each matched subcarrier pair is fixedas P = PT /N where PT is the total power transmission. Inthis paper, we will consider the capacity performances offour OFDM transmission strategies:

1. NOR scheme — no subcarrier matching and no powerallocation — the bits transmitted on subcarrier i at Swill be retransmitted on subcarrier i at R; the powerconsumption is equally allocated at S and R (equalP/2).

2. PA scheme — power allocation but no subcarriermatching — the bits transmitted on subcarrier i at S willbe retransmitted on subcarrier i at R; the power at Sand R is allocated to achieve the maximum capacityfor each subcarrier pair [6].

3. MA scheme — subcarrier matching but no power allo-cation — the bits transmitted on the ith worst subcarrierat S will be retransmitted on the ith worst one at R; thepower 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 subcarrierat S will be retransmitted on the ith worst one at R; thepower at S and R is allocated to maximize capacity foreach subcarrier pair.

2.2 The Capacity Formulation

The capacity of two-hop DF relaying scheme in the ithmatched 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 powerfor the subcarrier pair ith at S and R, with the constraintpS ,i + pR,α(i) = P. γS ,i and γR,α(i) are the power gain overnoise of the channel from S to R and R to D over the sub-carrier pair ith. In this paper, we assume that the channelsexperience the i.i.d. Rayleigh fading with the unit variance.The power spectral densities of noise are equal at S and Rwith zero mean and unit variance . Hence, γS ,i and γR,α(i) arethe exponential distributed random variable (r.v.) with theunit 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

CNORi = (1/2) log2

(1 + P min

{γS ,i, γR,i

}/2)

(2)

• PA scheme

Similar to the previous scheme, α(i) = i; however, the powerallocation is occurred. From (1), the maximal capacity isobtained 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

CPAi = (1/2) log2

[1 + PγS ,iγR,i

/(γS ,i + γR,i

)](4)

• MA scheme

It has been shown in [5], [6] that the optimal subcarriermatching for a maximum end-to-end overall capacity shouldbe that the kth worth subcarrier in Hop1 (S-R) is matched tothe 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 allocationis not achieved in this scheme. Therefore, the capacity ofthis 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 previousschemes where the optimal matching and power allocationare considered together. Thus, the optimal capacity for thematched 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 withthe unit variance; hence, the pdf f (x) and the cumulativedistribution function (CDF) F(x) of these r.v.s can be givenas

f (x) = e−x; F (x) = 1 − e−x (7)

We define the r.v. X(k) as the kth smallest value fromthe observed r.v. set X1, X2, ..., XN . Based on [7], the pdf andCDF of X(k) can be described as

f(k) (x) = NCk−1N−1

(1 − e−x)k−1 e−(N−k+1)x =

k−1∑i=0

uie−vi x

F(k) (x) =∫ x

0f(k) (z) dz =

∑k−1

i=0ui

(1 − e−vi x

)/vi (8)

where Cnm is the choose function of n and m, ui =

NCk−1N−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 twoi.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)

LETTER2479

Using this result, the capacity of the ith subcarrier pair canbe calculated as

CNORi = (1/2)

∫ ∞

0log2 (1 + Pz/2) fz (z) dz

= −e−4/PEi (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 bothMGFs 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 theclosed-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−2cK1(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

CPAi = (1/2)

∫ ∞

0log2 (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)

]

= 2k−1∑i=0

uie−vi x ×

⎡⎢⎢⎢⎢⎢⎢⎣1 −k−1∑j=0

uj

v j

(1 − e−v j x)

⎤⎥⎥⎥⎥⎥⎥⎦

= 2k−1∑i=0

uie−vi x − 2

k−1∑i=0

k−1∑j=0

uiu j

v je−(vi+v j)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)

∫ ∞

0log2 (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) =1x2

f(k)

(1x

)=

k−1∑i=0

uie−vi/x

x2

Φa(k) (s) = Φb(k) (s) =k−1∑i=0

ui2√vi sK1

(2√vi s

)(17)

Hence, the MGF of(a(k) + b(k)

)is achieved as

Φa(k)+b(k) (s)=k−1∑i, j=0

uiu j2√viv j sK1

(2√vi s

)K1

(2√v j s

)(18)

Due to applying the inverse Laplace transform, and thenote 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

2uiu jc√viv j

e−(vi+v j)c

×[2√viv jK0

(2√viv j

)+

(vi + v j

)K1

(2√viv j

)](19)

Finally, the capacity for the optimal matched subcarrierpair γ(k)

S ∼ γ(k)R with the power allocation between S and R is

given by

CMA−PA(k) = (1/2)

∫ ∞

0log2

(1+Pc(k)

)fc(k)

(c(k)

)dc(k) (20)

Then, the total capacity for each scheme can beachieved by taking the sum of all the subcarrier pair’s ca-pacities. Hence, the total capacity of the network is

CXtotal =

N∑i=1

CXi (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 pairin a dual-hop OFDM relay system with and without powerallocation fc(k) (x) and fq(k) (x), respectively. An OFDM sys-tem with N = 16 subcarriers is considered and the figureshows 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 khave the larger mean and the variance when comparing tothat of them for the smaller k. To validate the accuracy ofthe theoretical probability density function of the subcarriermatching systems, the derived simulation results were alsoobtained. 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

2480IEICE 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 ofrandom variable.

Fig. 2 Channel capacity against the total power PT (N = 16).

Fig. 3 Per channel capacity distribution for different total powertransmission 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 obtainedby comparing the capacity of NOR scheme to that of PAscheme. Similarly, the subcarrier matching can improve thecapacity by comparing the capacity of NOR scheme withthat of MA scheme. However, by comparing PA and MAschemes, the subcarrier matching method is preferred. Itmeans that MA scheme solves the “bottleneck problem” be-tween two hops; hence the higher capacity can be attainedwithout the power allocation. Finally, when both thesemethods are achieved we can get the highest capacity; itcan be observed as the capacities of the MA-PA scheme arehigher 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-hopOFDM relay systems with subcarrier matching. In simula-tions we selected N = 16 and SNR = 5, 10 dB. The capacityper channel without subcarrier matching is also shown as areference. As can be seen, the analytical results based onorder statistics exactly match with those of simulations.

5. Conclusion

In this letter, we proposed an exact analytical techniqueof evaluating the average capacity of a dual-hop decode-and-forward OFDM relay system with subcarrier mappingand power allocation for each subcarrier pair and the othernon-optimal schemes. Closed form density function expres-sions were derived for the end-to-end channel power gainsof all schemes. Monte Carlo simulation results and compar-isons with the analytical results were presented to confirmthe accuracy of the proposed analytical technique, and alsodemonstrate 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 andTechnology (No. 2009-0073895).

References

[1] M.O. Hasna and M.S. Alouini, “A performance study of dual-hoptransmissions 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 fadingchannels,” 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 systemsthrough cooperative relaying,” ETRI J., vol.29, no.1, pp.27–35, Feb.2007.

[4] H.A. Suraweera and J. Armstrong, “Performance of OFDM-baseddual-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,” IEEE65th, Vehicular Technology Conference, VTC2007-Spring, pp.2727–2731, 2007.

[7] A. Papoulis and S.U. Pillai, Probability, random variables, andstochastic processes, 4th ed., McGraw-Hill, Boston, 2002.

[8] I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table ofintegrals, series and products, 7th ed., Amsterdam, Boston, Elsevier,2007.

[9] H.K.G.E. Roberts, Table of Laplace Transforms, W.B. Saunders,1966.