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.

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. 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. 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

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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).
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