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Outage Analysis of AF OFDM Relaying Systems
with Power Amplifier Nonlinearity
Silas L. Silva and C. Alexandre R. Fernandes
Department of Computer Engineering, Federal University of Ceará
R. Estanislau Frota s/n, 62010-560, Centro, Sobral/CE, Brazil
Emails: silas.limasilva@gmail.com, alexandrefernandes@ufc.br.
C. Alexandre R. Fernandes is partially supported by CNPq/Brazil.
Abstract—The high peak-to-average power ratio (PAPR) is
one of the main problems of orthogonal frequency division
multiplexing (OFDM), which may cause the introduction of inter-
carrier interference (ICI) due to the presence of nonlinear power
amplifiers (PAs). On the other hand, cooperative diversity is a
very promising technology for the future wireless network. In this
paper, a performance analysis of an amplify-and-forward (AF)
cooperative OFDM systems accounting for nonlinear distortions
introduced by a nonlinear power amplifier (PA) is developed. In
particular, a close-form expression for a tight lower-bound of the
outage probability is presented and validated through computer
simulations. Our analysis is based on the sum of the mutual
information of all the subcarriers, contrarily to previous works
that have analyzed the performance of the nonlinear cooperative
OFDM systems based on a single subcarrier. Our analysis shows
in what situations the PA nonlinearity affects significantly the
outage probability.
I. INTRODUCTION
Cooperative communications have assumed a promising role
in the scenario of the mobile communications due to their
potential gains of coverage, spectral efficiency and capacity,
without needing multiple antennas on a single terminal [1],
as in multiple-input multiple-output (MIMO) systems. Several
cooperative relaying protocols have been proposed in the
literature, such as the amplify-and-forward (AF), decode-and-
forward (DF) and compress-and-forward (CF). Among them,
the AF protocol has been widely considered in situations
where the relays have limited signal processing resources.
On the other hand, the orthogonal frequency division mul-
tiplexing (OFDM) technology stands out in the current wire-
less communications due to its high spectral efficiency, and
robustness to intersymbol interference (ISI) and intercarrier
interference (ICI) [2]. Furthermore, OFDM has a simple
equalization and a low implementation complexity. OFDM has
become the basis of many wireless communication standards,
such as IEEE 802.11a, IEEE 802.16 and 3GPP LTE [3].
One of the main problems of OFDM systems is the high
peak-to-average power ratio (PAPR) of the transmitted signals.
This feature makes OFDM systems very sensitive to nonlinear
power amplifiers (PA), as the components of the transmitted
signal with high amplitude may operate in the nonlinear region
of the PA. This may cause the introduction of ICI, leading to
errors in the signal detection. This problem is usually more
significant in the uplink, as the mobile units have generally
stronger power constraints than the base stations. However, in
cooperative systems, a high PAPR is also an important issue
in the downlink, as the relay nodes may be mobile users or
small stations [4], [5], [6].
The effects of nonlinear distortions on the performance
of cooperative communication systems have been studied by
some authors. In [6], techniques for cancellation of nonlinear
PA distortions in cooperative OFDM communication systems
were proposed for AF relaying, but no performance analysis is
presented. In [7] and [8], bit error probability expressions were
developed for an OFDM AF cooperative diversity system,
assuming that both base station and relay have nonlinear
PAs, and that the relay-destination channel is time-invariant.
Furthermore, in [7] and [8], it is assumed that there is no
direct path between the source and the destination, and no
outage analysis is presented. In [4], an outage analysis of a
cooperative OFDM system was proposed considering that the
AF relay has a nonlinear PA. [4] showed that a nonlinear PA at
the relay affects the outage probability and the diversity order
only for high SNR thresholds. In [5], the analysis of [4] is
extended to the case where multiple relays are available and
a relay selection is performed. However, in [5], it is assumed
that the wireless channels between the source and the relay
have frequency-flat fading. In [9], an analysis of the outage
probability in a nonlinear cooperative OFDM system with
relay selection is performed. However, in [9], it is assumed that
there is no direct path between the source and the destination,
and only the source node has a nonlinear PA.
In the present work, a performance analysis of an AF
cooperative OFDM communication system is developed, ac-
counting for nonlinear distortions introduced by a nonlinear
PA at the relay. The system performance is evaluated based on
the outage probability. In particular, a close-form expression
for a tight lower-bound of the outage probability is presented
and the theoretical results are validated through computer
simulations. We assume that the wireless channels have fre-
quency selective Rayleigh fading and that the destination node
combines the signal received from the relay path and directly
from the source-destination link using the maximum ratio
combining (MRC) method.
Compared to the previous works that performed perfor-
mance analysis of nonlinear cooperative OFDM systems, the
present work has the differential of performing a performance
study based on the sum of the mutual information of all
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the subcarriers, taking the correlation among the subcarriers
into account. Indeed, in [4]-[9], the performance analysis
is based on a single subcarrier, which does not provide a
global evaluation of the OFDM link. An outage analysis of
a cooperative OFDM system based on the sum of the mutual
information is carried out in [10]. However, in [10], the effects
of the nonlinear distortions are not considered and the direct
path between the transmitter and the destination is not used.
The rest of the paper is organized as follows. Section 2
describes the system model considered in this work. In Section
3, the outage analysis is presented. In Section 4, we evaluate
the validity of the proposed outage analysis by means of
computer simulations and some conclusions and perspectives
are drawn in Section 5.
II. SYSTEM MODEL
Let us consider a cooperative OFDM system that provides
communication between a transmitting terminal, denoted by S
(source), and a receiver terminal, denoted by D (destination),
through a direct link between S and D, and through a relay
node, denoted by R (relay). The relay forwards to D the signal
received from S according to a well-defined protocol. In this
work, the AF protocol is used. It also is assumed that all the
wireless channels undergo frequency-selective Rayleigh fading
and that the length of the cyclic prefix is higher than or equal
to the maximum multipath delay of all the wireless channels.
The communication occurs in two orthogonal times slots.
In the first time slot, S transmits the information signal
simultaneously to R and D. At the second time slot, the relay
amplifies the received signal and send it to D. At the end,
terminal D combines the signals received from R and S using
the MRC method. The relay operates in half-duplex mode
and all the nodes are synchronized at symbol level. We also
consider that the node D has perfect channel state information
(CSI) of all links and that the node R has perfect CSI of the
SR link.
Moreover, it is assumed a downlink transmission where the
source has a linear PA while the relay has a nonlinear PA.
This consideration is based on the fact that the source is a
base station with weak power restrictions and the relay is a
small station or user terminal with strong power constraints.
In the discrete frequency domain, the signals received by D
and R during the firsttime slot and at the kth subcarrier are
expressed respectively by:
YSD(k) = H
′
SD(k)X(k) +NSD(k) (1)
and
YSR(k) =
√
PSHSR(k)X(k) +NSR(k), (2)
for 0 ≤ k ≤ N − 1, where N is the number of subcarriers,
H ′SD(k) =
√
PSHSD(k), HSD(k) and HSR(k) are the
frequency responses of the SD and SR channels, respectively,
X(k) is the unit power signal transmitted by S, PS is the
transmission power of the source, NSD(k) and NSR(k) are
the additive white Gaussian noise (AWGN) components of
the SD and SR channels, respectively, with variance N0. In
addition, X(k) is assumed to be uniformly distributed over a
phase shift keying (PSK) or quadrature amplitude modulation
(QAM) alphabet.
The relay multiplies the signal received at the kth sub-
carrier by a gain, denoted by G(k), and forwards it to
the destination. The signal transmitted by the relay, in the
discrete frequency domain, at the k-th subcarrier, is written as
XR(k) = G(k)YSR(k). In this article, we consider a variable
gain given by [14]: G(k) =
√
PR/(
√
PS |HSR(k)|2 +N0),
where PR is the transmission power of the relay. After that,
the Inverse Discrete Fourier Transform (IDFT) of XR(k) is
calculated, the cyclic prefix is appended and the signal is
passed through a nonlinear PA.
Consider a complex Gaussian random process with zero
mean subject to a nonlinear memoryless system. The Buss-
gang’s theorem establishes that the output of such a nonlinear
system can be expressed the sum of an attenuated version of
the input signal with a noise signal uncorrelated of the input
signal [11], [12]. Thus, as the IDFT of XR(k) is a complex
Gaussian random process [5], by assuming that the PA of the
relay can be modeled as memoryless nonlinear function, it is
possible to express the output of the relay’s PA in the discrete-
time domain as [12]:
ωR(xR(n)) = KRxR(n) + dR(n), (3)
where ωR(·) is the function that models the PA of the relay,
xR(n) is the IDFT of XR(k), KR is a fixed gain and dR(n)
is a nonlinear distortion with zero mean and uncorrelated with
the input signal xR(n). For some functions ωR(·), there are
analytical expressions for KR and for the variance of dR(n)
[12], [13].
Using (3), the signal received by D during the second time
slot and at the kth subcarrier can then be written as:
YRD(k) = HRD(k) [KRXR(n) +DR(n)] +NRD(k), (4)
where HRD(k) is the frequency response of the RD channel,
NRD(k) is the AWGN component andDR(k) is the frequency
domain version of dR(n). From (2) and (4), YRD(k) can be
re-expressed as:
YRD(k) = H
′
RD(k)X(k) +N
′
RD(k), (5)
where H ′RD(k) = KR
√
PSHSR(k)HRD(k)G(k) and
N ′RD(k) = KRHRD(k)G(k)NSR(k)
+HRD(k)DR(k) +NRD(k). (6)
The signals YRD(k) and YSD(k) are combined at the
destination node using the MRC method, the instantaneous
SNR of the resulting signal at the subcarrier k being given by:
γMRC(k) = γSD(k)+γSRD(k), where γSD(k) and γSRD(k)
stand for the instantaneous SNRs of the SD and global SRD
links, respectively. The global SRD link corresponds the series
cascade of the SR channel, relay’s PA and RD channel. The
SNR of the output of the MRC can be expressed as [4]:
γMRC(k) = γSD(k) +
γSR(k)γRD(NLD)(k)
γSR(k) + γRD(NLD)(k) + 1
, (7)
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where
γSD(k) =
PS |HSD(k)|2
N0
, γSR(k) =
PS |HSR(k)|2
N0
(8)
and
γRD(NLD)(k) =
K2RPR|HRD(k)|2
|HRD(k)|2σ2DR +N0
. (9)
with σ2DR being the variance of DR(k).
Note that γSD(k) and γSR(k) correspond to the instanta-
neous SNRs of SD and SR wireless channels without account
for the NLD distortions, while γRD(NLD)(k) considers the
effects of the RD wireless channel and the distortions due the
nonlinear PA of the relay.
In addition, to observe explicitly the effects of nonlinear
distortions, we can rewrite (9) as:
γRD(NLD)(k) =
[
1
γRD(k)
+
1
γNLD(R)
]−1
, (10)
where
γRD(k) =
K2RPR|HRD(k)|2
N0
and γNLD(R) =
K2RPR
σ2DR
.
(11)
Notice that γRD(k) corresponds to the instantaneous SNR
of RD wireless link without account for the NLD distortions,
while γNLD(R) is the SNR of the PA, without account for
the wireless channel. It is noteworthy mentioning that (10)
corresponds to the SNR of a two-hop transmission, while
γSRD(k) corresponds to the SNR of a three-hop transmission
[15], [4].
III. OUTAGE PROBABILITY
In this section, a closed-form expression for a lower bound
of the outage probability of the cooperative AF system
presented in Section II is developed. We consider that the
instantaneous mutual information of the OFDM link is the sum
of the mutual information associated with all subcarriers. Thus,
the mutual information in bits/second/Hz of the considered
cooperative system can be expressed by [16]
IAF =
1
2N
N−1
∑
k=0
log2 [1 + γMRC(k)] . (12)
The mutual information is multiplied by 1/2 due to the sharing
of the resources of the channel in two time slots.
In this work, we define the outage probability of a system
as the probability that the mutual information is less than or
equal to a given capacity threshold Rth, i.e. Pout(Rth) =
Prob[IAF ≤ Rth].
From the Jensen’s Inequality [10], we can write:
IAF ≤
1
2
log2
(
1
N
N−1
∑
k=0
[1 + γSD(k) + γSRD(k)]
)
. (13)
On the other hand, from [4], we have:
γSRD(k) ≤ min(γSR(k), γRD(k), γNLD(R)), (14)
where min(·) is a function that returns the smallest argument.
An upper bound for γSRD(k) lead to a lower bound for
Pout(Rth). Thus, (13) and (14) yield:
Pout(Rth) ≥ Prob
[
1
N
N−1
∑
k=0
γSD(k)
+
1
N
N−1
∑
k=0
min(γSR(k), γRD(k), γNLD(R)) ≤ 22Rth − 1
]
. (15)
Using the following relationship:
∑N−1
k=0 min(xk, yk, zk) ≤
min(
∑N−1
k=0 xk,
∑N−1
k=0 yk,
∑N−1
k=0 zk), a lower bound for
Pout(Rth) can be obtained as:
Pout(LB)(Rth) = Prob
[
ΓSD + ΓSRD ≤ 22Rth − 1
]
. (16)
where the random variables ΓSD and ΓSRD are given by
ΓSD =
1
N
N−1
∑
k=0
γSD(k) (17)
and
ΓSRD = min
(
ΓSR,ΓRD,ΓNLD(R)
)
(18)
with
ΓSR =
1
N
N−1
∑
k=0
γSR(k) and ΓRD =
1
N
N−1
∑
k=0
γRD(k). (19)
For unifying the notation, we defined ΓNLD(R) = γNLD(R).
We can rewrite (16) as :
Pout(LB)(Rth) =
∫ ∞
−∞
FΓSRD (γth − x)fΓSD (x)dx, (20)
where FΓSRD (·) is the cumulative distribution function (CDF)
of the random variable ΓSRD , fΓSD (·) is the probability
density function (PDF) of the random variable ΓSD and
γth = 2
2Rth − 1. FΓSRD (·) is given by:
FΓSRD(γth) = 1− (1 − FΓSR(γth))(1 − FΓRD (γth))
(1− FΓNLD(R)(γth)), (21)
where FΓSR(·), FΓRD (·) and FΓNLD(R)(·) are the CDFs of the
random variables ΓSR, ΓRD and ΓNLD(R), respectively.
The manipulation of the random variables ΓSD, ΓSR and
ΓRD using (17) and (19), requires a great effort as these
equations express ΓSD, ΓSR and ΓRD as sums of correlated
random variables. For simplifying the mathematical analysis,
we apply the Parseval’s Theorem in (17) and (19), so that
ΓSD, ΓSR and ΓRD are expressed as sums of uncorrelated
random variables, in the following way:
ΓSD =
LSD
∑
l=1
φSD(l), ΓSR =
LSR
∑
l=1
φSR(l) (22)
and
ΓRD =
LRD
∑
l=1
φRD(l), (23)
where φSD(l), φSR(l) and φRD(l) represent, respectively, the
instantaneous SNRs of the lth tap of the impulse responses
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of the SD, DR and RD channels, and LSD, LSR and LRD
represent the impulse response lengths of the SD, DR and
RD channels, respectively.
Assuming Rayleigh fading, the PDFs of φSD(l), φSR(l)
and φRD(l) are respectively given by:
fφSD(l)(x) =
e
− x
φSD(l)
φSD(l)
, fφSR(l)(x) =
e
− x
φSR(l)
φSR(l)
(24)
and
fφRD(l)(x) =
e
− x
φRD(l)
φRD(l)
, (25)
where φSD(l), φSR(l) and φRD(l) are the mean values of
φSD(i), φSR(l) and φRD(l), respectively.
The CDFs of ΓSR and ΓRD and the PDF of ΓSD can be
determined by their respective moment generating functions
(MGF). Given the PDFs set forth in (24) and (25), the MGFs
of ΓSD, ΓSR and ΓRD are expressed respectively by
MΓSD(s) =
LSD
∏
l=1
[
1
1− sφSD(l)
]
, (26)
MΓSR(s) =
LSR
∏
l=1
[
1
1− sφSR(l)
]
(27)
and
MΓRD(s) =
LRD
∏
l=1
[
1
1− sφRD(l)
]
. (28)
In the general case, obtaining the PDFs or CDFs ofΓSD,
ΓSR and ΓRD from their respective MGFs requires consid-
erable analytical work. We consider then the case where the
mean values of the SNRs associated with the impulse response
taps of the SR and RD channels are different, i.e.
φSR(i1) 6= φSR(i2), φRD(j1) 6= φRD(j2), φSR(i1) 6= φRD(j2)
(29)
for 1 ≤ i1, i2 ≤ LSR, i1 6= i2, 1 ≤ j1, j2 ≤ LRD and j1 6= j2.
In this situation, from (26), (27) and (28), we can write:
fΓSD(γth) =
LSD
∑
l=1


e
−
γth
φSD(l)
φSD(l)
ASD(l)

 , (30)
FΓSR(γth) =
LSR
∑
l=1
[(
1− e
−
γth
φSR(l)
)
ASR(l)
]
(31)
and
FΓRD (γth) =
LRD
∑
l=1
[(
1− e
−
γth
φRD(l)
)
ARD(l)
]
, (32)
where
Aa(l) =
La
∏
i=1,i6=l
1
1− φa(i)
φa(l)
, (33)
for a = SD, SR or RD. Moreover, as ΓNLD(R) is considered
to be a constant, its CDF is given by FΓNLD(R)(γth) = u(γth−
ΓNLD(R)).
By substituting (21) and (30)-(33) into (20) and making
some algebraic manipulations, omitted due to lack of space,
the lower bound of the outage probability is obtained:
Pout(LB)(Rth) = FΓSD (m) + BSD,SR + BSD,RD
−
LSD
∑
i=1
LSR
∑
j=1
LRD
∑
k=1
(ASD(i)ASR(l)ARD(k)
[
e
−m
φSD(i) − e
−γth
φSD(i) + φSR(j)φRD(k)



e
−γth
φSD(i) − e
−m
φSD(i) e
m
φSR(j) e
−γth
φSR(j) e
m
φRD(k) e
−γth
φRD(k)
φSD(i)φSR(j) + φSD(i)φRD(k) − φSR(j)φRD(k)



+
φSR(j)
(
e
−γth
φSD(i) − e
−m
φSD(i) e
m
φSR(j) e
−γth
φSR(j)
)
φSR(j) − φSD(i)
+
φRD(k)
(
e
−γth
φSD(i) − e
−m
φSD(i) e
m
φRD(k) e
−γth
φRD(k)
)
φRD(k) − φSD(i)












,
(34)
where
BSD,b =
LSD
∑
i=1
Lb
∑
j=1
(
ASD(i)Ab(l)
[
e
−m
φSD(i)
−e
−γth
φSD(i) +
φb(j)
(
e
−γth
φSD(i) − e
−m
φSD(i) e
m
φb(j) e
−γth
φb(j)
)
φb(j) − φSD(i)












,
(35)
for b = (SR or RD) and m = (max(γth − ΓNLD(R)), 0),
with max(·) being the function returns the highest number in
the argument.
It can be concluded form (34) that Pout(LB)(Rth) is af-
fected by PA nonlinearity only for γth > ΓNLD(R), i.e.
Rth > 0.5 log 2[1+ΓNLD(R)] which means that the nonlinear
PA increases the outage probability lower bound for high γth
(or high Rth), but it does not change Pout(LB)(Rth) for small
values of γth (or small Rth).
IV. SIMULATION RESULTS
In this section, the validity of the lower bound expression for
the outage probability developed in Section III is evaluated by
means of computer simulations. The PA of relay is modeled by
the soft clipper (soft limiter) model [12], with an amplitude
saturation equal to 1. Closed-form expressions for KR and
σ2DR are given in [12] for this PA model. Furthermore, we
used N = 64 subcarriers, QPSK (quadrature PSK) transmitted
signals and PS = PR = 0.5, which leads to ΓNLD(R) =
17.5dB. The length of all the channel impulse responses and
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0 5 10 15 20 25 30 35 40 45
10
−4
10
−3
10
−2
10
−1
10
0
SNR dB
O
u
ta
g
e
P
ro
b
a
b
il
it
y
Theoretical
Simulated
γth = 25dB
γth = 30dB
γth = 20dB
γth = 15dB
γth = 5dB
γth = 10dB
Fig. 1. Outage probability versus the mean SNR for several values of γth.
cyclic prefixes is equal to 4. Moreover, all the wireless links
have the same channel and noise variance. The results were
obtained via Monte Carlo simulations using a large number of
independent channel, data and noise realizations
Fig. 1 shows the outage probability versus the mean SNR at
the SD link, the outage probability being obtained by means of
simulations and using (34), for various values of γth = 2
2Rth−
1. It can be noted that the theoretical lower bound curves are
very close to the simulated curves in most of the tested cases.
Indeed, except when γth = 15dB, the gap between the curves
is approximatively 1dB in most of the cases. The higher gap
between the curves when γth = 15dB is due to the fact that
(14) is approximatively an equality when γth and ΓNLD(R)
do not have similar values [4]. Thus, when γth ∼= ΓNLD(R) =
17.5dB, then equality in (14) does not hold.
Fig. 2 shows outage probability versus the mean SNR
at the SD link, the outage probability being obtained by
means of simulations and using (34), considering linear and
nonlinear PAs, and various values of γth. As predicted in
Section III, it can be viewed that when γth > γNLD(R), the
nonlinear PA provides a significant increase on the outage
probability compared to the linear PA case. However, when
γth < γNLD(R), the difference between the linear and the
nonlinear PA cases becomes small. By the slope of the curves,
it can also be concluded that the PA nonlinearity affects
the order of diversity when γth < γNLD(R). Therefore, we
conclude that the system is severely affected by the nonlinear
PA only if γth > γNLD(R).
V. CONCLUSION
In this paper, the effects of nonlinear PA distortions on
the outage probability of an AF cooperative OFDM system
was studied. In particular, a closed-form expression for a tight
lower-bound of the outage probability was derived. Simulation
results have shown that the developed outage probability
expression has good accuracy in most of the tested cases.
It was also observed that the nonlinear distortions increase
significantly the outage probability only when for γth (or Rth)
is above a certain parameter that depends on the PA model.
0 5 10 15 20 25 30 35 40 45
10
−4
10
−3
10
−2
10
−1
10
0
SNR (dB)
O
u
ta
g
e
P
ro
b
a
b
il
it
y
Theor − Nonlinear PA
Simul − Nonlinear PA
Theor − Linear PA
Simul − Linear PA
γth = 35dB
γth = 25dB
γth = 15dB
γth = 5dB
Fig. 2. Outage probability versus the mean SNR for linear and nonlinear
PAs.
In this case, the order of diversity of the system is decreased.
In future works, we will extend the presented analysis
by making more generic assumptions and by considering
other relaying protocols and combining methods, as well as
simulation results in other environments should be presented.
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