Performance Analysis of Dual Hop Relaying over
Non-Identical Weibull Fading Channels
Salama S. Ikki, Student member IEEE and Mohamed H. Ahmed, Senior Member IEEE
Faculty of Engineering and Applied Sciences
Memorial University of Newfoundland
St. John’s, NL, Canada A1B 3X5
e-mail: firstname.lastname@example.org, email@example.com
Abstract—In this paper, we present closed form expressions
for tight lower bounds of the performance of dual-hop non-
regenerative relaying over independent non-identical Weibull
fading channels. Since it is hard to find a closed form expression
for the probability density function (PDF) of the signal-to-noise
ratio (SNR) for the Weibull fading distribution, we use an
approximate value instead. Novel closed form expressions for
the PDF, outage probability and the moments of the approximate
value of the SNR at the destination are derived. Also, the average
SNR and amount of fading are determined. Moreover, closed
form expressions (in terms of the tabulated Meijer’s G-function)
are found for the average symbol error probability (for several
modulations schemes) as well as the Shannon capacity. It should
be noted that the Meijer’s G-function is widely available in many
scientific software packages, such as MATHEMATICA R ? and
MAPLE R ?. Finally, simulations results are also shown to verify
the analytical results.
Index Terms—Error rate, outage probability, channel capacity,
dual-hop relaying, amplify-and-forward, Weibull fading channel.
Dual-hop relaying communication has a number of advan-
tages over direct-link transmission in terms of connectivity,
power saving, and channel capacity. Relaying techniques en-
able connectivity when traditional direct transmission is not
practical due to large path-loss and/or power constraints. The
relaying concept can be applied to cellular, wireless local area
network (WLAN) and hybrid networks.
The performance analysis of dual-hop wireless communi-
cation systems operating over fading channels has been an
important area of research in the past few years. The authors in
– have presented a useful semi-analytical framework for
the evaluation of the end-to-end outage probability and average
error probability of dual-hop wireless systems with non-
regenerative relaying over Rayleigh and Nakagami-m fading
channels. In – the authors have studied the performance
bounds for multi-hop relayed transmissions with fixed-gain re-
lays over Nakagami-(Rice), Nakagami-(Hoyt), and Nakagami-
m fading channels using the moments based approach. Re-
cently, the authors in  have proposed and characterized four
channel models for multi-hop wireless communications.
However, another well known fading channel model, namely
the Weibull model, has not yet received as much attention
as other fading distributions, despite the fact that it exhibits
an excellent fit to experimental fading channel measurements
for both indoor as well as outdoor environments , .
Considering related work on diversity combining in Weibull
fading, the authors in  have presented an analysis for the
evaluation of the generalized selection combining technique
performance over independent Weibull fading channels. Re-
cently, some other contributions dealing with switched and
selection diversity, as well as second-order statistics over
Weibull fading channels have been presented in –,
where useful performance criteria including the average output
SNR, outage probability, and the bit-error rate performance
have been studied.
In this paper, we focus on non-regenerative (amplify-and-
forward) dual-hop systems and study their end-to-end perfor-
mance over independent non-identical, Weibull fading chan-
nels. The main contribution of this paper is to derive closed
form expressions for the probability density function (PDF)
and the cumulative distribution function (CDF) of the approx-
imate value of the end-to-end SNR of this dual hop system.
Moreover, the average symbol error probability (ASEP) for
several modulations schemes and the average channel capacity
are expressed in terms of the tabulated Meijer’s G-function1
[14, eq. (9.301)].
The rest of this paper is organized as follows. In Section II,
the system and channel models are described. In Sections III,
the probability density function and cumulative distribution
function have been obtained . In Section IV and V Error
probability and channel capacity have been introduced. Results
obtained from numerical analysis and Monte-Carlo simulation
are compared in Section VI. Finally, conclusions are presented
in Section VII.
II. SYSTEM AND CHANNEL MODELS
A. System Model
As shown in Fig. 1, a source node (S) communicates
with the destination node (D) through a relay node (R). The
channel coefficients between S and R (h) and between R and
1Note that G[.] can be expressed in terms of more familiar generalized
hypergeometric functionspFq(.;.;.) [14, eq. (9.14/1)] using the transforma-
tion presented in [14, eq. (9.303)], with p and q being positive integers. In
addition, both G[.] andpFq(.;.;.) are included as built-in functions in most of
the popular mathematical software packages such as Maple or Mathematica.
978-1-4244-2517-4/09/$20.00 ©2009 IEEE
D (g) are flat Weibull fading coefficients. In addition, h and g
are mutually-independent and non-identical. We also assume
here without loss of generality that all additive white Gaussian
noise (AWGN) terms of the two links have zero mean and
equal variance N0.
Given that the relaying gain equals
 where Es is the transmitted signal energy; it is straight
forward to show that the end-to-end SNR at the destination
node can be written as
?Es/(h2+ N0) –
Fig. 1.Illustration of a multi branch cooperative diversity network.
γh+ γg+ 1
where γh= h2Es/N0is the instantaneous SNR of the source
signal at the relay and γg = g2Es/N0 is the instantaneous
SNR of the relayed signal by the relay at the destination. In
order to simplify the performance analysis calculations, (1)
should be expressed in a more mathematically tractable form.
To achieve it, we proposed in  a tight upper bound for
γout≤ min(γh,γg) = γub
Our subsequent analysis exclusively relies on γub, as this
upper bound has been shown to be quite accurate ,
. Moreover, we provide extensive simulation results to
complement the bounds. A lower bound can be formulated
as in  where γlb= 0.5min(γh,γg). As the lower bound
is different from the upper bound only by a factor of half, the
following analysis can easily be extended to the lower bound,
but is omitted here for brevity.
B. Channel Model
As mentioned above, we assume that the fading channel
coefficients (h and g) are Weibull random variables. Hence,
the PDF of h can be written as
where Ωhis the power scaling parameter given by Ω1/υh
E?h2?/Γ(1 + 1/υh), Γ(•) is the Gamma function [14, eq.
of fading and E(•) is the statistical average operator. For
special case υh= 1, the Rayleigh model may be considered.
(8.310.1)], υhis the fading parameter expressing the severity
The PDF of g is the same as in (3) after replacing the
subscript h with g. The PDF of the instantaneous signal-to-
noise ratio (SNR) of the S → R link, γh= h2Es/N0, can be
where ¯ γh = E?h2?Es/N0 is the average value of γh, and
link, γg, is the same as in (4) after replacing the subscript h
Finally, to capture the effect of the path-loss on the per-
formance metrics we use the following model, which is
widely accepted in the literature: E?h2?= (dS,D/dS,R)α, and
terminal i and j, and α is the power exponent path.
βh= Γ(1 + 1/υh). Also, the PDF of the SNR of the R → D
= (dS,D/dR,D)αwhere di,j is the distance between
III. STATISTICS OF THE END-TO-END SNR
A. Outage Probability
In noise limited systems, the probability of outage is defined
as the probability that the instantaneous SNR falls below a
specified threshold γth. This threshold is a minimum value of
the SNR, below which the quality of service is unsatisfactory.
Consequently, the outage probability is given by
Pout= Fγub(γth) = Pr(γub≤ γth)
where Fγub(γth) is the cumulative distribution function (CDF)
of γub, which can be calculated as
Pout=1 − Pr(γh> γth)Pr(γg> γth)
=1 − exp
B. PDF and the Moments
The PDF of γubcan be found by taking the derivative of
the CDF with respect to γth, yielding
In this paper and in order to get a closed form expression
of the nthmoments (μn= E(γn
υh= υg= υ and this is a valid assumption in most practical
applications , . This condition is required to facilitate
the mathematical solution2. Hence, the nthmoment of γub,
with the help of [14, eq. (6.455.1)], can be written as
ub)) of γub, we assume that
Γ(1 + n/υ)
2if υh ?= υg the performance metrics can be upper bounded and lower
bounded as υ = max(υh,υg) and υ = min(υh,υg), respectively.
If we have identical channel (i.e. ¯ γh = ¯ γg = ¯ γ) it can be
shown that (8) greatly simplifies to
μn=Γ(1 + n/υ)
By setting n = 1 in (8), the average total SNR (¯ γub) can be
obtained. Furthermore, the first two moments of γub can be
used in order to evaluate the amount of fading (AF) at the
destination [17, Chapter 1]. The AF is defined as the ratio of
the variance to the square mean of ¯ γub
?AF = μ2/¯ γ2
IV. ERROR PERFORMANCE ANALYSIS
The most straightforward approach to obtain the average
error probability, P (e), is to average the conditional symbol
error probability, P (e|γ), over the PDF of the end-to-end SNR
where P (e|γ) can be found as follows 
1) For coherent modulation (like M-ary phase shift keying
(M-PSK)), the error is in the form of A erfc?√Bγ?,
2) For non-coherent modulation (like M-ary differen-
tial PSK (DBPSK)), the error is in the form of
The particular values of A and B depend on the considered
modulation schemes and can be found in . P (e) is
obtained in closed-form expressions for each one of the above
1) P(e) for Coherent Modulation:
The integral in (10) can be evaluated in a closed form
as follows. Firstly, by expressing the exponential and
erfc(•) functions in terms of the Meijer’s G-function as
exp(−g (x)) = G1,0
Then, P(e) with the help of [18, eq. (24)] can be written
in a closed form as
P (e) =
P (e|γ )fγub(γ)dγ
where erfc(•) is the complementary error function [14,
where Λ(•,•) is defined as Λ(p,q)
1)/p,··· .,(q + p − 1)/p with p being positive integer
and q real constant and G[•] is the Meijer’s G-function,
while l and k are the smallest positive integers such that
For example, if υ = 1.25 we have to choose k = 4 and
l = 5.
If ¯ γh = ¯ γg = ¯ γ it can be shown that (13) greatly
simplifies to the compact form
P(e) = 2(β/¯ γ)υ υAlυ−1?k/π
2) P(e) for Non-Coherent modulation:
By expressing the exponential function as a Meijer’s G-
function as in (11), and with the help of [18, eq. (24)],
P(e) for non-coherent modulations can be written in a
closed form as
Λ(l,1 − υ)
Integers l and k must be chosen so that (14) holds. If
¯ γh= ¯ γg= ¯ γ it can be shown that (16) simplifies to the
Λ(l,1 − υ)
V. AVERAGE CHANNEL CAPACITY
The channel capacity, in the Shannon’s sense, is an im-
portant performance measure since it provides the maximum
achievable transmission rate under which the errors are recov-
erable. The average channel capacity can be expressed as
where BW is the transmitted signal bandwidth. The reason of
time slots for transmitting the data. By substituting (7) into
(18) the average channel capacity can be obtained in closed
log2(1 + γ)fγup(γ)dγ
2factor is that we need two orthogonal channels or two
VI. NUMERICAL AND SIMULATION RESULTS
An asymmetric network geometry is examined where R
is located on a straight line between S and D. Direct path
length S → D is normalized to be 1. In all presented results,
the path-loss exponent α is chosen to be equal to 3. In this
Section, we show numerical results of the analytical bit error
rate (BER) for binary phase shift keying (BPSK) modulation,
outage probability and the average channel capacity. We plot
the performance curves in terms of average BER, outage
probability and Shannon capacity versus the SNR of the
transmitted signal (Es/N0 dB). We also show the results of
the computer simulations for verification.
Fig. 2 shows the error performance for arbitrary values
of fading parameters for dS,R = 0.6. It can be seen form
Fig. 2 that our lower bound is tight enough especially for
medium and high SNR regime. For example, for SNR value
as low as 10 dB the difference between the exact BER and
the analytical BER is less than 7% and it is more accurate
for higher values of SNR. This shows that the analytical BER
expressions are almost exact. This trend (the tightness of our
bound) is valid at any different values of fading parameters and
distances. Also, Fig. 2 demonstrates that the relaying technique
improves the BER performance in comparison with the direct
transmission. These results are obvious because the relaying
technique benefits from the path-loss reduction.
05 10 15 202530
E(h2) = (1/0.6)3, E(g2) = (1/0.4)3, v = 1.8.
Analytical Lower Bound
systems over Weibull fading channels.
Error performance for dual-hop and single-hop communication
Fig. 3 illustrates the outage performance determined from
(6) for dS,R = 0.6. It is clear again that the difference
between the exact (from simulation) and analytical results
(lower bound) for Pout is very small for medium and high
values of Es/N0. Again here, the relaying technique improves
the outage at the destination due to the path loss reduction.
Finally, Fig. 4 also shows that our lower bound is tight
for the normalized average channel capacity in medium and
high SNR regime. Here, it should be mentioned that the
amount of resources required for the relaying technique is
E(h2) = (1/0.6)3, E(g2) = (1/0.4)3, v = 1.8.
Analytical Lower Bound
Fig. 3. probability for dual-hop and single-hop communication systems over
Weibull fading channels.
twice that for direct transmission since the relay does not
simultaneously transmit and receive. This is the reason why
the direct link capacity outperforms the channel capacity
for relaying technique. However, it is evident that at low
SNR region, the capacity of the dual-hop relaying slightly
outperforms that of the direct transmission.
It should be noted that for Figs. 2, 3 and 4 that the tightness
of the performance improves as Es/N0 increases; however,
the proposed lower bound slightly loses its tightness at low
Es/N0. This is due to the fact that the accuracy of end-to-end
SNR approximation (in (2)) improves as Es/N0increases.
05 10152025 3035 40
E(h2) = (1/0.6)3, E(g2) = (1/0.4)3, v = 1.8.
Analytical Upper Bound
systems over Weibull fading channels.
Channel Capacity for dual-hop and single-hop communication
We have analyzed the performance of dual-hop commu-
nication systems, operating over independent, but not neces-
sarily identically distributed, Weibull fading channels. Using Download full-text
the PDF formula of the end-to-end SNR, novel closed-form
expressions for the average output SNR, AF, error probability
(for a broad class of digital modulations) as well as the
Shannon average capacity were obtained. Computer simulation
results verified the accuracy and the correctness of the pro-
posed analysis. We also compared the performance of the dual-
hop relaying systems with that of the traditional single hop
transmission. Extending this work to analyze the performance
of (decode-and-forward and amplify-and-forward) cooperative
diversity networks over independent non-identical Weibull
channels is currently underway.
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