Performance analysis of maximal ratio combining over shadowedRice fading channels
13 Oct 2009pp 8387
TL;DR: New simple expressions for the probability density function and the cumulative distribution function of the sum of possibly correlated non identically distributed squared SR random variables are derived and a novel approach is introduced for Laplace inverting the moment generating function (MGF) of thesum.
Abstract: An accurate model for analyzing the performance of wireless land mobile satellite (LMS) communication systems is the shadowedRice (SR) fading model. In this paper, new simple expressions for the probability density function (PDF) and the cumulative distribution function (CDF) of the sum of possibly correlated non identically distributed squared SR random variables are derived. To this end, a novel approach is introduced for Laplace inverting the moment generating function (MGF) of the sum. Based on this approach the PDF is expressed in a simple infinite chisquared series form, from which the CDF is also easily obtained. The derived statistics are used to analyze the performance of maximal ratio combining (MRC) over SR fading channels and novel closedform expressions for various performance criteria such as the outage probability, the ergodic capacity and the bit error probability (BEP) are developed. The results of extensive Monte Carlo simulations are presented, which corroborate our theoretical analysis.
Summary (2 min read)
Jump to: [Introduction] – [A. Sum of independent squared SR RVs] – [C. Bit Error Probability] – [IV. PERFORMANCE EVALUATION] and [V. CONCLUSION]
Introduction
 Diversity combining techniques such as orthogonal spacetime block coding and maximal ratio combining (MRC) have gained considerable attention recently, due to their ability to combat fading in wireless communication systems.
 Motivated by the above, in this paper, new simple expressions for the PDF and the CDF of the sum of squared SR RVs are introduced, following a novel analytical approach.
 Moreover, the proposed analytical method can be generalized to other practical fading distributions, an issue, which due to space limitations will not be elaborated any further.
 New simple expressions of various performance metrics for MRC over SR fading channels are described in Section III.
A. Sum of independent squared SR RVs
 Here, the authors follow a method similar to that originally used in [1] for the sum of squared Gaussian RVs. (18) where L−1 stands for the inverse Laplace transform.
 Observe that (19) has a very simple form avoiding the use of complicated functions such as the con uent hypergeometric function, as is the case with the expression of the PDF proposed in [11].
 Where also the range of admissible values for parameter β is given.
C. Bit Error Probability
 In general, to calculate the BEP of a diversity scheme, averaging of the BEP for the additive white Gaussian (AWGN) channel with respect to the distribution of the instantaneous SNR per bit at the receiver side, is required.
 Eb is the transmitted energy per bit and N0 the noise density at each branch of the receiver.
 Additionally, due to the equivalence of OSTBC with MRC, similar expressions for the outage probability, ergodic capacity and BEP also hold for OSTBC over SR fading [7].
IV. PERFORMANCE EVALUATION
 Extensive Monte Carlo computer simulated results are used to verify the presented theoretical analysis under various fading conditions.
 For both fading schemes, n = 2 and n = 3 receive antennae are examined with the parameters of the ith fading channel selected as {Ωi, bi,mi}.
 For all the theoretical results concerning Scheme 1, the parameter β was taken equal to 0.5 and a number of 11 terms have been retained from the respective series expansions.
 Again, the close match of theoretical and simulations results can be veri ed.
V. CONCLUSION
 By applying an effective method to manipulate the Laplace transform of the PDF, new expressions for the PDF and CDF of the sum of squared SR RVs have been derived.
 Without setting any restriction to the parameters of the model, the proposed expressions are much simpler compared to already known ones.
 Based on these results, mathematically tractable closedform expressions for the outage probability, the ergodic capacity and the BEP of MRC over SR fading have been presented.
 The uniform convergence of the series in (19) can be proven by properly bounding the series truncation error.
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Performance Analysis of Maximal Ratio Combining
over ShadowedRice Fading Channels
George A. Ropokis
†‡
, Athanasios A. Rontogiannis
†
Kostas Berberidis
‡
and P. Takis Mathiopoulos
†
†
Institute for Space Applications and Remote Sensing, National Observatory of Athens, 15236, Athens, Greece
‡
Dept. of Computer Engineering & Informatics, University of Patras 26500, RioPatras, Greece
Email: {ropokis,tronto}@space.noa.gr, berberid@ceid.upatras.gr, mathio@space.noa.gr
Abstract—An accurate model for analyzing the performance
of wireless land mobile satellite (LMS) communication systems
is the shadowedRice (SR) fading model. In this paper, new
simple expressions for the probability density function (PDF)
and the cumulative distribution function (CDF) of the sum
of possibly correlated non identically distributed squared SR
random variables are derived. To this end, a novel approach is
introduced for Laplace inverting the moment generating function
(MGF) of the sum. Based on this approach the PDF is expressed
in a simple innite chisquared series form, from which the CDF
is also easily obtained. The derived statistics are used to analyze
the performance of maximal ratio combining (MRC) over SR
fading channels and novel closedform expressions for various
performance criteria such as the outage probability, the ergodic
capacity and the bit error probability (BEP) are developed. The
results of extensive Monte Carlo simulations are presented, which
corroborate our theoretical analysis.
I. INTRODUCTION
Diversity combining techniques such as orthogonal space
time block coding (OSTBC) and maximal ratio combining
(MRC) have gained considerable attention recently, due to their
ability to combat fading in wireless communication systems.
When OSTBC or MRC is employed the signal to noise ratio
(SNR) of the system is expressed as the sum of squared random
variables (RVs). Hence to analyze the performance of such
systems the probability density function (PDF) and cumulative
distribution function (CDF) of the sum of squared RVs must
be obtained in the simplest possible form, so as to facilitate
the calculation of several performance metrics.
A simple representation for the PDF of the sum of squared
RVs is based on innite series expansions of chisquared or
Gamma PDFs and has been widely used both in statistics
and in the analysis of wireless communications systems.
More specically, in [1],[2] and [3] the distribution of the
sum of squared Gaussian RVs is studied and several series
expansions for the PDF are proposed. Using a Gamma series
representation the PDF and CDF of the sum of independent
Gamma, i.e., squared Nakagami RVs are derived in [4] and
[5]. Based on these expressions the performance of MRC over
This paper is part of the 03ED838 research project, implemented within
the framework of the Reinforcement Programme of Human Research Man
power (PENED) and conanced by National and Community Funds (75%
from E.U.European Social Fund and 25% from the Greek Ministry of
DevelopmentGeneral Secretariat of Research and Technology). G. A. Ropokis
was funded in part by the Satellite Network of Excellence (SatNEx) project,
a Network of Excellence (NoE) funded by the the European Commitee (EC)
under FP6 program.
Nakagami fading channels is studied in [6]. In [7] the chi
squared series representation is adopted for the performance
analysis of OSTBC over Hoyt fading channels, while in [8]
the same series expression is applied for computing matched
lter bounds for BPSK over multipath Rician fading channels.
Yet another fading model that has attracted increasing
interest lately is the socalled shadowedRice (SR) model [9].
One such mathematically versatile model has been recently
proposed in [10]. It has been shown in [10] that this model
describes very accurately the land mobile satellite (LMS) chan
nel, which is expected to play a prominent role in future third
and fourth generation communication systems. Nevertheless,
to the best of our knowledge very few results related to the
SR distribution exist in the open technical literature. Only
recently in [11], an analysis for the sum of squared SR RVs
has been presented and the outage probability and capacity of
MRC over SR fading channels have been studied. However,
the closedform expression for the PDF proposed in [11] is
extremely complex, thus leading to expressions for the various
performance metrics, which are quite complicated and difcult
to use in practice.
Motivated by the above, in this paper, new simple expres
sions for the PDF and the CDF of the sum of squared SR
RVs are introduced, following a novel analytical approach. Our
approach is based on a proper manipulation of the Laplace
transform of the PDF, which was rst used in [1] and [2] for
the case of squared Gaussian RVs. The PDF is expressed in
a simple chisquared series form, which is proven to converge
uniformly. Then, the resulting PDF and CDF formulas are used
to analyze the performance of MRC over SR fading channels in
terms of the outage probability, the ergodic capacity and the bit
error probability (BEP). It should be emphasized that not only
the expressions for the PDF, CDF and the various performance
criteria are much simpler than those in [11], but also the fading
model considered in this work is more general too. Moreover,
the proposed analytical method can be generalized to other
practical fading distributions, an issue, which due to space
limitations will not be elaborated any further.
The outline of the paper is as follows. In Section II the
distribution of the sum of squared SR RVs is analyzed and
closedform expressions for the PDF and CDF are provided.
New simple expressions of various performance metrics for
MRC over SR fading channels are described in Section III.
Simulation results are presented in Section IV and concluding
9781424435593/09/$25.00
c
2009 IEEE IWSSC 2009
remarks in Section V.
II. D
ISTRIBUTION OF THE SUM OF SQUARED SR RVS
The SR fading model can be dened as a Rice fading
model, whose lineofsight (LOS) component is random. In
this work we adopt the SR model proposed in [10], in which
the amplitude of the LOS component is Nakagami distributed.
More specically, the complex baseband representation of this
SR fading channel model is given as follows [10]
a
F
= a
R
exp (jφ)+a
N
exp (jζ) (1)
where a
R
is a Rayleigh RV with average power 2b, φ is
uniformly distributed in [0, 2π), a
N
is Nakagami distributed
with parameters Ω and m,andζ is the nonrandom phase of
the LOS component of the channel. Then, the power r = a
F

2
of the fading process follows a squared SR distribution, whose
PDF is given by [10]
p
r
(r)=
2bm
2bm +Ω
m
1
2b
exp
−
r
2b
×
1
F
1
m, 1;
Ωr
2b (2bm +Ω)
(2)
where
1
F
1
(·, ·; ·) is the conuent hypergeometric function
[12]. It can then be shown that the moment generating function
(MGF) of r,isexpressedas[10]
M
r
(s)=
(1 − 2bs)
m−1
1 −
2b +
Ω
m
s
m
. (3)
Using this form of the MGF, the PDF and CDF of the sum of
independent and correlated squared SR RVs will be derived in
the following sections.
A. Sum of independent squared SR RVs
Let us rst dene the following RV
z =
n
i=1
r
i
(4)
where r
i
, i =1, 2,...,n are independent non identically
distributed squared SR RVs with parameters {Ω
i
,b
i
,m
i
}.Then
by utilizing the relation between the MGF of a RV and the
Laplace transform of its PDF [9], the Laplace transform of the
PDF of z is directly written from (3) and (4) as
L
z
(s)=
n
i=1
(1 + 2b
i
s)
m
i
−1
1+
2b
i
+
Ω
i
m
i
s
m
i
. (5)
Apparently, the PDF of z can be obtained by calculating the
inverse Laplace transform of L
z
(s) . Here, we follow a method
similar to that originally used in [1] for the sum of squared
Gaussian RVs. We show that this method is easily extended for
the case of squared SR RVs. Indeed, let us dene the function
θ (s)=
1
1+sβ
(6)
where β is an arbitrary positive parameter. Notice that for any
a>0 the following identity holds
1+as =1+sβ
a
β
+
a
β
−
a
β
=
a
βθ (s)
1 −
1 −
β
a
θ (s)
.
(7)
From (5) and (7), L
z
(s) is readily expressed as
L
z
(s)=Aθ
n
(s)
n
i=1
(1 − γ
i
θ (s))
m
i
−1
(1 − δ
i
θ (s))
m
i
(8)
where
A = β
n
n
i=1
(2b
i
)
m
i
−1
2b
i
+
Ω
i
m
i
m
i
(9)
and
γ
i
=1−
β
2b
i
,δ
i
=1−
β
2b
i
+
Ω
i
m
i
. (10)
The basic idea is to express L
z
(s) given in (8) as a series
expansion so that, through Laplace inversion to derive a simple
formula for the PDF. To this end, let us rst introduce the
function
L (θ)=
n
i=1
(1 − γ
i
θ)
m
i
−1
(1 − δ
i
θ)
m
i
(11)
whose logarithm is written as
ln L (θ)=
n
i=1
(m
i
− 1) ln (1 − γ
i
θ) −
n
i=1
m
i
ln (1 − δ
i
θ).
(12)
For θ satisfying θ<1/ max{max
i
{γ
i
}, max
i
{δi}},aseries
expansion for the logarithm of L (θ) exists and is given by
ln L (θ)=
∞
j=1
d
j
θ
j
j
(13)
where
d
j
=
n
i=1
m
i
δ
j
i
−
n
i=1
(m
i
− 1) γ
j
i
. (14)
Then according to [13, pp. 93], (11) can be rewritten as
L (θ)=
∞
i=0
c
i
θ
i
(15)
where the coefcients of the series are computed according to
the following recursive formula
c
0
= L(0) and c
i
=
1
i
i−1
l=0
d
i−l
c
l
for i>0. (16)
From (8) and (15), L
z
(s) is expressed in a series expansion
form, i.e.,
L
z
(s)=A
∞
i=0
c
i
θ
n+i
(s). (17)
Recall that for integer v
L
−1
{θ
v
(s)} =
z
v− 1
exp (−z/β)
β
v
(v − 1)!
(18)
where L
−1
stands for the inverse Laplace transform. Thus,
from (17) and (18) the PDF of z is nally given by the
following closedform expression
p
z
(z)=A
∞
i=0
c
i
z
n+i−1
exp (−z/β)
β
n+i
(n + i − 1)!
(19)
where A is given by (9) and c
i
are recursively computed from
(16). Observe that (19) has a very simple form avoiding the use
of complicated functions such as the conuent hypergeometric
function, as is the case with the expression of the PDF
proposed in [11]. Moreover, the analysis has been made for
a more general model than that adopted in [11], where it is
assumed that b
1
= b
2
= ...= b
n
.
Applying term by term integration in (19),theCDFofthe
sum of squared SR RVs is expressed as follows
P
z
(z)=A
∞
i=0
c
i
γ (n + i, z/β)
(n + i − 1)!
(20)
where γ(·, ·) is the lower incomplete Gamma function [12,
pp. 260]. Notice, that since n + i is an integer, γ(n + i, z/β)
can be written as a linear combination of elementary functions
leading to an extremely simple form for the CDF. To be able to
interchange integration and summation, so as to arrive in (20),
uniform convergence of the series in (19) is required. This
result is established in the Appendix, where also the range
of admissible values for parameter β is given. It should be
noted that β controls the convergence of the series, and thus
species the number of series terms that should be retained
for a prescribed accuracy.
B. Correlated SR RVs
The previous analysis can easily be extended for correlated
SR RVs obeying the correlated fading scenario employed in
[11]. Let us consider the following RV
z
C
=
n
i=1
r
i
(21)
where r
i
= a
R
i
exp(jφ
i
)+a
N
i
exp(jζ
i
)
2
, i =1, 2 ...,n
are squared SR RVs with parameter set {Ω
i
,b,m}. Under
the above mentioned scenario the Rayleigh RVs a
R
i
are
independent, while the RVs a
2
N
i
, i =1,...,n are correlated.
Then, by dening the matrix C whose (i, j)th element is the
square root of the correlation coefcient of the RVs r
i
and r
j
and the matrix D = diag
Ω
1
m
,...,
Ω
n
m
, it can be shown that
z
C
is equal in distribution with the following RV
z
I
=
n
i=1
r
i
(22)
where r
i
are independent squared SR RVs with parameters
{λ
i
m, b, m} with λ
i
being the ith eigenvalue of the matrix
DC. Thus, instead of analyzing the initial RV z
C
, an equiva
lent analysis can made for z
I
, as described above.
III. MRC
OVER SR FADING
As it is known, in a communications system employing n
receive antennae and an MRC combiner, the SNR z at the
receiving end can be expressed as
z =
n
i=1
z
t
h
i

2
(23)
where z
t
is the SNR at the transmitter side, i.e., the ratio of
the average transmitted power over the noise power, and h
i
the fading coefcient of the ith branch. Hence, assuming that
h
i
,i=1,...nare independent SR distributed RVs, it is easy
to see that
z
d
=
n
i=1
r
i
(24)
where
d
=
denotes equality with respect to distribution, and r
i
=
z
t
h
i

2
. By denoting with {Ω
h
i

,b
h
i

,m
h
i

} the parameters
of h
i
 and with {Ω
i
,b
i
,m
i
} the parameters of r
i
it is easy to
show that
{Ω
i
,b
i
,m
i
} = {z
t
Ω
h
i

,z
t
b
h
i

,m
h
i

}. (25)
Thus the PDF and CDF of z in (24) will be given by (19)
and (20) respectively with the coefcients c
i
calculated as in
(16). Based on the expressions of the PDF and CDF already
derived, in the following an analysis of MRC over SR fading
channels in terms of outage probability, ergodic capacity and
BEP is presented.
A. Outage Probability
The outage probability P
out
(z
0
) of a communication system
is dened as the probability that the SNR at the receiving end
drops under a predened threshold z
0
. Hence, for a maximal
ratio combiner operating over SR fading, P
out
(z
0
) can be
calculated from (20), i.e.,
P
out
(z
0
)=A
∞
i=0
c
i
γ (n + i, z
0
/β)
(n + i − 1)!
. (26)
It should be noticed that (26) is much simpler and easier
to calculate than the expression of the outage probability
proposed in [11], which involves a double innite series
expansion.
B. Ergodic Capacity
By denition, the capacity of an MRC system, expressed
in bits/sec/Hz is given by
C =log
2
(1 + z) (27)
with z dened as in (23). Moreover, the ergodic capacity is
dened as the average value of the capacity with respect to the
SNR. Hence, from (19) in the case of SR fading the ergodic
capacity is expressed as
C = A log
2
(e)
∞
i=0
c
i
β
n+i
(n + i − 1)!
I
n+i
(28)
with I
v
dened as
I
v
=
∞
0
z
v− 1
exp (−z/β)ln(1+z) dz. (29)
Since v is integer, I
v
takes the following closed form [14]
I
v
=(v − 1)! exp (1/β)
v
k=1
β
k
Γ(−v + k, 1/β) (30)
where Γ(·, ·) is the upper incomplete Gamma function [12,
pp. 260]. Again, by comparing (28) with [11, eq. (16)], the
advantage of (28) in terms of simplicity is clear.
C. Bit Error Probability
In general, to calculate the BEP of a diversity scheme,
averaging of the BEP for the additive white Gaussian (AWGN)
channel with respect to the distribution of the instantaneous
SNR per bit at the receiver side, is required. Following this
procedure and assuming BPSK or Graycoded QPSK, the bit
error probability takes the form
1
P
e
= A
∞
i=0
c
i
J
n+i
(31)
where c
i
’s are the coefcients of the series in (19) that result
by setting z
t
= E
b
/N
0
in (23). E
b
is the transmitted energy
per bit and N
0
the noise density at each branch of the receiver.
The integral J
v
is dened as
J
v
=
∞
0
z
v− 1
exp (−z/β)
β
v
(v − 1)!
Q
√
2z
dz (32)
where Q(·) denotes the Gaussian Qfunction. For v integer this
integral is expressed in closedform as [9, pp. 149150]
J
v
=
1
2
1 − μ
v− 1
k=0
2k
k
1 − μ
2
4
k
(33)
where
μ =
β
1+β
. (34)
Notice that similar results can be obtained for the performance
of MRC over correlated SR fading according to the analysis of
section II.B. Additionally, due to the equivalence of OSTBC
with MRC, similar expressions for the outage probability,
ergodic capacity and BEP also hold for OSTBC over SR fading
[7].
IV. P
ERFORMANCE EVALUATION
In this section, extensive Monte Carlo computer simulated
results are used to verify the presented theoretical analysis
under various fading conditions. Two different fading schemes
are considered for maximal ratio combiners employing up
to three receive antennae. In Scheme 1 the values of the
channel parameters are selected from the sets (Ω
1
, Ω
2
, Ω
3
)=
(0.278, 0.27, 0.3), (m
1
,m
2
,m
3
)=(5.21, 5.2, 5.25) and
1
As shown in [15], in case that Gray coding is employed, the BEP of M 
QAM coincides with the BEP of
√
M PAM.
0 5 10 15 20
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
SNR [dB]
Outage Probability
Scheme 1, n=2, Simulated
Scheme 1, n=2, Theoretical
Scheme 1, n=3, Simulated
Scheme 1, n=3, Theoretical
Scenario 2, n=2, Simulated
Scheme 2, n=2, Theoretical
Scheme 2, n=3, Simulated
Scheme 2, n=3, Theoretical
Fig. 1. Outage Probability as a function of the average transmitted SNR for
the two fading schemes (z
0
=1).
(b
1
,b
2
,b
3
)=(0.251, 0.251, 0.251), as in [11]. In Scheme
2, the values of the channel parameters are taken from
the sets (Ω
1
, Ω
2
, Ω
3
)=(0.2, 0.3, 0.4), (m
1
,m
2
,m
3
)=
(5.21, 3.2, 1.5) and (b
1
,b
2
,b
3
)=(0.1, 0.2, 0.3). Note that the
parameters of Scheme 2 have a larger spread and also b
i
= b
j
for i = j. For both fading schemes, n =2and n =3
receive antennae are examined with the parameters of the ith
fading channel selected as {Ω
i
,b
i
,m
i
}. For all the theoretical
results concerning Scheme 1, the parameter β was taken equal
to 0.5 and a number of 11 terms have been retained from
the respective series expansions. For Scheme 2 a parameter
β equal to 0.39 ensures a relatively fast convergence of the
series, which have been truncated at the 15th term.
In Fig. 1 the outage probability for the two schemes and
z
0
=1is plotted as a function of the transmit SNR z
t
, along
with Monte Carlo simulation results. It can be seen that for
both schemes the theoretical and simulations results almost
coincide thus verifying the validity of the derived expressions.
Additionally, in Fig. 2, the ergodic capacity versus the transmit
SNR for both fading schemes is plotted as calculated using
(28) along with simulation results. Again, the close match of
theoretical and simulations results can be veried. Finally, the
agreement between theoretical and simulations results can also
be observed in Fig. 3 where the BEP of BPSK for the two
fading schemes is plotted.
V. C
ONCLUSION
By applying an effective method to manipulate the Laplace
transform of the PDF, new expressions for the PDF and CDF
of the sum of squared SR RVs have been derived. Without
setting any restriction to the parameters of the model, the
proposed expressions are much simpler compared to already
known ones. Based on these results, mathematically tractable
closedform expressions for the outage probability, the ergodic
capacity and the BEP of MRC over SR fading have been
presented. The validity of our theoretical analysis has been
veried through extensive Monte Carlo Simulations.
0 5 10 15 20
10
0
10
1
SNR [dB]
Ergodic Capacity (bits/sec/Hz)
Scheme 1, n=2, Simulated
Scheme 1, n=2, Theoretical
Scheme 1, n=3, Simulated
Scheme 1, n=3, Theoretical
Scheme 2, n=2, Simulated
Scheme 2, n=2, Theoretical
Scheme 2, n=3, Simulated
Scheme 2, n=3, Theoretical
Fig. 2. Ergodic capacity as a function of the average transmitted SNR for
the two different fading schemes.
0 5 10 15 20
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
E
b
/N
0
[dB]
BEP
Scheme 1, n=2, Simulated
Scheme 1, n=2, Theoretical
Scheme 1, n=3, Simulated
Scheme 1, n=3, Theoretical
Scheme 2, n=2, Simulated
Scheme 2, n=2, Theoretical
Scheme 2, n=3, Simulated
Scheme 2, n=3, Theoretical
Fig. 3. BEP for BPSK for the two fading schemes.
APPENDIX
The uniform convergence of the series in (19) can be
proven by properly bounding the series truncation error. More
specically, since c
i
’s are the coefcients of the power series
expansion of L(θ) the following holds
c
i
=
L
(i)
(0)
i!
(35)
where L
(i)
(·) is the ith derivative of L (·). By employing
Cauchy’s inequality the absolute value c
i
can be bounded as
[16]
c
i
≤
L
0
(u)
u
i
(36)
where u is any positive value satisfying u<
1/ max{max
i
{γ
i
}, max
i
{δi}} and
L
0
(u)=max
θ=u
L (θ). (37)
By employing (36) a bound on the absolute value of the
truncation error ε(z) of (19) can be obtained as
ε (z) ≤ A
L
0
(u) u
n−1
exp
−
z
β
β
∞
i=N+1
(z/βu)
n+i−1
(n + i − 1)!
.
(38)
After some algebraic manipulations, (38) is rewritten as
ε (z) ≤A
L
0
(u) u
n−1
exp
−
z
β
β
×
exp
z
βu
−
n+N−1
i=0
(z/βu)
i
i!
.
(39)
By inspecting (39), it can be seen that uniform convergence of
p
z
(z), ∀r ≥ 0 is achieved as long as u>1 , or equivalently
as long as 0 <β<4min
i
{b
i
}.
R
EFERENCES
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Citations
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TL;DR: This paper derives approximate closedform expressions of the probability density function and cumulative distribution function of the received signaltonoise ratio of the MRC based receiver in SR fading LMS channels.
Abstract: In this paper, the maximal ratio combining (MRC) scheme in ShadowedRician (SR) fading land mobile satellite (LMS) channels is studied. The MRC scheme for SR fading LMS channels has been studied in existing literature; however, most of the existing analytical results are in the form of infinite power series, which are not in closedform. In this paper, we derive approximate closedform expressions of the probability density function and cumulative distribution function of the received signaltonoise ratio of the MRC based receiver in SR fading LMS channels. Then we provide approximate closedform expressions of the bit error rate (BER), outage probability, and capacity of the considered scheme. One of the derived closedform BER expressions is found useful for obtaining the analytical diversity order and coding gain of the considered MRC scheme.
113 citations
••
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TL;DR: Closeform expressions of the momentgenerating function (MGF) of the received signaltonoise ratio (SNR), bit error rate (BER), analytical diversity order, and capacity for the considered detector in correlated SR fading LMS links are derived.
Abstract: The problem of channel estimation and detection in shadowed Rician (SR) fading land mobile satellite (LMS) channels is studied. It is assumed that the satellite contains a single antenna and that the earth station has multiple antennas. In the proposed detector, joint channel estimation and detection of symbols is performed. It is shown by simulation and analysis that joint detection performs better than the existing decoupled detection. In this paper, we derive closedform expressions of the momentgenerating function (MGF) of the received signaltonoise ratio (SNR), bit error rate (BER), analytical diversity order, and capacity for the considered detector in correlated SR fading LMS links. Next, we consider independent and identically distributed (i.i.d.) SR fading LMS channels and derive approximate expressions of MGF, symbol error rate (SER), diversity order, and capacity of the satellite system with estimated channel gains for existing decoupled and proposed joint detectors. It is shown from the analysis and simulation that the channel estimation error does not affect the diversity order but reduces the capacity of the considered system.
53 citations
••
TL;DR: It is demonstrated in this paper that, by employing an additional transmit antenna at the satellite along with Alamouti code, the error rate of the LMS communication system can be significantly improved.
Abstract: In this paper, we study the transmission of orthogonal space–time block codes (OSTBCs) over a shadowed Rician (SR) land mobile satellite (LMS) link. Specifically, we derive the symbol error rate (SER), diversity order, and the average capacity of the OSTBCtransmissionbased scheme over the SR fading LMS communication systems. First, we consider the independent and identically distributed (i.i.d.) SR fading channels and derive the expressions of momentgenerating function (mgf) in terms of the MeijerG function. By substituting the MeijerG function in terms of hypergeometric function and then by considering the asymptotic approximation of hypergeometric function, the analytical diversity order of the scheme is obtained. The ergodic capacity of the scheme is derived by using the mgfbased approach. The effect of elevation angles over the SER performance of the scheme is analyzed, and it is observed that the performance of the system is improved with increasing values of elevation angles. Next, we consider the OSTBC transmission over correlated SR fading channels and derive the expressions of mgf, SER, diversity order, and average capacity of the considered setup. In particular, it is demonstrated in this paper that, by employing an additional transmit antenna at the satellite along with Alamouti code, the error rate of the LMS communication system can be significantly improved. It is also shown that the correlation has no impact on the diversity order of the system, but it adversely affects the capacity of the system.
47 citations
••
TL;DR: This study presents a comprehensive performance analysis of an energy detector over Gammashadowed Rician fading channels with the fluctuating lineofsight components following the Gamma distribution, shown to provide a remarkably accurate fading characterisation while leading to closedform expressions for important channel statistics.
Abstract: This study presents a comprehensive performance analysis of an energy detector over Gammashadowed Rician fading channels, namely Rician fading channels with the fluctuating lineofsight components following the Gamma distribution. This composite multipath/shadowing model has been shown to provide a remarkably accurate fading characterisation while leading to closedform expressions for important channel statistics. Rapidly convergent infinite series representations are firstly derived for the average probability of detection and the area under the receiver operating characteristic curve for the nodiversity reception case. These results are then extended to the case of maximal ratio, equal gain and selection diversity. To this end, novel analytical expressions for the statistics of the endtoend signaltonoise ratio of equal gain and selection diversity receivers, operating over Gammashadowed Rician fading channels are derived. Analytical results are substantiated by Monte Carlo simulation, as well as by extensive numerically evaluated results.
33 citations
••
TL;DR: It is shown from analysis and simulation that channel estimation error and correlation do not affect the diversity order but reduce the capacity of the considered system.
Abstract: The maximal ratio combining (MRC) scheme for ShadowedRician (SR) fading land mobile satellite (LMS) channels with estimated channel gains is studied, in this paper. It is assumed that satellite contains a single antenna and earth station (ES) has multiple antennas. For this setup, a maximumlikelihood decoder is derived under assumption that perfect channel state information (CSI) is known; then the perfect CSI is replaced with estimated CSI in decision metric at ES. The MRC scheme for SR fading LMS channels has been studied in existing literature mostly with perfect CSI assumption. We derive approximate closedform expressions of the probability density function, cumulative distribution, moment generating function (MGF), and analytical diversity order for the MRC based receiver in independent and identically distributed SR fading LMS channels with estimated channel gains. The author also provides approximate closedform expressions of the outage probability and capacity of the considered scheme with estimated channel gains. Next, the author considers correlated SR fading LMS links and derive approximate bit error rate of the satellite system with estimated channel gains. It is shown from analysis and simulation that channel estimation error and correlation do not affect the diversity order but reduce the capacity of the considered system.
21 citations
References
More filters
•
01 Jan 1970
17,608 citations
•
01 Jan 2004
TL;DR: The book gives many numerical illustrations expressed in large collections of system performance curves, allowing the researchers or system designers to perform tradeoff studies of the average bit error rate and symbol error rate.
Abstract: noncoherent communication systems, as well as a large variety of fading channel models typical of communication links often found in the real world, including single and multichannel reception with a large variety of types. The book gives many numerical illustrations expressed in large collections of system performance curves, allowing the researchers or system designers to perform tradeoff studies of the average bit error rate and symbol error rate. This book is a very good reference book for researchers and communication engineers and may also be a source for supplementary material of a graduate course on communication or signal processing. Nowadays, many new books attach a CDROM for more supplementary material. With the many numerical examples in this book, it appears that an attached CDROM would be ideal for this book. It would be even better to present the computer program in order to be interactive so that the readers can plug in their arbitrary parameters for the performance evaluation. —H. Hsu
6,469 citations
••
05 Nov 2004
2,299 citations
"Performance analysis of maximal rat..." refers background or methods in this paper
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[...]
...Yet another fading model that has attracted increasing interest lately is the socalled shadowedRice (SR) model [9]....
[...]
••
TL;DR: In this paper, the Shannon capacity of adaptive transmission techniques in conjunction with diversitycombining was studied. And the authors obtained closedform solutions for the Rayleigh fading channel capacity under three adaptive policies: optimal power and rate adaptation, constant power with optimal rate adaptation and channel inversion with fixed rate.
Abstract: We study the Shannon capacity of adaptive transmission techniques in conjunction with diversitycombining. This capacity provides an upper bound on spectral efficiency using these techniques. We obtain closedform solutions for the Rayleigh fading channel capacity under three adaptive policies: optimal power and rate adaptation, constant power with optimal rate adaptation, and channel inversion with fixed rate. Optimal power and rate adaptation yields a small increase in capacity over just rate adaptation, and this increase diminishes as the average received carriertonoise ratio (CNR) or the number of diversity branches increases. Channel inversion suffers the largest capacity penalty relative to the optimal technique, however, the penalty diminishes with increased diversity. Although diversity yields large capacity gains for all the techniques, the gain is most pronounced with channel inversion. For example, the capacity using channel inversion with twobranch diversity exceeds that of a singlebranch system using optimal rate and power adaptation. Since channel inversion is the least complex scheme to implement, there is a tradeoff between complexity and capacity for the various adaptation methods and diversitycombining techniques.
1,036 citations
"Performance analysis of maximal rat..." refers background in this paper
...Since v is integer, Iv takes the following closed form [14] Iv = (v − 1)! exp (1/β) v ∑...
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••
TL;DR: This work provides an exact and general closedform expression of the BER for onedimensional and twodimensional amplitude modulations, i.e., PAM and QAM, under an additive white Gaussian noise (AWGN) channel when Gray code bit mapping is employed.
Abstract: Quadrature amplitude modulation (QAM) is an attractive technique to achieve high rate transmission without increasing the bandwidth. A great deal of attention has been devoted to the study of bit error rate (BER) performance of QAM, and approximate expressions for the bit error probability of QAM have been developed in many places in the literature. However, the exact and general BER expression of QAM with an arbitrary constellation size has not been derived yet. We provide an exact and general closedform expression of the BER for onedimensional and twodimensional amplitude modulations, i.e., PAM and QAM, under an additive white Gaussian noise (AWGN) channel when Gray code bit mapping is employed. The provided BER expressions offer a convenient way to evaluate the performance of PAM and QAM systems for various cases of practical interest. Moreover, simple approximations can be found from our expressions, which are the same as the wellknown approximations, if only the dominant terms are considered.
1,007 citations
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