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Improving the Performance of Fading Channel
Simulators Using New Parameterization Method
Omar Alzoubi and Mohieldin Wainakh
Department of Communication Engineering, High Institute of Applied Sciences and Technology, Damascus, Syria
Email: omar_alzoubi@hotmail.com, wainakh@scs-net.org
Abstract—The modeling and simulation of mobile fading
channels can be done efficiently by using the finite sum of
weighted sinusoids with equally distributed phases known as
the concept of Rice method. In this paper, we evaluate the
statistical properties of Weibull-lognormal fading channel
model, such as Level Crossing Rate (LCR), Average
Duration of Fades (ADF), and Probability Density Function
(PDF). Several results are obtained by using different
methods to design the deterministic simulation model
parameters. New computation method of deterministic
simulation model parameters is presented. This procedure is
a combination of two methods, Method of Equal Areas and
Method of Exact Doppler Spread. It is called a combination
of MEA and MEDS. Comparisons between Autocorrelation
Functions (ACFs) of both reference and simulation models
are introduced for different methods. It is demonstrated by
several simulation results that statistical properties of
simulation and reference models will be much closer
according to the new method than MEDS and MEA. Finally,
the results indicate the superiority of the new method over
the MEDS and MEA with respect to LCR and ADF of
Weibull-lognormal fading channel model.
Index Terms—level crossing rate, average duration of fades,
autocorrelation functions
I. INTRODUCTION
The modeling of fading channels has a great
importance in the design, test and improvement the
performance of cellular radio communication systems.
The channel simulator must be efficient, flexible and
accurate. In addition, the statistical properties of the
channel simulator should be very close to the statistical
behavior of the desired reference model [1]. This will
depend on the design method of channel simulator.
Depending upon the radio propagation environment,
various multipath fading models are available in literature
[2]. Both of the advances on classical fading models and
a brief summary of some new fading models were
presented in [3].
Mobile radio channels are classified into two main
categories, namely frequency-nonselective and
frequency-selective channels. The first type is modeled
by using an appropriate stochastical models, such as
Rayleigh, Rice and Suzuki processes [4], whereas
frequency-selective channels can be modeled by using (n-
path) tap delay line model [5], which requires 2n
Manuscript received April 21, 2015; revised January 31, 2016.
coloured Gaussian processes. Therefore, computer
simulation models can be implemented by means of the
Rice method [6], which depends on approximation of the
coloured Gaussian processes by finite sum of weighted
sinusoids with phases uniformly distributed. Finding
proper design method for computing parameters of
simulation models provides deterministic processes at the
output of channel simulator with a statistical properties
closed to those of corresponding stochastic processes,
especially statistics of second order such as LCR and
ADF. Analytical expressions for these quantities have
been derived for Rayleigh and Rice [7], [8]. In this
research we are interested in modelling frequency-
nonselective channels, where Suzuki process [9] is
considered to be a more suitable model for non-selective
frequency cellular radio channels in many cases. Suzuki
process is obtained by multiplying Rayleigh and
Lognormal processes with each other. The reader can find
detailed information on the extended and modified
stochastic versions of Suzuki models in [9]. It is worth
mentioning that these stochastic models are not capable
of modeling non–uniform scattering. In order to combine
inhomogeneous diffuse scattering with shadow fading,
the Weibull-lognormal process appears as an appropriate
composite model [10]. The Weibull-lognormal model,
consists of three stochastic Gaussian processes, should be
designed by a proper method. There are many methods to
calculate parameters of simulation model (doppler
coefficients and discrete doppler frequencies), for
example the Method of Equal Areas (MEA) [1], [9],
which provides a satisfied approximation of the desired
statistics for Jakes Doppler power spectral density even
for a small number of sinusoids, but it fails and requires
large number of sinusoids for other types of Doppler
power spectral densities such as those with Gaussian
shapes [11]. Furthermore, it is found that MEA does not
result in a periodic ACF due to unequal distances
between discrete Doppler frequencies [12]. There is
another method called Method of Exact Doppler Spread
(MEDS), which compute parameters in such way the
Doppler spread is the same for both reference and
simulation models [13]. In this paper, new optimal design
method of simulation model parameters is presented. This
method is named a combination of MEA and MEDS.
Therefore, a brief review of reference models for
Rayleigh and Weibull-lognormal channel models and the
statistical properties for them is given in Sections II, III
respectively. After that in Section IV it is shown how the
International
Journal of Electronics and Electrical Engineering Vol. 4, No. 5, October 2016
©2016 Int. J. Electron. Electr. Eng. 443
doi: 10.18178/ijeee.4.5.443-448
deterministic simulation model for Weibull-lognormal is
obtained by using the concept of Rice’s sum of sinusoids.
For this purpose three computation methods of simulation
model parameters is presented in Section V. In Section
VI, the performance of the three methods is evaluated by
comparing statistical properties of reference and
simulation models over Weibull-lognormal channel
model, where it is observed that new method improved
statistical properties performance of simulation models
more than other methods, which will be reflected on the
performance of the simulator. Finally the conclusion is
given in Section VII.
II. RAYLEIGH CHANNEL MODEL
Rayleigh and Rice channels are the most important
channel models in mobile communications. Usually,
Rayleigh and Rice processes are preferred for modelling
fast-term fading, whereas slow-term fading is modelled
by a lognormal process. Rice process is proposed an
appropriate stochastic model for describing the envelope
of received signal in rural areas, where the Line of Sight
(LOS) is taken into consideration, whereas Rayleigh
process is considered suitable for urban regions, where
LOS is not exist [1], [9].
A Rayleigh process
()t
is obtained by taking the
absolute value of the zero-mean complex Gaussian
process
12
( ) ( ) ( )t t j t
, i.e.:
( ) ( )tt
(1)
where
()t
represents scattered component in the
received signal with uncorrelated real and imaginary parts,
and variances
2
0
var{ ( )} , 1,2
iti
.
A typical shape for the Doppler Power Spectral
Density (PSD) of the complex Gaussian processes is
given by the Jakes PSD [7]:
2
max
2
max max
max
2,
1 ( )
( ) 2 ( ) , 1,2
0
ii
off
f
f
S f S f i
f
ff
(2)
where
max
f
denotes the maximum Doppler frequency.
Taking the inverse Fourier transform of the Jakes PSD
results in the following ACF:
2
0 max
( ) 2 ( ) 2 2
ii o
r r J f
(3)
where
0.J
is the zeroth-order Bessel function of the
first kind. The PDF of Rayleigh process
()t
, which
describes the statistical signal variations, is given by the
Rayleigh distribution:
2
2
2
2,0
()
0, 0
o
x
o
xex
Px
x
(4)
LCR and ADF of Rayleigh Processes
()t
, which
belong to the statistical properties of the second degree,
are very important in assessing the performance of
channel simulators. LCR is defined as the rate (crossings
per second) at which the envelope
()t
crosses a given
level
r
in the positive (or negative) going direction. LCR
of Rayleigh process can be represented by [1], [9]:
( ) ( )
2
N r p r
(5)
It is obvious from (5) that LCR of Rayleigh process is
proportional to its PDF by the constant
which is the
reverse curve of ACF
(0), 1,2
ii
ri
. In the case of
isotropic scattering, where the ACF
()
ii
r
is given by
(3), the quantity
may be written as:
2
0 max
2( )f
(6)
On the other hand, ADF
()Tr
is the mean value of
the length of all time intervals over which the envelope
()t
remains below a given level
r
. In general, the ADF
is defined by [1], [9]:
()
() ()
Fr
Tr Nr
(7)
where
()Fr
the Cumulative Distribution Function
(CDF) of
()t
defined by
( ) [ ( ) ]
r
F r p t r
.
III. WEIBULL-LOGNORMAL CHANNEL MODEL
This model is used to simulate non–uniform scattering
with shadow fading, where the first one models the
possible scattering non–uniformities of the channel,
whereas the second accounts for the slow term variations
of the local mean due to shadowing. The Weibull-
lognormal process WL(t) is obtained by multiplying a
Weibull process W(t) by a lognormal one L(t). The
Weibull process results from a Rayleigh one R(t) as
2/
( ) ( )W t R t
and
a parameter expressing the fading
severity [10]. As we have seen in the Section II, the
Rayleigh process can be obtained by (1), whereas the
lognormal process is generated by a real valued Gaussian
process
3()t
with zero mean and unit variance as
3
( ) exp[ ( ) ]L t s t m
, where s and m determine the
type of shadowing environment. The stochastic Weibull-
lognormal process will be [10]:
1
22
1 2 3
( ) [ ( ) ( )] exp[ ( )]WL t a t t s t
(8)
where
2/
0
exp( )am
. The amplitude PDF
()
WL
P
of
WL(t) is given by [10]:
12
22
0
ln
( ) exp( ) exp( )
22
22
WL
x x z z
P x dz
as
sa
(9)
International
Journal of Electronics and Electrical Engineering Vol. 4, No. 5, October 2016
©2016 Int. J. Electron. Electr. Eng. 444
We note from (9) that the PDF of Weibull-lognormal
process depends on three parameters (
,,sa
), and then
this model shows high flexibility and includes Suzuki,
Weibull and Rayleigh models as special cases. The LCR
of Weibull-lognormal process was approximated in [10]
from the assumption of a slowly time varying lognormal
process compared to the Weibull one. So LCR is defined
by the equation [10]:
/2 22
max
/ 2 2 2
0
22ln
( ) exp( )exp( )
2
r
fx x z z
N x dz
s a a s
(10)
Now ADF easily can be found by using (10) with CDF
given by [10]:
2
22
0
1 1 ln
( ) 1 exp( )exp( )
22
2
r
x z z
F x dz
u a s
s
(11)
IV. DETERMINISTIC SIMULATION MODEL
An efficient simulator for Weibull-lognormal fading
channels is obtained by using the concept of Rice’s sum
of sinusoids [6]. According to that principle, we replace
the stochastic Gaussian processes
1()t
,
2()t
, and
3()t
of the reference model by the following
deterministic processes:
, , ,
1
( ) cos(2 ) 1,2,3
i
N
i i n i n i n
n
t c f t i
(12)
where
i
N
denotes the number of sinusoids. The
parameters
,in
c
,
,in
f
, and
,in
are called Doppler
coefficients, Doppler frequencies and Doppler phases
respectively. These parameters have to be computed
during the simulation setup phase, e.g., by one of the
methods described in the following sections. From the
fact that all parameters are known quantities, it follows
that
()
it
can be considered as a deterministic process.
In order the deterministic process
( ), 1,2
iti
, to be
uncorrelated, we define
21
1NN
[1], [9], thus we
obtain the three deterministic Gaussian processes
( ), 1,2,3
iti
. By analogy with (8), the received
envelope of Weibull-lognormal fading channels can be
modeled according to:
2 2 1/
1 2 3
( ) [ ( ) ( )] exp[ ( )]r t a t t s t
(13)
In general, the ACF of
1()t
and
2()t
are given by
[1], [9]:
2
,
,
1
( ) cos(2 ), 1,2
2
i
ii
Nin
in
n
c
r f i
(14)
V. COMPUTATION METHODS FOR THE MODEL
PARAMETERS
This section presents three different methods for the
determination of the Doppler coefficients
,in
c
and the
corresponding discrete Doppler frequencies
,in
f
. The
Doppler phases
,, (1,2,3)
in i
, are realizations of a
random variable uniformly distributed within the interval
(0,2 ]
[1], [9]. The procedures will be named by
Method of Equal Areas (MEA), Method of Exact Doppler
Spread (MEDS), and the new one is named by
Combination of MEDS and MEA method. Here, we will
not present in detail the simulation modeling employing
sum of sinusoids, but for the interested reader we refer to
[9] for detailed and well-presented analysis of the main
methods used in the sum of sinusoids simulation scheme.
A. Method of Equal Areas (MEA)
The Doppler coefficients
,in
c
have been designed in
terms of fulfilling the power constraint
22
00
. The
frequencies
,in
f
can be found by partitioning the Doppler
power spectral density of
()
it
into
i
N
sections of equal
power and using the upper frequency limits, related to
these areas. The Doppler coefficients
,in
c
and frequencies
,in
f
are computed by [1], [9], [12]:
,0
2
in i
cN
(15)
, max sin( )
2
in
i
n
ff N
(16)
respectively for
1,2,..., , 1,2
i
n N i
.
B. Method of Exact Doppler Spread (MEDS)
The MEDS is documented in [1], [9]. For the
computation of the gains
,in
c
, the same is valid as MEA.
Applying the MEDS results the frequencies
,in
f
in the
following relation:
, max
1
sin[ ( )]
22
in
i
f f n
N
(17)
respectively for
1,2,..., , 1,2
i
n N i
.
The discrete frequencies
3,n
f
of
3()t
are calculated
with a simple modification of MEDS method by means of
the relations [1], [9]:
3
3
3,
3
3
1
22
3, 3 3,
1
21 ( ) 0, 1,2,.., 1
2
n
c
N
N c n
n
f
nerf n N
N
f N f
(18)
As
3()t
has a Gaussian PSD
33
()Sf
defined as [1],
[9]:
33
2
2
1
( ) exp( )
2
2c
c
f
Sf
(19)
where
c
a parameter related to the 3dB-cut-off
frequency
c
f
according to
2 ln 2
cc
f
, whereas
c
f
is
International
Journal of Electronics and Electrical Engineering Vol. 4, No. 5, October 2016
©2016 Int. J. Electron. Electr. Eng. 445
much smaller than the maximum Doppler frequency
max
f
,
i.e.
maxc
ff
, so the frequency ratio
max
cc
f
Kf
is
much greater than one, i.e.
1
c
K
. In real worlds
channels
10
c
K
[1].
C. Combination of MEDS and MEA
The new method relies on the application of both
methods MEDS, MEA, but we must apply the method
MEDS on the first half number of sinusoids
2
i
N
.
Therefore,
1
,in
c
and
1
,in
f
are given as follows:
1
,0
2
in
i
cN
(20)
1
, max 1
1
sin[ ( )]
22
in
i
f f n
N
(21)
where
11,2,...., 2, 1,2
i
n N i
, then MEA method is
applied on the second half of sinusoids number:
2
,0
2
in
i
cN
(22)
2
, max 2
1
sin[ ( )]
22
in
i
f f n
N
(23)
where
2( 2) 1,( 2) 2,..., , 1,2
i i i
n N N N i
. Finally
the formulas for
,in
c
and
,in
f
according to combination of
MEDS and MEA are given by
12
, , ,
[ , ]
i n i n i n
c c c
and
12
, , ,
[ , ]
i n i n i n
f f f
respectively.
VI. COMPARISON OF STATISTICAL PROPERTIES
BETWEEN REFERENCE AND SIMULATION MODELS
The statistical properties of the reference model for
Weibull-lognormal fading channel are compared with the
corresponding simulation results. Assuming that
simulation model parameters have been found in one of
the previously described methods, In this case, the
parameters in (12) are known quantities and the ACF
()
ii
r
of the simulation model can be calculated for
1,2i
by means of (14), whereas the ACF
()
ii
r
of the
reference model is obtained from (3). Both of ACFs
()
ii
r
,
()
ii
r
are compared with
25, 1, 2
i
Ni
through Fig. 1, Fig. 2, and Fig. 3, where the computation
of the simulation model parameters was based on MEDS,
MEA, and combination of MEDS and MEA, respectively.
It is observed that fitting between ACFs is excellent at
least up to the
i
N th
zero-crossing of
()
ii
r
for all
methods, but the Fig. 3 shows the error has become less
between the reference and simulation models, especially
in the last three sinusoidal harmonics of the ACF in
comparison with Fig. 1 and Fig. 2. This means that ACFs
of both reference and simulation models are much closer
according to the new method than MEA and MEDS, and
of course this will affects the statistical properties later
when LCR and ADF are studied for Weibull-lognormal
model. From now the number of sinusoids was assumed
13
15NN
and
216N
for Weibull-lognormal
fading channel. The Fig. 4 shows the PDF of the received
envelope of Weibull-lognormal fading channel as a
function of
c
K
, with a parameter set defined as
0
=2.8,s 0.5, 1, 0.8, 91
max
m f Hz
. From Fig. 4
an excellent conformity between the reference and
simulation PDF’s is revealed, no matter what the value of
c
K
is. Because PDF of Weibull-lognormal process is
independent of time selectivity of the lognormal process.
In Fig. 5, Fig. 6, and Fig. 7, the normalized LCR of
deterministic Weibull-lognormal processes for all
introduced methods with
1
c
K
,
2
c
K
, and
3
c
K
is
shown, respectively. It can be observed clearly that LCR
of the simulation model is very close to that of the
reference model according to the combination of MEDS
and MEA in comparison with MEA and MEDs. The
corresponding graphs for the normalized ADF are
presented in Fig. 8, Fig. 9, and Fig. 10. In addition
increasing
c
K
improves the performance of LCR and
ADF of the simulation model. Finally, the results
documented in the Fig. 5-Fig. 10 indicate a superiority of
the combination of MEDS and MEA over the MEDS and
MEA with respect to LCR and ADF.
Figure 1. ACFs of reference and simulation models using MEA
Figure 2. ACFs of reference and simulation models using MEDS
International
Journal of Electronics and Electrical Engineering Vol. 4, No. 5, October 2016
©2016 Int. J. Electron. Electr. Eng. 446
Figure 3. ACFs of reference and simulation models using combination
of MEDS and MEA
Figure 4. The PDF of deterministic Weibull-lognormal processes with
variation of
c
K
Figure 5. The LCR of deterministic Weibull-lognormal processes
using MEDS, MEA, combination of MEDS and MEA with
1
c
K
Figure 6. The LCR of deterministic Weibull-lognormal processes
using MEDS, MEA, combination of MEDS and MEA with
2
c
K
Figure 7. The LCR of deterministic Weibull-lognormal processes
using MEDS, MEA, and combination of MEDS and MEA with
3
c
K
Figure 8. The ADF of deterministic Weibull-lognormal processes
using MEDS, MEA, and combination of MEDS and MEA with
1
c
K
Figure 9. The ADF of deterministic Weibull-lognormal processes
using MEDS, MEA, and combination of MEDS and MEA with
2
c
K
Figure 10. The ADF of deterministic Weibull-lognormal processes
using MEDS, MEA, and combination of MEDS and MEA with
3
c
K
International
Journal of Electronics and Electrical Engineering Vol. 4, No. 5, October 2016
©2016 Int. J. Electron. Electr. Eng. 447
VII. CONCLUSION
The concept of Rice’s sum of sinusoids is used to
design an efficient deterministic simulation models for
Weibull-Lognormal fading channels. A study of the
statistics of such types of simulation models was the topic
of the present paper, especially for the PDF, LCR, and
ADF. New computation method of deterministic
simulation model parameters is presented, this method is
called a combination of MEA and MEDS. The
performance of different parameter computation methods
is discussed and evaluated by comparing ACFs of both
reference and simulation models. It is observed that ACFs
of reference and simulation models are much closer
according to the new method than MEA and MEDS. In
addition, the new method gave us an excellent results
corresponding with PDF, LCR, and ADF of deterministic
simulation model. Therefore, the deterministic simulation
model, based on the combination of MEDS and MEA,
will be very close in its statistical properties to those of
the reference model. Finally, the improved performance
of the statistical properties of deterministic simulation
fading channel models lead to get a high accuracy and
efficiency fading channel simulator.
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[6] “Mathematical analysis of random noise,” Bell Syd.
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[12] M. Pätzold, U. Killat, and F. Laue, “A deterministic digital
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Omar Alzoubi was born on September 28,
1978, in Homs, Syria. He received his Master
degree in Communications Engineering in
2009 from Albaath University, Homs, Syria.
And he is currently a Ph.D. student in
Communication Engineering department in
High Institute of Applied Sciences and
Technology in Damascus, Syria.
Mohieldin Wainakh was born in January 27,
1948, in Konaitra, Syria. He received his PhD
degree in Cybernetics & Information Theory
in 1980 from Polytechnic Kiev-USSR.
Currently he is a head of Communication
Networks lab in High Institute of Applied
Sciences and Technology in Damascus, Syria.
His current research area includes digital
communication, statistics, and information
theory.
International
Journal of Electronics and Electrical Engineering Vol. 4, No. 5, October 2016
©2016 Int. J. Electron. Electr. Eng. 448
S. O. Rice,
