504
IEEE TRANSACTIONS ON BROADCASTING, VOL. 51, NO. 4, DECEMBER 2005
The Symmetrical Distribution: A General
Fading Distribution
Michel Daoud Yacoub, Gustavo Fraidenraich, Student Member, IEEE, Hermano Barros Tercius, and
Fábio Cézar Martins
Abstract—This paper specializes and parameterizes the general
result presented elsewhere in the literature in order to introduce,
Distribufully characterize, and investigate the Symmetrical
tion, a general distribution used to describe the statistical variation
of the envelope in a fast fading environment. It proposes estimators
for the involved parameters and uses field measurements to validate the distribution. In spite of its specialization, the Symmetrical
Distribution still includes, as special cases, important distributions such as Rayleigh, Rice, Hoyt, Nakagami-q, and One-Sided
Distribution has
Gaussian. The fact that the Symmetrical
one more parameter than the well-known distributions renders it
more flexible. Of course, in situations in which those distributions
included in it give good results a better fitting is given by the Symmetrical
Distribution. In addition, in many other situations
in which these distributions give poor results a good fitting may be
found through the Symmetrical
Distribution. More specifically, its nonmonomodal feature finds applications in several circumstances, examples of which are given in this paper.
Index Terms—Fast fading distributions, Hoyt distribution, onesided Gaussian distribution, Rayleigh distribution, rice distribution, short-term fading.
I. INTRODUCTION
T
HE propagation of energy in a mobile radio environment
is characterized by incident waves interacting with surface
irregularities via diffraction, scattering, reflection, and absorption. The interaction of the wave with the physical structures
generates a continuous distribution of partial waves [2], with
these waves showing amplitudes and phases varying according
to the physical properties of the surface. The propagated signal
then reaches the receiver through multiple paths, and the result
is a combined signal that fades rapidly, characterizing the short
term fading. For surfaces assumed to be of the Gaussian random
rough type, universal statistical laws can be derived in a parameterized form [2]. A great number of distributions exist that
well describe the statistics of the mobile radio signal. The long
term signal variation is well characterized by the Lognormal
distribution whereas the short term signal variation is described
by several other distributions such as Rayleigh, Rice, Hoyt,
Nakagami-q, Nakagami-m, and Weibull. Among the short term
distributions, the Nakagami-m distribution has been given a
special attention for its ease of manipulation and wide range of
Manuscript received November 24, 2003; revised March 10, 2005. This paper
was presented in part at the Personal Indoor and Mobile Radio Communications–PIMRC 2004, Barcelona, Spain, Sep. 2004 [1].
The authors are with the Department of Communications of the School
of Electrical and Computer Engineering, University of Campinas, CP 6101,
13083-970 Campinas, SP, Brazil.
Digital Object Identifier 10.1109/TBC.2005.859240
applicability [3]. Although, in general, it has been found that
the fading statistics of the mobile radio channel may well be
characterized by Nakagami-m, situations are easily found for
which other distributions such as Rice and Weibull yield better
results [4], [5]. More importantly, situations are encountered
for which no distributions seem to adequately fit experimental
data, though one or another may yield a moderate fitting.
Some researches [5] even question the use of the Nakagami-m
distribution because its tail does not seem to yield a good
fitting to experimental data, better fitting being found around
the mean or median. The well-known fading distributions have
been derived assuming a homogeneous diffuse scattering field,
resulting from randomly distributed point scatterers. With such
an assumption, the central limit theorem leads to complex
Gaussian processes. The assumption of a homogeneous diffuse scattering field is certainly an approximation because the
surfaces are spatially correlated characterizing a nonhomogeneous environment [2]. Finding general fading distributions
is indeed an old problem that still attracts the attention of the
communications researchers [6]–[13]. In [8], a general result
is presented in which the in-phase and quadrature components
of the fading envelope are dependent Gaussian variables with
different nonzero means and unequal variances. The classical
Rayleigh, Rice, Hoyt, Nakagami-q, and One-Sided Gaussian
density functions are special cases of this general result.
This paper specializes and parameterizes the general result
presented in [8] in order to introduce, fully characterize, and inDistribution. It proposes estimavestigate the Symmetrical
tors for the involved parameters and uses field measurements to
validate the distribution. In spite of its specialization, the SymDistribution still includes, as special cases, immetrical
portant distributions such as Rayleigh, Rice, Hoyt, Nakagami-q,
and One-Sided Gaussian. The fact that the Symmetrical
Distribution has one more parameter than the well-known distributions renders it more flexible. Of course, in situations in
which those distributions included in it give good results a better
Distribution. In addifitting is given by the Symmetrical
tion, in many other situations in which these distributions give
poor results a good fitting may be found through the SymmetDistribution. More specifically, its nonmonomodal
rical
feature finds applications in several circumstances, examples of
which are given in this paper.
This paper is structured as follows. Section II presents the
Symmetrical
distribution, Section III presents the derivation of the distribution, Section IV presents the estimators for
the parameters and , Section V presents the relations between the Symmetrical
and the other fading distributions,
Section VI presents the application of the
distribution,
0018-9316/$20.00 © 2005 IEEE
YACOUB et al. SYMMETRICAL
DISTRIBUTION: A GENERAL FADING DISTRIBUTION
Section VII presents some numerical examples and some plots
using field measurements and finally Section VIII presents the
conclusions.
505
with
2
DISTRIBUTION
II. THE SYMMETRICAL
(4)
where
0 is the ratio between the total power of the dominant components and the total power of the scattered waves,
0 is the ratio between the powers of the in-phase term
and
and quadrature term. In particular, the ratio of the power between the in-phase dominant component and the in-phase scattered wave is assumed to be identical to the corresponding ratio
of the quadrature term, consequently, the ratio of the scattered
energy in the in-phase and in-quadrature waves is identical to
the corresponding ratio of the dominant component. As can be
inferred from its derivation, which is shown in Section III, such
a condition leads the resulting distribution to be symmetrical
1. Thus, it suffices to explore it within one of the
around
1 or 1
. As can be seen from its defranges 0
inition, the parameter generalizes the concept of the Ricean
parameter for which the in-phase and quadrature powers of the
1 . In the same
scattered waves are assumed to be identical
way, the parameter defined here extends that of Hoyt, for it
takes into account a relation between the powers of the in-phase
and quadrature dominant components. The normalized envelope
probability density function is obtained as
2
1
1
where the
2
1
2
2
(6)
and
2
1
(7)
(8)
1
The phase distribution is obtained here in a closed-form
manner, as shown in (9) at the bottom of the page, where erf
is the Gaussian error function. The
moment
of
P can be attained in the usual integral manner or in a series
expansion given by
1
2
1
1
2
2
1
2
2
2
1
2
1
(5)
function, as defined here, is given by
1
1
(3)
4
In his classical paper [6], Nakagami departs from a very general fading model and carries out a series of simplifications, considered to be ‘‘sufficiently good enough for engineering problems [6], in order to arrive at the well-known Nakagami-m distribution. In the very general model, i.e. without laying hold of
the mentioned simplifications, the signal intensity at any observing point is assumed to be composed of a sum of independent random phasors, subject that the in-phase and quadrature components of the sum are normal, i.e., their terms satisfy the conditions of the Central Limit Theorem. These components, therefore, are dependent Gaussian variables with different
nonzero means and unequal variances. By choosing the in-phase
and quadrature axes parallel to the axes of the equiprobability
ellipses (i.e., by performing a convenient change of reference
phase), the covariance between the Gaussian terms vanishes.
The distribution can then be written in terms of independent
Gaussian variables with different nonzero means and unequal
variances. The derivation of the distribution in its very general
form is presented in [8], where it is shown in terms of the means
and variances of the Gaussian components.
It seems that very little attention has been given to this distribution, maybe because of its rather intricate form of presentation, or for lack of estimators, or, in general, for lack of full characterization. In a work from which the present paper is extracted
[14], this general fading distribution is parameterized in terms
of the envelope rms value and three power ratios: 1) in-phase
dominant component and in-phase scattered wave; 2) quadrature dominant component and quadrature scattered wave; and
3) in-phase term and quadrature term. By taking some specific,
but still wide-ranging conditions, simpler forms of the general
case can be achieved. In particular, for the power ratios of 1) and
Distribution is
2) assumed to be identical the Symmetrical
attained. For either one the ratios of 1) or 2) assumed to be nil,
Distribution [15] is accomplished. The
the Asymmetrical
concept of symmetry shall be clarified in the text. This paper
Distribution.
explores the Symmetrical
The Symmetrical
Distribution is a general fading distribution that can be used to represent the small-scale variation
of the fading signal. For a fading signal with envelope , phase
, and normalized envelope
, in which
is the rms value of , the Symmetrical
joint probability
density function
is written as
1
(2)
4
2
(1)
1
2
2
2
2
(10)
2
1
erf
(9)
2
2
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 51, NO. 4, DECEMBER 2005
where
is the Gauss hypergeometric function,
is the Gamma function. Of course,
. In the
same way, the cumulative probability function can be obtained
in the usual integral manner or in series expansion given by
2
2
1
where
function. The
pansion form as
2
2
1
2
2
1
1
2
1
1
2
2
(11)
is the incomplete Gamma
function can be written in a series ex-
1
2
Fig. 1.
0).
The Symmetrical 0 probability density function for a fixed ( =
Fig. 2.
0.5).
The Symmetrical 0 probability density function for a fixed ( =
(12)
where
is the modified Bessel function of -th order. Another expansion for it is shown in (13) at the bottom of the page,
is the discrete delta function and
. The
where
function presents some interesting properties related
to the Bessel functions. In particular,
0
and
0
, which is evident from (6) as well as from
(12). As for the calculations using the
function, it has
been observed that its integral form, as given in (6), is computationally more efficient and yields more accurate results than its
series forms (12) and (13).
Fig. 1 shows the various shapes of the Symmetrical
probability density function
for the in-phase scattered
wave and in-phase dominant component equal to zero, i.e.
0 and varying . Note that for
0 the Symmetrical
tends to the one-sided Gaussian distribution, as expected. With
the increase of , meaning that the deterministic component of
the signal is increasing, the density moves toward a bell-shaped
1 with
.
curve and tends to an impulse at
Fig. 2 shows the various shapes of the Symmetrical
probability density function
for the quadrature scattered
wave with twice the energy of the in-phase scattered wave and
the quadrature dominant component with twice the energy of
0.5 , and varying
the in-phase dominant component, i.e.
. In this case, for
0 the density coincides with that of Hoyt
(or Nakagami-q). Here, as before, as
an impulse tends
to occur at
1.
Fig. 3 shows the various shapes of the Symmetrical
probability density function
for the in-phase dominant
component equal to the in-phase scattered component and the
quadrature dominant component equal to the quadrature scattered wave
1 and varying . In this case, with
1 the
density coincides with the Rice.
DISTRIBUTION
III. DERIVATION OF THE
Distribution conThe fading model for the Symmetrical
siders a signal composed of multipath waves propagating in an
nonhomogeneous environment. The powers of the in-phase and
quadrature scattered waves are assumed to be arbitrary. In the
same way, the powers of the in-phase and quadrature dominant
components are also assumed to be arbitrary. These waves are
considered to be identically affected so that: (1) the ratio of the
scattered energy (and of the dominant energy) of the in-phase
and in-quadrature waves is constant (and equal to ), and (2) the
2
2
2
2
2
4
(13)
YACOUB et al. SYMMETRICAL
DISTRIBUTION: A GENERAL FADING DISTRIBUTION
507
and
(18)
For the symmetrical case, as explored here,
(19)
and accordingly
(20)
With the above definitions in (14)
Fig. 3.
1) .
The Symmetrical 0 probability density function for a fixed ( =
2
(21)
ratio of the dominant and the scattered energy for both in-phase
and in-quadrature waves is constant (and equal to ).
Let and be independent Gaussian wide sense stationary
processes of the in-phase and quadrature components of the
,
propagated wave, respectively. Assume that
, Var
, and Var
, where E (.)
and Var (.) are the mean and variance operators, respectively.
of and is given by
The joint distribution
where
(22)
Using the definitions given in (19) and (20) then (22) can be
1
1
. By carrying out
written as
the appropriate substitutions and after some algebraic manipufollows as in (1).
lations the density
and
given in (5) and (9), respectively,
The densities
follow directly by integration.
(14)
2
The envelope can be written in terms of the in-phase and
quadrature components of the fading signal as
IV. ESTIMATORS FOR THE PARAMETERS
of
, and
and
AND
Estimators for the parameters and can be obtained in
terms of the moments of the envelope. In particular, it can be
shown in (23) and (24) at the bottom ofthe page. By conveniently manipulating (23) and (24), the following are attained:
(15)
with
,
Then the joint density
is given in (14). From (15)
.
is given by
(16)
where
is the Jacobian of the transformation. Given
the physical model of the distribution, the following ratios are
defined
(25)
(17)
(26)
6
(23)
15
(24)
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 51, NO. 4, DECEMBER 2005
Equations (25) and (26) are the basis for estimating and .
Note that (25) yields four possible solutions for , but only two
will be nonnegative and real. Using one of these two possible
values for , (26) can be solved to obtain . Therefore, two posand
, are attained. Such
sible pairs of solutions,
an ambiguity can be resolved by means of the use of another moment (e.g., the first moment). In such a case: 1) estimate the first
in (10); 2) estimate the first
moment (E[P]) with the pair
in (10); 3) Estimate the first
moment ([P]) with the pair
moment of the measured data; 4) Compare the results of 1) with
3) and 2) with 3); 4) Choose the pair whose corresponding first
moment is closest to the one of the measured data. Of course, if
distribution in an exact
the data follows the Symmetrical
manner, then the smallest difference is nil.
Given a set of measured data for the fading envelope, the practical procedure in order to determinate the distributions parame,
, and
;
ters and is as follows: 1) Estimate
and P
in
2) Using P
(25) and (26) and are obtained. According to the definition
of these parameters as given previously, is the ratio between
the total power of the dominant components and the total power
of the scattered waves, and is the ratio between the powers of
the in-phase term and quadrature term. One important point to
raise is that, because the estimators of the Symmetrical
require the computation of higher order statistics, namely P
and P 1, a large quantity of data is required for an appropriate
convergence. In case a sufficient amount of data is not available,
it may be adequate to use one of the alternative fitting method
as described in Section VI.
V. THE SYMMETRICAL
DISTRIBUTION AND THE OTHER
FADING DISTRIBUTIONS
the Nakagami-m parameter the higher order statistics required is E[P
0
]
AND THE
Using such a definition for the Symmetrical
it can be shown that
1
21
1
1
OTHER
Distribution,
(27)
2
From (27) it can be seen that, apart from the case
0.5,
0 0 is the only possible solution, an infor which
curves can be found for
finite number of Symmetrical
pair may be found
the same parameter. An appropriate
that leads to the best Nagakami-m approximation. Interestingly,
it can be observed that the minimum of (27) is obtained for
0 0 , for which
0.5. In this case, the SymmetDistribution specializes into the One-Sided Gaussian,
rical
as does Nakagami- . Table I summarizes these relations for the
distribution deteriorate into some other
cases in which the
fading distributions in a exact manner. (The probability density
0 or
is
function for the limiting value of , i.e.
given in a simple closed formula in the Appendix.)
VI. APPLICATION OF THE
Distribution is a general fading distriThe Symmetrical
bution that includes the Rayleigh, Rice, Hoyt, Nakagami-q, and
One-Sided Gaussian distributions as special cases. It may also
approximate the Nakagami-m distribution.
The Hoyt distribution can be obtained from the Symmetrical
Distribution in an exact manner by setting
0 and
using the relation
, where is the Hoyt parameter. From the Hoyt distribution the One-Sided Gaussian is
1(
0 or
). In the same way, from
obtained for
the Hoyt distribution the Rayleigh distribution is obtained in an
0
1 . The Nakagami-q distribution
exact manner for
Distribution in an exact
can be obtained from Symmetrical
0 and using
, where is the respecmanner by setting
tive parameter. From the Nakagami-q the One-Sided Gaussian
0 or
(
0 or
). In the
can be obtained for
same way, from the Nakagami-q the Rayleigh distribution can
1
1 . The Rice Distribution
be obtained by setting
Distribution in an
can be obtained from the Symmetrical
exact manner by setting
1 and using
, where k is the
Rice parameter. From the Rice distribution the Rayleigh can be
0
0.
obtained for
The Nakagami parameter can be written in terms of and
by recognizing that such a parameter is the inverse of the norVar
.
malized variance of the squared envelope, i.e.
1For
TABLE I
THE SYMMETRICAL DISTRIBUTION
FADING DISTRIBUTIONS
DISTRIBUTION
The application of the Symmetrical
Distribution implies
the estimation of its parameters and (see Section IV). On
the other hand, it may be possible to use the Symmetrical
Distribution by estimating the parameter
and choosing the
pair satisfying (27) that leads to the best fit.
appropriate
In particular, given an , from (27) it can be seen that
1
1
(28)
2
or, equivalently,
1
2
1
(29)
0 and real. Therefore, for any
with the physical constraint
given (e.g., obtained from sample data) the parameter can
be arbitrarily chosen from (28) (or, equivalently, from (29)) and
the parameter estimated from (27) as
1
2
1
1
2
1
2 1
1
2
2
(30)
with the usual physical constraint
0 and real. Fig. 4 shows
the possible values of and for any given . Fig. 5 and Fig. 6,
respectively, depict a sample of the various shapes of the Symprobability density function
and probability
metrical
YACOUB et al. SYMMETRICAL
DISTRIBUTION: A GENERAL FADING DISTRIBUTION
Fig. 4. Possible values of and for any given m.
0
Fig. 5. The Symmetrical probability density function for the same
Nakagami parameter m (m = 1.25). The pair (; ) assumes the following
values [(0.0001, 3.435), (0.001, 3.426), (0.01, 3.336), (0.03, 3.148), (0.06,
2.892), (0.1, 2.592), (0.5, 1.123), (1, 0.809)].
distribution function
as a function of the normalized en1.25. It can
velope for the same Nakagami parameter
be seen that, although the normalized variance (parameter )
is kept constant for each figure, the curves are substantially different from each other. And this is particularly noticeable for the
distribution function, in which case the lower tail of the distribution may yield differences in the probability of some orders
curves
of magnitude. Note, in Fig. 6, that the Symmetrical
are always above the Nakagami curve.
VII. VALIDATION THROUGH FIELD MEASUREMENTS
A series of field trials was conducted at the University
of Campinas (Unicamp), Brazil, in order to investigate the
short term statistics of the fading signal. The reception setup
509
0
Fig. 6. The Symmetrical probability distribution function for the same
Nakagami parameter m (m = 1.25). The pair (; ) assumes the following
values [(0, 3.436), (0.0001, 3.435), (0.001, 3.426), (0.002, 3.416), (0.005,
3.386), (0.007, 3.366), (0.1, 2.592), (0.2, 2.011), (0.4, 1.321), (0.99,0.809)].
comprised a vertically polarized omnidirectional antenna, a
low noise amplifier, a spectrum analyzer, a data acquisition
apparatus, a notebook computer, and a distance transducer
[16]. The transmission consisted of a CW tone at 1.8 GHz
with transmitter and receiver placed within buildings (indoor
propagation). The spectrum analyzer was set to zero span and
centered at the desired frequency, and its video output used
as the input of the data acquisition and processing equipment.
The local mean was estimated through the moving average
method, the average being conveniently taken over samples
symmetrically adjacent to every point, a procedure widely
reported in the literature [17].
Through our measurements, it has been observed that the
Distribution finds its applications for the
Symmetrical
1 .
cases in which the Rice Distribution is also applicable
In the same way, it has found its applicability in those (less fre1 . On
quent) cases in which Hoyt is also applicable 0
the other hand, because of its versatility it yields excellent fitting
for the cases in which the fitting provided by these respective
distributions is only moderate. Fig. 7 shows some sample plots
illustrating the adjustment obtained by the Symmetrical
Distribution as compared to the Rice one. Note, in Fig. 7, that the
Rice distribution provides good fitting in both cases, although
Distribution gives a better adjustment.
the Symmetrical
We also show its fitting to experimental data other than those
obtained by the authors. We then turn our attention to [17], in
which a propagation measurement experiment at 10 GHz conducted in an indoor environment is reported. In order to adjust
Distribution to the data of [17], these
the Symmetrical
data were carefully extracted from Fig. 4(a) and Fig. 4(c) of
Distribution
[17]. Then the parameters of the Symmetrical
were adjusted to yield the best fitting following the procedure
described in Section VI. Fig. 8 compares the fitting of Symmetnormalized power
distributions to the
rical
empirical data reported [17]. For the upper curve, we note that
the Rice fitting is not adequate at the tail of the distribution but
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 51, NO. 4, DECEMBER 2005
TABLE II
MEAN ERROR DEVIATION
as opposed to the Rice one, the Symmetrical
Distribution
tends to reproduce or very closely follow the trends (concavity
and/or convexity) of the true curve.
VIII. CONCLUSIONS
0
Fig. 7. The Symmetrical distribution function adjusted to data of an
indoor propagation measurement experiment at 1.8 GHz conducted at Unicamp.
This paper has presented the Symmetrical
Distribution,
a general fading distribution that includes Rayleigh, Rice, Hoyt,
Nakagami-q, and One-Sided Gaussian as special cases. Simulation and field trials have been conducted in order to better
characterize this distribution. It has been found that SymmetDistribution finds its applications for the cases in
rical
which Hoyt as well as Rice distributions are also applicable.
Because of its versatility it yields excellent fitting for the cases
in which the fitting provided by these respective distributions
is only moderate. A numerical measure of the mean error deviation between the true curve and the distributions chosen to
fit the experimental data has been calculated. In all of them, the
distribution is lower than that for
error for the Symmetrical
the Rice distribution, although, the Rice distribution also gives
good results. It has been observed that the Symmetrical
Distribution tends to closely follow the shapes of the true distribution which, sometimes, does not comply with the monomodal
behavior of the well-known distributions.
APPENDIX
0
Fig. 8. The Symmetrical distribution function adjusted to data of an
indoor propagation measurement experiment at 10 GHz, as reported in [17].
The probability density function of P in the limitingcase
is given by
1
it yields a good fitting for higher values of the envelope. In condistribution provides a very good
trast, the Symmetrical
fitting for the lower values of the envelope and tends to keep
track of the change of concavity of the true curve. For the lower
curve the fitting provided by the Rice distribution is also very
good, but it does not follow the exact shape of the true curve as
Distribution.
does the Symmetrical
A numerical measure of the mean error deviation2 between
the true curve and the distributions chosen to fit the experimental
) has been calculated
data (namely, Rice and Symmetrical
for all of the cases. Table II shows these for the curves in Figs. 7
disand 8. In all of them, the error for the Symmetrical
tribution is lower than that for the Rice distribution, although,
the Rice distribution also gives good results. On the other hand,
2The mean error deviation between the measured data x and the theoretical
(y
value y (Rice or Symmetrical ) is defined as = (1=N )
x =x ), where N is the number of points. Other metrics have been tested, e.g.
presented a better
rms deviation, and in all of them the Symmetrical performance.
j
j 0
0
0
2
(31)
Its corresponding distribution is obtained as
1
2
1
erf
21
erf
1
21
(32)
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DISTRIBUTION: A GENERAL FADING DISTRIBUTION
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[17] A. F. Abouraddy and S. M. Elnoubi, “Statistical modeling of the indoor
radio channel at 10 GHz through propagation measurements—Part I:
Narrow-Band measurements and modeling,” IEEE Trans. Veh. Technol.,
vol. 49, no. 5, pp. 1491–1507, Sep. 2000.
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Michel Daoud Yacoub was born in Brazil in 1955.
He received the B.S.E.E. and the M.Sc. degrees from
the School of Electrical and Computer Engineering of
the State University of Campinas, UNICAMP, Brazil,
in 1978 and 1983, respectively, and the Ph.D. degree
from the University of Essex, U.K., in 1988. From
1978 to 1985, he worked as a Research Specialist at
the Research and Development Center of Telebrás,
Brazil, in the development of the Tropico digital exchange family. He joined the School of Electrical and
Computer Engineering, UNICAMP, in 1989, where
he is presently a Full Professor. He consults for several operating companies and
industries in the wireless communications area. He is the author of Foundations
of Mobile Radio Engineering (Boca Raton, FL: CRC, 1993), Wireless Technology: Protocols, Standards, and Techniques (Boca Raton, FL: CRC, 2001),
and the co-author of Telecommunications: Principles and Trends (São Paulo,
Brasil: Erica, 1997, in Portuguese). He holds two patents. His general research
interests include wireless communications.
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Gustavo Fraidenraich received the Electrical
Engineering degree from the Federal University of
Pernambuco, UFPE, Brazil, in 1997. He received his
M.Sc. degree from the State University of Campinas,
UNICAMP, Brazil, in 2002. He is currently pursuing
the Ph.D. degree and his research interests include
fading channels, diversity-combining systems and
Wireless Communications in general. Mr. Fraidenraich is also a Student Member of the IEEE.
Hermano Barros Tercius was born in Petrolina-PE,
Brazil, in 1979. He received the B.S. degree in
Electrical Engineering from the State University of
Campinas (UNICAMP), Brazil, in 2000. In 2001,
he worked as a Radio Frequency Engineer. During
this term, he worked in Lisbon, Portugal, designing
the Portugal Telecom’s third generation cellular
network, which was based on the WCDMA technology. He is currently pursuing his M.Sc. Degree
at the Wireless Technology Laboratory (Wisstek),
UNICAMP. Besides that, since November 2003
he has been working as a Telecommunication Engineer, designing mobile
communications systems and microwave radio links. His research interests
include field measurements and fast fading distribution of mobile signals.
Fábio Cézar Martins received the Electronic
Engineering degree from the National Institute
of Telecommunications (INATEL), Santa Rita do
Sapucaí, Brazil, in 1990. He received his M.Sc.
degree from the Aeronautical Institute of Technology/Aerospace Technical Center (ITA/CTA),
SJC, Sao Paulo, Brazil, 1996. He is currently pursuing the Ph.D. degree at the Wireless Technology
Laboratory (Wisstek), UNICAMP. His research
interests include field measurements, and Wireless
Communications in general. Actually he is Professor
at the State University of Londrina, (UEL), PR, Brazil.