On the approximation of the generalized-Κ distribution by a gamma distribution for modeling composite fading channels

Abstract

In wireless channels, multipath fading and shadowing occur simultaneously leading to the phenomenon referred to as composite fading. The use of the Nakagami probability density function (PDF) to model multipath fading and the Gamma PDF to model shadowing has led to the generalized-K model for composite fading. However, further derivations using the generalized K PDF are quite involved due to the computational and analytical difficulties associated with the arising special functions. In this paper, the approximation of the generalized-K PDF by a Gamma PDF using the moment matching method is explored. Subsequently, an adjustable form of the expressions obtained by matching the first two positive moments, to overcome the arising numerical and/or analytical limitations of higher order moment matching, is proposed. The optimal values of the adjustment factor for different integer and non-integer values of the multipath fading and shadowing parameters are given. Moreover, the approach introduced in this paper can be used to well-approximate the distribution of the sum of independent generalized-K random variables by a gamma distribution; the need for such results arises in various emerging distributed communication technologies and systems such as coordinated multipoint transmission and reception schemes including distributed antenna systems and cooperative relay networks.

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706IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010
On the Approximation of the Generalized-K
Distribution by a Gamma Distribution for
Modeling Composite Fading Channels
Saad Al-Ahmadi, Member, IEEE, and Halim Yanikomeroglu, Member, IEEE
Abstract—In wireless channels, multipath fading and shadow-
ing occur simultaneously leading to the phenomenon referred to
as composite fading. The use of the Nakagami probability density
function (PDF) to model multipath fading and the Gamma
PDF to model shadowing has led to the generalized-퐾 model
for composite fading. However, further derivations using the
generalized-퐾 PDF are quite involved due to the computational
and analytical difficulties associated with the arising special
functions. In this paper, the approximation of the generalized-
퐾 PDF by a Gamma PDF using the moment matching method
is explored. Subsequently, an adjustable form of the expressions
obtained by matching the first two positive moments, to overcome
the arising numerical and/or analytical limitations of higher
order moment matching, is proposed. The optimal values of the
adjustment factor for different integer and non-integer values
of the multipath fading and shadowing parameters are given.
Moreover, the approach introduced in this paper can be used
to well-approximate the distribution of the sum of independent
generalized-퐾 random variables by a Gamma distribution; the
need for such results arises in various emerging distributed
communication technologies and systems such as coordinated
multipoint transmission and reception schemes including dis-
tributed antenna systems and cooperative relay networks.
Index
generalized-퐾 distribution, moment matching, positive and
negative moments, lower and upper tails, network MIMO,
distributed antenna systems, radar and sonar.
Terms—Composite fading,Gammadistribution,
I. INTRODUCTION
M
lems including interference analysis in cellular systems and
performance analysis of network MIMO, distributed antenna
systems, and cooperative relay networks. The small-scale
multipath fading is often modeled using Rayleigh, Rician, or
Nakagami distribution. The latter one is versatile enough to
encompass the Rayleigh distribution as a special case and to
approximate the Rician distribution. The large-scale fading
(shadowing) is often modeled using a lognormal distribu-
tion (refer to [1] and the references therein). However, the
lognormal-based composite fading models [2, 3] do not lead
to closed-form expressions for the received signal power dis-
tribution which hampers further analytical derivations. As an
ODELING composite fading channels is essential for
the analysis of several wireless communication prob-
Manuscript received September 20, 2008; revised March 19, 2009 and May
28, 2009; accepted July 27, 2009. The associate editor coordinating the review
of this paper and approving it for publication was S. Gassemzadeh.
The authors are with the Department of Systems and Computer
Engineering, Carleton University, Ottawa, Canada (e-mail: {saahmadi,
halim}@sce.carleton.ca).
S. Al-Ahmadi’s work was supported by King Fahd University of Petroleum
and Minerals, Saudi Arabia.
Digital Object Identifier 10.1109/TWC.2010.02.081266
alternative, it has been proposed to use the Gamma distribution
to model large-scale fading where it has been observed that the
Gamma distribution fits the experimental data, and it closely
approximates the lognormal distribution [4-7].
The use of the Gamma distribution to model shadowing and
the Nakagami distribution to model the small-scale random
variations of the received signal envelope, has led to a closed-
form expression of the composite fading probability density
function (PDF) known as the Gamma-Gamma (generalized-
퐾) PDF. The Gamma-Gamma model was introduced to model
scattering in radar [8] and reverberation in sonar [9] and has
recently generated interest in wireless communications as well
[10-13]. However, further derivations using that model have
shown to be analytically difficult or computationally involved
due to the arising special functions.
In this paper, the approximation of the generalized-퐾
distribution by a Gamma distribution through matching both
positive and negative moments is explored. The obtained
results have shown that matching the higher order moments
leads to a good approximation, up to a certain level of accu-
racy, in both the upper and lower tail regions, and may lead
to lower and upper bounds on the approximated cumulative
distribution function (CDF). However, such a matching has
two main limitations: (i) it results in involved expressions that
are difficult to handle and complicated to draw insights from;
(ii) negative moments may not exist for small values of the
multipath fading and shadowing parameters. Subsequently, an
adjusted form of the expressions obtained by matching the first
two positive moments is introduced to closely approximate
the generalized-퐾 composite fading PDF by the simple and
tractable Gamma PDF. This region-wise approximation yields
sufficient accuracy for a broad range of integer and non-integer
values of the multipath fading and shadowing parameters.
Moreover, the introduced method can be used to approximate
the PDF of the sum of independent generalized-퐾 random
variables (RVs) in the lower and upper tail regions. Finally,
since the approximating Gamma model allows the use of the
closed-form expressions developed in literature for Nakagami
fading channels [14-15], some performance analysis applica-
tions, out of many, are stated.
II. THE COMPOSITE FADING MODEL AND RELATED
WORK
When the random variation of the envelope of the received
signal, due to small-scale multipath fading, is modeled by the
Nakagami distribution [16], the PDF of the received power 훾,
1536-1276/10$25.00 c ⃝ 2010 IEEE
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AL-AHMADI and YANIKOMEROGLU: ON THE APPROXIMATION OF THE GENERALIZED-K DISTRIBUTION BY A GAMMA DISTRIBUTION ...
707
conditioned on the average local power Ω, takes the form of
a Gamma distribution:
푝훾/Ω(푥) =
1
Γ(푚푚)Ω
(푚푚
)푚푚푥푚푚−1exp
(
−푚푚푥
Ω
)
, 푥 > 0,푚푚 ≥ 0.5,
(1)
where Γ(⋅) is the Gamma function and 푚푚is the Nakagami
multipath fading parameter.
The variation of the average local power, due to shadowing,
is usually modeled by the lognormal distribution [3]. However,
the analytically better tractable Gamma distribution has also
shown a good fit to data obtained through propagation mea-
surements [5, 6], besides it can approximate the lognormal
distribution for the relevant range of shadowing severity in
wireless channels [5, 7]:
푝Ω(푦) =
1
Γ(푚푠)
(푚푠
Ω0
)푚푠
푦푚푠−1exp
(
−푚푠
Ω0푦
)
, 푦 > 0,푚푠> 0.
(2)
In (2), 푚푠is the shadowing parameter and Ω0is the mean of
the received local power. Similar to the multipath parameter
푚푚, the severity of shadowing is inversely proportional to
푚푠 so that small values of 푚푠 indicate severe shadowing
conditions. In [5, 7], using the moment matching method
between the Gamma PDF in (2) and the lognormal PDF, it
was shown that 푚푠 =
standard deviation in the lognormal shadowing model.
Using (1) and (2), the PDF of 훾 can be derived as [8-10]1
2
Γ(푚푠)Γ(푚푚)푏푚푠+푚푚푥(푚푠+푚푚
× 퐾푚푠−푚푚(2푏√푥), 푥 > 0,푚푚≥ 0.5,푚푠> 0,
1
푒(휎푠/8.686)2−1where 휎푠 denotes the
푝훾(푥) =
2
)−1
(3)
where 퐾푚푠−푚푚(⋅) is the modified Bessel function of the
second kind and order (푚푠− 푚푚) and 푏 =
PDF in (3) appeared first in [8] where the instantaneous power
is assumed to follow a Gamma distribution whose mean is
also assumed to have a Gamma distribution. The PDF in
(3) is known as the generalized-퐾 model2and the McDaniel
model in wireless and sonar literature, respectively ([10] and
references therein).
The CDF of 훾, was derived in [13] as
[(푏2훾)푚푚1퐹2(푚푚;1 − 훼,1 + 푚푚;푏2훾)
−(푏2훾)푚푠1퐹2(푚푠;1 + 훼,1 + 푚푠;푏2훾)
Γ(푚푚)Γ(1 + 훼)Γ(푚푠+ 1)
√
푚푚푚푠
Ω0
. The
푃(훾) = 휋csc(휋훼)
Γ(푚푠)Γ(1 − 훼)Γ(푚푚+ 1)
]
,
(4)
where 훼 = 푚푠− 푚푚 and푝퐹푞 is the generalized hypergeo-
metric function for integer 푝 and 푞 [20].
However, as pointed out in [9], the computation of such a
CDF expression which contains the hyper-geometric function
1In fact, the distribution of the product of M independent Gamma RVs,
where the generalized-퐾 PDF corresponds to the case where M=2, was
derived in [17]. Moreover, the generalized-퐾 distribution belongs to the (Fox)
퐻-function distributions family [18].
2It should be highlighted here that in literature, the notion “generalized-퐾"
was used to denote another similar distribution [19].
term is not straightforward due to the associated numerical
instabilities that will require the use of approximations and
asymptotic expansions or the numerical inversion of the char-
acteristic function. Moreover, further derivations using the
characteristic function approach, such as the PDF of the sum
of 푁 generalized-퐾 RVs, are quite involved even for the
independent and identically distributed (i.i.d.) case due to the
difficulties associated with the Whittaker function [21].
III. APPROXIMATION USING THE MOMENT MATCHING
METHOD
An alternative approach, to avoid the analytical difficulties,
is to consider approximatingthe PDF in (3) by a more tractable
PDF using the moment matching method. We propose using
the Gamma distribution due to the following reasons: (i)
Gamma distribution is a Type-III Pearson distribution which
is widely used in fitting distributions for positive RVs by
matching the first and second moments [18], and (ii) since
the PDF in (3) corresponds to the product of two Gamma
RVs, one of the corresponding Gamma PDFs will dominate
for large values of 푚푚or 푚푠[7].
The 푛푡ℎmoment of the generalized-K distribution can be
derived as [13]
퐸[훾푛] = 휇푛=Γ(푚푚+ 푛)Γ(푚푠+ 푛)
Γ(푚푚)Γ(푚푠)
(
Ω0
푚푚푚푠
)푛
,
(5)
where 퐸[⋅] denotes the statistical expectation.
Denoting 푍 ∼ 풢(푘,휃) as a Gamma distributed RV with a
shape parameter 푘 and a scale parameter 휃, the PDF of 푍 is
given as [22]
푝푍(푥) =휃−푘
Γ(푘)푥푘−1exp(−푥/휃), 푥 > 0.
(6)
Furthermore, the 푛푡ℎmoment of the Gamma distribution
can be expressed as [22]
퐸[푧푛] = 휇(풢)
푛
=Γ(푘 + 푛)휃푛
Γ(푘)
.
(7)
Now, using the expressions in (5) and (7), the first, second,
and third moments of the generalized-퐾 distribution and the
approximating Gamma distribution can be matched as
푘휃=
=
Ω0,
퐾1Ω2
퐾2퐾1Ω3
(8)
(9)
(10)
휃2푘(푘 + 1)
0,
휃3푘(푘 + 1)(푘 + 2)=
0.
On the other hand, the negative moments, as defined in
[23], of the generalized-퐾 PDF and the Gamma PDF can be
expressed using the expressions in (5) and (7) as
휃(푘 − 1) = 퐾−1Ω0, 푘 > 1,푚푚> 1,푚푠> 1,
(11)
휃2(푘 − 2)(푘 − 1) = 퐾−1퐾−2Ω2
0, 푘 > 2,푚푚> 2,푚푠> 2,
(12)
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708IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010
where
퐾1
=(푚푚+1)(푚푠+1)
푚푚푚푠
=
푚푚푚푠
=(푚푚−1)(푚푠−1)
푚푚푚푠
=(푚푚−2)(푚푠−2)
푚푚푚푠
Matching different pairs of moments will result in the scale
and shape parameters for the approximating Gamma PDF as
shown in Table I. In Table I, 휃푖,푗and 푘푖,푗denote the scale and
shape parameters of the approximating Gamma PDF obtained
by matching the 푖푡ℎand the 푗푡ℎmoments, respectively.
Now, examining the expressions of the approximating
Gamma PDF parameters given in Table I, the following may
be stated:
∙ The scale parameter of the approximating Gamma PDF
obtained by matching the positive moments is larger than
the one obtained by matching the negative moments.
For example, it can be easily seen that 휃1,2 = 휃1,−1+
2
푚푚푚푠Ω0. Since the negative moments characterize a
distribution at the origin [23] (the lower tail for a positive
RV) and the positive moments characterize a distribution
at the upper tail, we may conclude that the generalized-
K PDF (CDF) can be approximated by a Gamma dis-
tribution whose scale and shape parameters depend on
the region of the PDF (CDF) of interest. Such a region-
wise (piece-wise) approximation was used in [24] to
well-approximate the sum of lognormal RVs by a single
lognormal RV.
∙ Matching moments for 푛≥2 will lead to involved expres-
sions as seen in Table I. Moreover, not including the first
positive moment in the moments matched results in an
approximating Gamma PDF that does not have the same
mean as the approximated generalized-퐾 PDF (i.e., the
generalized-퐾 PDF and the approximating Gamma PDF
have different average power values).
∙ Matching negative moments may not be possible for
small values of 푚푚and/or 푚푠as indicated in (11), (12)
and subsequent expressions in Table I.
∙ The scale and shape parameters of the approximating
Gamma distribution are dependent on the fading pa-
rameters in the sense that as 푚푚 and/or 푚푠 increase,
the difference between the predicted scale parameters
decreases and hence the difference between the approxi-
mating PDFs (CDFs) becomes small. So, for small values
of 푚푚 and/or 푚푠 (while 푚푚,푚푠 > 2), the difference
between the two approximating Gamma CDFs might be
large enough to bound the approximated CDF in the
lower tail region as seen in Fig. 1. On the other hand,
matching the lower order moments for large values of
푚푚and/or 푚푠does not result in a good approximation
as seen in Figs. 2 and 3 since the approximating CDFs
are too close to each other.
Note: In Figs. 1-3, the complementary cumulative distribution
function (CCDF), and particularly the region corresponding to
푃(푋 ≥ 푥) ≤ 0.1, is shown for the upper tail region to obtain
more illustrative results.
= 1 +
1
푚푚+
2
푚푚+
1
푚푚−
2
푚푚−
1
푚푠+
2
푚푠+
1
푚푠+
2
푚푠+
1
푚푚푚푠, (13a)
4
푚푚푚푠, (13b)
1
푚푚푚푠, (13c)
4
푚푚푚푠. (13d)
퐾2
(푚푚+2)(푚푠+2)
= 1 +
퐾−1
퐾−2
= 1 −
= 1 −
−3−2.5−2−1.5−1−0.500.5
−6
−4
−2
0
The lower tail of the CDFs
log(x)
log(P(X<x))


0.40.50.60.70.80.91 1.1
−4
−3
−2
−1
0
The upper tail of the CDFs
log(x)
log(P(X>x))


Exact
μ1, μ−1
μ1, μ2
μ1, μ3
Exact
μ1, μ−2
μ1, μ−1
μ1, μ2
Fig. 1.
Gamma RVs for 푚푚 = 2.5 and 푚푠 = 2.5 using the moment matching
method.
The log-log CDF plots of the generalized-퐾 and the approximating
−2−1.5−1−0.500.5
−6
−4
−2
0
The lower tail of the CDFs
log(x)
log(P(X<x))


0.30.40.50.60.70.80.9
−4
−3
−2
−1
0
The upper tail of the CDFs
log(x)
log(P(X>x))


Exact
μ1, μ−2
μ1, μ−1
μ1, μ2
Exact
μ1, μ−1
μ1, μ2
μ1, μ3
Fig. 2.
Gamma RVs for 푚푚= 7 and 푚푠= 4 using the moment matching method.
The log-log CDF plots of the generalized-퐾 and the approximating
IV. THE MOMENT MATCHING METHOD WITH
ADJUSTMENT
In order to bypass the limitations explained before on the
use of the moment matching for higher order moments, we
may consider an adjustable form for the scale and shape
parameters of the approximating Gamma PDF obtained by
matching only the first two positive moments since (i) these
expressions, as given in Table I, are simple and valid for all
values of 푚푚 and 푚푠, and (ii) the first positive moment is
included in the matching.
First, we may re-write the scale and shape parameters using
Table I as
[
=[AF] Ω0, 0 ≤ AF ≤ AF푚푎푥,
푘1,2
=
AF, 0 ≤ AF ≤ AF푚푎푥.
In the above, the term
푚푚+
of fading (AF) in the composite fading channel as derived
휃1,2
=
1
푚푚
+
1
푚푠
+
1
푚푚푚푠
]
Ω0
(14a)
1
(14b)
11
푚푠+
1
푚푚푚푠is the amount
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AL-AHMADI and YANIKOMEROGLU: ON THE APPROXIMATION OF THE GENERALIZED-K DISTRIBUTION BY A GAMMA DISTRIBUTION ...
709
TABLE I
EXPRESSIONS OF THE SCALE AND THE SHAPE PARAMETERS OF THE APPROXIMATING GAMMA PDF OBTAINED BY MOMENT MATCHING (FOR 퐾1, 퐾2,
퐾−1, 퐾−2, REFER TO (13A)-(13D))
Matched moments
휇1, 휇2
Scale parameter
휃1,2= (퐾1− 1)Ω0, 휃1,2> 0
(
4
Shape parameter
푘1,2=
퐾1−1, 푘1,2> 0
4
−3+√
−퐾2
(퐾2
푘1,−1=
푘1,−2=
푘−1,−2=
1
휇1, 휇3
휃1,3=
−3+√
9+8(퐾1퐾2−1)
)
Ω0
, 휃1,3> 0푘1,3=
9+8(퐾1퐾2−1),푘1,3> 0
)
2
퐾1−1
1
1−퐾−1, 푘1,−1> 1
4
3−√9+8(퐾−1퐾−2−1), 푘1,−2> 2
퐾−1Ω0
휃−1,−2+ 1, 푘−1,−2> 2
휇2, 휇3
휃2,3=
√
퐾1
(
푘2
2,3+푘2,3
)Ω0, 휃2,3> 0푘2,3=
(
2
퐾1+4+
√(퐾2
2
퐾1
)2
+8퐾2
2
퐾1
2
)
, 푘2,3> 0
휇1, 휇−1
휇1, 휇−2
휇−1, 휇−2
휃1,−1= (1 − 퐾−1)Ω0, 휃1,−1> 0
휃1,−2=(3−√9+8(퐾−1퐾−2−1))Ω0
(
4
)
, 휃1,−2> 0
휃−1,−2=
1
푚푚+
1
푚푠−
3
푚푚푚푠
Ω0, 푚푚 > 2,푚푠> 2,휃−1,−2> 0
−1−0.8−0.6−0.4−0.200.2
−6
−4
−2
0
The lower tail of the CDFs
log(x)
log(P(X<x))


Exact
μ1, μ−2
μ1, μ−1
μ1, μ2
0.25 0.30.350.40.45
log(x)
0.50.550.60.650.7
−4
−3
−2
−1
0
The upper tail of the CDFs
log(P(X>x))


Exact
μ1, μ−1
μ1, μ2
μ1, μ3
Fig. 3.
Gamma RVs for 푚푚 = 10 and 푚푠 = 10 using the moment matching
method.
The log-log CDF plots of the generalized-퐾 and the approximating
in [12]. The value of AF푚푎푥 is determined by the smallest
physical values of 푚푚 and 푚푠 which are non-zero in real
propagation channels; hence AF푚푎푥is finite.
The expressions of the scale and shape parameters given by
(14a) and (14b) result in poor approximation in the lower and
upper tail regions since matching only the first and second
moments will result in a good fit only around the mean.
To overcome this limitation, we may consider the following
adjustable form for the expressions in (14a) and (14b):
휃′
1,2= [AF − 휖]Ω0, 0 ≤ AF ≤ AF푚푎푥,휖0≤ 휖 ≤ AF, (15a)
푘′
1,2=
AF − 휖, 0 ≤ AF ≤ AF푚푎푥,휖0≤ 휖 ≤ AF. (15b)
Since the AF “added" to the scale parameter of the approx-
imating Gamma PDF should not exceed the original amount
of fading of the approximated PDF (i.e., 휖0 ≥ −AF), 휖 is
bounded as −AF ≤ 휖 ≤ AF. Due to the fact that the relevant
practical range of AF is from zero (for non-fading channels)
to 8 (for severe multipath fading and shadowing conditions
where 푚푚= 0.5 and 푚푠= 0.5)3, the relevant range of the
adjustment factor 휖 becomes −8 ≤ 휖 ≤ 8.
1
3Such small values of 푚푚 and 푚푠may take place in land mobile satellite
channels [26].
The adjustment factor can be computed using a numerical
measure of the difference between the approximated and
the approximating PDFs (CDFs). A common measure is the
absolute value of the difference between the approximated and
the approximating PDFs (CDFs) [22, 25] that is similar to the
well-known Kolmogorov distance between the CDFs of two
continuous distributions. For this purpose, the CDFs rather
than the PDFs are considered since the Gamma PDF goes to
infinity as 푥 → 0 for 푘 < 1 [22] which causes numerical
instabilities for comparison in the lower tail region.
The plots of the optimal adjustment factor, 휖표푝, versus the
multipath fading and shadowing parameters are shown in Figs.
4 and 5 for values of 푚푚 and 푚푠 ranging from 0.5 to 10.
The plots show that the adjustment factor decreases as either
or both 푚푚and 푚푠increase. The decrease of the adjustment
factor as both 푚푚 and 푚푠 increase is worth noting since it
indicates that the PDF of the product of two Gamma RVs
can be well-approximated, for the main body of the PDF, by
a Gamma PDF by matching the first two positive moments.
This is due to the fact that both PDFs approach the same
limiting PDF (the Dirac delta PDF) as fading diminishes. To
see that, the AF for equal values of the multipath fading and
shadowing parameters can be expressed as AF =2푚+1
푚 = 푚푚= 푚푠; clearly the amount of fading is approximately
2/푚 for moderate values of 푚 and converges to zero for very
large values of 푚. However, if a high degree of accuracy is
sought in the lower tail region, then the magnitude of the
adjustment factor increases as seen in Fig. 5. Similar plots can
be obtained for any region of interest and the corresponding
adjustment factor can be tabulated.
푚2 , where
V. ON THE APPROXIMATION OF THE PDF OF THE SUM OF
INDEPENDENT GENERALIZED-K RVS
The moment matching method can be used to approximate
the PDF (CDF) of the sum of 푁 generalized-퐾 RVs by a
Gamma PDF (CDF). However, matching the higher order mo-
ments is difficult since deriving or computing these moments
is involved or unfeasible [23, 24]. This motivates again the
use of an adjustable form for the scale and shape parameters
obtained by matching the first two positive moments.
We may start with N=2 where the first and second moments
of the sum of two independent RVs, 푧 = 푥 + 푦, can be
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710IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010
0
2
4
6
8
10
0
2
4
6
8
10
0
0.5
1
1.5
2
2.5
ms
mm
The optimal adjustment factor, εop
Fig. 4.
value of the difference between the approximated generalized-퐾 and the
approximating Gamma distributions over the whole CDF.
The plot of the adjustment factor that minimizes the absolute
0
2
4
6
8
10
0
2
4
6
8
10
0
1
2
3
4
5
6
ms
mm
The optimal adjustment factor, εop
Fig. 5.
of the difference between the generalized-퐾 and the approximating Gamma
distributions in the lower tail of the CDF (< 0.1).
The plot of the adjustment factor that minimizes the absolute value
expressed as [22]
퐸[푧] = 퐸[푥] + 퐸[푦],
(16a)
and
퐸[푧2]= 퐸[푥2]+ 퐸[푦2]+ 2퐸[푥]퐸[푦].
Matching the first and second moments of the sum of two in-
dependent generalized-퐾 RVs and the approximating Gamma
distribution results in
휃푠푢푚 =퐾1,푥Ω2
(Ω0,푥+ Ω0,푦)
=AF푥Ω2
(Ω0,푥+ Ω0,푦)
(16b)
0,푥+ 퐾1,푦Ω2
0,푦+ 2(Ω0,푥Ω0,푦) − (Ω0,푥+ Ω0,푦)2
0,푥+ AF푦Ω2
0,푦
, 휃푠푢푚 > 0,
(17a)
and
푘푠푢푚=
(Ω0,푥+ Ω0,푦)2
AF푥Ω2
0,푥+ AF푦Ω2
0,푦
, 푘푠푢푚> 0,
(17b)
where 퐾1,푥 and 퐾1,푦 denote the 퐾1 parameters (as defined
in (13-a)), Ω0,푥 and Ω0,푦 denote the values of the mean of
−4−3−2−1
log(x)
012
−6
−5
−4
−3
−2
−1
0
log(CDF)


Exact
Approx, ε=0.2
N=1
N=2
N=3
N=6
Fig. 6.
approximating Gamma RV for 푚푚 = 2, 푚푠 = 4 (휎푠 = 4.2 dB), 휖 = 0.2,
and different values of 푁.
The log-log CDF plots for the sum of generalized-퐾 RVs and the
the local power, and AF푥 and AF푦 denote the AF (as given
in (14a)) of the generalized-퐾 RVs 푥 and 푦, respectively.
The adjusted forms of (17a) and (17b) can be written as
휃′
푠푢푚=[AF푥− 휖푥]Ω2
0,푥+ [AF푦− 휖푦]Ω2
(Ω0,푥+ Ω0,푦)
0,푦
, 휃′
푠푢푚> 0,
(18a)
and
푘′
푠푢푚=
(Ω0,푥+ Ω0,푦)2
0,푥+ [AF푦− 휖푦]Ω2
[AF푥− 휖푥]Ω2
0,푦
, 푘′
푠푢푚> 0.
(18b)
In general, the expressions in (18a) and (18b) can be
generalized for the sum of 푁 independent generalized-퐾 RVs
as
∑푁
and
(∑푁
휃′
푠푢푚=
푖=1[AF푖− 휖푖]Ω2
∑푁
0,푖
푖=1Ω0,푖
, 휃′
푠푢푚> 0,
(19a)
푘′
푠푢푚=
푖=1Ω0,푖
)2
∑푁
푖=1[AF푖− 휖푖]Ω2
0,푖
, 푘′
푠푢푚> 0.
(19b)
For the i.i.d. case, the expressions in (19a) and (19b) simplify
to
휃′
푠푢푚= (AF − 휖)Ω0, 휃′
푠푢푚> 0,
(20a)
and
푘′
푠푢푚=
푁
AF − 휖, 푘′
푠푢푚> 0.
(20b)
Similar formulation can be carried out for the sum of corre-
lated generalized-퐾 RVs.
Three-dimensional plots of the adjustment factor versus the
composite fading parameters 푚푚and 푚푠can be produced for
different values of 푁. As an example, the plots of the lower
tail of the CDFs for 푚푚=2, 푚푠=4, and 푁= 1, 2, 3, and 6 are
given in Fig. 6 showing that an adjustment factor of 휖 = 0.2
results in almost identical CDFs, in the lower tail region, for
푁 = 6. Clearly, larger values of 휖 are needed for a more
accurate approximation for 푁=1, 2, and 3.
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