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706 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010

On the Approximation of the Generalized-KDistribution by a Gamma Distribution for

Modeling Composite Fading Channels

Saad Al-Ahmadi, Member, IEEE, and Halim Yanikomeroglu, Member, IEEE

AbstractIn wireless channels, multipath fading and shadow-ing occur simultaneously leading to the phenomenon referred toas composite fading. The use of the Nakagami probability densityfunction (PDF) to model multipath fading and the GammaPDF to model shadowing has led to the generalized- modelfor composite fading. However, further derivations using thegeneralized-PDF are quite involved due to the computationaland analytical difficulties associated with the arising specialfunctions. In this paper, the approximation of the generalized- PDF by a Gamma PDF using the moment matching methodis explored. Subsequently, an adjustable form of the expressions

obtained by matching the first two positive moments, to overcomethe arising numerical and/or analytical limitations of higherorder moment matching, is proposed. The optimal values of theadjustment factor for different integer and non-integer valuesof the multipath fading and shadowing parameters are given.Moreover, the approach introduced in this paper can be usedto well-approximate the distribution of the sum of independentgeneralized- random variables by a Gamma distribution; theneed for such results arises in various emerging distributedcommunication technologies and systems such as coordinatedmultipoint transmission and reception schemes including dis-tributed antenna systems and cooperative relay networks.

Index TermsComposite fading, Gamma distribution,generalized- distribution, moment matching, positive and

negative moments, lower and upper tails, network MIMO,distributed antenna systems, radar and sonar.

I. INTRODUCTION

MODELING composite fading channels is essential for

the analysis of several wireless communication prob-

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, orNakagami 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

Manuscript received September 20, 2008; revised March 19, 2009 and May28, 2009; accepted July 27, 2009. The associate editor coordinating the reviewof this paper and approving it for publication was S. Gassemzadeh.

The authors are with the Department of Systems and ComputerEngineering, Carleton University, Ottawa, Canada (e-mail: {saahmadi,halim}@sce.carleton.ca).

S. Al-Ahmadis work was supported by King Fahd University of Petroleumand 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-

) PDF. The Gamma-Gamma model was introduced to modelscattering in radar [8] and reverberation in sonar [9] and hasrecently generated interest in wireless communications as well

[10-13]. However, further derivations using that model haveshown to be analytically difficult or computationally involved

due to the arising special functions.

In this paper, the approximation of the generalized-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 cumulativedistribution 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 themultipath 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-composite fading PDF by the simple andtractable Gamma PDF. This region-wise approximation yields

sufficient accuracy for a broad range of integer and non-integervalues of the multipath fading and shadowing parameters.

Moreover, the introduced method can be used to approximatethe PDF of the sum of independent generalized- randomvariables (RVs) in the lower and upper tail regions. Finally,

since the approximating Gamma model allows the use of theclosed-form expressions developed in literature for Nakagami

fading channels [14-15], some performance analysis applica-

tions, out of many, are stated.

I I . THE C OMPOSITE FADINGM ODEL ANDR ELATED

WOR K

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,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 , takes the form ofa Gamma distribution:

() =

1

()

1 exp

> 0 05

(1)

where () is the Gamma function and is the Nakagamimultipath 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]:

() =

1

()

0

1 exp

0

>0 > 0

(2)

In (2), is the shadowing parameter and0 is the mean ofthe received local power. Similar to the multipath parameter

, the severity of shadowing is inversely proportional to so that small values of indicate severe shadowingconditions. In [5, 7], using the moment matching method

between the Gamma PDF in (2) and the lognormal PDF, it

was shown that = 1

(8686)21

where denotes thestandard deviation in the lognormal shadowing model.

Using (1) and (2), the PDF ofcan be derived as [8-10]1

() = 2

()()+(

+2 )1

(2

) >0 05 > 0(3)where () is the modified Bessel function of thesecond kind and order ( ) and =

0

. The

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- model2 and the McDanielmodel in wireless and sonar literature, respectively ([10] and

references therein).The CDF of, was derived in [13] as

() =csc() (2) 12(; 1 1 +; 2)()(1 )(+ 1)

(2) 12(; 1 + 1 +;

2)

()(1 +)(+ 1)

(4)

where = and is the generalized hypergeo-metric function for integer and [20].

However, as pointed out in [9], the computation of such aCDF expression which contains the hyper-geometric function

1In fact, the distribution of the product of M independent Gamma RVs,where the generalized- PDF corresponds to the case where M=2, wasderived in [17]. Moreover, the generalized-distribution belongs to the (Fox)

-function distributions family [18].2It should be highlighted here that in literature, the notion generalized-"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 generalized- RVs, are quite involved even for theindependent and identically distributed (i.i.d.) case due to the

difficulties associated with the Whittaker function [21].

III. APPROXIMATIONU SING THEM OMENTM ATCHING

METHOD

An alternative approach, to avoid the analytical difficulties,

is to consider approximating the PDF in (3) by a more tractable

PDF using the moment matching method. We propose usingthe 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 GammaRVs, one of the corresponding Gamma PDFs will dominate

for large values of or [7].

The moment of the generalized-K distribution can bederived as [13]

[] ==(+)(+)

()()

0

(5)

where [] denotes the statistical expectation.Denoting ( ) as a Gamma distributed RV with a

shape parameter and a scale parameter , the PDF of is

given as [22]

() =

() 1 exp() >0 (6)

Furthermore, the moment of the Gamma distributioncan be expressed as [22]

[] =() =(+)

() (7)

Now, using the expressions in (5) and (7), the first, second,

and third moments of the generalized-distribution and theapproximating Gamma distribution can be matched as

= 0 (8)

2(+ 1) = 120 (9)

3(+ 1)(+ 2) = 2130 (10)

On the other hand, the negative moments, as defined in

[23], of the generalized-PDF and the Gamma PDF can beexpressed using the expressions in (5) and (7) as

( 1) =10 >1 > 1 > 1 (11)

2( 2)( 1) =1220 >2 > 2 > 2(12)


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where

1 = (+1)(+1)

= 1 + 1

+ 1

+ 1

(13a)

2 = (+2)(+2)

= 1 + 2

+ 2

+ 4

(13b)

1 = (1)(1)

= 1 1

1

+ 1

(13c)

2 = (2)(2)

= 1 2

2

+ 4

(13d)

Matching different pairs of moments will result in the scaleand shape parameters for the approximating Gamma PDF as

shown in Table I. In Table I, and denote the scale andshape parameters of the approximating Gamma PDF obtained

by matching the and the moments, 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 PDFobtained by matching the positive moments is larger than

the one obtained by matching the negative moments.For example, it can be easily seen that 12 = 11+

2 0. Since the negative moments characterize adistribution 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 for2 will lead to involved expres-sions as seen in Table I. Moreover, not including the first

positive moment in the moments matched results in anapproximating Gamma PDF that does not have the same

mean as the approximated generalized- PDF (i.e., thegeneralized-PDF and the approximating Gamma PDFhave different average power values).

Matching negative moments may not be possible forsmall values of and/or as 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 and/or 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 and/or (while > 2), the differencebetween the two approximating Gamma CDFs might be

large enough to bound the approximated CDF in thelower tail region as seen in Fig. 1. On the other hand,

matching the lower order moments for large values of

and/or does not result in a good approximationas 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

(

)

01, is shown for the upper tail region to obtainmore illustrative results.

3 2.5 2 1.5 1 0.5 0 0.56

4

2

0The lower tail of the CDFs

log(x)

log(P(Xx))

Exact

1,

1

1,

2

1,

3

Exact

1,

2

1,

1

1,

2

Fig. 1. The log-log CDF plots of the generalized-and the approximatingGamma RVs for = 25 and = 25 using the moment matchingmethod.

2 1.5 1 0.5 0 0.56

4

2


log(x)

log(P(Xx))

Exact

1,

2

1,

1

1,

2

Exact

1,

1

1,

2

1,

3

Fig. 2. The log-log CDF plots of the generalized-and the approximatingGamma RVs for = 7and = 4using the moment matching method.

IV. THE MOMENT M ATCHINGM ETHOD 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 allvalues of and , and (ii) the first positive moment isincluded in the matching.

First, we may re-write the scale and shape parameters using

Table I as

12 =

1

+

1

+

1

0

= [AF] 0 0 AF AF (14a)12 =

1

AF 0 AF AF (14b)

In the above, the term 1

+ 1

+ 1

is the amount

of fading (AF) in the composite fading channel as derived


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TABLE IEXPRESSIONS OF THE SCALE AND THE SHAPE PARAMETERS OF THE APP ROXIMATINGGAMMAPD FOBTAINED BY MOMENT MATCHING(FO R 1 ,2,

1,2 , REFER TO( 13 A)-(13D))

Matched moments Scale parameter Shape parameter

1 2 12 = (1 1)0 12 >0 12 = 111 12 >0

1 3 13 =

3+

9+8(121)

0

4 13 >0 13=

4

3+9+8(121)

13 >0

2

3

23 =

1223+23

0

23

>0

23 =

221

+4

+

221

2+8

221

22211

23>

0

1 1 11 = (1 1)0 11 >0 11 = 111 11 >11 2 12 =

(39+8(121))0

4 12 >0 12 =

4

39+8(121)

12 >2

1 2 12 =

1

+ 1

3

0 >2 >2 12 >0 12 =

1012

+ 1 12 > 2

1 0.8 0.6 0.4 0.2 0 0.26

4

2


log(x)

log(P(Xx))

Exact

1,

1

1,

2

1,

3

Fig. 3. The log-log CDF plots of the generalized-and the approximatingGamma RVs for = 10 and = 10 using the moment matchingmethod.

in [12]. The value of AF is determined by the smallestphysical values of and which are non-zero in realpropagation channels; hence AF is 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):

12 = [AF ]0 0 AF AF 0 AF (15a)12=

1

AF 0 AF AF 0 AF (15b)

Since the AFadded" to the scale parameter of the approx-imating Gamma PDF should not exceed the original amount

of fading of the approximated PDF (i.e., 0 AF), isbounded asAF AF. Due to the fact that the relevantpractical range of AF is from zero (for non-fading channels)to 8 (for severe multipath fading and shadowing conditions

where = 05 and = 05)3, the relevant range of the

adjustment factor becomes8 8.3Such small values of and may take place in land mobile satellite

channels [26].

The adjustment factor can be computed using a numerical

measure of the difference between the approximated andthe 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 twocontinuous distributions. For this purpose, the CDFs rather

than the PDFs are considered since the Gamma PDF goes toinfinity as 0 for < 1 [22] which causes numericalinstabilities for comparison in the lower tail region.

The plots of the optimal adjustment factor, , versus themultipath fading and shadowing parameters are shown in Figs.

4 and 5 for values of and ranging from 0.5 to 10.The plots show that the adjustment factor decreases as either

or both and increase. The decrease of the adjustmentfactor as both and increase is worth noting since itindicates that the PDF of the product of two Gamma RVs

can be well-approximated, for the main body of the PDF, bya 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. Tosee that, the AF for equal values of the multipath fading and

shadowing parameters can be expressed as AF= 2+12

, where

= = ; clearly the amount of fading is approximately2for moderate values of and converges to zero for verylarge values of. However, if a high degree of accuracy issought in the lower tail region, then the magnitude of theadjustment 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.

V. ONT HE A PPROXIMATION OF THEPDF OF THES UM OF

INDEPENDENT GENERALIZED-KRVS

The moment matching method can be used to approximatethe PDF (CDF) of the sum of generalized- RVs by aGamma 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 withN=2 where the first and second moments

of the sum of two independent RVs, = + , can be


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02

46

810 0

24

68

100

0.5

1

1.5

2

2.5

ms

mm

Theoptimaladjustmentfactor,op

Fig. 4. The plot of the adjustment factor that minimizes the absolutevalue of the difference between the approximated generalized- and theapproximating Gamma distributions over the whole CDF.

02

46

810 0

24

6

810

0

1

2

3

4

5

6

ms

mm

Theoptimaladjustmentfactor,op

Fig. 5. The plot of the adjustment factor that minimizes the absolute valueof the difference between the generalized- and the approximating Gammadistributions in the lower tail of the CDF ( 0

(17a)

and

= (0+ 0)

2

AF20+ AF20

>0 (17b)

where 1 and 1 denote the 1 parameters (as definedin (13-a)), 0 and 0 denote the values of the mean of

4 3 2 1 0 1 26

5

4

3

2

1

0

log(x)

log(CDF)

Exact

Approx, =0.2

N=1

N=2

N=3

N=6

Fig. 6. The log-log CDF plots for the sum of generalized- RVs and theapproximating Gamma RV for = 2, = 4 ( = 42 dB), = 02,and different values of.

the local power, and AF and AF denote the AF (as givenin (14a)) of the generalized- RVs and , respectively.

The adjusted forms of (17a) and (17b) can be written as

=[AF ]20+ [AF ]20

(0+ 0) >0

(18a)

and

= (0+ 0)

2

[AF ]20+ [AF ]20 >0

(18b)

In general, the expressions in (18a) and (18b) can be

generalized for the sum of independent generalized-RVsas

=

=1[AF ]20

=10 > 0 (19a)

and

=

=10

2

=1[AF ]20 >0 (19b)

For the i.i.d. case, the expressions in (19a) and (19b) simplify

to

= (AF )0

>0 (20a)

and

=

AF >0 (20b)

Similar formulation can be carried out for the sum of corre-

lated generalized- RVs.Three-dimensional plots of the adjustment factor versus the

composite fading parameters and can be produced fordifferent values of. As an example, the plots of the lowertail of the CDFs for=2, =4, and= 1, 2, 3, and 6 aregiven in Fig. 6 showing that an adjustment factor of= 02results in almost identical CDFs, in the lower tail region, for

= 6. Clearly, larger values of are needed for a moreaccurate approximation for =1, 2, and 3.


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Remark: Another approach to approximate the PDF of the

sum of independent generalized-RVs can be based on thefact that the lower and upper tails of the PDF of the sum ofindependent positive RVs are due to the convolution of the

lower and upper tails of the corresponding individual PDFs,

respectively. So, the results obtained in Section IV can be used

to approximate the PDF of the sum of i.i.d. generalized-RVs by the PDF of the sum of the approximating i.i.d.

Gamma RVs. It is well-known that the sum of i.i.d. GammaRVs, with the same shape and scale parameters 12 and

12,

respectively, is another Gamma RV whose shape and scale

parameters are 12 and 12, respectively; these are the

same as the ones obtained in (20a) and (20b). For the non-

identically distributed case, the existing results in literatureon the distribution of the sum of independent non-identically

distributed Gamma RVs can be utilized [27-28].

V I . APPLICATIONS

The introduced region-wise approximation for the

generalized- distribution using the familiar Gammadistribution can be utilized in the performance analysis of

different communication schemes over composite fading

channels. So, using the closed-form expressions for the

different performance metrics that are already developed for

Nakagami fading channels, we present in this section examples

on the use of the proposed simplifying approximation to

compute some of these metrics.

A. Outage Probability

The outage probability corresponds to the probability that

the received signal power falls bellow a specific threshold

and can be expressed as

() = = 0

() (21)

In [29], an expression of the outage probability, for =1,alternative to the one given in [13] was developed; however,

the result in [29] is valid only for integer values of whereas the approximation introduced here applies for bothinteger and non-integer values of and including thecase < 1. So, the outage probability can be computed bythe simple CDF of the approximating Gamma distribution as

() =

()

(22)

where ( ) is the incomplete Gamma function defined as( ) =

0

1 [20, eq. 8.350.1].

B. Outage Capacity of SIMO Channels

The outage capacity for single-input multiple-output

(SIMO) channels is determined by the probability that the

instantaneous mutual information does not exceed a target rate

[30]:

() =

log2

1 + SNR

=1

(23)

where denotes the instantaneous power at the receiveantenna (out of antennas) and SNR is the signal-to-noise

ratio defined at the input. Using the result in Section V on the

PDF of the sum of independent generalized- RVs, theoutage capacity can be computed for different values ofand, and for different SNRs. Re-writing the expression in(23) as

() =

=1 2 1

SNR

=

21SNR

0

()

(24)the outage capacity can be computed using the familiar CDF

of the Gamma RV that approximates the CDF of the sum of

the independent generalized- RVs.

C. Bit Error Rate (BER)

Another common measure in performance analysis is the

bit error rate (BER) which can be expressed as

=

0

()() (25)

where()denotes the BER in an Additive White Gaussian

Noise (AWGN) channel. The BER for different modulationschemes can be computed using the approximating Gamma

PDF with the appropriate adjustment factor for each SNRvalue. So, the adjustment factor has to vary with the operating

SNR for the best match. The BER for differential phaseshift keying (DPSK) signaling is shown in Fig. 7 (see [12]).

The values of the adjustment factor used, at SNR=0, 5, 10,15, 20, 25, 30 dB are = 0.1, 0.21, 0.26, 0.31, 0.33, 0.34,0.36 (for =1 and =5), =0.15, 0.26, 0.35, 0.42, 0.50,0.53, 0.56 (for =2 and =2), and = 0.70, 0.95, 1.10,1.32, 1.42, 1.5, 1.57 (for =0.8 and =1.2). The valuesof the appropriate , in all cases, were found to lie in the

range 1 001 where 1 and 001 denotethe optimal adjustment parameters corresponding to thewhole CDF and to the lower portion of the CDF (< 001),respectively. These results show that (i) the BER will dependmore and more on the lower tail of the PDF as the SNR

increases [15], (ii) the region-wise approximation is more

needed for cases where the fading and shadowing parameters

are small and close to each other. In general, analytical

expressions of the BER for different transmission/reception

schemes, using the approximate Gamma model, are the same

as the ones obtained for Nakagami fading channels in [14].

D. Ergodic Capacity of SIMO Channels

The ergodic capacity of a SIMO channel can be expressed

as [30]

=h2 [log2(1 + SNRh2)]=0 log2(1 + SNR)h2()

(26)

where denotes the norm of the single channel vector h.Now, since the PDF of the sum of independent

generalized- RVs is approximated by a Gamma PDF, wemay write

= 1

()

0

log2(1 + SNR ) 1 exp

(27)Now using the obtained expression of the ergodic capacity for

Nakagami fading channels in [14], the ergodic capacity of a


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0 5 10 15 20 25 30

104

103

102

101

100

SNR (dB)

BER

Exact

Approx

mm

=0.8, ms=1.2

mm

=2, ms=2

mm

=1, ms=5

Fig. 7. The BER for DPSK signaling calculated using the proposedapproximation.

SIMO system over a generalized-composite fading channel

can be closely approximated, for integer values of, as

= log2()exp()1=0

( )

>0 (28)

where =

SNR and( )is the complementary incomplete

Gamma function as defined in [20, eq. 8.350.2].

Similar to the BER measure, the adjustment factor that

results in the best approximation of the ergodic capacity is

dependent on the operating SNR. However, the ergodic ca-pacity is not as sensitive as the BER to numerical inaccuracy.

The ergodic capacity of a heavily shadowed Rayleigh channel(=1) is shown in Fig. 8 where the loss in capacity, at highSNR, due to heavy shadowing is 1.66 bits/s/Hz as compared

to 0.83 bits/s/Hz for Rayleigh channels without shadowing[30]. In Fig. 8, the value of the adjustment factor (for =1)is chosen, for all SNRs, to be the average of1 and01(corresponding to the lower one-tenth portion of the CDF); i.e.,

= (1 + 01)2. For more severe shadowing conditionswith =0.5 (=9 dB), the loss increases to 2.6 bits/s/Hzand is well-predicted by the approximating Gamma PDF using

=2.1.

In Fig. 9, the ergodic capacity of the generalized-channelmodel and the approximating Gamma PDF for =2 and=1.0931 (as in [29]) is shown. Again the value of theadjustment factor, for =1, is chosen in the same way asin Fig. 8 showing that a sufficient accuracy for the ergodic

capacity, as compared to the BER, can be obtained through the

use of a single average value of. It can be seen that the use ofthe unadjusted values of the scale and shape parameters results

in a very good match for =4 since a small adjustment factoris needed. The obtained ergodic capacity for =4 is differentfrom the one in [29] since the latter does not correspond to

the sum ofi.i.d. generalized-RVs; it rather correspondsto the case when the multipath components are i.i.d. but theshadowing components are identically distributed and fully

correlated.

20 10 0 10 20 30 400

2

4

6

8

10

12

14

SNR (dB)

Ergodic

capacity(bits/s/Hz)

AWGN

mm

=1, no shadowing

Exact, mm

=1, ms=1

Approx, mm

=1, ms= 1, =1

Fig. 8. The ergodic capacity plot of a Rayleigh fading channel with andwithout shadowing.

5 0 5 10 15 20 25 300

2

4

6

8

10

12

SNR (dB)

Ergodiccapacity(bits/s/Hz)

Exact

Approx

N=4, =0

N=1, =0.4

Fig. 9. The ergodic capacity plot of a shadowed Nakagami channel with = 2 and = 10931 for =1 and =4.

VII. CONCLUSIONS

In this paper, we propose to approximate the generalized-

distribution by the familiar Gamma distribution throughthe use of the moment matching method. To avoid involved

expressions when matching the higher order moments andlimiting cases for small multipath fading and shadowingparameters, an adjusted form of the expressions of the pa-

rameters of the approximating Gamma distribution obtained

by matching the first two positive moments is proposed. The

obtained results show that the introduced adjustment results

in Gamma PDFs that closely approximate the generalized-

distribution in both the lower and upper tail regions andcan be further used to approximate the distribution of the

sum of independent generalized- RVs in these regions.This sufficiently accurate region-wise approximation using

the tractable Gamma distribution can significantly simplify

the performance analysis of composite fading channels usingmeasures such as probability of outage, outage capacity, and

ergodic capacity.


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ACKNOWLEDGMENT

The authors would like thank Sebastian Szyszkowicz for

his helpful discussions during early stages of this work. We

would like also to thank the anonymous reviewers for their

suggestions and comments.

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Saad Al-Ahmadi received his M.Sc. degree in electrical engineering fromKing Fahd University of Petroleum & Minerals (KFUPM), Dhahran, SaudiArabia in April 2002 and joined the the Department of Electrical Engineeringat KFUPM as a lecturer in August 2003. He is currently pursuing his Ph.Ddegree in electrical engineering at Carleton University, Ottawa, Canada. Hisresearch areas are statistical modeling of wireless channels and coordinatedmulti-point transmission and reception schemes in future cellular systems.

Halim Yanikomeroglu received a B.Sc. degreein Electrical and Electronics Engineering from theMiddle East Technical University, Ankara, Turkey,in 1990, and a M.A.Sc. degree in Electrical Engi-neering (now ECE) and a Ph.D. degree in Electricaland Computer Engineering from the University ofToronto, Canada, in 1992 and 1998, respectively. Hewas with the R&D Group of Marconi KominikasyonA.S., Ankara, Turkey, from 1993 to 1994.

Since 1998 Dr. Yanikomeroglu has been with theDepartment of Systems and Computer Engineeringat Carleton University, Ottawa, where he is now an Associate Professor.Dr. Yanikomeroglus research interests cover many aspects of the physical,medium access, and networking layers of wireless communications. Dr.Yanikomerogluss research is currently funded by Samsung (SAIT, Korea),Huawei (China), Communications Research Centre Canada (CRC), andNSERC. Dr. Yanikomeroglu is a recipient of the Carleton University ResearchAchievement Award 2009.

Dr. Yanikomeroglu has been involved in the steering committees andtechnical program committees of numerous international conferences; hehas also given 17 tutorials in such conferences. Dr. Yanikomeroglu is amember of the Steering Committee of the IEEE Wireless Communications andNetworking Conference (WCNC), and has been involved in the organizationof this conference over the years, including serving as the Technical ProgramCo-Chair of WCNC 2004 and the Technical Program Chair of WCNC 2008.Dr. Yanikomeroglu is the General Co-Chair of the IEEE Vehicular TechnologyConference to be held in Ottawa in September 2010 (VTC2010-Fall). Dr.Yanikomeroglu was an editor for IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS [2002-2005] and IEEE COMMUNICATIONSS URVEYS& TUTORIALS [2002-2003], and a guest editor for W ILEY JOURNAL ONWIRELESSCOMMUNICATIONS& MOBILECOMPUTING. He was an Officerof IEEEs Technical Committee on Personal Communications (Chair: 2005-06, Vice-Chair: 2003-04, Secretary: 2001-02), and he was also a member ofthe IEEE Communications Societys Technical Activities Council (2005-06).

Dr. Yanikomeroglu is also an adjunct professor at King Saud University,Riyadh, Saudi Arabia; he is a member of the Carleton University Senate, andhe is a registered Professional Engineer in the province of Ontario, Canada.

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