on the k and kg fading channels€¦ · serbian journal of electrical engineering vol. 6, no. 1,...

SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 6, No. 1, May 2009, 187-201

187

On the K and KG Fading Channels

Danijela Aleksić1, Mihajlo Stefanović2, Zoran Popović3, Dragan Radenković4, Jovan D. Ristić5

Abstract: In this paper, the composite Rayleigh-Gamma and Nakagami-Gamma distributions are considered. Multipath short-term fading and long-term fading (shadowing) affect wireless channels. A composite fading model was proposed for the modeling of shadowed channels, which resulted in a closed form solution for the probability density function (pdf). These composite Rayleigh-Gamma and Nakagami-Gamma are well-known as the K-distribution and KG-distribution, and are applied to cases where both micro- and macro-diversity schemes are implemented to mitigate short-term fading and shadowing, respectively. Thus, the composite pdf model offers significant improvement over approaches, which use lognormal pdf for shadowing. The results demonstrate the simplicity and usefulness of the composite pdf in the performance analyses of shadowed fading channels.

Keywords: Fading, Compound Gamma distribution, Nakagami-m, Rayleigh, RV, Shadowing.

1 Introduction The composite Rayleigh-Gamma and Nakagami-Gamma distributions are

mathematically tractable for the analytical evaluation, a bit error rate and an outage probability of communication systems. The composite Rayleigh–Gamma distribution is known as the K-distribution and the composite Nakagami-Gamma distribution is known as the KG-distribution. The composite Rayleigh-Gamma distribution closely approximates Rayleigh–lognormal distribution and the composite Nakagami-Gamma distribution closely approximates Nakagami-lognormal distribution.

The composite Rayleigh-lognormal and Nakagami-lognormal distributions are not mathematically tractable for analytical evaluations. Multipath fading and

1Technical college, Aleksandra Medvedeva 20, 18000 Niš, Serbia 2Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia, E-mail: [email protected] 3Technical Faculty Čačak, University of Kragujevac, Svetog Save 65, 32000 Čačak, Serbia; E-mail: [email protected] 4Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, Niš, Serbia 5Faculty of Technical Sciences, University of Priština, Kosovska Mitrovica

UDK: 621.391.822

D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić

188

shadow fading can be modeled by the composite Rayleigh-Gamma and Nakagami-Gamma distributions.

Using the K and KG fading models we can obtain closed form and easy use of expressions for the average BER.

The main drawback of the Rayleigh-lognormal distribution is its complicated mathematical form. Multipath fading and shadow fading affect wireless channels and the composite Rayleigh- Gamma and Nakagami-Gamma are proposed for the modeling of shadowed fading channels. This model is applied to the cases where both micro- and macro-diversity are used to mitigate short-term fading and long-term fading.

Channels suffer from short-term fading and long-term fading None of Rayleigh and Nakagami models leads to a closed form solution for

the probability density function of the signal for noise at the output of receiver. Recently, composite Rayleigh– Gamma and a composite Nakagami-Gamma distribution are proposed to describe the shadowed fading channels.

This model assumes a Nakagami or Rayleigh density function for the envelope of the received signal and the power of the envelope has gamma distribution. This composite model can be used to analyze the performance of wireless systems in shadowed fading channels.

A multipath short-term fading is mitigated using micro-diversity techniques and it will not be sufficient to mitigate channel degradation when shadowing is present. Long-term fading or shadowing fading is mitigated by using macro diversity techniques.

This paper is organized as follows. In the Section 2 the probability density function and the moment generating function (MGF) of Rayleigh-Gamma distribution are determined; in the Section 3 the bivariate Rayleigh-Gamma distribution is considered; in the Section 4 Nakagami-Gamma distribution is analyzed, and in the Section 5 bivariate Nakagami-Gamma distribution is calculated. In the Section 6 the conclusion is given.

2 Statistics of Rayleigh-Gamma Random Variable The conditional pdf of Rayleigh random variable is:

( )2

2Pr / e , 0ryrr y r

y

−= ≥ , (1)

where 2y r= , if 2 ,r rγ = = γ , and d 1d 2

r=

γ γ. We can obtain the conditional

pdf of squared Rayleigh random variable:


189

( ) ( )d 1/ / ed

yrP y P yy

γ−

γ γ = γ γ =γ

. (2)

Averaging the expression (2) with respect to y, we obtain Rayleigh distribution in the term:

( ) ( )0

( ) / dP P y Py y y∞

γ γ = γ γ∫ . (3)

The power of random variable γ has Gamma distribution:

010

1( ) e( )

yyc cPy y y y

c

−− −=

Γ. (4)

In ( )Py y , c is the order of the Gamma distribution and it is the measure of the shadowing present in the channel. By substituting (2) and (4) in (3) we can write:

0 01 200

0 0

1 1( ) e e d e d( ) ( )

y ycy y yc c cy yP y y y y y

y c c

γγ∞ ∞−− − −−− − −γ γ = =

Γ Γ∫ ∫ . (5)

Integral form of the modified Bessel function of second kind is:

( )21

0

e d 2 2x

xx x K

υ∞ β

− −γυ−υ

⎛ ⎞β= βγ⎜ ⎟γ⎝ ⎠

∫ . (6)

By substituting (6) in (5), we obtain pdf of Rayleigh in the equation:

1 ( 1)0 2

0 10

( ) 2 ( ) 2( )

c c

cyP y K

c y

− −

−

⎛ ⎞γγ γ = γ ⎜ ⎟⎜ ⎟Γ ⎝ ⎠

. (7)

The cumulative distribution function of the Rayleigh is:

0

( ) ( )dF v P x xγ

γ = γ∫ . (8)

Substituting (5) in (8) results in

020

0 0

( ) d e d( )

x ycy ycyF x y y

c

γ ∞− − −−γ γ =

Γ∫ ∫ . (9)

Changing the order of integrations can be written as:


190

0

0

0

0 0

20

0 0

20

0 0

10

0

1 10 0

0 0

00

( ) d e e d( )

d e e( )

d e 1 e( )

e d e d( ) ( )

( )

y xcyc y

y xcyc y

ycyc y

y yc cy y yc c

cc

yF y y xc

y y y yc

y y yc

y yy y y yc c

y yc

γ∞− − −−

γ∞− − −

−

γ∞− −−

γ∞ ∞− −− − −− −

−

γ γ = ⋅ =Γ

⎡ ⎤= ⋅ =⎢ ⎥Γ ⎢ ⎥⎣ ⎦

⎛ ⎞= ⋅ − =⎜ ⎟⎜ ⎟Γ ⎝ ⎠

= − =Γ Γ

= ΓΓ

∫ ∫

∫

∫

∫ ∫

0 20

0

2

0 0

( ) 2( ) 2( )

21 2 .( )

cc

c

c

c

yc y Kc y

Kc y y

− ⎛ ⎞γ− γ =⎜ ⎟⎜ ⎟Γ ⎝ ⎠

⎛ ⎞⎛ ⎞γ γ= − ⎜ ⎟⎜ ⎟ ⎜ ⎟Γ ⎝ ⎠ ⎝ ⎠

(10)

The moment generating function (MGF) of r is:

0

( ) e e ( )drs rsM s P∞

γ = = γ γ γ∫ . (11)

Substituting (5) in (11) and changing the order of integrations we can write:

( )

0

0

0

0

0

20

0 0

20

0 0

120

0 0

20

0

110

( ) d e d e e( )

d e d e( )

d e d e( )

d e( ) 1

= 1 e( )

ycy ys c y

ycyc sy

yc syyc

ycyc

ycyc

yM s y y yc

y y y ec

y y yc

y yy yc ys

y y ysc

γ γ∞ ∞− − − −γ −

γ∞ ∞− − −− γ

⎛ ⎞∞ ∞− −γ −− ⎜ ⎟− ⎝ ⎠

∞− −−

− −−−

⎡ ⎤γ = γ ⋅ =⎢ ⎥

Γ⎢ ⎥⎣ ⎦

= ⋅ γ =Γ

= ⋅ γ =Γ

= ⋅ =Γ −

−Γ

∫ ∫

∫ ∫

∫ ∫

∫

0

d .y∞

∫

(13)

Moment n-th order of γ is:


191

0

( )dn nnm P

∞

= γ = γ γ γ γ∫ . (14)

Substituting (5) in (14) and changing the order of integrations the moment of n-th order of γ becomes:

( )

0

0

0

0

20

0 0

20

0 0

2 10

0

100

0

d d e e( )

d e d e( )

d e 1( )

( )d e .( ) ( )

ycy yn c y

n

ycy yc n y

ycyc n

ycyc n n

ym y y yc

y y yc

y y y y nc

y c ny y yc c

γ γ∞ ∞− − − −−

γ γ∞ ∞− − − −−

∞− −− +

∞− −+ −

= γ γ ⋅ =Γ

= ⋅ γ ⋅ γ =Γ

= ⋅ Γ + =Γ

Γ += ⋅ =Γ Γ

∫ ∫

∫ ∫

∫

∫

(15)

The random variables 1γ and 2γ are Rayleigh distributed:

1 111 011

2 222 022

1011 1 1 1

1 0

1022 2 2 2

2 0

( ) e d ,( )

( ) e d .( )

ycy yc

ycy yc

yP y yc

yP y yc

γ∞− − −−

γ∞− − −−

γ γ =Γ

γ γ =Γ

∫

∫ (16)

The pdf of sum of 1γ and 2γ is:

( ) ( ) ( )

2 1 2 21 21 01 2 021 2

1 2 2 2 20

2 201 022 1 1 2 2

1 20 0 0

d

d d e d e .( ) ( )

y yc cy y y yc c

P P P

y y y y y yc c

γ

γ−γ γγ ∞ ∞− − − − − −− −

γ γ = γ γ γ γ γ γ =

= γΓ Γ

∫

∫ ∫ ∫ (17)

Changing the order of integrations, we can write:

1 2 2 21 21 01 02 1 021 2

1 2 1 21 21 01 021 2 2 1

2 201 021 1 2 2 2

1 2 0 0 0

2 201 02 1 21 1 2 2

1 2 2 10 0

( ) d e d e d e( ) ( )

d e d e e 1 .( ) ( )

y y yc cy y y y yc c

y y y yc cy y yc c y y

y yP y y y y yc c

y y y yy y y yc c y y

γγ γ∞ ∞− − − − − − −− −

γ∞ ∞− − − − − −γ− − −

γ γ = =Γ Γ

⎛ ⎞= −⎜ ⎟⎜ ⎟Γ Γ − ⎝ ⎠

∫ ∫ ∫

∫ ∫ (18)


192

3 The Bivariate Rayleigh-Gamma Distribution The conditional bivariate Rayleigh distribution is given by:

( ) ( ) ( )

2 21 2

1 2

1 2

11 1 21 2

1 2 1 2 0 1 21 2 1 2

24/ e , , 0.1 1

r ry y

r rr rr rP r r y y I r r

y y y y

⎛ ⎞− +⎜ ⎟⎜ ⎟−ρ⎝ ⎠

⎛ ⎞ρ= ≥⎜ ⎟⎜ ⎟− ρ −ρ⎝ ⎠

(19)

Let us now define two random variables 1γ and 2γ in terms of 1r and 2r by the transformation: 2

1 1rγ = and 22 2rγ = . (20)

The Jacobian of this transformation is:

1 2

14

J =γ γ

.

The conditional squared bivariate Rayleigh distribution is:

( ) ( ) ( )

1 2

1 2

11 1 2

1 2 1 2 1 2 01 2 1 2

21/ e1 1

y yP y y Iy y y y

⎛ ⎞γ γ− +⎜ ⎟−ρ⎝ ⎠

⎛ ⎞ρ γ γγ γ γ γ = ⎜ ⎟⎜ ⎟− ρ −ρ⎝ ⎠

. (21)

Assuming that 1y and 2y are i.d. gamma RV’s with parameters c and 0y , the joint pdf of 1y and 2y can be expressed as:

( ) ( ) ( ) ( )

( )

1 2

0 1

112

1 1 1 21 21 2 1 2 1 2 11

0 0 1

4e

( ) 1 1

cy yc

ycc

y yPy y y y y y I

c y y

−+− −−ρ

−+

⎛ ⎞ρρ= ⎜ ⎟⎜ ⎟Γ −ρ −ρ⎝ ⎠

, (22)

where 1ρ is the correlation between 1y and 2y , and 1cI − is the modified Bessel function of the first kind of order ( 1)c − . By averaging (21) with respect to 1y and 2y it follows,

( ) ( ) ( )1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 20 0

d d /P y y P y y Py y y y∞ ∞

γ γ γ γ = γ γ γ γ∫ ∫ . (23)

By substituting (21) and (22) in (23) it follows that:

( ) ( )( )

( )( )

( ) ( )( )

( )

( )

1 2

1 2

1 2

0 1

11 1 2

1 2 1 2 1 2 01 20 0 1 2

112

1 1 1 21 21 2 11

1 0 0 1

21d d e1 1

2e .

( ) 1 1

y y

cy yc

ycc

P y y Iy y y y

y yy y I

c y y

⎛ ⎞γ γ∞ ∞ − +⎜ ⎟−ρ ⎝ ⎠

−+− −−ρ

−+

⎛ ⎞ρ ⋅ γ γγ γ γ γ = ⋅⎜ ⎟⎜ ⎟− ρ −ρ⎝ ⎠

⎛ ⎞ρρ⋅ ⎜ ⎟⎜ ⎟Γ −ρ −ρ⎝ ⎠

∫ ∫ (24)


193

The infinite series representation for 0 ( ) I x and ( )nI x are:

1

1

2

2

2

0 20 1

2

0 2 2

1( ) ,2 ( !)

1( ) .2 ! ( 1)

i

i

i n

ni

xI xi

xI xi i n

∞

=

+∞

=

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟ Γ + +⎝ ⎠

∑

∑ (25)

Substituting (25) in (24) and changing the order of summations and integration can be written:

( )

( ) ( ) ( )( )

11 2

1 2

1

21 2 2

1 0 1

2

( 1)2

11 2 1 2 1

1 021

(1 ) 1 21 2 2

01 2 10 0

2 1( 1)1 2( 1) 1 1 2 12

1 20 2 2 0 1

1(1 ) ( )(1 )

1 1d d e( !) (1 )

e! ( ) 1

c

c

i

y y

i

i ccy y ii c y

i

Pc y

y yy y i

y yy y

i i c y

−

+

⎛ ⎞γ γ∞ ∞ ∞− +⎜ ⎟−ρ ⎝ ⎠

=

+ −−+ +∞−− + − −ρ

=

ργ γ γ γ = ⋅ ⋅

− ρ Γ −ρ

⎛ ⎞ρ γ γ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟− ρ⎝ ⎠

⎛ ⎞ρ⋅ =⎜ ⎟⎜ ⎟Γ + −ρ⎝ ⎠

∑∫ ∫

∑

( )

( )

( )( ) ( ) ( )

( ) ( )

1

1

2

2

1 1 2 2

1 1 0 2 1 01 2 1 2

1 212

1 2121

01 0 1

2 1

1

0 2 2 0 1

(1 ) (1 ) (1 ) (1 )2 21 1 2 2

0 0

1 11 1 1!

1! 1

d e d e .

ic

ci

i c

i

y yy y y yc i i c i i

c y i

i i c y

y y y y

−∞

+=

+ −∞

=

γ γ∞ ∞− − − −−ρ −ρ −ρ −ρ− + − − + −

⎛ ⎞ρ γ γρ= ⋅ ⋅⎜ ⎟⎜ ⎟− ρ Γ −ρ −ρ⎝ ⎠

⎛ ⎞ρ⋅ ⋅⎜ ⎟⎜ ⎟Γ + −ρ⎝ ⎠

⋅ ⋅ ⋅

∑

∑

∫ ∫

(26)

( ) ( )( )( ) ( ) ( )

( ) ( )( )( )

( ) ( ) ( ) ( )

1

1

2 1 2

2

1 2 1 2

1 2( 1)2

1 211 2 1 2 21

01 0 1

2 1 1

11

0 2 2 0 1 0 1 2

1 21 1

0 1 0 1

4 11 1 1!

11 1! 1 1

2 2 .1 1 1 1

ic

ci

i c c i i

i

c i i c i i

Pc y i

i i c y y

K Ky y

−∞

+=

+ − − + −∞

=

− + − − + −

⎛ ⎞ρ γ γργ γ γ γ = ⋅⎜ ⎟⎜ ⎟Γ −ρ −ρ −ρ⎝ ⎠

⎛ ⎞ ⎛ ⎞− ρρ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Γ + −ρ −ρ γ γ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞γ γ⋅ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− ρ −ρ −ρ −ρ⎝ ⎠ ⎝ ⎠

∑

∑


194

Fig. 1 – The probability of bivariate Rayleigh-Gamma distribution.

4 Nakagami – Gamma Distribution The conditional Nakagami–m distribution has the term:

( ) ( )2

2 12Pr / e , 0,m m r

m ymr y r rm y

−−⎛ ⎞

= ≥⎜ ⎟Γ ⎝ ⎠ (27)


195

where ( )mΓ is the Gamma function, y is an average signal power, and m an arbitrary fading security parameter with values from 0.5 through infinity. Transforming random variable r to γ , where is

2 , r rγ = = γ and d 1 d 2

r=

γ γ.

we obtain probability density function for squared Nakagami– m random variable as follow:

( ) ( )12/ e , 0

m mm ymP y

m y

− γ−⎛ ⎞

γ γ = γ γ ≥⎜ ⎟Γ ⎝ ⎠. (28)

Averaging ( )/P yγ γ with respect to y, using (6), it follows:

( )

( )

0

0

0

1 10

0

1 10

0

10 20

0

( ) / ( )d

2 e e d( ) ( )

d e( ) ( )

2 2 .( ) ( )

m ym cym cy

m ycy ym m m c

c c mm m

c m

P P y Py y y

ym y ym y c

y m y yc m

y mm my Kc m y

∞

∞ − −− γ− −

γ∞− − −− − + −

− −−

−

γ γ = γ γ =

⎛ ⎞= γ =⎜ ⎟Γ Γ⎝ ⎠

= γ ⋅ =Γ Γ

⎛ ⎞γ= γ ⋅ γ ⎜ ⎟⎜ ⎟Γ Γ ⎝ ⎠

∫

∫

∫ (29)

The cumulative distribution function of Nakagami-Gamma random variable is:

01 10

0 0 0

( ) ( )d d d e( ) ( )

mx yc my ym c my mF P x x x x y y

c m

γ γ ∞− − −− − −γ γ = γ = ⋅

Γ Γ∫ ∫ ∫ .

Changing the order of integrations, it follows:

( )

0

0

0

1 10

0 0

10

0

10

0

( ) d e e d( ) ( )

d e ( ) ,( )

1 e , d .( ) ( )

m

y mxc myc m m y

yc myc m

ycyc

y mF y y x xc m

y m y mxy y m mc m m y

y mxy m yc m y

γ∞− − −− − −

∞− −− −

∞− −−

γ γ = ⋅ =Γ Γ

⎡ ⎤⎛ ⎞⎛ ⎞= ⋅ Γ − γ =⎢ ⎥⎜ ⎟⎜ ⎟Γ Γ ⎝ ⎠ ⎝ ⎠⎣ ⎦

⎛ ⎞= − γ⎜ ⎟Γ Γ ⎝ ⎠

∫ ∫

∫

∫

(30)

The MGF of Nakagami–m random variables is:


196

01 10

0 0 0

( ) e e ( )d d e d e( ) ( )

m yc my ys s s m c my mM s P y y

c m

γ∞ ∞ ∞− − −γ γ γ − − −γ = = γ γ γ = γ ⋅ γ ⋅

Γ Γ∫ ∫ ∫ .

Changing the order of integrations, we can write:

0

0

0

1 10

0 0

10

0

10

0

( )( ) d e e d( ) ( )

( ) d e( ) ( )

( ) d e ( ) .( ) ( )

y mc myc m s m y

myc myc m

yc myc m m

y mM s y y ec m

y m yy yc m m ys

y m y y m ysc m

γ∞ ∞− − −− − γ −

∞− −− −

∞− −− − −

γ = ⋅ γ γ =Γ Γ

⎛ ⎞= ⋅ =⎜ ⎟Γ Γ −⎝ ⎠

= ⋅ −Γ Γ

∫ ∫

∫

∫

(31)

Moment of n-th order of Nakagami- m RV’s is:

01 10

0 0 0

( )d d d e( ) ( )

m yc my yn n n m c my mP y y

c m

γ∞ ∞ ∞− − −− − −γ = γ γ γ γ = γ γ γ ⋅

Γ Γ∫ ∫ ∫ .

Changing the order of the integrations, it follows:

0

0

0

1 10

0 0

10

0

10

0

0

d e d e( ) ( )

d e ( )( ) ( )

( ) d e( ) ( )( ) ( ) .

( ) ( )

y mc myn c m n m y

y n mc myc m

yc myc m

n

y m y yc m

y m yy y n mc m m

y m n m y yc mn m c n y

c m

γ∞ ∞− − −− − + −

+∞− −− −

∞− −− −

γ = ⋅ γ ⋅ γ =Γ Γ

⎛ ⎞= ⋅ Γ + =⎜ ⎟Γ Γ ⎝ ⎠

= Γ + ⋅ =Γ Γ

Γ + Γ +=

Γ Γ

∫ ∫

∫

∫ (32)

5 Bivariate Nakagami-Gamma Distribution

The bivariate Nakagami-Gamma distribution has the form:

( )2 2

1 2

1 2

1 2 1 2 1 2

11 1 21 2

11 ( 1)1 22

1 2 1 2

Pr /

24 ( ) e ,(1 )( ) (1 )( )

r rmm my y

mm

r r r y y

m r rm r r Iy ym y y y y

⎛ ⎞+ − +⎜ ⎟⎜ ⎟−ρ⎝ ⎠

−−

=

⎛ ⎞ρ= ⎜ ⎟⎜ ⎟− ρ⎝ ⎠Γ −ρ

(33)


197

where 2 21 1 2 2,y r y r= = , and

( )( ) ( )

2 21 2

2 21 2

cov

var var

r r

r r

+ρ =

+ is the power correlation

coefficient ( 0 1≤ ρ < ). Transformation of random variables 1r and 2r respectively into 1γ and 2γ , gives 2

1 1rγ = , 22 2rγ = and the Jacobian is:

1 2

14

J =γ γ

.

We can obtain the joint probability density function of shared Nakagami-m RV’s in the form:

( )1 2

1 2

1 2 1 2 1 2

11 1 21

1 2 111 22

1 2

/

2( ) e .

(1 )( )(1 )( )

mmy ym

mm

P y y

mm Iy ym y y

⎛ ⎞γ γ+ − +⎜ ⎟−ρ− ⎝ ⎠

−+

γ γ γ γ =

⎛ ⎞ρ γ γ= γ γ ⎜ ⎟⎜ ⎟− ρ⎝ ⎠Γ −ρ

(34)

Averaging the expression (34) with respect to 1y and 2y , it follows:

( ) ( ) ( )

( )

( ) ( )( )

1 2

1 2

1 2

0 1

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 20 0

1111 2 1 2

1 2 110 0 1 22

1 21 1 12 2 11 1 2 1 1 2

111 0 0

d d /

2d d e

(1 )( )(1 )( )

2e

( )(1 ) (1

m mmy y

mm

cy yc

ycc

P y y P y y Py y y y

m my y I

y ym y y

y y y yI

c y y

∞ ∞

⎛ ⎞γ γ−+∞ ∞ − +⎜ ⎟−ρ⎝ ⎠

−+

−+−

−−ρ

−+

γ γ γ γ = γ γ γ γ =

⎛ ⎞γ γ ρ γ γ= ⋅⎜ ⎟⎜ ⎟− ρ⎝ ⎠Γ −ρ

ρ ρ⋅Γ −ρ −ρ

∫ ∫

∫ ∫

1

.)

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(35)

Using the infinite series representations for modified Bessel functions and changing the order of summation and integration, we can write:

( ) ( ) ( )

( )

( ) ( )

( )( )( )

( ) ( )

1 2

1 2

1

1

1 2

0 1

2

11 1 ( 1) 11 2 21 2 1 2 1 2 1 2

0 0

2 11

1 2

0 1 1 1 2

1 1( 1) 12 2 11 1 2 1 1 21

1 0 2 2 0 1

d d e( )(1 )

1! (1 )

1e1 ! 1

m mmm y y

i mm

i

c y ycy

ci

mP y y y y

m

mi i m y y

y y y yc y i i c y

⎛ ⎞γ γ−+ ∞ ∞ − +⎜ ⎟− + −ρ⎝ ⎠

+ −−

=

− +−−

−ρ+

γ γγ γ γ γ = ⋅

Γ − ρ

⎛ ⎞ρ γ γ⋅ ⋅⎜ ⎟⎜ ⎟Γ + −ρ⎝ ⎠

⎛ ⎞ρ ρ⋅ ⋅ ⎜ ⎟⎜ ⎟Γ −ρ Γ + −ρ⎝ ⎠

∫ ∫

∑

22 1

0

i c+ −∞

=

=∑


198

( )( ) ( ) ( )

( ) ( )

( ) ( ) ( )

1

1

2

2

1 1 2 2

1 1 0 21 2 1 2

1 2 1( 1)11 21 2 1 21

101 0 1 1

2 1

1

0 2 2 0 1

1 1 1 11 11 1 2 2

0 0

1( )(1 ) ( ) 1 ! 1

1! 1

d e d e

i mcmm

ci

i c

i

m y m yy y yi i c m i i c m

m mm c y i i m

i i c y

y y y y

+ −−−+ ∞

+=

+ −∞

=

γ γ∞ ∞− − − −−ρ −ρ −ρ −− − + + − − − + + −

⎛ ⎞γ γ ρ γ γρ= ⋅⎜ ⎟⎜ ⎟Γ −ρ Γ −ρ Γ + −ρ⎝ ⎠

⎛ ⎞ρ⋅ ⋅⎜ ⎟⎜ ⎟Γ + −ρ⎝ ⎠

⋅ ⋅ ⋅

∑

∑

∫ ∫ ( )1 0yρ

( )

( )( ) ( ) ( ) ( )

( ) ( )

( )( )

1

1

2 1 2

2

1

1 2

1 2 1 2

1 2 1( 1)11 21 2 1 21

101 0 1 1

2 1

1

0 2 2 0 1 1 0 1 2

1 0

4 1( ) 1 ( ) 1 ! 1

1 1 1! 1 1

21 1

i mcmm

ci

i c c m i i

i

c m i i

P

m mm c y i i m

i i c y my

mK K

y

+ −−−+ ∞

+=

+ − − − +∞

=

− − +

γ γ γ γ =

⎛ ⎞γ γ ρ γ γρ= ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟Γ −ρ Γ −ρ Γ + −ρ⎝ ⎠

⎛ ⎞ ⎛ ⎞ρ −ρ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Γ + −ρ −ρ γ γ⎝ ⎠ ⎝ ⎠

⎛ ⎞γ⎜ ⎟⋅⎜ ⎟− ρ −ρ⎝ ⎠

∑

∑

( )( )2

1 21 0

2 .1 1c m i i

my− − +

⎛ ⎞γ⎜ ⎟⎜ ⎟− ρ −ρ⎝ ⎠

Fig. 2 – The probability of bivariate Nakagami-Gamma distribution.


199



Using the expression for the joint pdf of Nakagami RV’s, we can determine

the expressions for CDF, joint MGF and joint moments for Nakagami-m random variables.


200

6 Conclusion This paper considers the Rayleigh-Gamma and Nakagami-Gamma

distributions. Closed form of expressions for the probability density functions for Rayleigh-Gamma and Nakagami-Gamma random variables are obtained. In addition, the cumulative distribution functions, the generating functions and moments of these random variables are determined. These functions can be used to determine the outage probability and the bit error probability in digital communication systems operating over multipath and shadow fading channels. The Rayleigh distribution and shadow-fading channel are described with Rayleigh-Gamma distribution; the Nakagami-m distribution and shadow-fading channel are described with Nakagami-Gamma distribution. The results of this paper can be used to determine the performance analysis of macro- and micro-diversity systems. The macro-diversity systems are used to mitigate the shadowing and micro-diversity systems are used to mitigate the multipath fading.

7 References [1] M.K. Simon, M.S. Alouini: Digital Communications over Fading Channels, John Wiley &

Sons, 2000. [2] P.M. Shankar: Error Rates in Generalized Shadowed Fading Channels, Wireless Personal

Communications, Vol. 28, No. 3, 2004, pp. 233-238. [3] P.S. Bithas, N.C. Sagias, P.T. Mathiopoulos, G.K. Karagiannidis, A.A. Rontogiannis: On

the Performance Analysis of Digital Communications over Generalized-K Fading Channels, IEEE Communications Letters, Vol. 10, No. 5, 2006, pp. 353-355.

[4] P.M. Shankar: Outage Probabilities in Shadowed Fading Channels using a Compound pdf Model, IEE Proceedings-Communications, Vol. 152, No. 6, 2005, pp. 828-832.

[5] I.M. Kostic: Analytical Approach to Performance Analysis for Channel Subject to Shadowing and Fading, IEE Proceedings-Communications, Vol. 152, No. 6, 2005, pp. 821-827.

[6] A.M.D. Turkmani: Performance Evaluation of a Composite Microscopic Plus Macroscopic Diversity System, IEE Proceedings-Communications, Vol. 138, No. 1, 1991, pp. 15-20.

[7] E.K. Al-Hussaini, A.M. Al-Bassiouni, H.M. Mourad, H. Al-Shennawy: Composite Macroscopic and Microscopic Diversity of Sectorized Macrocellular and Microcellular Mobile Radio Systems Employing RAKE Receiver over Nakagami Fading Plus Lognormal Shadowing Channel, Wireless Personal Communications, Vol. 21, No. 3, 2002, pp. 309-328.

[8] W.C. Jeong, J.M. Chung, L. Dongfang: Performance Analysis of Macroscopic Diversity Combining of MIMO Signals in Mobile Communications, Vehicular Technology Conference, Vol. 3, 2003, pp. 1838-1842.

[9] P.M. Shankar: Performance Analysis of Diversity Combining Algorithms in Shadowed Fading Channels, Wireless Personal Communications, Vol. 37, No. 1-2, 2006, pp. 61-72.

[10] M. Abramowitz, I.A. Stegun: Handbook of Mathematical Functions, Dover, New York, 1972.


201

[11] V.A. Aa1o: Performance of Maximal-ratio Diversity Systems in a Correlated Nakagami-fading Environment, IEEE Transaction on Communications, Vol. 43, No. 8, 1995, pp. 2360-2369.

[12] M.S. Alouini, M.K. Simon: Dual Diversity over Correlated Lognormal Fading Channels, IEEE Transaction on Communications, Vol. 50, No. 12, 2002, pp. 1946-1959.

[13] P.S. Bithas, N.C. Sagias, P.T. Mathiopoulos: Dual Diversity over Correlated Ricean Fading Channels, Journal of Communications and networks, Vol. 9, No. 1, 2007, pp. 67-74.

[14] G.E. Corazza, F. Vatalaro: A Statistical Model for Land Mobile Satellite Channels and its Application to Nongeostationary Orbit Systems, IEEE Transactions on Vehicular Techno-logy, Vol. 43, No. 3, 1994, pp. 738-742.

[15] S. Yue, T.B.M.J. Ouarda, B. Bobee: A Review of Bivariate Gamma Distributions for Hydrological Applications, Journal of Hydrology, Vol. 246, No. 1-4, 2001, pp. 1-18.

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