On the K and K G Fading Channels

SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 6, No. 1, May 2009, 187-201 UDK: 621.391.822 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 1 Technical college, Aleksandra Medvedeva 20, 18000 Niš, Serbia 2 Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia, E-mail: [email protected] 3 Technical Faculty Čačak, University of Kragujevac, Svetog Save 65, 32000 Čačak, Serbia; E-mail: [email protected] 4 Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, Niš, Serbia 5 Faculty of Technical Sciences, University of Priština, Kosovska Mitrovica 187 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić 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: r2 2r − Pr ( r / y ) = e y , r ≥ 0 , (1) y dr 1 where y = r 2 , if γ = r 2 , r = γ , and = . We can obtain the conditional dγ 2 γ pdf of squared Rayleigh random variable: 188  On the K and KG Fading Channels γ Pγ ( γ / y ) = dr dγ Pγ ( )1 − γ/y = e y. y (2) Averaging the expression (2) with respect to y, we obtain Rayleigh distribution in the term: ∞ P γ ( γ ) = ∫ P γ ( γ / y )Py ( y ) d y . (3) 0 The power of random variable γ has Gamma distribution: y 1 − c c −1 − y0 Py ( y ) = y0 y e . (4) Γ (c ) 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: ∞ γ y γ y 1 − 1 − c c −1 − y0 y −c ∞ − − Pγ( γ ) = ∫ e y y0 y e d y = 0 ∫ y c − 2 e y y0 d y . (5) 0 y Γ (c ) Γ (c ) 0 Integral form of the modified Bessel function of second kind is: υ ∞ β ⎛ β ⎞2 ( ) − −γx ∫x υ−1 e x d x = 2 ⎜ ⎟ K υ 2 βγ . (6) 0 ⎝γ⎠ By substituting (6) in (5), we obtain pdf of Rayleigh in the equation: y0− c 1 ( c −1) ⎛ γ ⎞ Pγ (γ ) = 2 ( γy0 ) 2 K c −1 ⎜ 2 ⎟. (7) Γ (c ) ⎜ y0 ⎟⎠ ⎝ The cumulative distribution function of the Rayleigh is: γ F γ (v ) = ∫ P γ ( x ) d x . (8) 0 Substituting (5) in (8) results in γ ∞ x y y −c − − F γ ( γ ) = ∫ 0 d x ∫ y c − 2 e y y0 d y . (9) 0 Γ (c ) 0 Changing the order of integrations can be written as: 189 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić y γ x y0− c ∞ − − ∫ ∫ c−2 F γ(γ) = d y ⋅ y e y0 e y dx = Γ (c ) 0 0 γ y −c ∞ − y ⎡ −x ⎤ = 0 ∫ d y ⋅ y c − 2 e y0 y ⎢ e y ⎥ = Γ (c ) 0 ⎢⎣ ⎥⎦ 0 y −c ∞ − y ⎛ γ ⎞ Γ(c) ∫0 = 0 d y ⋅ y c −1 e y0 ⎜⎜1 − e y ⎟⎟ = ⎝ ⎠ y γ y (10) y −c ∞ − y −c ∞ − − = 0 ∫ y c −1 e y0 d y − 0 ∫ y c −1 e y y0 d y = Γ (c ) 0 Γ (c ) 0 y0− c c y −c c ⎛ γ ⎞ = y0 Γ(c) − 0 2( γy0 ) 2 K c ⎜ 2 ⎟= Γ (c ) Γ (c ) ⎜ y0 ⎟⎠ ⎝ c 2 ⎛ γ ⎞2 ⎛ γ ⎞ =1− ⎜ ⎟ K c ⎜⎜ 2 ⎟. Γ(c) ⎝ y0 ⎠ ⎝ y0 ⎟⎠ The moment generating function (MGF) of r is: ∞ M γ ( s ) = e rs = ∫ e rs P γ ( γ )d γ . (11) 0 Substituting (5) in (11) and changing the order of integrations we can write: ⎡∞ γs y0 −c ∞ γ y − − − ⎤ γ M γ ( s) = ⎢ ∫ d γ e ∫ d y ⋅ y c−2 e y y0 y e y ⎥= ⎣⎢ 0 Γ ( c ) 0 ⎦⎥ y γ y0− c ∞ − ∞ − ∫ ∫ c−2 y γs = d y ⋅ y e y0 d γ e e = Γ (c ) 0 0 y ⎛1 ⎞ y −c ∞ − ∞ − γ⎜ − s ⎟ = 0 ∫ d y ⋅ y c − 2 e y0 ∫ d γ e ⎝ y ⎠ = (13) Γ (c ) 0 0 y y0− c ∞ − y = ∫ Γ (c ) 0 d y ⋅ y c − 2 e y0 1 − ys = y y0− c ∞ c −1 − y (1 − ys ) e y0 d y. −1 = ∫ Γ (c ) 0 Moment n-th order of γ is: 190  On the K and KG Fading Channels ∞ mn = γ n = ∫ γ n P γ ( γ ) d γ . (14) 0 Substituting (5) in (14) and changing the order of integrations the moment of n-th order of γ becomes: ∞ γ y γ y0− c ∞ − − − mn = ∫ γ d γ n ∫ d y ⋅ y c−2 e y y0 y e y = 0 Γ (c ) 0 γ y γ y −c ∞ − − ∞ − = 0 ∫ d y ⋅ y c − 2 e y y0 ∫ d γ ⋅ γ n e y = Γ (c ) 0 0 y (15) y −c ∞ − = 0 ∫ d y ⋅ y c − 2 e y0 y n +1Γ ( n + 1) = Γ (c ) 0 y y −c ∞ − Γ (c + n ) n = 0 ∫ d y ⋅ y c + n −1 e y0 = y0 . Γ (c ) 0 Γ (c ) The random variables γ1 and γ 2 are Rayleigh distributed: − c1 ∞ γ y y01 − 1− 1 ∫ c1 −1 P γ1 ( γ1 ) = y1 e y1 y01 d y1 , Γ(c1 ) 0 γ y (16) − c2 ∞ y02 − 2− 2 ∫ c2 −1 Pγ 2 ( γ 2 ) = y2 e y2 y02 d y2 . Γ(c2 ) 0 The pdf of sum of γ1 and γ 2 is: γ P γ ( γ ) = ∫ P γ1 ( γ γ 2 )P γ 2 ( γ 2 ) d γ 2 = 0 γ γ−γ 2 y1 ∞ γ 2 y2 (17) − c1 − c2 ∞ y y − − − − =∫ 01 d γ 2 ∫ d y1 y1c1 − 2 e 02 y1 y01 ∫d y 2 y2 c2 − 2 e y2 y02 . 0 Γ(c1 ) Γ(c2 ) 0 0 Changing the order of integrations, we can write: − c2 ∞ γ y y γ γ y y − c1 y02 − − 1 ∞ − 2 − 2− 2 ∫ ∫ ∫ c1 − 2 c2 − 2 P γ ( γ ) = 01 d y1 y1 e y1 y01 d y 2 y 2 e y02 d y 2 e y1 y02 = Γ(c1 ) Γ(c2 ) 0 0 0 (18) y1 y2 ⎛ − γ y21− 2y1 ⎞ − c2 ∞ γ y y yy − c1 y01 y02 − − 1 ∞ − 2 Γ(c1 ) Γ(c2 ) ∫0 ∫0 d y2 y2 e y2 − y1 ⎜⎜ e c1 − 2 c2 − 2 = d y1 y1 e y1 y01 y02 − 1⎟ . ⎟ ⎝ ⎠ 191 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić 3 The Bivariate Rayleigh-Gamma Distribution The conditional bivariate Rayleigh distribution is given by: 1 ⎛r2 r 2 ⎞ 4r1r2 − ⎜ 1 + 2 ⎟ 1−ρ ⎜ y y ⎟ ⎛ 2 ρr1r2 ⎞ Pr1r2 ( r1r2 / y1 y2 ) = e ⎝ 1 2 ⎠ I0 ⎜ ⎟ , r1 , r2 ≥ 0. (19) y1 y2 (1 − ρ ) ⎜ (1 − ρ ) y y ⎝ 1 2 ⎟ ⎠ Let us now define two random variables γ1 and γ 2 in terms of r1 and r2 by the transformation: γ1 = r12 and γ 2 = r22 . (20) The Jacobian of this transformation is: 1 J = . 4 γ1 γ 2 The conditional squared bivariate Rayleigh distribution is: 1 ⎛γ γ ⎞ 1 − 1 2 ⎜ + ⎟ ⎛ 2 ρ γ1 γ 2 ⎞ P γ1 γ 2 ( γ1γ 2 / y1 y2 ) = 1−ρ y y e ⎝ 1 2 ⎠ I0 ⎜ ⎟. (21) y1 y2 (1 − ρ ) ⎜ (1 − ρ ) y y ⎝ 1 2 ⎟ ⎠ Assuming that y1 and y2 are i.d. gamma RV’s with parameters c and y0 , the joint pdf of y1 and y2 can be expressed as: c −1 y1 + y2 ρ1 2 c −1 − ⎛ 4ρ1 y1 y2 ⎞ Py1 y2 ( y1 y2 ) = ( y1 y2 ) 2 e y0 (1−ρ1 ) I c −1 ⎜ ⎟, (22) Γ(c) (1 − ρ ) y0 c +1 ⎜ y0 (1 − ρ1 ) ⎟ ⎝ ⎠ where ρ1 is the correlation between y1 and y2 , and I c −1 is the modified Bessel function of the first kind of order (c − 1) . By averaging (21) with respect to y1 and y2 it follows, ∞ ∞ P γ1γ 2 ( γ1γ 2 ) = ∫ d y1 ∫ d y2 P γ1γ 2 ( γ1γ 2 / y1 y2 ) Py1 y2 ( y1 y2 ) . (23) 0 0 By substituting (21) and (22) in (23) it follows that: 1 ⎛ γ1 γ 2 ⎞ ∞ ∞ 1 − ⎜ + ⎟ ⎛ 2 ρ ⋅ γ1 γ 2 ⎞ P γ1 γ 2 ( γ1 γ 2 ) = ∫ d y1 ∫ d y2 e ( )⎝ 1 2 ⎠ I0 ⎜ 1−ρ y y ⎟⋅ 0 0 y1 y2 (1 − ρ ) ⎜ (1 − ρ ) y y ⎝ 1 2 ⎟ ⎠ ( c −1) (24) y1 + y2 ρ1 2 ( c −1) − ⎛ 2 ρ1 y1 y2 ⎞ ⋅ ( y1 y2 ) 2 e y0 (1−ρ1 ) I c −1 ⎜ ⎟. Γ(c) (1 − ρ1 ) y0 c +1 ⎜ y0 (1 − ρ1 ) ⎟ ⎝ ⎠ 192  On the K and KG Fading Channels The infinite series representation for I 0 ( x) and I n ( x) are: ∞ 2i ⎛ x⎞ 1 1 I 0 ( x) = ∑ ⎜ ⎟ , i1 = 0 ⎝ 2 ⎠ (i1 !) 2 2 i2 + n (25) ∞ ⎛ x⎞ 1 I n ( x) = ∑ ⎜ ⎟ . i2 = 0 ⎝ 2 ⎠ i2 !Γ(i2 + n + 1) Substituting (25) in (24) and changing the order of summations and integration can be written: ( c −1) 1 ρ1 2 P γ1 γ 2 ( γ1 γ 2 ) = ⋅ ⋅ (1 − ρ) Γ(c)(1 − ρ1 ) y0c +1 1 ⎛ γ1 γ 2 ⎞ 2 i1 1 − (1−ρ ) ⎜⎝ y1 + y2 ⎟⎠ ∞ 1 ⎛ ρ γ1 γ 2 ⎞ ∞ ∞ ⋅ ∫ d y1 ∫ d y2 e ⋅∑ ⎜ 2 ⎜ ⎟ ⋅ y1 y2 i1 = 0 (i1 !) ⎝ (1 − ρ) ⎟ 0 0 ⎠ ( c −1) 2 i2 + c −1 ( y1 y2 ) y1 + y2 i2 + 1 − ∞ ⎛ ρ1 ⎞2 ⋅ ( y1 y2 ) y0 (1−ρ1 ) ∑ − i1 + ( c −1) 2 e ⎜ ⎟ = i2 = 0 i2 !Γ(i2 + c) ⎜⎝ y0 (1 − ρ1 ) ⎟⎠ (26) 1 ( c −1) 2 i1 1 ρ1 2 ∞ 1 ⎛ ρ γ1 γ 2 ⎞ c +1 ∑ = ⋅ ⎜ ⎟ ⋅ (1 − ρ ) Γ ( c )(1 − ρ1 ) y0 i1 =0 ( i1 !)2 ⎜⎝ (1 − ρ ) ⎟ ⎠ 2 i2 + c −1 ∞ 1 ⎛ ρ1 ⎞ ⋅∑ ⎜ ⎟ ⋅ i2 = 0 i2 !Γ ( i2 + c ) ⎝ y0 (1 − ρ1 ) ⎠ ⎜ ⎟ ∞ γ1 y1 ∞ γ2 y2 − − − − ⋅ ∫ d y1 ⋅ y1c −i1 + i2 − 2 e (1−ρ ) y1 (1−ρ1 ) y0 ∫ d y2 ⋅ y2 1 2 e c −i + i − 2 (1−ρ ) y2 (1−ρ1 ) y0 . 0 0 1 2 i1 ( c −1) 4ρ1 2 ∞ 1 ⎛ ρ γ1 γ 2 ⎞ c +1 ∑ P γ1 γ 2 ( γ1 γ 2 ) = ⎜ ⎟ ⋅ Γ ( c )(1 − ρ )(1 − ρ1 ) y0 i1 = 0 ( i1 !)2 ⎜⎝ (1 − ρ ) ⎟ ⎠ 2 i2 + c −1 c − i1 + i2 −1 ∞ 1 ⎛ ρ1 ⎞ ⎛ (1 − ρ1 ) 1 ⎞ ⋅∑ ⎜ ⎟ ⋅⎜ ⎟ ⋅ i2 = 0 i2 !Γ ( i2 + c ) ⎝ y0 (1 − ρ1 ) ⎠ ⎜ ⎟ ⎜ (1 − ρ ) y γ γ ⎟ ⎝ 0 1 2 ⎠ ⎛ γ1 ⎞ ⎛ γ2 ⎞ ⋅ K c −i1 + i2 −1 ⎜ 2 ⎟ K c −i1 + i2 −1 ⎜ 2 ⎟. ⎜ (1 − ρ ) y0 (1 − ρ1 ) ⎟ ⎜ (1 − ρ ) y0 (1 − ρ1 ) ⎟ ⎝ ⎠ ⎝ ⎠ 193 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić Fig. 1 – The probability of bivariate Rayleigh-Gamma distribution. 4 Nakagami – Gamma Distribution The conditional Nakagami–m distribution has the term: m m 2 ⎛ m ⎞ 2 m −1 − y r 2 Pr ( r / y ) = ⎜ ⎟ r e , r ≥ 0, (27) Γ (m) ⎝ y ⎠ 194  On the K and KG Fading Channels where Γ(m) 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 dr 1 γ = r 2 , r = γ and = . dγ 2 γ we obtain probability density function for squared Nakagami– m random variable as follow: m m 2 ⎛ m ⎞ m −1 − y γ Pγ ( γ / y ) = ⎜ ⎟ γ e , γ≥0. (28) Γ ( m) ⎝ y ⎠ Averaging P γ ( γ / y ) with respect to y, using (6), it follows: ∞ P γ ( γ ) = ∫ P γ ( γ / y )Py ( y ) d y = 0 ∞ m m y 2 ⎛ m ⎞ m −1 − y γ y0 − c c −1 − y0 =∫ ⎜ ⎟ γ e y e dy= 0 Γ ( m) ⎝ y ⎠ Γ (c ) ∞ mγ y (29) y0 − c − − = m γ ∫ d y ⋅ y − m + c −1 e m m −1 y y0 = Γ (c )Γ ( m ) 0 y0 − c c−m ⎛ mγ ⎞ = m m γ m −1 ⋅ 2 ( my0 γ ) 2 K c − m ⎜ 2 ⎟. Γ (c )Γ ( m ) ⎜ y0 ⎟⎠ ⎝ The cumulative distribution function of Nakagami-Gamma random variable is: γ γ ∞ mx y y0 − c m m − − F γ ( γ ) = ∫ P γ ( x)d x = ∫ x m −1 d x ∫ d y ⋅ y c − m −1 e y y0 . 0 Γ (c )Γ ( m ) 0 0 Changing the order of integrations, it follows: y γ mx y0− c m m ∞ − − F γ(γ) = ∫ d y ⋅ y c − m −1 e y0 ∫ x m −1 e dx= y Γ (c )Γ ( m ) 0 0 m y y0− c m m ∞ − ⎛y⎞ ⎡ ⎛ mx ⎞ ⎤ = ∫ Γ (c )Γ ( m ) 0 d y ⋅ y c − m −1 e y0 ⎜ ⎟ ⎢ Γ(m) − γ ⎜ m, ⎝m⎠ ⎣ ⎝ ⎟⎥ = y ⎠⎦ (30) y y0 − c ∞ c −1 − y0 ⎛ mx ⎞ Γ(c)Γ(m) ∫0 =1− y e γ ⎜ m, ⎟ d y. ⎝ y ⎠ The MGF of Nakagami–m random variables is: 195 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić ∞ ∞ mγ y y0− c m m m −1 ∞ − − M γ ( s ) = e = ∫ e P γ ( γ )d γ = ∫ d γ ⋅ e γs γs γ ∫ d y ⋅ y c − m −1 e γs y y0 . 0 0 Γ (c )Γ ( m ) 0 Changing the order of integrations, we can write: y mγ y0− c (m) m ∞ − ∞ − Γ(c)Γ(m) ∫0 ∫0 c − m −1 γs m −1 M γ(s) = d y ⋅ y e y0 e γ e y dγ = y m y − c ( m) m ∞ − ⎛ y ⎞ Γ(c)Γ(m) ∫0 c − m −1 = 0 d y ⋅ y e y0 ⎜ ⎟ = (31) ⎝ m − ys ⎠ y y − c ( m) m ∞ − Γ(c)Γ(m) ∫0 c − m −1 = 0 d y ⋅ y e y0 (m − ys ) − m . Moment of n-th order of Nakagami- m RV’s is: ∞ ∞ mγ y y − c m m m −1 ∞ − − γ = ∫ γ P γ ( γ )d γ = ∫ γ d γ 0 n n γ ∫ d y ⋅ y c − m −1 e n y y0 . 0 0 Γ (c )Γ ( m ) 0 Changing the order of the integrations, it follows: y γm y −c mm ∞ − ∞ − Γ(c)Γ(m) ∫0 ∫0 c − m −1 n + m −1 γ = 0n d y ⋅ y e y0 d γ ⋅ γ e y = y n+m y −c mm ∞ − y0 ⎛ y ⎞ ∫ c − m −1 = 0 d y ⋅ y e ⎜ ⎟ Γ ( n + m) = Γ (c )Γ ( m ) 0 ⎝m⎠ (32) ∞ y y −c mm − = 0 Γ(n + m) ∫ d y ⋅ y c − m −1 e y0 = Γ (c )Γ ( m ) 0 Γ ( n + m )Γ (c + n ) n = y0 . Γ (c )Γ ( m ) 5 Bivariate Nakagami-Gamma Distribution The bivariate Nakagami-Gamma distribution has the form: Pr1 r2 ( r1r2 / y1 y2 ) = m ⎛ r12 r22 ⎞ 4m m +1 ( r1r2 ) m − ⎜ + 1−ρ ⎜⎝ y1 y2 ⎟⎟ ⎛ 2m ρr1r2 ⎞ (33) = e ⎠ I m −1 ⎜ ⎟, 1 ( m −1) ⎜ (1 − ρ) y y ⎟ Γ( m) y1 y2 (1 − ρ)( y1 y2 ) 2 ⎝ 1 2 ⎠ 196  On the K and KG Fading Channels cov ( r12 + r2 2 ) where y1 = r , y2 = r2 , and ρ = 2 2 is the power correlation var ( r12 ) + var ( r2 2 ) 1 coefficient ( 0 ≤ ρ < 1 ). Transformation of random variables r1 and r2 respectively into γ1 and γ 2 , gives γ1 = r12 , γ 2 = r2 2 and the Jacobian is: 1 J = . 4 γ1 γ 2 We can obtain the joint probability density function of shared Nakagami-m RV’s in the form: P γ1γ 2 ( γ1γ 2 / y1 y2 ) = m ⎛ γ1 γ 2 ⎞ m m +1 m −1 − ⎜ + ⎟ 1−ρ ⎝ y1 y2 ⎠ ⎛ 2 m ρ γ1 γ 2 ⎞ (34) = ( γ1 γ 2 ) e I m −1 ⎜ ⎟. m +1 ⎜ (1 − ρ) y y ⎟ Γ( m)(1 − ρ)( y1 y2 ) 2 ⎝ 1 2 ⎠ Averaging the expression (34) with respect to y1 and y2 , it follows: ∞ ∞ P γ1γ 2 ( γ1γ 2 ) = ∫ d y1 ∫ d y2 P γ1γ 2 ( γ1γ 2 / y1 y2 ) Py1 y2 ( y1 y2 ) = 0 0 ( γ1 γ 2 ) m −1 m ⎛ γ1 γ 2 ⎞ ∞ ∞ m m +1 − ⎜ + ⎟ ⎛ 2 m ρ γ1 γ 2 ⎞ = ∫ d y1 ∫ d y2 1−ρ ⎝ y1 y2 ⎠ e I m −1 ⎜ ⎟⋅ (35) m +1 ⎜ (1 − ρ) y y ⎟ 0 0 Γ(m)(1 − ρ)( y1 y2 ) 2 ⎝ 1 2 ⎠ c −1 1 ( y1 y2 ) 2 ( c −1) y1 + y2 ρ1 2 ⎛ 2 ρ1 y1 y2 ⎞ − y0 (1−ρ1 ) ⋅ I c −1 ⎜ ⎟. e Γ(c)(1 − ρ1 ) y0 ⎜ y0 (1 − ρ1 ) ⎟ c +1 ⎝ ⎠ Using the infinite series representations for modified Bessel functions and changing the order of summation and integration, we can write: m m +1 ( γ1γ 2 ) m −1 ∞ ∞ m ⎛ γ1 γ 2 ⎞ 1 − ⎜ + ⎟ P γ1 γ 2 ( γ1 γ 2 ) = ∫d y ∫d y ( y y ) − ( m +1) 1−ρ ⎝ y1 y2 ⎠ 1 2 1 2 2 e ⋅ Γ(m)(1 − ρ) 0 0 2 i1 + m −1 m −1 1 ⎛ m ρ γ1 γ 2 ⎞ ⋅∑ ⎜ ⎟ ⋅ i1 = 0 i1 !Γ ( i1 + m ) ⎝ (1 − ρ) y1 y2 ⎠ ⎜ ⎟ 1 1 ( c −1) 2 i2 + c −1 ( y1 y2 ) 2 ( ) e− y (1−ρ ) ⋅ ∞ c −1 y1 + y2 ρ12 1 ⎛ ρ1 y1 y2 ⎞ ⋅ Γ ( c )(1 − ρ1 ) y0 c +1 ∑ 0 ⎜ i = 0 i2 !Γ ( i2 + c ) ⎝ ⎜ 1 ⎟ y0 (1 − ρ1 ) ⎟⎠ = 2 197 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić 1 2 i1 + m −1 ( c −1) m m +1 ( γ1γ 2 ) m −1 ρ12 ∞ 1 ⎛ m ρ γ1 γ 2 ⎞ c +1 ∑ = ⎜ ⎟ ⋅ Γ(m)(1 − ρ) Γ(c) (1 − ρ1 ) y0 i1 = 0 i1 !Γ ( i1 + m ) ⎜⎝ (1 − ρ ) ⎟ ⎠ 2 i2 + c −1 ∞ 1 ⎛ ρ1 ⎞ ⋅∑ ⎜ ⎟ ⋅ i2 = 0 i2 !Γ ( i2 + c ) ⎝ y0 (1 − ρ1 ) ⎠ ⎜ ⎟ ∞ m γ1 y1 ∞ mγ 2 y2 − − − − (1−ρ ) y1 (1−ρ1 ) y0 (1−ρ ) y2 (1−ρ1 ) y0 ⋅ ∫ d y1 ⋅ y 1 −1− i1 + i2 + c − m e ∫d y 2 ⋅ y2 −1− i1 + i2 + c − m e 0 0 P γ1 γ 2 ( γ1 γ 2 ) = 1 2 i1 + m −1 ( c −1) m m +1 ( γ1γ 2 ) m −1 4ρ12 ∞ 1 ⎛ m ρ γ1 γ 2 ⎞ c +1 ∑ = ⋅ ⋅ ⎜ ⎟ ⋅ Γ(m) (1 − ρ ) Γ(c) (1 − ρ1 ) y0 i1 = 0 i1 !Γ ( i1 + m ) ⎜⎝ (1 − ρ ) ⎟ ⎠ 2 i2 + c −1 c − m − i1 + i2 ∞ 1 ⎛ ρ1 ⎞ ⎛ 1− ρ 1 ⎞ ⋅∑ ⎜ ⎟ ⋅⎜ ⎟ ⋅ i2 = 0 i2 !Γ ( i2 + c ) ⎝ y0 (1 − ρ1 ) ⎠ ⎜ ⎟ ⎜ 1 − ρ1 my γ γ ⎟ ⎝ 0 1 2 ⎠ ⎛ mγ 1 ⎞ ⎛ mγ 2 ⎞ ⋅ K c − m −i1 + i2 ⎜ 2 ⎟ K c − m −i1 + i2 ⎜ 2 ⎟. ⎜ ⎝ (1 − ρ )(1 − ρ1 ) y0 ⎟⎠ ⎜ ⎝ (1 − ρ )(1 − ρ1 ) y0 ⎟⎠ Fig. 2 – The probability of bivariate Nakagami-Gamma distribution. 198  On the K and KG Fading Channels Fig. 3 – The probability of bivariate Nakagami-Gamma distribution. Fig. 4 – The probability of bivariate Nakagami-Gamma distribution. 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. 199 D. Aleksić, M. Stefanović, Z. Popović, D. Radenković, J.D. Ristić 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. 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