Modeling of Fading and Shadowing

Abstract

We presented various models to describe the statistical fluctuations in wireless channels. The models ranged from the simple Rayleigh ones, to cascaded ones, and to complex models such as those based on κ−μ and η− μ distributions. The models were compared in terms of their density functions, distribution functions, and quantitative measures such as error rates and outage probabilities. The shadowing was examined using the traditional lognormal model and approaches based on similarities between lognormal pdf and other density functions. We looked at the simultaneous existence of short-term fading and shadowing using the Nakagami-lognormal density function and approximations to it using the GK model and the Nakagami-N-gamma model. To complete the study of these models for fading, shadowing, and shadowed fading channels, we examined some second-order statistical properties for several models.

Appendix

We have seen the flexibility offered through the use of the Meijer G-functions in expressing the density functions, distribution functions, error rates, and outage probabilities. They will also be used extensively in Chaps. 5 and 6 as well. Even though they were introduced in Chap. 2, their properties and functionalities which make them very versatile in the study of communication systems were not discussed. In this section, we will provide an overview of the definition, properties, and characteristics of these functions (Mathai and Saxena 1973; Gradshteyn and Ryzhik 2007; Wolfram 2011). We will also examine the relationship among functions commonly encountered in communications such as the incomplete gamma functions, hypergeometric functions, Bessel functions, error functions, complementary error functions, etc. and the MeijerG functions. Furthermore, closed form expressions for the error rates in cascaded Nakagami channels will be given in terms of Meijer G-functions.

Most of the software packages such as Maple (Maple 2011), Mathematica (Wolfram 2011), and Matlab (Matlab 2011) provide computations involving Meijer G-functions. Mathematica also provides an excellent resource for understanding the properties of these functions. Additionally, Maple can provide a means to understand the relationships among Meijer G-functions and other functions.

The Meijer G-function G(x) is defined as (Gradshteyn and Ryzhik 2007)

$$ \begin{array}{lll}G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll}{{a_1}, \ldots, {a_n}, \ldots, {a_p}} \\{{b_1}, \ldots, {b_m}, \ldots, {b_q}}\end{array}}\right.} \right) = G_{p,q}^{m,n}\left(x\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.\right) \\= G_{p,q}^{m,n}(x) = G(x) \\= \frac{1}{{2\pi j}}\int {\frac{{\prod\nolimits_{i = 1}^m {\Gamma ({b_i} - s)} \prod\nolimits_{i = 0}^n {\Gamma (1 - {a_i} + s)} }}{{\prod\nolimits_{i = m + 1}^q {\Gamma (1 - {b_i} + s)} \prod\nolimits_{i = n + 1}^p {\Gamma ({a_i} - s)} }}} {x^s}{{d}}s.\end{array}$$

(4.254)

In most of the computational packages, the G(x) function is expressed as (Maple 2011)

$$ \begin{array}{lll}G_{p,q}^{m,n}\left( {x\left|{\begin{array}{lll} {{a_1}, \ldots, {a_n}, \ldots, {a_p}}\\{{b_1}, \ldots, {b_m}, \ldots, {b_q}} \\\end{array}} \right.} \right) ={{Meijer}}\,{{G}}\left(\left[\underbrace {\left[\underbrace{\begin{array}{lll} {{a_1},} & {{a_2,} \ldots,} & {{a_n}}\end{array}}_{n\;{{terms}}}\right],\left[\underbrace{\begin{array}{lll} {{a_{n + 1},}} & {{a_{n + 2},} \ldots,} & {{a_p}}\end{array}}_{(p -n)\;{{terms}}}\right]}_{p\;{{terms}}}\right],{\left[\underbrace{\left[\underbrace {\begin{array}{lll} {{b_1},} & {{b_2,}\ldots,} & {{b_m}}\end{array}}_{m\;{{terms}}}\right],\left[\underbrace{\begin{array}{lll} {{b_{m + 1,}}} & {{b_{m + 2},} \ldots,} & {{b_q}}\end{array}}_{(q -m)\;{{terms}}}\right]_{q\;{{terms}}}}\right],\, }x\right).\end{array}$$

(4.255)

A few properties of Meijer G-functions (Mathai and Saxena 1973; Gradshteyn and Ryzhik 2007; Wolfram 2011)

• (a) Multiplication G(w) with w ^k:

$$ {w^k}G_{pq}^{mn}\left( {w\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right) = G_{pq}^{mn}\left( {w\left| {\begin{array}{lll} {{a_p} + k} \\{{b_q} + k}\end{array}} \right.} \right). $$

(4.256)

• (b) Inversion of the argument

$$ G_{pq}^{mn}\left( {\frac{1}{w}\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right) = G_{qp}^{nm}\left( {w\left| {\begin{array}{lll} {1 - {b_q}} \\{1 - {a_p}}\end{array}} \right.} \right). $$

(4.257)

• (c) Transformations

$$ \begin{array}{lll} G_{p,q}^{m,n}\left( {z\left| {\begin{array}{lll} {{a_1}, \ldots, {a_n}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_n}, \ldots, {b_{q - 1}},{a_1}}\end{array}} \right.} \right) = \ \ G_{p - 1,q - 1}^{m,n - 1}\\\times\left( {z\left| {\begin{array}{lll} {{a_2}, \ldots, {a_n}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_n}, \ldots, {b_{q - 1}}}\end{array}} \right.} \right),\quad n,p,q \geqslant 1,\end{array}$$

(4.258)

$$ \begin{array}{lll}G_{p + 1,q + 1}^{m + 1,n}\left( {z\left| {\begin{array}{lll} {{a_1}, \ldots, {a_n}, \ldots, {a_p},1 - r} \\{0,{b_1},{b_2}, \ldots, {b_n}, \ldots, {b_q}}\end{array}} \right.} \right) = \ \ {( - 1)^r}G_{p + 1,q + 1}^{m,n + 1}\\\times\left( {z\left| {\begin{array}{lll} {1 - r,{a_1}, \ldots, {a_n}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_n}, \ldots, {b_q},1}\end{array}} \right.} \right),\quad r = 0,1,2, \ldots \,.\end{array}$$

(4.259)

• (d) Differentiation

$$ \begin{array}{lll}\frac{{\partial G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right)}}{{\partial x}} = \frac{1}{x}G_{p + 1,q + 1}^{m,n + 1}\left( {x\left| {\begin{array}{lll} {0,{a_1},{a_2}, \ldots, {a_n},{a_{n + 1}}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_m},1,{b_{m + 1}}, \ldots, {b_q}}\end{array}} \right.} \right) = G_{p+1,q + 1}^{m,n + 1}\left( {x\left| {\begin{array}{lll} { - 1,({a_1} - 1),({a_2} - 1), \ldots, ({a_n} - 1),({a_{n + 1}} - 1), \ldots, ({a_p} - 1)} \\{({b_1} - 1),({b_2} - 1), \ldots, ({b_m} - 1),0,({b_{m + 1}} - 1), \ldots, ({b_q} - 1)}\end{array}} \right.} \right),\end{array}$$

(4.260)

$$\begin{array}{lll} \frac{{\partial G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right)}}{{\partial x}} = ({a_1} - 1)G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {{a_1},{a_2}, \ldots, {a_n},{a_{n + 1}}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_m},{b_{m + 1}}, \ldots, {b_q}}\end{array}} \right.} \right) + G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {({a_1} - 1),{a_2}, \ldots, {a_n},{a_{n + 1}}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_m},{b_{m + 1}}, \ldots, {b_q}}\end{array}} \right.} \right),\end{array}$$

(4.261)

$$ \frac{{\partial \left\{ {{x^\alpha }G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right)} \right\}}}{{\partial x}} = {x^{\alpha - 1}}G_{p+1,q + 1}^{m,n + 1}\left( {x\left| {\begin{array}{lll} { - \alpha, {a_1},{a_2}, \ldots, {a_n},{a_{n + 1}}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_m},(1 - \alpha ),{b_{m + 1}}, \ldots, {b_q}}\end{array}} \right.} \right), $$

(4.262)

$$ \frac{{{\partial^\nu }G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right)}}{{\partial {x^\nu }}} = \frac{1}{{{x^\nu }}}G_{p+1,q + 1}^{m,n + 1}\left( {x\left| {\begin{array}{lll} {0,{a_1},{a_2}, \ldots, {a_n},{a_{n + 1}}, \ldots, {a_p}} \\{{b_1},{b_2}, \ldots, {b_m},\nu, {b_{m + 1}}, \ldots, {b_q}}\end{array}} \right.} \right). $$

(4.263)

• (e) Integration

$$ \int {{x^{\beta - 1}}} G_{p,q}^{m,n}\left( {ax\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right){{d}}x = {x^\beta }G_{p + 1,q + 1}^{m,n + 1}\left( {ax\left| {\begin{array}{lll} {(1 - \beta ),{a_p}} \\{{b_q}, - \beta }\end{array}} \right.} \right), $$

(4.264)

$$ \int {G_{p,q}^{m,n}\left( {x\left| {\begin{array}{lll} {{a_p}} \\{{b_q}}\end{array}} \right.} \right)} \,{{d}}x = G_{p + 1,q + 1}^{m,n + 1}\left( {x\left| {\begin{array}{lll} {1,({a_p} + 1)} \\{({b_q} + 1),0}\end{array}} \right.} \right), $$

(4.265)

$$ \begin{array}{lll}\int_0^\infty {G_{p,q}^{m,n}} \left( {\lambda x\left| {\begin{array}{lll} {{a_1}, \ldots, {a_n}, \ldots, {a_p}} \\{{b_1}, \ldots, {b_m}, \ldots, {b_p}}\end{array}} \right.} \right)G_{u,v}^{r,s}\left( {\beta x\left| {\begin{array}{lll} {{c_1}, \ldots, {c_s}, \ldots, {c_u}} \\{{d_1}, \ldots, {d_r}, \ldots, {d_v}}\end{array}} \right.} \right){{d}}x = \frac{1}{\lambda }G_{q + u,p + v}^{n + r,m + s}\left( {\frac{\beta }{\lambda }\left| {\begin{array}{lll} { - {b_1}, \ldots, - {b_m},{c_1}, \ldots, {c_s}, \ldots, {c_u}, - {b_{m + 1}}, \ldots, - {b_q}} \\{ - {a_1}, \ldots, - {a_n},{d_1}, \ldots, {d_r}, \ldots, {d_v}, - {a_{n + 1}}, \ldots, - {a_p}}\end{array}} \right.} \right).\end{array} $$

(4.266)

Relationships of Meijer G-functions to other functions

• 1. Exponential functions

$$ \frac{1}{a}\exp \left( { - \frac{x}{a}} \right) = \frac{1}{a}G_{0,1}^{1,0}\left( {\frac{x}{a}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right) = \left( {\frac{x}{a}} \right)\frac{1}{x}G_{0,1}^{1,0}\left( {\frac{x}{a}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right) = \frac{1}{x}G_{0,1}^{1,0}\left( {\frac{x}{a}\left| {\begin{array}{lll} - \\1\end{array}} \right.} \right). $$

(4.267)

To arrive at the result on the far right hand side of (4.267), we have made use of the identity in (4.256).

$$ \exp \left( { - \frac{x}{a}} \right) = G_{0,1}^{1,0}\left( {\frac{x}{a}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right), $$

(4.268)

$$ \exp \left( {\frac{x}{a}} \right) = G_{0,1}^{1,0}\left( { - \frac{x}{a}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right), $$

(4.269)

$$ 1 - \exp \left( { - \frac{x}{a}} \right) = 1 - G_{0,1}^{1,0}\left( {\frac{x}{a}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right) = G_{1,2}^{1,1}\left( {\frac{x}{a}\left| {\begin{array}{lll} 1 \\{1,0}\end{array}} \right.} \right). $$

(4.270)

• 2. Gaussian function

$$ \frac{1}{{\sqrt {{2\pi {\sigma^2}}} }}{{{e}}^{ - ({x^2}/2{\sigma^2})}} = \frac{1}{{\sqrt {{2\pi {\sigma^2}}} }}G_{0,1}^{1,0}\left( {\frac{{{x^2}}}{{2{\sigma^2}}}\left| {\begin{array}{lll} {} \\0\end{array}} \right.} \right). $$

(4.271)

• 3. Gaussian integrals

$$ \int_{ - \infty }^y {\frac{1}{{\sqrt {{2\pi }} }}} \exp \left( { - \frac{{{x^2}}}{2}} \right)\,{{d}}x = \frac{1}{2} + \frac{1}{{2\sqrt {\pi } }}G_{1,2}^{1,1}\left( {\frac{1}{2}{y^2}\left| {\begin{array}{lll} 1 \\{\frac{1}{2},0}\end{array}} \right.} \right),\quad y < 0, $$

(4.272)

$$ \int_{ - \infty }^{ - y} {\frac{1}{{\sqrt {{2\pi }} }}} \exp \left( { - \frac{{{x^2}}}{2}} \right)\,{{d}}x = \frac{1}{2} - \frac{1}{{2\sqrt {\pi } }}G_{1,2}^{1,1}\left( {\frac{1}{2}{y^2}\left| {\begin{array}{lll} 1 \\{\frac{1}{2},0}\end{array}} \right.} \right),\quad y < 0. $$

(4.273)

• 4. Complementary error functions (for bit error rate calculations)

$$ {{erfc}}\sqrt {x} = 1 - \frac{1}{{\sqrt {\pi } }}G_{1,2}^{1,1}\left( {x\left| {\begin{array}{lll} 1 \\{\frac{1}{2},0}\end{array}} \right.} \right)=\frac{1}{\sqrt\pi}G_{1,2}^{2,0}\left(x\left|\begin{array}{lll} 1 \\{\frac{1}{2},0}\end{array}\right.\right),$$

(4.274)

$$ \frac{1}{2}{{erfc}}\sqrt {x} = \frac{1}{2}\left\{ {1 - \frac{1}{{\sqrt {\pi } }}G_{1,2}^{1,1}\left( {x\left| {\begin{array}{lll} 1 \\{\frac{1}{2},0}\end{array}} \right.} \right)} \right\}, $$

(4.275)

$$ \frac{{d}}{{{{d}}x}}\left( {\frac{1}{2}{{erfc}}\sqrt {x} } \right) = - \frac{1}{2}\frac{1}{{\sqrt {{\pi x}} }}{{{e}}^{ - x}} = - \frac{1}{2}\left( {\frac{1}{{\sqrt {{\pi x}} }}} \right)G_{0,1}^{1,0}(x|0) = - \left( {\frac{1}{{2\sqrt {\pi } }}} \right)G_{0,1}^{1,0}\left( {x\left| { - \frac{1}{2}} \right.} \right), $$

(4.276)

$$ Q(\sqrt {{2z}} ) = \frac{1}{2}{{erfc(}}\sqrt {z} {)} = \frac{1}{2}\left\{ {1 - \frac{1}{{\sqrt {\pi } }}G_{1,2}^{1,1}\left( {x\left| {\begin{array}{lll} 1 \\{\frac{1}{2},0}\end{array}} \right.} \right)} \right\}. $$

(4.277)

• 5. Natural logarithm

$$ \ln \,(x) = (x - 1)G_{2,2}^{1,2}\left( {(x - 1)\left| {\begin{array}{lll} {0,0} \\{0, - 1}\end{array}} \right.} \right) = G_{2,2}^{1,2}\left( {(x - 1)\left| {\begin{array}{lll} {1,1} \\{1,0}\end{array}} \right.} \right), $$

(4.278)

$$ \ln \,(1 + x) = xG_{2,2}^{1,2}\left( {x\left| {\begin{array}{lll} {0,0} \\{0, - 1}\end{array}} \right.} \right) = G_{2,2}^{1,2}\left( {x\left| {\begin{array}{lll} {1,1} \\{1,0}\end{array}} \right.} \right). $$

(4.279)

• 6. Bessel functions

$$ {K_{m - c}}\left( {n\frac{x}{Z}} \right) = \frac{1}{2}G_{0,2}^{2,0}\left( {\frac{{{n^2}}}{4}{{\left( {\frac{x}{Z}} \right)}^2}\left| {\begin{array}{lll} - \\{\frac{1}{2}(m - c),\frac{1}{2}(c - m)}\end{array}} \right.} \right), $$

(4.280)

$$ {K_{m - c}}\left( {n\sqrt {{\frac{{xmc}}{Z}}} } \right) = \frac{1}{2}G_{0,2}^{2,0}\left( {\frac{{{n^2}}}{4}\frac{{xmc}}{Z}\left| {\begin{array}{lll} - \\{\frac{1}{2}(m - c),\frac{1}{2}(c - m)}\end{array}} \right.} \right), $$

(4.281)

$$ {J_n}(ax) = G_{0,2}^{1,0}\left( {\frac{1}{4}{a^2}{x^2}\left| {\begin{array}{lll} - \\{\frac{n}{2}, - \frac{n}{2}}\end{array}} \right.} \right), $$

(4.282)

$$ {I_n}(ax) = \frac{1}{{{{( - 1)}^{n/2}}}}G_{0,2}^{1,0}\left( { - \frac{1}{4}{a^2}{x^2}\left| {\begin{array}{lll} - \\{\frac{n}{2}, - \frac{n}{2}}\end{array}} \right.} \right),\quad n \ne 0, $$

(4.283)

$$ {I_0}(ax) = G_{0,2}^{1,0}\left( { - \frac{1}{4}{a^2}{x^2}\left| {\begin{array}{lll} - \\{0,0}\end{array}} \right.} \right). $$

(4.284)

• 7. Short-term fading faded channels

Probability density function of the amplitude (a) in a Nakagami faded channel is

$$ f(a) = 2{\left( {\frac{m}{{{Z_0}}}} \right)^m}\frac{{{a^{2m - 1}}}}{{\Gamma (m)}}\exp \left( { - \frac{m}{{{Z_0}}}a} \right) = \frac{2}{{a\Gamma (m)}}G_{0,1}^{1,0}\left( {\frac{m}{{{Z_0}}}{a^2}\left| {\begin{array}{lll} - \\m\end{array}} \right.} \right). $$

(4.285)

Probability density function of the SNR in a Nakagami channel is

$$ f(x) = {\left( {\frac{m}{{{Z_0}}}} \right)^m}\frac{{{x^{m - 1}}}}{{\Gamma (m)}}\exp \left( { - m\frac{x}{{{Z_0}}}} \right) = {\left( {\frac{m}{{{Z_0}}}} \right)^m}\frac{{{x^{m - 1}}}}{{\Gamma (m)}}G_{0,1}^{1,0}\left( {m\frac{x}{{{Z_0}}}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right) = \frac{{{x^{ - 1}}}}{{\Gamma (m)}}{\left( {\frac{{mx}}{{{Z_0}}}} \right)^m}G_{0,1}^{1,0}\left( {m\frac{x}{{{Z_0}}}\left| {\begin{array}{lll} - \\0\end{array}} \right.} \right) = \frac{1}{{\Gamma (m)x}}G_{0,1}^{1,0}\left( {m\frac{x}{{{Z_0}}}\left| {\begin{array}{lll} - \\m\end{array}} \right.} \right). $$

(4.286)

To arrive at the result on the far right hand side of (4.286), we have made use of the identity in (4.256). The average SNR is Z ₀.

The cumulative distribution function (CDF) of the SNR can take any one of the several forms expressed below. Note that γ(.,.) is the lower incomplete gamma function and Γ(.,.) is the upper incomplete gamma function (Gradshteyn and Ryzhik 2007). The hypergeometric function is represented by _p F _q (.).

$$ \int_0^x {{{\left( {\frac{m}{{{Z_0}}}} \right)}^m}\frac{{{y^{m - 1}}}}{{\Gamma (m)}}\exp \left( { - m\frac{y}{{{Z_0}}}} \right)} \,{{d}}y = \left\{ \begin{array}{lll}\displaystyle\frac{{\gamma(m,(mx/{Z_0}))}}{{\Gamma (m)}} = 1 - \frac{{\Gamma(m,(mx/{Z_0}))}}{{\Gamma (m)}}. \hfill \\\displaystyle\frac{{{{(mx/{Z_0})}^m}}}{{\Gamma (m)}}G_{1,2}^{1,1}\left( {\frac{{mx}}{{{Z_0}}}\left| {\begin{array}{lll} {1 - m} \\{0, - m}\end{array}} \right.} \right) = \frac{1}{{\Gamma (m)}}G_{1,2}^{1,1}\left( {\frac{{mx}}{{{Z_0}}}\left| {\begin{array}{lll} 1 \\{m,0}\end{array}} \right.} \right). \hfill \\1 - \displaystyle\frac{1}{{\Gamma (m)}}G_{1,2}^{2,0}\left( {\frac{m}{{{Z_0}}}x\left| {\begin{array}{lll} 1 \\{0,m}\end{array}} \right.} \right). \hfill \\\displaystyle\frac{1}{{m\Gamma (m)}}{\left( {\frac{{mx}}{{{Z_0}}}} \right)^m}{}_1{F_1}\left( {[m],[1 + m], - \frac{{mx}}{{{Z_0}}}} \right). \end{array}\right.$$

(4.287)

We can also express the density function of the SNR in a generalized gamma fading channel as

$$ f(z) = \frac{{\lambda {z^{\lambda m - 1}}}}{{\Gamma (m){\beta^m}}}\exp \left( { - \frac{{{z^\lambda }}}{\beta }} \right) = \frac{\lambda }{{\Gamma (m)z}}G_{0,1}^{1,0}\left( {\frac{{{z^\lambda }}}{\beta }\left| {\begin{array}{lll} - \\m\end{array}} \right.} \right). $$

(4.288)

The density function of the SNR in a Weibull fading channel can be obtained from (4.288) by putting m = 1. The probability density function of the SNR in a Rician faded channel (K ₀ is the Rician factor defined earlier) can be expressed as the product of two Meijer G-functions as

$$ \begin{array}{lll} f(z) = \displaystyle\frac{{{K_0} + 1}}{{{Z_{{Ri}}}}}\exp \left[ { - {K_0} - ({K_0} + 1)\frac{z}{{{Z_{{Ri}}}}}} \right]{I_0}\left( {2\sqrt {{\frac{{{K_0}({K_0} + 1)}}{{{Z_{{Ri}}}}}z}} } \right) \\= \left( {\displaystyle\frac{{{K_0} + 1}}{{{Z_{{Ri}}}}}} \right)G_{0,1}^{1,0}\left( {\left[ {{K_0} + ({K_0} + 1)\displaystyle\frac{z}{{{Z_{{Ri}}}}}} \right]\left| {\begin{array}{lll}- \\0\end{array}} \right.} \right) G_{0,2}^{1,0}\left( \left[ { - {K_0}({K_0} + 1)\displaystyle\frac{z}{{{Z_{{Ri}}}}}} \right]\left| {\begin{array}{lll} - \\{0,0} \end{array}} \right. \right).\end{array} $$

(4.289)

• 8. Shadowed fading channel: gamma–gamma pdf or generalized K pdf

$$ f(z) = \int_0^\infty {{{\left( {\frac{m}{y}} \right)}^m}\frac{{{z^{m - 1}}}}{{\Gamma (m)}}\exp \left( { - m\frac{z}{y}} \right)} \frac{{{y^{c - 1}}}}{{{b^c}\Gamma (c)}}\exp \left( { - \frac{y}{b}} \right)\,{{d}}y = \int_0^\infty {{{\left( {\frac{m}{y}} \right)}^m}\frac{{{z^{m - 1}}}}{{\Gamma (m)}}\exp \left( { - m\frac{z}{y}} \right)} \frac{{{y^{c - 1}}}}{{{{({Z_0}/c)}^c}\Gamma (c)}}\exp \left( { - \frac{y}{{({Z_0}/c)}}} \right)\,{{d}}y. $$

(4.290)

In terms of Meijer G-functions, (4.290) becomes

$$ f(z) = \int_0^\infty {\frac{1}{{z\Gamma (m)}}G_{0,1}^{1,0}\left( {\frac{{mz}}{y}\left| {\begin{array}{lll} - \\m\end{array}} \right.} \right)} \frac{1}{{y\Gamma (c)}}G_{0,1}^{1,0}\left( {\frac{y}{{({Z_0}/c)}}\left| {\begin{array}{lll} - \\c\end{array}} \right.} \right)\,{{d}}y. $$

(4.291)

The Meijer G-functions in (4.291) can be rewritten using the identities in (4.256) and (4.257) as

$$ \frac{1}{{z\Gamma (m)}}G_{0,1}^{1,0}\left( {\frac{{mz}}{y}\left| {\begin{array}{lll} - \\m\end{array}} \right.} \right) = \frac{1}{{z\Gamma (m)}}G_{1,0}^{0,1}\left( {\frac{y}{{mx}}\left| {\begin{array}{lll} {1 - m} \\-\end{array}} \right.} \right), $$

(4.292)

$$ \frac{1}{{y\Gamma (c)}}G_{0,1}^{1,0}\left( {\frac{{cy}}{{{Z_0}}}\left| {\begin{array}{lll} - \\c\end{array}} \right.} \right) = \left( {\frac{c}{{{Z_0}}}} \right)\frac{1}{{\Gamma (c)}}G_{0,1}^{1,0}\left( {\frac{{cy}}{{{Z_0}}}\left| {\begin{array}{lll} - \\{c - 1}\end{array}} \right.} \right). $$

(4.293)

Equation (4.291) now becomes

$$ f(z) = \int_0^\infty {\frac{1}{{z\Gamma (m)}}G_{1,0}^{0,1}\left( {\frac{y}{{mz}}\left| {\begin{array}{lll} {1 - m} \\-\end{array}} \right.} \right)} \left( {\frac{c}{{{Z_0}}}} \right)\frac{1}{{\Gamma (c)}}G_{0,1}^{1,0}\left( {\frac{{cy}}{{{Z_0}}}\left| {\begin{array}{lll} - \\{c - 1}\end{array}} \right.} \right)\,{{d}}y. $$

(4.294)

Using the integral of the product of Meijer G-function given in (4.266), (4.294) becomes

$$ f(z) = \left( {\frac{1}{{z\Gamma (m)\Gamma (c)}}} \right)(mz)\left( {\frac{c}{{{Z_0}}}} \right)G_{0,2}^{2,0}\left( {\frac{{mc}}{{{Z_0}}}z\left| {\begin{array}{lll} - \\{m - 1,c - 1}\end{array}} \right.} \right). $$

(4.295)

Using the identity in (4.256), (4.295) becomes

$$ f(z) = \frac{1}{{z\Gamma (m)\Gamma (c)}}G_{0,2}^{2,0}\left( {\frac{{mc}}{{{Z_0}}}z\left| {\begin{array}{lll} - \\{m,c}\end{array}} \right.} \right). $$

(4.296)

Using the Table of integrals (Gradshteyn and Ryzhik 2007), (4.290) becomes

$$ f(z) = \frac{2}{{\Gamma (m)\Gamma (c)}}{\left( {\frac{{mc}}{{{Z_0}}}} \right)^{(m + c)/2}}{z^{((m + c)/2) - 1}}{K_{m - c}}\left( {2\sqrt {{\frac{{mcz}}{{{Z_0}}}}} } \right). $$

(4.297)

The right hand side of (4.297) can also be obtained from the conversion from Meijer G-function to Bessel functions in (4.281).

CDF of the SNR in a GK channel

$$ \begin{array}{lll} {\int_0^z {\frac{2}{{\Gamma (m)\Gamma (c)}}{{\left( {\frac{{mc}}{{{Z_0}}}} \right)}^{(m + c)/2}}{y^{((m + c)/2) - 1}}{K_{m - c}}\left( {2\sqrt {{\frac{{mcy}}{{{Z_0}}}}} } \right)\,{{d}}y}} \\= \frac{{\Gamma (m - c)}}{{\Gamma (m)\Gamma (c + 1)}}{{\left( {\frac{{mcz}}{{{Z_0}}}} \right)}^c}{}_1{F_2}\left( {[c],[1 + c,1 - m + c],\frac{{mcz}}{{{Z_0}}}} \right) + \frac{{\Gamma (c - m)}}{{\Gamma (c)\Gamma (m + 1)}}{{\left( {\frac{{mcz}}{{{Z_0}}}} \right)}^m}{}_1{F_2}\left( {[m],[1 + m,1 - c + m],\frac{{mcz}}{{{Z_0}}}} \right) \\= \frac{1}{{\Gamma (m)\Gamma (c)}}G_{1,3}^{2,1}\left( {\frac{{mcz}}{{{Z_0}}}\left| {\begin{array}{lll} 1 \\{m,c,0}\end{array}} \right.} \right).\end{array} $$

(4.298)

• 9. Cascaded channels

Cascaded gamma pdf (nonidentical but independent) with

$$ f({x_k}) = \frac{1}{{b_k^{{m_k}}\Gamma ({m_k})}}x_k^{{m_k} - 1}\exp \left( { - \frac{{{x_k}}}{{{b_k}}}} \right),\quad k = 1,2, \ldots, N, $$

(4.299)

$$ Z = \prod\limits_{k = 1}^N {{X_k}}, $$

(4.300)

$$ {Z_0} = \prod\limits_{k = 1}^N {{b_k}} {m_k} = \prod\limits_{k = 1}^N {{m_k}} \prod\limits_{k = 1}^N {{b_k}}. $$

(4.301)

The pdf of the cascaded output (SNR) Z

$$ f(z) = \frac{1}{{z\prod\nolimits_{k = 1}^N {\Gamma ({m_k})} }}G_{0,N}^{N,0}\left( {\frac{{z\prod\nolimits_{k = 1}^N {{m_k}} }}{{{Z_0}}}\left| {\begin{array}{lll} - \\{{m_1},{m_2}, \ldots, {m_N}}\end{array}} \right.} \right). $$

(4.302)

CDF of the cascaded output

$$ F(z) = \frac{1}{{\prod\nolimits_{k = 1}^N {\Gamma ({m_k})} }}G_{1,N + 1}^{N,1}\left( {\frac{{z\prod\nolimits_{k = 1}^N {{m_k}} }}{{{Z_0}}}\left| {\begin{array}{lll} 1 \\{{m_1},{m_2}, \ldots, {m_N},0}\end{array}} \right.} \right). $$

(4.303)

For the case of N identical gamma channels,

$$ f(z) = \frac{1}{{z\Gamma {{(m)}^N}}}G_{0,N}^{N,0}\left( {\frac{{{m^N}}}{{{Z_0}}}z\left| {\begin{array}{lll} - \\{m,m, \ldots, m}\end{array}} \right.} \right), $$

(4.304)

$$ F(z) = \frac{1}{{\Gamma {{(m)}^N}}}G_{1,N + 1}^{N,1}\left( {\frac{{{m^N}}}{{{Z_0}}}z\left| {\begin{array}{lll} 1 \\{m,m, \ldots, m,0}\end{array}} \right.} \right). $$

(4.305)

• 10. Laplace transforms

$$ g(s) = \int_0^\infty {\frac{1}{{z\Gamma {{(m)}^N}}}G_{0,N}^{N,0}\left( {\frac{{{m^N}}}{{{Z_0}}}z\left| {\begin{array}{lll} - \\{m,m, \ldots, m}\end{array}} \right.} \right)\exp ( - sz)} \,{{d}}z = \left( {\frac{{s{Z_0}}}{{{m^N}}}} \right)\frac{1}{{\Gamma {{(m)}^N}}}G_{1,N}^{N,1}\left( {\frac{{{m^N}}}{{s{Z_0}}}\left| {\begin{array}{lll} 2 \\{(m + 1),(m + 1), \ldots, (m + 1)}\end{array}} \right.} \right). $$

(4.306)

Using the multiplication property from (4.256), the Laplace transform becomes

$$ g(s) = \int_0^\infty {\frac{1}{{z\Gamma {{(m)}^N}}}G_{0,N}^{N,0}\left( {\frac{{{m^N}}}{{{Z_0}}}z\left| {\begin{array}{lll} - \\{m,m, \ldots, m}\end{array}} \right.} \right)\exp ( - sz)} \,{{d}}z = \frac{1}{{\Gamma {{(m)}^N}}}G_{1,N}^{N,1}\left( {\frac{{{m^N}}}{{s{Z_0}}}\left| {\begin{array}{lll} 1 \\{m,m, \ldots, m}\end{array}} \right.} \right). $$

(4.307)

For the special case of N = 1 (gamma pdf or Nakagami channel), the Laplace transform simplifies to

$$ g(s) = \int_0^\infty {\frac{1}{{z\Gamma (m)}}G_{0,1}^{1,0}\left( {\frac{m}{{{Z_0}}}z\left| {\begin{array}{lll} - \\m\end{array}} \right.} \right)\exp ( - sz)} \,{{d}}z = \frac{1}{{{{(1 + ({Z_0}/m)s)}^m}}}. $$

(4.308)

• 11. Bit error rates (Coherent BPSK) for cascaded channels

Using the pdf

$$ \int_0^\infty {\frac{1}{2}{{erfc(}}\sqrt {z} {)}} \frac{1}{{\Gamma {{(m)}^N}z}}G_{0,N}^{N,0}\left( {\frac{{{m^N}z}}{{{Z_0}}}\left| {\begin{array}{lll} - \\{m,m, \ldots, m}\end{array}} \right.} \right){{d}}z = \frac{1}{2} - \frac{1}{{2\sqrt {\pi } \Gamma {{(m)}^N}}}G_{2,N + 1}^{N + 1,1}\left( {\frac{{{m^N}}}{{{Z_0}}}\left| {\begin{array}{lll} {\frac{1}{2},1} \\{0,m,m, \ldots, m}\end{array}} \right.} \right). $$

(4.309)

Using the CDF

$$ \int_0^\infty {\frac{1}{{2\sqrt {{\pi z}} }}\exp ( - z)} \frac{1}{{\Gamma {{(m)}^N}}}G_{1,N + 1}^{N,1}\left( {\frac{{{m^N}z}}{{{Z_0}}}\left| {\begin{array}{lll} 1 \\{m,m, \ldots, m,0}\end{array}} \right.} \right){{d}}z = \frac{1}{{2\sqrt {\pi } \Gamma {{(m)}^N}}}G_{2,N + 1}^{N,2}\left( {\frac{{{m^N}}}{{{Z_0}}}\left| {\begin{array}{lll} {\frac{1}{2},1} \\{m,m, \ldots, m,0}\end{array}} \right.} \right). $$

(4.310)

It can be shown through computation that (4.309) and (4.310) lead to the same error rates. For N = 1 (Nakagami channel, SNR), the bit error rate for coherent BPSK, we have (using the CDF)

$$ \begin{array}{lll}\int_0^\infty {\frac{1}{{2\sqrt {{\pi z}} }}\exp ( - z)} \frac{1}{{\Gamma (m)}}G_{1,2}^{1,1}\left( {\frac{{mz}}{{{Z_0}}}\left| {\begin{array}{lll} 1 \\{m,0}\end{array}} \right.} \right){{d}}z ={ \left( {\displaystyle\frac{1}{{2\sqrt {\pi } }}} \right)\displaystyle\frac{{\Gamma \left(m +({1}/{2})\right)}}{{m\Gamma (m)}}{\left( {\frac{m}{{{Z_0}}}} \right)^m}{}_2{F_1}\left( {\left[ {m,\frac{1}{2} + m} \right],[1 + m], - \frac{m}{{{Z_0}}}} \right) \hfill} = \left( {\displaystyle\frac{1}{{2\sqrt {\pi } }}} \right)\displaystyle\frac{1}{{\Gamma (m)}}G_{2,2}^{1,2}\left( {\frac{m}{{{Z_0}}}\left| {\begin{array}{lll} {1,\displaystyle\frac{1}{2}} \\{m,0}\end{array}} \right.} \right) \end{array}$$

(4.311)