A New Approximate Closed-Form Distribution and Performance Analysis of a Composite Weibull/Log-Normal Fading Channel

Abstract

The presence of both the fading and shadowing effects (also called composite multipath/shadowed fading) is often encountered in a realistic radio propagation scenario, thus, making it necessary to consider the simultaneous effect of fading and shadowing on the received signal. The multipath effect is captured using models such as Rician, Nakagami-m, Weibull distribution and shadowing effect is modeled using Log-normal distribution. In this paper we present the closed-form expression of composite (Weibull/log-normal shadowed) fading using the efficient tool proposed by Holtzman. Using this result, the closed-form expression of combined (time-shared) shadowed/unshadowed fading is presented. The performance measures of fading communication systems such as probability density function of signal to noise ratio, amount of fading, outage probability (P_out) and channel capacity are analyzed and expressed in closed form.

Appendix

Proof of (35)

$$I = \int\limits_{0}^{\infty } {\gamma^{n - 1} G_{2,2}^{1,2} \left[ {\gamma \left| {\begin{array}{*{20}c} {1,1} \\ {1,0} \\ \end{array} } \right.} \right]} G_{0,1}^{1,0} \left[ {a\gamma^{n} \left| {\begin{array}{*{20}c} {1,1} \\ {1,0} \\ \end{array} } \right.} \right]d\gamma$$

(37)

As per Generalization of classical Meijer’s integral from two G functions ([25], /07.34.21.0012.01)

$$\begin{aligned} \int\limits_{0}^{\infty } {\tau^{\alpha - 1} G_{u,v}^{s,t} \left( {\sigma \tau \left| {\begin{array}{*{20}c} {c_{1} ,c_{2} , \ldots ,c_{u} } \\ {d_{1} ,d_{2} , \ldots ,d_{v} } \\ \end{array} } \right.} \right)} G_{p,q}^{m,n} \left( {w\tau^{r} \left| {\begin{array}{*{20}c} {a_{1} ,a_{2} , \ldots ,a_{p} } \\ {b_{1} ,b_{2} , \ldots ,bq} \\ \end{array} } \right.} \right)d\gamma \hfill \\ = \sigma^{ - \alpha } H_{p + v,q + u}^{m + t,n + s} \left( {\frac{w}{{\sigma^{r} }}\left| {\begin{array}{*{20}c} {\left( {a_{1} ,1} \right), \ldots ,\left( {a_{n} ,1} \right),\left( {1 - \alpha - d_{1} ,r} \right), \ldots ,\left( {1 - \alpha - d_{v} ,r} \right),\left( {a_{n + 1} ,1} \right), \ldots ,,\left( {a_{p} ,1} \right)} \\ {\left( {b_{1} ,1} \right), \ldots ,\left( {b_{m} ,1} \right),\left( {1 - \alpha - c_{1} ,r} \right), \ldots ,\left( {1 - \alpha - c_{u} ,r} \right),\left( {b_{m + 1} ,1} \right), \ldots ,,\left( {b_{q} ,1} \right)} \\ \end{array} } \right.} \right)./;r > 0 \hfill \\ \end{aligned}$$

(38)

Using the result of (38), the integral of (37) can be easily written as

$$I = \left( 1 \right)^{ - n} H_{2,3}^{3,1} \left[ {a\left| {\begin{array}{*{20}c} {\left( { - n,n} \right),\left( {1 - n,n} \right)} \\ {\left( {0,1} \right),\left( { - n,n} \right),\left( { - n,n} \right)} \\ \end{array} } \right.} \right]$$

(39)

For any a, b, c, where $\text{Re} (b) > 0\,\&\, m \ge 1;$ ([26], /Eq. 1.66)

$$H_{p,q}^{m,n} \left[ {z\left| {\begin{array}{*{20}c} {\left( {a_{1} ,A_{1} } \right), \ldots ,\left( {a_{p} ,A_{p} } \right)} \\ {\left( {b,0} \right)\left( {a_{1} ,A_{1} } \right), \ldots ,\left( {a_{p} ,A_{p} } \right)} \\ \end{array} } \right.} \right] = \Gamma \left( b \right)H_{p,q - 1}^{m - 1,n} \left[ {z\left| {\begin{array}{*{20}c} {\left( {a_{1} ,A_{1} } \right), \ldots ,\left( {a_{p} ,A_{p} } \right)} \\ {\left( {a_{1} ,A_{1} } \right), \ldots ,\left( {a_{p} ,A_{p} } \right)} \\ \end{array} } \right.} \right]$$

(40)

Using (40), (6) can be easily written as

$$I = \Gamma \left( 1 \right)H_{2,2}^{2,1} \left[ {a\left| {\begin{array}{*{20}c} {\left( { - n,n} \right),\left( {1 - n,n} \right)} \\ {\left( { - n,n} \right),\left( { - n,n} \right)} \\ \end{array} } \right.} \right]$$

(41)

Now as per [27, Eq. 3.3],

$$H_{p,q}^{m,n} \left( {\left. {\begin{array}{*{20}c} {\left\{ {\left( {a_{j} ,c} \right)} \right\}_{j = 1, \ldots ,p} } \\ {\left\{ {\left( {a_{j} ,c} \right)} \right\}_{j = 1, \ldots ,p} } \\ \end{array} } \right|z} \right) = \frac{1}{c}G_{p,q}^{m,n} \left( {\left. {\begin{array}{*{20}c} {\left\{ {\left( {a_{j} ,c} \right)} \right\}_{j = 1, \ldots ,p} } \\ {\left\{ {\left( {a_{j} ,c} \right)} \right\}_{j = 1, \ldots ,p} } \\ \end{array} } \right|z^{{{1 \mathord{\left/ {\vphantom {1 {}}} \right. \kern-0pt} {}}c}} } \right)$$

(42)

Using (42), we obtain a simplified form of (41) in terms of Meijer-G function expressed as

$$I = \frac{1}{n}G_{2,2}^{2,1} \left[ {\left. {\left| {\begin{array}{*{20}c} { - n,1 - n} \\ { - n, - n} \\ \end{array} } \right.} \right|a^{{{1 \mathord{\left/ {\vphantom {1 {}}} \right. \kern-0pt} {}}n}} } \right]$$

(43)