An adaptive q-Lognormal model towards the computation of average channel capacity in slow fading channels

Abstract

The characterization of multipath fading and shadowing in wireless communication systems is essential towards the evaluation of various performance measures. It is well known that the statistical characterization of shadowing phenomena is captured by distributions viz., log-normal distribution, gamma distribution and other mixture distributions. However, it is observed that the log-normal distribution fails to characterize the outliers in the fading signal. The extreme fluctuations in the fading signal needs to be characterized efficiently for error free computation of the various performance metrics. In this context, this paper portrays an adaptive generalized Tsallis’ non-extensive q-Lognormal model towards the characterization of various fading channels. This model operates well with the synthesized fading signals and captures the wide range of tail fluctuations to adapt different fading scenarios. The significance and applicability of the proposed novel q-lognormal model in capturing the slow fading channels is validated using different statistical tests viz., chi-square test and symmetric JS measure. Furthermore, essential performance measures viz., the average channel capacity, closed form expression of cumulative distribution function (CDF) in terms of Gauss-Hypergeometric function ${}_2{F_1}\left[ {\mathrm{{a, b, c; z}}} \right]$, moment generating function, higher order moments corresponding to q-Lognormal channel capacity and coefficient of variation is evaluated corresponding to the proposed q-lognormal model performing extensive Monte-Carlo simulation techniques up to $O(10^7)$.

Appendices

Appendix

Proof of (Eq. 10)

The CDF of the proposed q-Lognormal distribution can be expressed as

$$\begin{aligned} \varTheta (\gamma ) = \int \limits _{ - \infty }^\gamma {{\varphi _q}(t)dt}. \end{aligned}$$

(46)

$$\begin{aligned} \varTheta (\gamma ) = \int \limits _{ - \infty }^\gamma {\frac{1}{{\xi t}}{{\left( {1 + \frac{{q - 1}}{{3 - q}}{{\left( {\frac{{ln(t) - \mu }}{\sigma }} \right) }^2}} \right) }^{\frac{1}{{1 - q}}}}dt}, \end{aligned}$$

(47)

where $\xi$ is the normalized constant as defined in Eq. 8. Replacing $ln(t)=x$, we obtain

$$\begin{aligned} \varTheta (\gamma ) = \frac{1}{\xi }\int \limits _{ - \infty }^{\ln \gamma } {{{\left( {1 + \frac{{q - 1}}{{3 - q}}{{\left( {\frac{{x - \mu }}{\sigma }} \right) }^2}} \right) }^{\frac{1}{{1 - q}}}}dx}. \end{aligned}$$

(48)

Now, substituting $\frac{{x - \mu }}{\sigma } = y$ we obtain

$$\begin{aligned} \varTheta (\gamma ) = \frac{\sigma }{\xi }\int \limits _{ - \infty }^{\frac{{\ln \gamma - \mu }}{\sigma }} {\left( {1 + \frac{{q - 1}}{{3 - q}}{{\left( y \right) }^2}} \right) ^{\frac{1}{{1 - q}}}}dy. \end{aligned}$$

(49)

Finally, after some mathematical computations, the CDF corresponding to the q-Lognormal distribution can be expressed in terms of Gauss Hypergeometric function [6] as

$$\begin{aligned} \varTheta (\gamma )= & {} \frac{1}{2}\nonumber \\&+ \frac{{\ln \gamma - \mu }}{\xi }{}_2{F_1}\left[ {\frac{1}{2},\frac{1}{{q - 1}},\frac{3}{2},\frac{{1 - q}}{{3 - q}}{{\left( {\frac{{\ln \gamma - \mu }}{\sigma }} \right) }^2}} \right] .\nonumber \\ \end{aligned}$$

(50)

Proof of (Eq. 25)

From Eq. 23, it is obtained

$$\begin{aligned} M(z) = \int \limits _0^\infty {f(\gamma )} {(1 + \gamma )^z}d\gamma . \end{aligned}$$

(51)

Applying the transformation $\mathop {\lim }\limits _{q \rightarrow 1}$ in Eq. 16 and substituting $ln(\gamma )=y$, the MGF is obtained as

$$\begin{aligned} M(z) = \frac{1}{{\sigma \sqrt{2\pi } }}\int \limits _{ - \infty }^\infty {\exp \left( { - \frac{{{{\left( {y - \mu } \right) }^2}}}{{2{\sigma ^2}}}} \right) } {(1 + \exp (y))^z}dy.\nonumber \\ \end{aligned}$$

(52)

On further simplification, we get

$$\begin{aligned} \begin{aligned} M(z) ={}&\frac{1}{{\sigma \sqrt{2\pi } }}\int \limits _0^\infty {\exp \left( { - \frac{{{{\left( {y + \mu } \right) }^2}}}{{2{\sigma ^2}}}} \right) } {(1 + \exp ( - y))^z}dy\\&+ \frac{1}{{\sigma \sqrt{2\pi } }}\int \limits _0^\infty {\exp \left( { - \frac{{{{\left( {y - \mu } \right) }^2}}}{{2{\sigma ^2}}} + yz} \right) } \\&(1+ \exp ( - y))^zdy. \end{aligned} \end{aligned}$$

(53)

Now, w.r.t $y>0$, $\exp (-y)<1$, it is obtained

$$\begin{aligned} {(1 + \exp ( - y))^z} = \sum \limits _{v = 0}^\infty {\frac{{{{(z)}_v}}}{{v!}}} \exp \left( { - yv} \right) , \end{aligned}$$

(54)

where ${{{(z)}_v}}$ is the lower factorial and is represented as ${(z)_v} = z(z - 1)(z - 2)...(z - v + 1)$ [9] .

Equation 53 along with Eq. 54 yields

$$\begin{aligned} \begin{aligned} M(z) ={}&\frac{1}{{\sigma \sqrt{2\pi } }}\sum \limits _{v = 0}^\infty {\frac{{{{(z)}_v}}}{{v!}}} \int \limits _0^\infty {\exp \left( { - yv} \right) } \exp \left( { - \frac{{{{\left( {y - \mu } \right) }^2}}}{{2{\sigma ^2}}} + yz} \right) dy \\&+ \frac{1}{{\sigma \sqrt{2\pi } }}\sum \limits _{v = 0}^\infty {\frac{{{{(z)}_v}}}{{v!}}} \int \limits _0^\infty {\exp \left( { - yv} \right) } \exp \left( { - \frac{{{{\left( {y + \mu } \right) }^2}}}{{2{\sigma ^2}}}} \right) dy. \end{aligned} \end{aligned}$$

(55)

Considering $\frac{1}{{\sigma \sqrt{2\pi } }}\int \limits _0^\infty {\exp \left( { - yv} \right) } \exp \left( { - \frac{{{{\left( {y + \mu } \right) }^2}}}{{2{\sigma ^2}}}} \right) dy = \sigma \sqrt{\frac{\pi }{2}} \mathrm {erfcx}\left( {\frac{{v\sigma }}{{\sqrt{2} }} + \frac{\mu }{{\sqrt{2} \sigma }}} \right) \exp \left( { - \frac{{{\mu ^2}}}{{2{\sigma ^2}}}} \right)$ and $\mathrm {erfcx}(t) = \exp ({t^2})\mathrm {erfc}(t)$, the MGF corresponding to the q-Lognormal channel capacity, $q\rightarrow 1$ is evaluated and is expressed in Eqs. 24 and  25.