Analysis of Outage Probability and Average Signal Power of Two Independent Block Fading Channel for a SISO System

Abstract

This paper deals with the design of a new system where the channel is subjected to a more generalized scenario by considering two-independent block fading (L = 2). Each block undergoes different fading characteristics for a particular time interval. This time interval is the coherence interval such that the channel coefficient is fixed during a fading block, and statistically independent from one block to another. The main focus of this work is to derive performance measures, such as outage probability and average signal power for Rayleigh, Nakagami, and Weibull fading channels. Once the mathematical expressions are derived, attention is focussed to obtain numerical results for L = 2 by plotting analytical graphs for the obtained expressions and are observed with proposed system parameters like average fading channel power and signal-to-noise ratio (SNR) in the case of a Rayleigh fading channel, Nakagami fading parameter m, fading power Ω, and SNR in the case of a Nakagami fading channel, and Weibull fading parameters k and λ in the case of a Weibull fading channel. The proposed system can thus be evaluated and the amount of degradation can be measured.

Appendix

Performance measures like outage probability and average signal power are derived for 3-independent block fading channel (L = 3) as follows:

1.1 Outage Probability

1.1.1 Rayleigh Fading Channel

The outage probability for 3-block Rayleigh fading channel is obtained similarly by substituting L = 3 in (3). It is given as

$$ P_{\text{out,3}}^{\text{ray}} = \frac{ - 1}{{v^{2} }}\exp \left( \frac{3}{v} \right)A - \frac{{\sqrt {2^{R} \pi } }}{\sqrt v }\exp \left( {\frac{{2 - 2^{R + 1} \sqrt {2^{R} } }}{v}} \right) - B + 1, $$

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where

$$ A = \left\{ {\int\limits_{1}^{\infty } {\exp \left( {\frac{{ - x_{1} }}{v}} \right)\left[ {\int\limits_{1}^{{\tfrac{{2^{3R} }}{{x_{1} }}}} {\left( {\frac{{ - x_{2} }}{v} - \frac{{2^{3R} }}{{x_{1} vx_{2} }}} \right)} dx_{2} } \right]dx_{1} } } \right\} \, ,\quad {\text{and}} \quad B = \int\limits_{0}^{1} {\exp } \left( {\frac{{ - 2^{3R} }}{{vx_{1} }} - \frac{{x_{1} }}{v}} \right)dx_{1} .v = \rho \left( {J_{T} } \right)P_{r} . $$

1.1.2 Nakagami Fading Channel

The outage probability for 3-block Nakagami fading channel is obtained similarly by substituting L = 3 in (3). It is given as

$$ P_{\text{out,3}}^{\text{nak}} = - uK_{1}^{3} \int\limits_{1}^{\infty } {\left( {x_{1} - 1} \right)}^{m - 1} \exp \left( {\frac{{ - x_{1} }}{u}} \right)\left\{ {\int\limits_{1}^{{\tfrac{{2^{3R} }}{{x_{1} }}}} {(x_{2} - 1)^{m - 1} \left( {\frac{{2^{3R} }}{{x_{1} x_{2} }} - 1} \right)}^{m - 1} \exp \left( {\frac{{ - x_{2} }}{u} - \frac{{u2^{3R} }}{{x_{1} x_{2} }}} \right)[B]dx_{2} } \right\}dx_{1} , $$

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where $u = \frac{{\rho (J_{T} )\Omega }}{m}$, $K_{1} = \frac{{e^{{\tfrac{1}{u}}} }}{{(\rho (J_{T} ))^{m} \Gamma (m)}}\left( {\frac{m}{\Omega }} \right)^{m}$

$$ B = \left\{ {1 + u(m - 1)\left( {B_{1} } \right)^{ - 1} + u^{2} (m - 1)(m - 2)\left( {B_{1} } \right)^{ - 2} + \cdots +\,\left( {B_{1} } \right)^{ - m + 1} u^{m - 1} (m - 1)(m - 2) \ldots (m - (m - 1))} \right\}, $$

and $B_{1} = \left( {\frac{{2^{3R} }}{{x_{2} }} - 1} \right)$.

1.1.3 Weibull Fading Channel

The outage probability for a 3-block Weibull fading channel is obtained similarly by substituting L = 3 in (3). It is given as

$$ P_{\text{out,3}}^{wb} = \frac{{k^{2} }}{{4\lambda^{k} (\rho (J_{T} ))^{\frac{k}{2}} }}\sum\limits_{n = 0}^{\infty } {\frac{{( - 1)^{n} }}{n!(1 + n)}\left( {\frac{1}{\mu }} \right)^{1 + n} } \int\limits_{1}^{\infty } {\left( {x_{1} - 1} \right)}^{{\frac{k}{2} - 1}} \exp \left( { - \left( {\frac{{x_{1} - 1}}{{\lambda^{2} \rho (J_{T} )}}} \right)^{\frac{k}{2}} } \right)[A]dx_{1} , $$

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where $\mu = \lambda^{2} \rho (J_{T} )$, $A = \int\nolimits_{0}^{{t_{\hbox{max} } }} {t^{\nu - 1} } \exp \left( { - t^{\nu } } \right)\left( {\frac{{2^{3R} }}{{x_{1} \left( {t\mu + 1} \right)}} - 1} \right)^{1 + n} dt$, $t_{\hbox{max} } = \frac{1}{\mu }\left( {\frac{{2^{3R} }}{{x_{1} }} - 1} \right)$, $t = \frac{{x_{2} - 1}}{\mu },$, $\nu = \frac{k}{2}$.

1.2 Average Signal Power

1.2.1 Rayleigh Fading Channel

The average signal power P _av for L = 3 is obtained from (17) as

$$ P_{\text{av,3}}^{\text{ray}} = \left\{ {\frac{ - 1}{{v^{2} }}\exp \left( \frac{3}{v} \right)A - \frac{{\sqrt {2^{R} \pi } }}{\sqrt v }\exp \left( {\frac{{2 - 2^{R + 1} \sqrt {2^{R} } }}{v}} \right) - B + 1} \right\}^{{ - \tfrac{1}{2}}} , $$

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where $v = \rho (J_{T} )P_{r}$, $A = \left\{ {\int\nolimits_{1}^{\infty } {\exp \left( {\frac{{ - x_{1} }}{v}} \right)\left[ {\int\nolimits_{1}^{{\tfrac{{2^{3R} }}{{x_{1} }}}} {\left( {\frac{{ - x_{2} }}{v} - \frac{{2^{3R} }}{{x_{1} vx_{2} }}} \right)} dx_{2} } \right]dx_{1} } } \right\}$ and $B = \int\nolimits_{0}^{1} {\exp } \left( {\frac{{ - 2^{3R} }}{{vx_{1} }} - \frac{{x_{1} }}{v}} \right)dx_{1}$.

1.2.2 Nakagami Fading Channel

The average signal power P _av is obtained from (17) as

$$ P_{\text{av,3}}^{\text{nak}} = \left\{ { - uK_{1}^{3} \int\limits_{1}^{\infty } {\left( {x_{1} - 1} \right)}^{m - 1} \exp \left( {\frac{{ - x_{1} }}{u}} \right)\left[ {\int\limits_{1}^{{\tfrac{{2^{3R} }}{{x_{1} }}}} {(x_{2} - 1)^{m - 1} \left( {\frac{{2^{3R} }}{{x_{1} x_{2} }} - 1} \right)}^{m - 1} \exp \left( {\frac{{ - x_{2} }}{u} - \frac{{u2^{3R} }}{{x_{1} x_{2} }}} \right)[B]dx_{2} } \right]} \right\}^{{ - \tfrac{1}{2}}} , $$

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where $u = \frac{{\rho (J_{T} )\Omega }}{m}$, $K_{1} = \frac{{e^{{\tfrac{1}{u}}} }}{{(\rho (J_{T} ))^{m} \Gamma (m)}}\left( {\frac{m}{\Omega }} \right)^{m}$, $B = \left\{ {1 + u(m - 1)\left( {B_{1} } \right)^{ - 1} + u^{2} (m - 1)(m - 2)\left( {B_{1} } \right)^{ - 2} + \cdots + \left( {B_{1} } \right)^{ - m + 1} u^{m - 1} (m - 1)(m - 2) \ldots (m - (m - 1))} \right\},$ and $B_{1} = \left( {\frac{{2^{3R} }}{{x_{2} }} - 1} \right)$.

1.2.3 Weibull Fading Channel

The average signal power P _av for L = 3, is obtained from (17) as

$$ P_{\text{av,3}}^{\text{wb}} = \left\{ {\frac{{k^{2} }}{{4\lambda^{k} \left( {\rho (J_{T} )} \right)^{{\tfrac{K}{2}}} }}\sum\limits_{n = 0}^{\infty } {\frac{{( - 1)^{n} }}{n!(1 + n)}\left( {\frac{1}{\mu }} \right)}^{1 + n} \int\limits_{1}^{\infty } {\left( {x_{1} - 1} \right)}^{{\tfrac{k}{2} - 1}} \exp \left( { - \left( {\frac{{x_{1} - 1}}{{\lambda^{2} \rho (J_{T} )}}} \right)^{{\tfrac{k}{2}}} } \right)\left[ A \right]dx_{1} } \right\}^{{ - \tfrac{1}{2}}} , $$

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where $\mu = \lambda^{2} \rho (J_{T} )$, $A = \int\nolimits_{0}^{{t_{\hbox{max} } }} {t^{\nu - 1} } \exp \left( { - t^{\nu } } \right)\left( {\frac{{2^{3R} }}{{x_{1} \left( {t\mu + 1} \right)}} - 1} \right)^{1 + n} dt$, $t_{\hbox{max} } = \frac{1}{\mu }\left( {\frac{{2^{3R} }}{{x_{1} }} - 1} \right)$, $t = \frac{{x_{2} - 1}}{\mu }$, $\nu = \frac{k}{2}$.