Investigation of multiple hop cooperative communication system over time-selective Nakagami-m fading channel

Abstract

Real-time communication is affected by practical constraints such as node mobility and imperfect channel state information. Most technical contributions in the area of cooperative communication neglect these practical constraints, thus, assumes ideal propagation scenarios. Such approximations make sense in low-rate systems, but can lead to very false results when examining future high data rate systems. In this work, closed-form expression of pairwise error probability (PEP) is derived for dual hop and multiple hop selective decode-and-forward relaying network over time-selective Nakagami-m fading links. Space–time block code is used in combination with multiple-input multiple-output to achieve the diversity gain along with cooperative diversity. To obtain the optimal source-relay power allocation factors, a mathematical framework has been proposed to improve the performance of the system in the power-constraint situation. Simulation results have been presented for various values of fading severity parameter and channel variance. Results for different node-mobility scenarios are also provided. Results show that the system performance improves by increasing the fading severity parameter and relay-to-destination (RD) channel gain. It is noted that the relaying network’s diversity order (DO) is the same as the non-cooperative network’s DO in a scenario when both source node (SN) and destination nodes (DNs) are static, and relay node (RN) is mobile. Also, if the RN and DN are static and the SN is mobile, DO will be equivalent to zero. At high signal-to-noise ratio (SNR), system performance with equal and optimal energy distribution can be seen approaching an asymptotic floor limit, because system performance independent of the power of the noise components. The effect of mobility of the DN on the average per-block PEP output is less important compared to the mobility of the SN. The scheme is restricted by an asymptotic error floor with a greater SNR regime for other node-mobility situations. The analytical results are verified by Monte Carlo simulations for M-ary signaling maximal ratio combiner cooperative system.

Appendix

$I_{1}$ can be evaluated in terms of the Gauss Hypergeometric function:

For solving $I_{1}$ let us change the variable by substitution:

$$ \cos^{2} \theta = t. $$

(67)

This leads to

$$ \sin^{2} \theta = 1 - t, $$

(68)

$$ 2\sin \theta \cos \theta \text{d}\theta = - \text{d}t. $$

(69)

Thus, the limits of the integral would change from 0 to 1 and the integrating variable d$\theta$ changes to

$$ \text{d}\theta = \frac{{ - \text{d}t}}{{2\sqrt t \sqrt {1 - t} }}. $$

(70)

Therefore, $I_{1}$ can now be expressed as

$$ I_{1} = \frac{a}{\pi }\int\limits_{0}^{1} {\left\{ {\frac{{t^{ - 0.50} (1 - t)^{{m_{SR} {\rm M}^{2} - 0.50}} (4{\rm M}^{2} R_{C} \eta_{SR} )^{{m_{SR} {\rm M}^{2} }} }}{{2\left( {\frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{m_{SR} }} + 4{\rm M}^{2} R_{C} \eta_{SR} - 4{\rm M}^{2} R_{C} \eta_{SR} t} \right)^{{m_{SR} {\rm M}^{2} }} }}} \right\}\text{d}t} . $$

(71)

After rearrangements and mathematical manipulations, this integral given above can be represented in the standard form as

$$ \begin{aligned} {\text{I}}_{1} & = \frac{{\left( {4{\rm M}R_{C} \eta_{SR} } \right)^{{m_{SR} {\rm M}}} }}{{2\pi \left( {4{\rm M}R_{C} \eta_{SR} { + }\frac{{\tilde{\delta }^{2}_{SR} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda_{ \ln }^{2} }}{{m_{SR} }}} \right)^{{m_{SR} {\rm M}}} }} \int\limits_{0}^{ 1} \prod\limits_{n = 1}^{N} \\ & \quad \times {\left\{ {{\text{t}}^{ - 0.50} (1 - t)^{{{\text{m}}_{\text{SR}} {\rm M} - 0.50}} \left( {1 - \frac{{4{\rm M}R_{C} \eta_{SR} }}{{4{\rm M}R_{C} \eta_{SR} { + }\frac{{\tilde{\delta }^{2}_{SR} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda_{ \ln }^{2} }}{{{\text{m}}_{SR} }}}}{\text{t}}} \right)^{{ - {\text{m}}_{\text{SR}} {\rm M}}} } \right\}} \text{d}{\text{t}} .\\ \end{aligned} $$

(72)

The Gauss hypergeometric function defined as,

$$ {}_{2}F_{1} (a,b;c;x) = \frac{{\left| \!{\overline {\, c \,}} \right. }}{{\left| \!{\overline {\, {c - a} \,}} \right. \times \left| \!{\overline {\, a \,}} \right. }}\int\limits_{0}^{1} {t^{b - 1} } (1 - t)^{c - b - 1} (1 - tx)^{ - a} \text{d}t. $$

(73)

Comparing (72) with the definition of the Gauss hypergeometric function given in (73), the parameters $a,b,c\,{\text{and}}\,x,$ respectively can be obtained as, $a = m_{SR} {\rm M}^{2} ,\,\,b = 0.50,\,\,c = 1 + m_{SR} {\rm M}^{2}$ and $x = \frac{{4{\rm M}^{2} R_{C} \eta_{SR} }}{{4{\rm M}^{2} R_{C} \eta_{SR} + \frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{m_{SR} }}}}.$

Using the identity given in (73) into (72), the $I_{1}$ can be expressed in the form of Gauss hypergeometric function as,

$$ I_{1} = \frac{a}{\pi } \times \left\{ \begin{aligned} & \frac{{(4{\rm M}^{2} R_{C} \eta_{SR} m_{SR} )^{{m_{SR} {\rm M}^{2} }} \varGamma (m_{SR} {\rm M}^{2} )}}{{2\left( {4{\rm M}^{2} R_{C} \eta_{SR} m_{SR} + P_{S} (\rho_{SR} )^{2(\tau - 1)} b\tilde{\delta }_{SR}^{2} \lambda^{2} } \right)^{{m_{SR} {\rm M}}} \varGamma (m_{SR} {\rm M}^{2} + 1)}} \\ & \quad \times {}_{2}F_{1} \left( {m_{SR} {\rm M}^{2} ,\frac{1}{2};1 + m_{SR} {\rm M}^{2} ;\frac{1}{{1 + \frac{{bP_{S} \tilde{\delta }_{SR}^{2} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{4{\rm M}^{2} R_{C} \eta_{SR} m_{SR} }}}}} \right) \\ \end{aligned} \right\}. $$

(74)

$I_{2}$ can be evaluated in terms of the Appell hypergeometric function of two variables:

Like the case of $I_{1}$, to solve $I_{2}$, let us change the variable by substituting $\sin^{2} \theta = t/2.$ Further manipulations of this substitution leads to

$$ 2\sin \theta \cos \theta \text{d}\theta = \text{d}t/2, $$

$$ \text{d}\theta = \frac{{\text{d}t}}{{4\sqrt {\frac{t}{2}} \sqrt {1 - \frac{t}{2}} }}, $$

After changing the limits of integration from 0 to 1, $I_{2}$ can now be expressed as

$$ \begin{aligned} I_{2} & = \frac{c}{\pi }\int\limits_{0}^{1} {\left( {\frac{{m_{SR} {\rm M}^{2} }}{{m_{SR} {\rm M}^{2} + \frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{2R_{C} \eta_{SR} t}}}}} \right)}^{{m_{SR} {\rm M}^{2} }} \frac{{\text{d}t}}{{4\sqrt {\frac{t}{2}} \sqrt {1 - \frac{t}{2}} }} \\ & = \frac{c}{\pi }\int\limits_{0}^{1} {\left( {1 + \frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{2R_{C} \eta_{SR} m_{SR} {\rm M}^{2} t}}} \right)^{{ - m_{SR} {\rm M}^{2} }} } \frac{{\text{d}t}}{{4\sqrt {\frac{t}{2}} \sqrt {1 - \frac{t}{2}} }}. \\ \end{aligned} $$

$$ \,\,\,\,\,\,\, = \frac{c}{{2\sqrt {2\pi } }}\int\limits_{0}^{1} {t^{{m_{SR} {\rm M}^{2} - 1/2}} \left( {t + \frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{m_{SR} {\rm M}^{2} 2R_{C} \eta_{SR} }}} \right)^{{ - m_{SR} {\rm M}^{2} }} } \frac{dt}{{\sqrt {1 - \frac{t}{2}} }}. $$

(75)

The expression above is compared with Appell hypergeometric function of two variables $x^{\prime}$ and $y^{\prime},$ defined as,

$$ \begin{aligned} & {\text{F}}_{\text{A}}^{(1)} (\chi_{1} ;\iota_{1} ,\iota^{\prime}_{1} ;\gamma^{\prime\prime};x^{\prime},y^{\prime}) = \frac{{\varGamma (\gamma^{\prime\prime})}}{{\varGamma (\chi_{1} )\varGamma (\gamma^{\prime\prime} - \chi_{1} )}}\\ &\quad \int\limits_{0}^{1} {\varTheta^{{\chi_{1} - 1}} \left( {1 - \varTheta } \right)^{{\gamma^{\prime\prime} - \chi_{1} - 1}} \left( {1 - \varTheta x^{\prime}} \right)^{{ - \iota_{1} }} } \left( {1 - \varTheta y^{\prime}} \right)^{{ - \iota^{\prime}_{1} }} \text{d}\varTheta . \end{aligned} $$

(76)

Thus, comparing the integral $I_{2}$ with the above definition of Appell hypergeometric function various parameters can be obtained as,

$$ \begin{aligned} & \chi_{1} = m_{SR} {\rm M}^{2} + \frac{1}{2},\iota_{1} = \frac{1}{2},\iota^{\prime}_{1} = m_{SR} {\rm M}^{2} ,\gamma^{\prime\prime} = m_{SR} {\rm M}^{2} + \frac{3}{2},x^{\prime} = \frac{1}{2}\,{\text{and}} \\ & y^{\prime} = \frac{{ - m_{SR} {\rm M}^{2} 2R_{C} \eta_{SR} t}}{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}, \\ & I_{2} = \frac{c}{{2\sqrt {2\pi } }}\left( {\frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{2R_{C} \eta_{SR} m_{SR} {\rm M}^{2} }}} \right)^{{m_{SR} {\rm M}^{2} }} \frac{{\varGamma \left( {m_{SR} {\rm M}^{2} + \frac{1}{2}} \right)}}{{\varGamma \left( {m_{SR} {\rm M}^{2} + \frac{3}{2}} \right)}}, \\ & \quad \times {\text{F}}_{\text{A}}^{(1)} \left( {m_{SR} {\rm M}^{2} + \frac{1}{2};\frac{1}{2},m_{SR} {\rm M}^{2} ;m_{SR} {\rm M}^{2} + \frac{3}{2};\frac{1}{2};\frac{{ - m_{SR} {\rm M}^{2} }}{{\frac{{b\tilde{\delta }_{SR}^{2} P_{S} (\rho_{SR} )^{2(\tau - 1)} \lambda^{2} }}{{2R_{C} \eta_{SR} }}}}} \right). \\ \end{aligned} $$

(77)