Impact of MIMO enabled relay on the performance of a hybrid satellite-terrestrial system

Abstract

In this paper, we investigate a hybrid satellite terrestrial system with multiple input and multiple output enabled fixed relay, which is more reliable than the conventional dual-hop multiple single input and single output relaying network. Providing a small number of antennas on a fixed relay is easier to undertake than setting up on a mobile terminal due to the size limitation. In this study, we consider the impact of cooperating fixed relay with multiple transmitting and receiving antennas on the hybrid satellite-terrestrial system. Amplify-and-forward cooperative protocol is used at the relay node while for signal combining at the relay node, maximum ratio combining and maximum ratio transmission techniques are used. Independent and non-identically distributed fading channels, shadowed Rician, and Nakagami-$m$, are assumed between the source-relay and relay-destination links respectively. An analytical approach is derived to evaluate the performance of the system in terms of outage probability, symbol error rate (SER) and ergodic capacity. The closed form expressions for the probability density function, cumulative distribution function, moment generating function, average SER of the total end-to-end signal-to-noise ratio are derived. We further derive the approximate closed form expression of ergodic capacity. These derived analytical expressions are applied to the general operating conditions together with available satellite channel data of various degrees of shadowing. The analytical results are compared with Monte Carlo simulations. The results show the improvement in the performance of the hybrid satellite-terrestrial system under various antenna arrangements at the relay node.

Appendices

Appendix 1: Proof of Theorem I

It is known that the sum of $N$ independent squared Rice random variables follow non-central chi square distribution with $N$ degree of freedom. The PDF expression given in (1) and the PDF of $\gamma _{rd}$ is expressed in (2). The end-to-end PDF of the proposed system with independent and non-identical fading distributions can be derived by using [26]

$$\begin{aligned} f_{\gamma _r}(\gamma )=\int _{\gamma }^{\infty }\frac{z^2}{(z-\gamma )^2}f_{\gamma _{\text {sr}}}\left( \frac{\gamma z}{z-\gamma }\right) f_{\gamma _r}(z)dz \end{aligned}$$

(33)

After carrying out some variable transformation, (33) can be written as

$$\begin{aligned} f_{\gamma _r}(\gamma )&= \frac{1}{\left( 2b_o\bar{\gamma }_{\text {sr}}\right) ^{N_r }\varGamma \left( N_r\right) }\left( \frac{2b_o m_o}{2b_om_o+\varOmega }\right) ^{N_rm_o }\frac{1}{\varGamma \left( m N_t\right) } \nonumber \\&\quad \times \left( \frac{m}{\bar{\gamma }_{\text {rd}}}\right) ^{m N_t}\exp \left[ -\left( \frac{1}{2b_o\bar{\gamma }_{\text {sr}}}+\frac{m}{\bar{\gamma }_{\text {rd}}}\right) \gamma \right] \gamma ^{N_r-1} \nonumber \\&\quad \times \int _0^{\infty }\left( 1+\frac{\gamma }{z}\right) ^{m N_t+N_r}z^{m N_t-1} \nonumber \\&\quad \times \exp \left[ -\left( \frac{\gamma ^2}{2b_o\bar{\gamma }_{\text {sr}}z}+\frac{m z}{\bar{\gamma }_{\text {rd}}}\right) \right] \nonumber \\&\quad \times \,_1F_1\left( N_rm_0,N_r,\frac{\varOmega \gamma (z+\gamma ) }{2b_o\bar{\gamma }_{\text {sr}}z\left( 2b_o m_o+\varOmega \right) }\right) dz\nonumber \\ \end{aligned}$$

(34)

Now substituting ${}_1F_1(\cdot ,\cdot ,\cdot )$ with a series expression by using [23, eq.(9.14.1)], (34) can be written as

$$\begin{aligned} f_{\gamma _r}(\gamma )\!&= \!\frac{1}{\left( 2b_o{\bar{\gamma }}_{\text {sr}}\right) ^{N_r }\varGamma \left( N_r\right) }\left( \frac{2b_o m_o}{2b_o m_o+\varOmega }\right) ^{N_rm_o }\frac{1}{\varGamma \left( m N_t\right) } \nonumber \\&\quad \times \left( \frac{m}{\bar{\gamma }_{\text {rd}}}\right) ^{m N_t} \exp \left[ -\left( \frac{1}{2b_o\bar{\gamma }_{\text {sr}}}+\frac{m}{\bar{\gamma }_{\text {rd}}}\right) \gamma \right] \sum _{i=0}^{\infty } \frac{1}{i!} \nonumber \\&\quad \times \frac{\varGamma \left( N_rm_o+i\right) \varGamma \left( N_r\right) }{\varGamma \left( N_r m_o\right) \varGamma \left( N_r+i\right) }\left( \frac{\varOmega }{2b_o\bar{\gamma }_{\text {sr}}\left( 2b_o m_o+\varOmega \right) }\right) ^i \nonumber \\&\quad \times \gamma ^{N_r+x-1}\int _0^{\infty }\left( 1+\frac{\gamma }{z}\right) ^{m N_t+N_r+x}z^{m N_t-1} \nonumber \\&\quad \times \exp \left[ -\left( \frac{\gamma ^2}{2b_o\bar{\gamma }_{\text {sr}}z}+\frac{m z}{\bar{\gamma }_{\text {rd}}}\right) \right] dz \end{aligned}$$

(35)

The term $\left( 1+\frac{\gamma }{z}\right) ^{m N_t+N_r+i}$ can be expanded according to [23, eq.(1.111)], we can reach

$$\begin{aligned} f_{\gamma _r}(\gamma )&= \frac{1}{\left( 2b_o\bar{\gamma }_{\text {sr}}\right) ^{N_r }\varGamma \left( N_r\right) }\left( \frac{2b_o m_o}{2b_o m_o+\varOmega }\right) ^{N_rm_o }\frac{1}{\varGamma \left( m N_t\right) } \nonumber \\&\quad \times \left( \frac{m}{\bar{\gamma }_{\text {rd}}}\right) ^{m N_t}\exp \left[ -\left( \frac{1}{2b_o\bar{\gamma }_{\text {sr}}}+\frac{m}{\bar{\gamma }_{\text {rd}}}\right) \gamma \right] \sum _{i=0}^{\infty } \frac{1}{i!} \nonumber \\&\quad \times \frac{\varGamma \left( N_rm_o\!+\!i\right) \varGamma \left( N_r\right) }{\varGamma \left( N_r m_o\right) \varGamma \left( N_r+i\right) }\left( \frac{\varOmega \text { }}{2b_o\bar{\gamma }_{\text {sr}}\left( 2b_o m_o\!+\!\varOmega \right) }\right) ^i \nonumber \\&\quad \times \sum _{j=0}^{m N_t+N_r+i} \left( {\begin{array}{c}m N_t+N_r+i\\ j\end{array}}\right) \gamma ^{N_r+i+j-1} \nonumber \\&\quad \times \int _0^{\infty }z^{m N_t-y-1}\exp \left[ -\left( \frac{\gamma ^2}{2b_o\bar{\gamma }_{\text {sr}}z}+\frac{m z}{\bar{\gamma }_{\text {rd}}}\right) \right] dz\nonumber \\ \end{aligned}$$

(36)

Now with the help of [23, eq.(3.471.9)], the combined PDF, $f_{\gamma _{r}}(\gamma )$, can be shown as (10).

Appendix 2: Derivation of relay path CDF $F_{\gamma _r}(\gamma )$

The PDF of SNR per symbol of $S\rightarrow R$ link is given in (1) and the CDF of $R\rightarrow D$ link for integer values of $m$ is given by

$$\begin{aligned} f_{\gamma _{\text {rd}}}(\gamma )= 1-\exp \left( -\frac{m \gamma _{\text {rd}}}{\bar{\gamma }_{\text {rd}}}\right) \sum _{n=0}^{m N_t-1}\frac{1}{n!}\left( \frac{m \gamma _{\text {rd}}}{\bar{\gamma }_{\text {rd}}}\right) ^n \end{aligned}$$

(37)

Then, the CDF of the end-to-end path can be calculated with the help of (9) as

$$\begin{aligned} F_{\gamma _r}(\gamma )&=\Pr \left( \gamma _r\le \gamma \right) =\Pr \left( \frac{\gamma _{\text {sr}}\gamma _{\text {rd}}}{\gamma _{\text {sr}}+\gamma _{\text {rd}}}\le \gamma \right) \nonumber \\&\quad =\int _0^{\gamma }\Pr \left( \gamma _{\text {rd}}\ge \frac{\gamma \gamma _{\text {sr}}}{\gamma _{\text {sr}}-\gamma }\right) f_{\gamma _{\text {sr}}}\left( \gamma _{\text {sr}}\right) d\gamma _{\text {sr}}\nonumber \\&\qquad + \int _{\gamma }^{\infty }\Pr \left( \gamma _{\text {rd}}\le \frac{\gamma \gamma _{\text {sr}}}{\gamma _{\text {sr}} -\gamma }\right) f_{\gamma _{\text {sr}}}\left( \gamma _{\text {sr}}\right) d\gamma _{\text {sr}}\nonumber \\&\quad = 1-\int _{\gamma }^{\infty }\Pr \left( \gamma _{\text {rd}}\ge \frac{\gamma \gamma _{\text {sr}}}{\gamma _{\text {sr}}-\gamma }\right) f_{\gamma _{\text {sr}}}\left( \gamma _{\text {sr}}\right) d\gamma _{\text {sr}}\nonumber \\&\quad =1-\frac{1}{\left( 2b_o\bar{\gamma }_{\text {sr}}\right) {}^{N_r }\varGamma \left( N_r\right) }\left( \frac{2b_o m_o}{2b_o m_o+\varOmega }\right) ^{N_rm_o} \nonumber \\&\quad \quad \times \sum _{n=0}^{m N_t-1}\frac{1}{n!}\left( \frac{m }{\bar{\gamma }_{\text {rd}}}\right) ^n \int _{\gamma }^{\infty }\gamma _{\text {sr}}^{N_r-1} \nonumber \\&\quad \quad \times \exp \left[ -\left( \frac{\gamma _{\text {sr}}}{2b_o\bar{\gamma }_{\text {sr}}\text { }}+\frac{m\gamma \gamma _{\text {sr}}}{\bar{\gamma }_{\text {rd}}\left( \gamma _{\text {sr}}-\gamma \right) }\right) \right] \nonumber \\&\quad \quad \times \left( \frac{\gamma \text { }\gamma _{\text {sr}}}{\gamma _{\text {sr}}-\gamma }\right) ^n\,_1F_1\left( N_rm_o,N,\varOmega \lambda \right) d\gamma _{\text {sr}} \end{aligned}$$

(38)

After integral substitution $x=\gamma _{sr}-\gamma$ and applying [23, eq.(9.14.1)] and [23, eq.(1.111)] we can reach

$$\begin{aligned} F_{\gamma _r}(\gamma )&=1-\frac{1}{\left( 2b_o\bar{\gamma }_{\text {sr}}\right) ^{N_r }}\left( \frac{2b_o m_o}{2b_o m_o+\varOmega }\right) ^{N_rm_o}\sum _{n=0}^{m N_t-1} \frac{1}{n!} \nonumber \\&\quad \times \left( \frac{m }{\bar{\gamma }_{\text {rd}}}\right) ^n\exp \left[ -\left( \frac{\gamma }{2b_o\bar{\gamma }_{\text {sr}}\text { }}+\frac{m \gamma }{\bar{\gamma }_{\text {rd}}}\right) \right] \sum _{i=0}^{\infty } \frac{1}{i!}\nonumber \\&\quad \times \frac{\varGamma \left( N_rm_o+i\right) }{\varGamma \left( N_r m_o\right) \varGamma \left( N_r+i\right) } \left( \frac{ \varOmega }{2b_o \bar{\gamma }_{\text {sr}}\left( 2b_om_o+\varOmega \right) }\right) ^i \nonumber \\&\quad \times \sum _{j=0}^{N_r+n+i-1} \left( {\begin{array}{c}N_r+n+i-1\\ j\end{array}}\right) \gamma ^{n+j} \nonumber \\&\quad \times \int _0^{\infty }\exp \left[ -\left( \frac{x}{2b_o\bar{\gamma }_{\text {sr}}}+\frac{m \gamma ^2}{x \bar{\gamma }_{\text {rd}}}\right) \right] x^{N_r+i-j-1}dx \nonumber \\ \end{aligned}$$

(39)

Now by using [23, Eq. (3.471.9)] and solving the above integral, we can get the closed form expression of CDF as given in (13).