Two Novel Adaptive Transmission Schemes in a Decode-and-Forward Relaying Network

Abstract

Adaptive transmission in a cooperative network with a half-duplex relay operating in decode-and-forward mode is considered. The main purpose of the paper is maximizing the spectral efficiency of the system, which is reduced by using half-duplex relaying, while the bit error performance is kept below an appropriate threshold. The source transmits its data adaptively using quadratic amplitude modulation. Two adaptive transmission schemes are proposed: the first scheme is named simple adaptive transmission scheme (SATS), and the second one is called high-performance spectral efficiency scheme (HPSES). The SATS has a low complexity system at the destination which does not combine received signals from the source and relay. However, the HPSES uses a linear combination at the destination which is a novel detector to take the possibility of error at the relay into account. Then, we derive exact closed-form expression for the average spectral efficiency and outage probability of the system and an approximate closed-form expression for the average bit error probability. The simulation results corroborates theoretical results. Furthermore, it is shown that despite much lower complexity, the performance of the SATS is close to other well-known schemes. Moreover, the HPSES outperforms other methods of adaptive transmissions in sense of the spectral efficiency and outage probability.

Appendices

Appendix 1: Joint Statistics of $\gamma _{\text {NC}}^{\text {SATS}}$ and $\gamma _{\text {C}}^{\text {SATS}}$

According to (8), it can be figured out that $\gamma _{\text {NC}}^{\text {SATS}}$ and $\gamma _{\text {C}}^{\text {SATS}}$ are independent. Therefore, their joint CDF is

$$\begin{aligned} {F_{\gamma _{\text {NC}}^{\text {SATS}},\gamma _{\text {C}}^{\text {SATS}}}}\left( {x,y} \right)&= \Pr \left\{ {\gamma _{\text {NC}}^{\text {SATS}} \le x,\gamma _{\text {C}}^{\text {SATS}} \le y} \right\} = \Pr \left\{ {{\gamma _{sd}} \le x} \right\} \Pr \left\{ {{\gamma _{\min }} \le y} \right\} \nonumber \\&=\Pr \left\{ {{\gamma _{sd}} \le x} \right\} \left( {1 - \Pr \left\{ {{\gamma _{sr}} \ge y} \right\} \Pr \left\{ {{\gamma _{rd}} \ge y} \right\} } \right) . \end{aligned}$$

(30)

Therefore, we can conclude

$$\begin{aligned} {F_{\gamma _{\text {NC}}^{\text {SATS}},\gamma _{\text {C}}^{\text {SATS}}}}\left( {x,y} \right) = \left( {1 - {e^{ - \frac{x}{{{{\bar{\gamma }}_{sd}}}}}}} \right) \left( {1 - {e^{ - y\left( {\frac{1}{{{{\bar{\gamma }}_{sr}}}} + \frac{1}{{{{\bar{\gamma }}_{rd}}}}} \right) }}} \right) U\left( x \right) U\left( y \right) . \end{aligned}$$

(31)

Furthermore, the joint PDF of $\gamma _{\text {NC}}^{\text {SATS}}$ and $\gamma _{\text {C}}^{\text {SATS}}$ can be calculated as

$$\begin{aligned} {f_{{\gamma _{\text {NC}}^{\text {SATS}}},{\gamma _{\text {C}}^{\text {SATS}}}}}\left( {x,y} \right) = \frac{{{d^2}{F_{{\gamma _{\text {NC}}^{\text {SATS}}},{\gamma _{\text {C}}^{\text {SATS}}}}}\left( {x,y} \right) }}{{dxdy}} \end{aligned}$$

(32)

which will result the same equation as (11).

Appendix 2: Joint Statistics of $\gamma _{\text {NC}}^{\text {HPSES}}$ and $\gamma _{\text {C}}^{\text {HPSES}}$

The joint CDF of $\gamma _{\text {NC}}^{\text {HPSES}}$ and $\gamma _{\text {C}}^{\text {HPSES}}$ can be obtained as

$$\begin{aligned} {F_{{\gamma _{\text {NC}}^{\text {HPSES}}},{\gamma _{\text {C}}^{\text {HPSES}}}}}\left( {x,y} \right)&= \Pr \left\{ {{\gamma _{\text {NC}}^{\text {HPSES}}} \le x,{\gamma _{\text {C}}^{\text {HPSES}}} \le y} \right\} = \Pr \left\{ {{\gamma _{sd}} \le x,{\gamma _{\min }} + {\gamma _{sd}} \le y} \right\} \nonumber \\&= \left\{ {\begin{array}{c} {\int \limits _0^y {\int \limits _0^{y - v} {{f_{{\gamma _{sd}},{\gamma _{\min }}}}\left( {v,u} \right) dudv} } ,\quad x \ge y \ge 0}\\ {\int \limits _0^x {\int \limits _0^{y - v} {{f_{{\gamma _{sd}},{\gamma _{\min }}}}\left( {v,u} \right) dudv} } ,\quad 0 \le x \le y} \end{array}} \right. \end{aligned}$$

(33)

where ${{f_{{\gamma _{sd}},{\gamma _{\min }}}}}$ is the joint PDF of ${{\gamma _{sd}}}$ and $\gamma _{\min }$. According to (4c), ${{\gamma _{sd}}}$ and $\gamma _{\min }$ are independent. As a result, the PDF of ${{\gamma _{\min }}}$ can be determined as

$$\begin{aligned} {f_{{\gamma _{\min }}}}\left( z \right) = \left( {\frac{1}{{{{\bar{\gamma }}_{sr}}}} + \frac{1}{{{{\bar{\gamma }}_{rd}}}}} \right) {e^{ - z\left( {\frac{1}{{{{\bar{\gamma }}_{sr}}}} + \frac{1}{{{{\bar{\gamma }}_{rd}}}}} \right) }}U\left( z \right) . \end{aligned}$$

(34)

By substituting (34) and the PDF of $\gamma _{sd}$ into (33), we can obtain ${F_{{\gamma _{\text {NC}}^{\text {HPSES}}},{\gamma _{\text {C}}^{\text {HPSES}}}}}\left( {x,y} \right)$ as

$$\begin{aligned} {F_{\gamma _{\text {NC}}^{\text {HPSES}},\gamma _{\text {C}}^{\text {HPSES}}}}\left( {x,y} \right) = \left\{ {\begin{array}{l} {\begin{array}{l} {1 - \frac{{{{\bar{\gamma }}_{sr}}{{\bar{\gamma }}_{rd}}}}{{{{\bar{\gamma }}_{sr}}{{\bar{\gamma }}_{rd}} - {{\bar{\gamma }}_{sd}}\left( {{{\bar{\gamma }}_{sr}} + {{\bar{\gamma }}_{rd}}} \right) }}{e^{ - y\left( {\frac{1}{{{{\bar{\gamma }}_{sr}}}} + \frac{1}{{{{\bar{\gamma }}_{rd}}}}} \right) }} + }\\ {\quad \;\;\;\;\;\;\quad \;\;\;\frac{{{{\bar{\gamma }}_{sd}}\left( {{{\bar{\gamma }}_{sr}} + {{\bar{\gamma }}_{rd}}} \right) }}{{{{\bar{\gamma }}_{sr}}{{\bar{\gamma }}_{rd}} - {{\bar{\gamma }}_{sd}}\left( {{{\bar{\gamma }}_{sr}} + {{\bar{\gamma }}_{rd}}} \right) }}{e^{ - y/{{\bar{\gamma }}_{sd}}}},\;\;\; x \ge y \ge 0} \end{array}}\\ {\begin{array}{l} {1 - {e^{ - x/{{\bar{\gamma }}_{sd}}}} - \frac{{{{\bar{\gamma }}_{sr}}{{\bar{\gamma }}_{rd}}}}{{{{\bar{\gamma }}_{sr}}{{\bar{\gamma }}_{rd}} - {{\bar{\gamma }}_{sd}}\left( {{{\bar{\gamma }}_{sr}} + {{\bar{\gamma }}_{rd}}} \right) }}{e^{ - y\left( {\frac{1}{{{{\bar{\gamma }}_{sr}}}} + \frac{1}{{{{\bar{\gamma }}_{rd}}}}} \right) }} \times }\\ {\quad \;\;\;\quad \;\;\;\;\;\left( {1 - {e^{ - x\left( {\frac{1}{{{{\bar{\gamma }}_{sd}}}} - \frac{1}{{{{\bar{\gamma }}_{sr}}}} - \frac{1}{{{{\bar{\gamma }}_{rd}}}}} \right) }}} \right) ,\;\;\; 0 \le x \le y} \end{array}} \end{array}} \right. . \end{aligned}$$

(35)

Calculation of the joint PDF of $\gamma _{\text {NC}}^{\text {HPSES}}$ and $\gamma _{\text {C}}^{\text {HPSES}}$ is straightforward.