A Threshold-Free LLR-Based Scheme to Minimize the BER for Decode-and-Forward Relaying

Abstract

In cooperative communications, the problem of error propagation has a detrimental effect on the diversity order of the wireless system. To mitigate such an effect, we present a relaying scheme that is based on the absolute value of the log-likelihood ratio (LLR) of the received message signals at both the relay node and the destination node. The calculated LLR values are then compared to each other and based on the result of the comparison, a decision is made on whether or not to activate the relay node. The proposed scheme does not rely on any threshold, and is thus simple in nature. A closed-form expression is derived for the bit-error-rate (BER) of the proposed scheme. The theoretical developments are validated by simulations. As a means for performance measurement, the proposed scheme is compared to its counterparts and is shown to provide a better BER performance at a much lower complexity. Furthermore, a closed-form expression of the outage probability is also derived.

Appendix 1: Derivation of \( \varvec{f}\left( {\varvec{Z}_{{\varvec{SR}}} } \right) \) and \( \varvec{f}\left( {\varvec{y}_{{\varvec{SD}}} } \right) \)

Employing a technique similar to [20], $f\left( {Z_{SR} } \right)$ is easily derived, with help from [21, Eq. (3.325)] as follows

$$ \begin{aligned} f\left( {Z_{SR} } \right) & = \mathop \int \limits_{0}^{\infty } f_{{Z_{\text{SR}} }} \left( {Z_{\text{SR}} |\left| {h_{\text{SR}} } \right|} \right)f_{{\left| {h_{\text{SR}} } \right|}} \left( {\left| {h_{\text{SR}} } \right|} \right){\text{d}}\left| {h_{\text{SR}} } \right| \\ & = \mathop \int \limits_{0}^{\infty } \frac{1}{{\sqrt {\pi \left| {h_{\text{SR}} } \right|^{2} \sigma_{\text{SR}}^{2} } }} \cdot {\text{Exp}}\left[ { - \frac{{\left( {Z_{\text{SR}} - \left| {h_{\text{SR}} } \right|^{2} \sqrt {E_{b} } } \right)^{2} }}{{\left| {h_{\text{SR}} } \right|^{2} \sigma_{\text{SR}}^{2} }}} \right] \cdot 2\frac{{\left| {h_{\text{SR}} } \right|}}{{\left( {d_{SR} } \right)^{ - a} }}{\text{Exp}}\left[ { - \frac{{\left| {h_{\text{SR}} } \right|^{2} }}{{\left( {d_{SR} } \right)^{ - a} }}} \right]{\text{d}}\left| {h_{\text{SR}} } \right| \\ & = \alpha_{1} {\text{Exp}}\left[ {\beta_{1} Z_{\text{SR}} } \right]{\text{Exp}}\left[ { - \chi_{1} \left| {Z_{\text{SR}} } \right|} \right]. \\ \end{aligned} $$

(29)

where $\alpha_{1} = 1/\sqrt {\left( {{\text{d}}_{SR} } \right)^{a} \sigma_{\text{SR}}^{2} \left( {\gamma_{\text{SR}} + 1} \right)}$, $\beta_{1} = 2\sqrt {E_{b} } /\sigma_{\text{SR}}^{2}$ and $\chi_{1} = 2\sqrt {E_{b} \left( {\gamma_{\text{SR}} + 1} \right)/\gamma_{\text{SR}} } /\sigma_{\text{SR}}^{2}$.

Similarly, we derive $f\left( {y_{SD} } \right)$, yielding

$$ f\left( {y_{SD} } \right) = \alpha_{2} {\text{Exp}}\left[ {\beta_{2} y_{\text{SD}} } \right]{\text{Exp}}\left[ { - \chi_{2} \left| {y_{\text{SD}} } \right|} \right], $$

(30)

where $\alpha_{2} = 1/\sqrt {\left( {{\text{d}}_{SD} } \right)^{a} \sigma_{\text{SD}}^{2} \left( {\gamma_{\text{SD}} + 1} \right)}$, $\beta_{2} = 2\sqrt {E_{b} } /\sigma_{\text{SD}}^{2}$ and $\chi_{2} = 2\sqrt {E_{b} \left( {\gamma_{\text{SD}} + 1} \right)/\gamma_{\text{SD}} } /\sigma_{\text{SD}}^{2}$.