Performance Improvement of Two-Hop Relay System Using Polarization Diversity

Abstract

With growing demands for mobile data, providing reliable communication with high quality of service has become a critical issue. Wireless relaying has been recognized as an effective solution to enhance system coverage, data rate, reliability and energy efficiency by introducing additional network node-relay. However, complex wireless channel characteristics, such as fading, path loss, noise and interference limits the performance of such systems in a manner that it is dominated by the worst channel. That is why additional techniques are necessary to be implemented in order to improve the characteristics of this bottleneck link, and performance of relay system as whole. Multi-antenna relaying has emerged as a promising technique to overcome bottleneck effect in relay system and provide performance enhancements, but the small size of network nodes limits their use at wireless terminals. In this paper we propose the implementation of polarization diversity, where one compact dual-polarized antenna is used instead of two adequately separated antennas. Maximal ratio combining of diversity signals in amplify-and-forward relaying systems applying variable gain is analysed. Mixed Rayleigh/Ricean fading environment is assumed, while two diversity signals are described as two correlated and non-identical Rayleigh fading channels. Novel analytical model for determining the performance of the analysed relay system expressed through the system’s outage and bit error rate is derived. Excellent match between Monte-Carlo simulations and numerical results confirms the validity of the proposed analytical model. Furthermore, the presented results illustrate the effect of various channels parameters on the system performance, such as Ricean K factor, average signal to noise ratios, diversity signals’ correlation and cross-polar discrimination. For a given performance requirements, the proposed solution can significantly improve relay system performance.

Appendix: Derivation of OP and BER Expressions

Starting from the integral definition of CDF of total SNR, given in (5) we get:

$$\begin{aligned} \begin{aligned} F\left( \gamma \right) =&1 - \frac{{\left( {1 + K} \right) {e^{ - K}}}}{{2\alpha {{\bar{\gamma }}_2}}}{e^{ - \frac{{\left( {1 + K} \right) \gamma }}{{{{\bar{\gamma }}_2}}}}}\int \limits _0^\infty \left[ {\left( {1 + \alpha } \right) {e^{ - \frac{{2\gamma \left( {\gamma + w} \right) }}{{w\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}}}}} - \left( {1 - \alpha } \right) {e^{ - \frac{{2\gamma \left( {\gamma + w} \right) }}{{w\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}}}}}} \right] \\&\times {I_0}\left( {2\sqrt{\frac{{K\left( {1 + K} \right) \left( {\gamma + w} \right) }}{{{{\bar{\gamma }}_2}}}} } \right) dw, \end{aligned} \end{aligned}$$

(11)

since $\int \limits _0^\infty p(\gamma ) d\gamma =1$.

Using the alternative representation of $I_{0}$, i.e. ${I_0}\left( z \right) = \sum \limits _{k = 0}^\infty {\frac{{{\left( z/2 \right) }^{2k}}}{ \left( k! \right) ^2}}$, we get:

$$\begin{aligned} \begin{aligned} F\left( \gamma \right) =&1 - \frac{{\left( {1 + K} \right) {e^{ - K}}}}{{2\alpha {{\bar{\gamma }}_2}}}{e^{ - \frac{{\left( {1 + K} \right) \gamma }}{{{{\bar{\gamma }}_2}}}}}\sum \limits _{n = 0}^\infty {\frac{{{{\left( {K\left( {1 + K} \right) } \right) }^n}}}{{{{\left( {n!} \right) }^2}{{\bar{\gamma }}_2}^n}}}\\&\times \left[ \left( {1 + \alpha } \right) {e^{ - \frac{{2\gamma }}{{\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}}}}}\int \limits _0^\infty {{e^{ - \frac{{2{\gamma ^2}}}{{w\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}}}}}{e^{ - \frac{{\left( {1 + K} \right) w}}{{{{\bar{\gamma }}_2}}}}}{{\left( {\gamma + w} \right) }^n}dw } \right. \\&\left. - \left( {1 - \alpha } \right) {e^{ - \frac{{2\gamma }}{{\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}}}}}\int \limits _0^\infty {{e^{ - \frac{{2{\gamma ^2}}}{{w\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}}}}}{e^{ - \frac{{\left( {1 + K} \right) w}}{{{{\bar{\gamma }}_2}}}}}{{\left( {\gamma + w} \right) }^n}dw} \right] , \end{aligned} \end{aligned}$$

(12)

Rewriting $(\gamma +w)^{n}=\sum \limits _{m=0}^{n}\left( {\begin{array}{c}n\\ m\end{array}}\right) \gamma ^{m}w^{n-m}$, the following expression is obtained:

$$\begin{aligned} \begin{aligned} F\left( \gamma \right) =&1 - \frac{{\left( {1 + K} \right) {e^{ - K}}}}{{2\alpha {{\bar{\gamma }}_2}}}{e^{ - \frac{{\left( {1 + K} \right) \gamma }}{{{{\bar{\gamma }}_2}}}}}\sum \limits _{n = 0}^\infty {\frac{{{{\left( {K\left( {1 + K} \right) } \right) }^n}}}{{{{\left( {n!} \right) }^2}{{\bar{\gamma }}_2}^n}}} \sum \limits _{m = 0}^n {\left( {\begin{array}{c}n\\ m\end{array}}\right) {\gamma ^m}} \\&\times \left[ {\left( {1 + \alpha } \right) {e^{ - \frac{{2\gamma }}{{\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}}}}}\int \limits _0^\infty {{e^{ - \frac{{2{\gamma ^2}}}{{w\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}}}}}{e^{ - \frac{{\left( {1 + K} \right) w}}{{{{\bar{\gamma }}_2}}}}}{w^{n - m}}dw} } \right. \\&\left. { - \left( {1 - \alpha } \right) {e^{ - \frac{{2\gamma }}{{\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}}}}}\int \limits _0^\infty {{e^{ - \frac{{2{\gamma ^2}}}{{w\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}}}}}{e^{ - \frac{{\left( {1 + K} \right) w}}{{{{\bar{\gamma }}_2}}}}}{w^{n - m}}dw} } \right] . \end{aligned} \end{aligned}$$

(13)

With the aid of [28, Eq. (3.471–9)], i.e. $\int \limits _0^\infty {{x^{\nu - 1}}{e^{ - \frac{\beta }{x} - \gamma x}}dx = 2{{\left( {\frac{\beta }{\gamma }} \right) }^{\frac{\nu }{2}}}{K_\nu }\left( {2\sqrt{\beta \gamma } } \right) }$, having $\nu =n-m+1$, $\beta =\frac{2\gamma ^{2}}{(1\pm \alpha )\bar{\gamma }_1}$ and $\gamma =\frac{1+K}{\bar{\gamma }_2}$ we finally obtain the expression given in (6).

Similarly, starting from the BER definition (7), we easily get (9). Furthermore, it is known that $\frac{1}{\sqrt{2\pi }}\int \limits _0^{\infty }e^{-x^{2}/2}=\frac{1}{2}$, thus:

$$\begin{aligned} \begin{aligned}&{P_b}\left( e \right) = \frac{1}{2} - \frac{{\left( {1 + K} \right) {e^{ - K}}}}{{\sqrt{2\pi } \alpha {{\bar{\gamma }}_2}}}\sum \limits _{n = 0}^\infty {\frac{{{K^n}}}{{{{\left( {n!} \right) }^2}}}} \sum \limits _{m = 0}^n\left( {\begin{array}{c}n\\ m\end{array}}\right) {\left( {\frac{{1 + K}}{{{{\bar{\gamma }}_2}}}} \right) ^{\frac{{n + m - 1}}{2}}}{\left( {\frac{2}{{{{\bar{\gamma }}_1}}}} \right) ^{\frac{{n - m + 1}}{2}}}\times \\&\left[ {\frac{1}{{\left( {1 + \alpha } \right) {^{\frac{{n - m - 1}}{2}}}}}\int \limits _0^\infty {{e^{ - \frac{{{x^2}}}{2}}}{e^{ - \left( {\frac{2}{{\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}}} + \frac{{\left( {1 + K} \right) }}{{{{\bar{\gamma }}_2}}}} \right) \frac{{{x^2}}}{\beta }}}{{\left( {\frac{{{x^2}}}{\beta }} \right) }^{n + 1}}} {K_{n - m + 1}}\left( {\frac{{{x^2}}}{\beta }\sqrt{\frac{{8\left( {1 + K} \right) }}{{\left( {1 + \alpha } \right) {{\bar{\gamma }}_1}{{\bar{\gamma }}_2}}}} } \right) }dx \right. \\&-\left. {\frac{1}{{\left( {1 - \alpha } \right) {^{\frac{{n - m - 1}}{2}}}}}\int \limits _0^\infty {{e^{ - \frac{{{x^2}}}{2}}}{e^{ - \left( {\frac{2}{{\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}}} + \frac{{\left( {1 + K} \right) }}{{{{\bar{\gamma }}_2}}}} \right) \frac{{{x^2}}}{\beta }}}{{\left( {\frac{{{x^2}}}{\beta }} \right) }^{n + 1}}} {K_{n - m + 1}}\left( {\frac{{{x^2}}}{\beta }\sqrt{\frac{{8\left( {1 + K} \right) }}{{\left( {1 - \alpha } \right) {{\bar{\gamma }}_1}{{\bar{\gamma }}_2}}}} } \right) } dx\right] \end{aligned} \end{aligned}$$

(14)

Introducing the change of variables $\frac{x^{2}}{\beta }=t$ ($dx=\frac{\sqrt{\beta }}{2\sqrt{t}}dt$), and using [28, Eq. (6.621–3)], with the following parameters of defined integral: $\mu =n+\frac{3}{2}$, $\alpha =\frac{\beta }{2}+\frac{(1+K)}{\bar{\gamma }_2}+\frac{2}{(1\pm \alpha )\bar{\gamma }_1}$, $\nu =n-m+1$ and $\beta =2\sqrt{\frac{2(1+K)}{(1\pm \alpha )\bar{\gamma }_1\bar{\gamma }_2}}$, we get average BER as in (10).