Performance of DF Based Dual-Hop Dual-Path Hybrid RF/FSO Cooperative System

Abstract

In this paper, we investigate the performance of an asymmetric radio frequency/free space optical (RF/FSO) hybrid satellite–terrestrial cooperative system with two relays utilizing a non-linear decoder (NLD) and a low-complexity piecewise linear (PL) decoder. The satellite transmits data over the RF links to the relays on the ground and the relays follow the decode-and-forward (DF) protocol and transmit the decoded data to a distant destination over FSO links assuming the constant transmit power constraint. The RF and FSO channels of the asymmetric system are assumed to follow the Shadowed-Rician and Gamma–Gamma distributions, respectively. The relays are assumed to have hybrid RF/FSO capabilities and utilize subcarrier intensity modulation scheme for transmitting the decoded signal to the destination. A NLD is derived at the destination of considered system for M-ary phase-shift keying and utilizing the derived NLD, we obtain the error rate and its upper bound. Further, in order to reduce the decoding complexity a PL decoder is derived, which is a sub-optimal decoder and works very close to the NLD. Utilizing the derived PL decoder we obtain a closed form expression of bit error rate for the considered system. Lastly, we show the effect of imperfect channel estimation and optimized power distribution of the relays on the performance of the DF based FSO cooperative system with two relays.

Appendix: Proof of Theorem 1

Let us state some useful transformation results before providing the Proof of Theorem 1.

• If $W\sim {\mathcal {N}}(\mu ,\sigma ^2)$, where ${\mathcal {N}}(\mu ,\sigma ^2)$ denotes the real valued normal distribution with mean $\mu$ and variance $\sigma ^2$, then $S=\exp (W)\sim {\text {Log}}{\mathcal {N}}(\mu ,\sigma ^2)$, where ${\text {Log}}{\mathcal {N}}(.,.)$ denote log-normal distribution.

• If $S=\exp (W)\sim {\text {Log}}{\mathcal {N}}(\mu ,\sigma ^2)$, then $aS\sim {\text {Log}}{\mathcal {N}}(\ln (a)+\mu ,\sigma ^2)$, where a is a constant.

• If $S_p=\exp (W_p)\sim {\text {Log}}{\mathcal {N}}(\mu _p,\sigma _p^2)$, $p=1,2,\dots ,N$, where $W_p\sim {\mathcal {N}}(\mu _p,\sigma ^2_p)$ are fully correlated random variables, then $S=\sum _{p=1}^{N}S_p$ approximately follows the log-normal distribution [24] i.e., $S\sim {\text {Log}}{\mathcal {N}}(\mu _s,\sigma _s^2)$, where

  $$\begin{aligned} \mu _s=\ln \left[ \sum _{p=1}^{N}e^{\mu _p+\frac{\sigma _p^2}{2}}\right] -\frac{\sigma _s^2}{2}, \end{aligned}$$

  (30)

  and

  $$\begin{aligned} \sigma _{s}^{2}=\ln \left[ \frac{ \begin{array}{c} \sum _{p=1}^{N} e^{2\mu _p+\sigma _p^{2}}(e^{\sigma _p^{2}}-1)+2\sum _{p=1}^{N - 1} \sum _{q=p + 1}^{N} \\ \times e^{\mu _p+\frac{\sigma _p^{2}}{2}+\mu _q+\frac{\sigma _q^{2}}{2}}(e^{\sigma _p\sigma _q}-1) \end{array} }{\left( \sum _{p=1}^{N}e^{\mu _p+\frac{\sigma _p^{2}}{2}}\right) ^2}+1\right] , \end{aligned}$$

  (31)

• If $S_1\sim {\text {Log}}{\mathcal {N}}(\mu _{s_1},\sigma _{s_1}^2)$ and $S_2\sim {\text {Log}}{\mathcal {N}}(\mu _{s_2},\sigma _{s_2}^2)$ are fully correlated log-normal distributed random variables, then $S_1/S_2\sim {\text {Log}}{\mathcal {N}}(\mu _{s_1}-\mu _{s_2},\sigma _{s_1}^2+\sigma _{s_2}^2+2\sigma _{s_1}\sigma _{s_2})$ [25].

It is observed from (14) that $\kappa _i$, $i=1,2,3,4$ are real gaussian and fully correlated random variables. Utilizing the above stated transformations in (13) we can say that $X_i\sim {\text {Log}}{\mathcal {N}}(\mu _{x_i},\sigma _{x_i}^2)$ and $Y_i\sim {\text {Log}}{\mathcal {N}}(\mu _{y_i},\sigma _{y_i}^2)$, $i=1,2,3,4$. Thus the mean and variance of $X_i$ will be:

$$\begin{aligned} \mu _{x_1}& =\ln (\epsilon _1(1-\epsilon _2))-\frac{1}{N_d}\left[ P_{R_1}I_1^2(1+2\hat{z}_{R_1}) + P_{R_2}I_2^2(1-2\hat{z}_{R_2}) \right. \\&\left. \quad -2 \sqrt{P_{R_1}P_{R_2}}I_1I_2(1+\hat{z}_{R_1}-\hat{z}_{R_2})\right] , \\ \mu _{x_2}& =\ln ((1-\epsilon _1)\epsilon _2)-\frac{1}{N_d}\left[ P_{R_1}I_1^2(1-2\hat{z}_{R_1}) + P_{R_2}I_2^2(1+2\hat{z}_{R_2}) \right. \\&\left. \quad - 2\sqrt{P_{R_1}P_{R_2}}I_1I_2(1-\hat{z}_{R_1}+\hat{z}_{R_2})\right] , \\ \mu _{x_3}& =\ln (\epsilon _1\epsilon _2)-\frac{1}{N_d}\left[ P_{R_1}I_1^2(1+2\hat{z}_{R_1}) + P_{R_2}I_2^2(1+2\hat{z}_{R_2}) \right. \\&\left. \quad + 2\sqrt{P_{R_1}P_{R_2}}I_1I_2(1+\hat{z}_{R_1}+\hat{z}_{R_2})\right] , \\ \mu _{x_4}& =\ln ((1 - \epsilon _1)(1 - \epsilon _2)) - \frac{1}{N_d}\left[ P_{R_1}I_1^2(1 - 2\hat{z}_{R_1}) + P_{R_2}I_2^2(1 - 2\hat{z}_{R_2}) \right. \\&\left. \quad - 2\sqrt{P_{R_1}P_{R_2}}I_1I_2(\hat{z}_{R_1} + \hat{z}_{R_2} - 1)\right] , \end{aligned}$$

(32)

$$\begin{aligned} \sigma _{x_1}^2& =-\frac{2}{N_d}(\sqrt{P}_{R_1} I_1 - \sqrt{P}_{R_2} I_2)^2,\;\;\sigma _{x_2}^2=-\frac{2}{N_d}(\sqrt{P}_{R_2} I_2 -\sqrt{P}_{R_1} I_1)^2 \\ \sigma _{x_3}^2& =-\frac{2}{N_d}(\sqrt{P}_{R_1}I_1+\sqrt{P}_{R_2}I_2)^2,\;\;\sigma _{x_4}^2=-\frac{2}{N_d}(\sqrt{P}_{R_2} I_2 +\sqrt{P}_{R_1} I_1)^2, \end{aligned}$$

(33)

Similarly, the mean and variance of $Y_i$ can be obtained. Let $X\triangleq \sum _{i=1}^{4}X_i$ and $Y\triangleq \sum _{i=1}^{4}Y_i$, thus we can say that $X\sim {\text {Log}}{\mathcal {N}}(\mu _x,\sigma _x^2)$ and $Y\sim {\text {Log}}{\mathcal {N}}(\mu _y,\sigma _y^2)$. Further, with the use of given transformations in (12), it is observed that the LLR decoder approximately follows the real valued normal distribution with mean $(\mu _x-\mu _y)$ and variance $(\sigma _x^2+\sigma _y^2+2\sigma _x\sigma _y)$.

Based on the correct and erroneous decoding at both relays, there can be four possible events with probability of occurrence ${\text {Pr}}({\mathcal {D}}_i|h_{1},h_{2})$ given in Table 1. Therefore, the average probability of error conditioned over the channel coefficients, $I_1$ and $I_2$ is given in (19).