BER Analysis for BPSK Based SIM–FSO Communication System Over Strong Atmospheric Turbulence with Spatial Diversity and Pointing Errors

Abstract

Free space optical communication (FSO) has much attention in recent years for the applications viz. inter-satellite, deep space communications, inter and intra chip communications. The performance of FSO systems sternly suffers from atmospheric turbulence due to the random nature of weather conditions. Spatial diversity is an emerging technique for improving the performance of the system over strong atmospheric turbulences. In this paper, the error rate performance of binary phase shift keying based subcarrier intensity modulated free space optical (SIM–FSO) communication system over gamma–gamma channel with pointing errors is investigated. Novel closed-form analytical expressions are derived for the average bit error rate of single-input multiple-output FSO (SIMO–FSO) system with various combining schemes. The error rate performance of SISO and SIMO–FSO systems are compared in terms of 2D and 3D plots.

Appendices

Appendix 1: Proof of BER of SISO [Eq. (13)]

The probability of average BER expressed by Eq. (12) is reproduced

$$\begin{aligned} P_{e,SISO} =\frac{\alpha \beta \xi ^{2}}{A_0 h_l {\Gamma }(\alpha ){\Gamma }(\beta )}\times \mathop \int \nolimits _0^\infty 0.5 \hbox { erf}c\left( {\frac{h\gamma }{2\sigma _n }} \right) \times G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha \beta h}{A_0 h_l }} \right| _{-1+\xi ^{2},\alpha -1,\beta -1}^{\xi ^{2}} } \right] dh\nonumber \\ \end{aligned}$$

(33)

The complementary error function $\hbox {erfc}\left( \cdot \right)$ can be expressed as Meijer G function using Eq. (34). Using this identity, Eq. (33) reduces to (35)

$$\begin{aligned} \hbox {erfc}\left( {\sqrt{x}} \right)&= \frac{1}{\sqrt{\pi }}G_{1,2}^{2,0} \left[ {\left. x \right| _{0,1/2}^{1} } \right] \end{aligned}$$

(34)

$$\begin{aligned} P_{e,SISO}&= \frac{\alpha \beta \xi ^{2}}{A_0 h_l {\Gamma }(\alpha ){\Gamma }(\beta )}\times \mathop \int \nolimits _0^\infty \frac{1}{2\sqrt{\pi }}G_{1,2}^{2,0} \left[ {\left. {\frac{\gamma ^{2}h^{2}}{4\sigma _n^2 }} \right| _{0,1/2}^{1} } \right] \nonumber \\&\times G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha \beta h}{A_0 h_l }} \right| _{-1+\xi ^{2},\alpha -1,\beta -1}^{\xi ^{2}} } \right] dh \end{aligned}$$

(35)

By using [31, Eq. (21)] in (35), Eq. (13) can be obtained.

Appendix 2: Proof of BER of SIMO with OC [Eq. (20)]

The strong atmospheric channel model (Eq. (9)) and the average BER of SIMO–FSO with OC (Eq. (19)) are reproduced respectively

$$\begin{aligned} f_{h_n } \left( {h_n } \right)&= \frac{\alpha _n \beta _n \xi _n^2 }{A_{0_n } h_{l_n } {\Gamma }(\alpha _n ){\Gamma }(\beta _n )}G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] \end{aligned}$$

(36)

$$\begin{aligned} P_{e,OC}&\approx \frac{1}{12}\mathop \prod \limits _{n=1}^N \mathop \int \nolimits _0^\infty f_{h_n } \left( {h_n } \right) e^{-\left( {\frac{\gamma ^{2}h_n^2 }{4N\sigma _n^2 }} \right) }dh_n\nonumber \\&+\,\frac{1}{4}\mathop \prod \limits _{n=1}^N \mathop \int \nolimits _0^\infty f_{h_n } \left( {h_n } \right) e^{-\left( {\frac{\gamma ^{2}h_n^2 }{3N\sigma _n^2 }} \right) }dh_n \end{aligned}$$

(37)

By applying Eq. (36) in (37)

$$\begin{aligned} P_{e,OC}&\approx \frac{1}{12}\mathop \prod \limits _{n=1}^N \mathop \int \nolimits _0^\infty \frac{\alpha _n \beta _n \xi _n^2 }{A_{0_n } h_{l_n } {\Gamma }\left( {\alpha _n } \right) {\Gamma }\left( {\beta _n } \right) }G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] e^{-\left( {\frac{\gamma ^{2}h_n^2 }{4N\sigma _n^2 }} \right) }dh_n\nonumber \\&+\,\frac{1}{4}\mathop \prod \limits _{n=1}^N \mathop \int \nolimits _0^\infty \frac{\alpha _n \beta _n \xi _n^2 }{A_{0_n } h_{l_n } {\Gamma }(\alpha _n ){\Gamma }(\beta _n )}G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] e^{-\left( {\frac{\gamma ^{2}h_n^2 }{3N\sigma _n^2 }} \right) }dh_n \nonumber \\ \end{aligned}$$

(38)

An exponential function $\hbox {exp}\left( \cdot \right)$ can be expressed as Meijer G function using Eq. (39). Using this identity, Eq. (38) can be simplified to (40).

$$\begin{aligned} \hbox {exp}\left( {-x} \right)&= G_{0,1}^{1,0} \left[ {\left. x \right| _{0}^{\cdot } } \right] \end{aligned}$$

(39)

$$\begin{aligned} P_{e,OC}&\approx \frac{1}{12}\mathop \prod \limits _{n=1}^N \frac{\alpha _n \beta _n \xi _n^2 }{A_{0_n } h_{l_n } {\Gamma }\left( {\alpha _n } \right) {\Gamma }\left( {\beta _n } \right) }\mathop \int \nolimits _0^\infty G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] \nonumber \\&\quad \quad G_{0,1}^{1,0} \left[ {\left. {\frac{\gamma ^{2}h_n^2 }{4N\sigma _n^2 }} \right| _{0}^- } \right] dh_n\nonumber \\&+\,\frac{1}{4}\mathop \prod \limits _{n=1}^N \frac{\alpha _n \beta _n \xi _n^2 }{A_{0_n } h_{l_n } {\Gamma }\left( {\alpha _n } \right) {\Gamma }\left( {\beta _n } \right) }\mathop \int \nolimits _0^\infty G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] \nonumber \\&\quad \quad G_{0,1}^{1,0} \left[ {\left. {\frac{\gamma ^{2}h_n^2 }{3N\sigma _n^2 }} \right| _{0}^- } \right] dh_n \end{aligned}$$

(40)

By using [31, Eq. (21)] in (40), Eq. (20) can be obtained.

Appendix 3: Proof of BER with EGC [Eq. (25)]

The average BER of SIMO–FSO with EGC (Eq. (24)) is reproduced

$$\begin{aligned} P_{e,EGC} =0.5\times \mathop \int \nolimits _0^\infty \hbox {erfc}\left( {\frac{\gamma }{2N\sigma _n }\mathop \sum \limits _{n=1}^N h_n } \right) f_h \left( h \right) dh \end{aligned}$$

(41)

By applying Eq. (36) in (41)

$$\begin{aligned} P_{e,EGC}&= 0.5\times \frac{\alpha _n \beta _n \xi _n^2 }{A_{0_n } h_{l_n } {\Gamma }(\alpha _n ){\Gamma }(\beta _n )}\mathop \int \nolimits _0^\infty \hbox {erfc}\left( {\frac{\gamma }{2N\sigma _n }\mathop \sum \limits _{n=1}^N h_n } \right) \nonumber \\&G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] dh_n \end{aligned}$$

(42)

The complementary error function $\hbox {erfc}\left( \cdot \right)$ can be expressed as Meijer G function using Eq. (34). Using this identity, Eq. (42) reduces to (43).

$$\begin{aligned} P_{e,EGC}&= \frac{\alpha _n \beta _n \xi _n^2 }{2\sqrt{\pi }A_{0_n } h_{l_n } {\Gamma }(\alpha _n ){\Gamma }(\beta _n )}\mathop \prod \limits _{n=1}^N \mathop \int \nolimits _0^\infty G_{1,3}^{3,0} \left[ {\left. {\frac{\alpha _n \beta _n h_n }{A_{0_n } h_{l_n } }} \right| _{-1+\xi _n^2 ,\alpha _n -1,\beta _n -1}^{\xi _n^2 } } \right] \nonumber \\&G_{1,2}^{2,0} \left[ {\left. {\frac{\gamma ^{2}h_n^2 }{4N\sigma _n^2 }} \right| _{0,0.5}^{1} } \right] dh_n \end{aligned}$$

(43)

By using [31, Eq. (21)] in (43), Eq. (25) can be obtained.