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.

1 Introduction

Free space optical (FSO) communication provides high data rate and secure communication with low maintenance cost and less deployment time [1]. FSO is the best complement technology to radio frequency (RF) communication, due to huge capacity and greater bandwidth, superior protection against interference, lesser power consumption and compact transceiver architecture [2]. The performance of the FSO systems purely depends on the external atmospheric conditions caused by the variations of the refractive index. Various channel models are developed to evaluate the impact of atmospheric turbulence such as, log-normal, negative exponential and gamma–gamma model [3, 4]. The log-normal distribution is valid only in weak turbulence and the gamma–gamma distribution is found to be the most suitable for moderate to strong turbulence channels [5]. The strong atmospheric channel model with the mutual effect of atmospheric turbulence and pointing error effects are investigated in [6–8]. The wireless optical link impaired by absorption, scattering, diffraction and atmospheric turbulence. The random variations in the refractive index of atmosphere cause random fluctuations in both amplitude and phase of the transmitting optical beam which result in fading, beam spread, and angular spread [9]. As per previous studies, the various solutions to mitigate the effects of atmospheric turbulence are error control coding in conjunction with interleaving, maximum likelihood sequence detection and spatial diversity [10, 11].

The different modulation formats used in FSO systems are on-off keying (OOK), binary phase shift keying (BPSK) [4], pulse position modulation (PPM) [12], differential phase shift keying (DPSK) [13], polarization shift keying (PolSK) [14] and subcarrier intensity modulation (SIM) [3, 4]. The OOK is the simplest and widely used modulation format in FSO systems. But, it requires an adaptive threshold to perform optimally and very sensitive to atmospheric turbulence [3]. The drawbacks of OOK modulation are overcome by PPM; however it requires complex transceiver design and suffers from poor bandwidth efficiency. The modulation scheme plays a vital role to decide the power efficiency in FSO systems. The phase angular modulation formats, such as BPSK and DPSK are highly sensitive to the phase noise effects [15]. PolSK bids greater immunity to the phase noise and the atmospheric turbulence. The SIM was first proposed in the optical wireless communications (OWC) application in [16] and then it is employed with various phase-shift keying (PSK) modulations over different atmospheric turbulence channels. Recently, the performances of FSO systems are analyzed including the combined effect of atmospheric turbulence and misalignment fading over strong atmospheric channels [6, 7]. A multiple-input multiple-output FSO (MIMO–FSO) system improves the bit error rate (BER) performance over strong atmospheric turbulence channels [17]. Single-input multiple-output FSO (SIMO–FSO) systems with diversity techniques can offer signal redundancy which mitigates the fading effects and significantly improve the transmission rate [18]. In [19], the outage probability of MIMO–FSO system is investigated.

The tradeoff between BER and link distance by compensating the phase noise using OFDM pilot subcarriers in Coherent Optical OFDM Systems is studied in [20]. A very high bitrate is achieved through an all-optical fiber distributed indoor system providing wireless cells in [21]. The availability for FSO and MMW systems for the region of Uruguay (South America) is studied in [22]. In [23], the performance of wireless underwater communications for different ocean water types is analyzed.

In a recent work, [24], SIM based DPSK scheme in FSO system is considered and its BER performance is evaluated over MIMO–FSO links which follow the gamma–gamma distribution. The performance analysis of multi-beam FSO system employed in the diversity or multiplexing scheme is studied in [18]. In this paper, the BER performance of BPSK based SIM–FSO system over gamma–gamma distributed strong atmospheric channel with pointing error is investigated. The novel closed form BER expressions are derived for FSO links with multiple apertures at the receiver.

The rest of the paper is organized as follows: Sect. 2 discusses the system and channel model used. In Sect. 3 and 4, the average BER for SISO and SIMO are discussed. Section 5 describes the numerical results with graphical analysis. Finally, concluding remarks are highlighted in Sect. 6.

2 System and Channel model

2.1 System Model

An FSO communication system with $M$ transmission apertures and $N$ reception apertures over a discrete time ergodic channel with additive white Gaussian noise (AWGN) is considered. The binary input data are modulated by using BPSK–SIM and transmitted through the telescope into strong atmospheric turbulence. The received electrical signal at the $n^{th}$ receiver aperture is given by [25]

$$\begin{aligned} r_n =x\gamma \sum \limits _{m=1}^M h_{mn} +v_n , \quad n=1,2,\ldots ..N \end{aligned}$$

(1)

where $x$ is the transmitted signal, $\gamma$ is the detector responsivity, $h_{mn}$ is the irradiance from the $m^{th}$ transmitter and $n^{th}$ receiver, $v_n$ is the AWGN with zero mean and variance $\sigma _v =N_0 /2$. The irradiance ${\prime }h_{mn} {\prime }$ models the optical intensity fluctuations resulting from atmospheric loss, turbulence and fading is given as [6]

$$\begin{aligned} h_{mn} =h_{l_{mn} } h_{s_{mn} } h_{p_{mn} } \end{aligned}$$

(2)

where $h_{l_{mn} }$ is the attenuation due to beam extinction and path loss from the $m^{th}$ transmitter and $n^{th}$ receiver, $h_{s_{mn} }$ due to scintillation effects and $h_{p_{mn} }$ due to the geometric spread and pointing errors.

2.2 Channel Model

The atmospheric path loss is modeled by Beers-Lambert law is given as [26]

$$\begin{aligned} h_{l_{mn} } =e^{-\sigma L_{mn} } \end{aligned}$$

(3)

where $\sigma$ is the attenuation coefficient and $L_{mn}$ is the propagation distance of $m^{th}$ transmitter and the $n^{th}$ receiver.

The strong atmospheric turbulence can be modeled by gamma–gamma distribution with scintillation parameters $\alpha$ and $\beta$, which are indicated as functions of the Rytov variance and a geometry factor. The probability density function (PDF) of the gamma–gamma distribution is given by [26]

$$\begin{aligned} f_{h_{s_{mn} } } \left( {h_{s_{mn} } } \right) =\frac{2(\alpha _{mn} \beta _{mn} )^{(\alpha _{mn} +\beta _{mn} )/2}}{{\Gamma }(\alpha _{mn} ){\Gamma }(\beta _{mn} )}h_{s_{i,j} }^{\left( {\alpha +\beta } \right) /2-1} K_{\left( {\alpha _{mn} -\beta _{mn} } \right) } \left( {2\sqrt{\alpha _{mn} \beta _{mn} h_{s_{i,j} } }} \right) \end{aligned}$$

(4)

where $\alpha$ and $\beta$ are the effective number of large and small scale turbulent eddies, ${\Gamma }(\cdot )$ is the gamma function and $K_{\left( {\alpha -\beta } \right) }$ is the modified Bessel function of the second kind of order $(\alpha -\beta )$.

The signal fading due to pointing errors can be defined based on a circular detection aperture of radius $r$ and a Gaussian beam. The PDF of $h_p$ is given by [6]

$$\begin{aligned} f_{h_p } \left( {h_p } \right) =\frac{\xi _{mn}^2 }{A_0 ^{\xi _{mn}^2 }}h_p ^{\xi _{mn}^2 -1},0\le h_p \le A_0 \end{aligned}$$

(5)

where $A_0 =\left[ {erf\left( v \right) } \right] ^{2}$ is the fraction of the collected power at $r=0$. The Gauss error function $erf\left( \cdot \right)$ is defined as $erf\left( x \right) =\frac{2}{\sqrt{\pi }}\mathop \int \nolimits _0^x e^{-t^{2}}dt$. The radial distance is denoted as $r$ and $\xi$ is the ratio between the equivalent beam radius at the receiver and the pointing error displacement (jitter) standard deviation at the receiver.

The combined channel distribution for strong atmospheric turbulence regime is given by [27]

$$\begin{aligned}&f_{h_{i,j} } \left( {h_{i,j} } \right) =\nonumber \\&\quad \frac{2\xi _{mn}^2 (\alpha _{mn} \beta _{mn} )^{(\alpha _{mn} +\beta _{mn} )/2}}{\left( {A_0 h_{l_{i,j} } } \right) ^{\xi _{mn}^2 }{\Gamma }(\alpha _{mn} ){\Gamma }(\beta _{mn} )}h_{i,j}^{\xi _{mn}^2 -1}\times \mathop \int \nolimits _{h_{i,j} /A_0 h_{l_{i,j} } }^\infty h_{s_{i,j} }^{\left( {\alpha _{mn} +\beta _{mn} } \right) /2-1-\xi _{mn}^2 } K_{\left( {\alpha _{mn} -\beta _{mn} } \right) } \left( {2\sqrt{\alpha _{mn} \beta _{mn} h_{s_{i,j} } }} \right) dh_{s_{i,j} }\qquad \quad \end{aligned}$$

(6)

where $\alpha$ and $\beta$ are the effective number of large and small scale turbulent eddies, ${\Gamma }(\cdot )$ is the gamma function, $K_{\left( {\alpha -\beta } \right) }$ is the modified Bessel function of the second kind of order $\left( {\alpha -\beta } \right)$. The effective number of large and small scale turbulent eddies $\alpha$ and $\beta$ for a spherical wave are given by [27],

$$\begin{aligned} \alpha =\left[ {exp\left( {\frac{0.49\delta _n^2 }{\left( {1+0.18d^{2}+0.56\delta _n^{12/15} } \right) ^{7/6}}} \right) -1} \right] ^{-1}\end{aligned}$$

(7)

$$\begin{aligned} \beta =\left[ {exp\left( {\frac{0.51\delta _n^2 \left( {1+0.69\delta _n^{12/15} } \right) ^{-5/6}}{\left( {1+0.9d^{2}+0.62d^{2}\delta _n^{12/15} } \right) ^{5/6}}} \right) -1} \right] ^{-1} \end{aligned}$$

(8)

where $d=\sqrt{kD^{2}/4L}$, $k=2\pi /\lambda$, is the optical wave number, with $\lambda$ being the operational, $L$ is the length of the optical link and $D$ is the receiver’s aperture diameter. The parameter $\delta _n^2$ is the Rytov variance and is given as, $\delta _n^2 =0.5C_n^2 k^{7/6}L^{11/6}$ and the $C_n^2$ represents the refractive index structure parameter.

The strong atmospheric turbulence is modeled using gamma–gamma distribution channel with pointing errors. The probability density function (PDF) of the considered channel model is given by using the Meijer G function [28]

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

(9)

where $\alpha _{mn}$ and $\beta _{mn}$ represent the effective number of large and small scale turbulent eddies, ${\Gamma }(\cdot )$ is the gamma function.

3 Average BER for SISO–FSO Links

For a coherent BPSK based SIM–FSO communication system, the probability of conditional BER depending on the intensity fluctuation [4] can be expressed as

$$\begin{aligned} P_{ec,SISO} \left( h \right) =Q\left( {\frac{h\gamma }{\sqrt{2}\sigma _n }} \right) =0.5\times \hbox {erfc}\left( {\frac{h\gamma }{2\sigma _n }} \right) \end{aligned}$$

(10)

where $\gamma$ is the photo detector responsivity, $\sigma _n^2$ is the variance of the channel noise and $Q\left( \cdot \right)$ is the Gaussian $Q$ function related to the complementary error function $\hbox {erfc}\left( \cdot \right)$ as $2Q\left( {\sqrt{2}x} \right) =\hbox {erfc}\left( x \right)$.

For a strong atmospheric turbulence channel with gamma–gamma distribution, the average BER $P_e$, can be obtained by averaging Eq. (10) over the PDF of $h$ can be computed as

$$\begin{aligned} P_{e,SISO} =\mathop \int \nolimits _0^\infty P_{ec,SISO} \left( h \right) f_h \left( h \right) dh \end{aligned}$$

(11)

By using Eqs. (9) and (10) in (11), the average BER can be obtained as

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

(12)

By expressing the $\hbox {erfc}(\cdot )$ as Meijer G function [29, Eq. (8.4.14.2)], the average BER of BPSK–SIM [30] can be expressed in a closed-form by utilizing [31, Eq. (21)]. The proof is available in Appendix 1.

$$\begin{aligned} P_{e,SISO} =\frac{2^{\alpha +\beta -4}\xi ^{2}}{\sqrt{\pi ^{3}}{\Gamma }(\alpha ){\Gamma }(\beta )}G_{7,4}^{2,6} \left[ {\left. {\frac{4\gamma ^{2}A_0^2 h_l^2 }{\sigma _n^2 \alpha ^{2}\beta ^{2}}} \right| _{0,\frac{1}{2},\frac{-\xi ^{2}}{2},\frac{-\xi ^{2}+1}{2}}^{\frac{1-\xi ^{2}}{2},\frac{2-\xi ^{2}}{2},\frac{1-\alpha }{2},\frac{2-\alpha }{2},\frac{1-\beta }{2},\frac{2-\beta }{2},1} } \right] \end{aligned}$$

(13)

The above average BER expression is expressed in terms of average SNR can be expressed as

$$\begin{aligned} P_{e,SISO} =\frac{2^{{\alpha }+{\beta }-4}{\xi }^{2}}{\sqrt{{\pi }^{3}}{\Gamma }({\alpha }){\Gamma }({\beta })}{G}_{7,4}^{2,6} \left[ {\left. {\frac{8SNR \cdot \hbox {A}_0^2 }{{\alpha }^{2}{\beta }^{2}}} \right| _{0,\frac{1}{2},\frac{-{\xi }^{2}}{2},\frac{-\xi ^{2}+1}{2}}^{\frac{1-{\xi }^{2}}{2},\frac{2-{\xi }^{2}}{2},\frac{1-{\alpha }}{2},\frac{2-{\alpha }}{2},\frac{1-{\beta }}{2},\frac{2-{\beta }}{2},1} } \right] \end{aligned}$$

(14)

4 Average BER for SIMO–FSO Links

The primary solution for a strong atmospheric turbulence channel is spatial diversity technique, which can be applied either at the transmitter (MISO) or at the receiver (SIMO) or at both the sides (MIMO). The average BER for MIMO–FSO links can be calculated from [25]

$$\begin{aligned} P_{e,MIMO} =\mathop \int \nolimits _0^\infty f_{\varvec{h}} \left( {\varvec{h}} \right) Q\left( {\frac{\gamma }{MN\sqrt{2}\sigma _n }\sqrt{\mathop \sum \nolimits _{n=1}^N \left( {\mathop \sum \nolimits _{m=1}^M h_{mn} } \right) ^{2}}} \right) d{\varvec{h}} \end{aligned}$$

(15)

where $f_{\varvec{h}} \left( {\varvec{h}} \right)$ is the joint PDF of vector ${\varvec{h}}=(h_{11} ,h_{12} ,\ldots h_{MN} )$ of length MN.

To improve the BER performance in strong atmospheric turbulence, multiple receiver apertures that are added at the receiver mitigate the turbulence induced irradiance fluctuation. Introducing diversity in the receiver is known as SIMO. The various diversity techniques used in the receiver are, optimal combining (OC), equal gain combining (EGC) and selection combining (SC).

4.1 Optimal Combining

The electrical SNR of the FSO system with OC can be described as

$$\begin{aligned} SNR_{OC} =\frac{\gamma ^{2}}{2N\sigma _n^2 } \sum \limits _{n=1}^N h_n^2 \end{aligned}$$

(16)

The conditional BER of the system can be computed as

$$\begin{aligned} P_{ec,OC} \left( {\varvec{h}} \right) =Q\left( {\sqrt{SNR_{OC} }} \right) =Q\left( {\frac{\gamma }{\sqrt{2N}\sigma _n }\sqrt{\mathop \sum \nolimits _{n=1}^N h_n^2 }} \right) \end{aligned}$$

(17)

The OC diversity is applied at the receiver, then the average BER of SIMO–FSO is written as

$$\begin{aligned} P_{e,OC} =\mathop \int \nolimits _0^\infty Q\left( {\frac{\gamma }{\sqrt{2N}\sigma _n }\sqrt{\mathop \sum \nolimits _{n=1}^N h_n^2 }} \right) f_{\varvec{h}} \left( {\varvec{h}} \right) d{\varvec{h}} \end{aligned}$$

(18)

The integral in Eq. (18) is simplified by using the approximation of $Q$-function [32, Eq. (14)] to obtain the average BER as

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

(19)

By applying Eq. (9) in (19) and express the $\hbox {exp}(\cdot )$ as Meijer G function [31, Eq. (11)], the average BER of BPSK–SIM [30] can be evaluated using [31, Eq. (21)] as

$$\begin{aligned} P_{e,OC}&\approx \frac{1}{12}\mathop \prod \limits _{n=1}^N \frac{2^{\alpha _{n}+\beta _{n}-3}\xi _{n}^{2}}{{\pi \Gamma }\left( \alpha _{n}\right) {\Gamma }\left( \beta _{n}\right) }G_{6,3}^{1,6} \left[ {\left. {\frac{4\gamma ^{2}A_0^2 h_l^2 }{N\sigma _n^2 \alpha _{n}^{2}\beta _{n}^{2}}} \right| _{0,\frac{-\xi _{n}^{2}}{2},\frac{1-\xi _{n}^{2}}{2}}^{\frac{1-\xi _{n}^{2}}{2},\frac{2-\xi _{n}^{2}}{2},\frac{1-\alpha _{n} }{2},\frac{2-\alpha _{n}}{2},\frac{1-\beta _{n}}{2},\frac{2-\beta _{n}}{2}} } \right] \nonumber \\&+\,\frac{1}{4}\mathop \prod \limits _{n=1}^N \frac{2^{\alpha _{n}+\beta _{n}-3}\xi _{n}^{2}}{{\pi \Gamma }\left( \alpha _{n}\right) {\Gamma }\left( \beta _{n}\right) }G_{6,3}^{1,6} \left[ {\left. {\frac{16\gamma ^{2}A_0^2 h_l^2 }{3N\sigma _n^2 \alpha _{n}^{2}\beta _{n}^{2}}} \right| _{0,\frac{-\xi _{n}^{2}}{2},\frac{1-\xi _{n}^{2}}{2}}^{\frac{1-\xi _{n}^{2}}{2},\frac{2-\xi _{n}^{2}}{2},\frac{1-\alpha _{n}}{2},\frac{2-\alpha _{n}}{2},\frac{1-\beta _{n}}{2},\frac{2-\beta _{n}}{2}} } \right] \quad \quad \quad \end{aligned}$$

(20)

The proof is available in Appendix 2.

In terms average SNR, the Eq. (20) can be rewritten as

$$\begin{aligned} P_{e,OC}&\approx \frac{1}{12}\mathop \prod \limits _{n=1}^N \frac{2^{\alpha _{n}+\beta _{n}-3}\xi _{n}^{2}}{{\pi \Gamma }\left( \alpha _{n}\right) {\Gamma }\left( \beta _{n}\right) }G_{6,3}^{1,6} \left[ {\left. {\frac{8SNR \cdot A_0^2 }{N\alpha _{n}^{2}\beta _{n}^{2}}} \right| _{0,\frac{-\xi _{n}^{2}}{2},\frac{1-\xi _{n}^{2}}{2}}^{\frac{1-\xi _{n}^{2}}{2},\frac{2-\xi _{n}^{2}}{2},\frac{1-\alpha _{n}}{2},\frac{2-\alpha _{n}}{2},\frac{1-\beta _{n}}{2},\frac{2-\beta _{n}}{2}} } \right] \nonumber \\&+\,\frac{1}{4}\mathop \prod \limits _{n=1}^N \frac{2^{\alpha _{n}+\beta _{n}-3}\xi _{n}^{2}}{{\pi \Gamma }\left( \alpha _{n}\right) {\Gamma }\left( \beta _{n}\right) }G_{6,3}^{1,6} \left[ {\left. {\frac{32SNR \cdot A_0^2 }{3N\alpha _{n}^{2}\beta _{n}^{2}}} \right| _{\qquad \quad 0,\frac{-\xi _{n}^{2}}{2},\frac{1-\xi _{n}^{2}}{2}}^{\frac{1-\xi _{n}^{2}}{2},\frac{2-\xi _{n}^{2}}{2},\frac{1-\alpha _{n}}{2},\frac{2-\alpha _{n}}{2},\frac{1-\beta _{n}}{2},\frac{2-\beta _{n}}{2}} } \right] \quad \quad \quad \end{aligned}$$

(21)

4.2 Equal Gain Combining

An FSO system with EGC diversity consists of $N$ number of photo detectors in the receiver. The $N$ photo detector currents are combined and sent to the demodulator. The electrical SNR of the FSO system with EGC can be described as

$$\begin{aligned} SNR_{EGC} =\frac{\gamma ^{2}}{2N^{2}\sigma _n^2 }\left( {\mathop \sum \limits _{n=1}^N h_n } \right) ^{2} \end{aligned}$$

(22)

The probability of conditional BER can be defined as

$$\begin{aligned} P_{ec,EGC} ({\varvec{h}})=Q\left( {\sqrt{SNR_{EGC} }} \right) =0.5\times \hbox {erfc}\left( {\frac{\gamma }{2N\sigma _n }\mathop \sum \limits _{n=1}^N h_n } \right) \end{aligned}$$

(23)

The average BER of SIMO–FSO system with EGC diversity can be expressed from Eq. (11) as

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

(24)

The above integral can be evaluated by expressing the complementary error function as Meijer G function [29, Eq. (8.4.14.2)] and using [31, Eq. (21)]. The proof is available in Appendix 3.

$$\begin{aligned} P_{e,EGC} =\mathop \prod \limits _{n=1}^N \frac{2^{\alpha _{n} +\beta _{n} -4}\xi _{n}^{2}}{\sqrt{\pi ^{3}}{\Gamma }(\alpha _{n}){\Gamma }(\beta _{n})}G_{7,4}^{2,6} \left[ {\left. {\frac{4\gamma ^{2}A_0^2 h_l^2 }{N^{2}\sigma _n^2 \alpha ^{2}\beta _{n}^{2}}} \right| _{\qquad \quad 0,\frac{1}{2},\frac{-\xi _{n}^{2}}{2},\frac{-\xi _{n}^{2+1}}{2}}^{\frac{1-\xi _{n}^{2}}{2},\frac{2-\xi _{n}^{2}}{2},\frac{1-\alpha _{n}}{2},\frac{2-\alpha _{n}}{2},\frac{1-\beta _{n}}{2},\frac{2-\beta _{n}}{2},1} } \right] \end{aligned}$$

(25)

4.3 Selection Combining

This is the simplest and least complex receiver diversity technique among the SIMO receiver combining schemes. In this technique, the aperture receiving maximum SNR is considered. The other apertures having less $SNR$ are discarded. The selection is made according to [25]

$$\begin{aligned} h_{SC} =\hbox {max}(h_1 ,h_2 ,\ldots .h_n ) \end{aligned}$$

(26)

The electrical SNR of the FSO system with SC can be described as

$$\begin{aligned} SNR_{SC} =\frac{\left( {\gamma h_{SC} } \right) ^{2}}{2N\sigma _n^2 } \end{aligned}$$

(27)

The probability of conditional BER can be defined as

$$\begin{aligned} P_{ec,SC} \left( {h_{SC} } \right) =Q\left( {\sqrt{SNR_{SC} }} \right) =0.5\times \hbox {erfc}\left( {\frac{\gamma h_{SC} }{2\sqrt{N}\sigma _n }} \right) \end{aligned}$$

(28)

The average BER of SIMO–FSO system with EGC diversity can be expressed as

$$\begin{aligned} P_{e,SC} =0.5\times \mathop \int \nolimits _0^\infty \hbox {erfc}\left( {\frac{\gamma h_{SC} }{2\sqrt{N}\sigma _n }} \right) f_h \left( {h_{SC} } \right) dh_{SC} \end{aligned}$$

(29)

The PDF of $h_{SC}$ can be expressed as

$$\begin{aligned} f_{h_{SC} } \left( {h_{SC} } \right) =\frac{d}{dh_{SC} }F_{h_{SC} } \left( {h_{SC} } \right) =\mathop \sum \limits _{i=1}^N \mathop \prod \limits _{j=1,j\ne i}^N f_{h_i } \left( {h_{SC} } \right) F_{h_j } \left( {h_{SC} } \right) \end{aligned}$$

(30)

By applying Eq. (30) in (29), the average BER can be calculated as

$$\begin{aligned} P_{e,SC} =0.5\times \mathop \sum \limits _{i=1}^N \mathop \prod \limits _{j=1,j\ne i}^N \mathop \int \nolimits _0^\infty \hbox {erfc}\left( {\frac{\gamma h_{SC} }{2\sqrt{N}\sigma _n }} \right) f_{h_i } \left( {h_{SC} } \right) F_{h_j } \left( {h_{SC} } \right) dh_{SC} \end{aligned}$$

(31)

Where the cumulative distribution function (CDF), $F_h \left( {h_{SC} } \right)$ can be evaluated as [7]

$$\begin{aligned} F_{h_{SC} } \left( {h_{SC} } \right) =\frac{\xi ^{2}}{{\Gamma }(\alpha ){\Gamma }(\beta )}G_{2,4}^{3,1} \left[ {\left. {\frac{\alpha \beta }{A_0 h_l }h_{SC} } \right| _{\xi ^{2},\alpha ,\beta ,0}^{1,\xi ^{2}+1} } \right] \end{aligned}$$

(32)

The integral in Eq. (31) can be evaluated by the Gaussian quadrature rule (GQR).

5 Numerical Results and Discussions

In this section, certain numerical results based on the analytical expressions derived in Sects. 3 and 4 are presented. For the numerical evaluations, the following FSO system parameters are considered.

Noise standard deviation $\sigma _n =10^{-7}\hbox {A}/\hbox {Hz}$, photo detector responsivity $\gamma =0.5\hbox {A}/\hbox {W}$, beam radius $w_L \cong 2.5 \hbox { m}$ at $1 \hbox { km}$ distance and jitter standard deviation $\sigma _s \cong 30\hbox { cm}$.

The simulation and analytical results of the BPSK–SIM based SISO and SIMO–FSO system are plotted for three apertures at the receiver ($N=3)$ and for $\alpha =4,1$ and $\beta =1$ in Fig. 1. It is clearly depicted that the average BER performance of the SIMO system is improved, compared to the SISO–FSO system. From Fig. 1, it can be noticed that the higher values of $\alpha$ and $\beta$ (strong turbulence) gives a better performance compared with lower values of $\alpha$ and $\beta$ (very strong turbulence). Among the various diversity techniques considered, a better BER performance can be obtained for SC with $\alpha =4$ and $\beta =1$. Also Fig. 1 demonstrates that the average BER is improved along with responsivity of the photo detector. The preferred sensitivity of the photodetetor is around 0.4–0.6 A/W.

Fig. 1

BER against responsivity for SISO and SIMO–FSO systems with $\alpha =4$, $\beta =1$ and $\alpha =1,\beta =1$

In Figs. 2 and 3, the two different values of $\alpha =4,1$ and $\beta =1$ represent strong and very strong turbulence respectively. It can be seen from Figs. 1, 2, 3 and 4, by increasing the number of receiver apertures, significant average BER performance can be improved with respect to responsivity of the photo detector. A better BER performance I achieved through multiple receiver apertures.

Fig. 2

BER against responsivity for SIMO–FSO system with $\alpha =4$, $\beta =1$ and $N=4,5$

Fig. 3

BER against responsivity for SIMO–FSO system with $\alpha =1$, $\beta =1$ and $N=4,5$

Fig. 4

BER against responsivity for SIMO–FSO system with $\alpha =4$, $\beta =1$ and $N=6,7$

Figure 5 shows that a better average BER performance is achieved by using the SIMO system in terms of average SNR. It can be noticed that the average BER performance can be improved, by increasing the number of receive apertures. From the Figs. 5, 6, 7, 8 and 9, it is inferred that the FSO system with SC diversity ($N=7,\alpha =4,\beta =1)$ gives a better error rate performance of $10^{-6}$ for $SNR=40$ dB.

Fig. 5

BER against SNR for SISO and SIMO–FSO systems with $\alpha =4$, $\beta =1$ and $\alpha =1,\beta =1$

Fig. 6

BER against SNR for SIMO–FSO system with $\alpha =4$, $\beta =1$ and $N=4,5$

Fig. 7

BER against SNR for SIMO–FSO system with $\alpha =1$, $\beta =1$ and $N=4,5$

Fig. 8

BER against SNR for SIMO–FSO systems with $\alpha =4$, $\beta =1$ and $N=6,7$

Fig. 9

BER against SNR for SIMO–FSO system with $\alpha =1$, $\beta =1$ and $N=6,7$

The 3D plots shown in Fig. 10 depicts the trade-off between the average BER against the effective number of large scale $\left( \alpha \right)$ and small scale $\left( \beta \right)$ turbulent eddies for SISO, SIMO with OC, EGC and SC combining schemes respectively. The 3D graphs are plotted only for the values of $\alpha =\beta =4$. From Fig. 10, it is inferred that a better error rate performances are achieved for large values of $\alpha =\beta =4$. The BER values are 0.15,$5\times 10^{-3}$, $10^{-4}$ and $10^{-5}$ for SISO, SIMO with OC, EGC and SC diversity using seven receiver apertures respectively ($N=7)$. From the aforementioned values, SIMO with diversity techniques providing a better BER performance compared with SISO system.

Fig. 10

3D plot for BER variation with respect to $\alpha$ and $\beta$ for a SISO, b OC, c EGC, d SC

6 Conclusions

The novel exact closed-form expressions for the average BER of BPSK-SIM based SISO and SIMO–FSO systems over a strong atmospheric channel with pointing errors are derived. It is observed that SC provides the best BER performance compared to other considered combining schemes. It is also shown that a large number of receiving apertures offer a better BER performance. In addition, this work presents the simulation of BER performances of SIMO with a maximum of seven receiver apertures. A better BER performance is achieved by using SIMO with SC combining is $10^{-5}$ at $SNR=30$ dB.

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.