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Performance analysis of FSO links employing a transmit and receive diversity-based operating system under Málaga turbulence channels with pointing errors

Akinchan Das, Banibrata Bag, Chayanika Bose, and Aniruddha Chandra

Open Access Open Access

Abstract

This paper represents the performance analysis of a transmit and receive diversity-based free-space-optical (FSO) communication system employing Alamouti space-time-block-code (STBC) and the switch-and-examine combining (SEC) technique at the transmitter end receiver section. The generalized Málaga statistical distribution has been considered to realize the effect of atmospheric turbulence with pointing error. A new unified analytical framework has been employed to obtain the closed-form expressions for various performance measures like outage probability, average bit error rate (BER), and average capacity of the said system. The results indicate that the performance gets better as the number of receiving antennas increases but only within the low signal to noise radio (SNR) region. Further, the optimum switching threshold scheme has been implemented for this proposed multi input multi output (MIMO) FSO system to achieve improved and optimum performances. However, applying the optimum switching threshold for SEC, the performance improvement is observed in the system’s outage and BER performance but not significantly in average capacity. Finally, the derived analytical expressions for each performance metric are validated through the corresponding Monte-Carlo simulations presented in various graphical plots.

© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The large unlicensed bandwidth and excellent throughput compared to RF link [1] has gained extended and noteworthy attention to the FSO communication system. Atmospheric Turbulence (AT) is the main barrier to communicating between the transmitter and receiver of the FSO transmission system. The received signal fluctuates randomly due to the effect of turbulence, and consequently, the system’s performance is affected severely [2], [3]. Apart from this atmospheric turbulence, the misalignment between the transmitter $(T_X)$ and receiver $(R_X)$ telescopes diminishes the received signal power at the receiver section of the system. Typically, receiving antenna has been installed at the top of any high building or tower. Naturally, the buildings sway due to different natural hazards like thermal expansion, earthquakes, wind, and heavy rain causes fluctuation of the intensity and phase of the received optical signal at the receiver telescope. This phenomenon of fluctuation of the received signal is known as misalignment fading or pointing error. This type of errors may be mitigated by using either a narrow laser beam or using the telescopes’ aperture diameter, smaller in size. Many authors have introduced the mathematical formulation to establish the random behaviour of pointing errors [4–7]. Furthermore, many of them have manifested the performance analysis in terms of bit-error-rate (BER), outage probability (OP) or channel capacity considering both the atmospheric turbulence and pointing error. [8,9].

In the last couple of decades, different distribution functions have been chosen to model the varying atmospheric turbulence conditions successfully. E.g. Log-normal and Negative-exponential distributions are selected for weak and strong turbulence conditions whereas, gamma-gamma distribution is widely accepted for modelling strong atmospheric turbulence. Recently, the same domain researchers are showing importance in converging different distributions to a single generalized distribution function that is suitable and well-accepted to model everchanging atmospheric turbulence conditions. One such type of distribution is Málaga (${\mathcal M}$) distribution [10] which have been implemented by the authors in [11–13] in the FSO communication system and presented their derived mathematical closed-form expressions for different performance metrics. The main advantages of ${\mathcal M}$-distribution lie in modelling other different statistical distribution [14].

On the other hand, many authors have proposed various techniques available to the literature to mitigate the performance degradation of the FSO communication system caused by atmospheric turbulence. Out of these possible solutions, transmitter and receiver diversity techniques are very promising [15–17] and a viable solution to this hindrance. However, we have used the Alamouti STBC technique as transmit diversity technique at the transmitter side, proving to provide full diversity order. At the receiver side, we have used the switch-and-examine combining (SEC) scheme as receiving diversity technique which is very useful for its less hardware requirement [18].

1.1 State of the art

The Málaga distribution is a suitable distribution technique proposed by Navas et al. They have also provided the relations of other distribution techniques with Málaga distribution to uphold the unified property of this distribution [19]. Trigui et al., in [20] have revealed that Málaga distribution can be employed to define the atmospheric effects into the channel of the FSO communication system quite nicely. Authors in [21] employed Málaga distribution with pointing error to study the performance of FSO communication in terms of secrecy outage probability and average secrecy capacity. Authors in [12] applied Málaga distribution with dual-hop RF-FSO system and presented the different performances metrics, like SER, outage probability with variable gain relay and the ergodic capacity of the system. They have also studied the performance implementing switch-and-examine combining (SEC) receiving diversity technique [12]. The switch-and-examine combing (SEC) technique has been appropriately discussed by the authors in [22]. Alamouti [23] introduced the Alamouti space-time block code (STBC) transmitting diversity technique to get a maximum diversity gain with low complexity. The authors in [24] have measured the performance in terms of the average bit error rate of the system using repetition code and OSTBC. In [18,25], We have already reported some critical insights on the MIMO FSO system implementing both $(T_X)$ and $(R_X)$ diversity technique using Alamouti STBC and switch-and-examine combing, respectively. Moreover, we summarised of our literature survey in Table 1 based on recent work under this area of research.

1.2 Contributions

This paper investigated the $2\times L$ MIMO FSO communication system using Alamouti STBC as transmit diversity and switch-and-examine combining as receiving diversity techniques. Furthermore, the Málaga statistical distribution has been considered for modelling the FSO channels in the presence of pointing errors. Precisely, our contributions are:

  • • Derived the analytical closed-forms of some common but essential performance matrices of the said MIMO FSO system considering Málaga distributed FSO channel with pointing error impairments. These analytical closed-forms are expressed in terms of efficiently computable Meijer’s$G$-function.
  • • The optimum switching threshold also listed for $L$ number of $(R_X)$ antennas ($L$=2, 3, 4, 5 and 6) at different average SNR per branch to find the average capacity and average BER in the presence of the receiver diversity.
  • • We have presented and compared the system’s performances in terms of the outage probability,average bit error rate and average capacity when the system accepts the fixed switching threshold with the system that uses the optimal switching threshold during the implementation of the SEC scheme.
  • • All the closed-form analytical expressions are verified through Monte Carlo simulations.

Tables Icon

Table 1. MIMO FSO analysis: state of the art

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1.3 Organisation

The rest of the paper is organized as follows. We have described our proposed system model and the channel characterisation in Section 2. The statistical characterization of instant SNR and end-to-end SNR of the proposed system model has been described for different configurations like single-input single-output (SISO), single-input multi-output (SIMO), multi-input single-output (MISO) and multi-input and multi-output (MIMO) in Section 3. Next, in Section 4, we have presented all the derivations for getting the exact closed-form expressions of the proposed system’s different performance metrics. Section 5 represents the performances of the said system using various graphical plots of the numerical results obtained in the previous section, along with the simulation results. Finally, Section 6 concludes the paper with a summary.

2. Channel and system models

In this paper we have deal with $2 \times {L}$ MIMO FSO communication system. To implement the MIMO technique, we have considered multiple LASER sources at the transmitter end, and multiple telescopes at the receiver side. At transmitter terminal LASER sources are controlled by the Alamouti STBC transmit diversity technique and SEC receiver diversity techniques is employed to switches the multiple telescopes at the receiver terminal of the proposed MIMO FSO system. To model the FSO link at the different atmospheric turbulence intensity, we have used a very new and generic distribution model called ${{\cal {M}}}$-distribution or Málaga distribution. Besides the atmospheric turbulence describe by Málaga distribution, the intensity of the received optical signal also fluctuates due to the other two crucial parameters, path loss and pointing error. In this regard, the received signal at the receiver can be expressed as $I =I_{a}I_{pl}I_{p}$, where $I_{a}$ indicates the atmospheric turbulence component and $I_{pl}$, $I_{p}$ are represents the atmospheric path loss and misalignment fading component respectively.

Tables Icon

Table 2. Description of the fading parameters

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2.1 Modelling the atmospheric turbulence

Here, considering the FSO links experience with the atmospheric turbulence characterised by the Málaga statistical model, the standard mathematical equation of the corresponding PDF with irradiance $I_{a}$, is given by [8]:

$$\begin{aligned}f_{I_a}(I_a)& = {\rm {\cal A}}\sum\limits_{k_t = 1}^{\beta _t} {a_k} (I_a)^{\displaystyle{{\alpha _t + k_t} \over 2}-1} \\& \times K_{\alpha _t-k_t}\left( {2\sqrt {\displaystyle{{\alpha _t\beta _tI_a} \over {g_t\beta _t + \overline {\Omega _t} }}} } \right) \end{aligned}$$
where
$$\mathcal{A} \triangleq \frac{2(\alpha_{t})^{\frac{\alpha_{t}}{2}}}{(g_{t})^{1+\frac{\alpha_{t}}{2}}\Gamma(\alpha_{t})}\left(\frac{g_{t}\beta_{t} }{g_{t}\beta_{t} + \bar{\Omega_{t}}}\right)^{\beta_{t}+\frac{\alpha_{t}}{2}}$$
and
$$a_{k} \triangleq \left(_{k_{t}-1}^{\beta_{t}-1}\right)\frac{\left(g_{t}\beta_{t} + \bar{\Omega_{t}}\right)^{1-\frac{k_{t}}{2}}}{({k_{t}}-1)!}\left(\frac{\bar{\Omega_{t}}}{g_{t}}\right)^{{k_{t}}-1}\left(\frac{\alpha_{t}}{\beta_{t}}\right)^{\frac{{k_{t}}}{2}}$$
$$\bar{\Omega_{t}} = \Omega_{t}+2b_{0}\rho_{t}+2\sqrt{2b_{0}\rho_{t}\Omega_{t}}\cos(\Phi_{A}-\Phi_{B})$$

In Table  2, we have summarised the name of the parameter and with its symbol which are extensively used in (1) to (4) in this context.

2.2 Atmospheric path loss

The atmospheric path loss typically follows the principle of light absorption and scattering in the communication channel, which can be expressed using exponential Beer-Lambert law for length $x$, between transmitter and receiver of the FSO system as [26]

$$I_{pl}(x) = \exp(-\mu x)$$
where $\mu$ is the linear attenuation coefficient, which is again defined by $\mu =\sigma c$, where, $\sigma$ represents the concentration of molecules coefficient and $c$ is the concentration of the attenuating species in the medium. However, for a long period of transmission, the path loss component $I_{pl}$ is deterministic [5], but rest two parameters are randomly distributed.

2.3 Misalignment fading or pointing error model

Figure 1 represents the pointing error at the receiver apertures due to the not correctly aligned transmitter and receiver telescopes of the FSO system. Consider the intensity of the transmitted optical beam followed Gaussian spatial intensity profile with the waist $W_{0}$ and the beam waist of the receiver end $(W_{z})$ after the propagation length $Z$ is given by [31]

$$W_{z} = W_{0} \left\{\left(\Omega+\Lambda_{0}\right)\left(1+1.63\sigma_{R}^{12/5}\Lambda_{1}\right)\right\}^{1/2}$$
where $\Omega = 1-\frac {Z}{F_{0}}, \Lambda _{0} = \frac {2Z}{kW_{0}^{2}},\Lambda _{1} = \frac {\Lambda _{0}}{\Omega ^{2}+\Lambda _{0}^{2}}$.

The radius of curvature is represented by $F_{0}$ in cm and $k$ indicating the wave number which is the function of wavelength $(k = \frac {2\pi }{\lambda })$ in nm. The probability density function for $I_{p}$ can be expressed as [5]

$$I_{p}(r) = \mathcal{A_{0}}\exp\left(-\frac{2r^{2}}{W_{zeq}^{2}} \right)$$
where ${\mathcal A_{0}}$ is the fraction of received power at $r = 0$ and $W_{zeq}^{2}$ is called the equivalent beam width. The mathematical representation of the those symbolic notion is expressed by ${\mathcal A_{0}} = \left (\text {erf} (v)\right )^{2}$ and $W_{zeq}^{2} = W_{z}^{2}\frac {\sqrt {\pi }\text {erf}(v)}{2v\exp (-v^{2})}$, where $v= \sqrt {\frac {\pi }{2}}\frac {a}{W_{z}}$ . On the other hand, the radial displacement follows the characteristics of the Rayleigh distribution, which can be represented as follows, with $r$ being a positive numerical value.
$$f_{r}(r) = \frac{r}{\sigma_{s}^{2}}\exp \left(-\frac{r^{2}}{2\sigma_{s}^{2}}\right)$$
where $\sigma _{s}^{2}$ is depth of standard deviation at the receiver side. Finally with the help of (7) and (8), we may find the PDF for $I_{p}$ as
$$f_{I_{p}}(I_{p}) = \frac{\gamma^{2}}{\mathcal{A_{0}}^{\gamma^{2}}}I_{p}^{\gamma^{2}-1}, 0\leq I_{p}\leq \mathcal{A_{0}}$$
where $\gamma = \frac {W_{zeq}}{2\sigma _{s}}$ is the ratio between equivalent beam width and the pointing error displacement standard deviation at the receiver end of the system.

2.4 Realization of the combined attenuation

Now to find the unconditional probability density function of the atmospheric turbulence channel including all types of losses, the corresponding PDF $f_{I}(I)$ of the FSO link may be expressed using (1) and (9) as

$$f_{I}(I) = \int f_{I|I_{a}}(I|I_{a})f_{I_{a}}(I_{a}) \mathrm{d} I_{a}$$
where $f_{I|I_{a}}(I|I_{a})$ is the joint conditional probability which dependents on the misalignment fading and can be written as
$$f_{I|I_{a}}(I|I_{a}) = \frac{\gamma^{2}}{\mathcal{A_{0}}^{\gamma^{2}}I_{a}I_{pl}} \left(\frac{I}{I_{a}I_{pl}}\right)^{\gamma^{2}-1}, 0\leq I\leq \mathcal{A_{0}}I_{a}I_{pl}$$
Now substituting the value of $f_{I|I_{a}}(I|I_{a})$ in (10), we may write
$$f_{I}(I) = \int_{\frac{I}{\mathcal{A_{0}}I_{pl}}}^{\infty}\frac{\gamma^{2}}{\mathcal{A_{0}}^{\gamma^{2}}I_{a}I_{pl}} \left(\frac{I}{I_{a}I_{pl}}\right)^{\gamma^{2}-1}f_{I_{a}}(I_{a})\mathrm{d}I_{a}$$
Again, using (1) in the above equation it can be expressed as
$$f_{I}(I) = \frac{(I)^{\gamma^{2}-1}\gamma^{2}}{(\mathcal{A_{0}}I_{pl})^{\gamma^{2}}} \mathcal{A}\sum_{k_{t} = 1}^{\beta_{t}}a_{k} \int_{\frac{I}{\mathcal{A_{0}}I_{pl}}}^{\infty} (I_{a})^{\frac{\alpha_{t}+k_{t}}{2}-\gamma^{2}-1}K_{\alpha_{t}-{k_{t}}}\left(2\sqrt{\frac{\alpha_{t}\beta_{t} I_{a}}{g_{t}\beta_{t} + \bar{\Omega_{t}}}}\right)\mathrm{d}I_{a}$$
Using [32, eq. (8.4.23.1,2.24.2.3 and 8.2.2.15)], the closed form expression of the above equation may be represented as [8]
$$f_{I}(I) = \frac{\gamma^{2}\mathcal{A}}{2I}\sum_{k_{t} = 1}^{\beta_{t}}b_{k} {G^{\thinspace{3}\ {0}}_{\thinspace{1}\ {3}}} \left[ \frac{\alpha_{t}\beta_{t} }{g_{t}\beta_{t} + \bar{\Omega_{t}}}\frac{I}{\mathcal{A_{0}}I_{pl}}\Bigr|^{\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}}}\right]$$
where $b_{k} = a_{k}\left (\alpha _{t}\beta _{t}/g_{t}\beta _{t} + \bar {\Omega _{t}}\right )^{-\frac {(\alpha _{t}+{k_{t}})}{2}}$ is the constant component and it is typically depends upon the positive integer value of $\beta _{t}$.

3. Statistical characterization of the end-to-end SNR

Figure 2 shows the joint internal architecture of our proposed system with Alamouti transmits diversity at the transmitter end and the "SEC" receiver diversity at the receiver end with $L$ number of receiving antennas.

 figure: Fig. 1.

Fig. 1. (a) shows that receiver aperture absence of pointing error and (b) indicating presence of pointing error.

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 figure: Fig. 2.

Fig. 2. Schematic diagram of switch-and-examine combing (SEC) technique

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According to the Alamouti STBC scheme, let us consider ${s_{1}}$ and ${s_{2}}$ are the couples of information that would be transmitted from $T_{X_{1}}$ and $T_{X_{2}}$ transmitting antenna respectively to the receiving end during the first time slot followed by in the next time slot, $T_{X_{1}}$ and $T_{X_{2}}$ send the data bit as ${-s_{2}}^{*}$ and ${s_{1}}^{*}$ respectively. Mathematically the received signals $r_{X_{j}}$ at the receiver end at different $L$ receiving antennas during two consecutive time slots can be formulated as [23]

$$r_{1j} = h_{1j} s_1 + h_{2j} s_2 + N_{1j}$$
$$r_{2j} ={-}h_{1j} s_2^{*} + h_{2j} s_1^{*} + N_{2j}$$
The mathematical notation $h_{ij}$ presents channel coefficient between $T_{X_i}$ and $R_{X_j}$; $(i \in \{1,2\}, j \in \{1,2, \ldots, L\})$ and they are affected by zero-mean Gaussian noise, $N_{ij}$, with variance $N_0$ that are statistically independent of the channel fading. Next, according to our proposed model, the transmitted signals are received by the L-branch space-time (ST) combiner, which typically acts as an Alamouti decoder. Finally, the combiner produces the estimated output pair as follows.
$$y_{1j} = \hat{h}_{1j}^{*} r_{1j} + \hat{h}_{2j} r_{2j}^{*}$$
$$y_{2j} ={-}\hat{h}_{2j} r_{2j}^{*} + \hat{h}_{2j}^{*} r_{1j}$$
where $\hat {h}_{ij}$ is an estimate of $h_{ij}$.

Throughout our investigation, IM/DD detection technique has been applied and considered the effect of pointing error at the receiver terminal. The probability density function of such a system with the end-to-end SNR of FSO links may be defined after some mathematical manipulation as [8]

$$f_{\xi}(\xi) = \frac{\gamma^{2}\mathcal{A}}{4\xi}\sum_{k_{t} = 1}^{\beta_{t}}b_{k} {G^{\thinspace{3}\ {0}}_{\thinspace{1}\ {3}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\xi}{\overline{\xi}}}\Bigr|^{\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}}}\right]$$
where ${\mathcal B_{0}} = \gamma ^{2}\alpha _{t} \beta _{t} (g+\bar {\Omega })/\{(\gamma ^{2}+1)(g \beta _{t} +\bar {\Omega })\}$ , $\xi$ is the instantaneous SNR and $\overline {\xi }$ is the average SNR of the system.

The following equation represents the corresponding cumulative distribution functions (CDF) considering the pointing error impairments.

$$F_{\xi}(\xi) = \frac{\gamma^{2}\mathcal{A}}{2}\sum_{k_{t} = 1}^{\beta_{t}}b_{k} {G^{\thinspace{3}\ {1}}_{\thinspace{2}\ {4}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\xi}{\overline{\xi}}}\Bigr|^{1,\ \gamma^{2}+1}_{\gamma^{2},\ \alpha_{t},\ {k_{t}},\ 0}\right]$$
Next, we need to find the moment generating function (MGF) of the given system to find the system’s performances later in our discussion. We may follow the below equation to derive the corresponding MGF from its PDF as given in (18).

${{\cal {M}}}_{\xi }\left (s\right ) = E\{\exp (-s\xi )\} = \int _{0}^{\infty }\exp (-s\xi )f_{\xi }(\xi ) d\xi$. Now applying [32, eq. (8.4.23.1)] in (17), we may find the following equation for estimating the MGF of the given system.

$$\begin{aligned} {{\cal{M}}}_{\xi}\left(s\right) & = \frac{\gamma^{2}\mathcal{A}}{8\pi}\sum_{k_{t} = 1}^{\beta_{t}}b_{k}(2)^{\alpha_{t}+{k_{t}}-1} {G^{\thinspace{6}\ {1}}_{\thinspace{3}\ {6}}} \left[\frac{\mathcal{B_{0}}^{2}}{16s\overline{\xi}}\Bigr|^{\mathcal{C}}_{\mathcal{D}}\right] \end{aligned}$$
where ${\mathcal C}\in \left \{ 1,\frac {\gamma ^{2}+1}{2},\frac {\gamma ^{2}+2}{2}\right \}$ and ${\mathcal D}\in \left \{ \frac {\gamma ^{2}}{2},\frac {\gamma ^{2}+1}{2},\frac {\alpha _{t}}{2},\frac {\alpha _{t}+1}{2},\frac {{k_{t}}}{2},\frac {{k_{t}}+1}{2}\right \}$

3.1 SIMO channel statistics

Firstly, considering $L$-branch SEC system at the receiver end, the CDF of the combined output SNR our estimated SIMO system [33, eq. (9.340)] is

$$\begin{aligned} F_{\zeta_{SEC}}(\zeta) = \left\{ \begin{array}{l l} [F_{\zeta}(\zeta_{th})]^{L-1} F_{\zeta}\left(\zeta\right) & ; \zeta < \zeta_{th} \\ \sum_{j=0}^{L-1} \left[F_{\zeta}\left(\zeta\right) - F_{\zeta}\left(\zeta_{th}\right)\right] \\ \times\left[F_{\zeta}\left(\zeta_{th}\right)\right]^{j} + \left[F_{\zeta}\left(\zeta_{th}\right)\right]^{\textit{L}} & ; \zeta \geq \zeta_{th} \end{array} \right. \end{aligned}$$
where, $F_{\zeta }\left (\cdot \right )$ represents the common CDF of each receiver branch of the system as in (18), and the predetermined switching threshold is represented by $\zeta _{th}$.

The corresponding MGF can be obtained through the following equation as [33, eq. (9.342)]

$${{\cal{M}}}_{\zeta_{SEC}}(s) = [F_{\zeta}(\zeta_{th})]^{L-1} {{\cal{M}}}_{\zeta}(s) + \sum_{j=0}^{L-2} [F_{\zeta}(\zeta_{th})]^{j} \tilde{{{\cal{M}}}}_{\zeta_{SEC}}(s) $$
where ${{\cal {M}}}_{\zeta }(\cdot )$ is MGF of the individual receiver branch as in (19) and $\tilde {{{\cal {M}}}}_{\zeta _{SEC}}(s) = \int _{\zeta _{th}}^{\infty }\exp \left (s\zeta \right )f_\zeta \left (\zeta \right )\mathrm {d}\zeta$.

Now, to solve $\tilde {{{\cal {M}}}}_{\zeta _{SEC}}(s)$ for SISO channel statistics, the simple way is to use the infinite series expansion of the exponential term as [34, eq. (1.211.1)]

$$\exp(s\zeta) = \sum_{q=0}^{\infty} \frac{s^{q}\zeta^{q}}{q!}$$
Therefore, the corresponding MGF may be expressed using (17) and (22) as $\tilde {{{\cal {M}}}}_{\zeta _{SEC}}(s) = \Psi _{p}(s)$ may following.
$$\begin{aligned} \Psi_{p}(s) & = \frac{\gamma^{2}\mathcal{A}}{4\xi}\sum_{k_{t} = 1}^{\beta_{t}}b_{k} \sum _{q=0}^{\infty}\frac{s^{q}}{q!} \int_{\xi_t}^{\infty} \xi^{q+\left(\frac{\alpha_{t}+{k_{t}}}{4}\right)-1} {G^{\thinspace{3}\ {0}}_{\thinspace{1}\ {3}}} \left[\mathcal{B_{0}}\sqrt{\frac{\xi}{\overline{\xi}}}\Bigr|^{\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}}}\right] \mathrm{d} \xi \end{aligned}$$
Now using [35, eq. (07.34.21.0085.01)], the simplified closed-form expression becomes
$$\Psi_{p}(s) = \frac{\gamma^{2}\mathcal{A}}{8\pi}\sum _{q = 0}^{\infty}\sum_{k_{t} = 1}^{\beta_{t}}\frac{(s\xi_{th})^{q}}{q!}b_{k}(2)^{\alpha_{t}+{k_{t}}-1} {G^{\thinspace{7}\ {0}}_{\thinspace{3}\ {7}}} \left[\frac{\mathcal{B_{0}}^{2}\xi_{th}}{16\overline{\xi}}\Bigr|^{\mathcal{U}}_{\mathcal{V}}\right]$$
where ${\mathcal U} \in \left \{ \frac {\gamma ^{2}+1}{2},\frac {\gamma ^{2}+2}{2},1-q\right \}$ and ${\mathcal V} \in \left \{-q, \frac {\gamma ^{2}}{2},\frac {\gamma ^{2}+1}{2},\frac {\alpha _{t}}{2},\frac {\alpha _{t}+1}{2},\frac {{k_{t}}}{2},\frac {{k_{t}}+1}{2}\right \}$

3.2 MISO and MIMO channel statistics

In the MISO system, the instantaneous SNR of the proposed system would be the sum of each SNR of the $L^{th}$ branches. Therefore the overall MGF, in this case, can be calculated in a very straightforward way, only by multiplying the individual MGFs of the SISO system.

Hence, the MGF of $2 \times L$ MIMO system employing both the technique, Alamouti STBC and SEC can be formulated as,

$${{\cal{M}}}^{Al}_{\zeta_{SEC}}\left(s\right) = {{\cal{M}}}_{\zeta_{SEC}}\left(s\right) \times {{\cal{M}}}_{\zeta_{SEC}}\left(s\right).$$
where ${{\cal {M}}}_{\zeta _{SEC}}(\cdot )$ denotes the MGF of the $1 \times L$ (SIMO) system as given in (21).

4. Performance analysis

This segment has examined the performances of the proposed system through some familiar and relevant performance metrics like the outage probability, average BER, and average capacity. The closed-form expressions for those performance metrics have been derived here.

4.1 Outage probability

Usually, we declare that any system is in an outage state when the instantaneous SNR, $(\zeta )$ falls below a predefined threshold SNR value, $(\zeta _{th})$ defined as

$$P_{out} = Pr (\zeta < \zeta_{th})= \int_{0}^{\zeta_{th}} f_{\zeta_{SEC}} (\zeta )\mathrm{d}\zeta$$
where $\zeta _{th}$ is the predefined threshold SNR value and $f_{\zeta _{SEC}}(\zeta )$ is the PDF of $L$-branch SEC system, which can be expressed as [33]
$$\begin{aligned} f_{\zeta_{SEC}}(\zeta) = \left\{ \begin{array}{l l} f_{\zeta}(\zeta )[F_{\zeta}(\zeta_{th})]^{L-1} & ; \zeta < \zeta_{th} \\ \\ f_{\zeta}(\zeta) \sum_{j=0}^{L-1} \left[F_{\zeta}\left(\zeta_{th}\right)\right]^{j} & ; \zeta \geq \zeta_{th} \end{array} \right. \end{aligned}$$
where $f_{\zeta }(\cdot )$ is the probability density function of the system, in our article which is defined in (17). Next, using (27), the outage probability of the said system can be expressed as
$$P_{out} = \left[F_{\zeta}(\zeta_{th})\right]^{L-1} \int_{0}^{\zeta_{th}}f_{\zeta}(\zeta)\mathrm{d}\zeta + \sum_{j=0}^{L-1} \left[F_{\zeta}\left(\zeta_{th}\right)\right]^{j} \int_{\zeta_{th}}^{\zeta_{0}}f_{\zeta} (\zeta)\mathrm{d}\zeta$$
Now to evaluate the $P_{out}$, we need to find the closed-form expressions of the integration parts in the above equation. Therefore, let consider ${\mathcal I}_{1p} =\int _{0}^{\zeta _{th}}f_{\zeta }(\zeta )d\zeta$ and ${\mathcal I}_{2p} = \int _{\zeta _{th}}^{\zeta _{0}}f_{\zeta } (\zeta )d\zeta$. In case of ${\mathcal I}_{1p}$, it can be expressed using (17) and [36, eq. (26)] as
$$\mathcal{I}_{1p} = \frac{\gamma^{2}\mathcal{A}}{2}\sum_{k_{t} = 1}^{\beta_{t}}b_{k} {G^{\thinspace{3}\ {1}}_{\thinspace{2}\ {4}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\zeta_{th}}{\overline{\xi}}}\Bigr|^{1,\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}},0}\right]$$
and in case of ${\mathcal I}_{2p}$, using (17) and taking help of [37, eq. (07.34.21.0001.01)] and [32, eq.(8.2.2.15)], ${\mathcal I}_{2p}$, it can be expressed as
$$\frac{\gamma^{2}\mathcal{A}}{2}\sum_{k_{t} = 1}^{\beta_{t}}b_{k}{G^{\thinspace{3}\ {1}}_{\thinspace{2}\ {4}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\zeta_{0}}{\overline{\xi}}}\Bigr|^{1,\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}},0}\right] - \frac{\gamma^{2}\mathcal{A}}{2}\sum_{k_{t} = 1}^{\beta_{t}}b_{k}{G^{\thinspace{3}\ {1}}_{\thinspace{2}\ {4}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\zeta_{th}}{\overline{\xi}}}\Bigr|^{1,\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}},0}\right]$$

4.2 Average BER

This section evaluates the analytical expression for average bit error rate by employing Alamouti STBC and SEC in the system as a transmit and receiving diversity. Here, we have used the integral form of Gaussian Q-function rather than normal Q-function to avoid complex mathematical calculation. The associated mathematical form of the average BER employing OOK modulation is as follows [38]:

$$P_{e}=\frac{1}{\pi} \int_{0}^{\frac{\pi}{2}}{{\cal{M}}}^{Al}_{\zeta_{SEC}}\left(-\frac{1}{2\sin^{2}\theta}\right)d\theta$$
where
$${{\cal{M}}}^{Al}_{\zeta_{SEC}}\left(-\frac{1}{2\sin^{2}\theta}\right) = \bigg[{{\cal{M}}}_{\zeta_{SEC}}\left(s\right)\bigg]^{2} \bigg |_{s =\left(-\frac{1}{2\sin^{2}\theta}\right)}$$

4.3 Average Capacity

According to Shannonś capacity formula, the average capacity of the system can be expressed in terms of instantaneous SNR of the given system as [39], $C(\zeta ) = B\log _{2}(1+\zeta )$, where $B$ is the baseband signal bandwidth,for this particular considered unit bandwidth. Now, to find the average capacity of our aimed MIMO FSO system, we may achieve it by averaging the $C(\zeta )$ over the PDF of the SEC combiner output SNR. Hence, analytically, the average capacity $(C)$ of the system can be written as:

$$C =\int_{0} ^{\infty}C(\zeta)f_{\zeta_{SEC}}(\zeta)\mathrm{d}\zeta$$
where $f_{\zeta _{SEC}}(\zeta )$ is the PDF of the end-to-end SNR at SEC output. Now, with the help of equation of $C(\zeta )$ and using (27) the average capacity $(C)$ of the proposed MIMO FSO system can be rewritten as follows
$$\begin{aligned} C & = B\log _2e\left[ {{[F_\zeta (\zeta _{th})]}^{L-1}-\sum\limits_{j = 0}^{L-1} {{\left[ {F_\zeta \left( {\zeta _{th}} \right)} \right]}^j} } \right] \times \\ & \int_0^{\zeta _{th}} \Phi (\zeta ){\rm d}\zeta + B\log _2e\sum\limits_{j = 0}^{L-1} {{\left[ {F_\zeta \left( {\zeta _{th}} \right)} \right]}^j} \int_0^\infty \Phi (\zeta ){\rm d}\zeta \end{aligned}$$
where $\Phi (\zeta ) = \ln (1+\zeta )f_{\zeta }(\zeta )$.

To find the numerical value of $C$, we need to find out the closed-form expressions of the two integral components in the above equation, and thus we have followed the subsequent steps.

In case of first integral component in (33), we may write

$$\int_{0} ^{\zeta_{th}}\Phi(\zeta) \mathrm{d}\zeta = \int_{0}^{\xi_{th}}\ln(1+\xi)f_{\xi}(\xi)d\xi$$
Substituting the value of $f_{\xi }(\xi )$ in (17), the above equation may be rewritten as
$$\int_{0} ^{\zeta_{th}}\Phi(\zeta) \mathrm{d}\zeta = \frac{\gamma^{2}\mathcal{A}}{4}\sum_{k_t=1}^{\beta_{t}}b_{k} \int_{0}^{\xi_{th}}\frac{1}{\xi}\ln(1+\xi) {G^{\thinspace{3}\ {0}}_{\thinspace{1}\ {3}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\xi}{\overline{\xi}}}\Bigr|^{\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}}}\right]d\xi$$
Now, we may express the equivalent power series of $\ln (1+\xi )$ as
$$\ln(1+\xi) = \sum_{k=1}^{\infty}\frac{\left({-}1\right)^{k+1}}{k!} \xi^{k} =\xi-\frac{\xi^{2}}{2}+\frac{\xi^{3}}{3}+O\left(\xi^{4}\right).$$
Therefore, replacing (36) in (36) and with the help of [36, eq.(26)] we get the closed-form of (37) as
$$\int_{0} ^{\zeta_{th}}\Phi(\zeta) \mathrm{d}\zeta = \frac{\gamma^{2}\mathcal{A}}{4} \sum_{k_t=1}^{\beta_{t}} \sum_{k=1}^{\infty}\frac{\left({-}1\right)^{k+1}}{k!}b_{k} (\xi_{th})^{k} {G^{\thinspace{3}\ {1}}_{\thinspace{2}\ {4}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\xi_{th}}{\overline{\xi}}}\Bigr|^{1-2k,\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}},-2k}\right]$$
Similarly, in case of second integral component in (33), we may write
$$\int_{0}^{\infty}\Phi(\zeta) \mathrm{d}\zeta = \frac{\gamma^{2}\mathcal{A}}{4}\sum_{k_t=1}^{\beta_{t}} b_{k} \int_{0}^{\infty} \frac{1}{\xi}\ln(1+\xi) {G^{\thinspace{3}\ {0}}_{\thinspace{1}\ {3}}} \left[ \mathcal{B_{0}}\sqrt{\frac{\xi}{\overline{\xi}}}\Bigr|^{\gamma^{2}+1}_{\gamma^{2},\alpha_{t},{k_{t}}}\right]d\xi$$
Now to find the closed from expression of the above equation, we may replace $\ln (1+\xi )$ by its equivalent Meijer’s$G$function [[32], eq.(8.4.6.5)] as below:
$$\ln(1+\xi) = {G^{\thinspace{1}\ {2}}_{\thinspace{2}\ {2}}} \left[ \xi\Bigr|^{1,1}_{1,0}\right]$$
Therefore, using (39) and with the help of [32, eq.(2.24.1.1)], (38) can be expressed as
$$\int_{0}^{\infty}\Phi(\zeta) \mathrm{d}\zeta = \frac{\gamma^{2}\mathcal{A}}{8\pi}\sum_{k_t=1}^{\beta_{t}}b_{k} (2)^{\alpha_{t}+{k_{t}}-1} {G^{\thinspace{8}\ {1}}_{\thinspace{4}\ {8}}} \left[ \frac{\mathcal{B_{0}}^{2}}{16 \overline{\xi}}\Bigr|^{\mathcal{S}}_{\mathcal{T}}\right]$$
where ${\mathcal S} \in \left \{0,1,\frac {\gamma ^{2}+1}{2},\frac {\gamma ^{2}+2}{2}\right \}$ and ${\mathcal T} \in \left \{ \frac {\gamma ^{2}}{2},\frac {\gamma ^{2}+1}{2},\frac {\alpha _{t}}{2},\frac {\alpha _{t}+1}{2},\frac {{k_{t}}}{2},\frac {{k_{t}}+1}{2},0,0\right \}$

5. Numerical and simulation results

We have investigated the performance of the MIMO FSO system using the Alamouti STBC and SEC scheme in the presence of pointing errors. To present the outcomes of our investigation, we have provided several graphical plots and correlation charts regarding different performance metrics. The accuracy of the proposed mathematical approximation and the derived closed-form expressions are validated through corresponding Monte Carlo simulation results. In this regard, we have used the two well-known mathematical software, namely Wolfram Mathematica, for finding the numerical results and MATLAB to perform the Monte Carlo simulations. All simulation parameters used in this context are listed in Table 3.

Tables Icon

Table 3. Simulation parameters

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Fig. 3 represents the outage probability of our proposed MIMO FSO system for different number of $R_X$ antenna with fixed outage threshold ($\zeta _{th}$ = 3 dB) and link switching threshold at ($\zeta _{0}$ = 2 dB), considering pointing errors $\gamma = 0.9129$. The figure confirms that the outage probability could not be improved further by increasing the number of $R_X$ antennas after a certain SNR value (near $40$ dB). This limitation can be avoided in keeping both the link switching threshold and outage threshold same ($\zeta _{0}$ = $\zeta _{th}$ = 3 dB) as shown in Fig. 4. Further, Fig. 5 represents the system’s outage probability under different amount of pointing errors present at the receiver, which indicates that the performance of the system deteriorates as the effect of pointing error increases.

 figure: Fig. 3.

Fig. 3. Outage Probability for MIMO FSO system using SEC with L= 3,4,5,6 $R_X$ antennas using fixed switching threshold $\zeta _{th} = 3$ dB and optimal switching threshold $\zeta _{0} = 2$ dB respectively when pointing error ($\gamma$) is $0.9129$

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 figure: Fig. 4.

Fig. 4. Outage Probability for MIMO FSO system using SEC with L= 3,4,5,6 $R_X$ using fixed switching threshold and optimal switching thresholds are of same value as $\zeta _{th} = \zeta _{0} = 3$ dB at $\gamma = 0.9129$

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 figure: Fig. 5.

Fig. 5. Outage Probability of the system considering $\textit {L}= 6$ $R_X$ antennas and fixed switching threshold and optimal switching thresholds are of same value as $\zeta _{th} = \zeta _{0} = 3$ dB under different pointing error values at $\gamma = 0.9129, 0.6890$ and $0.5422$

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Next, we have analyzed another important performance metrics, called Bit Error Rate (BER), of the said system.

First of all, Fig. 6 represents the comparisons of average BER for i) single FSO link ii) SIMO FSO system with $1 \times 2$ iii) MISO FSO system with $2 \times 1$ and iv) MIMO FSO system with $2 \times 2$ which exhibit that MIMO system offers much better performance than the other mentioned FSO systems.

 figure: Fig. 6.

Fig. 6. A Comparison of ABER using $\alpha _{t} = 2.296$, $\beta _{t} = 2$ and $\gamma = 0.9129$ under various configuration of FSO system.

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However, Fig. 7 represents the BER of the system using fixed threshold of $2$ dB and pointing error $\gamma = 0.9129$. This figure also exhibits that the system experiences the same constraint, i.e., no improvement is noticed for the BER performance after a certain SNR despite increasing $L$. As we can see, the curves are overlapping with each other after SNR $30$ dB. Therefore, it is prominent that the BER could not be improved further beyond $35$ dB average SNR.

 figure: Fig. 7.

Fig. 7. BER for MIMO FSO system using SEC with L=3,4,5,6 $R_X$ antennas using fixed switching threshold (2 dB) with pointing error $\gamma = 0.9129$

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Thus to eliminate this limitation, we have used optimum threshold values against each SNR as tabulated in Table 4 for different pointing errors. In this regard, we have obtained the same optimum threshold values for all $R_X$ antennas.

Tables Icon

Table 4. Optimum switching threshold $(\xi _{th})$ values for all $R_X$ antennas at different average SNR per branch for different pointing error

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Finally, using these optimum threshold values, the BER performance of the system for all $R_X$ is depicted in Fig. 8.This figure indicates that the limitation could be overcome using the optimum threshold values. In [29] authors achieved the ABER of $\sim$ $2\times 10^{-4}$ at an average SNR of 30 dB using SC-based MIMO FSO system with pointing error under Gamma-Gamma turbulence channel ( considering low turbulence at $\alpha _{t} = 9$ and $\beta _{t} = 8$ and the pointing error of 11.23). Contrary, as shown in Fig. 7, our proposed SEC based MIMO FSO system provides the same amount of ABER at an average SNR of $\sim$ 25 dB under strong atmospheric turbulence and pointing error conditions. Therefore our proposed scheme provides better performance. Moreover, from Fig. 6 and Fig. 8, it is observed that our proposed MIMO-FSO system is performed better in terms of ABER than the single FSO link under any transmit diversity order. Say for $2\times 4$ MIMO FSO system offer $2.94755\times 10^{-6}$ ABER with average SNR 35 dB while a single FSO link offers only $0.0181108$ ABER, so the amount of improvement is nearly 217$\%$. However, the BER performance of the system using optimum threshold values deteriorates with increasing severity of pointing error, which is shown in Fig. 9.

 figure: Fig. 8.

Fig. 8. BER for MIMO FSO system using SEC with L=3,4,5,6 $R_X$ antennas using different optimal switching threshold with pointing error $\gamma = 0.9129$

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 figure: Fig. 9.

Fig. 9. BER for MIMO FSO system using SEC with L=6 $R_X$ antennas using different threshold values and under different pointing error values of $0.9129$, $0.6890$ and $0.5422$

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Comparing Fig. 7 and Fig. 8, we get to notice that while Fig. 7 shows the same average BER of $7.06\times 10^{-5}$ at $30$ dB for L=5 and L=6, Fig. 7 shows the average BER of $1.14\times 10^{-5}$ and $2.20 \times 10^{-6}$ for L=5 and L=6 respectively. Additionally, in Fig. 10, we have shown the improvement against BER performance when we have used the optimum threshold instead of the fixed switching threshold. Moreover, Fig. 11 shows the average BER improvement for different diversity order (L= 4, 5 and 6), which shows that the improvement percentage increases as the number of SEC branch increases in the case of an optimum threshold.

 figure: Fig. 10.

Fig. 10. Improvement of ABER from fixed switching threshold to optimal switching threshold for MIMO FSO system under different average SNR using pointing error $\gamma = 0.9129$ with diversity order L = 6.

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 figure: Fig. 11.

Fig. 11. Improvement of ABER from fixed switching threshold to optimal switching threshold for MIMO FSO system under different average SNR using pointing error $\gamma = 0.9129$ with different diversity order.

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Next, we have analyzed the average capacity of the proposed MIMO FSO system. Fig. 12 presents the average capacity versus average SNR plot of the system with $\gamma = 0.9129$ under a fixed switching threshold of $2$ dB, keeping other parameters related to atmospheric turbulence the same as previous.

 figure: Fig. 12.

Fig. 12. Average capacity of 1$\times$1 FSO and MIMO FSO system under different receiver diversity order with fixed switching threshold (2dB) and uses of pointing error $\gamma = 0.9129$.

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In [15] authors have reported the average capacity of the MRC based MIMO FSO system is $\sim$ 6.5 bits/s/Hz at average SNR of 20 dB in the absence of pointing error under Gamma-Gamma turbulence channel. In Fig. 12 our proposed system provides $\sim$ 4.8 bits/s/Hz at the same average SNR in the presence of pointing error condition with Málaga turbulence channel. Therefore, our proposed scheme provides a lesser average capacity due to the presence of pointing errors. However, if we compare it in the absence of pointing error, it provides $\sim$ 8 bits/s/Hz [25] at the same average SNR, which is better than the other. On the other hand the average capacity of the $1 \times 1$ FSO system is little bit better at high SNR region (after $\sim$ 18 dB SNR) then the MIMO FSO system under presence of pointing error condition. From Fig. 12, it is evident that the average capacity of the system improves with increasing the number of receivers path (L) within the SNR region $0$ dB to $18$ dB.However, beyond that, the enhancement in the average capacity is less significant.

In order to improve the capacity performance, we have calculated the optimum threshold values for each SNR as given in Table 5 for different $R_X$ antennas at pointing error $\gamma = 0.9129$.

Tables Icon

Table 5. Optimum switching threshold $(\xi _{th})$ values for different number of $R_X$ antennas at different average SNR per branch for pointing error $\gamma = 0.9129$

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Eventually, Fig. 13 shows the average capacity of the system when it uses the optimum switching threshold as given in Table 5. Analyzing the graphical representations of Fig. 12 and Fig. 13, no notable improvement is seen on the average capacity of the system using the optimum switching threshold.

 figure: Fig. 13.

Fig. 13. Average capacity for MIMO FSO for different diversity order using optimum switching threshold at $\gamma = 0.9129$.

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Additionally, Fig. 14 shows the rate of change in the improvement of the average capacity of the system when it operates using the optimum switching threshold at L=6, which shows that beyond $18$ dB average SNR, the improvement is not remarkable.

 figure: Fig. 14.

Fig. 14. Improvement of average capacity from fixed switching threshold to optimal switching threshold for MIMO FSO system under different average SNR using pointing error $\gamma = 0.9129$ with diversity order L = 6.

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Moreover, Fig. 15 shows the improvement of the average capacity for a different number of receivers path (L) at different average SNR. Therefore, another remarkable observation is that the improvement difference for different diversity order decreases at higher SNR region. Finally, Fig. 16 represents the graphical presentation of the average capacity of the proposed MIMO FSO system for L= 6 $R_X$ antennas in the presence of different severity of pointing errors, namely ’More Pointing Error’, ’Less Pointing Error’ and ’No pointing Error’. This particular figure exhibits that the average capacity is much lesser in the presence of a pointing error regime.

 figure: Fig. 15.

Fig. 15. Improvement of average capacity from fixed switching threshold to optimal switching threshold for MIMO FSO system using pointing error $\gamma = 0.9129$ with different diversity order.

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 figure: Fig. 16.

Fig. 16. Average Capacity for MIMO FSO system using SEC with L= 6 $R_X$ antennas using different threshold when $\alpha _{t}$ = 2.296 and corresponding $\beta _{t}$ = 2 with various pointing error

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6. Conclusion

This paper has derived and expressed the closed-form expressions for the associated performance metrics of the said MIMO FSO system under the Málaga turbulence channel. Here Alamouti STBC scheme has been applied at the transmitter side, and switch-and-examine combining (SEC) has been applied at the receiver side as a transmit and receive diversity technique, respectively, in the presence of pointing error. The analytical results of the performance matrices are validated by Monte Carlo simulation. We have observed that the performance of the said system improves as the number of receiver diversity increases. Also, we have observed that the optimal switching threshold improves the outage probability and BER performance. The performance improvement regarding average capacity of the system remains constant for all receiver path at high SNR. However, applying the optimal switching threshold for SEC receiving diversity technique, the performance improvement is observed only in the low SNR region and not significantly improves at the high SNR region. We have presented different graphical plots for the different performance matrices considering pointing error which is one of the most crucial parameters in defining the quality of service of an FSO system. Lastly, we may conclude that this proposed system offers a better performance using transmit and received diversity with pointing error impairments and hope the outcome of this investigation would add some important aspect in the future research on optical wireless communication.

Disclosures

AD, BB: HIT Haldia (E), CB: Jadavpur University (E), AC: NIT Durgapur (F, E).

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (16)
Fig. 1.
Fig. 1. (a) shows that receiver aperture absence of pointing error and (b) indicating presence of pointing error.

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Fig. 2.
Fig. 2. Schematic diagram of switch-and-examine combing (SEC) technique

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Fig. 3.
Fig. 3. Outage Probability for MIMO FSO system using SEC with L= 3,4,5,6 $R_X$ antennas using fixed switching threshold $\zeta _{th} = 3$ dB and optimal switching threshold $\zeta _{0} = 2$ dB respectively when pointing error ($\gamma$) is $0.9129$

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Fig. 4.
Fig. 4. Outage Probability for MIMO FSO system using SEC with L= 3,4,5,6 $R_X$ using fixed switching threshold and optimal switching thresholds are of same value as $\zeta _{th} = \zeta _{0} = 3$ dB at $\gamma = 0.9129$

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Fig. 5.
Fig. 5. Outage Probability of the system considering $\textit {L}= 6$ $R_X$ antennas and fixed switching threshold and optimal switching thresholds are of same value as $\zeta _{th} = \zeta _{0} = 3$ dB under different pointing error values at $\gamma = 0.9129, 0.6890$ and $0.5422$

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Fig. 6.
Fig. 6. A Comparison of ABER using $\alpha _{t} = 2.296$, $\beta _{t} = 2$ and $\gamma = 0.9129$ under various configuration of FSO system.

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Fig. 7.
Fig. 7. BER for MIMO FSO system using SEC with L=3,4,5,6 $R_X$ antennas using fixed switching threshold (2 dB) with pointing error $\gamma = 0.9129$

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Fig. 8.
Fig. 8. BER for MIMO FSO system using SEC with L=3,4,5,6 $R_X$ antennas using different optimal switching threshold with pointing error $\gamma = 0.9129$

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Fig. 9.
Fig. 9. BER for MIMO FSO system using SEC with L=6 $R_X$ antennas using different threshold values and under different pointing error values of $0.9129$, $0.6890$ and $0.5422$

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Fig. 10.
Fig. 10. Improvement of ABER from fixed switching threshold to optimal switching threshold for MIMO FSO system under different average SNR using pointing error $\gamma = 0.9129$ with diversity order L = 6.

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Fig. 11.
Fig. 11. Improvement of ABER from fixed switching threshold to optimal switching threshold for MIMO FSO system under different average SNR using pointing error $\gamma = 0.9129$ with different diversity order.

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Fig. 12.
Fig. 12. Average capacity of 1$\times$1 FSO and MIMO FSO system under different receiver diversity order with fixed switching threshold (2dB) and uses of pointing error $\gamma = 0.9129$.

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Fig. 13.
Fig. 13. Average capacity for MIMO FSO for different diversity order using optimum switching threshold at $\gamma = 0.9129$.

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Fig. 14.
Fig. 14. Improvement of average capacity from fixed switching threshold to optimal switching threshold for MIMO FSO system under different average SNR using pointing error $\gamma = 0.9129$ with diversity order L = 6.

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Fig. 15.
Fig. 15. Improvement of average capacity from fixed switching threshold to optimal switching threshold for MIMO FSO system using pointing error $\gamma = 0.9129$ with different diversity order.

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Fig. 16.
Fig. 16. Average Capacity for MIMO FSO system using SEC with L= 6 $R_X$ antennas using different threshold when $\alpha _{t}$ = 2.296 and corresponding $\beta _{t}$ = 2 with various pointing error

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Tables (5)

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Table 1. MIMO FSO analysis: state of the art

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Table 2. Description of the fading parameters

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Table 3. Simulation parameters

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Table 4. Optimum switching threshold (ξth) values for all RX antennas at different average SNR per branch for different pointing error

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Table 5. Optimum switching threshold (ξth) values for different number of RX antennas at different average SNR per branch for pointing error γ=0.9129

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Equations (43)

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fIa(Ia)=A∑kt=1βtak(Ia)αt+kt2−1×Kαt−kt(2αtβtIagtβt+Ωt¯)
A≜2(αt)αt2(gt)1+αt2Γ(αt)(gtβtgtβt+Ωt¯)βt+αt2
ak≜(kt−1βt−1)(gtβt+Ωt¯)1−kt2(kt−1)!(Ωt¯gt)kt−1(αtβt)kt2
Ωt¯=Ωt+2b0ρt+22b0ρtΩtcos⁡(ΦA−ΦB)
Ipl(x)=exp⁡(−μx)
Wz=W0{(Ω+Λ0)(1+1.63σR12/5Λ1)}1/2
Ip(r)=A0exp⁡(−2r2Wzeq2)
fr(r)=rσs2exp⁡(−r22σs2)
fIp(Ip)=γ2A0γ2Ipγ2−1,0≤Ip≤A0
fI(I)=∫fI|Ia(I|Ia)fIa(Ia)dIa
fI|Ia(I|Ia)=γ2A0γ2IaIpl(IIaIpl)γ2−1,0≤I≤A0IaIpl
fI(I)=∫IA0Ipl∞γ2A0γ2IaIpl(IIaIpl)γ2−1fIa(Ia)dIa
fI(I)=(I)γ2−1γ2(A0Ipl)γ2A∑kt=1βtak∫IA0Ipl∞(Ia)αt+kt2−γ2−1Kαt−kt(2αtβtIagtβt+Ωt¯)dIa
fI(I)=γ2A2I∑kt=1βtbkG1 33 0[αtβtgtβt+Ωt¯IA0Ipl|γ2,αt,ktγ2+1]
r1j=h1js1+h2js2+N1j
r2j=−h1js2∗+h2js1∗+N2j
y1j=h^1j∗r1j+h^2jr2j∗
y2j=−h^2jr2j∗+h^2j∗r1j
fξ(ξ)=γ2A4ξ∑kt=1βtbkG1 33 0[B0ξξ¯|γ2,αt,ktγ2+1]
Fξ(ξ)=γ2A2∑kt=1βtbkG2 43 1[B0ξξ¯|γ2, αt, kt, 01, γ2+1]
Mξ(s)=γ2A8π∑kt=1βtbk(2)αt+kt−1G3 66 1[B0216sξ¯|DC]
FζSEC(ζ)={[Fζ(ζth)]L−1Fζ(ζ);ζ<ζth∑j=0L−1[Fζ(ζ)−Fζ(ζth)]×[Fζ(ζth)]j+[Fζ(ζth)]L;ζ≥ζth
MζSEC(s)=[Fζ(ζth)]L−1Mζ(s)+∑j=0L−2[Fζ(ζth)]jM~ζSEC(s)
exp⁡(sζ)=∑q=0∞sqζqq!
Ψp(s)=γ2A4ξ∑kt=1βtbk∑q=0∞sqq!∫ξt∞ξq+(αt+kt4)−1G1 33 0[B0ξξ¯|γ2,αt,ktγ2+1]dξ
Ψp(s)=γ2A8π∑q=0∞∑kt=1βt(sξth)qq!bk(2)αt+kt−1G3 77 0[B02ξth16ξ¯|VU]
MζSECAl(s)=MζSEC(s)×MζSEC(s).
Pout=Pr(ζ<ζth)=∫0ζthfζSEC(ζ)dζ
fζSEC(ζ)={fζ(ζ)[Fζ(ζth)]L−1;ζ<ζthfζ(ζ)∑j=0L−1[Fζ(ζth)]j;ζ≥ζth
Pout=[Fζ(ζth)]L−1∫0ζthfζ(ζ)dζ+∑j=0L−1[Fζ(ζth)]j∫ζthζ0fζ(ζ)dζ
I1p=γ2A2∑kt=1βtbkG2 43 1[B0ζthξ¯|γ2,αt,kt,01,γ2+1]
γ2A2∑kt=1βtbkG2 43 1[B0ζ0ξ¯|γ2,αt,kt,01,γ2+1]−γ2A2∑kt=1βtbkG2 43 1[B0ζthξ¯|γ2,αt,kt,01,γ2+1]
Pe=1π∫0π2MζSECAl(−12sin2⁡θ)dθ
MζSECAl(−12sin2⁡θ)=[MζSEC(s)]2|s=(−12sin2⁡θ)
C=∫0∞C(ζ)fζSEC(ζ)dζ
C=Blog2⁡e[[Fζ(ζth)]L−1−∑j=0L−1[Fζ(ζth)]j]×∫0ζthΦ(ζ)dζ+Blog2⁡e∑j=0L−1[Fζ(ζth)]j∫0∞Φ(ζ)dζ
∫0ζthΦ(ζ)dζ=∫0ξthln⁡(1+ξ)fξ(ξ)dξ
∫0ζthΦ(ζ)dζ=γ2A4∑kt=1βtbk∫0ξth1ξln⁡(1+ξ)G1 33 0[B0ξξ¯|γ2,αt,ktγ2+1]dξ
ln⁡(1+ξ)=∑k=1∞(−1)k+1k!ξk=ξ−ξ22+ξ33+O(ξ4).
∫0ζthΦ(ζ)dζ=γ2A4∑kt=1βt∑k=1∞(−1)k+1k!bk(ξth)kG2 43 1[B0ξthξ¯|γ2,αt,kt,−2k1−2k,γ2+1]
∫0∞Φ(ζ)dζ=γ2A4∑kt=1βtbk∫0∞1ξln⁡(1+ξ)G1 33 0[B0ξξ¯|γ2,αt,ktγ2+1]dξ
ln⁡(1+ξ)=G2 21 2[ξ|1,01,1]
∫0∞Φ(ζ)dζ=γ2A8π∑kt=1βtbk(2)αt+kt−1G4 88 1[B0216ξ¯|TS]
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