Outage Performance Estimation of a MISO FSO System with OOK Signaling and Alamouti Type Space-Time Coding

Abstract

In this work, multi input and single output optical wireless communication (OWC) links are analyzed, assuming the channel experiences fading due to atmospheric turbulence. OOK signaling and Alamouti-type space-time coding with intensity modulation and direct detection is assumed for weak to strong turbulence conditions modeled with the gamma–gamma distribution and closed form analytical expressions for their outage probability are derived for the first time, to the best of our knowledge. Finally, the corresponding numerical results are obtained, through the obtained mathematical expressions. Performance of a multi input and single output channel is analyzed for its outage probability as a function of power to noise ratio and normalized threshold. Three different cases for strong, medium, and weak turbulences are investigated.

1. Introduction

In the last few years, the widespread usage of wireless communication systems has been observed throughout the entire globe. A large number of new wireless devices are being connected to the already existing wireless network. The capacity of the radio frequency (RF) channel will soon fall short of servicing all the new devices. Along with the low capacity of the RF channel, it also has a higher cost as its frequency bands are licensed and need to be purchased for data communication. Thus, other types of wireless links are being studied and feasible frequency bands in the electromagnetic spectrum are being investigated in order to be used for communication links.

A solution for this exponential demand are high data rates which can be obtained using terrestrial free space optical (FSO) links. FSO channels utilize visible, infrared, or ultraviolet bands from the electromagnetic spectrum to transport information. These systems are attractive due to their very high capacity and consequently their high achievable data rates, no licensing requirements, low deployment time, and inherent security. FSO channels can achieve data rates up to serval gigabytes per second over a few kilometers links. In 2018, researchers [1] achieved a record-breaking speed of 13.16 Tbps over a distance of 10.45 km using a terrestrial FSO communication link. Although OWC provides very high data rates for a relative low cost, it still has drawbacks. One problem is the attenuation due to gas molecules, vapors, pollutants, dust, fog, and other suspended particles in the atmosphere. These particles cause fluctuations in the received signal’s irradiance, which is called the scintillation effect, and the channel’s coherence time is an order larger than the block transmission time. For this type of fading process, the outage performance by means of the outage probability, is a very significant metric for analyzing and investigating the link’s availability.

Untill now, the number of probability measures has been used to model the channel gain and evaluate the impact of atmospheric turbulence on it [2,3,4,5]. The optical signal through air faces many types of attenuations; one of the reasons can be fog. The effect of fog on optical attenuation on the micro-level is investigated, considering the surface area of fog droplets [6]. The adverse effect of fading can be mitigated by using a multi input and multi output (MIMO) OWC system, as explored in [7,8]. Furthermore, Alamouti-like coding, which is used mainly for real and complex bipolar modulation, can be altered to be used for unipolar modulations like on-off keying (OOK) and pulse position modulation (PPM), which is mostly used for OWC with direct detection [9]. The maximal diversity order and the probability of outage of MIMO OWC channels were studied in [10]. The performance of MIMO OWC with space time multiplexing was investigated in [11], while MIMO OWC channels with Alamouti coding and the switch-and-examine combining technique were studied in [12]. In addition, the outage probability for single input and single output (SISO) systems was presented in [13]. The outage probability of a multi input and single output (MISO) downlink was investigated using non-orthogonal multiple access also known as NOMA. Outage probability of the MISO channel was compared for different power allocations in [14].

The aim of this work is to analyze the outage performance of a MISO OOK FSO system with intensity fluctuations due to the atmospheric turbulence effect modeled by the statistical model of the gamma–gamma distribution and Alamouti-like coding. An investigation of the MISO system can be helpful in laying the foundation for a multipath effect in OWC, considering the path delays. Additionally, it is worth mentioning here that the intensity modulation/direct detection (IM/DD) systems like OOK modulation were chosen here because they are one of the most popular formats which are used for FSO links. To analyze the MISO system, closed form expressions for outage performance are derived from this work. In Figure 1, the 2 × 1 MISO system under consideration is presented. By assuming the channel coherence time to be of the order of 1–100 ms, the channel remains constant over hundreds of thousands up to millions of consecutive bits for typical optical signaling rates. [15].

The remainder of this work is organized as follows: Section 2 presents an overview of the model of an optical MISO system working in fading conditions and Alamouti-type coding. Section 3 outlines the process adopted to derive the probability of outage and Section 4 concludes the paper.

2. System Designing

In optical wireless communication, OOK or PPM are mostly used for the IM/DD of data, and threshold-based photo-detection is applied at the receiver. Other modulation schemes like phase shift keying (PSK), quadrature amplitude modulation (QAM), etc, which are widely used in radio frequency (RF) communication systems, are used more rarely in OWC due to the fact that the phase detection of an optical signal would require coherent optical transducers. On the other hand, the spectral efficiency (SE) of the OOK is low and thus other techniques with higher SE can be used, such as the orthogonal frequency-division multiplexing (OFDM) and single carrier frequency division multiplexing (SCFDM) [16]. However, each specific technique is selected on the basis of the user application and requirements depending on their advantages and disadvantages [17].

OWC links are subjected to both mean and power constraints dictated by source power conservations and human eye safety considerations. These constraints translate to the following conditions on optical intensity signal $X,$ which is inherently a power signal [10]:

$$\begin{array}{c} X \ge 0, \mathrm{N}\mathrm{o}\mathrm{n}-\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y} \\ E\left[X\right] \le \epsilon , \mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n} \mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}\\ Prob \left(X>A\right)=0, \mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k} \mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}\end{array}\}$$

(1)

where $A$ stands for the value of the signal’s peak power. The above constraints on the optical intensity signal dictate modification in the design of modulation as well as error correction codes for optical wireless systems.

2.1. Channel Modeling

We consider a communication system with the $N_r$ receiver and $N_t$ transmitter antennas. The transmission vector at time $k$ is a $N_t\times 1$ vector represented by:

$$S\left(k\right)={\left[s_1\left(k\right),s_2\left(k\right),s_3\left(k\right), \dots s_{N_t}\left(k\right)\right]}^T k=1, 2, \dots$$

(2)

where $S^T$ is the transpose of vector $S$ and $s_i\in \left[0, A\right]$. Each element $s_i\left(k\right)$ of the transmission vector is assumed to be drawn from a digital constellation. Channel state $H$ is assumed to be fixed for the transmission of symbols of block length $B$. Received signal $y_{j }\left(n\right)$ at the receiver antenna $j$ at the time $k$ is given as:

$$y_j\left(k\right)=H{s}_i\left(k\right)+n_j\left(k\right) i=1, 2, \dots , N_t$$

(3)

The received vector $Y$ is a $N_r\times 1$ vector and is written as below:

$$Y={\left[y_1\left(n\right), y_2\left(n\right), y_3\left(n\right), \dots , y_{N_r}\left(n\right)\right]}^T$$

(4)

Channel state $H$ is of order $N_r\times N_t$.

$$H=\left[\begin{array}{ccc}{h}_{11}& \cdots & h_{1\times N_t}\\ ⋮& \ddots & ⋮\\ h_{N_r\times 1}& \cdots & h_{N_r\times N_t}\end{array}\right]$$

(5)

For simplicity reasons here, two sources and one receiver have been investigated. In this case, (5) can be reduced to:

$$H=\left[h_{11}{h}_{12}\right]$$

(6)

The noise is assumed to be Gaussian with zero mean and variance ${\sigma }^2$ [9]. The noise samples at the receiver are independent and uncorrelated with one another. The gains $h_{11}$ and $h_{12}$ are complex Gaussian random variables with 0 and $2\pi$ uniformly distributed phases.

2.2. Alamouti-Type Space-Time Coding

One of the most elegant space-time codes, the Alamouti, was adapted by Simon and Vilnrotter [10] for OWC MIMO systems. The modified Alamouti code was shown to be amenable to maximum likelihood detection. For the $2\times 1$ OWC MISO systems, the flat fading channel gain for transmitter 1 and 2 to the receiver links is $h_{11}$ and $h_{12},$ respectively. It is assumed that channels gains remain constant for two symbol durations, which is a fair assumption since fading due to scintillation is a slow process.

In the first time slot, the source 1 and 2 sends $x_1$ and $x_2$, respectively. In the succeeding time slot, transmitters send ${\overline{x}}_2$ and $x_1$, respectively. $x_1$ and $x_2$ are selected from the signal set $S=s_1,s_2$. Different from conventional Alamouti coding, no complex conjugate notation is required with $x_1$ and $x_2$ when dealing with real signals for OOK. In the case of OOK, the signal set can be defined in terms of the below mentioned waveforms:

$$\begin{gathered} s_1=0, 0\le t\le T \\ {s}_2=A, 0\le t\le T \end{gathered}$$

(7)

where A is a positive constant describing the optical signal strength and T is the signal duration. For a binary PPM, the signal set consists of the waveforms described as [10]:

$$\begin{gathered} s_1=\{\begin{array}{c} 0 0\le t\le T/2\\ A T/2\le t\le T\end{array} \\ {s}_2=\{\begin{array}{c} A 0\le t\le T/2\\ 0 T/2\le t\le T\end{array} \end{gathered}$$

(8)

It is worth nothing that both for the OOK and binary PPM signal sets, the following relation holds:

$$s_i=-s_j+A, j\ne i$$

(9)

with real valued signals. The complements are defined as:

$$\begin{gathered} \mathrm{if} x_i=s_1 \\ \mathrm{then} {\overline{x}}_i\triangleq s_2 \\ {\overline{x}}_i=-s_1+A \\ {\overline{x}}_i=-x_i+A \end{gathered}$$

(10)

$\mathrm{if} x_i=s_2$, then similarly,

$${\overline{x}}_i\triangleq -x_i+A$$

(11)

Consequent to the foregoing discussion signal at the receiver in the first and second time slot are [10]:

$$\begin{gathered} y_1=h_{11}{x}_1+h_{12}{x}_2+n_1 \\ {y}_2=h_{11}{\overline{x}}_2+h_{12}{x}_1+n_2 \end{gathered}$$

(12)

where $n_1$ and $n_2$ are zero mean Gaussian variables with variance ${\sigma }^2$. Since ${\overline{x}}_2=-x_2+A$, the above equation can be recast as:

$$\begin{gathered} y_1=h_{11}{x}_1+h_{12}{x}_2+n_1 \\ {y}_2-h_{11}A=-h_{11}{x}_2+h_{12}{x}_1+n_2 \end{gathered}$$

(13)

As the receiver has knowledge of the transmitted optical signal strength, that is A, and the channel state $h_{11}$ and $h_{12}$, it can make an estimate ${\tilde{x}}_1$ and ${\tilde{x}}_2$ from the received signals $y_1$ and $y_2$. Knowledge of the transmitted optical signal strength at the receiver is not an additional requirement since in OOK we already know it to choose the detector with a specific threshold value [18].

$$\begin{gathered} {\tilde{x}}_1=h_{11}{y}_1+h_{12}{y}_2-h_{11}{h}_{12}A \\ {\tilde{x}}_2=h_{12}{y}_1-h_{12}{y}_2-h_{11}^{2}A \end{gathered}$$

(14)

By substituting (13) into (14), we get:

$$\begin{gathered} {\tilde{x}}_1=\left(h_{11}^2+h_{12}^2\right)x_1+h_{11}{n}_1+h_{12}{n}_2=\left(h_{11}^2+h_{12}^2\right)x_1+{\tilde{N}}_1 \\ {\tilde{x}}_2=\left(h_{11}^2+h_{12}^2\right)x_2+h_{12}{n}_1-h_{11}{n}_2=\left(h_{11}^2+h_{12}^2\right)x_2+{\tilde{N}}_2 \end{gathered}$$

(15)

where ${\tilde{N}}_1$ and ${\tilde{N}}_2$ are independent and uncorrelated Gaussian noise samples.

2.3. The Gamma–Gamma Turbulence Model

For the OWC system fading gain, $h_{ij}\in \left[0,1\right]$ can be modeled by the gamma–gamma distribution which has proven to be a very accurate stochastic model for a wide range of turbulence conditions [15]. The probability density function (PDF) of the specific distribution, as a function of the normalized irradiance, I, is given as [9]:

$$f_h\left(\mathrm{h}\right)=\frac{2{\left(\alpha \beta \right)}^{\frac{\left(\alpha +\beta \right)}{2}}{h}^{\frac{\left(\alpha +\beta -2\right)}{2}}}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta h}\right)$$

(16)

where the parameters $\alpha >0$ and $\beta >0$ can be estimated from the link’s characteristics and are given as [9]:

$$\begin{gathered} \alpha ={\left[\exp \left(\frac{0.49 {\sigma }_r^2{\left(1+0.69{\sigma }_r^{\frac{12}{5}}\right)}^{-5/6}}{{\left(1+0.18d^2+0.56{\sigma }_r^{\frac{12}{5}}\right)}^{7/6}}-1\right)\right]}^{-1} \\ \beta ={\left[\exp \left(\frac{0.51 {\sigma }_r^2}{{\left(1+0.9d^2+0.62d^2{\sigma }_r^{\frac{12}{5}}\right)}^{5/6}}-1\right)\right]}^{-1} \end{gathered}$$

(17)

where ${\sigma }_r=23.17 C_n^2{k}^{7/6} L^{11/6}$ is the Rytov variance, $d\triangleq \sqrt{\frac{k{D}^2}{4L}}$ stands for the receiver’s aperture diameter, $k\triangleq \frac{2\pi }{\lambda }$ is the optical wave number, L is the link’s length, $C_n^2$ represents the refractive index structure parameter, while $\lambda$ and $D$ are the operational wavelength and antenna’s aperture diameter, respectively.

3. Outage Probability of the OWC MISO Channel

The outage performance represents the probability that SNR, which arrives at the receiver, falls below a certain threshold level, ${\gamma }_{th}$, [10]. Letzepis and Guillen in [19] evaluated the signal-to-noise exponents and calculated the outage probability for the MIMO FSO channel equal gain combining (ECG) at the receiver.

The outage probability (OP) is defined as [10]:

$$P_{out}=F_{\gamma }\left({\gamma }_{th}\right)$$

(18)

where $F_{\gamma }\left({\gamma }_{th}\right)$ is the cumulative distribution function (CDF) of the corresponding distribution model of Equation (16).

The main idea here is that the outage probability can be estimated by developing $p(h_{i,j}^2)$ from $p\left(h_{i,j}\right)$, using the Mellin transform [20]. Then, the moment generating function (MGF) for h is obtained and the equation for the probability of the MIMO channel through convolution in terms of the product of MGF of combining channels. Then, by taking the inverse Laplace of the resultant MGF divided by $s$ gives the CDF, which depicts the OP.

Assuming the same level of the signal is sent by transmitters 1 and 2, $x ϵ$ (0, A), the signal at the receiver’s input is given as:

$$y_1=(h_{11}^2+h_{12}^2)x+\tilde{N}=H^{2}x+\tilde{N}$$

(19)

In addition, by estimating the peak power to noise ratio (PTN) as $\xi =HA/{\sigma }^2$, the OP of (18), is given as:

$$P_{out}=F_{\xi }\left({\xi }_{th}\right)$$

(20)

In order to determine $F_{\xi }\left({\xi }_{th}\right)$ the PDF for the random variable, $H=h_{11}^2+h_{12}^2$, should be estimated. By expressing the PDF of (16) through the corresponding Meijer-G function [9], we conclude that:

$$p\left(h_{11}\right)=\frac{{\alpha }_1{\beta }_1}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\beta }_1\right)}{G}_{0,2}^{2,0}\left({\alpha }_1{\beta }_{1}h|\begin{array}{c}-\hfill \\ {\alpha }_1-1,{\beta }_1-1\end{array}\right)$$

(21)

$p\left(h_{11}^2\right)$ is simply the multiplication of $p\left(h_{11}\right)$ with itself, which forms the two sided Laplace transform. So, it can be calculated using the Mellin transformation i.e., [21]:

$$ℳ(h_{11}^2)=ℳ\left(h_{11}\right)ℳ\left(h_{11}\right)$$

(22)

as,

$$p(h_{11}^2)=\frac{M_1^2}{K_1^2}{G}_{0,4}^{4,0}\left({\alpha }_1{\beta }_{1}h|\begin{array}{c}-\hfill \\ {\alpha }_1-1,{\beta }_1-1,{\alpha }_1-1,{\beta }_1-1\end{array}\right)$$

(23)

Similarly, for $p\left(h_{12}^2\right):$

$$p(h_{12}^2)=\frac{M_2^2}{K_2^2}{G}_{0,4}^{4,0}\left({\alpha }_2{\beta }_{2}h|\begin{array}{c}-\hfill \\ {\alpha }_2-1,{\beta }_2-1,{\alpha }_2-1,{\beta }_2-1\end{array}\right)$$

(24)

where,

$$\begin{gathered} M_1={\alpha }_1{\beta }_1 \\ {M}_2={\alpha }_2{\beta }_2 \\ {K}_1=\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\beta }_1\right) \\ {K}_2=\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\beta }_2\right) \end{gathered}$$

(25)

where ${\alpha }_{\mathrm{i}},{\beta }_{\mathrm{i}}$ stand for the parameters of the gamma–gamma distribution, for the transmitter’s “1”, i.e., $i=1$, and transmitter’s “2”, i.e., $i=2$, receiver paths. Moreover, as $H=h_{11}^2+h_{12}^2$, it is concluded that:

$$ℒ\left[p\left(H\right)\right]=ℒ[p(h_{11}^2)]\times ℒ[p(h_{12}^2)]$$

(26)

with $ℒ[.]$ being a moment generating function $MGF=ℒ\left[p\left(h\right)\right]$,

$$ℒ[p(h_{11}^2)]=\frac{M_1^2}{K_1^2}{G}_{1,4}^{4,1}\left(\frac{M_1}{s}|\begin{array}{c}1\hfill \\ {\alpha }_1,{\beta }_1,{\alpha }_1,{\beta }_1\end{array}\right)$$

(27)

and

$$ℒ[p(h_{12}^2)]=\frac{M_2^2}{K_2^2}{G}_{1,4}^{4,1}\left(\frac{M_2}{s}|\begin{array}{c}1\hfill \\ {\alpha }_2,{\beta }_2,{\alpha }_2,{\beta }_2\end{array}\right)$$

(28)

Therefore, from (26) it is obtained that:

$$ℒ\left[p\left(H\right)\right]=\frac{M_1^2{M}_2^2}{K_1^2{K}_2^2}{G}_{1,4}^{4,1}\left(\frac{M_1}{u}|\begin{array}{c}1\hfill \\ {\alpha }_1,{\beta }_1,{\alpha }_1,{\beta }_1\end{array}\right)\times G_{1,4}^{4,1}\left(\frac{M_2}{w}|\begin{array}{c}1\hfill \\ {\alpha }_2,{\beta }_2,{\alpha }_2,{\beta }_2\end{array}\right)$$

(29)

with u and w being the variables of MGF.

Then, by estimating the Meijer-G multiplication [21], the (29) takes the form:

$$ℒ\left[p_H\left(z\right)\right]=\frac{M_1^2{M}_2^2}{K_1^2{K}_2^2}{G}_{2,8}^{8,2}\left(\frac{M_1{M}_2}{uw}|\begin{array}{c}1,1\hfill \\ {\alpha }_1,{\beta }_1,{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,{\alpha }_2,{\beta }_2\end{array}\right)$$

(30)

and its CDF is:

$$ℒ\left[F_H\left(z\right)\right]=\frac{M_1^2{M}_2^2}{K_1^2{K}_2^{2}uw}{G}_{2,8}^{8,2}\left(\frac{M_1{M}_2}{uw}|\begin{array}{c}1,1\hfill \\ {\alpha }_1,{\beta }_1,{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,{\alpha }_2,{\beta }_2\end{array}\right)$$

(31)

Next, by taking the inverse Laplace transform of (31) [20]:

$$F_H\left(z\right)=\frac{M_1^2{M}_2^2}{K_1^2{K}_2^2}{G}_{2,9}^{8,2}\left(\frac{h}{M_1{M}_2}|\begin{array}{c}1,1\hfill \\ {\alpha }_1,{\beta }_1,{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,{\alpha }_2,{\beta }_2,0\end{array}\right)$$

(32)

and from (18) and (32) the OP is given as:

$$P_{out}=F_{\gamma }\left({\gamma }_{th}\right)=\frac{M_1^2{M}_2^2\sigma }{K_1^2{K}_2^{2}A}{G}_{2,9}^{8,2}\left(\frac{{\sigma }^2{\gamma }_{th}}{M_1{M}_2{A}^2}|\begin{array}{c}1,1\hfill \\ {\alpha }_1,{\beta }_1,{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,{\alpha }_2,{\beta }_2,0\end{array}\right)$$

(33)

From the (33), the outage probability of a MISO system using Alamouti-like coding with IM/DD can be estimated.

4. Numerical Results

In this section, probability analysis is presented for the $2\times 1$ MISO system by plotting the derived outage probability equation for typical values of parameters. The outage probability is observed as a function of PTN and the normalized threshold. The gamma–gamma distribution is suitable for the study of the irradiance fluctuations at the receiver for the cases of weak to strong turbulence conditions [9]. Thus, three cases are investigated here with the parameters of the gamma–gamma model being $\left(\alpha =2.296 , \beta =2\right)$, $\left(\alpha =4.2 , \beta =3\right),$ and $\left(\alpha =8 , \beta =4\right)$, for weak, moderate, and strong turbulence conditions, respectively. The fixed gain on each path $h_{11}$ and $h_{12}$ has been normalized to the unity without loss of generality. Figure 2 gives the probability of outage as a function of PTN, which is an alternative of the single to noise ratio (SNR), for three values of thresholds, ${\gamma }_{th}$, i.e., $20 \mathrm{dB}$, $26 \mathrm{dB},$ and $32 \mathrm{dB}$.

From Figure 2, it can be seen that the outage probability of the system decreases as PTN increases. For all three cases, the chances for the system to go into outage reduces and hence the performance is improved. Moreover, performance of the channel with the lower fixed threshold ${\gamma }_{th}$ value surmounts the channel with the higher fixed threshold ${\gamma }_{th}$. This is because for the higher threshold value, the channel falls easier into the insufficient power range for operation.

Figure 3 gives the probability of outage versus normalized threshold for three values of PTN, i.e., $20 \mathrm{dB}$, $26 \mathrm{dB},$ and $32 \mathrm{dB}$. It can be seen in Figure 3 that the probability of outage increases along with the normalized threshold. The three cases show the same behavior. The system performs better in the weak turbulence regime compared to the strong and medium turbulences, as expected. This behavior is also evident from the plots in Figure 2. Moreover, the channel assumed with the higher PTN performs slightly better than the channel with lower PTN. This result is obtained because the better power signal allows less noise interference, hence easy retrieval of the original signal from the received signal. Similar results can be seen in [13] for the MIMO channel with the switch-and-examine combining technique. The performance of this system can also be compared with the outcomes of [9], where the simple point-to-point (PtP) link has been studied, to visualize the significance of using Alamouti space-time coding and how it improves the outage probability.

Additionally, it should be mentioned here that the equalization technique can be used in practice for compensating the channel’s distortions, but its study is beyond the scope of this work. However, the total scheme with various equalization techniques can be used and studied in the future, with either decision feedback (DF) or feed-forward (FF) and investigated for OOK to cater the linear impairments which are caused by non-linear devices, limited bandwidth, and chromatic dispersion [22].

5. Conclusions

The MIMO and MISO techniques have been adopted in the modern FSO links in order to improve their availability characteristics and performance. Quantification of the performance metric for such systems is an open problem and needs reliable results. In this work, the availability performance of a MISO system is analyzed by means of its OP. Novel analytical closed form expressions have been derived for the outage probability estimation in terms of the PTN threshold and its average value to investigate its behavior for various atmospheric turbulence conditions modeled with the gamma–gamma distribution. The obtained OP mathematical expressions were derived for the OWC MISO which uses Alamouti space-time coding with IM/DD. Their outcomes are shown through the corresponding figures where it proves that the probability of the signal to go into outage reduces with better PTN and stronger receivers with lower PTN thresholds. The derived expressions could be used in practice for the designing and implementation of very efficient and realistic FSO links.