## Abstract

Atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the performance over free-space optical (FSO) links. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. In this paper, the outage probability as a performance measure for multiple-input/multiple-output (MIMO) FSO communication systems with intensity modulation and direct detection (IM/DD) over strong atmospheric turbulence channels with pointing errors is analyzed. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results show that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver. Simulation results are further demonstrated to confirm the analytical results. Additionally, since proper FSO transmission requires transmitters with accurate control of their beamwidth, asymptotic expressions here obtained for different diversity techniques are used to find the optimum beamwidth that minimizes the outage performance.

© 2011 OSA

## 1. Introduction

Optical wireless communications using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications [1], providing an unregulated spectral segment and high security. Recently, the use of atmospheric free-space optical (FSO) transmission is being specially interesting to solve the *“last mile” problem*, as well as a supplement to radio-frequency (RF) links [2, 3]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance [4, 5]. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [6–11]. In [12, 13], selection transmit diversity is proposed for FSO links over strong turbulence channels, where the transmit diversity technique based on the selection of the optical path with a greater value of irradiance has shown to be able to extract full diversity as well as providing better performance compared to general FSO space-time codes (STCs) designs, such as conventional orthogonal space-time block codes (OSTBCs) and repetition codes (RCs). The combined effect of atmospheric and misalignment fading is analyzed in the case of single-input/single-output (SISO) FSO channels [14, 15]. In [16], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for SISO links. In [17, 18] the error rate performance for uncoded and coded FSO links over strong turbulence and misalignment fading channels is studied. The capacity calculation and the analysis of error rate performance for FSO links over log-normal and gamma-gamma turbulence and misalignment fading channels is presented in [19]. In [20] the error rate performance of PPM on terrestrial FSO Links with gamma-gamma turbulence and pointing errors is analyzed. However, to the best of our knowledge, the performance of MIMO FSO channels under the effects of atmospheric and misalignment fading has only been studied in [21], assuming repetition coding on the transmitter side and equal gain combining on the receiver side. In this work [21], the study of the outage probability and diversity gain has been considered for multiple-input/multiple-output (MIMO) FSO communication systems impaired by log-normal atmospheric turbulence and misalignment fading, showing that the diversity gain is independent of the number of transmitters and receivers, being conditioned only by misalignment parameters.

In this paper, the outage probability as a performance measure for MIMO FSO communication systems using IM/DD over strong atmospheric turbulence channels with pointing errors is analyzed, assuming different configurations as repetition coding and transmit laser selection on the transmitter side and equal gain combining and selection combining on the receiver side. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model, as in [16–18, 21], where the effect of beam width, detector size and jitter variance is considered. In this strong turbulence FSO scenario, obtained results show that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, showing the same slope of the outage performance versus average signal-to-noise ratio (SNR) as in a similar FSO scenario where misalignment fading is not considered. However, different coding gain, i.e. the horizontal shift in the outage performance in the limit of large SNR, is achieved as a consequence of the severity of the pointing errors effects and MIMO configuration assumed. Additionally, a significant improvement in performance is demonstrated when MIMO FSO links based on transmit laser selection with equal gain combining are adopted. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also on-off keying (OOK) formats with any pulse shape, corroborating the advantage of using pulses with high peak-to-average optical power ratio (PAOPR). In contrast to [21], wherein it is concluded that the diversity gain for MIMO FSO communication systems impaired by log-normal atmospheric turbulence and misalignment fading is conditioned only by misalignment parameters, it is shown in this paper that the outage diversity of MIMO FSO links over strong turbulence and misalignment fading channels is independent of pointing errors when transmitters are adequately designed, providing an accurate control of their beamwidth in order to guarantee an equivalent beam radius at the receiver of at least twice the value of the pointing error displacement standard deviation.

## 2. System and channel model

We adopt a multiple-input/multiple-output array based on *L* laser sources, assumed to be intensity-modulated only and all pointed towards a distant array of *M* photodetectors, assumed to be ideal noncoherent (direct-detection) receivers. The transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. The use of infrared technologies based on IM/DD links is considered, where the instantaneous current *y _{lm}*(

*t*) in the

*m*th receiving photodetector corresponding to the information signal transmitted from the

*l*th laser can be written as

*η*is the detector responsivity, assumed hereinafter to be the unity,

*X*≜

*x*(

*t*) represents the optical power supplied by the

*l*th source and

*I*≜

_{lm}*i*(

_{lm}*t*) the equivalent real-valued fading gain (irradiance) through the optical channel between the

*l*th transmit and the

*m*th receive aperture. Additionaly, the fading experienced between source-detector pairs

*I*is assumed to be statistically independent.

_{lm}*Z*≜

_{m}*z*(

_{m}*t*) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the

*m*th detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance ${\sigma}_{m}^{2}\hspace{0.17em}=\hspace{0.17em}{N}_{0}/2M$, i.e.

*Z*∼

_{m}*N*(0,

*N*

_{0}/2

*M*), independent of the on/off state of the received bit. The factor

*M*is consequence of the fact that the sum of the

*M*receive aperture areas is the same as the aperture area of a system with no receive diversity, wherein the noise is usually modelled as AWGN with zero mean and variance

*σ*

^{2}=

*N*

_{0}/2. This allows the systems to be compared fairly [6]. Since the transmitted signal is an intensity,

*X*must satisfy ∀

*t x*(

*t*) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of

*X*is limited. The received electrical signal

*Y*≜

_{lm}*y*(

_{lm}*t*), however, can assume negative amplitude values. We use

*Y*,

_{lm}*X*,

*I*and

_{lm}*Z*to denote random variables and

_{m}*y*(

_{lm}*t*),

*x*(

*t*),

*i*(

_{lm}*t*) and

*z*(

_{m}*t*) their corresponding realizations.

The irradiance is susceptible to either strong atmospheric turbulence conditions and pointing error effects. In this case, it is considered to be a product of two independent random variables, i.e.
${I}_{\mathit{\text{lm}}}\hspace{0.17em}=\hspace{0.17em}{I}_{\mathit{\text{lm}}}^{(a)}\hspace{0.17em}{I}_{\mathit{\text{lm}}}^{(p)}$, representing
${I}_{\mathit{\text{lm}}}^{(a)}$ and
${I}_{\mathit{\text{lm}}}^{(p)}$ the attenuation due to atmospheric turbulence and the attenuation due to geometric spread and pointing errors, respectively, between *l*th transmitter and *m*th receiver. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [19]. Considering a limiting case of strong turbulence conditions [4, 22, 23], the turbulence-induced fading is modelled as a multiplicative random process which follows the negative exponential distribution, whose probability density function (PDF) is given by

*ω*, on the receiver plane at distance

_{z}*z*from the transmitter and a circular receive aperture of radius

*r*, the PDF of ${I}_{\mathit{\text{lm}}}^{(p)}$ is given by

*φ*=

*ω*/2

_{zeq}*σ*is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver, ${\omega}_{{z}_{\mathit{\text{eq}}}}^{2}\hspace{0.17em}=\hspace{0.17em}{\omega}_{z}^{2}\sqrt{\pi}\text{erf}(v)/2v\text{exp}(-{v}^{2})$, $v\hspace{0.17em}=\hspace{0.17em}\sqrt{\pi}r/\sqrt{2}{\omega}_{z}$,

_{s}*A*

_{0}= [erf(

*v*)]

^{2}and erf(·) is the error function [25, eqn. (7.1.1)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being ${\sigma}_{s}^{2}$ the jitter variance at the receiver. Using the previous PDFs for turbulence and misalignment fading, a closed-form expression of the combined PDF of

*I*was derived in [18] as

_{lm}*M*> 1) since it depends on locations of the transmitter and receiver, requiring to take into account the relative position for each receive aperture regarding initial pointing in undisturbed position. This represents a fixed pointing error called boresight. As previously indicated, we adopt a MIMO array wherein the transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. For the sake of simplicity, it is here assumed that the receivers are placed very closely together and beam radius is large enough so that the relative displacement for each receiver regarding the initial pointing can be considered negligible, and hence the pointing error model in Eq. (3) can be approximately applied. This assumption can be reinforced by the fact that strong turbulence channels are more robust to boresight [19]. Additionally, as shown in the appendix, we consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12, 23].

## 3. Outage performance analysis

In this section, the outage probability as a performance measure is evaluated for different MIMO FSO configurations over strong atmospheric turbulence channels with pointing errors. Together with the average error rate, outage probability, *P*
_{out}, is another standard performance criterion of communications systems operating over fading channels [26]. It is defined as the probability that the instantaneous combined SNR, *γ _{T}*, falls below a certain specified threshold,

*γ*, which represents a protection value of the SNR above which the quality of the channel is satisfactory, i.e.,

_{th}*f*(

_{γ}_{T}*γ*) and

_{T}*F*(

_{γ}_{T}*γ*) are the PDF and the cumulative distribution function (CDF) of

_{T}*γ*, respectively. Next, the outage performance of a SISO FSO system is firstly evaluated in order to be considered as a benchmark in the analysis of the remaining MIMO configurations, corresponding to a multiple-input/single-output (MISO) system with a

_{T}*L*× 1 array, a single-input/multiple-output (SIMO) system with a 1 ×

*M*array and, finally, a MIMO FSO system with a

*L*×

*M*array.

#### 3.1. SISO FSO system

In this subsection, the outage probability for the FSO system model previously presented with *L* = *M* = 1 is evaluated. According to Eq. (1) and the OOK signaling described in the appendix, a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of *d*, the statistical channel model corresponding to the SISO configuration can be written as

*γ*represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats,

*T*parameter is the bit period,

_{b}*P*

_{opt}is the average optical power transmitted and

*ξ*represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in the appendix. Using Eq. (5), the outage probability for a SISO FSO system can be written as

*I*,

*F*(

_{I}*i*), can be easily derived from Eq. (4) as

*ξ*= 1 are used, assuming different values of normalized beamwidth,

*ω*/

_{z}*r*, and normalized jitter,

*σ*/

_{s}*r*(i.e.

*ω*/

_{z}*r*= 5 with

*σ*/

_{s}*r*= {1,3,4,5} and

*ω*/

_{z}*r*= 10 with

*σ*/

_{s}*r*= {1,4,7,11}). Outage simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (8) for different misalignment fading conditions.

Additionally, we can take advantage of this series expansion in order to quantify the outage probability at high SNR, showing that the asymptotic performance of this metric as a function of the average SNR is characterized by two parameters: the diversity and coding gains. Therefore, for large enough SNR (*γ* → ∞), the asymptotic outage performance can be expressed after some algebraic manipulations as

*φ*, as follows

*O*(

_{c}*γ*/

*γ*))

_{th}^{–Od}, where

*O*and

_{d}*O*denote outage diversity and coding gain, respectively [26, 28]. At high SNR, if asymptotically the outage probability behaves as (

_{c}*O*(

_{c}*γ*/

*γ*))

_{th}^{–Od}, the outage diversity

*O*determines the slope of the outage performance versus average SNR curve in a log-log scale and the coding gain

_{d}*O*(in decibels) determines the shift of the curve in SNR. Additionally, as previously reported by the authors [12, 13, 23], a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10 log

_{c}_{10}

*ξ*decibels.

At this point, it can be convenient to compare with the outage performance obtained in a similar context when misalignment fading is not present. In this case, knowing that the CDF corresponding to the negative exponential distribution in Eq. (2) is given by *F*
_{I(a)} (*i*) = 1 − exp(−*i*) and the fact that the series expansion for the exponential function can be simplified by exp(*z*) ∝ 1 + *z* + *O*(*z*
^{2}), an asymptotic expression can be easily derived as
${P}_{\text{out}}^{\mathit{\text{exp}}}\hspace{0.17em}\doteq \hspace{0.17em}({\gamma}_{\mathit{\text{th}}}/\gamma \xi ){)}^{1/2}$. From these results, it can be deduced that *φ* is a key parameter in order to optimize the outage diversity. So, it is shown that the diversity order is independent of the pointing errors when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. *φ* > 1. Once this condition is satisfied, taking into account the coding gain in Eq. (11*a*), the impact of the pointing error effects translates into a coding gain disadvantage, *D*[*dB*], relative to strong atmospheric turbulence without misalignment fading given by

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (5, 1), (

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (10,1) and (

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (10,4), respectively.

#### 3.2. MISO FSO system with a L ×1 array

In this subsection, the outage probability for the FSO system model previously presented where the information signal is transmitted via *L* apertures and received by only one aperture (i.e. *M* = 1) is evaluated. Here, two different transmit diversity strategies are considered: repetition coding (RC) and transmit laser selection (TLS).

### 3.2.1. Transmit diversity using repetition coding

It is assumed that the same symbol is transmitted from the *L* transmitters at a given bit period [6, 10, 29]. According to Eq. (1), the statistical channel model corresponding to this MISO configuration can be written as

*I*represents the equivalent irradiance through the optical channel between the

_{l}*l*th transmit aperture and the photodetector. Here, the division by

*L*is considered so as to maintain the average optical power in the air at a constant level of

*P*

_{opt}, being transmitted by each laser an average optical power of

*P*

_{opt}/

*L*. In this way, the total transmit power is the same as in a FSO system with no transmit diversity. The resulting received electrical SNR can be defined as

*L*variates, the PDF in Eq. (4) is approximated by a single polynomial term as

*f*(

_{Ilm}*i*) ≈

*ai*, based on the fact that the asymptotic behavior of the system performance is dominated by the behavior of the PDF near the origin, i.e.

^{b}*f*(

_{Ilm}*i*) at

*i*→ 0 determines high SNR performance [28]. Using the series expansion corresponding to the upper incomplete Gamma function [27, eqn. (8.354.2)], different asympotic expressions for Eq. (4), depending on the value of

*φ*, can be written as

*F*(

_{IT}*i*), of the combined variate can be easily derived as

*ξ*= 1 are used, assuming a normalized beamwidth of

*ω*/

_{z}*r*= 5 and different values of normalized jitter of

*σ*/

_{s}*r*= 1 and

*σ*/

_{s}*r*= 4.

### 3.2.2. Transmit diversity using laser selection

When channel side information is not only available at the receiver but also on the transmitter, an alternative transmit diversity scheme can be adopted. Following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [12], our MISO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the TLS scheme, *I _{max}*, can be written as

*I*= max

_{max}

_{l}_{=1,2,…}

_{L}*I*. In this way, having a SISO system as a reference, the statistical channel model corresponding to this MISO configuration can be written as

_{l}*P*

_{out}, the division by

*L*is not included in Eq. (19) because of not more than one laser is simultaneously used. According to order statistics [30], the CDF,

*F*(

_{Imax}*i*), of the resulting channel irradiance,

*I*, is given by

_{max}*F*(

_{Imax}*i*) = [

*F*(

_{I}*i*)]

*and, hence, the outage probability can be expressed as*

^{L}*A*[

_{TLS}*dB*], relative to the RC scheme given by

*φ*< 1 and the value of

*φ*is decreasing.

#### 3.3. SIMO FSO system with a 1 × M array

In this subsection, the outage probability for the FSO system model previously presented is evaluated when the information signal is transmitted via only one aperture (i.e. *L* = 1) and received by *M* apertures. Here, two different receive diversity strategies are considered: equal gain combining (EGC) and selection combining (SC).

### 3.3.1. Receive diversity using equal gain combining

Following this combining technique, the receiver adds the information corresponding to *M* photodetectors and, hence, according to Eq. (1), the statistical channel model for this SIMO configuration can be written as

*I*represents the equivalent irradiance through the optical channel between the transmit aperture and the

_{m}*m*th photodetector. Here, the division by

*M*is considered to ensure that the area of the receive aperture in SISO links has the same size as in the sum of

*M*receive aperture areas of SIMO links [6]. It can be noted that the resulting expression is equivalent to the expression in Eq. (13), obtained for MISO FSO links assuming RC at the transmitter side [10]. Therefore, we can interchange

*L*by

*M*in Eq. (15) and Eq. (18) to obtain the outage probability and its corresponding asympotic expressions for SIMO FSO links using equal gain combining.

### 3.3.2. Receive diversity using selection combining

This type of combining is based on the selection of the receive aperture corresponding to the optical path with a greater value of fading gain (irradiance) [6, 10]. In this way, in a similar way as previously presented for MISO FSO links with transmit laser selection, our SIMO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the SC scheme, *I _{max}*, can be written as

*I*= max

_{max}

_{m}_{=1,2,···}

_{M}*I*and, hence, the statistical channel model can be written as

_{m}*M*represents the same role as previously explained when EGC is adopted. It can be observed that the noise variance is given by

*N*

_{0}/2

*M*since only one of the

*M*receive apertures is used. Taking into account Eq. (7), the resulting received electrical SNR can be defined as

*φ*, can be easily derived from Eq. (11) as

*φ*, corroborating the superiority of EGC scheme when

*φ*> 1 and the superiority of both SC scheme or EGC scheme when

*φ*< 1, depending on the values of

*M*and

*φ*. This improvement in performance of the EGC scheme when

*φ*> 1 translates into a coding gain advantage,

*A*[

_{EGC}*dB*], relative to the SC scheme as follows

*M*= 2. In a similar way, the superiority of SC scheme when

*φ*< 1 translates into a coding gain advantage,

*A*[

_{SC}*dB*], relative to the EGC scheme as follows

*φ*and when a small number of receive apertures is considered, being only justified the adoption of this receive diversity technique in the context of very severe pointing errors. As depicted in Fig. 3 for a normalized beamwidth of

*ω*/

_{z}*r*= 10, better performance for the EGC scheme is again achieved as the number of receive apertures increases and the value of the normalized jitter,

*σ*/

_{s}*r*, is lower and lower. So, the value of

*φ*becomes higher and higher, and hence it is closer to the unity.

#### 3.4. MIMO FSO system with a L ×M array

According to results in previous subsections, the outage probability for MIMO FSO IM/DD links is here evaluated when transmit laser selection and equal gain combining are the diversity strategies adopted in the transmit and receive side, respectively. The statistical channel model for this MIMO configuration can be written as

*I*represents the equivalent irradiance through the optical channels between the

_{m}*L*transmit apertures and the

*m*th photodetector, based on the selection of the optical path with a greater value of irradiance, i.e.

*I*= max

_{m}

_{l}_{=1,2,···}

_{L}*I*. The resulting received electrical SNR can be defined as

_{lm}*I*represents the sum of

_{T}*M*variates following the

*L*th-order statistics distribution for

*L*observations from the distribution in Eq. (4). As in previous subsections, in order to address the cumbersome statistical problem related to the CDF of the sum of these

*M*variates and making use of Eq. (16) , we use an approximated expression for the PDF of

*I*, which is given by

_{m}*F*(

_{IT}*i*), of the combined variate can be easily derived as

*M*= 1 and

*L*= 1, respectively. In this way, it can be noted that expressions in Eq. (35) are equivalent to the expressions in Eq. (21) when a MISO FSO system with transmit laser selection is assumed. In the same way, expressions in Eq. (35) can also be reduced to those corresponding to SIMO FSO system with equal gain combining, being equivalent to the expressions in Eq. (18) when

*L*is interchanged by

*M*. As previously commented in section 3.3.1, expressions corresponding to a SIMO FSO system using equal gain combining are equivalent to those corresponding to a MISO FSO system assuming RC at the transmitter side. Monte Carlo simulation results for Eq. (32) together with the asymptotic expressions in Eq. (35) are illustrated in Fig. 4, where rectangular pulse shapes with

*ξ*= 1 are used, assuming a normalized beamwidth of

*ω*/

_{z}*r*= 5 with a normalized jitter of

*σ*/

_{s}*r*= 1 and a normalized beamwidth of

*ω*/

_{z}*r*= 10 with a normalized jitter of

*σ*/

_{s}*r*= 7 and values of

*L*= {2, 4} and

*M*= {2, 4}.

## 4. Outage optimization

From obtained results, it can be concluded that the outage diversity is independent of the pointing error effects when the equivalent beam radius value at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. *φ* > 1, being *φ* a main parameter to consider in order to optimize the outage performance. Once this condition is satisfied, comparing different diversity techniques and their corresponding asymptotic expressions, it can be noted that better performance is achieved when increasing the number of transmit apertures, i.e. *L*, instead of the number of receive apertures, i.e. *M*, in order to guarantee a same diversity order, as shown in Fig. 4 when assuming a normalized beamwidth of *ω _{z}*/

*r*= 5 with a normalized jitter of

*σ*/

_{s}*r*= 1 and a diversity order of 8 with values of {

*L*,

*M*} = {4,2} and {

*L*,

*M*} = {2,4}. Taking into account expressions in Eq. (35

*a*) and Eq. (21

*a*), this can be quantified as a coding gain disadvantage,

*D*[

_{MIMO}*dB*], for the MIMO FSO system relative to the MISO FSO system using laser selection as follows

*O*and

_{d}*M*denote outage diversity and the number of receive apertures, respectively. In agreement with this expression, it can be seen in Fig. 4 that a difference of 2.13 decibels is achieved for a diversity order of 8 and values of

*M*= 2 and

*M*= 4 when a normalized beamwidth of

*ω*/

_{z}*r*= 5 with a normalized jitter of

*σ*/

_{s}*r*= 1 is assumed. Lastly, since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth,

*ω*/

_{z}*r*, that gives the minimum outage [16, 17, 31]. From Eq. (35

*a*), it can be observed that this is equivalent to minimize the expression in Eq. (12) corresponding to the SISO configuration, being independent of the MIMO configuration adopted. In this way, the optimum beamwidth can be achieved using numerical optimization methods for different values of normalized jitter,

*σ*/

_{s}*r*[32]. Numerical results for the optimum beamwidth are illustrated in Fig. 5 for a range of values in the normalized jitter from

*σ*/

_{s}*r*= 1 to

*σ*/

_{s}*r*= 10 in discrete steps of 0.5 when the stochastic function minimizer simulated annealing is used. From this figure, it can be deduced that the outage optimization provides numerical results following a linear performance. This leads to easily obtain a first-degree polynomial as follows

*σ*/

_{s}*r*= 1 and

*σ*/

_{s}*r*= 10, respectively. As can be seen in this figure, it is clearly depicted that the approximative analytical expression remains very accurate to numerical results. The use of this expression is shown in Fig. 4, where results assuming the optimum beamwidth corresponding to values of jitter of

*σ*/

_{s}*r*= 1 and

*σ*/

_{s}*r*= 7 are also included for

*L*= 4 and

*M*= 2. As expected, a greater improvement in performance is achieved when

*σ*/

_{s}*r*= 7 compared to the improvement in performance corresponding to the normalized jitter of

*σ*/

_{s}*r*= 1 since not only coding gain but also diversity gain is obtained, making outage diversity performance independent of the impact of misalignment fading.

## 5. Conclusions

In this paper, the outage probability as a performance measure for MIMO FSO communication systems using IM/DD over strong atmospheric turbulence channels with pointing errors is analyzed, assuming different configurations as repetition coding and transmit laser selection on the transmitter side and equal gain combining and selection combining on the receiver side. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. In this strong turbulence FSO scenario, obtained results show that the diversity order is independent of the pointing error effects when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, showing the same slope of the outage performance versus average SNR as in a similar FSO scenario where misalignment fading is not considered. However, different coding gain, i.e. the horizontal shift in the outage performance in the limit of large SNR, is achieved as a consequence of the severity of the pointing errors effects and MIMO configuration assumed. Additionally, a significant improvement in performance is demonstrated when MIMO FSO links based on transmit laser selection with equal gain combining are adopted. Simulation results are further demonstrated to confirm the analytical results. The asympotic expressions here obtained are especially useful for performance analysis of MIMO FSO systems, providing an easier understanding of performance limiting factors when operating under different misalignment fading conditions and various diversity techniques are adopted. From the relevant results here obtained, investigating in this FSO scenario the impact on the diversity order of different receive configurations, e.g., rectangular and circular arrays of receive apertures, wherein the displacement for each photodetector regarding the initial pointing is considered as well as the possible correlation among their corresponding shifting, is an interesting topic for future research.

## Appendix

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12, 23]. A new basis function *ϕ* (*t*) is defined as
$\varphi \hspace{0.17em}(t)\hspace{0.17em}=\hspace{0.17em}g(t)/\sqrt{{E}_{g}}$ where *g*(*t*) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤ *g*(*t*) ≤ 1 in the bit period and 0 otherwise, and
${E}_{g}\hspace{0.17em}=\hspace{0.17em}{\int}_{-\infty}^{\infty}{g}^{2}\hspace{0.17em}(t)\mathit{\text{dt}}$ is the electrical energy. In this way, an expression for the optical intensity can be written as

*G*(

*f*= 0) represents the Fourier transform of

*g*(

*t*) evaluated at frequency

*f*= 0, i.e. the area of the employed pulse shape, and

*T*parameter is the bit period. The random variable (RV)

_{b}*a*follows a Bernoulli distribution with parameter 1/2, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is

_{k}*P*

_{opt}, defining a constellation of two equiprobable points (

*x*

_{0}= 0 and

*x*

_{1}=

*d*) in a one-dimensional space with an Euclidean distance of $d\hspace{0.17em}=\hspace{0.17em}2{P}_{\text{opt}}\sqrt{{T}_{b}\xi}$ where

*ξ*=

*T*

_{b}*E*/

_{g}*G*

^{2}(

*f*= 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse.

## Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments that helped to improve the presentation of the paper. The authors are grateful for financial support from the Junta de Andalucía (research group “ Communications Engineering ( TIC-0102)”).

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