Prof Karim

Prof Murat

Prof Ayman

Dr Kostas Peppas

Dr Chadi

Dr Majid

Ismail Ben

Diversity Techniques for Correlated

ρ = 0.5 and σ x = 0.1. . . . . . . . . . .

Résumé

Au cours de la dernière décennie, les communications optiques en espace libre (FSO) ont pris de l'ampleur dans les deux domaines académiques et industriels. L'importance de FSO s'appuie sur la possibilité de faire un système de transmission économique et écologique avec un débit élevé et sans licence à l'opposition des systèmes de transmission radiofréquences (RF). Dans la plupart des travaux antécédents sur les systèmes multiémetteurs, seulement les canaux décorrélés ont été considérés. Un canal décorrélé nécessite un espace suffisant entre les émetteurs. Cette condition devient difficile et non-réalisable dans certaines applications. Pour cette raison, nos études se focalisent sur les performances des codes à répétition RC (Repitition Codes) et les codes OSTBC (Orthogonal Space-Time Block Codes) dans des canaux lognormaux corrélés en utilisant une modulation d'intensité et une détection directe (IM/DD). En addition, les effets des différentes conditions météorologiques sur le taux d'erreur moyen (ABER) sont étudiés. Les systèmes FSO à multi-entrées/ multi-sorties MIMO (Multiple-Input Multiple-Output) avec une modulation SSK (Space Shift Keying) ont été abordés. Les résultats obtenus montrent que la SSK est supérieure aux RC avec une modulation d'impulsion (Multiple Pulse Amplitude Modulation) pour toute efficacité spectrale égale ou supérieure à 4 bit/s/Hz. Nous avons aussi analysé les performances d'un système à sauts multiples (Multi-Hop) et des relais à transmission directe (forward relays). Nos simulations montrent que le système ainsi considéré est efficace pour atténuer les effets météorologiques et les pertes géométriques dans les systèmes de communication FSO. Nous avons montré qu'un tel système avec plusieurs entrées et une sortie (MISO, i.e. multiple-input single-output) à sauts multiples est supérieur à un système MISO avec un lien direct (direct link) avec une forte atténuation. Pour satisfaire la demande croissante des réseaux de communication à débits élevés, la communauté scientifique s'intéresse de plus en plus aux systèmes FSO avec des relais full-duplex (FD). Pour ces derniers systèmes, nous avons étudié la probabilité d'erreur moyenne (ABER) et nous avons analysé leurs performances. En considérant des différentes conditions de transmission, les performances de relais FD ont été comparées à celles d'un système avec un lien direct ou des relais half-duplex. Les résultats obtenus montrent que les relais FD ont le minimum ABER. En conséquence, les résultats obtenus dans cette thèse sont très prometteurs pour la prochaine génération de FSO.

Abstract

Free-space optical (FSO) communication has been the subject of ongoing research activities and commercial attention in the past few years. Such attention is driven by the promise of high data rate, license-free operation, and cheap and ecological friendly means of communications alternative to congested radio frequency communications. In most previous work considering multiple transmitters, uncorrelated channel conditions have been considered. An uncorrelated channel requires sufficient spacing between transmitters. However, this can be difficult and may not be always feasible in some applications. Thereby, this thesis studies repetition codes (RCs) and orthogonal spacetime block codes performance in correlated log-normal FSO channels using intensity modulation and direct detection (IM/DD). Furthermore, the effect of different weather conditions on the average bit error rate (ABER) performance of the FSO links is studied. Multiple-input multiple-output (MIMO) FSO communication systems using space shift keying (SSK) modulation have been also analyzed. Obtained results show that SSK is a potential technique for spectral efficiencies equal or greater than 4 bits/s/Hz as compared to RCs with multiple pulse amplitude modulations. The performance analysis of a multi-hop decode and forward relays for FSO communication system using IM/DD is also considered in this thesis. It is shown that multi-hop is an efficient technique to mitigate atmospheric turbulence and different weather attenuation effects and geometric losses in FSO communication systems. Our simulation results show that multiple-input single-output (MISO) multi-hop FSO systems are superior to direct link and MISO systems over links exhibiting high attenuation. Meeting the growing demand for higher data rates communication networks, a system with full-duplex (FD) relays is considered. For such a system, the outage probability and the ABER performance are analyzed under different turbulence conditions, misalignment error and path loss effects. FD relays are compared with the direct link and half-duplex relays. Obtained results show that FD relays have the lowest ABER and the outage probability as compared to the two other systems. Finally, the obtained results in this thesis are very promising towards the next generation of FSO systems.

1.2

The main axes of the research. . . Correlated LN with σ x = 0.1, µ x = -σ 2 x , N t = 2 and α = 0 dB. . . . . . 54

3.6

Correlated LN with σ x = 0.1, µ x = -σ 2 x , N t = 2 and light fog. . . . . . 55

3.7

Correlated LN with σ x = 0.1, µ x = -σ 2 x , N t = 3 and α = 0 dB. . . . . . 55

3.8

Correlated LN with σ x = 0.1, µ x = -σ 2 x , N t = 3 and light fog. . . . . . 56

3.9

Correlated LN with σ x = 0.374, µ x = -σ 2 x , N t = 2, and α = 0 dB. . . . 56 3.10 Correlated LN with σ x = 0.3, µ x = -σ 2

x , N t = 2, and clear air. . . . . . 57 3.11 Correlated LN with σ x = 0.374, µ x = -σ 2

x , N t = 3 and α = 0 dB. . . . . 57 3.12 Correlated LN with σ x = 0.3, µ x = -σ 2

x , N t = 3 and clear air. . . . . . . 58 3.13 Correlated LN using repetition codes with σ x = 0.1, µ x = -σ 2

x , ρ = 0.25, N t = 3 for dierent weather conditions. . . . . . . . . . . . . . . . . . . . 58 3.14 Histogram of the absolute dierence of two LN RVs (σ x = 0.37 and µ x = -σ 2

x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Histogram of the absolute dierence of two LN RVs (σ x = 0.1 and µ x = -σ 2

x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . However, practical deployment of FSO faces major challenges that degrade the performance signicantly. High sensitivity to harsh weather conditions such as snow, haze, rain, smoke, fog and dust is shown to impact the performance signicantly [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Datsikas | Serial freespace optical relaying communications over gamma-gamma atmospheric turbulence channels[END_REF]. In addition, background noise as a result of exposure to direct or indirect sunlight or articial lights is another major issue to be considered. Background noise can be eliminated using optical lters [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF]. Geometric losses caused by beam divergence is shown to degrade the performance of FSO systems as well [START_REF] Wainright | Wavelength diversity in free-space optics to alleviate fog eects[END_REF]. An ecient solution is to narrow the transmitted beam [START_REF] Bloom | Understanding the performance of free-space optics[END_REF]. However, such solution causes a misalignment problem, which can lead to a communication interruption and requires automatic pointing and tracking at the receiver [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF]. Also, atmospheric turbulence-induced fading, which is the uctua-Introduction tion of the received signal due to inhomogeneity in temperature and pressure, leads to signicant performance degradation [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF].

These turbulences are the main reason for fading in FSO links. Such fading is caused by multipath propagation in RF systems [START_REF] Datsikas | Serial freespace optical relaying communications over gamma-gamma atmospheric turbulence channels[END_REF][START_REF] He | Bit-interleaved coded modulation for hybrid RF/FSO systems[END_REF]. RF and FSO are aected dierently by atmospheric and weather conditions. FSO links suer from high attenuation in the presence of fog but are less aected by rain. In contrast, fog has practically no impact on RF systems while rain signicantly increases link attenuation [START_REF] He | Bit-interleaved coded modulation for hybrid RF/FSO systems[END_REF]. Many statistical models have been proposed to describe the atmospheric turbulence in FSO which give good agreement between theoretical and experimental data. The most widely used of are log-normal distribution (LN) for weak-moderate [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF][START_REF] Abaza | Diversity techniques for a free-space optical communication system in correlated log-normal channels[END_REF], Gamma-Gamma (G-G) distribution for moderate-strong [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF] and negative exponential distribution [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF][START_REF] Abaza | BER performance of M-ary PPM free-space optical communications with channel fading[END_REF] for strong saturated turbulence regimes. Moreover, generalized distributions such as exponentiated Weibull (EW) [START_REF] Barrios | Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves[END_REF] and M-distribution [START_REF] Jurado-Navas | Impact of pointing errors on the performance of generalized atmospheric optical channels[END_REF] can provide excellent agreement with the probability density function (PDF) of irradiance for both simulation and experiment data in weak-to-strong turbulence regimes.

Thesis Contributions

The main objective of this thesis is to propose reliable solutions for the previously mentioned challenges of FSO. In summary, the main contributions of this thesis are:

1. Showing that correlation among multiple transmitters leads to signal-to-noise ratio (SNR) losses. Separating the transmitters by few centimeters can decrease the correlation. However, the required separation may be dicult in practice as available space for the transmitters may not be sucient for this requirement [START_REF] Zhu | Maximum-likelihood spatial-diversity reception on correlated turbulent free-space optical channels[END_REF].

For this reason, correlated LN channels, as well as path loss due to weather eects using intensity modulation and direct detection schemes, have been also studied.

Additionally, an approximated average bit error rate (ABER) expressions for repetition codes (RCs) and orthogonal space-time block codes (OSTBCs) are derived.

Under the considered scenarios, results show that RCs outperform OSTBCs.

Thesis Organization

Figure 1.2 shows the main three axes considered in the thesis to mitigate various challenges facing FSO. Firstly, the existing works in the three axes are discussed in Subsections 2.6.4-2.6.6. Secondly, the obtained contribution of the rst axis, multiple-input multiple-output (MIMO), is considered in Chapter 3. Thirdly, the achieved contribution of the second axis, spatial modulation, is considered where a special case of spatial modulation called space shift keying is employed in Chapter 4. Finally, the third axis, relay assisted is considered in its two schemes: multi-hop and cooperative relays in Chapter 5 and Chapter 6, respectively. In this chapter, the key technologies, motivation, major challenges, channel models and the practical results of free-space optical (FSO) communication systems are discussed. This chapter also considers dierent modulation techniques, forward error correction techniques, spatial diversity, aperture averaging and multi-hop relays as well as cooperative relays to improve FSO communication systems. Figure 2.1 shows the main three axes considered in the thesis to mitigate various challenges facing FSO. The existing works in the three axes are discussed in Subsections 2.6.4-2.6.6.

relay assisted communications spatial modulation

relay assisted communications spatial modulation

Key Technologies

Wireless optical communications transmit information through light. Light source converts the electrical signal to light signal which propagates through the optical channel till a photodetector. The photodetector converts again the light signal to electrical signal. Wireless optical can be classied into indoor wireless optical communications, underwater wireless optical communications, satellite wireless optical communications and free-space optical communications [2].

24

State of the Art

Indoor wireless optical communications

Indoor wireless optical communication can be classied into infrared and visible light communications (VLC). VLC has great attention nowadays due to its ability to communicate and illuminate simultaneously. White illumination using light-emitting diodes (LEDs) is preferred for VLC, due to their long lifetimes and energy eciencies (at least 10 times greater than incandescent bulbs). In addition, by as early as 2018, majority of lighting installations are expected to be LED-based. This green technology can be used in places where radio frequency (RF) is prevented such as hospitals and airplanes [START_REF] Hranilovic | Visible light communications : the road to standardization and commercialization (Part 1)[END_REF][START_REF] Grobe | High-speed visible light communication systems[END_REF].

Moreover, it can be used for positioning of mobile devices in indoor environments as supermarkets, shopping malls, and museums (using the camera sensor as the receiver).

The market for mobile indoor positioning in the retail sector is expected to reach $5 billion by 2018 [START_REF] Jovicic | Visible light communication : opportunities, challenges and the path to market[END_REF]. Also, sale navigation using VLC technology [START_REF] Lim | Entertainment lighting control network standardization to support VLC services[END_REF] is very interesting application as depicted in Figure 2.2. In addition, VLC can be used for smart automotive lighting using the already existing LEDs in the car to decrease the percentage of accidents [START_REF] Yu | Smart automotive lighting for vehicle safety[END_REF]. for the incorrect detection of LED color [START_REF] Karunatilaka | LED based indoor visible light communications : state of the art[END_REF].

Key Technologies 25

High eciency LEDs named organic light emitting diodes (OLEDs), contains multilms such as carbon and hydrogen between two conductors, has been introduced since 1990 [START_REF] Burroughes | Light-emitting diodes based on conjugated polymers[END_REF]. The advantages of OLEDs are low cost, no restriction on size, bright, power ecient and light weight. However, OLEDs have many challenges such as degradation of the organic layer and OLEDs have a capacitor-like behavior so huge bandwidth required small photoactive area while small OLEDs does not support the required illumination.

These challenges limit the data rate being in the range of 2.7 Mbps [START_REF] Haigh | Visible light communications using organic light emitting diodes[END_REF]. Finally, OLEDs for VLC still needs a lot of developments to be commercialized.

Underwater wireless optical communications

Recently, wireless optical communication has been deployed for underwater robot [START_REF] Tian | Design and development of an LED-based optical communication system for autonomous underwater robots[END_REF], see

Free-space optical communications

FSO communications or outdoor wireless optical usually referred to wireless optical communications among buildings see Figure 2.5. In this thesis, we focus on FSO using lasers.

Motivation

In fact, the information carrying capacity of a communication system increases by using higher carrier frequency. In RF systems, the allowable bandwidth can be up to 20% of the carrier frequency for a licensed system. Even 1% of the optical bandwidth for the carrier frequency of ≈ 10 16 Hz can achieve a bandwidth of 100 THz. Therefore, optical systems can handle almost 10 5 times actual RF systems [START_REF] Williams | RF and optical communications: A comparison of high data rate returns from deep space in the 2020 timeframe[END_REF]. However, available electronics can not provide such huge bandwidth at the moment. Thus, techniques such as wavelength division multiplexing and spatial diversity should be used.

The fast growing demand for high data rates in conjunction with the scarcity of RF band necessitate considering alternative solutions. Fiber optic network can achieve these demands, however, it requires an expensive infrastructure. Therefore, FSO communication is a leading technology that has been at the forefront of both research and commercial activities over the past years.

Major Challenges

Major Challenges

FSO advantages did not come without cost. The transmitted optical signal is aected by several challenges such as geometric loss, misalignment loss, weather attenuation loss, and background noise and atmospheric turbulence.

Geometric loss

The geometric loss is due to the divergence of the optical beam through propagating which is aected by the divergence angle, the propagation distance, and the receiver aperture size [START_REF] Wainright | Wavelength diversity in free-space optics to alleviate fog eects[END_REF].

Misalignment loss

Misalignment loss occurs due to an error in tracking system or building sway. The main reasons for building sway are the wind, a small earthquake, and vibrations [START_REF] Ansari | Impact of pointing errors on the performance of mixed RF/FSO dual-hop transmission systems[END_REF].

For short range communications, a laser with a wide divergence angle can be used to reduce the eect of misalignment loss. However, for long-range communications (i.e.

several kilometers), a laser with narrow divergence angle must be used to avoid severe geometric loss and automatic pointing and tracking adjustment must be employed [START_REF] Bloom | Understanding the performance of free-space optics[END_REF].

Weather attenuation loss

Dierent weather attenuation losses such as haze, rain, snow, dust, and smoke and fog cause severe attenuation for the transmitted signal [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Datsikas | Serial freespace optical relaying communications over gamma-gamma atmospheric turbulence channels[END_REF]. RF and FSO are dierently aected by weather eects. FSO links suer from the highest attenuation in the presence State of the Art of fog but are less aected by the rain. In contrast, RF links at 10 GHz aren't aected by the fog while the rain signicantly increases the attenuation. This property leads to the design of hybrid RF/FSO systems [START_REF] He | Bit-interleaved coded modulation for hybrid RF/FSO systems[END_REF] and mixed RF/FSO [START_REF] Ansari | Impact of pointing errors on the performance of mixed RF/FSO dual-hop transmission systems[END_REF][START_REF] Zhang | Unied performance analysis of mixed radio frequency/free-space optical dual-hop transmission systems[END_REF]. Hybrid RF/FSO uses FSO and RF parallel between two nodes [START_REF] He | Bit-interleaved coded modulation for hybrid RF/FSO systems[END_REF]. However, in mixed RF/FSO, RF and FSO are deployed together for dual-hop communication systems [START_REF] Zhang | Unied performance analysis of mixed radio frequency/free-space optical dual-hop transmission systems[END_REF]. The attenuation coecient (α in dB/km) of dierent weather conditions are provided in Table 2.1.

Atmospheric turbulence

Atmospheric turbulence-induced fading can be dened as a random uctuation in both amplitude and phase of the received signal due to the inhomogeneity in temperature and pressure. Such turbulence causes signicant performance degradation in FSO [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF]. The turbulence is the main reason for the fading in FSO. While in RF links, the fading is due to the multipath propagation [START_REF] Datsikas | Serial freespace optical relaying communications over gamma-gamma atmospheric turbulence channels[END_REF][START_REF] He | Bit-interleaved coded modulation for hybrid RF/FSO systems[END_REF]. The atmospheric turbulence has a severe eect on the performance of FSO systems especially for long distances [START_REF] Andrews | Laser Beam Scintillation with Applications[END_REF]. The turbulence eect depends on the link distance, the wavelength of the light source and

the refractive index constant, C 2 n (m -2/3 ) [14]. C 2
n is strongly aected by temperature so it has the peak value at noon and it has the minimum value at midnight [START_REF] Henniger | An introduction to free-space optical communications[END_REF].

2.5

Atmospheric Turbulence Channels

Experimental results

In [START_REF] Phillips | Measured statistics of laser-light scattering in atmospheric turbulence[END_REF], a FSO experiment was carried out using a 632.8 nm wavelength laser, a beam divergence of 1 mrad, a power of 15 mW and the distances were 183 m and Atmospheric Turbulence Channels

Simulation results

Simulation results for FSO channels assuming plane wave [START_REF] Bracher | Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation[END_REF] and spherical wave [START_REF] Hill | Probability distribution of irradiance for the onset of strong scintillation[END_REF] are carried out. The simulations carried out by approximating a three-dimensional random medium as a collection of two-dimensional random phase screens that are transverse to the direction of wave propagation. These simulation results have excellent agreements with the FSO experiment carried out in [START_REF] Churnside | Probability density of irradiance scintillations for strong path-integrated refractive turbulence[END_REF].

Proposed analytical models

In [START_REF] Al-Habash | Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media[END_REF], Gamma-Gamma (G-G) distribution is proposed as an eciency model for moderate-strong turbulence and has a good agreement with simulation results for both plane wave [START_REF] Bracher | Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation[END_REF] and spherical wave [START_REF] Hill | Probability distribution of irradiance for the onset of strong scintillation[END_REF]. In G-G distribution, the received light intensity is the product of two independent Gamma random variables, where the irradiance uctuations are the product of small-scale and large-scale uctuations.

Double Weibull distribution is proved in [START_REF] Chatzidiamantis | New results on turbulence modeling for free-space optical systems[END_REF] to be more accurate than G-G distribution for moderate-strong turbulence using the same set of measured data of [START_REF] Al-Habash | Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media[END_REF].

However, G-G distribution has simpler calculation complexity. Additionally, Double Generalized Gamma (Double GG) model is proposed in [START_REF] Kashani | A novel statistical channel model for turbulence-induced fading in free-space optical systems[END_REF], where the received light intensity is the product of two independent Generalized Gamma random variables. Compared with G-G and Double Weibull using the same set of measured data of [START_REF] Al-Habash | Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media[END_REF] for the comparison, Double GG is more accurate in the strong turbulence using spherical wave model and in moderate turbulence using plane wave model [2].

Moreover, generalized distributions such as exponentiated Weibull (EW) [START_REF] Barrios | Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves[END_REF], Mdistribution [START_REF] Jurado-Navas | Impact of pointing errors on the performance of generalized atmospheric optical channels[END_REF] and Generalized homodyned K distribution [START_REF] Peppas | Free space optical communication with spatial modulation and coherent detection over H-K atmospheric turbulence channels[END_REF] provide excellent agreements with the probability density function (PDF) of irradiance for both simulated and experimented data in weak-to-strong turbulence regimes.

Energy and spectral eciencies

There are two important factors relative to the choice of a modulation scheme as energy eciency and spectral eciency. Energy eciency refers to the minimum bit error rate (BER) at a target data rate for a given transmit energy irrespectively of the occupied bandwidth. However, it does not take into account the implementation complexity. Spectral eciency, on the other hand, refers to the information transmission rate for a given bandwidth without taking the required transmit energy into account [2,[START_REF] Ghassemlooy | Mobile and Wireless Communications Network Layer and Circuit Level Design[END_REF].

Modulations suitable for IM/DD systems

On-o keying (OOK), a binary level modulation scheme, is the most commonly used IM technique due to its simple implementation. In OOK signaling, modulated data is represented by the presence (on) or absence (o ) of a light pulse in each symbol interval. At the receiver side and for an optimal signal detection, the instantaneous channel fading coecient should be estimated to perform dynamic thresholding [START_REF] Ghassemlooy | Mobile and Wireless Communications Network Layer and Circuit Level Design[END_REF]. In addition, OOK has relatively poor energy and spectral eciency [START_REF] Karout | Power ecient subcarrier modulation for intensity modulated channels[END_REF]. Another modulation technique is the pulse position modulation (PPM), which is an energy ecient modulation technique that does not require dynamic thresholds for optimal detection compared with OOK [START_REF] Wilson | Freespace optical mimo transmission with q-ary ppm[END_REF].

1. It is important to note that the intensity modulation and the amplitude modulation are not identical as the spectrum for intensity modulation contains harmonics of the modulation waveform [START_REF] Cox | Analog Optic Links : Theory and Practice[END_REF].

In comparison with PPM, multiple PPM (MPPM) is proposed in [START_REF] Sugiyama | MPPM : A method for improving the bandutilization ecency in optical PPM[END_REF] to achieve a higher spectral eciency at the expense of demodulation complexity. It is important to note that although optical communications have a large bandwidth and high spectral eciency modulation, it requires high-speed electronic circuits which are dicult to be designed and implemented. Under the constraint on peak transmiting power, MPPM outperforms PPM in signal-to-noise ratio (SNR) gain [START_REF] Wilson | Freespace optical mimo transmission with q-ary ppm[END_REF][START_REF] Xu | Coded PPM and multipulse PPM and iterative detection for Free-Space optical links[END_REF]. Compared with PPM, pulse width modulation (PWM) has higher spectral eciency at the expense of higher average power requirements [START_REF] Fan | Comparison of pulse position modulation and pulse width modulation for application in optical communications[END_REF].

Digital pulse interval modulation (DPIM) sends a pulse followed by a number of empty slots which depends on the information bits. To avoid the problem of consecutive on pulses, an additional guard slot is added to each symbol. The major drawback of DPIM is the inferior error performance compared with PPM and MPPM as they require symbol synchronization [START_REF] Ghassemlooy | Digital pulse interval modulation for optical communications[END_REF]. Other modulation techniques based on PWM and PPM are dierential PPM (DPPM) and pulse-position-pulse-width modulation (PPMPWM). In DPPM, the slots after the on pulse are discarded so every DPPM symbol ends with a pulse which decreases the complexity of the synchronization at the receiver. Discarding the empty slots improves the spectral eciency of the system. However, variable rate encoding and decoding have practical drawbacks [START_REF] Ghassemlooy | Optical Wireless Communications : System and Channel Modelling with MATLAB[END_REF][START_REF] Hemmati | Deep Space Optical Communications[END_REF]. In PPMPWM, the energy eciency and the spectral eciency are in mid-way between PWM and PPM [START_REF] Fan | Comparison of pulse position modulation and pulse width modulation for application in optical communications[END_REF].

Multiple pulse amplitude modulation (M -PAM) can be used in FSO to obtain high spectral eciency compared with OOK at the expense of energy eciency and system complexity as it requires a laser with variable emission intensity [START_REF] Hranilovic | Wireless Optical Communication Systems[END_REF].

Subcarrier intensity modulation

In subcarrier intensity modulation (SIM), the data is rst modulated onto a RF signal then the signal is transmitted using the IM of an optical source. The concept of the SIM is to transmit parallel data streams through a number of non-orthogonal overlapping subcarriers [START_REF] Hwang | SIM/SM-aided free-space optical communication with receiver diversity[END_REF]. Compared with coherent modulation at the same spectral eciency, SIM has cost eective implementation at the expense of energy eciency as a result of DC bias added to the signal to avoid negative amplitudes [2,[START_REF] Hwang | SIM/SM-aided free-space optical communication with receiver diversity[END_REF]. The DC bias problem can be decreased by using a variable bias for each symbol [START_REF] Popoola | Performance of sub-carrier modulated free-space optical communication link in negative exponential atmospheric turbulence environment[END_REF].

Polarization modulation

Polarization modulation (PM) encodes the information as dierent states of the polarization of the laser source by an external modulator. At the receiver, based on the extraction of stoke parameters of received light, the signal can be recovered. PM has high mitigation to atmospheric turbulence as the polarization states are less aected by atmospheric turbulence compared with amplitude and phase. Moreover, PM has a high immunity to the phase noise and non-linearity of lasers. This advantage makes it a perfect solution for long-range FSO applications. However, PM requires perfect transmitter-receiver polarization alignment [START_REF] Karbassian | Transceiver architecture for incoherent optical CDMA network based on polarization modulation[END_REF][START_REF] Zhao | Circle polarization shift keying with direct detection for free-space optical communication[END_REF][START_REF] Wang | Employing circle polarization shift keying in free space optical communication with gamma-gamma atmospheric turbulence channel[END_REF].

State of the Art 2.6.1.6 Coherent modulation Compared with IM/DD systems, coherent systems can send the information in the amplitude, phase, or polarization of the optical signal which increases the spectral efciency of the system. At the receiver, the signal is mixed with a local oscillator (LO) signal then the received signal is amplied. For this reason, coherent detection increases the receiver sensitivity and decreases the shot noise at the expense of the complexity of the system [START_REF] Belmonte | Capacity of coherent free-space optical links using diversity combining techniques[END_REF].

Commercial FSO systems are based on IM/DD due to its simplicity and low cost.

However, the great evolution of digital signal processing of high-speed electronic circuits leads to turn into coherent systems [START_REF] Kuschnerov | DSP for coherent single-carrier receivers[END_REF].

The coherent detection can be divided into heterodyne and homodyne detection.

In homodyne detection, the frequency of the LO is exactly the same as the incoming radiation which makes it very expensive. While in heterodyne detection, the incoming radiation, and the LO frequencies are dierent [START_REF] Ghassemlooy | Mobile and Wireless Communications Network Layer and Circuit Level Design[END_REF]. FSO link employing homodyne binary phase shift keying (BPSK) has been established between two of the Canary islands to transmit 5.625 Gbps over a distance of 142 km [START_REF] Lange | 142 km, 5.625 Gbps free-space optical link based on homodyne BPSK modulation[END_REF].

Finally, modulation techniques alone are not sucient in enhancing the perofmrance of FSO communication systems. Other techniques must be considered such as forward error correction, spatial diversity, aperture averaging, multi-hop relay systems and cooperative relay systems.

Forward error correction techniques

One useful mitigation tool for atmospheric turbulence is forward error correction (FEC). Here, we discuss some FEC codes used for FSO communication systems.

Reed-Solomon (RS) codes

The total code length of RS (n, V ) is dened by n and the message length is dened by V . In [START_REF] Yan | Improved performance of M-ary PPM freespace optical communication systems in atmospheric turbulence due to forward error correction[END_REF], the authors employed 256-ary PPM using dierent RS codes in the presence of atmospheric turbulence to keep the average bit error rate (ABER) below 10 -9 . The authors used RS codes due to its easily implemented decoder and its alphabet size is easily matched to M-ary PPM. The results are employed over a weak turbulence channel using PIN receivers and with a data rate of 1 Gbps. Results show that truncated RS (255, 223) and RS (255, 207) outperform the standard RS (255, 239).

M -ary PPM and RS (255, 127) are employed in the presence of ambient noise and moderate fog conditions [START_REF] Muhammad | Reed Solomon coded PPM for terristal FSO links[END_REF]. Results show a signicant mitigation for both challenges by increasing the modulation level of PPM. In [START_REF] Muhammad | Comparison of hard-decision and soft-decision channel coded M-ary PPM performance over free space optical links[END_REF], the authors have shown that increasing the modulation level of the M-ary PPM improves the power eciency of FSO system at the expense of bandwidth and synchronization complexity.

Turbo codes and low-density parity-check codes

In [START_REF] Muhammad | Comparison of hard-decision and soft-decision channel coded M-ary PPM performance over free space optical links[END_REF], the authors compared the performance of RS (255, 127) with dierent rates of Turbo codes. They found that RS coded PPM might still be the preferable choice for practical implementation due to the decoding complexity of Turbo codes. Other works have considered the use of low-density parity-check (LDPC) codes for FSO systems over atmospheric turbulence channels such as [START_REF] Djordjevic | LDPCcoded MIMO optical communication over the atmospheric turbulence channel[END_REF].

FEC drawbacks for FSO

Although FEC achieves good results in mitigating atmospheric turbulence, however, using them for high data rates are not practical. As for high data rates, the complexity of encoder and decoder and the delay will not be acceptable [START_REF] Pham | A survey of performance improvement methods for free-space optical communication systems[END_REF].

Aperture averaging

Aperture averaging is another simple solution to reduce the fading eect, which uses a relatively large aperture at the receiver side to average over intensity uctuations. However, the data rate of the system decreases as large parasitic capacitance is required [START_REF] Khalighi | Fading reduction by aperture averaging and spatial diversity in optical wireless systems[END_REF].

Spatial diversity

Spatial diversity is considered as a promising solution to combat atmospheric turbulences and maximize the data rate of the systems [START_REF] Abbas | ul Islam, Performance enhancement of multi-input multi-output (MIMO) system with diversity[END_REF][START_REF] Mesleh | Indoor MIMO optical wireless communication using spatial modulation[END_REF]. By using multiple apertures at the transmitter and/or the receiver sides, spatial diversity has the potential to mitigate atmospheric turbulence eects, enhance the performance of the FSO links and overcome the limitations on the transmit optical power. Also, the possibility of laser beam blockage by obstacles (e.g., birds or moving vehicles) is further reduced and longer distances can be covered in harsh weather conditions [START_REF] Mesleh | Indoor MIMO optical wireless communication using spatial modulation[END_REF][START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF][START_REF] Safari | Diversity gain for near-eld MISO atmospheric optical communications[END_REF][START_REF] Abaza | MIMO techniques for high data rate free space optical communication system in log-normal channel[END_REF].

State of the Art 2.6.4.1 Single-input multiple-output For single-input multiple-output (SIMO) systems, equal gain combing (EGC) has the same performance of the optimal maximal ratio combining (MRC) for weak turbulence channels [START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF]. Also, at moderate turbulence channels, EGC has only a lower SNR gain of 0.3 dB compared with MRC at ABER of 10 -5 while having the advantage of lower implementation complexity and simple analytical model. Moreover, the ABER expression of multiple-input multiple-output (MIMO) FSO system using EGC in LN channel is similar to that of single-input single-output (SISO) link with the dierence of dividing the variance of the LN channel by the number of sub-channels.

Multiple-input single-output

For the multiple-input single-output (MISO) systems, the conventional orthogonal space-time block codes (OSTBCs), as used in RF systems, should be modied to deal with IM/DD techniques because the output of the transmitter must be unipolar [START_REF] Simon | Alamouti-type space-time coding for free-space optical communication with direct detection[END_REF][START_REF] Premaratne | Orthogonal space-time block codes for freespace IM/DD optical links[END_REF]. To achieve this, the negative part of a modulated signal is represented by the 1's complement (i.e., bitwiseNot) of a positive signal [START_REF] Simon | Alamouti-type space-time coding for free-space optical communication with direct detection[END_REF]. However, it has been shown in [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF] that if we are not limited by practical implementation considerations such as time synchronization, repetition codes (RCs) outperform their counterpart OSTBCs using IM/DD in uncorrelated LN channels. In a RCs system, the same signal is transmitted simultaneously from available transmitters. Yet, coherent and dierential OSTBCs FSO systems are shown to outperform their counterpart RCs systems at the expense of laborious receivers [START_REF] Bayaki | Performance and design of coherent and dierential space-time coded FSO systems[END_REF]. Moreover, transmit laser selection (TLS) system, which requires channel knowledge at the transmitter and at the receiver, outperforms the ABER performance of RCs and OSTBCs at the expense of increases system complexity [START_REF] Garcia-Zambrana | Selection transmit diversity for FSO links over strong atmospheric turbulence channels[END_REF].

Multiple-input multiple-output

Spatial modulation (SM) MIMO system for indoor optical wireless communication has been studied in [START_REF] Mesleh | Indoor MIMO optical wireless communication using spatial modulation[END_REF]. In SM, a group of data bits modulate a symbol from certain constellation diagram and another group activates one of the transmitter beams. A line-of-sight (LOS) indoor wireless optical communication system using SM is proposed in [START_REF] Fath | Spatial modulation applied to optical wireless communications in indoor LOS environments[END_REF]. According to [START_REF] Fath | Spatial modulation applied to optical wireless communications in indoor LOS environments[END_REF], SM outperforms its counterpart RC system with M -PAM for spectral eciencies larger than 4 bits/s/Hz.

Another system is space shift keying (SSK), which is a special case of SM where only the index of the active transmitter is transmitted rather than the data symbols. A study conducted in [START_REF] Yang | Contrasting space-time schemes for MIMO FSO systems with non-coherent modulation[END_REF] shows that OOK RC outperforms OSTBCs, spatial multiplexing (SMux) and SSK for FSO communications using IM/DD over G-G channels. However, these results are given only for a spectral eciency of 1 bit/s/Hz. A detailed survey of SM and SSK systems are discussed in [START_REF] Renzo | Spatial modulation for generalized mimo : challenges, opportunities, and implementation[END_REF].

In [START_REF] Ozbilgin | Optical Spatial Pulse Position Amplitude Modulation over Atmospheric Turbulence Channels[END_REF], spatial pulse position amplitude modulation (SPPAM) is proposed for FSO links. It is shown that SPPAM outperforms SSK for spectral eciencies of 2 bits/s/Hz or less.

Suggested Solutions 35

Tight upper bounds are derived for the error probability for coherent SM FSO systems over Generalized H-K distribution [START_REF] Peppas | Free space optical communication with spatial modulation and coherent detection over H-K atmospheric turbulence channels[END_REF]. In [START_REF] Lee | Selection diversity-aided subcarrier intensity modulation/spatial modulation for free-space optical communication[END_REF], hybrid SIM and SM system are proposed and the results show that the proposed system outperforms SIM system at the spectral eciency of 4 bits/s/Hz using less number of receivers.

Correlation eects

In the case of uncorrelated channels, the spacing among transmitters is assumed to be larger than the fading correlation length. In addition, the total length of the transmitters must not exceed the capture zone of the receiver. Which is equal to θ f m, where θ f is the eld of view (FOV) of the receiver (in rad) and is the link distance (in m) [START_REF] Simon | Alamouti-type space-time coding for free-space optical communication with direct detection[END_REF]. However, the last assumption might be dicult in practice as available space for the transmitters may not be sucient for this requirement [START_REF] Zhu | Maximum-likelihood spatial-diversity reception on correlated turbulent free-space optical channels[END_REF]. A correlation model called the exponential model [START_REF] Karagiannidis | An ecient approach to multivariate Nakagami-m distribution using green's matrix approximation[END_REF], corresponds to the scenario of a multichannel transmission from linearly equispaced transmitters, can be considered to calculate the correlation [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF]. A simple and general method to approximate the PDF of correlated and uncorrelated LN channels using moment generating function (MGF) is derived in [START_REF] Mehta | Approximating the sum of correlated lognormal or lognormal-Rice random variables[END_REF].

A unied framework determining the spatial correlation for both transmit and receive units is presented in [START_REF] Ozbilgin | Inter-aperturecorrelationinMIMOfreespaceopticalsystems[END_REF]. This framework is applicable to dierent wave types, turbulence conditions and aperture size.

In [START_REF] Yang | Approximation to the sum of two correlated gamma-gamma variates and its applications in freespace optical communications[END_REF], for a SIMO system with two receive apertures, the sum of correlated G-G random variables is approximated by an α -µ distribution [START_REF] Yacoub | The α -µ distribution : a physical fading model for the Stacy distribution[END_REF] in order to evaluate the BER performance at the receiver. This idea is then generalized to the case of MIMO in [START_REF] Yang | Fading correlation and analytical performance evaluation of the space-diversity free-space optical communications system[END_REF]. The Padé's approximation method approximates a function in one variable [START_REF] Karagiannidis | Moments-based approach to the performance analysis of equal gain diversity in Nakagami-m fading[END_REF] and it is used in [START_REF] Yang | Iet communications, Performance analysis of space-diversity FSO systems over the correlated Gamma-Gamma fading channel using Pade approximation method[END_REF] to obtain the PDF of the sum of correlated G-G random variables from their MGF, which is then used to evaluate the system performance analytically.

However, due to the limitation of the Padé's approximation, this method cannot be used for very low BERs.

Finally, spatial diversity can mitigate turbulence eects, however, it can not mitigate the path loss eects. Other techniques such as multi-hop relay systems and cooperative relay systems must be considered for path loss eects.

Our contributions for spatial diversity are discussed in Chapter 3 and Chapter 4.

Multi-hop relay systems

To further mitigate turbulence and path losses eects, multi-hop system is used due to its advantage of shorter hops that yields signicant performance improvements.

Conventional multi-hop relay systems

In [START_REF] Akella | Error analysis of multihop free-space optical communication[END_REF], the authors introduce multi-hop relays for FSO. They showed that multihop relays are more ecient than single-hop at the same BER. However, their results only considered weather attenuation without atmospheric turbulence eects. The outage probability (OP) of multi-hop FSO systems using amplify-and-forward (AF) over strong State of the Art turbulent channels is investigated in [START_REF] Karagiannidis | Outage probability of relayed free space optical communication systems[END_REF]. In [START_REF] Tsiftsis | Multihop free-space optical communications over strong turbulence channels[END_REF], the OP of multi-hop FSO systems using AF or decode-and-forward (DF) over G-G channels is investigated and the ABER of dual-hop DF systems is derived.

The physical layer of cooperative relays (CRs) FSO systems are proposed in [START_REF] Safari | Relay-assisted free-space optical communication[END_REF].

The OP of both CRs and multi-hop systems using DF and AF systems are derived. In addition, atmospheric turbulence and path loss attenuation are considered. Moreover, the results of [START_REF] Safari | Relay-assisted free-space optical communication[END_REF] show that multi-hop systems outperform CRs, however, CRs can create an articial broadcasting. Asymptotical relative diversity order (ARDO) of DF FSO for multi-hop systems and CRs is derived in [102]. Results show that ARDO of multi-hop systems is greater than CRs.

In [103], the OP of DF multi-hop system considering weak turbulence, path loss, and misalignment error is derived. In [START_REF] Safari | Multi-hop relaying over the atmospheric poisson channel : outage analysis and optimization[END_REF], the OP of a multi-hop DF FSO system over a Poisson channel degraded by LN turbulence channel is calculated. The eect of dierent weather conditions on multi-hop FSO systems is studied [START_REF] Vavoulas | Weather eects on FSO network connectivity[END_REF]. By neglecting background radiation and dark currents, power allocation for multi-hop FSO system is discussed in [START_REF] Abou-Rjeily | Power allocation for quantum-limited multihop free-space optical communication systems[END_REF]. The ABER of multi-hop FSO systems over G-G and misalignment fading channels is derived in [START_REF] Dang | Average BER analysis of multihop FSO systems over strong turbulence and misalignment fading Channels[END_REF] and the OP of the same system is derived in [START_REF] Hu | Outage probability analysis of multi-hop free space optical communications over strong turbulence channels[END_REF].

All-optical multi-hop relay systems

All-optical FSO relaying is proposed in [START_REF] Kazemlou | All-optical multihop free-space optical communication systems[END_REF], without the need for optical-to-electrical and electrical-to-optical conversions required in conventional FSO relays. In all-optical FSO systems [START_REF] Kazemlou | All-optical multihop free-space optical communication systems[END_REF], DF systems outperform AF systems. Moreover, it is shown in [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF], that FSO systems with all-optical relays achieve a better performance compared with systems having conventional FSO relay.

All-optical AF FSO systems, using an erbium doped ber amplier (EDFA) combined with an optical hard-limiter (OHL) over G-G channels, are employed in [START_REF] Trinh | All-optical AF relaying FSO systems using EDFA combined with OHL over gamma-gamma channels[END_REF]. OHL is used to remove accumulated background noise. Then, closed-form ABER expression is derived for the system. Moreover, results show that using OHL leads to achieve the same performance of many relays with fewer relays.

Coherent multi-hop relay systems

Multi-hop coherent FSO systems are employed over turbulent channels. Moreover, the diversity and multiplexing gains are derived [START_REF] Aghajanzadeh | Multi-hop coherent free-space optical communications over atmospheric turbulence channels[END_REF]. The OP of employing APD receivers for CRs and multi-hop systems over LN channels using BPSK is derived in [START_REF] Pham | Performance analysis of multihop FSO systems using APD receivers over log-normal channels[END_REF] for DF relays and in [START_REF] Pham | Performance of APD-based amplify-and forward multihop FSO systems over turbulence channels[END_REF] for AF relays. In [START_REF] Peppas | Capacity analysis of dual amplify-and-forward relayed free-space optical communication systems over turbulence channels with pointing errors[END_REF], approximated closed-form, upper bound and asymptotic expressions for the capacity of dual-hop AF FSO communication systems over G-G channels with pointing errors are derived.

Closed-form expressions for capacity and average symbol error rate (SER) of Mary phase shift keying (MPSK) for dual-hop AF FSO systems over G-G channels are derived [START_REF] Aggarwal | A novel MGF-based approach to analyze SIM-based relayed FSO systems[END_REF]. In [START_REF] Tang | Multihop free-space optical communications over turbulence channels with pointing errors using heterodyne detection[END_REF], the OP and the average SER of multi-hop FSO communications over G-G channels with pointing errors, using heterodyne detection and dierential phase-shift keying (DPSK) modulation are derived.

Suggested Solutions 37

The ABER and the OP of a dual-hop DF relay using BPSK over G-G channels are derived in [START_REF] Aggarwal | Performance analysis of optical wireless communication system with a decode and forward relay[END_REF], using the series representation of MGF of G-G distribution which was originally proposed by [START_REF] Bayaki | Performance analysis of MIMO freespace optical systems in Gamma-Gamma fading[END_REF]. The lower bound OP for dual-hop AF coherent FSO systems is derived in [START_REF] Chae | Outage probability analysis of a coherent FSO amplifyand-forward relaying system[END_REF]. Moreover, the results show that the outage performance of AF is comparable with that of DF for coherent FSO systems. In [START_REF] Zedini | On the performance of multihop heterodyne FSO systems with pointing errors[END_REF], closed-form and asymptotic for ABER and ergodic capacity are derived for multi-hop AF heterodyne FSO system over atmospheric turbulence with pointing errors.

Subcarrier intensity modulation multi-hop relay systems

The ergodic capacity of SIM dual-hop FSO systems is derived over G-G channels with pointing error for DF systems [START_REF] Aggarwal | Ergodic capacity of SIM based DF relayed optical wireless communication systems[END_REF] and in [START_REF]Exact capacity of amplify-and-forward relayed optical wireless communication systems[END_REF] for variable and xed gains of AF systems.

Optimum placement for multi-hop relay systems

The optimum placement for multi-hop relays is determined being equidistance between the source and the destination while for CRs, it is found being closer to the source and it is also a function of link distance, number of relays and SNRs [START_REF] Kashani | Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems[END_REF]. The optimum placement for multi-hop relays is determined in the presence of obstacles blocking the direct link [START_REF] Zhu | Optimal FSO relay nodes placement with link obstacles and infeasible regions[END_REF].

Photon counting receiver for multi-hop relay systems

Multi-hop FSO system using count-and-forward (CF) relay with photon counting receiver (PCR), that counts the number of received photons, is proposed in [START_REF] Gong | Non-line of sight optical wireless relaying with the photon counting receiver : a count-and-forward protocol[END_REF]. The relay employs OOK and uses discrete power levels corresponding to the number of received photons. Results show that using CF with PCR outperforms the performance of the hard-decision forwarding scheme.

Beam size variation at the receiver for multi-hop relay systems

The eect of beam size variation at the receiver due to turbulence is calculated for multi-hop FSO systems using a spatially coherent Gaussian beam model [START_REF] Pham | Eects of atmospheric turbulence and misalignment fading on performance of serial-relaying M-ary pulse-position modulation free-space optical systems with partially coherent Gaussian beam[END_REF].

Hybrid spatial diversity and multi-hop relay systems

The OP of SIMO multi-hop systems using EGC is proposed over correlated G-G channels [START_REF] Pham | Optical wireless communications over fading channels : spatial diversity or multihop relaying[END_REF]. Both AF and DF relays are considered for the proposed system. Results

show that employing fewer relays with SIMO have higher SNR gain than employing many relays without SIMO. Extended results are proposed in [START_REF] Abou-Rjeily | Serial relaying over gamma-gamma MIMO FSO links : diversity order and aperture allocation[END_REF] for MIMO multihop systems, where the OP, the diversity order and power allocation of DF relays are considered over G-G channels.

State of the Art 2.6.5.9 Hybrid FSO and RF links for multi-hop relay systems Mixed dual-hop AF systems using RF link in the rst hop represented by Nakagamim channel and using FSO link represented by G-G channel with pointing error is proposed in [START_REF] Anees | Performance of an amplify-and-forward dualhop asymmetric RF-FSO communication system[END_REF], where SIM is used for FSO transmission. Moreover, CDF, PDF, MGF, OP, ABER and the capacity are derived for the proposed dual-hop AF RF-FSO communication system. The same system of dual-hop AF RF-FSO is employed in [START_REF] Zedini | On the performance of hybrid line of sight RF and RF-FSO xed gain dual-hop transmission systems[END_REF] with an extra direct RF link, where both IM/DD and coherent modulation techniques are employed, and MRC and selection combining (SC) are employed at the destination side.

In this subsection, the superiority of multi-hop relay systems in mitigating the effects of turbulence and path loss eects are discussed. Our contributions for multi-hop systems are discussed in Chapter 5.

Cooperative relay systems

Cooperative relays systems as a powerful mitigation tool for the atmospheric turbulence and path loss eects are discussed in this subsection. Moreover, articial broadcasting can be obtained using cooperative relays [START_REF] Safari | Relay-assisted free-space optical communication[END_REF]. In cooperative relays, perfect synchronization is required to ensure that all the signals arrive simultaneously at the destinations. In practice, perfect synchronization is a cumbersome task especially for a large number of relays [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF].

Transmission relays

CRs can be classied into all-active transmission and selective transmission systems.

In all-active transmission, relays forward received signals to the destination [START_REF] Safari | Relay-assisted free-space optical communication[END_REF]. However, in selective transmission, a relay only forward the received message based on the max-min SNR [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF]. The selective transmission is considered an asynchronous low complexity system and outperforms the performance of the OP of all-active transmission.

The selection process, based on the max-min SNR, is achieved according to [START_REF] Cui | Relay selection for two-way full duplex relay networks with amplify-and-forward protocol[END_REF]:

j = max i {1, • • • , N re } (min(h 2 SR i , h 2 R i D )), (2.1)
where N re is the number of relays, j is the index of the best relay, i is a relay index, h SR i and h R i D are the channel fading coecients between the source and ith relay, R i , and between R i and the destination, respectively.

Types of DF cooperative relay systems

In [START_REF] Karimi | BER analysis of cooperative systems in freespace optical networks[END_REF], three CRs for FSO communication systems using direct link are discussed as bit detect and forward (BDF), adaptive bit detect and forward (ABDF) and adaptive decode and forward (ADF). At BDF, the relay forwards the received bit regardless its correctness. However, at ABDF both the relay and the direct link re-forward the received bit with low SNR. While in ADF, the relay corrects the false bit then re-forward it. If the relay fails to correct the bit, ADF will behave like ABDF. Results show that the proposed systems outperform both the direct link and MISO systems.

In the absence of background noise, BDF has the same performance of ABDF. While, in the presence of background noise, ABDF has more SNR gain compared to BDF [START_REF] Abou-Rjeily | Communications in Computer and Information Science[END_REF].

Moreover, results show that using a single feedback bit of RF has signicant SNR gain especially in low SNRs.

In [135], the author showed that the diversity order for N re relay system using BDF and ABDF is equal to Nre 2 + 1 and N re + 1, respectively. Where the function a , rounds a to the smallest integer greater than a. A novel CRs FSO scheme, called, improved adaptive decode-and-forward (IADF) is proposed [START_REF] Zhu | A distance-dependent free-space optical cooperative communication system[END_REF]. The proposed system transmits two frames into four time slots. In the rst time slot, the source transmits the message to the relay. In the second time slot, the relay transmits the message to the destination. The same thing happens for the second frame in the third and fourth time slots. However, if the relay has failed to receive the message correctly in any case, the source performs an XOR operation on the rst frame with the second frame and sends the resultant directly to the destination. In addition, the diversity order is derived for the proposed system as a function of the propagation distance.

Conventional cooperative relay systems

CRs for DF FSO communication system using a single relay IM/DD in the absence of background noise is employed. The considered system achieve the same ABER compared with direct and MISO systems, that suers from correlation at low SNR [START_REF] Abou-Rjeily | Cooperative diversity for free-space optical communications : transceiver design and performance analysis[END_REF]. Closedform expressions of the outage performance for multi-hop and CRs FSO systems over M-distribution with nonzero boresight pointing errors are derived [START_REF] Wang | Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors[END_REF]. The boresight is the displacement between the center of the detector and the center of the received beam.

Asymptotic expressions for OP are derived for DF IM/DD CRs FSO system using repetition coding over G-G channels with pointing error and weather attenuation effects [START_REF] Castillo-Vazquez | Outage performance of DF relay-assisted FSO communications using time-diversity[END_REF]. Where the destination receives two copies of the message over three time slots. The considered system outperforms the OP of both MISO and conventional CRs systems.

All-optical cooperative relay systems

All-optical CRs systems employing AF relays using PCR over LN channels with path loss eects are considered [START_REF] Karimi | Free space optical communications via optical amplify-and-forward relaying[END_REF]. The proposed system has less SNR at the same ABER as compared with the multiple-transmitter system especially for long distances and/or high attenuation factors.

All-optical CRs systems using PCR over LN channels with path loss eects are considered for full channel state information (CSI) and semi-blind CSI [START_REF] Wang | Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors[END_REF]. However, both cases have comparable performance, semi-blind CSI has lower complexity. The OP of all-optical relay selection FSO systems using PCR over G-G turbulent channels is derived for both full CSI and semi-blind relays [START_REF] Mostaani | Relay selection in FSO systems with all-optical relaying over Gamma-Gamma turbulence channels[END_REF]. All-optical relay selection FSO systems have SNR gain up to 11.5 dB compared with all-active relays for a system using four relays.

State of the Art 2.6.6.5 Subcarrier intensity modulation cooperative relay systems CRs for SIM FSO communication systems using DF relays and dierential modulation are employed over G-G channels [START_REF] Bhatnagar | Average BER analysis of dierential modulation in DF cooperative communication system over gamma-gamma fading FSO links[END_REF]. Moreover, the ABER, the diversity order and coding gain are derived. The OP of CRs BPSK-SIM for FSO systems over G-G turbulent channels with pointing errors is derived [START_REF] Prabu | Outage analysis of relay-assisted BPSK-SIM based FSO systems over strong atmospheric turbulence with pointing errors[END_REF].

Selected relay for cooperative relay systems

In [START_REF] Celik | On relay selection for cooperative free-space optical communication[END_REF], the OP of relay selection for FSO systems using DF relays over LN channels with path loss eects is discussed. At OP of 10 -5 , a selected relay depend on the SNR of all the links, has less power by 6 dB than a selected relay depend on the SNR of relay to destination (R-D). DF CRs in a FSO IM/DD communications system is employed over G-G channels and MRC is used at the receiver [START_REF] Aghajanzadeh | Outage performance and DMT analysis of DF parallel relaying in FSO IM/DD communications[END_REF]. Then the OP, diversity and coding gain are derived.

In [START_REF] He | Opportunistic cooperation for FSO links aided by decode and forward relay[END_REF], opportunistic CRs for FSO links are proposed, where the relay participate only if the source to relay (S-R) link is strong enough. In the case that S-R and R-D links has the same performance, opportunistic CRs has the same OP of conventional CRs.

While if S-R link is weak, opportunistic CRs has better OP compared with conventional CRs.

The ABER for CRs for FSO communication systems using selective transmission is obtained for LN and Rayleigh channels [START_REF] Abou-Rjeily | Performance analysis of selective relaying in cooperative freespace optical systems[END_REF]. Partial relay section according to the best S-R path is employed and an approximated ABER of G-G channels is derived [START_REF] Bhatnagar | Average BER analysis of relay selection based decode-andforward cooperative communication over gamma-gamma fading FSO links[END_REF].

The ABER and the diversity order of relaying selection of DF CRs FSO over G-G channels with pointing errors are derived [START_REF] Boluda-Ruiz | Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels[END_REF]. The ABER performance of a single relay selection according to max-min criterion of SNR for dual-hop parallel DF FSO over G-G channels employing adaptive subcarrier quadrature amplitude modulation is obtained using a power series approximation method [START_REF] Vu | Selection decode-and-forward cooperative FSO systems with adaptive rate strategy over gamma-gamma fading chan-References 125 nels[END_REF].

Closed-form ABER of selective CRs for DF FSO system using SIM for BPSK over generalized M-distributed turbulent channel is derived [START_REF] Sharma | On the performance of selective transmission based DF FSO system with generalized M-distributed turbulence[END_REF]. Moreover, the diversity order for various distributions, as a special case of generalized M-distributed turbulent channel, is derived. The ABER and the OP of selective transmission for multi-hop parallel DF relaying using BPSK are derived for EW fading channels using the Gauss-Laguerre quadrature rule [START_REF] Wang | Performance analysis of multi-hop parallel free-space optical systems over exponentiated Weibull fading channels[END_REF].

Power allocation for cooperative relay systems

Power allocation for quantum-limited CRs FSO systems is discussed in [START_REF] Abou-Rjeily | Cooperative FSO systems : performance analysis and optimal power allocation[END_REF]. In the absence of CSI, the ABER performance is better if the power is distributed among the links. While in the presence of CSI, the ABER performance can be signicantly enhanced by allocating all the power to the strongest link.

2.6.6.8 Multiple-source cooperation relaying Multiple-source cooperation relaying for FSO systems over strong turbulence is proposed [START_REF] Zhou | Network-coded multiple-source cooperation aided relaying for free-space optical transmission[END_REF]. Where many users send their information to a single relay then this relay forwards all the messages to the nal destination. Compared with conventional CRs, the proposed system requires less equipment cost.

Optimum placement for cooperative relay systems

The optimum placement for CRs is found being closer to the source and function of link distance, the number of relays and SNRs [START_REF] Kashani | Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems[END_REF]. Asymptotic ABER of CR for FSO communication system using EGC over G-G channels with pointing error is derived [START_REF] Garcia-Zambrana | Bit detect and forward relaying for FSO links using equal gain combining over gamma-gamma atmospheric turbulence channels with pointing errors[END_REF]. This also results in that the diversity order is independent on pointing error in some cases, however, the diversity order strongly depends on the placement of the relay.

The ABER and the diversity order of relay placement in ADF CRs for FSO systems over G-G channels with pointing errors are derived [START_REF] Boluda-Ruiz | Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications[END_REF]. Unlike [START_REF] Kashani | Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems[END_REF], in the presence of pointing errors, the authors [START_REF] Boluda-Ruiz | Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications[END_REF] show that the diversity order is better when the relay nodes are closer to the destination node. Moreover, the authors demonstrate that ADF system based on S-R selection always outperforms ADF system based on R-D regardless the position and number of relays.

Multi-hop parallel relaying

In [START_REF] Kashani | Outage performance and diversity gain analysis of free-space optical multi-hop parallel relaying[END_REF], OP and diversity gain of multi-hop parallel relaying over LN channels are derived. Multi-hop parallel relaying has less power compared with multi-hop and parallel relaying at the same OP. Multi-hop parallel relaying, where only one selected participating path (i.e. the strongest channel gain) is discussed [START_REF] Zhalehpour | Outage performance analysis of mth best path selection protocol in FSO communications[END_REF]. In addition, the second and the third strongest paths are considered in a case where the rst strongest path is not available due to scheduling and load balancing issues. This results of the OP of the second and the third strongest paths outperforms multi-hop system using the same number of relays per path.

Mean time between failures for cooperative relay systems

In [START_REF] Dubey | Bit error rate and reliability analysis of cooperative communication in free-space optical systems[END_REF], results of mean time between failures (MTBF) was 12 years for a SISO FSO system using low data rate RF back up, while MTBF was 20 years for CRs FSO system using low data rate RF back up.

Inter-relay cooperation of cooperative relay systems

The OP and the diversity of inter-relay cooperation for all-active and selection CRs FSO systems are introduced and derived [START_REF] Abou-Rjeily | All-active and selective FSO relaying : do we need inter-relay cooperation[END_REF]. The inter-relay cooperation has less power compared with conventional CRs for certain relay positions. Moreover, power allocation and BER of inter-relay cooperation are derived in [START_REF] Abou-Rjeily | Inter-relay cooperation : a new paradigm for enhanced relay-assisted FSO communications[END_REF].

State of the Art 2.6.6.13 Two-way relay systems Two-way relays (TWRs) using DF for FSO systems is employed in [START_REF] Tang | Performance analysis of a two-way network-coded free space optical relay scheme over strong turbulence channels[END_REF] over G-G channels, where two users communicate using three time slots. Thus achieving higher data rate, TWR using two time slots to communicate is investigated in [START_REF] Sharma | Bi-directional decode-XOR-forward relaying over Mdistributed free space optical links[END_REF][START_REF] Puri | Analysis of spectrally ecient two-way relay assisted free space optical systems in atmospheric turbulence with path loss[END_REF]. In the rst time slot, the two sources send their information to the relay. The relay decodes the received signals and performs an XOR operation on the decoded signals. The resultant signal is then forwarded to each user in the second time slot. The user can obtain its intended received signal by performing an XOR operation with the received message.

Moreover, the SER and the OP are derived over M-channel [START_REF] Sharma | Bi-directional decode-XOR-forward relaying over Mdistributed free space optical links[END_REF] and LN channel [START_REF] Puri | Analysis of spectrally ecient two-way relay assisted free space optical systems in atmospheric turbulence with path loss[END_REF].

In [START_REF] Sharma | Relay assisted bi-directional communication in generalized turbulence fading[END_REF], TWR DF FSO is employed over M-distribution FSO channels, in addition the ABER and the OP are derived in the presence of pointing errors. Multi-user TWR is proposed in [START_REF] Puri | Multiple user pair scheduling in TWR-FSO systems in presence of building sway[END_REF], where the received signal is aected by atmospheric turbulence, path loss and building sway. Partial relay selection for TWR coherent FSO using AF is employed over G-G channels and aected by path losses and pointing errors [START_REF] Puri | Asymptotic analysis of TWR assisted FSO links with partial dual-relay selection[END_REF].

Our contributions for cooperative relay systems are given in Chapter 6. The techniques discussed in this section, make FSO communication systems reliable means of providing wireless communications.

Practical Results

In this section, we selected some practical results for FSO communication systems:

1. FSONA, one of the leading FSO companies, provides Crédit Agricole French bank with FSO links of 10 Gbps. Four 2.5 Gbps links were deployed instead of a ber optic link for their new building in Paris. This link provides service for more than 1600 employees [START_REF]FSONA Provides Crédit Agricole With Innovative 10 Gigabit FSO Link[END_REF].

2. FSONA provides a mobile backhaul connectivity for the fourth generation longterm evolution (LTE) services to customers in Lebanon of 1.25 Gbps without the delay of a laying cable [START_REF]FSONA Networks And Touch Team Up To Deliver 4G LTE Services To Mobile Customers In Lebanon[END_REF]. 4. In [START_REF] Libich | Experimental verication of an all-optical dual-hop 10 Gbit/s free-space optics link under turbulence regimes[END_REF], experimental results are employed for an all-optical dual-hop FSO system and 10 Gbps is achieved for a link of 500 m.

According to these encouraging results, we suggests the market of FSO communication systems with will be increased in the coming years.

Summary

In this chapter, a review of the state of the art for FSO communication systems has been discussed. We can conclude that spatial diversity and relay-assisted FSO communication systems are potential candidates for the next-generation communications networks ; as spatial diversity combats atmospheric turbulence and maximizes the data rate of the systems. In addition, relay-assisted systems mitigate turbulence and path losses eects due to its advantage of shorter hops that yields signicant performance improvements. In Chapter 3, spatial diversity is considered and the correlation eects among transmitters are discussed for correlated LN channels for RCs and OSTBCs. Introduction

Performance analysis of free-space optical (FSO) communication systems assuming uncorrelated channel conditions have gained signicant attention in the literature [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF].

This assumption, however, requires sucient spacing among transmitters and can not be satised in some applications. Thereby, this chapter considers correlated log-normal Table 3.1 summarizes the contribution of the most relevant works in the literature and our work by specifying the considered channel models. Figure 3.1 shows the main three axes considered in the thesis to mitigate various challenges facing FSO. In this chapter, the obtained contribution of the rst axis, multiple-input multiple-output (MIMO), is considered where a special case of MIMO, when the receiver has a single aperture, called multiple-input single-output (MISO) is employed in this chapter.

System Model

The synoptic diagram of the considered system model is depicted in Figure 3.2. The investigated system has N t transmitters and a single receiver. On-o keying (OOK) pulsed modulation is employed to modulate the source bits then a MISO encoder is used. In the RCs system, the encoder transmits the same signal simultaneously from the N t transmitters. In OSTBCs system, the encoder considers full rate real orthogonal [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF]. In [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF], a linear transformation of G matrix to OOK matrix, S ∈ C p×Nt , through S = U+G 2 is proposed, where U denotes a matrix of ones and p is the number of transmitted symbols.

The encoded real symbols modulate the synchronized laser diodes. The transmitted light is aected by a correlated LN channel and weather attenuation eects. Moreover, at the receiver side an additive white Gaussian noise (AWGN) is added. For the OSTBC system, the received signal vector, r ∈ C p×1 , is given by r = η S I + n,

(3.1)
where η, S ∈ C p×Nt , I ∈ C Nt×1 and n ∈ C p×1 are the optical-to-electrical conversion coecient, the codeword matrix, the light intensity vector and AWGN vector, respectively.

The received signal for RC system is given by where s(t) denotes the transmitted symbols ∈ {0, 1} and n(t) is an AWGN with zero mean and variance σ 2

r(t) = η Nt i=1 s(t)I i + n(t),
n . The shot noise, caused by background light, is assumed to dominate other noise sources such as thermal and dark noises [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF]. I i denotes the received signal light intensity from the i th transmitter to the receiver after channel eect which can be evaluated according to [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF] as follows

I i = β I h i , (3.3)
with I being the received signal light intensity without considering the channel eect, h i = e 2x i is the channel irradiance from transmitter i to the receiver with x i being modeled as spatially correlated identically distributed (c.i.d.) Gaussian random variables (RVs) with mean µ x and variance σ 2

x .

The path loss is modeled as [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF] β = 10 -α /10 ,

(3.4)
where α is the weather-dependent attenuation coecient (in dB/km) and is the link distance (in km). The values of α for dierent weather conditions are provided in Table 2.1.

Hence, h i is a LN RVs with PDF given by [START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF]

f h i (h i ) = 1 h i 8πσ 2 x × exp - (ln(h i ) -2µ x ) 2 8σ 2 x . (3.5)
To ensure that the fading channel does not attenuate or amplify the average power, the fading coecients are normalized as E[h i ] = e 2(µx+σ 2 x ) = 1. where E[.] is the mean.

At the receiver side, for RCs maximum-likelihood (ML) decoder is considered to decode the received signals. However, for OSTBCs the transpose of I vector must be multiplied by the received signal (rI T ) before using ML decoder [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF].

Performance Analysis

Orthogonal Space-Time Block Codes

The PDF of the instantaneous SNR γ i , 1 ≤ i ≤ N t , of correlated jointly LN channels under path loss eect can be calculated as [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF]

f γ i (γ 1 , • • • , γ Nt ) = exp -1 32 Z (Γ) -1 Z T 4 Nt (2π) Nt (det[Γ])Π Nt i=1 γ i , (3.6)
where

Z = ln γ 1 β 2 γ 1 • • • ln γ N t β 2 γ N t -4µ
x , γ i is the average SNR, (•) -1 is the inverse of a matrix, (•) T is the transpose of a matrix and det [.] is the determinant of a matrix. The spatial covariance matrix Γ coecients, modeling the spatial correlation among transmitters, are given by [START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF] Γ

ij = σ 2 x × ρ |i-j| , (3.7)
where |.| stands for the absolute value, (ρ ≤ 1) is the correlation coecient and i and j are the row and the column indices of the covariance matrix coecients, respectively. According to [START_REF] Zhu | Maximum-likelihood spatial-diversity reception on correlated turbulent free-space optical channels[END_REF], ρ is a function of the separation distance, d, between each two successive transmitters,

ρ = exp - d d o 5/3 , (3.8)
where the correlation length d o ≈ √ λ with λ being the wavelength. This correlation model is called the exponential model [START_REF] Karagiannidis | An ecient approach to multivariate Nakagami-m distribution using green's matrix approximation[END_REF] and corresponds to the scenario of a multichannel transmission from linearly equispaced transmitters. From (3.8), the correlation of combined signals decays by increasing the spacing between the transmitters.

The instantaneous SNR γ i in Equation (3.6) is given by [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF],

γ i = (ηI i ) 2 N o . (3.9)
where N o is the noise spectral density. The conditional bit error rate (BER) for FSO system using OSTBCs can be calculated as [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF] Pr(e|h

) = Q   E b β 2 2N o Nt i=1 h 2 i   (3.10)
The Gaussian-Q function is dened as [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF] Q

(y) = 1 √ 2π ∞ y exp - z 2 2 dz (3.11) = 1 π π 2 0 exp - y 2 2 sin 2 θ dθ.
The average electrical energy E b of the transmitted pulse at each transmitter is given by [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF].

E b = η 2 I 2 N 2 t .
(3.12)

To guarantee that the total power of the MISO system is equal to the power of a single-input single-output (SISO) system, N 2 t is added to Equation (3.12) 1 .

1. In FSO, the energy of the symbols is proportional to the square of the optical power [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF].

Performance Analysis 49 Substituting Equation (3.12) into Equation (3.10) and using Equation (3.3) yields

Pr(e|h) = Q   η 2 2N o (N t ) 2 Nt i=1 I 2 i   . (3.13)
Using Equation (3.9), the conditional BER with respect to the SNR becomes

Pr(e|h) = Q   Nt i=1 γ i 2(N t ) 2   . (3.14)
Using the PDF of the instantaneous SNR γ i in Equation (3.6), the ABER of OSTBCs is given by

ABER = ∞ 0 • • • ∞ 0 Q   Nt i=1 γ i 2(N t ) 2   × f γ i (γ 1 , • • • , γ Nt ) dγ 1 • • • dγ Nt . (3.15)
According to [START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF], the MGF is dened by

Ψ γ (-s) = ∞ 0 • • • ∞ 0 f γ i (γ 1 , • • • , γ Nt ) × exp -s Nt i=1 γ i dγ 1 • • • dγ Nt , s ∈ R, (3.16)
With the use of Equations (3.3), (3.5) and (3.9), and after few algebraic manipulations, the PDF of instantaneous branch SNR for the given branch i ,γ i , becomes

f γ i (γ i ) = 1 32πσ 2 x γ i exp   - ln γ i β 2 γ i + 4σ 2 x 2 32σ 2 x    (3.17)
The MGF of the RVs γ i is then given by

Ψ γ (-s) = ∞ 0 exp(-sγ i ) 1 32πσ 2 x γ i × exp   - ln γ i β 2 γ i + 4σ 2 x 2 32σ 2 x    dγ i . (3.18) Making the change of variable z =   ln γ i β 2 γ i +4σ 2 x √ 32σ 2 x   yields Ψ γ (-s) = 1 √ π ∞ -∞ exp(-z 2 ) × exp -sβ 2 γ i exp 32σ 2 x z -4σ 2 x dz. ( 3
∞ -∞ exp(-z 2 )g(z)dz ≈ N i=1 w i g(z i ), (3.20)
where z i and w i are the roots and the weights of the Hermite's polynomial, respectively and N is the order of the approximation. It was shown in [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF] that accurate results for LN channels can be obtained if N ≥ 20.

Applying Equation (3.20) on Equation (3.19) yields

Ψ γ (-s) ≈ 1 √ π N i=1 w i exp -sβ 2 γ i exp( 32σ 2 x z i -4σ 2 x ) (3.21)
In [START_REF] Mehta | Approximating the sum of correlated lognormal or lognormal-Rice random variables[END_REF], the authors found that N = 12 is sucient for the approximation of Equation (3.21).

Using Equation (3.21) and substitute σ x by ćij which is the (i, j) th coecient of Γ sq = Γ 1/2 , the general form of the MGF in correlated LN channel under path loss eect becomes

Ψ γ (-s) ≈ N n 1 =1 • • • N n N t =1 Nt i=1 w n i √ π ×exp   -s Nt i=1 β 2 γ i   exp   √ 32 Nt j=1 ćij z n j -4σ 2 x       , (3.22)
The values of w n i and z nj of the N th order Hermite's polynomial are tabulated in [174, Table 25.10 ].

Using the denition of the Q-function in Equation (3.11) on Equation (3.15), the ABER of OSTBCs system over correlated LN channel under path loss eect can be written as

ABER = 1 π ∞ 0 • • • ∞ 0 π 2 0 exp - Nt i=1 γ i 4N 2 t sin 2 θ × f γ i (γ 1 , • • • , γ Nt ) dθ dγ 1 • • • dγ Nt . (3.23)
The ABER in Equation (3.23) can be written using the MGF approach in Equation (3.16) as

ABER = 1 π π 2 0 Ψ γ - 1 4N 2 t sin 2 θ dθ. (3.24)
Using Equation (3.22) and assuming that all the average SNR from the transmitters are equal, i.e., γ i = γ, ∀i = 1, • • • , N t , an approximate novel expression of the ABER of OSTBCs over correlated LN channel under path loss eect is derived as

ABER ≈ N n 1 =1 • • • N n N t =1 Nt i=1 w n i √ π × Q    β 2 γ 2N 2 t Nt i=1   exp   √ 32 Nt j=1 ćij z n j -4σ 2 x        .
(3.25)

Repetition Codes

In the case of RCs, the conditional BER of MISO FSO system is given by [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF] Pr(e|h

) = Q   E b β 2 2N o Nt i=1 h i   . ( 3
ABER = 1 π ∞ 0 • • • ∞ 0 π 2 0 exp   - Nt i=1 √ γ i 2 4N 2 t sin 2 θ   × f γ i (γ 1 , • • • , γ Nt ) dθ dγ 1 • • • dγ Nt . (3.27)
It is important to note that the MGF approach cannot be used to represent Equation (3.27), since it is directly proportional to ( √ γ i ) 2 . However, under the condition that σ 2

x 1, the following approximation can be used

E 1 = Nt i=1 √ γ i 2 ≈ E 2 = N t Nt i=1 γ i . (3.28)
Extensive Monte Carlo simulations of 10 6 realizations of the random variables were generated to verify the accuracy of our approximation in Equation (3.28). The relative estimation error percent, δ = 100 Please note that the maximum value of σ x for FSO LN channels corresponds to the maximum scintillation index (SI) value for LN channels, which is ≤ 0.75 [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF]. The relation between SI and σ x is dened in [START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF] as σ x = √ ln(SI+1) 2

× E 1 -E 2 E 1 < 10%, is calculated for dierent σ x
. Hence, σ x should be ≤ 0.374.

ABER ≈ N n 1 =1 • • • N n N t =1 Nt i=1 w n i √ π × Q    β 2 γ 2N t Nt i=1   exp   √ 32 Nt j=1 ćij z n j -4σ 2 x        . (3.30)
It is important to note here that Equations (3.25) and (3.30) can also applied for uncorrelated LN channels by setting ρ ≈ 0 in ćij .

Numerical Results and Discussions

In this section, a typically ABER for IM/DD FSO system using OOK of 10 -9 is targeted, λ = 1550 nm and = 1 km. The transmitters branches are assumed to be Diversity Techniques for Correlated Log-normal Channels In Figure 3.3, the relation between the correlation eect and the separation distance among the transmit units is studied. The selected wavelength is λ = 1550 nm and different propagating distances are considered. Simulation results show that if a correlation coecient ρ ≤ 10 -3 is to be maintained, the distance among the transmit units should be at least 3 times the correlation length.

In Figure 3.5, a correlated LN channel with σ x = 0.1 and N t = 2 is employed. It is revealed that RCs are more power ecient than OSTBCs by about 3 dB for the two considered cases, ρ = 0.25 and 0.5. In Figure 3.6, a light fog environment with σ x = 0.1 and α = 20 dB/km is considered. These parameters are chosen considering the inversely proportional relationship between the turbulence strength and attenuation. It is very unlikely that strong turbulence occurs during a fog event [START_REF] Bushuev | Analysis of the performance of a wireless optical multiinput to multi-output communication system[END_REF]. The light fog eects lead to a severe eect on the performance of SNR gain as depicted in Figure 3.6.

In Figures 3.7 and 3.8, results demonstrate that RCs with N t = 3 has more SNR gain by 0.6 dB and 0.5 dB for ρ = 0.25 and 0.5 as compared with RCs with N t = 2, respectively. On the other hand, increasing the number of transmitters from two to three, degrades the performance of OSTBCs by 1 dB and 1.3 dB for 0.25 and 0.5, respectively.

For LN channels with small σ x , the LN channels behave like Gaussian channels. Moreover, in Gaussian channels there is no diversity gain due to the use of multiple transmitter. This is the reason of OSTBCs with N t = 2, has better SNR gain than OSTBCs with N t = 3.

In Figure 3.9, moderate turbulence represented by LN channel of σ x = 0.374 with N t = 2 is considered. This, in turn, results in degradation in the ABER performance of FSO system severely. The required SNR to achieve the target ABER increases by 18.9 dB and 21 dB for RCs and by 19.3 dB and 21.5 dB for OSTBCs codes with ρ = 0.25 and 0.5, respectively compared to σ x = 0.1 with N t = 2 and α = 0 dB. This also results in RCs outperform OSTBCs by about 3 dB.

In Figure 3.10, a clear air environment of a LN channel σ x = 0.3 and α = 0.43 dB/km with N t = 2 is considered. It can be seen that light fog eect in Figure 3.6 is higher than increasing the atmospheric turbulence on the ABER performance of the In Figure 3.13, the severe eect of dierent weather conditions on the FSO links using three transmitters are investigated using repetition codes with α = 0.1 dB/km and ρ = 0.25. Finally, the eect of increasing correlation on the ABER performance is depicted on Figure 3.14 using RCs for two transmitters with σ x = 0.3 and by considering clear air condition.

Summary

In this chapter, MISO FSO systems with OOK modulation using IM/DD are considered. The eects of both correlated LN channels with dierent path loss eects are also considered. The PDF of correlated LN channels is expressed using the MGF approach.

Based on the MGF approach, approximated ABER expressions for both OSTBCs and links are studied where it is shown that fog degrades the FSO performance signicantly.

In Chapter 4, a high spectral ecient FSO link based on space shift keying (SSK) system is presented and compared with the state of the art of RC using multiple pulse position modulation.

Performance Analysis of Space Shift Keying

Over Turbulent Channels

Introduction

Enhancing the spectral eciency is the ultimate goal of all wireless techniques to cope up with the ever increasing demand of high data rate and better quality of service.

To obtain high spectral eciency, space shift keying (SSK) is considered in this chapter.

We investigated the average bit error rate (ABER) performance of free-space optical (FSO) communication links for SSK over log-normal (LN) and negative exponential (NE) atmospheric turbulence channels.

SSK is compared with the state-of-the-art repetition codes (RCs) and a single-input single-output (SISO) systems using multiple pulse amplitude modulations (M -PAM).

Simulation results show that the signal-to-noise ratio (SNR) gain of SSK largely increases with greater spectral eciencies or higher turbulence eects. It is so dicult to use a probability density function (PDF) approach to obtain ABER expressions. Hence, a moment generation function (MGF) approach is used and an exact MGF for NE channel and an approximate MGF for LN channels are derived. Based on MGFs, a tight bounds for ABER expressions are derived. Finally, extensive Monte Carlo simulations have been run to validate the analytical analysis.

Table 4.1 summarizes the contribution of the most relevant works in the literature and our work by specifying the considered channel models. Figure 4.1 shows the main three axes considered in the thesis to mitigate various challenges facing FSO. The achieved contribution of the second axis, spatial modulation, is considered where a special case of spatial modulation called space shift keying is employed in this chapter.

System Model

A system with SSK modulation consisting of N t transmitters and N r receivers is illustrated in Figure 4.2. In our system, log 2 (N t ) bits are transmitted at each particular time instant and are mapped to the spatial location, g, of one of the existing N t transmitters. Hence, only one transmitter becomes active at any particular time instant [START_REF] Mesleh | Spatial modulation[END_REF].

The transmitted light propagates over a NE channel for a strong turbulence or a LN channel for a moderate turbulence. Furthermore, it is assumed that the received signal [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF] and it can be modeled as an Additive White Gaussian Noise (AWGN).

The received signal vector r is given by

r = ηHs + n, (4.1)
where r ∈ C Nr×1 with the received signal at the jth branch, r j , η is the optical to electrical conversion coecient, H ∈ C Nr×Nt is the channel irradiance matrix with channel irradiance entries, h jg ≥ 0 denoting the channel irradiance between the active transmitter g = 2 p ; 1 ≤ p ≤ log 2 (N t ) and receiver j = 1,

I PAM i = I M -1 (i -1), i = 1, 2, • • • , M, (4.2)
where I denotes the average light intensity and M is the signal constellation size.

Channel state information (CSI) is considered available at the receiver side. Therefore, the maximum likelihood (ML) decoder decides for the signal vector ŝ which minimises the Euclidean distance between the received signal vector r and all potential received signals leading to [START_REF] Fath | Spatial modulation applied to optical wireless communications in indoor LOS environments[END_REF] ŝ = argmin

s ||r -ηHs|| 2 F , (4.3)
where ||.|| F is the Frobenius norm [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF].

For a strong turbulence condition, the channel irradiance h jg can be modeled as an independent and identically distributed (i.i.d.) NE random variables (RVs) with a PDF given by [START_REF] Popoola | BPSK subcarrier intensity modulated freespace optical communications in atmospheric turbulence[END_REF] f h jg (h jg ) = exp(-h jg ).

(4.4)

However in moderate turbulence, h jg can be modeled as i.i.d. LN RVs with a PDF

given by [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF] f h jg (h jg ) = 1

h jg 8πσ 2 x × exp - (ln(h jg ) -2µ x ) 2 8σ 2 x , (4.5)
where h jg = exp (2x jg ) with x jg being modeled as an i.i.d. Gaussian RV with mean µ x and variance σ 2

x . To ensure that the fading channel does not attenuate or amplify the average power, the fading irradiances are normalized as E[h jg ] = e 2(µx+σ 2 x ) = 1 (i.e., µ x = -σ 2

x ).

Performance Analysis

A tight upper bound of the bit error rate (BER) of SSK can be calculated by [START_REF] Fath | Spatial modulation applied to optical wireless communications in indoor LOS environments[END_REF] BER U SSK ≤

1 N t log 2 (N t ) Nt g=1 Nt i=1 d H (b g , b i ) Q   1 N r γ log 2 (N t ) 2 Nr j=1 | h jg -ĥji | 2   , (4.6)
where Q(.) is the Gaussian-Q function as dened in Equation (3.11), with γ = (ηI) 2 /N o being the average SNR [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF]. The Hamming distance d H (b g , b i ) counts the number of bit errors between received symbol, b i , and transmitted symbol, b g [START_REF] Mesleh | Indoor MIMO optical wireless communication using spatial modulation[END_REF]. Besides, the 1/N r term is added to make sure that the total area of the receivers becomes the same as the single-input single-output (SISO) receiver aperture [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF], and ĥji is the channel irradiance between the receiver j and the estimated active transmitter i.

A tight lower bound of the BER of RC M -PAM using maximum ratio combining (MRC) at the receiver is given by [179]

BER L RC ≥ 2(M -1) M log 2 (M ) Q    1 N r N t (M -1) γ log 2 (M )N r 2 Nr j=1   Nt g=1 h jg   2    . (4.7)
The term 1/N t is added to ensure that the total power for the N t transmitters is equivalent to the active transmitter of SSK system [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF] and the term √ N r is due to the use of MRC where the noise variance in each aperture is equal to N o /2N r [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF].

4.4

Average BER Probability of SSK Over Turbulent Channels Indoor optical wireless channel is mainly aected by multi-path, while FSO communication is aected by atmospheric turbulence which causes scintillation of the laser beam due to temperature and pressure variations. As such, concluding similar behavior of SSK system [START_REF] Fath | Spatial modulation applied to optical wireless communications in indoor LOS environments[END_REF][START_REF] Fath | Performance comparison of MIMO techniques for optical wireless communications in indoor environments[END_REF] over both channels is not straightforward. Also, the derived expressions for indoor channel can not be directly used in FSO channels. For this reason, we derived an exact MGF of the absolute dierence of two NE RVs and an approximated MGF of the absolute dierence of two LN RVs. The derived MGFs are used to derive tight bound of the ABER of SSK over LN and NE channels.

Negative exponential channels

If X and Y are two independent RVs and U = X -Y . It is well known that the PDF of U is equal to the cross-correlation between the two PDFs as in [START_REF] Shynk | Probability, Random Variables, and Random Processes : Theory and Signal Processing Applications[END_REF]Equation (4.160)]

f U (u) = ∞ -∞ f X (u + y)f Y (y)dy. (4.8)
Let X ≥ 0 and Y ≥ 0 be i.i.d. NE RVs, then it can be shown in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]Equation (3.310) The MGF is dened by [START_REF] Aggarwal | Performance analysis of optical wireless communication system with a decode and forward relay[END_REF] Ψ γ (-s) = ∞ 0 exp(-sγ)f γ (γ)dγ.

] that f U (u) = exp(-|u|)/2. If Z = |U | =| h jg -ĥji | then f Z (z) = f U (z) + f U (-z) = exp(-z).
Ψ γ (-s) = π sγ exp 1 4sγ Q 1 2sγ
, s > 0.

(4.11)

Since the RVs are i.i.d., the MGF of their dierence becomes independent of the transmitter indices (i and g). Therefore, the Q function can be taken outside the summation in Equation (4.6) and the two-fold summation can be replaced by [START_REF] Renzo | Bit error probability of SM-MIMO over generalized fading channels[END_REF] 1

N t log 2 (N t ) Nt g=1 Nt i=1 d H (b g , b i Nt log 2 (Nt)/2 ) = N t 2 .
(4.12)

with the use of Q(x) ≈ 1 12 e -x 2 2 + 1 4 e -2x 2 3 which is accurate for x > 0.5 [START_REF] Chiani | New exponential bounds and approximations for the computation of error probability in fading channels[END_REF] and Equation (4.12), Equation (4.6) can be rewritten as

BER U SSK ≤ N t 24 exp - Nr j=1 γ j log 2 (N t ) 4N 2 r + N t 8 exp - Nr j=1 γ j log 2 (N t ) 3N 2 r . (4.13)
The upper bound of the ABER probability of SSK over NE channels can be obtained as

ABER U SSK = ∞ 0 • • • ∞ 0 BER U SSK f γ (γ)dγ,
Ψ T (-s) = Nr j=1 Ψ γ j (-s) = Ψ γ (-s) Nr . (4.15)
The upper bound of the ABER using Equations (4.11), (4.12) and (4.13), is derived as

ABER U SSK ≤ N t c 2 1 12 Ψ γ - log 2 (N t ) 4N 2 r Nr + 1 4 Ψ γ - log 2 (N t ) 3N 2 r Nr , (4.16)
where c is added to improve the tight bound of the ABER of SSK over LN and NE channels. Extensive simulations and dierent values of N t and N r showed that c = 0.8 will improve the tight bound for N r > 2. SSK uses only an active transmitter at any time instance, so it cannot provide transmit diversity.

LN channels (moderate turbulence)

If X and Y are i.i.d. LN RVs, there exists no closed-form expression for the PDF f Z (z) of Z = |X -Y |. Hence, obtaining the ABER of SSK system over LN channel is analytically not possible. However, we assume that f Z (z) for moderate turbulence can be approximately modeled by Gamma-Gamma (G-G) distribution. G-G channel represents the eective number of large-scale and small-scale eddies denoted by a and b, respectively [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF]. To demonstrate the accuracy of this approximation, Figure 4.4 shows a comparison between the exact and approximate PDF of the absolute dierence of two LN RVs of σ x = 0.37. In our study, extensive simulations are used to obtain accurate estimates of a and b. The MGF of G-G distribution using the series representation which was originally proposed by [START_REF] Bayaki | Performance analysis of MIMO freespace optical systems in Gamma-Gamma fading[END_REF], Ψ I (-s), and after a few manipulations the MGF of the SNR of G-G [START_REF] Aggarwal | Performance analysis of optical wireless communication system with a decode and forward relay[END_REF] is given by,

Ψ γ (-s) = ∞ p=0 d p (b, a)Γ b + p 2 (sγ) -b+p 2 + d p (a, b)Γ a + p 2 (sγ) -a+p 2 , (4.17)
where Γ(w) is the gamma function dened as Γ = ∞ 0 t w-1 exp(-t)dt and d p (f, v) is given as

d p (f, v) = π(vf ) f +p 2 sin[π(v -f )]p!Γ(v)Γ(f )Γ(f -v + p + 1)
. By conducting many simulations, a saturation eect around 300 was observed for the innite sum in Equation (4.17), which still has accurate numerical results. In this case, a tight bound for the ABER can be provided by substituting Equation (4.17

LN channels (weak turbulence)

As mentioned in the previous section, f Z (z) must be approximately modeled by another PDF. In the case of weak turbulence (i.e., σ x = 0.1 and µ x = -σ 2

x ), it is derived in [START_REF] Carmona | Pricing and hedging spread options[END_REF] that f U (u) can be approximated by a normal distribution, with the addition of the variances and the dierence of the means then f Z (z) can be written as

f Z (z) ≈ 2 16πσ 2 x exp - z 2 16σ 2 x (4.19)
The histogram of the absolute dierence of two LN RVs, for the weak turbulence case, is depicted in Figure 4.5. However the derivation of ABER for weak turbulence is omitted from this chapter as the superiority of the proposed SSK is only for moderate to strong turbulence.

Numerical Results and Discussions

In the analysis, the derived ABER performance of SSK and the semi-analytical ABER of RC M -PAM using Equation (4.7) are compared with respect to that of SNR.

Dierent scenarios over NE or LN channels are employed while considering similar spectral eciency for compared systems. Hence, M -PAM and SSK provide spectral eciency of log 2 (M ) and log 2 (N t ) bits/s/Hz, respectively.

µ x = -σ 2 x ).
It is shown that the upper bound in Equation (4.6) is loose at low SNR values but tightens at pragmatic values. This is in accordance with other obtained results using similar bound as in [START_REF] Fath | Spatial modulation applied to optical wireless communications in indoor LOS environments[END_REF][START_REF] Fath | Performance comparison of MIMO techniques for optical wireless communications in indoor environments[END_REF]. For NE, it is worth mentioning that the ABER of SISO is considered as a benchmark using Equation (4.7) with N t = N r = 1 as in [START_REF] Nistazakis | Performance estimation of free space optical links over negative exponential atmospheric turbulence channels[END_REF]. For LN channels, the ABER of SISO is considered as in [START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF]. While using M -PAM for fair comparison with the same spectral eciency of Furthermore, the performance of commercially available MIMO systems using RCs with N t = N r = 2 and employing OOK modulation is compared with that of the proposed SSK system. For this purpose and for the sake of fair comparison, the power of OOK 1 is normalized by log 2 (M ).

To achieve a spectral eciency of 3 bits/s/Hz, a 8 × 8 MIMO system is considered and the simulation results are reported in Figure 4.6. Analytical and simulation results show close match for a wide range of SNR values which support the analysis.

Additionally, the same gure shows that SSK outperforms RC system for SNR < 28 dB.

This is due to the benets of SSK over RC M -PAM, namely, increased SNR gain along with greater spectral eciency or higher turbulence eects. This, while RC systems show better performance at higher SNR due to the high diversity gain. Increasing the number of receivers to 10 instead of 8 enhances the performance of SSK systems as 1. γ is proportional to the square of the intensity modulation and direct detection (IM/DD) optical power compared to RC for high SNR. The SSK system gains about 2 dB in SNR at ABER of 10 -6 .

Figure 4.7 shows the results of increasing the spectral eciency to 4 bits/s/Hz. This increase is made by considering a 16 × 16 MIMO setup. SSK system shows superior performance as compared to RC and a gain of about 6.5 dB is achieved at ABER of 10 -6 . Simulation results in Figure 4.7 depict the case of an ABER of 10 -9 and shows the accuracy of the Q-function approximation [START_REF] Zhang | Block error rate of optical wireless communication systems over atmospheric turbulence channels[END_REF]. Obtained results show that at an ABER of 10 -6 , SSK gains 4 dB and 7.5 dB over RC 16-PAM and RC OOK (2 × 2), respectively. The SNR corresponding to RC illustrated in Figure 4.8 is lower than that of the RC illustrated in Figure 4.7 by 0.3 dB, while SSK system is shown to perform better in strong turbulent channels as compared to moderate turbulent channels. This is mainly because the ABER of SSK system is not determined by the actual channel realizations as in the case of RCs and SISO, rather Since, the SSK uses a single transmitter at each time instant and all others are o for that time. Hence, single-stream-based low-complexity ML detection is employed. As such, the synchronization and the complexity are signicantly reduced ; this fact enables the use of large number of transmitters. Figure 4.9 depicts the lack of the SNR gain of SSK system compared to RC M -PAM system over weak turbulence channels at the same ABER. Adaptive selection between SSK and RCs can be obtained according to the channel state information at the transmitter. In this case, the SNR gain will be enhanced at the expenses of creating a feedback link between transmitter and receiver which increases overall system complexity.

Summary

Data volumes in wireless multimedia applications (e.g., video on demand and interactive video gaming) is increased by a factor of ten every ve years associated with an increase of energy consumption [START_REF] Elkashlan | Green media : the future of wireless multimedia networks[END_REF]. Moreover, the main goal for wireless communication companies is to cut energy costs therefore our analyzed system will be important for the next generation of FSO. Our proposed link achieves triple or quadruple data rate without changing both the energy eciency of the link and the bandwidth, increases the propagation distance and decreases the complexity of the receiver. For the current FSO companies, they already multiplied the number of apertures by four. In the next FSO generation, the companies may multiply it again by four according to the need of the market.

In this chapter, the ABER performance of FSO links with SSK over NE and LN atmospheric turbulence channels are investigated. Tight upper and lower bounds for ABER expressions corresponding to NE channels and LN channels, respectively are obtained. In comparison to SISO and RC M -PAM techniques, our results are in favor of SSK FSO links for high spectral eciencies applications or channel high turbulence eects. Moreover, increasing the number of receivers is shown to have yield lower SNR in SSK systems as compared to RC M -PAM systems. In Chapter 5, MISO multi-hop decode-and-forward FSO system, considering dierent turbulence conditions and path loss attenuations, is proposed and compared to direct link and MISO systems. Table 5.1 summarizes the contribution of the most relevant works in the literature and our work by specifying the considered channel models. Figure 5.1 shows the main three axes considered in the thesis to mitigate various challenges facing FSO. The third axis, relay-assisted multi-hop scenario is considered in this chapter.

System Description

Single-Input Single-Output

A single-input single-output (SISO) scheme implementation of FSO system is depicted in Figure 5.2. On-o keying (OOK) signals are transmitted through laser over a LN channel where path losses result from dierent weather conditions and geometric loss are considered. The received signal at the photodetector input is given by [START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF] r(t) = s(t) η I + n(t), where s(t) represents the transmitted electrical signal, η is the optical-to-electrical conversion coecient, I stands for the received signal light intensity given by [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF],

I = β n I h, (5.2)
System Description 75 with I being the received signal light intensity without considering the channel eect, h = e 2x is the channel irradiance from the transmitter to the receiver with x being modeled as independent and identically distributed (i.i.d.) Gaussian random variables (RVs) with a mean µ x and a variance σ 2

x and β n is the normalized path loss coecient, β n = β( )/β d ( ) with β( ) is the path loss coecient and β d ( ) being the path loss of direct link in clear air conditions. n(t) in Equation (5.1) is an additive white Gaussian noise (AWGN), with zero mean and variance σ 2 n = N o /2, resulting from mainly background noise [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF].

Due to beam divergence loss, the transmit beam diverges and illuminates the static receiver. Assuming a static receiver, without laser tracking methods, spherical wave propagation is considered [START_REF] Henniger | An introduction to free-space optical communications[END_REF]. The channel variance is calculated based on the Rytov theory as a function of the distance [START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF] σ 2

x ( ) = 0.124

k 7/6 C 2 n 11/6 , (5.3)
with being the link distance per hop (in km), k is the wave number, C 2 n is the refractive index constant and σ x ( ) ≤ 0.37 for FSO communication as discussed in subsection 3.3.2. Usually C 2 n is assumed to be independent on distance and its typical values vary from 10 -17 to 10

-13 m -2/3 [2].
The path losses can be calculated by combining weather attenuation [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF] with geometric losses [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] as

β( ) = 10 -α /10 × D 2 R (D T + θ T ) 2 , (5.4)
with α is the weather-dependent attenuation coecient (in dB/km), D R and D T are the receiver and transmitter aperture diameters (in m), and θ T is the optical beam divergence angle (in mrad). The values of α, for dierent weather conditions, are provided in Table 2.1. Acceptable values for D R , D T and θ T are given in [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF][START_REF] Chatzidiamantis | Relay selection in relay-assisted free space optical systems[END_REF].

Hence, h becomes a LN RVs with probability density functions (PDF) given by [START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF] f h (h) = 1

h 8πσ 2 x × exp - (ln(h) -2µ x ) 2 8σ 2 x . h > 0 (5.5)
To ensure that the fading channel does not attenuate or amplify the average power, the fading coecients, throughout the chapter, are normalized as

E[h] = e 2(µx+σ 2 x ) = 1, (i.e., µ x = -σ 2 x ).
Using Equation (5.2), the instantaneous SNR at the receiver input [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF], γ, can be obtained as

γ = (ηI) 2 N o = η 2 β 2 n I 2 h 2 N o = β 2 n γ h 2 , (5.6)
with γ = (ηI) 2 /N o being the average SNR [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF].

After performing a simple transformation of the RVs in Equation (5.5) and by using Equation (5.6), the PDF of the instantaneous SNR becomes (see Appendix B.3)

f γ (γ) = 1 32πσ 2 x γ exp   - ln γ β 2 n γ + 4σ 2 x 2 32σ 2 x    .
(

The conditional bit error rate (BER) of a FSO communication system using OOK modulation is given 1 by [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF] Pr(e|γ

) = Q γ 2 , (5.8)
where Q(.) is the Gaussian-Q function as dened in Equation (3.11). Using Q(x) ≈

1 12 e -x 2 2 + 1 4 e -2x 2 3
which is accurate for x > 0.5 [START_REF] Chiani | New exponential bounds and approximations for the computation of error probability in fading channels[END_REF] to Equation (5.8) yields

Pr(e|γ) ≈ 1 12 exp - γ 4 + 1 4 exp - γ 3 , (5.9)
which is an approximated closed-form expression of the BER for FSO SISO systems using OOK modulation. Now, to consider the impact of LN channel and path losses eects on the BER, the ABER can be obtained by averaging the BER Equation (5.9) using Equation (5.7) as

ABER = ∞ 0 Pr(e|γ)f γ (γ)dγ.
(

The above integral can be approximated by using Hermite polynomial of Equation (3.20) yield an approximated closed-form solution given by

ABER ≈ 1 12 √ π N i=1 w i exp - β 2 n γ e -4σ 2 x +z i √ 32σ 2 x 4 + 1 4 √ π N i=1 w i exp - β 2 n γ e -4σ 2 x +z i √ 32σ 2 x 3 , (5.11)
where z i and w i are the zeros and the weights of the Hermite polynomial [174, page 924] of order N , respectively. It has been shown in [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF], that accurate results for LN channels can be obtained when N ≥ 20.

At the receiver side, maximum-likelihood (ML) decoder is considered to decode the received signals [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF].

MISO Using Repetition Codes

A FSO MISO system, with N t transmitters and using RCs, is depicted in Figure 5.3. In RCs system, OOK signals are transmitted simultaneously through N t lasers over correlated LN channel with similar statistics as SISO scheme. The analytical model of MISO using repetition codes is discussed in details in sections 3.2 and 3.3. However, in this subsection, normalized path loss β n is used.

Both correlated and uncorrelated eects among the transmitters are considered in this chapter. The assumption of uncorrelated channel paths assumes the spacing between 1. Many wireless optical books such as [START_REF] Hranilovic | Wireless Optical Communication Systems[END_REF] dene Pr(e|γ) as Q( √ γ), to be compatible with the probability of error of OOK in radio frequency digital communications system. In our work, we followed the notation adopted in [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF] to compare our work with their works. √ λ with λ being the wavelength [START_REF] Anguita | Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link[END_REF]. However, in many cases the available space for the transmitters may not be sucient for such requirement [START_REF] Zhu | Maximum-likelihood spatial-diversity reception on correlated turbulent free-space optical channels[END_REF]. Also, the total spacing among the transmitters must be within the eld of view (FOV) of the receiver to avoid serious degradation of the ABER performance as shown in [80, Figure 1]. In fact, the spacing should be less than θ f meter, where θ f is the FOV of the receiver in milliradians. (mrad) [START_REF] Simon | Alamouti-type space-time coding for free-space optical communication with direct detection[END_REF]. For example, if = 2 km and θ f ∼ = 5 mrad, then the length of transmitters, (d × (N t -1)), should be less than 10 m apart [START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF][START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF].

Multi-Hop DF Relaying

In this subsection, an approximate ABER closed-form expression for FSO multi-hop system using DF relays, depicted in hops between the source and the destination and r k (t) is the received signal at the kth hop given by,

r k (t) = s(t) η I k + n k (t), k = 1, • • • , K.
(5.12)

In case of SISO and MISO schemes, one time slot is required for transmission. However, in this scheme K time slots are required. In order to achieve similar spectral eciency as compared to these systems, 2 (K) -ary pulse amplitude modulation (PAM) modulation is considered. Besides, in DF system, every relay receives the signal, regenerates it and send it again either to the next relay or to the nal destination. However, shorter communication links are created and the impact of turbulence eect σ k and path loss β k are signicantly reduced. The calculation of σ k and β k are the same as given in Equation ( 5.3) and Equation (5.4), respectively.

The upper bound BER for multi-hop DF is given by [START_REF] Akella | Error analysis of multihop free-space optical communication[END_REF] BER ≤ 1 -K k=1

(1 -BER k ).

(

Assuming all hops having the same statistical behavior, an approximated BER can be calculated by [START_REF] Morgado | Endto-end average BER in multihop wireless networks over fading channels[END_REF] BER ≈

1 2 1 -(1 -2BER k ) K .
(

The conditional BER of M -PAM modulation is derived in Appendix B. 4 as

Pr(e|γ) ≈ 2(M -1) M log 2 (M ) Q P 2 log 2 (M ) (M -1) 2 Rσ 2 n , (5.15)
where P is the optical power and R is the bit rate. Following the same notation of Equation (5.8), γ is dened as

γ = 2P 2 σ 2 n R . (5.16)
Now, applying the approximate Q function on Equation (5.15) and substituting in Equation (5.16) gives

Pr(e|γ) ≈ 2(M -1) M log 2 (M ) 1 12 exp - γ log 2 (M ) 4(M -1) 2 + 1 4 exp - γ log 2 (M ) 3(M -1) 2 . (5.17)
It is important to note that Equation (5.17) is a general expression of the conditional BER of FSO systems using M -PAM scheme and the expression for OOK in Equation (5.9) can be derived as a special case by setting M = 2.

Using Equation (5.17), the kth hop ABER of M -PAM FSO communication system under a LN channel and path loss attenuation eects can be calculated following similar steps as discussed above to obtain Equation (5.11) and is given by

ABER k ≈ G 12 √ π N i=1 w i exp - log 2 (M )β 2 kn γ e -4σ 2 k +z i √ 32σ 2 k 4(M -1) 2 (5.18) + G 4 √ π N i=1 w i exp - log 2 (M )β 2 kn γ e -4σ 2 k +z i √ 32σ 2 k 3(M -1) 2 ,
where G = 2(M -1) M log 2 (M ) , and β kn is the normalized path loss coecient for multi-hop system with respect to the direct link in clear air conditions,

β kn = β k /β d ( ).
Assuming all ABER k are equal, substituting Equation (5.18) in Equations (5.14) and (5.13), the nal approximated closed form expression of the ABER of the multihop system and the upper bound ABER expressions are obtained, respectively.

MISO Multi-Hop DF Relaying

In this subsection, an approximate ABER closed-form expression for FSO MISO multi-hop system using DF relays, with N t transmitters and using RCs, as depicted in Figure 5.5, is derived. This scheme is a hybrid MISO and multi-hop system that benets from the advantages of both systems to mitigate the impact of FSO channel turbulence and attenuations. Let us consider (K -1) relays where each relay receives the signal and re-transmits it simultaneously using N t transmitters with RCs. The number of hops is assumed to be similar to the number of transmitters for each relay. The received signal at each hop is given by,

x(t) Rx M-PAM Modulator

r k (t) = s(t) η Nt i=1 I ki + n k (t), k = 1, • • • , K. (5.19)
Using Equation (5.18) and substitute σ x by ćij which is the (i, j) th coecient of Γ sq = Γ 1/2 , the kth hop ABER of M -PAM MISO multi-hop FSO communication Performance Analysis of MISO Multi-hop Over Log-normal Channels system is

ABER k ≈ N n 1 =1 • • • N n N t =1 Nt i=1 w n i √ π × G 12 exp   - log 2 (M )β 2 kn γ 4(M -1) 2 N t Nt i=1   exp   √ 32 Nt j=1 ćij x n j -4σ 2 k       + N n 1 =1 • • • N n N t =1 Nt i=1 w n i √ π × G 4 exp   - log 2 (M )β 2 kn γ 3(M -1) 2 N t Nt i=1   exp   √ 32 Nt j=1 ćij x n j -4σ 2 k       , (5.20)
where N t is added to Equation (5.20) to guarantee that the total power of the MISO system is equal to the power of a SISO system.

The spatial covariance matrix Γ can be calculated as

Γij = σ 2 k × ρ |i-j| .
(

Substituting Equation (5.20) in Equation (5.14) and Equation (5.13), the nal approximated closed form of ABER of the MISO multi-hop system and the upper bound ABER expressions are obtained, respectively.

Numerical Results and Discussions

In this section, numerical results are presented to validate the derived analysis in this chapter. The considered parameters are provided in Table 5.2. Derived analytical results are corroborated via Monte Carlo simulations and both results are shown to match closely for wide and pragmatic SNR values. In the obtained simulation results, 10 7 bits are transmitted for each depicted SNR value with 1000 iterations to achieve the target ABER of 10 -9 . Throughout this chapter, relays are placed equidistant to each other to obtain the optimum performance as derived in [START_REF] Kashani | Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems[END_REF].

In Figure 5.6, a clear air condition with moderate turbulence is assumed and a single relay is placed mid-distance between the source and the destination in a multihop system. The spectral eciency of M -PAM is log 2 (M ). Hence, in order to achieve the spectral eciency of SISO and MISO systems, 4-PAM modulation was employed.

The obtained results clearly demonstrate the signicant performance enhancement of SISO multi-hop system even in clear air conditions. SISO multi-hop system has SNR gains of about 11 dB and 22 dB at the target ABER as compared to uncorrelated MISO and SISO systems, respectively. A correlation of ρ = 0.3 is shown to degrade the performance of MISO system by about 3 dB. Moreover, correlated and uncorrelated MISO multi-hop outperforms SISO multi-hop system by 3.4 dB and 5 dB, respectively.

Performance Analysis of MISO Multi-hop Over Log-normal Channels

There is an inversely proportional relationship between the turbulence strength and attenuation such as it is very unlikely that strong turbulence occurs during a fog event [START_REF] Bushuev | Analysis of the performance of a wireless optical multiinput to multi-output communication system[END_REF]. So, the impact of light fog with weak turbulence is considered in Figure 5.7.

While light fog degrades the performance of all systems, it has a major impact on SISO and MISO systems as compared to multi-hop systems due to multi-hop systems have shorter hops that yields signicant performance improvements. As compared to clear air condition results, shown in Figure 5.6, 35 dB and 31 dB, losses are reported at the target ABER for MISO and SISO systems, respectively. However, the losses are less than 19 dB in multi-hop systems.

Consequently, SISO multi-hop system still outperform other systems and SNR gains of about 33 dB and 37 dB can be achieved when compared to MISO and SISO systems, respectively. Results demonstrate that uncorrelated MISO multi-hop system has SNR gains of 1.75 dB and 2.5 dB compared to correlated MISO multi-hop system and SISO multi-hop system, respectively. It is worth mentioning that the very high power required by direct link and spatial diversity link can not be achieved at practice. As the behavior of the photodiode becomes non-linear then the photodiode will be damaged as the received optical power will be absorbed as heat. Relay Selection For Full-Duplex FSO Relays Over Turbulent Channels Monte Carlo simulations corroborate the correctness of the obtained analytical results. Table 6.1 summarizes the contribution of the most relevant works in the literature and our work by specifying the considered channel models. Figure 6.1 shows the main three axes considered in the thesis to mitigate various challenges facing FSO. The third axis, a relay-assisted cooperative relays scenario is considered in this chapter.

6.2

System And Channel Models

System Model

The considered system is depicted in Figure 6.2, where two users communicate through N re relay nodes. Only a single relay, with index j, among existing N re relays participates in the communication process. The selected relay should have the highest receive end-to-end SNR and that is known to both users [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF]135,[START_REF] Vu | Selection decode-and-forward cooperative FSO systems with adaptive rate strategy over gamma-gamma fading chan-References 125 nels[END_REF][START_REF] Wang | Performance analysis of multi-hop parallel free-space optical systems over exponentiated Weibull fading channels[END_REF]. The selection process is achieved according to Equation (2.1).

In the rst time slot, t 1 , each user transmits his data to the preselected relay. The It is assumed that all relays have full channel knowledge and the channel is symmetrical [START_REF] Sharma | Bi-directional decode-XOR-forward relaying over Mdistributed free space optical links[END_REF]. The received signals are aected by an additive white Gaussian noise (AWGN) with zero mean and variance σ 2 n = N o /2 resulting mainly from background noise [START_REF] Safari | Do we really need OSTBC for free-space optical communication with direct detection[END_REF].

The normalized channel coecient of the considered system can be formulated as follows [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF]:

h = h a h p β, (6.1)
where h a and h p are the channel fading coecients due to atmospheric turbulence and misalignment error, respectively. The path losses, β, can be calculated by combining weather attenuation [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF] with geometric losses [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] as Equation (5.4).

The general model of misalignment fading proposed in [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF] is considered hereinafter

h p ≈ A o exp -2r 2 w 2 zeq , (6.2)
where r is the radial displacement which is modeled by Rayleigh distribution, the equi-System And Channel Models 89 Reference Channel Model Contributions [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] gamma-gamma

(G-G)
derive the OP of an asynchronous low-complexity cooperative relaying [START_REF] Bhatnagar | Average BER analysis of relay selection based decode-andforward cooperative communication over gamma-gamma fading FSO links[END_REF] the ABER expression is derived using the power series expansion [START_REF] Cao | Average bit error rate of multi-hop parallel decode-and-forward-based FSO cooperative system with the max-min criterion under the gamma-gamma distribution[END_REF] the ABER of DF relays using Gauss-Laguerre quadrature rule is derived [START_REF] Boluda-Ruiz | Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels[END_REF] the ABER and the diversity order of DF relays considering misalignment error are derived [START_REF] Puri | Asymptotic analysis of TWR assisted FSO links with partial dual-relay selection[END_REF] partial relay selection for FD coherent system is employed while considering the eects of path loss and misalignment error [START_REF] Abou-Rjeily | Performance analysis of selective relaying in cooperative freespace optical systems[END_REF] LN & Rayleigh the ABER expressions are derived [START_REF] Sharma | Relay assisted bi-directional communication in generalized turbulence fading[END_REF] M-distribution The radial displacement can be modeled as

f r (r) = r σ 2 s exp - r 2σ 2 s , r > 0 (6.3)
where σ 2 s is the jitter variance at the receiver.

LN Channels

The probability density function (PDF) of LN channels considering path loss eects and misalignment errors is given by [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF] f

h (h) = ξ 2 h (ξ 2 -1) 2(A o β n ) ξ 2 erfc   ln h Aoβn + q √ 8σ x   exp 2σ 2 x ξ 2 (1 + ξ 2 ) , (6.4)
where erfc(.) is the complementary error function (see Appendix C.1), β n denotes the normalized path loss coecient, β n = β/β k with β k being the path loss for a single hop, q = 2σ 2

x (1 + 2ξ 2 ) and ξ =

w 2 zeq 2σ 2 s
is the ratio between the equivalent beam width at the receiver and the pointing error displacement standard deviation at the receiver.

The channel fading coecient h a = exp (2x), with x being an independent and identically distributed (i.i.d.) Gaussian random variable (RV) with a mean µ x and a variance σ 2

x . To ensure that the fading channel does not attenuate or amplify the average power, the fading coecients are normalized by considering E[h a ] = e 2(µx+σ 2 x ) = 1 (i.e., µ x = -σ 2

x ) [76]. Hence, a plane wave propagation is assumed and the log-amplitude variance is calculated based on the Rytov theory as a function of the distance [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF] σ 2

x = 0.30545 k 7/6 C 2 n 11/6 , (

with k = 2π/λ is the wave number, λ is the wavelength and C 2 n is the refractive index constant (m -2/3 ).

G-G Channels

The PDF of G-G channels is given by [START_REF] Feng | Outage performance for parallel relay-assisted free-space optical communications in strong turbulence with pointing errors[END_REF] f

h (h) = abξ 2 A o β n Γ(a)Γ(b) G 3,0 1,3 abh β n A o ξ 2 ξ 2 -1, a -1, b -1 , (6.6)
where G m,o p,w [.] is the Meijer's G-function (see Appendix C.1), Γ(.) is the Gamma function (see Equation 4.17), a and b are the eective number of large-scale and small-scale eddies of scattering environment, respectively. Their values for plane wave are given as [START_REF] Andrews | Laser Beam Scintillation with Applications[END_REF]

a =   exp   0.49σ 2 R (1 + 1.11σ 12 5 R ) 7 6   -1   -1 , (6.7) b =   exp   0.51σ 2 R (1 + 0.69σ 12 5 R ) 7 6   -1   -1 , (6.8)
and the Rytov variance, σ 2 R , is given as [38] σ 2 R = 1.23 k 7/6 C 2 n 11/6 . (6.9)

6.3

Outage Performance Analysis

LN Channels

The OP at each node can be obtained directly from the CDF of SNR, F γ (γ), [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] P out,direct = Pr(γ < γ th ) = F γ (γ th ), (6.10) Outage Performance Analysis 91 where γ = h 2 γ is the SNR, γ is the average SNR. If SNR exceeds γ th , no outage occurs and the signal can be decoded with an arbitrarily low error probability at the receiver.

For LN channels, the OP is derived as [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF] P out,direct = Pr h <

1 ρP M = F γ (γ th ) (6.11) = 1 2 exp ξ 2 ψ -2σ 2 x ξ 4 erfc ψ √ 8σ x + 1 2 erfc 4σ 2 x ξ 2 -ψ √ 8σ x , where P M = γ γ th
is the power margin and ψ = ln

1 Aoβn ρP M + 2σ 2 x (1 + 2ξ 2 ).
The power of HD systems, such as direct link and HD relays, are assumed to be half the power of FD systems in order to achieve a fair comparison using the same equipment of Figure 6.2 [START_REF] Riihonen | Rate-interference trade-o between duplex modes in decode-and-forward relaying[END_REF]. Hence, the power margin is multiplied by a constant, ρ. The value of ρ is unity for FD systems and one-half for HD systems.

Since the SNRs for the two hops are independent and without loss of generality, the relay is assumed being located at equidistant from the source and the destination, and equal average SNRs per hop are also assumed, the OP for a dual-hop system is given by [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] P

out = 1 -[1 -P out,direct ] 2 .
(6.12)

Due to the assumption that the relays have full channel knowledge and the network is symmetrical [START_REF] Sharma | Bi-directional decode-XOR-forward relaying over Mdistributed free space optical links[END_REF], Equation (6.12) can be used for both FD and HD relays taking into consideration the power constraint.

The values of β n and σ 2

x in dual-hop scheme can be decreased as the distance can be divided by two which enhances the performance of multi-hop systems. If multiple branches of dual-hop exist, as in Figure 6.2, the OP for the best relay selection as in Equation (2.1) can be calculated as [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] P

out = 1 -[1 -P out,direct ] 2 Nre = F γ (γ th ). (6.13)
It is evident that a diversity gain of N re is achieved for a dual-hop system with the best relay selection.

G-G Channels

For G-G channels, the OP at each node is derived as [START_REF] Aggarwal | Performance analysis of optical wireless communication system with a decode and forward relay[END_REF] P

out,direct = ξ 2 Γ(a)Γ(b) × G 3,1 2,4 ab β n ρP M 1, ξ 2 + 1 ξ 2 , a, b, 0 = F γ (γ th ), (6.14)
The OP for the best relay selection can be calculated as [START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF] P

out = 1 -[1 -(P out,direct )] 2 Nre = F γ (γ th ). (6.15)
For dual-hop system, the values of a, b and β n should be changed as the link distance is divided by two.

Relay Selection For Full-Duplex FSO Relays Over Turbulent Channels

Performance Analysis

The ABER can be calculated directly using the CDF approach as [START_REF] Zhao | Symbol error rate of selection amplify-andforward relay systems[END_REF] ABER

= e E [Q( √ cγ )] = e √ 2π ∞ 0 F γ y 2 c exp - y 2 2 dy, (6.16)
where Q(.) is the Gaussian-Q function as dened in Equation (3.11), E[.] denotes the average over channel fading distributions, and c and e are constants determined by the modulation format. In this study, multiple pulse amplitude modulation (M -PAM)

using intensity modulation with direct detection (IM/DD) is considered as in [START_REF] Fath | Performance comparison of MIMO techniques for optical wireless communications in indoor environments[END_REF]. The spectral eciency of M -PAM is equal to log 2 (M ) bits/s/Hz [START_REF] Fath | Performance comparison of MIMO techniques for optical wireless communications in indoor environments[END_REF]. Hence, 2-PAM is considered for direct link (1 bit per slot) and FD relays (2 bits from two users per 2 slots), while 4-PAM is considered for HD relays (2 bits per 2 slots) to maintain similar spectral eciency [START_REF] Louie | Practical physical layer network coding for two-way relay channels : performance analysis and comparison[END_REF].

The conditional bit error probability (BEP) of M -PAM is given by [START_REF] Hranilovic | Wireless Optical Communication Systems[END_REF] Pr

(e|γ) ≈ 2(M -1) M log 2 (M ) Q γ log 2 (M ) 2(M -1) 2 . ( 6.17)
The values of e and c can be calculated using Eq. (6.17 The ABER of the direct link over LN channels can be derived by substituting Equation (6.11) into Equation (6.16) as

ABER d = e √ 2π ∞ 0 1 2 exp ξ 2 ψ -2σ 2 x ξ 4 erfc ψ √ 8σ x exp - y 2 2 dy + e √ 2π ∞ 0 1 2 erfc 4σ 2 x ξ 2 - ψ √ 8σ x exp - y 2 2 dy, (6.18)
where ψ = ln y Aoβn ρ√ γc + 2σ 2

x (1 + 2ξ 2 ).

After performing a simple transformation of x = y 2 2 , Equation (6.18) is easily obtai- ned as

ABER d = e 2 √ π ∞ 0 1 2 √ x exp ξ 2 ψ -2σ 2 x ξ 4 erfc ψ √ 8σ x exp (-x) dx + e 2 √ π ∞ 0 1 2 √ x erfc 4σ 2 x ξ 2 - ψ √ 8σ x exp (-x) dx, (6.19)
Numerical Results and Discussions 93 where ψ = ln √ 2x Aoβn ρ√ γc + 2σ 2

x (1 + 2ξ 2 ).

The integration in Equation (6.19) can be computed using the generalized Gauss-Laguerre quadrature function as [START_REF] Concus | Tables for the evaluation of ∞ 0 x β e -x f (x)dx by Gauss-Laguerre quadrature[END_REF] ∞ 0

z f exp(-z)g(z)dz ≈ N i=1 ẁi g(z i ), (6.20)
where N is the order of the approximation and zi and ẁi are the roots and the weights of the generalized Laguerre polynomial of degree f , respectively. The rst hundred values of zi and ẁi are well tabulated in [197]. Thus Equation (6.19) can be expressed by truncated series as

ABER d ≈ e 2 √ π N i=1 ẁi 2 exp ξ 2 ψ -2σ 2 x ξ 4 erfc ψ √ 8σ x + e 2 √ π N i=1 ẁi 2 erfc 4σ 2 x ξ 2 - ψ √ 8σ x ≈ e 2 √ π N i=1 ẁi Ì, (6.21)
where ψ = ln

√ 2z i Aoβn ρ√ γc + 2σ 2 x (1 + 2ξ 2 ) and Ì = 1 2 exp ξ 2 ψ -2σ 2 x ξ 4 erfc ψ √ 8σx + 1 2 erfc 4σ 2 x ξ 2 - ψ √ 8σx
.

For a dual-hop system, assuming equal average SNRs per hop, an approximated ABER can be calculated by [START_REF] Morgado | Endto-end average BER in multihop wireless networks over fading channels[END_REF] ABER

≈ 1 2 1 -(1 -2 ABER d ) 2 . (6.22)
Following the same steps of Equation (6.21), an approximated ABER expression of the best relay selection for LN channels can be calculated by substituting the CDF, F γ (γ th ), of Equation (6.13) into Equation (6.16) and using Equation (6.20) as

ABER ≈ e 2 √ π N i=1 ẁi 1 -1 -Ì 2 Nre
. (6.23)

Numerical Results and Discussions

In the presented results, a target ABER of 10 -9 , OP for 10 -15 and equal hop distance are considered. Derived analytical results are corroborated via Monte Carlo simulations.

In the obtained simulation results, 10 7 bits are transmitted for each depicted SNR value and the Gauss-Laguerre quadrature approximation order is N ≤ 50. Table 6.2 shows the system parameters under investigation which are used in various FSO communication systems [START_REF] Farid | Outage capacity optimization for free-space optical links with pointing errors[END_REF][START_REF] Chatzidiamantis | Relay selection protocols for relay-assisted free-space optical systems[END_REF][START_REF] Soni | Impact of beam divergence on the performance of free space optical system[END_REF]. Using Table 6.2, atmospheric turbulence conditions are calculated by Equations (6.5), (6.7), (6.8) and (6.9) and are presented in The most common models are log-normal (LN) channels for weak-to-moderate turbulence, gamma-gamma (G-G) channels for moderate-to-strong turbulence and negative exponential (NE) channels for saturated strong turbulence.

To mitigate the severe eect of atmospheric turbulence, many promising solutions such as multiple-input multiple-output (MIMO), spatial modulation and relay assisted technique have been proposed in the literature as shown in Figure 7.1 and these three axis are considered in the thesis as follows.

In this study, special spatial correlation for multiple-input single-output (MISO) 100 Conclusions and Future Work FSO systems with on-o keying (OOK) modulation using intensity modulation and direct detection (IM/DD) is considered. In the case of spatial correlation, signal-to-noise ratio (SNR) gain decreases with decreasing the separation distance among transmitter units or receiver units. Hence, the importance of spatial correlation appears when there is a spacing limitation in the transmitter or in the receiver side. LN channels are considered for weak-to-moderate turbulence and the probability density function (PDF) of correlated LN channels is expressed according to an approach based on moment generating functions. In our approach, approximated average bit error rate (ABER) expressions for both orthogonal space-time block codes (OSTBCs) and repetition codes (RCs) in correlated LN channels with dierent path loss eects are obtained.

Our results show that RCs always have better SNR gain than OSTBCs by at least 3 dB. In addition, increasing the number of transmitters for RCs always enhances the SNR. On the other hand, increasing the number of transmitters for OSTBCs in weak turbulence degrades the SNR gain.

The demand for higher data rates in wireless systems is essential to sustain the increasing demand for bandwidth-hungry applications, such as for video applications. Our results show that space shift keying (SSK) is a good candidate for FSO communication systems for the next generation of FSO in order to keep pace with progress of wireless devices and simultaneously decrease the energy cost as well as the receiver complexity. Tight upper and lower bounds for ABER expressions corresponding to NE and LN channels respectively are obtained for SSK FSO communication systems. Our results demonstrate that SSK has more SNR gain than the single-input single-output (SISO) system as well as recent RC multi-pulse amplitude modulation (M -PAM) techniques, especially for strong turbulence and high spectral eciency. In addition, the diversity order of SSK system depends on the receiver diversity order. Therefore, by increasing the number of receivers we can enhance the SNR gain compared to RC M -PAM systems.

The relay-assisted technique, in particular, has recently attracted attention for overcoming several challenges such as atmospheric turbulence, path loss attenuation, and line-of-sight propagation. MISO multi-hop decode-and-forward (DF) FSO system is proposed and the analytical model is derived. Our results show the superior performance of the proposed system as compared with SISO multi-hop system using M -PAM, SISO or MISO using OOK at the same spectral eciency. Our results also show that the proposed system can mitigate turbulence eects, weather attenuation eects and beam divergence loss. It is worth mentioning that MISO systems mitigate only turbulence eects as they do not have the advantage of shorter hops. In addition, increasing the number of relays leads to more SNR gain as compared with the increasing number of transmitters in MISO systems.

The selection of a single relay based on the max-min SNR criterion of FSO communication systems is investigated for both full-duplex (FD) and half-duplex (HD) relays.

The outage probability (OP) of the considered system, using dual-hop DF for both LN and G-G channels, takes into consideration that path loss and misalignment errors are obtained. In addition, approximated ABER expressions using Gauss-Laguerre quadrature rule and assuming full channel state information (CSI) are derived for FD relays, HD relays, and direct link. FD relays have better spectral eciency as compared to Recommendations for Future Work 101 HD relays. Therefore, FD relays can be considered as a good candidate for the next generation of FSO communication systems due to its superiority in outage probability and in ABER compared to HD relays.

7.2

Recommendations for Future Work

The results of this dissertation point to several intriguing paths, which can be pursued in future work. The instantaneous signal-to-noise ratio (SNR) γ i is given by [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF],

γ i = (ηI i ) 2 N o (B.2)
The received signal light intensity from the i th transmitter to the receiver after channel eect which can be evaluated according to [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF] as follows where h i = e 2x i is the channel irradiance from transmitter i to the receiver with x i being modeled as spatially correlated identically distributed (c.i.d.) Gaussian random variables (RVs) with mean µ x and variance σ 2

I i = β I o h i (B.
x .

Based on Equation (B.4), γ i can be written as To see the impact of σ x on this approximation, we assume that h i can be modeled as an independent and identically distributed (i.i.d.) log-normal RVs, the mean of the fading coecients is

γ i = ch 2 i (B.
E[h i ] = e 2(µx+σ 2
x ) (B.10)

We assume that log-normal parameters are equal for all channels. To ensure that the fading channel does not attenuate or amplify the average power, the fading coecients are normalized as E[h i ] = e 2(µx+σ 2 x ) = 1 (i.e., µ x = -σ 2

x ).

The second moment of log-normal channel is For correlated and uncorrelated cases, if σ 2

E[h 2 i ] = e 4(µx+2σ
x 1, which is the case for FSO communications, then e 4σ 2

x ≈ 1. Please note that the maximum value of σ x for FSO LN channels corresponds to the maximum scintillation index (SI) value for LN channels, which is ≤ 0.75 [START_REF] Kiasaleh | Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence[END_REF]. The relation between SI and σ x is dened in [START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF] The PDF of log-normal (LN) RVs h is given by [START_REF] Moradi | A switched diversity approach for multi-receiving optical wireless systems[END_REF] f h (h) = 1

h 8πσ 2 x × exp - (ln(h) -2µ x ) 2 8σ 2 x . h > 0 (B.16)
To ensure that the fading channel does not attenuate or amplify the average power, the fading coecients are normalized as E[h] = e 2(µx+σ 2 x ) = 1, (i.e., µ x = -σ 2

x ).

The instantaneous SNR at the receiver input, γ, is given by [START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF],

γ = (ηI) 2 N o , (B.17

)
where I stands for the received signal light intensity and is given by [START_REF] Bayaki | EDFA-based all-optical relaying in free-space optical systems[END_REF][START_REF] Navidpour | BER performance of free-space optical transmission with spatial diversity[END_REF],

I

C.2 The complementary error function

This function is dened in the fourth equation in section 8.250 in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF] erfc(x) = 1 -erf(x). Γ (a j -s)

x s ds.

(C.3)
where this integration is called line integration, an empty product is interpreted as 1 and i is a complex number.

Special cases include:

G 1,

RESUME

Au cours de la dernière décennie, les communications optiques en espace libre (FSO) ont pris de l'ampleur dans les deux domaines académiques et industriels. L'importance de FSO s'appuie sur la possibilité de faire un système de transmission économique et écologique avec un débit élevé et sans licence à l'opposition des systèmes de transmission radiofréquences (RF

ABSTRACT

Free-space optical (FSO) communication has been the subject of ongoing research activities and commercial attention in the past few years. Such attention is driven by the promise of high data rate, licensefree operation, and cheap and ecological friendly means of communications alternative to congested radio frequency communications. In most previous work considering multiple transmitters, uncorrelated channel conditions have been considered. An uncorrelated channel requires sucient spacing between transmitters. However, this can be dicult and may not be always feasible in some applications. Thereby, this thesis studies repetition codes (RCs) and orthogonal space-time block codes performance in correlated log-normal FSO channels using intensity modulation and direct detection (IM/DD). Furthermore, the eect of dierent weather conditions on the average bit error rate (ABER) performance of the FSO links is studied. Multiple-input multiple-output (MIMO) FSO communication systems using space shift keying (SSK) modulation have been also analyzed. Obtained results show that SSK is a potential technique for spectral eciencies equal or greater than 4 bits/s/Hz as compared to RCs with multiple pulse amplitude modulations. The performance analysis of a multi-hop decode and forward relays for FSO communication system using IM/DD is also considered in this thesis. It is shown that multi-hop is an ecient technique to mitigate atmospheric turbulence and dierent weather attenuation eects and geometric losses in FSO communication systems. Our simulation results show that multiple-input single-output (MISO) multi-hop FSO systems are superior to direct link and MISO systems over links exhibiting high attenuation. Meeting the growing demand for higher data rates communication networks, a system with full-duplex (FD) relays is considered. For such a system, the outage probability and the ABER performance are analyzed under dierent turbulence conditions, misalignment error and path loss eects. FD relays are compared with the direct link and half-duplex relays. Obtained results show that FD relays have the lowest ABER and the outage probability as compared to the two other systems. Finally, the obtained results in this thesis are very promising towards the next generation of FSO systems.

  RCs over correlated LN with σ x = 0.3, µ x = -σ 2 x , N t = 2, and clear air for dierent correlation coecients. . . . . . . . . . . . . . . . . . . . . . 59 4.1 The main axes of the research. . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Synoptic diagram of the proposed model. . . . . . . . . . . . . . . . . . 63 4.3 Histogram of the absolute dierence of the two NE RVs. . . . . . . . . . 65 4.4

68 4 . 6 Fhr 2 nStandard deviation for a hop σ 2 Rvariance σ 2 s 2 x

 462222 Comparison of space shift keying and repetition codes with spectral eciency = 3 bits/s/Hz in a NE channel (strong turbulence) with N t = 8 and the number of receivers N r [8 10]. . . . . . . . . . . . . . . . . . . 69 4.7 Comparison of space shift keying and repetition codes with spectral efciency = 4 bits/s/Hz in a NE channel (strong turbulence), N t = N r = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.8 Comparison of space shift keying and repetition codes with spectral eciency = 4 bits/s/Hz in a LN channel with σ x = 0.37 (moderate turbulence), N t = N r = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . shift keying and repetition codes with spectral eciency = 4 bits/s/Hz in a LN channel with σ x = 0.1 (weak turbulence), N t = N r = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.10 A comparison between upper bound SSK using Q-function and using the approximation of Q-function on space shift keying. . . . . . . . . . . . . 725.1 The main axes of the research. . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 FSO SISO system using OOK modulation. . . . . . . . . . . . . . . . . . 74 5.3 MISO scheme using RCs. . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.4 Multi-hop DF relay scheme. . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.5 MISO Multi-hop DF relay scheme. . . . . . . . . . . . . . . . . . . . . . 79 5.6 ABER of single-relay multi-hop using one or two transmitters with 4-PAM modulation, two transmitters MISO using RCs and OOK, and SISO using OOK modulation FSO systems in LN fading channel and clear air conditions (C 2 n = 5 × 10 -14 m -2/3 , α = 0.43 dB/km). From Equation (3.8), ρ in Equations (3.29) and (5.20) can be approximated as zero when d d o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.7 ABER of single-relay multi-hop using one or two transmitters with 4-PAM modulation, two transmitters MISO using RCs and OOK, and SISO using OOK modulation FSO systems in LN fading channel and light fog (C 2 n = 1.7 × 10 -14 m -2/3 , α = 20 dB/km). . . . . . . . . . . . . . . 82 5.8 ABER of two-relays multi-hop using one or three transmitters with 8-PAM modulation, three transmitters MISO using RCs and OOK, and SISO using OOK modulation FSO systems in LN fading channel and clear air (C 2 n = 5 × 10 -14 m -2/3 , α = 0.43 dB/km). . . . . . . . . . . . 83 5.9 ABER of two-relays multi-hop using one or three transmitters with 8-PAM modulation, three transmitters MISO using RCs and OOK, and SISO using OOK modulation FSO systems in LN fading channel and light fog (C 2 n = 1.7 × 10 -14 m -2/3 , α = 20 dB/km). . . . . . . . . . . . 84 6.1 The main axes of the research. . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Block diagram of the proposed FD FSO communication system. . . . . . 88 6.3 OP for FSO over weak fading channel and clear air conditions (C 2 n = 0.5 × 10 -14 m -2/3 , α = 0.43 dB/km). . . . . . . . . . . . . . . . . . . . 95 6.4 OP for FSO over moderate fading channel and clear air conditions (C 2 n = 2 × 10 -14 m -2/3 , α = 0.43 dB/km). . . . . . . . . . . . . . . . . . . . . 95 6.5 OP for FSO over strong fading channel and clear air conditions (C 2 n = 5 × 10 -14 m -2/3 , α = 0.43 dB/km). . . . . . . . . . . . . . . . . . . . . 96 6.6 ABER of FSO over weak fading channel and clear air conditions (C 2 n = 0.5 × 10 -14 m -2/3 , α = 0.43 dB/km). . . . . . . . . . . . . . . . . . . . 96 6.7 ABER of FSO over moderate fading channel and clear air conditions (C 2 n = 2 × 10 -14 m -2/3 , α = 0.43 dB/km). . . . . . . . . . . . . . . . . . . . . 97 7.1 The main axes of the research. . . . . . . . . . . . . . . . . . . . . . . . Average electrical energy of the transmitted pulse at each transmitter Channel irradiance from transmitter to the receiver Symbols h R i D Channel fading coecients between ith relay and the destination h SR i Channel fading coecients between the source and ith relay h jg Channel irradiance between transmitter g and receiver j h a Channel fading coecients due to atmospheric turbulence h i Channel irradiance from transmitter i to the receiver h p Channel fading coecients due to misalignment error I Received signal light intensity from the transmitter to the receiver I i Received signal light intensity from the i th transmitter to the receiver I PAM i Intensity levels of pulse amplitude modulation K Radial displacement which is modeled by Rayleigh distribution r(t) Received signal r j Received signal at each branch r k (t) Received signal at the kth hop s(t) Weights of N th order Hermite polynomial w i Weights of the Hermite polynomial Symbols w z Beam waist (in m) (radius calculated at e -2 ) x Independent and identically distributed Gaussian random variables X Roots of N th order Hermite polynomial z i Roots of the Hermite polynomial ρ Constant α Attenuation coecient (in dB/km) β Path loss β kn Normalized path loss for a hop β d ( ) Path loss of direct link in clear weather conditions β k Signal-to-noise ratio of transmitter i λ Wavelength of the transmitter Γ Spatial covariance matrix Γ sq Square root spatial covariance matrix µ x Mean of an independent and identically distributed Gaussian random variables γ i Average signal-to-noise ratio of transmitter i Ψ(.) Moment generating function ρ Correlation coecient σ Variance of an additive white Gaussian noise σ k Rytov Jitter variance at the receiver σ Variance of an independent and identically distributed Gaussian random variables θ f Field of view of the receiver (in mrad)θ T Optical beam divergence angle (in mrad) -space optical (FSO) communication was rst invented by Alexander Graham Bell in 1880[1]. He employed an experiment called "photophone" that converts voice sounds into telephone signals and transmits the signal through sunlight to an optical receiver at a distance of 213 m. Bell considered this optical technology the greatest invention he had ever made, greater than the telephone. However, it never came out as a commercial product, but it demonstrated the basic principle of FSO communications. After the invention of the laser, wireless optical commination has prompted great attention.The fast growing demand for high data rates along with the scarcity of radio frequency (RF) spectrum necessitate the need to consider alternative wireless transmission technologies such as FSO. FSO, unlike ber optic networks, does not require an expensive infrastructure. Due to this and other benets FSO communication has become the leading subject for both research and commercial activities over the past years. Moreover, FSO utilizes an unlicensed spectrum and is a green communication technology, which constitutes a perfect solution for wide-range applications. FSO systems have typical applications such as backhaul for wireless cellular networks, ber back up, disaster recovery for damaged landlines, connecting buildings with ultra-high data rate, broadcasting high-denition video and satellite communications[2, 3, 4, 5, 6,[START_REF] Kedar | Urban optical wireless communication networks : The main challenges and possible solutions[END_REF]. Employing dense wavelength division multiplexing (DWDM), FSO communication provides the world record transmission of 1.6 Tbps using 16 channels each operating at 100 Gbps over a link of 80 m[START_REF] Parca | Optical wireless transmission at 1.6-Tbit/s (16 × 100 Gbit/s) for next-generation convergent urban infrastructure[END_REF].

3 . 4 .

 34 Mitigating turbulence and path losses eects, the multi-hop system is investigated due to its advantage of shorter hops that yields signicant performance improvements. Decode and forward (DF) multiple-input single-output (MISO) multi-hop FSO systems are proposed and the obtained results show the superiority of the Thesis Contributions 19 proposed system over a direct link and MISO systems considering correlation effects at the transmitter. Analyzing full-duplex (FD) relays under dierent turbulence conditions, misalignment error and path loss eects. FD relays are compared with the direct link and half-duplex (HD) relays. Our results show that FD relays have the lowest ABER and the outage probability (OP).

Figure 1 . 1

 11 Figure 1.1 Summary of thesis contributions.These contributions are summarised in Figure1.1. We consider two challenges facing FSO: weather attenuation and geometric loss, and atmospheric turbulence. In the case of weather attenuation and geometric loss, these challenges lead to power loss and relayassisted FSO systems can mitigate these eects. Two schemes are employed as follows:

Figure 1 . 2

 12 Figure 1.2 The main axes of the research.

Figure 2 . 1

 21 Figure 2.1 The main axes of the research.

Figure 2 . 2

 22 Figure 2.2 Sale navigation using VLC technology in Korea [25]. In September 2011, IEEE ratied 802.15.7, a wireless personal area network standard for VLC at data rates ranging from 11 kbps to 96 Mbps [25]. VLC faces a lot of challenges such as increasing the temperature which reduces the LEDs lifetime, dimming of the light source for low data rates, path loss attenuation, sunlight and articial light eects, uctuation of the brightness of light and eye constrains safety and incorrect detection of LED color [27]. Dierent potential solutions are proposed in the literature to mitigate these challenges such as special lters for uctuation of the brightness of the light, advanced modulation techniques for dimming of the light source and feedback correction

Figure 2 . 3 .

 23 LEDs are preferred than laser for robot communications as LEDs have a small size and low price. The authors achieved 20-30 meters propagation distance with average power consumption of 500 mW. The interesting thing is the cost of wireless optical communication devices is only 70 dollars. Dierent data rates between 9600 bps to 38400 bps were carried out to ensure the reliability of the system.

Figure 2 . 3

 23 Figure 2.3 Underwater robots using LED-based communications [30].

2. 2 . 3

 23 Space wireless optical communicationsNASA employed wireless optical communications link between Moon and Earth for a propagation distance of 384600 km at 622 Mbps[START_REF] Boroson | The lunar laser communications demonstration[END_REF]. German Space Agency, named DLR-Space, employed wireless optical communications link between two low Earth orbit (LEO) satellites (TerraSAR-X and NFIRE) see Figure2.4. This link is considered the State of the Art rst coherent link for wireless optical satellite communication with BPSK and homodyne detection for about 8000 km and 5.6 Gbps is achieved[START_REF] Gutlich | German roadmap on optical communication in space[END_REF].

Figure 2 . 4

 24 Figure 2.4 Wireless optical communication between satellites [33].

Figure 2 . 5

 25 Figure 2.5 Free-space optical communication network [2].

2

 2 Concatenated Reed-Solomon codes Two concatenated codes, RS (255, 239) + RS (255, 239) and RS (255, 239) + RS (255, 223), are compared with the standard RS (255, 239) of international telecommunication union -telecommunications (ITU-T) for optical communications. The results show that the two concatenated codes outperform the standard RS (255, 239) at an average bit error rate (ABER) 10 -8 [70]. Using 256-PPM with concatenated RS (255, 239) + RS (255, 223) helps to upgrade the existing uncoded 256-PPM FSO system to work on strong turbulence rather than being limited to weak turbulence at ABER of 10-9 [18]. The same results appear using avalanche photodetectors (APDs) or PIN receivers[START_REF] Abaza | BER performance of M-ary PPM free-space optical communications with channel fading[END_REF].

3 .

 3 Northern Storm, one of the leading FSO companies, installed a FSO link in California city of 10.31 Gbps with 1 Gbps RF backup for a distance of 238 m. For the considered system, an availability of 99.999% is achieved in all weather conditions[171].

(

  LN) FSO channels with weather attenuation eects. Approximate analytical expressions using moment generating function (MGF) of the joint probability density function (PDF) of correlated LN channels are developed.Using MGF approximation, the average bit error rate (ABER) expressions for repetition codes (RCs) and orthogonal space-time block codes (OSTBCs) using intensity modulation and direct detection (IM/DD) schemes in correlated LN channels are derived. Results demonstrate that RCs have higher signal-to-noise ratio (SNR) gain than OSTBCs in correlated channel conditions. In addition, the eect of light fog and clear air conditions on the ABER performance of the FSO communication systems are studied. Our Monte Carlo simulations corroborate the validity of the derived mathematical analysis.

46 Diversity

 46

Figure 3 . 1

 31 Figure 3.1 The main axes of the research.

Figure 3 . 2

 32 Figure 3.2 Synoptic diagram of the proposed model.

. 26 )

 26 Numerical Results and Discussions 51Using the alternative denition of the Q-function in Equation(3.11), the ABER of RCs system over correlated LN channel under path loss eect is given by

  values. The results are shown in Figure 3.14. Results clearly demonstrate that the proposed approximation is valid and leads to a negligible error (< 10%) with practical σ x values. For more details on the proposed approximation, see Appendices B.1 and B.2.

  identically distributed, low and high correlation values are considered (ρ = 0.25 and 0.5). Analytical results with N = 20 are compared to Monte Carlo simulations. Results demonstrate close match for a wide range of SNR values.

FSO

  system. The required SNR to achieve the target ABER decreases by 25.74 dB and 24.88 dB for RCs and by 25.94 dB and 24.34 dB for OSTBCs with ρ = 0.25 and 0.5, respectively.

Figure 3 .Figure 3 . 9 ,

 339 Figure 3.11 shows results for N t = 3, σ x = 0.374 and α = 0. An interesting observation is the behavior of OSTBCs results, where increasing the number of transmitters enhances the performance unlike weak turbulence case. As compared to the results in Figure 3.9, OSTBCs show a SNR gain of 1.95 dB and 2 dB for ρ = 0.25 and 0.5, respectively. The required SNR for moderate turbulence increases by 14.5 dB and 17.7 dB for RCs and 15.4 dB and 17.6 dB for OSTBCs codes with ρ = 0.25 and 0.5, respectively compared to σ x = 0.1 with N t = 3 and α = 0 dB.Figure3.12 shows results for N t = 3, σ x = 0.3 and α = 0.43 dB/km. As compared

Figure 3 .Figure 3 . 3

 333 Figure 3.3 Correlation coecient versus distance among transmit units for dierence propagation distances.

  RCs in correlated LN channels are obtained. It is shown that increasing the number of transmitters always enhances the ABER performance of RCs, but it does not necessarily improve the SNR gain in OSTBCs.Additionally, RCs are shown to always has higher SNR gain than OSTBCs approximately by 3 dB. Finally, the severe eects of dierent weather conditions on the FSO Diversity Techniques for Correlated Log-normal Channels

Figure 3 . 4

 34 Figure 3.4 Error due to the proposed approximation.

5 Figure 3 . 5

 535 Figure 3.5 Correlated LN with σ x = 0.1, µ x = -σ 2 x , N t = 2 and α = 0 dB.

Figure 3 . 6

 36 Figure 3.6 Correlated LN with σ x = 0.1, µ x = -σ 2 x , N t = 2 and light fog.

Figure 3 . 7 5 Figure 3 . 8

 37538 Figure 3.7 Correlated LN with σ x = 0.1, µ x = -σ 2x , N t = 3 and α = 0 dB.

5 Figure 3 . 9

 539 Figure 3.9 Correlated LN with σ x = 0.374, µ x = -σ 2x , N t = 2, and α = 0 dB.

Figure 3 .

 3 Figure 3.10 Correlated LN with σ x = 0.3, µ x = -σ 2 x , N t = 2, and clear air.

Figure 3 . 58 Diversity 5 Figure 3 .

 35853 Figure 3.11 Correlated LN with σ x = 0.374, µ x = -σ 2x , N t = 3 and α = 0 dB.

Figure 3 .

 3 Figure 3.13 Correlated LN using repetition codes with σ x = 0.1, µ x = -σ 2x , ρ = 0.25, N t = 3 for dierent weather conditions.

1 Figure 3 .

 13 Figure 3.14 RCs over correlated LN with σ x = 0.3, µ x = -σ 2 x , N t = 2, and clear air

62 Performance

 62

Figure 4 . 1

 41 Figure 4.1 The main axes of the research.

  The histogram of the absolute dierence of the two NE RVs and exact NE PDF are depicted in Figure4.3. The PDF of the electrical SNR, γ = z 2 γ, can be obtained from the PDF of Z f γ (γ)

  of the absolute difference of two negative exponential RVs

Figure 4 . 3

 43 Figure 4.3 Histogram of the absolute dierence of the two NE RVs.

( 4 . 10 )

 410 Using[START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF] Equation (3.322/2)], the MGF of Equation (4.10) can be derived as

(4. 14 )

 14 where γ = γ 1 • • • γ Nr and since the RVs are i.i.d., the total MGF is the product of the individual MGFs as

Figure 4 .

 4 4 shows that G-G distribution with a = 3.3 and b = 1.1 achieve accurate approximation of f Z (z).

  -Gamma PDF (a = 3.3 and b = 1.1) Exact

Figure 4 . 4

 44 Figure 4.4 Histogram of the absolute dierence of two LN RVs (σ x = 0.37 and µ x = -σ 2 x ).

  absolute difference of two log-normal RVs

Figure 4 . 5

 45 Figure 4.5 Histogram of the absolute dierence of two LN RVs (σ x = 0.1 and

8 Figure 4 . 6

 846 Figure 4.6 Comparison of space shift keying and repetition codes with spectral eciency = 3 bits/s/Hz in a NE channel (strong turbulence) with N t = 8 and the number of receivers N r [8 10].

Figure 4 .

 4 8 shows the case of a moderate turbulence in a LN channel with σ x = 0.37 as an example of a moderate turbulence. Based on the approximation, the results of which are shown in Figure 4.4, the MGF of G-G is used to derive a tight bound ABER for SSK over LN channel. This approximation leads to a narrow gap between the upper bound Equation (4.6) and the derived lower bound of SSK. This is unlike the results shown in Figures 4.6

  and 4.7, which are based on an exact MGF of the absolute dierence of NE RVs.

  SSK (4.6) Lower Bound RC-16PAM (4.7) Derived Upper Bound SSK (4.16) SISO-16PAM (4.7) 2 × 2 RC OOK -1 bit/s/Hz (4.7)

Figure 4 . 7

 47 Figure 4.7 Comparison of space shift keying and repetition codes with spectral eciency = 4 bits/s/Hz in a NE channel (strong turbulence), N t = N r = 16.

Finally, Figure 4 .

 4 10 shows the accuracy of the Q approximation by comparing the conditional BER of Equation (4.6) with Equation (4.13) for parametric SNR values using spectral eciency = 4 bits/s/Hz in a NE channel (strong turbulence), N t = N r = 16.

  [dB] Lower Bound RC 16-PAM (4.7) Upper Bound SSK (4.6) Simulations Derived Lower Bound SSK SISO 16PAM (4.7) 2 × 2 RC OOK -1 bit/s/Hz (4.7)

Figure 4 . 8

 48 Figure 4.8 Comparison of space shift keying and repetition codes with spectral eciency = 4 bits/s/Hz in a LN channel with σ x = 0.37 (moderate turbulence), N t = N r = 16.

  [dB] Lower Bound RC 16-PAM (4.7) Upper Bound SSK(4.6)

Figure 4 . 9

 49 Figure 4.9 Comparison of space shift keying and repetition codes with spectral eciency = 4 bits/s/Hz in a LN channel with σ x = 0.1 (weak turbulence), N t = N r = 16.

6 )

 6 Using Q-function Upper Bound SSK (4.13) Using Approximation of Q-function

Figure 4 .

 4 Figure 4.10 A comparison between upper bound SSK using Q-function and using the approximation of Q-function on space shift keying.

Figure 5 . 1

 51 Figure 5.1 The main axes of the research.

Tx N t - 1 Figure 5 . 3

 153 Figure 5.3 MISO scheme using RCs.

Figure 5 . 4 ,Figure 5 . 4

 5454 Figure 5.4 Multi-hop DF relay scheme.

Figure 5 .

 5 Figure 5.5 MISO Multi-hop DF relay scheme.

2 n 1 . 7 ×Figure 5 . 6

 21756 Figure 5.6 ABER of single-relay multi-hop using one or two transmitters with 4-PAM modulation, two transmitters MISO using RCs and OOK, and SISO using OOK modulation FSO systems in LN fading channel and clear air conditions (C 2 n = 5 × 10 -14 m -2/3 , α = 0.43 dB/km). From Equation (3.8), ρ in Equations (3.29) and (5.20) can be approximated as zero when d d o .

  Hop (5.14 & 5.18) Repetition 2x1, ρ = 0.3 (3.29) Approx. MISO Dual Hop, ρ = 0.3 (5.14 & 5.20) Approx. MISO Dual Hop, ρ = 0 (5.14 & 5.20) Monte Carlo Simulation

Figure 5 . 7 Figure 5 . 8

 5758 Figure 5.7 ABER of single-relay multi-hop using one or two transmitters with 4-PAM modulation, two transmitters MISO using RCs and OOK, and SISO using OOK modulation FSO systems in LN fading channel and light fog (C 2 n = 1.7 × 10 -14 m -2/3 , α = 20 dB/km).

6 . 1

 61 IntroductionThis chapter investigates the performance of a best relay selection, based on the max-min signal-to-noise ratio criterion for dual-hop free-space optical (FSO) full-duplex (FD) relays communication system. Decode-and-forward (DF) relays over log-normal (LN) channels for weak-to-moderate turbulence and gamma-gamma (G-G) channels for strong turbulence are considered. We assume that the relays have full channel knowledge and the channel is symmetrical. Considering path loss eects and misalignment errors, the outage probability (OP) of the selection is obtained for both half-duplex (HD) and FD relays using the cumulative distribution function (CDF) of the best selection for LN and G-G random variables. Moreover, the average bit error rate (ABER) expressions for FSO communication system over LN channels are derived with the help of Gauss-Laguerre's quadrature rule for HD relays, FD relays and direct link. Our results show that FD relays have lowest ABER and OP compared with the direct link and HD relays.

  user's transceiver steers the signal in the direction of the selected relay. The relay has 88 Relay Selection For Full-Duplex FSO Relays Over Turbulent Channels

Figure 6 . 1

 61 Figure 6.1 The main axes of the research.

FD

  DF relays re employed and the ABER and the OP are OP are derived while considering the eects of path loss and misalignment error for FD and HD relays valent beam width (in m), w zeq , is calculated as w 2 zeq = (w 2 z √ πerf(v))/(2v exp(-v 2 )), v = ( √ πD R )/(2 √ 2w z ), A o = [erf(v)] 2 , erf(.) is the error function (see Appendix C.1), w z is the beam waist (in m) (radius calculated at e -2 ) and D R is the receiver aperture diameters (in m).

  ) as (e, c) = (1, 0.5) for 2-PAM and (e, c) = (0.75, 0.1111) for 4-PAM.It is worth mentioning that M -PAM has dierent intensity levels as in Equation (4.2). Channel state information (CSI) is assumed available at the receiver side and a maximum likelihood (ML) decoder is used to decode the received signals as Equation (4.3).

5 ×

 5 10 -14 m -2/3 , 2 × 10 -14 m -2/3 , 5 × 10 -14 m -2/3

  ξ = 3.3377 HD dual-hop FD dual-hop FD 3 relays select-max FD 2 relays select-max HD 3 relays select-max HD 2 relays select-max

Figure 6 . 3

 63 Figure 6.3 OP for FSO over weak fading channel and clear air conditions (C 2 n = 0.5 × 10 -14 m -2/3 , α = 0.43 dB/km).

  ξ = 3.3377 HD dual-hop FD dual-hop HD 2 relays select-max FD 2 relays select-max HD 3 relays select-max FD 3 relays select-max Direct, ξ → ∞ Simulations

Figure 6 . 4

 64 Figure 6.4 OP for FSO over moderate fading channel and clear air conditions (C 2 n = 2 × 10 -14 m -2/3 , α = 0.43 dB/km).

Figure 6 . 5 Figure 6 . 6 Figure 6 . 7

 656667 Figure 6.5 OP for FSO over strong fading channel and clear air conditions (C 2 n = 5 × 10 -14 m -2/3 , α = 0.43 dB/km).

Figure 7 . 1

 71 Figure 7.1 The main axes of the research.Free-space optical (FSO) systems are gaining signicant popularity due to their ability to provide high bandwidth and cost-eectiveness for a variety of applications in wireless communication systems. While promising, FSO communication systems still face challenges including atmospheric turbulence, misalignment error, and harsh weather conditions that degrade their performance.Atmospheric turbulence results from the variations of the index of refraction due to inhomogeneities in temperature and pressure causes uctuation of the received signal. Several statistical models of atmospheric turbulence are available in the literature.

2 . 3 .

 23 Optimum Relay Number: The optimum number of relays given the maximum diversity should be investigated. The analytical model for that optimum number should be developed under dierent turbulence channels, a dierent conguration of relays and dierent misalignment error cases. Blind and Semi-Blind Channel Estimation: At present the FSO systems in this thesis are losing some of their capacity using CSI. To overcome this issue, blind and semi-blind channel estimation may be considered. That may increase the spectral eciency of FSO systems at the expenses of SNR gain.4. Light-Emitting Diodes (LEDs) Communications: Extensive research and measurement should be carried out on outdoor LEDs communications in order to increase the data rate from just a few Mbps to several Gbps. Furthermore, non line of sight (LOS) wireless optical communications should be also considered.5. Mixed FSO and Indoor Wireless Optical Communications:We can also investigate mixed systems using both advantages of FSO and indoor wireless optical communications. FSO can be used for last mile high data rate applications instead of heavy infrastructure required by optical ber. In addition, indoor wireless optical communications can be applied for both illuminations and high date rate green applications.B.1Approximation of Equation (3.27)Here, we would like to investigate the proposed approximation of Equation(3.28)

3 )

 3 Substituting Equation (B.3) into Equation (B.2) getsγ i = (ηβ I o h i ) 2 N o (B.4)

5 )E 2 = N t c Nt i=1 h 2 i(B. 7 )

 5227 where c is a constant. Substituting Equation (B.5) on Equation (B.1) yields we can simplify c from E 1 and E 2 as

as σ x = √ ln(SI+1) 2 .

 2 Hence, σ x should be ≤ 0.374. In this case, E[E1] ≈ E[E2] that explain the impact of σ x on the proposed approximation of Equation(3.27). simulations with 10 6 realizations, the results show that the proposed approximation of Equation (3.27) can be applied with acceptance error percentage as shown in Figure B.2. In addition, Figure B.3, shows the eect of increasing the number of realizations on the proposed approximation. In this code, correlated normal RVs are obtained by multiplying the transpose of Cholesky factorization of the correlation matrix (U') by the uncorrelated normal RVs (Ra).Matlab Code of Equation (3

Figure B. 2

 2 Figure B.2 Impact of the correlation coecient on the proposed approximation.

Figure B. 3

 3 Figure B.3 Impact of Monte Carlo simulations realizations on the proposed approximation for ρ = 0.5 and σ x = 0.1.

  Equations (B.20) and (B.22) in Equation (B.21) and using µ x = -σ 2 x , we can obtain Equation (B.15).

Γ ( 1 -Γ( 1 -

 11 This function is dened in Equation (9.301) in[START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]. Meijer's G-function introduced by C. S. Meijer in 1936. Many special functions can be written as G functions such as logarithms, exponentials, trigonometric functions, hypergeometric functions and Bessel functions[START_REF] Prudnikov | More special functions[END_REF]. G m,o p,w x a 1 , . . . , a p b 1 , . . . , b w a j + s) w j=m+1 b j + s) p j=o+1

  . . . . . . . . . . . . . . . . . . . . 20 2.1 The main axes of the research. . . . . . . . . . . . . . . . . . . . . . . . The main axes of the research. . . . . . . . . . . . . . . . . . . . . . . .

				23
	2.2	Sale navigation using VLC technology in Korea. . . . . . . . . . . . . . .	24
	2.3	Underwater robots using LED-based communications. . . . . . . . . . .	25
	2.4	Wireless optical communication between satellites. . . . . . . . . . . . .	26
	2.5	Free-space optical communication network.	. . . . . . . . . . . . . . . .	27
	3.1			46
	3.2	Synoptic diagram of the proposed model.

. . . . . . . . . . . . . . . . . 47 3.3 Correlation coecient versus distance among transmit units for dierence propagation distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Error due to the proposed approximation. . . . . . . . . . . . . . . . . . 54 3.5

Table 2

 2

	.1 Weather attenuation coecient of FSO
	Weather conditions	α [dB/km]
	Clear air	0.43
	Haze	4.2
	Moderate rain (12.5 mm/h)	5.8
	Heavy rain (25 mm/h)	9.2
	Light fog	20
	Moderate fog	42.2
	Heavy fog	125
	2.4.4 Background noise
	Background noise, due to undesirable background radiations, results of exposure of
	the receiver to direct or indirect sunlight or articial lights. This noise reduces the SNR
	gain but it can be eliminated using bandpass optical lter before photo-detection. Direct
	sunlight statistically occurs less than 1 hour per year [13, 37].

Table 3 .

 3 1 Comparison among our MISO scheme & the closed literature works

	Reference	Channel Model	Contributions
	[80]	negative-exponential	a modied version of Alamouti's code for IM/DD
	[91]	correlated LN	derive approximated MGF for correlated LN
	[11]	LN	RCs outperform OSTBCs using IM/DD
	[83]	K distribution	transmit laser selection outperform RCs
	Our work	correlated LN (to model correlation among transmitters)	impact of spatial correlation on RCs and OSTBCs
	matrices, G,

for N t ≤ 8 for OSTBCs

Table 4 .

 4 1 Comparison between our SSK scheme & the closed literature works

	Reference	Channel Model	Contributions
	[75] [84]	indoor optical wireless (multipath)	applying spatial modulation (SM) for indoor optical communications SM outperforms RCs for spectral eciencies ≥ 4 bit/s/Hz
	[85]	gamma-gamma	RCs outperforms SSK for spectral eciency = 1 bit/s/Hz
			spatial pulse position amplitude
	[87]	LN & gamma-gamma	modulation outperforms SSK
			for spectral eciencies ≥ 2 bit/s/Hz
	Our work	LN & NE (to model weak-to-strong turbulence)	SSK outperforms RCs over moderate-to-strong turbulence for spectral eciencies ≥ 4 bit/s/Hz
	suers from shot, thermal and dark noises. The shot noise is assumed to dominate other
	noise sources

Table 5 .

 5 1 Comparison between our MISO multi-hop scheme & the closed literature

	works Reference Channel Model			Contributions
	[140]			LN		cooperative optical amplify and forward relay outperforms MISO using repetition codes (RCs)
	[137]		LN & Rayleigh		cooperative decode and forward relay outperforms MISO using RCs
							cooperative decode and forward relay
	[156]		gamma-gamma	outperforms MISO using transmit laser selection
							under certain relay location
	Our work	correlated LN (to model correlation among transmitters)		MISO multi-hop outperforms MISO using RCs
	Source bits	OOK Modulator	s(t)	Tx		Rx	r(t)	decoder ML	Estimated bits
						l	+
							n(t)
			Figure 5.2 FSO SISO system using OOK modulation.

Table 5 .

 5

		2 System conguration
	Parameter	Symbol	Value
	The target ABER

Table 5 .

 5

			3 Summary of SNR gains of Figures 5.6-5.7
	Weather condition		SISO	2 × 1 MISO ρ = 0.3	2 × 1 MISO ρ = 0	Dual-hop relay	2 × 1 MISO ρ = 0.3 dual-hop relay	2 × 1 MISO ρ = 0 dual-hop relay
	Clear air +	0 dB	8 dB	11 dB	22 dB	25.4 dB	27 dB
	moderate
	turbulence
	Light fog +	-31 dB	-27 dB		6 dB	7.75 dB	8.5 dB
	weak
	turbulence
	Weather condition	Table 5.4 Summary of SNR gains of Figures 5.8-5.9 SISO 3 × 1 MISO ρ = 0.3 3 × 1 MISO ρ = 0 3 × 1 Triple-hop MISO ρ = 0.3 relay triple-hop relay	3 × 1 MISO ρ = 0 triple-hop relay
	Clear air +	0 dB	9 dB	13 dB	27 dB	30.2 dB	31.4 dB
	moderate
	turbulence
	Light fog +	-31 dB	-25 dB		16 dB	17.5 dB	18 dB
	weak
	turbulence

Table 6 .

 6 1 Comparison between our FD best relay selection & the closed literature works

Table 6

 6

.3.

The OP for HD relays, FD relays and direct link for FSO links over weak, moderate and strong turbulence channels are depicted in Figures 6.3, 6.4 and 6.5 respectively.

Table 6 .

 6

		2 System conguration
	Parameter	Symbol	Value
	Wavelength

Table 6 .

 6

		3 Atmospheric conditions
	Atmospheric turbulence Link type σ x a	b	σ 2 R
	Weak turbulence	Direct link Dual-hop	0.16 0.08
	Moderate turbulence	Direct link Dual-hop	0.31 0.17
	Strong turbulence	Direct link Dual-hop	4.4 8.9	3.15 7.77	1 0.28
	Figure 6.3 shows that FD relays, with a single relay or multiple relays, outperforms
	their counterparts HD relays and direct link systems. Furthermore, it can be noticed
	that the performance of FD relays with the best relay selection is enhanced by increasing
	the number of relays. Performance gains of about 3 dB can be noticed as compared to
	HD relays.
	Similar behaviors can be noticed as well in Figures 6.4 and 6.5, where FD relays are
	shown to outperform HD relays and direct link systems. It is worth mentioning that the
	performance of FD relays degrades by about 2 dB for moderate turbulence and by more
	than 7 dB for strong turbulence. HD relays performance is shown to degrade as well.
	FD relays outperform HD relays by about 3 dB for moderate and strong turbulence.
	Moreover, Figures 6.3 and 6.5 show that misalignment error eect is more sensitive to
	weak turbulence than moderate and strong turbulence.
	For the negligible misalignment error case when (ξ → ∞), Equation (6.11) matches
	with the formula obtained in [101, Equation 21] and Equation (6.14) is consistent with
	the formula obtained in [131, Equation 54].

The ABER of FD relays, HD relays and direct link for FSO links over weak and moderate turbulent channels with respect to the average SNR are shown in

Figures 6.6

  1. Massive MIMO Systems: Extending the basic work presented in Chapter 3 by employing massive MIMO systems as in 5G radio frequency (RF) systems. This needs a study of the correlation eects among transmit and receive units. Our approximation in Equation (3.28) to calculate the ABER expression for correlated LN channels will not be applicable for massive MIMO systems. The alternative analytical model worth investigating for massive MIMO future systems in order to achieve both higher spectral eciency and high a SNR gain.

  2 x ) = e 4σ 2

	(B.13)
	The mean of Equation (B.9) is
	E[E2] = N 2 t E[h 2 i ] = N 2 t e 4σ 2

x (B.11)

The mean of Equation (B.8) is

E[E1] = N t E[h 2 i ] + (N t (N t -1)) ρe 4σ 2 x -ρ + E[h i ] 2 E[h i h j ] (B.12)

Substituting Equation (B.11) and E[h i ] = 1 on Equation (B.12), yields

E[E1] = N t e 4σ 2 x + (N t (N t -1)) ρe 4σ 2 x -ρ + 1 x (B.14)

  = β n I h, Equation (B.[START_REF] Jurado-Navas | Impact of pointing errors on the performance of generalized atmospheric optical channels[END_REF]) is a bijective function as h > 0 ; in this case, we know[START_REF] Mansour | Probabilités et statistiques pour les ingénieurs : cours, exercices et programmation[END_REF] that

					(B.18)
	Substituting Equation (B.18) in Equation (B.17), yields
	γ =	η 2 β 2 n I N o 2 h 2	= β 2 n γ h 2 ,
	Using Equation (B.19), we can conclude:
		h =	γ β 2 n	γ	(B.20)

(B.

[START_REF] Barrios | Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves[END_REF]

with γ = (ηI) 2 /N o being the average SNR

[START_REF] Tsiftsis | Optical wireless links with spatial diversity over strong atmospheric turbulence channels[END_REF]

.

  ). Dans la plupart des travaux antécédents sur les systèmes multi-émetteurs, seulement les canaux décorrélés ont été considérés. Un canal décorrélé nécessite un espace susant entre les émetteurs. Cette condition devient dicile et nonréalisable dans certaines applications. Pour cette raison, nos études se focalisent sur les performances des codes à répétition RC (Repitition Codes) et les codes OSTBC (Orthogonal Space-Time Block Codes) dans des canaux log-normaux corrélés en utilisant une modulation d'intensité et une détection directe (IM/DD). En addition, les eets des diérentes conditions météorologiques sur le taux d'erreur moyen (ABER) sont étudiés. Les systèmes FSO à multi-entrées/ multi-sorties MIMO (Multiple-Input Multiple-Output) avec une modulation SSK (Space Shift Keying) ont été abordés. Les résultats obtenus montrent que la SSK est supérieure aux RC avec une modulation d'impulsion (Multiple Pulse Amplitude Modulation) pour toute ecacité spectrale égale ou supérieure à 4 bit/s/Hz. Nous avons aussi analysé les performances d'un système à sauts multiples (Multi-Hop) et des relais à transmission directe (forward relays). Nos simulations montrent que le système ainsi considéré est ecace pour atténuer les eets météorologiques et les pertes géométriques dans les systèmes de communication FSO. Nous avons montré qu'un tel système avec plusieurs entrées et une sortie (MISO, i.e. multiple-input single-output) à sauts multiples est supérieur à un système MISO avec un lien direct (direct link) avec une forte atténuation. Pour satisfaire la demande croissante des réseaux de communication à débits élevés, la communauté scientique s'intéresse de plus en plus aux systèmes FSO avec des relais fullduplex (FD). Pour ces derniers systèmes, nous avons étudié la probabilité d'erreur moyenne (ABER) et nous avons analysé leurs performances. En considérant des diérentes conditions de transmission, les performances de relais FD ont été comparées à celles d'un système avec un lien direct ou des relais half-duplex. Les résultats obtenus montrent que les relais FD ont le minimum ABER. En conséquence, les résultats obtenus dans cette thèse sont très prometteurs pour la prochaine génération de FSO.

This function is dened in the rst equation in section 8.250 in[START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]

d'Opale for honoring me by accepting to be rapporteurs of my dissertation. I also want to thank Prof. Gilles BUREL from Moustafa HUSSEIN from Arab Academy for Science, Technology and Maritime Transport (AASTMT), Egypt for accepting to be examiners of my dissertation. This work has been carried out at Sensor Networks and Cellular Systems (SNCS) Research Center, University of Tabuk. The nancial support provided by SNCS is gratefully acknowledged. I would also like to thank Prof. el-Hadi M. AGGOUNE.

List of Figures

Summary

In this chapter, MISO multi-hop DF FSO systems are considered. Results demonstrate that MISO multi-hop system using M -PAM has superior improvement than other existing systems such as SISO multi-hop system using M -PAM, SISO and MISO using OOK at the same spectral eciency. The impact of weak turbulence with light fog and moderate turbulence with clear air conditions on the performance of considered FSO systems are analyzed. Multi-hop systems have the ability to mitigate both turbulence and path loss eects such as weather and geometric loss attenuation. However, MISO systems can only mitigate turbulence eect.

Consequently, increasing the number of relays is shown to signicantly enhance the overall system performance. This is while increasing the number of transmitters in MISO systems is shown to lead to small enhancements in SNRs. In Chapter 6, full-duplex (FD) relays are analyzed under dierent turbulence conditions, misalignment error and path loss eects. FD relays are compared with the direct link and half-duplex relays. Our results show that FD relays have lowest ABER as well as the outage probability. In this section, the conditional bit error rate (BER) of M -ary pulse amplitude modulation (PAM) modulation is derived as

The symbol probability of error using a union bound approximation is given by [START_REF] Haykin | Digital Communications[END_REF] Pr

where N is the average number of constellation points away from any constellation point and d min is the minimum Euclidean distance between two points in the signal constellation.

In the case of M -ary PAM, N = M -1 and M -ary PAM has one dimensional basis function where M constellation points are equally spaced on the axis, d min = 2P √ Ts M -1 [58].

where T s is the symbol duration and P is the optical power. In addition, (2/M ) is multiplied with Equation (B.24) to ensure that the average optical power of each symbol is xed at P .

Now, Equation (B.24) becomes

Pr sym ≈ 2(M -1)

The probability of symbol error, Pr sym , can be converted to a probability of bit error, Pr(e|γ), by multiplying Pr sym by a factor of (1/ log 2 M ) as in [START_REF] Proakis | Digital Communications[END_REF] The probability of bit error becomes where R is the bit rate.

Appendix C

The denitions of commonly used special functions are given.

C.