# Error Probability of a Coherent \(M\)-ary PSK FSO System Influenced by Phase Noise
## 1 Introduction  
### Background and Related Work  
Mobile network data traffic grew 40 percent between the first quarters of 2021 and 2022, respectively. According to [1], at the end of 2021, there were around 8.2 billion mobile subscriptions, and this number is expected to increase to around 9.1 billion by the end of 2027. During the same time, the share of mobile broadband subscriptions will increase from 84 to 93 percent. Implementation of the fifth generation (5G) mobile systems has been enabling broadband data transmission. These systems should support high user data rates (up to 10 Gbps) and high reliability (about 99.999), low energy consumption and low latency (less than 1 ms), as well as high security levels and density of user terminals. These demands will be achieved by using a small-cell concept with a combination of millimeter wave (mmWave) technology. Besides designing antenna systems for mmWave 5G wireless communication services [2], special attention should also be paid to designing links between core network and access nodes (backhaul), as well as links between central unit and multiple remote distributed units (fronthaul). The designing of backhaul and fronthaul parts of 5G and beyond networks becomes an important challenge [3, 4]. Besidesoptical fiber and millimeter waves, free-space optics (FSO) technology has been recognized to be able to fulfill the requirements of 5G and beyond backhaul and fronthaul networks. Generally, FSO is a promising technology for first-mile and last-mile links because of larger bandwidth, greater immunity to electromagnetic interference, better security and greater flexibility compared with traditional radio-frequency systems [5]. The signal propagation over an FSO link is influenced by several impairments. The intensity of optical signal propagating over an FSO channel is a random process due to turbulence-induced scintillation. Actually, inhomogeneity in atmospheric temperature and pressure cause variations of refractive index along the optical beam propagation path. The resulting atmospheric turbulence influences the signal phase and intensity fluctuations [6]. The stochastic variations of optical signal irradiance can be statistically characterized by different distributions, such as \(K\), Gamma-Gamma, Fisher-Snedecor and exponential Weibull distribution [5; 6; 7; 8]. In addition, Malaga (\(\mathcal{M}\)) distribution was proposed for modeling the atmospheric turbulence channel when there is a light line-of-sight component (LOS), as well as scattered components by the eddies on the propagation axis and off-axis eddies. Some other distributions are particular cases of Malaga (\(\mathcal{M}\)) distribution for specific numerical values of its parameters [9; 10]. In [11], the authors used the so-called mixture Gamma distribution to study the outage and error performance of FSO systems. The mixture Gamma distribution was initially proposed in [12] with the aim to approximate more complex Malaga (\(\mathcal{M}\)) distributions in estimating FSO system performance. Besides turbulence-induced scintillation, misalignment fading (also known as pointing errors) has the strong influence on FSO system performance as well. Pointing errors are caused by building sway, thermal expansion and wind. This phenomenon consists of boresight and jitter. Boresight exists when there is the fixed displacement between transmitter laser and receiver detector. Jitter corresponds to random offset of received beam center at the detector surface. Both effects were considered in [12], while the effect of jitter was emphasized in [13; 14]. Nowadays, FSO systems are mostly based on intensity modulation with direct detection (IM/DD) and on-off keying [5]. To achieve the best performance of these systems, it is necessary to adapt the decision threshold under varying turbulence conditions. In order to avoid adapting a threshold, subcarrier intensity modulation (SIM) technology was proposed for FSO links and has been extensively studied [5].