## 4 Numerical Results  
In the analysis that follows, we employ some experimental measurements of a real-world FSO link. By properly using the analytical expressions derived in the previous section, we can numerically verify the joint influence of \(M\)(alaga) fading and non-zero boresight pointing errors on the PSK performance for both single and multi-hop FSO configurations along with or without phase noise impairments. More specifically, we adopt the parameter values used by [PERSON] et al. and [PERSON] et al. in [15; 17]. The \(M\)(alaga) turbulence values are obtained by experimental measurements performed at the University of Waseda, Japan, on 15 October 2009 [27]. The specific link has been operated at \(\lambda=785\)\(\upmu\)m at a height of 25 m above sea level, and a receiver aperture diameter of 0.1 m. Additionally, the link length is 1 km, while the optical power used for data transmission is 11.5 dBm, with a responsivity of 0.8A/W [27]. We have to bear in mind that the strength of the atmospheric turbulence effect depends on the Rytov variance, which is defined as \(\sigma_{R}^{2}=1.23\,C_{n}^{2}\,\kappa^{7/6}L^{11/6}\), where \(C_{n}^{2}\) represents the index of the refraction structure parameter of the atmosphere, \(\kappa=2\pi/\lambda\) is the optical wave number, and \(  
the performance comparison between the first two figures reveals the negative-side effects of phase noise on FSO transmissions. Figure 3 exemplifies the same situation as in Figure 2 through Equation (32), but now under a moderate turbulence regime governed by \(\rho=0.75\). Indeed, by assuming identical 4-PSK multi-hop configurations along with the same phase noise and non-zero boresight pointing error parameters,  
Figure 1: Average symbol error probability (ASEP) versus the average electrical signal-to-noise ratio(SNR) of single-hop or multi-hop free-space optics (FSO) configurations for weak turbulence strengths, a weak amount of non-zero boresight pointing errors, and various phase shift keying (PSK) modulation formats with negligible phase noise impairments. Figure 2: ASEP versus average electrical SNR of multi-hop FSO configurations for weak turbulence strengths, weak and strong amounts of non-zero boresight pointing errors, and 4-PSK modulation formats with moderate and strong phase noise impairments. but over moderate, and thus stronger \(M\)(alaga) turbulence conditions, we observe that the ASEP significantly degrades. This highly demonstrates that a practical FSO system is subject to a significant ASEP increase as the atmospheric turbulence effect gets stronger. The latter observation is also emphasized in Figure 4, where both weak and strong \(M\)(alaga) turbulent channel states are illustrated for dual-hop and triple-hop DF relaying implementations. By considering weak phase noise effects and moderate non-zero boresight misalignment, we note that for strong turbulence conditions, the use of 4-PSK signals leads to enhanced ASEP results compared with the 8-PSK scheme. Thus, in this Figure, the ASEP degradation is verified due to higher order PSK formats along with stronger turbulence-induced fading. Figure 4: ASEP versus average electrical SNR of multi-hop FSO configurations for weak and strong turbulence strengths, a moderate amount of non-zero boresight pointing errors, and 4-PSK as well as 8-PSK modulation formats with weak phase noise impairments. Figure 3: ASEP versus average electrical SNR of multi-hop FSO configurations for moderate turbulence strengths, weak and strong amounts of non-zero boresight pointing errors, and 4-PSK modulation formats with moderate and strong phase noise impairments. Finally, by using Equations (41) and (42), Figure 5 depicts the reliability by means of the MOD per hour performance metric for both single-hop and dual-hop realizations along with weak turbulence effects and weak-to-moderate non-zero boresight misalignments over a wide SNR range that is normalized by the related threshold, i.e., \(\mu_{n,NZB}/\gamma_{n,th}\).