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Free-space optical communication systems over fading channels Yang, Fan 2015

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Free-Space Optical CommunicationSystems Over Fading ChannelsbyFan YangB.Sc., South China University of Technology, 2010M.Sc., Southeast University, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)January 2016c© Fan Yang, 2015AbstractFree-space optical (FSO) communication systems can provide larger band-width and rapid deployment for communication links. Such systems do notinterfere with existing radio frequency (RF) systems and can make com-munication more secure. However, the performance of FSO communicationsystems is highly dependent on its channel conditions. The atmosphericchannels can impose attenuation and scintillation effects on the communi-cation link, and these effects can hinder the correct detection of informationon receiver side.In this thesis, we focus on the performance analysis of terrestrial FSOsystems over atmospheric fading channels. One successful channel model tofit the experiment data is the lognormal-Rician model, but its widely adop-tion is impeded by its analytically intractable probability density function(PDF). Therefore we use Pade´ approximants method to obtain accurate ap-proximations of the PDF, cumulative density function, and moment generat-ing function of lognormal-Rician distribution. Simple closed-form bit-errorrate (BER) expression are obtained for binary phase-shift keying (BPSK)modulation with maximum ratio combining (MRC) reception and for binarydifferential phase-shift keying (DPSK) with selection combing (SC) recep-tion. Asymptotic error rate analysis for BPSK and DPSK is also presentedto reveal the performance behavior in large signal-to-noise ratio regimes.The pointing error effects in FSO systems can also contribute to channelimpairments. In order to study the influence of pointing error on systemperformance, we propose a statistical model for pointing error with nonzeroboresight error, which takes into account of laser beamwidth, detector aper-ture size, and jitter variance. A novel closed-form PDF is derived for thispointing error model. Furthermore, we obtain closed-form PDF and seriesPDF, respectively, for the composite lognormal and Gamma-Gamma tur-bulence channels with nonzero boresight pointing errors. We conduct errorrate analysis of on-off keying signaling with intensity modulation and directdetection over the lognormal and Gamma-Gamma fading channels. TheBER results are presented in highly accurate converging series. Asymptoticerror rate analysis and outage probability of such a system are also presentediiAbstractbased on the derived composite PDFs. It is shown that the boresight errorcan only affect the coding gain, while the diversity order is determined byeither the atmospheric fading effect or the pointing error effect, dependingon which effect is more dominant.For subcarrier intensity modulated FSO systems, the carrier phase es-timation error (CPE) would degrade the system performance. We studythe BER performance of subcarrier M -ary phase-shift keying systems withcarrier phase errors (CPE) in lognormal turbulence channels. The CPEis modeled as a Tikhonov random variable. The CPE induced asymptoticnoise reference losses for the studied systems are quantified analytically byintroducing the lognormal-Nakagami fading as an auxiliary channel model.One effective counter fading technique is spatial diversity, which requiresmultiple apertures at transmitter or receiver side. We first conduct a diver-sity analysis on single-branch FSO systems over atmospheric fading chan-nels. We find that the diversity order of an FSO system is usually de-termined by small scale effects in its fading channels when the irradiancefluctuation can be modeled as a modulation process (K, lognormal-Rician,Gamma-Gamma and M distribution). Based on this observation and thefact that lognormal channel does not have valid diversity order, we proposea lognormal-Nakagami model to facilitate asymptotic analysis on lognormalchannels. Using such an approach, we study different multi-branch FSO sys-tems over correlated lognormal fading channels that may have nonidenticalvariance. We discover that the correlation among the lognormal channelscan impose large signal-to-noise ratio (SNR) penalty to system bit-error rateperformance, compared to that of a similar system with independent log-normal channels. This property is not shared with the other commonly usedfading channels. In addition, we also derive a compact expression for theasymptotic relative diversity order (ARDO) between an L-branch combin-ing system over correlated lognormal channels and a single-branch system.It is found that the ARDO is related to the number of diversity branchesas well as entry-wise norm of the covariance matrix of the logarithm of thelognormal channel states. While maximal ratio combing (MRC), equal gaincombining (EGC) and selection combining (SC) result in the same ARDO,we find that the coding gain difference between MRC and EGC is negligible,but SC suffers a 10 log(L) dB loss.iiiPrefaceThis thesis is based on the research work conducted in the School ofEngineering at UBC Okanagan campus under the supervision of Dr. JulianCheng.Chapter 4 of this thesis is partially published in J5 and C4. In thesepublications, I proposed the research topic and conducted all the analysisand simulation. Dr. T. A. Tsiftsis helped with proofreading the manuscript.Chapter 5 contains a part of the results in J2. In this journal paper, Iconducted the system performance analysis and evaluation on asymptoticperformance loss between binary phase-shift keying and quadrature phase-shift keying modulation scheme. Dr. X. Song established the system modeland performed the analytical analysis and numerical evaluation of the systemperformance. In Chapter 5 of this thesis, we generalize the analysis to M-aryphase-shift keying.Refereed Journal PublicationsJ1. B. Zhu, F. Yang, J. Cheng and L. Wu, “Performance Bounds forDiversity Receptions Over Arbitrarily Correlated Nakagami-m FadingChannels,” Accepted for publication in IEEE Trans. Wireless Com-mun., 2015.J2. X. Song, F. Yang, J. Cheng, N. Al-Dhahir, and Z. Xu, “SubcarrierPhase-Shift Keying Systems With Phase Errors in Lognormal Turbu-lence Channels,” IEEE/OSA J. Lightwave Technol., vol.33, pp. 1896-1904, May 2015. (Part of Chapter 5)J3. X. Song, F. Yang, J. Cheng, and M.-S. Alouini, “Asymptotic SERPerformance Comparison of MPSK and MDPSK in Wireless FadingChannels,” IEEE Wireless Commun. Lett., vol. 4, pp. 18-21, Feb.2015.J4. X. Song, F. Yang, J. Cheng, and M.-S. Alouini, “BER of SubcarrierMPSK/MDPSKModulated OWC Systems in Gamma-Gamma Turbu-ivPrefacelence,” IEEE/OSA J. Lightwave Technol., vol. 33, no. 1, pp. 161-170,Jan. 2015.J5. F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-Space Optical Com-munication With Nonzero Boresight Pointing Errors,” IEEE Trans.Commun., vol. 62, pp. 713-725, Feb. 2014. (Part of Chapter 4)J6. X. Song, F. Yang, and J. Cheng, “Subcarrier Intensity ModulatedOptical Wireless Communications in Atmospheric Turbulence withPointing Errors,” IEEE/OSA J. Opt. Commun. Netw., vol. 5, pp.349-358, Apr. 2013.J7. F. Yang and J. Cheng, “Coherent Free-Space Optical Communica-tions in Lognormal-Rician Turbulence,” IEEE Commun. Lett., vol.16, pp. 1872-1875, Nov. 2012. (Part of Chapter 3)Refereed Conference PublicationsC1. F. Yang and J. Cheng, “Recent Results on Correlated LognormalFading Channels,” Accepted for publication in International Confer-ence on Computing, Networking and Communications (ICNC), 2015.(Part of Chapter 6)C2. B. Zhu, F. Yang, J. Cheng and L. Wu, “Performance Bounds for MRCand SC Over Nakagami-m Fading Channels With Arbitrary Correla-tion,” Accepted for publication in International Conference on Com-puting, Networking and Communications (ICNC), 2015.C3. X. Song, F. Yang, J. Cheng, andM.-S. Alouini, “Subcarrier MPSK/MDPSKModulated Optical Wireless Communications in Lognormal Turbu-lence,” Proceedings of 2015 IEEE Wireless Communications and Net-working Conference (WCNC), New Orleans, LA, Mar. 9-12, 2015.C4. F. Yang, J. Cheng, and T. A. Tsiftsis, “Free-Space Optical Commu-nications With Generalized Pointing Errors,” Proceedings of the IEEEInternational Conference on Communications (ICC), Budapest, Hun-gary, June 9-13, 2013. (Part of Chapter 4)C5. X. Song, F. Yang, and J. Cheng, “Subcarrier BPSK Modulated FSOCommunications With Pointing Errors,” Proceedings of the IEEEWire-less Communications and Networking Conference (WCNC), Shanghai,China, Apr. 7-10, 2013.vPrefaceRefereed Journal Publications (submitted)SJ1. F. Yang and J. Cheng, “Asymptotic Performance Analysis of Free-Space Optical Communication over Correlated Lognormal Channels,”Submitted to IEEE Trans. Wireless Commun., 2015. (Part of Chapter6)viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Organization and Contributions . . . . . . . . . . . . . 10Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . 132.1 FSO Communication System Model . . . . . . . . . . . . . . 132.2 IM/DD FSO Systems . . . . . . . . . . . . . . . . . . . . . . 142.3 Coherent FSO Systems . . . . . . . . . . . . . . . . . . . . . . 152.4 Atmospheric Turbulence Channel Models . . . . . . . . . . . 162.4.1 Lognormal Fading Model . . . . . . . . . . . . . . . . 172.4.2 Gamma-Gamma Fading Model . . . . . . . . . . . . . 172.4.3 Lognormal-Rician Fading Model . . . . . . . . . . . . 182.5 Error Rate Performance Analysis . . . . . . . . . . . . . . . . 19viiTABLE OF CONTENTS2.5.1 Bit Error Rate . . . . . . . . . . . . . . . . . . . . . . 202.5.2 Outage Probability . . . . . . . . . . . . . . . . . . . . 202.5.3 Asymptotic Error Rate . . . . . . . . . . . . . . . . . 202.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Chapter 3: Performance Analysis of FSO Communicationsover Lognormal-Rician Fading Channels . . . . . . 223.1 Pade´ Approximants . . . . . . . . . . . . . . . . . . . . . . . 223.2 Approximating PDF and CDF of Output SNR . . . . . . . . 243.3 Error Rate Analysis . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 BER for BPSK and DPSK with Spatial Diversity . . . 253.3.2 Asymptotic Error Rate Analysis . . . . . . . . . . . . 273.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Chapter 4: FSO Communication with Nonzero Boresight Point-ing Error . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Nonzero Boresight Pointing Errors Model . . . . . . . . . . . 334.1.1 Pointing Errors . . . . . . . . . . . . . . . . . . . . . . 344.1.2 Composite PDF with Generalized Pointing Error . . . 354.2 Error Rate Performance . . . . . . . . . . . . . . . . . . . . . 374.2.1 Bit-Error Rate . . . . . . . . . . . . . . . . . . . . . . 374.2.2 Asymptotic Error Rate Analysis . . . . . . . . . . . . 404.2.3 Outage Probability . . . . . . . . . . . . . . . . . . . . 424.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Chapter 5: Performance of SubcarrierM-ary PSK with PhaseRecovery Error over Lognormal Fading Channels 535.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.1 Phase Error . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Asymptotic Noise Reference Loss Analysis . . . . . . . . . . . 555.2.1 Subcarrier MPSK System . . . . . . . . . . . . . . . . 555.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Chapter 6: Asymptotic Performance Analysis of FSO Com-munication over Correlated Lognormal Fading Chan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 63viiiTABLE OF CONTENTS6.2 Diversity Analysis of FSO Systems . . . . . . . . . . . . . . . 666.3 FSO Systems over Correlated Lognormal Fading Channels . . 736.3.1 Dual-branch System . . . . . . . . . . . . . . . . . . . 736.3.2 Multiple-Branch System . . . . . . . . . . . . . . . . . 746.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Chapter 7: Conclusions . . . . . . . . . . . . . . . . . . . . . . . 827.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 827.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.2.1 FSO system with pointing errors . . . . . . . . . . . . 847.2.2 FSO Networks over Correlated Lognormal Fading Chan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Appendix A: Analytical and Numerical Results of fhp(hp) . . . . . 94Appendix B: Derivation of the moments of hp . . . . . . . . . . . . 96Appendix C: Gamma-Gamma Composite PDF . . . . . . . . . . . 98Appendix D: Bound on approximation error . . . . . . . . . . . . . 101Appendix E: Proof of convergence of series . . . . . . . . . . . . . 103Appendix F: Approximation error . . . . . . . . . . . . . . . . . . 105Appendix G: PDF of received instantaneous SNR of multiple-branchsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 107Appendix H: ARDO of MRC . . . . . . . . . . . . . . . . . . . . . 113ixList of TablesTable 4.1 System Settings . . . . . . . . . . . . . . . . . . . . . . 50Table 4.2 Weather Conditions . . . . . . . . . . . . . . . . . . . 50Table 6.1 SNR and BER offset between MRC, EGC and SC sys-tem over lognormal fading channels (σ21 = σ22 = 4). . . 80Table 6.2 SNR and BER offset between MRC, EGC and SC sys-tem over lognormal fading channels (σ21 = σ22 = 0.64). . 80Table D.1 Values of B for different RB values . . . . . . . . . . . 102Table F.1 Minimum Required Pt(dBm) forε(Pe)Pe< 10−6 andε(Pout)Pout< 10−6 . . . . . . . . . . . . . . . . . . . . . . . 106xList of FiguresFigure 2.1 Block diagram of an FSO system. . . . . . . . . . . . 13Figure 3.1 The exact and approximate CDF in different lognormal-Rician parameters. . . . . . . . . . . . . . . . . . . . . 28Figure 3.2 The exact and approximate BER of BPSK MRC andBER of DPSK SC (L = 1, 2, 3) in lognormal-Rician(r = 5, σ2z = 0.4) turbulence. . . . . . . . . . . . . . . 29Figure 3.3 Asymptotic error analysis of BPSK over multi-branchMRC (L = 1, 2, 3) in lognormal-Rician (r = 5, σ2z =0.4) turbulence. . . . . . . . . . . . . . . . . . . . . . 30Figure 3.4 Asymptotic error analysis of DPSK over multi-branchSC (L = 1, 2, 3) in lognormal-Rician (r = 5, σ2z = 0.4)turbulence. . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 4.1 BER performance of IM/DD OOK over the lognormalfading with zero and nonzero boresight pointing errors. 43Figure 4.2 BER for the composite lognormal channel (σ2R = 0.01, s/a =2) with different jitter values. . . . . . . . . . . . . . . 44Figure 4.3 BER for the composite lognormal channel (σ2R = 0.01, s/a =2, σs/a = 1.5) with different beamwidth values. . . . . 45Figure 4.4 Outage probability of an FSO system over the lognor-mal fading with zero and nonzero boresight pointingerrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.5 BER performance of IM/DD OOK over the Gamma-Gamma fading with zero and nonzero boresight point-ing errors. . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 4.6 Outage probability of an FSO system over the Gamma-Gamma fading with zero and nonzero boresight point-ing errors. . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.7 The SNR penalty factor induced by boresight errorin different turbulence conditions. . . . . . . . . . . . 49xiLIST OF FIGURESFigure 5.1 Asymptotic noisy reference loss of subcarrier BPSKsystem over the lognormal-Nakagami channel withdifferent PLL parameter C values. . . . . . . . . . . . 60Figure 5.2 Asymptotic noisy reference loss of subcarrier QPSKsystem over the lognormal-Nakagami channel and thelognormal channels with different PLL parameter Cvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 6.1 The BER of an IM/DD OOK system over the Gamma-Gamma fading channel with parameters α = 4, β = 3. 69Figure 6.2 The BER of an IM/DD OOK system over the Gamma-Gamma fading channel with parameters α = 4, β = 1.3. 70Figure 6.3 The BER of an IM/DD OOK system over the lognormal-Nakagami fading channel with parameters σ2 = 0.09,m =3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 6.4 The BER of an IM/DD OOK system over the lognormal-Nakagami fading channel with parameters σ2 = 0.09,m =1.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 6.5 The RDO between a dual-branch system in the log-normal fading channels with σ21 = σ22 = 0.49 and asingle-branch system with σ2 = 0.49. . . . . . . . . . 76Figure 6.6 The RDO between a dual-branch system in the log-normal fading channels with σ21 = 0.49, σ22 = 0.64 anda single-branch system with σ2 = 0.49. . . . . . . . . 77Figure 6.7 The RDO between a three-branch system in the log-normal fading channels and a single-branch systemwith σ2 = 0.49. . . . . . . . . . . . . . . . . . . . . . . 78Figure A.1 Comparison of the analytical PDF in (4.5) and the ex-act PDF of hp under various system settings (wz/a =10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95xiiList of AcronymsARDO Asymptotic Relative Diversity OrderAWGN Additive White Gaussian NoiseBER Bit-Error RateBPSK Binary Phase-Shift KeyingCDF Cumulative Distribution FunctionCPE Carrier Phase ErrorDPSK Differential Phase-Shift KeyingEGC Equal Gain CombiningFSO Free-Space Opticali.i.d. Independent and Identically DistributedIM/DD Intensity Modulation with Direct DetectionLOS Line-of-SightMGF Mmoment Generating FunctionMIMO Multiple-Input Multiple-OutputMPSK M -ary Phase-Shift KeyingMRC Maximum Ratio CombiningOOK On-Off KeyingPA Pade´ ApproximantsPDF Probability Density FunctionPPM Pulse Position ModulationxiiiList of AcronymsRF Radio FrequencyRV Random VariableSC Selection CombinigSIM Subcarrier Intensity ModulationSNR Signal-to-Noise RatioxivList of SymbolsSymbols Definitionsℜ[·] The real part of a complex quantityj j2 = −1△f Effective noise bandwidth of a receiverσ2R The Rytov varianceC2n The index of refraction structure parameterσ2si The scintillation indexΓ(·) The Gamma functionKν(·) The modified Bessel function of the second kind oforder νE[·] The statistical expectation operationδ(·) The Dirac delta functionp! The factorial of a positive integer pZ The set of all integersMR(·) The moment generating function of a RV Rerfc(x) The complementary error functionQ(x) The Gaussian Q-functionIν(·) The modified Bessel function of the first kind withorder νMu,v(·) The Whittaker functionlog(·) The log function with base 10erf(·) The Gauss error function(Nn)The generalized binomial coefficientxvList of Symbols(·)i The Pochhammer symbol standing for a fallingfactorialR The set of all real numbersPr[·] The probability of an eventℑ{·} The imaginary part of a complex quantity| · | The absolute value of the argument⌊·⌉ The nearest integer to the argumentx ∗ y The convolution of x and yxviAcknowledgementsI want to express my deepest gratitude to my supervisor Dr. JulianCheng for his constant guidance and encouragement. I am deeply influencedby his work enthusiasm and scientific rigor.I want to thank Dr. Hongchuan Yang from University of Victoria forserving as my external examiner. I would also like to thank Dr. ShawnWang, Dr. Jonathan Holzman, and Dr. Jahangir Hossian for being mycommittee members. I really appreciate their valuable time. Besides, Iwould like to give special thanks to Dr. Bingcheng Zhu, Dr. Xuegui Songand Dr. Luanxia Yang for insightful discussions and valuable suggestionson my research work.Here I would like to thank my fiance Wendi Zhang for accompanying meon my Ph.D. adventure. She helps me restart the journey after detour, andrevive my strength after defeat.I would like to thank my friends who helped me when I was in Kelownaand I miss the fun we had together.Finally, I would like to thank my parents and grandparents for theirunconditional love and support over the years. I would not have made itwithout them.xviiTo My ParentsxviiiChapter 1Introduction1.1 Background and MotivationFree-space optical (FSO) communication system is a type of communi-cation system that transfers information with a high frequency carrier atoptical spectrum. The transmitter launches a narrow beam of light modu-lated by the signal. Then the light transmits through the atmosphere andis received and detected at the receiver. Such a system has several notableadvantages:− Potentially larger bandwidth than the radio frequency (RF) system− Narrowness of the transmitted beam for power concentration and se-cure communication− Reduced system complexity and easy to deployFSO systems have a long history. The first FSO system prototype is the“photophone” built by Alexander Bell and his assistant Charles Tainter onJune 21, 1880. In this experiment, Mr. Tainter transmitted a wireless voicetelephone message of a 213m distance from the roof of the Franklin Schoolto the window of Bell’s laboratory. The modulation of the transmitted lightbeam was accomplished by a mirror made to vibrate by a person’s voice:the very thin mirror would alternate between concave and convex forms,thus focusing or dispersing the light from the light source. The brightnessof the transmitted light beam was observed from the location of the receiver;therefore, it is varied in accordance with the audio-frequency variations act-ing upon the mirror on transmitter side. In 1960s, the invention of lasersboosted the development of FSO systems. However the technology lost itsmarket for civilian uses when optical fiber networks were introduced andwidely adopted, and the main applications for FSO system are for militarypurpose and deep space communication. In late 20th century, the mainreason for the recession of FSO market are:− Existing radio frequency (RF) systems can handle the demand of userat that time.11.1. Background and Motivation− The reliability of FSO system is determined by the atmospheric con-dition, and it can not be assured under extreme weather conditions.− The pointing and tracking devices for FSO system add complexity tosystem design.Now with the demand for larger bandwidth and rapid deployment, FSOfinds its civilian applications. The development of optoelectronics deviceand extensive research on FSO systems also promotes the FSO technique asan add-on or alternative communication infrastructure to RF system. Manyhybrid RF/FSO system are commercially available now. For instance, FSOsystem are being deployed for ultra low latency networks with high networkcapacity, and such networks are used in high frequency trading applications[1]. The company Facebook is applying free space optical technology to loworbit satellites and solar powered drones to connect people in developingcountries to the internet [2]. The future of FSO systems is bright, while thechallenges still remain:− The atmospherical attenuation can affect the link significantly.− The high directionality of the transmitted beam requires accurate ac-quisition and pointing.− The atmospheric turbulence can degrade the system performance.The attenuation of FSO channel is determined by the weather condition.At clear weather conditions, the attenuation is approximately 6.5 dB/km,and at a fog event, the attenuation can be 115 dB/km or even 173 dB/km[3]. Therefore the fog can usually cause outage of the FSO system and thelink range of FSO is limited.The scintillation of FSO channel is caused by thermally induced changesin the index of refraction of the air along the transmit path. The time scaleof these fluctuations is of the order of milliseconds, approximately equal tothe time that it takes a volume of air (having the same size as that of thebeam) to move across the path. Therefore the time scale is related to thewind speed [3].Pointing and alignment can also affect the performance of FSO systems.The narrow beam from transmitter has to be aligned with the receiver forproper detection; otherwise, the link can not be established. A typical FSOtransceiver transmits one or more beams of light, each of which is 5 ∼ 8cmin diameter at the transmitter and typically spreads to roughly 1 ∼ 5m indiameter at a range of 1 km and it is important that both the transmitted21.2. Literature Reviewbeam of light and the receive field of view cone encompass the transceiver atthe opposite end of the link [3]. Because of the building sway and beam wan-der effects, the accurate pointing can not be easily achieved. Active pointingdevice is usually used for accurate alignment, which greatly increases the costand complexity of the system. Therefore many FSO systems have relativelywider beam and operate with the initial pointing at first installation time(or requires manual monthly calibration).The performance of FSO system depends on various system parame-ters including link range, atmospherical model, transmit power, pointingmethod, etc. Therefore understanding how these system parameters affectthe overall performance of FSO system is critical in FSO system design. Ourresearch will focus on analyzing the effects of different system parameterson the performance of FSO systems. It will include the performance of FSOsystems over different fading channels, and the effect of pointing error andphase error on performance of FSO systems.1.2 Literature ReviewFSO communication systems gain much interest because of its capabil-ity for meeting the growing demand of high-data-rate connection and itsrapid deployment[4]. Some terrestrial FSO products provide data rates onthe order of Gbps, which is much greater than those of digital subscriberlines or coaxial cables [5]. Besides, the installation of an FSO system onlyrequires days, making it flexible and effective for deployment. Recently, theapplications of FSO communication systems include high data rate hybridnetworks (also known as RF/FSO hybrid communication system) for highspeed connection, ultra low latency networks for stock market trading [1],and fast deployed network for communication recovery.The FSO systems can be categorized by two types: intensity modu-lation with direct detection (IM/DD) systems and coherent systems. InIM/DD system, the lens system and photodetector operate to detect theinstantaneous power in the collected field when it arrives the receiver. Incoherent systems, the collected field is optically mixed with a local gener-ated field through a front end mirror before the photodetector. The on-offkeying (OOK) modulation is widely used for IM/DD FSO systems, since op-tical communication systems with higher order modulation are complex toimplement [6, 7]. In [8], the authors described several communication tech-niques to mitigate turbulence-induced intensity fluctuations for an IM/DDOOK system. In [9], the building sway problem was studied for an FSO31.2. Literature Reviewsystem with OOK modulation. In [10], the authors presented error rateperformance bounds for an OOK FSO communication systems over K fad-ing channels. In [11], the pointing error effect on a OOK FSO system wasinvestigated and a statistical model for pointing error factor was derived.In [7], the author studied a multiple-input multiple-output (MIMO) FSOlink over K turbulence channels, and IM/DD with OOK modulation is as-sumed. In [12], the authors analyzed the performance of an OOK FSOsystem with Hoyt distributed misalignments. In [13], the authors conductedexperimental evaluation of error performance for IM/DD FSO communi-cation links with different modulation schemes, which include OOK, pulseposition modulation (PPM) and binary phase-shift keying (BPSK). In co-herent FSO systems, the provision of phase information allows a varietyof digital modulation formats in comparison to irradiance modulation withdirect detection IM/DD. In such systems, the signal can be amplitude, fre-quency or phase modulated on the optical carrier. The received signals canbe made shot-noise-limited through the use of a local oscillator. Such coher-ent FSO systems offer excellent background noise rejection capability [14],higher sensitivity, and improved spectral efficiency. While there has been anumber of studies on IM/DD FSO systems [8], [10], there exists relativelyfewer analysis of coherent FSO systems. In [15], Kiasaleh introduced anexact bit-error rate (BER) expression for FSO communication links withdifferential phase-shift keying (DPSK). Belmonte and Kahn analyzed theperformance of M -ary phase-shift keying with lognormal turbulence [16].In [17], Niu et al. analyzed a coherent FSO systems under K-distributedstrong turbulence conditions. In [18], Sandalidis et al. analyzed a coherentFSO system in the presence of pointing errors.The main challenge in FSO communication system design is that thedeleterious effects in atmospherical channel can severely degrade the perfor-mance of such systems. The atmospheric attenuation can sometimes causeoutage of an FSO system, which also considerably limits its link range. Atclear weather conditions, the atmospheric attenuation is approximately 6.5dB/km, while the attenuation can be 115 dB/km or even 173 dB/km at afog event [3]. Therefore many FSO products operate at the link range lessthan 1km. Another adverse effect in atmospheric channel is scintillation,which is caused by thermally induced changes in the index of refraction ofthe air along the beam transmit path. As a result, the received irradiance atreceiver will randomly fluctuate. Such fluctuation can dramatically degradethe performance of FSO systems [13, 19]. The atmospheric attenuation canbe treated as a constant factor under the same weather condition, while thescintillation is considered as a random factor. Therefore we focus our study41.2. Literature Reviewon the scintillation effects rather than the attenuation. The scintillationeffects can be characterized by many statistical models describing the dis-tribution of channel states. The lognormal distribution is widely acceptedto describe weak scintillation, while the negative exponential distribution isused to describe the limiting case of saturated scintillation. The probabil-ity density function (PDF) proposed for strong scintillation include the Kdistribution [20], [21], the lognormally modulated exponential distribution[22], and the generalized Gamma distribution [23], [24]. Furthermore, someuniversal PDFs, including the I −K distribution [25], [26], the lognormal-Rician distribution [27], and the Gamma-Gamma distribution [28], havebeen proposed to model both weak and strong scintillation. Among thesePDF models, the lognormal-Rician model has excellent agreement with ex-perimental data for both weak and strong scintillation [29]. Despite thepractical importance of the lognormal-Rician turbulence channel, there ex-ists few performance study of FSO systems assuming this turbulence model.This is because the lognormal-Rician turbulence model is mathematicallyintractable. Therefore we propose to provide the engineering and scientificcommunities with a simple and accurate solution for evaluating the perfor-mance of FSO systems under the lognormal-Rician turbulence model.In addition to the scintillation effects, pointing and alignment can alsoaffect the performance of FSO systems. Because of the building sway andbeam wander effects, the accurate pointing cannot be easily achieved. In ter-restrial FSO communication systems, the transceivers are often positionedat the top of tall buildings to obtain a line of sight. The building sway,building vibration and thermal expansion of building can result in pointingerrors that consist of two components: boresight and jitter. The boresightis the fixed displacement between beam center and center of the detector.Although typical terrestrial FSO systems are initially installed with nearzero boresight error, the boresight is still considerable due to the thermalexpansion of the building. In [30], the pointing errors are recorded over threedays in a TerraLink laser communication system. The radial displacementshows a cyclical pattern every 24 hours, which suggests the thermal expan-sion of the building plays an important role in pointing errors. The reportedboresight can be as high as 0.3mrad [30]. In [31], the values of boresightin the range of 0.0039 − 0.0117mrad are used. In [32], the author consid-ered boresight up to 0.03mrad in a horizontal FSO link. The jitter is therandom offset of the beam center at detector plane, which is mainly causedby building sway and building vibration. The typical value of jitter stan-dard deviation in a terrestrial FSO system is below 0.3mrad [30]. In [31],the jitter standard deviations of 0.0033 − 0.0100mrad are used. In [32], the51.2. Literature Reviewauthor considered jitter standard deviation up to 0.01mrad. In satellite-to-ground and intersatellite communications, the transmitter and receiver havehigh relative velocity, and there is mechanical noise due to satellite-basedmotion and gimbal friction [33]. Thus, it is difficult to realize perfect track-ing, while jitter and boresight can also arise as residual pointing error. In[9], the author proposed a mathematical model to minimize the transmitterpower and the beam divergence angle in an urban optical wireless commu-nication system with pointing errors caused by building sway. In [34], amaximum-likelihood estimator was developed to estimate the boresight andjitter component of the pointing error. In this system model, a point de-tector and nonzero boresight component are assumed. In a follow-up work[32], the same authors further considered the effects of atmospheric tur-bulence for the lognormal and Gamma-Gamma fading channels, and theyadopted a wave-optics based approach to evaluate the channel capacity. In[11], a statistical pointing error model was proposed by considering the laserbeamwidth, jitter variance, and detector size. In this model, a closed-formPDF of the pointing error loss factor was derived by assuming that the point-ing error has zero boresight. In [11], the size of detector area was taken intoconsideration. When the size of the detector area becomes negligibly small,this pointing error loss factor coincides with the one derived for the pointdetector in earlier works [9], [35]. In [36], the error rate performance of FSOlinks over K-distributed turbulence channels in the presence of pointing er-ror is studied. The average bit-error rate (BER) is presented in closed-formusing the Meijer G-function. In a related work [18], the authors studiedthe BER performance of a heterodyne differential phase shift keying opti-cal wireless communication system in the presence of pointing error overthe Gamma-Gamma turbulence channels. In [37], two optimization mod-els were proposed to mitigate the pointing error effects with zero boresightby taking into account the transmitter power, wavelength, transmitter andreceiver telescope gains. In [38], M -ary PPM was investigated with impair-ments from atmospheric turbulence and pointing error with zero boresight.In [39], asymptotic BER performance is analyzed for FSO communicationsystems using transmit laser selection over atmospheric turbulence channelswith the same pointing error model developed in [11]. A statistical chan-nel model was recently proposed for MIMO FSO communication systemsover atmospheric fading channels with pointing errors [40]. Both outageprobability and diversity order were studied, and it was found that the di-versity order is determined by pointing error effects other than the numberof transmitters or receivers. More recently, the pointing error model in [11]was generalized by modeling the radial displacement distance with a Hoyt61.2. Literature Reviewor Nakagami-q distribution [12], which allows the horizontal and elevationjitters to have nonidentical jitter standard deviations. The pointing errormodel developed in [11] is widely used in the literature [18, 36, 38, 39]. Inthis model, the boresight component of pointing error is assumed to be zero,and both horizontal and elevation displacements are assumed to follow anindependent, identically distributed zero-mean Gaussian distribution [35].As a result, the random radial displacement at the receiver is Rayleigh dis-tributed. While most aforementioned literatures fucus on pointing errorswith zero boresight, our proposed study will focus on the effect of nonzeroboresight component on performance of terrestrial FSO systems.Besides the scintillation and pointing error effects, the carrier phase er-ror (CPE) can degrade the performance of an subcarrier intensity modula-tion (SIM) FSO system. Coherent modulation schemes are usually used forSIM FSO systems, which requires carrier phase recovery for the subcarriersignal [41]. In such system, the received optical intensity is first being con-verted to electrical signal, then the phase of the modulated signal is beinglocked/estimated by a phase locked loop (PLL), where there may exist cer-tain errors on the estimated phase. In [42], the error rate performance ofan SIM system with DPSK and M -ary phase-shift keying (MPSK) over thelognormal turbulence channels is analyzed. In [43–45], the authors studiedBER performance of a BPSK SIM system over lognormal, negative expo-nential, and Gamma-Gamma fading channels. The BER performance of aBPSK SIM system over K-distributed turbulence channels was studied in[46]. In [47–49], the authors studied the error rate performance of SIM sys-tems with various PSK modulations over different atmospheric turbulencechannels. While the above mentioned literatures emphasize on the errorrate performance of SIM systems over different atmospheric fading chan-nels, there is few work studying the effect of carrier phase synchronizationerror on such systems. In fact, there exists certain errors in the carrierphase tracking for SIM systems using coherent modulation. Therefore, wepropose to study the BER performance of CPE impaired SIM systems overatmospheric fading channels.In order to combat scintillation, FSO designers often employ multi-branch reception (or transmitter) diversity technique [50]. Such techniquecan significantly reduce the probability of system outage caused by deepfading in the channel. Using multiple apertures can also increase the totaltransmit laser power while guarantee eye safety. Therefore the receiver canobtain higher power which allows the system to operate at higher signal-to-noise ratio or longer transmission distances. Aperture averaging can alsobe used for fading mitigation when the aperture size is much larger than71.2. Literature Reviewthe correlation length [5]. However, it is not always feasible to make theaperture large enough, which justifies using spatial diversity at the receiverside [6]. There are several papers discussing the performance of FSO sys-tems using spatial diversity. For examples, the author in [51] analyzed FSOsystems with spatial diversity reception over independent and identicallydistributed (i.i.d.) lognormal fading channels. The system performance wasstudied under selection combinig (SC) and equal gain combining (EGC) re-ceptions. In [52], the outage probability and the power gain of MIMO FSOsystems are derived assuming i.i.d. lognormal fading channels. The per-formance of FSO channels using spatial diversity was investigated in [53].The authors adopted i.i.d. lognormal fading channels and compared differ-ent reception techniques including aperture averaging, linear combining, andadaptive optics. They found that the maximum ratio combining (MRC) andEGC have similar performance in lognormal fading channels; however, theyprovided no analytical proof. In [54], the authors studied a MIMO FSOsystem with repetition Q-ary PPM. Both Rayleigh and lognormal fadingmodels are considered for the system assuming independent fading. TheBER for a spatial repetition code was analyzed in [7] for strong atmosphericturbulence. Closed-form expressions for the average BER of single-inputmultiple-output FSO systems over i.i.d. K-distributed fading channels wasobtained. In [40], a generalized statistical model for MIMO FSO channelsimpaired by atmospheric and misalignment fading was developed. The de-rived model was then used to study the outage probability and diversity gainof FSO channels over i.i.d. lognormal fading channels. For FSO systemswith spatial diversity, channel independence is often desired for optimumperformance. The correlation length for such systems can be approximatedby√λL, where λ denotes the optical carrier’s wavelength and L the linkdistance [6]. In order to guarantee channel independence, the detectors arepreferred to be placed as far apart as possible [8]. However, in practice wemay not always have sufficient spacing between optical apertures, and usu-ally the receiver apertures are closely placed for better power concentration[51]. Besides, FSO links are highly directive that the channels between thetransmitter and receiver are likely to be dependent. Therefore, the FSO sys-tems with spatial diversity will likely experience correlated fading channels[8, 54–56]. As aforementioned, lognormal model is a common fading channelmodel for FSO systems with short link distance (several hundred meters)or under weak turbulence condition [57]. Lognormal is an important fadingmodel because it fits empirical fading measurements well in many transmis-sion scenarios of practical interest [58]. For example, the lognormal fadingcan characterize the shadowing effects in outdoor RF communications and81.2. Literature Reviewdescribe indoor radio propagation environments. The lognormal fading isalso suitable for describing ultra-wideband channels [59]. The performanceof an MIMO FSO system over correlated lognormal fading channels was an-alyzed in [8]. Assuming identically distributed lognormal channel model, theauthors found that the diversity gain will be reduced by correlation when thespacing between receivers is not much greater than the fading correlationlength. In order to reduce the diversity gain penalty caused by correla-tion, the authors further proposed a maximum-likelihood detection schemefor spatial diversity reception. In [6], the authors investigated the BERperformance of FSO links with spatial diversity over lognormal atmosphericturbulence fading channels. An approximation for the sum of correlated log-normal random variables was used to study the EGC system over correlatedlognormal channels with identical variance. In [55], the authors studied acooperative diversity FSO system over Rayleigh and lognormal fading chan-nels. They concluded that when the receivers in MIMO FSO systems sufferfrom correlation, relays can be a practical alternative to achieve full spatialdiversity. Diversity analysis of communication systems can give us insightsinto how the parameters affect system performance over fading channels,and show which parameters dominate the system performance. By con-ducting diversity analysis, we can usually obtain two system performancemetrics: diversity order and coding gain, which can be considered as simplecriteria for system optimization [60]. The diversity order1 will indicate howfast the BER decreases with increasing average signal-to-noise ratio (SNR)in high SNR region, and the coding gain denotes the amount of shift of theBER curve relative to the benchmark curve. There are many prior worksconducting diversity analysis on FSO systems over various fading channels[7, 17, 39, 40, 61–63]. A modern FSO system may employ multiple trans-mitters and receivers at various locations (or multi-hop relays); therefore,the beams may experience different channel conditions. For example, thebeams propagating on a horizontal path above a parking lot will experiencedifferent amount of fading than those above grass or other terrain. There-fore the terrestrial FSO communication systems employed in urban areamay experience different amount of fading on different branches, due to thefact that the beam paths cross complex terrains. Although the beams maybe close to each other, the thermal gradient and different wind level on thebeam path may cause the beams to have nonidentical distribution. Other1The diversity order used in this paper describes the rate of descending BER withincreasing SNR in a loglog plot, which may have different definition with those used intraditional RF field.91.3. Thesis Organization and Contributionsexamples include systems with closely located beam paths with different linkdistances, and systems involving relay nodes. Therefore the multiple chan-nels of beams may have nonidentical distribution and be correlated witheach other. However, there exists no prior works on diversity analysis ofFSO systems over correlated lognormal fading channels with nonidenticaldistribution.1.3 Thesis Organization and ContributionsIn this thesis, we present the research work conducted on the followingfour topics:− Performance of FSO systems over lognormal-Rician channels− Effect of pointing errors on FSO systems over fading channels− Effect of phase errors on FSO systems over lognormal channels− Effect of correlation on FSO systems over lognormal channelsWe discuss each topic with one chapter. The summary and contributions ofeach chapter are as follows.In Chapter 1, we provide an introduction to FSO communication sys-tems. The history of FSO is also discussed. The challenges in FSO systemsare presented and related research works are reviewed.Chapter 2 presents some background knowledge for the FSO systemsstudied in this thesis. Two types of FSO systems: Coherent and IM/DDsystems, are discussed. We also present the performance factors of an FSOsystem and introduce atmospheric turbulence channel models including log-normal, Gamma-Gamma and lognormal-Rician fading models. The systemmodel and receiver SNR statistics are also discussed, which will be used bythe derivations in subsequent chapters.In Chapter 3, we analyze the error rate performance of a coherent FSOsystem over lognormal-Rician turbulence channels. The PDF, cumulativedistribution function (CDF) and mmoment generating function (MGF) oflognormal-Rician distributed irradiance are approximated using pade´ ap-proximants (PA). We obtain highly accurate closed-form approximate BERfor BPSK with MRC reception using an MGF approach, and we deriveclosed-form approximate BERs for DPSK with SC reception using a PDFapproximated by PA. We then present an asymptotic error rate analysis toillustrate error rate performance under large SNR condition.101.3. Thesis Organization and ContributionsIn Chapter 4, we propose a nonzero boresight pointing error model for anFSO system by taking into account of beamwidth, detector aperture, and jit-ter variance. A closed-form PDF has been derived for the nonzero boresightpointing error model. We derive closed-form composite PDF for the lognor-mal fading and highly accurate series based PDF for the Gamma-Gammafading. Highly accurate convergent series BER expressions are obtained.Through the asymptotic BER analysis and numerical results, we observethat the boresight error causes an SNR penalty factor on error rate perfor-mance at high SNR. By examining the asymptotic BER curves, we find thatthe diversity order of the FSO system over the composite lognormal fadingchannel is solely determined by the pointing error parameter γ2, and theboresight component does not affect the diversity order. While in the com-posite Gamma-Gamma fading channel, the diversity order is determined byeither the Gamma-Gamma fading effect or the pointing error effect.In Chapter 5, we study the effects of CPE on a SIM system. TheBER performance of such system with M -ary phase-shift keying modula-tion is investigated under the lognormal turbulence channels. We use theTikhonov model for CPE with the assumption that the SNR in receiver’sPLL is proportional to the instantaneous receiver SNR. Using an auxiliarylognormal-Nakagami fading model, we obtain the CPE induced asymptoticnoise reference losses of the FSO system over the lognormal channels.In Chapter 6, we study a multiple-branch FSO systems over correlatedlognormal fading channels. The diversity of FSO systems over compos-ite fading channels is also analyzed, and we find that the diversity of thesystem is determined by the lowest diversity random variable (RV) in theproduct form. Based on this observation and the fact that lognormal chan-nel does not have valid diversity order, we propose a lognormal-Nakagamimodel to facilitate asymptotic analysis on lognormal channels. Using thisapproach, we study different multi-branch FSO systems over correlated log-normal fading channels that may have nonidentical variance. We discoverthat the correlation among the lognormal channels can impose large SNRpenalty to system BER performance, compared to that of a similar systemwith independent lognormal channels. This property is not shared with theother commonly used fading channels. In addition, we also derive a com-pact expression for the asymptotic relative diversity order (ARDO) betweenan L-branch combining system over correlated lognormal channels and asingle-branch system. It is found that the ARDO is related to the numberof diversity branches as well as entry-wise norm of the covariance matrixof the logarithm of the lognormal channel states. The derived ARDO canprovide FSO system designers with a simple asymptotic performance metric111.3. Thesis Organization and Contributionsto allow asymptotic performance comparison between different FSO sys-tems under various lognormal channel conditions. The MRC, EGC andSC combining techniques were also compared in the context of correlatedlognormal channels. We found that the MRC and EGC have similar per-formance but SC suffers a 10 log(L)dB loss, which suggests that EGC isdesirable for multiple-branch FSO systems in terms of better performanceand complexity tradeoff.In Chapter 7, we summarize the works presented in this thesis, anddiscuss some future research topics that are closely related to the researchwork in previous chapters.12Chapter 2BackgroundIn this chapter, we first introduce the general FSO system model. De-pending on the receiver structure, we discuss two categories of FSO systems:the IM/DD FSO system and the coherent FSO system. For IM/DD systems,the OOK modulation and SIM modulation are also discussed. Then we pro-vide some basic knowledge on the atmospheric fading effects, and presentthree widely accepted atmospheric fading channel models. In the end, wediscuss three system performance metrics including BER, outage probabil-ity and asymptotic BER, which are used for evaluating the performance ofFSO systems in following chapters.2.1 FSO Communication System ModelIn an FSO communication system, a laser source first transmits a nar-row beam of modulated light at the transmitter. Then the modulated lightbeam propagates through free space before being received by the receiver.Fig. 2.1 shows a simplified block diagram of such system, which employs twosets of transceivers for a full-duplex link. For communication purpose, theinformation can be modulated onto the amplitude, frequency, phase or in-tensity of the optical carrier. The optical carrier is then transmitted throughthe atmospheric channel. At the receiver, the incident optical field is col-lected by the lens system, and being converted to electrical signal by theFigure 2.1: Block diagram of an FSO system.132.2. IM/DD FSO Systemsphotodetector. The electrical signal is further processed to recover informa-tion. Based on the receiver structure, the FSO communication systems canbe categorized by two types: IM/DD systems and coherent systems. Thereceiver of an IM/DD FSO system often consists a power detector that re-sponds only to the instantaneous power of the collected field. Such receiveris used when the signal is intensity modulated at the transmitter end, andit is often called a direct detection receiver. The structure of an intensitymodulation transmitter with a direct detection receiver is acronymized asIM/DD. Another type of receiver is the heterodyning receiver used in co-herent systems. In such a receiver, the received optical field is mixed witha locally generated field before the photodetector. Heterodyning receiver isused when the signal is amplitude, frequency or phase modulated on the op-tical carrier at the transmitter end, and it is often called a coherent receiver.The coherent receiver is more complex than the detection receiver, becausethe coherent receiver requires the local oscillator laser to be tuned to thesame frequency and phase as the received carrier.2.2 IM/DD FSO SystemsAn IM/DD FSO system collects the transmitted optical field that is di-rectly imaged through the receiver lens system onto the photodetector. Thephotodector is a power detecting device, which responds to the instantaneousintensity of the received optical field. At the transmitter, the informationcan be directly modulated or subcarrier-modulated on the intensity of theoptical field. Assuming a turbulence free channel, the received instantaneousintensity can be expressed asI(t) = Ir(1 + km(t)) (2.1)where Ir is the average received intensity, m(t) is the modulating signaland k is a scaling factor. Since the transmitting intensity is a nonnegativequantity, |km(t)| ≤ 1 is required.In IM/DD FSO systems, the OOK modulation is widely used becauseof its simplicity. The modulating signal can be 1 for a logic “on” or −1 fora logic “off” with k = 1 [64]. At the nth symbol duration, we havem(t) ={1, nT < t < (n + 1)T “on”−1, nT < t < (n + 1)T “off” (2.2)where T denotes the symbol duration.142.3. Coherent FSO SystemsAnother modulation scheme for IM/DD systems is the subcarrier mod-ulation. In such modulation scheme, the modulating signal can have theformm(t) = ℜ[(∑iai exp(j(2pifi + θi)))exp(j2pifc)](2.3)where ℜ[z] gives the real part of z, ai, fi and θi respectively denote theamplitude, frequency and phase of ith base band equivalent signal, and fcis the carrier frequency.In an IM/DD FSO system, the main noise source are background noiseand circuit thermal noise, and we can express the receiver SNR asγ =(RAkI)2(qRNb +Nc)2∆f= CsI2 (2.4)where R denotes the photodetector responsivity, A is the detector area, q isthe electronic charge, ∆f denotes the noise equivalent bandwidth, Nb andNc respectively denote the power spectrum density of background noise andcircuit thermal noise, and Cs =(RA)2(qRNb+Nc)2∆fcan be treated as a constantwhen the system parameters are set.2.3 Coherent FSO SystemsA coherent FSO system mixes the received optical field with a localfield generated by a local oscillator at the photodetector to downconvertthe optical carrier to an intermediate frequency carrier. For simplicity, wediscuss a coherent FSO system with BPSK modulation. At the photodector,the incident mixed optical field is being detected and a beat term containingboth the amplitude and phase of optical carrier and local field is generatedi(t) = 2aLas(t)cos((ωo − ωL)t+ θs(t)− θL) (2.5)where as(t), ωo and θs(t) respectively denote the amplitude, frequency andphase of the received optical carrier, aL, ωL and θL respectively representthe amplitude, frequency and phase of the local field. After the filteringprocess, the received signal power can be expressed asPsignal = E[(Ri(t))2] = 2R2PLPs (2.6)where E[·] denotes the expectation, PL = a2LA is the local field power term,Ps = E[a2s(t)]A is the received optical field power term. Considering the152.4. Atmospheric Turbulence Channel Modelsshot noise, background noise and circuit thermal noise, we can express theinstantaneous SNR for coherent FSO systems asγ =2R2PLPs2qR∆fPL + 2qR∆fNb + 2∆fNc. (2.7)In order to suppress the circuit thermal noise, we can use a strong localsource with large PL, and assuming negligible background noise, we cansimplify the SNR in (2.7) asγ ≈ Rq∆fPs = CcI (2.8)where Cc =RAq∆f is a deterministic value for particular FSO system.2.4 Atmospheric Turbulence Channel ModelsThe atmospheric channel can impose attenuation and scintillation effecton the light beam propagating through it.The attenuation of atmospheric channels is determined by the weathercondition. Under clear weather conditions, the attenuation is approximately6.5 dB/km, and at a fog event, the attenuation can be 115 dB/km or even173 dB/km [3]. Therefore the fog can usually cause outage of the FSOsystem and the link range of FSO is limited.The scintillation effect of atmospheric channels is caused by thermallyinduced fluctuations in the index of refraction of the air along the transmitpath. The time scale of these fluctuations is on the order of milliseconds,approximately equal to the time that takes a volume of air (having the samesize as that of the beam) to move across the path. Therefore the time scaleis related to the wind speed [3]. We can use scintillation index to describethe strength of turbulence induced fading, which is given asσ2si =Var(I)(E[I])2=E(I2)(E[I])2− 1. (2.9)The scintillation index is the normalized variance of the intensity and is usedas a measure of scintillation. Another parameter related to the strength ofthe turbulence is Rytov variance, which approaches the scintillation indexunder weak turbulence conditionsσ2R = 1.23k7/6C2nz11/6 (2.10)162.4. Atmospheric Turbulence Channel Modelswhere C2n is the index of refraction structure parameter of atmosphere,k = 2pi/λ is the optical wavenumber with λ being the wavelength, and zdenotes the link distance. Depending on the value of Rytov variance, we canapproximately categorize the turbulence regime as follows [65]: the weak tur-bulence regime (σ2R < 0.3), the moderate turbulence regime (0.3 ≤ σ2R < 5),and the strong turbulence regime (σ2R ≥ 5).Under different levels of scintillation, there are different statistic modelsto describe the distribution of channel states. For weak turbulence con-ditions, the most widely accepted model is lognormal turbulence model.For moderate to strong turbulence conditions, Gamma-Gamma turbulencemodel is often used (describing a much wider irradiance fluctuations rangeswith the K-distributed turbulence model being its special case).2.4.1 Lognormal Fading ModelLognormal model is often used for FSO systems with short link dis-tance (several hundred meters) or under weak turbulence conditions [57].Lognormal is an important fading model because it fits empirical fadingmeasurements well in many transmission scenarios of practical interest [58].A lognormal RV h has the PDFfh(h) =12h√2piσ2Xexp(−(lnh+ 2σ2X)28σ2X)(2.11)where σ2X is the log-amplitude variance given by [11]σ2X ≈ σ2R/4 = 0.31k7/6C2nz11/6. (2.12)The parameters of the lognormal fading model can be measured directly forFSO systems [66].2.4.2 Gamma-Gamma Fading ModelFor medium to strong turbulence conditions, we can use the Gamma-Gamma turbulence model to characterize the atmospheric fading. A Gamma-Gamma distributed RV can be constructed by multiplying a Gamma RVwith another Gamma RV, which denote, respectively, the fading processcaused by small scale and large scale eddies. The PDF of a Gamma-GammaRV h is given byfh(h) =2(αβ)(α+β)/2Γ(α)Γ(β)hα+β2−1Kα−β(2√αβh)(2.13)172.4. Atmospheric Turbulence Channel Modelswhere Γ(·) is the Gamma function, and Kα−β(·) is the modified Bessel func-tion of the second kind of order α− β. The two shaping parameters α andβ are directly related to the effective number of large scale cells and smallscale cells of the scattering process respectively, and are related to the Rytovvariance σ2R. Assuming plane wave propagation with negligible inner scale,the parameters α and β are, respectively, given by [28]α =[exp(0.49σ2R(1 + 1.11σ12/5R )7/6)− 1]−1(2.14)andβ =[exp(0.51σ2R(1 + 0.69σ12/5R )5/6)− 1]−1. (2.15)2.4.3 Lognormal-Rician Fading ModelAnother important turbulence model is lognormal-Rician model [22, 62].It fits the experimental data better than the Gamma-Gamma model underboth weak and strong scintillation. The FSO communication inherently hasa line-of-sight (LOS) link; therefore, a Rician model with shadowing is anappropriate channel model. The amplitude PDF of the optical wave at thereceiver can be described aspZ(Z) =∫ ∞0p(Z|S)pS(S)dS (2.16)where p(Z|S) represents the Rician PDF conditioned on a shadowing S,which is assumed to be lognormal distributed. Mathematically, it can beshown thatZ = RS (2.17)where R and S are, respectively, the Rician and lognormal RV. We letR = |UC + UG| and S = exp(χ), where UC is a real deterministic quantityrepresenting the LOS component, UG is a circular complex Gaussian RVwith zero mean, and χ is a real Gaussian RV. The channel gain is given byh = Z2 [67], and it can be expressed as [22]h = |UC + UG|2 exp(2χ) (2.18)where exp(2χ) is another lognormal RV, and |UC + UG|2 is a noncentralchi-square RV with degree of freedom of two. Then the resulting PDF of182.5. Error Rate Performance Analysischannel gain h is known as the lognormal-Rician [22]fh(h) =∫∞0 dz1+rz exp(−r − (1+r)hz)h0(2√h√r(r+1)z)× 1√2piσzzexp[−12(ln z+ 12σ2zσz)2] (2.19)where r is the coherence parameter defined by r = |UC |2/E[|UG|2], σ2z isthe variance of the logarithm of the lognormal modulation factor exp(2χ),and I0(·) is the zero-order modified Bessel function of the first kind. Thelognormal-Rician is a general scintillation model, and it includes severalwell-known PDFs as its special cases. For example, when the coherence pa-rameter r approaches infinity, the lognormal-Rician distribution specializesto lognormal distribution, whose MGF does not exist on the domain R, withPDF [29]fh(h) =1√2piσzhexp−12(lnh+ 12σ2zσz)2 . (2.20)The lognormal-Rician model can also be used to describe strong turbulencewhen r approaches 0, and the resulting lognormally modulated exponentialdistribution has the PDF [22], [29]fh(h) =1√2piσz∫ ∞0dz1z2exp[−hz− 12σ2z(ln z +12σ2z)2]. (2.21)2.5 Error Rate Performance AnalysisThe error rate performance analysis is important for FSO system de-sign. It can provide the designers with standard performance metrics ofthe system, which includes but not limit to the BER, outage probabilityand asymptotic error rate. For simplicity, we conduct the error rate per-formance analysis on an IM/DD FSO system with OOK modulation. Sincethe channel coherence time is on the order of msec and the data rate isassumed to be on the order of Gbps, we can therefore adopt a slow fadingchannel model. Assuming additive white gaussian noise (AWGN) for thenoise source and unit detector responsitivity, we can express the receivedsignal y at the detector asy = hx+ n (2.22)where x is the transmit intensity being either 0 or 2Pt where Pt is theaverage transmitted optical power, h is the channel state, n is zero-mean192.5. Error Rate Performance AnalysisAWGN with variance σ2n. We can express the instantaneous SNR of thesystem asγ =2P 2t h2σ2n= γ¯h2 (2.23)where γ¯ =2P 2tσ2ndenotes the average SNR.2.5.1 Bit Error RateConditioned on the instantaneous SNR, the BER of IM/DD system withOOK modulation isPe(e|γ) = Q(√γ2)=12erfc(Pth√2σn)(2.24)where Q(·) is the Gaussian Q-function and erfc(·) is the complementary errorfunction that has definition erfc(x) = 2√pi∫∞x exp(−t2)dt. The average BERcan be obtained asPe =∫ ∞0Pe(e|γ)fγ(γ)dh. (2.25)2.5.2 Outage ProbabilityWe define the outage probability as the probability that the decodingerror probability cannot be made arbitrarily small when the transmitterencodes the data at rate R. With channel state h, the outage probability isgiven asPout = Prob(log2(1 + |h|2γ¯) < R) (2.26)which can be calculated asPout =∫ h00fh(h)dh (2.27)where h0 =√(2R − 1)/γ¯.2.5.3 Asymptotic Error RateAt asymptotically high average SNR, average symbol error probabilityof an uncoded system in fading channels can be accurately approximatedas P∞e = (Gcγ¯)−Gd , where Gd is the diversity order indicating how fast theBER decreases with γ¯ in high average SNR region, and Gc is the coding gaindetermining the shift of the BER curve in γ¯ relative to the benchmark curve202.6. Summaryγ¯−Gd . The asymptotic BER P∞e can reveal the behavior of BER in highaverage SNR region, which is helpful in conceptual understanding of per-formance limiting factors in communications over fading channels [60]. Thediversity order and coding gain are determined from instantaneous SNR’sPDF through its behavior near the origin. When h→ 0, the PDF of channelgain can be expanded into power series as [60]fh(h) = aht + gt(h) (2.28)where gt(h) satisfies limh→0gt(h)ht = 0. Considering (2.23), the diversity orderand coding gain are obtained as [60]Gd =t+ 12(2.29)andGc =(2t−32 aΓ(t2 + 1)√pi(t+12) )− 2t+1 . (2.30)2.6 SummaryIn this chapter, we provided essential background knowledge for therest of the thesis. We first introduced IM/DD and coherent FSO systems,which are adopted as system models in the following chapters. We alsopresented the atmospheric channel models including lognormal, Gamma-Gamma, lognormal-Rician models. In the end, we discussed the BER, out-age probability and asymptotic error rate analysis for the performance studylater conducted in this thesis.21Chapter 3Performance Analysis ofFSO Communications overLognormal-Rician FadingChannelsIn this chapter, we study a coherent FSO communication system overthe lognormal-Rician turbulence channels. By using PA, we obtain accurateapproximations of the PDF, CDF, and MGF of lognormal-Rician distribu-tion. We use the MGF approach to derive a closed-form BER expression forBPSK and DPSK with MRC reception. Asymptotic error rate analysis isalso presented to reveal the performance behavior of such systems in largeSNR regimes.3.1 Pade´ ApproximantsThe PA method was first introduced to obtain a rational approximationto the power series representation of MGFs in [68]. PA can be used toapproximate infinite power series that are either not guaranteed to convergeor converge slowly. The approximation is given in terms of a simple rationalfunction of arbitrary numerator and denominator orders. This techniquewas successfully used in analyzing pre-detection EGC diversity systems incorrelated Nakagami-m fading channels [69], and in studying the Weibullfading channels [70]. An accurate approximation of MGF of a RV can beefficiently obtianed using PA, and the process can be encapsulated as follows.Starting with the definition of MGF of a RV X, MX(s) = E[esX ], weseek power expansions of MX(s) around both s → 0 and s → ∞. A PA of223.1. Pade´ Approximantsthe MGF is then given by [68]P[A/B](J,K) (s) =A∑i=0aisi1 +B∑i=1bisi(3.1)where the superscripts A and B are, respectively, the specified order forthe numerator polynomial and the denominator polynomial in the rationalfunction, and the subscripts J,K are respectively the number of coefficientsas s→ 0 and s→∞. When s→ 0, we haveP[A/B](J,K) (s) ≈J∑i=0cisi. (3.2)When s→∞, we haveP[A/B](J,K) (s) ≈K∑i=1dis−i. (3.3)Typically, we require A+B = J +K [68]. In order to obtain ci, we can useMX(s) =∞∑i=0µii!sn, s→ 0 (3.4)where µi is the ith moment of X. In practice, moments of all orders may notbe available or estimated with high accuracy. Therefore a truncated serieswith only a finite number of N moments is consideredMX(s) =N∑i=0µii!si + gi(s), s→ 0. (3.5)The coefficients di in (3.1) can be found by usingMX(s) =∞∑i=0f(i)X (0)si+1, s→∞ (3.6)where the f(i)X (0) denotes the ith order derivative of the PDF of X evaluatedat origin. In practice, the information about the derivative of PDF at origin233.2. Approximating PDF and CDF of Output SNRcan be limited, i.e. with only Mth order derivatives, the truncated serieshas the formMX(s) =M∑i=0f(i)X (0)si+1+ o((s−1)i+1), s→∞. (3.7)The truncated series in (3.5) and (3.7) respectively define the behavior ofthe MGF of X for small s and large s.Matching the powers of s in the left and right hand side of (3.1) givesaj =j∑i=0bicj−i, 0 ≤ j ≤ J,{aj = 0, j > Abi = 0, i > B(3.8)aA−l =l∑i=0di+1bB−l+i, 0 ≤ l ≤ K − 1,{aj = 0, j > Abi = 0, i > B(3.9)where the values of ci and di can be found in (3.5) and (3.7).Intuitively, more power series coefficients provide more information ats→ 0, then the right tail of PDF is better approximated because the regions→ 0 corresponds to x→∞ in its PDF fX(x). Likewise, the region s→∞corresponds to x → 0, accurate approximation of the PDF fX(x) near itsorigin can be obtained by PA. By the theory of asymptotic analysis, weexpect excellent estimation of error rates in large SNR regime.3.2 Approximating PDF and CDF of OutputSNRRecall the discussion in Chapter 2, we can express the output SNR persymbol of a coherent FSO system as [15], [17]γ =RAq∆fI = CI (3.10)where C = RAq∆f is a multiplicative constant for a given FSO system. Ofimportance to the on-going investigation is the fact that the SNR in (3.10)is proportional to I and is independent of the local oscillator power. Withoutloss of generality we set the constant C to unity. The MGF of the single-branch output SNR, which is proportional to the instantaneous intensity Ifollowing lognormal-Rician distribution, can be approximated by PA with243.3. Error Rate AnalysisMI(s) ≈ P [A/B](J,K) (s). The PDF of intensity I can be obtained using inverseLaplace transformfI(I) = L−1(P[A/B](J,K)(s)). (3.11)To facilitate the inverse Laplace transform on MGF, we can first apply theresidue inversion formula on the approximate MGF to obtainMI(s) ≈ P [A/B](J,K) (s) =A∑i=0aisi1 +B∑i=1bisi=B∑k=1λks+ pk(3.12)where pk and λk are, respectively, the poles and the direct terms of thepartial fraction expansion ofA∑i=0aisi1+B∑i=1bisi. Then we apply the inverse Laplacetransform to obtain the PDF and CDF respectively asfI(I) =B∑k=1λk exp(−pkI), pk > 0 (3.13)andFI(I) = 1−B∑k=1λkpkexp(−pkI). (3.14)3.3 Error Rate AnalysisIn this section, we use the approximate PDF, CDF and MGF of inten-sity I to study the error rate performance of a coherent FSO system withBPSK and DPSK modulation. In order to facilitate our analysis, we furthernormalize I by setting E[I] = 1.3.3.1 BER for BPSK and DPSK with Spatial DiversityThe average BER over a turbulence channel can be expressed as Pb =∫∞0 Pb(γ|I)f(I)dI , where Pb(γ|I) denotes the conditional bit error probabil-ity. For coherent BPSK, we havePb(γ|I) = Q(√2γ) = Q(√2γ¯I). (3.15)253.3. Error Rate AnalysisFor noncoherent DPSK, we havePb(γ|I) = 12exp(−γ¯I). (3.16)Assuming an FSO link with L receivers, we denote the instantaneousSNR in the lth diversity branch by γl, l = 1, 2, · · · , L. Since the noise termsin these branches are independent, the SNR at the output of the maximumratio combiner isγMRC =L∑l=1γl = γ¯L∑l=1Il (3.17)where Il is the instantaneous received optical intensity in the lth branch. Ifthe turbulence is independent for all L branches, the MGF of γMRC can beexpressed using the approximated MGF of Il asMγMRC (s) =L∏l=1MIl(γ¯s). (3.18)The BER for BPSK over i.i.d. turbulence with L branch MRC reception (wecan write MIl =MI , l = 1, 2, · · · , L) is found to bePe,BPSK =1pi∫ pi20[MI(− γ¯sin2θ)]Ldθ. (3.19)Substituting (3.12) into (3.19), we obtainPe,BPSK ≈LB∑k=1λk2pk(1−√γ¯γ¯ + pk)(3.20)where pk and λk are, respectively, the poles and the direct terms of thepartial fraction expansion of the MGF of γMRC in (3.18).The BER for DPSK over i.i.d. turbulence with L branch SC receptioncan be obtained asPe,DPSK =∫ ∞012exp(−γ¯I)L [FI(I)]L−1 fI(I)dI. (3.21)Substituting (3.13) and (3.14) into (3.21), we have the approximate BER asPe,DPSK ≈ L2B∑j=1 ∑n0+n1+···+nB=L−1(L− 1)!λjB∏k=1(−λkpk)nkB∑i=1(pini) + γ¯ + pj (3.22)where pk and λk are, respectively, the poles and the direct terms of thepartial fraction expansion of the MGF in (3.12).263.3. Error Rate Analysis3.3.2 Asymptotic Error Rate AnalysisThe lognormal-Rician PDF shown in (2.19) has nonzero value at originfI(0) = limI→0fI(I) = (1 + r) exp(−r + σ2z) (3.23)and the first order derivative near origin isf ′I(0) = limI→0+f ′I(I) = (r − 1)(1 + r)2 exp(−r + 3σ2z). (3.24)Therefore the Maclaurin series of fI(I) has the form fI(I) = fI(0)+f′I(0)I+g1(I). Using (3.23) and (3.24), we can find the diversity order and codinggain in single branch transmission to beGd = 1,Gc =2kf(0)(3.25)where k is a constant determined by the type of digital modulation scheme(i.e. k = 2 for coherent BPSK, k = 1 for orthogonal coherent binary fre-quency shift keying). More generally, we approximate the PDF of outputSNR γMRC with L branch MRC reception asfγMRC (x) =fI(0)L(L− 1)!xL−1 + gL−1(x) (3.26)where fI(0) is given in (3.23). The diversity order and coding gain can be,respectively, obtained asGd = L (3.27)andGc = k[2L−1[fI(0)]LΓ(L+ 1/2)√piL!]− 1L. (3.28)Thus the asymptotic BER of coherent BPSK FSO with MRC receptionbecomesPe,asym,BPSK =2L−1[fI(0)]LΓ(L+ 1/2)√piL!(2γ¯)−L. (3.29)For noncoherent DPSK with SC reception, the PDF of the output SNRγSC with L branch SC reception over the lognormal-Rician fading has theformfγSC(x) = LfI(0)LxL−1 + gL−1(x). (3.30)273.4. Numerical Results0 1 2 3 4 500.10.20.30.40.50.60.70.80.91xCumulative Density Function  ExactApproximationr=5, σz2=0.5r=5, σz2=0.4r=5, σz2=0.2Figure 3.1: The exact and approximate CDF in different lognormal-Ricianparameters.From (3.30), the diversity order and coding gain can be respectively obtainedas Gd = L andGc =[L2[fI(0)]LΓ(L)]− 1L. (3.31)Thus, the corresponding asymptotic error rate at large SNR becomesPe,asym,DPSK =L2[fI(0)]LΓ(L)(γ¯)−L. (3.32)3.4 Numerical ResultsIn this section we present some numerical study of coherent FSO sys-tems over the lognormal-Rician distributed turbulence. Considering thatthe higher order derivatives of PDF f(i)I (0) can be difficult to obtain, we283.4. Numerical Results0 5 10 15 20 25 30 3510−1510−1010−5100SNR (dB)Bit Error Rate  Exact (BPSK)Approximation (BPSK)Exact (DPSK)Approximation (DPSK)L=1L=2L=3Figure 3.2: The exact and approximate BER of BPSK MRC and BER ofDPSK SC (L = 1, 2, 3) in lognormal-Rician (r = 5, σ2z = 0.4) turbulence.293.4. Numerical Results0 5 10 15 20 25 30 3510−1510−1010−5100SNR (dB)Bit Error Rate  ExactAsymptoticL=1L=2L=3Figure 3.3: Asymptotic error analysis of BPSK over multi-branch MRC(L = 1, 2, 3) in lognormal-Rician (r = 5, σ2z = 0.4) turbulence.303.4. Numerical Results0 5 10 15 20 25 30 3510−1510−1010−5100SNR (dB)Bit Error Rate  ExactAsymptoticL=2L=3L=1Figure 3.4: Asymptotic error analysis of DPSK over multi-branch SC (L =1, 2, 3) in lognormal-Rician (r = 5, σ2z = 0.4) turbulence.313.5. Summarychoose K = 1 in (3.1). Moreover we find that 18 power series coefficientsare sufficient to guarantee accuracy. We use the two-point (s → 0, s → ∞)subdiagonal (i.e., A = B − 1) PA with A = 9, B = 10, and A+B − 1 = 18power series coefficients in (3.7), i.e. P[9/10](18,1) (s).Figure 3.1 plots the exact CDF curve of the output SNR obtained from(2.19) and its approximation obtained from (3.14) under different lognormal-Rician parameter settings. Since the CDF of lognormal-Rician distributioncan be sensitive to the parameter σ2z , we choose three distinct variance pa-rameters σ2z = 0.2, 0.4, 0.5 with coherence parameter r = 5. It can be seenfrom Fig. 3.1 that the approximate CDF is highly accurate.In Fig. 3.2, we present the BER performance of an FSO system over thelognormal-Rician fading channels. The exact BER for BPSK and DPSK arerespectively calculated using (3.19) and (3.21) via numerical integrations,while the approximate BER for BPSK and DPSK are respectively estimatedusing (3.20) and (3.22). We can observe that the approximation is accurateover a wide range of SNR values and it approaches the exact BER curve inlarge SNR region, which agrees with our expectation.Figures. 3.3 and 3.4 plot the exact BERs for coherent FSO systemswith BPSK MRC reception and with DPSK SC reception. Both figuresvalidate the asymptotic error rates expressions developed in Section 3.3.2.It is interesting to observe that while the diversity order over a lognormalfading channel is undefined, the diversity order of a coherent FSO system inlognormal-Rician fading channel is well defined to be the number of diversitybranches.3.5 SummaryIn this chapter, we analyzed the error rate performance of a coherentFSO system over the lognormal-Rician turbulence channels using a PA ap-proach. We obtained closed-form BER expression for BPSK and DPSKFSO systems with MRC reception. Asymptotic error rate analysis is alsopresented. Our results suggested that PA is a powerful analytical tool foranalyzing an optical communication system in a turbulence channel that ismathematically intractable. Our analysis further showed that the PA ap-proach is particularly useful in obtaining highly accurate small error rateestimation.32Chapter 4FSO Communication withNonzero Boresight PointingErrorIn this chapter, we study the effect of pointing errors on the perfor-mance of FSO communication systems. A statistical model is investigatedfor pointing errors with nonzero boresight by taking into account the laserbeamwidth, detector aperture size, and jitter variance. A novel closed-formPDF is derived for this new nonzero boresight pointing error model. Fur-thermore, we obtain closed-form PDF for the composite lognormal turbu-lence channels and finite series approximate PDF for the composite Gamma-Gamma turbulence channels, which are suitable for terrestrial FSO appli-cations impaired by building sway. We conduct error rate analysis of OOKsignaling with IM/DD over the lognormal and Gamma-Gamma fading chan-nels. Asymptotic error rate analysis and outage probability of such a systemare also presented based on the derived composite PDFs.4.1 Nonzero Boresight Pointing Errors ModelAssuming AWGN for the thermal/shot noise and unit detector respon-sitivity, we can express the received signal y at the detector asy = hx+ n (4.1)where x is the transmit intensity being either 0 or 2Pt where Pt is the averagetransmitted optical power, h is the channel state, n is zero-mean AWGN withvariance σ2n. The channel state h can be expressed as h = hlhpha, where hlrepresents the path loss which is a constant at given weather condition andlink distance, hp is the pointing error loss factor, and ha is the atmosphericfading loss factor. Note that pointing error loss factor hp and atmosphericfading loss factor ha are both RVs.334.1. Nonzero Boresight Pointing Errors Model4.1.1 Pointing ErrorsWhen a Gaussian beam propagates through distance z from the trans-mitter to a circular detector with aperture radius a, and the instantaneousradial displacement between the beam centroid and the detector center is r,the fraction of the collected power at receiver can be approximated as [11]hp(r; z) ≈ A0 exp(− 2r2w2zeq)(4.2)where A0 is the fraction of the collected power at r = 0, and wzeq is theequivalent beamwidth. We have A0 = [erf(v)]2 and w2zeq = w2z√pierf(v)2v exp(−v2) ,where v =√pi/2 aωz is the ratio between aperture radius and beamwidth,and erf(x) = 2√pi∫ x0 e−t2dt is the error function. The beamwidth wz canbe approximated by wz = θz, where θ is the transmit divergence angledescribing the increase in beam radius with distance from the transmitter.For example, at a range of 1 km, a 1 mrad divergence produces a beam radiusof 1m at receiver. The approximation in (4.2) is accurate when wz/a > 6,which is satisfied in typical terrestrial FSO communication systems.At the receiver aperture plane, we can express the radial displacementvector as r = [rx, ry]T , where rx and ry, respectively, denote the displace-ments located along the horizontal and elevation axes at the detector plane.We consider a nonzero boresight error in addition to the random jitters,and model rx and ry as nonzero mean Gaussian distributed RVs, i.e., rx ∼N (µx, σ2x), ry ∼ N (µy, σ2y). Then the radial displacement r = |r| =√r2x + r2yfollows the Beckmann distribution [71]fr(r) =r2piσxσy×∫ 2pi0exp(−(r cosφ− µx)22σ2x− (r sinφ− µy)22σ2y)dφ.(4.3)The Beckmann distribution is also known as lognormal-Rician distribution[22, 62], which is used to describe the PDF of fading channels in general. It isa versatile model which applies to a variety of distributions. For examples,it can specialize to Rayleigh when µx = µy = 0, σx = σy; Rician withµ2x + µ2y 6= 0, σx = σy; Hoyt distribution when µx = µy = 0, σx 6= σy [12];and single-sided Gaussian when µx = µy = σx = 0, σy 6= 0 [72].In satellite FSO communication systems, it is widely accepted that thejitter variance is the same for both horizontal and elevation axes [73], [74]. In344.1. Nonzero Boresight Pointing Errors Modelterrestrial FSO systems, however, the jitter is mainly caused by turbulenceand building motion. Since the turbulence cells randomly appear on thebeam path, and the building might be considered to sway in orthogonaland parallel directions to the beam path with equal probabilities, we cantherefore assume σ2x = σ2y = σ2s [11], [18],[40]. As a result, the PDF of radialdisplacement r in (4.3) becomes Ricianfr(r) =rσ2sexp(−(r2 + s2)2σ2s)I0(rsσ2s)(4.4)where s =√µ2x + µ2y is the boresight displacement. From (4.2) and (4.4),we derive the PDF of nonzero boresight pointing error asfhp(hp) =γ2 exp(−s22σ2s)Aγ20hγ2−1p I0 sσ2s√−w2zeq ln hpA02 ,0 ≤ hp ≤ A0(4.5)where γ = wzeq/2σs is the ratio between the equivalent beamwidth and jitterstandard deviation, which is a measure of the severity of the pointing erroreffect. If we consider zero boresight error with s = 0, our pointing errormodel in (4.5) specializes to the one in [11]. Our analytical result in (4.5)is accurate when wz/a > 6. In Appendix A, we compare our derived PDFof hp in (4.5) with the exact PDF, which is obtained numerically, withoutusing (4.2) to show the accuracy of our analytical model. In Appendix B,we derive the moments of hp asE[hnp]=An0γ2n+ γ2exp(− ns2(n+ γ2)2σ2s). (4.6)4.1.2 Composite PDF with Generalized Pointing ErrorThe PDF of channel gain h = hlhpha can be calculated as [11]fh(h) =∫fh|ha(h|ha)fha(ha)dha=∫1hahlfhp(hhahl)fha(ha)dha.(4.7)In weak turbulence conditions, we use lognormal fading model to char-acterize the atmospheric turbulence fading. We substitute (2.11) into (4.7),354.1. Nonzero Boresight Pointing Errors Modeland after some mathematical manipulation, the composite PDF of lognor-mal fading with the pointing error model in (4.5) can be obtained asfLN(h) =γ2 exp(ua)2(A0hl)γ2hγ2−1erfc(ln hA0hl + ubuc)(4.8)where we have ua =s2σ2s+ 2σ2Xγ2 + 2σ2Xγ4, ub =6s2ω2zeq+ 2σ2X + 4σ2Xγ2, uc =√8(4s2σ2sω4zeq+ σ2X). It is worthy to note that our PDF in (4.8) can specializeto eq. (14) in [11] when we set the boresight to zero (s = 0).Since hl is a constant, while hp and ha are independent RVs, The nthmoment of h can be obtained as E[hn] = hnl E[hnp ]E[hna ]. With the derivedmoments for hp in (4.6) and the moments of the lognormal RV, we haveE[hn] =(A0hl)nγ2n+ γ2× exp(−2σ2Xn+ 2σ2Xn2 −ns2(n+ γ2)2σ2s).(4.9)For medium to strong turbulence conditions, we model ha as a Gamma-Gamma distributed RV. After some mathematical derivation shown in Ap-pendix C, we obtain a finite series approximation of the composite PDF asfGG(h) ≈ f˜GG(h) =J∑j=0{1j!(αβA0hl)j(vj(α, β)hβ−1+j − vj(β, α)hα−1+j)} (4.10a)wherevj(α, β) =γ2pi(αβA0hl)βexp(− s22σ2s− s2γ2/σ2s2β−2γ2+2j)sin−1((α− β)pi)Γ(α)Γ(β)Γ(j − (α− β) + 1)| − (β − γ2 + j)|(4.10b)and J = ⌊γ2 − α⌋. Therefore (4.10a) is an approximation2 of the com-posite PDF of Gamma-Gamma fading with pointing error. Our compositePDF in (4.10a) is applicable under the condition γ2 > α. Using a rough2This finite series approximation is accurate for strong turbulence conditions, but itcan be inaccurate for weak turbulence conditions. A large boresight and jitter can alsomake the series inaccurate.364.2. Error Rate Performanceapproximation, we can show that the condition γ2 > α corresponds tow2z4σ2s> max{6, 2.31σ4/5R }. Now we present some examples where this con-dition holds: When σ2R = 5.0 and the jitter angle is 0.3mrad, the transmitdivergence is larger than 1.28mrad. When σ2R = 25.0 and the transmit beamdivergence is 2mrad, the jitter angle is less than 0.35mrad. In terrestrial FSOsystems with link distance less than 5km, the system parameters have thefollowing typical values: 5 ∼ 20cm for the receiver diameter, 2 ∼ 10mradfor the transmit divergence, and 0 ∼ 0.3mrad for the boresight and jitterangle [3, 30]. It can be shown that such systems generally operate under thecondition γ2 > α. However, in applications with long link range or narrowbeam divergence, our PDF in (4.10a) is not applicable since γ2 > α maynot hold in such applications. In the rest of this chapter, we assume γ2 > αunless otherwise stated. In (4.10a), the parameters α, β are required to sat-isfy (α − β) /∈ Z. When (α − β) ∈ Z, one can add a small value ε to α tosatisfy the condition (α− β) /∈ Z [61]. The value of ε is empirically chosen,which should be small (i.e. 10−3) that will not change the Gamma-Gammadistribution dramatically. For example, if we have α = 2.12, β = 1.12, wecan add 0.001 to α, and use α = 2.121 in the Gamma-Gamma model.Similar to the moments obtained for composite lognormal channel, using(4.6) and the moments for the Gamma-Gamma RV we can obtain the nthmoment of h asE[hn] =(A0hl)nγ2Γ(α+ n)Γ(β + n)(n+ γ2)Γ(α)Γ(β)(αβ)nexp(− ns2(n+ γ2)2σ2s). (4.11)4.2 Error Rate Performance4.2.1 Bit-Error RateWe derive the average BER of lognormal fading with nonzero boresightpointing error asPe,LN =γ2 exp(ua)4(A0hl)γ2×∫ ∞0hγ2−1erfc(ln hA0hl + ubuc)erfc(Pt√2σnh)dh.(4.12)Using a change of variable rule, eq. (4.12) can be expressed asPe,LN =γ2uc4exp(ua − γ2ub)×∫ ∞−∞exp(γ2ucx)erfc(x)erfc(PtA0hl√2σn exp(ub − ucx))dx.(4.13)374.2. Error Rate PerformanceBy introducing an auxiliary parameter B and partitioning the integrationinterval in (4.13) into [−∞, B] and [B,∞], we can rewrite (4.13)Pe,LN =γ2uc4exp(ua − γ2ub)×∫ B−∞exp(γ2ucx)erfc (x) erfc(PtA0hl√2σnexp (ucx− b))dx+RB(4.14)where RB is the approximation error given byRB =γ2uc4exp(ua − γ2ub)×∫ ∞Bexp(γ2ucx)erfc(x)erfc(PtA0hl√2σn exp(ub − ucx))dx.(4.15)In Appendix D, it is shown that the approximation error RB can be upperbounded byRB <√piγ2uc8exp(ua − γ2ub + γ4u2c4)erfc(B − γ2uc2). (4.16)We quantify RB under various system parameters and calculate the valuesof B for different RB . The results are presented in Appendix D, showingthat the approximation error RB decreases rapidly with increasing B. Wecan always adjust the value of B to make RB arbitrarily small. Therefore,eq. (4.14) can be accurately approximated asPe,LN ≈ P˜e,LN = γ2uc4exp(ua − γ2ub)×∫ B−∞exp(γ2ucx)erfc(x)erfc(PtA0hl√2σn exp(ub − ucx))dx.(4.17)Using a series expansion of the complementary error function, which is [75,eq. (06.27.06.0002.01)]erfc(z) = 1− 2√pi∞∑k=0(−1)kz2k+1k!(2k + 1)(4.18)and an integral identity [75, eq. (06.27.21.0011.01)]∫ebzerfc(az)dz =1b(ebzerfc(az)− e b24a2 erf(b2a− az))(4.19)384.2. Error Rate Performancewe derive an infinite series expression of the BER in (4.17) asP˜e,LN =γ2uc4exp(ua − γ2ub)×{1γ2uc[exp(γ2ucB)erfc(B) + exp(γ4u2c4)erfc(γ2uc2−B)]− 2√pi∞∑j=0[(−1)jj!(2j + 1)(Pt√2σnA0hl)2j+1 exp(−ub(2j + 1))(2j + 1 + γ2)ucSj]}(4.20)whereSj =exp((γ2 + 2j + 1)ucB)erfc(B)+ exp((γ2 + 2j + 1)2u2c/4)erfc((γ2 + 2j + 1)uc/2−B).(4.21)The infinite series in (4.20) can be rigorously shown to be convergent, anda detailed proof is presented in Appendix E.For the Gamma-Gamma fading with nonzero boresight pointing error,we substitute (4.10a) into (2.25) to find its BER. Using an integral formula[75, eq. (06.27.21.0132.01)]∫ ∞0tα−1erfc(t) =1√piαΓ(α+ 12)(4.22)we can obtain the approximate BER in terms of a finite series asPe,GG ≈ P˜e,GG =12√piJ∑j=0 1j!(2αβA0hl)j2βΓ(β+j+12)β + jvj(α, β)(2P 2tσ2n)−β+j2−2αΓ(α+j+12)α+ jvj(β, α)(2P 2tσ2n)−α+j2 .(4.23)We comment (4.23) is obtained without additional approximations overthose in (4.10a). The approximation error defined as ε(Pe) = |Pe − P˜e|is discussed in Appendix F. When Pt is beyond the minimum required valuein Table IV, we can guarantee the relative errorε(Pe,GG)Pe,GGandε(Pout,GG)Pout,GGless394.2. Error Rate Performancethan 10−6, where Pout,GG denotes the outage probability of system in thecomposite Gamma-Gamma fading case and ε(Pout,GG) denotes the approxi-mation error |Pout,GG−P˜out,GG|. With the setting s/a = 1.0 and σs/a = 1.0,the minimum required Pt is below −10 dBm, which can be satisfied for prac-tical FSO systems.4.2.2 Asymptotic Error Rate AnalysisWe first derive the power series expansion of the PDF near its origin, thenwe obtain the diversity order and coding gain from (2.29) and (2.30). Theasymptotic BER is obtained from P∞e = (Gc · SNR)−Gd . For the compositelognormal fading with pointing error, we obtain the PDF of the channel gainh near the origin h→ 0 asfLN(h) =γ2(A0hl)γ2exp (ua)hγ2−1 + gγ2−1(h). (4.24)From (4.24), we obtain the diversity order as Gd =γ22 =w2zeq8σ2s. This indi-cates that the diversity order is determined by the ratio between equivalentbeamwidth and jitter standard deviation. More specifically, the boresightcomponent of pointing error does not affect the diversity order in lognormalfading channel. The coding gain can also be obtained from (4.24) asGc =2γ2−1Γ(γ22 +12)exp (ua)√pi(A0hl)γ2−2γ2. (4.25)With SNR =2P 2tσ2n, we can present the asymptotic BER asP∞e,LN =2γ2−1Γ(γ22 +12)exp (ua)√pi(A0hl)γ2(2P 2tσ2n)− γ22. (4.26)For the Gamma-Gamma composite fading channels, we derive the powerseries expansion of PDF near the origin in Appendix C, and it is given byfGG(h) = v0(α, β)hβ−1 + gβ−1(h) (4.27)where v0(α, β) follows the definition in (4.10b) and we have assumed γ2 > α.From (4.27), we obtain the diversity order as Gd = β/2 and the coding gain404.2. Error Rate PerformanceasGc =[2β−1Γ(β2 +12)exp(− s22σ2s +−s2γ2/σ2s2β−2γ2)(αβA0hl)βΓ(α)Γ(β) sin((α− β)pi)Γ(−(α − β) + 1)|γ2 − β|β× Γ(β + 12)√piγ2]− 2β.(4.28)Thus, the asymptotic BER can be expressed asP∞e,GG =2β−1Γ(β2 +12)exp(− s22σ2s+ −s2γ2/σ2s2β−2γ2)(αβA0hl)βΓ(α)Γ(β) sin((α− β)pi)Γ(−(α − β) + 1)|γ2 − β|β× Γ(β + 12)√piγ2(2P 2tσ2n)−β2.(4.29)Note that the diversity order is Gd = β/2 when γ2 > α, which implies thatthe Gamma-Gamma fading effect is more dominant than the pointing erroreffect with respect to the BER performance at high average SNR region.However, when γ2 < α, the diversity order will depend on both γ2 and theboresight s. An explicit expression of diversity order for nonzero boresightpointing case is difficult to obtain. For the special case with zero boresight,the diversity order is given as min{γ2/2, β/2} [76].To evaluate the performance loss caused by boresight error, we define anSNR penalty factor in dB asSNRboresight = 10 log10[SNRP∞e ,nonzeroboresightSNRP∞e ,zeroboresight](4.30)which represents the constant SNR loss caused by boresight error at certainerror probability P∞e when SNR is asymptotically large. From (4.26) and(4.29), we obtain the SNR penalty factor for the composite lognormal andthe composite Gamma-Gamma fading, respectively, asSNRboresight,LN =80ln 10[sωzeq]2(4.31)andSNRboresight,GG =40ln 10 s√ω2zeq − 4σ2sβ2 (4.32)414.2. Error Rate Performancewhere we have assumed γ2 > α, that is w2zeq > 4σ2sβ in (4.32). The resultsin (4.31) and (4.32) show that, for both composite lognormal and com-posite Gamma-Gamma cases, the boresight error imposes a constant SNRpenalty on the error rate performance when SNR is large. As expected,larger beamwidth ωzeq can mitigate the adverse impact of the boresight er-ror. However, we can not arbitrarily increase the beamwidth, because it willreduce the total received power. Moreover the maximum level of transmit-ted power is also limited by the heat dissipation and eye safety requirementsat the transmitter, thus we can not increase the power too insensitively.4.2.3 Outage ProbabilityFor weak turbulence condition, substituting (4.8) into (2.27) and usingthe integral identity in (4.19) we obtain the outage probability asPout,LN =12[(h0A0hl)γ2exp(ua)erfc(ln h0A0hl + ubuc)+exp(ua − ubγ2 + u2cγ44)erfc(ucγ22− lnh0A0hl+ ubuc)].(4.33)For medium to strong turbulence conditions, by substituting (4.10a) into(2.27), we obtain the outage probability asPout,GG ≈ P˜out,GG =exp(− s22σ2s)γ2piΓ(α)Γ(β) sin((α− β)pi)×J∑j=0(αβA0hl)j(αβA0hl)βexp( −s2γ2/σ2s2β−2γ2+2j)Γ(j − (α− β) + 1)j!| − (β − γ2 + j)|(β + j)hβ+j0−(αβA0hl)αexp( −s2γ2/σ2s2α−2γ2+2j)Γ(j + (α− β) + 1)j!| − (α− γ2 + j)|(α + j)hα+j0 .(4.34)It is worthy to mention that we can obtain the diversity order using theoutage probability derived in (4.33) and (4.34), the derivation follows thatin [40]. The results are Gd,LN = γ2/2 and Gd,GG = β/2, which coincidewith the ones obtained using the power series expansion approach.424.2. Error Rate Performance−5 0 5 1010−810−710−610−510−410−310−210−1100Pt (dBm)BER  Exact, σR2=0.05Exact, σR2=0.2Seriess/a=0,3Figure 4.1: BER performance of IM/DD OOK over the lognormal fadingwith zero and nonzero boresight pointing errors.434.2. Error Rate Performance−5 0 5 1010−810−710−610−510−410−310−210−1100Pt (dBm)BER  Exact, σs/a=1.5Exact, σs/a=2Exact, σs/a=3SeriesAsymptoticFigure 4.2: BER for the composite lognormal channel (σ2R = 0.01, s/a = 2)with different jitter values.444.2. Error Rate Performance−5 0 5 10 1510−810−710−610−510−410−310−210−1100Pt (dBm)BER  Exact, wz/a=8Exact, wz/a=11Exact, wz/a=14SeriesAsymptoticFigure 4.3: BER for the composite lognormal channel (σ2R = 0.01, s/a =2, σs/a = 1.5) with different beamwidth values.454.2. Error Rate Performance−5 0 5 1010−810−710−610−510−410−310−210−1100Pt (dBm)Outage probability  Exact, σR2=0.05Exact, σR2=0.2Seriess/a=0,3Figure 4.4: Outage probability of an FSO system over the lognormal fadingwith zero and nonzero boresight pointing errors.464.2. Error Rate Performance−35 −30 −25 −20 −15 −10 −5 0 5 1010−810−610−410−2100Pt (dBm)BER  Exact, σR2=0.6Exact, σR2=2.0Series (Markers)Asymptotics/a=0,3Figure 4.5: BER performance of IM/DD OOK over the Gamma-Gammafading with zero and nonzero boresight pointing errors.474.2. Error Rate Performance−15 −10 −5 0 510−810−710−610−510−410−310−210−1100Pt (dBm)Outage probability  Exact, σR2=0.6Exact, σR2=2.0Seriess/a=0,3Figure 4.6: Outage probability of an FSO system over the Gamma-Gammafading with zero and nonzero boresight pointing errors.484.2. Error Rate Performance0 1 2 3 4 50123456789Normalized boresight displacementSNR penalty factor (dB)  LognormalGamma−Gamma, σR2=0.6, α=5.41, β=3.78Gamma−Gamma, σR2=2.0, α=3.99, β=1.70Figure 4.7: The SNR penalty factor induced by boresight error in differentturbulence conditions.494.2. Error Rate PerformanceTable 4.1: System SettingsParameter ValueTransmission rate 1 GbpsTransmitted power (Pt) 10 dBmNoise standard deviation (σn) σn = 10−7 A/HzReceiver Diameter (2a) 20 cmNoise standard deviation (σn) σn = 10−7 A/HzLink distance 1 kmTransmit Divergence at 1/e 1 mradCorresponding beam radius (ωz) at 1 km ≈ 100 cmBoresight angle 0.1 mradCorresponding boresight displacement (s) at 1 km ≈ 10 cmJitter angle 0.1 mradCorresponding jitter standard deviation (σs) at 1 km ≈ 10 cmTable 4.2: Weather ConditionsWeather Path loss (hl) Fading model σ2R ParametersLight fog 0.008 Lognormal0.05 σ2X = 0.01250.1 σ2X = 0.0250.2 σ2X = 0.05Clear 0.9 Gamma-Gamma0.6 α = 4.15, β = 3.782.0 α = 2.21, β = 1.705.0 α = 2.12, β = 1.24504.3. Numerical Results4.3 Numerical ResultsIn this section, we adopt the system settings shown in Table 4.2.3, whichare used in many practical terrestrial FSO communication systems [3, 30,77]. Under two typical weather conditions shown in Table 4.2.3 [11], wecarry out the error rate performance of an FSO communication system. Inorder to investigate the degradation effects induced by the boresight error,we use two different values of normalized boresight displacement s/a = 0, 3.In weak turbulence regime, we study the BER of an FSO link over thelognormal fading channel with nonzero boresight pointing errors. We cal-culate the exact BER from (4.12), the approximate BER from (4.20), andthe asymptotic BER from (4.26). The BER curves are plotted in Fig. 4.1against the transmitted optical power Pt. From Fig. 4.1, we can see howthe boresight displacement s affects the BER performance of the system.We can infer from the result that for a Gaussian beam the energy collectedat receiver aperture decreases with increasing boresight errors, and there-fore the BER performance deteriorates with a large boresight displacement.However, if we assume a tophat beam profile, the boresight would not affectthe collected power at the receiver as long as the boresight displacement sis smaller than the difference of distance between the beamwidth at receiverand the receiver aperture.From the asymptotic BER curves in Fig. 4.2 and Fig. 4.3, we find thatthe diversity order is determined by the equivalent beamwidth as well as thejitter variance. A similar finding is reported in [40] for a MIMO FSO systemwith zero boresight. With larger beamwidth or smaller jitter variance, thediversity order of the system becomes larger.The outage probability of an FSO system with code rate R0 = 0.5 (bitsper channel use) over the composite lognormal fading channel is presented inFig. 4.4, where we use (4.33) to calculate outage probability. It can be seenthat the outage probability of the system worsens with increasing boresighterrors.In medium to strong turbulence regimes, we study the BER of an FSOlink over the Gamma-Gamma fading channel with nonzero boresight point-ing errors. We calculate the exact BER from (2.25), the approximate BERfrom (4.23) (with J = ⌊γ2−α⌋), and the asymptotic BER from (4.29). TheBER curves are plotted in Fig. 4.5 against the transmitted power Pt. It canbe seen that our series solution developed in (4.23) can accurately approx-imate the exact BER when Pt is beyond a certain threshold (see AppendixF). However, for small values of Pt, the series approximation in (4.23) canbe inaccurate. This is the limitation of the series approach for the Gamma-514.4. SummaryGamma channels with pointing errors, and the same limitation can also beseen from Fig. 2 of [12]. As expected, the BER performance worsens whenthe boresight displacement s becomes larger. Moreover, the boresight errorcauses a horizontal shift of the BER curve, resulting in an SNR penaltyfactor for the error rate performance. From the asymptotic curves in Fig.4.5 we can find that the boresight displacement does not affect the diversityorder of the system.Assuming a code rate of R0 = 0.5 (bits per channel use), we present theexact outage probability obtained from (2.27), and the approximate outageprobability calculated using (4.34) in Fig. 4.6, and it can be seen thatthe approximate outage probability is accurate. (The threshold of Pt foraccurate outage probability approximation is shown in Appendix F.)In Fig. 4.7, we plot the SNR penalty factor versus the normalized bore-sight displacement s in both composite lognormal and composite Gamma-Gamma cases. The result indicates that the boresight displacement haslarger penalty factor on SNR when the turbulence is weaker.4.4 SummaryIn this chapter, a nonzero boresight pointing error model was inves-tigated by considering beamwidth, detector aperture, and jitter variance.Our derivation is based on the assumption that the boresight component ofpointing error effects is not negligible. A closed-form PDF was derived forthe nonzero boresight pointing error model. We derived closed-form compos-ite PDF for the lognormal fading and highly accurate series based PDF forthe Gamma-Gamma fading. Highly accurate convergent series BER expres-sions were obtained. Through the asymptotic BER analysis and numericalresults, we observed that the boresight error causes an SNR penalty factoron error rate performance at high SNR. By examining the asymptotic BERcurves, we found that the diversity order of the FSO system over the com-posite lognormal fading channel is solely determined by the pointing errorparameter γ2, which means that the boresight component does not affectthe diversity order. While in the composite Gamma-Gamma fading channel,the diversity order is determined by either the Gamma-Gamma fading effector the pointing error effect.52Chapter 5Performance of SubcarrierM-ary PSK with PhaseRecovery Error overLognormal Fading ChannelsIn this chapter, we study the BER performance of subcarrier MPSK sys-tems with CPE over lognormal turbulence channels. Since the traditionalasymptotic analysis techniques cannot be applied directly on FSO commu-nication systems over the lognormal channels, we introduce an auxiliary RVapproach to analyze the asymptotic performance of such systems. We de-rive exact asymptotic noise reference loss expressions for subcarrier MPSKsystems with CPE over the lognormal channels. Our analysis quantifies theperformance degradation introduced by the CPE.5.1 System ModelFor a subcarrier FSO system, the instantaneous receiver SNR is definedas the ratio of the time-averaged AC photocurrent power to the total noisevariance, and it can be expressed as [48]γ =(PRξ)2σ2nI2 = γI2 (5.1)where P is the average transmitter power, R is the photodetector responsiv-ity, ξ is the modulation index, σ2n is the noise variance, I is the turbulenceinduced channel gain, and γ is the average SNR assuming normalized meanof channel gain. The SNR γ defined in (5.1) is SNR per symbol. For anM -ary modulation scheme, we have γ = γb log2M and γ = γb log2M withγb and γb denoting the instantaneous SNR per bit and the electrical SNRper bit, respectively.535.1. System Model5.1.1 Phase ErrorIn this chapter, the CPE is assumed to be derived from the pilot toneusing a PLL in the presence of AWGN and fading. Thus, the CPE followsa Tikhonov distribution conditioned on the channel gain [78, 79]. The PLLSNR ρ can be expressed as ρ = Pc/(N0BL) where Pc is the power allocatedto the carrier phase recovery pilot, N0 is the noise spectral density, and BLis the loop bandwidth. We further assume that a fixed fraction (ς) of theavailable total power Pt is allocated to the pilot (i.e., Pc = ςPt) and theremaining fraction is for data detection. Therefore, we have (1 − ς)Pt =Eb/Tb where Eb is the energy per bit and Tb is the bit interval. In a fadingenvironment, it is straightforward to obtain [78]ρ =ς1− ς1BLTbI2EbN0=ς1− ς1BLTbγb. (5.2)From (5.2), we observe that the PLL SNR ρ is proportional to the instan-taneous receiver SNR γb, i.e., ρ = Cγb where C is a constant defined asC = ς/[(1 − ς)BLTb] with typical value around 10 [78, 80]. The PDF of theCPE conditioned on the fading is given by [79]fΘ(θ|ρ) = exp (ρ cos θ)2piI0(ρ), −pi ≤ θ ≤ pi. (5.3)Substituting ρ = Cγb into (5.3), we obtainfΘ(θ|γb) = exp (Cγb cos θ)2piI0(Cγb). (5.4)With a specific fading model, we can average the conditional PDF of theCPE over the PDF of SNR per bit γb and obtain the PDF of Θ asfΘ(θ) =∫ ∞0exp (Cγb cos θ)2piI0(Cγb)fγb(γb)dγb (5.5)where fγb(γb) is the PDF of the instantaneous SNR per bit γb. It is importantto note that the condition ρ ≫ 1 holds in practice; therefore, the varianceof the CPE σ2θ can be approximated by σ2θ ≈ 1/ρ [78, 81, 82]. We will makethis assumption in our analysis.545.2. Asymptotic Noise Reference Loss Analysis5.2 Asymptotic Noise Reference Loss Analysis5.2.1 Subcarrier MPSK SystemFor nonideal coherent detection of MPSK and assuming Gray mapping,we can express the conditional BER as [83]Pb,MPSK(θ, γb) =12 log2MM/2−1∑n=0erfc(√log2Mγb sin((2n+ 1)piM+ θ))(5.6)where θ denotes the CPE. The average BER of a subcarrier MPSK systemover turbulence channels can be obtained asPb,MPSK =∫ ∞0∫ pi−piPb,MPSK(θ, γb)fΘ(θ|γb)fγb(γb)dθdγb. (5.7)We can use (5.7) to calculate the average BER of a subcarrier MPSKsystem. The performance gap between systems with phase error and thatwithout phase error can also be obtained using (5.7). We define the noisyreference loss SNRL as the amount of additional SNR (compared with theideal coherent detection) required to achieve a specific BER in the presenceof CPE [78]. In this chapter, we obtain the asymptotic noisy reference loss,SNR∞L , when the SNR is asymptotically large. This asymptotic measure-ment can give insights of the system performance.In a lognormal turbulence environment, the receiver SNR γb is anotherlognormal RV having PDFfγb,LN(γb) =12√2piσγbexp(−(ln γb − ln γb + σ2)28σ2)(5.8)where ln γb is a Gaussian RV with mean σ2 − ln γb and variance 4σ2. Us-ing (5.7) and (5.8), we can evaluate the BER of subcarrier MPSK systemsin the lognormal turbulence channels. However, it is well-known that theasymptotic BER analysis cannot be performed on a lognormal channel di-rectly because the diversity order of such a channel is undefined. In orderto investigate the asymptotic noisy reference loss of subcarrier BPSK inthe lognormal channels, we introduce the lognormal-Nakagami fading as anauxiliary channel model where the receiver SNR follows a lognormal-Gammadistribution. Since the lognormal-Gamma distribution approaches a lognor-mal distribution as the channel parameter m approaches infinity, the results555.2. Asymptotic Noise Reference Loss Analysisobtained for the lognormal-Nakagami channels will reveal the performancecharacteristics of the lognormal channels when m→∞.In a lognormal-Nakagami fading environment, the receiver SNR γb fol-lows a lognormal-Gamma distribution with PDFfγb,LG(γb) =12√2piσΩ∫ ∞0mmγm−1bΩmΓ(m)exp(−mγbΩ)× exp(−(lnΩ− ln γb + σ2)28σ2)dΩ(5.9)where m is the Nakagami-m fading parameter and Ω is the second momentof a Nakagami-m RV. When m→∞, we can show thatlimm→∞ fγb,LG(γb) =12√2piσγbexp(−(ln γb − ln γb + σ2)28σ2)(5.10)which is the lognormal PDF given in (5.8).Substituting u = Ω/γb into (5.9), we rewrite the lognormal-Gamma PDFasfγb,LG(γb) =mmγm−1b γ−mb2√2piσΓ(m)∫ ∞01um+1exp(−mγbuγb)× exp(−(lnu+ σ2)28σ2)du.(5.11)When γb approaches ∞, we obtain from (5.7) and (5.11) the asymptoticBER of subcarrier MPSK in the lognormal-Nakagami channels asP∞b,MPSK =∫ ∞0∫ pi−piPb,MPSK(θ, γb)fΘ(θ|γb)fγb(γb)dθdγb=gcG(m)4pi log2Mγ−mb(5.12)wheregc =mm2√2piσΓ(m)∫ ∞01um+1exp(−(lnu+ σ2)28σ2)du (5.13)and G(x) is defined asG(x) =∫ ∞0∫ pi−piexp (Cγb cos θ) γbx−1I0(Cγb)×M/2−1∑n=0erfc(√log2Mγb sin((2n+ 1)piM+ θ))dθdγb.(5.14)565.2. Asymptotic Noise Reference Loss AnalysisFor the systems without phase error, we can express the conditional BERasPb,MPSK(γ) =1log2MM−1∑k=1dkPk(γ) (5.15)wheredk = 2∣∣∣∣ kM −⌊kM⌉∣∣∣∣+ 2 log2 M∑i=2∣∣∣∣ k2i −⌊k2i⌉∣∣∣∣ (5.16)andPk(γ) =12pi[∫ mkpi0exp(−Ak log2Mγbsin2 φ)dφ−∫ nkpi0exp(−Bk log2Mγbsin2 φ)dφ](5.17)where we havemk =(M − 2k + 1)/M,nk =(M − 2k − 1)/M,Ak =sin2 ((2k − 1)pi/M) ,Bk =sin2 ((2k + 1)pi/M) .(5.18)The asymptotic BER can be obtained asP∞b,MPSK =12pi log2MM−1∑k=1dk (P∞k (mk, Ak)− P∞k (nk, Bk)) (5.19)whereP∞k (mk, Ak) =∫ mkpi0∫ ∞0exp(−Ak log2Mγbsin2 φ)gcγm−1γmdγdφ=gcΓ (m) g(mk, sin2 φ,m)(log2MAk)mγ−m(5.20)and whereg (η, u(φ), v) ,∫ ηpi0(u(φ))v dφ. (5.21)575.2. Asymptotic Noise Reference Loss AnalysisUsing (5.12) and (5.19), we obtain the asymptotic noisy reference overthe LG channel asSNR∞L,M,LG(m) =1m10 log(gcG3(m)2∑M−1k=1 dk(P∞k (mk, Ak)− P∞k (nk, Bk))) .(5.22)After some mathematical manipulation and keeping only the dominantterms, we can rewrite (5.22) asSNR∞L,M,LG(m)≈ 1m10 log(gcG3(m)2d1 (P∞1 (m1, A1)− P∞1 (n1, B1)))=1m10 log gcG3(m)2gcΓ(m)∫ M−1Mpi0(sin θ√log2M sin( piM ))2mdθ=1m10 log∑M/2−1n=0∫ pi2−pi2√2piCΓ(m)((2n+1)piM+θ)(C−C cos θ+log2M sin2((2n+1)piM+θ))mdθ2Γ(m)∫ M−1Mpi0(sin θ√log2 M sin( piM ))2mdθ .(5.23)In order to approach the lognormal fading channel, we let m→∞ in (5.23),and the asymptotic noisy reference over the lognormal channel can be ob-tained asSNR∞L,M,LN = 10 log qmax1log2 M sin2 piM (5.24)where we haveqmax =max 1(C −C cos θ + log2M sin2 ( (2n+1)piM + θ)) ,n = 0, 1, · · · , M2− 1; θ ∈ (−pi2,pi2).(5.25)Therefore for the BPSK case, we can obtain the asymptotic noisy refer-ence loss from (5.24) asSNR∞L,2,LN = 10 log(11)= 0dB. (5.26)585.3. Numerical ResultsFrom (5.26), we can find that the CPE introduced asymptotic noisy referenceloss is 0 dB for subcarrier BPSK system over the lognormal channels.For QPSK system, we can obtain the asymptotic noisy reference lossfrom (5.24) asSNR∞L,4,LN = 10 log(qmax) = 10 log(1C + 1 + sin (2θˆ)− C cos θˆ)(5.27)whereθˆ = sin−1(C −√C2 + 328). (5.28)Using (5.24), we can obtain the asymptotic noisy reference loss for arbitraryM and C. For M = 8, C = 15, we haveSNR∞L,8,LN = 10 log3.085913 sin(pi8)= 1.3218 dB. (5.29)For M = 8, C = 10, we haveSNR∞L,8,LN = 10 log3.505613 sin(pi8)= 1.8756 dB. (5.30)For M = 16, C = 15, we haveSNR∞L,16,LN = 10 log9.980114 sin( pi16)= 1.8167 dB. (5.31)For M = 16, C = 10, we haveSNR∞L,16,LN = 10 log11.701414 sin( pi16)= 2.5077 dB. (5.32)From the above results, we can see that the noisy reference loss increaseswith higher modulation order and smaller PLL SNR coefficient C values.5.3 Numerical ResultsIn this section, for simplicity, we will present the noisy reference loss ofa BPSK (M = 2) and a QPSK (M = 4) subcarrier FSO systems.Figure. 5.1 presents the asymptotic noisy reference loss of subcarrierBPSK system over the lognormal-Nakagami channels. From (5.26), we ex-pect the asymptotic noisy reference loss to be 0 dB, which can be observed595.3. Numerical Results0 5 10 15 20 25 30−0.500.511.522.53mNoisy reference loss (dB)  C = 7C = 9C = 11LognormalFigure 5.1: Asymptotic noisy reference loss of subcarrier BPSK system overthe lognormal-Nakagami channel with different PLL parameter C values.605.3. Numerical Results0 5 10 15 20 25 3000.511.522.53mNoisy reference loss (dB)  C = 7C = 9C = 11LognormalFigure 5.2: Asymptotic noisy reference loss of subcarrier QPSK system overthe lognormal-Nakagami channel and the lognormal channels with differentPLL parameter C values.615.4. Summaryfrom Fig. 5.1. The noisy reference loss tends to approach 0 dB with increas-ing channel parameter m.In Fig. 5.2, we present the asymptotic noisy reference loss of subcarrierQPSK system over the lognormal-Nakagami channels with different PLLparameter C values. We observe from Fig. 5.2 that the asymptotic noisyreference losses for the lognormal-Nakagami channel approach the ones ob-tained in (5.27) with increasing channel parameter m, which validate ourasymptotic noisy reference loss results for lognormal channels.5.4 SummaryIn this chapter, we studied the BER performance of subcarrier MPSKsystems with CPE over lognormal fading channels. We quantified the CPEinduced asymptotic noise reference loss over the lognormal channels. Ourresults showed that the CPE induced performance degradation for a sub-carrier BPSK system over the lognormal channel is negligible. The CPEinduced performance degradation increases with higher modulation orderand smaller PLL SNR coefficient C values.62Chapter 6Asymptotic PerformanceAnalysis of FSOCommunication overCorrelated LognormalFading ChannelsIn this chapter, we analyze the performance of a multi-branch FSO sys-tem over correlated lognormal fading channels. We propose a lognormal-Nakagami model to facilitate asymptotic analysis on lognormal channels.Using such an approach, we study different multi-branch FSO systems overcorrelated lognormal fading channels that may have nonidentical variance.We discover that the correlation among the lognormal channels can imposelarge asymptotic SNR penalty to system BER performance. We also de-rive a compact expression for the ARDO between an L-branch combiningsystem over correlated lognormal channels and a single-branch system. ForMRC, EGC and SC receptions, we obtain the coding gain differences onsuch systems.6.1 System ModelIn this chapter, we consider an L-branch OOK FSO system with IM/DD.Assuming AWGN for the thermal/shot noise [7, 36, 40] and unit detectorresponsitivity, we can express the received signal yl at the lth detector asyl = hlI + nl, l = 1, 2, · · · , L (6.1)where I is the transmit intensity being either 0 or 2Pt, where Pt is theaverage transmitted optical power, hl is the lth channel gain, nl is zero-mean AWGN with variance σ2n. Therefore we denote the average SNR at a636.1. System Modelsingle receiver as γ¯ =2P 2tσ2n, and the instantaneous SNR at the lth receivercan be written asγl = γ¯h2l . (6.2)Assuming weak turbulence, we can model the lth channel gain hl ashl = exp(Vl) (6.3)where Vl, l = 1, 2, · · · , L are Gaussian distributed RVs with mean µl andvariance σ2l . Therefore the channel gain hl follows the lognormal distributionwith PDFfhl(hl) =1hlσl√2piexp(−(lnhl − µl)22σ2l)(6.4)and the instantaneous SNR has the PDFfγl(γl) =12σlγl√2piexp(−(ln γl + 2µl − ln γl)28σ2l). (6.5)For normalized received intensity, we have µl = −σ2l2 . The parameters ofthe lognormal fading model can be measured directly for FSO systems [66].For an L-branch FSO system, we use the covariance matrix to model thecorrelation between receive apertures, which is given byΣ =σ21,1 σ21,2 · · · σ21,Lσ22,1 σ22,2 · · · σ22,L....... . ....σ2L,1 σ2L,2 · · · σ2L,L (6.6)where σ2i,j = E[(Vi−µi)(Vj−µj)]. When every branch has identical varianceσ2, we have Σ = σ2R where R is the correlation matrix of RVs Vl, l =1, 2, · · · , L.In order to perform diversity analysis on systems over lognormal fad-ing channels, we have to obtain a Taylor series expansion of the lognormalPDF near its origin [60]. Since lognormal PDF does not have a Taylorseries expansion at the origin, it is infeasible to perform the conventionalasymptotic analysis directly on the lognormal channel. Therefore, we intro-duce the lognormal-Nakagami fading as an auxiliary channel model wherethe receiver instantaneous SNR follows a lognormal-Gamma (LG) distri-bution. Note that such model is an auxiliary channel model rather thana experimentally validated model, and the reason for choosing lognormal-Nakagami is that the Nakagami component is easy for us to manipulate,646.1. System Modeli.e., it will approach a Dirac delta function when m → ∞. Moreover, wewill have the diversity of system over lognormal-Nakagami fading tend to∞ when m→∞, which corresponds to lognormal channel’s behavior. Thelognormal-Nakagami random variable z for the channel state can be con-structed by multiplying a lognormal RV x by a Nakagami-m RV y. Unlikethe lognormal PDF, the lognormal-Gamma PDF indeed has a Taylor seriesexpansion at the origin; therefore, one can perform asymptotic analysis onsuch a channel. It will be shown in Section 6.2 that the diversity order ofthe lognormal-Nakagami channel is just the Nakagami fading parameter m.As m approaches ∞, the lognormal-Gamma PDF approaches that of a log-normal [84, 85]. The lognormal RV x follows the PDF shown in (6.4). TheNakagami RV y’s PDF isfNaka(y) =2mmΓ(m)Ωmy2m−1 exp(−mΩy2)(6.7)where m is the shape parameter, Ω is the spread parameter and Γ(·) is theGamma function. For normalization purpose, we often have Ω = 1. Forthe lognormal-Nakagami channels with channel gain h = z, the receivedinstantaneous SNR γ follows a lognormal-Gamma distribution with PDFfLG(γ) =∫ ∞0mmγm−1ΩmΓ(m)exp(−mγΩ)× 12σΩ√2piexp(−(lnΩ + 2µ − ln γ)28σ2)dΩ. (6.8)When m→∞, we can show thatlimm→∞mmγm−1ΩmΓ(m)exp(−mγΩ)= δ( γΩ− 1)(6.9)where δ(·) is the Dirac delta function. Applying (6.9) to (6.8), we obtainlimm→∞ fLG(γ) =12σγ√2piexp(−(ln γ + 2µ− ln γ)28σ2)(6.10)which corresponds the lognormal PDF given in (6.5). In the remainderof this chapter, we will use this lognormal-Nakagami composite model tostudy the asymptotic performance of multiple-branch FSO systems over thecorrelated lognormal channels.656.2. Diversity Analysis of FSO Systems6.2 Diversity Analysis of FSO SystemsIn this section, we will present useful results on the diversity of FSOsystems over fading channels. These results will facilitate the analysis inSection 6.3.Lemma 6.1. For a channel state h having asymptotic PDF3 fH,asym(h) =aht+ gt(h), if the received instantaneous SNR has the form γ = Chk, whereC is a constant and k is a positive number, the asymptotic PDF of theinstantaneous SNR can be obtained as fγ,asym(γ) =akCt+1kγt+1k−1+g t+1k−1(γ).Proof. This result can be easily obtained by using a change of variable ruleh =( γC) 1k in fH,asym(h) = aht + gt(h).Lemma 6.1 will be frequently used in this chapter to convert the asymp-totic PDF of channel state to that of received instantaneous SNR and viceversa.Proposition 6.2. For an L-branch system with independent received instan-taneous SNRs γi, i = 1, 2, · · · , L, the MRC combiner output instantaneousSNR γ = γ1+γ2+ · · ·+γL has the diversity order of Gd,1+Gd,2+ · · ·+Gd,L,where Gd,i is the diversity order on the ith branch.Proof. We first discuss the two-branch case (L = 2) and the multi-branchcase can be proved iteratively by using the result of the two-branch case.For two received SNRs γ1, γ2 having PDFs f1(γ1), f2(γ2) and asymptoticPDFsf1,asym(γ1) =a1γt11 + gt1(γ1),f2,asym(γ2) =a2γt22 + gt2(γ2)(6.11)the MRC output instantaneous SNR γ = γ1 + γ2 has the PDFfγ,MRC(γ) =∫ γ0f1(γ − a)f2(a)da. (6.12)When γ → 0, we have a→ 0 and (γ − a)→ 0; therefore, we can obtain theasymptotic PDF of γ using the asymptotic PDFs of γ1 and γ2 via convolutionasfγ,asym(γ) =∫ γ0f1,asym(γ − a)f2,asym(a)da. (6.13)3Hereafter we use the term asymptotic PDF to denote the PDF near origin.666.2. Diversity Analysis of FSO SystemsSubstituting (6.11) into (6.13) and using Newton’s generalised binomial the-orem, we can obtain the asymptotic PDF of γ asfγ,asym = a1a2( ∞∑k=0(−1)k(t1k)′t2 + k + 1)γt1+t2+1 + gt1+t2+1(γ) (6.14)where(rk)′= r(r−1)···(r−k+1)k! . Therefore we haveGd = (t1 + t2 + 1) + 1 = (t1 + 1) + (t2 + 1) = Gd,1 +Gd,2. (6.15)For the L-branch case, we can construct γ = ((· · · (γ1 + γ2) + γ3) + γ4) +· · · + γL−1) + γL) and use the two-branch result iteratively to obtain Gd =L∑i=1Gd,i.Proposition 6.3. When the received instantaneous SNR can be expressedas γ = x · y, where x and y are independent random variables, the di-versity order of the system is defined by the one with lower diversity order:min{Gd,x, Gd,y}, where Gd,x and Gd,y are the diversity orders of system hav-ing instantaneous SNR x and y respectively.Proof. We first define the asymptotic PDFs of x and y asfX,asym(x) =a1xt1 + gt1(x),fY,asym(y) =a2yt2 + gt2(y).(6.16)For the t1 > t2 case, we can use the product distribution rule to obtain thePDF of received instantaneous SNR asfγ(γ) =∫ ∞0fX(x)fY(γx) 1xdx (6.17)where we have used the fact that the SNR is always a positive quantity.Partitioning the interval [0,∞] into [0, ε] and [ε,∞], where ε is an arbitrarilysmall positive quantity, we can rewrite the PDF of γ as a sum of two parts:fγ(γ) =∫ ε0fX(x)fY(γx) 1xdx+∫ ∞εfX(x)fY(γx) 1xdx (6.18)When γ → 0, we havefγ,asym(γ) =∫ ε0fX,asym(x)fY,asym(γx) 1xdx+∫ ∞εfX(x)fY,asym(γx) 1xdx.(6.19)676.2. Diversity Analysis of FSO SystemsSubstituting (6.16) in (6.19), we can getfγ,asym(γ) =(a1a2t1 − t2 εt1−t2 + a2∫ ∞ε1xt2+1fX(x)dx)γt2 + gt2(γ). (6.20)Let ε→ 0, we can rewrite (6.20) asfγ,asym(γ) = a2Mlog(X)(−t2 − 1)γt2 + gt2(γ) (6.21)where MX(t) is the moment-generating function (MGF) of RV x. From(6.20), we have Gγ = t2 +1 = min{Gd,1, Gd,2}. For the case t1 < t2, we canuse an equivalent product distribution rulefγ(γ) =∫ ∞0fX(γy)fY (y)1ydy (6.22)and follow the derivation in the t1 > t2 case to obtainfγ,asym(γ) = a1Mlog(Y )(−t1 − 1)γt1 + gt1(γ) (6.23)which shows that Gγ = t1 + 1 = min{Gd,1, Gd,2}. Therefore, the proof ofProposition 6.3 is complete.Corollary 6.4. When the received instantaneous SNR can be expressed asa product of multiple independent RVs, the diversity order of the system isdefined by the one with the lowest diversity order.Proof. We can obtain this corollary by using the derivation for the two RVscase (Proposition 6.3) iteratively.Proposition 6.3 and Corollary 6.4 facilitate the diversity analysis of sys-tems with composite fading channel or with channel state that can be de-composed to multiplication of independent random variables. Based on ex-tended Rytov theory, many atmospheric fading models that can be describedby the modulation process having the channel gain of the form h = XY ,and the received instantaneous SNR will have the same form [5]. The RVX is assumed to arise from large-scale turbulent eddy effects and Y fromstatistically independent small-scale eddy effects. Such product form can beconsidered as one random process modulating another: the mean of one dis-tribution (fading) is modulated by another distribution (shadowing). Thesecomposite channels include, but are not limited to, lognormal-Nakagami, K,lognormal-Rician, Gamma-Gamma andM fading channels in FSO systems.We also find that in these atmospheric fading models, the diversity order is686.2. Diversity Analysis of FSO Systems−10 −5 0 5 10 15 20 25 3010−1210−1010−810−610−410−2100Transmit power Pt (dBm)BER  x, α = 4y, β = 3zFigure 6.1: The BER of an IM/DD OOK system over the Gamma-Gammafading channel with parameters α = 4, β = 3.usually determined by small scale effects (fading). Moreover, this proposi-tion can be used on FSO systems with composite fading and pointing errorchannels [86], as well as Rayleigh, Rice, Nakagami fading with lognormalshadowing channels in radio frequency systems.Here we assume that the thermal noise is the dominant noise source,and it is modeled by a zero mean Gaussian distribution. In the rest ofthis chapter, unless otherwise stated, we set its variance as σ2n = 10−6.We now use the Gamma-Gamma distribution and the lognormal-Nakagamidistribution to exemplify Proposition 6.3.The Gamma-Gamma RV z can be expressed as the multiplication of twoGamma RVs, namely, z = x · y. In Fig. 6.1, we have α = 4, β = 3 andthe BER of IM/DD OOK systems with three channel states correspondingto RVs x, y, z are shown. In Fig. 6.2, we have α = 4, β = 1.3. It can beseen that the diversity order of the system having channel state z is thesame as that of the system having channel state y, which is Gd = Gy =min{Gd,x, Gd,y}.696.2. Diversity Analysis of FSO Systems−10 −5 0 5 10 15 20 25 3010−1210−1010−810−610−410−2100Transmit power Pt (dBm)BER  x, α = 4y, β = 1.3zFigure 6.2: The BER of an IM/DD OOK system over the Gamma-Gammafading channel with parameters α = 4, β = 1.3.706.2. Diversity Analysis of FSO Systems−10 −5 0 5 10 15 20 25 3010−1010−810−610−410−2100Transmit power Pt (dBm)BER  x, σ2 = 0.09y, m = 3zFigure 6.3: The BER of an IM/DD OOK system over the lognormal-Nakagami fading channel with parameters σ2 = 0.09,m = 3.716.2. Diversity Analysis of FSO Systems−10 −5 0 5 10 15 20 25 3010−1010−810−610−410−2100Transmit power Pt (dBm)BER  x, σ2 = 0.09y, m = 1.8zFigure 6.4: The BER of an IM/DD OOK system over the lognormal-Nakagami fading channel with parameters σ2 = 0.09,m = 1.8.726.3. FSO Systems over Correlated Lognormal Fading ChannelsIn the lognormal-Nakagami case, we have z = x · y, where x followsthe lognormal distribution and y follows the Nakagami-m distribution. TheBER of the IM/DD OOK systems having channel states x, y, z are shownin Fig. 6.3 with σ2 = 0.09,m = 3 and Fig. 6.4 with σ2 = 0.09,m = 1.8. Itcan be seen that the diversity order of the system is determined by the mparameter, which is Gd = Gd,y = min{Gd,x, Gd,y}.Therefore, we can observe that the diversity order of the system ismin{Gd,x, Gd,y}, which is predicted by Proposition 6.3.6.3 FSO Systems over Correlated LognormalFading ChannelsIn order to investigate the performance characteristics of correlated log-normal channels, we use the lognormal-Nakagami fading model introducedin Section 6.1. First we consider a dual-branch system, then we generalizeit to the multiple-branch case.6.3.1 Dual-branch SystemFor a dual-branch system, we model the two channel states h1 and h2 ash1 =x1 · y1,h2 =x2 · y2(6.24)where y1 and y2 are i.i.d. Nakagami-m RVs; x1 and x2 are correlated log-normal RVs with PDFflogn,logn(x1, x2) =12piσ1σ2√1−ρ2x1x2×exp− 12(1−ρ2)(lnx1−µ1)2σ21+(lnx2−µ2)2σ22−2ρ(ln x1−µ1)(lnx2−µ2)σ1σ2 (6.25)where ρ is the correlation coefficient between x1 and x2. Based on thederivation in Appendix G.1.1, we can obtain the asymptotic PDF of theinstantaneous SNR γ at the output of MRC asfγ,asym(γ) =m2mΓ(m)2S1,1γ2m−1 exp(m2 (m+ 1)(σ21 + σ22) +m2ρσ1σ2) (6.26)736.3. FSO Systems over Correlated Lognormal Fading Channelswhere Sp,q =(∑pm−1i=0 (−1)i (pm−1i )qm+i), p, q = 1, 2, · · · . The diversity ordercan be obtained from (6.26) asGd = 2m. (6.27)The coding gain is obtained asGc=222m−2Γ(2m+ 12)m2m√pimΓ(m)2m−1∑i=0 (−1)i m− 1im+i× exp (m2 (m+ 1)(σ21 + σ22) +m2ρσ1σ2)− 12m(6.28)where(nk)= n!(n−k)!k! .It is clear from (6.27) and (6.28) that the diversity order is determinedonly by the parameter m in Nakagami distribution, while the coding gainis affected by the correlation ρ. Therefore, the relative coding gain can beobtained asGc,correlatedGc,independent= exp(−m2ρσ1σ2)= −2.17mρσ1σ2 dB. (6.29)From (6.29), we conclude that the SNR penalty (w.r.t. the independentbranches) induced by correlation trends to be infinity when m→∞.6.3.2 Multiple-Branch SystemLemma 6.5. The asymptotic PDF of L-branch MRC, EGC and SC systemover the correlated lognormal-Nakagami channels can be expressed asfγ,asym(γ) =mLmcΓ(m)LγLm−1 exp(12m((L∑i=1σ2i)+m||Σ||1))(6.30)where ||Σ||1 denotes the entry-wise norm of the covariance matrix Σ in (6.6),which is defined as ||Σ||1 =L∑i=1L∑j=1|σ2i,j|, and we havec =L−1∏i=1S1,i MRC,c = 2L−1LLmL−1∏i=1S2,2i EGC,c = Lm1−L SC.(6.31)746.3. FSO Systems over Correlated Lognormal Fading ChannelsProof. See Appendix A.From (6.30), we can obtain the SNR penalty induced by correlation onmulti-branch systems asGc,correlatedGc,independent= exp(−m4(||Σ||1 − tr(Σ)))= −1.09m (||Σ||1 − tr(Σ))(6.32)where tr(A) denotes the trace of an square matrix A. It can be seen from(6.32) that the SNR penalty induced by correlation is also infinity whenm → ∞ for the multi-branch systems over the correlated lognormal fadingchannels. In [6], the authors also reached a similar conclusion based only onnumerical observations.In order to compare the asymptotic performance of different FSO sys-tems over different lognormal fading channels, we define the asymptoticrelative diversity order asARDO , limγ¯→∞log(Pe,1(γ¯))log(Pe,L(γ¯)). (6.33)Proposition 6.6. The asymptotic relative diversity order between L-branchMRC system with covariance matrix Σ of the logarithm of lognormal channelstates and single-branch system with lognormal fading variance of σ2 is L2σ2divided by the entry-wise norm of the covariance matrix of L-branch system,which is ARDO = L2σ2||Σ||1 . When every branch of L-branch system has identi-cal variance of that of single-branch system, we have ARDO = L2||R||1 whereR is the correlation matrix of the logarithm of lognormal channel states inan L-branch system.Proof. See Appendix H.Note that when each branch of the L-branch system has identical vari-ance of that of single channel system, we have 1 ≤ ARDO ≤ L. The leftequality sign can be achieved when the channels are fully correlated (all-onescorrelation matrix), and the right equality sign can be achieved when thechannels are uncorrelated (identity correlation matrix). It is worthy to notethat the ARDO between different systems can be obtained using the ARDOderived from Proposition 6.6.In Fig. 6.5 and Fig. 6.6, we use a dual-branch system to exemplifyProposition 6.6. Fig. 6.5 presents the RDO between a dual-branch systemhaving identical lognormal fading (σ21 = σ22 = 0.49) and a single-branchsystem (σ2 = 0.49). The the ARDO calculated by ARDO = L2||R||1 is also756.3. FSO Systems over Correlated Lognormal Fading Channels0 10 20 30 40 50 60 70 801.051.251.542Transmit power (dBm)RDO  ρ = 0ρ = 0.3ρ = 0.6ρ = 0.9Figure 6.5: The RDO between a dual-branch system in the lognormal fadingchannels with σ21 = σ22 = 0.49 and a single-branch system with σ2 = 0.49.shown in Fig. 6.5 as the dot lines. In Fig. 6.6, the RDO is presented for adual-branch system having nonidentical lognormal fading (σ21 = 0.49, σ22 =0.64) and a single-branch system (σ2 = 0.49). In Fig. 6.7, we presentthe RDO between a three-branch system and a single-branch system (σ2 =0.49). For simplicity, we set the covariance matrix of the three-branch systemasΣ = σ2 1 ρ ρ2ρ 1 ρρ2 ρ 1 . (6.34)Due to the computational limitations in mathematical softwares, some RDOscan not be calculated when the transmit power is large. From the obtainedresults, we can see that the RDO tends to approach the ARDO given inProposition 6.6.Proposition 6.7. The coding gain difference between the L-branch MRCand EGC systems over correlated lognormal channels tends to be 0dB, that766.3. FSO Systems over Correlated Lognormal Fading Channels0 10 20 30 40 50 60 70 800.921.091.341.73Transmit power (dBm)RDO  ρ = 0ρ = 0.3ρ = 0.6ρ = 0.9Figure 6.6: The RDO between a dual-branch system in the lognormal fadingchannels with σ21 = 0.49, σ22 = 0.64 and a single-branch system with σ2 =0.49.is limγ¯→∞Gc,EGCGc,MRC= 0dB,while the asymptotic BER difference is 3.01(L−1)dB,that is limγ¯→∞Pe,EGC,asymPe,MRC,asym= 2L−1. The coding gain difference between the MRCand SC systems is limγ¯→∞Gc,SCGc,MRC= 1L = −10 log(L)dB.Proof. This proposition can be obtained by comparing the results in Lemma6.5. First we have Pe = (Gcγ¯)−Gd , and from (6.30) we can obtain thatGd =Lm,Gc =2(2Lm−1Γ(Lm+ 12)mLmc√piLmΓ(m)Lexp(12m((L∑i=1σ2i)+m||Σ||1)))− 1Lm.(6.35)776.3. FSO Systems over Correlated Lognormal Fading Channels0 10 20 30 40 50 60 70 801.091.472.053Transmit power (dBm)RDO  ρ = 0ρ = 0.3ρ = 0.6ρ = 0.9Figure 6.7: The RDO between a three-branch system in the lognormal fadingchannels and a single-branch system with σ2 = 0.49.Therefore we haveGd,MRC = Gd,EGC = Gd,SC,Gc,EGCGc,MRC=(cEGCcMRC)− 1Lm,Gc,SCGc,MRC=(cSCcMRC)− 1Lm.(6.36)Recall that Sp,q =∑pm−1i=0 (−1)i (pm−1i )qm+i , which can be shown to have thefollowing equivalent form:Sp,q =1∫0(1− a)pm−1aqm−1da. (6.37)786.3. FSO Systems over Correlated Lognormal Fading ChannelsFor m→∞, we havelimm→∞Sp,q1m =||(1− a)paq||∞=max{(1 − a)paq, a ∈ [0, 1]}=ppqq(p+ q)p+q.(6.38)Therefore for a sufficiently large value of m, we haveSp,q ≈(ppqq(p + q)p+q)m. (6.39)When m→∞, we can use (6.39), (6.31) and (6.36) to obtainlimm→∞Gc,EGCGc,MRC= limm→∞(2L−1)− 1Lm = 1,limm→∞Gc,SCGc,MRC= limm→∞(LLm)− 1Lm = L−1.(6.40)While the coding gain difference of MRC and EGC tends to be 0dB, theBER difference in large average SNR region is a constant and it is given bylimγ¯→∞Pe,EGCPe,MRC=(Gc,EGCGc,MRC)−Lm=(cEGCcMRC)−Lm= 2L−1.(6.41)With the results in (6.40) and (6.41), we have completed the proof.For a dual-branch system, the asymptotic BER difference between MRCand EGC is 3.01dB, which represents the vertical distance between the BERplots of MRC and EGC systems. The coding gain difference between MRCand SC is 3.01dB, which represents the horizontal distance between theBER plots of MRC and SC systems. These results suggest that the MRCand EGC has similar diversity performance while SC suffers certain SNRloss. Similar results have also been observed in the numerical results sectionof [6, 52]. In Tables I and II, we present the results on the SNR and BERoffset among the MRC, EGC and SC systems. It can be seen that the resultstend to match those stated in Proposition 6.7.796.3. FSO Systems over Correlated Lognormal Fading ChannelsTable 6.1: SNR and BER offset between MRC, EGC and SC system overlognormal fading channels (σ21 = σ22 = 4).Transmit Power (dB) 0 50 100 150 200SNR offset: EGC/MRC (dB) -0.76 -0.22 -0.12 -0.08 -0.06SNR offset: SC/MRC (dB) -0.56 -2.06 -2.38 -2.56 -2.67BER offset: EGC/MRC (dB) -0.34 -1.23 -1.36 -1.41 -1.43BER offset: SC/MRC (dB) -0.23 -10.76 -26.25 -42.58 -59.22Table 6.2: SNR and BER offset between MRC, EGC and SC system overlognormal fading channels (σ21 = σ22 = 0.64).Transmit Power (dB) 0 20 40 60 80SNR offset: EGC/MRC (dB) -0.17 -0.07 -0.04 -0.03 -0.02SNR offset: SC/MRC (dB) -1.90 -2.57 -2.72 -2.80 -2.83BER offset: EGC/MRC (dB) -0.51 -1.15 -1.29 -1.36 -1.39BER offset: SC/MRC (dB) -3.47 -38.10 -79.33 -121.54 -164.12806.4. Summary6.4 SummaryIn this chapter, we studied a multiple-branch FSO system over correlatedlognormal fading channels. The diversity of FSO systems over compositefading channels was also analyzed, and we found that the diversity of thesystem is determined by the lowest diversity RV in the product form. Formultiple-branch FSO systems, we found that the correlation of lognormalchannels can cause large SNR penalty. We also showed that the ARDObetween L-branch FSO system and a single-branch FSO system over thelognormal fading channel is L2σ2 divided by the entry-wise norm of the co-variance matrix Σ. The derived ARDO can provide FSO system designerswith a simple asymptotic performance metric to allow asymptotic perfor-mance comparison between different FSO systems under various lognormalchannel conditions. The MRC, EGC and SC combining techniques were alsocompared in the context of correlated lognormal channels. We found thatthe MRC and EGC have similar performance but SC suffers a 10 log(L)dBloss, which suggests that EGC is desirable for multiple-branch FSO systemsin terms of better performance and complexity tradeoff.81Chapter 7ConclusionsIn this chapter, we summarize the results obtained in this thesis. Also,we present several potential future research topics which are related to ouraccomplished work.7.1 Summary of ResultsIn this thesis, we conducted analytical performance evaluation of FSOsystems over atmospheric fading channels. The obtained results can providethe FSO system designers with insights into the performance of such sys-tems, which include the system performance over different atmospheric fad-ing channels, the effects of adverse factors on system performance, and the ef-fects of spatial diversity and correlation on system performance. First, we in-vestigated the FSO systems over lognormal, Gamma-Gamma and lognormal-Rician fading channels. Then we studied the pointing error effects and phaseerror effects, which may cause performance degradation in practical FSOsystems. In the end, we analyzed an FSO system with multiple branch re-ception over correlated lognormal fading channels. Here we summarize theresults obtained in each chapters as follows.In Chapter 3, we analyzed the error rate performance of a coherent FSOsystem over the lognormal-Rician turbulence channels using a PA approach.Closed-form BER expressions for BPSK and DPSK FSO systems with MRCreception were obtained. We also presented the asymptotic error rate anal-ysis. We proved that PA is a powerful analytical tool for analyzing opticalcommunication systems in lognormal-Rician channel that is analytically in-tractable. Our analysis further showed that the PA approach is particularlyuseful in obtaining highly accurate small error rate estimation.In Chapter 4, we proposed a nonzero boresight pointing error model byconsidering beamwidth, detector aperture, and jitter variance. We assumethat the boresight component of pointing error effects is not negligible, andthe sway on x-axis and y-axis are Gaussian distributed. We derived closed-form PDF for the nonzero boresight pointing error model. Based on such apointing error model, we derived closed-form composite PDF for the lognor-827.1. Summary of Resultsmal fading and accurate series based PDF for the Gamma-Gamma fading.Highly accurate convergent series BER expressions were obtained. Throughthe asymptotic BER analysis and numerical results, we observed that theboresight error causes an SNR penalty factor on error rate performanceat high SNR. By examining the asymptotic BER curves, we found that thediversity order of the FSO system over the composite lognormal fading chan-nel is solely determined by the pointing error parameter γ2, which meansthat the boresight component does not affect the diversity order. Whilein the composite Gamma-Gamma fading channel, the diversity order is de-termined by either the Gamma-Gamma fading effect or the pointing erroreffect, depending on which is stronger.In Chapter 5, we studied the BER performance of a subcarrier MPSKsystem with CPE over lognormal fading channels. The CPE induced asymp-totic noise reference loss over the lognormal channels was quantified. In thederivation, we proposed to use an auxiliary RV method to approach thelognormal channel with a lognormal-Nakagami channel. Our results showedthat the CPE induced performance degradation for a subcarrier BPSK sys-tem over the lognormal channel is negligible. We also found that the CPEinduced performance degradation increases with higher modulation orderand lower PLL SNR coefficient C.In Chapter 6, we studied an FSO system with spatial diversity receptionover correlated lognormal fading channels. The diversity of such systemover composite fading channels was also analyzed, and we found that thediversity is determined by the lowest diversity RV in the product form. Wealso found that the correlation of lognormal channels can cause large SNRpenalty for multiple-branch FSO systems. We obtained that the ARDObetween L-branch FSO system and a single-branch FSO system over thelognormal fading channel is L2σ2 divided by the entry-wise norm of the co-variance matrix Σ. The derived ARDO can provide FSO system designerswith a simple asymptotic performance metric to allow asymptotic perfor-mance comparison between different FSO systems under various lognormalchannel conditions. The MRC, EGC and SC combining techniques were alsocompared in the context of correlated lognormal channels. We observed thatthe MRC and EGC have similar performance but SC suffers a 10 log(L)dBloss, which suggests that EGC is desirable for multiple-branch FSO systemsdue to its near optimal performance and moderate complexity.837.2. Future Work7.2 Future Work7.2.1 FSO system with pointing errorsIn Chapter 4, we proposed a nonzero boresight pointing error model byassuming Gaussian distribution on x-axis and y-axis sway. In practical FSOsystems, the displacement between beam center and receiver center may fol-low other distributions, such as uniform distribution, arcsin distribution andtruncated normal distribution. Therefore it is useful to investigate the effectsof pointing errors with different distributions on the system performance.On the other hand, in the thesis we only obtain the diversity order for FSOsystems with nonzero boresight pointing errors over the Gamma-Gammafading channels when γ2 > α, which implies that the Gamma-Gamma fad-ing effect is more dominant than the pointing error effect with respect to theBER performance at high average SNR region. However, when γ2 < α, thediversity order will depend on both γ2 and the boresight s. An explicit ex-pression of diversity order for nonzero boresight pointing case has not beenobtained yet. It is interesting to investigate the case of γ2 < α for such asystem model.7.2.2 FSO Networks over Correlated Lognormal FadingChannelsIn Chapter 6, we conducted the asymptotic performance analysis on FSOsystems over lognormal fading channels. 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Zwillinger, Table of Integrals, Series, and Products.Elsevier Science, 2000. → pages 96, 97, 99, 101[88] J. M. Wozencraft and I. M. Jacobs, Principles of Communication En-gineering. Waveland Press, Incorporated, 1990. → pages 10393AppendicesAppendix AAnalytical and NumericalResults of fhp(hp)When a Gaussian beam propagates through distance z from the trans-mitter to a circular detector with aperture radius a, the fraction of thecollected power at receiver is [11]hp(r; z) = G(r) ,∫ a−a∫ √a2−x′2−√a2−x′22piw2zexp(−2(x′ − r)2 + y′2w2z)dy′dx′(A.0.1)where r is the instantaneous radial displacement between the beam centroidand the detector center. We use the approximation given in (4.2) to derivethe analytical PDF of hp in (4.5), and the approximation in (4.2) has nor-malized mean-squared error less than 10−3 when wz/a > 6. Using (A.0.1),we obtain the exact PDF of hp asfhp(hp) = fr(G−1(hp)) · dG−1(hp)dhp. (A.0.2)Since there is no explicit expression of G−1(hp), we calculate it numerically.In Fig. A.1, we compare the analytical PDF of hp in (4.5) as well asthe exact PDF of hp without using (4.2) under various system settings.It is shown that our analytical model in (4.5) is accurate even with largeboresight (s/a = 5) and large jitter (σs/a = 3).94Appendix A. Analytical and Numerical Results of fhp(hp)2 4 6 8 10 12 14 16 18x 10−310−1010−810−610−410−2100102hpPDF  AnalyticalNumericals/a=5,σs/a=3s/a=0,σs/a=3s/a=5,σs/a=1 s/a=0,σs/a=1Figure A.1: Comparison of the analytical PDF in (4.5) and the exact PDFof hp under various system settings (wz/a = 10).95Appendix BDerivation of the moments ofhpThe moments of the generalized pointing error hp are given byE[hnp]=∫ A00hnpfhp(hp)dhp. (B.0.1)Substituting (4.5) into (B.0.1), we obtainE[hnp]=∫ A00γ2 exp(− s22σ2s)Aγ20hn+γ2−1p I0(sνσ2s)dhp (B.0.2)where ν =√−ω2zeq lnhpA02 . Using a series representation of I0(·) [87, Eq.(8.445)],we haveE[hnp]=γ2 exp(− s22σ2s)Aγ20∫ A00hn+γ2−1p∞∑m=01(m!)2(sν2σ2s)2mdhp.(B.0.3)Since each term of the series is non-negative and the infinite series uniformlyconverges to I0(sνσ2s), we can swap the integral and infinite summation and96Appendix B. Derivation of the moments of hpwrite (B.0.3) asE[hnp]=γ2 exp(− s22σ2s)Aγ20×∞∑m=01(m!)2∫ A00hn+γ2−1p(s24σ4s· −ω2zeq lnhpA02)mdhp=γ2 exp(− s22σ2s)Aγ20×∞∑m=0(−ω2zeqs28σ4s)m(m!)2An+γ20∫ 10xn+γ2−1(lnx)mdx.(B.0.4)Applying an integral identity [87, Eq.(4.294.10)] to (B.0.4), we haveE[hnp]= γ2 exp(− s22σ2s)An0∞∑m=0(γ2s22σ2s (n+γ2))mm!1(n+ γ2)=An0γ2n+ γ2exp(− ns2(n+ γ2)2σ2s).(B.0.5)97Appendix CGamma-Gamma CompositePDFC.1 Composite PDF ApproximationWe substitute (2.13) into (4.7) and write the composite PDF of theGamma-Gamma fading with nonzero boresight pointing error asfGG(h) =2γ2 exp(−s2/2σ2s)(αβ)(α+β)/2(A0hl)γ2Γ(α)Γ(β)hγ2−1×∫ ∞h/(A0hl)hα+β2−γ2−1a I0 sσ2s√−w2zeq2ln(hhaA0hl)Kα−β(2√αβha)dha.(C.1.1)Applying a change of variable rule x =√−w2zeq2 ln(hhaA0hl), eq. (C.1.1)can be expressed asfGG(h) =8γ2 exp(−s2/2σ2s)(αβ)(α+β)/2(A0hl)α+β2 Γ(α)Γ(β)w2zeq×∫ ∞0x exp(2x2w2zeq(α+ β2− γ2))I0(sσ2sx)W (h)dx(C.1.2)where W (h) = hα+β2−1Kα−β(2√αβhA0hlexp(x2w2zeq)). Using a series expan-sion of the modified Bessel function of the second kind [75, eq. (3.4.6.2.1)]Kv(x) =pi2 sin(piv)∞∑p=0[(x/2)2p−vΓ(p − v + 1)p! −(x/2)2p+vΓ(p+ v + 1)p!](C.1.3)98Appendix C. Gamma-Gamma Composite PDFwhere it requires v /∈ Z and |x| <∞, we can express W (h) asW (h) =pi2 sin(pi(α− β))∞∑j=0(αβA0hlexp(2x2w2zeq))j−α−β2Γ(j − (α− β) + 1)j! hj+β−1−(αβA0hlexp(2x2w2zeq))j+α−β2Γ(j + (α− β) + 1)j! hj+α−1 .(C.1.4)Substituting (C.1.4) into (C.1.2), and, after some manipulation, we havefGG(h) =4piγ2 exp(−s2/2σ2s)(αβ)(α+β)/2(A0hl)(α+β)/2Γ(α)Γ(β) sin((α − β)pi)w2zeqh(α+β)/2−1×∞∑j=0{1Γ(j − (α− β) + 1)j!(αβhA0hl)j−(α−β)/2G(β − γ2 + jw2zeq,sσ2s)− 1Γ(j + (α− β) + 1)j!(αβhA0hl)j+(α−β)/2G(α− γ2 + jw2zeq,sσ2s)}(C.1.5)where G(a, b) =∫∞0 x exp(2ax2)I0(bx)dx. In the following derivation, weuse an integral identity [87, eq. (6.643.2)]∫ ∞0xu−12 e−αxI2v(2β√x)dx=Γ(u+ v + 12)Γ(2v + 1)β−1 exp(β22α)α−uM−u,v(β2α) (C.1.6)where Mu,v(·) is the Whittaker function. Hence, by using another the iden-tity [75, eq. (07.44.03.0041.01)]Mm−12,m2(z) = exp(−z2)z1−m2 m!(exp(z)−m−1∑k=0zkk!),m ≥ 0 (C.1.7)99Appendix C. Gamma-Gamma Composite PDFwe can represent (C.1.5) as (12). In our derivation, the integral identity(C.1.3) requires that (α− β) /∈ Z, and the integral identity (C.1.6) requiresthat the summation index j in (C.1.5) must be lower than or equal to J =⌊γ2 − α⌋, where ⌊x⌋ denotes the largest integer not greater than x. Suchtruncation method is also used in [12] to estimate the BER of systems overthe Gamma-Gamma channels with Hoyt distributed pointing error. Dueto the truncation of the infinite series and the constraint on the number ofterms j ≤ ⌊γ2 − α⌋, our series PDF in (12) may not converge to the exactPDF for weak turbulence. However, beyond certain threshold of transmitpower Pt (shown in Appendix F), the approximate BER is accurate sincethe PDF fGG(h) near the origin can be accurately described by (12).C.2 PDF Near the OriginTo obtain the power series expansion of the Gamma-Gamma compositePDF near the origin, we can express W (h) aslimh→0W (h) =pi(αβA0hlexp(2x2w2zeq))−α−β22 sin(pi(α − β))Γ(−(α − β) + 1)hβ−1 + gβ−1(h).(C.2.1)From (C.2.1) and (C.1.2), we can derive the PDF near the origin aslimh→0fGG(h) =γ2pi(αβA0hl)βexp(− s22σ2s −s2γ2/σ2s2β−2γ2)sin−1((α− β)pi)Γ(α)Γ(β)Γ(−(α − β) + 1)| − (β − γ2)| hβ−1+ gβ−1(h).(C.2.2)100Appendix DBound on approximationerrorBy applying an upper bound erfc(x) < exp(−x2) to (4.15), we can upperbound RB asRB <γ2uc4exp(ua − γ2ub)×∫ ∞Bexp(γ2ucx− x2 − P2t2σ2nA20h2l exp(ucx− b))dx<γ2uc4exp(ua − γ2ub) ∫ ∞Bexp(γ2ucx− x2)dx.(D.0.1)Using an integral identity [87, Eq.(2.33.1)]∫exp(−(ax2 + 2bx+ c))dx =12√piaexp(b2 − aca)erf(√ax+b√a) (D.0.2)we derive the upper bound for RB asRB <√piγ2uc8exp(ua − γ2(ub − γ2u2c4))erfc(B − γ2uc2).(D.0.3)For different RB , the required values of B, which are calculated by(D.0.3), are shown in Table D.1, where we set wz/a = 10, s/a = 1.0, σs/a =1.0. It is found that the approximation error RB decreases rapidly withincreasing B, and therefore we can adjust the value of B to make RB arbi-trarily small.101Appendix D. Bound on approximation errorTable D.1: Values of B for different RB valuesParameters RB = 10−6 RB = 10−8 RB = 10−10σ2R = 0.05 7.60 8.18 8.69σ2R = 0.1 9.28 9.86 10.36σ2R = 0.2 11.65 12.22 12.73102Appendix EProof of convergence of seriesThe series in (4.20) contains the expressionPS =2√pi∞∑j=0(−1)jj!(2j + 1)(PtA0hl√2σn)2j+1 exp(−ub(2j + 1))(2j + 1 + γ2)ucSj.(E.0.1)We now use the ratio test to assess the convergence of this series. Theabsolute ratio between two consecutive terms is∣∣∣∣aj+1aj∣∣∣∣ =(2j + 1)(PtA0hl√2σn)2exp(−2ub)(2j + 1 + γ2)(j + 1)(2j + 3)(2j + 3 + γ2)· Sj+1Sj.(E.0.2)By applying the bounds of the erfc(·) function [88]1√pix(1− 12x2)exp(−x2) < erfc(x) < 1√pixexp(−x2) (E.0.3)to Sj+1 (upper bound) and Sj (lower bound), we haveSj+1Sj<1B +1(γ2+2j+3)uc2−B exp(2Buc)1B(1− 12B2)+ 1(γ2+2j+1)uc2−B(1− 12((γ2+2j+1)uc2−B)2) . (E.0.4)Applying the limit operation to both sides of (E.0.4) and noting that Sj andSj+1 are positive, we have0 < limj→∞Sj+1Sj<11− 12B2. (E.0.5)103Appendix E. Proof of convergence of seriesFor the rest of the terms in (E.0.2), we havelimj→∞(2j + 1)(PtA0hl√2σn)2exp(−2ub)(2j + 1 + γ2)(j + 1)(2j + 3)(2j + 3 + γ2)= 0. (E.0.6)Since limits in (E.0.5) and (E.0.6) both exist, we havelimj→∞∣∣∣∣aj+1aj∣∣∣∣ = limj→∞ (2j + 1)(PtA0hl√2σn)2(2j + 1 + γ2)exp(2ub)(j + 1)(2j + 3)(2k + 3 + γ2)· limj→∞Sj+1Sj= 0.(E.0.7)Thus, we have limj→∞∣∣∣aj+1aj ∣∣∣ = 0. We conclude that the series PS in (4.20) isconvergent.104Appendix FApproximation errorAfter the numerical evaluation of the approximation error in (4.23) underdifferent system parameters, we find our series solutions in (4.23) and (4.34)may not approach to the exact results when Pt is small. This is a limitationof the series approach. In Table F.1, we show the minimum required Pt thatcan guarantee the relative error ε(Pe)Pe < 10−6 and ε(Pout)Pout < 10−6 for threerepresentative turbulence conditions. From Table F, we observe that theminimum required Pt becomes larger with increasing boresight or jitter.105Appendix F. Approximation errorTable F.1: Minimum Required Pt(dBm) forε(Pe)Pe< 10−6 and ε(Pout)Pout < 10−6Parameters s/a = 1.0,σs/a = 1.0s/a = 1.0,σs/a = 1.5s/a = 2.0,σs/a = 1.5σ2R = 0.6;α = 5.41,β = 3.78−14, −18 −1, −5 9, 5σ2R = 2.0;α = 3.99,β = 1.70−19, −23 −8, −12 6, 2σ2R = 4.0;α = 4.34,β = 1.31−19, −23 −13, −17 −9, −12106Appendix GPDF of receivedinstantaneous SNR ofmultiple-branch systemG.1 MRCG.1.1 Dual BranchIn order to obtain the joint PDF of received instantaneous SNR (Forsimplicity here, we assume γ = h2) of system having channel state z1, z2in (6.24), we first introduce the PDF of a product of two independent RVsZ = X · Y :fZ(z) =∞∫−∞fX(x)fY( zx) 1|x|dx (G.1.1)where X,Y are two independent, continuous random variables, described byprobability density functions fX(x) and fY (y). Therefore using (6.7), (6.25)and (G.1.1), we can obtain the joint PDF of the received SNRs γ1, γ2 asfγ1,γ2(γ1, γ2) =∞∫0∞∫0fY1(γ1x1)fY2(γ2x2)fX1,X2(x1, x2)1x1x2dx1dx2(G.1.2)where we have used the fact that γ1 and γ2 are both positive variables. TheMRC has the combiner output instantaneous SNR γ = γ1 + γ2, which has107Appendix G. PDF of received instantaneous SNR of multiple-branch systemthe following PDFfγ(γ) =γ∫0fγ1,γ2(γ − a, a)da=γ∫0∞∫0∞∫0fY1(γ − ax1)fY2(ax2)fX1,X2(x1, x2)1x1x2dx1dx2da=γ∫0∞∫0∞∫0m2m exp(−m(γ−ax1+ ax2))Γ(m)2(γ − ax1)m−1( ax2)m−1× fX1,X2(x1, x2)1x1x2dx1dx2da.(G.1.3)For the PDF near the origin, we have the asymptotic PDF when γ → 0 asfγ,asym(γ) =γ∫0∞∫0∞∫0m2mΓ(m)2(γ − ax1)m−1( ax2)m−1fX1,X2(x1, x2)1x1x2dx1dx2da.(G.1.4)Using a binomial expansion theorem, we can simplify the asymptotic PDFin (G.1.4) tofγ,asym(γ)=∞∫0∞∫0m2mΓ(m)2(m−1∑i=0(−1)i(m−1i)m+ i)γ2m−1fX1,X2(x1, x2)1xm1 xm2dx1dx2=m2mΓ(m)2(m−1∑i=0(−1)i(m−1i)m+ i)γ2m−1∞∫0∞∫0fX1,X2(x1, x2)1xm1 xm2dx1dx2(G.1.5)where the double integral in (G.1.5) can be expressed by the MGF of amultivariate Gaussian distribution, which has two normal variables beingthe natural logarithm of x1, x2 in (6.24). Therefore the asymptotic PDF108Appendix G. PDF of received instantaneous SNR of multiple-branch systemnear the origin can be further simplified tofγ,asym(γ)=m2mΓ(m)2(m−1∑i=0(−1)i(m−1i)m+ i)γ2m−1Mlog(x1),log(x2)(−m,−m)=m2mΓ(m)2(m−1∑i=0(−1)i(m−1i)m+ i)γ2m−1 exp(m2(m+ 1)(σ21 + σ22) +m2ρσ1σ2).(G.1.6)G.1.2 Multiple-BranchFollowing the derivation of the two-branch case, we can write the jointPDF of L-branch received instantaneous SNR asfγ1,··· ,γL(γ1, · · · , γL) =∞∫0· · ·∞∫0fY1(γ1x1)· · · fYL(γLxL)fX1,··· ,XL(x1, · · · , xL)x1 · · · xL dx1 · · · dxL.(G.1.7)The output instantaneous SNR of MRC has the PDFfγ(γ) =γ∫0a1∫0· · ·aL−2∫0fγ1,γ2,··· ,γL(γ − a1, a1 − a2, · · · , aL−1)da1da2 · · · daL−2daL−1.(G.1.8)When γ → 0, we can get the asymptotic PDF from (G.1.8) asfγ,asym(γ) =γ∫0· · ·aL−2∫0∞∫0· · ·∞∫0mLmΓ(m)L(γ − a1x1)m−1· · ·(aLxL)m−1× fX1,··· ,XL(x1, · · · , xL)x1 · · · xL dx1 · · · dxLda1 · · · daL−1(G.1.9)109Appendix G. PDF of received instantaneous SNR of multiple-branch systemwhich can be simplified tofγ,asym(γ)=mLmΓ(m)L∞∫0· · ·∞∫0(γ − a1)m−1 · · · (aL)m−1da1 · · · daL−1×γ∫0· · ·aL−2∫0fX1,··· ,XL(x1, · · · , xL)xm1 · · · xmLdx1 · · · dxL=mLmΓ(m)L∞∫0· · ·∞∫0(γ − a1)m−1 · · · (aL)m−1da1 · · · daL−1×Mlog(x1),··· ,log(xL)(−m, · · · ,−m)(G.1.10)where Mx1,··· ,xL(t1, · · · , tL) is the MGF of a multivariate distribution func-tion fX1,··· ,XL(x1, · · · , xL). Since x′is, i = 1, · · · , L follow the lognormal dis-tribution, we haveMlog(x1),··· ,log(xL)(t1, · · · , tL) = exp(µT t+12tTΣt)(G.1.11)where T denotes matrix transpose, µ = [µ1, · · · , µL]T , t = [t1, · · · , tL]Tand Σ denotes the covariance matrix of log(x1), · · · , log(xL). Substituting(G.1.11) into (G.1.10) and using a binomial expansion theorem, we canobtain the asymptotic PDF asfγ,asym(γ)=mLmΓ(m)L(m−1∑i=0(−1)i(m−1i )Lm+ i)(m−1∑i=0(−1)i(m−1i )(L− 1)m+ i)· · ·(m−1∑i=0(−1)i(m−1i )m+ i)× γLm−1Mlog(x1),··· ,log(xL)(−m, · · · ,−m)=mLmΓ(m)LL−1∏j=1(m−1∑i=0(−1)i(m−1i )jm+ i)× γLm−1 exp(12m((L∑i=1σ2i)+m||Σ||1))=mLmΓ(m)LL−1∏j=1S1,j γLm−1 exp(12m((L∑i=1σ2i)+m||Σ||1)).(G.1.12)110Appendix G. PDF of received instantaneous SNR of multiple-branch systemG.2 EGCThe asymptotic PDF of instantaneous SNR can be obtained by followingderivation similar to the MRC case. In EGC, the received instantaneousSNR has the formγ =(h1 + h2 + · · ·+ hL)2L(G.2.1)where h′is are channel states following the lognormal-Nakagami distribution.The instantaneous SNR in the MRC case isγ = h21 + h22 + · · ·+ h2L. (G.2.2)From (G.2.1) and (G.2.2), we can see that instantaneous SNR of EGC isthe scaled squared of the summation of lognormal-Nakagami RVs, while theoutput instantaneous SNR of MRC is the summation of lognormal-GammaRVs (lognormal-Nakagami squared). Substituting the lognormal-Nakagamidistribution into (G.1.8) and, after some mathematica manipulation, we canobtain the asymptotic PDF asfγ,EGC,asym(γ) =2L−1(Lm)LmΓ(m)LL−1∏i=1S2,2iγLm−1 exp(12m((L∑i=1σ2i)+m||Σ||1)).(G.2.3)G.3 SCFor the SC system, we can first derive the cumulative density function(CDF) of received instantaneous SNR and then obtain the PDF using theCDF or outage probability. The outage probability of L-branch SC systemover the correlated lognormal-Nakagami channels can be expressed asPout,asym = Pr(γ1 < γth, γ2 < γth, · · · , γL < γth). (G.3.1)Using (G.1.7), we havePout,asym =γth∫0· · ·γth∫0fγ1,··· ,γL(γ1, · · · , γL)dγ1 · · · dγL=mL(m−1)Γ(m)L(γthγ¯)Lmexp(12m((L∑i=1σ2i)+m||Σ||1)).(G.3.2)111Appendix G. PDF of received instantaneous SNR of multiple-branch systemFrom (G.3.2), we can obtain the PDF of instantaneous SNR at the outputof SC asfγ,SC,asym(γ) =LmL(m−1)+1Γ(m)LγLm−1 exp(12m((L∑i=1σ2i)+m||Σ||1)).(G.3.3)112Appendix HARDO of MRCFrom the PDF in (G.1.12), we observe that the Nakagami parameter mand the number of branch L determine the diversity order, while the pa-rameters in lognormal PDF only appear in the exponential component. Thediversity order is dominated by the Nakagami RV because the lognormalPDF has undefined diversity order near its origin. Therefore the informa-tion of lognormal PDF is only contained in the exponential component in(G.1.12). In order to obtain the ARDO of MRC combined lognormal channelstates, we have to compensate the impact of the Nakagami variable on thediversity order of the system. Therefore we can setm1 = LmL, which makesthe Nakagami contributed diversity order to be identical for single-branchand multiple-branch systems. Then we can cancel out the contribution ofthe Nakagami variable by setting mL →∞, and obtain the ARDO asARDO = limmL→∞(12LmL(σ21 + LmLσ21))12mL((L∑i=1σ2i)+mL||Σ||1)=L2σ21||Σ||1 .(H.0.1)113

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