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On the performance of non-adaptive and adaptive optical wireless communications in atmospheric turbulence Hassan, Md. Zoheb 2013

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On the Performance of Non-Adaptiveand Adaptive Optical WirelessCommunications in AtmosphericTurbulencebyMd. Zoheb HassanB.Sc. Hons., Bangladesh University of Engineering and Technology, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)September 2013c? Md. Zoheb Hassan, 2013AbstractOptical wireless communication is an attractive solution to the ?lastmile? bottleneck problem with additional benefits of having rapid deploy-ment, enhanced security, protocol transparency, and unlicensed transmis-sion bandwidth. However, atmospheric turbulence induced signal fading isa major performance degrading factor for any outdoor optical wireless com-munication systems. Since the atmospheric turbulence induced fading variesslowly, adaptive transmission is an effective fading mitigation solution foran outdoor optical wireless communication system. In this thesis, we pri-marily focus on the performance analysis of optical wireless communicationsystems employing different adaptive transmission schemes.We first study the average symbol error rate for a nonadaptive subcar-rier intensity modulated optical wireless communication systems employinggeneral order rectangular quadrature amplitude modulation. We considerthree different turbulence channel models, i.e., the Gamma-Gamma chan-nel, the K-distributed channel, and the negative exponential channel withdifferent levels of turbulence. Exact average symbol error rate expressionsare derived using a series expansion of the modified Bessel function. In ad-dition, detailed truncation error analysis and asymptotic error rate analysisare also presented. Numerical results demonstrate that our series solutionsare highly accurate and efficient.Next we investigate a variable-rate, constant-power adaptive subcarrierintensity modulation employing M -ary phase shift keying and rectangu-lar quadrature amplitude modulation for optical wireless communicationover the Gamma-Gamma turbulence channels. The adaptive schemes of-fer efficient utilization of optical wireless communication channel capacityby adapting the modulation order according to the received signal-to-noiseiiAbstractratio and a pre-defined target bit-error rate requirement. Highly accurateseries solutions are presented for the achievable spectral efficiency, averagebit-error rate, and outage probability using a series expansion approach ofthe modified Bessel function. In addition, asymptotic bit-error rate andoutage probability analyses are presented. Our asymptotic bit-error rateanalysis shows that the diversity order of both non-adaptive and adaptivesystems depends only on the smaller channel parameter of the Gamma-Gamma turbulence. Numerical results demonstrate high accuracy of ourseries solutions with a finite number of terms and an improved spectral effi-ciency achieved by the adaptive systems without increasing the transmitterpower or sacrificing bit-error rate requirements.Finally, ergodic capacity is investigated for the optical wireless commu-nications employing subcarrier intensity modulation with direct detection,and coherent systems with and without polarization multiplexing over theGamma-Gamma turbulence channels. We consider three different adaptivetransmission schemes: (i) variable-power, variable-rate adaptive transmis-sion, (ii) complete channel inversion with fixed rate, and (iii) truncated chan-nel inversion with fixed rate. For the considered systems, highly accurateseries expressions for ergodic capacity are derived using a series expansion ofthe modified Bessel function and the Mellin transformation of the Gamma-Gamma random variable. Our asymptotic analysis reveals that the highSNR ergodic capacities of coherent, intensity modulated, and polarizationmultiplexing systems gain 0.33 bits/s/Hz, 0.66 bits/s/Hz, and 0.66 bits/s/Hzrespectively with 1 dB increase of average transmitted optical power. Nu-merical results indicate that a polarization control error less than 10? hasa little influence on the capacity performance of polarization multiplexingsystems.iiiPrefaceThis thesis is based on the research work conducted in the School ofEngineering at The University of British Columbias Okanagan campus un-der the supervision of Profs. Jahangir Hossain and Julian Cheng. Bothpublished and submitted works are contained in this thesis.Chapter 3 of this thesis has been partially published in the Proceed-ings of IEEE GLOBECOM 2012 and in the IEEE/OSA Journal of OpticalCommunications and Networking. I am the principle investigator of the an-alytical evaluations and numerical simulations of these publications. Alongwith Prof. Cheng, Mr. Xuegui Song, a Phd. student of Prof. Cheng, alsohelped to prepare the manuscripts by checking the validity of analytical andnumerical results, and proofreading the presentations.Chapter 4 of this thesis has been partially published in the IEEE Trans-actions on Communications, and in the Proceedings of IEEE ICC 2013,and IEEE CWIT 2013. A part of chapter 4 is also partially submitted tothe IEEE ICNC 2014. I like to specify that the research idea, analyticalevaluations, and numerical simulations related to these publications are theresults of my own work. My supervisors helped me check the validity of theanalytical and numerical findings and proofread the manuscripts.Chapter 5 of this thesis has been partially published in the OSA OpticsExpress, and in the proceedings of the IEEE WOCC 2013. I am the prin-ciple contributor of the research idea, analytical evaluations, and numericalsimulations related to these publications. My supervisors helped me to checkthe validity of the analytical and numerical findings and to proofread themanuscripts.A list of my publications at The University of British Columbia is pro-vided in the following.ivPrefaceJournal Papers Published1. M. Z. Hassan, M. J. Hossain, and J. Cheng, ?Ergodic capacity compar-ison of optical wireless communications using adaptive transmissions,?OSA Optics Express, vol. 21, pp. 20346-20362, Aug. 2013 (part ofChapter 5).2. Md. Z. Hassan, Md. J. Hossain, and J. Cheng, ?Performance of non-adaptive and adaptive subcarrier intensity modulations in Gamma-Gamma turbulence,? IEEE Trans. Commun. vol. 61, pp. 2946-2957,Jul. 2013 (part of Chapter 4).3. M. Z. Hassan, X. Song, and J. Cheng, ?Subcarrier intensity modulatedwireless optical communications with rectangular QAM,? IEEE/OSAJ. Opt. Commun. Netw., vol. 6, pp. 522-532, Jun. 2012 (part ofChapter 3).4. X. Song, M. Z. Hassan, and J. Cheng,?Subcarrier DQPSK modu-lated optical wireless communications in atmospheric turbulence,? IETElectronics Letters, vol. 48, pp. 1224-1225, Sep. 2012.Conference Papers Published1. M. Z. Hassan, M. J. Hossain, and J. Cheng,?Exact BER Analysisof Subcarrier QAM and PSK Intensity Modulations in Strong Tur-bulence,? Accepted in International Conference on Computing, Net-working and Communications (ICNC), Honolulu, USA, Feb. 3-6, 2014(part of chapter 4).2. M. Z. Hassan, M. J. Hossain, and J. Cheng, ?Ergodic Capacity ofCoherent Optical Wireless Communications in Gamma-Gamma Tur-bulence,? in Proceedings of the 22nd Annual Wireless and OpticalCommunications Conference (WOCC), Chongqing, China, May 16-18, 2013 (part of chapter 5).vPreface3. M. Z. Hassan, M. J. Hossain, and J. Cheng, ?Performance of adap-tive subcarrier QAM intensity modulation in Gamma-Gamma Turbu-lence,? in Proceedings of IEEE International Conference on Commu-nications (ICC), Budapest, Hungary, June 9-13, 2013 (part of chapter4).4. M. Z. Hassan, M. J. Hossain, and J. Cheng, ?Performance of MIMOAdaptive Subcarrier QAM Intensity Modulation in Gamma-GammaTurbulence,? in Proceedings 13th Canadian Workshop on InformationTheory (CWIT), Toronto, ON, Canada, Jun. 18-21, 2013 (part ofchapter 4).5. M. Z. Hassan, X. Song, and J. Cheng, ?Error rate analysis of subcar-rier intensity modulation using rectangular QAM in Gamma-Gammaturbulence,? in Proceedings of the IEEE Global Communications Con-ference (GLOBECOM), Anaheim, CA, USA, Dec. 3-7, 2012 (part ofchapter 3).viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xviChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Outline and Contributions . . . . . . . . . . . . . . . . 8Chapter 2: Background of Optical Wireless CommunicationSystems and Channel Models . . . . . . . . . . . . 122.1 Classification of OWC Systems Based on Detection Techniques 122.1.1 Coherent OWC System . . . . . . . . . . . . . . . . . 122.1.2 POLMUX OWC System with Coherent Detection . . 142.1.3 OWC System with Subcarrier Intensity Modulation/DirectDetection . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Atmospheric Turbulence Models . . . . . . . . . . . . . . . . 172.2.1 Gamma-Gamma Model . . . . . . . . . . . . . . . . . 172.2.2 K-Distributed Model . . . . . . . . . . . . . . . . . . . 18viiTABLE OF CONTENTS2.2.3 Negative Exponential Model . . . . . . . . . . . . . . 192.3 Adaptive Transmissions over Fading Channels . . . . . . . . . 202.3.1 Constant-Power Variable-Rate Adaptive TransmissionScheme . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Variable-Power Variable-Rate Adaptive TransmissionScheme . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.3 Variable-Power Fixed-Rate Adaptive Transmission . . 26Chapter 3: Subcarrier Intensity Modulated Wireless OpticalCommunications With Rectangular QAM . . . . . 283.1 Error Rate Analysis of SIM Using Rectangular QAM . . . . . 283.1.1 Subcarrier QAM in Gamma-Gamma Turbulence Chan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Subcarrier QAM in K-Distributed Turbulence Channels 343.1.3 Subcarrier QAM in Negative Exponential TurbulenceChannels . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.4 Truncation Error Analysis for Subcarrier QAM . . . . 373.1.5 Asymptotic Error Rate Analysis for Subcarrier QAM 393.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Chapter 4: Performance of Non-Adaptive and Adaptive Sub-carrier Intensity Modulations in Gamma-GammaTurbulence . . . . . . . . . . . . . . . . . . . . . . . 524.1 Performance Analysis of Non-Adaptive SIM System . . . . . 524.1.1 BER of Subcarrier R-QAM in Gamma-Gamma Tur-bulence Channels . . . . . . . . . . . . . . . . . . . . . 534.1.2 BER of Subcarrier M -PSK in Gamma-Gamma Tur-bulence Channels . . . . . . . . . . . . . . . . . . . . . 574.1.3 Asymptotic BER . . . . . . . . . . . . . . . . . . . . . 594.2 Adaptive Modulation Strategy . . . . . . . . . . . . . . . . . 604.3 Performance Analysis of Adaptive SIM . . . . . . . . . . . . 614.3.1 Achievable Spectral Efficiency . . . . . . . . . . . . . . 61viiiTABLE OF CONTENTS4.3.2 BER of R-QAM Based Adaptive SIM . . . . . . . . . 624.3.3 BER of M -PSK Based Adaptive SIM . . . . . . . . . 644.3.4 Asymptotic BER . . . . . . . . . . . . . . . . . . . . . 674.3.5 Outage Probability . . . . . . . . . . . . . . . . . . . . 694.3.6 Adaptive MIMO Based SIM Systems . . . . . . . . . . 694.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 744.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Chapter 5: Ergodic Capacity Comparison of Optical Wire-less Communications Using Adaptive Transmis-sions . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.1 Ergodic Capacity of Coherent OWC System . . . . . . . . . . 895.1.1 Variable-Power, Variable-Rate Adaptive Transmission 895.1.2 Complete Channel Inversion with Fixed Rate . . . . . 935.1.3 Truncated Channel Inversion with Fixed Rate . . . . . 945.2 Ergodic Capacity of Coherent POLMUX OWC System . . . . 955.2.1 Variable-Power, Variable-Rate Adaptive Transmission 955.2.2 Complete Channel Inversion with Fixed Rate . . . . . 965.2.3 Truncated Channel Inversion with Fixed Rate . . . . . 975.3 Ergodic Capacity of Subcarrier IM/DD OWC System . . . . 975.3.1 Variable-Power, Variable-Rate Adaptive Transmission 975.3.2 Channel Inversion with Fixed Rate . . . . . . . . . . . 1005.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 1015.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Chapter 6: Conclusions . . . . . . . . . . . . . . . . . . . . . . . 1116.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . 1116.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Appendix A: ASER Analysis of Subcarrier R-QAM Using DirectIntegration Approach . . . . . . . . . . . . . . . . . . 125ixTABLE OF CONTENTSAppendix B: Truncation Error Analysis for the BER of R-QAMBased Adaptive SIM . . . . . . . . . . . . . . . . . . 130Appendix C: Truncation Error Analysis for the BER of AdaptiveM -PSK SIM . . . . . . . . . . . . . . . . . . . . . . . 134Appendix D: Mellin Transformation of the Gamma-Gamma Ran-dom Variable . . . . . . . . . . . . . . . . . . . . . . . 137xList of FiguresFigure 3.1 ASER of subcarrier 4 ? 4 QAM over the Gamma-Gamma channels with difference levels of turbulence. 42Figure 3.2 ASER of subcarrier 8 ? 8 QAM over the Gamma-Gamma channels with different levels of turbulence. . 43Figure 3.3 ASER of subcarrier 4?4 QAM over the K-distributedturbulence channels. . . . . . . . . . . . . . . . . . . . 44Figure 3.4 ASER of subcarrier 8?8 QAM over the K-distributedturbulence channels. . . . . . . . . . . . . . . . . . . . 45Figure 3.5 ASER of subcarrier 4?4 and 8?8 QAM over negativeexponential turbulence channel. . . . . . . . . . . . . 46Figure 3.6 Absolute truncation error of subcarrier 4 ? 4 QAMover a moderate Gamma-Gamma turbulence channel(? = 2.50, ? = 2.06) using different values of J . . . . . 47Figure 3.7 Absolute truncation error of subcarrier 8 ? 8 QAMover a K-distributed fading channel (? = 1.99) usingdifferent values of J . . . . . . . . . . . . . . . . . . . . 48Figure 3.8 Comparison of ASER of subcarrier 4 ? 4 QAM overthe Gamma-Gamma turbulence channel using (3.18)and eq. (24) in [30]. . . . . . . . . . . . . . . . . . . . 49Figure 4.1 BER of R-QAM based non-adaptive SIM over theGamma-Gamma turbulence channels. . . . . . . . . . 75Figure 4.2 BER of 32-PSK based non-adaptive SIM over theGamma-Gamma turbulence channels. . . . . . . . . . 76xiLIST OF FIGURESFigure 4.3 Absolute truncation error for the BER of 32 R-QAMnon-adaptive SIM over a moderate Gamma-Gammaturbulence channel. . . . . . . . . . . . . . . . . . . . 77Figure 4.4 ASE of R-QAM based adaptive SIM over the Gamma-Gamma turbulence channels. . . . . . . . . . . . . . . 78Figure 4.5 ASE ofM -PSK based adaptive SIM over the Gamma-Gamma turbulence channels. . . . . . . . . . . . . . . 79Figure 4.6 ASE of MIMO (T transmit lasers and P photo detec-tors) adaptive SIM with N = 8 regions over a strongGamma-Gamma turbulence channel. . . . . . . . . . 81Figure 4.7 BER of R-QAM based adaptive SIM over the Gamma-Gamma turbulence channels. . . . . . . . . . . . . . . 82Figure 4.8 BER of an adaptive SIM with N = 5 regions over astrong Gamma-Gamma turbulence channel. . . . . . . 83Figure 4.9 BER of an adaptive SIM with N = 5 regions over amoderate Gamma-Gamma turbulence channel. . . . . 84Figure 4.10 Outage probability of an adaptive SIM over a strongGamma-Gamma turbulence channel. . . . . . . . . . 85Figure 5.1 Ergodic capacity of Coherent OWC over a strongGamma-Gamma turbulence channel wtih ? = 2.04and ? = 1.10. . . . . . . . . . . . . . . . . . . . . . . . 102Figure 5.2 Ergodic capacity of Coherent POLMUX OWC over astrong Gamma-Gamma turbulence channel wtih ? =2.04 and ? = 1.10. . . . . . . . . . . . . . . . . . . . . 103Figure 5.3 Ergodic Capacity of Coherent OWC over a moderateGamma-Gamma turbulence channel with ? = 2.50and ? = 2.06. . . . . . . . . . . . . . . . . . . . . . . . 104Figure 5.4 Ergodic Capacity of Coherent POLMUX OWC overa moderate Gamma-Gamma turbulence channel with? = 2.50 and ? = 2.06. . . . . . . . . . . . . . . . . . 105xiiLIST OF FIGURESFigure 5.5 Ergodic capacity comparison among the subcarrierIM/DD, and coherent VPVR adaptive transmissionscheme with and without POLMUX over with ? =2.04 and ? = 1.10. . . . . . . . . . . . . . . . . . . . . 107Figure 5.6 Ergodic capacity comparison among the subcarrierIM/DD, and coherent CCIFR adaptive transmissionscheme with and without POLMUX over a 700 mweak turbulence channel with ? = 4.43 and ? = 4.39. 108xiiiList of AcronymsAcronyms DefinitionsAWGN Additive White Gaussian NoiseBER Bit-Error RateBPSK Binary Phase Shift KeyingCCIFR Complete Channel Inversion With Fixed RateCDF Cumulative Distribution FunctionCPVR Constant-Power, Variable-RateDPSK Differential Phase Shift KeyingESA European Space AgencyFASCODE The Fast Atmospheric Signature CodeFSO Free Space OpticalGbps Gigabit per secondIM/DD Intensity Modulation with Direct DetectionLANs Local Area NetworksLOS Line-of-SightMANs Metropolitan Area NetworksMLCD Mars Laser Communication DemonstrationM -PSK M -ary Phase Shift KeyingMODTRAN Moderate Resolution Atmospheric TransmissionNASA National Aeronautics and Space AdministrationNCFSK Noncoherent frequency shift keyingOOK On-Off KeyingOWC Optical Wireless CommunicationsPDF Probability Density FunctionPOLMUX Polarization MultiplexingPPM Pulse Position ModulationxivList of AcronymsQAM Quadrature Amplitude ModulationRF Radio FrequencyR-QAM Rectangular Quadrature Amplitude ModulationRV Random VariableSIM Subcarrier Intensity ModulationSILEX Semiconductor-laser Inter-satellite Link ExperimentSNR Signal-to-Noise RatioWANs Wide Area NetworksVPVR Variable-Power, Variable-RateTCIFR Truncated Channel Inversion With Fixed RatexvAcknowledgementsI am deeply grateful to my supervisors Prof. Jahangir Hossain and Prof.Julian Cheng for their enthusiasm, guidance, advice, encouragement, andsupport. They granted me a great flexibility and freedom in my researchwork. They taught me academic knowledge and research skills. I will con-tinue to be inuenced by their rigorous scholarship, clarity in thinking, andprofessional integrity. It is my honor to study and research under theirsupervision.I would like to express my thanks to Prof. Paramjit Gill to serve as myexternal examiner. I would also like to thank Prof. Jonathan Holzman andProf. Stephen O?Leary for serving on the committee. I really appreciatetheir valuable time and constructive comments on my thesis.I owe many people for their generosity and support during my MAScstudy at The University of British Columbia. I would like to thank my dearcolleagues Luanxia Yang, Xuegui Song, Mingbo Niu, Nianxin Tang, QuianZhang, and Fan Yang for sharing their academic experiences and construc-tive viewpoints generously with me during our discussions. Special thanksgo to my dear friends Sharear Tanzil, Sukanto Mondal, and Talha Malik forsharing my excitement and encouraging me when I was in frustration duringthis journey. I would also like to thank all of my other friends in Kelownafor offering some nice moments.Finally, I would like to thank my parents and younger sister for theirpatience, understanding, and support all these years. All my achievementswould not have been possible without their constant encouragement andsupport.xviChapter 1Introduction1.1 Background and MotivationThe growing bandwidth demand due to the surge of bandwidth hungrydigital multimedia applications is the driving force behind the research inoptical communications. The significant portion of optical communicationincludes transmission of information carrying lightwave signal through thefiber optics. Optical fiber offers enormous bandwidth which translates intohigh data rate transmission up to hundreds of Gbps. Such advantage makesthe fiber optics as the most robust and efficient communication link to con-nect the cities, countries, and continents. However, high installation cost,high infrastructure complexity, and large installation time precludes the useof fiber optics in access networks. That?s why it is not surprising that fiberoptics are primarily installed for the long-haul applications in order to com-pensate the increased infrastructure cost. Due to bandwidth limitation ofthe conventional access networks, there is a significant bandwidth gap be-tween the local area networks (LANs) and the metropolitan area networks(MANs) or the wide area networks (WANs) which is known as the so called?last-mile connectivity? problem. Consequently, a high bandwidth bridgeis required between the LAN and the MAN/WAN. That bridge is the op-tical wireless communication (OWC) or free space optical communication(FSO) system which combines the speed of light and flexibility of wirelesscommunications. OWC is a wireless broadband technology that provideshigh speed data connection between the two nodes of a line-of-sight (LOS)transmission link by transmitting the modulated optical beam of visible orinfrared light through the atmosphere. In addition to the data speed equiv-alent to the fiber optics, OWC offers other benefits as well. Cost effective11.1. Background and Motivationrapid deployment, enhanced security (thanks to the narrow beam width ofthe optical signal), protocol transparency, and freedom from spectrum li-censing regulations are the additional attractive features of OWC systems[1, 2].The photophone, patented by Alexander G. Bell in 1860 was the firstpractical demonstration of optical communications through atmosphere [3].In this patent the sun radiation was modulated by voice and transmittedover a distance of 213 meters. The invention of laser in 1960s revolutionizedthe OWC technology. The original white paper on OWC, written by Dr.Erhard Kube regarded as the father of FSO technology, ?Information trans-mission by light beams through the atmosphere,? was published in GermanNachrichtentechnik, June 1968. In 1970, NEC corporation developed a firstcommercial laser link using a 0.6338 ?m He-Ne laser over a distance of 14Kms [4]. Since then, OWC is continuously being researched by the mili-tary organizations for the covert communications. OWC became popularin military applications for the following two reasons [5]. The first one isthe availability of significant larger bandwidth compared to other RF mili-tary applications. The RF military applications provide transmission datarate on the order of several Mbps where the FSO based military applica-tions are capable to provide Gbps range transmission data rate [5]. Thesecond advantage of the OWC basd military applications is the enhancedsecurity provided by the OWC due to the narrow beamwidth, and highlyLOS nature of the OWC systems which makes the jamming difficult. OWCbecame also popular in deep space applications. For example, OWC wasused by National Aeronautics and Space Administration (NASA) and Eu-ropean Space Agency (ESA) respectively in their programmes like the MarsLaser Communication Demonstration (MLCD) and the Semiconductor-laserInter-satellite Link Experiment (SILEX) [6, 7]. Over the past decade, thematurity of optoelectronics devices led to a rebirth of OWC systems in civilapplications. It is also necessary to introduce new access technology in thefuture heterogenous wireless communication network in order to support theincreasing bandwidth demand. This fact coupled with the prerecorded suc-cess of the OWC systems in military applications fueled the OWC research21.1. Background and Motivationfor the civil applications. Several successful filed trials were carried out inthe last few years in different parts of the world (more details on these fieldtrials can be found in [8] and the references therein). Commercially avail-able OWC equipment can support 1.25 Gbps data transmission over a linkdistance of 500 m to 2 Km [9]. The practical demonstration of a 10 GbpsOWC link was also reported in the literature [10]. Since the performance ofOWC link is highly dependent on weather condition, in order to enhance therobustness of the OWC link a hybrid RF/OWC link with an RF/microwavebackup was also reported in the literature [11].There are three atmospheric factors those affect OWC link performance:absorbtion, scattering, and atmospheric turbulence induced irradiance fluc-tuation (also known as the scintillation). Absorbtion and scattering con-tribute to the reduction of optical signal power. Such effects are determin-istic, and can be accurately predicted by the software packages like MOD-TRAN or FASCODE as a function of optical wavelength and propagationdistance [12]. On the other hand, atmospheric turbulence induced irradiancefluctuation is a random effect which is due to the random atmospheric re-fractive index variations caused by inhomogeneities in both temperature andpressure. Temperature changes in the air on the order of 1?K cause refractiveindex change on the order of several parts per million [13]. The interactionbetween the transmitted laser beam and the refractive index varying prop-agation medium results in irradiance fluctuation or fading in the receiver.Typically a deep fade can last up to 1-100 ms and result in loss of 109 consec-utive information bits for a transmission rate of 10 Gbps. Consequently, foroutdoor OWC links scintillation is the most serious performance degradingfactor, and it needs to be mitigated. Since atmospheric turbulence channelis a slowly varying channel, adaptive transmission can be an effective fadingmitigation solution. By varying basic transmission parameters according tothe channel fading states, adaptive transmission offers efficient utilizationof the channel capacity and improves the link performance. In this thesiswe mainly focus on the performance analysis of different adaptive OWCsystems over the commonly used atmospheric turbulence channels.31.2. Literature Review1.2 Literature ReviewOwing to its simplicity, on-off keying (OOK) based intensity modula-tions/direct detection (IM/DD) has been employed in commercial OWCsystems [14]. However, because of random intensity fluctuation, an OOKIM/DD system requires adaptive thresholds to achieve its optimal perfor-mance. Such a system, while feasible, is costly to implement and is alsosubject to channel estimation error. As a result, practical OOK IM/DDsystems for OWC applications are implemented with a predetermined de-tection threshold, resulting irreducible error floors [15]. As an alternativeto OOK IM/DD, pulse position modulation (PPM) has been proposed forwireless optical communications. PPM does not require adaptive detectionthresholds and it is mainly used in deep space communications [16, 17].However, PPM requires tight synchronization (which increases complexityof transceiver design) and suffers from poor bandwidth efficiency. Anotherattractive alternative to the OOK IM/DD is subcarrier intensity modula-tions (SIM) which was first proposed for OWC systems by Huang et al.in [18]. The authors analyzed error rate performance of differential phaseshift keying (DPSK) and M -ary phase shift keying (PSK) modulations overlognormal fading channels [18]. In [15], Li et al. analyzed error rate perfor-mance of coded and un-coded M -ary PSK modulations over the lognormalfading channels. Later, Popoola et al. studied bit-error rate (BER) perfor-mance of binary PSK (BPSK) subcarrier intensity modulated OWC systemover lognormal, Gamma-Gamma, and negative exponential turbulence chan-nels [19, 20]. However, none of these works presented closed-form error rateexpressions. In [21], Chatzidiamantis et al. evaluated error rate performanceof subcarrier intensity modulations employing adaptive PSK modulations.For the non-adaptive case, they obtained BER of subcarrier BPSK systemin terms of Meijer?s G-function. Although their analytical BER expressionis in closed form, it does not reveal any insights. In [22], Park et al. eval-uated average BER of an Alamouti scheme in Gamma-Gamma turbulencechannels using a moment generating function (MGF) approach and a seriesrepresentation of the modified Bessel function [23]. However, their analysis41.2. Literature Reviewwas limited to the BPSK modulation and they did not carry out detailedtruncation error analysis. In another work [24], Samimi et al. studied BERperformance of BPSK SIM over the K-distributed fading channel. The au-thors evaluated the BER performance by approximating the K-distributedprobability density function (PDF) as a weighted sum of three negative ex-ponential PDFs. However, such an approximation is valid only for ? ? 3(where ? is a channel parameter for the K-distributed model) and is not use-ful for practical range of ? value, i.e., ? ? (1, 2) [57]. In a recent work [25],Song et al. presented a closed-form error rate expression of subcarrier inten-sity modulations using BPSK, Q-ary PSK (QPSK), DPSK and noncoherentfrequency shift keying (NCFSK) modulations over the Gamma-Gamma tur-bulence fading channels using a direct integration approach by employingthe same series expansion of the modified Bessel function [23]. They alsostudied truncation error analysis and asymptotic behavior of error rate per-formance. The same authors also developed series based symbol error rate(SER) expressions for the M -ary PSK based SIM over the K-distributedand negative exponential turbulence channels [26].Besides PSK modulation, quadrature amplitude modulation (QAM) hasalso gained attention for subcarrier intensity modulations in OWC com-munications. The main advantage of QAM is its high spectral efficiency[27]. In [28], Cvijetic et al. presented simulation results on symbol errorrate performance of free space optical (FSO) orthogonal frequency divisionmultiplexing (OFDM) system employing 4-PSK, 16-QAM and 64-QAM. In[34], Djordjevic et al. presented SER of an LDPC coded OFDM employing16-QAM. In [30], Peppas et al. evaluated average symbol error rate (ASER)of a SIM employing general order rectangular QAM over the lognormal andthe Gamma-Gamma turbulence channels. However, their closed-from errorrate expressions are expressed in terms of a complicated Meijer?s G-functionwhich does not reveal any insights into system. Further, they used an expo-nential approximation of the Gaussian Q-function which resulted noticeableperformance gap between the approximated and simulated results (see Figs.3-6 in [30]).Several fading mitigation techniques have been proposed for the OWC51.2. Literature Reviewsystems. In [31], Uysal et al. proposed error control coding in conjunctionwith interleaving to mitigate strong turbulence fading. However, due toslow fading nature of OWC channels, such system requires large interleaversthat can be impractical to implement. Zhu and Kahn proposed maximum-likelihood sequence detection (MLSD) technique for lognormal turbulencechannels [14]. Due to computational complexity of the proposed technique,only suboptimal MLSD techniques can be used in practice [32]. Spatial di-versity using multiple apertures for transmission and/or reception offers asignificant bit-error rate (BER) performance improvement [23]. However,increasing number of apertures also increases implementation cost and com-plexity, and effectiveness of spatial diversity is also restricted by the spatialcorrelation among multiple apertures.Adaptive transmission technique is a promising technique which takesadvantage of time-varying nature of channel by varying basic transmissionparameters according to the channel fading states. This technique allowshigher data rate transmission under the favourable channel conditions, andit achieves smooth reduction of data rate when responding to channel degra-dation. Such a system provides higher spectral efficiency without increasingtransmitted power or sacrificing BER requirements. Adaptive transmissiontechnique for OWC systems was first studied in [33]. Djordjevic studiedlow density pairty check coded variable-power variable-rate adaptive OWCsystem employing M -ary pulse amplitude modulation (PAM) [34], but theauthor did not derive any closed-form expressions for the BER. Chatzidia-mantis et al. analyzed performance of single-input single-output (SISO) andmultiple-input multiple-output (MIMO) M -PSK based non-adaptive andadaptive SIM systems over the lognormal and the Gamma-Gamma turbu-lence channels [21]. However, they carried out the analysis using an approxi-mate conditional BER of M -PSK, which is only valid for high signal-to-noiseratio (SNR) values. It is shown in this thesis that in a strong turbulence con-dition such approximation creates noticeable performance gap between theexact and approximate BER of non-adaptive SIM. Moreover, the achievablespectral efficiency (ASE) in [21] was presented in terms of a complicatedMeijer?s G-function, which does not reveal any insight into the system per-61.2. Literature Reviewformance. In addition, no closed-form BER expressions were presented in[21] for the SISO and MIMO adaptive SIM systems.For an adaptive transmission, ergodic capacity, also known as averagechannel capacity, is an important information-theoretic performance metricwhich defines an upper bound for the achievable transmission rate with anegligible error over the fading channels [36, 37]. Recent works on ergodiccapacity of OWC systems with coherent detection and intensity modula-tion with direct detection (IM/DD) are reviewed as follows. The authors in[38, 39] study the ergodic capacity and outage capacity of the coherent OWCsystems with lognormal amplitude fluctuations and Gaussian phase fluctua-tions. The authors in [40] develop closed-form ergodic capacity expressionsfor a coherent OWC system with heterodyne detection by considering thecombined impact of the Gamma-Gamma turbulence and pointing errors.Closed-form ergodic capacity expressions are derived for the IM/DD OWCsystems over the Gamma-Gamma, K-distributed, and I-K distributed at-mospheric turbulence channels in [41?44]. However, all the aforementionedclosed-form ergodic capacity expressions are limited to the constant-power,variable-rate (CPVR) adaptive transmission policy where the channel stateinformation (CSI) is available only at the receiver. Recently the authors in[45] present closed-form ergdoic capacity expressions for the variable-power,variable-rate (VPVR), CPVR, and truncated channel inversion with fixedrate (TCIFR) adaptive transmissions over the lognormal channels, which iscommonly used to describe the irradiance fluctuation in a weak turbulence.Previously the authors in [34, 46] study ergodic capacity of the VPVR,CPVR, complete channel inversion with fixed rate (CCIFR), and TCIFRadaptive transmissions for subcarrier IM/DD OWC systems in Gamma-Gamma turbulence. However, the authors did not present any closed-formexpression for the ergodic capacity. In [47], the authors also derive ergodiccapacity expressions of the VPVR, CPVR, CCIFR and TCIFR adaptivetransmissions over the generalized-K fading channels (which is similar tothe Gamma-Gamma fading channels) in terms of series solutions. However,their derived expressions for VPVR, CPVR and TCIFR adaptive trans-missions are limited only to the integer values of m (a parameter of the71.3. Thesis Outline and Contributionsgeneralized-K fading channel). In most of practical scenarios, the parame-ters of the Gamma-Gamma turbulence channels take non-integer values. Assuch the closed-form expressions developed in [47] for integer value of thegeneralized-K fading channel parameter can not be used to evaluate capac-ity performance of VPVR and TCIFR schemes over the Gamma-Gammaturbulence channels. In [48], the author also derives closed-form ergodiccapacity expressions for the VPVR, CPVR, CCIFR and TCIFR adaptivetransmission policies for subcarrier IM/DD over the Gamma-Gamma turbu-lence. However such closed-form results are presented in terms of the specialMeijer?s G-function, which does not reveal any insight into the system per-formance. OWC systems employing polarization multiplexing (POLMUX)with coherent detection has also gained attention because of its ability toboost the data throughput. In [49], the authors analyze and experimentallyverify a novel POLMUX system with coherent detection over a lognormalturbulence channel. In a related work, the authors in [50] analyze BERperformance of a coherent OWC system with POLMUX in Gamma-Gammaturbulence and Gaussian phase noise. However, to the best of authors?knowledge, no prior work has been reported on the comparison of ergodiccapacities of subcarrier IM/DD, and coherent systems with and withoutPOLMUX when adaptive transmissions are employed.1.3 Thesis Outline and ContributionsThe thesis is arranged into six chapters. Chapter 1 presents backgroundknowledge on the history and development of OWC systems. In addition,this chapter provides a detailed literature review related to the rest of thisthesis.Chapter 2 presents the required technical background for the entire the-sis. We first introduce different OWC systems, for instance, SIM with directdetection and coherent detection with and without POLMUX. Next we dis-cuss different atmospheric turbulence models, namely, the Gamma-Gamma,K-distributed, and negative exponential turbulence. Finally, we presentsome basic concepts of the adaptive transmission over the fading channels.81.3. Thesis Outline and ContributionsChapter 3 provides highly accurate series expression for the ASER ofsubcarrier intensity modulated OWC systems employing general order rect-angular QAM (R-QAM). Our analytical solutions provide noticeable perfor-mance improvement from the approximate error rate estimation proposed in[30] in strong turbulence condition. Although R-QAM is a suboptimal mod-ulation scheme compared to other QAM constellations of QAM, R-QAM canbe easily generated and demodulated [68]. For this reason we have chosenR-QAM in our analysis. In this chapter, using the same series expansionformula [23], we study ASER performance of a subcarrier intensity modu-lated OWC system employing general order R-QAM. The Gamma-Gamma,the K-distributed, and the negative exponential channel models are consid-ered for different levels of turbulence fading. Exact error rate expressionsare derived for the Gamma-Gamma turbulence channels in terms of an in-finite series using an MGF approach. Inspired by the relation among thethree turbulence models, we obtain highly accurate error rate expressionsfor both K-distributed and negative exponential turbulence channels. Wealso carry out a detailed study of truncation error analysis and asymptoticanalysis of our series solutions. Our numerical simulations will be presentedto demonstrate high accuracy and high efficiency of our series solutions.In Chapter 4, we investigate the performance of non-adaptive and adap-tive SIM systems over the Gamma-Gamma turbulence channels. In partic-ular, we make the following contributions. We derive highly accurate seriesexpressions for the exact BER of non-adaptive and adaptive SIM systemsemploying M -PSK and R-QAM modulations. Our analytical results arepresented in terms of a fast-converging infinite series, and a truncation er-ror analysis of the series solutions is also provided. Highly accurate seriessolutions for the ASE and outage probability are also derived. In addition,closed-form expressions for the asymptotic BER and outage probability areprovided.The motivation of Chapter 5 is to analyze and compare the ergodiccapacity of the subcarrier IM/DD and coherent OWC systems with andwithout POLMUX using different adaptive transmission strategies whenthe CSI is available at both transmitter and receiver. In particular, we91.3. Thesis Outline and Contributionsconsider VPVR, CCIFR, and TCIFR adaptive transmission policies overthe Gamma-Gamma turbulence channels. For all the adaptive transmissionschemes, we assume the receiver can accurately estimate the channel, andan accurate channel estimation is available at the transmitter via a reliablefeedback channel. Highly accurate ergodic capacity expressions are derivedusing a series expansion of the modified Bessel function and the Mellin trans-formation of the Gamma-Gamma random variable. Unlike [47], our seriessolutions for VPVR and TCIFR schemes work well when both parametersof the Gamma-Gamma turbulence channels take non-integer values. In ad-dition, we carry out the asymptotic analysis of our ergodic capacity expres-sions and reveal some interesting insights into the capacity performance ofthe considered systems over the Gamma-Gamma turbulence channels. Wealso present a channel capacity comparison among the subcarrier IM/DD,and coherent OWC systems with and without POLMUX under an averagetransmitted optical power constraint. Although for non-adaptive transmis-sion, the ergodic capacity is achievable with the assumption of independentturbulence/fading across time, capacity associated with adaptive transmis-sion schemes namely, VPVR, CCIFR, and TCIFR is achievable without suchindependent assumption. While the detailed reason can be found in [76],for completeness of this thesis, we discuss it here briefly. For the VPVRtransmission, the time-varying channel can effectively be reduced to a setof time-invariant additive white Gaussian noise (AWGN) channels in par-allel [76]. For each of these time-invariant channels, a capacity achievingencoder/decoder pair can be used in a time multiplexed fashion. As suchthere is no need to have the channel turbulence to be independent. Theaverage capacity of the time-varying channel corresponds to the sum of ca-pacities of the time-invariant channels weighted by the probabilities of hav-ing these time-invariant channels over time. For CCIFR transmission, sincethe transmitter adjusts the transmit signal power according to the channelturbulence in order to maintain a constant received SNR and transmits ata fixed rate, the channel appears to the encoder/decoder pair as a time-invariant AWGN channel. The capacity achieving encoder/decoder pair forthe AWGN channel can be used and there is no need to have the channel101.3. Thesis Outline and Contributionsturbulence to be independent. For TCIFR transmission, the transmitteradjusts the transmit signal power according to the channel turbulence inorder to maintain a constant received power and transmits at a fixed rateas long as the received SNR is above a certain threshold, otherwise, thetransmission is suspended. Similar argument as CCIFR can be made forTCIFR transmission. Therefore, our derived capacity expressions for theabove mentioned three adaptive transmission schemes are valid without in-dependent assumption of atmospheric turbulence over time. Since OWCsystems transmit in the range of Gbits/sec, ergodicity assumption for suchsystems may not be valid. However, this is a standard assumption usedin the literature [40?44]. This assumption corresponds to an ideal scenarioand obtained results based on this assumption can be considered as theo-retical upper bounds on the systems? performance. Yet these results allowus to compare different systems under consideration and reveal insight intosystem?s performance improvement as well as design.Chapter 6 summarizes the entire thesis and lists our contributions. Inaddition, some future works related to our current research are also sug-gested.11Chapter 2Background of OpticalWireless CommunicationSystems and Channel ModelsIn this chapter, we first introduce different detection techniques in OWCsystems. We then present background knowledge on some commonly usedatmospheric turbulence channel models. Finally, we review different adap-tive transmission schemes over the fading channels.2.1 Classification of OWC Systems Based onDetection Techniques2.1.1 Coherent OWC SystemIn a coherent OWC system, the information is modulated on the opti-cal carrier amplitude and phase for transmission. After being transmittedthrough a free-space channel, the received optical signal is coherently mixedwith the optical beam generated by a local oscillator (LO), and the com-bined optical beam is incident on the photodetector. We assume that thephase noises from the turbulence and narrow-linewidth lasers change slowly,and they can be compensated using a co-transmitted carrier?s phase esti-mation circuit [12]. This assumption is reasonable because the atmosphericturbulence channel is a slowly varying fading channel with channel coher-ence time on the order of milliseconds [38], and the narrow-linewidth lasershave linewidths on the order of 10 kHz [52]. Consequently, the accumulatedphase noises introduced by the atmospheric turbulence and/or lasers have122.1. Classification of OWC Systems Based on Detection Techniquesmillisecond variation on the timescales. In order to compensate such phasenoises the PLL circuits should have a loop bandwidth (also known as theresponse rate of the PLL circuit) on the order of KHz [52]. Note that apractical PLL can work successfully up to 10 MHz [53]. Consequently, theaccumulated phase noises due to the atmospheric turbulence and/or laserscan be tracked and corrected almost perfectly following the photodetection[49]. Therefore, in the subsequent analysis we do not consider the impact ofphase noises on the coherent detection. The optical power incident on thephotodetector can be expressed asPr(t) = Ps + PLO + 2?PsPLO cos(?IF t+ ?) (2.1)where Ps is the received optical signal power, PLO is the LO power, ? isthe phase information associated with the modulation order, and ?IF =?c??LO is the intermediate frequency, where ?c and ?LO denote the carrierfrequency and the LO frequency, respectively. The photocurrent generatedby the photodetector can be written as ir,c(t) = iDC + iAC(t) + nc(t) whereiDC = R(Ps +PLO) and iAC(t) = 2R?PsPLO cos(?IF t+?) are the DC andAC terms, respectively, R is the responsivity of phtotodetector, and nc(t)is a shot noise limited additive zero-mean white Gaussian noise (AWGN)process with variance ?2c . In a practical coherent OWC receiver, we havePLO  Ps, and hence, the DC term of the photocurrent is dominated bythe term RPLO. The shot noise generated by the LO is dominant comparedto the background irradiance generated shot noise and/or receiver thermalnoise. The shot noise variance can be written as ?2c = 2qRPLO4f , whereq is the electronic charge, and 4f is the noise equivalent bandwidth ofphotodetector. The SNR at the input of an electrical demodulator can bewritten as the ratio of the time-averaged AC photocurrent power to the totalnoise variance. The received optical power Ps can be written as Ps = AI,where A is the photodetector area and I is the received optical irradiance.Assuming the mean of optical irradiance I is unity, the instantaneous SNR132.1. Classification of OWC Systems Based on Detection Techniquesat the input of an electrical demodulator can be obtained as [51]?c =?i2AC(t)?2qRPLO4f=RAq4fI = ?cI (2.2)where ?c is the average SNR per symbol. The average received power, Ps,can be expressed as Ps = gPt where g is a constant path-loss factor, and Ptis the average transmitted optical power. The average SNR of the coherentOWC system can be written in terms of the average transmitted opticalpower as?c =Rq4fP s =Rq4fgP t. (2.3)2.1.2 POLMUX OWC System with Coherent DetectionIn a coherent POLMUX OWC system, the optical signal from a con-tinuous wave laser source is splitted into two orthogonal polarizations by apolarization beam splitter (PBS). The outputs of PBS are amplitude and/orphase modulated and are then combined by a polarization beam combiner(PBC) before being transmitted. Since atmospheric turbulence channel istypically a weak depolarizing channel [75], the fluctuation of the state ofpolarization (SOP) of the transmitted optical beam is assumed to be ina controllable order [50]. The electric field of the received optical signal atthe front end of optical receiver can be written as Er(t) = [Ex,r(t), Ey,r(t)]Twhere the subscripts x and y denote the two orthogonal channels, and whereEx,r(t) =?Ps2ej(?ct+?)Ey,r(t) =?Ps2ej(?ct+?).(2.4)At the receiver, a polarization controller (PC) is employed to adjust theSOP of the received optical beam. The adjusted optical beam is splittedinto two orthogonal channels using a second PBS. The output of the second142.1. Classification of OWC Systems Based on Detection TechniquesPBS can be written as [50, eq. 3][Ex(t)Ey(t)]=[cos  ? sin sin  cos ][Ex,r(t)Ey,r(t)](2.5)where  is the polarization control error which describes the SOP differencebetween the light after PBC at the transmitter and the light before PBSat the receiver. Following [49, Fig. 5], a post photodetection digital signalprocessing aided coherent receiver is employed to recover the transmittedsymbols from the two outputs of the PBS. For a fair comparison, we assumethe LO power is equally distributed between two orthogonally polarizedmodes. We express the instantaneous SNR per orthogonal channel as [49,eq. 20]?x =R2PsPLO(cos + sin )2qRPLO4f= ?1?pI?y =R2PsPLO(cos ? sin )2qRPLO4f= ?2?pI(2.6)where ?p =?c2 =RA2q4f is the normalized average SNR per channel, ?1 =cos + sin , and ?2 = cos ? sin . For a system with fully equalized cross-polarization interference, we have  = 0? or ?1 = ?2 = 1.2.1.3 OWC System with Subcarrier IntensityModulation/Direct DetectionIn a SIM OWC link, an RF signal, s(t), pre-modulated with data source1,is used to modulate the irradiance of a continuous wave optical beam at thelaser transmitter after being properly biased. For an atmospheric turbulencechannel, the received photocurrent after direct detection using photodetectorcan be expressed asir(t) = RI(t)A [1 + ?s(t)] + n(t) (2.7)1In this thesis we consider only R-QAM and M -PSK as a pre-modulation scheme for theSIM systems. However, other pre-modulation schemes for example PAM is also possible.152.1. Classification of OWC Systems Based on Detection Techniqueswhere ? is the modulation index satisfying the condition ?1 ? ?s(t) ? 1in order to avoid overmodulation, I(t) is assumed to be a stationary ran-dom process for the received irradiance fluctuation caused by atmosphericturbulence, and n(t) is the noise term caused by background radiation (i.e.,ambient light) and/or thermal noise, and it is modeled as an AWGN processwith variance ?2n. The sample I = I(t)|t=t0 at a time instant t = t0 givesthe random variable (RV) I. Normalizing the power of s(t) to unity, theinstantaneous SNR at the input of electrical demodulator can be written as[51]?s =(RA?)224f(qRIb + 2kbTk/RL)I2 = ?I2 (2.8)where Ib is the background light irradiance, kb is Boltzman constant, Tk isthe temperature in Kelvin, RL is the load resistance, and ? is the electricalSNR (per symbol) assuming unit mean received irradiance, i.e., E[I] = 1.Using (2.18), we have the following relationship between the average SNRand electrical SNR of the SIM systems?s = ?E[I2] = ?(?+ 1)(? + 1)??(2.9)and we relate ?s to the instantaneous SNR ?s by?s =???s(?+ 1)(? + 1)I2. (2.10)The average SNR can also be expressed as ?s =(R?)224f(qRIb+2kbTk/RL)E[P 2s ].Using (2.18) and the relation P s = gP t, we write average SNR of the sub-carrier IM/DD systems in terms of the average transmitted optical poweras?s =(R?)2(1 + 1?)(1 +1? )24f(qRIb + 2kbTk/RL)g2P2t . (2.11)162.2. Atmospheric Turbulence Models2.2 Atmospheric Turbulence ModelsSeveral statistical models have been proposed to describe turbulence in-duced irradiance fluctuation. According to [54], the lognormal turbulencemodel is widely accepted for describing irradiance fluctuations in weak tur-bulence conditions; the K-distributed turbulence model describes irradiancefluctuation in strong turbulence conditions; and the negative exponentialturbulence model is suitable for describing irradiance fluctuation in sat-urated turbulence conditions. The Gamma-Gamma distribution recentlyemerges as a useful turbulence model as it has an excellent fit with mea-sured data over a wide range of turbulence conditions [54]. It can be shownthat the K-distributed and negative exponential models can be obtained asspecial cases from the Gamma-Gamma model.2.2.1 Gamma-Gamma ModelAccording to the Gamma-Gamma model, the irradiance I can be mod-eled as weak eddies induced irradiance fluctuation modulated by strong ed-dies induced irradiance fluctuation, i.e. I = XY where X and Y are respec-tively the strong and the weak eddies induced irradiance fluctuation obeyingthe Gamma probability density functions (PDFs)fX(x) =??x??1?(?)exp(??x) (2.12)andfY (y) =??y??1?(?)exp(??y). (2.13)Here ? is the effective number of large scale eddies and ? is the effectivenumber of small scale eddies. By fixing X and letting Y = I/X, we obtainthe following conditional PDFfI/X=x(I/x) =1x??( Ix)??1?(?)exp(??Ix). (2.14)172.2. Atmospheric Turbulence ModelsAveraging (2.14) with respect to the Gamma PDF in (2.12), one obtains theGamma-Gamma PDF asfI(I) = EX [fI/X=x(I/x)]=2?(?)?(?)(??)?+?2 I?+?2 ?1K???(2???I).(2.15)where ?(?) is the Gamma function [63, eq. 8.310] and K???(?) is the modifiedBessel function of the second kind of order ? ? ? [63, eq. 8.432(9)]. Theshaping parameters ? and ? are related to the Rytov variance, and underan assumption of plane wave propagation with negligible inner scale (whichcorresponds to long propagation distance and small detector area), they canbe determined by [55]? =[exp(0.49?2I(1 + 1.11?12/5I )7/6)? 1]?1(2.16)and? =[exp(0.51?2I(1 + 0.69?12/5I )5/6)? 1]?1(2.17)where ?2I is the Rytov variance. The k-th moment of the Gamma-Gammadistribution is defined as E[Ik] = ?(?+k)?(?+k)?(?)?(?) (1?? )k [56]. Consequently, thescintillation index (?2SI), an important parameter to measure the turbulencelevel, is given by?2SI =E[I2](E[I])2? 1 =1?+1?+1??. (2.18)2.2.2 K-Distributed ModelA K-distributed RV I for received irradiance is assumed to be a productof two independent RVs W and V obeying the exponential and GammaPDFs,fW (w) = exp(?w) (2.19)182.2. Atmospheric Turbulence ModelsandfV (v) =??v??1?(?)exp(??v). (2.20)Conditioning V and letting W = I/V , we obtain the conditional PDFfI/V=v(I/v) =1vexp(?Iv). (2.21)Averaging (2.21) with respect to the PDF of Gamma RV in (2.20), oneobtains PDF of K-distributed irradiance asfI(I) =? ?01vexp(?Iv)fV (v) dv=2?(?)??+12 I??12 K??1(2??I).(2.22)By setting ? = 1 in (2.15), one obtains (2.22). From (2.18), forK-distributedchannel one obtains ? = 2/(?2SI ? 1) where the value of ?2SI is confined tothe range (2, 3) [57], and as a result the channel parameter ? is restrictedto (1, 2) [67].2.2.3 Negative Exponential ModelThe PDF of a negative exponential random variable can be expressed asfI(I) = ? exp(??I) (2.23)where ? > 0 is the mean irradiance. Without loss of generality we assumeE[I] = 1? = 1. The PDF of the Gamma RV given in (2.20) converges to aDirac delta function at 1 when ? approaches to ?, i.e.lim???fV (v) = ?(v ? 1). (2.24)192.3. Adaptive Transmissions over Fading ChannelsBy considering the limiting PDF in (2.24), we can evaluate PDF of K-distributed irradiance from (2.22) asfI(I) = lim???? ?01vexp(?Iv)fv(v) dv=? ?01vexp(?Iv)?(v ? 1) dv= exp(?I).(2.25)From the above derivation, it is obvious that the negative exponential is aspecial case of the K-distributed channel when ? ? ?. Such relationshipwill be useful to obtain accurate symbol error probability for the negativeexponential channels.2.3 Adaptive Transmissions over FadingChannelsAdaptive modulation is a promising fading mitigation solution whichwas originally proposed for the RF wireless communications [58?60]. In RFwireless communications, signal propagates through a time-varying randomchannel due to multipath fading and/or shadowing. Fixed modulation (non-adaptive) with typical fading mitigation solutions such as an increased linkbudget margin or interleaving with channel coding are usually designed rel-ative to the the worst case propagation scenario. The BER performance ofsuch system gets improved with the improvement of SNR while the spectralefficiency remains constant. On the other hand, most practical applicationsrequire a maximum BER and there is normally no reason for providing asmaller BER than required. Consequently, the fixed modulation results inpoor utilization of the full channel capacity under negligible or shallow fad-ing conditions. In contrast, adapting certain transmission parameters suchas constellation size, symbol duration, coding rate, transmit power or anycombinations of these parameters based on the channel fading conditionsleads to a better utilization of the channel capacity [71]. The objective of202.3. Adaptive Transmissions over Fading Channelsthe adaptive transmission is to utilize the randomness of the channel tovary the basic transmission parameters while maintaining a required BERperformance at the receiver. The spectral efficiency of the adaptive trans-mission increases with SNR without degrading the BER performance fromthe required value. However, there are two main challenges for the adaptivetransmission schemes over the fading channels. The first one is the require-ment of an accurate channel estimation at the transmitter via a reliablefeedback channel from the receiver. The second one is that the adaptivemodulation is well suited for a slowly varying flat fading channel where thechannel coherence time is much larger than the symbol duration. An exam-ple of such channel is composite multipath/shadow fading channel betweenthe fixed-to-fixed or the fixed to low speed mobile terminals. However, ifthe channel changes very rapidly the adaptive transmission can not adaptto the channel variations since frequent feedback on the CSI is necessaryfrom the receiver to the transmitter, and transmission parameters need tobe adapted frequently. For this reason, adaptive transmission does not workwell for the communications between the high speed mobile terminals. Sincethe atmospheric turbulence induced fading slowly varies with the channelcoherence time on the order of milliseconds, adaptive transmissions can bean effective fading mitigation solution for the OWC systems. In this sectionwe review some basic concepts of the adaptive transmissions over the slowlyvarying flat fading channels which will be used later in this thesis. In particu-lar, we discuss the following adaptive transmission schemes: constant-powervariable-rate (CPVR), variable-power variable-rate (VPVR), and variable-power fixed-rate adaptive transmission schemes.2.3.1 Constant-Power Variable-Rate AdaptiveTransmission SchemeIn a CPVR adaptive transmission scheme, the transmission power is keptconstant and the transmission data rate is adapted according to the chan-nel in order to achieve the target BER performance at the receiver. Thedata rate can be adapted by adapting the symbol duration while using a212.3. Adaptive Transmissions over Fading Channelsfixed modulation (e.g. BPSK) or by adapting the constellation size with afixed symbol duration. In practical transmission scenarios, it is difficult toadapt the symbol duration since it varies the transmission bandwidth andcomplicates the bandwidth sharing. On the other hand, constellation sizeadaption based on channel quality is relatively easy, and this technique isused in the current communication systems such as GSM, IS-136 EDGE,and 802.11a wireless LANS [59]. In a CPVR adaptive transmission scheme,the transmitter periodically sends the known ?channel sounding? sequencesto the receiver, and the receiver accurately estimates the received SNR levelsby observing the ?channel sounding? sequences. Since a slowly time vary-ing frequency flat fading channel is considered so the data symbols insertedafter the pilot symbols experience same channel fading within a duration ofchannel coherence time. Based on the received SNR estimation, a decisiondevice at the receiver selects the modulation order to be used for transmit-ting the data symbols, configures its demodulator accordingly, and confirmsthe adaptive transmitter about the decision via a reliable feedback channel.The objective of the above adaptive transmission technique is to maximizethe number of transmitted bits per symbol interval by using the largestpossible modulation order under a target BER requirement Po. Hence, theproblem can be formally formulated asmaxMlog2M, subject to Pb(M,?) ? Po (2.26)where Pb(M,?) is the conditional BER of an M -ary modulation schemeover the AWGN channel. To fulfill the above design goal, the constella-tion assignment procedure is described as follow. The received SNR rangeis divided into N + 1 regions with region boundaries (SNR thresholds){?1, ?2, ?3, ? ? ??, ?n}, and the modulation order Mn is assigned to the nthregion, [?n, ?n+1). If the received SNR is estimated to be in the nth region,modualtion order Mn is transmitted. The region boundaries {?n} are setto the required values of SNR in order to achieve the target BER Po overthe AWGN channels. If the received SNR falls below the SNR thresholdof the lowest possible modulation order, the system can not maintain the222.3. Adaptive Transmissions over Fading Channelsrequired BER performance even using the lowest possible modulation or-der. Consequently, the transmission will be terminated and the adaptivesystem will suffer from an outage. Two important performance metrics ofan adaptive transmission scheme are the spectral efficiency and the averageBER. Assuming ideal Nyquist data pulses for each constellation, the spectralefficiency of a CPVR adaptive transmission scheme is defined asRB=N?n=1log2Mn? ?n+1?nf?(?) d? (2.27)where R and B are, respectively, the transmission data rate and bandwidthand f?(?) is the PDF of SNR ?. In the literature two definitions of the aver-age BER for the adaptive transmission schemes are available. In this thesiswe use the following definition of average BER for the adaptive transmissionschemes [60, eq. 4]BER =E[Number of error bits per transmission]E[Number of bits per transmission]. (2.28)where E[?] is the expectation operator. The above mentioned adaptive trans-mission scheme is based on the instantaneous BER constraint where theinstantaneous BER is lower than the target BER at all instantaneous SNRsexcept the region boundaries. Consequently, for this adaptive transmissionscheme the average BER will also be lower than the target BER. It is alsopossible to design a CPVR adaptive transmission scheme based on the av-erage BER constraint, i.e., BER = Po. However, it is difficult to obtain theregion boundaries for this case, and only suboptimal solution exists for thisoptimization problem [58]. In order to avoid the complexity of computingregion boundaries, in this thesis we consider the CPVR adaptive transmis-sions scheme with an instantaneous BER constraint.232.3. Adaptive Transmissions over Fading Channels2.3.2 Variable-Power Variable-Rate Adaptive TransmissionSchemeIn a VPVR adaptive transmission scheme, both transmit power andtransmission rate is varied according to the channel fading states. In or-der to facilitate the performance analysis of VPVR adaptive transmissionsscheme, we first assume that channel exhibits slowly varying flat fading.We also assume that the channel power gain g[i] at an instantaneous timeindex i is perfectly known both at the receiver and the transmitter. Thismeans an ideal channel estimator is assumed to be present in the receiver,and an ideal feedback channel is assumed to exist from the receiver to thetransmitter. The average transmit power, the variance of the noise in thereceiver, and the average channel power gain are respectively denoted byP , ?2, and g. Without loss of generality we assume g = 1. Assuming thetransmit power equals to P , the instantaneous SNR is given by ?[i] = Pg[i]?2 .However, this definition of SNR is incapable of reflecting the influence of thevarying transmit power on the received instantaneous SNR. In a variable-power adaptive transmission scheme the transmit power varies accordingto ?[i], and it is denoted by P (?[i]). The instantaneous received SNR isthen given by ?[i]P (?[i])P[58]. Since the channel fading remains constant fora large block of symbols (block fading channels), we can drop the index iwithout loss of generality. In a VPVR adaptive transmission scheme, thetransmit power is adapted according to ? under an average transmit powerconstraint given by [59, eq. 4.8]? ?0P (?)f?(?) d? ? P . (2.29)With this constraint, the average capacity can be computed from [59, eq.4.9]C = maxP (?):?P (?)f? d?=PB? ?0log2(1 +?P (?)P)f?(?) d?. (2.30)242.3. Adaptive Transmissions over Fading ChannelsIn order to find the optimum power allocation P (?), we form the LagrangianJ(P (?)) =? ?0B log2(1 +?P (?)P)f?(?) d???? ?0P (?)f?(?) d?. (2.31)Differentiating (2.31) with respect to P (?) and setting the derivative equalto zero, we obtain?J(P (?))?P (?)=[(B/ ln 21 + ?P (?)/P)?P? ?]f?(?) = 0. (2.32)Since P (?) > 0, from (2.32) we obtain the optimal power allocation asP (?)P={1?o? 1? if ? ? ?o0 if ? < ?o.(2.33)where ?o is the cutoff SNR below which the transmission is suspended. Sub-stituting (2.33) into (2.29) and replacing the upper bound with an equalitysign (for using the maximum available power), we obtain? ??o(1?o?1?)f?(?) d? = 1. (2.34)For a valid cutoff SNR, ?o must satisfy (2.34). From (2.34), it is not possibleto obtain a closed-form solution of ?o in terms of the channel parametersand average SNR for the commonly used fading distributions. As a result,the value of ?o can only be obtained numerically [76].The optimum power allocation given by (2.33) is also known as thewater-filling policy. This refers to the fact that if we draw a curve of thisoptimal power allocation, the curve 1/? looks like the bottom of a bowlwhere the power is poured into the bowl to a constant water level 1/?o. Thismeans for a given ? the allocated power is equal to 1/?o ? 1/?. This powerallocation offers an opportunity to take the advantage of the randomness ofchannel. The VPVR system allocates more transmit power and higher datarate when the channel is good (? is large), it reduces the transmit power anddata rate when the channel condition is bad (? is small), and it suspends the252.3. Adaptive Transmissions over Fading Channelstransmission when the SNR falls below the cutoff ?o. Substituting (2.33)into (2.30), we obtain an integral expression of the average channel capacityasC = B? ??olog2(??o)f?(?) d?. (2.35)2.3.3 Variable-Power Fixed-Rate Adaptive TransmissionVariable-power fixed-rate is a suboptimal adaptive transmission schemewhere the transmission rate is kept constant and the transmit power isadapted according to the channel. The motivation of adapting transmitpower alone is to compensate the SNR variation due to fading. The objectiveof such transmission is to maintain a constant SNR and a constant BER atthe receiver regardless of the channel conditions. As a result, upon poweradaption the channel is inverted in such a way so that the channel appears tobe an AWGN channel to the modulator and the demodulator. Consequently,for this particular transmission scheme the design complexity of encoder anddecoder is similar to that of an AWGN channel [76]. The power allocationfor channel inversion is given by [59]P (?)P=??(2.36)where ? is the constant SNR that needs to be maintained at the receiver.Applying (2.36) to (2.29) and replacing the upper bound with equality, weobtain ? = 1E[1?] . The average capacity is obtained asC = B log2(1 + ?) = B log2(1 +1??0 ??1f?(?) d?). (2.37)The above VPFR adaptive transmission scheme is known as complete chan-nel inversion with fixed rate (CCIFR) scheme. CCIFR scheme always main-tains a fixed data rate regardless of the channel conditions. However, inextreme fading environments CCIFR schemes exhibit a large SNR penaltyfrom the optimal adaptive transmission schemes [61]. This is because the262.3. Adaptive Transmissions over Fading ChannelsCCIFR scheme has to maintain a constant data rate in all fading environ-ments. The capacity of the channel inversion schemes can be improvedby suspending the transmission in bad fading states, and by resuming thetransmission at a higher data in the good fading states. Such a system isknown as the truncated channel inversion with fixed rate (TCIFR) scheme.TCIFR scheme improves the channel capacity performance with a certaincost of loss of some data due to the channel outage. However, the practicalapplications can always tolerate an outage probability up to a certain limit.In a TCIFR scheme, the transmit power is adapted according to the channelonly when the SNR is above a ceratin threshold ?0. The power adaption isgiven by [59, eq. 4.19]P (?)P={?? if ? ? ?00 if ? < ?0.(2.38)Since the channel is only used when ? ? ?0, the power constraint given by(2.29) results in ? = 1E?0[1?] and the average channel capacity is obtained asC = B log2(1 +1???0??1f?(?) d?). (2.39)The threshold ?0 is selected either by maintaining a certain outage proba-bility, Po, given by Po = Pr[? < ?0], or by maximizing (2.39) for all possiblevalues of ?0 for a given average SNR and/or transmit power [76]. The chan-nel capacity obtained by maximizing (2.39) for all possible values of ?0 willstill be less than the channel capacity the VPVR scheme since TCIFR is asuboptimal scheme. However, the TCIFR scheme has a lower implementa-tion complexity compared to the optimal schemes with water-filling powerallocation [59].27Chapter 3Subcarrier IntensityModulated Wireless OpticalCommunications WithRectangular QAMIn this chapter, ASER is first developed for a subcarrier intensity modu-lated OWC system employing R-QAM over the Gamma-Gamma turbulencechannels. From these results, ASERs are obtained in closed-form for the K-distributed and the negative exponential turbulence channels. Truncationerror analysis is also presented along with asymptotic error rate analysis.3.1 Error Rate Analysis of SIM UsingRectangular QAM3.1.1 Subcarrier QAM in Gamma-Gamma TurbulenceChannelsASER for SIM over a turbulence channel can be calculated byPe =? ?0Pe(?)f?(?) d? (3.1)where Pe(?) is the conditional error probability and f?(?) is the PDF of SNR?. For an MI ?MQ rectangular QAM, the conditional error probability is283.1. Error Rate Analysis of SIM Using Rectangular QAM[62, eq. 9]Pe(?) = 2(1?1MI)Q (AI??) + 2(1?1MQ)Q (AQ??)? 4(1?1MI)(1?1MQ)Q (AI??)Q (AQ??) .(3.2)HereQ(?) is the GaussianQ-function defined asQ(z) =??z exp(?t2/2)/?2pi dt,AI and AQ are given by [30]AI =?6/[(M2I ? 1) + r2(M2Q ? 1)](3.3)andAQ =?6r2/[(M2I ? 1) + r2(M2Q ? 1)](3.4)where r is the in-phase to quadrature phase decision distance ratio definedas r = dI/dQ, where dI and dQ are respectively the in-phase and quadraturedecision distances of an MI ?MQ R-QAM constellation. To facilitate ourerror rate analysis, we use the following power series representation of themodified Bessel function [25, eq. 4]K?(x) =pi2 sin(pi?)??p=0[(x/2)2p???(p? ? + 1)p!?(x/2)2p+??(p+ ? + 1)p!](3.5)where ? /? Z, |x| < ?. Using (3.5), the relations ? = ?I2, and fX(X) =fI(?X)/(2?X) assuming X = I2, we obtain the moment generating func-tion (MGF) of SNR ? asM?(s) , E [exp (?s?)]=B(?? ?, 1? ?+ ?)2?(?)?(?)??p=0[ap(?, ?)s? p+?2 ??p+?2?ap(?, ?)s? p+?2 ??p+?2].(3.6)293.1. Error Rate Analysis of SIM Using Rectangular QAMIn obtaining (3.6) we have used the following integral identity [63, eq.3.326(2)]? ?0xm exp(?bxn) dx =?(m+1n )nbm+1n(3.7)and the identity pisin(pix) = ?(x)?(1?x) [26]. In (3.6), B(x, y) = ?(x)?(y)/?(x+y) is the Beta function [63, eq. 8.384(1)] and ap(x, y) is defined asap(x, y) =(xy)p+y?(p? x+ y + 1)p!?(p+ y2). (3.8)To continue the error rate evaluation, we use an alternative representationof the Gaussian Q-function [64, eq. 4.2]Q(x) =1pi? pi20exp(?x22 sin2 ?)d? (3.9)and an integral representation of the product of two Gaussian Q-functions[64, eq. 4.8]Q(x)Q(y) =12pi? pi2?tan?1( yx )0exp(?x22 sin2 ?)d?+12pi? tan?1( yx )0exp(?y22 sin2 ?)d?.(3.10)Substituting (4.4) into (3.1) and using (3.9), (3.10) and the definition ofMGF given in (3.6), we obtain ASER for rectangular QAM as303.1. Error Rate Analysis of SIM Using Rectangular QAMPe =2pi(1?1MI)? pi20M?(A2I2 sin2 ?)d?? ?? ?e1+2pi(1?1MQ)? pi20M?(A2Q2 sin2 ?)d?? ?? ?e2?2pi(1?1MI)(1?1MQ)? pi2?tan?1(AIAQ)0M?(A2Q2 sin2 ?)d?? ?? ?e3?2pi(1?1MI)(1?1MQ)? tan?1( AIAQ)0M?(A2I2 sin2 ?)d?? ?? ?e4.(3.11)Using (3.6) and the following integral property [63, eq. 3.621(1)]gp(12, x)=? pi20sinp+x ? d?= 2p+x?1B(p+ x+ 12,p+ x+ 12) (3.12)we can showe1 =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)???p=0[ap(?, ?)gp(12, ?)A?(p+?)I(?2)? p+?2?ap(?, ?)gp(12, ?)A?(p+?)I(?2)? p+?2].(3.13)313.1. Error Rate Analysis of SIM Using Rectangular QAMSimilarly, we can obtain e2 ase2 =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MQ)???p=0[ap(?, ?)gp(12, ?)A?(p+?)Q(?2)? p+?2?ap(?, ?)gp(12, ?)A?(p+?)Q(?2)? p+?2].(3.14)In order to evaluate e3 and e4 in (3.11), we first define tan?1(AI/AQ) = npior n = 1pi tan?1(AI/AQ) where n is a real number. When r ? 1 (which isthe usual case for MI ?MQ rectangular QAM ), we find n is confined tothe range 0 < n ? 14 . We also define pi/2? tan?1(AI/AQ) = mpi so that msatisfies m = 12?n. Using (3.6) and the following integral property obtainedby Mathematicagp(?, x) =? ?pi0sinp+x ? d? =?pi?(1+p+x2 )2?(1 + p+x2 )? cos(?pi)2F1(12,1? p? x2;32; cos2(?pi)) (3.15)we obtain e3 ase3 =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)(1?1MQ)???p=0[ap(?, ?)gp(m,?)A?(p+?)Q(?2)? p+?2?ap(?, ?)gp(m,?)A?(p+?)Q(?2)? p+?2].(3.16)323.1. Error Rate Analysis of SIM Using Rectangular QAMIn (3.15), 2F1(? , ? ; ? ; ?) is the Hypergeometric function [63, eq. (9.111)].Similarly, e4 can be obtained ase4 =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)(1?1MQ)???p=0[ap(?, ?)gp(n, ?)A?(p+?)I(?2)? p+?2?ap(?, ?)gp(n, ?)A?(p+?)I(?2)? p+?2].(3.17)It is seen that e1, e2, e3 and e4 involve only the Beta function, Gammafunction and Hypergeometric function. Consequently, they can be evaluatedeasily by any scientific software without using numerical integration. With(3.13), (3.14), (3.16) and (3.17) we obtain ASER for SIM employing generalorder rectangular QAM over the Gamma-Gamma turbulence channels asPe,??? = e1 + e2 ? e3 ? e4. (3.18)In the above analysis, we have assumed r ? 1. The derived expressionscan also be used for r < 1 by exchanging MI with MQ and by exchangingAI with AQ. Although an MGF based approach has been adopted for thederivation, it is also possible to obtain a numerically equivalent ASER ex-pression through direct integration, and for completeness such an analysisis presented in Appendix A. When using the direct integration approach wehave to distinguish between R-QAM with r 6= 1 and R-QAM with r = 1,and such consideration is not required in our MGF approach. Hence, theMGF approach is preferred because it provides more compact results thana direct integration approach.333.1. Error Rate Analysis of SIM Using Rectangular QAM3.1.2 Subcarrier QAM in K-Distributed TurbulenceChannelsFrom (3.18) we can obtain ASER for K-distributed channels asPe,K = Pe,??? |?=1= e1|?=1 + e2|?=1 ? e3|?=1 ? e4|?=1.(3.19)wheree1|?=1 =?(2? ?)pi(?? 1)(1?1MI)???p=0[ap(?, 1)gp(12, 1)A?(p+1)I(?2)? p+12?ap(1, ?)gp(12, ?)A?(p+?)I(?2)? p+?2],(3.20)e2|?=1 =?(2? ?)pi(?? 1)(1?1MQ)???p=0[ap(?, 1)gp(12, 1)A?(p+1)Q(?2)? p+12?ap(1, ?)gp(12, ?)A?(p+?)Q(?2)? p+?2],(3.21)e3|?=1 =?(2? ?)pi(?? 1)(1?1MI)(1?1MQ)???p=0[ap(?, 1)gp(m, 1)A?(p+1)Q(?2)? p+12?ap(1, ?)gp(m,?)A?(p+?)Q(?2)? p+?2],(3.22)343.1. Error Rate Analysis of SIM Using Rectangular QAMande4|?=1 =?(2? ?)pi(?? 1)(1?1MI)(1?1MQ)???p=0[ap(?, 1)gp(n, 1)A?(p+1)I(?2)? p+12?ap(1, ?)gp(n, ?)A?(p+?)I(?2)? p+?2].(3.23)3.1.3 Subcarrier QAM in Negative Exponential TurbulenceChannelsSince the negative exponential channel is the limiting case of the K-distributed channel when ? approaches infinity. We can obtain ASER fornegative exponential channel asPe,NE = lim???Pe,K= lim???e1|?=1 + lim???e2|?=1 ? lim???e3|?=1 ? lim???e4|?=1.(3.24)To evaluate ASER for the NE channel we are required to compute each limitof R.H.S in (3.24). To evaluate lim??? e1|?=1, we first observe that when? ? ? the second term of the summation in (3.20) becomes negligible for353.1. Error Rate Analysis of SIM Using Rectangular QAMall values of p. Based on this observation we obtainlim???e1|?=1 = lim????(2? ?)pi(?? 1)(1?1MI)???p=0?p+1?(p+12 )?(p? ?+ 2)p!gp(12, 1)A?(p+1)I(?2)? p+12=1pi(1?1MI)? lim?????p=0(?1)p?p+1?(p+12 )?pk=0(k ? ?+ 1)p!gp(12, 1)A?(p+1)I(?2)? p+12=1pi(1?1MI) ??p=0(?1)p?(p+12 )p!gp(12, 1)A?(p+1)I(?2)? p+12.(3.25)Similarly, we can obtainlim???e2|?=1 =1pi(1?1MQ) ??p=0(?1)p?(p+12 )p!gp(12, 1)A?(p+1)Q(?2)? p+12,(3.26)lim???e3|?=1 =1pi(1?1MQ)(1?1MQ)???p=0(?1)p?(p+12 )p!gp(m, 1)A?(p+1)Q(?2)? p+12(3.27)andlim???e4|?=1 =1pi(1?1MQ)(1?1MQ)???p=0(?1)p?(p+12 )p!gp(n, 1)A?(p+1)I(?2)? p+12.(3.28)363.1. Error Rate Analysis of SIM Using Rectangular QAMSubstituting (3.25), (3.26), (3.27) and (3.28) into (3.24), we obtain a closed-form expression for ASER of a subcarrier QAM over the negative exponentialturbulence channels.3.1.4 Truncation Error Analysis for Subcarrier QAMASER of subcarrier QAM over the Gamma-Gamma turbulence channelscan be obtained from (3.18) where each term of R.H.S. contains summationof infinite terms. For practical evaluation, we have to consider only sum-mation of finite terms. As a result, a truncation error is introduced dueto elimination of infinite terms after the first J + 1 terms. To evaluate thetruncation error of e1 in (3.18), we first define it ase1,J =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)???p=J+1[ap(?, ?)gp(12, ?)A?(p+?)I(?2)? p+?2?ap(?, ?)gp(12, ?)A?(p+?)I(?2)? p+?2].(3.29)In order to facilitate the truncation error analysis, we rewrite (3.29) ase1,J =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI) ??p=J+11p!(?2??AI??)p?[up(?, ?,AI ,12)? up(?, ?,AI ,12)] (3.30)where up(x, y, z, ?) is defined byup(x, y, z, ?) =?(p+y2 )?(p? x+ y + 1)gp(?, y)(?2xyz??)y. (3.31)By Taylor series expansion of the exponential function we can simplify thesummation term in (3.30) and obtain an upper bound for truncation error373.1. Error Rate Analysis of SIM Using Rectangular QAMof e1 asupe1,J =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)exp(?2??AI??)?maxp>J????up(?, ?,AI ,12)? up(?, ?,AI ,12)???? .(3.32)Similarly, we can also obtain upper bounds for truncation error of e2, e3 ande4 asupe2,J =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MQ)exp(?2??AQ??)?maxp>J????up(?, ?,AQ,12)? up(?, ?,AQ,12)???? ,(3.33)upe3,J =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)(1?1MQ)? exp(?2??AQ??)maxp>J|up(?, ?,AQ,m)? up(?, ?,AQ,m)|(3.34)andupe4,J =B(?? ?, 1? ?+ ?)pi?(?)?(?)(1?1MI)(1?1MQ)? exp(?2??AI??)maxp>J|up(?, ?,AI , n)? up(?, ?,AI , n)| .(3.35)Using (3.32)-(3.35) and (3.18) we obtain the following upper bound fortruncation error of subcarrier QAM over Gamma-Gamma turbulence fadingchannels asJ ? upe1,J+ upe2,J ? upe3,J? upe4,J . (3.36)In order to prove that (3.36) is valid we need to prove that the maximumvalue of |up(?, ?,AI , ?)? up(?, ?,AI , ?)| exists where ? can be equal to 12 ,383.1. Error Rate Analysis of SIM Using Rectangular QAMm or n. To prove this, we first note from (4.47) that the value of gp(?, y)is always finite for any value of p, ? and y, which can be confirmed from(3.15). We also note?( p+y2 )?(p?x+y+1) decreases with an increase of p and becomeszero when p approaches infinity. Based on these observations we concludeup(?, ?,AI , ?) or up(?, ?,AI , ?) approaches zero when p approaches infinity.Consequently, the maximum value of |up(?, ?,AI , ?)?up(?, ?,AI , ?)| exists;therefore, we say the truncation error J diminishes with increasing valuesof J . Also, by investigating (3.36), we find the truncation error decreasesrapidly with an increase of average SNR ?. This suggests that our seriessolutions are highly accurate in large SNR region, and it allows us to performasymptotic error rate analysis.3.1.5 Asymptotic Error Rate Analysis for Subcarrier QAMIn optical wireless communications over the Gamma-Gamma turbulencechannels, we have ? > ? in most scenarios [56]. Therefore, without losingof generality, we assume ? > ? in the study of asymptotic error rate perfor-mance. Results for ? < ? can be readily derived since the parameters ? and? are symmetric in the Gamma-Gamma model. From (3.13), (3.14), (3.16)and (3.17) we find that (?2 )? p+?2 decreases faster than (?2 )? p+?2 when theaverage SNR ? increases. As a result, when the average SNR ? approachesinfinity, the leading terms in (3.13), (3.14), (3.16) and (3.17) become domi-nant. Therefore asymptotic values of e1, e2, e3 and e4 in large SNR regimescan be directly obtained ase1,asym =B(?? ?, 1? ?+ ?)pi?(?)?(?)(??)??(?2 ))?(? ? ?+ 1)??pi?(1+?2 )2?(1 + ?2 )(1?1MI)A??I(?2)??2,(3.37)393.1. Error Rate Analysis of SIM Using Rectangular QAMe2,asym =B(?? ?, 1? ?+ ?)pi?(?)?(?)(??)??(?2 ))?(? ? ?+ 1)??pi?(1+?2 )2?(1 + ?2 )(1?1MQ)A??Q(?2)??2,(3.38)e3,asym =B(?? ?, 1? ?+ ?)pi?(?)?(?)(??)??(?2 ))?(? ? ?+ 1)?(1?1MI)(1?1MQ)g0(m,?)A??Q(?2)??2,(3.39)ande4,asym =B(?? ?, 1? ?+ ?)pi?(?)?(?)(??)??(?2 ))?(? ? ?+ 1)?(1?1MI)(1?1MQ)g0(n, ?)A??I(?2)??2.(3.40)Using (3.37)-(3.40) we obtain the asymptotic ASER of subcarrier QAM forthe Gamma-Gamma turbulence channels asPe,asym = e1,asym + e2,asym ? e3,asym ? e4,asym. (3.41)Substituting ? = 1 into (3.41), one obtains asymptotic ASER of subcarrierQAM for the K-distributed turbulence channels. Similarly, taking ? ??, one obtains asymptotic ASER for the negative exponential turbulencechannels.For the Gamma-Gamma turbulence channels, when there is a large dif-ference between values of ? and ?, the asymptotic error rate in (3.41) willconverge to the exact error rate rapidly. On the other hand, when the dif-ference between values of ? and ? is small, (?2 )? p+?2 decreases at a similarrate of (?2 )? p+?2 . The resulting asymptotic error rate will converge to the ex-act error rate slowly. From (3.37)-(3.41), we also observe that the diversity403.2. Numerical Resultsorder depends only on the smaller channel parameter ?.3.2 Numerical ResultsIn this section, we compare exact ASER of subcarrier QAM with ASERobtained by closed-form solutions in Section 3.1. The exact ASER is com-puted by numerical integration of (3.1) for different fading channel models.Closed-form error rates are evaluated using the series solutions in (3.18),(3.19) and (3.24) by eliminating the infinite terms after the first J+1 terms.In this paper for all numerical experiments we have chosen J = 30. Althoughwe have presented here only 4?4 subcarrier QAM (16-QAM) and 8?8 sub-carrier QAM (64-QAM) with r = 1 (since these QAM formats are oftenused in practice), our theoretical analysis has been validated by numericalevaluation for other configurations of subcarrier QAM.Fig. 3.1 plots ASER of a subcarrier 4? 4 QAM for the Gamma-Gammachannels with different levels of turbulence. The results presented in Fig.3.1 show excellent agreement between the exact error rate and the approx-imate error rate obtained by our series solution. Fig. 3.2 plots ASER of asubcarrier 8 ? 8 QAM for the same Gamma-Gamma turbulence channels.We also find excellent agreement between the exact error rate and the ap-proximate error rate obtained by series solutions. Also in both Figs. 3.1 and3.2, asymptotic error rates are shown for the strong (? = 2.04, ? = 1.10),moderate (? = 2.50, ? = 2.06), and weak (? = 4.43, ? = 4.39) Gamma-Gamma turbulence. Although for strong and moderate Gamma-Gammaturbulence, the asymptotic error rate converges to the exact error rate athigh values of SNR, it does not converge to the exact error rate for theweak Gamma-Gamma turbulence. According to the prediction of (3.41),when the difference between ? and ? becomes small, asymptotic error rateconverges to the exact error rate slowly. Since in weak turbulence conditionthe values of ? and ? are almost similar, asymptotic error rate does notconverge to the exact error rate even at high values of SNR.In Figs. 3.3 and 3.4, ASERs of a 4 ? 4 subcarrier QAM and an 8 ? 8subcarrier QAM are presented for the K-distributed channel with different413.2. Numerical Results0 10 20 30 40 50 60 70 8010?1010?810?610?410?2100Electrical SNR per bit (dB)Average symbol error rate  Approx,?=2.04,?=1.10Exact,?=2.04,?=1.10Approx,?=2.50,?=2.06Exact,?=2.50,?=2.06Approx,?=4.43,?=4.39Exact,?=4.43,?=4.39AsymptoticFigure 3.1: ASER of subcarrier 4? 4 QAM over the Gamma-Gamma chan-nels with difference levels of turbulence.423.2. Numerical Results0 10 20 30 40 50 60 70 8010?1010?810?610?410?2100Electrical SNR per bit (dB)Average symbol error rate  Approx,?=2.04,?=1.10Exact,?=2.04,?=1.10Approx,?=2.50,?=2.06Exact,?=2.50,?=2.06Approx,?=4.43,?=4.39Exact,?=4.43,?=4.39AsymptoticFigure 3.2: ASER of subcarrier 8? 8 QAM over the Gamma-Gamma chan-nels with different levels of turbulence.433.2. Numerical Results0 10 20 30 40 50 60 70 8010?310?210?1100Electrical SNR per bit (dB)Average symbol error rate  Approx,?=1.11Exact,?=1.11Approx,?=1.53Exact,?=1.53Approx,?=1.99Exact,?=1.99AsymptoticFigure 3.3: ASER of subcarrier 4?4 QAM over theK-distributed turbulencechannels.443.2. Numerical Results0 10 20 30 40 50 60 70 8010?310?210?1100Electrical SNR per bit (dB)Average symbol error rate  Approx,?=1.11Exact,?=1.11Approx,?=1.53Exact,?=1.53Approx,?=1.99Exact,?=1.99AsymptoticFigure 3.4: ASER of subcarrier 8?8 QAM over theK-distributed turbulencechannels.453.2. Numerical Results0 10 20 30 40 50 60 70 8010?310?210?1100Electrical SNR per bit (dB)Average symbol error rate  Approx,4 ? 4 QAMExact,4 ? 4 QAMApprox, 8 ? 8 QAMExact, 8 ? 8 QAMAsymptoticFigure 3.5: ASER of subcarrier 4 ? 4 and 8 ? 8 QAM over negative expo-nential turbulence channel.463.2. Numerical Results0 10 20 30 40 50 60 70 8010?2010?1510?1010?5100105Electrical SNR per bit (dB)Absolute Truncation Error  ?J,J=8?J,J=15?J,J=30Figure 3.6: Absolute truncation error of subcarrier 4?4 QAM over a moder-ate Gamma-Gamma turbulence channel (? = 2.50, ? = 2.06) using differentvalues of J .473.2. Numerical Results0 10 20 30 40 50 60 70 8010?1410?1210?1010?810?610?410?2100102Electrical SNR per bit (dB)Absolute Truncation Error  ?J,J=12?J,J=15?J,J=25Figure 3.7: Absolute truncation error of subcarrier 8 ? 8 QAM over a K-distributed fading channel (? = 1.99) using different values of J .483.2. Numerical Results0 10 20 30 40 50 60 70 8010?1210?1010?810?610?410?2100Average Electrical SNR in dBAverage Symbol Error Rate  Approx, eq.(28) with J=30Exact,eq. (11)Approx, eq. (24) of [30]?=4.43,?=4.39?=2.50,?=2.06?=2.04,?=1.10Figure 3.8: Comparison of ASER of subcarrier 4?4 QAM over the Gamma-Gamma turbulence channel using (3.18) and eq. (24) in [30].493.2. Numerical Resultsvalues of ?. In both figures we see an excellent agreement between the exacterror rate and our series solution. We also observe in both figures that theasymptotic error rate converges to the exact error rate for ? = 1.99 fasterthan for ? = 1.53. The rate of convergence of asymptotic error rate toexact error rate is slower when ? becomes close to unity. As a result, theasymptotic error rate converges to the exact error rate only at high value ofSNR which is illustrated in both Figs. 3.3 and 3.4.In Fig. 3.5, we have presented ASER of a 4? 4 and an 8? 8 subcarrierQAM over the negative exponential channel. From Fig. 3.5 we again observethat the exact ASER matches well with the error rate obtained by our seriessolution. For both QAM formats, asymptotic error rate converges to exacterror rate with moderate values of SNR. The convergence rate of asymptoticerror rate to exact error rate for the NE channel is faster than that for theK-distributed channel. Since for NE channel ? ? ?, the term (?2 )? p+?2decreases much faster than (?2 )? p+12 ; and consequently, the leading termbecomes dominant in moderate SNR region.In Fig. 3.6 we have plotted absolute truncation error for 4?4 QAM overmoderate Gamma-Gamma turbulence channel (with ? = 2.50, ? = 2.06)using J = 8, 15, and 30 terms. From Fig. 3.6 we see that the truncationerror diminishes with increasing values of J for the same SNR . We alsoobserve, the truncation error decreases rapidly with increasing values ofSNR. To obtain a truncation error of 10?10 at 20 dB SNR we need J = 30terms, while to produce the same truncation error at 25 dB SNR we needonly J = 8 terms. These behaviors of our series solutions are consistentwith the predictions from (3.36).In Fig. 3.7 we have presented the absolute truncation error for an 8? 8subcarrier QAM over the K-distributed channel (with ? = 1.99) using J =12, 15, and 25 terms. From Fig. 3.7 we also see that the truncation errordiminishes with increase of J as well as SNR. For example, at 0 dB SNR weneed J = 25 terms to achieve 10?8 truncation error where the same level oftruncation error is achieved using only J = 12 terms when SNR is 14 dB.This suggests our series solutions are highly accurate in large SNR region.In Fig. 3.8 we have presented comparison of ASER of 4 ? 4 subcarrier503.3. SummaryQAM over the Gamma-Gamma turbulence channel using (3.18) and the de-rived expression of [30, eq. 24]. From this figure, we find our proposed seriessolution shows better performance than the derived expression of [30, eq. 24]in terms of agreement with the exact ASER. We also find that the perfor-mance difference between our series solutions and the derived expression of[30, eq. 24] diminishes with a decrease of turbulence strength.3.3 SummaryIn this chapter we have developed highly accurate ASER series expres-sions for an R-QAM SIM over the Gamma-Gamma, K-distributed, and neg-ative exponential turbulence channels. We have also presented asymptoticASER expression, and it can be used for rapid error rate estimation in largeSNR regimes. Numerical results suggest that an efficient fading mitigationsolution must be employed in order to achieve a reliable OWC link over astrong atmospheric turbulence channel.51Chapter 4Performance ofNon-Adaptive and AdaptiveSubcarrier IntensityModulations inGamma-Gamma TurbulenceWe have concluded in Chapter 3 that a suitable fading mitigation solu-tion is necessary in order to achieve a reliable OWC link in strong turbulenceconditions. In this chapter we introduce a constant-power, variable-rateadaptive SIM as a fading mitigation solution. We first study the BER perfor-mance of R-QAM and M -PSK based non-adaptive SIMs over the Gamma-Gamma turbulence channels. After briefly introducing adaptive modulationstrategy, we carry out the performance analysis of R-QAM and M -PSKbased adaptive SIMs with and without diversity. Finally, we present someselected numerical examples to show the accuracy of our derived expressionsand the advantage of using adaptive transmission as a fading mitigation so-lution.4.1 Performance Analysis of Non-Adaptive SIMSystemIn this section, average BER expressions are developed for non-adaptiveSIM systems employing R-QAM and M -PSK over the Gamma-Gamma tur-524.1. Performance Analysis of Non-Adaptive SIM Systembulence channels. The average BER of a non-adaptive SIM system in anatmospheric turbulence channel is calculated byP b =? ?0Pb(?)f?(?) d? (4.1)where Pb(?) is the conditional BER over an AWGN channel and it dependson the chosen modulation scheme, and f?(?) is the PDF of SNR ?. Using aseries expansion of the modified Bessel function of second kind [25, eq. 4],we obtain the PDF of SNR ? asf?(?) =B(?? ?, 1? ?+ ?)2?(?)?(?)??p=0[a?p(?, ?)(??) p+?2??1?a?p(?, ?)(??) p+?2??1] (4.2)where a?p(x, y) is defined asa?p(x, y) =(xy)p+y?(p? x+ y + 1)p!. (4.3)4.1.1 BER of Subcarrier R-QAM in Gamma-GammaTurbulence ChannelsFor a Gray coded I ? J R-QAM (with modulation order M = I ? J)2,conditional BER over the AWGN channel is given by [69, eq. 22]PQAMb (I, J, ?) =1log2(I ? J)??1Ilog2 I?k=1(1?2?k)I?1?i=0?i(k, I)erfc ((2i+ 1)D(I, J)??)+1Jlog2 J?l=1(1?2?l)J?1?j=0?j(l, J)erfc ((2j + 1)D(I, J)??)?? .(4.4)2Without causing notational confusions, the symbol I in the ensuing analysis denotesthe in-phase dimension of the R-QAM constellation.534.1. Performance Analysis of Non-Adaptive SIM Systemwhere erfc(?) is the complementary error function defined as erfc(z)= 2?pi??z exp(?t2) dt, and ?i(k, I), ?j(l, J), and D(I, J) are given, respec-tively, as?i(k, I) = (?1)?i?2k?1I??(2k?1 ??i ? 2k?1I+12?), (4.5)?j(l, J) = (?1)?j?2l?1J??(2l?1 ??j ? 2l?1J+12?)(4.6)andD(I, J) =?3I2 + J2 ? 2. (4.7)In (4.5) and (4.6), bxc denotes the largest integer not greater than x. Ap-plying (4.2) and (4.4) to (4.1) and using an integral identity [65, eq. 06. 27.21. 0132. 01], we obtain the BER of a non-adaptive SIM employing Graycoded R-QAM over the Gamma-Gamma turbulence channel asPQAMb (I, J) =B(?? ?, 1? ?+ ?)?pi?(?)?(?)1log2(I ? J)(Pb,1 + Pb,2) (4.8)wherePb,1 =1Ilog2 I?k=1(1?2?k)I?1?i=0?i(k, I)??p=0[bp(?, ?)??p+?2(2i+ 1)p+?(D(I, J))p+??bp(?, ?)??p+?2(2i+ 1)p+?(D(I, J))p+?] (4.9)andPb,2 =1Jlog2 J?l=1(1?2?l)J?1?j=0?j(l, J)??p=0[bp(?, ?)??p+?2(2j + 1)p+?(D(I, J))p+??bp(?, ?)??p+?2(2j + 1)p+?(D(I, J))p+?].(4.10)544.1. Performance Analysis of Non-Adaptive SIM SystemIn (4.9) and (4.10), bp(x, y) = a?p(x, y)?( p+y+12 )p+y . The BER of R-QAM basednon-adaptive SIM can be calculated from (4.8) where each term of R.H.S.contains summation of infinite terms. For practical evaluation, we consideronly summation of finite terms. As a result, a truncation error is introduceddue to elimination of infinite terms after the first K + 1 terms. To evaluatethe truncation error of Pb,1 in (4.8), we first define it asPb,1(K) =1Ilog2 I?k=1(1?2?k)I?1?i=0?i(k, I)??p=K+1[bp(?, ?)??p+?2(2i+ 1)p+?(D(I, J))p+??bp(?, ?)??p+?2(2i+ 1)p+?(D(I, J))p+?].(4.11)In order to facilitate the truncation error analysis, we rewrite (4.11) asPb,1(K) =1Ilog2 I?k=1(1?2?k)I?1?i=0?i(k, I)??p=K+11p!(??(2i+ 1)D(I, J)??)p? [?p(?, ?, i)? ?p(?, ?, i)](4.12)where ?p(x, y, z) is defined as?p(x, y, z) =?(p+y+12)?(p? x+ y + 1)(p+ y)(xy(2z + 1)D(I, J)??)y. (4.13)By Taylor series expansion of the exponential function we can simplify thesummation term in (4.12) and obtain an upper bound for the truncationerror of Pb,1 asupPb,1(K) =1Ilog2 I?k=1(1?2?k)I?1?i=0?i(k, I) exp(??(2i+ 1)D(I, J)??)?maxp>K?????p(?, ?, i)? ?p(?, ?, i)????.(4.14)554.1. Performance Analysis of Non-Adaptive SIM SystemSimilarly, we can also obtain an upper bound for the truncation error of Pb,2asupPb,2(K) =1Jlog2 J?l=1(1?2?l)J?1?j=0?j(l, J) exp(??(2j + 1)D(I, J)??)?maxp>K?????p(?, ?, j)? ?p(?, ?, j)????.(4.15)Using (4.8), (4.14), and (4.15), we obtain an upper bound for the truncationerror for the BER of R-QAM based non-adaptive SIM asK,QAM ?B(?? ?, 1? ?+ ?)?pi?(?)?(?)1log2(I ? J)(upPb,1(K) + upPb,2(K)). (4.16)In order to show (4.16) is a decaying upper bound, we need to prove thatthe maximum value of?????p(?, ?, ?) ? ?p(?, ?, ?)???? exists where ? is eitheri or j. To prove this, we note from (4.13) that?( p+y+12 )?(p?x+y+1)(p+y) decreaseswith an increase of p and becomes zero when p approaches infinity. Basedon these observations we conclude both ?p(?, ?, ?) and ?p(?, ?, ?) approachzero when p approaches infinity. Also, for sufficiently large K values (K ? 30is sufficient for typical values of ? and ?),?????p(?, ?, ?)??p(?, ?, ?)???? decreasesmonotonically if p > K. Consequently, the maximum value of?????p(?, ?, ?)??p(?, ?, ?)???? exists; therefore, we conclude that the truncation error K,QAMdiminishes with increasing K values. Also, by inspecting (4.14)-(4.16), weobserve that the truncation error decreases rapidly with an increase of SNR?. This observation suggests our series solutions with finite number of termsare highly accurate in large SNR regimes.564.1. Performance Analysis of Non-Adaptive SIM System4.1.2 BER of Subcarrier M-PSK in Gamma-GammaTurbulence ChannelsFor a Gray coded M -PSK modulation, conditional BER over AWGNchannel is given by [70]P PSKb (M,?) =1log2MM?1?k=1d(k)P (k,M, ?) (4.17)where d(k) is the average distance spectrum defined asd(k) = 2????kM??kM?????+ 2log2M?i=2????k2i??k2i????? (4.18)and where P (k,M, ?) is the probability of detecting the kth (k = 1, 2, ???,M?1) symbol given the zeroth symbol is transmitted (assuming all symbols areequally likely to be transmitted), and it is defined as [64, eq. 8.29]P (k,M, ?) =12pi[? mkpi0exp(?A2(k,M)sin2 ??)d??? nkpi0exp(?B2(k,M)sin2 ??)d?].(4.19)In (4.18), bxe rounds x to the nearest integer. In (4.19), we have mk = 1?2k?1M , nk = 1?2k+1M , A(k,M) = sin((2k?1)piM), and B(k,M) = sin((2k+1)piM).Substituting (4.17) into (4.1) and using the definition of MGF, we obtainthe BER of an M -PSK based non-adaptive SIM over the Gamma-Gamma574.1. Performance Analysis of Non-Adaptive SIM Systemturbulence channels asPPSKb (M) =1log2MM?1?k=1d(k)?????12pi? mkpi0M?(A2(k,M)sin2 ?)d?? ?? ?Pb,3?12pi? nkpi0M?(B2(k,M)sin2 ?)d?? ?? ?Pb,4?????.(4.20)where M?(?) is the MGF of SNR given by (3.6). Using (3.15), we can furtherexpress Pb,3 and Pb,4 asPb,3 =B(?? ?, 1? ?+ ?)4pi?(?)?(?)??p=0[ap(?, ?)gp(mk, ?)(A(k,M))?(p+?)??p+?2?ap(?, ?)gp(mk, ?)(A(k,M))?(p+?)??p+?2](4.21)andPb,4 =B(?? ?, 1? ?+ ?)4pi?(?)?(?)??p=0[ap(?, ?)gp(nk, ?)(B(k,M))?(p+?)??p+?2?ap(?, ?)gp(nk, ?)(B(k,M))?(p+?)??p+?2].(4.22)With (4.20), (4.21) and (4.22), we obtain a highly accurate series BERexpression of M -PSK based non-adaptive SIM asPPSKb (M) =1log2M?M?1k=1 d(k)(Pb,3 ? Pb,4). Since both Pb,3 and Pb,4 con-tain summation of infinite terms, for practical evaluation they need to betruncated by eliminating infinite terms after the first K + 1 terms. Conse-quently, a truncation error, K,PSK ,???PPSKb (M)? PPSKb,K (M)??? is introducedwhere PPSKb,K (M) is the truncated version of PPSKb (M) with the first K + 1terms. Using a similar argument for K,QAM , we can show K,PSK decays584.1. Performance Analysis of Non-Adaptive SIM Systemwith increasing values of K and/or ?.4.1.3 Asymptotic BERSubcarrier R-QAMIn (4.9) and (4.10), ??p+?2 decreases faster than ??p+?2 as the SNR ?increases since typically ? > ? > 0 for the Gamma-Gamma turbulencechannels [56]. As a result, when the SNR ? approaches infinity, the leadingterms in (4.9) and (4.10) with p = 0 become dominant. Therefore, asymp-totic values of Pb,1 and Pb,2 in large SNR regimes can be directly obtainedasP asymb,1 =1Ilog2 I?k=1(1?2?k)I?1?i=0?i(k, I)(??)??(?+12)???2?(? ? ?+ 1)(2i+ 1)?(D(I, J))??(4.23)andP asymb,2 =1Jlog2 J?l=1(1?2?l)J?1?j=0?j(l, J)(??)??(?+12)???2?(? ? ?+ 1)(2j + 1)?(D(I, J))??. (4.24)Using (4.8), (4.23) and (4.24), asymptotic BER of R-QAM based non-adaptive SIM can be obtained asPQAM,asymb (I, J) =B(?? ?, 1? ?+ ?)?pi?(?)?(?)1log2(I ? J)(P asymb,1 + Pasymb,2 ). (4.25)For the Gamma-Gamma turbulence channels, when there is a large differencebetween values of ? and ? (for example, the negative exponential turbulenceis a special case of the Gamma-Gamma turbulence when ? ? ? and ? =1), the asymptotic BER in (4.25) will rapidly converge to the exact BERobtained in (4.8). On the other hand, when the difference between values of? and ? is small, ??p+?2 decreases at a similar rate of ??p+?2 . The resultingasymptotic BER will slowly converge to the exact BER. From (4.23)-(4.25),we also observe that the diversity order of the system depends only on thesmaller channel parameter ?.594.2. Adaptive Modulation StrategySubcarrier M-PSKUsing a similar argument in obtaining (4.25), asymptotic BER of M -PSK based non-adaptive SIM in large SNR regimes can be obtained asPPSK,asymb (M) =1log2MM?1?k=1d(k)(P asymb,3 ? Pasymb,4 ) (4.26)whereP asymb,3 =B(?? ?, 1? ?+ ?)4pi?(?)?(?)(??)??(?2)?(? ? ?+ 1)g0(mk, ?) (A(k,M))?? ???2(4.27)andP asymb,4 =B(?? ?, 1? ?+ ?)4pi?(?)?(?)(??)??(?2)?(? ? ?+ 1)g0(nk, ?) (B(k,M))?? ???2 .(4.28)4.2 Adaptive Modulation StrategyThe objective of a constant-power, variable-rate adaptive transmissiontechnique is to maximize the number of transmitted bits per symbol in-terval by using the largest possible modulation order while maintaining apre-defined target BER Po. In practice, the receiver selects a modulationorder from N available choices {M1,M2, ? ? ?MN}, depending on the values ofreceiver estimated SNR3 ?? and the target BER requirement Po. Specifically,the range of SNR is divided into N+1 regions, and each region is associatedwith a modulation order, Mn, according to the following rule for R-QAM3Assuming perfect estimation of SNR at the receiver, we use ?? = ? throughout the restof the chapter.604.3. Performance Analysis of Adaptive SIMbased adaptive SIM [71]M = Mn = InJn = 2nif ?n ? ?? < ?n+1 n = 1, 2, ? ? ?, N.(4.29)If n is even In = Jn = 2n2 , and if n is odd In = 2n+12 and Jn = 2n?12 . ForM -PSK based adaptive SIM system, the rule in (4.29) is modified toM = Mn = 2n if ?n ? ?? < ?n+1, n = 1, 2, ? ? ?, N. (4.30)The region boundaries {?n} and {?n} are set to the required values of SNRin order to achieve the target BER Po over the AWGN channels using (4.4)and (4.17) for R-QAM and M -PSK respectively. We also set ?N+1 = ?,and ?N+1 = ? [21]. Since (4.4) and (4.17) are non-invertible, the regionboundaries (SNR thresholds) for Gray coded R-QAM and M -PSK can onlybe numerically obtained as a function of modulation order and the targetBER using (4.4) and (4.17), respectively. If the received SNR falls belowthe SNR threshold of the lowest possible modulation order, i.e., ?? < ?1 (forR-QAM) and ?? < ?1 (for M -PSK), the transmission will be terminated andthe adaptive SIM system will suffer an outage. We comment that ?1 = ?1since ?1 and ?1 are respectively the SNR thresholds for 2-QAM and BPSK.Consequently, both adaptive systems will have the same outage probability.4.3 Performance Analysis of Adaptive SIM4.3.1 Achievable Spectral EfficiencyFor the constant-power, adaptive discrete rate SIM assuming ideal Nyquistdata pulses for each constellation [76], the ASE is the data rate transmittedper unit bandwidth and it is defined as [71]S =RW=N?n=1an log2Mn (4.31)614.3. Performance Analysis of Adaptive SIMwhere R and W represent the transmitted data rate and bandwidth mea-sured in bits/s and in Hz, respectively. In (4.31), an is the probability thatthe received SNR falls in the nth region, and it is defined asan = Pr[?n ? ? < ?n+1] =? ?n+1?nf?(?) d? = F?(?n+1)? F?(?n) (4.32)where F?(?) is the cumulative density function (CDF) of SNR. From (4.2),CDF of SNR ? can be obtained asF?(?) =B(?? ?, 1? ?+ ?)?(?)?(?)??p=0[a?p(?, ?)p+ ?(??) p+?2?a?p(?, ?)p+ ?(??) p+?2].(4.33)Hence, with (4.32) and Mn = 2n, (4.31) can be written asS =N?n=1n[F?(?n+1)? F?(?n)] = N ?N?n=1F?(?n) (4.34)where in obtaining (4.34) we have used the fact F?(?N+1) = 1. Note that(4.34) can be easily modified for M -PSK based adaptive SIM by replacing?n with ?n4. It is obvious from (4.34), for asymptotically large SNR, i.e.,when ? ? ?, the ASE becomes N , the spectral efficiency of the largestavailable modulation order in an adaptive system.4.3.2 BER of R-QAM Based Adaptive SIMThe BER of a constant-power, adaptive discrete rate system can becalculated as the ratio of the average number of erroneous bits to the totalaverage number of transmitted bits [71]. Hence, the BER of R-QAM basedadaptive SIM isBER =?Nn=1 ?Pb?QAMn log2Mn?Nn=1 an log2Mn(4.35)4For M -PSK based adaptive SIM, ASE obtained in (4.34) is numerically equivalent to[21, eq. 32].624.3. Performance Analysis of Adaptive SIMwhere ?Pb?QAMn is the BER of an Mn (In ? Jn) order R-QAM transmitted inthe SNR range [?n ?n+1), and it is defined as?Pb?QAMn =? ?n+1?nPQAMb (In, Jn, ?)f?(?) d?. (4.36)Applying (4.2) and (4.4) to (4.36) and using an integral identity [65, eq.06.27.21.0005.01], we can evaluate ?Pb?QAMn as?Pb?QAMn =B(?? ?, 1? ?+ ?)?(?)?(?)1log2(In ? Jn)(?1 + ?2) (4.37)where?1 =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)??p=0[a?p(?, ?)wp(?n, ?n+1, ?, i, In, Jn)?? p+?2?a?p(?, ?)wp(?n, ?n+1, ?, i, In, Jn)?? p+?2](4.38)and?2 =1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?j(l, Jn)??p=0[a?p(?, ?)wp(?n, ?n+1, ?, j, In, Jn)?? p+?2?a?p(?, ?)wp(?n, ?n+1, ?, j, In, Jn)?? p+?2].(4.39)634.3. Performance Analysis of Adaptive SIMHere wp(?L, ?U , y, z, I, J) in (4.38) and (4.39) is defined aswp(?L, ?U , y, z, I, J)=1p+ y[(??U )p+yerfc ((2z + 1)D(I, J)??U )?(??L)p+yerfc ((2z + 1)D(I, J)??L)]?[(2z + 1)D(I, J)]?(p+y)?pi(p+ y)[?(p+ y + 12, (2z + 1)2D2(I, J)?U)??(p+ y + 12, (2z + 1)2D2(I, J)?L)].(4.40)In (4.40), D(I, J) is defined in (4.7) and ?(?, ?) denotes the upper incompleteGamma function [63, eq. 8.350(2)]. Finally, eqs. (4.35), (4.37), (4.38) and(4.39) allow one to estimate the BER performance of R-QAM based adap-tive SIM. In (4.37), both ?1 and ?2 contain summation of infinite terms. Forpractical evaluation, we consider only summation of finite terms. Conse-quently, a truncation error is introduced due to elimination of infinite termsafter the first K + 1 terms. Truncation error analysis of (4.37) is presentedin the Appendix B.4.3.3 BER of M-PSK Based Adaptive SIMBER of M -PSK based adaptive SIM can be determined from (4.35) byusing an = F?(?n+1)?F?(?n), and by replacing ?Pb?QAMn with ?Pb?PSKn . Here?Pb?PSKn is the BER of an Mn order PSK transmitted in the SNR range[?n ?n+1), and it is defined as?Pb?PSKn =? ?n+1?nP PSKb (Mn, ?)f?(?) d?. (4.41)Applying (4.2) and (4.17) to (4.41), we obtain ?Pb?PSKn as?Pb?PSKn =B(?? ?, 1? ?+ ?)4pi?(?)?(?) log2MnMn?1?k=1d(k)(e1 ? e2) (4.42)644.3. Performance Analysis of Adaptive SIMwheree1 =? ?n+1?n? mkpi0exp(?A2(k,Mn)sin2 ??)f?(?) d? d? (4.43)ande2 =? ?n+1?n? nkpi0exp(?B2(k,Mn)sin2 ??)f?(?) d? d?. (4.44)Using an integral identity [63, eq. 2.33(10)], e1 and e2 can be further ex-pressed ase1 =??p=0[a?p(?, ?)up(?,Mn, ?n, ?n+1)?? p+?2?a?p(?, ?)uP (?,Mn, ?n, ?n+1)?? p+?2](4.45)ande2 =??p=0[a?p(?, ?)vp(?,Mn, ?n, ?n+1)?? p+?2?a?p(?, ?)vp(?,Mn, ?n, ?n+1)?? p+?2].(4.46)In (4.45) and (4.46), up(y,M, ?L, ?U ) and vp(y,M, ?L, ?U ) are respectivelydefined asup(y,M, ?L, ?U ) =? mkpi0?????(p+y2 ,A2(k,M)sin2 ??L)(A2(k,M)sin2 ?) p+y2??(p+y2 ,A2(k,M)sin2 ??U)(A2(k,M)sin2 ?) p+y2???? d?(4.47)654.3. Performance Analysis of Adaptive SIMandvp(y,M, ?L, ?U ) =? nkpi0?????(p+y2 ,B2(k,M)sin2 ??L)(B2(k,M)sin2 ?) p+y2??(p+y2 ,B2(k,M)sin2 ??U)(B2(k,M)sin2 ?) p+y2???? d?.(4.48)We observe from (4.47) and (4.48) that definite integrals are constant forany given p, y, M , k, ?L and ?U values. Generally, the integral (4.47) and(4.48) have no closed-form solutions and they are computed efficiently by astandard software for any given parameters p, y, M , k, ?L and ?U . Hence,eqs. (4.35), (4.42), (4.45) and (4.46) allow one to estimate the BER per-formance M -PSK based adaptive SIM over the Gamma-Gamma turbulencechannels. In order to facilitate the truncation error analysis for e1 and e2,we first derive upper bounds of e1 and e2. Using the following factup(y,M, ?L, ?U ) ? uupp (y,M, ?L, ?U ),1Ap+y(k,M)(?(p+ y2, A2(k,M)?L)??(p+ y2, A2(k,M)?U))? mkpi0sinp+y ? d?=1Ap+y(k,M)(?(p+ y2, A2(k,M)?L)??(p+ y2, A2(k,M)?U))gp(mk, y)(4.49)we obtain upper bound of e1 aseup1 =??p=0[a?p(?, ?)uupp (?,Mn, ?n, ?n+1)?? p+?2?a?p(?, ?)uupP (?,Mn, ?n, ?n+1)?? p+?2].(4.50)664.3. Performance Analysis of Adaptive SIMSimilarly e2 can be upper bounded byeup2 =??p=0[a?p(?, ?)vupp (?,Mn, ?n, ?n+1)?? p+?2?a?p(?, ?)vupp (?,Mn, ?n, ?n+1)?? p+?2](4.51)where vupp (?,Mn, ?n, ?n+1) is defined asvupp (y,M, ?L, ?U ) =1Bp+y(k,M)(?(p+ y2, A2(k,M)?L)??(p+ y2, A2(k,M)?U))gp(nk, y).(4.52)Using (4.50) and (4.51), the truncation error analysis of e1 and e2 is pre-sented in the Appendix C.4.3.4 Asymptotic BERR-QAM Based Adaptive SIMIn order to calculate asymptotic BER of an adaptive SIM, it is necessaryto determine the asymptotic values of both numerator and denominator of(4.35) when the SNR ? approaches infinity. Assuming ? > ? and using asimilar argument in obtaining (4.25), asymptotic value of ?Pb?QAMn for R-QAM based adaptive SIM in large SNR regimes can be obtained as?Pb?QAM,asymn =B(?? ?, 1? ?+ ?)?(?)?(?)1log2(In ? Jn)(?asym1 + ?asym2 ) (4.53)where?asym1 =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)(??)??(? ? ?+ 1)w0(?n, ?n+1, ?, i, In, Jn)???2(4.54)674.3. Performance Analysis of Adaptive SIMand?asym2 =1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?j(l, Jn)(??)??(? ? ?+ 1)w0(?n, ?n+1, ?, j, In, Jn)???2 .(4.55)Using (4.32) and Mn = 2n, we can write the denominator of (4.35) as?Nn=1 an log2Mn = N ??Nn=1 F?(?n), and it becomes N when ? ? ?.Therefore, we obtain the asymptotic BER of R-QAM based adaptive SIMin large SNR regimes asBERasym=?Nn=1 ?Pb?QAM,asymn log2MnN. (4.56)M-PSK Based Adaptive SIMAsymptotic BER of M -PSK based adaptive SIM in large SNR regimescan be obtained from (4.56) by replacing ?Pb?QAM,asymn with ?Pb?PSK,asymn ,where ?Pb?PSK,asymn is the asymptotic value of ?Pb?PSKn in large SNR regimes,and it is defined as?Pb?PSK,asymn =B(?? ?, 1? ?+ ?)4pi?(?)?(?) log2MMn?1?k=1d(k)(easym1 ? easym2 ) (4.57)whereeasym1 =(??)??(? ? ?+ 1)u0(?,Mn, ?n, ?n+1)???2 (4.58)andeasym2 =(??)??(? ? ?+ 1)v0(?,Mn, ?n, ?n+1)???2 . (4.59)In obtaining (4.57), we have assumed ? > ? and used the similar argumentfor obtaining (4.23). From (4.53)-(4.59), it is obvious that the diversity orderof the adaptive system also depends only on the smaller channel parameterof the Gamma-Gamma turbulence channel.684.3. Performance Analysis of Adaptive SIM4.3.5 Outage ProbabilitySince no data will be transmitted when the received SNR ?? falls be-low ?1(or ?1), the proposed adaptive system have an outage probability,P adpo (?1), given byP adpo (?1) = Pr(? < ?1) =? ?10f?(?) d? = F?(?1)=B(?? ?, 1? ?+ ?)?(?)?(?)??p=0[a?p(?, ?)p+ ?(?1?) p+?2?a?p(?, ?)p+ ?(?1?) p+?2].(4.60)The asymptotic outage probability of the adaptive SIM in large SNR regimescan be obtained asP adp,asymo (?1) =?(?? ?)(??)??(?)?(?)?(?1?)?2. (4.61)4.3.6 Adaptive MIMO Based SIM SystemsWe consider a MIMO OWC system with T transmitter lasers and P pho-todetectors. At the transmitter end, a repetition coding is adopted acrossthe T lasers, i.e., the T continuous wave laser beams are simultaneouslymodulated with identical properly biased RF subcarrier signal, s(t), whichis pre-modulated by the data source. For a fair comparison we assume thatboth MIMO based and SISO based SIM systems have the same transmitpower Pt that is divided equally among T laser beams. Without loss ofgenerality, we normalize Pt to unity. We also assume that the P photode-tectors have the same responsivity R and photodetector area Ap = A/P ,where A is the area of a single photodetector in a SISO OWC system. Thus,the total receiver aperture area in a MIMO OWC system is equal to thereceiver aperture area of a SISO system. With direct detection, the received694.3. Performance Analysis of Adaptive SIMphotocurrent at the pth photodetector can be expressed as [21]ip(t) =RA [1 + ?s(t)]TPT?k=1Ikp(t) +1?Pnp(t), p = 1, 2, ? ? ?, P (4.62)where Ikp(t) denotes the instantaneous optical irradiance between the kthlaser and pth photodetector, and np(t) represents zero mean AWGN processwith variance ?2n. The sample Ikp = Ikp(t)|t=t0 at a time instant t = t0gives the RV Ikp. We assume that the irradiance Ikp?s are independentand identically distributed (i.i.d) RVs. Such an assumption is realistic byplacing the transmitter and the receiver apertures just few centimeters apart[13]. Assuming equal gain combining (EGC) at the receiver, the combinedreceived photocurrent can be expressed asi(t) =RA [1 + ?s(t)]TPT?k=1P?p=1Ikp? ?? ?IT+P?p=11?Pnp(t)? ?? ?z(t)=RA [1 + ?s(t)]TPIT + z(t)(4.63)where z(t) is a zero mean AWGN process with variance ?2z = ?2n. Nor-malizing the power of s(t) to unity, the instantaneous SNR at the input ofelectrical demodulator can be written as?T =(RA?)2T 2P 2?2nI2T =?T 2P 2I2T (4.64)where ? is the average electrical SNR assuming T = P = 1. The proposedconstant-power variable-rate adaptive SIM can be directly applied to theMIMO configuration where the decision on the transmitted modulation or-der will be based on ?T . Using [73, eq. 13], we can obtain the PDF of RVIT asfIT (IT ) =TP?m=0(TPm) ??p=0cp(TP ?m,m)(??)2Gd?(2Gd)I2Gd?1T (4.65)704.3. Performance Analysis of Adaptive SIMwhere cp(i, j) = e(i)p (?, ?) ? e(j)p (?, ?), Gd =p+(TP?m)?+m?2 , and ? de-notes convolutional operator. Here, e(i)p (x, y) is calculated by convolvingep(x, y) with itself i? 1 times, e.g. e(2)p (x, y) = ep(x, y) ? ep(x, y). Note thate(1)p (x, y) = 1 and e(0)p (x, y) = 1. Also ep(x, y) is defined asep(x, y) =pi?(p+ y)sin(pi(x? y))?(x)?(y)?(p? x+ y + 1)p!. (4.66)Using (4.64) and (4.65), we obtain the PDF of SNR ?T asf?T (?T ) =12TP?m=0(TPm) ??p=0cp(TP ?m,m)?(2Gd)(?o?)?Gd?Gd?1T (4.67)where ?o = 1?2?2T 2P 2 .Achievable Spectral EfficiencyThe ASE of a MIMO based adaptive SIM with T transmit and P receiveapertures over the Gamma-Gamma turbulence channels can be written asSMIMO =N?n=1n(F?T (?n+1)? F?T (?n)) = N ?N?n=1F?T (?n) (4.68)where F?T (?) is the CDF of ?T which can be directly obtained from (4.67)asF?T (?T ) =TP?m=0(TPm) ??p=0cp(TP ?m,m)2Gd?(2Gd)(?o?)?Gd?GdT . (4.69)BER of R-QAM Based MIMO Adaptive SIMBER of an R-QAM based MIMO adaptive SIM over the Gamma-Gammaturbulence channels can be obtained from (4.35) where an = F?T (?n+1) ?F?T (?n), and where ?Pb?n is given by?Pb?n =? ?n+1?nPb(In, Jn, ?T )f?T (?T ) d?T . (4.70)714.3. Performance Analysis of Adaptive SIMApplying (4.4) and (4.67) to (4.70) and using an integral identity [65, eq.06.27.21.0005.01], we can evaluate ?Pb?n as?Pb?n =1log2(In ? Jn)(?d1 + ?d2 ) (4.71)where?d1 =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)TP?m=0(TPm) ??p=0cp(TP ?m,m)?(2Gd)? wdp(?n, ?n+1, Gd, i, In, Jn)(?o?)?Gd(4.72)and?d2 =1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?j(l, Jn)TP?m=0(TPm) ??p=0cp(TP ?m,m)?(2Gd)? wdp(?n, ?n+1, Gd, j, In, Jn)(?o?)?Gd .(4.73)Here in (4.72) and (4.73), wdp(?L, ?U , Gd, z, I, J) can be obtained from (4.40)by replacing p+ y with 2Gd.BER of M-PSK Based MIMO Adaptive SIMBER of an M -PSK based MIMO adaptive SIM can be determined from(4.35) where an = F?T (?n+1)?F?T (?n), and where ?Pb?n can be written as?Pb?n =? ?n+1?nPb(Mn, ?T )f?T (?T ) d?T . (4.74)Applying (4.17) and (4.67) to (4.74), and using an integral identity [63, eq.2.33(10)], we obtain ?Pb?n as?Pb?n =14pi log2MMn?1?k=1d(k)(ed1 ? ed2) (4.75)724.3. Performance Analysis of Adaptive SIMed1 =TP?m=0(TPm) ??p=0cp(TP ?m,m)?(2Gd)udp(Gd,Mn, ?n, ?N+1)(?o?)?Gd (4.76)anded2 =Tp?m=0(TPm) ??p=0cp(TP ?m,m)?(2Gd)vdp(Gd,Mn, ?n, ?N+1)(?o?)?Gd (4.77)where udp(Gd,M, ?L, ?U ) and vdp(Gd,M, ?L, ?U ) can respectively be obtainedfrom (4.47) and (4.48) by replacing p+ y with 2Gd.Asymptotic BERAsymptotic BER of an R-QAM based MIMO adaptive SIM can be ob-tained from (4.56) where ?Pb?QAM,asymn is given by?Pb?QAM,asymn =B(?? ?, 1? ?+ ?)?(?)?(?)1log2(In ? Jn)(?d,asym1 + ?d,asym2 ) (4.78)where?d,asym1 =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)cp(TP, 0)?(TP?)? w0(?n, ?n+1,TP?2, i, In, Jn)(?o?)?TP?2(4.79)and?d,asym2 =1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?j(l, Jn)cp(TP, 0)?(TP?)? w0(?n, ?n+1,TP?2, j, In, Jn)(?o?)?TP?2 .(4.80)Similarly, we can obtain asymptotic BER of M -PSK based adaptive SIM.From (4.78)-(4.80), we also observe that the diversity order of the MIMOadaptive SIM system depends on the number of transmit lasers T , numberof photodetectors P , and the smaller channel parameter ?.734.4. Numerical Results4.4 Numerical ResultsIn this section we present selected numerical examples and compare per-formance of non-adaptive and adaptive SIM system over different Gamma-Gamma turbulence conditions. Closed-from series solutions of ASE, BER,and outage probability derived in previous sections are evaluated by elimi-nating infinite terms after the first K+1 terms, and the corresponding exactsolutions are evaluated using numerical integration. We have found thatK = 30 is sufficient to obtain desired accuracy for typical values of ? and ?.We also consider two representative turbulence conditions for the Gamma-Gamma channels with ? = 2.04, ? = 1.10 as strong and ? = 2.50, ? = 2.06as moderate turbulence. Although we have used the target BER Po = 10?6for performance evaluation of adaptive SIM systems, our theoretical analysishas been validated by numerical evaluation for other target BER require-ments.In Fig. 4.1, BER performance of 4, 32 and 128 R-QAM based non-adaptive SIM is presented for the strong and moderate Gamma-Gammaturbulence channels5. From Fig. 4.1, we observe an excellent agreementbetween the exact BER evaluated using direct integration and the BER ob-tained by our series solutions with finite number of terms in (4.8). Asymp-totic BERs are also presented in Fig. 4.1 and we observe that the asymptoticBERs converge to the exact BERs at high SNR, as expected. The conver-gence rate is faster for ? = 2.04, ? = 1.10 compared to ? = 2.50, ? = 2.06which is consistent with the prediction from (4.25). Fig. 4.2 presents theBER performance of non-adaptive SIM employing 32-PSK over the strongand moderate Gamma-Gamma turbulence channels. Here, we again see thatthe exact BER obtained using direct integration matches well with the BERobtained using our series solutions with finite number of terms. BER of M -PSK based non-adaptive SIM obtained using [21, eq. 15] is also presented inFig. 4.2. It is observed that there is a small noticeable performance gap inlow SNR regimes between the BER approximation presented in [21, eq. 15]5In Figs. 4.1 and 4.2, Approx means the BER obtained using the series solutions withfinite number of terms.744.4. Numerical Results0 10 20 30 40 50 60 70 8010?810?710?610?510?410?310?210?1100Electrical SNR per bit (dB)Average BER  Approx, 4?QAMExact, 4?QAMApprox, 32?QAMExact, 32?QAMApprox, 128?QAMExact, 128?QAMAsymptotic?=2.04,?=1.10?=2.50,?=2.06Figure 4.1: BER of R-QAM based non-adaptive SIM over the Gamma-Gamma turbulence channels.754.4. Numerical Results0 10 20 30 40 50 60 70 8010?610?510?410?310?210?1100 Electrical SNR per bit (dB)Average BER  ?=2.04,?=1.10, Exact?=2.04,?=1.10, Approx?=2.04,?=1.10, [21,eq.15]?=2.50,?=2.06, Exact?=2.50,?=2.06, Approx?=2.50,?=2.06, [21,eq.15]AsymptoticFigure 4.2: BER of 32-PSK based non-adaptive SIM over the Gamma-Gamma turbulence channels.764.4. Numerical Results0 10 20 30 40 50 60 70 8010?1510?1010?5100Electrical SNR per bit (dB)Absolute Truncation Error  ?K,K=8?K,K=15?K,K=30Figure 4.3: Absolute truncation error for the BER of 32 R-QAM non-adaptive SIM over a moderate Gamma-Gamma turbulence channel.and the BER obtained in this work. This is because the BER approxima-tion for non-adaptive M -PSK over the Gamma-Gamma turbulence channelspresented in [21, eq. 15] was derived using an approximate conditional BERexpression that is only valid at high SNR. It is also depicted in Fig. 4.2 thatthe performance gap is more noticeable in the strong turbulence condition.In a strong turbulence condition the probability of having high instantaneousSNR is lower compared to a moderate turbulence condition for a given aver-age SNR. As a result, in the strong turbulence condition more performancegap is observed between the exact BER and BER obtained using [21, eq.15].In Fig. 4.3 we plot the absolute truncation error for the BER of a32 R-QAM non-adaptive SIM system over the moderate Gamma-Gammaturbulence condition using K = 8, 15, and 30 terms. As predicated by774.4. Numerical Results0 20 40 60 80 100 120012345678910Electrical SNR (dB)Achievable Spectral Efficiency(bits/s/Hz)  ?=2.04,? =1.10?=2.50,?=2.06Non?adaptive 2?QAM ?=2.04,? =1.10 Non?adaptive 2?QAM ?=2.50,?=2.06N=9N=7 N=5Adaptive SIMFigure 4.4: ASE of R-QAM based adaptive SIM over the Gamma-Gammaturbulence channels.(4.16), the truncation error rapidly diminishes as the values of K and SNRincrease. For example, at 0 dB SNR it requires K = 30 terms in order toachieve a truncation error of 10?9 order where the same level of truncationerror is achieved only with K = 8 terms at 20 dB SNR. This suggests thatour series solutions of BER are highly accurate in large SNR regimes. Atlow SNR, our series solutions require relatively higher number of terms toconverge.In Fig. 4.4, ASE is presented for R-QAM based adaptive SIM withN =5,7 and 9 regions6 over the strong and moderate Gamma-Gamma turbulencechannels along with the spectral efficiency of 2-QAM based non-adaptiveSIM. In Fig. 4.5 we plot the ASE of M -PSK based adaptive SIM with N =5,6An adaptive SIM with n regions means that 2, 4, 8, ? ? ?, 2n modulation orders areavailable.784.4. Numerical Results0 20 40 60 80 100 120012345678910Electrical SNR (dB)Achievable Spectral Efficiency(bits/s/Hz)  ?=2.04,?=1.10?=2.50,2.06Non?Adaptive BPSK ?=2.04,?=1.10Non?Adaptive BPSK ?=2.50,?=2.06N=9N=7 N=5Adaptive SIMFigure 4.5: ASE of M -PSK based adaptive SIM over the Gamma-Gammaturbulence channels.794.4. Numerical Results7 and 9 regions in the strong and moderate turbulence conditions along withthe spectral efficiency of BPSK based non-adaptive SIM. Spectral efficienciesof 2-QAM and BPSK based non-adaptive SIM are found by determining therequired values of SNR respectively from (4.8) and (4.20) with target BER ofPo = 10?6. From both Figs. 4.4 and 4.5 it is obvious that the adaptive SIMoffers large spectral efficiency gain compared to the non-adaptive SIM in thestrong turbulence conditions. However, spectral efficiency gain gets reducedas the turbulence strength decreases. This is because at moderate turbulenceconditions non-adaptive SIM requires less average SNR to achieve the targetBER. We also observe that the adaptive SIM systems can achieve higherASE in a moderate turbulence condition in the low SNR regimes. Thisresult is expected since for a less-faded channel the receiver tends to pick ahigher order modulation while maintaining the minimum BER requirements.However, in the large SNR regimes, ASE obtained by the adaptive SIM isindependent of channel parameters, and it approaches the spectral efficiencyof the largest available modulation order. This result is also expected, sincein this SNR regime the receiver always picks the largest available modulationorder independent of turbulence condition. The last observation can also beexplained from (4.34) as the SNR ? approaches infinity, the asymptotic valueASE can be obtained as S = N which is independent of channel parameters.Figure 4.6 illustrates ASE of R-QAM and M -PSK based MIMO adaptiveSIM over a strong Gamma-Gamma turbulence channel with N = 8 regions,and with different numbers of transmit and receive apertures. This figureclearly depicts that for low to medium values of SNR, MIMO adaptive SIMoffers better ASE than the SISO adaptive SIM. For asymptotically large SNRvalues, both systems provide similar ASE. This can be explained by the fol-lowing argument. The MIMO systems mitigate fading and improve channelquality. This allows the receiver of a MIMO adaptive SIM to pick up a largermodulation order compared to that of a SISO adaptive system. Hence, inlow to medium SNR regimes the MIMO adaptive SIM achieves higher ASEthan the SISO adaptive SIM for the same average SNR. However, when theSNR becomes asymptotically large, both MIMO and SISO adaptive systems804.4. Numerical Results0 20 40 60 80 100 1200123456789Electrical SNR per branch (dB)Achievable Spectral Efficiency (bits/s/Hz)  T=1,P=1T=1,P=2T=2,P=2MPSK Based Adaptive SIMR?QAM Based Adaptive SIMFigure 4.6: ASE of MIMO (T transmit lasers and P photo detectors) adap-tive SIM with N = 8 regions over a strong Gamma-Gamma turbulencechannel.814.4. Numerical Results0 10 20 30 40 50 60 70 8010?1410?1210?1010?810?610?410?2100 Electrical SNR in dBAverage BER  Non?Adaptive BPSK SIMAdaptive SIM,Exact Adaptive SIM, Approx,?=2.04,?=1.10Adaptive SIM, Approx, ?=2.50,?=2.06AsymptoticN=3N=9?=2.04,?=1.10?=2.50,?=2.06Figure 4.7: BER of R-QAM based adaptive SIM over the Gamma-Gammaturbulence channels.have favourable channel conditions. Consequently, the receiver always picksthe largest available modulation order. Therefore, at asymptotically largeSNR values ASEs of both SISO and MIMO adaptive SIM converge. Wealso observe that the R-QAM based adaptive SIM provides improved ASEthan the M -PSK based adaptive SIM for low to moderate SNR values. TheSNR thresholds of R-QAM based adaptive SIM are smaller than that ofM -PSK based adaptive SIM for M ? 8. As a result, receiver of R-QAMbased adaptive SIM tends to pick a larger modulation compared to that ofM -PSK based modulation. Hence, in low to moderate values of SNR R-QAM based adaptive SIM achieves larger ASE. However, when the SNR isasymptotically large, receivers of both adaptive SIM select the largest avail-able modulation order. Consequently, ASEs of both adaptive SIM convergein this SNR regime.824.4. Numerical Results0 10 20 30 40 50 60 70 8010?1310?1210?1110?1010?910?810?710?6 Electrical SNR per branch (dB)Average BER  ExactApprox, Adaptive R?QAM SIMApprox,Adaptive MPSK SIMT=1,P=1T=2,P=2Figure 4.8: BER of an adaptive SIM with N = 5 regions over a strongGamma-Gamma turbulence channel.834.4. Numerical Results0 10 20 30 40 50 60 70 8010?1810?1610?1410?1210?1010?810?6Electrical SNR per branch (dB)Average BER  ExactApprox, Adaptive R?QAM SIMApprox,Adaptive MPSK SIMT=1,P=1T=2,P=2Figure 4.9: BER of an adaptive SIM with N = 5 regions over a moderateGamma-Gamma turbulence channel.844.4. Numerical Results0 10 20 30 40 50 60 70 8010?1010?810?610?410?2100Electrical SNR per branch (dB)Outage Probability  Approx, T=1,P=1Exact, T=1,P=1Approx, T=1,P=2Exact, T=1,P=2Approx, T=2,P=2Exact, T=2,P=2AsymptoticFigure 4.10: Outage probability of an adaptive SIM over a strong Gamma-Gamma turbulence channel.854.4. Numerical ResultsFigure 4.7 presents the BER performance of R-QAM based adaptive SIMwith N = 3 and 9 regions over the strong and moderate Gamma-Gammaturbulence channels7 along with the BER of non-adaptive BPSK SIM. FromFig. 4.7, excellent agreement is observed between the exact BER evaluatedusing direct integration and the BER obtained by our series solutions withfinite number of terms using (4.35) and (4.37). It is also depicted thatthe adaptive SIM system always offers a BER less than the target BER,satisfying the basic design goal of the adaptive SIM system. Also adaptiveSIM always offers an improved BER performance compared to the non-adaptive BPSK SIM. Asymptotic BERs are also presented in Fig. 4.7, andwe observe that the asymptotic BERs converge to the exact BERs at highSNR, as expected. The convergence rate is faster for ? = 2.04, ? = 1.10compared to ? = 2.50, ? = 2.06 which is consistent with the prediction from(4.56).Figure 4.8 presents the BER performance of R-QAM and M -PSK basedadaptive SIM with N = 5 regions over a strong Gamma-Gamma turbulencechannel. Figure 4.9 presents the BER performance of the same systemsover a moderate Gamma-Gamma turbulence channel. From both Figs. 4.8and 4.9, we observe an excellent agreement between the approximate and theexact BER of both adaptive SIM systems. We also observe that the R-QAMbased adaptive SIM outperforms the M -PSK based adaptive SIM in termsof BER performance. This is an expected outcome, because for M ? 8 anR-QAM always offers better BER performance than an M -PSK modulationover a fading channel. Finally, as expected, by adopting MIMO the effectof turbulence fading is substantially reduced, and the BER performance issignificantly improved.In Fig. 4.10, we plot the outage probability of an adaptive SIM over astrong Gamma-Gamma turbulence channel. Excellent agreement is observedbetween the exact outage probability and outage probability obtained byour series solutions. We also observe that the MIMO adaptive SIM offersimproved outage probability performance compared to the SISO adaptive7In Fig. 4.7, Approx means the BER obtained using the series solution with finitenumber of terms.864.5. SummarySIM, as expected. Asymptotic outage probabilities are also shown in thisfigure, and they converge to the exact outage probability at high values ofSNR.4.5 SummaryA detailed performance study has been carried out for both SIMO andMIMO constant-power, variable-rate adaptive SIMs employing R-QAM andM -PSK over the Gamma-Gamma turbulence channels. Numerical resultsdemonstrate that the adaptive SIM simultaneously offers a significant spec-tral efficiency gain and a BER performance less than the target BER. Con-sequently, adaptive SIM is an excellent fading mitigation solution for theOWC systems.87Chapter 5Ergodic CapacityComparison of OpticalWireless CommunicationsUsing AdaptiveTransmissionsIn Chapter 4 we have showed that a constant-power, variable-rate adap-tive SIM offers significant spectral efficiency gain without sacrificing theBER performance. In this chapter, we study the ergodic capacity perfor-mance of other adaptive SIM systems. In particular, we consider VPVR,CCIFR, and TCIFR adaptive transmission policies over the Gamma-Gammaturbulence channels. Highly accurate ergodic capacity expressions are de-rived using a series expansion of the modified Bessel function and the Mellintransformation of the Gamma-Gamma random variable. We also analyzethe ergodic capacity of adaptive coherent OWC systems with and withoutPOLMUX. Finally, we numerically compare the ergodic capacity of differentOWC systems under the same average transmitted optical power constraint.885.1. Ergodic Capacity of Coherent OWC System5.1 Ergodic Capacity of Coherent OWC System5.1.1 Variable-Power, Variable-Rate Adaptive TransmissionIn a VPVR adaptive transmission scheme, the transmitter simultane-ously adapts the power and data rate in order to maintain a target BERat the receiver for all SNRs. With this adaptive transmission policy, morepower and higher data rates are allocated when the channel condition isgood, and the transmission is terminated when the received SNR falls belowa cutoff level ?o. For a VPVR adaptive scheme, the ergodic capacity (inbits/s/Hz) of a fading channel can be calculated by [76, eq. 7]?C?cohVPVR =? ??olog2(?c?o)f?c(?c) d?c =1ln 2? ??oln(?c?o)f?c(?c) d?c.(5.1)Under an average transmit power constraint, the cutoff level ?o must satisfy[76, eq. 6]? ??o(1?o?1?c)f?c(?c) d?c = 1 (5.2)where in (5.1) and (5.2) f?c(?c) is the PDF of the SNR ?c. Using a seriesexpansion of the modified Bessel function of second kind [74, eq. 15], weobtain the PDF and the CDF of ?c, respectively, asf?c(?c) =?(?, ?)??p=0[ap(?, ?)(?c?c)p+???1c ? ap(?, ?)(?c?c)p+???1c](5.3)andF?c(?c) =?(?, ?)??p=0[ap(?, ?)p+ ?(?c?c)p+??ap(?, ?)p+ ?(?c?c)p+?]. (5.4)where ?(?, ?) , ?(???)?(1??+?)?(?)?(?) and ap(x, y) ,(xy)p+y?(p?x+y+1)p! .895.1. Ergodic Capacity of Coherent OWC SystemComputation of Cutoff SNR ?oEq. (5.2) can be rewritten asF?c(?o) + ?o?????? ?0??1c f?c(?c) d?c? ?? ?r1,c(?,?,?c)?? ?o0??1c f?c(?c) d?c? ?? ?r2,c(?,?,?o,?c)?????+ ?o = 1. (5.5)Substituting ?c = ?cI into the first integral of (5.2), we can write r1,c(?, ?, ?c)asr1,c(?, ?, ?c) =1?c? ?0I?1fI(I) dI =1?cE[I?1] (5.6)where E[I?1] is the first negative integer moment of the Gamma-GammaRV I. The Mellin transformation of a positive RV X provides all moments ofX including positive integer, negative integer, and fractional moments [77].Using the Mellin transformation of the Gamma-Gamma RV (see AppendixD), we obtain r1,c(?, ?, ?c) asr1,c(?, ?, ?c) =1?c?(?? 1)?(? ? 1)???(?)?(?). (5.7)Substituting (5.3) into the second integral of (5.2), after some algebraicmanipulation, we can evaluate r2,c(?, ?, ?o, ?c) asr2,c(?, ?, ?o, ?c) =?(?, ?)??p=0[ap(?, ?)(p+ ? ? 1)?o(?o?c)p+??ap(?, ?)(p+ ?? 1)?o(?o?c)p+?].(5.8)Finally, applying (5.4), (5.6), and (5.8) to (5.5) one obtains a series expres-sion involving the cutoff SNR ?o. For min{?, ?} > 1, such expression can benumerically solved along with (5.2) in order to compute the cutoff SNR ?ofor a given average SNR. From (5.4), (5.7) and (5.8), when ?c ?? we have905.1. Ergodic Capacity of Coherent OWC SystemF?c(?) ? 0, r1,c(?, ?, ?c) ? 0, and r2,c(?, ?, ?o, ?c) ? 0. As a result, whenthe average SNR in (5.5) approaches infinity, ?o approaches unity. There-fore, for the Gamma-Gamma turbulence channels the value of the cutoffSNR, ?o, is restricted to [0, 1].Computation of Ergodic CapacityIn order to facilitate the computation of ergodic capacity of the VPVRadaptive scheme, we rewrite (5.1) as?C?cohVPVR =1ln 2?????? ?0ln (?c) f?c(?c) d?c? ?? ?g1,c(?,?,?c)?? ?o0ln (?c) f?c(?c) d?c? ?? ?g2,c(?,?,?o,?c)?????? log2(?o)(1? F?c(?o)).(5.9)Here g1,c(?, ?, ?c) in (5.9) is the expectation of ln ?c, i.e., E[ln ?c]. In order toestimate the expected value of ln ?c, we first define Z = ln ?c. The momentgenerating function of Z isMZ(s) = E[esz] = E[es ln ?c ] = E[?sc ]. (5.10)The expected value of ln ?c can be obtained asE[ln ?c] = E[Z] =dMZ(s)ds????s=0=dE[?sc ]ds????s=0. (5.11)It can be easily shown that if ?c = ?cI, the k-th moment of ?c is E[?kc ] =?ckE[Ik], where E[Ik] is the k-th moment of RV I and it is given by [56, eq.3]E[Ik] =?(?+ k)?(? + k)?(?)?(?)(1??)k. (5.12)915.1. Ergodic Capacity of Coherent OWC SystemUsing [65, eq. 06.05.20.0001.01] we can derive g1,c(?, ?, ?c) asg1,c(?, ?, ?c) = ?(?) + ?(?) + ln ?c ? ln(??) (5.13)where ?(?) is the Euler?s digamma function [63, eq. 8.360(1)]. In order toevaluate g2,c(?, ?, ?o, ?c) in (5.9), we first recall the fact that limz?0 za ln z =0 for any real non-negative z and a. Substituting (5.3) into the secondintegral of (5.9) and using [63, eq. 2.723(1)], we obtain g2,c(?, ?, ?o, ?c) asg2,c(?, ?, ?o, ?c) =?(?, ?)??p=0[ap(?, ?)wp(?, ?, ?o)(?o?c)(p+?)?ap(?, ?)wp(?, ?, ?o)(?o?c)(p+?)] (5.14)where wp(x, y, ?o) ,ln ?op+y ?1(p+y)2 . To evaluate (5.14) numerically, we con-sider only a summation of finite K + 1 terms. Consequently, a truncationerror is introduced due to elimination of infinite terms after the first K + 1terms. Using a similar method described in [74], we can derive an upperbound for the truncation error and show that the truncation error rapidlydecreases with increasing K values and/or average SNR ?c. Finally, using(5.13) and (5.14) one obtains the ergodic capacity for the coherent VPVRadaptive OWC system as?C?cohVPVR =1ln 2(g1,c(?, ?, ?c)? g2,c(?, ?, ?o, ?c))? log2(?o)(1? F?c(?o)).(5.15)Asymptotic High SNR CapacityWhen the average SNR ?c approaches infinity, the value of the cutoffSNR ?o approaches unity, and consequently, log2(?o) approaches zero. Also,it can be shown that in large SNR regimes g2,c(?, ?, ?o, ?c)  g1,c(?, ?, ?c)since in large SNR regimes(1?c)? ln ?c. Hence, for asymptotically highaverage SNR we can evaluate the ergodic capacity of a coherent VPVR925.1. Ergodic Capacity of Coherent OWC Systemadaptive OWC system as?C?coh,asymVPVR =?(?) + ?(?)ln 2? log2(??) + 3.3 log10 ?c=?(?) + ?(?)ln 2? log2(??) + log2(Rgq4f)+ 0.33(10 log10 P t)(5.16)where we have used (2.3) to obtain the last equality. Eq. (5.16) revels thatthe ergodic capacity of the coherent VPVR system gains 0.33 bits/s/Hz with1 dB increase of average transmitted optical power.5.1.2 Complete Channel Inversion with Fixed RateUnder this adaptive transmission scheme, the transmitter adapts thetransmit power (under an average transmit power constraint) according tothe channel fading state in order to maintain a constant SNR at the recevier,i.e., inverts the channel fading while maintaining a constant transmissionrate. The ergodic capacity (in bits/s/Hz) of a CCIFR adaptive transmissionscheme is given by [76, eq. 9]?C?cohCCIFR = log2(1 +1??0 ??1c f?c(?c) d?c). (5.17)The integral??0 ??1c f?c(?c) d?c has already been evaluated in (5.7). As aresult, we can express the ergodic capacity of the CCIFR scheme as?C?cohCCIFR = log2(1 +(?? 1)(? ? 1)???c). (5.18)It can be shown that for a given channel capacity the CCIFR schemes overthe Gamma-Gamma turbulence channels exhibit SNR penalty of10 log10(??(??1)(??1))dB from the AWGN channel. Eq. (5.18) is similar tothe ergodic capacity of the CCIFR scheme over the generalized-K fadingchannel given by [47, eq. 29]. However, our derivation approach is differentfrom [47] and our capcity expression in (5.18) does not require that ? be935.1. Ergodic Capacity of Coherent OWC Systeman integer. The asymptotic capacity in large SNR regimes can be obtainedas ?C?coh,asymCCIFR = log2((??1)(??1)??)+ log2(Rgq4f)+ 0.33(10 log10 P t). Hence,CCIFR scheme also gains 0.33 bits/s/Hz with 1 dB increase of averagetransmitted optical power.5.1.3 Truncated Channel Inversion with Fixed RateSince the CCIFR scheme exhibits a large channel capacity penalty in se-vere fading, a modified channel inversion scheme is proposed where only thetransmit power is adaptive according to the channel fading state providedthat the received SNR is above a certain cutoff SNR ?coh. The channelwill not be used if the received SNR falls below ?coh. For this adaptivetransmission policy, the ergodic capacity (in bits/s/Hz) is given by [76, eq.12]?C?cohTCIFR = log2(1 +1???coh??1c f?c(?c) d?c)(1? P coho (?coh))(5.19)where P coho (?coh) is the outage probability given by Pcoho (?coh) = F?c(?coh).We can evaluate the integral???coh??1c f?c(?c) d?c as? ??coh??1c f?c(?c) d?c = r1,c(?, ?, ?c)? r2,c(?, ?, ?coh, ?c) (5.20)where r1,c(?, ?, ?) and r2,c(?, ?, ?, ?) are obtained from (5.6) and (5.8), respec-tively. Here, the cutoff level ?coh is selected to maximize the channel capacityin (5.19) for a given average SNR. Our numerical result shows that the cut-off SNR that maximizes the channel capacity of a coherent TCIFR systemincreases with an increase of average SNR. Recall that the cutoff SNR ofa coherent VPVR system is restricted to the range [0, 1]. Hence, due toincreasing cutoff SNR values, coherent TCIFR systems exhibit higher out-age probability compared to the coherent VPVR systems. Also note that,for coherent systems, eqs. (5.18) and (5.20) are valid when min{?, ?} > 1,which is typically satisfied for the Gamma-Gamma turbulence channels [35].945.2. Ergodic Capacity of Coherent POLMUX OWC System5.2 Ergodic Capacity of Coherent POLMUXOWC System5.2.1 Variable-Power, Variable-Rate Adaptive TransmissionIn a coherent VPVR POLMUX scheme the transmit power on each chan-nel is adapted subject to an average transmit power constraint per channel.For a fair comparison, it is assumed that the total average transmit powerof a coherent OWC system with and without polarization are same, and thetotal average transmit power is equally allocated between the two orthog-onal channels in a POLMUX system. Under this assumption, the ergodiccapacity of a coherent POLMUX VPVR scheme can be expressed as?C?polVPVR =? ??alog2(?x?a)f?x(?x) d?x? ?? ??C?pol,xVPVR+? ??blog2(?y?b)f?y(?y) d?y? ?? ??C?pol,yVPVR(5.21)where f?x(?x) and f?x(?x) are respectively PDFs of ?x and ?y; ?a and ?b arethe cutoff SNRs for the two orthogonal channels satisfying? ??a(1?a?1?x)f?x(?x) d?x = 1 (5.22)and? ??b(1?b?1?y)f?y(?y) d?y = 1. (5.23)An asymptotic expansion of (5.22) and (5.23) reveals that both ?a and ?b areconfined to the range [0, 1]. Using a similar method applied to the coherentOWC system, it can be shown?C?pol,xVPVR =1ln 2(g1,c(?, ?, ?1?p)? g2,c (?, ?, ?a, ?1?p))? log2 (?a) (1? F?x (?a))(5.24)955.2. Ergodic Capacity of Coherent POLMUX OWC Systemand?C?pol,yVPVR =1ln 2(g1,c(?, ?, ?2?p)? g2,c (?, ?, ?b, ?2?p))? log2 (?b)(1? F?y (?b)) (5.25)where F?x(?) and F?y(?) are respectively the CDFs of ?x and ?y, and theycan be obtained by substituting ?c in (5.4) with ?1?p and ?2?p, respectively.Using a similar argument for obtaining (5.16), the asymptotic capacity athigh SNR can be expressed as?C?pol,asymVPVR =2?(?) + 2?(?)ln 2? 2 log2(??) + log2(?1?2)+ 2 log2(Rg2q4f)+ 0.66(10 log10 P t).(5.26)From (5.26), we observe that the ergodic capacity of coherent POLMUXVPVR adaptive scheme gains 0.66 bits/s/Hz with 1 dB increase of averagetransmitted optical power.5.2.2 Complete Channel Inversion with Fixed RateThe ergodic capacity of coherent POLMUX CCIFR scheme is obtainedas?C?polCCIFR = log2(1 +(?? 1)(? ? 1)???1?p)+ log2(1 +(?? 1)(? ? 1)???2?p).(5.27)The asymptotic capacity at high SNR is expressed as?C?coh,asymCCIFR = 2 log2((?? 1)(? ? 1)??)+ log2(?1?2)+ 2 log2(Rg2q4f)+ 0.66(10 log10 P t).(5.28)965.3. Ergodic Capacity of Subcarrier IM/DD OWC System5.2.3 Truncated Channel Inversion with Fixed RateThe ergodic capacity of coherent POLMUX TCIFR scheme is obtainedas?C?polTCIFR = log2(1 +1r1,c(?, ?, ?1?p)? r2,c(?, ?, ?pol,x, ?1?p))? (1? F?x(?pol,x))+ log2(1 +1r1,c(?, ?, ?2?p)? r2,c(?, ?, ?pol,y, ?2?p))?(1? F?y(?pol,y)).(5.29)For a given average SNR or average transmitted power the cutoff SNRs ?pol,xand ?pol,y are selected in order to maximizelog2(1 + 1???pol,i??1i f?i (?i) d?i)(1? F?i(?pol,i)) where i = x, y. Similar to thecoherent OWC, eqs. (5.27) and (5.29) are valid for min{?, ?} > 1.5.3 Ergodic Capacity of Subcarrier IM/DDOWC System5.3.1 Variable-Power, Variable-Rate Adaptive TransmissionThe ergodic capacity of a subcarrier IM/DD VPVR adaptive OWC sys-tem is given by?C?IMVPVR =1ln 2?????? ?0ln (?s) f?s(?s) d?s? ?? ?g1,s?? ?e0ln (?s) f?s(?s) d?s? ?? ?g2,s?????? log2(?e)(1? F?s(?e))(5.30)where ?e is the cutoff SNR satisfying the condition? ??e(1?e?1?s)f?s(?s) d?s = 1. (5.31)975.3. Ergodic Capacity of Subcarrier IM/DD OWC SystemIn (5.30) and (5.31), f?s(?s) and F?s(?) are respectively the PDF and CDFof the SNR of the subcarrier IM/DD OWC systems, and they are given byf?s(?s) =?(?, ?)2??p=0[bp(?, ?)(?s?s) p+?2??1s ? bp(?, ?)(?s?s) p+?2??1s](5.32)andF?s(?s) =?(?, ?)??p=0[bp(?, ?)p+ ?(?s?s) p+?2?bp(?, ?)p+ ?(?s?s) p+?2](5.33)where in (5.32) and (5.33) bp(x, y) , ap(x, y)[(x+1)(y+1)xy] p+y2. Similar to thecoherent VPVR OWC systems, an asymptotic expansion of (5.31) revealsthat for subcarrier IM/DD VPVR adaptive system the value of ?e is alsoconfined to the range [0, 1]. Using a similar approach in deriving (5.13), wecan evaluate g1,s asg1,s = 2?(?) + 2?(?) + ln ?s ? 2 ln(??). (5.34)Following (5.14), we obtain g2,s asg2,s =?(?, ?)2??p=0[bp(?, ?)vp(?, ?, ?e)(?e?s) p+?2?bp(?, ?)vp(?, ?, ?e)(?e?s) p+?2] (5.35)where vp(x, y, ?e) ,2 ln ?ep+y ?4(p+y)2 . Finally, using (5.34) and (5.35) wecan evaluate the ergodic capacity of the subcarrier IM/DD VPVR adaptiveOWC system as?C?IMVPVR =1ln 2(g1,s ? g2,s)? log2(?e)(1? F?s(?e)). (5.36)985.3. Ergodic Capacity of Subcarrier IM/DD OWC SystemUsing a similar argument in obtaining (5.16), we obtain the ergodic capacityof the subcarrier IM/DD VPVR system at high SNR as?C?IM,asymVPVR =2?(?) + 2?(?)ln 2? 2 log2(??)+ log2((R?g)2(1 + 1?)(1 +1? )24f(qRIb + 2kbTk/RL))+ 0.66(10 log10 P t).(5.37)We observe from Eq. (5.37) that the ergodic capacity of subcarrier IM/DDVPVR system gains 0.66 bits/s/Hz with 1 dB increase of average trans-mitted optical power. In high SNR region with 1 dB increase of averagetransmitted optical power, the subcarrier IM/DD systems provide higherspectral efficiency improvement than the coherent OWC systems withoutPOLMUX. This can be explained by the following analytical arguments.The average SNR of the coherent OWC systems is proportional to the av-erage transmitted optical power whereas the average SNR of the subcarrierIM/DD systems is proportional to square of the the average transmitted op-tical power, as observed from Eqs. (2.3) and (2.11) respectively. Therefore,the high SNR ergodic capacity of the subcarrier IM/DD VPVR scheme hasa steeper slope (2 times) with increasing average transmitted optical powercompared to the high SNR ergodic capacity of the coherent VPVR scheme.Due to this steeper slope, in high SNR region subcarrier IM/DD systemsprovide higher spectral efficiency improvement with 1 dB increase in aver-age transmitted optical power than the coherent detection systems. As aresult, although the coherent systems significantly outperform the subcarrierIM/DD systems, the capacity performance gap between the coherent andsubcarrier IM/DD systems is reduced with the increase of average trans-mitted optical power. Such a observation is also depicted by our numericalresults.995.3. Ergodic Capacity of Subcarrier IM/DD OWC System5.3.2 Channel Inversion with Fixed RateThe ergodic capacity of a subcarrier IM/DD CCIFR scheme is given by?C?IMCCIFR = log2(1 +1??0 ??1s f?s(?s) d?s). (5.38)Substituting ?s =???s(?+1)(?+1)I2 into (5.38) and using Mellin transforma-tion of Gamma-Gamma RV, the ergodic capacity of the subcarrier IM/DDCCIFR scheme can be expressed as?C?IMCCIFR = log2(1 +?(?)?(?)??(?+ 1)(? + 1)?(?? 2)?(? ? 2)?s). (5.39)An asymptotic analysis of (5.39) reveals that ergodic capacity of subcarrierIM/DD CCIFR scheme gains 0.66 bits/s/Hz with 1 dB increase of averagetransmitted optical power. The ergodic capacity of a subcarrier IM/DDTCIFR scheme is given by?C?IMTCIFR = max?th>0log2(1 +1???th??1s f?s(?s) d?s)(1? P IMo (?th)) (5.40)where ?th is the cutoff SNR below which no power adaption is accomplished,and P IMo (?th) is the outage probability of the IM/DD OWC systems thatis given by P IMo (?th) = F?s(?th). The integral???th??1s f?s(?s) d?s can beexpressed as? ??th??1s f?s(?s) d?s =? ?0??1s f?s(?s) d?s? ?? ?r1,s?? ?th0??1s f?s(?s) d?s? ?? ?r2,s(?th)(5.41)wherer1,s =(?+ 1)(? + 1)?s?(?? 2)?(? ? 2)???(?)?(?)(5.42)1005.4. Numerical Resultsandr2,s(?th) =?(?, ?)??p=0[bp(?, ?)(p+ ? ? 2)?e(?th?s) p+?2?bp(?, ?)(p+ ?? 2)?e(?th?s) p+?2].(5.43)For the subcarrier IM/DD systems, eqs. (5.39), (5.42), and (5.43) are validwhen min{?, ?} > 2.5.4 Numerical ResultsIn this section, we compare the ergodic capacity of the subcarrier IM/DDand coherent OWC systems with and without POLMUX using our series so-lutions with the exact ergodic capacity obtained using numerical integration.All series solutions are calculated using the first 31 terms. We consider thefollowing turbulent OWC scenarios where the path loss factor is empiricallyexpressed in terms of visibility, and the turbulence strength is assumed toincrease with propagation distance [35] : 1) a 2 Km haze optical channel(in strong turbulence with ? = 2.04 and ? = 1.10) with 4.35 dB/Km pathloss; 2) a 900 m light smoke optical channel (in moderate turbulence with? = 2.50 and ? = 2.06) with 9.56 dB/Km path loss; 3) a 700 m lightfog optical channel (in weak turbulence ? = 4.43 and ? = 4.39) with 11.5dB/Km path loss. In order to generate the numerical plots of capacityversus the average transmitted optical power, we also make the followingassumptions [35]: the modulation index  = 0.85, the photodetector respon-sivity R = 0.75 A/W, load resistance RL = 50 ?, the bit duration T = 1nswith approximate transmission bandwidth 1 GHz, thermal noise variance3.3? 10?13 Amp2, and the background noise variance 10?15 Amp2. Assum-ing a typical LO power 10?2 W [78], the local oscillator-induced shot noisevariance is 5? 10?12 Amp2. Also, in our numerical results we have selectedthe range of the average transmitted optical power from ?12 dBm to 6 dBmso that the assumption PLO  Ps remains valid.Figures 5.1 and 5.2 respectively present the ergodic capacity of the co-1015.4. Numerical Results0 5 10 15 20 25 30 35 40 45012345678910Average SNR, ?c (dB)Ergodic Capcity in bits/s/Hz  Approx,VPVRExact,VPVRAprrox,TIFRExact,TCIFRApprox,CCIFRExact,CCIFRAsymptoticFigure 5.1: Ergodic capacity of Coherent OWC over a strong Gamma-Gamma turbulence channel wtih ? = 2.04 and ? = 1.10.1025.4. Numerical Results0 5 10 15 20 25 30 35 40051015Normalized Average SNR, ?p (dB) per channelErgodic Capcity in bits/s/Hz  Exact, ?=0oApprox, ?=0oExact, ?=10oApprox, ?=10oExact, ?=20oApprox, ?=20oExact, ?=30oApprox, ?=30oAsymptoicVPVRCCIFRFigure 5.2: Ergodic capacity of Coherent POLMUX OWC over a strongGamma-Gamma turbulence channel wtih ? = 2.04 and ? = 1.10.1035.4. Numerical Results0 5 10 15 20 25 30 35 40012345678910Average SNR, ?c (dB)Ergodic Capcity in bits/s/Hz  Approx,VPVRExact,VPVRAprrox,TCIFRExact,TCIFRApprox,CCIFRExact,CCIFRAsymptoticFigure 5.3: Ergodic Capacity of Coherent OWC over a moderate Gamma-Gamma turbulence channel with ? = 2.50 and ? = 2.06.1045.4. Numerical Results0 5 10 15 20 25 30 35 40 450510152025Normalized Average SNR, ?p (dB) per channelErgodic Capcity in bits/s/Hz  Exact, ?=0oApprox, ?=0oExact, ?=10oApprox, ?=10oExact, ?=30oApprox, ?=30oAsymptoticVPVRCCIFRFigure 5.4: Ergodic Capacity of Coherent POLMUX OWC over a moderateGamma-Gamma turbulence channel with ? = 2.50 and ? = 2.06.1055.4. Numerical Resultsherent and coherent POLMUX OWC systems over a strong Gamma-Gammaturbulence channel. In both figures, excellent agreement is observed betweenthe exact channel capacity and capacity obtained by our series approxima-tions. From these figures, it is obvious that the VPVR scheme achievesthe largest channel capacity for both the coherent and coherent POLMUXOWC systems. From Fig. 5.2 it is depicted that due to the polarizationcontrol error the ergodic capacity of a coherent POLMUX system gets re-duced. However, a polarization control error with  = 10? has no noticeableinfluence on the ergodic capacity performance. Asymptotic channel capaci-ties for the VPVR and CCIFR schemes are also shown in both figures, andthey converge to the exact capacity at large SNR values, as expected.Figures 5.3 and 5.4 plot the ergodic capacities of the coherent and coher-ent POLMUX OWC systems over a moderate Gamma-Gamma turbulencechannel. Excellent agreement is also observed between the exact channel ca-pacity and the capacity obtained by our series approximations. In both fig-ures we observe that the performance gap between different adaptive trans-mission schemes gets decreased with a decrease of turbulence strength. FromFigs. 5.3 and 5.4, the SNR gap between the VPVR and CCIFR schemesis approximately 3 dB for both the coherent and coherent POLMUX OWCsystems when ? = 2.50 and ? = 2.06. This result agrees with the pre-dicted SNR gap of 3.06 dB between the VPVR and CCIFR schemes fromour asymptotic ergodic capacity analysis. Asymptotic capacities are alsoshown in this figure, and they converge to the exact ergodic capacity when?c ? 20 dB or ?p ? 20 dB.Figure 5.5 compares the ergodic capacities of the subcarrier IM/DD andcoherent POLMUX VPVR schemes over a strong Gamma-Gamma turbu-lence channel, while Fig. 5.6 compares the ergodic capacities of the sub-carrier IM/DD and coherent POLMUX CCIFR schemes over a weak tur-bulence channel with the same average transmitted power constraint. Sincethe noise sources for the coherent and subcarrier IM/DD systems are dif-ferent, it is therefore not appropriate to compare their capacity in terms ofaverage SNR. This is why we compare the channel capacity among differ-ent OWC systems in terms of the same average transmitted optical power1065.4. Numerical Results?12 ?10 ?8 ?6 ?4 ?2 0 2 4 6510152025303540Average Transmitted Optical Power, Pt (dBm)Ergodic Capcity in bits/s/Hz  Approx, ?=2.04,?=1.10Exact,?=2.04,?=1.10?=0o and10o?=30oCoherent POLMUX VPVRIM/DD VPVRCoherent VPVRFigure 5.5: Ergodic capacity comparison among the subcarrier IM/DD, andcoherent VPVR adaptive transmission scheme with and without POLMUXover with ? = 2.04 and ? = 1.10.1075.4. Numerical Results?12 ?10 ?8 ?6 ?4 ?2 0 2 40510152025303540Average Transmitted Optical Power, Pt (dBm)Ergodic Capcity in bits/s/Hz  Approx, ?=4.43,?=4.39Exact,?=4.43,?=4.39Coherent POLMUX CCIFR?=0o and 10o?=30oCoherent CCIFRIM/DD CCIFRFigure 5.6: Ergodic capacity comparison among the subcarrier IM/DD, andcoherent CCIFR adaptive transmission scheme with and without POLMUXover a 700 m weak turbulence channel with ? = 4.43 and ? = 4.39.x1085.5. Summaryconstraint. Figs. 5.5 and 5.6 clearly demonstrate that the coherent POL-MUX schemes significantly outperform the coherent and subcarrier IM/DDschemes for the same average transmitted optical power. For example, atan average transmitted optical power of ?12 dBm, the coherent POLMUXVPVR scheme achieves 26.28 bits/s/Hz and 25.37 bits/s/Hz channel capac-ity with polarization control error  = 0? and  = 30?, respectively. Onthe other hand, for the same average transmitted optical power, the coher-ent and subcarrier IM/DD VPVR schemes attain channel capacity of 14.19bits/s/Hz and 5.03 bits/s/Hz, respectively. From Fig. 5.6, at an averagetransmitted optical power of ?4 dBm coherent POLMUX CCIFR schemeachieves 32.74 bits/s/Hz and 31.83 bits/s/Hz channel capacity with polar-ization control error  = 0? and  = 30?, respectively. On the other hand,for the same average transmitted optical power, the coherent and subcar-rier IM/DD CCIFR schemes obtain channel capacity of 17.42 bits/s/Hz and9.738 bits/s/Hz, respectively. Both Figs. 5.5 and 5.6 illustrate that theperformance gap between the coherent and the subcarrier IM/DD systemsgets narrowed with an increase of transmitted optical power. However, theperformance gap between coherent POLMUX and subcarrier IM/DD sys-tems remains constant with an increase of transmitted optical power. Thisis because both coherent POLMUX and subcarrier IM/DD systems havethe same slope of 0.66 bits/s/Hz, and the coherent systems have a slope of0.33 bits/s/Hz with respect to the dB values of average transmitted opticalpower.5.5 SummaryIn this chapter, highly accurate series ergodic capacity expressions aredeveloped for the different OWC systems namely, coherent detection withand without POLMUX and subcarrier IM/DD employing adaptive trans-mission schemes. The asymptotic analysis of the developed series capacityexpressions provides useful engineering insights at high SNR regimes. Thechannel capacity comparison among different OWC systems subject to thesame average transmitted optical power reveals that even in the presence of1095.5. Summarypolarization control error the coherent POLMUX OWC system significantlyoutperforms the coherent and subcarrier IM/DD OWC systems.110Chapter 6ConclusionsIn this chapter, we summarize the contributions of this dissertation, andpropose some future work related to the adaptive OWC systems.6.1 Summary of ContributionsFor OWC systems, SIM is a suitable alternative to the conventional OOKwith fixed detection threshold. On the other hand, due to slow fading na-ture of atmospheric turbulence channel, adaptive transmission is an efficientfading mitigation solution for the OWC systems. In this thesis, we haveanalyzed the performance of the non-adaptive and adaptive SIM over thecommonly used atmospheric channels. In addition, performance compari-son among different adaptive OWC systems has been presented. Specificcontributions are summarized as follows.In Chapter 3, we have developed highly accurate ASER expression usingseries solution for a SIM OWC communication system employing generalorder rectangular QAM. In particular, exact series ASER expressions aredeveloped for the Gamma-Gamma, the K-distributed and the negative ex-ponential channels along with truncation error and asymptotic error rateanalysis. Our truncation error analysis shows that the developed seriesASER expressions converge rapidly to the exact error rates, especially inlarge SNR regimes. From the asymptotic error rate analysis the diversityorder of the considered system is obtained as ?/2.In Chapter 4, we have developed highly accurate ASE, BER and outageprobability expressions in terms of series solutions for both non-adaptive andadaptive SIM systems employing R-QAM and M -PSK over the Gamma-Gamma turbulence channels. Our truncation error analysis demonstrates1116.2. Future Workthat our series BER solutions rapidly converge to the exact BER, especiallyin large SNR regimes. Our asymptotic BER analysis shows that the diversityorder of the both non-adaptive and adaptive system depends only on thesmaller channel parameter ?. Our numerical results show that the R-QAMbased adaptive SIM offers improved ASE and BER performance comparedto the M -PSK based adaptive SIM.In Chapter 5, we have developed highly accurate series ergodic capacityexpressions for the subcarrier IM/DD and coherent OWC systems with andwithout POLMUX employing different adaptive transmission schemes. Inaddition, asymptotic ergodic capacity expressions are developed which fa-cilities rapid estimation of channel capacity in large SNR regimes. Besides,asymptotic capacity also provides some important insights into the capacityperformance of the OWC systems in large SNR regimes. Our ergodic capac-ity analysis reveals that the coherent POLMUX OWC systems significantlyoutperform the coherent and SIM OWC systems even in the presence ofpolarization control error.6.2 Future WorkFor the presented constant-power, variable-rate adaptive SIM the trans-mission parameters are adapted based on instantaneous received SNR. Suchadaption requires an increased feedback rate. Due to slow fading natureof atmospheric turbulence, link adaption based on average SNR is also aninteresting future research option. Such adaptive transmission will reducethe feedback rate, and will lead to a less complex transmission scheme.For the analysis of adaptive transmissions, we assumed that the channelis perfectly known at the transmitter. However, this is an ideal assumptionsince in practical transmission scenarios channel estimation can be erro-neous. Hence, the impact of imperfect channel estimation on the perfor-mance of adaptive SIM is also an interesting research topic. 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Cressie, A. S. Davis, J. L. Folks, and G. E. Policello II,?Themoment-generating function and negative integer moments,? The Amer-ican Statistician, vol. 35, pp. 148-150, Aug. 1981. ? pages 90, 137[78] J. M. Hunt, F. Holmes, and F. Amizajerdian, ?Optimum local oscialltorlevels for coherent detection using photodetectors,? Appl. Opt., vol. 27,pp. 3135-3141, Aug. 1988. ? pages 101[79] P. Galambos and I. Simonelli, Products of Random Variables: Applica-tions to Problems of Physics and to Arithmetical Functions. NewYork:Marcel Dekker Inc., 2004. ? pages 137123Appendix124Appendix AASER Analysis of SubcarrierR-QAM Using DirectIntegration ApproachFrom (4.4) the conditional error probability for MI ?MQ rectangularQAM can be written as (for r 6= 1)Pe(I) =(1?1MI)erfc(AI??2I)+1MI(1?1MQ)erfc(AQ??2I)+(1?1MI)(1?1MQ)erfc(AQ??2I)erf(AI??2I).(A.1)To obtain (A.1), we have used the equities Q(x) = 12erfc(x2 ) and erfc(x) =1?erf(x). Averaging (A.1) with respect to the PDF of the Gamma-GammaRV in (2.15), we obtain the average ASER for subcarrier QAM as Pe =e1A + e2A + e3A, where e1A, e2A and e3A are given bye1A =(1?1MI)? ?0erfc(AI??2I)fI(I) dI (A.2)ande2A =1MI(1?1MQ)? ?0erfc(AQ??2I)fI(I) dI (A.3)125Appendix A. ASER Analysis of Subcarrier R-QAM Using Direct Integration Approachande3A =(1?1MI)(1?1MQ)? ?0erfc(AQ??2I)erf(AI??2I)fI(I) dI.(A.4)Applying (2.15) and (3.5) to (A.2), e1A can be evaluated ase1A =(1?1MI)B(?? ?, 1? ?+ ?)?pi?(?)?(?)???p=0[bp(?, ?)A?(p+?)I(?2)? p+?2?bp(?, ?)A?(p+?)I(?2)? p+?2](A.5)where B(x, y) is the Beta function [63, eq. 8.384(1)] and bp(x, y) is definedbybp(x, y) =(xy)p+y?(p? x+ y + 1)p!?(p+y+12 )p+ y. (A.6)Similarly, we can derive e2A ase2A =1MI(1?1MQ)B(?? ?, 1? ?+ ?)?pi?(?)?(?)???p=0[bp(?, ?)A?(p+?)Q(?2)? p+?2?bp(?, ?)A?(p+?)Q(?2)? p+?2].(A.7)In obtaining (A.5) and (A.7), we have used the following integral identity[65, eq. 06.27.21.0132.01]? ?0ta?1erfc(t) dt =1?pia?(a+ 12). (A.8)126Appendix A. ASER Analysis of Subcarrier R-QAM Using Direct Integration ApproachSubstituting (2.15) and (3.5) into (A.4), we obtain e3A ase3A =(1?1MI)(1?1MQ)B(?? ?, 1? ?+ ?)?(?)?(?)???p=0[wp(?, ?)(?2)? p+?2? wp(?, ?)(?2)? p+?2] (A.9)where we have used the following integral identity [66, eq. 3.9(16)]? ?0xp?1erfc(ax)erf(bx)dx =2bap+1?(1 + p2)pi(p+ 1)? 2F1(12,p+ 12,p+ 22;32,p+ 32;?b2a2), a > b, p > ?1(A.10)and where 2F1(? , ? , ? ; ? , ? ; ?) is the generalized Hypergeometric function [63,eq. 9.111], and wp(x, y) is defined aswp(x, y) =(xy)p+y?(p? x+ y + 1)p!2AIAp+y+1Q?(1 + p+y2 )pi(p+ y + 1)? F(12,p+ y + 12,p+ y + 22;32,p+ y + 32;?A2IA2Q).(A.11)In obtaining (A.5), (A.7) and (A.9) we have assumed r > 1. However,exchanging MI with MQ and AI with AQ in above results, it is also possibleto obtain ASER of subcarrier QAM when r < 1. For MI ?MQ rectangularQAM with r = 1, we have AI = AQ. Subsequently the conditional errorrate can be written asPe(I) = 2(2?1MI?1MQ)Q(AI??I)? 4(1?1MI)(1?1MQ)Q2(AI??I).(A.12)Averaging (A.12) with respect to the PDF of the Gamma-Gamma turbu-lence channel, we find ASER as Pe = e4A ? e5A, where e4A and e5A are127Appendix A. ASER Analysis of Subcarrier R-QAM Using Direct Integration Approachdefined ase4A = 2(2?1MI?1MQ)? ?0Q(AI??I)fI(I) dI (A.13)ande5A = 4(1?1MI)(1?1MQ)? ?0Q2(AI??I)fI(I) dI. (A.14)Using a procedure similar to (A.5)-(A.7), e4A can be evaluated ase4A =(2?1MI?1MQ)B(?? ?, 1? ?+ ?)?pi?(?)?(?)???p=0[bp(?, ?)A?(p+?)I(?2)? p+?2?bp(?, ?)A?(p+?)I(?2)? p+?2].(A.15)In order to evaluate e5A, we use the following alternative expression forsquare of the Gaussian Q-function [64, eq. 4.9]Q2(x) =1pi? pi40exp(?x22 sin2 ?)d?. (A.16)Applying (2.15), (3.5) and (A.16) to (A.14) and using an integral identity[63, eq. 3.326(2)], we evaluate e5A ase5A =(1?1MI)(1?1MQ)2B(?? ?, 1? ?+ ?)pi?(?)?(?)???p=0[vp(?, ?)A?(p+?)I(?2)? p+?2?vp(?, ?)A?(p+?)I(?2)? p+?2](A.17)128Appendix A. ASER Analysis of Subcarrier R-QAM Using Direct Integration Approachwhere vp(x, y) is defined asvp(x, y) =(xy)p+y?(p? x+ y + 1)p!?(p+ y2)?F (12 ,p+y+12 ;p+y+32 ;12)2p+y+12 (p+ y + 1).(A.18)In deriving of (A.17) we have also used the following integral property [25]? pi40sinx ? d? =F (12 ,x+12 ;x+32 ;12)2x+12 (x+ 1). (A.19)Note that (A.19) can also be obtained from (3.15) by setting p = 0 and? = 14 .129Appendix BTruncation Error Analysisfor the BER of R-QAMBased Adaptive SIMWe define the truncation error of ?1 in (4.37)?1(K) =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)??p=K+1[a?p(?, ?)wp(?n, ?n+1, ?, i, In, Jn)?? p+?2?a?p(?, ?)wp(?n, ?n+1, ?, i, In, Jn)?? p+?2].(B.1)130Appendix B. Truncation Error Analysis for the BER of R-QAM Based Adaptive SIMIn order to facilitate the truncation error analysis, we rewrite (B.1) as?1(K) =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)??p=K+11p!(????n+1??)p? [u1,p(?, ?, i, ?n+1)? u1,p(?, ?, i, ?n+1)]?1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)??p=K+11p!(????n??)p? [u1,p(?, ?, i, ?n)? u1,p(?, ?, i, ?n)]?1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)??p=K+11p!(??(2i+ 1)D(In, Jn)??)p? [u2,p(?, ?, i, ?n+1)? u2,p(?, ?, i, ?n+1)]+1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In)??p=K+11p!(??(2i+ 1)D(In, Jn)??)p? [u2,p(?, ?, i, ?n)? u2,p(?, ?, i, ?n)](B.2)whereu1,p(x, y, z, ?) =(xy????)y erfc((2z + 1)D(In, Jn)??)?(p? x+ y + 1)(p+ y)(B.3)andu2,p(x, y, z, ?) =(xy(2z + 1)D(In, Jn)??)y ?(p+y+12 , (2z + 1)2D2(In, Jn)?)?pi?(p? x+ y + 1)(p+ y).(B.4)131Appendix B. Truncation Error Analysis for the BER of R-QAM Based Adaptive SIMUsing the Taylor series expansion of the exponential function, we obtain anupper bound for the truncation error of ?1 asup?1 (K) =1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In) exp(????n+1??)?maxp>K????u1,p(?, ?, i, ?n+1)? u1,p(?, ?, i, ?n+1)?????1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In) exp(????n??)?maxp>K????u1,p(?, ?, i, ?n)? u1,p(?, ?, i, ?n)?????1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In) exp(??(2i+ 1)D(In, Jn)??)?maxp>K????u2,p(?, ?, i, ?n+1)? u2,p(?, ?, i, ?n+1)????+1Inlog2 In?k=1(1?2?k)In?1?i=0?i(k, In) exp(??(2i+ 1)D(In, Jn)??)?maxp>K????u2,p(?, ?, i, ?n)? u2,p(?, ?, i, ?n)????.(B.5)132Appendix B. Truncation Error Analysis for the BER of R-QAM Based Adaptive SIMSimilarly, we obtain an upper bound for the truncation error of ?2 in (4.37)asup?2 (K) =1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?i(l, Jn) exp(????n+1??)?maxp>K????u1,p(?, ?, j, ?n+1)? u1,p(?, ?, j, ?n+1)?????1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?i(l, Jn) exp(????n??)?maxp>K????u1,p(?, ?, j, ?n)? u1,p(?, ?, j, ?n)?????1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?i(l, Jn) exp(??(2j + 1)D(In, Jn)??)?maxp>K????u2,p(?, ?, j, ?n+1)? u2,p(?, ?, j, ?n+1)????+1Jnlog2 Jn?l=1(1?2?l)Jn?1?j=0?i(l, Jn) exp(??(2j + 1)D(In, Jn)??)?maxp>K????u2,p(?, ?, j, ?n)? u2,p(?, ?, j, ?n)????.(B.6)Hence, we obtain an upper bound for the truncation error ?Pb?QAMn asadpK,QAM ?B(?? ?, 1? ?+ ?)?pi?(?)?(?)1log2(In ? Jn)(up?1 (K) + up?2 (K)). (B.7)From (B.3) and (B.4), we observe that both u1,p(x, y, z, ?) and u2,p(x, y, z, ?)decrease with an increase of p and become zero when p approaches infin-ity. Consequently, maximum values of |u1,p(?, ?, z, ?)? u1,p(?, ?, z, ?)| and|u2,p(?, ?, z, ?) ? u2,p(?, ?, z, ?)| exist where z is either i or j, and ? is ei-ther ?n+1 or ?n. As a result, we conclude that the truncation error adpK,QAMdiminishes with increasing K values. Also, by inspecting (B.5)-(B.7), thetruncation error rapidly diminishes with an increase of SNR ?.133Appendix CTruncation Error Analysisfor the BER of AdaptiveM-PSK SIMThe truncation error of e1 in (4.45) due to elimination of infinite termsafter the first K + 1 terms is upper bounded by e1(K) ? eup1 (K) whereeup1 (K) is the truncation error of eup1 in (4.50) due to the elimination ofinfinite terms after the first K + 1 terms. We define the truncation error ofeup1 aseup1 (K) =??p=K+1[a?p(?, ?)uupp (?,Mn, ?n, ?n+1)?? p+?2?a?p(?, ?)uupP (?,Mn, ?n, ?n+1)?? p+?2].(C.1)In order to facilitate the truncation error analysis, we rewrite (C.1) aseup1 (K) =??p=K+11p!(??A(k,M)??)p[sp(?, ?, ?n, ?n+1)? sp(?, ?, ?n, ?n+1)](C.2)134Appendix C. Truncation Error Analysis for the BER of Adaptive M -PSK SIMwhere sp(x, y, ?L, ?U ) is defined assp(x, y, ?L, ?U ) =(xyA(k,M)??)y?(?(p+y2 , A2(k,M)?L)? ?(p+y2 , A2(k,M)?U))?(p? x+ y + 1)gp(mk, y).(C.3)By Taylor series expansion of the exponential function we can simplify thesummation term in (C.2) and obtain an upper bound for the truncationerror of eup1 asueup1(K) = exp(??A(k,M)??)?maxp>K????sp(?, ?, ?n, ?n+1)?sp(?, ?, ?n, ?n+1)????.(C.4)From (C.3), it can be shown that there is a po ? K such that bothsp(?, ?, ?n, ?n+1) and sp(?, ?, ?n, ?n+1) monotonically decrease if p > po.For sufficiently large K values (K ? 30 is sufficient for typical ? and ? val-ues), po = K holds. Consequently, the maximum value of????sp(?, ?, ?n, ?n+1)?sp(?, ?, ?n, ?n+1)???? exists; therefore, we conclude the truncation error eup1 (K)diminishes with increasing K values. Also, by inspecting (C.4), we observethat the truncation error rapidly decreases with an increase of SNR ?. Sim-ilarly we can show that the truncation error of e2 is upper bounded bye2(K) ? eup2 (K) ? ueup2(K), where ueup2(K) is defined asueup1(K) = exp(??B(k,M)??)?maxp>K????rp(?, ?, ?n, ?n+1)? rp(?, ?, ?n, ?n+1)????(C.5)whererp(x, y, ?L, ?U ) =(xyB(k,M)??)y?(?(p+y2 , B2(k,M)?L)? ?(p+y2 , B2(k,M)?U))?(p? x+ y + 1)gp(nk, y).(C.6)135Appendix C. Truncation Error Analysis for the BER of Adaptive M -PSK SIMFrom (C.5) we can also show that the truncation error of e2 rapidly decreaseswith increase of K values and/or SNR ?. This suggests our series solutionsare highly accurate in large SNR regimes.136Appendix DMellin Transformation of theGamma-Gamma RandomVariableMellin transformation of a RV I is defined as MI(z) = E[Iz?1] =??0 Iz?1fI(I) dI. Assuming I is a Gamma-Gamma RV, we have shownI = XY where X and Y are two independent Gamma RVs with PDFs re-spectively given by (2.12) and (2.13). Since the Mellin transformation ofa product of two independent RVs is a product of Mellin transformationsof the two RVs [79], we obtain MI(z) = MX(z)MY (z). Using an integralidentity [63, eq. 3.326(2)], we obtain the Mellin transformations of RVs Xand Y , respectively, asMX(z) =?(?+ z ? 1)?(?)1?z?1(D.1)andMY (z) =?(? + z ? 1)?(?)1?z?1. (D.2)Finally, the Mellin transformation of I becomesMI(z) =?(?+ z ? 1)?(? + z ? 1)?(?)?(?)(1??)z?1. (D.3)It follows that E[I?1] and E[I?2], which are required by (5.6) and (5.42),can be obtained by letting z = 0 and z = ?1 in (D.3) [77].137

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