Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Free-space optical communication systems using on-off keying in atmospheric turbulence Yang, Luanxia 2015

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2015_september_yang_luanxia.pdf [ 1.02MB ]
Metadata
JSON: 24-1.0166401.json
JSON-LD: 24-1.0166401-ld.json
RDF/XML (Pretty): 24-1.0166401-rdf.xml
RDF/JSON: 24-1.0166401-rdf.json
Turtle: 24-1.0166401-turtle.txt
N-Triples: 24-1.0166401-rdf-ntriples.txt
Original Record: 24-1.0166401-source.json
Full Text
24-1.0166401-fulltext.txt
Citation
24-1.0166401.ris

Full Text

FREE-SPACE OPTICALCOMMUNICATION SYSTEMS USINGON-OFF KEYING IN ATMOSPHERICTURBULENCEbyLuanxia YangB.Eng., Beijing University of Chemical Technology, P. R. China, 2009M.Sc., The University of Edinburgh, United Kingdom, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)July 2015c Luanxia Yang, 2015AbstractIn this thesis, we focus on the bit-error rate (BER) performance improvements for free-space opti-cal (FSO) communication links operating over atmospheric turbulence channels using on-off key-ing (OOK). Laser beams employed in these links are subject to scintillation, during their propaga-tion through atmospheric channels, and this can lead to significant BER performance degradation.Such systems can suffer from irreducible error floors that result from the use of demodulationwith fixed and unoptimized detection thresholds. The resulting error floors are analyzed for thegeneral case of low and high state offsets (i.e., nonzero extinction ratios). To improve the BERperformance, there are three techniques developed in this thesis.The first technique employs electrical signal-to-noise ratio (SNR) optimized detection. Thesystem uses the electrical SNRs to implement adaptive detection thresholds and eliminate the errorfloors. The system can accommodate operation with nonzero extinction ratios, as it uses the methodof moments and maximum likelihood estimation techniques to estimate the low and high stateoffsets and electrical SNR.The second technique employs source information transformation. Using source informationtransformation can also eliminate error floors, and it can detect the OOK signal without knowledgeof the instantaneous channel state information and probability density function of the turbulencemodel. It is shown that source information transformation can achieve comparable performance tothe idealized adaptive detection system, with greatly reduced implementation complexity.The third technique employs convolutional code. Using convolutional code can mitigate theeffects of turbulence induced fading. The BER performance is analyzed for FSO systems usingconvolutional code and OOK. Through our analysis, it is shown that using convolutional code canimprove the BER performance of an FSO system significantly.iiPrefaceThis thesis is based on [J1-J2, SJ1, SJ2]. My supervisors, Dr. Julian Cheng and Dr. JonathanF. Holzman, co-authored all of the publications and supervised all of my research work. I amresponsible for all theories derived, as well as the manuscript preparation and revisions.Refereed Journal PublicationsJ1. L. Yang, J. Cheng, and J. F. Holzman, “Performance of convolutional coded subcarrier inten-sity modulation over Gamma-Gamma turbulence channels,” IEEE Communications Letters,vol. 17, pp. 2332-2335, December 2012.J2. L. Yang, J. Cheng, and J. F. Holzman, “Maximum likelihood estimation of the lognormal-Rician FSO channel model,” IEEE Photonics Technology Letters, Accepted for publicationin IEEE Photonics Technology Letters, 2015.Refereed Conference PublicationsC1. L. Yang, J. Cheng, and J. F. Holzman,“Optical communications over lognormal fading chan-nels using OOK,” Proceedings of the International Workshop on Optical Wireless Commu-nications (IWOW), Newcastle Upon Tyne, UK, October, 2013.C2. L. Yang, J. Cheng, and J. F. Holzman, “Electrical-SNR-optimized detection thresholds forOOK IM/DD optical wireless communications,” Proceedings of the IEEE Canadian Work-shop on Information Theory (CWIT), pp. 186-189, Toronto, Canada, June, 2013.iiiPrefaceC3. L. Yang, J. Cheng, and J. F. Holzman, “Performance of convolutional coded OOK IM/DDsystems over strong turbulence channels,” Proceedings of the IEEE International Conferenceon Computing, Networking and Communications (ICNC), pp. 35-39, San Diego, USA,January, 2013.Refereed Journal Publications (submitted)SJ1. L. Yang, J. Cheng, and J. F. Holzman, “Estimation of electrical SNR for FSO communica-tions,” IEEE Photonics Technology Letters, submitted for publication.SJ2. L. Yang, J. Cheng, and J. F. Holzman, “Optical wireless communications over lognormalfading channels using OOKwith nonzero extinction ratios,” IEEE/OSA Journal of LightwaveTechnology, submitted for publication.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Organization and Contributions . . . . . . . . . . . . . . . . . . . . . . . 52 OOK IM/DD System and Turbulence Channel Models . . . . . . . . . . . . . . . . 82.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Additive Noise at the Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13vTable of Contents2.3 Atmospheric Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Lognormal Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 K-distributed Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Negative Exponential Turbulence . . . . . . . . . . . . . . . . . . . . . . 162.3.4 Gamma-Gamma Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.5 Lognormal-Rician Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 OOK IM/DD Systems with Nonzero Extinction Ratios and Electrical-SNR-OptimizedDetection Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 OOK with Fixed and Unoptimized Detection Thresholds . . . . . . . . . . . . . . 193.2 OOK with Electrical-SNR-Optimized Detection Thresholds . . . . . . . . . . . . 233.2.1 Electrical-SNR-Optimized Detection Based on a Known Turbulence pdf . 243.2.2 Electrical-SNR-Optimized Detection Based on an Unknown Turbulencepdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Channel Parameters and Electrical SNR Estimation . . . . . . . . . . . . . . . . . . 364.1 MLE for the Lognormal-Rician Shaping Parameter Estimation . . . . . . . . . . . 364.2 Electrical SNR Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Method of Moments Estimation . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . 434.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 OOK IM/DD System with Source Information Transformation . . . . . . . . . . . . 545.1 System and Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 The Probability Density Function of the Detection Threshold . . . . . . . . . . . . 58viTable of Contents5.3 The Upper Bound on the Average BER . . . . . . . . . . . . . . . . . . . . . . . 625.3.1 The Error Caused In The Detection At The Receiver . . . . . . . . . . . . 625.3.2 Average BER of the Output Binary Sequence . . . . . . . . . . . . . . . . 655.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 OOK IM/DD Systems with Convolutional Code . . . . . . . . . . . . . . . . . . . . 746.1 Bit-By-Bit Interleaved Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.1.1 Pairwise Codeword Error Probability Calculation . . . . . . . . . . . . . . 746.1.2 Truncation Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 776.1.3 Asymptotic Analysis of PEP . . . . . . . . . . . . . . . . . . . . . . . . . 786.1.4 Upper Bound on Average BER . . . . . . . . . . . . . . . . . . . . . . . 786.2 Quasi-static Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.1 Summary of Accomplished Work . . . . . . . . . . . . . . . . . . . . . . . . . . 887.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Appendix A: The CF and MGF od lognormal pdf . . . . . . . . . . . . . . . . . . . 99viiList of Tables3.1 Error floor expressions for FSO systems employing fixed detection thresholds ofTth = (1+x )E[I] over a lognormal fading channel with s = 0.25. . . . . . . . . . 223.2 Error floor expressions for various turbulence channel models. . . . . . . . . . . . 233.3 Comparison of detection thresholds using an exact and approximated lognormalpdf with s = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1 The conditional probability of the received (2M  1)-nary number is hˆl given thetransmitted (2M1)-nary number is hl . . . . . . . . . . . . . . . . . . . . . . . . 675.2 The conditional probability of the received (2M  1)-nary number is hˆl given thetransmitted (2M1)-nary number is hl . . . . . . . . . . . . . . . . . . . . . . . . 68viiiList of Figures2.1 Block diagram of an FSO system through an atmospheric turbulence channels. . . . 93.1 The likelihood functions f (r|s0) and f (r|s1) with s = 0.25 and x = 0.2 when g = 2dB and g = 8 dB. The likelihood functions are a result of the convolution of thelognormal pdf and the Gaussian pdf. . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 BERs of OOK modulated systems using fixed detection thresholds Tth, electrical-SNR-optimized detection thresholds and adaptive detection thresholds over a log-normal turbulence channel with s = 0.25 and x = 0. . . . . . . . . . . . . . . . . 293.3 BERs of OOK modulated systems using fixed detection thresholds Tth, electrical-SNR-optimized detection thresholds and adaptive detection thresholds over a log-normal turbulence channel with s = 0.25 and x = 0.2. . . . . . . . . . . . . . . . 303.4 Comparison of an approximated pdf using J = 3 sample moments and an exact pdffor a lognormal fading channel with s = 0.25. . . . . . . . . . . . . . . . . . . . . 323.5 The absolute error between the approximated pdf and the exact lognormal pdf withs = 0.25 and J = 3,6,10 sample moments. . . . . . . . . . . . . . . . . . . . . . 333.6 Comparison of BERs obtained by the approximated pdf and the exact lognormalpdf with x = 0 and J = 3 sample moments. . . . . . . . . . . . . . . . . . . . . . 344.1 MSE and NMSE performance of the maximum likelihood estimators for the lognormal-Rician parameters r and s2z with s2z = 0.25. . . . . . . . . . . . . . . . . . . . . . 504.2 MSE and NMSE performance of the maximum likelihood estimators for the lognormal-Rician parameters r and s2z with r = 4. . . . . . . . . . . . . . . . . . . . . . . . . 51ixList of Figures4.3 Comparison of MoME and MLE normalized sample variance for different train-ing sequence lengths over a lognormal turbulence channel with s = 0.25. Thenormalized MSE is computed over M = 1⇥104 trials. . . . . . . . . . . . . . . . 524.4 Comparison of BERs obtained by the estimated electrical SNR and the exact elec-trical SNR with x = 0 and J = 3 sample moments. . . . . . . . . . . . . . . . . . 535.1 Block diagram of the transmitter for the system using source information transfor-mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Block diagram of the source information transformation. . . . . . . . . . . . . . . 575.3 Block diagram of the receiver for the system using source information transformation. 595.4 Comparison of the derived and simulated pdfs for the detection threshold Tth overa lognormal fading channel with s = 0.25 and M = 3. . . . . . . . . . . . . . . . 725.5 The simulated BER and BER upper bounds of the system using source informationtransformation over lognormal turbulence channels (with s = 0.25, s = 0.5, x = 0and M = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.1 Block diagram of a coded FSO system through quasi-static atmospheric turbulencechannels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 The PEP of coded IM/DD systems using OOK and SIM-BPSK versus averageSNR operating over Gamma-Gamma turbulence channels. Results are shown forweak (a = 4.62,b = 4.24) and strong (a = 2.14,b = 1.21) turbulence conditionsusing series, exact, and approximate solutions. . . . . . . . . . . . . . . . . . . . . 856.3 The BER of uncoded and coded IM/DD systems (with perfect interleaving) andupper bounds on average BER of convolutional coded (Rc = 1/2,Kc = 3) OOKIM/DD systems versus average SNR over Gamma-Gamma turbulence channels.Results are for weak (a = 4.62,b = 4.24) and strong (a = 2.14,b = 1.21) turbu-lence conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86xList of Figures6.4 The simulated BER and upper bounds (dmax = 40) on average BER of terminatedconvolutional coded (Rc = 1/2,Kc = 3,B = 999998) OOK IM/DD systems ver-sus average SNR over quasi-static Gamma-Gamma turbulence channels with andwithout block interleaving. Results are for weak (a = 4.62,b = 4.24) and strong(a = 2.14,b = 1.21) turbulence conditions. . . . . . . . . . . . . . . . . . . . . . 87xiList of AcronymsAcronyms DefinitionsAWGN Additive White Gaussian NoiseBER Bit-Error RateBPSK Binary Phase-Shift KeyingCDF Cumulative Distribution FunctionCF Characteristic FunctionCRLB Cramer-Rao Lower BoundCSI Chanel State InformationEM Expectation-MaximizationFSO Free-Space OpticalGbit/s Gigabit per Secondi.i.d. Independent and Identically DistributedIM/DD Intensity Modulation with Direct DetectionMGF Moment Generating FunctionMLE Maximum Likelihood EstimationMoME Method of Moments EstimationMSE Mean Squared ErrorxiiList of AcronymsNMSE Normalized Mean Squared ErrorOOK On-Off KeyingOWC Optical Wireless Communicationspdf Probability Density FunctionPEP Pairwise Error ProbabilityPPM Pulse Position ModulationRF Radio FrequencyRV Random VariableSI Scintillation IndexSIM Subcarrier Intensity ModulationSNR Signal-to-Noise RatioxiiiList of SymbolsSymbols Definitions4 f Effective noise bandwidth of a receivers2R The Rytov varianceC2n The index of refraction structure parameters2si The scintillation indexln(·) The log function with base eG(·) The Gamma functionKn(·) The modified Bessel function of the second kind oforder nE[·] The statistical expectation operationVar[·] The statistical variance operationCov[·, ·] The statistical covariance operationd (·) The Dirac delta functionp! The factorial of a positive integer pZ The set of all integerserfc(·) The complementary error functionQ(·) The Gaussian Q-functionxivList of SymbolsIn(·) The modified Bessel function of the first kind withorder nmFn(· · · ; · · · ; ·) The generalized Hypergeometric functionB(·, ·) The Beta functionNnThe generalized binomial coefficientR The set of all real numbersFR(·) The CF of a RV Rx⇤ y The convolution of x and yx y The exclusive OR gate with inputs x and yRe[·] The real part of a numberIm[·] The imaginary part of a numberxvAcknowledgementsI am deeply grateful to my supervisors Dr. Julian Cheng and Dr. Jonathan F. Holzman for theirenthusiasm, guidance, advice, encouragement, support, and friendship. It is my honor to study andresearch under their supervision. I will continue to be influenced by their rigorous scholarship,clarity in thinking, and professional integrity.I would like to thank Dr. Yunfei Chen from the University of Warwick in United Kingdom forserving as my external examiner. It is my great honor to have such an expert on my committee. Iwould also like to thank Dr. Thomas Johnson, Dr. Jahangir Hossian and Dr. Shawn Wang for theirwillingness to serve on the thesis examination committee. I really appreciate their valuable timeand constructive comments on my thesis.I owe many people for their generosity and support during my Ph.D. study at the University ofBritish Columbia. I would like to thank my dear colleagues for sharing their academic experiencesand constructive viewpoints generously with me during our discussions. I would also like to thankmy dear friends for sharing in my excitement and encouraging me when I was frustrated duringthis journey.Finally, I would like to thank my parents for their patience, understanding, support, and loveover all these years. All my achievements would not have been possible without their constantencouragement and support.xviChapter 1Introduction1.1 Background and MotivationOptical wireless communication (OWC) links have certain advantages over radio frequency links.Examples of such advantages include low deployment cost, high link security, and freedom ofspectral license regulations. An outdoor OWC system, in particular, uses free-space as the trans-mission medium, and is also known as free-space optical (FSO) communication [1]. However,FSO communication can suffer from fog/cloud coverage and harsh weather conditions. These at-mospheric effects can degrade the system reliability and performance [2]. Ultimately, rain, snow,sleet, fog, dust, heat, etc. can affect our viewing of distant objects, and these factors can also affectthe transmission of laser beams through the atmosphere.Absorption, scattering, and refractive-index fluctuations (i.e., optical turbulence1) are threeprimary atmospheric processes that affect optical wave propagation through the atmosphere. Typi-cally, absorption and scattering are grouped together under the topic of extinction, which is definedby a reduction or attenuation in the amount of radiation passing through the atmosphere. Botheffects are well characterized and can be effectively modeled and compensated for by softwarepackages such as LOWTRAN, FASCODE, MODTRAN, HITRAN, and LNPCWIN as a functionof wavelength l [3]. On the other hand, optical turbulence is a random effect, which is generallyconsidered to be the most serious optical effect on a propagating laser beam through the atmo-sphere.It is well known that the performance of FSO systems can be significantly degraded by turbulence-1Optical turbulence is a subset of atmospheric turbulence. However, we will not distinguish between these twoterms and use them interchangeably in this thesis.11.1. Background and Motivationinduced fading. Turbulence-induced fading can lead to power losses at the photodetector and ran-dom fluctuations of the received signal. The performance degradation is especially pronouncedfor FSO systems using irradiance modulation and direct detection (IM/DD) with on-off keying(OOK), and it is these systems that are the primary focus of this thesis.In this thesis, we focus on bit-error rate (BER) performance improvements for FSO commu-nication links operating with OOK over atmospheric turbulence channels. To improve the BERperformance, there are three techniques developed in this thesis.The first technique, for improving the BER performance of FSO links using OOK, employselectrical signal-to-noise ratio (SNR) optimized detection. The electrical-SNR-optimized detec-tion requires the perfect knowledge of the probability density function (pdf) of the turbulenceand electrical SNR. Such knowledge is often quantified by way of mathematical models for theturbulence-induced fading [4], [5]. Among the turbulence-induced fading2 models introduced sofar, it has been well accepted that the lognormal distribution characterizes FSO fading channelsunder weak turbulence conditions over several hundred meters, or longer, depending on the tem-perature, wind strength, altitude, humidity, and atmospheric pressure [6], [7]. The K-distributioncharacterizes FSO fading channels under strong turbulence conditions over several kilometres [8].The negative exponential distribution characterizes FSO fading channels in the limiting case ofsaturated scintillation [9]. There also exist generalized models for use over a broad range of weak-to-strong turbulence conditions. The Gamma-Gamma distribution is used, but it can underestimateeffects of small- and large-scale scintillations and can suffer from decreased accuracy [10]. Thelognormal-Rician distribution can be used, and it has been found to offer two advantages. First,heuristic analyses of wave propagation through turbulence show that the lognormal-Rician fadingdistribution accurately characterizes experimental data [11]. Second, the lognormal-Rician fadingdistribution is highly adaptable over a wide range of weak-to-strong turbulence conditions throughits parameters [12]. However, the application of the lognormal-Rician fading distribution has beenlimited, as it does not have a tractable closed-form pdf.2In the rest of the thesis, turbulence-induced fading is referred as (atmospheric) turbulence for simplicity unlessstated otherwise.21.1. Background and MotivationGiven the potential of the lognormal-Rician distribution, for accurately characterizing FSOfading channels, there have been efforts to characterize this distribution with estimated shapingparameters. In [13], the authors applied a physical model of turbulence-induced scattering to es-timate the shaping parameters of the lognormal-Rician fading distribution. It should be noted,however, that this approach depends heavily upon estimated parameters in a physical model ofthe turbulence-induced scattering, and such parameters are often either unavailable or lacking inaccuracy. For computational simplicity, the authors applied the Tatarskii model to characterizerefractive index fluctuations and geometrical optics to characterize turbulent eddies, but the un-derlying assumptions of this approach can lead to notable inaccuracies, as discussed in [11]. In[14], the authors introduced the generalized method of moments approach to estimate the shapingparameters of the lognormal-Rician distribution. It should be noted, however, that this approachdemands a large number of data samples, on the order of 106 data samples, and this impedes itsimplementation in FSO communications. For a standard FSO link, experiencing quasi-static tur-bulence fading on a typical millisecond timescale, the system would exhibit latency on the order of1⇥106 millisecond = 1000 seconds. This duration is unacceptably long for FSO communications,as typical FSO channels exhibit stationary statistics, i.e., constant channel model parameters, on thetimescale of several minutes. Clearly, FSO systems applying channel estimation with a lognormal-Rician fading distribution need a more rapid estimation of the shaping parameters.The above turbulence models can be used to characterize FSO channels, for the implementa-tion of adaptive detection thresholds in OOK IM/DD systems. FSO systems operating withoutsuch adaptive detection thresholds would simply apply fixed detection thresholds, and such mod-ulation is often adopted by current commercial FSO products. In [2], [15], the authors studiedthe performance of OOK IM/DD systems using fixed detection thresholds through atmosphereturbulence channels, and it was found that these unoptimized systems suffered from irreducibleerror floors. Existing commercial FSO communication systems have employed high transmissionpowers to overcome the effects of atmospheric turbulence, but this practice results in high cost.With this in mind, there have been many recent efforts to implement OOK IM/DD systems with31.1. Background and Motivationeffective adaptive detection. In [2], [16], the authors applied adaptive detection with assumed per-fect knowledge of the instantaneous channel state information (CSI), as the instantaneous SNR isused to detect each data symbol. In this case, the receiver merely requires computation of a simplemathematical expression, and the BER remains at a minimal value. However, perfect knowledgeof CSI is challenging to realize in practice. In [17], the authors investigated blind detection, i.e.,detection assuming the absence of instantaneous CSI and a statistical channel description at the re-ceiver, for OOK in a FSO system. This method leads to a decision delay as the receiver is requiredto compute the detection threshold using all the received statistics. In [18], [19], the authors con-sidered sequence detection for OOK in an FSO system, in which block-wise decisions are madeusing an observation window of N bit intervals. Unfortunately, such an algorithm had significantcomputational requirements. In general, the above methods have obvious practical concerns forOOK IM/DD operation with nanosecond data symbol durations (i.e., Gbps rates) and millisecondturbulence coherence times, as rapid detection threshold adjustments are needed on the timescaleof the millisecond turbulence coherence times [2], [17], [19], [20].To accommodate practical concerns for adaptive detection, the electrical-SNR-optimized de-tection system was proposed in [21], [22]. Such a system offers a compromise between the practi-cal advantages of operation with fixed detection threshold, as only slow adaptations are needed todefine the detection thresholds, and the performance advantages of operation with adaptive detec-tion thresholds, as it avoids irreducible error floors [23]. The electrical-SNR-optimized detectionthresholds only need to change over the especially long timescales, of seconds or minutes, overwhich a stationary turbulence channel assumption is applied (as the electrical SNR remains con-stant over these timescales) [6]. The electrical-SNR-optimized system can, therefore, reduce theimplementation complexity, compared to that of the idealized system using adaptive threshold de-tection. Unfortunately, existing electrical-SNR-optimized systems make an assumption of perfectknowledge of the electrical SNR and turbulence pdf.The second technique, for improving the BER performance of FSO links using OOK, is sourceinformation transformation. In [24], the authors introduced pilot-symbol (PS) assisted modulation41.2. Thesis Organization and Contributions(PSAM) to mitigate the turbulence fading and improve the system performance. The PS providesthe receiver with explicit turbulence fading references for detection and helps mitigate the fadingeffects. However, PSAM causes delays in the receiver as it is necessary to store the whole framebefore decoding. In [25], it was demonstrated that an FSO system can use two laser wavelengthsat the transmitter and two photodetectors at the receiver operating in a differential mode withexcellent BER performance for an OOK IM/DD system with a detection threshold fixed at zero.Unfortunately, this scheme suffers from low throughput, as two lasers are used to transmit the sameinformation in each symbol duration.The third technique, for improving the BER performance of FSO links using OOK, employserror control coding. In particular, convolutional coding is considered here, as it can be effective inmitigating fading effects and improving the BER performance [26]. For such convolutional codes,error bounds are widely used to analyze the error rate performance. In [26], [27], the authors de-rived approximate upper bounds on pairwise error probability (PEP) as well as approximate upperbound expressions on the average BER for OOK IM/DD systems over the lognormal turbulencechannels using various coding schemes (include convolutional code). In [28], the authors studiedthe PEP of OOK OWC systems within temporally correlated K-distributed turbulence. They de-rived an upper bound on the PEP for the channel and then applied the union-bound technique inconjunction with the derived PEP bound to obtain upper bounds on the BER performance. In [29],the authors extended their work on PEP for coded OOK OWC systems to the Gamma-Gammaturbulence channels. In [31], the authors also derived an approximate PEP expression for codedFSO links over the Gamma-Gamma turbulence channels. However, an accurate approximation ofPEP was not obtained, and the computation of the upper bound for the average BER is complex.1.2 Thesis Organization and ContributionsThis thesis consists of seven chapters. A summary of each chapter and its contributions are givenas follows.In Chapter 1, we present background knowledge of FSO history and its development. The51.2. Thesis Organization and Contributionsmotivation of the research in this thesis is justified. We also provide a comprehensive review ofFSO literature related to the research topics of this thesis.Chapter 2 presents essential technical background for the entire thesis. A brief description ofan OOK IM/DD system is given. The additive noise at the receiver is also discussed. Then, somebackground knowledge on atmospheric turbulence channels is presented, and five atmosphericturbulence channel models are reviewed.In Chapter 3, we investigate the BER of OOK IM/DD FSO systems employing various detec-tion thresholds, as OOK IM/DD systems with fixed detection thresholds lead to irreducible errorfloors. The expressions of irreducible error loors are derived and expressed as comulative distribu-tion functions (CDFs) of the channel irradiance. We apply the electrical-SNR-optimized detectionsystem, as such a system can eliminate irreducible error floors without requiring perfect knowledgeof the instantaneous CSI and turbulence pdf.In Chapter 4, maximum-likelihood estimation (MLE) is applied to characterize the lognormal-Rician turbulence model parameters, and the expectation-maximization (EM) algorithm is used tocompute maximum likelihood estimates of the unknown parameters. Electrical SNR estimation isalso investigated for FSO communication systems using IM/DD over lognormal fading channels.Both method of moments estimation (MoME) and MLE are studied for electrical SNR estimation.The MSE is used to examine the performance of the estimators.In Chapter 5, FSO communication using OOK and source information transformation is pro-posed. This system can detect the OOK signal without knowledge of the instantaneous CSI andpdf of the turbulence model. The pdf of the detection threshold and an upper bound on the av-erage BER are derived. Numerical studies ultimately show that the proposed system can achievecomparable performance to the idealized adaptive detection system, with a greatly reduced levelof implementation complexity and a SNR penalty factor of only 1.85 dB at a BER of 2.17⇥107for a lognormal turbulence channel with a scintillation level of s = 0.25.In Chapter 6, we investigate IM/DD systems employing OOK and subcarrier intensity modu-lation (SIM) and binary phase-shift keying (BPSK) with convolutional code. We analyze the error61.2. Thesis Organization and Contributionsrate performance of OOK IM/DD systems operating over weak and strong turbulence conditionsand compare the BER performance of OOK to that of SIM-BPSK systems. A highly accurateconvergent series solution is derived for the PEP of the OOK IM/DD system. The solution es-tablishes a simplified upper bound on the average BER. For quasi-static fading channels, we alsostudy the BER performance of a convolutional coded system using block interleaving where eachblock experiences independent fading.Finally, we summarize the thesis in Chapter 7 and suggest some future research topics forfurther study.7Chapter 2OOK IM/DD System and TurbulenceChannel ModelsIn this chapter, we present a brief description of the OOK IM/DD system and the additive noisemodel at the receiver. Background information on atmospheric turbulence channels is presentedby way of five atmospheric turbulence channel models.2.1 System ModelFigure 2.1 shows the block diagram of an FSO system operating through the atmosphere. Thetransmitter is composed of a source encoder, an optical modulator and a transmitting telescope. Atthe transmitter, the source information bits are input into the source encoder, and encoded into anelectrical signal. After being properly biased, the electrical signal modulates a laser beam. At theend of transmitter, there is a telescope to control the direction and size of the laser beam. At thereceiver, a telescope is used to collect and focus the received optical beam onto the photodetectorfor optoelectronic conversion. The electrical signal is then decoded.For OOK IM/DD operation with the above system, the signal in the baseband to be transmittedcan be written ass(t) =Âiaig(t iTp) (2.1)where ai 2 {1,1} is the data bit, and Tp is the symbol duration. In (2.1), the pulse shaping isdefined as g(t) = 1 for 0 < t < Tp, and g(t) = 0 otherwise. The transmitted intensity has a bias82.1. System ModelFigure 2.1: Block diagram of an FSO system through an atmospheric turbulence channels.92.1. System Modelterm of unity added to ensure non-negative values and it can be expressed assˆ(t) = 1+Âiaig(t iTp). (2.2)The signal sˆ(t) is transmitted through an atmospheric turbulence channel and is distorted by a mul-tiplicative irradiance process I(u, t), where u is used to describe the space. The received electricalsignal after photodetection can be written asr(t) = R[(1+x )I(u, t)+ÂiI(u, t)aig(t iTp)]+n(t). (2.3)The photodetector responsivity, without loss of generality, is assumed to be R = 1. In (2.3), thepositive parameter x is the low and high state offset that quantifies a nonzero extinction ratio [32].Nonzero extinction ratios are due to practical considerations for semiconductor laser transmitters,which often operate with finite power levels for the low and high states. Typical values of x arebetween 0.1053 and 0.2857 [33]. When x 6= 0, the low and high states of the received electricalsignal are affected by turbulence. When x = 0, the received electrical signal simplifies to theclassical model discussed in [15].In (2.3), I(u, t) is assumed to be a stationary random process for signal scintillation causedby atmospheric turbulence, and n(t) is additive white Gaussian noise (AWGN) due to thermalnoise and/or ambient shot noise. Using a p-i-n photodiode and following [21], the shot noise isassumed to be dominated by the ambient shot noise. (Both ambient shot noise and thermal noiseare statistically independent of the desired signal.) The total noise power is s2g = s2s +s2T , wheres2s and s2T denote the respective ambient shot noise power and the thermal noise power.The received signal is sampled at time Tp. The sample I(u, t = Tp) is a random variable (RV)I, and the sample n(t = Tp) is a RV N having zero mean and variance s2g = N0/2, where N0 is thenoise power spectral density. If “0” is transmitted, s0 is true and the laser is in the low state, sothe sample for demodulation is r|s0 = x I+N. If “1” is transmitted, s1 is true and the laser is in thehigh state, so the sample for demodulation is r|s1 = (2+ x )I +N. It is important to note that the102.2. Additive Noise at the Receivernonzero state offset x leads to turbulence dependence for the received signal when s0 or s1 is true.2.2 Additive Noise at the ReceiverOptical receivers convert incident power into electric current through a photodetector. Besidesturbulence effects, two major types of additive noise, thermal noise and shot noise can affect thereceived signal photocurrent. The additive noise at the receiver is related to the type of photode-tector that is used, and we will focus on the p-i-n photodiode in this thesis. We will briefly reviewthe two types of additive noise, and then discuss the resulting SNR for the optical receivers in therest of this section.2.2.1 Thermal NoiseThermal noise (also known as Johnson noise [34] or Nyquist noise [35]) exists at a finite tem-perature. It is due to random thermal motion of electrons and atoms in a resistor, which createsa random voltage signal across its terminals. Mathematically, thermal noise can be modeled asa stationary Gaussian random process with a spectral density that is frequency independent wellinto the terahertz spectrum. Therefore, it is considered to be white noise. The spectral density ofthermal noise is given by [32]ST ( f ) =2kBTRL(2.4)where kB is the Boltzmann constant, T is the absolute temperature in Kelvins, and RL is the loadresistance. The spectral density ST ( f ) is two-sided and the photocurrent noise variance can beobtained as by [32] s2T = Z ••ST ( f )|H( f )|2d f =Z 4 f4 fST ( f )d f =44 f kBTRL(2.5)where H( f ) is the frequency reponse of the filter at the receiver and we assume |H( f )| = 1, andthe 4 f is the effective noise bandwidth of the receiver. It is worth noting that the thermal noise112.2. Additive Noise at the Receiverspectral density ST ( f ) and the resulting photocurrent noise variance do not depend on the receivedaverage photocurrent.2.2.2 Shot NoiseShot noise is due to photons and electron quantization (because light and electric current consist ofquantized ‘packets’). Contributions from signal photons lead to quantum noise, and contributionsfrom electrons in the semiconductor lead to dark noise. Shot noise was first introduced in 1918by Schottky [36] who studied fluctuations of current in vacuum tubes and has been thoroughlyinvestigated since then [37], [38].The energy associated with particles comes in discrete steps. A photon with a frequency nwill have an energy of hn where h is the Planck’s constant. It is therefore not possible to havea continuous flow of energy. Instead, the energy comes as bursts of particles that are witnessedas quantum noise fluctuations. Mathematically, quantum noise is a stationary random processfollowing Poisson statistics, which in practice is often approximated by Gaussian statistics. Forour purposes, quantum noise is assumed to be white noise with a constant spectral density due tothe received average photocurrent, i.e., Sq( f ) = qIp, where q is the electronic charge and Ip is thereceived average photocurrent. The noise variance can be obtained as [32]s2q = Z ••Sq( f )|H( f )|2d f =Z 4 f4 fST ( f )d f = 24 f qIp. (2.6)Dark noise is present when no light is incident on the photodetector. This dark noise currentis due to the semiconductor material in the photodetector. Electrons and holes are liberated dueto thermal effects in the semiconductor, as these carriers overcome the bandgap. This results in atime-averaged dark current Id with spectral density Sd( f ) = qId and variances2d = Z ••Sd( f )|H( f )|2d f =Z 4 f4 fSd( f )d f = 24 f qId. (2.7)122.3. Atmospheric Turbulence ModelsTherefore, the total variance of shot noise can be obtained ass2s = s2q +s2d = 24 f q(Ip + Id). (2.8)2.2.3 Signal-to-Noise RatioThe performance of an optical receiver depends on the SNR which is defined as the ratio of signalpower to total noise power. The SNR of a receiver with a p-i-n photodiode is considered here. Thesignal power is proportional to the photocurrent squared, while the noise contributions are fromthe thermal noise and shot noise, i.e., s2n = s2T +s2s . Therefore, the SNR can be expressed asSNR =I2ps2n = I2p24 f q(Ip + Id)+44 f KBT/RL . (2.9)It is worth noting that in the p-i-n receivers for FSO systems, thermal noise tends to dominatebecause the incident signal power and the dark current are relatively low.2.3 Atmospheric Turbulence ModelsIn an FSO system, the transmitted signals are typically subject to atmospheric turbulence over theatmospheric transmission links. The random variation in signal irradiance due to atmospheric tur-bulence caused by inhomogeneities in both temperature and pressure of the atmosphere is a majorsource of degradation of FSO system performances. To predict and mitigate such performancedegradation caused by atmospheric turbulence, researchers have studied FSO channels extensivelyand proposed different atmospheric turbulence models [4].The common statistical models that are used to characterize atmospheric turbulence channelsare the lognormal, K, negative exponential, Gamma-Gamma, and lognormal-Rician models [4].The lognormal distribution characterizes weak turbulence and is suitable for characterizing FSOcommunications in clear sky links over several hundred meters [39]. The K-distribution is suitablefor describing strong turbulence over links that are several kilometres in length [8]. The negative132.3. Atmospheric Turbulence Modelsexponential distribution describes the limiting case of saturated scintillation [40]. The Gamma-Gamma distribution and lognormal-Rician are two generalized models that can be applied to awide range of turbulence conditions [41], [11].2.3.1 Lognormal TurbulenceFor the lognormal channel model, the optical irradiance I is given byI = exp(X) (2.10)where X is a Gaussian RV with mean µ and variance s2. Consequently, I follows a lognormaldistribution with a pdf given by [39]fI(I) =1p2ps I exp✓(ln Iµ)22s2 ◆ , I > 0. (2.11)Normalizing the mean, i.e., E[I] = 1, where E[·] is the expectation operation, the pdf of I can bewritten asfI(I) =1p2ps I exp✓(ln I+s2/2)22s2 ◆ , I > 0. (2.12)The parameter s is the scintillation level, and its value is typically less than 0.75 [15]. Turbulenceeffects on the performance are minimal when scintillation levels are below s = 0.1, so the investi-gated electrical SNR in this thesis is characterized for typical scintillation levels ranging from 0.1to 0.75 [21], [42].2.3.2 K-distributed TurbulenceThe K-distributed turbulence model is a widely accepted turbulence model, and it can be used todescribe the irradiance fluctuations under strong turbulence conditions. It can be shown that the K-distributed RV I is a product of two independent RVs Ix and Iy, where Ix and Iy follow exponential142.3. Atmospheric Turbulence Modelsand Gamma distributions, respectively [43], with pdfsf (Ix) = exp(Ix), Ix > 0 (2.13)andf (Iy) =aIa1yG(a) exp(aIy), Iy > 0 (2.14)where G(·) is the Gamma function, and a is the effective number of discrete refractive scatters andit takes a value in (1,2) [44].The K-distribution can be derived as follows. Conditioning on Iy and substituting Ix = I/Iy into(2.13), we can obtain the conditional pdf of I asfI|Iy(I|Iy) =1Iyexp✓IIy◆, I > 0. (2.15)To obtain the pdf of I, we take the expectation of (2.15) with respect to the pdf of Iy. The pdf of Ibecomes [45]fI(I) =Z •0fI(I|Iy) f (Iy)dIy=2G(a)a a+12 I a12 Ka1(2paI) (2.16)where Ka1(·) is the modified Bessel function of the second kind with order a  1. The mthmoment of I is given by [43]E[Im] =m!G(m+a)amG(a) . (2.17)The scintillation index of K-distributed turbulence iss2SI , E[I2](E[I])2 1 = 1+ 2a (2.18)where s2SI is within (2,3).152.3. Atmospheric Turbulence Models2.3.3 Negative Exponential TurbulenceIn the limiting case of strong irradiance fluctuations (i.e., in the saturation regime and beyond)where the propagation distances span several kilometres, the number of independent scatteringsbecomes large [46], [47]. This saturation regime is also called the fully developed speckle regime.The amplitude fluctuation of the field traversing the random medium under this condition is exper-imentally verified to obey the Rayleigh distribution [48], [49]. Thus, the irradiance I follows thenegative exponential distribution whose pdf is given asfNE(I) =1I0exp✓II0◆, I > 0 (2.19)where I0 = E[I] is the mean irradiance. Without loss of generality, we can normalize I by settingE[I] = 1. In the saturation regime, the value of the scintillation index approaches unity.2.3.4 Gamma-Gamma TurbulenceThe Gamma-Gamma distribution is a useful and flexible turbulence model because it can describea wide range of turbulence conditions [41]. The pdf of the Gamma-Gamma distributed opticalirradiance isfI(I) =2G(a)G(b )(ab )a+b2 I a+b2 1⇥Kab ⇣2pab I⌘ , I > 0. (2.20)Assuming spherical wave propagation, the parameters a and b are related to the atmosphericconditions according to [29]a = "exp 0.49c2(1+0.18d2 +0.56c 125 ) 76 !1#1 (2.21)and b = "exp 0.51c2(1+0.69c 125 ) 56(1+0.9d2 +0.62c 125 ) 56 !1#1 (2.22)162.3. Atmospheric Turbulence Modelswhere c2 = 0.5C2nk 76L 116 and d = 0.5k1/2DL1/2. Here, the parameterC2n is the altitude-dependentindex of the refractive structure parameter that varies from 1017m23 for weak turbulence to1013m23 for strong turbulence, l is the optical wavelength and k = 2p/l is the wavenumber.The optical link distance is L, and D is the diameter of the receiver aperture.2.3.5 Lognormal-Rician TurbulenceFor the FSO system of interest, there is an assumption of perfect background noise rejection, fromnarrowband optical, electronic, and/or spatial filtering [39]. For the resulting lognormal-Ricianchannel model, the optical irradiance I can be obtained by I = |UC +UG|2 exp(2c), where UC isa real deterministic quantity, UG is a circular complex Gaussian RV with zero mean, c is a realGaussian RV, |UC +UG| is a Rician RV, |UC +UG|2 is a noncentral chi-square RV with a degreeof freedom of two, and exp(2c) is a lognormal RV. Consequently, I follows a lognormal-Riciandistribution with a pdf given by [13]fI(I) =(1+ r)erp2psz Z •0 dzz2 I0 2(1+ r)rz I1/2!⇥ exp 1+ rzI12s2z ✓lnz+ 12s2z ◆2! (2.23)where z represents exp(2c), r = |UC|2/E[|UG|2] is the coherence parameter, s2z is the varianceof the logarithm of the irradiance modulation factor z, and I0(·) is the zero-order modified Besselfunction of the first kind.As noted in [4], it is not generally known how to directly relate the above two empirical pa-rameters to the physical characteristics of atmospheric conditions, but it is possible to characterizetrends in the two parameters, with respect to the Rytov variance, s2R = 0.5k 76L 116 C2n . The charac-teristic trends in the two parameters are seen in [13], for variations in the Rytov variance, s2R, i.e.,variations in L and/orC2n . In the limit of zero inner scale, the parameter r decreases as s2R increases,while the parameter s2z is approximately equal to the Rytov variance for small s2R, reaches a peakvalue of approximately 0.58 for s2R ⇡ 8, and decreases slowly to approximately 0.4 for large s2R.172.4. SummaryWhen the coherence parameter r approaches infinity, the lognormal-Rician distribution specializesto the lognormal distribution with a pdf of [13], [14]fI(I) =1p2pszI exp  12s2z ✓ln I+ 12s2z ◆2! . (2.24)When r approaches 0, the lognormal-Rician distribution specializes to the lognormally modulatedexponential distribution, characterizing strong scintillation, with a pdf of [13], [50]fI(I) =1p2psz Z •0 dzz2 exp  Iz  12s2z ✓lnz+ 12s2z ◆2! . (2.25)The nth moment of the lognormal-Rician RV I is known to be [13]E[In] =(n!)2(1+ r)nexp✓n(n1)2s2z ◆ nÂk=0rk(n k)!(k!)2. (2.26)2.4 SummaryIn this chapter, we presented the essential background knowledge needed for technical contentin the remainder of this thesis. A brief description of an OOK IM/DD system was provided.The additive noise at the receiver was also discussed. Then, some background knowledge onatmospheric turbulence channels was presented, according to five commonly-used atmosphericturbulence channel models.18Chapter 3OOK IM/DD Systems with NonzeroExtinction Ratios andElectrical-SNR-Optimized DetectionThresholdsIn this chapter, we study the error rate performance of OOK IM/DD FSO systems employingvarious detection thresholds. Such systems, when implemented with fixed detection thresholds,lead to irreducible error floors. The expressions for the reducible error floors are derived andexpressed as cumulative distribution functions of the channel irradiance. We then investigate anelectrical-SNR-optimized detection system, as such a system can eliminate irreducible error floorswithout requiring perfect knowledge of the instantaneous CSI and the turbulence pdf.3.1 OOK with Fixed and Unoptimized Detection ThresholdsIn the low state, the received signal (r = x I+N) is a sum of two RVs, N and Is, where Is = x I. SinceN and Is are assumed to be independent, the pdf of the received low state signal is the convolution193.1. OOK with Fixed and Unoptimized Detection Thresholdsof the marginal pdfs of Is and N, according tof (r|s0) =1x fI✓ rx ◆⇤ fN(r)=Z •01p2psx exp0B@⇣ln xx + s22 ⌘22s2 1CA 1p2psg exp (r x)22s2g ! dx (3.1)where ⇤ denotes the convolution operation, and fN(r) = 1p2psg exp⇣ r22s2g ⌘ denotes the noise pdf.In the high state, the pdf of the received signal (r = (2+ x )I +N) can be defined in a similarmanner according tof (r|s1) =12+x fI✓ r2+x ◆⇤ fN(r)=Z •01p2psx exp0B@⇣ln x2+x + s22 ⌘22s2 1CA 1p2psg exp (r x)22s2g ! dx. (3.2)For a given fixed detection threshold Tth, the probability of false alarm PF and probability ofmiss PM can be written as the respective expressionsPF =Z •Tthf (r|s0)dr =Z •Tth1x fI✓ rx ◆⇤ fN(r)dr (3.3)andPM =Z Tth0f (r|s1)dr =Z Tth012+x fI✓ r2+x ◆⇤ fN(r)dr. (3.4)Assuming that p1 represents the a priori probability that “1” is sent, one can write the BER for203.1. OOK with Fixed and Unoptimized Detection ThresholdsOOK using a fixed detection threshold Tth asPe = (1 p1)PF + p1PM=(1 p1)exp⇣s28 ⌘p2ps Z •0 pxx3/2 exp  ln2 xx2s2 !Q✓Tth xsg ◆ dx+p1 exp⇣s28 ⌘p2ps Z •0 p2+xx3/2 exp  ln2 x2+x2s2 !Q✓xTthsg ◆ dx=(1 p1)exp⇣s28 ⌘p2ps Z •0 pxx3/2 exp  ln2 xx2s2 !Q⇣p2g(Tth x)⌘ dx+p1 exp⇣s28 ⌘p2ps Z •0 p2+xx3/2 exp  ln2 x2+x2s2 !Q⇣p2g(xTth)⌘ dx (3.5)where Q(x) = 1p2p R •x e t22 dt is the Gaussian Q-function, and we have denoted the electrical SNRby g = (E[I])2/N0 [21], or simply g = 1/N0 under a normalized mean assumption.In the large SNR regime, when g approaches infinity or equivalently when s2g = N0/2 ap-proaches zero, the Gaussian distribution approaches a Dirac delta function d (·). Hence, one canhavelimg!• fN(r) = d (r) (3.6)andlimg!• 1a fI ⇣ ra⌘⇤ fN(r) = 1a fI ⇣ ra⌘ (3.7)where a is a constant taking either x or 2+ x . When the electrical SNR is asymptotically large,and we assume that limits and integrals are reversible, it follows thatlimg!•PF = Z •Tth 1x fI✓ rx ◆ dr = 1FI✓Tthx ◆ (3.8)andlimg!•PM = Z Tth0 12+x fI✓ r2+x ◆ dr = FI✓ Tth2+x ◆ (3.9)213.1. OOK with Fixed and Unoptimized Detection ThresholdsTable 3.1: Error floor expressions for FSO systems employing fixed detection thresholds of Tth =(1+x )E[I] over a lognormal fading channel with s = 0.25.x 0.15 0.18 0.2 0.25Theoretical error floor 0.0044 0.0049 0.0054 0.0065Simulated error floor 0.0045 0.0049 0.0054 0.0066where FI(·) represents the CDF of the irradiance I. Therefore, the false alarm probability and missprobability in a large SNR regime are determined by the CDF of the irradiance evaluated at Tth/xand Tth/(2+x ), respectively. Substituting (3.8) and (3.9) into (3.5) giveslimg!•Pe = limg!•(1 p1)PF + p1PM=(1 p1)Q✓lnTth lnx +s2/2s ◆+ p1Q✓ ln(2+x ) lnTths2/2s ◆ (3.10)which is the error floor for an OOK IM/DD system with a fixed detection threshold through log-normal turbulence channels. As seen from (3.10), the error floor depends on both Tth and x , andtypically one chooses the fixed detection threshold as Tth = (E[r|s1] +E[r|s0])/2. When x = 0,Tth = E[I] and the analytical error floor expression in (3.10) is equivalent to [15, eq. (20)], whichwas derived under an assumption of a normalized second moment, i.e., E[I2] = 1. When x 6= 0, itis simple to show that Tth = (1+x )E[I].It is important to note that the error floor varies with the offset x . For a lognormal turbulencechannel with s = 0.25 and an equal a priori data symbol probability, the predicted error floors areshown in Table 3.1 for different values of x . It is seen that an increase of x results in a higher errorfloor. The theoretical error floors are verified with simulated BER limits with the results shown inTable 3.1.Following the same approach, one can predict the error floors for different turbulence channelmodels based on the corresponding CDFs. The resulting error floors are summarized in Table 3.2,223.2. OOK with Electrical-SNR-Optimized Detection ThresholdsTable 3.2: Error floor expressions for various turbulence channel models.Turbulence Channel Error FloorsLognormal (1 p1)Q⇣lnTthlnx+s2/2s ⌘+p1Q⇣ln(2+x )lnTths2/2s ⌘K-distribution (1 p1){1 12 [h(1,a,Tth,x )+h(a,1,Tth,x )]}+p1[h(1,a,Tth,2+x )+h(a,1,Tth,2+x )]/2Gamma-Gamma (1 p1){1 12 [h(b ,a,Tth,x )+h(a,b ,Tth,x )]}+p1[h(b ,a,Tth,2+x )+h(a,b ,Tth,2+x )]/2Negative Exponential (1 p1)exp⇣Tthxµ⌘+ p1 h1 exp⇣ Tth(2+x )µ⌘iwhere the function h(x,y,z,w) is defined ash(x,y,z,w) =⇣xyzw⌘x G(y x)G(x+1)G(y)1F2⇣x;x+1,x y+1;xyzw⌘(3.11)and where 1F2(·; ·, ·; ·) is the generalized hypergeometric function [51].3.2 OOK with Electrical-SNR-Optimized DetectionThresholdsA performance trade-off can be established between operation with fixed detection thresholds(which can suffer from irreducible error floors) and operation with adaptive detection thresholds(which requires knowledge of the instantaneous SNR for each data symbol). With this in mind, weconsider a system with electrical-SNR-optimized detection thresholds, as it offers a compromisebetween the practical advantages of operation with fixed detection thresholds and the performanceadvantages of operation with adaptive detection thresholds. Our approach considers the optimiza-tion problemargminTthPe = argminTth[(1 p1)PF + p1PM]. (3.12)233.2. OOK with Electrical-SNR-Optimized Detection ThresholdsFrom (3.5) and (3.12), it is clear that our electrical-SNR-optimized detection requires knowledgeof Tth, x , and the underlying turbulence model. To find the detection threshold that minimizes theBER at a given electrical SNR, we take the derivative of (3.5) with respect to Tth and set it to zero,i.e., ddTth Pe = 0. This gives(1 p1) f (Tth|s0)+ p1 f (Tth|s1) = 0 (3.13)where f (Tth|s0) and f (Tth|s1) are the likelihood functions evaluated at Tth.3.2.1 Electrical-SNR-Optimized Detection Based on a Known TurbulencepdfAssuming perfect knowledge of the pdf of the lognormal turbulence model, we substitute (3.1) and(3.2) into (3.13) and have (1 p1)Z •0pxx3/2exp ln2 xx2s2 !expg(x22xTth) dx+ p1Z •0p2+xx3/2exp ln2 x2+x2s2 !expg(x22xTth) dx = 0. (3.14)For a given electrical SNR g , the electrical-SNR-optimized detection threshold can be calculatednumerically from (3.14). The location of the electrical-SNR-optimized detection threshold lies atthe intersection of two scaled likelihood functions: (1 p1) f (r|s0) and p1 f (r|s1). As shown inFig. 3.1, when the electrical SNR approaches infinity, the total area underneath the intersected pdfs,i.e., (1 p1)PF + p1PM, will become infinitely small. The proposed electrical-SNR-optimized de-tection can therefore be used to eliminate the error floors caused by a receiver using fixed detectionthresholds. Our numerical results in Section 3.3 will be used to support these analytical arguments.243.2. OOK with Electrical-SNR-Optimized Detection Thresholds−3 −2 −1 0 1 2 3 4 500.10.20.30.40.50.60.70.80.91Variable rLikelihood Functions  SNR = 2 dBSNR = 8 dBFigure 3.1: The likelihood functions f (r|s0) and f (r|s1) with s = 0.25 and x = 0.2 when g = 2dB and g = 8 dB. The likelihood functions are a result of the convolution of the lognormal pdf andthe Gaussian pdf.253.2. OOK with Electrical-SNR-Optimized Detection Thresholds3.2.2 Electrical-SNR-Optimized Detection Based on an UnknownTurbulence pdfTo accommodate the fact that the FSO receiver may not always know the underlying characteristicsof the turbulence model, the turbulence distribution can be approximated by sample moments.The approximated turbulence distribution can then be used to derive the electrical-SNR-optimizeddetection threshold.The density functions of numerous statistical models on the positive half-line can be approxi-mated by a sum of Laguerre polynomials [52], [53]. Using this approach, one can approximate thepdf of I as [54]fI(I)⇡Iv exp(I/c)cv+1G(v+1)•Âj=0d jL j(v, I/c) (3.15)where Lj (v, I/c) is a Laguerre polynomial of order j in I/c and is written asLj✓v,Ic◆=jÂk=0(1)kG(v+ j+1)k!( j k)!G(v+ j k+1)✓Ic◆ jk(3.16)and d j = jÂk=0(1)kj!G(v+1)k!( j k)!G(v+ j k+1)µ Ic [ j k] (3.17)where the jth moment of I is denoted by µI[ j]. In (3.15), the parameters c = µI [2]µ2I [1]µI [1] and v =µI [1]c  1 are chosen to have the mean and variance of the Gamma RV I0, whose pdf, fI0(I) =Iv exp(I/c)cv+1G(v+1) , matches that of the RV I. From (3.15), the corresponding characteristic function (CF)and moment generating function (MGF) for the RV I can also be obtained. The detailed derivationsare given in Appendix A. These analytical expressions can be used to estimate the performance ofan FSO system over lognormal fading. Substituting (3.15) into the last equalities of (3.1) and (3.2)263.2. OOK with Electrical-SNR-Optimized Detection ThresholdsTable 3.3: Comparison of detection thresholds using an exact and approximated lognormal pdfwith s = 0.25.SNR (dB) Thresholds with Thresholds with Sampleexact pdf approximated pdf variance0 0.9497 0.9505 3.11⇥1084 0.8633 0.8637 2.27⇥1088 0.7528 0.7496 1.89⇥10812 0.6302 0.6214 2.56⇥10816 0.5087 0.4984 1.08⇥10720 0.3981 0.4697 8.62⇥10624 0.3036 0.5239 1.03⇥105yields the likelihood functionsf (r|s0) =1xp2psgcv+1 •Âj=0d j Z •0 ✓ xx ◆v1 exp✓ xxc◆⇥ expg(r x)2Lj✓v, xx ◆ dx (3.18)andf (r|s1) =1(2+x )p2psgcv+1 •Âj=0d j Z •0 ✓ x2+x ◆v1 exp✓ x(2+x )c◆⇥ expg(r x)2Lj✓v, x2+x ◆ dx. (3.19)Substituting (3.18) and (3.19) into (3.13) yields1 p1x •Âj=0d j Z •0 ✓ xx ◆v1 exp✓ xxc◆expg(x22xTth)Lj✓v, xx ◆ dx+p12+x •Âj=0d j Z •0 ✓ x2+x ◆v1 exp✓ x(2+x )c◆expg(x22xTth)⇥Lj✓v,x2+x ◆ dx = 0. (3.20)The detection threshold can be obtained numerically with respect to a given offset x and electrical273.3. Numerical ResultsSNR from (3.20). A comparison of the electrical-SNR-optimized detection thresholds, acquired bythe approximated and exact lognormal pdfs, are presented in Table 3.3. The thresholds are obtainedby averaging 10 calculated detection thresholds. As shown from Table 3.3, the approximated pdfcan be used to calculate the detection threshold with high accuracy when the electrical SNR is lessthan 16 dB. For higher values of SNR, the calculated detection thresholds lose accuracy, and thecorresponding BER curve deviates from the BER curve obtained with perfect knowledge of thelognormal pdf. This discrepancy occurs because the Laguerre-polynomial-based pdf approxima-tion can not accurately describe the behaviours of the lognormal pdf near the origin. Fortunately,this inaccuracy does not concern most practical FSO systems, as they typically operate at relativelylow SNR values [55].3.3 Numerical ResultsFigures 3.2 and 3.3 show the BER versus electrical SNR when the OOK modulated system usesfixed detection thresholds of Tth = 1 and Tth = 1.2. For expository purposes, the parameters areset as s = 0.25 and x = 0,0.2. It is observed that the BER curves obtained by using Monte Carlosimulation show excellent agreement with the derived error floors in large SNR regimes and theerror floors decrease for lower fixed detection thresholds.To eliminate the error floors and improve the performance, the system with electrical-SNR-optimized detection thresholds is used. The BERs for the systemwith the electrical-SNR-optimizeddetection thresholds are shown in Figs. 3.2 (with no state offset, x = 0) and 3.3 (with a finite stateoffset, x = 0.2), along with the BERs for the system with the adaptive detection thresholds. Bothelectrical-SNR-optimized detection thresholds are obtained by using the approximated lognormalpdf with J = 3 sample moments.It is seen from Figs. 3.2 and 3.3 that the electrical-SNR-optimized and adaptive detectionthreshold results exhibit no error floors, for increasing electrical SNR values, and that there existsan SNR penalty factor between the system with electrical-SNR-optimized detection thresholds andthe optimum OOK system using adaptive detection thresholds. For example, in Fig. 3.2, the OOK283.3. Numerical Results0 5 10 15 20 25 30 3510−410−310−210−1100Electrical SNR, γ (dB)Bit Error Rate  Tth=1Tth=1.20Electrical−SNR−optimized detection Adaptive detection Figure 3.2: BERs of OOK modulated systems using fixed detection thresholds Tth, electrical-SNR-optimized detection thresholds and adaptive detection thresholds over a lognormal turbulencechannel with s = 0.25 and x = 0.293.3. Numerical Results0 5 10 15 20 25 30 35 4010−410−310−210−1Electrical SNR, γ (dB)Bit Error Rate  Tth = 1Tth = 1.20Electrical−SNR−optimized detectionAdaptive detectionFigure 3.3: BERs of OOK modulated systems using fixed detection thresholds Tth, electrical-SNR-optimized detection thresholds and adaptive detection thresholds over a lognormal turbulencechannel with s = 0.25 and x = 0.2.303.3. Numerical Resultsmodulated system using adaptive detection thresholds requires an SNR of 13 dB to attain a BERof 105, while the system using electrical-SNR-optimized detection thresholds requires an SNRof 15.3 dB to achieve the same BER performance. The corresponding SNR penalty factor in Fig.3.2 for the system using an electrical-SNR-optimized detection threshold, is 2.3 dB at BER of105. The corresponding SNR penalty factor in Fig. 3.3 for the system using an electrical-SNR-optimized detection threshold, increases to 4.5 dB when x = 0.2. This performance difference canbe factored into the ultimate FSO system design to offset the complexity of implementing systemswith adaptive detection thresholds (and their need for knowledge of the instantaneous SNR).It is also important to point out that the BER performance achieved by the electrical-SNR-optimized system does not require rapid adjustment of the detection threshold. Since practicalFSO systems typically operate at constant transmit power, the detection threshold only needs to becalculated once over durations of seconds or even minutes. The electrical-SNR-optimized systemcan therefore reduce the implementation complexity, compared to that of the idealized systemusing adaptive threshold detection.In Fig. 3.4, the approximated lognormal pdf using J = 3 sample moments is compared withthe exact lognormal pdf, for s = 0.25. The absolute errors between these pdfs are shown explicitlyin Fig. 3.5. The approximated lognormal pdf shows good agreement with the exact lognormal pdfwhen s = 0.25. However, for higher s values (s > 0.75), the approximation of the lognormal pdfbecomes inaccurate as integer moments can not uniquely determine the lognormal pdf. Fortunately,such scintillation levels are not encounted in practice [56]. A comparison of absolute errors fromthe pdf approximations using different numbers of sample moments is also given in Fig. 3.5.Clearly, larger numbers of sample moments can reduce the absolute error, but this comes at the costof higher computational complexity. In general, a higher scintillation level s will require higherorder sample moments and the resulting approximation can become increasingly inaccurate. TheLaguerre-polynomial-based approximation is accurate for the 0.1 to 0.5 range of s values that isof interest to FSO applications [21], [42].In Fig. 3.6, we compare the BER performance between the approximated lognormal pdf, for313.3. Numerical Results0 0.5 1 1.5 2 2.5 3 3.5 400.511.522.5Variable IProbability Density Function  Exact pdfApproximated pdfFigure 3.4: Comparison of an approximated pdf using J = 3 sample moments and an exact pdf fora lognormal fading channel with s = 0.25.323.3. Numerical Results0 1 2 3 4 5 600.0020.0040.0060.0080.010.0120.0140.0160.018Variable xAbsolute Error  10 moments6 moments3 momentsFigure 3.5: The absolute error between the approximated pdf and the exact lognormal pdf withs = 0.25 and J = 3,6,10 sample moments.333.3. Numerical Results0 5 10 15 2010−810−610−410−2100Electrical SNR, γ (dB)Bit Error Rate  Exact pdf, σ=0.25Approximated pdf, σ=0.25Exact pdf, σ=0.35Approximated pdf, σ=0.35Figure 3.6: Comparison of BERs obtained by the approximated pdf and the exact lognormal pdfwith x = 0 and J = 3 sample moments.343.4. Summarydifferent values of s with J = 3 sample moments, and the exact lognormal pdf. The two simulatederror rate curves show good agreement over a wide range of SNR values. For the large SNR regime,the BER results from the approximate pdf have reduced accuracy, because the approximated pdfbased on Laguerre polynomials is unable to characterize the behaviours of the lognormal pdf nearthe origin.3.4 SummaryIt is known that FSO systems operating with OOK and fixed detection thresholds can suffer fromirreducible error floors and power inefficiency. With this in mind, the resulting error floors were an-alyzed here (and validated with simulations) for lognormal turbulence channels and quantified forthe general case having low and high state offsets, i.e., for nonzero extinction ratios. It was shownthat the error floors can be eliminated by using electrical-SNR-optimized detection thresholds thatminimize the average BER. The electrical-SNR-optimized system with the Laguerre-polynomials-based approximate pdf for the turbulence was found to be effective for typical FSO systems, whichoperate at relatively low SNR values, as it yields near-optimal BER performance.35Chapter 4Channel Parameters and Electrical SNREstimationIn this chapter, the on-going challenges are addressed for the application of the lognormal-Ricianturbulence model to FSO communication systems. MLE is applied to characterize the lognormal-Rician turbulence model parameters, and the EM algorithm is used to compute the MLE of theunknown parameters. As an FSO system that applies electrical-SNR-optimized detection musthave knowledge of the electrical SNR, electrical SNR estimation is also investigated for FSOcommunication systems using IM/DD over the lognormal fading channels. Both MoME and MLEare studied for this electrical SNR estimation. The MSE is used to examine the performance of theestimators.4.1 MLE for the Lognormal-Rician Shaping ParameterEstimationThe maximum likelihood principle is the most popular approach to obtain practical estimators. Itsperformance is optimal for large quantities of data, and it yields an approximation of the minimum-variance unbiased estimator.Assuming that we have K independent and identically distributed (i.i.d.) observations of thelognormal-Rician distribution, I = [I[0] ... I[K1]]T , with the unknown vectors q = [s2z r]T ,364.1. MLE for the Lognormal-Rician Shaping Parameter Estimationthe distribution of I can be written asfI(I ;q ) =K1’l=0(1+ r)erp2psz Z •0 dzz2 I0 2(1+ r)rz I[l]1/2!⇥ exp 1+ rzI[l]12s2z ✓lnz+ 12s2z ◆2! . (4.1)The MLE of the unknown vector q is then obtained by maximizing the log-likelihood functionL(I ;q )= ln f (I ;q )=K1Âl=0ln (1+ r)erp2psz Z •0 dzz2 I0 2(1+ r)rz I[l]1/2!⇥ exp 1+ rzI[l]12s2z ✓lnz+ 12s2z ◆2!! . (4.2)It is difficult to obtain closed-form estimates of the lognormal-Rician parameters, due to theintegral form of the density function in (2.23), so the ensuring analysis uses the EM algorithmto find the MLE of q [57]. This method, although iterative in nature, is guaranteed under mildconditions to converge and produce a local maximum [58]. The vector z = [z[0] ... z[K1]]T isfirst defined, where each element of the vector follows the lognormal pdff (z[l];sz) = 1p2pszz[l] exp  12s2z ✓lnz[l]+ 12s2z ◆2! , l = 0,1, . . . ,K1. (4.3)374.1. MLE for the Lognormal-Rician Shaping Parameter EstimationBy treating z as the unobserved data, we can write the complete-data log-likelihood function asL(I ,z;q ) =K1Âl=0ln f (z[l];sz)+K1Âl=0ln f (I[l]|z[l];r)=K( lnq(2ps2z )+ ln(1+ r) 52K K1Âl=0 lnz[l]s2z812s2z 1K K1Âl=0 (lnz[l])2 1K K1Âl=0 1+ rz[l] I[l]r+1KK1Âl=0ln I0 2(1+ r)rz[l]I[l]1/2!) (4.4)wheref (I[l]|z[l];r) =1+ rz[l]I0 2(1+ r)rz[l]I[l]1/2!exp✓r1+ rz[l]I[l]◆. (4.5)The resulting complete-data sufficient statistics areT1(z) =1KK1Âl=0lnz[l] (4.6a)T2(z) =1KK1Âl=0(lnz[l])2 (4.6b)T3(z) =1KK1Âl=01+ rz[l]I[l]+1KK1Âl=0ln I0 2(1+ r)rz[l]I[l]1/2!. (4.6c)The initial value of s2z is obtained by fitting the simple lognormal turbulence model, wheresˆ2z (0) = 2K ÂK1l=0 ln I[l]. The initial value of the coherence parameter r can be obtained by solvingthe following polynomial equation [14, eq. (6)][(rˆ(0))2 +4rˆ(0) +2]3[1+(rˆ(0))3][(rˆ(0))3 +9(rˆ(0))2 +18rˆ(0) +6]= 1K ÂK1l=0 I2[l]31K ÂK1l=0 I3[l].(4.7)Iterations are then carried out with the expectation-step (E-step) and maximization-step (M-step)[58].384.1. MLE for the Lognormal-Rician Shaping Parameter EstimationE-step: The E-step is carried out by computingT1⇣z; qˆ ( j)⌘= 1KK1Âl=0Ez|Ihlnz[l]I ; qˆ ( j) i (4.8)T2⇣z; qˆ ( j)⌘= 1KK1Âl=0Ez|Ih(lnz[l])2I ; qˆ ( j) i (4.9)T3⇣z; qˆ ( j)⌘= 1KK1Âl=0(1+ r)Ez|Ihz1[l]I ; qˆ ( j) i I[l] (4.10)+1KK1Âl=0Ez|I"ln I0 2(1+ r)rz[l]I[l]1/2! I ; qˆ ( j)#where qˆ ( j) = [sˆ2z ( j) rˆ( j)]T is an estimate of q in the jth iteration. For the computations carriedout as part of the E-step, it is noted that the expectation expressions in (8)-(10) are all functions ofz[l], so the conditional expectations in (8)-(10) can be expressed asEz|Ihg(z[l])I ; qˆ ( j) i= Z g(z[l]) f (z[l] I ; qˆ ( j) )dz[l] (4.11)where f (z[l]|I[l]; qˆ ( j)) = f (I[l]|z[l];qˆ ( j)) f (z[l];qˆ ( j))f (I[l];qˆ ( j)) and wheref (I[l]|z[l]; qˆ ( j)) = exp rˆ( j) 1+ rˆ( j)z[l]I[l]!⇥1+ rˆ( j)z[l]I00@2"(1+ rˆ( j))rˆ( j)z[l]I[l]#1/21A (4.12)f (z[l]; qˆ ( j)) = 1p2psˆ ( j)z z[l]⇥ exp0B@ 12⇣sˆ ( j)z ⌘2 lnz[l]+ 12 ⇣sˆ ( j)z ⌘221CA (4.13)394.2. Electrical SNR Estimationandf (I[l]; qˆ ( j)) = (1+ rˆ( j))erˆ( j)p2psˆ ( j)z Z •0 I00@2"(1+ rˆ( j))rˆ( j)z[l] I[l]#1/21A⇥ exp0B@1+ rz[l]I[l]12⇣sˆ ( j)z ⌘2 lnz[l]+ 12 ⇣sˆ ( j)z ⌘221CA dzz[l]2 . (4.14)M-step: The M-step is carried out by computingsˆ2z ( j+1) = T2⇣z; qˆ ( j)⌘⇣T1⇣z; qˆ ( j)⌘⌘2 (4.15)and finding rˆ( j+1) such that it maximizesrˆ( j+1) = argmaxrnT3⇣z; qˆ ( j)⌘o (4.16)where qˆ ( j+1) = [sˆ2z ( j+1) rˆ( j+1)]T is the new estimate of q . For the EM algorithm, the conditionalexpectation of the complete data is nondecreasing until it reaches a fixed point. This fixed point isthe MLE of q , i.e., qˆML = [sˆ2z,ML rˆML]T .4.2 Electrical SNR EstimationAs the electrical-SNR-optimized detection threshold introduced in Chapter 3 requires knowledgeof the state offset x and electrical SNR g , it is necessary to estimate x and g . MoME and MLE areused for such estimation in this section.It is assumed that there are 2L sampled signals during the observation interval. The vectorsR = [r[0] ... r[2L1]]T , I f = [I[0] ... I[2L1]]T , and N = [n[0] ... n[2L1]]T representthe received signal vector, fading coefficient vector, and noise vector, respectively. Assuming thata training sequence of length 2L is transmitted with L consecutive 1’s followed by L consecutive404.2. Electrical SNR Estimation0’s, one can write the received signal at the lth bit interval when bit 1 is transmitted asr[l]|s1 = (2+x )I[l]+n[l], l = 0,1, ...,L1 (4.17)where I[l] and n[l] represent the fading coefficient and noise during the lth bit interval, respectively.Similarly, if L 0’s are transmitted, the received signal at the kth bit interval can be written asr[k]|s0 = x I[k]+n[k], k = L,L+1, ...,2L1. (4.18)4.2.1 Method of Moments EstimationUsing (4.17) and (4.18), one can obtain the estimation of x asxˆ = 1LÂ2L1k=L r[k]|s012LÂL1l=0 r[l]|s1 12LÂ2L1k=L r[k]|s0. (4.19)To assess the performance of the moment estimator xˆ , approximate expressions can be de-rived for the mean and variance of xˆ when the sample size is asymptotically large. Assumingthe statistics T = [T1 T2]T , where T1 = 1LÂL1l=0 r[l]|s1 and T2 = 1LÂ2L1k=L r[k]|s0, one can obtain thecovariance matrix asCT =0B@ Var[T1] Cov[T1,T2]Cov[T2,T1] Var[T2]1CA=0B@ 1L [s2g +(2+x )2Var[I]] 00 1L(s2g +x 2Var[I]) 1CA . (4.20)Here, Var[·] denotes the variance, and Cov[·, ·] denotes covariance of two RVs. The estimator xˆcan be rewritten as xˆ D= j(T) = 2T2T1T2. (4.21)The estimator in (4.19) is consistent, i.e., xˆ Pr! x as L ! •, and is asymptotically Gaussian414.2. Electrical SNR Estimationdistributed, i.e.,pL(xˆ  x ) L! N (0,s2xˆ ). Performing a first-order Taylor expansion of j(·)about the point T = E[T] gives [59]xˆ ⇡ j(T) T=E[T] + 2Âi=1 ∂j∂Ti T=E[T] (TiE[Ti]) (4.22)where E[T] = [(2+x )E[I] xE[I]]T . Taking the expectation of (4.22) givesE[xˆ ]⇡ j(T)T=E[T]= x (4.23)and the asymptotic variance of xˆ can be expressed as [58]s2xˆ =Var[xˆ ] = ∂j∂Ti TT=E[T]CT ∂j∂Ti T=E[T]=s2g [x 2 +(2+x )2]+2x 2(2+x )2Var[I]4L(E[I])2.(4.24)Using (4.17) and (4.18), one can obtain the estimation of the turbulence mean m = exp(µ +s2/2) and N0, respectively, asmˆ =12LL1Âl=0r[l]|s112L2L1Âk=Lr[k]|s0 (4.25)andNˆ0 = 2(mˆ|s1)2 µˆr[2]|s0 (mˆ|s0)2 µˆr[2]|s1(mˆ|s1)2 (mˆ|s0)2 (4.26)wheremˆ|s1 =1LL1Âl=0r[l]|s1 (4.27)µˆr[2]|s0 = 1L 2L1Âk=L r2[k]|s0 (4.28)424.2. Electrical SNR Estimationmˆ|s0 =1L2L1Âk=Lr[k]|s0 (4.29)and µˆr[2]|s1 = 1L L1Âl=0 r2[l]|s1. (4.30)Using (4.25) and (4.26), one can obtain the estimation of g asgˆ = mˆ2Nˆ0=(mˆ|s1 mˆ|s0)2h(mˆ|s1)2 (mˆ|s0)2i8h(mˆ|s1)2 µˆr[2]|s0 (mˆ|s0)2 µˆr[2]|s1i . (4.31)4.2.2 Maximum Likelihood EstimationFor the MLE, we transmit a training sequence consisting of 2L consecutive 1’s. Assuming thereceived signal model is the same as (4.17), one can write the pdf of the received signal as [23]f (r[k]|s1;q g) = fI (r[k])⇤ fN(r[k])=Z •01p2psx exp (lnx ln(2+x )µ)22s2 !⇥1ppN0 exp✓(r[k] x)2N0 ◆ dx (4.32)where q g = [µ s2 N0 x ]T denotes the unknown vector, and fN(r[k]) = 1ppN0 exp⇣ r2[k]N0 ⌘ isthe noise pdf. Assuming that the components of the received signal vector R are independent, wecan write the pdf of the received signal when s1 is true asf (R;q g) =2L1’k=0f (r[k]|s1;q g)=2L1’k=0Z •01p2psx exp (lnx ln(2+x )µ)22s2 !⇥1ppN0 exp✓(r[k] x)2N0 ◆ dx. (4.33)434.2. Electrical SNR EstimationThe MLE of the unknown vector q g is obtained by maximizing the log-likelihood functionL(R;q g) = ln f (R;q g)= ln2L1’k=0fI (r[k])⇤ fN(r[k])=2L1Âk=0lnZ •01p2psx exp (lnx ln(2+x )µ)22s2 !⇥1ppN0 exp✓(r[k] x)2N0 ◆ dx. (4.34)Taking the derivative of (4.34) with respect to the unknown parameter and setting it equal to zero,we can obtain the MLE of the unknown vector q g . As it is difficult to obtain a closed-formexpression for each unknown parameter, the EM algorithm can be implemented numerically todetermine the MLE.In order to simplify the problem, we decompose the original data sets into the independent datasets y1[k] = I[k] and y2[k] = n[k], where y1[k] and y2[k] are the complete data, and they are relatedto the original data as r[k] = y1[k] + y2[k]. Instead of maximizing ln f (R;q g), we can maximizeln f (Y;q g), where Y= [y1 y2]T , y1 = [y1[0] ... y1[2L1]]T and y2 = [y2[0] ... y2[2L1]]T .Since y1[k] = I[k], we haveln f (y1[k];q g) = ln 1p2ps2y1[k] exp✓(lny1[k] ln(2+x )µ)22s2 ◆!= lnp2ps2 lny1[k] (lny1[k] ln(2+x )µ)22s2 . (4.35)Similarly, we haveln f (y2[k];q g) = ln✓ 1ppN0 exp✓y22[k]N0 ◆◆= lnppN0 y22[k]N0 . (4.36)Assuming qˆ ( j)g = [µˆ( j) (sˆ2)( j) (Nˆ0)( j) (x )( j)]T is an estimate of q g in the jth iteration, each444.2. Electrical SNR Estimationiteration of the EM algorithm can be carried out with the following E- and M-steps.E-step: This step determines the conditional expectation of the complete dataU(q g , qˆ ( j)g ) = EY|R;qˆ ( j)g [ln f (Y;q g)]= Ey1|R;qˆ ( j)g [ln f (y1;q g)]+Ey2|R;qˆ ( j)g [ln f (y2;q g)]=Zln f (y1;q g) f (y1|R; qˆ ( j)g )dy1 +Z ln f (y2;q g) f (y2|R; qˆ ( j)g )dy2. (4.37)where we havef (y1|R; qˆ ( j)g ) = f (R|y1; qˆ ( j)g ) f (y1; qˆ ( j)g )f (R; qˆ ( j)g ) (4.38)andf (y2|R; qˆ ( j)g ) = f (R|y2; qˆ ( j)g ) f (y2; qˆ ( j)g )f (R; qˆ ( j)g ) (4.39)and wheref (R|y1; qˆ ( j)g ) =2L1’k=01p2p(sˆ2)( j)(r[k] y1[k])⇥ exp (ln(r[k] y1[k]) ln(2+(x )( j)) µˆ( j))22(sˆ2)( j) ! (4.40)andf (R|y2; qˆ ( j)g ) = 2L1’k=01pp(N0)( j) exp✓(r[k] y2[k])2(N0)( j) ◆ . (4.41)M-step: This step maximizes (4.37) with respect to q g , by way ofq ( j+1)g = argmaxq g U(q g , qˆ ( j)g ) (4.42)where qˆ ( j+1)g is the new estimate of q g . For the EM algorithm, the conditional expectation of the454.3. Numerical Resultscomplete data is nondecreasing until it reaches a fixed point. This fixed point is the MLE of q gqˆ g,ML = [µˆML sˆ2ML Nˆ0,ML xˆML]T . (4.43)Based on the invariance property of theMLE, we obtain theMLE of µI as µˆI,ML = exp⇣µˆML + sˆ2ML2 ⌘.The MLE of g can be obtained asgˆML = (µˆI,ML)2Nˆ0,ML=⇣exp⇣µˆML + sˆ2ML2 ⌘⌘2Nˆ0,ML. (4.44)The Crame´r-Rao lower bound (CRLB) of gˆ can be calculated using [58]Var[gˆ]  ∂g∂µ ∂g∂s2 ∂g∂N0 ∂g∂x I1(q g) ∂g∂µ ∂g∂s2 ∂g∂N0 ∂g∂x T (4.45)where I(q g) is the Fisher information matrix that can be found asI(q g) = 2666666664E h∂ 2 ln f (R;q g )∂m2 i E h∂ 2 ln f (R;q g )∂m∂s2 iE h∂ 2 ln f (R;q g )∂m∂N0 i E h∂ 2 ln f (R;q g )∂m∂x iE h∂ 2 ln f (R;q g )∂s2∂m i E h∂ 2 ln f (R;q g )(∂s2)2 iE h∂ 2 ln f (R;q g )∂s2∂N0 i E h∂ 2 ln f (R;q g )∂s2∂x iE h∂ 2 ln f (R;q g )∂N0∂m i E h∂ 2 ln f (R;q g )∂N0∂s2 iE h∂ 2 ln f (R;q g )∂N20 i E h∂ 2 ln f (R;q g )∂N0∂x iEh∂ 2 ln f (R;q g )∂x∂m i E h∂ 2 ln f (R;q g )∂x∂s2 iE h∂ 2 ln f (R;q g )∂x∂N0 i E h∂ 2 ln f (R;q g )∂x 2 i3777777775 . (4.46)4.3 Numerical ResultsTo evaluate the estimator performance, the MSE of the estimator qˆ is studied. The MSE isMSE[qˆ ] = Var[qˆ ] + (E[qˆ ] q )2, where Var[qˆ ] is the variance of the estimator, i.e., Var[qˆ ] =1M1ÂM1i=0 (qˆ i ¯ˆq )2, ¯ˆq is the sample mean of the estimator, and E[qˆ ] = 1M ÂM1i=0 qˆ i is the mean ofthe estimator [60]. The simulation uses K = 1,000 data samples to estimate the lognormal-Ricianparameters and M = 100 trials to calculate the MSE of the estimator.In Fig. 4.1, we present the simulatedMSE and NMSE performance of rˆ and sˆ2z when s2z = 0.25and r ranges from 1 to 9. The NMSE is defined as the MSE scaled by the true value of the464.3. Numerical Resultsestimator. The performance trends at or above r = 2 are especially noteworthy. Increasing thevalue of r decreases the NMSE of rˆ but it does not change the NMSE of sˆ2z to any great extent.Thus, changes to the value of r have minimal effects on the estimation performance of s2z , and theMLE is insensitive to the value of r. The same conclusion can be seen for the MSE of rˆ and sˆ2z ,which remain relatively flat as r increases.In Fig. 4.2, we present the simulated MSE and NMSE performance of rˆ and sˆ2z when r = 4and s2z ranges from 0.1 to 0.8. From the figure, we note that the MSE of sˆ2z increases with thevalue of s2z while the MSE performance curve of rˆ stays flat with changing values of s2z . It can beseen that the MLE performance of the lognormal-Rician parameter s2z is insensitive to the value ofr but sensitive to the value of s2z , while the MLE performance of the lognormal-Rician parameterr is insensitive to both the values of r and s2z .Overall, the results for the MSE and NMSE in Figs. 4.1 and 4.2 are indicative of accurateestimation. The MSE and NMSE performance is comparable with that of the prior study [14],albeit with three orders of magnitude fewer data samples being required for the method proposedhere.In order to evaluate the estimator performance, the sample variance of the electrical SNR es-timator is compared with the CRLB. The variance of the electrical SNR estimator is given bysˆ2gˆ = 1M1 M1Âi=0 (gˆi ¯ˆg)2 (4.47)where gˆi is the estimation by using MoME or MSE at the ith trials, M represents the total numberof trails, and ¯ˆg is the sample mean of the electrical SNR estimator. In order to assess the estimator,Monte Carlo simulations are used to obtain sˆ2gˆ . In the simulation, different training sequencelengths are used to estimate the mean and noise variance, M = 1⇥104 trials are used to calculate thevariance of the electrical SNR estimator, and x is set at 0.2. Figure 4.3 plots the normalized samplevariance of the electrical SNR estimator, which is defined as the sample variance scaled by g , versusthe average electrical SNR. It is shown that the normalized sample variance for MLE attains thenormalized CRLB, which is obtained by scaling the CRLB by g . However, there is a discrepency474.4. Summarybetween the normalized sample variance for MoME and the normalized CRLB for SNR valuesgreater than 12 dB due to the inaccurate estimation of the noise variance. It can be shown thatthe discrepency between the normalized sample variance for MoME and the normalized CRLBwill disappear when x = 0. In this case, the received signal is characterized completely by thenoise when 0 is transmitted. Thus, the noise variance can be accurately estimated by transmittinga training sequence with consecutive 0’s. (When x 6= 0 and 0 is transmitted, the received signal isthe noise as well as the fading coefficient term, and this leads to inaccurate estimation of the noisevariance if a training sequence is transmitted with consecutive 0’s.)In Fig. 4.4, we compare the BER performance between the estimated electrical SNR, fordifferent values of s , and the exact electrical SNR. The pdf of the lognormal turbulence modelis approximated by using Laguerre-polynomials with J = 3 sample moments. The two simulatederror rate curves show good agreement over a wide range of SNR values.4.4 SummaryThe challenges were addressed in this chapter for the application of the lognormal-Rician turbu-lence model to FSO communications. The proposed technique used MLE, to estimate the pa-rameters of the lognormal-Rician fading channels and the EM algorithm, to compute the MLE ofthe unknown parameters. The performance was simulated in terms of MSE. Numerical resultsshowed that MLE with the EM algorithm can effectively characterize FSO communications overlognormal-Rician fading channels, given a wide range of turbulence conditions. Accurate estima-tion was shown with reduced demands on the quantity of data samples.For FSO communication systems, the reduced demand on the number of data samples for theproposed method leads to a shortened latency, being on the order of 103⇥103 seconds = 1 second,for turbulence-induced fading on a millisecond timescale. (In contrast, the method introduced in[14] demands a large number of data samples, on the order of 106, and it becomes necessaryto operate with a latency on the order of 106⇥ 103 seconds = 103 seconds.) The second-longlatency enabled by the work in this chapter is sufficiently short to support electrical-SNR-optimized484.4. Summaryadaptive detection in FSO channels exhibiting stationary statistics, i.e., constant channel modelparameters, over several seconds or minutes. To the authors’ best knowledge, this is the firstpractical implementation capable of electrical-SNR-optimized adaptive detection using the highly-accurate lognormal-Rician turbulence model.Electrical SNR estimation was studied for FSO systems using IM/DD over lognormal fadingchannels. Training sequences based on MoME and MLE were proposed. It was found that MoMEcould produce a closed-form expression for the estimator, while MLE requires numerical compu-tation to produce the estimator. The estimator performance was examined in terms of MSE. MonteCarlo simulations were used to assess the performance of the proposed estimators. The proposedestimators can be effective tools for future FSO systems implementing electrical-SNR-optimizeddetection.494.4. Summary1 2 3 4 5 6 7 8 910−310−2rMSE or NMSE  MSE of rMSE of σz2NMSE of rNMSE of σz2Figure 4.1: MSE and NMSE performance of the maximum likelihood estimators for the lognormal-Rician parameters r and s2z with s2z = 0.25.504.4. Summary0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810−510−410−310−2σ2zMSE or NMSE  MSE of rMSE of σ2zNMSE of rNMSE of σz2Figure 4.2: MSE and NMSE performance of the maximum likelihood estimators for the lognormal-Rician parameters r and s2z with r = 4.514.4. Summary0 5 10 15 20 25 3010−410−310−210−1100101102Electrical SNR, γ (dB)Normalized Variance of the Estimators  MoMEMLENormalized CRLBL=1,000L=10,000Figure 4.3: Comparison of MoME and MLE normalized sample variance for different trainingsequence lengths over a lognormal turbulence channel with s = 0.25. The normalized MSE iscomputed over M = 1⇥104 trials.524.4. Summary0 2 4 6 8 10 12 14 16 18 2010−810−710−610−510−410−310−210−1100Electrical SNR (dB)Bit Error Rate  Exact SNR, σ=0.25Estimated SNR, σ=0.25Exact SNR, σ=0.35Estimated SNR, σ=0.35Figure 4.4: Comparison of BERs obtained by the estimated electrical SNR and the exact electricalSNR with x = 0 and J = 3 sample moments.53Chapter 5OOK IM/DD System with SourceInformation TransformationIn this chapter, an FSO communication system using OOK and source information transformationis proposed. This system can detect the OOK signal without knowledge of the instantaneousCSI and pdf of the turbulence model. The pdf of the detection threshold and an upper boundon the average BER are derived. Numerical studies show that the proposed system can achievecomparable performance to the idealized adaptive detection system, with a greatly reduced level ofimplementation complexity and an SNR performance loss of only 1.85 dB at a BER of 2.17⇥107for a lognormal turbulence channel with s = 0.25.5.1 System and Channel ModelsWe consider an IM/DD system with M laser source transmitters and M photodetectors operatingthrough atmospheric turbulence channels.The operation of the proposed scheme is described as follows. At the transmitter, which isshown in Fig. 5.1, there are M distinct optical wavelengths, l1,l2, . . . ,lM, assigned to the Mlaser transmitters. Each wavelength is used to transmit an independent information sequence, withsource information transformation used to ensure that one or more lasers transmit bit “1” duringeach symbol duration. When M = 2, the proposed system can almost double the multiplexing gainachieved in the system of [25] using double-laser differential signaling. For source informationtransformation, we first convert a binary information sequence of length L to a (2M  1)-nary545.1. System and Channel ModelsFigure 5.1: Block diagram of the transmitter for the system using source information transforma-tion.555.1. System and Channel Modelsinformation sequence of length J as shown in Fig. 5.2. This mapping can be written asT1 : {0,1}L ! {0, . . . ,2M1}J. (5.1)Then we map each element of the (2M1)-nary sequence into an M-bit binary sequence that doesnot contain the all-zero binary sequence. The resulting M-bit binary sequence after the serial-to-parallel conversion determines, among M transmitted lasers, which link transmits bit “0” andwhich link transmits bit “1”. For example, when M = 3, we map the seven elements of the 7-nary sequence (0,1,2,3,4,5,6) to the binary sequence {001,010,011,100,101,110, 111}. Thismapping can be written asT2 : {0, . . . ,2M1}J ! {0,1}JM. (5.2)The mapping described in (5.1) and (5.2), which we call source information transformation, willensure that the M received signals have an explicit turbulence fading reference for detection ineach symbol duration, meaning that at least one laser is on (bit “1” is transmitted). When bit “1”is transmitted, the signal will suffer turbulence distortion. It is desirable to select values of L and Jthat make the mapping of T1 be a one-to-one mapping, i.e.,2L = (2M1)J. (5.3)However, the above equality is difficult to achieve in practice for arbitrary values of L and J. Toapproximate the ideal case of (5.3), we considermin(L,J)⇥(2M1)J2L⇤(5.4)Subject to 2L  (2M1)J.Since there might be more than one pair (L,J) that satisfies (5.4), we will choose the smallest pair565.1. System and Channel ModelsFigure 5.2: Block diagram of the source information transformation.(L,J) for our system, i.e., the value of L+ J is the smallest among the pairs which satisfy (5.4), asit is desirable to minimize the system delay. For example, when M = 3, we select L = 14 and J = 5by using a computer search.At the receiver, as shown in Fig. 5.3, diffractive optical elements and/or narrowband opticalfilters are used to separate the prescribed wavelengths for the detection of the M transmitted sig-nals. In Fig. 5.3, we use the acronym PD to represent the photodetector. After the M parallelphotodetectors and the parallel-to-serial conversion, in each symbol duration, a value of one-halfof the largest received signal is used to define the detection threshold for the M received signals. Ifall of the M-bit binary sequence are demodulated as bit “0”, which may happen due to the noise,this is an incorrect decision (since an all-zero binary sequence is not transmitted for our system),and we will assume the source transmits 00 . . .01. The demodulated JM-bit binary sequence willbe mapped to a (2M1)-nary sequence of length J, and then this (2M1)-nary sequence of lengthJ will ultimately be converted back to a binary information sequence of length L.At the mth transmitter, the transmitted intensity can be expressed assm(t) = 1+Âiai,mgm(t iTp), m = 1,2, . . . ,M (5.5)where ai,m 2 {1,1} is the ith data in the mth transmitter, and Tp is the symbol duration. In(5.5), pulse shaping in the mth transmitter is defined as gm(t) = 1 for 0 < t < Tp, and gm(t) =0 otherwise. These M signals are transmitted through atmospheric turbulence channels and aredistorted by a multiplicative process I(u, t). We have assumed that the channel fading is the samefor all the wavelengths in each symbol duration. This assumption can be achieved by ensuring that575.2. The Probability Density Function of the Detection Thresholdthe transmitter wavelengths are sufficiently close to each other (being separated only by tens ofnanometers), which will ensure that the transmitter beams are spatially overlapped and experiencethe same atmospheric turbulence distortion [25].At the mth receiver, the received signal after the photodetection can be written asrm(t) = R[(1+x )I(u, t)+ I(u, t)Âiai,mgm(t iTp)]+nm(t), m = 1,2, . . . ,M. (5.6)Without loss of generality, the photodetector responsivity R is assumed to be unity. In (5.6), thepositive parameter x is the low and high state offset that quantifies a nonzero extinction ratio,I(u, t) is assumed to be a stationary random process for signal scintillation caused by atmosphericturbulence, and nm(t) is AWGN due to thermal noise and/or ambient shot noise in mth receiver.Using a p-i-n photodiode and following [21], the shot noise is assumed to be dominated by ambientshot noise. (Both ambient shot noise and thermal noise are statistically independent of the desiredsignal.) The total noise power is s2g = s2s +s2T , where s2s and s2T denote the respective ambientshot noise power and the thermal noise power.The mth received signal is sampled at time Tp. The sample I(u, t = Tp) is a RV I, and the samplenm(t = Tp) is a RV nm having zero mean and variance s2g . When bit “0” is transmitted, s0 becomestrue and the laser is off. The demodulation sample is rm|s0 = nm. When bit “1” is transmitted, s1becomes true and the laser is on. The demodulation sample is rm|s1 = I+nm.5.2 The Probability Density Function of the DetectionThresholdWith perfect knowledge of the instantaneous CSI, the minimum error probability is provided bythe ML based decision threshold expressed by [22]Tth =s0(I0 + I1)+s1I0s0 +s1 (5.7)585.2. The Probability Density Function of the Detection ThresholdFigure 5.3: Block diagram of the receiver for the system using source information transformation.595.2. The Probability Density Function of the Detection Thresholdwhere s1 and s0 are the standard deviations of the noise currents for bits “1” and “0”, respectively;I1 and I0 are averages of the generated currents at the receiver for bits “1” and “0”. For simplicity,we assume s0 = s1 = sg, I0 = 0 and I1 = I. The ML-based detection threshold is Tth = I/2, whichis an adaptive detection threshold and varies with the fading coefficient. Note that this is complexto realize in practice, as it requires perfect knowledge of the instantaneous CSI for each symboldetection. However, when the average SNR (denoted by g) approaches infinity, or for a noiselesssystem, we havelimg!•max{r1,r2, . . . ,rM} = I. (5.8)Thus, we can intuitively set the detection threshold for the system to beTth =max{r1,r2, . . . ,rM}2. (5.9)The most important feature of the detection threshold proposed in (5.9) is that it only depends onthe received signal, and unlike a coherent OOK detection scheme, an estimate of the CSI is notrequired.We now derive the pdf of the detection threshold Tth in (5.9). In a symbol duration, we firstconsider the case for which k branches transmit bit “1”, where k = 1,2, . . . ,M, and the rest of theMk branches transmit bit “0”. Without loss of generality, we assume the first k branches transmitbit “1”, and the rest of the M k branches transmit bit “0”. The conditional CDF of Tth can bewritten asFTth(tth|I,k) = P✓max{I+n1, . . . , I+nk,nk+1, . . . ,nM}2< tth◆= P✓I+n12< tth, . . . ,I+nk2< tth,nk+12< tth, . . . ,nM2< tth◆= P(n1 < 2tth I, . . . ,nk < 2tth I,nk+1 < 2tth, . . . ,nM < 2tth).(5.10)605.2. The Probability Density Function of the Detection ThresholdSince all the noise components n1,n2, . . . ,nM are assumed to i.i.d., we haveP(n1 < 2tth I, . . . ,nk < 2tth I,nk+1 < 2tth, . . . ,nM < 2tth)= P(n1 < 2tth I) . . .P(nk < 2tth I)P(nk+1 < 2tth) . . .P(nM < 2tth)= [P(n1 < 2tth I)]k[P(nM < 2tth)]Mk.(5.11)It follows thatFTth(tth|I,k) = [P(n1 < 2tth I)]k[P(nM < 2tth)]Mk= [F(2tth I)]k[F(2tth)]Mk(5.12)where F(x) = R x• 1p2p exp⇣ r22 ⌘dr is the CDF of a standard Gaussian RV. The pdf of Tth condi-tioned on k branches transmitting bits “1”s and I can be written asfTth(tth|I,k) =ddtthFTth(tth|I,k) =2k✓F✓2tth Isg ◆◆k1✓F✓2tthsg ◆◆Mk fN(2tth I)+2(M k)✓F✓2tth Isg ◆◆k✓F✓2tthsg ◆◆Mk1 fN(2tth)(5.13)where fN(x) = 1p2psg exp⇣ x22s2g ⌘ denotes the pdf of the noise term. The pdf of Tth conditioned onI can be obtained asfTth(tth|I) =MÂk=1fTth(tth|I,k)p(k)=fN(2Tth I)2M1(MF✓2Tthsg ◆+F✓2Tth Isg ◆M1+✓F✓2Tthsg ◆◆M1)+ fN(2Tth)2M1 (M✓F✓2Tthsg ◆+F✓2Tthsg ◆◆M1+✓F✓2Tthsg ◆◆M2(M1)F✓2Tthsg ◆MF✓2Tthsg ◆) (5.14)615.3. The Upper Bound on the Average BERwhere p(k) = (Mk )2M is the probability that there are k branches transmitting bit “1”. Averaging (5.14)with respect to the fading coefficient I, one can obtain the pdf of Tth asfTth(tth) = EI [ fTth(tth|I)] (5.15)where EI[·] represents the statistical expectation with respect to I.5.3 The Upper Bound on the Average BERAs it is challenging to find the exact BER expression for our proposed system, we will find anupper bound on the average BER. For expository purposes, we first analyze the error caused in thedetection process. The error caused in the detection process, before the (2M  1)-nary sequenceconversion at the receiver, is the turbulence induced error when the binary sequence is transmittedthrough the turbulence channel. We then analyze the average BER of the output binary sequence.5.3.1 The Error Caused In The Detection At The ReceiverIf k branches transmit bit “1”, and the rest of the M k branches transmit bit “0”. Without lossof generality, we assume the first k branches transmit bit “1” values and the detection thresholdbecomesT˜th =max{I+n1, . . . , I+nk,nk+1, . . . ,nM}2. (5.16)We define N =n[1] ... n[M]Tas the noise vector, and Nk =n[1] ... n[k1] n[k+1] ...n[M]iTas the noise vector without the kth noise component nk. The probability of having incorrect625.3. The Upper Bound on the Average BERdetection in one or more links can be written asP(e|k) =1MEN1⇥EI⇥PI+n1 < T˜thN1 , I⇤⇤+ · · ·+ENk⇥EI⇥PI+nk < T˜thNk , I⇤⇤+ENk+1⇥EI⇥Pnk+1 > T˜thNk+1 , I⇤⇤+ · · ·+ENM⇥EI⇥PnM > T˜thNM , I⇤⇤ . (5.17)Since all components of the noise vector N are i.i.d., for k1 6= k2, where k1,k2 2 {1,2, . . . ,M}, wehaveENk1hEIhP⇣I+nk1 < T˜thNk1 , I⌘ii= ENk2 hEI hP⇣ I+nk2 < T˜thNk2 , I⌘ii (5.18)andENk1hEIhP⇣nk1 > T˜thNk1 , I⌘ii= ENk2 hEI hP⇣nk2 > T˜thNk2 , I⌘ii . (5.19)Thus, the probability of having incorrect detection in one or more links can be written asP(e|k) =kMEN1⇥EI⇥PI+n1 < T˜thN1 , I⇤⇤+M kMENM⇥EI⇥PnM > T˜thNM , I⇤⇤ . (5.20)635.3. The Upper Bound on the Average BERThe first term in (5.20) can be upper-bounded asPI+n1 < T˜thN1 , I= P✓ I+n1 < max{I+n1, . . . , I+nk,nk+1, · · · ,nM}2 N1 , I◆= P✓⇢I+n1 <I+n12[ · · ·[⇢I+n1 <I+nk2[nI+n1 <nk+12o[ · · ·[nI+n1 <nM2oN1 , I⌘ P✓I+n1 <I+n12 I◆+P✓ I+n1 < I+n22 n2, I◆+ · · ·+P✓I+n1 <I+nk2nk, I◆+P⇣ I+n1 < nk+12 nk+1, I⌘+ · · ·+P⇣I+n1 <nM2nM, I⌘= P(n1 <I| I)+P✓n1 <n2 I2n2, I◆+ · · ·+P✓n1 < nk I2 nk, I◆+P⇣n1 <nk+12 Ink+1, I⌘+ · · ·+P⇣n1 < nM2  InM, I⌘= P(n1 <I)+(k1)P✓n1 <n2 I2n2, I◆+(M k)P⇣n1 < nM2  InM, I⌘ . (5.21)The second term in (5.20) can be upper-bounded asPnM > T˜thNM , I= P✓nM > max{I+n1, . . . , I+nk,nk+1, . . . ,nM}2 NM , I◆= P✓⇢nM >I+n12\ · · ·\⇢nM >I+nk2\nnM >nk+12o\ · · ·\nnM >nM2oNM , I⌘ P✓nM >I+n12nM, I◆ . (5.22)Subtituting (5.21) and (5.22) into (5.20), we haveP(e|k) <kM⇢EI [P(n1 <I)]+(k1)En2EIP✓n1 <n2 I2n2, I◆+ (M k)EnMhEIhP⇣n1 <nM2 InM, I⌘iio+M kMEn1EIP✓nM >I+n12n1, I◆ . (5.23)645.3. The Upper Bound on the Average BERThe upper bound on the average BER for the binary sequence transmitted through the turbulencechannel with M transmit lasers is obtained asPe2 =MÂk=1P(e|k)p(k)<MÂk=1Mk2M1⇢kMEI [P(n1 <I)]+(k1)En2EIP✓n1 <n2 I2n2, I◆+ (M k)EnMhEIhP⇣n1 <nM2 InM, I⌘ii+M kMEn1EIP✓nM >I+n12n1, I◆=MÂk=1Mk2M1⇢kM✓EIQ✓Isg◆+(k1)En2 EI Q✓ In22sg ◆+(M k)EnMEIQ✓2InM2sg ◆◆+ M kM En1 EI Q✓ I+n12sg ◆ . (5.24)It is difficult to find a closed-form expression of (5.24) that contains a double integral; however,this integral can be evaluated numerically with high accuracy.5.3.2 Average BER of the Output Binary SequenceAt the transmitter, a binary sequence aL . . .a2a1 is converted into a (2M1)-nary sequence hJhJ1 . . .h2h1. This (2M1)-nary sequence of length J is mapped to a binary sequence of length JM. At thereceiver, after the demodulation, we will map a binary sequence of length JM to a (2M 1)-narysequence hˆJhˆJ1 . . . hˆ2hˆ1. The decimal value of a (2M1)-nary sequence hˆJhˆJ1 . . . hˆ2hˆ1 of lengthJ can be calculated asX = (2M1)J1hˆJ + · · ·+(2M1)hˆ2 + hˆ1 =JÂj=1(2M1) j1hˆ j. (5.25)To convert the (2M1)-nary sequence hˆJhˆJ1 . . . hˆ2hˆ1 to a binary sequence, denoted as aˆL+1aˆL . . . aˆ2aˆ1,the lowest element of the binary sequence isaˆ1 = mod(X ,2) (5.26)655.3. The Upper Bound on the Average BERwhere mod(·) yields the remainder after division of X by 2. Similarly, the second element of thebinary sequence isaˆ2 = mod✓X aˆ12,2◆. (5.27)The third element of the binary sequence isaˆ3 = mod✓X2aˆ2 aˆ122,2◆. (5.28)Thus, the lth element of the binary sequence isaˆl = mod✓X2l2aˆl1 · · ·2aˆ2 aˆ12l1,2◆= mod XÂl1i=1 2i1aˆi2l1,2!= mod ÂJj=1(2M1) j1hˆ jÂl1i=1 2i1aˆi2l1,2! (5.29)oraˆl =$ÂJj=1(2M1) j1hˆ jÂl1i=1 2i1aˆi2l1%(5.30)where b·c is the floor function that returns the largest integer, and this integer is less than or equalto the argument. We comment that when a (2M  1)-nary sequence of length J is converted toa binary sequence of length L+ 1; however, at the transmitter, we convert a binary informationsequence of length L to a (2M  1)-nary information sequence of length J. Thus, at the receiver,we will ignore aˆL+1 and use aˆL . . . aˆ2aˆ1 as our binary information sequence.To calculate the probability of having an incorrect decision in the binary sequence aˆL . . . aˆ2aˆ1,for expository purposes, we consider M = 3. When M = 3, we will convert L = 14 binary infor-mation bits a14 . . .a2a1 to a 7-nary sequence with a length of J = 5 (h5hJ1 . . .h2h1). Then we mapthe 7-nary sequence of length of J = 5 into a binary sequence of length JM = 15. At the receiver,after the demodulation, we will map each M = 3 binary bits to an element of the 7-nary sequence.665.3. The Upper Bound on the Average BERTable 5.1: The conditional probability of the received (2M1)-nary number is hˆl given the trans-mitted (2M1)-nary number is hl .hˆl 0 1 2 3 4 5 6P(hˆl|hl = 0) P(0|0) P(1|0) P(2|0) P(3|0) P(4|0) P(5|0) P(6|0)P(hˆl|hl = 1) P(0|1) P(1|1) P(2|1) P(3|1) P(4|1) P(5|1) P(6|1)P(hˆl|hl = 2) P(0|2) P(1|2) P(2|2) P(3|2) P(4|2) P(5|2) P(6|2)P(hˆl|hl = 3) P(0|3) P(1|3) P(2|3) P(3|3) P(4|3) P(5|3) P(6|3)P(hˆl|hl = 4) P(0|4) P(1|4) P(2|4) P(3|4) P(4|4) P(5|4) P(6|4)P(hˆl|hl = 5) P(0|5) P(1|5) P(2|5) P(3|5) P(4|5) P(5|5) P(6|5)P(hˆl|hl = 6) P(0|6) P(1|6) P(2|6) P(3|6) P(4|6) P(5|6) P(6|6)The mapping can be seen as follows:000! 0001! 0010! 1011! 2100! 3101! 4110! 5111! 6.(5.31)Table 5.1 shows the conditional probability of the received (2M  1)-nary number is hˆl given thetransmitted (2M1)-nary number is hl . The conditional probability in Table 5.1 can be calculatedby using the bit error probability for the binary bit transmitted through the turbulence channel, i.e.,P(0|0) = P(000|001)+P(001|001) = (1Pe2)2Pe2 +(1Pe2)3. This is shown in Table 5.2. InTabel 5.2, Pe2 is the error probability for the binary bit transmitted through the turbulence channel,and its upper-bound has been derived in accordance with (5.24).Since each element of hˆ5 . . . hˆ2hˆ1 and h5 . . .h2h1 is independent, the conditional probability for675.3. The Upper Bound on the Average BERTable5.2:Theconditionalprobabilityofthereceived(2M1)-narynumberishˆ lgiventhetransmitted(2M1)-narynumberish l.hˆ l0123456P(hˆl|hl=0)(1P e2)2P e2+(1P e2)3(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)P2 e 2P(hˆl|hl=1)(1P e2)2P e2+(1P e2)P2 e 2(1P e2)3(1P e2)P2 e 2(1P e2)P2 e 2P3 e 2(1P e2)2P e2(1P e2)P2 e 2P(hˆl|hl=2)(1P e2)2P e2+(1P e2)P2 e 2(1P e2)2P e2(1P e2)3P3 e 2(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)2P e2P(hˆl|hl=3)(1P e2)2P e2+(1P e2)P2 e 2(1P e2)P2 e 2P3 e 2(1P e2)3(1P e2)2P e2(1P e2)2P e2(1P e2)P2 e 2P(hˆl|hl=4)(1P e2)2P e2+(1P e2)P2 e 2P3 e 2(1P e2)P2 e 2(1P e2)2P e2(1P e2)3(1P e2)P2 e 2(1P e2)P2 e 2P(hˆl|hl=5)P3 e 2+(1P e2)P2 e 2(1P e2)2P e2(1P e2)P2 e 2(1P e2)2P e2(1P e2)P2 e 2(1P e2)3(1P e2)P2 e 2P(hˆl|hl=6)P3 e 2+(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)2P e2(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)P2 e 2(1P e2)3685.3. The Upper Bound on the Average BERthe received (2M1)-nary sequence is hˆ5 . . . hˆ2hˆ1 given the transmitted (2M1)-nary sequence ish5 . . .h2h1 isP(hˆ5 . . . hˆ2hˆ1|h5 . . .h2h1) =P5j=1P(hˆ j|h5 . . .h2h1) =P5j=1P(hˆ j|h j). (5.32)The conditional probability of the received binary sequence is aˆL . . . aˆ2aˆ1 given the transmittedbinary sequence aL . . .a2a1 can be written asP(aˆ14 . . . aˆ2aˆ1|aL . . .a2a1) = P(hˆ5 . . . hˆ2hˆ1|h5 . . .h2h1) =P5j=1P(hˆ j|h j). (5.33)If the transmitted binary sequence is a14 . . .a2a1, and the received binary sequence is aˆ14 . . . aˆ2aˆ1,the conditional error probability of this system given a14 . . .a2a1 and aˆ14 . . . aˆ2aˆ1 isP(e|a14 . . .a2a1, aˆ14 . . . aˆ2aˆ1) =Â14l=1 al aˆl14(5.34)where implements an exclusive OR. Thus, the BER of this system with M = 3 can be written asP(e) = Âa14...a2a1,aˆ14...aˆ2aˆ1P(e|a14 . . .a2a1, aˆ14 . . . aˆ2aˆ1)P(a14 . . .a2a1, aˆ14 . . . aˆ2aˆ1)= Âa14...a2a1,aˆ14...aˆ2aˆ1Â14l=1 al aˆl14P(a14 . . .a2a1, aˆ14 . . . aˆ2aˆ1)= Âa14...a2a1,aˆ14...aˆ2aˆ1Â14l=1 al aˆl14P(aˆ14 . . . aˆ2aˆ1|a14 . . .a2a1)P(a14 . . .a2a1).(5.35)In (5.35), the elements of a14 . . .a2a1 are independent, so we have P(a14 . . .a2a1) = 1214 . Substitut-ing (5.33) into (5.35), we haveP(e) = Âa14...a2a1,aˆ14...aˆ2aˆ1Â14l=1 al aˆl14P5j=1P(hˆ j|h j)P(a14 . . .a2a1)=1214 Âa14...a2a1,aˆ14...aˆ2aˆ1Â14l=1 al aˆl14P5j=1P(hˆ j|h j).(5.36)695.4. Numerical ResultsIn general, the BER for the proposed system with M transmitted lasers can be written asP(e) =12L ÂaL...a2a1,aˆL...aˆ2aˆ1ÂLl=1 al aˆl14PJj=1P(hˆ j|h j). (5.37)5.4 Numerical ResultsIn this section, the pdf of the detection threshold is first verified, and the BER performance of theproposed system is numerically studied.In Fig. 5.4, the derived pdf of the detection threshold Tth is compared with the simulated pdf.For expository purposes, we let M = 3 and x = 0. The simulated pdf is obtained by using MonteCarlo computer simulations with 104 trials. The derived pdf shows excellent agreement with thesimulated pdf.In Fig. 5.5, we plot the BER versus electrical SNR when the OOK IM/DD system uses a fixeddetection threshold of Tth = 0.5. Note that an error floor appears in the large SNR regime. Thesystem using source information transformantion can eliminate the error floor, although its BERperformace is worse than that of the OOK IM/DD system using fixed detection threshold in lowSNR regimes. This is due to the fact that a value of one-half of the largest received signal is used todefine the detection threshold for the M received signals in each symbol duration. This detectionthreshold is only optimum when the electrical SNR approaches infinity and/or there is no noise.In the low SNR regimes, the detection threshold is not an optimum detection threshold for ourproposed system, due to the noise influence, and the BER of our proposed system becomes worsethan that of the OOK IM/DD system using a fixed detection threshold.In Fig. 5.5, we also plot the upper bounds on the average BER for the proposed system overlognormal fading channels with different turbulence conditions. Simulated BER curves are alsoused to verify the analytical BER upper bound solutions. The upper bound is tight when M =3. However, as we have used the union upper bound technique, it can be shown that the upperbound becomes loose with increased M. It is seen from Fig. 5.5 that the OOK modulated systemusing idealized adaptive detection thresholds with a lognormal turbulence model having s = 0.25705.5. Summaryrequires an SNR of 22 dB to attain a BER of 2.17⇥107, while the proposed system requires anSNR of 23.85 dB to achieve the same BER performance. Thus, the corresponding SNR penaltyfactor for the system using OOK and source information transformation in a lognormal turbulencechannel with s = 0.25 is only 1.85 dB at BER of 2.17⇥ 107. This performance difference canbe factored into the ultimate FSO system design to offset the complexity of implementing systemswith adaptive detection thresholds (and their need for knowledge of the instantaneous CSI).5.5 SummaryAn FSO communication system using OOK and source information transformation has been pro-posed. It was shown that such a system can achieve good BER performance, without the need forknowledge of the instantaneous CSI and pdf of the turbulence model. We derived an analyticalexpression for the pdf of the detection threshold and developed an upper bound on the averageBER. Numerical studies ultimately showed that the proposed system achieves comparable perfor-mance to the idealized adaptive detection system, with a greatly reduced level of implementationcomplexity and a SNR penalty factor of only 1.85 dB at a BER of 2.17⇥ 107 for a lognormalturbulence channel with s = 0.25.715.5. Summary0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.810−410−310−210−1100101Variable tthProbability Density Function  Derived pdfSimulated pdfFigure 5.4: Comparison of the derived and simulated pdfs for the detection threshold Tth over alognormal fading channel with s = 0.25 and M = 3.725.5. Summary0 2 4 6 8 10 12 14 16 18 20 22 2410−810−710−610−510−410−310−210−1100Average SNR (dB)Bit Error Rate  BER upper boundIdealized adaptive detectionSimulated BERFixed threshold detection, Tth=0.5σ=0.5σ=0.25Figure 5.5: The simulated BER and BER upper bounds of the system using source informationtransformation over lognormal turbulence channels (with s = 0.25, s = 0.5, x = 0 and M = 3).73Chapter 6OOK IM/DD Systems with ConvolutionalCodeIn this chapter, we investigate IM/DD systems employing OOK and SIM-BPSK with convolutionalcode. We analyze the error rate performance of OOK IM/DD systems operating over weak andstrong turbulence conditions and compare the BER performance of OOK to that of SIM-BPSKsystems. A highly accurate convergent series solution is derived for the PEP of the OOK IM/DDsystem. The solution establishes a simplified upper bound on the average BER. For quasi-staticfading channels, we also study the BER performance of a convolutional coded system using blockinterleaving where each block experiences independent fading.6.1 Bit-By-Bit Interleaved Channels6.1.1 Pairwise Codeword Error Probability CalculationThe PEP for a coded OOK IM/DD system conditioning on a sequence of M fading coefficientsI = [I1, I2, ..., IM] is [29]P(C, Cˆ|I) = Q0@se(C, Cˆ)2N01A (6.1)where Cˆ= [cˆ1, cˆ2, ..., cˆM] is the chosen incorrect sequence when the coded sequenceC= [c1,c2, ...,cM]is transmitted. In (6.1), e(C, Cˆ) = EbÂk2W I2k is the energy difference of these two coded se-quences, where the set W contains the indices of bit locations in which the sequences C and Cˆdiffer, and Eb is the transmitted bit energy.746.1. Bit-By-Bit Interleaved ChannelsUsing an alternative expression of the GaussianQ-function, i.e., Q(x)= 1p R p20 exp⇣ x22sin2 q ⌘ dq ,one can write the conditional PEP as [29, eq. (7)]P(C, Cˆ|I) =1p Z p20 exp  Eb4N0 sin2q Âk2W I2k! dq=1p Z p20 ’k2Wexp✓ g4sin2q I2k◆ dq (6.2)where g = Eb/N0 denotes the SNR per bit. Following [29], we assume perfect bit-by-bit interleav-ing and average (6.2) with respect to the independent fading coefficients. The PEP isP(C, Cˆ) =1p Z p20 E "’k2Wexp✓ gI2k4sin2q◆# dq=1p Z p20 ’k2WE exp✓ gI2k4sin2q◆ dq=1p Z p20 ’k2WZ •0 exp✓ gI2k4sin2q◆ fI(Ik)dIk dq=1p Z p20 Z •0 exp✓ gI24sin2q◆ fI(I)dI|W| dq (6.3)where |W| denotes the cardinality of set W and represents the number of error events. Applying anaccurate series representation of the Gamma-Gamma pdf [61]fI(I) =G(ab )G(b a +1)G(a)G(b )⇥•Âp=0"(ab )p+b I p+b1G(pa +b +1)p!  (ab )p+a I p+a1G(p+ab +1)p!#to (6.3) yieldsP(C, Cˆ) =(2l (a,b ))|W|p⇥Z p20(•Âp=0[ap(a,b )gp(b )+ap(b ,a)gp(a)])|W| dq (6.4)756.1. Bit-By-Bit Interleaved Channelswhere ap(x,y)=G(xy)G(yx+1)(xy)p+yG(px+y+1)p! , l (a,b )= 12G(a)G(b ) , and gp(x)= R •0 I p+x1 exp⇣ gI2sin2 q ⌘ dI =G( p+x2 )2( g¯sin2 q ) p+x2 [51, eq. 3.326(2)]. Substituting ap(x,y) and gp(x) into (6.4) givesP(C, Cˆ) =(l (a,b ))|W|p Z p20 ( •Âp=0 g p+b2(2ab sinq)(p+b )⇥bp(a,b )+ g p+a2(2ab sinq)(p+a)bp(b ,a))|W| dq (6.5)where bp(x,y) =G( p+y2 )G(xy)G(yx+1)G(px+y+1)p! . Applying two power series identities [51, eqs. (0.314),(0.316)] gives  •Âk=0akxk!n=•Âk=0ckxk (6.6)where c0 = an0 and cm =1ma0 Âmk=1(knm+ k)akcmk, and•Âk=0akxk•Âk=0bkxk =•Âk=0ckxk (6.7)where ck = Âkn=0 anbkn, we then obtainP(C, Cˆ) =(l (a,b ))|W|p |W|Âm=0✓|W|m ◆ •Âp=0Cp(|W|m,m)⇥✓ pg2ab ◆p|W|(a1)m Z p20 (sinq)p+(|W|m)b+am+1 dq=(l (a,b ))|W|2pp |W|Âm=0✓|W|m ◆ •Âp=0Cp(|W|m,m)⇥✓ pg2ab ◆p(|W|m)bam G⇣ p+(|W|m)b+am+12 ⌘G⇣1+ p+(|W|m)b+am2 ⌘ (6.8)where Cp(i, j) , b(i)p (a,b ) ⇤ b( j)p (b ,a), and b(i)p (a,b ) is calculated by convolving bp(a,b ) withitself i 1 times, i.e., b(2)p (a,b ) = bp(a,b ) ⇤ bp(a,b ). The last equality of (6.8) is obtained by766.1. Bit-By-Bit Interleaved Channelsusing the integral identity [61, eq. (20)]Z p20(sinq)p+(|W|m)b+am+1 dq = ppG⇣ p+(|W|m)b+am+12 ⌘2G⇣1+ p+(|W|m)b+am2 ⌘ . (6.9)6.1.2 Truncation Error AnalysisTruncation error is introduced when we approximate the infinite series in (6.8) with the first Pterms. We define the truncation error ase(P), (l (a,b ))|W|2pp |W|Âm=0✓|W|m ◆ •Âp=P+1up(m)✓2abpg ◆p (6.10)whereup(m),Cp(|W|m,m)G⇣p+(|W|m)b+am+12⌘G⇣1+ p+(|W|m)b+am2 ⌘⇥✓2abpg ◆(|W|m)b+am . (6.11)It can be shown that the infinite series solution in (6.8) is a converging series by verifying that thetruncation error e(P) decreases as P increases. Following [62], we use the Taylor series expansionof xn/(1 x) = •j=n x j and obtain an upper bound of the truncation error ase(P) (l (a,b ))|W|2pp(pg/2ab 1)(pg/2ab )P⇥|W|Âm=0✓|W|m◆maxp>P{up(m)}.(6.12)If a> b and p approaches•, G(p+b)G(p+a) approaches zero. After examining the first two terms in (6.11),we note that up(m) approaches 0 when p approaches •. Therefore, the truncation error e(P) willdiminish when P increases. We also note from (6.12) that the truncation error diminishes rapidlywith increasing g . This suggests that the PEP solution in (6.8) is a convergent series and our series776.1. Bit-By-Bit Interleaved Channelssolution is accurate with high SNR values.6.1.3 Asymptotic Analysis of PEPWithout loss of generality, we assume a > b > 0 so that the term in (6.5) with g p+a2 decreasesfaster than the term with g p+b2 for increasing g [61]. When p approaches •, G( p+y2 )G(p+yx+1) in (6.5)approaches 0. The PEP expression in (6.5) can therefore be approximated at asymptotically highSNR byP•(C, Cˆ)=1p Z p20 "l (a,b )b0(a,b )✓ pg2ab ◆b (sinq)b#|W| dq=1p "l (a,b )b0(a,b )✓ pg2ab ◆b#|W| Z p20 (sinq)b |W| dq=G⇣b |W|+12⌘2ppG⇣1+ b |W|2 ⌘ "l (a,b )b0(a,b )✓ pg2ab ◆b#|W| .6.1.4 Upper Bound on Average BERSince the convolutional code is linear, the set of distances of the code sequences with respect tothe all-zero sequence is the same as the set of distances with respect to any other code sequences.Thus, without loss of generality, we assume an all-zero sequence is transmitted. For a (Kc,kc,nc)convolutional code where Kc is the constraint length and kc/nc is the code rate, the transfer functioncan be written as [63, 64]T (D,N) =•Âd=d f reeadDdN f (d) (6.13)where d represents the number of different bits between the selected path and the all-zero path,d f ree is the free distance of the convolutional code, ad denotes the number of nonzero paths withdistance d from the all-zero path that merge with the all-zero path for the first time, D denotes thedistance of the particular path from the all-zero path, N indicates the transition caused by the input786.1. Bit-By-Bit Interleaved Channelsbit “1”, and f (d) determines the number of bit errors for the path corresponding to the term N.By taking the derivative of T (D,N) with respect to N and setting N = 1, we obtain∂T (D,N)∂N N=1 = •Âd=d f ree BdDd (6.14)where Bd = ad f (d) is the total number of errors on all paths of distance d.Accordingly, an upper bound on the average BER based on PEP can be obtained as [65]Pub 1kcÂCp(C)•Âd=d f reeBdP(C, Cˆ) (6.15)where P(C) is the probability that the codeword C is transmitted. Using the transfer function tech-nique for (6.15) and the alternative form for the Q-function, i.e., Q(x) = 1p R p20 exp⇣ x22sin2 q ⌘ dq ,one can obtain an upper bound on the average BER as [65]Pb 1p Z p/20 1kc ∂T (D(q),N)∂N N=1 dq (6.16)where T (D(q),N) is the transfer function of the convolutional code, and D(q) is given by [29]D(q) = Z •0exp✓g4sin2q I2◆ fI(I)dI. (6.17)From (6.3) and (6.17), we note that the PEP equals a single integral of D(q) to the power of |W|.Substituting (6.4) into (6.17), we obtain D(q) asD(q) = l (a,b ) •Âp=0"bp(a,b )✓ pg2ab sinq◆(p+b )+bp(b ,a)✓ pg2ab sinq◆(p+a)# . (6.18)For expository purposes, we set the parameters of the convolutional code to Kc = 3, kc = 1and nc = 2, with the two function generators being g0 = [1 1 1] and g1 = [1 0 1]. Such a796.2. Quasi-static Fading Channelsnonsystematic convolutional code will not produce catastrophic errors. The corresponding transferfunction becomesT (D(q),N) = D5(q)N12ND(q) . (6.19)Substituting (6.19) into (6.16) givesPb 1p Z p20 D5(q)(12D(q))2 dq . (6.20)Finally, applying (6.18) to (6.20), we obtain a simplified expression of the upper bound on theaverage BER for a (3,1,2) convolutional coded OOK IM/DD system.6.2 Quasi-static Fading ChannelsSince the coherence time of an atmospheric turbulence channel is on the order of milliseconds [26]and the data rate of a typical FSO system is on the order of Gb/s, the ideal bit-by-bit interleavingassumed in Section 6.1 is difficult to achieve. Therefore, it is of practical importance to considera coded FSO system in a realistic quasi-static fading channel where the same fading coefficientaffects a block of data symbols. Following [66], we consider a convolutional code with rate Rc =kc/nc for a quasi-static fading channel3 using a block interleaver of depth n. For this system,which is shown in Fig. 6.1, (B+Kc1)kc information bits are convolutionally encoded into (B+Kc1)nc coded bits. The (B+Kc1)nc coded bits are denoted by xl j, where l = 1,2, . . . ,nc andj = 1,2, . . . ,B+ kc1. Each coded bit is placed on one of n blocks where each block experiencesindependent fading, i.e., Il is the fading coefficient for lth block. Using interleaving will result inlatency on a millisecond timescale or longer, but this latency can be reduced. A recent example byresearchers at the MIT Lincoln Lab showed this with interleavers and forward error correction for3When the codeword is affected by the same fading coefficient, the channel is called quasi-static. However, we cantransform a quasi-static channel into a block fading channel by way of block interleaving [67]. In such a system, weplace each coded bit in different blocks, such that each block experiences independent fading from that of neighbouringblocks and bits within a block suffer the same fading coefficient.806.2. Quasi-static Fading ChannelsFigure 6.1: Block diagram of a coded FSO system through quasi-static atmospheric turbulencechannels.OTU-1 (2.66 Gb/s) and OTU-2 (10.70 Gb/s) over a 5.4 km optical wireless communication link[68].Assuming first that the fading coefficients I˜ = [I˜1, I˜2, ..., I˜n] for nc blocks are fixed, and usingthe same technique as discussed in Section 6.1, the conditional union upper bound on the BER canbe obtained asPe(I˜)1kc•Âd=d fc(d)P2(d|I˜) (6.21)where d = [d1,d2, ...,dn] are the Hamming distances of the n blocks, d f = [d1 f ,d2 f , ...,dn f ] arethe free component distances of the n blocks, P2(d|I˜) is the conditional PEP given by P2(d|I˜) =Q⇣q2gÂncl=1 dl I˜2l ⌘, c(d) =•i=1 ia(d, i) are the coefficients obtained from the generalized transferfunction of the code, and a(d, i) is the number of error events with distance vector d [66]. Theaverage BER over the quasi-static Gamma-Gamma turbulence channel can be upper-bounded by[66]Pe ZI˜min"12,1kc•Âd=d fc(d)P2(d|I˜)#fI˜(I˜)dI˜ (6.22)where the fading coefficients I˜ for n blocks are assumed to be independent, i.e., fI˜(I˜) =’ncl=1 fI˜l(I˜l),and where fI˜l(I˜l) is the Gamma-Gamma pdf given by (2.20).816.3. Numerical Results6.3 Numerical ResultsIn this section, the PEP and BER performances are numerically studied for IM/DD systems em-ploying OOK with convolutional code. The PEP performance of OOK is compared with that ofSIM-BPSK. The BER performance is then compared for uncoded and coded OOK systems.In Fig. 6.2, the PEP results of coded IM/DD systems are shown for OOK and SIM-BPSKsystems operating over Gamma-Gamma turbulence channels with weak (a = 4.62,b = 4.24)and strong (a = 2.14,b = 1.21) turbulence conditions. For the coded SIM-BPSK system withe(C, Cˆ) = 4EbÂk2W I2k [26], we evaluate the PEP performance numerically for comparison pur-poses. The length of the error event is chosen to be |W| = 3. Our series approximation of PEP(with P = 60) shows excellent agreement with the exact PEP for weak and strong turbulence con-ditions. The PEP of the coded SIM-BPSK has better BER performance compared to that of acoded OOK system. As expected, the PEP is better in weak turbulence conditions. For example,at g = 20 dB, the PEP for strong turbulence (a = 2.14,b = 1.21) is 2.18⇥104, and the PEP forweak turbulence (a = 4.62,b = 4.24) is 3.57⇥ 108. For comparison, we also plot in Fig. 6.2the approximate PEP for coded OOK (diamonds) and SIM-BPSK (stars) systems by using [29, eq.(17)]. The approximate PEP results of [29] show reasonable agreement with the exact PEP resultsfor large SNRs, but the approximations are less accurate at the lower SNRs. The series representa-tion of the Gamma-Gamma pdf relies on a series expansion of Kn(x) where n is non-integer. As aresult, our series PEP expression requires (ab ) /2 Z. When (ab ) 2 Z, one can use the smallconstant adjustment method to circumvent this minor restriction [69]. In Fig. 6.2, as expected,asymptotic PEPs approach exact PEPs faster for strong turbulence conditions compared to thoseof weak turbulence conditions. A similar asymptotic behaviour for series solutions is seen in [61]for uncoded SIM-BPSK systems.In Fig. 6.3, we compare the simulated BER performance of uncoded and coded IM/DD systems(with perfect interleaving) employing OOK versus average SNR operating over Gamma-Gammaturbulence channels for weak (a = 4.62,b = 4.24) and strong (a = 2.14,b = 1.21) turbulenceconditions. The simulated BER results of the uncoded system have been numerically verified with826.4. SummaryBER results of the exact analytical expression. It is seen that strong turbulence can significantlydegrade the error rate performance of an uncoded system. However, the convolutional code canbe used to introduce coding gain and improve the BER performance in these strong turbulenceconditions. For example, when g = 20 dB (a = 2.14,b = 1.21), the BER for an uncoded systemcan be reduced from 8.48⇥ 102 to 1.02⇥ 105 with a convolutional coded (Rc = 1/2,Kc = 3)system. In Fig. 6.3, we also plot the exact upper bounds of BER and the upper bounds obtained bya series solution. The simplified series solution shows excellent agreement with the exact solutionobtained by using (6.17) and (6.20). Simulated BER curves are also used to verify the analyticalBER upper bound solutions.In Fig. 6.4, we plot the upper bounds on the average BER for a convolutional coded (Rc =1/2,Kc = 3,B= 999998) IM/DD system employing OOK over quasi-static Gamma-Gamma fadingchannels with weak (a = 4.62,b = 4.24) and strong (a = 2.14,b = 1.21) turbulence conditions.The block length B is chosen to represent Gb/s transmission with a one millisecond coherencetime. The upper bounds obtained from (6.22) are truncated at dmax = 40, where only the errorevents having the total distance d1 + d2  dmax are considered. The simulated BERs in Fig. 6.4,for quasi-static channels, clearly demonstrate the benefits of block-interleaved convolutional codeover a convolutional coded system without interleaving.6.4 SummaryIn this chapter, we derived an accurate series PEP expression for convolutional coded OOK IM/DDFSO systems using a series representation of the Gamma-Gamma pdf for ideal bit-by-bit inter-leaved channels. This novel PEP expression can facilitate rapid calculation of upper bounds onthe average SNR in weak-to-strong Gamma-Gamma distributed turbulence conditions. We alsostudied BER performance of a convolutional coded system using a block interleaver over realisticquasi-static Gamma-Gamma turbulence channels. While our analysis has been presented by wayof an OOK based IM/DD FSO system, the same analysis can be easily extended to an SIM-BPSKbased FSO system. For an SIM-BPSK based FSO system, we only need to change the energy836.4. Summarydifference of these two coded sequence to e(C, Cˆ) = 4EbÂk2W I2k .846.4. Summary0 5 10 15 20 25 3010−1210−1010−810−610−410−2100Average SNR (dB)Pairwise Error Probability  Series, OOKExact, OOKApprox, OOK [5, eq. (17)]Series, SIM−BPSKExact, SIM−BPSKApprox, SIM−BPSK [5, eq. (17)]Asymptotic, SIM−BPSKα=4.62β=4.24α=2.14β=1.21Figure 6.2: The PEP of coded IM/DD systems using OOK and SIM-BPSK versus average SNRoperating over Gamma-Gamma turbulence channels. Results are shown for weak (a = 4.62,b =4.24) and strong (a = 2.14,b = 1.21) turbulence conditions using series, exact, and approximatesolutions.856.4. Summary0 5 10 15 2010−1010−810−610−410−2100Average SNR (dB)Bit Error Rate  Series, α=2.14, β=1.21Exact, α=2.14, β=1.21Simulated, α=2.14, β=1.21Series, α=4.62, β=4.24Exact, α=4.62, β=4.24Simulated, α=4.62, β=4.24uncoded systemcoded systemFigure 6.3: The BER of uncoded and coded IM/DD systems (with perfect interleaving) and upperbounds on average BER of convolutional coded (Rc = 1/2,Kc = 3) OOK IM/DD systems versusaverage SNR over Gamma-Gamma turbulence channels. Results are for weak (a = 4.62,b =4.24) and strong (a = 2.14,b = 1.21) turbulence conditions.866.4. Summary0 5 10 15 2010−510−410−310−210−1100Average SNR (dB)Bit Error Rate  BER Upper BoundSimulated BER, Block InterleavingSimulated BER, Without Interleavingα=4.62β=4.24α=2.14β=1.21Figure 6.4: The simulated BER and upper bounds (dmax = 40) on average BER of terminatedconvolutional coded (Rc = 1/2,Kc = 3,B = 999998) OOK IM/DD systems versus average SNRover quasi-static Gamma-Gamma turbulence channels with and without block interleaving. Resultsare for weak (a = 4.62,b = 4.24) and strong (a = 2.14,b = 1.21) turbulence conditions.87Chapter 7ConclusionsIn this chapter, we conclude the thesis by summarizing the accomplished work and suggestingsome potential further research topics.7.1 Summary of Accomplished WorkIn this thesis, we investigated the BER performance of OOK IM/DD systems using different detec-tion thresholds over atmospheric turbulence channels. Such investigations can be used as guide-lines for practical FSO system design. Besides the BER performance investigation, turbulencechannel parameter and electrical SNR estimation were investigated. The proposed estimators canbe effective tools for future FSO systems implementing electrical-SNR-optimized detection. Wealso introduced coding techniques to mitigate the effects of turbulence induced fading and showedanalytically that the convolutional coded OOK IM/DD systems have greatly improved BER per-formance.We will summarize the accomplished work as follows:• In Chapter 3, we studied the error rate performance of OOK IM/DD systems using fixed,electrical-SNR-optimized and idealized adaptive detection thresholds. We obtained errorfloor expressions for various turbulence channel models for OOK IM/DD systems usingfixed detection thresholds. We approximated the turbulence pdf by a sum of Laguerre poly-nomials. The electrical-SNR-optimized system with the Laguerre-polynomials-based ap-proximate pdf for the turbulence was found to be effective for typical FSO systems, whichoperate at relatively low SNR values, as it yields comparable BER performance to that of theelectrical-SNR-optimized system with perfect knowledge of turbulence pdf.887.2. Suggested Future Work• In Chapter 4, we studied the MoME and MLE method. We used MLE to estimate the param-eters of the lognormal-Rician fading channels, and used the EM algorithm to compute theMLE of the unknown parameters. Electrical SNR estimation was also studied for FSO sys-tems using IM/DD over the lognormal fading channels. Training sequence based MoME andMLE were investigated. It was found that MoME could produce a closed-form expressionfor the estimator, while MLE requires numerical computation to produce the estimator.• Chapter 5 investigated FSO communication systems using OOK and source informationtransformation. It was shown that such a system can achieve good BER performance withoutthe need for knowledge of the instantaneous CSI and pdf of the turbulence model. We alsoderived an analytical expression for the pdf of the detection threshold and developed a tightupper bound on the average BER.• In Chapter 6, we derived an accurate series PEP expression for convolutional coded OOKIM/DD FSO systems for ideal bit-by-bit interleaved channels using a series representationof the Gamma-Gamma pdf. This novel PEP expression can facilitate rapid calculation ofupper bounds on the average SNR in weak-to-strong Gamma-Gamma distributed turbulenceconditions. We also studied BER performance of a convolutional coded system using a blockinterleaver over realistic quasi-static Gamma-Gamma turbulence channels.7.2 Suggested Future WorkIn Chapter 5, we presented FSO communication systems using OOK and source informationtransformation. However, we assumed that the largest of the M signals contain the CSI, i.e.,limg!•max{r1,r2, . . . ,rM} ⇡ I, and defined the detection threshold as a value of one-half of thelargest received signal. The BER performance for the system with this detection threshold wasacceptable only when the SNR is sufficiently large. Thus, it would be of future interest to furtheroptimize the detection threshold for operation in low SNR regimes.In Chapter 6, we investigated the BER performance of FSO communication systems using897.2. Suggested Future WorkOOK and convolutional code, but there are other effective coding techniques. Turbo codes are afamily of powerful error-correcting codes. Turbo codes have an impressive near-Shannon-limit forerror correcting performance. Thus, it would be of future interest to investigate the BER perfor-mance of FSO communication systems using OOK and turbo code.Overall, in this thesis, we introduced new implementations and thorough performance analysesfor OOK IM/DD systems using various detection schemes operating over atmospheric turbulencechannels. These findings can support future developments and innovations in FSO systems.90Bibliography[1] V. W. S. Chan, “Free-space optical communications,” IEEE/OSA Journal of LightwaveTechnology, vol. 24, pp. 4750–4762, Dec. 2006.[2] J. H. Shapiro and R. C. Harney, “Burst-mode atmospheric optical communication,” inProc.1980 National Telecommunications Conference, 1980, pp. 27.5.1–27.5.7.[3] M. Niu, Coherent Optical Wireless Communications over Atmospheric Turbulence Channels.The University of British Columbia: Ph.D. Thesis, 2012.[4] L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. UpperSaddle River: Prentice Hall, 1998.[5] S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels. New York: Plenum,1988.[6] D. A. DeWolf, “Are strong irradiance fluctuations log normal or rayleigh distributed,” Journalof the Optical Society of America A, vol. 57, pp. 787–797, June 1967.[7] A. Prokesˇ, “Modeling of atmospheric turbulence effect on terrestrial fso link,” Radioengi-neering, vol. 18, pp. 42–47, Apr. 2009.[8] M. Niu, J. Cheng, and J. F. Holzman, “Exact error rate analysis of equal gain and selec-tion diversity for coherent free-space optical systems on strong turbulence channels,” OpticsExpress, vol. 18, pp. 13915–13926, June 2010.[9] K. P. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, “Simple, accurate formulafor the average bit error probability of multiple-input multiple-output free-space optical links91Chapter 7. Bibliographyover negative exponential turbulence channels,” Optics Letter, vol. 37, pp. 3243–3245, Aug.2012.[10] L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications.Bellingham, WA: SPIE Press, 2001.[11] R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strongscintillation,” Journal of the Optical Society of America A, vol. 14, pp. 1530–1540, July 1997.[12] J. F. Paris, “Advances in the statistical characterization of fading: from 2005 to present,”International Journal of Antennas and Propagation, vol. 2014, pp. 1–5, Jun. 2014.[13] J. H. Churnside and S. F. Clifford, “Log-normal rician probability-density function of opticalscintillations in the turbulent atmosphere,” Journal of the Optical Society of America A, vol.4, pp. 727–733, Oct. 1987.[14] X. Song and J. Cheng, “Joint estimation of the lognormal-rician atmospheric turbulencemodel by the generalized method of moments,” Optical Communication, vol. 285, pp. 4727–4732, Nov. 2012.[15] J. Li, J. Q. Liu, and D. P. Tayler, “Optical communication using subcarrier psk intensity mod-ulation through atmospheric turbulence channels,” IEEE Transactions on Communications,vol. 55, pp. 1598–1606, Aug. 2007.[16] H. Moradi, M. Falahpour, H. H. Refai, P. G. LoPresti, and M. Atiquzzaman, “BER analysisof optical wireless signals through lognormal fading channels with perfect CSI,” in Proc.17thInternational Conference on Telecommunications, Doha, Qatar, Apr. 2010, pp. 493–497.[17] M. L. B. Riediger, R. Schober, and L. Lampe, “Blind detection of on-off keying for free-space optical communications,” in Proc. Canadian Conference on Electrical and ComputerEngineering, Niagara Falls, Canada, May 2008, pp. 1361–1364.92Chapter 7. Bibliography[18] X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection forfree-space optical communication through atmospheric turbulence channels,” IEEE Transac-tions on Communications, vol. 51, pp. 509–516, Mar. 2003.[19] M. L. B. Riediger, R. Schober, and L. Lampe, “Fast multiple-symbol detection for free-spaceoptical communications,” IEEE Transactions on Communications, vol. 57, pp. 1119–1128,Apr. 2009.[20] H. R. Burris, “Laboratory implementation of an adaptive thresholding system for free-spaceoptical communication receivers with signal dependent noise,” Proceedings of SPIE, vol.5892, pp. 1–20, Aug. 2005.[21] X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulencechannels,” IEEE Transactions on Communications, vol. 50, pp. 1293–1300, Oct. 2002.[22] H. Moradi, H. H. Refai, and P. G. LoPresti, “Thresholding-based optimal detection of wire-less optical signals,” IEEE/OSA Journal of Optical Communications and Networking, vol. 2,pp. 689–700, Sept. 2010.[23] L.Yang, J. Cheng, and J. F. Holzman, “Electrical-snr-optimized detection threshold for ookim/dd optical wireless communications,” in Proc. Canadian Workshop Information Theory,Toronto, ON, Canada, June 2013, pp. 186–189.[24] X. Zhu and J. M. Kahn, “Pilot-symbol assisted modulation for correlated turbulent free-spaceoptical channels,” Proceedings of SPIE, vol. 4489, pp. 138–145, Jan. 2002.[25] M. Khalighi, F. Xu, Y. Jaafar, and S. Bourennane, “Double-laser differential signaling forreducing the effect of background radiation in free-space optical systems,” IEEE/OSA Journalof Optical Communications and Networking, vol. 3, pp. 145–154, Feb. 2011.[26] X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communicationsthrough atmospheric turbulence channels,” IEEE Transactions on Communications, vol. 51,pp. 1233–1239, Aug. 2003.93Chapter 7. Bibliography[27] X. Zhu and J. M. Kahn, “Pairwise codeword error probability for coded free-space opticalcommunication through atmospheric turbulence channels,” in Proc. International Conferenceof Commununications, Helsinki, Finland, June 2001, pp. 161–164.[28] M. Uysal, M. Navidpour, and J. Li, “Error rate performance of coded free-space optical linksover strong turbulence channels,” IEEE Communications Letters, vol. 8, pp. 635–637, Oct.2004.[29] M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space opticallinks over gamma-gamma atmospheric turbulence channels,” IEEE Transactions on WirelessCommunications, vol. 5, pp. 1229–1233, June 2006.[30] M. Uysal and J. Li, “Error performance analysis of coded wireless optical links over atmo-spheric turbulence channels,” in Proc. IEEE Wireless Communication Networking Confer-ence, Atlanta, GA, Mar. 2004, pp. 2405–2410.[31] W. Gappmair and M. Flohberger, “Error performance of coded fso links in turbulent atmo-sphere modeled by gamma-gamma distributions,” IEEE Transactions on Wireless Communi-cations, vol. 8, pp. 2209–2213, May 2009.[32] G. P. Agrawal, Fiber-Optic Communication Systems, fourth ed. New York: Wiley, 2010.[33] Agilent Technologies. (2001). Measuring extinction ratio of optical transmitters. [online].Available: http://literature.cdn.keysight.com/litweb/pdf/5966-4316E.pdf[34] J. B. Johnson, “Thermal agitation of electricity in conductors,” Physical Review, vol. 32, pp.97–109, July 1928.[35] H. Nyquist, “Thermal agitation of electric charge in conductors,” Physical Review, vol. 32,pp. 110–113, July 1928.[36] W. Schottky, “u¨ber spontane stromschwankungen in verschiedenen elektrizita¨tsleitern,” An-nalen der Physik, vol. 362, pp. 541–567, 1918.94Chapter 7. Bibliography[37] D. K. C. MacDonald, Noise and Fluctuations: An Introduction. New York: Dover Publica-tions, 2006.[38] F. N. H. Robinson, Noise and Fluctuations in Electronic Devices and Circuits. Oxford:Clarendon Press, 1974.[39] E. J. Lee and V. W. S. Chan, “Part 1: Optical communication over the clear turbulent atmo-spheric channel using diversity,” IEEE Journal on Selected Areas in Communications, vol.22, pp. 1896–1906, Nov. 2004.[40] W. O. Popoola and Z. Ghassemlooy, “Bpsk subcarrier intensity modulated free-space opticalcommunications in atmospheric turbulence,” IEEE/OSA Journal of Lightwave Technology,vol. 27, pp. 967–973, Apr. 2009.[41] A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradianceprobability density function of a laser beam propagating through turbulent media,” OpticsEngineering, vol. 40, pp. 1554–1562, Aug. 2001.[42] M. Uysal, J. Li, and M. Yu, “Signal detection in optical communications through the at-mospheric turbulence channel,” IEEE Transactions on Wireless Communications, vol. 5, pp.1229–1233, June 2006.[43] E. Jakeman and P. N. Pusey, “A model for non-Reyleigh sea echo,” IEEE Transactions onAntennas and Propagation, vol. AP-24, pp. 806–814, Nov. 1976.[44] K. Kiasaleh, “Performance of coherent dpsk free-space optical communication systems in k-distributed turbulence,” IEEE Transactions on Communications, vol. 54, pp. 604–607, Apr.2006.[45] E. Jakeman, “On the statistics of K-distributed noise,” Journal of Physics A: Mathematicaland General, vol. 13, pp. 31–48, Jan. 1980.95Chapter 7. Bibliography[46] Z. Sodnik, B. Furch, and H. Lutz, “Free-space laser communication activities in europe:Silex and beyond,” in Proc. IEEE Lasers and Electro-Optics Society, pp. 78–79, Oct. 2006.[47] H. Hemmati, “Interplanetary laser communications,” Optics and Photonics News, vol. 18,pp. 22–27, Nov. 2007.[48] G. R. Osche, Optical Detection Theory for Laser Applications. New Jersey: Wiley, 2002.[49] A. Garcia-Zambrana, “Error rate performance for stbc in free-space optical communicationsthrough strong atmospheric turbulence,” IEEE Communications Letters, vol. 11, pp. 390–392, May 2007.[50] J. H. Churnside and R. G. Frehlich, “Experimental evaluation of lognormally modulatedrician and ik models of optical scintillation in the atmosphere,” Optics Express, vol. 6, pp.1760–1766, Nov. 1989.[51] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, sixth ed. SanDiego: Academic Press, 2000.[52] S. B. Provost, “Moment-based density approximants,” The Mathematica Journal, vol. 9, pp.727–756, May 2005.[53] R. Price, “An orthonormal laguerre expansion yielding rices envelope density function fortwo sine waves in noise,” IEEE Transactions on Information Theory, vol. 34, pp. 1375–1382,Nov. 1988.[54] L. Devroye, “On random variate generation when only moments or fourier coefficients areknown,” Mathematics and Computers in Simulation, vol. 31, pp. 71–89, Feb. 1989.[55] A. A. Farid and S. Hranilovic, “Capacity bounds for wireless optical intensity channels withgaussian noise,” IEEE Transactions on Information Theory, vol. 36, pp. 6066–6077, Dec.2010.96Chapter 7. Bibliography[56] I. I. Kim, E. Woodbridge, V. Chan, and B. R. Strickland, “Scintillation measurements per-formed during the limited-visiblity lasercom experiment,” Proceedings of SPIE, vol. 3266,pp. 209–220, Jan. 1998.[57] A. Doganduˇic´ and J. Jin, “Maximum likelihood estimation of statistical properties of com-posite gamma-lognormal fading channel,” IEEE Transactions on Signal Processing, vol. 52,pp. 2940–2945, Oct. 2004.[58] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Upper SaddleRiver: Prentice Hall, 1993.[59] E. L. Lehmann, Elements of Large-Sample Theory. New York: Springer-Verlag, 1999.[60] D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgnchannel,” IEEE Transactions on Communications, vol. 48, pp. 1681–1691, Oct. 2000.[61] X. Song and J. Cheng, “Performance of subcarrier intensity modulated mimo wireless opticalcommunications,” in Proc. 26th Queen’s Biennial Symposium on Communications, Kingston,ON, Canada, May 2012, pp. 5–9.[62] X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulation for wirelessoptical communication,” IEEE Communications Letters, vol. 16, pp. 5–9, Apr. 2012.[63] A. J. Viterbi, “Convolutional codes and their performance in communication systems,” IEEETransactions on Communications Technology, vol. 19, pp. 751–772, Oct. 1971.[64] J. G. Proakis and M. Salehi, Digital Communications, fifth ed. New York: McGraw-Hill,2007.[65] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels: A UnifiedApproach to Performance Analysis, first ed. New York: John Wiley & Sons, 2000.97Chapter 7. Bibliography[66] E. Malkama¨ki and H. Leib, “Evaluating the performance of convolutional codes over blockfading channels,” IEEE Transactions on Information Theory, vol. 45, pp. 771–781, Mar.1999.[67] H. E. Gamal and A. R. Hammons, “On the design of algebraic space-time codes for mimoblock-fading channels,” IEEE Transactions on Information Theory, vol. 49, pp. 151–163,Nov. 2003.[68] J. A. Greco, “Design of the high-speed framing, fec, and interleaving hardware used in a5.4km free-space optical communication experiment,” Proceedings of SPIE, vol. 7464, pp.746409–1–746409–7, Aug. 2009.[69] E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of mimo free-space opticalsystems in gamma-gamma fading,” IEEE Transactions on Communications, vol. 57, pp.3415–3424, Nov. 2009.98Appendices AAppendix A: The CF and MGF odlognormal pdfThe CF of a RV I is the Fourier transform of its pdf, fI(I), and it is defined byFI(w) = Z ••fI(I)exp( jwI)dI (A.1)orFI(w) = Re[FI(w)]+ jIm[FI(w)] (A.2)where j2 = 1. In (A.2), Re[·] and Im[·] denote the real and imaginary parts, respectively. Bothcan be written, respectively, asRe[FI(w)] = Z •0fI(I)cos(wI)dI (A.3)andIm[FI(w)] = Z •0fI(I)sin(wI)dI. (A.4)Using (3.15), one can approximate (A.3) asRe[FI(w)]⇡Z •0Iv exp(I/c)cv+1•Ân=0dnLn✓v, Ic◆cos(wI)dI=1cv+1•Ân=0dn Z •0Iv exp(I/c)Ln✓v,Ic◆cos(wI)dI. (A.5)99Appendices A. Appendix A: The CF and MGF od lognormal pdfSubstituting (3.16) into (A.5), one hasRe[FI(w)]⇡ 1cv+1 •Ân=0dn nÂk=0 (1)kG(a)k!(n k)!G(a k)⇥Z •0Iv exp(I/c)✓Ic◆nkcos(wI)dI=•Ân=0dn nÂk=0(1)kG(a)k!(n k)!G(a k)(1+ c2w2)ak2⇥ cos((a k)arctan(cw)) (A.6)where a = v+n+1. In deriving the last equality of (A.6), an integral identity [51, eq. 3.944(6)]has been used.Similarly, substituting (3.15) and (3.16) into (A.4) and using an integral identity [51, eq.3.944(5)], one obtainsIm[FI(w)]⇡ •Ân=0dn nÂk=0(1)kG(a)k!(n k)!G(a k)(1+ c2w2)ak2⇥ sin((a k)arctan(cw)). (A.7)The approximate CF is then found to beFI(w)⇡ •Ân=0dn nÂk=0(1)kG(a)k!(n k)!G(a k)(1+ c2w2)ak2⇥ [cos((a k)arctan(cw))+ j sin((a k)arctan(cw))]. (A.8)Using an integral identity [51, eq. 3.326(2)], one can obtain the MGF asMI(s)⇡•Ân=0dn nÂk=0(1)kG(v+n+1)k!(n k)!(1 sc)v+nk+1. (A.9)100

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0166401/manifest

Comment

Related Items