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Performance analysis of subcarrier quadrature phase-shift keying systems with I/Q imbalance over Gamma-Gamma… Zhu, Changle 2016

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dyrzormunwy Anulysis oz guvwurriyreuuxrutury dhusyAghizt Kyyinggystyms kith ICe Imvulunwy cvyrGummuAGummu Fuxing WhunnylsbyChangle ZhuB.Eng., Zhejiang University, P. R. China, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)June 2016c© Changle Zhu, 2016The undersigned certify that they have read, and recommend to the College of Graduate Studiesfor acceptance, a thesis entitled:Performance Analysis of Subcarrier Quadrature Phase-Shift Keying Systems With I/Q ImbalanceOver Gamma-Gamma Fading ChannelsSubmitted by Changle Zhu in partial fulfillment of the requirements of thedegree of Master of Applied Science .Dr. Julian Cheng, School of Engineering, The University of British ColumbiaSipefvisof, Pfofessof (plyusy print numy unx zuwulty/swhool uvovy thy liny)Dr. Chen Feng, School of Engineering, The University of British ColumbiaSipefvisofm Coaaittee Meabef, drozyssor (plyusy print numy unx zuwulty/swhool in thy liny uvovy)Dr. Jonathan F. Holzman, School of Engineering, The University of British ColumbiaSipefvisofm Coaaittee Meabef, drozyssor (plyusy print numy unx zuwulty/swhool in thy liny uvovy)Dr. Lutz Lampe, Electrical and Computer Engineering, The University of British ColumbiaUbivefsitm ElUaibef, drozyssor (plyusy print numy unx zuwulty/swhool in thy liny uvovy)EltefbUl ElUaibef, drozyssor (plyusy print numy unx univyrsity in thy liny uvovy)June 20, 2016(Duty suvmittyx to Grux gtuxiys)iiAvstruwtThe error rate performance is studied for subcarrier intensity modulated (SIM) quadraturephase-shift keying (QPSK) system with in-phase and quadrature (I/Q) imbalance over Gamma-Gamma fading channels. In free-space optical (FSO) communication system, the transmittedsignals are typically affected by the atmospheric turbulence over the transmission links. In orderto study the system performance analytically, different statistical distributions have been proposedto describe the random variation in signal irradiance due to the scintillation caused by the inhomo-geneities in both temperature and pressure along the transmission path. Besides the atmosphericturbulence, other sources can also introduce performance degradation to an FSO system. Usingquadrature conversion, the performance of the practical system is affected by the phase and am-plitude offsets in the two branches. This phenomenon is referred to as I/Q imbalance. Usingdirect-conversion transceivers, the I/Q imbalance is unavoidable due to the considerable mismatch-es between the circuit components. This imbalance can happen at transmitter end, receiver end,or both ends of the transceiver.First, we study the error rate performance for a SIM QPSK system with transmitter I/Q im-balance over the Gamma-Gamma fading channels. Then, the error rate performance of a SIMQPSK system with receiver I/Q imbalance is investigated. Finally, the error rate performance isanalyzed for a SIM QPSK system with I/Q imbalance at both ends of the transceiver over theGamma-Gamma fading channels. Closed-form symbol error rate (SER) expressions are derived bytaking into account of both the I/Q imbalance and the fading. It is discovered that the value ofthe receiver amplitude imbalance is irrelevant to the SER of a subcarrier QPSK system. Trunca-tion error analyses are carried out to ensure the accuracy of the approximate series solutions andsupport the asymptotic analyses. We also present the numerical results to show the performanceimprovement using a calibrated transceiver. We will treat these three cases separately since theyiii4UfgeTcghave slightly different models.ivdryzuwyThis thesis is based on [C1, SJ1, SC1]. My supervisor, Dr. Julian Cheng, co-authored all thepublications and supervised all of my research work. I am responsible for the relevant manuscriptpreparation and revisions. The following statements explain the co-authorship of the publicationsand submitted manuscripts incorporated in this thesis.Chapter 3 contains my work from one conference paper [C1]. I established the system modeland performed the analytical error rate performance analysis of the system. I also carried out theasymptotic analysis and the numerical evaluation of the system error rate performance.Chapter 4 contains my work from one submitted conference paper [SC1]. The submittedmanuscript [SC1] is co-authored with Dr. Naofal Al-Dhahir. I reviewed the literature on re-ceiver I/Q imbalance and established the system model. Then I performed the analytical error rateperformance analysis of the system. I also carried out the numerical evaluation of the system errorrate performance. Dr. Al-Dhahir provided input to the system error rate performance analysis.Chapter 5 contains my work from one submitted journal paper [SJ1]. The submitted manuscript[SJ1] is co-authored with Dr. Naofal Al-Dhahir. I reviewed the literature on two-sided I/Q imbal-ance and established the system model. Then I performed the analytical error rate performanceanalysis of the system. I also carried out the numerical evaluation of the system error rate perfor-mance. Dr. Al-Dhahir provided input to the system error rate performance analysis.Jefereed Cgfferefce PmZdacalagfkC). C. Rhm and J. Cheng, “Performance analysis of subcarrier QPSK systems with transmitterI/Q imbalance over Gamma-Gamma fading channels,” Ikhceedbggl hf mae 21ma BbeggbZe Lrf-ihlbnf hg ChffngbcZmbhgl (BLC), Kelowna, BC, Canada, June 5-8, 2016. The material isincorporated in Chapter 3.vCeefTceJefereed Bgmrfad PmZdacalagfk (kmZealled)KB). C. Rhm, J. Cheng, and N. Al-Dhahir, “Performance analysis of subcarrier QPSK with two-sided I/Q imbalance over Gamma-Gamma fading channels,” BEEE MkZgl’ Pbkeeell Chffng’,submitted for publication. The material is incorporated in Chapter 5.Jefereed Cgfferefce PmZdacalagfk (kmZealled)KC). C. Rhm, J. Cheng, and N. Al-Dhahir, “Error rate analysis of subcarrier QPSK with receiv-er I/Q imbalances over Gamma-Gamma fading channels,” Ikhceedbggl hf mae BEEE GehbZeMeeechffngbcZmbhgl Chgfekegce (GEOBECOF), submitted for publication. The material isincorporated in Chapter 4.vihuvly oz Wontynts9Zklracl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aaaPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nLaZde gf Cgfleflk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . naaDakl gf Fagmrek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pDakl gf 9crgfyek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . paaaDakl gf KyeZgdk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pn9ccfgodedgeeeflk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pnaaDedacalagf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .pnaaaChahler )2 Aflrgdmclagf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . )1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Organization and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 4Chahler 22 Kyklee Mgded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /2.1 Subcarrier FSO System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Additive Noise at the Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The Gamma-Gamma Fading Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 11viiG45LE BF 6BAGEAGF2.4 Statistics of the Channel Gain and Receiver SNR . . . . . . . . . . . . . . . . . . . . 122.5 I/Q Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Chahler 32 KmZcarraer IPKC Kyklee Oalh Lrafkealler A/I AeZadafce Gner lheGaeea-Gaeea Fadafg Chaffedk . . . . . . . . . . . . . . . . . . . . . . ).3.1 Conditional SER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.1 Ideal QPSK System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Transmitter I/Q Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Average SER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 Error Rate of an Ideal QPSK System . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Error Rate with Transmitter I/Q Imbalance . . . . . . . . . . . . . . . . . . . 233.3 Truncation Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Asymptotic SER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Chahler 42 KmZcarraer IPKC Kyklee Oalh Jeceaner A/I AeZadafce Gner lheGaeea-Gaeea Fadafg Chaffedk . . . . . . . . . . . . . . . . . . . . . . 3)4.1 Conditional SER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Average SER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Truncation Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Asymptotic Average SER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Chahler -2 KmZcarraer IPKC Kyklee Oalh Log-Kaded A/I AeZadafce Gner lheGaeea-Gaeea Fadafg Chaffedk . . . . . . . . . . . . . . . . . . . . . . 4-5.1 Conditional SER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Average SER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 Truncation Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53viiiG45LE BF 6BAGEAGF5.4 Asymptotic Average SER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Chahler .2 Cgfcdmkagfk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..6.1 Summary of Accomplished Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67:aZdaggrahhy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19hhefdap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /.Appendix A: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Appendix B: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix C: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Appendix D: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Appendix E: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81ixList oz FigurysFigure 2.1 Block diagram of a subcarrier FSO system. . . . . . . . . . . . . . . . . . . . 8Figure 2.2 Block diagram of an ideal QPSK system. . . . . . . . . . . . . . . . . . . . . 14Figure 2.3 Block diagram of a subcarrier QPSK system with I/Q imbalance at both sides. 15Figure 3.1 Block diagram of a QPSK system with transmitter I/Q imbalance. . . . . . . 19Figure 3.2 Average SER of a subcarrier QPSK system over the Gamma-Gamma chan-nels with transmitter I/Q imbalance with different levels of turbulence. Thetransmitter phase imbalance is set to T = 0:35. The transmitter amplitudeimbalance is set to T = 0:15. . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 3.3 Average SER of a subcarrier QPSK system with and without transmitterI/Q imbalance over the Gamma-Gamma channels having different levels ofturbulence. The transmitter phase imbalance is set to T = 0:35. Thetransmitter amplitude imbalance is set to T = 0:15. . . . . . . . . . . . . . 29Figure 4.1 Block diagram of the a subcarrier QPSK system with receiver I/Q imbalance. 32Figure 4.2 Average SER of a subcarrier QPSK system over the Gamma-Gamma fadingchannels with receiver I/Q imbalance in a moderate ( = 2:50P  = 2:06)turbulence condition. The phase imbalances are set to R = 0:10P 0:20P 0:35and the amplitude imbalances are set to R = 0:05P 0:10P 0:15. . . . . . . . . 41Figure 4.3 Average SER of a subcarrier QPSK system with receiver I/Q imbalance overthe Gamma-Gamma fading channels having different levels of turbulence.Phase imbalance is set to R = 0:35. . . . . . . . . . . . . . . . . . . . . . . . 42xLIFG BF FIGUEEFFigure 4.4 Average SER of a subcarrier QPSK system with and without receiver I/Qimbalance over the Gamma-Gamma fading channels having different levelsof turbulence. Phase imbalance is set to R = 0:35. . . . . . . . . . . . . . . 43Figure 5.1 Average SERs of a subcarrier QPSK system over the Gamma-Gamma chan-nels with I/Q imbalance at both transmitter and receiver ends in a moderate( = 2:50P  = 2:06) turbulence condition. The transmitter phase imbal-ance is set to T = 0:35 and the transmitter amplitude imbalance is set toT = 0:15. The receiver phase imbalances are set to R = 0:10P 0:20P 0:35and the receiver amplitude imbalances are set to R = 0:05P 0:10P 0:15. . . . . 57Figure 5.2 Average SERs of a subcarrier QPSK system over the Gamma-Gamma chan-nels with I/Q imbalance at both transmitter and receiver ends in a moderate( = 2:50P  = 2:06) turbulence condition. The transmitter phase imbalancesare set to T = 0:10P 0:20P 0:35 and the transmitter amplitude imbalance isset to T = 0:15. The receiver phase imbalance is set to R = 0:35 and thereceiver amplitude imbalances are set to R = 0:05P 0:10P 0:15. . . . . . . . . 58Figure 5.3 Average SERs of a subcarrier QPSK system over the Gamma-Gamma chan-nels with I/Q imbalance at both transmitter and receiver ends in a moderate( = 2:50P  = 2:06) turbulence condition. The transmitter phase imbalanceis set to T = 0:35 and the transmitter amplitude imbalances are set toT = 0:05P 0:10P 0:15. The receiver phase imbalance is set to R = 0:35 andthe receiver amplitude imbalances are set to R = 0:05P 0:10P 0:15. . . . . . . 59Figure 5.4 Average SERs of a subcarrier QPSK system with I/Q imbalance at bothsides over the Gamma-Gamma channels having different levels of turbulence.The transmitter phase imbalance is set to T = 0:35 and the transmitteramplitude imbalance is set to T = 0:15. The receiver phase imbalance isset to R = 0:35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60xiLIFG BF FIGUEEFFigure 5.5 Absolute truncation error of a subcarrier QPSK system over the Gamma-Gamma fading channels with I/Q imbalance at transmitter and receiverends in a moderate ( = 2:50P  = 2:06) turbulence condition using differentvalues of J . The transmitter phase imbalance is set to T = 0:35 and thetransmitter amplitude imbalance is set to T = 0:15. The receiver phaseimbalance is set to R = 0:35. . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 5.6 Average SERs of a subcarrier QPSK system with and without I/Q imbal-ance over the Gamma-Gamma fading channels having different levels ofturbulence. The transmitter phase imbalance is set to T = 0:35 and thetransmitter amplitude imbalance is set to T = 0:15. The receiver phaseimbalance is set to R = 0:35. . . . . . . . . . . . . . . . . . . . . . . . . . . 63xiiList oz AwronymsAwronyms DynitionsADC Analog-to-Digital ConverterAWGN Additive White Gaussian NoiseBER Bit-Error RateBPSK Binary Phase-Shift KeyingCDF Cumulative Distribution FunctionCMOS Complementary Metal-Oxide SemiconductorDAC Digital-to-Analog ConverterDC Direct CurrentDPSK Differential Phase-Shift KeyingDQPSK Differential Quadrature Phase-Shift KeyingFD Frequency DependentFI Frequency IndependentFSO Free-Space OpticalI/Q In-phase and QuadratureIF Intermediate FrequencyMGF Moment Generating FunctionNCFSK Noncoherent Frequency-Shift KeyingxiiiLifg bf 4cebalmfOFDM Orthogonal Frequency Division MultiplexingOOK On-Off KeyingOWC Optical Wireless CommunicationsPDF Probability Density FunctionPIN Positive-Intrinsic-NegativePPM Pulse Position ModulationPSK Phase-Shift KeyingQAM Quadrature Amplitude ModulationQPSK Quadrature Phase-Shift KeyingRF Radio FrequencyRV Random VariableSER Symbol-Error RateSIM Subcarrier Intensity ModulatedSINR Signal-to-Interference-plus-Noise RatioSNR Signal-to-Noise RatioxivList oz gymvolsgymvols Dynitions△f Effective noise bandwidth of a receiverΓ(·) The Gamma functionK(·) The modified Bessel function of the second kind of order 2R Rytov varianceX2n Index of refraction structure parameterp! Factorial of a non-negative integer pZ Set of all integersMR(·) Moment generating function of a random variable gZ[·] Statistical expectation operationj j2 = −1f(x) Gaussian f-function defined as f(x) = 1√2∫∞x exp(− t22 )yt2F1(· P · ; · ; ·) Gauss Hypergeometric functionerfc(x) Complementary error functionRe [·] Real part of a numberlog(·) Logarithmic function with base 10r∗ Complex conjugate of rCov[·P ·] Statistical covariance operationxvLifg bf FlmUblfΦ(x) Cumulative distribution function of the standard normal distributioni (hP v) Owen’s i -function defined as i (hP v) = 12∫ a0exp[− 12h2(1+t2)]1+t2ytLP {· } Low-pass filtering procedureB(·P ·) Beta function defined as B(xP y) = Γ(x)Γ(y)Γ(x+y)xviAwknowlyxgymyntsI am deeply grateful to my thesis supervisor Dr. Julian Cheng for his enthusiasm, guidance,advice, encouragement, support, and friendship. I will continue to be influenced by his rigorousscholarship, clarity in thinking, and professional integrity.I would like to thank Dr. Lutz Lampe for his willingness to serve as my external examiner. Iwould also like to thank Dr. Chen Feng and Dr. Jonathan F. Holzman for their willingness to serveon the committee. I really appreciate their valuable time and constructive comments on my thesis.I would also like to express my thanks to Dr. Naofal Al-Dhahir from University of Texas at Dallasfor his feedback, constructive comments, and valuable suggestions on my research work.I owe many people for their generosity and support during my study at the University of BritishColumbia. I would like to thank my dear colleagues for sharing their academic experiences andconstructive viewpoints generously with me during our discussions. I would also like to thank mydear friends for sharing in my excitement and encouraging me when I was in frustration during thisjourney.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.xviiTo My ParentsxviiiWhuptyr EIntroxuwtionEBE Buwkgrounx unx aotivutionOutdoor optical wireless communications (OWC), also known as free-space optical (FSO) com-munications, has drawn significant research interests in recent years due to its advantages of lowercosts, larger available bandwidths, better security, greater deployment flexibility, and a reducedtime-to-market [1, Chapter 1] [2, Chapter 1]. These advantages make FSO a promising solutionfor solving the “first-mile” and “last-mile” problems [3]. The on-off keying (OOK) and the pulseposition modulation (PPM) are commonly used signal modulation schemes for the OWC systems[4, 5]. As an alternative modulation scheme, subcarrier intensity modulated (SIM) FSO systemwas first proposed in [6]. The SIM FSO system has gained more attention due to its ability toachieve higher throughput than the OOK system and lower implementation complexity than thePPM system [7].However, the FSO system suffers from the cloud coverage and harsh weather conditions. Theinduced atmospheric effects degrade the system performance. Also, rain, snow, sleet, fog, dust, heat,etc. can affect our viewing of distant objects [4]. Therefore, these factors can affect the transmissionof laser beams through the atmosphere. Absorption, scattering, and refractive-index fluctuations(i.e., optical turbulence1) are three primary atmospheric processes that affect the optical wavepropagation through the atmosphere [8]. Typically, we can group the absorption and the scatteringtogether under the topic of extinction which is defined as the reduction or attenuation in theamount of the radiation passing through the atmosphere. Both of them are deterministic effectsthat are well studied, and their effects can be predicted using software packages such as LOWTRAN,FASCODE, MODTRAN, HITRAN, and LNPCWIN as a function of wavelength  [55, Chapter 1].1cptiwul turvulynwy is u suvsyt oz utmosphyriw turvulynwy. Howyvyr, wy will not xistinguish vytwyyn thysy twotyrminologiys unx usy thym intyrwhungyuvly in this thysis.11.1. 5TckgebhaW TaW MbgiiTgibaOn the other hand, the optical turbulence is a random effect which can be considered to be themajor source of the performance degradation through the FSO link.In order to study the system performance analytically, different statistical distributions havebeen proposed to describe the random variation in signal irradiance due to the scintillation causedby the inhomogeneities in both temperature and pressure along the transmission path [10]. Amongthese distributions, the Gamma-Gamma distribution has the ability to describe both the weak andstrong turbulence conditions. The Gamma-Gamma model is based on the assumption that thetransmitted signals experience both the small-scale and large-scale fluctuations which are modeledby two Gamma random variables (RVs) [9, 11]. The performances of the SIM FSO systems havebeen studied extensively and the early works can be found in [3, 12–16]. In [12], Popoola em Ze.studied the bit-error rate (BER) performance of a subcarrier binary phase-shift keying (BPSK)system. However, they did not provide a closed-form expression. Then, the adaptive subcarrierphase-shift keying (PSK) system over the Gamma-Gamma fading channels was investigated in [13].The results are presented in terms of the Meijer’s G-function, which does not reveal any insightsinto the subcarrier systems [4]. More recently, using a direct integration approach, Song em Ze’studied the error rate of the subcarrier FSO systems employing BPSK, quadrature phase-shiftkeying (QPSK), differential phase-shift keying (DPSK), and noncoherent frequency-shift keying(NCFSK) over the Gamma-Gamma fading channels [14]. In addition, the BER performance ofthe subcarrier differential quadrature phase-shift keying (DQPSK) FSO systems over the Gamma-Gamma fading channels was studied in [15]. In [16], Hassan em Ze. studied the BER performanceof the subcarrier OWC systems using non-adaptive and adaptive PSK modulation schemes overthe Gamma-Gamma fading channels. Besides the subcarrier PSK systems, the symbol error rate(SER) performance of the SIM FSO system over the Gamma-Gamma channels with rectangularquadrature amplitude modulation (QAM) was investigated in [3].Furthermore, the performance of the SIM FSO system is affected by the imperfection of theanalog front-end. For the analog front-end, there are different transceiver architectures includingheterodyne architecture, direct-conversion architecture (also called zero-intermediate frequency (IF)or homodyne conversion), image-rejection architecture and low-IF architecture [17, Chapter 4][18, Chapter 3]. Using a direct-conversion transceiver, the signals are directly up-converted from21.1. 5TckgebhaW TaW MbgiiTgibabaseband to radio frequency (RF) at the transmitter end and down-converted from RF to basebandat the receiver end [17]. The direct-conversion transceiver was more popular in the past decadedue to its better image-rejection, simpler structure and higher integrability [17, Chapter 4] [19].However, the direct-conversion transceiver is sensitive to the front-end impairment including directcurrent (DC) offset, in-phase and quadrature (I/Q) imbalance, even-order distortion and oscillatorleakage [19, 20]. In this thesis, we will focus on the impacts of the I/Q imbalance using the direct-conversion transceiver.For quadrature modulation such as QPSK and QAM, quadrature oscillators play a critical rolein up-conversion and down-conversion. For perfect quadrature oscillators, an in-phase local carriercos(2.fxt) and its 90 degree phase-shift quadrature local carrier − sin(2.fxt) are generated [21].However, in a practical QPSK system, the mismatches between the amplitudes of the mixers andthe errors in the 90 degree phase-shift lead to imbalances in the amplitudes and phases of the I andQ branches. This phenomenon is described as I/Q imbalance or I/Q mismatch in the literature[17, 21]. The I/Q imbalance is unavoidable since the current integrated circuit technologies suchas the low-cost complementary metal-oxide semiconductor (CMOS) technology have considerablemismatches between the circuit components [22]. The mismatches could be caused by the edgeeffects, implantation and surface-state charges, oxide effects, and mobility effects [23]. Also, I/Qimbalance tends to be larger in the direct-conversion architecture than in the heterodyne architec-ture [17]. As shown in [17, 24–26], the decoded signals are severely distorted by the I/Q imbalance.Thus, the I/Q imbalance cannot be ignored and must be taken into account in the performanceanalysis of the communication systems. In addition, in an orthogonal frequency division multiplex-ing (OFDM) system, the OFDM carriers are interfered by their image carriers [22]. Therefore, theOFDM system is more sensitive to the I/Q imbalance than the single carrier system [27]. SinceFSO communication channels are mainly frequency-nonselective, this thesis will primarily focus onsingle subcarrier communication using a benchmark QPSK modulation [28, Chapter 3].Since the quadrature oscillators are used at both sides of the QPSK system, I/Q imbalance canoccur at both transmitter and receiver ends of the QPSK system [18, Chapter 3]. The impacts of thetransmitter I/Q imbalance, the receiver I/Q imbalance and the I/Q imbalance at both transmitterand receiver ends need to be discussed separately, since these three cases have slightly different31.2. Ghefif BegTaimTgiba TaW 6bageiUhgibafmodels [18]. The impacts of the transmitter and receiver imperfections on the performance ofthe coherent optical QPSK communication systems were shown in [25]. However, they did notprovide an analytical tool to quantify the performance degradation. In [29], Wang em Ze. studiedthe monobit digital receivers for a QPSK system with I/Q imbalance. The performance of anOFDM system with I/Q imbalance at both transmitter and receiver ends is studied in [27]. In[30], Zareian em Ze. analyzed the BER performance of the QAM OFDM systems in the presence ofthe I/Q imbalance. However, they did not consider the interference between the OFDM carriersin the performance analysis. The exact signal-to-interference-plus-noise ratio (SINR) analysis ofan OFDM system with I/Q imbalance at both transmitter and receiver ends was investigated in[31]. Using the SINR metric, the outage probability of an OFDM system over the c∗Nakagami-mchannels was studied in [20, 32]. In [33], Zou em Ze. studied the performance of an OFDM systemwith I/Q imbalance over the Rayleigh fading channels. Also, the authors in [34] and [35] investigatedthe performance of the DPSK and DQPSK systems with I/Q imbalance over the Rayleigh fadingchannels. To this end, the majority of the literature has focused on how to mitigate the I/Qimbalance [22, 36–43]. Different methods have been proposed to mitigate the transmitter or thereceiver I/Q imbalance [22, 36, 37, 39, 40]. Joint compensation of transmitter and receiver I/Qimbalance in the OFDM system was proposed in [38]. Also, the compensation algorithms havebeen proposed to further reduce the impacts of the transmitter or receiver I/Q imbalances usingthe low-IF topology [41, 42]. In addition, the digital signal processing methods were proposed tomitigate the I/Q imbalance (refer to [43, 44], and the references therein). To our best knowledge,no prior work has studied the impacts of I/Q imbalance on the performance of subcarrier QPSKsystems over the atmospheric turbulence channels.EBF hhysis crgunizution unx WontrivutionsThis thesis consists of six chapters. A summary of each chapter and its contributions arepresented as follows:In Chapter 1, we present some background knowledge of the FSO system and I/Q imbalance.The motivation of this research is justified. We also provide a comprehensive review of FSO andI/Q imbalance literature related to the research topics of this thesis.41.2. Ghefif BegTaimTgiba TaW 6bageiUhgibafChapter 2 presents essential technical background for the entire thesis. First, we provide abrief description of a subcarrier FSO system. The additive noise is also investigated. Then, thecharacteristics of the Gamma-Gamma fading channels are introduced for the ensuing analysis inthis thesis. Finally, the I/Q imbalance at the analog front-end is described briefly.In Chapter 3, we study the SER performance of the subcarrier QPSK system over the Gamma-Gamma fading channels with transmitter I/Q imbalance only. The conditional SER expression isobtained at first. Using the moment generating function (MGF) of the Gamma-Gamma distribu-tion, we derive the average SER of the subcarrier QPSK systems with transmitter I/Q imbalance.Closed-form error rate expression is derived in terms of an infinite series. For practical evaluation,we present an accurate approximate series expression of the average SER by eliminating the infiniteterms except the first J terms. We also carry out a truncation error analysis and an asymptoticanalysis.In Chapter 4, we perform an analytical analysis of the subcarrier QPSK system over the Gamma-Gamma fading channels with receiver I/Q imbalance only. The crosstalk of the quadrature branchescauses the correlation between the I-component and Q-component signals. The exact conditionalSER expression is derived as a function of the Gaussian f-function and the Owen’s i -function.Using the MGF of the Gamma-Gamma distribution, the closed-form average SER expression isobtained in terms of an infinite series. We discover that the level of the amplitude I/Q imbalance isirrelevant to the average SER performance. Then, we present an approximate series expression ofthe average SER with finite terms. We also carry out a truncation error analysis and an asymptoticanalysis for the obtained SER expression.Chapter 5 investigates the error rate performance of the subcarrier QPSK system with I/Qimbalance at both transmitter and receiver ends over the Gamma-Gamma fading channels. Inthe presence of the receiver I/Q imbalance, the crosstalk of the quadrature branches leads to thecorrelation between the I-component and Q-component signals. Consequently, the conditional SERcan be derived in terms of the Gaussian f-function and the Owen’s i -function. Using the MGFof the Gamma-Gamma distribution, the closed-form average SER expression is obtained in termsof an infinite series. We discover that the level of the receiver amplitude imbalance is irrelevantto the SER performance of the subcarrier QPSK system. Then, we present an approximate series51.2. Ghefif BegTaimTgiba TaW 6bageiUhgibafexpression of the average SER for the practical evaluation. In addition, we carry out a truncationerror analysis and an asymptotic analysis. Numerical results are presented to verify the accuracyof the analysis and show the performance improvement using a calibrated transceiver.Finally, we summarize the thesis in Chapter 6 and suggest some further research topics relatedto this thesis.6Whuptyr Fgystym aoxylIn this chapter, we provide a brief description of the subcarrier FSO system first. The additivenoise at the receiver is also discussed. Then, some background knowledge and the statistics of theGamma-Gamma fading channel are presented. Finally, we briefly describe the I/Q imbalance atthe analog front-end.FBE guvwurriyr Fgc gystymFigure 2.1 presents the block diagram of a subcarrier FSO system. In such a system, an RFsignal m(t) is modulated with the data at first. After proper DC biasing, the modulated RF signalis used to modulate the irradiance of a continuous wave optical laser beam [45]. The transmittedirradiance can be written as [3]iI(t) = e [1 + m(t)] (2.1)where e is the average transmitter power and  is the modulation index satisfying −1 ≤ m(t) ≤ 1in order to avoid the overmodulation.At the receiver end, the received optical power is converted into the electrical signal throughdirect detection at the photodetector. We assume a positive-intrinsic-negative (PIN) photodetectorin this thesis. In most practical systems, the background noise (dominated by ambient shot noiseand/or thermal noise) at the receiver can be modeled as additive white Gaussian noise (AWGN)[5]. The photocurrent at the output of the photodetector is given byiG(t) = egI(t)[1 + m(t)] + np(t) (2.2)where g is the photodetector responsivity in amperes per watt; I(t) is the instantaneous channel72.1. FhUcTeeiee FFB FlfgemFigure 2.1: Block diagram of a subcarrier FSO system.gain; np(t) is an AWGN random process, and it becomes an AWGN RV with mean zero andvariance 2p after sampling. In an FSO system, we can treat I(t) as a constant over a number ofconsecutive symbol durations since the typical data rate of an FSO system is hundreds to thousandsof Mbps while the coherence time of the turbulence channel is on the order of msec [5, 46, 47]. Afterremoving the DC bias and demodulating through an electrical demodulator, the sampled electricalsignal is used to recover the transmitted information [4, 6].For a subcarrier FSO system, the instantaneous receiver signal-to-noise ratio (SNR) can bedefined as the ratio of the time-averaged photocurrent power to the variance of the noise [14], andit can be written as [45, 48] =(eg)22pI2 (2.3)where I is the turbulence induced channel gain obtained by sampling the instantaneous channelgain I(t). In (2.3),  is the SNR per symbol.82.2. 4WWigiie Abife Tg ghe EeceiieeFBF Axxitivy boisy ut thy fywyivyrAt the receiver end, the optical receiver converts the incident power into the electric currentthrough a photodiode. In addition to the turbulence, the received signals are affected by additivenoise. There are two major types of additive noise: thermal noise and shot noise. The additivenoise at the receiver is also related the type of used photodiode, and we will focus on the PINphotodiode in this thesis. We will give a brief review of the two types of additive noise. Then theSNR in optical receivers will be investigated.FBFBE hhyrmul boisyThermal noise (also known as Johnson noise [49] or Nyquist noise [50]) is generated by randomthermal motion of the electrons and atoms in a resistor. The random thermal motion creates arandom voltage signal across its terminals. Mathematically, thermal noise can be modeled as astationary Gaussian random process with a spectral density that is frequency independent wellinto the terahertz spectrum [4]. Thus, it can be considered to be white noise. We can obtain thespectral density of the thermal noise as [51]ht(f) =2kBiga(2.4)where kB is the Boltzmann constant; i is the absolute temperature in Kelvin; ga is the loadresistance. Therefore, the thermal noise variance can be obtained as [2, Chapter 2]2t =∫ ∞−∞ht(f)|H(f)|2yf = 4△fkBiga(2.5)where H(f) is the frequency response of the filter at the receiver and we assume |H(f)| = 1; △fis the effective noise bandwidth of the receiver. We can observe that the thermal noise spectraldensity ht(f) and the resulting photocurrent noise variance do not depend on the received averagephotocurrent.92.2. 4WWigiie Abife Tg ghe EeceiieeFBFBF ghot boisySince light and electric current consist of the movement of quantized ‘packets’, shot noise isgenerated due to the photon and electron quantization error. Photon quantization induces quantumnoise and electron quantization induces dark noise. The shot noise was first introduced in 1918by Schottky [52], who studied fluctuations of current in vacuum tubes, and has been extensivelyinvestigated in [53, Chapter 3].The energy associated with the particles comes in discrete steps. A photon with a frequency ,will have an energy of hp, where hp is the Planck’s constant. Therefore, it is not possible to havea continuous flow of energy. Instead, the energy comes as bursts of particles that are witnessedas the quantum noise. Mathematically, quantum noise is a stationary random process followingthe Poisson statistics. In practice, the quantum noise can be approximated by Gaussian statistics.In this thesis, quantum noise is assumed to be white noise with a constant spectral density. Thespectral density of the quantum noise is given by hq(f) = qIp where q is the electronic chargeand Ip is the received average photocurrent. The quantum noise variance can be obtained as [2,Chapter 4]2q =∫ ∞−∞hq(f)yf = 2△fqIp: (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. Due to the thermal effects in thesemiconductor, electrons and holes are liberated as these carriers overcome the bandgap. Thisphenomenon results in a time-averaged dark current Iy with spectral density hy(f) = qIy andvariance2y =∫ ∞−∞hy(f)yf = 2△fqIy: (2.7)Therefore, we can obtain the total variance of shot noise as2s = 2q + 2y = 2△fq(Ip + Iy): (2.8)102.3. Ghe GTmmT-GTmmT FTWiag 6hTaaelFBFBG gignulAtoAboisy futioThe performance of an optical system depends on the SNR which is defined as the ratio of thesignal power to the total noise power. The signal power is proportional to the received photocurrentsquared. Using the thermal noise variance and the shot noise variance, we can obtain the noisepower as 2p = 2t + 2s . Therefore, the SNR can be expressed asSNR =I2p2p=I2p2△fq(Ip + Iy) + 4△fKBiRga : (2.9)In a subcarrier FSO system, the received photocurrent is given in (2.2) so that I2p = (egI)2, and(2.9) specializes to (2.3). In a PIN photodiode, thermal noise tends to dominate since the incidentsignal power and the dark current are relatively low [4].FBG hhy GummuAGummu Fuxing WhunnylIn order to analytically study the system performance over the atmospheric turbulence channels,several statistical distributions have been proposed to model the irradiance fluctuation under theturbulence environment. Among these distributions, the Gamma-Gamma distribution is capable ofmodeling the irradiance of both the weak and strong turbulence conditions. In the Gamma-Gammafading channel, the irradiance I can be modeled as the weak eddies induced irradiance fluctuationmodulated by the strong eddies induced irradiance. We can obtain the irradiance I by I = mnwhere m and n are, respectively, the strong and the weak eddies induced irradiance fluctuation.RVs m and n follow the Gamma distribution having the probability density functions (pdfs)fm(x) =x−1Γ()exp(−x) (2.10)andfn (y) =y−1Γ()exp(−y) (2.11)112.4. FgTgifgicf bf ghe 6hTaael GTia TaW Eeceiiee FAEwhere Γ (·) is the Gamma function;  and  are, respectively, the effective numbers for the large-scale and small-scale eddies. Conditioned on m, we can obtain the conditional pdf asfI|m (I|m = x) =(Ix)−1xΓ()exp(−Ix): (2.12)Averaging (2.12) w.r.t. the Gamma pdf of m in (2.10), we can obtain the pdf of the irradianceI as [54, 55]fGG(I) =2()+2 I+2−1Γ()Γ()K−(2√I) (2.13)where K− (·) is the modified Bessel function of the second kind of order  − . The positiveshaping parameters  and  are related to the Rytov variance, and they can be determined by [54] =exp 0:492R(1 + 1:1112R5R)7R6− 1−1(2.14)and =exp 0:512R(1 + 0:6912R5R)5R6− 1−1(2.15)where 2R is the Rytov variance. The Rytov variance is an important parameter in the literatureon atmospheric turbulence studies, and it is defined as [54]2R = 1:23X2nkn7+a11+p (2.16)where X2n is the index of refraction structure parameter; kn = 2.R is the optical wave number (is the wavelength); ap is the propagation path length between transmitter and receiver.FB4 gtutistiws oz thy Whunnyl Guin unx fywyivyr gbfIn a Gamma-Gamma turbulence environment, the pdf of the irradiance I is given by (2.13).Using the following series expansion of the modified Bessel function of the second kind [56, eq.122.4. FgTgifgicf bf ghe 6hTaael GTia TaW Eeceiiee FAE(03.04.06.0002.01)]K(x) =.2 sin(.)∞∑p=0[(xR2)2p−Γ(p− + 1)p! −(xR2)2p+Γ(p+ + 1)p!](2.17)with  R∈ ZP |x| Q∞, we can obtain a series expression of the Gamma-Gamma pdf as [14]fGG(I) =∞∑p=0[vp(P )Ip+−1 + vp(P )Ip+−1](2.18)wherevp(P ) =.()p+Γ()Γ()Γ(p− +  + 1)p! sin [(− ).] : (2.19)Using an integral identity [57, eq. 3.326(2)]∫ ∞0xm exp (−txn) yx = Γ(m+1n)ntm+1n(2.20)the MGF of I can be calculated as [14]MIPGG(s) = Z[e−sI]=∞∑p=0[vp(P )Γ(p+ )s−(p+) + vp(P )Γ(p+ )s−(p+)] (2.21)where Z [·] denotes the statistical expectation operation.Using (2.9) and (2.18), the SNR  can be shown to be another Gamma-Gamma RV having pdf[14]fPGG() =12∞∑p=0[vp(P )() p+2−1 + vp(P )() p+2−1]: (2.22)Therefore, the MGF of  can be derived asMPGG(s) = Z[e−s ]=∞∑p=0vp(P )Γ(p+2)2(s)−p+2 +vp(P )Γ(p+2)2(s)−p+2 : (2.23)132.5. I/Q ImUTlTaceFigure 2.2: Block diagram of an ideal QPSK system.FBI ICe ImvulunwyFor an ideal QPSK system, shown in Fig. 2.2, an in-phase local carrier cos(2.fxt) and its90 degree phase-shift quadrature local carrier − sin(2.fxt) are generated where fx is the carrierfrequency [21]. As shown in Fig. 2.3, in a practical QPSK system, the mismatches between theamplitudes of the mixers and the errors in the 90 degree phase-shift lead to imbalances in theamplitudes and phases of the I and Q branches. This phenomenon is described as I/Q imbalanceor I/Q mismatch in the literature [21] [17, Chapter 4]. In addition, the amplitude and phase offsetscan occur at both transmitter and receiver ends of the QPSK system. Using direct-conversiontransceivers, the I/Q imbalance is unavoidable since the current integrated circuit technologies suchas the low-cost complementary metal-oxide semiconductor (CMOS) technology have considerablemismatches between the circuit components due to the fabrication process variations including thedoping concentration, oxide thickness, mobility, and geometrical sizes over the chips [22]. There aretwo different sources for the I/Q imbalance, one is from the mixer, which is frequency independent(FI); the other is from the mismatched analog filters in the I and Q branches, which is frequencydependent (FD) [58, Chapter 2]. However, for narrowband systems, the I/Q imbalance can beassumed to be FI [58, Chapter 3] [59]. Since we will study the QPSK system in this thesis, wewill focus on the FI I/Q imbalance in the ensuing discussion. The quadrature mixers with higher142.6. FhmmTelFigure 2.3: Block diagram of a subcarrier QPSK system with I/Q imbalance at both sides.carrier frequency will experience greater I/Q imbalance [17]. For example, a delay mismatch of10ps between two mixers cause a phase imbalance of 21:6 degree at 6 GHz and 3:6 degree at 1 GHz[17].FB6 gummuryIn this chapter, we presented essential technical background knowledge for the entire thesis.First, we provided a brief description of a subcarrier FSO system. The additive noise was alsoinvestigated. Then, the characteristics of the Gamma-Gamma fading channels were introduced forthe ensuing analysis in this thesis. Finally, the I/Q imbalance in analog front-end was describedbriefly.15Whuptyr Gguvwurriyr edgK gystym kithhrunsmittyr ICe Imvulunwy cvyr thyGummuAGummu Fuxing WhunnylsIn this chapter, we study the SER performance of a subcarrier QPSK system over the Gamma-Gamma fading channels with mkZglfbmmek I/Q imbalance. The conditional SER expression is ob-tained at first. Using the MGF of the Gamma-Gamma distribution, we derive the average SERexpression of a subcarrier QPSK systems with transmitter I/Q imbalance. Closed-form error rateexpression is derived in terms of an infinite series. For practical evaluation, we present an accurateapproximate series expression of the average SER by eliminating the infinite terms except the firstJ terms. We also carry out a truncation error analysis and an asymptotic analysis.GBE Wonxitionul gEfIn this section, we first present the conditional SER of an ideal QPSK system in order toestablish our notational convention. Then, we derive the conditional SER of a QPSK system withtransmitter I/Q imbalance.GBEBE Ixyul edgK gystymFor an ideal QPSK system shown in Fig. 2.2, the transmitted complex data can be written asA(k) = AI(k) + jAQ(k) (3.1)163.1. 6baWigibaTl FEEwhere k is the index of the data sequence; j is the imaginary unit satisfying j2 = −1; AI(k) = ±1and AQ(k) = ±1, respectively, denote the I-component data and Q-component data.We assume perfect synchronization with an analog-to-digital converter (ADC) and a digital-to-analog converter (DAC) in this thesis. For brevity, the data sequence index k will be omitted inthe ensuing analysis. With up-conversion, the transmitted complex baseband representation of theup-converted signal can be written ass(t) = sI(t) + jsQ(t) (3.2)where sI(t) and sQ(t) are, respectively, the transmitted baseband I-component signal and Q-component signal. In an ideal QPSK system, the equalities sI(t) = AI and sQ(t) = AQ canbe obtained.After the transmitted signal goes through the AWGN and fading channels, the received complexbaseband signal isr(t) = Is(t) + n(t) (3.3)where n(t) is an AWGN random process, and it becomes an AWGN RV with mean zero and variance2n after sampling. The noise term n(t) can be decomposed asn(t) = nI(t) + jnQ(t) (3.4)where nI(t) and nQ(t) are the baseband I-component noise and Q-component noise, respectively.nI(t) and nQ(t) are independent AWGN random processes, and they become AWGN RVs withmean zero and variances 2 = 2nR2 after sampling [28, Chapter 4].Substituting (3.2) and (3.4) into (3.3), we obtainr(t) = rI(t) + jrQ(t) (3.5)where rI(t) = IsI(t) + nI(t) and rQ(t) = IsQ(t) + nQ(t) are the received baseband I-componentsignal and Q-component signal, respectively.At the receiver front-end, the received signal will undergo down-conversion and low-pass filtering173.1. 6baWigibaTl FEEbefore detection. Therefore, the pre-detection complex signal is given byrˆ(t) = rˆI(t) + jrˆQ(t) (3.6)where rˆI(t) = IsˆI(t) + nˆI(t) and rˆQ(t) = IsˆQ(t) + nˆQ(t), respectively, denote the received pre-detection I-component signal and Q-component signal; sˆI(t) is the pre-detection I-component signal;sˆQ(t) is the pre-detection Q-component signal; nˆI(t) is the pre-detection I-component noise andnˆQ(t) is the pre-detection Q-component noise. In an ideal QPSK system without I/Q imbalance, wehave nˆI ∼ N (0P 2) and nˆQ ∼ N (0P 2) where nˆI and nˆQ are the sampled pre-detection I-componentnoise and Q-component noise respectively.In order to facilitate comparison between an ideal QPSK system and a QPSK system with I/Qimbalance, the normalized SNR2 is defined ast =I22= tI2 (3.7)where t is the unfaded normalized SNR.As a result, the conditional SER of an ideal QPSK system can be obtained as [28, Chapter 4][60, Chapter 5]eePId(t) = 2f (√t)−f2 (√t) (3.8)where the subscript Id denotes an ideal QPSK system; f(x) is the Gaussian f-function defined asf(x) = 1√2∫∞x exp(− t22 )yt.GBEBF hrunsmittyr ICe ImvulunwyFigure 3.1 shows the block diagram of a subcarrier QPSK system with transmitter I/Q imbal-ance. In this figure, the transmitter in-phase local carrier is (1 + T ) cos(2.fxt − T R2) and thetransmitter quadrature local carrier is −(1 − T ) sin(2.fxt + T R2) where T is the transmitteramplitude imbalance and T is the transmitter phase imbalance [18, Chapter 3].In the presence of transmitter I/Q imbalance, the received baseband I-component signal and2Without loss oz gynyrulity, thy proxuwt oz thy photoxytywtor rysponsivity unx thy moxulution inxyfi R is nor-mulizyx to unity.183.1. 6baWigibaTl FEEFigure 3.1: Block diagram of a QPSK system with transmitter I/Q imbalance.Q-component signal can be, respectively, obtained asrI(t) = I [(1 + T ) cos(T R2)AI − (1− T ) sin(T R2)AQ] + nI(t) (3.9)andrQ(t) = I [(1− T ) cos(T R2)AQ − (1 + T ) sin(T R2)AI ] + nQ(t): (3.10)It can be shown that the pre-detection I/Q component noise terms nˆI(t) and nˆQ(t) are bgde-iegdegm AWGN random processes, and they become bgdeiegdegm AWGN RVs with mean zero andvariances 2 = 2nR2 after sampling [28, Chapter 4]. Therefore, when conditioned on IPAI P AQ,RVs rˆI and rˆQ are also bgdeiegdegm and Gaussian-distributed with meansrˆI = I [(1 + T ) cos(T R2)AI − (1− T ) sin(T R2)AQ] (3.11)andrˆQ = I [(1− T ) cos(T R2)AQ − (1 + T ) sin(T R2)AI ] (3.12)and variances 2rˆI = 2rˆQ= 2. The proof can be found in Appendix A.193.1. 6baWigibaTl FEEFor a QPSK system with transmitter I/Q imbalance, the conditional SER is given byeePIx(t) =14[eePIxP1+j(t) + eePIxP−1−j(t) + eePIxP1−j(t) + eePIxP−1+j(t)] (3.13)where the subscript Tx denotes a subcarrier QPSK system with transmitter I/Q imbalance.For A = 1 + j (i.e. AI = 1, AQ = 1), it is straightforward to derive the means of rˆI and rˆQ asrˆI = I[(1 + T ) cos(T R2)− (1− T ) sin(T R2)] (3.14)andrˆQ = I[(1− T ) cos(T R2)− (1 + T ) sin(T R2)] (3.15)and their variances as 2rˆI = 2rˆQ= 2.Thus, the conditional SER is given byeePIxP1+j(t) = f (√thIxP1) +f (√tlIxP1)−f (√thIxP1)f (√tlIxP1) (3.16)wherehIxP1 , (1 + T ) cos(T R2)− (1− T ) sin(T R2) (3.17)andlIxP1 , (1− T ) cos(T R2)− (1 + T ) sin(T R2): (3.18)Similarly, for A = 1− j (i.e. AI = 1, AQ = −1), the conditional SER can be derived aseePIxP1−j(t) = f (√thIxP2) +f (√tlIxP2)−f (√thIxP2)f (√tlIxP2) (3.19)wherehIxP2 , (1 + T ) cos(T R2) + (1− T ) sin(T R2) (3.20)andlIxP2 , (1− T ) cos(T R2) + (1 + T ) sin(T R2): (3.21)203.2. 4ieeTge FEE 4aTllfifDue to the symmetry of the QPSK signals, it is straightforward to show that eePIxP−1−j(t) =eePIxP1+j(t) and eePIxP−1+j(t) = eePIxP1−j(t). Therefore, the conditional SER can be obtained aseePIx(t) =12f (√thIxP1) +12f (√tlIxP1) +12f (√thIxP2) +12f (√tlIxP2)− 12f (√thIxP1)f (√tlIxP1)− 12f (√thIxP2)f (√tlIxP2) :(3.22)It can be shown that when T = 0P T = 0, i.e. no I/Q imbalance occurs, eq. (3.22) is reduced tothe conditional SER expression of an ideal QPSK system shown in (3.8).GBF Avyrugy gEf AnulysisIn Chapter 2, we have introduced the pdf of the irradiance I in a series expression as (2.18).According to (3.7), the normalized SNR t is another Gamma-Gamma RV having pdf [14]ftPGG(t) =12∞∑p=0[vp(P )(tt) p+2−1t + vp(P )(tt) p+2−1t]: (3.23)Following the similar procedure to obtain (2.23), the MGF of t is given by [14]MtPGG(s) = Z[e−st ]=12∞∑p=0[vp(P )Γ(p+ 2)(st)− p+2 + vp(P )Γ(p+ 2)(st)− p+2]:(3.24)GBFBE Error futy oz un Ixyul edgK gystymThe average SER of an ideal QPSK system over the Gamma-Gamma fading channels is givenby3eePQPSKPIdPGG =∫ ∞0ee(t)ftPGG(t)yt (3.25)where ee(t) is the conditional SER shown in (3.8). Thus, the average SER can be calculated aseePQPSKPIdPGG = 2∫ ∞0f(√t)ftPGG(t)yt︸ ︷︷ ︸eI−∫ ∞0f2(√t)ftPGG(t)yt︸ ︷︷ ︸eII: (3.26)3In this thysis, wy ussumy un unwoxyx systym unx thy normulizyx gbf t is pry-fiyx.213.2. 4ieeTge FEE 4aTllfifWith the definition of the Gauss Hypergeometric function 2F 1(· P · ; · ; ·) [57, eq. (9.111)], it isstraightforward to obtain the integral identity as (see proof in Appendix B)∫ 20(sin )xy = 2F 1(1− x2P12;32; 1): (3.27)Using (3.27), eI can be obtained aseI =12.∞∑p=0[2p+2 vp(P )Γ(p+ 2)t− p+2 2F 1(1− p− 2P12;32; 1)+2p+2 vp(P )Γ(p+ 2)t− p+2 2F 1(1− p− 2P12;32; 1)]:(3.28)With the help of an integral identity [61, eq. (06.27.21.0132.01)]∫ ∞0erfc (t) tm−1yt =1√.mΓ(m+ 12)Re[m] S 0 (3.29)where erfc(x) is the complementary error function defined as erfc(x) = 2√∫∞x exp(−t2)yt and Re [·]denotes the real part. eI has an alternative expression aseI =12√.∞∑p=02 p+2 vp(P )Γ(p++12)p+ t− p+2 + 2p+2vp(P )Γ(p++12)p+ t− p+2 : (3.30)Eqs. (3.28) and (3.30) are numerically equivalent to [14, eq. (6)].Also, we can obtain eII as [14]eII =12.∞∑p=0[vp(P )√2(1 + p+ )Γ(p+ 2)t− p+2 2F 1(1 + p+ 2P12;3 + p+ 2;12)+vp(P )√2(1 + p+ )Γ(p+ 2)t− p+2 2F 1(1 + p+ 2P12;3 + p+ 2;12)]:(3.31)With (3.30) and (3.31), a series expression of the average SER of an ideal subcarrier QPSKsystem can be written aseePQPSKPIdPGG = 2eI − eII: (3.32)We can observe that eI and eII involve only the Gamma function and the Gauss Hypergeometric223.2. 4ieeTge FEE 4aTllfiffunction. Therefore, eePQPSKPIdPGG can be evaluated using standard scientific software such as Matlabwithout requiring numerical integration.GBFBF Error futy with hrunsmittyr ICe ImvulunwyFor a subcarrier QPSK system with transmitter I/Q imbalance, the average SER over theGamma-Gamma fading channels is given by4eePQPSKPIxPGG =∫ ∞0eePIx(t)ftPGG(t)yt (3.33)where eePIx(t) is the conditional SER shown in (3.22). Therefore, we obtain the average SER aseePQPSKPIxPGG =12∫ ∞0(f(√thIxP1) +f(√tlIxP1) +f(√thIxP2) +f(√tlIxP2)) ftPGG(t)yt︸ ︷︷ ︸eIx;I− 12∫ ∞0(f(√thIxP1)f(√tlIxP1) +f(√thIxP2)f(√tlIxP2)) ftPGG(t)yt︸ ︷︷ ︸eIx;II:(3.34)For the empirical values of the transmitter amplitude imbalance T and phase imbalance T usedin [21], the values of hIxP1, lIxP1, hIxP2 and lIxP2 are all positive.Similar to (3.30), it is straightforward to show thateIxPI =12√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)IxP1 + l−(p+)IxP1 + h−(p+)IxP2 + l−(p+)IxP2)+12√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)IxP1 + l−(p+)IxP1 + h−(p+)IxP2 + l−(p+)IxP2):(3.35)Using the following integral identity (see proof in Appendix C)∫ tan−1(hl )0(sin )xy =(h2h2 + l2)x+12 1x+ 12F 1(12Px+ 12;x+ 32;h2h2 + l2)(3.36)eIxPII can be presented aseIxPII = g1(P ) + g1(P ) (3.37)4In this thysis, wy ussumy un unwoxyx systym unx thy normulizyx gbf t is pry-fiyx.233.3. GehacTgiba Eeebe 4aTllfifwhere g1(P ) is given byg1(P ) =14.∞∑p=0vp(P )Γ(p+2)p+  + 1t− p+2×√ h2IxP1h2IxP1 + l2IxP1(2h2IxP1 + l2IxP1) p+22F 1(p+  + 12P12;p+  + 32;h2IxP1h2IxP1 + l2IxP1)+√l2IxP1h2IxP1 + l2IxP1(2h2IxP1 + l2IxP1) p+22F 1(p+  + 12P12;p+  + 32;l2IxP1h2IxP1 + l2IxP1)+√h2IxP2h2IxP2 + l2IxP2(2h2IxP2 + l2IxP2) p+22F 1(p+  + 12P12;p+  + 32;h2IxP2h2IxP2 + l2IxP2)+√l2IxP2h2IxP2 + l2IxP2(2h2IxP2 + l2IxP2) p+22F 1(p+  + 12P12;p+  + 32;l2IxP2h2IxP2 + l2IxP2) :(3.38)Thus, a series expression of the average SER of a subcarrier QPSK system with transmitterI/Q imbalance can be written aseePQPSKPIxPGG =12(eIxPI − eIxPII) : (3.39)When T = 0 and T = 0, i.e. no imbalance occurs, eq. (3.39) reduces to the average SERexpression of an ideal QPSK system given by (3.32). Also, we observe that eIxPI and eIxPII involveonly the Gamma function and the Gauss Hypergeometric function. Thus, eePQPSKPIxPGG can beevaluated using standard scientific software such as Matlab without requiring numerical integration.GBG hrunwution Error AnulysisThe average SER of a subcarrier QPSK system over the Gamma-Gamma fading channels withtransmitter I/Q imbalance can be obtained from (3.39). However, each term of the right-hand sideof (3.39) contains summations of infinite terms. For practical evaluation, we need an expression offinite terms. Therefore, truncation error is introduced to represent the error due to the eliminationof infinite terms except the first J + 1 terms [3].243.3. GehacTgiba Eeebe 4aTllfifTo evaluate the truncation error of (3.39), we define two error factors as"JPIxP1(h) , Λ(P )∞∑p=J+11p!(√2√th)p[upPIxP1(P P h)− upPIxP1(P P h)] (3.40)and"JPIxP2(hP l) , Λ(P )∞∑p=J+11p!(√2√th)p[upPIxP2(P P hP l)− upPIxP2(P P hP l)] (3.41)whereΛ(P ) , .Γ()Γ() sin [(− ).] (3.42)upPIxP1(P P h) ,Γ(p++12 )(√2√th)Γ(p− +  + 1)(p+ )(3.43)andupPIxP2(P P hP l) ,Γ(p+2 )(√2√th)Γ(p− +  + 1)∫ tan−1(hl )0(sin )p+y:(3.44)With (3.40) and (3.41), the truncation error of (3.39) is given by"JPIx =14√.("JPIxP1(hIxP1) + "JPIxP1(lIxP1) + "JPIxP1(hIxP2) + "JPIxP1(lIxP2))− 18.("JPIxP2(hIxP1P lIxP1) + "JPIxP2(lIxP1P hIxP1) + "JPIxP2(hIxP2P lIxP2) + "JPIxP2(lIxP2P hIxP2)) :(3.45)Using the Taylor series expansion of the exponential function, the upper bounds of (3.40) and(3.41) can be, respectively, derived as"JPIxP1(h) ≤ Λ(P ) exp(√2√th)maxp>J{upPIxP1(P P h)− upPIxP1(P P h)} (3.46)and"JPIxP2(hP l) ≤ Λ(P ) exp(√2√th)maxp>J{upPIxP2(P P hP l)− upPIxP2(P P hP l)}: (3.47)After examining (3.43), it can be noted thatΓ( p+y+12)Γ(p−x+y+1)(p+y) decreases with an increase of p and253.4. 4flmcgbgic FEE 4aTllfifbecomes 0 when p approaches ∞. Hence, when the unfaded normalized SNR t increases, "JPIxP1diminishes rapidly. Similarly, with the bounded characteristic of∫ tan−1(hl )0 (sin )xy, "JPIxP2 di-minishes rapidly with an increase of t. Thus, the truncation error "JPIx diminishes rapidly withincreasing value of t. This suggests that the finite series solutions in high SNR regimes is accurateand can allow an asymptotic analysis.GB4 Asymptotiw gEf AnulysisIn an FSO system over the Gamma-Gamma fading channels, typically we have  S  S 0 inmost scenarios [62]. Therefore, the term t2− p+2 decreases faster than the term t2− p+2 in (3.32) forthe same p value when t increases. Consequently, the leading term of the series in (3.32) becomesdominant when t approaches ∞. Therefore, with the identity .Rsin(.x) = Γ(x)Γ(1 − x), theaverage SER of an ideal subcarrier QPSK system in high SNR regimes can be approximated bye∞ePQPSKPIdPGG =()Γ(− )t−2Γ()Γ()22 Γ(+12)√.−Γ(2)2F 1(1+2 P12 ;3+2 ;12)2√2.(1 + ) : (3.48)Similarly, for a subcarrier QPSK system with transmitter I/Q imbalance, we can calculate theaverage SER in high SNR regimes ase∞ePQPSKPIxPGG =()Γ(− )Γ(+12)4√.Γ()Γ()(t2)−2 (h−IxP1 + l−IxP1 + h−IxP2 + l−IxP2)−()Γ(− )Γ(2)8.(1 + )Γ()Γ()(t2)−2×√ h2IxP1h2IxP1 + l2IxP1(1h2IxP1 + l2IxP1)22F 1(1 + 2P12;3 + 2;h2IxP1h2IxP1 + l2IxP1)+√l2IxP1h2IxP1 + l2IxP1(1h2IxP1 + l2IxP1)22F 1(1 + 2P12;3 + 2;l2IxP1h2IxP1 + l2IxP1)+√h2IxP2h2IxP2 + l2IxP2(1h2IxP2 + l2IxP2)22F 1(1 + 2P12;3 + 2;h2IxP2h2IxP2 + l2IxP2)+√l2IxP2h2IxP2 + l2IxP2(1h2IxP2 + l2IxP2)22F 1(1 + 2P12;3 + 2;l2IxP2h2IxP2 + l2IxP2) :(3.49)263.5. AhmeeicTl EefhlgfFor the Gamma-Gamma fading channels, the asymptotic average SER shown in (3.48) and (3.49)converge to the exact value rapidly when the difference between  and  is large. From (3.48)and (3.49), we can also observe that the ideal subcarrier QPSK system and the subcarrier QPSKsystem with transmitter I/Q imbalance have the same diversity order of 2 .To continue the evaluation, we first define iasy and iasyPIx asiasy , e∞ePQPSKPIdPGGΓ()Γ()()Γ(− )t2iasyPIx , e∞ePQPSKPIxPGGΓ()Γ()()Γ(− )t2 :(3.50)Then we introduce the asymptotic SNR penalty factor SNR∞Ix-Id as the required amount of additionalSNR of the system with transmitter I/Q imbalance to achieve the same SER performance of theideal system in high SNR regimes [45]. From (3.48) and (3.49), the asymptotic SNR penalty factorfor a subcarrier QPSK system is given bySNR∞Ix-Id =210 log(iasyPIxiasy)dB (3.51)where log (·) is the logarithmic function with base 10. It can be seen that the SNR penalty factordepends on the smaller channel parameter  only. We comment that transmitter I/Q imbalance willnot affect the diversity order, but only introduce a SNR penalty. Thus, we use (3.51) to quantifythis SNR penalty.GBI bumyriwul fysultsFigure 3.2 presents SER curves of a subcarrier QPSK system with transmitter I/Q imbalanceover the Gamma-Gamma fading channels having different levels of turbulence. The transmitterphase imbalance is set to T = 0:35. The transmitter amplitude imbalance is set to T = 0:15.Excellent agreement between the error rate obtained by numerical integration and the approximateerror rate obtained by eliminating infinite terms in series solutions can be found in Fig. 3.2.Asymptotic error rates are shown for strong ( = 2:04P  = 1:10), moderate ( = 2:50P  = 2:06)and weak ( = 4:43P  = 4:39) turbulence conditions. It can be observed that the asymptotic error273.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 6010−1010−910−810−710−610−510−410−310−210−1100Unfaded normalized SNR (dB)Average symbol error rate  Approx,α=2.04,β=1.10Numerical,α=2.04,β=1.10Approx,α=2.50,β=2.06Numerical,α=2.50,β=2.06Approx,α=4.43,β=4.39Numerical,α=4.43,β=4.39AsymptoticFigure 3.2: Average SER of a subcarrier QPSK system over the Gamma-Gamma channels withtransmitter I/Q imbalance with different levels of turbulence. The transmitter phase imbalance isset to T = 0:35. The transmitter amplitude imbalance is set to T = 0:15.283.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 60 7010−1210−1010−810−610−410−2100Unfaded normalized SNR (dB)Average symbol error rate  Tx I/Q imbalanceno imbalanceα=2.50, β=2.06α=2.04, β=1.10α=4.43, β=4.39Figure 3.3: Average SER of a subcarrier QPSK system with and without transmitter I/Q imbalanceover the Gamma-Gamma channels having different levels of turbulence. The transmitter phaseimbalance is set to T = 0:35. The transmitter amplitude imbalance is set to T = 0:15.293.6. FhmmTelrate converges to the error rate obtained by numerical integration faster in the strong ( = 2:04P  =1:10) turbulence condition than in the moderate ( = 2:50P  = 2:06) turbulence condition in highSNR regimes. However, in the weak ( = 4:43P  = 4:39) turbulence condition, the asymptoticerror rate does not converge to the error rate obtained by numerical integration in Fig. 3.2. Thiscan be explained that in the weak ( = 4:43P  = 4:39) turbulence condition, values of  and  arenearly equal, the convergence is too slow to be recognized in Fig. 3.2.Figure 3.3 presents the average SER of a subcarrier QPSK system with and without transmitterI/Q imbalance over the Gamma-Gamma fading channels having different levels of turbulence. Fora subcarrier QPSK system with transmitter I/Q imbalance, the SNR penalty is approximately 1:8dB at the SER level of 10−10 in the weak ( = 4:43P  = 4:39) turbulence condition, which agreeswith the prediction of 1:7 dB obtained by (3.51). Also, in the moderate ( = 2:50P  = 2:06)turbulence condition, the SNR penalty is approximately 1:0 dB at the SER level of 10−5, whichagrees with the prediction of 1:1 dB. In the strong ( = 2:04P  = 1:10) turbulence condition, theSNR penalty is approximately 0:80 dB at the SER level of 10−3, which agrees with the predictionof 0:78 dB. We observe that the SNR penalty decreases as the turbulence becomes stronger. Thiscan be explained that in the strong turbulence condition, the average SER is dominated by theimpacts of fading.GB6 gummuryIn this chapter, we developed highly accurate series error rate expression for a subcarrier QP-SK system with transmitter I/Q imbalance over the Gamma-Gamma turbulence channels. Thetruncation analysis demonstrated that the developed series solution can converge to the error rateobtained by numerical integration rapidly, especially in large SNR regimes. From the asymptoticanalysis, we noted that the transmitter I/Q imbalance will not affect the diversity order. Thepresented asymptotic analysis showed that the diversity order and the asymptotic SNR penaltyfactor of a subcarrier QPSK system with transmitter I/Q imbalance depend on the smaller channelparameter  only. The numerical results demonstrated that the SNR penalty decreases as theturbulence becomes stronger.30Whuptyr 4guvwurriyr edgK gystym kithfywyivyr ICe Imvulunwy cvyr thyGummuAGummu Fuxing WhunnylsIn this chapter, we perform an analytical analysis of a subcarrier QPSK system over the Gamma-Gamma fading channels with keceboek I/Q imbalance. The crosstalk of the quadrature branchescauses the correlation between the I-component and Q-component signals. The exact conditionalSER expression is derived as a function of the Gaussian f-function and the Owen’s i -function.Using the MGF of the Gamma-Gamma distribution, the closed-form average SER expression isobtained in terms of an infinite series. We discover that the level of the amplitude I/Q imbalanceis irrelevant to the average SER. Then, an approximate series expression of the average SER withfinite terms is presented. We also carry out a truncation error analysis and an asymptotic analysisfor the obtained SER expression.4BE Wonxitionul gEfFigure 4.1 shows the block diagram of a subcarrier QPSK system with receiver I/Q imbalance.Similar to (3.2), the transmitted baseband I-component and Q-component signals are, respectively,obtained as sI(t) = AI and sQ(t) = AQ. Therefore, the received baseband I-component andQ-component signals are, respectively, derived as rI(t) = IAI + nI(t) and rQ(t) = IAQ + nQ(t).To continue the evaluation, we introduce the received passband RF signal r′(t) asr′(t) = 2Re [r(t) exp (j2.fxt)] = r(t) exp (j2.fxt) + r∗(t) exp (−j2.fxt) (4.1)314.1. 6baWigibaTl FEEFigure 4.1: Block diagram of the a subcarrier QPSK system with receiver I/Q imbalance.where r∗(t) is the complex conjugate.At the receiver front-end, the down-conversion procedure can be modeled by multiplying thereceived passband RF signal r′(t) by the complex expression of quadrature oscillators y(t) [22]. Inthe presence of receiver I/Q imbalance, the receiver in-phase local carrier is (1+R) cos(2.fxt−RR2)and the receiver quadrature local carrier is −(1 − R) sin(2.fxt + RR2) where R is the receiveramplitude imbalance and R is the receiver phase imbalance [21]. Therefore, the complex expressionof quadrature oscillators y(t) can be derived asy(t) =(1 + R) cos(2.fxt− RR2) + j (−(1− R) sin(2.fxt+ RR2))= (cos(RR2) + jR sin(RR2)) exp(−j2.fxt) + (R cos(RR2)− j sin(RR2)) exp(j2.fxt):(4.2)Thus, after multiplying r′(t) by y(t) and low-pass filtering the 4.fxt terms, the pre-detectioncomplex signal rˆ(t) can be derived as [21, 37, 43, 63–65]rˆ(t) = [cos(RR2) + jR sin(RR2)] r(t) + [R cos(RR2)− j sin(RR2)] r∗(t)= [cos(RR2) + jR sin(RR2)] (rI(t) + jrQ(t)) + [R cos(RR2)− j sin(RR2)] (rI(t)− jrQ(t)):(4.3)Using (3.6), the pre-detection I-component and Q-component signals can be, respectively, obtained324.1. 6baWigibaTl FEEasrˆI(t) =I(1 + R) [cos(RR2)AI − sin(RR2)AQ]+ (1 + R) cos(RR2)nI(t)− (1 + R) sin(RR2)nQ(t)(4.4)andrˆQ(t) =I(1− R) [cos(RR2)AQ − sin(RR2)AI ]+ (1− R) cos(RR2)nQ(t)− (1− R) sin(RR2)nI(t):(4.5)Consequently, we can obtain the pre-detection I-component noise and Q-component noise as nˆI(t) =(1 + R) cos(RR2)nI(t) − (1 + R) sin(RR2)nQ(t) and nˆQ(t) = (1 − R) cos(RR2)nQ(t) − (1 −R) sin(RR2)nI(t) respectively. It can be shown that the pre-detection I/Q component noise termsnˆI(t) and nˆQ(t) are chkkeeZmed Gaussian random processes, and they become chkkeeZmed AWGN RVsafter sampling [66, Chapter 5]. Therefore, when conditioned on IP AI P AQ, RVs rˆI and rˆQ are alsochkkeeZmed and Gaussian-distributed with meansrˆI = I[(1 + R) cos(RR2)AI − (1 + R) sin(RR2)AQ] (4.6)andrˆQ = I[(1− R) cos(RR2)AQ − (1− R) sin(RR2)AI ] (4.7)and their variances can be, respectively, obtained as 2rˆI = (1 + R)22 and 2rˆQ = (1− R)22. Inaddition, we can obtain the correlation coefficient of rˆI and rˆQ as/ =Cov [rˆI P rˆQ]rˆIrˆQ= − sin R (4.8)where Cov [·P ·] denotes the statistical covariance operation.Thus, the joint pdf of rˆI and rˆQ is given by [66, Chapter 5]fGx(rˆI P rˆQ) =12.rˆIrˆQ√1− /2× exp{− 12(1− /2)[(rˆI − rˆI )22rˆI− 2/(rˆI − rˆI )(rˆQ − rˆQ)rˆIrˆQ+(rˆQ − rˆQ)22rˆQ]} (4.9)where the subscript Rx denotes a subcarrier QPSK system with receiver I/Q imbalance.334.1. 6baWigibaTl FEEFor a QPSK system with receiver I/Q imbalance, the conditional SER is given byeePGx(t) =14[eePGxP1+j(t) + eePGxP−1−j(t) + eePGxP1−j(t) + eePGxP−1+j(t)] : (4.10)For A = 1 + j (i.e. AI = 1, AQ = 1), it is straightforward to derive the means of rˆI and rˆQ asrˆI = I[(1 + R) cos(RR2)− (1 + R) sin(RR2)] (4.11)andrˆQ = I[(1− R) cos(RR2)− (1− R) sin(RR2)]: (4.12)The variances of rˆI and rˆQ are, respectively, 2rˆI= (1 + R)22 and 2rˆQ = (1 − R)22. Thecorrelation coefficient of rˆI and rˆQ is / = − sin R.Thus, the conditional symbol correct rate can be derived asexPGxP1+j(t) = BQ(rˆIrˆIPrˆQrˆQ;− sin R)(4.13)whereBQ(hP l; /) ,∫ l−∞∫ h−∞12.√1− /2 exp[− 12(1− /2)(u2 − 2/uv + v2)] yuyv (4.14)is the joint cumulative distribution function (CDF) of a bivariate Gaussian RV with zero meansand unity variances [67].According to [67], we can obtain BQ(hP l; /) asBQ(hP l; /) =12Φ(h) +12Φ(l)− i(hPl − /hh√1− /2)−i(lPh− /ll√1− /2)if hl S 0 or if hl = 0P h or l ≥ 012Φ(h) +12Φ(l)− i(hPl − /hh√1− /2)−i(lPh− /ll√1− /2)− 12if hl Q 0 or if hl = 0P h or l Q 0(4.15)344.1. 6baWigibaTl FEEwhere Φ(x) is the CDF of the standard normal distribution with Φ(x) = 1√2∫ x−∞ exp(− t22 )yt;i (hP v) is the Owen’s i -function [67] defined asi (hP v) =12.∫ a0exp[−12h2(1 + t2)]1 + t2yt: (4.16)For the empirical values of I/Q imbalance used in [21], we can obtain that 0 Q R Q 1.Therefore, the conditional SER can be obtained aseePGxP1+j(t) = f(√thGxP1) + 2i (√thGxP1P vGxP1) (4.17)where hGxP1 , cos(RR2)− sin(RR2) and vGxP1 ,√1+sin R1−sin R .Similarly, for A = 1− j (i.e. AI = 1, AQ = −1), we can derive the means of rˆI and rˆQ asrˆI = I(1 + R)[cos(RR2) + sin(RR2)] (4.18)andrˆQ = I(1− R)[− cos(RR2)− sin(RR2)]: (4.19)The variances of rˆI and rˆQ are, respectively, 2rˆI= (1 + R)22 and 2rˆQ = (1 − R)22. Thecorrelation coefficient of rˆI and rˆQ is / = − sin R.Thus, the conditional symbol correct rate can be derived asexPGxP1−j(t) =∫ −SrQSrQ−∞∫ ∞−SrISrI12.√1− /2 exp[− 12(1− /2)(u2 − 2/uv + v2)] yuyv=Φ(−rˆQrˆQ)−BQ(−rˆIrˆIP−rˆQrˆQ;− sin R):(4.20)Using (4.15), the conditional SER can be derived aseePGxP1−j(t) = f(√thGxP2) + 2i (√thGxP2P vGxP2) (4.21)where hGxP2 , cos(RR2) + sin(RR2) and vGxP2 ,√1−sin R1+sin R.Due to the symmetry of the QPSK signals, it is straightforward to show that eePGxP−1−j(t) =354.2. 4ieeTge FEE 4aTllfifeePGxP1+j(t) and eePGxP−1+j(t) = eePGxP1−j(t). Thus, the conditional SER can be obtained aseePGx(t) =12f(√thGxP1) +12f(√thGxP2) + i (√thGxP1P vGxP1) + i (√thGxP2P vGxP2) : (4.22)From (4.22), we can observe that the conditional SER of a subcarrier QPSK system with receiverI/Q imbalance does not rely on the value of the amplitude imbalance R. It can be shown thatwhen R = 0P R = 0, i.e. no I/Q imbalance occurs, eq. (4.22) is reduced to the conditional SERexpression of an ideal QPSK system shown in (3.8), as expected.4BF Avyrugy gEf AnulysisUsing the conditional SER shown in (4.22), the average SER of a subcarrier QPSK system withreceiver I/Q imbalance over the Gamma-Gamma fading channels is obtained by5eePQPSKPGxPGG =∫ ∞0eePGx(t)ftPGG(t)yt: (4.23)Substituting (4.22) into (4.23), we obtain the average SER aseePQPSKPGxPGG =12∫ ∞0f(√thGxP1) +f(√thGxP2)ftPGG(t)yt︸ ︷︷ ︸eGx;I+∫ ∞0(i (√thGxP1P vGxP1) + i (√thGxP2P vGxP2)) ftPGG(t)yt︸ ︷︷ ︸eGx;II:(4.24)For the empirical value of the phase imbalance R used in [21], values of hGxP1, vGxP1, hGXP2 and vGxP2are all positive.Similar to (3.30), we obtain eGxPI aseGxPI =12√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)GxP1 + h−(p+)GxP2)+12√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)GxP1 + h−(p+)GxP2):(4.25)5In this thysis, wy ussumy un unwoxyx systym unx thy normulizyx gbf t is pry-fiyx.364.3. GehacTgiba Eeebe 4aTllfifWith the definition of the Gauss Hypergeometric function 2F 1(· P · ; · ; ·) [57, eq. (9.111)], wecan obtain ∫ ∞0i(x√xP v)xm−1yx (x S 0P v S 0)=v2.(2x2)mΓ(m)2F 1(m+ 1P12;32;−v2) (4.26)which is proved in Appendix D. Using (4.26), eGxPII can be expressed aseGxPII =14.∞∑p=0vp(P )Γ(p+ 2)(t2)− p+2×[vGxP1hp+GxP12F 1(p+  + 22P12;32;−v2GxP1)+vGxP2hp+GxP22F 1(p+  + 22P12;32;−v2GxP2)]+14.∞∑p=0vp(P )Γ(p+ 2)(t2)− p+2×[vGxP1hp+GxP12F 1(p+ + 22P12;32;−v2GxP1)+vGxP2hp+GxP22F 1(p+ + 22P12;32;−v2GxP2)]:(4.27)With (4.25) and (4.27), a series expression of the average SER of a subcarrier QPSK system isgiven byeePQPSKPGxPGG =12eGxPI + eGxPII: (4.28)We can observe from (4.25) and (4.27) that the average SER of a subcarrier QPSK system withreceiver I/Q imbalance depends on the phase imbalance R only. Since eGxPI and eGxPII involveonly the Gamma function and the Gauss Hypergeometric function, eePQPSKPGxPGG can be calculatedusing standard scientific software such as Matlab without requiring numerical integration. WhenR = 0P R = 0, i.e. no I/Q imbalance occurs, eq. (4.28) is reduced to the average SER expressionof an ideal QPSK system given by (3.32).4BG hrunwution Error AnulysisIn Section 4.2, we obtain the average SER of a subcarrier QPSK system with receiver I/Qimbalance over the Gamma-Gamma fading channels as summations of infinite terms. However,for practical evaluation, we require an expression of finite terms. Therefore, truncation error isintroduced to represent the error due to the elimination of infinite terms except the first J + 1374.3. GehacTgiba Eeebe 4aTllfifterms [14].Similar to Section 3.3, in order to evaluate the truncation error of (4.28), we define two errorfactors as"JPGxP1(h) ,Λ(P )∞∑p=J+11p!(√2√th)p[upPGxP1(P P h)− upPGxP1(P P h)] (4.29)and"JPGxP2(hP v) ,Λ(P )∞∑p=J+11p!(√2√th)p[upPGxP2(P P hP v)− upPGxP2(P P hP v)] (4.30)where Λ(P ) is defined in (3.42); upPGxP1(P P h) and upPGxP2(P P hP v) are, respectively, defined asupPGxP1(P P h) ,Γ(p++12 )(√2√th)Γ(p− +  + 1)(p+ )(4.31)andupPGxP2(P P hP v) ,Γ(p+2 )(√2√th)Γ(p− +  + 1)∫ a0(1 + t2)− p++22 yt:(4.32)With (4.29) and (4.30), the truncation error of (4.28) is given by"JPGx =14√.("JPGxP1(hGxP1) + "JPGxP1(hGxP2)) +14.("JPGxP2(hGxP1P vGxP1) + "JPGxP2(hGxP2P vGxP2)) :(4.33)Using the Taylor series expansion of the exponential function, we obtain the upper bounds of(4.29) and (4.30) as"JPGxP1(h) ≤ Λ(P ) exp(√2√th)maxp>J{upPGxP1(P P h)− upPGxP1(P P h)} (4.34)and"JPGxP2(hP v) ≤ Λ(P ) exp(√2√th)maxp>J{upPGxP2(P P hP v)− upPGxP2(P P hP v)}: (4.35)384.4. 4flmcgbgic 4ieeTge FEE 4aTllfifAfter examining (4.31), we observe thatΓ( p+y+12)Γ(p−x+y+1)(p+y) decreases with an increase of p and becomes0 when p approaches ∞. Hence, when the unfaded normalized SNR t increases, "JPGxP1 diminishesrapidly. Similarly, since the value of∫ a0(1 + t2)− p++22 yt is bounded, "JPGxP2(hP v) diminishes rapidlywith an increase of t. Thus, by investigating (4.33), we find that the truncation error "JPGx decreasesrapidly with an increase of the unfaded normalized SNR t. This suggests that we can expect anaccurate approximate series solution and perform an asymptotic analysis.4B4 Asymptotiw Avyrugy gEf AnulysisIn an FSO system, we typically have  S  S 0 for the Gamma-Gamma distribution [14].Therefore, the term t2− p+2 decreases faster than the term t2− p+2 in (4.25) and (4.27) for thesame p value as t increases. Consequently, the leading terms of the series in (4.25) and (4.27)become dominant when t approaches ∞. Thus, for a subcarrier QPSK system with receiver I/Qimbalance, the average SER in high SNR regimes can be approximated bye∞ePQPSKPGxPGG=()Γ(− )Γ(+12)4√.Γ()Γ()(t2)−2 (h−GxP1 + h−GxP2)+()Γ(− )Γ(2)4.Γ()Γ()(t2)−2×[vGxP1hGxP12F 1(2 + 2P12;32;−v2GxP1)+vGxP2hGxP22F 1(2 + 2P12;32;−v2GxP2)]:(4.36)It can be noted that the asymptotic average SER expression (4.36) converges to the exact valuerapidly when the difference between  and  is large. From (3.48) and (4.36), we can observethat the asymptotic average SERs of the ideal subcarrier QPSK system and the subcarrier QPSKsystem with receiver I/Q imbalance have the same diversity order of 2 .To continue the evaluation, we first define iasyPGx asiasyPGx , e∞ePQPSKPGxPGGΓ()Γ()()Γ(− )t2 : (4.37)Then we introduce the SNR penalty factor SNR∞Gx-Id as the required amount of additional SNR of394.5. AhmeeicTl Eefhlgfthe system with receiver I/Q imbalance to achieve the same SER performance of the ideal system.Using (4.37), the asymptotic SNR penalty factor for a QPSK system is given bySNR∞Gx-Id =210 log(iasyPGxiasy)dB (4.38)where iasy is defined in (3.50). We can observe that the SNR penalty factor depends on the smallerchannel parameter  only. We comment that receiver I/Q imbalance will not affect the diversityorder, but only introduce a SNR penalty. Thus, we use (4.38) to quantify this SNR penalty.4BI bumyriwul fysultsFigure 4.2 presents SER curves of a subcarrier QPSK system over the Gamma-Gamma fadingchannels with receiver I/Q imbalance having different levels of imbalance in a moderate ( =2:50P  = 2:06) turbulence condition. We can observe that the average SERs of subcarrier QPSKsystem depend on the phase imbalance R only. This observation agrees with the prediction from(4.28). Similar results can be obtained under strong ( = 2:04P  = 1:10) and weak ( = 4:43P  =4:39) turbulence conditions.Figure 4.3 presents SER curves of a subcarrier QPSK system with receiver I/Q imbalance overthe Gamma-Gamma fading channels having different levels of turbulence. The phase imbalance isset to R = 0:35 according to the empirical value of I/Q imbalance used in [21]. Excellent agreementbetween the error rate obtained by numerical integration and the approximate error rate obtainedby the approximate series solutions can be found in Fig. 4.3. Asymptotic error rates are shownfor strong ( = 2:04P  = 1:10), moderate ( = 2:50P  = 2:06) and weak ( = 4:43P  = 4:39)turbulence conditions. We can observe that the asymptotic error rate converges to the error rateobtained by numerical integration faster in the strong ( = 2:04P  = 1:10) turbulence conditionthan in the moderate ( = 2:50P  = 2:06) turbulence condition in high SNR regimes. However,in the weak ( = 4:43P  = 4:39) turbulence condition with nearly equal values of  and , theconvergence is too slow to be recognized in Fig. 4.3.Figure 4.4 presents the average SERs of a subcarrier QPSK system with and without receiverI/Q imbalance over the Gamma-Gamma fading channels having different levels of turbulence. For404.5. AhmeeicTl Eefhlgf15 15.5 16 16.5 1710−1Unfaded normalized SNR (dB)Average symbol error rate  αR=0.15αR=0.10αR=0.05θR=0.10θR=0.20 θR=0.35Figure 4.2: Average SER of a subcarrier QPSK system over the Gamma-Gamma fading chan-nels with receiver I/Q imbalance in a moderate ( = 2:50P  = 2:06) turbulence condition.The phase imbalances are set to R = 0:10P 0:20P 0:35 and the amplitude imbalances are set toR = 0:05P 0:10P 0:15.414.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 6010−1010−910−810−710−610−510−410−310−210−1100Unfaded normalized SNR (dB)Average symbol error rate  Approx,α=2.04, β=1.10Numerical,α=2.04, β=1.10Approx,α=2.50, β=2.06Numerical,α=2.50, β=2.06Approx,α=4.43, β=4.39Numerical,α=4.43, β=4.39AsymptoticFigure 4.3: Average SER of a subcarrier QPSK system with receiver I/Q imbalance over theGamma-Gamma fading channels having different levels of turbulence. Phase imbalance is set toR = 0:35.424.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 6010−1010−910−810−710−610−510−410−310−210−1100Unfaded normalized SNR (dB)Average symbol error rate  α=2.04, β=1.10α=2.50, β=2.06α=4.43, β=4.39Rx I/Q imbalanceno imbalanceFigure 4.4: Average SER of a subcarrier QPSK system with and without receiver I/Q imbalanceover the Gamma-Gamma fading channels having different levels of turbulence. Phase imbalance isset to R = 0:35.434.6. FhmmTela subcarrier QPSK system with receiver I/Q imbalance, the SNR penalty is approximately 0:90dB at the SER level of 10−10 in the weak ( = 4:43P  = 4:39) turbulence condition, which agreeswith the prediction of 0:85 dB obtained by (4.38). In the moderate ( = 2:50P  = 2:06) turbulencecondition, the SNR penalty is approximately 0:70 dB at the SER level of 10−5, which agrees withthe prediction of 0:60 dB. Also, the SNR penalty is approximately 0:50 dB at the SER level of 10−3in the strong ( = 2:04P  = 1:10) turbulence condition, which agrees with the prediction of 0:48dB. We can observe that the SNR penalty decreases as the turbulence becomes stronger. This canbe explained that in the strong turbulence condition, the average SNR is dominated by the impactsof fading.4B6 gummuryIn this chapter, we developed highly accurate series error rate expression for subcarrier QPSKsystem with receiver I/Q imbalance over the Gamma-Gamma fading channels. We noted thatthe crosstalk of the quadrature branches introduces correlation between the I-component and theQ-component signals. Our theoretical and simulation results showed that the receiver amplitudeimbalance is irrelevant to the value of the performance degradation of the QPSK system. Also, thepresented asymptotic analysis showed that the receiver I/Q imbalance will not affect the diversityorder, but only introduce a SNR penalty. It can be noted that the diversity order and the asymptoticSNR penalty factor of a subcarrier QPSK system with receiver I/Q imbalance depend on the smallerchannel parameter  only. For the empirical values of I/Q imbalance, we observed that the SNRloss is less than 1 dB. In addition, the numerical results showed that the average SNR is dominatedby the impacts of fading in the strong turbulence condition.44Whuptyr Iguvwurriyr edgK gystym kithhwoAgixyx ICe Imvulunwy cvyr thyGummuAGummu Fuxing WhunnylsIn this chapter, we investigate the error rate performance of a subcarrier QPSK system with I/Qimbalance at bhma mkZglfbmmek Zgd keceboek ends over the Gamma-Gamma fading channels. In thepresence of receiver I/Q imbalance, the crosstalk of the quadrature branches leads to correlationbetween the I-component and Q-component signals. Consequently, the conditional SER can bederived in terms of a function of the Gaussian f-function and the Owen’s i -function. A closed-form average SER expression is obtained in terms of infinite series. We discover that the levelof the receiver amplitude imbalance is irrelevant to the SER performance. Then, we present anapproximate series expression of the average SER for the practical evaluation. We also carry out atruncation error analysis and an asymptotic analysis. Numerical results are presented to verify theaccuracy of the analysis and show the performance improvement using a calibrated transceiver.IBE Wonxitionul gEfFigure 2.3 shows the block diagram of a subcarrier QPSK system with I/Q imbalance at bothtransmitter and receiver ends. In the presence of the transmitter I/Q imbalance, following thesimilar procedure in Section 3.1.2, the transmitted baseband I-component and Q-component signalscan be, respectively, obtained assI(t) = (1 + T ) cos(T R2)AI − (1− T ) sin(T R2)AQ (5.1)455.1. 6baWigibaTl FEEandsQ(t) = (1− T ) cos(T R2)AQ − (1 + T ) sin(T R2)AI : (5.2)Consequently, similar to (3.9) and (3.10), the received baseband I-component and Q-componentsignals can be, respectively, derived asrI(t) = I [(1 + T ) cos(T R2)AI − (1− T ) sin(T R2)AQ] + nI(t) (5.3)andrQ(t) = I [(1− T ) cos(T R2)AQ − (1 + T ) sin(T R2)AI ] + nQ(t): (5.4)In the presence of receiver I/Q imbalance, substituting (5.3) and (5.4) into (4.3), with somealgebraic and trigonometric manipulations, the pre-detection I-component signal and Q-componentsignal can be, respectively, obtained asrˆI(t) =(1 + R) cos(RR2)rI(t)− (1 + R) sin(RR2)rQ(t)=I(1 + R)[(1 + T ) cos(RR2− T R2)AI − (1− T ) sin(RR2 + T R2)AQ]+ (1 + R) cos(RR2)nI(t)− (1 + R) sin(RR2)nQ(t)(5.5)andrˆQ(t) =(1− R) cos(RR2)rQ(t)− (1− R) sin(RR2)rI(t)=I(1− R)[(1− T ) cos(RR2− T R2)AQ − (1 + T ) sin(RR2 + T R2)AI ]+ (1− R) cos(RR2)nQ(t)− (1− R) sin(RR2)nI(t):(5.6)Therefore, the noise terms nˆI(t) and nˆQ(t) are, respectively, given bynˆI(t) = (1 + R) cos(RR2)nI(t)− (1 + R) sin(RR2)nQ(t) (5.7)andnˆQ(t) = (1− R) cos(RR2)nQ(t)− (1− R) sin(RR2)nI(t): (5.8)Thus, it can be shown that the pre-detection I/Q component noise terms nˆI(t) and nˆQ(t) are465.1. 6baWigibaTl FEEchkkeeZmed Gaussian random processes, and they become chkkeeZmed AWGN RVs after sampling [66,Chapter 5]. Therefore, when conditioned on IP AI P AQ, RVs rˆI and rˆQ are also chkkeeZmed andGaussian-distributed with meansrˆI = I(1 + R)[(1 + T ) cos(RR2− T R2)AI − (1− T ) sin(RR2 + T R2)AQ] (5.9)andrˆQ = I(1− R)[(1− T ) cos(RR2− T R2)AQ − (1 + T ) sin(RR2 + T R2)AI ] (5.10)and their variances can be obtained as 2rˆI = (1+R)22 and 2rˆQ = (1−R)22. Similar to (4.8),we can obtain the correlation coefficient of rˆI and rˆQ as/ =Cov [rˆI P rˆQ]rˆIrˆQ= − sin R: (5.11)Hence, the joint pdf of rˆI and rˆQ is given by [66, Chapter 5]fIG(rˆI P rˆQ) =12.rˆIrˆQ√1− /2× exp{− 12(1− /2)[(rˆI − rˆI )22rˆI− 2/(rˆI − rˆI )(rˆQ − rˆQ)rˆIrˆQ+(rˆQ − rˆQ)22rˆQ]}(5.12)where the subscript TR denotes a subcarrier QPSK system with I/Q imbalance at both transmitterand receiver ends.For a QPSK system with I/Q imbalance at both transmitter and receiver ends, the conditionalSER is given byeePIG(t) =14[eePIGP1+j(t) + eePIGP−1−j(t) + eePIGP1−j(t) + eePIGP−1+j(t)] : (5.13)For A = 1 + j (i.e. AI = 1, AQ = 1), it is straightforward to derive the means of rˆI and rˆQ asrˆI = I(1 + R)[(1 + T ) cos(RR2− T R2)− (1− T ) sin(RR2 + T R2)] (5.14)475.1. 6baWigibaTl FEEandrˆQ = I(1− R)[(1− T ) cos(RR2− T R2)− (1 + T ) sin(RR2 + T R2)]: (5.15)The variances of rˆI and rˆQ are, respectively, 2rˆI= (1 + R)22 and 2rˆQ = (1 − R)22. Thecorrelation coefficient of rˆI and rˆQ is / = − sin R.Thus, the conditional symbol correct rate can be derived asexPIGP1+j(t) =∫ ∞−SrQSrQ∫ ∞−SrISrI12.√1− /2 exp[− 12(1− /2)(u2 − 2/uv + v2)] yuyv=BQ(rˆIrˆIPrˆQrˆQ;− sin R):(5.16)For the empirical values of I/Q imbalance used in [21], the inequality 0 Q R Q 1 can beobtained. Using (4.15), the conditional SER can be written aseePIGP1+j(t) =1− exPIGP1+j(t)=12f (√thIGP1) +12f (√tlIGP1) + i (√thIGP1P vIGP1) + i (√tlIGP1P bIGP1)(5.17)where hIGP1, lIGP1, vIGP1 and bIGP1 are, respectively, defined ashIGP1 , (1 + T ) cos(RR2− T R2)− (1− T ) sin(RR2 + T R2) (5.18)lIGP1 , (1− T ) cos(RR2− T R2)− (1 + T ) sin(RR2 + T R2) (5.19)vIGP1 ,lIGP1 + sin RhIGP1hIGP1√1− sin2 R(5.20)andbIGP1 ,hIGP1 + sin RlIGP1lIGP1√1− sin2 R: (5.21)Similarly, for A = 1− j (i.e. AI = 1, AQ = −1), we can derive the means of rˆI and rˆQ asrˆI = I(1 + R)[(1 + T ) cos(RR2− T R2) + (1− T ) sin(RR2 + T R2)] (5.22)485.1. 6baWigibaTl FEEandrˆQ = I(1− R)[−(1− T ) cos(RR2− T R2)− (1 + T ) sin(RR2 + T R2)]: (5.23)The variances of rˆI and rˆQ are, respectively, 2rˆI= (1 + R)22 and 2rˆQ = (1 − R)22. Thecorrelation coefficient of rˆI and rˆQ is / = − sin R.Thus, the conditional symbol correct rate can be derived asexPIGP1−j(t) =∫ −SrQSrQ−∞∫ ∞−SrISrI12.√1− /2 exp[− 12(1− /2)(u2 − 2/uv + v2)] yuyv=Φ(−rˆQrˆQ)−BQ(−rˆIrˆIP−rˆQrˆQ;− sin R):(5.24)Using (4.15), the conditional SER can be derived aseePIGP1−j(t) =1− exPIGP1−j(t)=12f (√thIGP2) +12f (√tlIGP2) + i (√thIGP2P vIGP2) + i (√tlIGP2P bIGP2)(5.25)where hIGP2, lIGP2, vIGP2 and bIGP2 are, respectively, defined ashIGP2 , (1 + T ) cos(RR2− T R2) + (1− T ) sin(RR2 + T R2) (5.26)lIGP2 , (1− T ) cos(RR2− T R2) + (1 + T ) sin(RR2 + T R2) (5.27)vIGP2 ,lIGP2 − sin RhIGP2hIGP2√1− sin2 R(5.28)andbIGP2 ,hIGP2 − sin RlIGP2lIGP2√1− sin2 R: (5.29)Due to the symmetry of the QPSK signals, we can obtain eePIGP−1−j(t) and eePIGP−1+j(t) aseePIGP−1−j(t) = eePIGP1+j(t) and eePIGP−1+j(t) = eePIGP1−j(t) respectively (see proof in Appendix495.2. 4ieeTge FEE 4aTllfifE). Therefore, the conditional SER can be obtained aseePIG(t) =14f (√thIGP1) +14f (√tlIGP1) +12i (√thIGP1P vIGP1) +12i (√tlIGP1P bIGP1)+14f (√thIGP2) +14f (√tlIGP2) +12i (√thIGP2P vIGP2) +12i (√tlIGP2P bIGP2) :(5.30)From (5.30), we observe that the conditional SER does not rely on the value of the receiveramplitude imbalance R. It can be noted that when T = 0, T = 0, R = 0 and R = 0, i.e. noI/Q imbalance occurs, eq. (5.30) is reduced to the conditional SER expression of an ideal QPSKsystem shown in (3.37). Also, when R = 0 and R = 0, i.e. transmitter I/Q imbalance only,eq. (5.30) reduces to the conditional SER expression of a QPSK system with transmitter I/Qimbalance given by (3.22). In addition, when T = 0 and T = 0, i.e. receiver I/Q imbalanceonly, eq. (5.30) is reduced to the conditional SER expression of a QPSK system with receiver I/Qimbalance shown in (4.22). Therefore, we conclude that (5.30) is a generalized conditional SERexpression of a QPSK system with I/Q imbalance.IBF Avyrugy gEf AnulysisThe average SER of a subcarrier QPSK system with I/Q imbalance at both transmitter andreceiver ends over the Gamma-Gamma fading channels is obtained by6eePQPSKPIGPGG =∫ ∞0eePIG(t)ftPGG(t)yt (5.31)where eePIG(t) is the conditional SER shown in (5.30).Substituting (5.30) into (5.31), the average SER is given byeePQPSKPIGPGG =14∫ ∞0(f(√thIGP1) +f(√tlIGP1) +f(√thIGP2) +f(√tlIGP2)) ftPGG(t)yt︸ ︷︷ ︸eIG;I+12∫ ∞0fT (tP hIGP1P lIGP1P hIGP2P lIGP2)ftPGG(t)yt︸ ︷︷ ︸eIG;II(5.32)+In this thysis, wy ussumy un unwoxyx systym unx thy normulizyx gbf t is pry-fiyx.505.2. 4ieeTge FEE 4aTllfifwhere fT (tP hIGP1P lIGP1P hIGP2P lIGP2) is defined asfT (tP hIGP1P lIGP1P hIGP2P lIGP2),i (√thIGP1P vIGP1) + i (√tlIGP1P bIGP1) + i (√thIGP2P vIGP2) + i (√tlIGP2P bIGP2) :(5.33)For the empirical values of I/Q imbalance used in [21], the values of hIGP1, lIGP1, hIGP2, lIGP2, vIGP1,bIGP1, vIGP2 and bIGP2 are all positive.Using the MGF of the Gamma-Gamma distribution (3.24), similar to the derivation of (3.30),it is straightforward to show thateIGPI =12√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)IGP1 + l−(p+)IGP1 + h−(p+)IGP2 + l−(p+)IGP2)+12√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)IGP1 + l−(p+)IGP1 + h−(p+)IGP2 + l−(p+)IGP2):(5.34)Using the integral identity (4.26), eIGPII can be calculated aseIGPII = g1(P ) + g1(P ) (5.35)whereg1(P ) =14.∞∑p=0vp(P )Γ(p+ 2)t− p+2×vIGP1( 2h2IGP1) p+22F 1(p+  + 22P12;32;−v2IGP1)+ bIGP1(2l2IGP1) p+22F 1(p+  + 22P12;32;−b2IGP1)+ vIGP2(2h2IGP2) p+22F 1(p+  + 22P12;32;−v2IGP2)+ bIGP2(2l2IGP2) p+22F 1(p+  + 22P12;32;−b2IGP2) :(5.36)With (5.34) and (5.35), the average SER of a subcarrier QPSK system with I/Q imbalance at515.2. 4ieeTge FEE 4aTllfifboth transmitter and receiver ends is derived aseePQPSKPIGPGG =14eIGPI +12eIGPII: (5.37)When an ideal QPSK system is assumed, i.e. T = 0, T = 0, R = 0 and R = 0, it canbe shown that (5.37) is reduced to the average SER expression of an ideal QPSK system shownin (3.32). We can also observe that eIGPI and eIGPII involve only the Gamma function and theGauss Hypergeometric function. Therefore, eePQPSKPIGPGG can be evaluated using standard scientificsoftware such as Matlab without requiring numerical integration.For a subcarrier QPSK system with transmitter I/Q imbalance only, i.e. R = 0 and R = 0,we can obtain the average SER aseePQPSKPIxPGG = e′IxPI + e′IxPII: (5.38)We can obtain e′IxPI ase′IxPI =18√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)IxP1 + l−(p+)IxP1 + h−(p+)IxP2 + l−(p+)IxP2)+18√.∞∑p=0vp(P )Γ(p++12)p+ (t2)− p+2 (h−(p+)IxP1 + l−(p+)IxP1 + h−(p+)IxP2 + l−(p+)IxP2) (5.39)where hIxP1, lIxP1, hIxP2 and lIxP2 are, respectively, defined in (3.17), (3.18), (3.20) and (3.21). Also,e′IxPII can be derived ase′IxPII = g2(P ) + g2(P ) (5.40)525.3. GehacTgiba Eeebe 4aTllfifwhereg2(P ) =18.∞∑p=0vp(P )Γ(p+ 2)t− p+2×vIxP1( 2h2IxP1) p+22F 1(p+  + 22P12;32;−v2IxP1)+ bIxP1(2l2IxP1) p+22F 1(p+  + 22P12;32;−b2IxP1)+ vIxP2(2h2IxP2) p+22F 1(p+  + 22P12;32;−v2IxP2)+ bIxP2(2l2IxP2) p+22F 1(p+  + 22P12;32;−b2IxP2)(5.41)with vIxP1, bIxP1, vIxP2 and bIxP2 defined as vIxP1 , lIxP1RhIxP1, bIxP1 , hIxP1RlIxP1, vIxP2 , lIxP2RhIxP2and bIxP2 , hIxP2RlIxP2 respectively. Eq. (5.38) is numerically equivalent to the average SERexpression of a subcarrier QPSK system with transmitter I/Q imbalance shown in (3.39).Similarly, when T = 0 and T = 0, i.e. receiver I/Q imbalance only, the average SER expres-sion (5.37) is reduced to the average SER expression of a subcarrier QPSK system with receiverI/Q imbalance given by (4.28). Therefore, we conclude that (5.37) is a generalized average SERexpression of the subcarrier QPSK system with I/Q imbalance over the Gamma-Gamma fadingchannels. From (5.37) and (4.28), we also observe that the average SER does not rely on the valueof the receiver amplitude imbalance R.IBG hrunwution Error AnulysisIn Section 5.2, we have derived the average SER expression of a subcarrier QPSK system overthe Gamma-Gamma fading channels with I/Q imbalance at both transmitter and receiver endsin (5.37). We can observe that each term of the right-hand side of (5.37) contains summationsof infinite terms. However, for practical evaluation, it is required to have expressions of finitesummation terms. Thus, we introduce the truncation error to represent the error due to theelimination of the infinite terms except the first J + 1 terms [3].Similar to Section 3.3, in order to evaluate the truncation error of (5.37), two error factors are535.3. GehacTgiba Eeebe 4aTllfifdefined as"JPIGP1(h) , Λ(P )∞∑p=J+11p!(√2√th)p[upPIGP1(P P h)− upPIGP1(P P h)] (5.42)and"JPIGP2(hP v) , Λ(P )∞∑p=J+11p!(√2√th)p[upPIGP2(P P hP v)− upPIGP2(P P hP v)] (5.43)where Λ(P ) is defined in (3.42); upPIGP1(P P h) and upPIGP2(P P hP v) are, respectively, defined asupPIGP1(P P h) ,Γ(p++12 )Γ(p− +  + 1)(p+ )(√2√th)(5.44)andupPIGP2(P P hP v) ,Γ(p+2 )Γ(p− +  + 1)(√2√th) ∫ a0(1 + t2)− p++22 yt: (5.45)With (5.42) and (5.43), the truncation error of (5.37) is given by"JPIG =18√.("JPIGP1(hIGP1) + "JPIGP1(lIGP1) + "JPIGP1(hIGP2) + "JPIGP1(lIGP2))+18.("JPIGP2(hIGP1P vIGP1) + "JPIGP2(lIGP1P bIGP1) + "JPIGP2(hIGP2P vIGP2) + "JPIGP2(lIGP2P bIGP2)) :(5.46)Using the Taylor series expansion of the exponential function, the upper bounds of (5.42) and(5.43) can be derived as"JPIGP1(h) ≤ Λ(P ) exp(√2√th)×maxp>J{upPIGP1(P P h)− upPIGP1(P P h)}(5.47)and"JPIGP2(hP v) ≤ Λ(P ) exp(√2√th)×maxp>J{upPIGP2(P P hP v)− upPIGP2(P P hP v)}:(5.48)After examining (5.44), it can be noted thatΓ( p+y+12)Γ(p−x+y+1)(p+y) decreases with an increase of p and545.4. 4flmcgbgic 4ieeTge FEE 4aTllfifbecomes 0 when p approaches ∞. Thus, both upPIGP1(P P h) and upPIGP1(P P h) approach 0 whenp approaches ∞. Therefore, upPIGP1(P P h) − upPIGP1(P P h) has a finite maximum value. Hence,when the unfaded normalized SNR t increases, "JPIGP1(h) diminishes rapidly. Similarly, we alsonote in (5.45) thatΓ( p+2)Γ(p−++1) decreases with an increase of p and becomes 0 when p approaches∞.With the bounded characteristic of∫ a0(1 + t2)− p++22 yt, we observe that "JPIGP2(hP v) diminishesrapidly with an increase of t. Thus, the truncation error "JPIG diminishes rapidly with increasingvalues of t. This suggests that our finite series solutions are highly accurate in high SNR regimesand can allow an asymptotic error rate analysis.IB4 Asymptotiw Avyrugy gEf AnulysisIn an FSO system over the Gamma-Gamma fading channels, typically we have  S  S 0[62]. Therefore, the term t2− p+2 decreases faster than the term t2− p+2 in (5.34) and (5.35) for thesame p value as t increases. Therefore, the leading terms of the series in (5.34) and (5.35) becomedominant when t approaches∞, and the asymptotic average SER of the subcarrier QPSK systemwith I/Q imbalance at both transmitter and receiver ends can be obtained ase∞ePQPSKPIGPGG=()Γ(− )Γ(+12)8√.Γ()Γ()(t2)−2 (h−IGP1 + l−IGP1 + h−IGP2 + l−IGP2)+()Γ(− )Γ(2)8.Γ()Γ()(t2)−2×vIGP1( 1h2IGP1)22F 1(2 + 2P12;32;−v2IGP1)+ bIGP1(1l2IGP1)22F 1(2 + 2P12;32;−b2IGP1)+ vIGP2(1h2IGP2)22F 1(2 + 2P12;32;−v2IGP2)+ bIGP2(1l2IGP2)22F 1(2 + 2P12;32;−b2IGP2) :(5.49)For a subcarrier QPSK system with transmitter I/Q imbalance only, i.e. R = 0 and R = 0,555.4. 4flmcgbgic 4ieeTge FEE 4aTllfifwe can calculate the average SER of the subcarrier QPSK system in high SNR regimes ase∞ePQPSKPIxPGG=()Γ(− )Γ(+12)8√.Γ()Γ()(t2)−2 (h−IxP1 + l−IxP1 + h−IxP2 + l−IxP2)+()Γ(− )Γ(2)8.Γ()Γ()(t2)−2×vIxP1( 1h2IxP1)22F 1(2 + 2P12;32;−v2IxP1)+ bIxP1(1l2IxP1)22F 1(2 + 2P12;32;−b2IxP1)+ vIxP2(1h2IxP2)22F 1(2 + 2P12;32;−v2IxP2)+ bIxP2(1l2IxP2)22F 1(2 + 2P12;32;−b2IxP2) :(5.50)Eq. (5.50) is numerically equivalent to (3.49).Similarly, for a subcarrier QPSK system with receiver I/Q imbalance only, i.e. T = 0 andT = 0, eq. (5.49) is reduced to (4.36). Therefore, we conclude that (5.49) is a generalizedasymptotic average SER expression of a subcarrier QPSK system with I/Q imbalance over theGamma-Gamma fading channels.For the Gamma-Gamma fading channels, the asymptotic average SERs shown in (5.49) and(5.50) converge to the exact value rapidly when the difference between  and  is large. From(3.48), (5.49), (3.49) and (4.36), we can observe that the asymptotic average SERs have the samediversity order of 2 . For ensuing analysis, we first define iasyPIG asiasyPIG , e∞ePQPSKPIGPGGΓ()Γ()()Γ(− )t2 : (5.51)Then we introduce the SNR penalty factor SNR∞IG-Id as the required amount of additional SNR ofthe system with I/Q imbalance at both transmitter and receiver ends to achieve the same SERperformance of the ideal system[45]. Using (5.51), we can calculate the asymptotic SNR penaltyfactor for a subcarrier QPSK system asSNR∞IG-Id =210 log(iasyPIGiasy)dB (5.52)565.5. AhmeeicTl Eefhlgf30 31 32 33 34 35 36 37 38 39 4010−310−2Unfaded normalized SNR (dB)Average symbol error rate  αR=0.15αR=0.10αR=0.05θR=0.20θR=0.10 θR=0.35Figure 5.1: Average SERs of a subcarrier QPSK system over the Gamma-Gamma channels withI/Q imbalance at both transmitter and receiver ends in a moderate ( = 2:50P  = 2:06) turbulencecondition. The transmitter phase imbalance is set to T = 0:35 and the transmitter amplitudeimbalance is set to T = 0:15. The receiver phase imbalances are set to R = 0:10P 0:20P 0:35 andthe receiver amplitude imbalances are set to R = 0:05P 0:10P 0:15.where iasy is defined in (3.50). It can be noticed that the SNR penalty factor depends on thesmaller channel parameter  only. We comment that I/Q imbalance does not affect the diversityorder, but only introduces a SNR penalty. Therefore, we can use (5.52) to quantify this SNRpenalty.IBI bumyriwul fysultsFigures 5.1, 5.2 and 5.3 present SER curves of a subcarrier QPSK system over the Gamma-Gamma fading channels with I/Q imbalance at both transmitter and receiver ends having different575.5. AhmeeicTl Eefhlgf30 31 32 33 34 35 36 37 38 39 4010−310−2Unfaded normalized SNR (dB)Average symbol error rate  αR=0.15αR=0.10αR=0.05θT=0.35θT=0.20θT=0.10Figure 5.2: Average SERs of a subcarrier QPSK system over the Gamma-Gamma channels withI/Q imbalance at both transmitter and receiver ends in a moderate ( = 2:50P  = 2:06) turbulencecondition. The transmitter phase imbalances are set to T = 0:10P 0:20P 0:35 and the transmitteramplitude imbalance is set to T = 0:15. The receiver phase imbalance is set to R = 0:35 and thereceiver amplitude imbalances are set to R = 0:05P 0:10P 0:15.585.5. AhmeeicTl Eefhlgf30 31 32 33 34 35 36 37 38 39 4010−310−2Unfaded normalized SNR (dB)Average symbol error rate  αR=0.15αR=0.10αR=0.05αT=0.10αT=0.15αT=0.05Figure 5.3: Average SERs of a subcarrier QPSK system over the Gamma-Gamma channels withI/Q imbalance at both transmitter and receiver ends in a moderate ( = 2:50P  = 2:06) turbulencecondition. The transmitter phase imbalance is set to T = 0:35 and the transmitter amplitudeimbalances are set to T = 0:05P 0:10P 0:15. The receiver phase imbalance is set to R = 0:35 andthe receiver amplitude imbalances are set to R = 0:05P 0:10P 0:15.595.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 6010−1010−910−810−710−610−510−410−310−210−1100Unfaded normalized SNR (dB)Average symbol error rate  Approx,α=2.04, β=1.10Numerical,α=2.04, β=1.10Approx,α=2.50, β=2.06Numerical,α=2.50, β=2.06Approx,α=4.43, β=4.39Numerical,α=4.43, β=4.39AsymptoticFigure 5.4: Average SERs of a subcarrier QPSK system with I/Q imbalance at both sides over theGamma-Gamma channels having different levels of turbulence. The transmitter phase imbalance isset to T = 0:35 and the transmitter amplitude imbalance is set to T = 0:15. The receiver phaseimbalance is set to R = 0:35.levels of imbalance in a moderate ( = 2:50P  = 2:06) turbulence condition. It can be observedthat the value of the receiver amplitude imbalance is irrelevant to the average SER of a subcarrierQPSK system. This observation agrees with the prediction from (5.37). In Figs. 5.1, 5.2 and 5.3,we can also observe that the average SER increases as the level of imbalance increases. Similarresults can be obtained in a strong ( = 2:04P  = 1:10) and a weak ( = 4:43P  = 4:39) turbulenceconditions.Figure 5.4 presents SER curves of a subcarrier QPSK system with I/Q imbalance at bothtransmitter and receiver ends over the Gamma-Gamma fading channels having different levels ofturbulence. The transmitter phase imbalance is set to T = 0:35 and the transmitter amplitude605.5. AhmeeicTl Eefhlgfimbalance is set to T = 0:15. The receiver phase imbalance is set to R = 0:35. In Fig. 5.4,we can find excellent agreement between the error rate obtained by numerical integration andthe approximate error rate obtained by the series solutions. Asymptotic error rates are shownfor strong ( = 2:04P  = 1:10), moderate ( = 2:50P  = 2:06) and weak ( = 4:43P  = 4:39)turbulence conditions. It can be observed that the asymptotic error rate converges to the error rateobtained by numerical integration faster in the strong ( = 2:04P  = 1:10) turbulence conditionthan in the moderate ( = 2:50P  = 2:06) turbulence condition in high SNR regimes. However, theconvergence of the asymptotic error rate to the error rate obtained by numerical integration in theweak ( = 4:43P  = 4:39) turbulence condition is not shown in the figure. This can be explainedthat in the weak ( = 4:43P  = 4:39) turbulence condition with nearly equal values of  and ,the convergence is too slow to be recognized in Fig. 5.4.Figure 5.5 presents the absolute truncation error of a subcarrier QPSK system over the Gamma-Gamma fading channels with I/Q imbalance at both transmitter and receiver ends in a moderate( = 2:50P  = 2:06) turbulence condition using J = 8P 15P 30 terms. We can observe that thetruncation error diminishes with increasing values of J for the same SNR. To obtain a truncationerror of 10−10 at 10 dB SNR, we need J = 30 terms while we only need J = 15 terms to achievethe same truncation error at 15 dB SNR. Also, the truncation error decreases as the SNR increasesfor the same value of J .Figure 5.6 presents the average SERs of a subcarrier QPSK system with and without I/Qimbalance over the Gamma-Gamma channels having different levels of turbulence. For a subcarrierQPSK system with I/Q imbalance at both transmitter and receiver ends, the SNR penalty isapproximately 4:2 dB at the SER level of 10−10 in the weak ( = 4:43P  = 4:39) turbulencecondition, which agrees with the prediction of 4:3 dB obtained by (5.52). Also, in the moderate( = 2:50P  = 2:06) turbulence condition, the SNR penalty is approximately 2:7 dB at the SERlevel of 10−5, which agrees with the prediction of 2:8 dB. In the strong ( = 2:04P  = 1:10)turbulence condition, the SNR penalty is approximately 2:1 dB at the SER level of 10−3, whichagrees with the prediction of 2:0 dB. It can be observed that the SNR penalty decreases as theturbulence becomes stronger. This can be explained that in the strong turbulence condition, theaverage SNR is dominated by the impacts of fading. In Fig. 5.6, we also observe that under the615.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 6010−1610−1410−1210−1010−810−610−410−2100102104Unfaded normalized SNR (dB)Absolute truncation error  J=8J=15J=30Figure 5.5: Absolute truncation error of a subcarrier QPSK system over the Gamma-Gamma fadingchannels with I/Q imbalance at transmitter and receiver ends in a moderate ( = 2:50P  = 2:06)turbulence condition using different values of J . The transmitter phase imbalance is set to T = 0:35and the transmitter amplitude imbalance is set to T = 0:15. The receiver phase imbalance is setto R = 0:35.625.5. AhmeeicTl Eefhlgf0 10 20 30 40 50 6010−1010−910−810−710−610−510−410−310−210−1100Unfaded normalized SNR (dB)Average symbol error rate  Tx I/Q ImbalanceRx I/Q ImbalanceTx and Rx I/Q ImbalanceNo ImbalanceCalibrated Transceiverα=2.04, β=1.10α=4.43, β=4.39α=2.50, β=2.06Figure 5.6: Average SERs of a subcarrier QPSK system with and without I/Q imbalance overthe Gamma-Gamma fading channels having different levels of turbulence. The transmitter phaseimbalance is set to T = 0:35 and the transmitter amplitude imbalance is set to T = 0:15. Thereceiver phase imbalance is set to R = 0:35.635.6. FhmmTelsame level of I/Q imbalance, the SNR penalty of a subcarrier QPSK system with I/Q imbalanceat both transmitter and receiver ends is larger than the sum of the SNR penalties of a subcarrierQPSK system with transmitter I/Q imbalance only and a subcarrier QPSK system with receiver I/Qimbalance only. For modern technology, using a calibrated transceiver, the level of I/Q imbalancecan be reduced to the phase imbalance 0:05 and the amplitude imbalance 0:025 [68]. As shownin Fig. 5.6, using the calibrated transceiver, the SNR penalties can be reduced to less than 0:1dB in strong ( = 2:04P  = 1:10), moderate ( = 2:50P  = 2:06) and weak ( = 4:43P  =4:39) turbulence conditions. Therefore, we conclude that the calibrated transceiver can effectivelymitigate the impacts of the I/Q imbalance on the error rate performance of a subcarrier QPSKsystem over the Gamma-Gamma fading channels.IB6 gummuryIn this chapter, we developed error rate expression for subcarrier QPSK system with I/Q im-balance at both transmitter and receiver ends over the Gamma-Gamma fading channels using anMGF approach. The crosstalk of the quadrature branches introduces correlation between the I-component and the Q-component signals. Our theoretical and simulation results showed that thevalue of the receiver amplitude imbalance is irrelevant to the performance degradation of the QPSKsystem. The presented asymptotic analysis showed that the receiver I/Q imbalance will not affectthe diversity order, but only introduce a SNR penalty. It can be noted that the diversity orderand the asymptotic SNR penalty factor of a subcarrier QPSK system with I/Q imbalance at bothtransmitter and receiver ends depend on the smaller channel parameter  only. We also observedthat under the same level of I/Q imbalance, the SNR penalty of a subcarrier QPSK system withI/Q imbalance at both transmitter and receiver ends is larger than the sum of the SNR penaltiesof a subcarrier QPSK system with transmitter I/Q imbalance only and a subcarrier QPSK systemwith receiver I/Q imbalance only. In addition, the numerical results showed that the average S-NR is dominated by the impacts of fading in the strong turbulence condition. Using a calibratedtransceiver, the impacts of the I/Q imbalance can be effectively mitigated. This chapter providesa frame work of analyzing error rate performance of linear modulations in the presence of I/Qimbalance. While our analysis is presented in the context of FSO communication, it can also be645.6. FhmmTelapplied to mobile communication scenario. The analysis can deal with the general asymmetric I/Qimbalance where there is typically more I/Q imbalance at the user terminal (due to form-factorconstraints) than at the basestation.65Whuptyr 6WonwlusionsIn this chapter, we conclude the thesis by summarizing the accomplished work and suggestingsome potential further research topics.6BE gummury oz Awwomplishyx korkIn this thesis, we developed analytical performance evaluation tools for subcarrier QPSK systemswith I/Q imbalance over the Gamma-Gamma fading channels. Besides the turbulence inducedfading, the SER performance of subcarrier QPSK systems is also affected by the I/Q imbalanceat the analog front-end. As the I/Q imbalance can occur at the transmitter, receiver, or bothends of the system, we discussed these three cases separately since they lead to slightly differentmodels. Such analytical tools can be used to investigate the performance of the subcarrier QPSKsystem at the early stage of system design. In addition, we carried out asymptotic analysis forsubcarrier QPSK systems with I/Q imbalance. These asymptotic results can be used as guidelinefor practical subcarrier QPSK system design. In order to conclude the thesis, we will summarizethe accomplished work as follows:− In Chapter 3, we studied the performance of the subcarrier QPSK system over the Gamma-Gamma turbulence channels with transmitter I/Q imbalance only. Since I/Q imbalanceoccurs at the transmitter only, we can observe that the pre-detection noise terms remainindependent. Afterwards, we developed an analytical tool that can be used to quantify theperformance degradation of a subcarrier QPSK signal in the presence of both transmitter I/Qimbalance and the Gamma-Gamma fading.− In Chapter 4, we investigated the performance of the subcarrier QPSK system over theGamma-Gamma fading channels with receiver I/Q imbalance only. The crosstalk of quadra-666.2. FhggefgeW Fhghee Wbekture branches causes correlation between the I-component and Q-component signals. Ananalytical tool was developed to quantify the performance degradation of a subcarrier QPSKsignal impaired by both the receiver I/Q imbalance and the Gamma-Gamma fading. The re-sults revealed that the amplitude I/Q imbalance is irrelevant to the performance degradation.− Chapter 5 investigated the performance of the subcarrier QPSK system over the Gamma-Gamma turbulence channels with I/Q imbalance at both transmitter and receiver ends.Due to the crosstalk of the quadrature branches, we can observe correlation between theI-component and Q-component signals. Under the generalized system model, we developedan analytical tool that can be used to quantify the performance degradation of a subcarrierQPSK system in the presence of both the I/Q imbalance and the Gamma-Gamma fading.With proper values of the parameters, we can also quantify the performance degradationof a subcarrier QPSK system over the Gamma-Gamma fading channels with transmitter orreceiver I/Q imbalance only. We observed that the receiver amplitude I/Q imbalance is ir-relevant to the performance degradation. Also, it can be noted that under the same level ofI/Q imbalance, the SNR penalty of a subcarrier QPSK system with I/Q imbalance at bothtransmitter and receiver ends is larger than the sum of the SNR penalties of a subcarrier QP-SK system with transmitter I/Q imbalance only and a subcarrier QPSK system with receiverI/Q imbalance only. Using a calibrated transceiver, we can conclude that the impacts of I/Qimbalance on error rate performance are effectively mitigated.6BF guggystyx Futury korkIn this thesis, we studied the SIM FSO system with QPSK scheme. As QPSK is a benchmarkscheme of the quadrature modulation, using our results as a starting point, it is of interest toinvestigate the effects of I/Q imbalance on the error rate performance of higher-order QAM as wellas adaptive QAM in different fading models.In this thesis, we studied the FI I/Q imbalance only. However, in the wideband communicationsystems, the impacts of the FD I/Q imbalance cannot be ignored and must be taken into accountin the performance analysis. Thus, it would be of future interest to analyze the performance of a676.2. FhggefgeW Fhghee Wbekwideband system with FD I/Q imbalance over atmospheric turbulence channels.In a QPSK OFDM systems, the kth carrier is interfered with its mirror carrier in the presenceof I/Q imbalance. Hence, the theoretical analysis of an SIM FSO QPSK OFDM system with I/Qimbalance over the Gamma-Gamma channels involves the joint CDF of the quadrivariate normaldistribution. Therefore, more mathematical tools are needed for the theoretical analysis of theperformance degradation. Thus, it would be of future interest to investigate the SER performanceof a subcarrier QPSK OFDM communication system with I/Q imbalance over the atmosphericturbulence channels.68Bivliogruphy[1] H. Willebrand and B. S. 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Vassiliou em Ze’, “A single-chip digitally calibrated 5.15-5.825-GHz 0.18-mCMOS transceiverfor 802.11a wireless LAN,” BEEE C’ Lhebd-LmZme Cbkcnbml, vol. 38, no. 12, pp. 2221–2231, Dec.2003. → pages 6475Appynxifi76Appynxifi ARecall the decomposition of baseband complex noise n(t) in (3.4)n(t) = nI(t) + jnQ(t): (A.1)The pre-detection I/Q component noise terms nI(t) and nQ(t) are independent AWGN randomprocesses, and they become AWGN RVs with mean zero and variances 2 = 2nR2 after sampling[28, Chapter 4]. With down-conversion and low-pass filtering, nˆI(t) and nˆQ(t) can be, respectively,written asnˆI(t) = LP {2 (nI(t) cos(2.fxt)− nQ(t) cos(2.fxt)) cos 2.fxt}= nI(t)nˆQ(t) = LP {2 (nI(t) cos(2.fxt)− nQ(t) cos(2.fxt)) (− sin 2.fxt)}= nQ(t)(A.2)where LP {· } represents the low-pass filtering procedure.Therefore, the sampled noise terms nˆI and nˆQ are independent Gaussian RVs with mean zeroand variances 2.77Appynxifi BIt can be shown that∫ 20(sin )x y =12∫ 02(sin )x−1(cos )−1(−2) cos  sin y=12∫ 10(1− t)x−12 t− 12yt (let t = cos2 )=∫ 10 t− 12 (1− t)0(1− t)x−12 ytB(12 P 1)(B.1)where B(xP y) is the Beta function defined as B(xP y) = Γ(x)Γ(y)Γ(x+y) [57, eq. (8.384.1)]. Using thedefinition of the Gauss Hypergeometric function [57, eq. (9.111)], we obtain (3.27).78Appynxifi WLet y = cos2 , one can obtain∫ tan−1(hl )0(sin )xy =12∫ 1l2h2+l2(1− y)x−12 y− 12yy=12(h2h2 + l2)x+12∫ 10tx−12 (1− h2h2 + l2t)−12yt(let t =h2 + l2h2(1− y))=(h2h2 + l2)x+12∫ 10 tx−12 (1− t)0(1− h2h2+l2t)−12yt(x+ 1)B(x+12 P 1) :(C.1)Using the definition of the Gauss Hypergeometric function [57, eq. (9.111)], it is straightforwardto obtain (3.36).79Appynxifi DUsing the Owen’s i -function defined in (4.16), we can obtain∫ ∞0i(x√xP v)xm−1yx (x S 0P v S 0)=∫ ∞0(12.∫ a0exp[−12x2x(1 + t2)]1 + t2yt)xm−1yx=12.∫ a011 + t2∫ ∞0exp[−12x2x(1 + t2)]xm−1yxyt:(D.1)With the help of an integral identity [57, eq. (3.326.2)]∫ ∞0xm exp (−txn) yx = Γ(m+1n)ntm+1n(D.2)it can be shown that ∫ ∞0i(x√xP v)xm−1yx (x S 0P v S 0)=Γ(m)2.(2x2)m ∫ a0(1 + t2)−m−1yt=Γ(m)2.(2x2)m v2∫ 10(1 + v2t)−m−1t− 12yt(let t =t2v2)=vΓ(m)2.(2x2)m ∫ 10 t− 12(1− t)0 (1 + v2t)−m−1 ytB(12 P 1) :(D.3)Finally using the definition of the Gauss Hypergeometric function [57, eq. (9.111)], we canobtain (4.26).80Appynxifi EFor A = −1− j (i.e. AI = −1, AQ = −1), we can obtain the means of rˆI and rˆQ asrˆI = I(1 + R)[−(1 + T ) cos(RR2− T R2) + (1− T ) sin(RR2 + T R2)] (E.1)andrˆQ = I(1 + R)[−(1 + T ) cos(RR2− T R2) + (1− T ) sin(RR2 + T R2)]: (E.2)The variances of rˆI and rˆQ are, respectively, 2rˆI= (1 + R)22 and 2rˆQ = (1 − R)22. Thecorrelation coefficient of rˆI and rˆQ is / = − sin R.Using the definition of BQ(hP l; /) in (4.14), the conditional symbol correct rate can be derivedasexPIGP−1−j(t) =∫ −SrQSrQ−∞∫ −SrISrI−∞12.√1− /2 exp[− 12(1− /2)(u2 − 2/uv + v2)] yuyv=BQ(−rˆIrˆIP−rˆQrˆQ;− sin R):(E.3)Therefore, the conditional SER can be calculated aseePIGP−1−j(t) =1− exPIGP−1−j(t)=12f (√thIGP1) +12f (√tlIGP1) + i (√thIGP1P vIGP1) + i (√tlIGP1P bIGP1)=eePIGP1+j(t):(E.4)Similarly, for A = −1 + j (i.e. AI = −1, AQ = 1), we can derive the means of rˆI and rˆQ asrˆI = I(1 + R)[−(1 + T ) cos(RR2− T R2)− (1− T ) sin(RR2 + T R2)] (E.5)andrˆQ = I(1 + R)[(1 + T ) cos(RR2− T R2) + (1− T ) sin(RR2 + T R2)]: (E.6)814cceaWik E.The variances of rˆI and rˆQ are, respectively, 2rˆI= (1 + R)22 and 2rˆQ = (1 − R)22. Thecorrelation coefficient of rˆI and rˆQ is / = − sin R.Thus, the conditional symbol correct rate can be derived asexPIGP−1+j(t) =∫ ∞−SrQSrQ∫ −SrISrI−∞12.√1− /2 exp[− 12(1− /2)(u2 − 2/uv + v2)] yuyv=Φ(−rˆIrˆI)−BQ(−rˆIrˆIP−rˆQrˆQ;− sin R)=12Φ (√thIGP2) +12Φ (√tlIGP2)− i (√thIGP2P vIGP2)− i (√tlIGP2P bIGP2) :(E.7)Using (E.7), after some algebraic manipulations, the conditional SER can be derived aseePIGP−1+j(t) =1− exPIGP−1+j(t)=12f (√thIGP2) +12f (√tlIGP2) + i (√thIGP2P vIGP2) + i (√tlIGP2P bIGP2)=eePIGP1−j(t):(E.8)82

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