The Pennsylvania State University. The Graduate School. College of Engineering OPTICAL WIRELESS COMMUNICATIONS: THEORY AND APPLICATIONS

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering OPTICAL WIRELESS COMMUNICATIONS: THEORY AND APPLICATIONS A Dissertation in Electrical Engineering by Mohammadreza Aminikashani 2016 Mohammadreza Aminikashani Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2016

2 The dissertation of Mohammadreza Aminikashani was reviewed and approved by the following: Mohsen Kavehrad W.L. Weiss Chair Professor of Electrical Engineering Dissertation Advisor, Chair of Committee Timothy Kane Professor of Electrical Engineering Julio Urbina Associate Professor of Electrical Engineering Jamal Rostami Associate Professor of Energy and Mineral Engineering Kultegin Aydin Professor of Electrical Engineering Head of the Department of Electrical Engineering Signatures are on file in the Graduate School. ii

3 Abstract This dissertation focuses on optical communications having recently attracted significant attentions as a promising complementary technique for radio frequency (RF) in both short- and long-range communications. These systems offer significant technical and operational advantages such as higher capacity, virtually unlimited reuse, unregulated spectrum and robustness to electromagnetic interference. Optical wireless communication (OWC) can be used both indoors and outdoors. Part of the dissertation contains novel results on terrestrial free-space optical (FSO) communications. FSO communication is a line-of sight technique that uses lasers for high rate wireless communication over distances up to several kilometers. In comparison to RF counterparts, a FSO link has a very high optical bandwidth available, allowing aggregate data rates on the order of Tera bits per second (1 Tera bits per second is 1000 Giga bites per second). However, FSO suffers limitations. The major limitation of the terrestrial FSO communication systems is the atmospheric turbulence, which produces fluctuations in the irradiance of the transmitted optical beam, as a result of random variations in the refractive index through the link. The existence of atmospheric-induced turbulence degrades the performance of FSO links particularly with a transmission distance longer than 1 kilometer. The identification of a tractable probability density function (pdf) to describe atmospheric turbulence under all irradiance fluctuation regimes is crucial in order to study the reliability of a terrestrial FSO system. This dissertation addresses this daunting problem and proposes a novel statistical model that accurately describes turbulence-induced fading under all irradiance conditions and unifies most of the proposed statistical models derived until now in the literature. The proposed model is important for the research community working on FSO communications because it allows them to fully capitalize on the potentials of currently used FSO iii

4 systems. Furthermore, utilizing this new statistical channel model, closed-form expressions for the diversity gain and the error rate performance of FSO links with spatial diversity are derived. In addition to addressing ways to improve outdoor FSO communication systems, this dissertation addresses some major challenges in indoor visible light communication (VLC). VLC is an advantageous technique that is proposed for wireless indoor communications. In VLC systems, the existence of multiple paths between the transmitter and receiver causes multipath distortion, particularly in links using non-directional transmitters and receivers, or in links relying upon non-line of-sight propagation. This multipath distortion can lead to intersymbol interference (ISI) at high bit rates. Multicarrier modulation usually implemented by orthogonal frequency division multiplexing (OFDM) can be used to mitigate ISI and multipath dispersion. Nevertheless, the performance of VLC systems employing OFDM modulation is significantly affected by nonlinear characteristic of light-emitting diode (LED) due to the large peak-to-average power ratio (PAPR) of OFDM signal. In other words, signal amplitudes below the LED turn-on-voltage and above the LED saturation point are clipped. This dissertation targets these important issues and successfully addresses them by developing some techniques to reduce high PAPR of optical OFDM signal and determining the optimum operating characteristics of LEDs for combined lighting and communications applications. VLC can also provide a practical solution for indoor positioning as global positioning system (GPS) does not provide an accurate and rapid indoor positioning since GPS radio signals are attenuated and scattered by walls of large buildings and other objects. A practical VLC system would be likely to deploy the same configuration for both positioning and communication purposes where high speed data rates are desired. This dissertation also proposes a novel OFDM VLC system that provides a high data rate transmission and can be used for both indoor positioning and communications where the multipath reflections are taken into account. Description of an experimental demonstration is also part of the dissertation where a software defined radio (SDR) was employed as the primary hardware and software interface to verify some of the results of the topics discussed earlier. iv

5 Table of Contents List of Figures List of Tables List of Abbreviations List of Symbols Acknowledgments ix xiii xiv xvii xx Chapter 1 Introduction Motivation Objectives Organization Chapter 2 Statistical Channel Model for Turbulence-Induced Fading in Free-Space Optical Systems Introduction Double GG Distribution Verification of the Proposed Channel Model Plane Wave Spherical Wave Performance Evaluation BER Analysis of SISO FSO System Outage Probability Analysis of SISO FSO System Conclusions v

6 Chapter 3 Statistical Analysis of Sum of Double Generalized Gamma Variates Introduction An Upper-Bound for the Distribution of the Sum of Double GG Variates An Approximate Distribution of the Sum of Double GG Variates Verification of the Mathematical Analysis Conclusions Chapter 4 Performance Evaluation of Free-Space Optical Links with Spatial Diversity System Model Performance Analysis of of MIMO FSO Links Performance Analysis of FSO Links with Receive Diversity SIMO FSO Links with Optimal Combining SIMO FSO Links with Selection Combining SIMO FSO Links with Equal Gain Combining Performance Analysis of FSO Links with Transmit Diversity MISO FSO Links Performance Comparison of Diversity Techniques Conclusions Chapter 5 Performance Evaluation of Single- and Multi-carrier Modulation Schemes for Indoor Visible Light Communication Systems Introduction System Model of ACO-OFDM System Model of ACO-SCFDE Peak-to-Average Power Ratio Performance Analysis Conclusions Chapter 6 Robust Timing Synchronization for AC OFDM Based Optical Wireless Communications Introduction New Timing Synchronization for AC Based OFDM Systems vi

7 6.2.1 Timing Synchronization for ACO-OFDM Timing Synchronization for PAM-DMT Timing Synchronization DHT Based Optical OFDM Simulation Results Experimental Results Conclusions Chapter 7 Indoor Location Estimation with Optical-based OFDM Communications Introduction System Configuration System Model Optical Wireless Channel OFDM Transmitter and Receiver Positioning Algorithm Simulation and Analysis Performance Comparison of Single- and Multi-carrier Modulation Schemes Effect of Signal Power on the Positioning Accuracy Effect of Modulation Order on the Positioning Accuracy Effect of Number of Subcarriers on the Positioning Accuracy Conclusions Chapter 8 Conclusions and Future Work Conclusions Future Work Publication List Appendix A Special Cases of Double GG Distribution 120 Appendix B Special Cases of BER Expression of SISO FSO System over Double GG Channel 123 Appendix C Special Cases of Outage Probability Expression of SISO FSO System over Double GG Channel 125 vii

8 Appendix D Special Cases of BER Expression of SIMO FSO Links over Double GG Channel 127 Bibliography 129 viii

9 List of Figures 2.1 Pdfs of the scaled log-irradiance for a plane wave assuming weak irradiance fluctuations Pdfs of the scaled log-irradiance for a plane wave assuming moderate irradiance fluctuations Pdfs of the scaled log-irradiance for a plane wave assuming strong irradiance fluctuations Pdfs of the scaled log-irradiance for a spherical wave assuming weak irradiance fluctuations Pdfs of the scaled log-irradiance for a spherical wave assuming moderate irradiance fluctuations Pdfs of the scaled log-irradiance for a spherical wave assuming strong irradiance fluctuations Average BER as a function of γ a) Plane wave - σ 2 Rytov = 2, l 0 /R 0 = 0.5, b) Plane wave-σ 2 Rytov = 25, l 0 /R 0 = 1, c) Spherical wave - σ 2 Rytov = 2, l 0 /R 0 = 0, d) Spherical wave - σ 2 Rytov = 5, l 0 /R 0 = Outage probability as a function of γ/γ th for a) Plane wave - σ 2 Rytov = 2, l 0 /R 0 = 0.5, b) Plane wave-σ 2 Rytov = 25, l 0 /R 0 = 1, c) Spherical wave - σ 2 Rytov = 2, l 0 /R 0 = 0, d) Spherical wave - σ 2 Rytov = 5, l 0 /R 0 = Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming plane wave and moderate irradiance fluctuations (channel a) Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming plane wave and strong irradiance fluctuations (channel b) Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming spherical wave and moderate irradiance fluctuations (channel c) Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming spherical wave and strong irradiance fluctuations (channel d) ix

10 4.1 Comparison of the average BER between SISO and SIMO with optimal combing for plane wave with σrytov 2 = 25 and l 0 /R 0 = Comparison of the average BER between SISO and SIMO with optimal combing for spherical wave with σrytov 2 = 2 and l 0 /R 0 = Bounded, approximate and exact outage probability of EGC and SISO for a) Plane wave with σrytov 2 = 2 and l 0 /R 0 = 0.5, b) Plane wave with σrytove 2 = 25 and I 0 /R 0 = Bounded, approximate and exact outage probability of EGC and SISO for a) Spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0, b) Spherical wave with σrytov 2 = 5 and l 0 /R 0 = Bounded, approximate and exact BER of EGC and SISO for a) Plane wave with σrytov 2 = 2 and l 0 /R 0 = 0.5, b) Plane wave with σrytove 2 = 25 and I 0 /R 0 = Bounded, approximate and exact BER of EGC and SISO for a) Spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0, b) Spherical wave with σrytov 2 = 5 and l 0 /R 0 = Exact, approximate and asymptotic BER of EGC over two i.n.i.d. atmospheric turbulence channels defined as plane wave with σrytov 2 = 2 and l 0 /R 0 = 0.5, and plane wave with σrytove 2 = 25 and I 0 /R 0 = Exact, approximate and asymptotic BER of EGC over two i.n.i.d. atmospheric turbulence channels defined as spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0, and spherical wave with σrytov 2 = 5 and l 0 /R 0 = Comparison of the average BER between SISO and different diversity techniques for plane wave assuming i.i.d. turbulent channel with σrytove 2 = 25 and I 0 /R 0 = Comparison of the average BER between SISO and different diversity techniques for spherical wave assuming i.i.d. turbulent channel with σrytov 2 = 2 and l 0 /R 0 = Comparison of the OC, EGC and SC receivers for SIMO FSO links over two i.n.i.d. atmospheric turbulence channels defined as plane wave with σrytov 2 = 2 and l 0 /R 0 = 0.5, and plane wave with σrytove 2 = 25 and I 0 /R 0 = Comparison of the OC, EGC and SC receivers for SIMO FSO links over two i.n.i.d. atmospheric turbulence channels defined as spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0, and spherical wave with σrytov 2 = 5 and l 0 /R 0 = ACO-OFDM transmitter and receiver configuration ACO-SCFDE transmitter and receiver configuration x

11 5.3 CCDF of PAPR comparison of ACO-OFDM and ACO-SCFDE for L = CCDF of PAPR comparison of ACO-OFDM and ACO-SCFDE for L = Impulse response of the indoor diffuse channel Transfer characteristics of OPTEK, OVSPxBCR4 1-Watt white LED. (a) Fifth-order polynomial fit to the data. (b) The curve from the data sheet BER comparison of ACO-OFDM and ACO-SCFDE for bias point of 3.2V BER of ACO-OFDM for M = 16 for different bias points BER comparison of uncoded and coded ACO-OFDM and ACO- SCFDE for M = Normalized SNR versus normalized bandwidth/bit-rate required to achieve BER of a) Average of Schmidl s and Park s timing metrics with modified training symbol suitable for ACO-OFDM. b) Average of Tian s timing metrics for ACO-OFDM. c) Average of timing metrics for bipolar correlation method for ACO-OFDM. d) Average of timing metrics for bipolar correlation method for PAM-DMT systems a) Schematic of the experimental setup. b) Real implementation with software defined radio systems Average of timing metrics for bipolar correlation method for consecutive ACO-OFDM symbols with a) L = 256 and 4-QAM modulation b) L = 256 and 16-QAM modulation c) L = 512 and 4-QAM modulation b) L = 512 and 16-QAM modulation System configuration The contributions from different orders of reflections to the total impulse response of a location at the center of the room (weak scatterings and multipath reflections) The contributions from different orders of reflections to the total impulse response of a location at the edge of the room (medium scatterings and multipath reflections) The contributions from different orders of reflections to the total impulse response of a location at the corner of the room (strong scatterings and multipath reflections) OFDM transmitter and receiver configuration for both positioning and communication purposes xi

12 7.6 Positioning error distribution for OFDM system with 4-QAM modulation, L = 512 and P te,k = 5 dbm Positioning error distribution for OOK modulation with P te,k = 5 dbm Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 512 and P te,k = 5 dbm Histogram of positioning errors for OOK modulation with P te,k = 5 dbm Positioning error distribution for OFDM system with 4-QAM modulation, L = 512 and P te,k = -10 dbm Positioning error distribution for OFDM system with 4-QAM modulation, L = 512 and P te,k = 20 dbm Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 512 and P te,k = -10 dbm Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 512 and P te,k = 20 dbm Positioning error distribution for OFDM system with 16-QAM modulation, L = 512 and P te,k = 5 dbm Positioning error distribution for OFDM system with 64-QAM modulation, L = 512 and P te,k = 5 dbm Histogram of positioning errors for OFDM system with 16-QAM modulation, L = 512 and P te,k = 5 dbm Histogram of positioning errors for OFDM system with 64-QAM modulation, L = 512 and P te,k = 5 dbm Positioning error distribution for OFDM system with 4-QAM modulation, L = 64 and P te,k = 5 dbm Positioning error distribution for OFDM system with 4-QAM modulation, L = 256 and P te,k = 5 dbm Positioning error distribution for OFDM system with 4-QAM modulation, L = 1024 and P te,k = 5 dbm Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 64 and P te,k = 5 dbm Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 256 and P te,k = 5 dbm Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 1024 and P te,k = 5 dbm xii

13 List of Tables 2.1 NRMSE for different statistical models and turbulence conditions defined in Figs. 2.1 to Room configuration under consideration Optical and electrical characteristics of OPTEK, OVSPxBCR4 1- Watt white LED System parameters Positioning error for single- and multi-carrier modulation schemes. 102 xiii

14 List of Abbreviations AC asymmetrically clipped. ACO-OFDM asymmetrically clipped optical OFDM. AoA angle of arrival. AWGN Additive White Gaussian noise. BER bit error rate. BICM bit-interleaved coded modulation. ccdf complementary cumulative distribution function. cdf cumulative distribution function. CDMMC combined deterministic and modified Monte Carlo. CP cyclic prefix. CPC compound parabolic concentrator. CSI channel state information. DHT discrete Hartley transform. Double GG Double Generalized Gamma. EGC equal gain combining. FFT fast Fourier transform. xiv

15 FOV field of view. FSO free-space optical. GG Generalized Gamma. GPS global positioning system. i.i.d. independent and identically distributed. i.n.i.d. independent but not necessarily identically distributed. IBT inter-block interference. ICT inter-carrier interference. ID identification. IFFT inverse fast Fourier transform. IM/DD intensity modulation and direct detection. IR infrared. ISI intersymbol interference. LBS location based services. LED light-emitting diode. LOS line-of-sight. MIMO multiple-input multiple-output. MISO multiple-input single-output. MMC modified Monte Carlo. MMSE minimum mean square error equalization. NRMSE normalized root-mean-square error. OC optimal gain combining. OFDM orthogonal frequency division multiplexing. xv

16 OOK on-off keying. OWC optical wireless communication. PAM-DMT PAM-modulated discrete multitone. PAPR peak-to-average power ratio. PC personal computer. PD photo diode. pdf probability density function. RF radio frequency. RMS root mean square. RSS received signal strength. RV random variable. SC selection combining. SCFDE single carrier frequency domain equalization. SDR software defined radio. SIMO single-input multiple-output. SISO single-input single-output. SNR signal to noise ratio. U-OFDM unipolar OFDM. UV ultraviole. VLC visible light communication. xvi

17 List of Symbols I U x U y γ i m i Ω i Irradiance of the received optical wave Large-scale irradiance fluctuations Small-scale irradiance fluctuations Generalized Gamma (GG) distribution parameter GG distribution parameter GG distribution parameter G m,n p,q [.] Meijer s G-function σx 2 σy 2 σrytov 2 R 0 l 0 β0 2 σi 2 R t γ th γ Large-scale fluctuations variance Small-scale fluctuations variance Rytov parameter Fresnel zone Finite inner scale Rytov scintillation index of a spherical wave Scintillation index Targeted transmission rate Signal to noise ratio (SNR) threshold Instantaneous electrical SNR xvii

18 γ P e P out ν σn 2 y n x η erfc (.) Q (.) K ν (.) A N G N P t,k P r,k H k (0) A r ψ k T s (ψ k ) g (ψ k ) ϕ k d k m H Average electrical SNR Bit error rate (BER) probability Outage probability Additive White Gaussian noise (AWGN) Noise variance Received signal at the n th receive aperture Information bits Optical-to-electrical conversion coefficient Complementary error function Gaussian Q-function Modified Bessel function of the second kind Arithmetic means Geometric means Transmitted optical power from the k th LED bulb Received optical power from the k th transmitter Channel DC gain Physical area of the detector Angle of incidence with respect to the receiver axis Gain of optical filter Concentrator gain Angle of irradiance with respect to the transmitter perpendicular axis Distance between transmitter k and receiver Lambertian order Transmitter height xviii

19 h n r Ψ c P k,i Receiver height Refractive index of the concentrator FOV of the concentrator Power attenuation of the i th symbol transmitted from the k th transmitter (x c, y c ) Receiver coordinates (x ck, y c k) k th transmitter coordinates xix

20 Acknowledgments I would like to thank all those people who have made this dissertation possible and because of whom my Ph.D. experience has been one that I will cherish forever. Foremost, I would like to sincerely thank my supervisor, Prof. Mohsen Kavehrad. It is hard to find words to describe how much I appreciate all he has done for me as a supervisor and mentor. His inspiration and guidance have motivated me throughout the course of my Ph.D. studies. I am truly grateful to the members of my dissertation committee, Professors Timothy Kane, Julio Urbina and Jamal Rostami for their time in evaluating this dissertation and providing me valuable feedback. I would like to specially thank Prof. Murat Uysal for his helpful suggestions and comments on Chapters 2 and 4. I am also thankful to all my colleagues in CICTR group for their help and support. I sincerely thank the Penn State Graduate School for selecting me for the Alumni Association Dissertation Award. I would like to thank Prof. Kavehrad once again for nominating me for this award. I am also grateful to graduate committee at the Department of Electrical Engineering who supported my nomination. Finally, I would like to thank my beloved wife, Sepideh, for her unconditional love, kindness, and patience, and my parents and siblings whose love, encouragement and prayer helped me overcome difficulties throughout my life. Without them, nothing would have been possible. xx

21 To my wife, Sepideh, and my dear parents. xxi

22 Chapter 1 Introduction 1.1 Motivation Optical wireless communication (OWC) has been extensively studied recently as a powerful and promising complementary and/or alternative to the existing radio frequency (RF) solutions for a wide range of applications [1 4]. These systems enable wireless connectivity within wavelengths ranging from infrared (IR) to ultraviole (UV) including the visible light spectrum. They offer many attractive features such as higher capacity, robustness to electromagnetic interference, excellent security and low cost deployment. OWC comprises two main categories, indoor and outdoor systems. Indoor OWC is free from major outdoor environmental degradations such as rain, snow, building sway, and atmospheric turbulence, and usually characterized by short transmission range. Outdoor OWC commonly referred to as free-space optical (FSO) communication is categorized as satellite-based and terrestrial-based links. Terrestrial FSO communication uses lasers or light-emitting diodes (LEDs) to optically transmit information through atmosphere [5, 6]. These systems provide 1

23 high data rates comparable to fiber optics while they offer much more flexibility in (re)deployment. Since they operate in unregulated spectrum, no licensing fee is required making them also a cost-effective solution [1, 7, 8]. With their unique features and advantages, FSO systems have attracted attention initially as a "last mile" solution and can be used in a wide array of applications including cellular backhaul, inter-building connections in enterprise/campus environments, video surveillance/monitoring, fiber back-up, redundant link in disaster recovery and relief efforts among others. On the other hand, IR and visible light bands are commonly used for indoor OWC applications. OWC systems operating in the visible band ( nm) are commonly referred to as visible light communication (VLC) [9,10]. The novelty of this technique is its dual usage. VLC makes use of light emitted from LED lamps to both transmit data and emit light. LED lights are now favorable to incandescent lamps because of their greater energy efficiency. As a strong candidate for highspeed wireless networks of the next generation, VLC offers many advantages over its RF counterparts summarized as Visible light, together with IR and UV spectral band, provides unregulated and unlimited bandwidth promising a practical solution to the current spectrum crunch issue. The frequency band can be reused among different rooms and a secure communication is easily achieved as light waves do not penetrate into solid walls and therefore are confined in a room. Unlike the use of WiFi, the wireless technique most often currently implemented, use of VLC does not involve the interference on the RF since optical wireless utilizes optical carrier to convey data through media. Therefore, it 2

24 is most suitable for RF-restricted environments such as hospitals, aircrafts, military installation and factory floors. The installation of a VLC wireless network is easy and cost effective as illumination infrastructure exists. VLC is also a strong candidate for indoor positioning systems considering global positioning system (GPS) does not perform well for indoor environments since a satellite signal suffers from severe attenuation when penetrating into solid walls [11 13]. Indoor positioning can be used for a number of applications such as guiding users inside large buildings, detecting location of products inside large warehouses and location based services (LBS). 1.2 Objectives Despite the key advantages of OWC, its widespread use has been hindered by a number of major issues. This dissertation targets and addresses some of these important issues for both indoor and outdoor wireless communication. The performance of terrestrial FSO systems can be significantly diminished by turbulence-induced fading (also called as scintillation) resulting from beam propagation through the atmosphere. Since the calculation of detection and fade probabilities are primarily based on the probability density function (pdf) of this random phenomenon, identification of a unifying statistical model which is valid under all range of turbulence conditions is vital. This dissertation addresses this very challenging issue and proposes a novel and unifying pdf which accurately describes irradiance fluctuations over atmospheric channels under a wide range of turbulence conditions. The proposed model called Double Generalized Gamma 3

25 (Double GG) distribution outperforms existing turbulence channel models in the literature. Using this new statistical channel model, we derive closed-form expressions for the outage probability and the average bit error as well as corresponding asymptotic expressions of FSO systems over turbulence channels. We demonstrate that our derived expressions cover many existing results in the literature earlier reported for other channel models as special cases. Spatial diversity techniques provide a promising approach to mitigate turbulenceinduced fading. In this dissertation, we obtain a closed-form upper-bound and a novel and accurate approximation for the distribution of a sum of Double GG random variables (RVs). Then, capitalizing on the derived distribution, we study the performance of FSO links with spatial diversity over atmospheric turbulence channels described by Double GG distribution. On the other hand, VLC systems suffer from multipath distortion due to dispersion of the optical signal caused by reflections from various sources inside a room. This dispersion leads to intersymbol interference (ISI) at high data rates which reduces signal to noise ratio (SNR) and severely impairs the link performance. We investigate and compare the performance of single- and multi-carrier modulation schemes used to combat ISI effects and improve the performance of indoor VLC systems taking into account both nonlinear characteristics of LED and dispersive nature of optical wireless channel. We also present a robust timing synchronization scheme suitable for asymmetrically clipped (AC) orthogonal frequency division multiplexing (OFDM) based optical wireless systems. An experimental demonstration of an indoor VLC system using software defined radio (SDR) devices as the primary hardware is also described to verify the theoretical results and further demonstrate the usefulness of our methods. 4

26 Finally, a novel OFDM VLC system is proposed which can be utilized for both communications and indoor positioning. We calculate the positioning errors in all the locations of a room and compare them with those using single carrier modulation scheme. The impact of different system parameters on the positioning accuracy of the proposed OFDM VLC system is further investigated. 1.3 Organization The dissertation organization follows the steps mention in the previous sub-section. In the next Chapter, we propose Double GG distribution to characterize turbulenceinduced fading and confirm the accuracy of our model through comparisons with simulation data for plane and spherical waves. We also present the derivation of bit error rate (BER) and outage probability for single-input single-output (SISO) FSO systems over Double GG channel. In Chapter 3, an upper bound along with an approximate expression for the distribution of the sum of Double GG variates is presented. We confirm the accuracy of the derived expressions through comparisons with Monte-Carlo simulation results for plane and spherical wave propagation. In Chapter 4, we introduce the multiple-input multiple-output (MIMO) FSO system model and provide the BER expressions for single-input multiple-output (SIMO), multiple-input single-output (MISO) and MIMO FSO links. We present numerical results to confirm the accuracy of the derived expressions and demonstrate the advantages of employing spatial diversity over SISO links. In Chapter 5, we briefly describe indoor VLC asymmetrically clipped optical OFDM (ACO-OFDM) and single carrier frequency domain equalization (SCFDE) systems and compare their peak-to-average power ratio (PAPR) performance. We study the impact of LED bias point on the performance of multi-carrier modulation schemes. We also 5

27 investigate and compare the performance of single- and multi-carrier modulation schemes and show that bit-interleaved coded modulation (BICM) can combat signal degradation due to LED nonlinearity and ISI. In Chapter 6, we present a novel and robust timing synchronization method that works perfectly for all AC systems, namely ACO-OFDM, PAM-modulated discrete multitone (PAM-DMT) and discrete discrete Hartley transform (DHT) based optical OFDM systems, and can also be used for channel estimation simultaneously. We verify the accuracy of the proposed method through simulations and experimental results. In Chapter 7, an OFDM VLC system is proposed which can be employed for both communications and indoor positioning. We present numerical results on the positioning accuracy of the proposed OFDM VLC system and compare its performance with that of its on-off keying (OOK) counterpart. The effect of different OFDM system parameters on the positioning accuracy is also investigated. Finally, Chapter 8 concludes and summarizes this thesis. 6

28 Chapter 2 Statistical Channel Model for Turbulence-Induced Fading in Free-Space Optical Systems In this Chapter, we propose a new probability distribution function which accurately describes turbulence-induced fading under a wide range of turbulence conditions. The proposed model, termed Double Generalized Gamma (Double GG), is based on a doubly stochastic theory of scintillation and developed via the product of two Generalized Gamma (GG) distributions. The proposed Double GG distribution generalizes many existing turbulence channel models and provides an excellent fit to the published plane and spherical waves simulation data. Using this new statistical channel model, we derive closed-form expressions for the outage probability and the average bit error rate (BER) as well as corresponding asymptotic expressions of free-space optical (FSO) communication systems over turbulence channels. We demonstrate that our derived expressions cover many existing results in the literature earlier reported for Gamma-Gamma, Double-Weibull 7

29 and K channels as special cases. 2.1 Introduction A major performance limiting factor in terrestrial FSO systems is atmospheric turbulence-induced fading (also called as scintillation) [14]. Inhomogenities in the temperature and the pressure of the atmosphere result in variations of the refractive index and cause atmospheric turbulence. This manifests itself as random fluctuations in the received signal and severely degrades the FSO system performance particularly over long ranges. In the literature, several statistical models have been proposed in an effort to model this random phenomenon. Historically, log-normal distribution has been the most widely used model for the probability density function (pdf) of the random irradiance over atmospheric channels [15 17]. This pdf model is however only applicable to weak turbulence conditions. As the strength of turbulence increases, lognormal statistics exhibit large deviations compared to experimental data. Moreover, lognormal pdf underestimates the behavior in the tails as compared with measurement results. Since the calculation of detection probabilities for a communication system is primarily based on the tails of the pdf, underestimating this region significantly affects the accuracy of performance analysis. In an effort to address the shortcomings of the lognormal distribution, other statistical models have been further proposed to describe atmospheric turbulence channels under a wide range of turbulence conditions. These include the Negative Exponential/Gamma model (also known widely as the K channel) [18], I-K distribution [19], log-normal Rician channel (also known as Beckman) [20], Gamma- Gamma [21], M distribution [22] and Double-Weibull [23]. Particularly worth 8

30 mentioning is the Gamma-Gamma model [21], [24] which has been widely used in the literature for the performance analysis of FSO systems, see e.g., [25, 26], along with the log-normal model. This model builds upon a two-parameter distribution and considers irradiance fluctuations as the product of small-scale and large-scale fluctuations, where both are governed by independent gamma distributions. In a more recent work by Chatzidiamantis et al. in [23], the Double-Weibull distribution was proposed as a new model for atmospheric turbulence channels. Similar to the Gamma-Gamma model, it is based on the theory of doubly stochastic scintillation and considers irradiance fluctuations as the product of small-scale and large-scale fluctuations which are both Weibull distributed. It is shown in [23] that Double-Weibull is more accurate than the Gamma-Gamma particularly for the cases of moderate and strong turbulence. In this Chapter, we propose a new and unifying statistical model, named Double GG, for the irradiance fluctuations. The proposed model is valid under all range of turbulence conditions (weak to strong) and contains most of the existing statistical models for the irradiance fluctuations in the literature as special cases. Furthermore, we provide comparison of the proposed model with Gamma-Gamma and Double-Weibull models. For this purpose, we use the set of simulation data from [27,28] for plane and spherical waves 1. Our model demonstrates an excellent match to the simulation data and is clearly superior over the other two models which show discrepancy from the simulation data in some cases. In the second 1 The simulation data in [27, 28] was obtained through phase screen approach which consists of approximating a three-dimensional random medium as a collection of equally spaced, two-dimensional, random phase screens that are transverse to the direction of wave propagation. In [27], it was discussed in detail that such a numerical simulation approach contains all the essential physics for accurately predicting the pdf of irradiance (or, equivalently, log-normal irradiance), and shown that the simulation results provide an excellent match to known experimental measurements reported in [29] for both plane and spherical waves. The same set of simulation data was also used in [21] and [23] which respectively introduced Gamma-Gamma and Double-Weibull distributions as turbulence channel models. 9

31 part of this Chapter, we use this new channel model to derive closed-form expressions for the BER and the outage probability of single-input single-output (SISO) FSO systems with intensity modulation and direct detection (IM/DD). Our performance results can be seen as a generalization of the results in [30 33]. 2.2 Double GG Distribution The irradiance of the received optical wave can be modeled as [21], [23] I = U x U y, where U x and U y are statistically independent random processes arising respectively from large-scale and small scale turbulent eddies. We assume that both large-scale and small-scale irradiance fluctuations are governed by GG distributions [34, Eq. (1)]. The pdfs of U x GG (γ 1, m 1, Ω 1 ) and U y GG (γ 2, m 2, Ω 2 ) are given by f Ux (U x ) = f Uy (U y ) = γ 1 U m 1γ 1 1 ( x (Ω 1 /m 1 ) m 1 Γ (m 1 ) exp m ) 1 U γ 1 x Ω 1 γ 2 U m 2γ 2 1 ( y (Ω 2 /m 2 ) m 2 Γ (m 2 ) exp m ) 2 U γ 2 y Ω 2 (2.1) (2.2) where γ i > 0, m i 0.5 and Ω i i = 1, 2 are the GG parameters. The pdf of I can be written as f I (I) = 0 f Ux (I U y ) f Uy (U y ) du y (2.3) where f Ux (I U y ) is obtained as f Ux (I U y ) = γ 1 (I/U y ) m 1γ 1 1 ( U y (Ω 1 /m 1 ) m 1 Γ (m 1 ) exp m ( ) γ1 ) 1 I. (2.4) Ω 1 U y 10

32 The integration in (2.3) yields f I (I) = γ 2pp m2 1/2 q m1 1/2 (2π) 1 (p+q)/2 I 1 Γ (m 1 ) Γ (m 2 ) ( ) p G 0,p+q Ω2 p p q q Ω q 1 p+q,0 I γ 2 m q 1m p (q : 1 m 1), (p : 1 m 2 ) 2 (2.5) where G m,n p,q [.] is the Meijer s G-function 2 defined in [35, Eq.(9.301)], p and q are positive integer numbers that satisfy p/q = γ 1 /γ 2 and (j; x) is defined as (j; x) x/j,..., (x + j 1)/j. (2.6) We name this new distribution as Double GG. Employing [36, Eq. (10)] and after some simplifications, the cumulative distribution function (cdf) of Double GG distribution can be obtained as F I (I) = pm 2 1/2 q m1 1/2 (2π) 1 (p+q)/2 Γ (m 1 ) Γ (m 2 ) ( G p+q,1 I γ 2 ) p m q 1m p 1 2 1,p+q+1 Ω 2 p p q q Ω q. 1 (q : m 1 ), (p : m 2 ), 0 (2.7) The distribution parameters γ i and Ω i i = 1, 2 of the Double GG model can be identified using the first and second order moments of small and large scale irradiance fluctuations. The latter are directly tied to the atmospheric parameters. Without loss of generality, we assume E (I) = 1 or equivalently E (U x ) = 1 and 2 Meijer s G-function is a standard built-in function in mathematical software packages such as MATLAB, MAPLE and MATHEMATICA. If required, this function can be also expressed in terms of the generalized hypergeometric functions using [35, Eqs.( )]. 11

33 E (U y ) = 1. The second moment of irradiance is expressed as E ( I 2) = E ( U 2 x ) E ( U 2 y ) = ( 1 + σ 2 x ) ( 1 + σ 2 y ) (2.8) where σ 2 x and σ 2 y are respectively normalized variances of U x and U y. The n th moment of U x (similarly U y ) is given by E (U n x ) = ( Ω1 m 1 ) n/γ1 Γ (m 1 + n/γ 1 ). (2.9) Γ (m 1 ) Therefore, by inserting the second order moment obtained from (2.9) in (2.8), and considering that E (I) = 1, we have σx 2 = Γ (m 1 + 2/γ 1 ) Γ (m 1 ) 1, (2.10a) Γ 2 (m 1 + 1/γ 1 ) σy 2 = Γ (m 2 + 2/γ 2 ) Γ (m 2 ) 1, (2.10b) Γ 2 (m 2 + 1/γ 2 ) ( ) γi Γ (m i ) Ω i = m i, i = 1, 2 (2.11) Γ (m i + 1/γ i ) where m i is a distribution shaping parameter and found using curve fitting on the simulated/measured channel data. Note that in (2.10a) and (2.10b), the variances of small- and large-scale fluctuations (i.e., σ 2 x and σ 2 y) are directly tied to the atmospheric conditions. Particularly, assuming a plane wave when inner scale effects are considered, the variances for the large- and the small-scale scintillations 12

34 are given by [14, Eqs and 9.55] σx 2 = exp 0.16σRytov η l η l σRytovη 2 7/6 l 1/ η l σRytovη 2 7/6 l 7/ η l σ 2 Rytovη 7/6 l σy 2 = exp 0.51σRytov 2 ) 5/6 ( σ 12/5 Rytov 7/6 1, (2.12) 1 (2.13) where σ 2 Rytov is Rytov parameter, η l = (R 0 /l 0 ), and R 0 /l 0 denotes the ratio of Fresnel zone to finite inner scale. For spherical waves in the absence of inner scale, σ 2 x and σ 2 y are given by [14, Eqs and 9.70] σx β0 2 = exp ( 12/ β 0 σy β0 2 = exp ( 12/ β 0 ) 7/6 ) 5/6 1, (2.14) 1 (2.15) where β 2 0 is the Rytov scintillation index of a spherical wave given by β 2 0 = σ 2 Rytov/ σ 2 (l 0 /R 0 ). (2.16) 13

35 In (2.16), σ 2 (l 0 /R 0 ) is defined as σ 2 (l 0 /R 0 ) [ (1 ) ( = /η 2 11/12 l sin ( η ) 3 tan 1 l (9 + η 2 l )1/4 sin sin (9 + ηl 2 )7/24 3 ( 5 4 tan 1 η l 3 ( 11 6 tan 1 η l 3 ) ) ) ] 8.75η 5/6 l. (2.17) In the presence of a finite inner scale, the small-scale scintillation is again described by (2.15) and the large-scale variance is given by [14, Eq. 78] σx 2 = exp 0.04β ( ( ( 8.56η l η l β 2 0η 7/ η l β 2 0η 7/ η l β 2 0η 7/6 l l l ) 7/12 ) 7/6 ) 1/2 1. (2.18) Therefore, the parameters of the Double GG distribution are readily deduced from these expressions using only values of the refractive index structure parameter and inner scale according to the atmospheric conditions. The scintillation index can be further calculated as σi 2 = E (I2 ) E(I) 2 1 = ( ) ( ) 1 + σx σ 2 y 1. (2.19) We should emphasize that this distribution is very generic since it includes some commonly used fading models as special cases. As demonstrated in Appendix A, for γ i 0, m i, Double GG pdf coincides with the log-normal pdf. For γ i = 1, Ω i = 1, it reduces to Gamma-Gamma while for m i = 1, it becomes 14

36 Table 2.1: NRMSE for different statistical models and turbulence conditions defined in Figs. 2.1 to 2.6 Gamma-Gamma [21] Double-Weibull [23] Double GG (Proposed) Fig % 7% 0.6% Fig % 1.2% 0.8% Fig % 1% 0.8% Fig % 10% 0.3% Fig % 8.7% 1.5% Fig % 2.4% 1.7% Double-Weibull. For γ i = 1, Ω i = 1, m 2 = 1, it coincides with the K channel. 2.3 Verification of the Proposed Channel Model In this section, we compare the Double GG distribution model with simulation data of plane and spherical waves provided respectively in [27] and [28]. In [27], Flatté et al. carried out exhaustive numerical simulations and published the results assuming plane wave propagation through homogeneous and isotropic Kolmogorov turbulence. In [28], Hill and Frehlich presented the simulation data for the propagation of a spherical wave through homogeneous and isotropic atmospheric turbulence. The turbulence severity is characterized by Rytov variance (σrytov) 2 which is proportional to the scintillation index [14]. We emphasize that the same data set was also employed in [21] and [23] which have introduced the Gamma-Gamma and Double-Weibull fading models Plane Wave Figs. 2.1 to 2.3 compare the Gamma-Gamma, Double-Weibull and Double GG models under a wide range of turbulence conditions (weak to strong) assuming 15

37 Figure 2.1: Pdfs of the scaled log-irradiance for a plane wave assuming weak irradiance fluctuations. plane wave propagation. In these figures, the vertical axis of the figure represents the log-irradiance pdf multiplied by the square root of variance. The logarithm of irradiance was particularly chosen to illustrate the high and low irradiance tails [27]. Thus, sensitivity to the small irradiance fades is increased, while sensitivity to large irradiance peaks is decreased. The pdf plots were also scaled by subtracting the mean value to center all distributions on zero and dividing by the square root of variance. In Fig. 2.1, we assume weak turbulence conditions which are characterized by σrytov 2 = 0.1 and l 0 /R 0 = 0.5. The values of the variances of small and large scale fluctuations, (σx 2 and σy) 2 are calculated from (2.12) and (2.13). Using (2.10a), (2.10b) and (2.11), the Double GG parameters are obtained as γ 1 = 2.1, γ 2 = 2.1, m 1 = 4, m 2 = 4.5, Ω 1 = and Ω 2 = 1.06 where p = q = 1. 16

38 Figure 2.2: Pdfs of the scaled log-irradiance for a plane wave assuming moderate irradiance fluctuations. We further employ normalized root-mean-square error (NRMSE) as a statistical goodness of fit test. Table 2.1 provides the NRMSE results for different statistical models. According to Table 2.1 and Fig. 2.1, both Gamma-Gamma and Double- Weibull distributions fail to match the simulation data. On the other hand, the proposed Double GG distribution follows closely the simulation data. In Fig. 2.2, we assume moderate irradiance fluctuations which are characterized by σrytov 2 = 2 and l 0 /R 0 = 0.5. The parameters of the Double GG distribution for this case are obtained as γ 1 = , γ 2 = , m 1 = 0.55, m 2 = 2.35, Ω 1 = and Ω 2 = In the calculations, p and q are chosen as p = 28 and q = 11 in order to satisfy p/q = γ 1 /γ 2. Among the three distributions under consideration, the proposed Double GG model provides the best fit to the 17

39 Figure 2.3: Pdfs of the scaled log-irradiance for a plane wave assuming strong irradiance fluctuations. simulation data. It is apparent that Gamma-Gamma fails to match the simulation data particularly in the tails. As Table 2.1 demonstrates, the accuracy of Double Weibull is better than that of Gamma-Gamma, but slightly inferior to our proposed distribution. In Fig. 2.3, we assume strong irradiance fluctuations which are characterized by σrytove 2 = 25 and I 0 /R 0 = 1. The parameters of the Double GG distribution are calculated as γ 1 = , γ 2 = , m 1 = 0.5, m 2 = 1.8, Ω 1 = and Ω 2 = where p and q are chosen as 17 and 7 respectively. The Double GG model again provides an excellent match to the simulation data and as it is clear from Table 2.1, its accuracy is better than Gamma-Gamma and Double Weibull. 18

40 Figure 2.4: Pdfs of the scaled log-irradiance for a spherical wave assuming weak irradiance fluctuations Spherical Wave Figs. 2.4 to 2.6 compare the Gamma-Gamma, Double-Weibull and Double GG models under weak, moderate and strong turbulence conditions assuming spherical wave propagation. These pdfs are plotted as a function of (ln I + 0.5σ 2 )/σ [28], where σ is the square root of the variance of ln I. The y-axis depicts the logirradiance pdf multiplied by σ. In Fig. 2.4, we consider spherical wave propagation and assume weak turbulence which are characterized by σrytov 2 = 0.06 and l 0 /R 0 = 0. The parameters of Double GG are evaluated using the variances of small and large scale fluctuations, (σy 2 and σx) 2 for spherical waves. The values of these variances are given by (2.14), (2.15) and (2.18). Therefore, employing (2.10a), (2.10b) and (2.11), we obtain 19

41 Figure 2.5: Pdfs of the scaled log-irradiance for a spherical wave assuming moderate irradiance fluctuations. m 1 = 34.24, m 2 = 32.79, γ 1 = γ 2 = Ω 1 = Ω 2 = 1 where p and q are equal to 1. It can be noted that in this case, the Double GG coincides with the Gamma- Gamma distribution. It is apparent that both Gamma-Gamma and Double GG distributions provide an excellent match to the simulation data while the Double- Weibull distribution fails to match the simulation data. In Fig. 2.5, we assume moderate irradiance fluctuations, which are characterized by σrytov 2 = 2 and l 0 /R 0 = 0. The parameters of the Double GG model for this case are calculated as γ 1 = , γ 2 = , m 1 = 2.65, m 2 = 0.85, Ω 1 = and Ω 2 = where p and q are selected as 7 and 11 respectively. It is clearly observed that the Double GG model provides a better fit with simulation data, especially for small irradiance values. In Fig. 2.6, we assume strong irradiance fluctuations which are characterized by 20

42 Figure 2.6: Pdfs of the scaled log-irradiance for a spherical wave assuming strong irradiance fluctuations. σrytov 2 = 5 and l 0 /R 0 = 1. The parameters of the Double GG model are calculated as γ 1 = , γ 2 = , m 1 = 3.2, m 2 = 2.8, Ω 1 = and Ω 2 = where p and q are chosen as 7 and 11 respectively. It is apparent from this figure and Table 2.1 that both Gamma-Gamma and Double-Weibull distributions fail to match the simulation data. On the other hand, the proposed Double GG distribution follows closely the simulation data. 2.4 Performance Evaluation BER Analysis of SISO FSO System In this section, we present the BER performance analysis of an FSO system with on-off keying (OOK) over the proposed Double GG channel. The received optical 21

43 signal is written as y = ηix + ν (2.20) where x represents the information bits and can be either 0 or 1, ν is the Additive White Gaussian noise (AWGN) term with zero mean and variance σn 2 = N 0 /2, η is the optical-to-electrical conversion coefficient and I is the normalized irradiance whose pdf follows (2.5). For the system under consideration, the instantaneous electrical signal to noise ratio (SNR) can be defined as γ = (ηi) 2 /N 0. (2.21) Therefore, the average electrical SNR is obtained as γ = η 2 /N 0 since E (I) = 1. Conditioned on the irradiance, the instantaneous BER for OOK is given by [32] P e,ins = 0.5 erfc ( ) ηi 2 N 0 (2.22) where erfc (.) stands for the complementary error function defined as erfc(x) = 2 e t2 dt. (2.23) π x The average BER can be then calculated by averaging (2.22) over the distribution of I, i.e., P e,siso = 0 f I (I) [ 0.5 erfc ( ηi 2 N 0 )] di. (2.24) The above integral can be evaluated in closed-form by expressing the erfc (.) integrand via a Meijer s G-function using [37, Eq. ( )], [37, Eq. ( )] and 22

44 [38, Eq. (21)]. Thus, a closed-form solution is obtained as γ 2 k m 1+m 2 p m2+1/2 q m 1 1/2 P e,siso =, (2.25) lγ (m 1 ) Γ (m 2 ) (2π) l+k(p+q) 1 2 ( G k(p+q),2l m q 1m p ) ( ) k 2 (4l) l 2l,k(p+q)+l p P Ω p 2q q Ω q 1 γ l k (l : 1), l : 1 2, k(p+q). J k (q : 1 m 1 ), J k (p : 1 m 2 ), (l : 0) In (2.25) k and l are positive integer numbers that satisfy pγ 2 /2 = l/k and J ξ (y, x) is defined as ( J ξ (y, x) = ξ, y x ) (, ξ, y 1 x ) (,..., ξ, 1 x ). (2.26) y y y The derived BER expression in (2.25) can be seen as a generalization of earlier BER results in the literature. If we insert γ i = 1 and Ω i = 1 in (2.25), we obtain the BER expression derived in [31, Eq. (9)] under the assumption of Gamma- Gamma channel. Setting m i = 1 in (2.25), we obtain Eq (15) of [23] derived for Double-Weibull channel. On the other hand, for γ i = 1, Ω i = 1 and m 2 = 1, (2.25) reduces to (12) of [32] reported for the K-channel. Appendix B provides the details on these. In an effort to have some further insights into system performance, we investigate the asymptotical BER performance in the following. For large SNR values, the asymptotic BER behavior is dominated by the behavior of the pdf near the origin, i.e. f I (I) at I 0 [39]. Thus, employing series expansion corresponding to the Meijer s G-function [40, Eq. ( )], the Double GG distribution 23

45 given in (2.5) can be approximated by a single polynomial term as f I (I) A p+q j=1 j k Γ (b j b k )I pγ 2 min{ m 1 q, m 2 p } 1 (2.27) where A is obtained as A = γ 2pp m 2 1/2 q m 1 1/2 (2π) 1 (p+q)/2 Γ (m 1 ) Γ (m 2 ) ( m q 1m p ) min { m 1 q, m 2 p } 2 (qω 1 ) q (pω 2 ) p. (2.28) In (2.27), b k and b j are defined as b k = min b j {1 (q : 1 m 1 ), 1 (p : 1 m 2 )} \ min { m1 q, m } 2, (2.29) p { m1 q, m } 2. (2.30) p Therefore, based on (2.24), the average BER can be well approximated by P e,siso A p+q j=1 j k ( ) pγ2 b 2 k Γ ((1 + pγ 2 b k ) /2) Γ (b j b k ) γ 2. (2.31) πpγ 2 b k From (2.31), it can be readily deduced that the diversity order of SISO FSO system is given by 0.5pγ 2 min {m 1 /q, m 2 /p}. It is observed from Fig. 2.7a that an SNR of 51.1 db is required to achieve a BER of 10 3 for a plane wave in moderate turbulence conditions. For stronger turbulence conditions, the required SNR to achieve the same BER performance is 68.2 db as seen from Fig. 2.7b. For spherical waves, SNRs of 49.8 db and 63.8 are respectively required for moderate and strong turbulence conditions. Comparison with the expressions presented for other channel models reveals that the Gamma- 24

46 (a) (b) (c) (d) Figure 2.7: Average BER as a function of γ a) Plane wave - σ 2 Rytov = 2, l 0 /R 0 = 0.5, b) Plane wave-σ 2 Rytov = 25, l 0 /R 0 = 1, c) Spherical wave - σ 2 Rytov = 2, l 0 /R 0 = 0, d) Spherical wave - σ 2 Rytov = 5, l 0 /R 0 = 1 Gamma model significantly overestimates the performance. Also, the superiority of Double GG is more obvious for spherical wave. As observed from Figs. 2.7c and 2.7d, the performance plots of Double-Weibull and Gamma Gamma considerably diverge particularly for strong turbulence conditions. Furthermore, it can be clearly seen that the asymptotic results are in excellent agreement with exact analytical results within a wide range of SNR showing the accuracy and usefulness of the derived asymptotic expression given in (2.31). 25

47 2.4.2 Outage Probability Analysis of SISO FSO System Denote R t as a targeted transmission rate and assume γ th = C 1 (R t ) as the corresponding SNR threshold in terms of the instantaneous channel capacity for a particular channel realization. Therefore, the outage probability is calculated by P out (R t ) = P r (γ < γ th ) [41]. If SNR exceeds γ th, no outage happens and the receiver can decode the signal with arbitrarily low error probability. Employing (2.21), I can be expressed as I = γ/ γ. After the transformation of the random variable (RV), I, the cdf of γ can be easily derived from (2.7) and setting γ = γ th therein, we obtain the outage probability as P out = F γ (γ th ) = pm 2 1/2 q m1 1/2 (2π) 1 (p+q)/2 Γ (m 1 ) Γ (m 2 ) ( ) pγ2 /2 G p+q,1 γth m q 1m p 1 2 1,p+q+1 γ (Ω 2 p) p (Ω 1 q) q. (q : m 1 ), (p : m 2 ), 0 (2.32) (2.32) can be seen as a generalization of earlier outage analysis results in the literature. Specifically, if we insert γ i = 1 and Ω i = 1 in (2.32), we obtain the outage probability expression reported in [30, Eq. (15)] for Gamma-Gamma channel. Setting m i = 1 in (2.32), we recover Eq (16) of [23] derived for Double-Weibull channel. Similarly, for γ i = 1, Ω i = 1 and m 2 = 1, (2.32) reduces to (3) of [42] reported for the K channel. Appendix C provides the details on these. Based on the derived expression in (2.32), Figs. 2.8a to 2.8d present the outage probabilities of a SISO FSO system as a function of the normalized outage threshold (i.e., γ/γ th ) for different degrees of turbulence severity. We adopt the same parameters used in Figs. 2.2 and 2.3 and Figs. 2.5 and 2.6 and consider the following four cases: a) Plane wave with σrytov 2 = 2 and l 0 /R 0 = 0.5, b) Plane wave 26

48 (a) (b) (c) (d) Figure 2.8: Outage probability as a function of γ/γ th for a) Plane wave - σ 2 Rytov = 2, l 0 /R 0 = 0.5, b) Plane wave-σ 2 Rytov = 25, l 0 /R 0 = 1, c) Spherical wave - σ 2 Rytov = 2, l 0 /R 0 = 0, d) Spherical wave - σ 2 Rytov = 5, l 0 /R 0 = 1 with σrytov 2 = 25 and l 0 /R 0 = 1, c) Spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0, d) Spherical wave with σrytov 2 = 5 and l 0 /R 0 = 1. It is observed from Fig. 2.8a that an SNR of 37.8 db is required to achieve a targeted outage probability of As the turbulence strength gets stronger (see Fig. 2.8b), the required SNR to maintain the same performance climbs up to 50.5 db. Similarly, for spherical waves, SNRs of 36.8 db and 50.9 db are respectively required for moderate and strong turbulence conditions. In these figures, we further include the outage 27

49 results for Double-Weibull and Gamma-Gamma for comparison purposes. As expected from the earlier comparisons of their pdfs, the outage performance over Double-Weibull and Double GG for plane wave (See Figs. 2.8a and 2.8b) are similar while the Gamma-Gamma model overestimates the outage performance. On the other hand, the superiority of Double GG is more obvious for spherical wave (See Figs. 2.8c and 2.8d), particularly for strong turbulence conditions, where the outage performance plots of Double-Weibull and Gamma Gamma significantly deviate. 2.5 Conclusions In this Chapter, we have introduced a new channel model, so called Double GG, which accurately describes irradiance fluctuations over atmospheric channels under a wide range of turbulence conditions. It is based on the theory of doubly stochastic scintillation and considers irradiance fluctuations as the product of small-scale and large-scale fluctuations which are both GG distributed. We have obtained closedform expressions for the pdf and cdf in terms of Meijer s G-function. Comparisons with the Gamma Gamma and Double-Weibull have shown that the new model provides an accurate fit with numerical simulation data for both plane and spherical waves. Using the new channel model, we have obtained closed-form expressions for the BER and the outage probability of SISO FSO systems. We have demonstrated that our derived expressions cover many existing results in the literature earlier reported for Gamma-Gamma, Double-Weibull and K channels as special cases. Based on the asymptotical performance analysis, we have further derived diversity gains for SISO FSO systems under consideration. 28

50 Chapter 3 Statistical Analysis of Sum of Double Generalized Gamma Variates 3.1 Introduction A unifying statistical distribution named Double Generalized Gamma (Double GG) was proposed in Chapter 2 which generalizes many existing turbulence models in a closed-form expression and covers all turbulence conditions. An analytical solution for the distribution of the sum of Double GG random variables (RVs) is a very cumbersome task if not impossible. Thus, in this Chapter, we derive an upper bound and a novel and accurate approximation for the distribution of the sum of N Double GG distributed RVs. Particularly, a useful expression for the distribution of the product of N Double GG distributed RVs is first obtained. Then, based on a well-known inequality between arithmetic and geometric means, a closed-form union upper bound for the distribution of the sum of Double GG distributed RVs is derived. As the upper bound exhibits large deviations compared to Monte-Carlo simulation results for sufficiently large values of N, we modify the upper bound 29

51 and propose a novel and accurate approximation of the distribution of the sum of Double GG distributed RVs. Extensive numerical and computer simulation results verify the tightness of the proposed bound and the accuracy of the approximate expression. 3.2 An Upper-Bound for the Distribution of the Sum of Double GG Variates Let {I n } N n=1 be N statistically independent but not necessarily identically distributed (i.n.i.d.) Double GG RVs whose probability density function (pdf) follows (2.5). We define a new RV R as N R I n. (3.1) n=1 The Double GG distribution considers irradiance fluctuations as the product of small-scale and large-scale fluctuations which are both governed by Generalized Gamma (GG) distributions, i.e. I n = U x,n U y,n, where U x,n and U y,n are statistically independent arising respectively from large-scale and small scale turbulent eddies. Thus, R can be expressed as the product of 2N GG RVs U l, i.e., R = 2N l=1 U l. (3.2) The pdf of R can be obtained using the statistical model proposed in [36] as f R (r) = αξ r Gβ,0 0,β r α ω (3.3) J α (γ 1:2N, m 1:2N ) 30

52 where ξ, ω and J α (γ 1:2N, m 1:2N ) are defined as ξ = ( ) 2N β 2N (α/γ l ) m l 1/2 2π, (3.4) l=1 Γ (m l ) ( ) α/γl 2N αωl ω =, (3.5) m l γ l l=1 J α (γ 1:2N, m 1:2N ) ( α / γ1 ; m 1 ), ( α / γ2 ; m 2 ),..., ( α / γ2n ; m 2N ). (3.6) In (3.3), α and β are two positive integers defined as α 2N β α l=1 2N k l, (3.7) l=1 1 γ l (3.8) under the constraint that l l = 1 l k i (3.9) γ l i=1 is a positive integer with k i being also a positive integer. The cumulative distribution function (cdf) of R can be derived from (3.3) as F R (r) = ξg β,1 1,β+1 r α ω 1. (3.10) J α (γ 1:2N, m 1:2N ), 0 Considering (3.2), the n th moment of R can be calculated utilizing the n th moment of U l as E (R n ) = ( 2N Ωl l=1 m l ) n/γl Γ (m l + n/γ l ). (3.11) Γ (m l ) The well-known inequality between arithmetic and geometric means, i.e. A N 31

53 G N, with A N = 1 N n I n (3.12) n=1 and G N = N n=1 I 1/N n (3.13) is used to obtain a lower-bound for RV Z defined as the sum of Double GG RVs, i.e., Z N n=1 I n as Z NR 1/N. (3.14) Considering Eqs. (3.10) and (3.14), the cdf of Z is upper bounded as F Z (z) ξg β,1 1,β+1 (z/n ) αn ω 1. (3.15) J α (γ 1:2N, m 1:2N ), An Approximate Distribution of the Sum of Double GG Variates As we will show later in Section 3.4, the proposed bound given in (3.15) does not provide satisfactory accuracy for sufficiently large values of N. Thus, we obtain an approximation for the distribution of the sum of N statistically i.n.i.d. Double GG RVs in the following. Let us define a new RV W such that Z = NW R 1/N. Thus, an estimate of Z can be obtained as Z = N W R 1/N (3.16) 32

54 where W = E [W ] is given by 1 W = 1 E [R 1/N ] = 2N l=1 ( ml Ω l Utilizing (3.10), the cdf of Z is derived as ) 1/Nγl Γ (m l ) Γ (m l + 1/Nγ l ). (3.17) F Z ( z) = ξg β,1 ( 1,β+1 ω 1 ) z αn 1 W N. (3.18) J α (γ 1:2N, m 1:2N ), 0 Note that for W = 1, (3.18) becomes the upper bound for the cdf of Z that is given by (3.15). By taking the first derivative of (3.18) with respect to z, the approximate pdf of Z can be obtained in closed-form as f Z ( z) = Nαξ z Gβ,0 0,β ( ω 1 ) z αn W N. (3.19) J α (γ 1:2N, m 1:2N ) We should emphasize that (3.18) (similarly (3.19)) is very generic since it includes the cdf expression for the sum of some commonly used RVs used to describe turbulence-induced fading as special cases. For γ l 0, m l, (3.18) coincides with the cdf of the sum of N log-normal RVs. For γ l = 1, Ω l = 1, it reduces to the cdf of the sum of N Gamma-Gamma RVs while for m l = 1, it becomes the cdf of the sum of N Double-Weibull RVs. For γ l = 1, Ω l = 1, m 2l = 1, it coincides with the sum of N K distributed RVs 2. 1 Note that without loss of generality, it is assumed in Chapter 2 that E [I n ] = 1 in calculating Eqs. (2.11), (2.10a) and (2.10b) 2 By following the same approach, the pdf for the same spacial cases can be obtained from (3.19). 33

55 3.4 Verification of the Mathematical Analysis In this section, the previous mathematical analysis for the cdf of the sum of i.n.i.d. Double GG RVs is verified and evaluated. We consider the following four scenarios of atmospheric turbulence conditions reported in Chapter 2 Channel a: Plane wave and moderate irradiance fluctuations with γ 1 = , γ 2 = , m 1 = 0.55, m 2 = 2.35, Ω 1 = , Ω 2 = , p = 28 and q = 11 Channel b: Plane wave and strong irradiance fluctuations with γ 1 = , γ 2 = , m 1 = 0.5, m 2 = 1.8, Ω 1 = , Ω 2 = , p = 17 and q = 7. Channel c: Spherical wave and moderate irradiance fluctuations with γ 1 = , γ 2 = , m 1 = 2.65, m 2 = 0.85, Ω 1 = and Ω 2 = , p = 7 and q = 11. Channel d: Spherical wave and strong irradiance fluctuations with γ 1 = , γ 2 = , m 1 = 3.2, m 2 = 2.8, Ω 1 = and Ω 2 = , p = 7 and q = 11. Figs. 3.1 to 3.4 present the analytical results which have been obtained through (3.15) and (3.18) assuming N = 2, 3, 4. In order to verify the tightness of the proposed bound and the accuracy of the approximation, Monte-Carlo simulation results for the exact cdf are also included in each figure. In Figs. 3.1 and 3.2, we assume plane wave propagation with moderate (channel a) and strong (channel b) irradiance fluctuations respectively. For channel a, W is calculated through (3.17) as 1.31, 1.46 and 1.54 for N = 2, 3, 4 respectively. 34

56 Figure 3.1: Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming plane wave and moderate irradiance fluctuations (channel a) Figure 3.2: Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming plane wave and strong irradiance fluctuations (channel b) 35

57 Figure 3.3: Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming spherical wave and moderate irradiance fluctuations (channel c) Similarly for channel b, W is respectively obtained as 1.5, 1.76 and 1.92 for N = 2, 3, 4. As clearly seen from Figs. 3.1 and 3.2, our approximate expressions provide an excellent match to the simulation results for all values of N and under all range of turbulence conditions. However, the upper bound provides good accuracy for small values of N, and exhibits large deviations compared to the simulation results as N increases. Comparing Figs. 3.1 and 3.2, it is further observed that as the turbulence condition becomes weaker, the upper bound shows better accuracy. Figs. 3.3 and 3.4 illustrate the numerical results assuming spherical wave propagation with moderate (channel c) and strong (channel d) irradiance fluctuations respectively. For channel c, we obtain W = 1.31, 1.45, 1.53 for N = 2, 3, 4 respectively. For channel d, W is respectively calculated as 1.74, 2.14 and 2.39 for 36

58 Figure 3.4: Bounded, approximate and exact cdf of the sum of Double GG distributed RVs assuming spherical wave and strong irradiance fluctuations (channel d) N = 2, 3, 4. It is apparent from Figs. 3.3 and 3.4 that the proposed approximation again follows closely the exact simulation results for all values of N and under all range of turbulence conditions. Similar to the earlier observations on the upper bound for the plane wave propagation, the less the value of N and the weaker the turbulence condition, the better accuracy is observed. 3.5 Conclusions In this Chapter, we have derived a closed-form union upper bound and accurate approximate expression for the distribution of the sum of N i.n.i.d. Double GG distributed RVs in terms of Meijers G-function. Comparing with the computer 37

59 simulation results, we have shown that the proposed approximate expression provides an excellent match to the simulation results for all values of N and under all range of turbulence conditions. 38

60 Chapter 4 Performance Evaluation of Free-Space Optical Links with Spatial Diversity In Chapter 2, we derived the bit error rate (BER) performance of single-input single-output (SISO) free-space optical (FSO) link over Double Generalized Gamma (Double GG) channels. As it can be noticed from Section 2.4, the performance of a SISO FSO link over moderate and strong atmospheric turbulence is quite poor. To address this issue, multiple transmit and/or receive apertures can be employed and the performance can be improved via diversity gains. In this Chapter, we extend our performance analysis to evaluate the performance of FSO links with spatial diversity. Spatial diversity techniques provide a promising approach to mitigate turbulence-induced fading. Some earlier results on FSO systems with spatial diversity over log-normal, K, negative exponential and Gamma-Gamma channels can be found in [26,32,33,43,44]. In this Chapter, we study the error rate performance of multiple-input multiple-output (MIMO), single-input multiple-output (SIMO) 39

61 and multiple-input single-output (MISO) systems employing intensity modulation and direct detection (IM/DD) with on-off keying (OOK) over independent but not necessarily identically distributed (i.n.i.d.) Double GG turbulence channels. 4.1 System Model We consider an FSO system employing IM/DD with OOK where the information signal is transmitted via M apertures and received by N apertures over the Double GG channel. The received signal at the n th receive aperture is then given by M y n = ηx I mn + υ n, n = 1,..., N (4.1) m=1 where x represents the information bits and can be either 0 or 1, υ n is the Additive White Gaussian noise (AWGN) term with zero mean and variance συ 2 n = N 0 /2, and η is the optical-to-electrical conversion coefficient. Here, I mn is the normalized irradiance from the m th transmitter to the n th receiver whose probability density function (pdf) and cumulative distribution function (cdf) follow (2.5) and (2.7) respectively. 4.2 Performance Analysis of of MIMO FSO Links The optimum decision metric for OOK is given by [33] P (y on,i mn ) on P (y off,i mn ) (4.2) off where y = (y 1, y 2,..., y N ) is the received signal vector. Following the same approach as [32, 33], the conditional bit error probabilities are given by (see [33] for details 40

62 of derivation) 1 P e (off I mn ) = P e (on I mn ) = Q γ MN 2 ( N M ) 2 I mn (4.3) n=1 m=1 where Q (.) is the Gaussian Q-function defined as Q (x) = ( 1/ 2π ) x exp ( t2 /2 )dt [45, 46]. Therefore, the average error rate can be expressed as P e,mimo = I 1 f I (I) Q γ MN 2 ( N M ) 2 I mn di (4.4) n=1 m=1 where f I (I) is the joint pdf of vector I = (I 11, I 12,..., I MN ). The factor M in (4.4) ensures that the total transmitted powers of diversity system and SISO link are equal for a fair comparison. On the other hand, the factor N is used to ensure that sum of the N receive aperture areas is the same as the area of the receive aperture of the SISO link. The integral expressed in (4.4) does not yield a closed-form solution even for simpler turbulence distributions; however, it can be calculated through multidimensional Gaussian quadrature rule (GQR)techniques [47]. In the following, we investigate the receive and the transmit diversity as special cases to have further insight into the performance of FSO links with spatial diversity. 41

63 4.3 Performance Analysis of FSO Links with Receive Diversity SIMO FSO Links with Optimal Combining In this Section, we assume that multiple receive apertures are employed and present the BER derivations under the assumption that optimal gain combining (OC) with perfect channel state information (CSI) is used where the variance of the noise in each receiver is given by σn 2 = N 0 /2N. Therefore, replacing M = 1 in (4.4) we obtain P e,simo (OC) = I f I (I) Q γ 2N N In 2 di. (4.5) n=1 (4.5) does not yield a closed-form solution and requires N-dimensional integration. Nevertheless, the Q-function can be well-approximated as [48] Q(x) e x2 2 /12 + e 2x2 3 /4. (4.6) Thus the average BER can be obtained as P e,simo (OC) N n=1 0 N n=1 0 ( ) γ f In (I n ) exp 4N I2 n di n ( γ f In (I n ) exp 3N I2 n ) di n. (4.7) The above integral can be evaluated by first expressing the exponential function in terms of the Meijer G-function presented in [38, eq. (11)] as exp ( x) = G 1,0 [ ] 0,1 x 0. (4.8) 42

64 Then, a closed-form expression for (4.7) is obtained using [38, Eq. (21)] as where Λ (n, c) is defined as P e,simo (OC) 1 12 N Λ (n, 4) + 1 N Λ (n, 3) (4.9) n=1 4 n=1 Λ (n, c) = α n ln 0.5 k m 1,n+m 2,n n (4.10) 2(2π) 0.5(ln 1+(kn 1)(pn+qn)) G k n(p n +q n ),l n (cn) l n ω n kn ln ln l n,k n(p n+q n) γ l k n k n(p n+q n) n (l n, 1). J kn (q n, 1 m 1,n ), J kn (p n, 1 m 2,n ) In (4.10), l n and k n are positive integer numbers that satisfy p n γ 2,n /2 = l n /k n, and α n and ω n n {1, 2,..., N} are defined as α n = γ 2,np m 2,n+1/2 n q m 1,n 1/2 n (2π) 1 (p n+q n )/2 Γ (m 1,n ) Γ (m 2,n ), (4.11) ω n = ( ) Ω 2,n p n m 1 pn ( 2,n qn Ω 1,n m1,n) 1 qn. (4.12) As detailed in Appendix D, the derived expression in (4.9) includes the previously reported result in [32] for K channel as a special case. Based on the approximation in (2.27), the corresponding closed-form asymptotic solution for (4.7) can be obtained as P e,simo (OC)_asy 1 12 where Λ asy (n, c) is defined as N Λ asy (n, 4) + 1 N Λ asy (n, 3) (4.13) n=1 4 n=1 p n +q n Λ asy (n, c) = α n j=1 j k Γ (b j,n b k,n )Γ (p n γ 2,n b k,n ) 43 ( cn ) pn γ 2,n b k,n 2( γ) pnγ 2,nb k,n. (4.14)

65 Figure 4.1: Comparison of the average BER between SISO and SIMO with optimal combing for plane wave with σ 2 Rytov = 25 and l 0 /R 0 = 1. Therefore, the diversity order of FSO links with N receive apertures employing optimal gain combining is obtained as 0.5 N n=1 p n γ 2,n min { m 1,n /q n, m 2,n /p n }. Figs. 4.1 and 4.2 illustrate the BER performance of the SIMO FSO system under consideration assuming following two cases a) Plane wave with σrytov 2 = 25 and l 0 /R 0 = 1, b) Spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0. We present approximate analytical results which have been obtained through (4.9) and (4.13) along with the Monte-Carlo simulation of (4.5). As clearly seen from Figs. 4.1 and 4.2, our approximate expressions provide an excellent match to simulation results. As a benchmark, the BER of SISO FSO link is also included in these figures. It is observed that receive diversity significantly improve the performance. For instance, at a target bit error rate of 10 3, we observe performance improvements 44

66 Figure 4.2: Comparison of the average BER between SISO and SIMO with optimal combing for spherical wave with σ 2 Rytov = 2 and l 0 /R 0 = 0. of 26.8 db and 39.6 db respectively for with N = 2 and 3 with respect to the SISO transmission for the plane wave scenario. Similarly, for the spherical wave scenario, at a BER of 10 3, performance improvements of 19 db and 25.1 db are achieved for SIMO links with N = 2 and 3 compared to the SISO deployment. It can be further observed that asymptotic bounds on the BER become tighter at high enough signal to noise ratios (SNRs) confirming the accuracy and usefulness of the asymptotic expression given in (4.13) SIMO FSO Links with Selection Combining As an alternative to OC, we also consider selection combining (SC) which is the least complicated of the combining schemes since it only processes one of the 45

67 diversity apertures. Specifically, the SC chooses the aperture with the maximum received irradiance (or electrical SNR). Therefore, the pdf of the output of SC receiver can be obtained as f Imax (I max ) = df I max (I max ) di max = N N n=1 k=1,k n The average BER can be then expressed as P e,simo (SC) = N N n=1 k=1,k n 0 f In (I max )F Ik (I max ). (4.15) γ f In (I max )F Ik (I max ) Q I max di max (4.16) 2N which can be efficiently calculated through numerical means [49] SIMO FSO Links with Equal Gain Combining Capitalizing on the mathematical analysis in Chapter 3, we now study the performance of an FSO system with receive diversity employing equal gain combining (EGC) where the receiver adds the receiver branches. The received signal is then given by N y = ηx I n + υ n, n = 1,..., N. (4.17) n=1 Outage Probability Analysis: The outage probability is defined as the probability that the instantaneous SNR of the received signal falls below a predefined threshold, that is P out = P r (γ EGC < γ th ) = F γegc (γ th ). (4.18) For the system under consideration, the instantaneous electrical SNR can be calu- 46

68 lated as [50] γ EGC = ( ) η N 2 I n i=1 N 2 N 0. (4.19) Since E (I n ) = 1, the corresponding average electrical SNR is obtained as γ EGC = γ = η 2 /N 0. (4.20) Using Eqs. (4.19) and (4.20), Z, defined in Chapter 3 as Z expressed as N n=1 I n, can be γegc Z = N. (4.21) γ EGC Thus, the approximate cdf of γ EGC can be easily derived from (3.18) and setting γ EGC = γ th therein, an accurate closed-form approximation for the outage probability is obtained as P out = F γegc (γ th ) = ξg β,1 ( 1,β+1 ω 1 γ th γ EGC W 2 ) αn/2 1. (4.22) J α (γ 1:2N, m 1:2N ), 0 We should emphasize that if we insert W = 1 in (4.22), the upper bound expression for the outage probability is obtained. Moreover, by setting N = 1 that results in W = 1, we recover the exact expression for the outage probability of SISO reported in (2.32). (4.22) can be seen as a generalization of outage analysis of SIMO FSO systems with EGC over other atmospheric turbulence models in the literature. Setting γ l = 1 and Ω l = 1 in (4.22), we obtain the outage probability expression under the assumption of Gamma-Gamma channel. If we insert m l = 1 in (4.22), we derive the outage expression for Double-Weibull channel. On the other hand, for γ l = 1, Ω l = 1 and m 2l = 1, the outage expression for K-channel is obtained. 47

69 (a) (b) Figure 4.3: Bounded, approximate and exact outage probability of EGC and SISO for a) Plane wave with σ 2 Rytov = 2 and l 0 /R 0 = 0.5, b) Plane wave with σ 2 Rytove = 25 and I 0 /R 0 = 1. 48

70 (a) (b) Figure 4.4: Bounded, approximate and exact outage probability of EGC and SISO for a) Spherical wave with σrytov 2 = 2 and l 0 /R 0 = 0, b) Spherical wave with σrytov 2 = 5 and l 0 /R 0 = 1. 49

71 Figs. 4.3 and 4.4 present the outage probabilities as a function of the normalized outage threshold (i.e., γ EGC /γ th ) for different degrees of turbulence severity and values of N based on the derived expression in (4.22). In these figures, the same parameters used in Figs. 3.1 to 3.4 are adopted, and the outage results obtained by Monte Carlo simulations are further included in order to verify the tightness of the upper bound and the accuracy of (4.22). As expected from the earlier results discussed in Section 3.4, the upper bound provides satisfactory accuracy only for small values of N and weak turbulence conditions. However, the numerical results of (4.22) are nearly equivalent to the simulated ones representing the exact outage performance for all values of N and under all range of turbulence conditions. For instance, at a target outage probability of 10 4, the gaps between the exact and the approximate curves are 0.25 db, 0.08 db and 0.45 db respectively for N = 2, 3 and 4 in Fig. 4.4b. It is also evident that for N = 1, the upper bound, approximate and exact curves are identical. In addition, the obtained results clearly show that outage performance significantly improves with increasing of N. BER Performance Analysis: Following the same approach as [32, 33], the conditional bit error probabilities are given by P e,simo (EGC) (off I n ) = P e,simo (EGC) (on I n ) = 1 2 erfc ( γegc 2N ) N I n. (4.23) n=1 Therefore, the average error rate can be expressed as P e,simo (EGC) = 1 2 I f I (I) erfc ( γegc 2N ) N I n di. (4.24) n=1 The integral expressed in (4.24) does not yield a closed-form solution even for simpler turbulence distributions. However, an accurate closed-form approximation 50

72 of (4.24) can be obtained as P e,simo (EGC) 1 = f 2 Z ( z) erfc 0 ) ( γegc z 2N d z. (4.25) The above integral can be evaluated in closed-form by first expressing the erfc (.) in terms of the Meijer G-function presented in [38, eq. (11)] as erfc ( ) 1 x = π G 2,0 1,2 x 1. (4.26) 0, 1/2 Then, a closed-form expression for (4.25) is obtained using [38, Eq. (21)] as P e,simo (EGC) = Nαξq λ 2 2s ( 2π ) s+(q 1)β, 2s,qβ+s G qβ,2s H e γ s EGC (s, 1), (s, 1/2 ) (4.27) K q (α/γ 1:2N, m 1:2N ), (s, 0) where s and q are positive integer numbers that satisfy s/q = αn/2, and H, λ and K q (α/γ 1:2N, m 1:2N ) are defined as H e (4sN 2 ) s (ωq β ) q ( W N ) αnq, (4.28) K q (α/γ 1:2N, m 1:2N ) (4.29) J α (γ 1:2N, m 1:2N ) q = 1, { } Jα (γ 1:2N,m 1:2N ), J α(γ 1:2N,m 1:2N ) q = 2 51

73 (a) (b) Figure 4.5: Bounded, approximate and exact BER of EGC and SISO for a) Plane wave with σ 2 Rytov = 2 and l 0 /R 0 = 0.5, b) Plane wave with σ 2 Rytove = 25 and I 0 /R 0 = 1. 52

74 λ 2N l=1 m l N + 1. (4.30) Note that for N = 1 leading to W = 1, (4.27) reduces to (2.25) reported as the BER expression of the SISO link. Furthermore, setting W = 1 in (4.27), the upper bound expression for the BER is obtained. The derived approximate BER expression in (4.27) for SIMO FSO systems with EGC can be seen as a generalization of BER results over other atmospheric turbulence models as well. Specially, if we insert γ l = 1 and Ω l = 1 in (4.27), we obtain an approximate BER expression over Gamma-Gamma channel. Setting m i = 1 in (4.27), we obtain an approximate BER for Double Weibull channel. Similarly, for γ l = 1, Ω l = 1 and m 2l = 1, an approximate BER expression for K-channel is obtained. Figs. 4.5 and 4.6 demonstrate the average BER over independent and identically distributed (i.i.d.) Double GG channels assuming both plane and spherical wave propagation, respectively. In order to verify the tightness of the bound and accuracy of the approximate expression, we present analytical results obtained through (4.27) along with the Monte-Carlo simulation of (4.24). As a benchmark, the average BER of SISO FSO link obtained through (2.25) is also included in these figures 1. As clearly seen from Figs. 4.5 and 4.6, the approximate expression provides an excellent match to the simulation data which represent the exact BER. For instance, at a target bit error rate of 10 4, the gaps between the exact and the approximate curves are 0.16 db, 0.3 db and 0.6 db respectively for N = 2, 3 and 4 in Fig. 4.5b. This observation clearly demonstrates the accuracy of the proposed approximation. It is also illustrated that the upper bound becomes tighter as the number of receive apertures decreases. This is expected as the upper bound and the exact BER curves coincide for N = 1. In addition, we observe that 1 The same results can be obtained by inserting N = 1 and therefore W = 1 in (4.27). 53

75 (a) (b) Figure 4.6: Bounded, approximate and exact BER of EGC and SISO for a) Spherical wave with σ 2 Rytov = 2 and l 0 /R 0 = 0, b) Spherical wave with σ 2 Rytov = 5 and l 0 /R 0 = 1. 54

76 multiple receive apertures deployment employing EGC significantly improves the performance. Specially, in Fig. 4.5b, at a target bit error rate of 10 4, we observe performance improvements of 23.4 db, 33.4 db and 41.4 db for SIMO FSO links with N = 2, 3 and 4 receive apertures employing EGC respectively with respect to the SISO transmission. Although Meijer s G-function can be expressed in terms of more tractable generalized hypergeometric functions, (4.27) appears to be complex and the impact of the basic system and channel parameters on performance is not very clear. However, as discussed in Section 2.4.1, for large SNR values, the asymptotic behavior of the system performance is dominated by the behavior of the pdf near the origin, i.e. f Z ( z) at z 0 [39]. Thus, employing a series expansion corresponding to the Meijer s G-function [40, Eq. ( )], f Z ( z) given in (3.19) can be approximated by a single polynomial term as f Z ( z) Nαξ (ω ( ) ) αn min{ m 1 γ 1 α W,, m 2N γ 2N N α } β j=1 j k Γ (c j c k ) z N min{m 1γ 1,,m 2N γ 2N } 1 (4.31) where c k and c j are defined as { m1 γ 1 c k = min α,, m } 2Nγ 2N, (4.32) α { c j { ( α / γ2 ; m 2 ),..., ( α m1 γ 1 / γ2n ; m 2N )} \ min α,, m } 2Nγ 2N. (4.33) α It must be noted that (4.31) is only valid for i.n.i.d. Double GG turbulence channels. Based on Eqs. (4.25) and (4.31) and at high SNRs, the average BER can be 55

77 Figure 4.7: Exact, approximate and asymptotic BER of EGC over two i.n.i.d. atmospheric turbulence channels defined as plane wave with σ 2 Rytov = 2 and l 0 /R 0 = 0.5, and plane wave with σ 2 Rytove = 25 and I 0 /R 0 = 1. well approximated as P e,simo (EGC) Γ ((1 + Nαc k) /2) ξ 2 πc k ( ω ( W N ) αn ) ck β j=1 j k ( ) Nαck 2N Γ (c j c k ). (4.34) γegc Therefore, the diversity order of FSO links with N receive apertures employing EGC is obtained as 0.5N min {m 1 γ 1, m 2N γ 2N }. Figs. 4.7 and 4.8 illustrate the BER performance of SIMO FSO links employing EGC receivers over i.n.i.d. Double GG channels. Similar to i.i.d. results, our approximate closed-form expression yields a very close match to the simulation results. For example, in Fig. 4.7 at a BER of 10 4 in SIMO links with N = 2, the 56

78 Figure 4.8: Exact, approximate and asymptotic BER of EGC over two i.n.i.d. atmospheric turbulence channels defined as spherical wave with σ 2 Rytov = 2 and l 0 /R 0 = 0, and spherical wave with σ 2 Rytov = 5 and l 0 /R 0 = 1. difference between the exact and the approximate curve is 2.4 db. In Fig. 4.8 and at a BER of 10 4, this gap is 1.1 db. It can be further observed that asymptotic bounds on the BER become tighter at high enough SNR values confirming the accuracy and usefulness of the asymptotic expression given in (4.34). 57

79 4.4 Performance Analysis of FSO Links with Transmit Diversity MISO FSO Links In this Section, we assume that multiple transmit apertures are employed. Therefore, replacing N = 1 in (4.4) we obtain P e,miso = I f I (I) Q ( γ M 2 ) M I m di. (4.35) m=1 It should be noted that (4.35) is equivalent to (4.24) obtained for the SIMO FSO links with EGC. Thus, the mathematical analysis developed in Section can be use to evaluate the performance of MISO FSO links. 4.5 Performance Comparison of Diversity Techniques In this Section, we present and compare the BER performance results of FSO systems employing different diversity techniques over Double GG channels assuming both plane and spherical wave propagation. The performance improvements over SISO systems are further quantified. Figs. 4.9 and 4.10 present the average BER over i.i.d. turbulent channels assuming plane wave propagation with strong irradiance fluctuations and spherical wave propagation with moderate irradiance fluctuations, respectively. As a benchmark, the average BER of SISO FSO link obtained through (2.25) is also included 58

80 Figure 4.9: Comparison of the average BER between SISO and different diversity techniques for plane wave assuming i.i.d. turbulent channel with σ 2 Rytove = 25 and I 0 /R 0 = 1. in these figures. As clearly seen from Figs. 4.9 and 4.10, the diversity techniques significantly improve the performance. it is also illustrated that EGC receivers yield nearly the same performance as OC receivers. For example, in Fig. 4.10, for N = 2 the performance difference between OC and EGC receivers is merely 0.4 db at a BER of Also as expected, SIMO FSO links employing OC and EGC outperform SC counterpart. Figs and 4.12 demonstrate the BER performance of SIMO FSO links employing OC, EGC and SC receivers over i.n.i.d. Double GG channels. Similar to i.i.d. results, EGC receiver demonstrates nearly the same performance as OC receiver. We further compare the performance of i.n.i.d. case with respect to i.i.d. case presented in Figs. 4.9 and For example, to achieve a BER of 10 5 in 59

81 Figure 4.10: Comparison of the average BER between SISO and different diversity techniques for spherical wave assuming i.i.d. turbulent channel with σ 2 Rytov = 2 and l 0 /R 0 = 0. SIMO links with N = 2 over i.n.i.d. channels assuming plane wave propagation, we need 8.2 db, 8.5 db and 4.9 db less in comparison to i.i.d. case respectively for OC, EGC and SC receivers. Note that in Fig. 4.9, we assume that both of the two channels between the transmitter and receivers are described by a plane wave with strong irradiance fluctuations. Thus, since in Fig. 4.11, one of the channels is less severe, we need less SNR in comparison to i.i.d. case to obtain the same BER. On the other hand, to achieve a BER of 10 5 for SIMO links with N = 2 over i.n.i.d. assuming spherical wave propagation, we need 6.8 db more for OC and EGC receivers and 11.6 db more for SC receiver in comparison to i.i.d. channels. Note that in Fig. 4.10, both of the two channels between the transmitter and receivers are described by a spherical wave with moderate irradiance fluctuations. 60

82 Figure 4.11: Comparison of the OC, EGC and SC receivers for SIMO FSO links over two i.n.i.d. atmospheric turbulence channels defined as plane wave with σrytov 2 = 2 and l 0 /R 0 = 0.5, and plane wave with σrytove 2 = 25 and I 0 /R 0 = 1. Therefore, as one of the channels is more severe than those channels in Fig. 4.10, we need more SNR in comparison to i.i.d. case to achieve the same performance. 4.6 Conclusions In this Chapter, we have investigated the BER performance of FSO links with spatial diversity over atmospheric turbulence channels described by the Double GG distribution. We have obtained efficient and unified closed-form expressions for the BER of SIMO FSO systems with OC and EGC receivers along with outage probability of the EGC receiver which generalize existing results as special cases. We have further obtained the diversity gains for SIMO FSO systems employing OC 61

83 Figure 4.12: Comparison of the OC, EGC and SC receivers for SIMO FSO links over two i.n.i.d. atmospheric turbulence channels defined as spherical wave with σ 2 Rytov = 2 and l 0 /R 0 = 0, and spherical wave with σ 2 Rytov = 5 and l 0 /R 0 = 1. and EGC receivers based on the asymptotical performance analysis. For MIMO and SIMO FSO systems with SC receiver, we have presented BER performance based on numerical calculations of the integral expressions. Our numerical results have demonstrated that spatial diversity schemes can significantly improve the system performance and bring impressive performance gains over SISO systems. Our comparisons among SIMO FSO links employing OC, EGC and SC receivers have further demonstrated that EGC scheme presents a favorable trade-off between complexity and performance. 62

84 Chapter 5 Performance Evaluation of Singleand Multi-carrier Modulation Schemes for Indoor Visible Light Communication Systems In this Chapter, we investigate and compare the performance of single- and multicarrier modulation schemes for indoor visible light communication (VLC). Particularly, the performances of single carrier frequency domain equalization (SCFDE), orthogonal frequency division multiplexing (OFDM) and on-off keying (OOK) with minimum mean square error equalization (MMSE) are analyzed in order to mitigate the effect of multipath distortion of the indoor optical channel where nonlinearity distortion of light-emitting diode (LED) transfer function is taken into account. Our results indicate that SCFDE system, in contrast to OFDM system, does not suffer from high peak-to-average power ratio (PAPR) and can outperform OFDM and OOK systems. We further investigate the impact of LED bias point on 63

85 the performance of OFDM systems and show that biasing LED with the optimum value can significantly enhance the performance of the system. Bit-interleaved coded modulation (BICM) is also considered for OFDM and SCFDE systems to further compensate signal degradation due to intersymbol interference (ISI) and LED nonlinearity. 5.1 Introduction OFDM has been proposed in the literature to combat ISI caused by multipath reflections [51 56]. OFDM is capable of employing very low-complexity equalization with single-tap equalizers in the frequency domain, and allows adaptive modulation and power allocation. There have been several OFDM techniques for VLC systems using intensity modulation and direct detection (IM/DD) including DCclipped OFDM [55], asymmetrically clipped optical OFDM (ACO-OFDM) [56], PAM-modulated discrete multitone (PAM-DMT) [57], Flip-OFDM [58] and unipolar OFDM (U-OFDM) [59]. In DC-clipped OFDM, a DC bias is added to the signal to make it unipolar and suitable for optical transmission. Hard-clipping on the negative signal peaks is used in order to reduce the DC bias required. The other techniques have been proposed to remove the biasing requirement and therefore improve the energy efficiency of DC-clipped OFDM. Particularly, ACO-OFDM and PAM-DMT clip the entire negative excursion of the waveform. In ACO-OFDM, to avoid the impairment from clipping noise, only odd subcarriers are modulated by information symbols. In PAM-DMT, only the imaginary parts of the subcarriers are modulated such that clipping noise falls only on the real part of each subcarrier and becomes orthogonal to the desired signal. On the other hand, U-OFDM and Flip-OFDM extract the negative and positive samples from the real bipolar OFDM 64

86 symbol and separately transmit these two components over two successive OFDM frames where the polarity of the negative samples is inverted before transmission. As discussed in [60], all of these four non-biasing OFDM approaches exhibit the same performance in an Additive White Gaussian noise (AWGN) channel. Among these schemes, ACO-OFDM has been shown to be more efficient in terms of optical power than the systems that use DC-biasing as it utilizes a large dynamic range of the LED. Therefore, it is considered in this work. There exist several investigations analyzing different OFDM techniques and comparing them with SCFDE [61 63] or single carrier modulation [64, 65]. To the best of our knowledge, these previous studies were built on the assumption of ideal AWGN channels or did not consider the nonlinear characteristics of LED. In this Chapter, we analyze and compare performance of the aforementioned techniques along with OOK with MMSE which is commonly used in IM/DD communication systems considering an off-the-shelf LED model and a multipath channel. Moreover, BICM is considered for OFDM and SCFDE systems to further combat signal degradation due to LED nonlinearity and ISI. 5.2 System Model of ACO-OFDM ACO-OFDM is a form of OFDM that modulates the intensity of an LED. Because ACO-OFDM modulation employs IM/DD, the time-domain transmitted signal must be real and positive. The block diagram of an ACO-OFDM system is depicted in Fig The information stream is first parsed into a block of L/4 complex data symbols denoted by X = [ ] T X 0, X 1,...X L/4 1 where the symbols are drawn from constellations such as M-QAM or M-PSK where M is the constellation size. To ensure a real output signal used to modulate the LED intensity, 65

87 Figure 5.1: ACO-OFDM transmitter and receiver configuration. ACO-OFDM subcarriers must have Hermitian symmetry. In ACO-OFDM, only odd subcarriers are modulated, and this results in avoiding the impairment from clipping noise. Therefore, the complex symbols are mapped onto a L 1 vector as S = [0, X 0, 0, X 1,..., 0, X L/4 1, 0, X L/4 1, 0,..., X 1, 0, X 0, 0] T where (.) denotes the complex conjugate and (.) T indicates the transpose of a vector. An L-point inverse fast Fourier transform (IFFT) is then applied on the vector S to build the time domain signal s. A cyclic prefix (CP) is added to s turning the linear convolution with the channel into a circular one to mitigate multipath dispersion. To make the transmitted signal unipolar, all the negative values are clipped to zero. It is proven in [56] that since only the odd subcarriers are used to carry the data symbols, the clipping does not affect the data-carrying subcarriers, but only reduces their amplitude by a factor of two. The unipolar signal is then converted to analog and filtered to modulate the intensity of an LED. At the receiver, the signal is converted back to digital. CP is then removed and the electrical OFDM signal is demodulated by taking a L fast Fourier transform (FFT) and equalized with a single-tap equalizer on each 66

88 Figure 5.2: ACO-SCFDE transmitter and receiver configuration. subcarrier to compensate for channel distortion. The even subcarriers are then discarded and the transmitted data is recovered by a hard or soft decision. The extraction of odd subcarriers along with the equalization are represented by the Demapping block in Fig System Model of ACO-SCFDE SCFDE is a special technique which is compatible with any of OFDM techniques. In this work, we apply ACO-OFDM to SCFDE to achieve ACO-SCFDE with low PAPR. The block diagram of an ACO-SCFDE system is depicted in Fig ACO-SCFDE and ACO-OFDM are the same except that in ACO-SCFDE, an extra L/4-point FFT and IFFT are used at the transmitter and the receiver respectively resulting in a single carrier transmission instead of multicarrier. As it will be shown latter, the additional complexity of the extra FFT and IFFT blocks is offset by the fact that SCFDE has lower PAPR and better bit error rate (BER) performance than its OFDM counterpart when the signal is sent through the non-linear LED. 67

89 Figure 5.3: CCDF of PAPR comparison of ACO-OFDM and ACO-SCFDE for L = Peak-to-Average Power Ratio In this Section, the PAPR of ACO-OFDM signals is analyzed and compared with that of ACO-SCFDE. The PAPR is defined as the maximum power of transmitted signal divided by the average power, that is P AP R = max s2 (n) E [s 2 (n)] (5.1) where E [.] denotes expectation. Due to the large number of subcarriers and occasional constructive combining of them, OFDM systems have a large dynamic signal range and exhibit a very high PAPR. Thus, the OFDM signal will be clipped when passed through a nonlinear LED at the transmitter end which results in degrading 68

90 Figure 5.4: CCDF of PAPR comparison of ACO-OFDM and ACO-SCFDE for L = 256. the BER performance. SCFDE can be used as a promising alternative technique for OFDM to reduce the PAPR and combat the effect of nonlinear characteristics of the LED. PAPR is usually presented in terms of a complementary cumulative distribution function (ccdf) which is the probability that PAPR is higher than a certain PAPR value P AP R 0, i.e. Pr (P AP R > P AP R 0 ). Figs. 5.3 and 5.4 demonstrate the ccdf of PAPR for L = 64 and 256 subcarriers respectively, calculated by Monte Carlo simulation for different modulation constellations. We notice that ACO-SCFDE has a lower PAPR as compared to ACO-OFDM system for the same number of subcarriers. We also observe that the PAPR increases with increasing L for all of the constellations. 69

91 Table 5.1: Room configuration under consideration. Room Length 6 m Width 5 m Height 3 m Reflectivity ρ North 0.8 ρ South 0.8 ρ East 0.8 ρ West 0.8 ρ Ceiling 0.8 ρ Floor 0.3 Source Mode 1 Azimuth 0 Elevation 90 x, y, z 0.1 m, 0.2 m, 3 m Receiver Area 1 CM 2 FOV 85 x, y, z 2.5 m, 2.5 m, 1 m 5.5 Performance Analysis Simulations are conducted assuming indoor optical multipath channel where the transmitter and receiver are placed in a room whose configuration is summarized in Table 5.1. The methodology developed by Barry et al [66] is employed to simulate the impulse response of the channel where 10 reflections are taken into account. Fig. 5.5 presents the impulse respond of a diffuse channel. We assume an OFDM signal whose average electrical power before modulating the LED is varied from -10 dbm to 30 dbm, and the power of AWGN is -10 dbm. Thus, the simulated electrical signal to noise ratio (SNR) ranges from 0 db to 40 db matching the reported SNR values for indoor optical wireless communication (OWC) systems [67,68]. A number of subcarriers of L = 64 with M-QAM modulation are also assumed. Furthermore, OPTEK, OVSPxBCR4 1-Watt white LED is considered in simulations whose optical and electrical characteristics are given 70

92 Figure 5.5: Impulse response of the indoor diffuse channel. Table 5.2: Optical and electrical characteristics of OPTEK, OVSPxBCR4 1-Watt white LED. Symbol Parameter MIN TYP MAX Units V F Forward Voltage V Φ Luminous Flux lm Θ 1/2 50% Power Angle 120 deg in Table 5.2. A polynomial order of five is used to realistically model measured transfer function. Fig. 5.6 demonstrates the non-linear Transfer characteristics of the LED from the data sheet and using the polynomial function. We first compare the BER performance of ACO-SCFDE and ACO-OFDM. Fig. 5.7 presents the BER performance of ACO-OFDM and ACO-SCFDE for different modulation orders and LED bias point of 3.2V. As the results indicate, SCFDE exhibits better BER performance in the optical multipath channel. Furthermore, we investigate the impact of LED bias point on the performance 71

93 (a) (b) Figure 5.6: Transfer characteristics of OPTEK, OVSPxBCR4 1-Watt white LED. (a) Fifth-order polynomial fit to the data. (b) The curve from the data sheet. Figure 5.7: BER comparison of ACO-OFDM and ACO-SCFDE for bias point of 3.2V. 72

94 Figure 5.8: BER of ACO-OFDM for M = 16 for different bias points. of ACO-OFDM systems. According to the data sheet of the LED used in the simulations, three different bias points (3V, 3.2V and 3.5V) are considered. Fig. 5.8 demonstrates BER performance of an ACO-OFDM system with M = 16 and different LED bias points. As it can be clearly seen, the nonlinearity of LED has a significant impact on the performance of optical OFDM systems. It is also observed that there is an optimum LED bias point which is 3.2V for the case under consideration from which deviation can significantly deteriorate the system performance. BICM [69] is also considered for OFDM and SCFDEs systems to further compensate signal degradation due to ISI and LED nonlinearity. To demonstrate the usefulness of BICM, we assume that the information sequence is first encoded by a 73

95 Figure 5.9: BER comparison of uncoded and coded ACO-OFDM and ACO-SCFDE for M = 16. rate 1/2 convolutional encoder with generator matrix g = (5, 7), constraint length of 3 and minimum Hamming distance of 5. The coded information is then interleaved by a bitwise interleaver. At the receiver, the Viterbi soft-decoder [70] and the de-interleaver are used. Fig. 5.9 shows the BER of uncoded and coded ACO-OFDM and ACO-SCFDE for the indoor VLC under consideration. As it can be clearly observed, BICM can significantly enhance the system performance. However, the achieved gains come at the cost of significant reduction in the data rate due to the insertion of coded bits. Finally, we compare the performance of ACO-OFDM, ACO-SCFDE and OOK modulation with MMSE over an indoor VLC medium. The performance compari- 74

96 son is done in terms of normalized SNR and normalized bandwidth/bit-rate relative to OOK [55]. According to [55] and [65], we define the modulation bandwidth as the position of the first spectral null. To make a fair comparison between different modulation schemes, the normalized bandwidth of the signal is calculated as the modulation bandwidth which is normalized relative to OOK of the same transmitted data rate. For ACO-OFDM and ACO-SCFDE, first null occurs at a normalized frequency of 1 + 2/L. Thus, the normalized bandwidth/bit-rate is obtained as 2 (1 + 2/L) / log M 2 for ACO-OFDM and ACO-SCFDE. Fig shows normalized SNR required for a BER of 10 9 as a function of normalized bandwidth/bit-rate for OOK, ACO-OFDM and ACO-SCFDE. We observe while ACO-OFDM and ACO-SCFDE with 4-QAM modulation of each subcarrier require approximately the same bandwidth as OOK, they are more efficient in terms of power. Particularly, ACO-OFDM and ACO-SCFDE are 2.7 db and 3.7 db more efficient than OOK, respectively. For the higher orders of M, OOK outperforms ACO-OFDM and ACO-SCFDE but it requires greater bandwidth. 5.6 Conclusions We have evaluated and compared the performance of IM/DD single- and multicarrier modulation schemes for indoor VLC systems taking into account both nonlinear characteristics of LED and dispersive nature of optical wireless channel. We have shown through the use of simulation that SCFDE system has a lower PAPR than its counterpart OFDM system and outperforms OOK and OFDM systems and therefore is a promising modulation technique for indoor VLC systems. We have also investigated the performance of OFDM systems for different LED bias points and shown that significant gain can be achieved by biasing LED with the 75

97 Figure 5.10: Normalized SNR versus normalized bandwidth/bit-rate required to achieve BER of optimum value. BICM technique has been further considered to combat signal degradation due to LED nonlinearity and dispersive nature of the channel. 76

98 Chapter 6 Robust Timing Synchronization for AC OFDM Based Optical Wireless Communications In this Chapter, a novel timing synchronization technique suitable for asymmetrically clipped (AC) orthogonal frequency division multiplexing (OFDM) based optical intensity modulation and direct detection (IM/DD) wireless systems is presented. We demonstrate that the proposed technique can be directly applied to asymmetrically clipped optical OFDM (ACO-OFDM), PAM-modulated discrete multitone (PAM-DMT) and discrete Hartley transform (DHT) based optical OFDM systems. In contrast to existing OFDM timing synchronization methods which are either not suitable for AC OFDM techniques due to unipolar nature of output signal or perform poorly, our proposed method is suitable for AC OFDM schemes and outperforms all other available techniques. Both numerical and experimental results confirm the accuracy of the proposed method. 77

99 6.1 Introduction It is well known that OFDM systems are sensitive to carrier frequency offset and timing synchronization errors. For optical based OFDM systems employing IM/DD timing, synchronization errors result in performance deterioration. Therefore, an efficient timing synchronization scheme with an excellent accuracy that can be used for all AC systems needs to be proposed. A vast number of papers have been published in the literature on timing synchronization schemes for radio frequency (RF) based OFDM systems [71 75]. These techniques cannot be directly adopted for optical OFDM systems employing IM/DD since the output signal is unipolar in these systems. Thus, a new timing synchronization scheme that can be utilized for IM/DD systems needs to be proposed. Tian et al. recently proposed a technique tailored specifically to ACO-OFDM in [76]. Their proposed scheme may not work for other AC systems and the detection accuracy of this scheme also depends on the choice of training symbol used. Some training symbols may not yield perfect accuracy even at high signal to noise ratio (SNR) and without noise and multipath. In [77], a method using symmetry property of ACO-OFDM output signal in time domain with some additional redundancy was presented. However, the channel cannot be estimated using this technique. In this chapter, we address this daunting problem and present a novel and robust timing synchronization method working accurately for all AC systems, i.e. ACO-OFDM, PAM-DMT and DHT based optical OFDM, and can also be used for channel estimation at the same time. 78

100 6.2 New Timing Synchronization for AC Based OFDM Systems Timing Synchronization for ACO-OFDM As detailed in [78], our proposed method utilizes a very important characteristic of ACO-OFDM output waveform which has a format [C clip D clip ] where C represents the first L/2 samples, D = C represents negative part of first L/2 samples of unclipped output time domain ACO-OFDM and L is the number of subcarriers. C clip and D clip represent the first and the second half of the clipped output symbol. This clearly demonstrates that the negative parts of the first L/2 samples of unclipped time domain symbol are present in the second half of L/2 samples of clipped symbol. Therefore, a bipolar signal of length L/2 with these two halves can be easily reconstructed that is identical to the original unclipped bipolar signal of length L/2. The bipolar signal is constructed as r BP (l) = C clip (l) D clip (l) (6.1) where 0 l L/2 1 and subscript BP represents bipolar. This reconstructed bipolar signal is then employed to carry out correlation with a local copy of training symbol p (l) known at the receiver to accurately estimate the starting location of OFDM symbols. We use timing metric obtained by M (d) = 1 K K 1 l=0 r BP (l + d) p (l + d), K = 1, 2,..., L/2 (6.2) 79

101 where K is the cross-correlation length that can be set based on the desired performance. A higher value of K offers a better performance. The maximum of this timing metric shows the starting location of OFDM training symbol. Without loss of generality, we assume throughout this Chapter that average electrical power of ACO-OFDM output training symbols before clipping is unity, i.e., E {p 2 (l)} = Timing Synchronization for PAM-DMT The output waveform of PAM-DMT has a format [0 C clip 0 D mirror clip ] where C represents the first L/2 1 samples excluding the first sample and D = C represents negative of first L/2 1 samples of unclipped PAM-DMT output symbol. C clip and D clip represent clipped version of C and D, respectively. In this case, the second half of output symbol includes a mirror image of negative samples of the first half. Therefore, we can reconstruct the received bipolar signal as r BP (l) = C clip (l) ( D mirror clip (l) ) mirror (6.3) where 0 l L/2 1. This bipolar signal is correlated with a local copy of training symbol to detect the starting of OFDM training symbol. The maximum of the same timing metric used by ACO-OFDM given in (6.2) along with the bipolar signal reconstructed using (6.3) is employed. 80

102 6.3 Timing Synchronization DHT Based Optical OFDM Output waveform of DHT based optical OFDM has the same format as that of ACO-OFDM, i.e. [C clip D clip ]. Therefore, the same method used for ACO-OFDM is adopted to reconstruct the bipolar signal. We also use the same timing metric given in (6.2) to find the starting location of DHT based optical OFDM symbol. 6.4 Simulation Results A total of 10,000 random training symbols are used with inverse fast Fourier transform (IFFT) size of L = 256 and cyclic prefix (CP) length of L/8. To generate more realistic results, each training symbol was followed and preceded by another random ACO-OFDM or PAM-DMT symbol. Fig. 6.1a presents the average of the timing metric for Schmidl s and Park s methods proposed for RF based OFDM [73, 74] with modified training symbols suitable for ACO-OFDM. It can be clearly seen that Schmidl s timing metric demonstrates a flat region during the length of CP of the training symbol. However, Park s method does not suffer from this flat region but has four distinct peaks one of which is at the correct timing instant. Fig. 6.1b demonstrates the average of the timing metric for Tian s method proposed for ACO-OFDM systems [76]. It can be observed that in addition to the main peak at the correct timing instance of d = L/2, there is another peak at d = 0. As there is not a large difference between these two peaks, this can reduce correct detection probability especially at low SNRs. Figs. 6.1c and 6.1d show the average of the timing metric for bipolar correlation method with K = L/2 81

103 (a) (b) (c) (d) Figure 6.1: a) Average of Schmidl s and Park s timing metrics with modified training symbol suitable for ACO-OFDM. b) Average of Tian s timing metrics for ACO-OFDM. c) Average of timing metrics for bipolar correlation method for ACO- OFDM. d) Average of timing metrics for bipolar correlation method for PAM-DMT systems. and K = L/2 1 for ACO-OFDM and PAM-DMT schemes respectively. From these figures, we can clearly see that for both ACO-OFDM and PAM-DMT a peak occurs at the correct location at d = 0. For ACO-OFDM system, there are two other negative peaks at d = ±L/2 caused by the negative correlation of the first and second half of the reference signal with the received signal. Since we are using the maximum of the timing metric, those peaks are ignored and cannot result in any uncertainty in the detection of the correct location. For both ACO-OFDM and PAM-DMT schemes, there is a small peak occurring at d = L. This is due to the correlation of the local training symbol with the CP of the received training 82

104 symbol. The magnitude of this peak depends on the size of CP. Since CP length is usually small compared to the length of useful part of symbol, the magnitude of this peak is small compared to the main peak and thus does not cause any uncertainty in correct location of the beginning of the training symbol and does not result in erroneous detections. Note that, a plot of the average of the timing metric for DHT based OFDM system with K = L/2 is identical to that obtained for ACO-OFDM. Therefore, to avoid repetition of results, we do not show results for DHT based OFDM system. 6.5 Experimental Results An experimental test-bed is the perfect setup to verify the results discussed earlier. We employ USRP210, a software defined radio (SDR), as the primary hardware and software interface for the test-bed. The experimental setup is shown in Fig As can be seen, the baseband data from the host personal computer (PC) properly processed goes through the first USRP210 kit, where it is converted to RF signal. Instead of feeding this signal to an antenna, it is fed to a driver circuit, which in turn drives the light-emitting diode (LED) transmitter which is an OSRAM OSTAR Phosphor White LED. The LED transmitter is thus properly modulated by the data from the host PC and the light propagates through the channel. The intensity of the light is detected by the Si PIN photodiode, which produces a current signal proportional to this intensity. The signal is then amplified by an amplifier and fed to a second USRP210 that sends it to the target PC as baseband data. The data is further processed at the target PC. Figs. 6.3a to 6.3d demonstrate the timing metric for the bipolar correlation method with K = L/2 and CP length of L/8 where L = 256, 512. The symbols are drawn from 4-and 16-QAM constellations. 83

105 (a) (b) Figure 6.2: a) Schematic of the experimental setup. b) Real implementation with software defined radio systems. As can be clearly observed, the experimental results validate the simulation results discussed earlier. 84

106 (a) (b) (c) (d) Figure 6.3: Average of timing metrics for bipolar correlation method for consecutive ACO-OFDM symbols with a) L = 256 and 4-QAM modulation b) L = 256 and 16-QAM modulation c) L = 512 and 4-QAM modulation b) L = 512 and 16-QAM modulation. 85

107 6.6 Conclusions We have presented a novel and robust timing synchronization technique that can be applied to all AC based OFDM systems using IM/DD. Our proposed timing metric was based on the correlation of a local copy of the training symbol with a bipolar signal reconstructed from unipolar received signal. In contrast to existing timing synchronization methods, no special format of the training symbols is required. Simulations and experimental results have been presented to confirm the accuracy of the proposed method. 86

108 Chapter 7 Indoor Location Estimation with Optical-based OFDM Communications Orthogonal frequency division multiplexing (OFDM) has been applied to indoor wireless optical communications in order to mitigate the effect of multipath distortion of the optical channel as well as increasing data rate. In this Chapter, a novel OFDM visible light communication (VLC) system is proposed which can be utilized for both communications and indoor positioning. A positioning algorithm based on power attenuation is used to estimate the receiver coordinates. We further calculate the positioning errors in all the locations of a room and compare them with those using single carrier modulation scheme, i.e., on-off keying (OOK) modulation. We demonstrate that our proposed OFDM positioning system outperforms by 74% its conventional counterpart. Finally, we investigate the impact of different system parameters on the positioning accuracy of the proposed OFDM VLC system. 87

109 7.1 Introduction In current indoor visible light positioning systems, several algorithms have been proposed to calculate the receiver coordinates. In one approach, a photo diode (PD) is employed to detect received signal strength (RSS) information. The distance between transmitter and receiver is then estimated based on the power attenuation, and the receiver coordinates are calculated by lateration algorithm [11,79]. In another approach, RSS information is pre-detected by a PD for each location and stored as fingerprint in the offline stage. By matching the stored fingerprints with the RSS feature of the current location, the receiver location is estimated in the online stage [80]. In [12], proximity positioning concept has been used relying on a grid of transmitters as reference points, each of which has a known coordinate. The mobile receiver is assigned the same coordinates as the reference point sending the strongest signal. Image sensor is another form of receiver which detects angle of arrival (AoA) information for the angulation algorithm used to calculate the receiver location [81]. Other techniques have been also proposed for VLC systems to improve the indoor positioning performance. In [82], a two phase hybrid RSS/AoA algorithm for indoor localization using VLC has been proposed. In [83], Gaussian mixture sigma point particle filter technique has been applied to increase the accuracy of the estimated coordinates. Accelerometer has been employed in [84] such that the information on the receiver height is not required. To the best of our knowledge, the previous studies have been built on the assumption of a low speed single carrier modulation or/and have not considered the multipath reflections. However, a practical VLC system would be likely to deploy the same configuration for both positioning and communication purposes where high speed 88

110 data rates are desired. Furthermore, it has been shown in [85 87] that multipath reflections can severely degrade the positioning accuracy especially in the corner and the edge areas of a room. In this Chapter, to mitigate the multipath reflections as well as providing a high data rate transmission, we propose an OFDM VLC system that can be used for both indoor positioning and communications. The positioning algorithm employed is based on RSS information detected by a PD and the lateration technique. We show that our proposed system can achieve an excellent accuracy even in dispersive optical channels and for very low signal power values. 7.2 System Configuration System Model We consider a typical room shown in Fig. 7.1 with dimensions of 6 m 6 m 3.5 m where four light-emitting diode (LED) bulbs are located at height of 3.3 m with a rectangular layout. Data are transmitted from these LED bulbs after they are modulated by driver circuits. Each LED bulb has an identification (ID) denoting its coordinates which is included in the transmitted data. A PD as the receiver is located at the height of 1.2 m and has a field of view (FOV) of 70 and a receiving area of 1 cm 2. The room configuration is summarized in Table 7.1. Furthermore, strict time domain multiplexing is used where the entire OFDM frequency spectrum is assigned to a single LED transmitter for at least one OFDM symbol including a cyclic prefix (CP). 89

111 Figure 7.1: System configuration Optical Wireless Channel We assume an indoor optical multipath channel where transmitters and a receiver are placed in the room shown in Fig The baseband channel model including noise is expressed as y (t) = ηx (t) h (t) + ν (t) (7.1) where y (t) is the received electrical signal, η is the photodetector responsivity, h (t) is the multipath impulse response of the optical channel, and ν (t) denotes ambient light shot noise and thermal noise. Combined deterministic and modified Monte Carlo (CDMMC) method recently developed by Chowdhury et al [88] is used to simulate the impulse response of the optical wireless channel. Deterministic approaches [66,89] proposed to approximate impulse response of indoor optical wireless channels divide the reflecting surfaces 90

112 Table 7.1: System parameters Room dimensions Reflection coefficients length: 6 m ρ wall : 0.66 width: 6 m ρ Ceiling : 0.35 height: 3.5 m ρ F loor : 0.60 Transmitters (Sources) Receiver Wavelength: 420 nm Area (A r ): m 2 Height (H): 3.3 m Height (h): 1.2 m Lambertian mode (m): 1 Elevation: +90 Elevation: -90 Azimuth: 0 Azimuth: 0 FOV (Ψ c ): 70 Coordinates: (2,2) (2,4) (4,2) (4,4) Power for "1"/ "0": 5 W/3 W Figure 7.2: The contributions from different orders of reflections to the total impulse response of a location at the center of the room (weak scatterings and multipath reflections). 91

113 Figure 7.3: The contributions from different orders of reflections to the total impulse response of a location at the edge of the room (medium scatterings and multipath reflections). into small elements and provide the best accuracy, but at the cost of high computing time. On the other hand, modified Monte Carlo (MMC) approaches [90, 91] calculate the impulse responses very fast, but the calculated impulse responses are not as temporally smooth when compared to deterministic approaches. The algorithm in [88] takes advantage of both deterministic and MMC methods. In particular, the contribution of the first reflections to the total impulse response is calculated by a deterministic method for high accuracy, while an MMC method is employed to calculate the second and higher order reflections and achieve a lower execution time. We consider the line-of-sight (LOS) and first three reflections to simulate the impulse response of the channel. For each transmitter, we generate 4096 different channels by placing the receiver in different locations within the room with the 92

114 Figure 7.4: The contributions from different orders of reflections to the total impulse response of a location at the corner of the room (strong scatterings and multipath reflections). same height (i.e., 1.2 m). Figs. 7.2 to 7.4 demonstrate the contributions from different orders of reflections to the total impulse responses for three exemplary locations inside the room representing weak to strong scatterings and multipath reflections OFDM Transmitter and Receiver Different OFDM techniques have been proposed for optical wireless communications in the literature. For the sake of brevity, asymmetrically clipped optical OFDM (ACO-OFDM) is considered in this Chapter as it utilizes a large dynamic range of LED and thus is more efficient in terms of optical power than systems using DC-biasing. However, the generalization to other techniques is very straight- 93

115 Figure 7.5: OFDM transmitter and receiver configuration for both positioning and communication purposes. forward. A block diagram of an ACO-OFDM communication and positioning system is depicted in Fig The data and LED ID code are combined as the input bits which is parsed into a set of L/4 complex data symbols denoted by X = [ ] T X 0, X 1,...X L/4 1 where L is the number of subcarriers, and (.) T indicates the transpose of a vector. These symbols are drawn from constellations such as M-QAM or M-PSK where M is the constellation size. For VLC systems using intensity modulation and direct detection (IM/DD), a real valued signal is required to modulate the LED intensity. Thus, ACO-OFDM subcarriers must have Hermitian symmetry. As discussed in Section 5.2, in ACO-OFDM, impairment from clipping noise is avoided by mapping the complex input symbols onto an L 1 vector S as S = [ 0, X 0, 0, X 1,..., 0, X L 1, 0, X L 1, 0,..., X 1, 0, X 0, 0 ] T (7.2) where (.) denotes the complex conjugate of a vector. An L-point inverse fast Fourier transform (IFFT) is then applied creating the time domain signal x. A 94

116 CP is added to the real valued output signal turning the linear convolution with the channel into a circular one to mitigate inter-carrier interference (ICT) as well as inter-block interference (IBT). All the negative values of the transmitted signal are clipped to zero to make it unipolar and suitable for optical transmission. This clipping operation does not affect the data-carrying subcarriers but decreases their amplitude to exactly a half. The clipped signal is then converted to analog and finally modulates the intensity of an LED. At the receiver, the signal is detected by a PD and then converted back to a digital signal. The CP is removed and an L-point fast Fourier transform (FFT) is applied on the electrical OFDM signal. The training sequence is employed for synchronization and channel estimation as discussed in Chapter 6. A single tap equalizer is then used for each subcarrier to compensate for channel distortion and the transmitted symbols are recovered from the odd subcarriers and denoted by ˆX = [ ˆX0, ˆX 1,... ˆX ] T L/4 1. The LED ID is decoded and the transmitter coordinates are obtained which are fed to the positioning block along with the estimated channel DC gain as shown in Fig The receiver coordinates are finally estimated by employing the positioning algorithm detailed in the following section. 7.3 Positioning Algorithm For the system under consideration, the received optical power from the k th transmitter, k = 1, 2, 3 and 4, can be expressed as P r,k = H k (0)P t,k (7.3) 95

117 where P t,k denotes the transmitted optical power from the k th LED bulb, and H k (0) is the channel DC gain that can be obtained as [13] H k (0) = m + 1 A 2πd 2 r cos m (ϕ k )T s (ψ k )g(ψ k ) cos(ψ k ). (7.4) k In (7.4), A r is the physical area of the detector, ψ k is the angle of incidence with respect to the receiver axis, T s (ψ k ) is the gain of optical filter, g (ψ k ) is the concentrator gain, ϕ k is the angle of irradiance with respect to the transmitter perpendicular axis, d k is the distance between transmitter k and receiver, and m is the Lambertian order. Assuming that both receiver and transmitter axes are perpendicular to the ceiling, ϕ k and ψ k are equal and can be estimated as cos (ψ k ) = cos (ϕ k ) = (H h) /d k (7.5) where H and h are the transmitter and receiver heights, respectively. For a compound parabolic concentrator (CPC), g (ψ k ) is defined as n 2 r g (ψ k ) =, 0 ψ sin 2 (Ψ c ) k Ψ c (7.6) 0, ψ k > Ψ c where n r and Ψ c respectively denote the refractive index and the FOV of the concentrator. For the proposed OFDM system, the channel DC gain can be well estimated 96

118 as 1 Hk (0) = 4 L/4 P k,i (7.7) L where P k,i is the power attenuation of the i th symbol transmitted from the k th transmitter and is obtained using the training symbols used for synchronization as P k,i = i=1 ˆX k,i X k,i. (7.8) Considering (7.3)-(7.8), d k can be calculated as d m+3 k = (m + 1) A rt s (ψ k ) g (ψ k ) (H h) m+1 2π H k. (7.9) Horizontal distance between the k th transmitter and the receiver can be estimated as r k = d k 2 (H h) 2. (7.10) Then, according to the lateration algorithm [92, 93], a set of four quadratic equations can be formed as follows (x c x c1 ) 2 + (y c y c 1) 2 = r1 2 (x c x c2 ) 2 + (y c y c 2) 2 = r2 2 (7.11) (x c x c3 ) 2 + (y c y c 3) 2 = r 2 3 (x c x c4 ) 2 + (y c y c 4) 2 = r4 2 where (x c, y c ) is the receiver coordinates to be estimated and (x ck, y c k) is the k th 1 Note that for the other OFDM techniques, it is only required to estimate the channel DC gain similarly using the sample mean of the power attenuation of the transmitted symbols. Thus, our proposed system can be easily utilized for other OFDM techniques as well. 97

119 transmitter coordinates obtained from the recovered LED ID in a two-dimensional space. By subtracting the first equation from the last three equations, we obtain ( xc1 x cj ) xc + ( y c 1 y c j ) yc = ( r 2 j r 2 1 x c 2 j + x c 2 1 y c 2 j + y c 2 1 ) /2 (7.12) where j = 2, 3 and 4. (7.12) can be formed in a matrix format as AX = B where A, B and X are defined as x c2 x C 1 y c 2 y c 1 A = x c3 x c1 y c 3 y c 1, (7.13) x c4 x c1 y c 4 y c 1 (r B = r ) + (x c2 + y c2 ) (x c1 + y c1 ) (r r ) + (x c3 + y c3 ) (x c1 + y c1 ), (7.14) (r1 2 r4) (x c4 + y c4 ) (x c1 + y c1 ) X = [x c y c ] T. (7.15) The estimated receiver coordinates can then be obtained by the linear least squares estimation approach as [92] ˆX = (A T A) 1 A T B. (7.16) 7.4 Simulation and Analysis In this section, we present numerical results for the proposed indoor VLC system. In the following, we consider an OFDM system with a number of subcarriers of L = 64, 256, 512 or 1024 where the symbols are drawn from an M-QAM modulation 98

120 constellation. We set the CP length three times of the root mean square (RMS) delay spread of the worst impulse response and assume a data with minimum rate of 25 Mbps 2. The sum of ambient light shot noise and receiver thermal noise is modeled as real baseband Additive White Gaussian noise (AWGN) with zero mean and power of -10 dbm [67, 68]. Furthermore, to take LED nonlinearity into account, OPTEK, OVSPxBCR4 1-Watt white LED is considered in simulations whose optical and electrical characteristics are given in Table 5.2. A polynomial order of five is used to realistically model the measured transfer function. The four OPTEK LEDs are biased at 3.2V Performance Comparison of Single- and Multi-carrier Modulation Schemes In this subsection, the positioning performance of the proposed OFDM system is compared with the performance of those using single carrier modulation scheme, i.e., OOK. We assume that the average electrical power of the transmitted signal before modulating each LED is P te,k =5 dbm. For OOK modulation, the positioning algorithm discussed in [13] is used 3. Fig. 7.6 demonstrates the positioning error distribution over the room for an indoor OFDM VLC system with 4-QAM modulation and the FFT size of 512. As it can be seen, the positioning errors are very small for the most locations inside the room, but become larger when the receiver approaches the corners and edges due to the severity of the multipath reflections. Fig. 7.7, on the other hand, shows the positioning error distribution over the 2 The data rate is 25 Mbps for 4-QAM modulation scheme. For higher-order modulations (i.e., M > 4), the bit rate can be achieved as R = 25 Mbps log 2 (M). 3 The positioning algorithm used for OOK is quite similar to the one described in Section 7.3. The two algorithms only differ in the estimation method of the signal attenuation. 99

121 Figure 7.6: Positioning error distribution for OFDM system with 4-QAM modulation, L = 512 and P te,k = 5 dbm. Figure 7.7: Positioning error distribution for OOK modulation with P te,k = 5 dbm. 100

122 Figure 7.8: Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 512 and P te,k = 5 dbm. room for an indoor VLC system employing OOK modulation with the same data rate as that of the OFDM system with 4-QAM modulation (i.e., 25 Mbps). As observed, the positioning errors are relatively small within the rectangle shown in Fig. 7.7 where the LED bulbs are located right above its corners. However, the positioning error becomes significantly larger when the receiver moves toward the corners and edges as the effect of the multipath reflections increases. Figs. 7.8 and 7.9 present the histograms of the positioning errors for OFDM and OOK modulation schemes, respectively. For OFDM modulation, most of the positioning errors are less than 0.1 m and only a few of them are more than 1 m corresponding to the corner area. However, for OOK modulation, the positioning errors are widely spread from zero to around 2.3 m, and only a few of them are less than 0.1 m that correspond to the central area. From Figs. 7.6 to 7.9, it can be clearly seen that the OFDM system outperforms its OOK counterpart. Table 7.2 summarizes and compares the positioning errors of OFDM and OOK 101

123 Figure 7.9: Histogram of positioning errors for OOK modulation with P te,k = 5 dbm. Table 7.2: Positioning error for single- and multi-carrier modulation schemes Positioning error (m) OFDM modulation (m) OOK modulation (m) Corner (0, 0) Edge (3 m, 0) Center (3 m, 3 m) RMS error of the rectangular area RMS error of the whole room modulation schemes. As seen, OFDM modulation provides a much better positioning accuracy than OOK modulation for all the locations inside the room. Particularly, the RMS error is 0.08 m for the rectangular area covered perfectly by the four LED bulbs when OFDM modulation is used while it is 0.43 m for OOK modulation. The total RMS errors are m and 1.01 m for OFDM and OOK modulation schemes as the rectangular area covered by LED bulbs is only 102

124 Figure 7.10: Positioning error distribution for OFDM system with 4-QAM modulation, L = 512 and P te,k = -10 dbm. 11.1% of the total area. Thus, OFDM modulation decreases the RMS error by 74% compared to OOK modulation. It should be noted that the average positioning accuracy for can be increased by optimizing the layout design of the LED bulbs in future Effect of Signal Power on the Positioning Accuracy In this Subsection, we investigate the effect of the average electrical power of the transmitted signal on the positioning accuracy of the proposed OFDM VLC system. We consider an OFDM system with 4-QAM modulation and L = 512. Figs and 7.11 present the positioning error distribution for the OFDM system with transmitted signal power values of -10 dbm and 20 dbm, respectively. The total RMS error is calculated as m for the OFDM system with P te,k =

125 Figure 7.11: Positioning error distribution for OFDM system with 4-QAM modulation, L = 512 and P te,k = 20 dbm. dbm and m for P te,k = 20 dbm. Figs and 7.13 further demonstrate the corresponding histograms of the positioning errors for different transmitted signal power values. It is apparent from Figs and 7.12 that our proposed OFDM positioning system works satisfactorily even at very low transmitted signal power values resulting from dimming and shadowing effects 4. Furthermore, according to Figs. 7.6, 7.8 and 7.10 to 7.13 and as expected, increasing the average electrical power of the transmitted signal results in a better performance. However, at very high power values, nonlinearity distortion effects dominate the performance and the positioning accuracy decreases. It is the main reason the performance of the VLC system with P te,k = 20 dbm is slightly worse than that of P te,k = 5 dbm presented earlier. Therefore, for an 4 Note that the total RMS error for OOK modulation with P te,k = -10 dbm is calculated as 1.32 m. 104

126 Figure 7.12: Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 512 and P te,k = -10 dbm. Figure 7.13: Histogram of positioning errors for OFDM system with 4-QAM modulation, L = 512 and P te,k = 20 dbm. 105

127 Figure 7.14: Positioning error distribution for OFDM system with 16-QAM modulation, L = 512 and P te,k = 5 dbm. OFDM indoor VLC positioning system, there is an optimum power value that depends on the LED characteristics Effect of Modulation Order on the Positioning Accuracy Here, we analyze the impact of the modulation order on the positioning performance. Figs and 7.15 show the positioning error distribution of the OFDM system with the FFT size of 512 and P te,k = 5 dbm employing 16- and 64-QAM modulation, respectively. The corresponding histograms of the positioning errors are shown in Figs and The total RMS error is obtained as m for 16-QAM and m for 64-QAM. By comparing these RMS error values with the one calculated for 4-QAM modulation, we observe that all three systems 106

128 Figure 7.15: Positioning error distribution for OFDM system with 64-QAM modulation, L = 512 and P te,k = 5 dbm. Figure 7.16: Histogram of positioning errors for OFDM system with 16-QAM modulation, L = 512 and P te,k = 5 dbm. 107

129 Figure 7.17: Histogram of positioning errors for OFDM system with 64-QAM modulation, L = 512 and P te,k = 5 dbm. yield nearly the same positioning performance. Thus, the constellation size does not have a significant effect on the positioning performance of the proposed OFDM VLC system although the communication performance obviously deteriorates with increasing the constellation size. The numerical results clearly show that our proposed channel DC gain estimation works perfectly for high-order constellations as well Effect of Number of Subcarriers on the Positioning Accuracy Finally, we investigate the effect of number of total subcarriers (i.e. the FFT size) on the positioning accuracy. We consider an OFDM system with 4-QAM and P te,k = 5 dbm. Figs to 7.23 illustrate the positioning error distribution for differ- 108

130 Figure 7.18: Positioning error distribution for OFDM system with 4-QAM modulation, L = 64 and P te,k = 5 dbm. Figure 7.19: Positioning error distribution for OFDM system with 4-QAM modulation, L = 256 and P te,k = 5 dbm. 109

131 Figure 7.20: Positioning error distribution for OFDM system with 4-QAM modulation, L = 1024 and P te,k = 5 dbm. ent FFT sizes providing sufficiently narrow-banded sub-channels along with their corresponding error histograms. The total RMS errors are respectively calculated as m, m and m for L = 64, 256 and Considering the results presented earlier for the FFT size of 512, it is observed that increasing the number of subcarriers results in a better positioning performance as it improves the estimation of the channel DC gain, i.e., Eq. (7.7). However, the peak-to-average power ratio (PAPR) also increases with increasing the FFT size [94]. Therefore, for sufficiently large values of L, the positioning performance is slightly degraded. 110