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Performance analysis and design of multiple-input multiple-output and multi-hop free-space optical communication… Bayaki, Ehsan 2011

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PERFORMANCE ANALYSIS AND DESIGN OF MULTIPLE–INPUT MULTIPLE–OUTPUT AND MULTI–HOP FREE–SPACE OPTICAL COMMUNICATION SYSTEMS by Ehsan Bayaki M.Sc., Chalmers University of Technology, 2006 B.Sc., Sharif University of Technology, 2002  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2011 c Ehsan Bayaki, 2011  ii  Abstract Free–space optical (FSO) communication has recently gained a lot of interest as an attractive solution for high–rate last–mile terrestrial applications. FSO has many attractive features including the use of unlicensed parts of the electromagnetic spectrum, ease of deployment, cost efficiency, high security, and high data rates. There are however challenges in the design of FSO communication systems. Specifically, the weather–dependent optical wireless channel introduces attenuation and intensity variations known as turbulence–induced fading, which impose severe challenges for reliable data transmission. Meanwhile, the limitation in transmit power due to eye– safety regulations adds yet another design constraint. In this thesis, we first consider the performance analysis of FSO systems subject to Gamma–Gamma fading. The Gamma–Gamma probability density function (pdf) includes a modified Bessel function that precludes simple closed–form expressions. We employ a series representation of the modified Bessel function and derive closed–form expressions for the pairwise error probability (PEP) of FSO systems. We then study the performance of multiple–input multiple–output (MIMO) FSO systems for general space–time codes (STCs) for both direct and coherent/differential detection. We develop comprehensive models for both detection schemes and also use the derived models for a fair comparison between different detection schemes. For performance analysis only, we limit the number of transmit apertures to two and  Abstract  iii  derive the asymptotic PEP in closed form. Moreover, design criteria are established which are used to prove the quasi–optimality of repetition and Alamouti STCs for direct and coherent/differential detection in Gamma–Gamma fading, respectively. Finally, we investigate dual–hop and multi–hop FSO systems employing electrical and all–optical relays. Erbium–doped fiber amplifiers (EDFAs) are assumed for all– optical relaying and comprehensive signal and noise models are then derived. We show that all–optical relays outperform electrical relays unless the number of relays is very large. Furthermore, we conclude that, for a fixed source–destination distance, performance improves as the number of hops increases up to a certain point. Adding more relays will then result in performance degradation for both amplification schemes.  iv  Preface This thesis is based on the research work conducted at the Department of Electrical and Computer Engineering at UBC under the supervision of Prof. Robert Schober. Chapter 3 of this thesis has been partially published in the Proceedings of IEEE GLOBECOM 2008 and IEEE Transactions on Communications (cf. the list of articles and papers below). The research related to these publications was conducted by myself under the supervision of Prof. Schober and Prof. Ranjan K. Mallik of the Indian Institute of Technology. The research idea, the analytical evaluations, and simulation programs are all the result of my work. My supervisors helped with discussions, checking the validity of my analytical and simulation results, and proofreading the respective conference paper and journal article. Chapter 5 has been presented in a conference and has been also partially submitted as a journal article (cf. the list of articles and papers below). During the underlying research project, besides my supervisor, Dr. Diomidis Michalopoulos, who is a postdoctoral fellow in our group also helped me with my research. I should specify that the research idea related to these publications was originally proposed by me and all analytical and simulation results were also provided by myself. Dr. Michalopoulos and Prof. Schober helped me by having discussions during meetings, checking the validity of my simulation and analytical approaches, and proofreading the respective publications.  Preface  v  In the following, a list of articles and papers where the results in this thesis have been partially published/submitted is provided.  Journal Articles 1. (Part of Chapter 4) E. Bayaki and R. Schober. Performance and Design of Coherent and Differential Space–Time Coded FSO Systems. Submitted, Sep. 2011. 2. (Part of Chapter 5) E. Bayaki, D. S. Michalopoulos, and R. Schober. EDFA– Based All–Optical Relaying in Free–Space Optical Systems. Submitted, Apr. 2011. 3. (Part of Chapter 4) E. Bayaki and R. Schober. On Space–Time Coding for Free–Space Optical Systems. IEEE Transactions on Communications, vol. 58, no. 1, pp. 58-62, Jan. 2010. 4. (Part of Chapter 3) E. Bayaki, R. Schober, and R. K. Mallik. Performance Analysis of MIMO Free–Space Optical Systems in Gamma–Gamma Fading. IEEE Transactions on Communications, vol. 57, no. 11, pp. 3415-3424, Nov. 2009.  Preface  vi  Conference Papers 1. (Part of Chapter 5) E. Bayaki, D. S. Michalopoulos, and R. Schober. EDFA– Based All–Optical Relaying in Free-Space Optical Systems. In Proceedings of IEEE Vehicular Technology Conference (VTC, Spring), Budapest, Hungary, May 2011. 2. (Part of Chapter 3) E. Bayaki, R. Schober, and R. K. Mallik. Performance Analysis of Free–Space Optical Systems in Gamma–Gamma Fading. In Proceedings of IEEE Global Communications Conference (GLOBECOM), New Orleans, LA, USA, Dec. 2008.  vii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xvi  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction  xx  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  FSO Communication Systems . . . . . . . . . . . . . . . . . . . . . .  2  1.2  History of FSO Communications . . . . . . . . . . . . . . . . . . . . .  3  1.3  Design Challenges in FSO Systems . . . . . . . . . . . . . . . . . . .  4  1.4  FSO Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . .  6  1.5  Thesis Contributions and Organization . . . . . . . . . . . . . . . . .  9  Table of Contents 2 FSO Fading Distributions and Detection Schemes 2.1  2.2  viii . . . . . . . . .  13  Fading Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .  13  2.1.1  Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . .  14  2.1.2  Gamma–Gamma Distribution . . . . . . . . . . . . . . . . . .  15  FSO Detection Schemes . . . . . . . . . . . . . . . . . . . . . . . . .  19  2.2.1  Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . .  20  2.2.2  Coherent Detection  26  . . . . . . . . . . . . . . . . . . . . . . .  3 Performance Analysis of MIMO FSO Systems in ...  . . . . . . . .  32  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  34  3.1.1  Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . .  34  3.1.2  Series Representation of Gamma–Gamma Distribution . . . .  36  3.1.3  Receiver Structure and Error Rate . . . . . . . . . . . . . . .  37  Performance Analysis for SISO Channels . . . . . . . . . . . . . . . .  39  3.2.1  PEP in SISO Channels . . . . . . . . . . . . . . . . . . . . . .  39  3.2.2  Diversity and Coding Gain . . . . . . . . . . . . . . . . . . . .  40  3.2.3  Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .  41  Performance Analysis for MIMO Channels . . . . . . . . . . . . . . .  42  3.3.1  EGC Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . .  42  3.3.2  MRC Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . .  45  3.3.3  Comparison of EGC and MRC  . . . . . . . . . . . . . . . . .  48  3.4  Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . . .  49  3.5  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  3.1  3.2  3.3  4 Performance and Design of Space–Time Coded FSO Systems 4.1  . .  60  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  61  MIMO FSO with Direct Detection . . . . . . . . . . . . . . .  62  System Model 4.1.1  Table of Contents 4.1.2 4.2  4.3  ix  MIMO FSO with Coherent and Differential Detection  . . . .  65  . . . . . . . . .  68  4.2.1  Direct Detection STCs . . . . . . . . . . . . . . . . . . . . . .  69  4.2.2  Coherent STCs . . . . . . . . . . . . . . . . . . . . . . . . . .  71  4.2.3  Differential STCs . . . . . . . . . . . . . . . . . . . . . . . . .  71  Asymptotic Performance Analysis for General STCs  . . . . . . . . . . . . . . . . . . . . .  72  4.3.1  Direct Detection STC Design . . . . . . . . . . . . . . . . . .  73  4.3.2  Coherent and Differential Detection STC Design  . . . . . . .  76  Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . . .  77  4.4.1  Direct Detection Results . . . . . . . . . . . . . . . . . . . . .  78  4.4.2  Coherent/Differential Detection Results . . . . . . . . . . . . .  79  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84  5 Multi–Hop FSO Communication Systems . . . . . . . . . . . . . . .  85  4.4  4.5  FSO Space–Time Code Design  5.1  Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  87  5.2  Dual–Hop FSO Model . . . . . . . . . . . . . . . . . . . . . . . . . .  88  5.2.1  Optical Amplification  . . . . . . . . . . . . . . . . . . . . . .  89  5.2.2  Electrical Amplification . . . . . . . . . . . . . . . . . . . . .  94  Outage Analysis of Dual–Hop All–Optical Relaying . . . . . . . . . .  96  5.3.1  Fixed Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .  97  5.3.2  Variable Gain . . . . . . . . . . . . . . . . . . . . . . . . . . .  99  5.3  5.4  5.5  Multi–Hop FSO Relay Model . . . . . . . . . . . . . . . . . . . . . . 101 5.4.1  Optical Amplification  5.4.2  Electrical Amplification . . . . . . . . . . . . . . . . . . . . . 104  5.4.3  Outage Analysis of Multi–Hop All–Optical Relaying  Numercial Results  . . . . . . . . . . . . . . . . . . . . . . 102  . . . . . 105  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106  Table of Contents 5.6  x  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114  6 Summary of Thesis and Future Work  . . . . . . . . . . . . . . . . . 115  6.1  Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 115  6.2  Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2.1  MIMO FSO STC Design with M > 2 in Gamma–Gamma Fading117  6.2.2  Performance Analysis of MIMO FSO Systems with Pointing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118  6.2.3  Parallel All–Optical Relaying in FSO Systems . . . . . . . . . 119  6.2.4  Cooperative Diversity in FSO Systems . . . . . . . . . . . . . 119  6.2.5  Multi–Hop FSO Systems Subject to Gamma–Gamma Fading . 120  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 A A Useful Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B Bound on Approximation Error . . . . . . . . . . . . . . . . . . . . . 131  xi  List of Tables 3.1  Required J to achieve a relative truncation error ε(J)/Pe (1) = |Pe (1)− Pe (1, J)|/Pe (1) of less than 10−9 for M = 1, N = 1 and EGC with M = 1, N = 3, respectively. . . . . . . . . . . . . . . . . . . . . . . .  53  4.1  System Parameters [38, 47, 48]. . . . . . . . . . . . . . . . . . . . . .  81  5.1  Coefficients of the noise variance expression in (5.30) and means, vari2 ances, and covariances in (5.34) for fixed gain amplification. Beq =  (2Be Bo − Be2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2  99  Coefficients of the noise variance expression in (5.36) for variable gain amplification. Cb,ase = Pb + nsp hνBo and Cr,ase = Pr + nsp hνBo . . . . 100  5.3  Means, variances, and covariances in (5.38) for variable gain amplification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101  5.4  Coefficients of the noise variance in (5.54) for multi–hop variable gain all–optical relaying. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105  5.5  Multi–hop Relaying System Parameters [38, 47, 48]. . . . . . . . . . . 106  xii  List of Figures 2.1  Lognormal fading distribution for different link distances in haze (Cn2 = 1.7 × 10−14 , λ = 1550 nm [43]). . . . . . . . . . . . . . . . . . . . . .  2.2  Values of α and β vs. link distance in Gamma–Gamma fading and haze (Cn2 = 1.7 × 10−14 , λ = 1550 nm, and D/L → 0 [22]). . . . . . .  2.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24  Block diagram of a coherent detection receiver (PD: photodetector, A: transimpedance amplifier, LO: local oscillator). . . . . . . . . . . . . .  3.1  23  Power spectrum of the signal-background beat noise (PSD: power spectral density).  2.7  21  Power spectrum of the background-background beat noise (PSD: power spectral density). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.6  18  Block diagram of a direct detection receiver (PD: photodetector, A: transimpedance amplifier). . . . . . . . . . . . . . . . . . . . . . . . .  2.5  17  Gamma–Gamma fading distribution for different link distances in haze (Cn2 = 1.7 × 10−14 , λ = 1550 nm [43]). . . . . . . . . . . . . . . . . . .  2.4  16  27  Block diagram of a MIMO FSO system with M transmit lasers and N photodetectors. The footprint of each laser illuminates the whole receiver array (PD: photodetector). . . . . . . . . . . . . . . . . . . .  34  List of Figures 3.2  BER of OOK vs. SNR γ. SISO (M = N = 1), Gamma–Gamma fading, β = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.3  xiii  49  Normalized approximation error ε(J)/Pe (1) = |Pe (1) − Pe (1, J)|/Pe (1) and corresponding upper bound (B.2) vs. truncation constant J. SISO (M = N = 1), OOK, Gamma–Gamma fading, β = 2. . . . . . . . . .  3.4  51  BER of OOK vs. SNR γ. SISO (M = N = 1) and MIMO (M = 1, N = 3) with EGC and MRC, Gamma–Gamma fading, α = 3.1, β = 2.  52  3.5  Performance gain G(M, N ) of MRC over EGC vs. N . . . . . . . . . .  54  3.6  Performance gain G(M, N ) of MRC over EGC vs. link distance L. α and β from (4.30) and (4.36), respectively. λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. . . . . . . . . . . . . . . . . . . . . . .  3.7  55  BER of OOK vs. link distance L. α and β from (4.30) and (4.36), respectively. SNR = 20 dB, λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  3.8  BER of 2–PPM vs. SNR γ. . . . . . . . . . . . . . . . . . . . . . . . .  57  3.9  BER of two different MPPM formats vs. SNR γ. EGC, spherical wave propagation, L = 5 km, λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  58  3.10 BER of 2–PPM and OPPM (S = 5, w = 2) vs. SNR γ. Spherical wave propagation, L = 5 km, λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1  59  BER vs. SNR γdd for M = 2 lasers and Gamma–Gamma fading parameters α = 2.5 and β = 1. . . . . . . . . . . . . . . . . . . . . . . .  77  List of Figures 4.2  xiv  Coding gain advantage A(N µ) of RC over OSTBC vs. link distance L. Spherical wave propagation, wavelength λ = 1550 nm, index of refraction structure parameter Cn2 = 1.7 · 10−14 , and D/L → 0 (D: receiver aperture, L: link distance), M = 2 lasers, and S = 2 pulse intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.3  79  BER of FSO systems with coherent heterodyne, differential heterodyne, and direct detection schemes in Gamma-Gamma fading (α = 2.5, β = 1). The 4 × 4 space-time block code from [65] is employed for coherent/differential detection and M = 4. Repetition coding is used for direct detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.4  81  BER of FSO systems with coherent heterodyne, differential heterodyne, and direct detection schemes in Gamma-Gamma fading (α = 2.5, β = 1). The more bandwidth efficient 8-PPM and 8-PSK modulation schemes are applied to direct and coherent/differential detection FSO systems, respectively. . . . . . . . . . . . . . . . . . . . . . . . .  4.5  BER of TSC [30] and BBH [64] STTCs in Gamma-Gamma fading and haze (Cn2 = 1.7 × 10−14 ) for a heterodyne detection receiver. . . . . .  5.1  83  Block diagram of FSO AF relay systems with (a) optical amplification using an EDFA and (b) electrical amplification (PD: photodetector). .  5.2  82  87  Outage probability of electrical and optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). dsd = 5 km. A single variable gain relay is placed dsr = 3.5 km from the source node. Source and relay transmit with equal power Pt = Pr = P/2. . . . . . . . . . . 107  List of Figures 5.3  xv  Outage probability of EDFA-based all–optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). Fixed gain and variable gain relaying are compared with direct transmission for different relay locations. Source and relay transmit with equal power Pt = Pr = P/2. dsd = 5 km. Markers indicate simulation results. . . . 109  5.4  Outage probability of EDFA-based fixed gain all–optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). Source and relay transmit with equal power Pt = Pr = P/2. dsd = 5 km. Results were obtained from analysis. . . . . . . . . . . . . . . . . 110  5.5  Outage probability of EDFA-based variable gain all–optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). 100 × ρ % of the total available power are allocated to the source. P Tb = −125 dBJ. dsd = 5 km. Results were obtained from analysis. . 111  5.6  Outage probability of EDFA-based multi–hop variable gain all–optical relaying in lognormal fading and clear air (Cn2 = 5 × 10−14 , aattn = 0.43 dB/km). Power is equally divided between the transmitting nodes (source and relays). All nodes have identical distances from their nearest neighbors. Markers indicate simulation results. . . . . . . . . . . . 112  5.7  Outage probability of multi–hop variable gain all–optical and electrical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). Power is equally divided between the transmitting nodes (source and relays). All nodes have identical distances from their nearest neighbors. Results were obtained from analysis. . . . . . . . . . . 113  xvi  List of Abbreviations AF  Amplify–and–Forward  ASE  Amplified Spontaneous Emission  AWGN  Additive White Gaussian Noise  BER  Bit Error Rate  BPSK  Binary Phase Shift Keying  CSI  Channel State Information  DF  Decode–and–Forward  EDFA  Erbium–Doped Fiber Amplifier  EGC  Equal Gain Combining  EMI  Electromagnetic Interference  FSO  Free–Space Optical  Gbps  Giga Bits per Second  i.i.d.  Independent and Identically Distributed  IM/DD  Intensity Modulation/Direct Detection  LED  Light Emitting Diode  LOS  Line–of–Sight  Mbps  Mega Bits per Second  MGF  Moment Generating Function  MIMO  Multiple–Input Multiple–Output  List of Abbreviations ML  Maximum Likelihood  MPAM  M–ary Pulse Amplitude Modulation  MPPM  Multipulse Pulse Position Modulation  MPSK  M–ary Phase Shift Keying  MRC  Maximal Ratio Combining  OOK  On–Off Keying  OPPM  Overlapping Pulse Position Modulation  OSTBC  Orthogonal Space–Time Block Code  pdf  Probability Density Function  PDFA  Praseodymium–Doped Fiber Amplifier  PEP  Pairwise Error Probability  PPM  Pulse Position Modulation  RC  Repetition Coding  RF  Radio Frequency  RV  Random Variable  SI  Scintillation Index  SISO  Single–Input Single–Output  SLA  Semiconductor Laser Amplifier  SNR  Signal–to–Noise Ratio  STBC  Space–Time Block Code  STC  Space–Time Code  STTC  Space–Time Trellis Code  xvii  xviii  Notation In this dissertation, bold upper case and lower case letters denote matrices and vectors, respectively. The remaining notation and operators used in this thesis are listed as follows: E{·}  Statistical expectation of a random variable  Q(·)  Gaussian Q-function, Q(x)  ln(·)  Natural logarithm function  lnN (µ, σ 2 )  Lognormal distribution with parameters µ and σ 2  |·|  Absolute value of a complex number  || · ||  L2 –norm of a vector or Frobenius norm of a matrix  (·)T  Transposition  (·)H  Hermitian transposition  (y)(x)  y is convolved x − 1 times with itself  C S×M  Matrix C of size S × M  cmn  The element in row m and column n of matrix C  ℜ{·}  Real part of a complex number  tr{·}  Trace of a matrix  Pr{·}  Probability of an event  Γ(·)  Gamma function, Γ(x)  Kν (·)  νth order modified Bessel function of the second kind  √1 2π  ∞ −t2 /2 e dt x  ∞ −t x−1 e t dt 0  Notation  xix  F (·, ·; ·; ·) . A=B  Asymptotic equivalence of A and B  o(·)  A function f (x) is o(x) if limx→0  IX  X × X Identity matrix  c¯  Bit complement of c  Hypergeometric function  f (x) x  =0  xx  Acknowledgments First and foremost, I would like to express my deepest gratitude to my supervisor, Prof. Robert Schober for his constant guidance, technical insight, and encouragement throughout the thesis work. I am greatly indebted to him for his commitment, support, and understanding from the initial to the final steps of my PhD research. His energy, drive, and enthusiasm have been a great source of motivation for me in my studies and in my life. I would also like to thank the members of my supervisory committee, Prof. Lutz Lampe, Prof. Victor Leung, Prof. Z. Jane Wang, and Prof. Vincent Wong, and examining committee members, Prof. Shahriar Mirabbasi and Prof. Son Vuong for their valuable feedback and suggestions and also for their time and effort. I would specially like to thank my external examiner, Prof. John Barry of Georgia Institute of Technology for taking the time to read my thesis and his valuable comments and suggestions on my research work and dissertation. My best wishes go to the colleagues in the Communications Theory Group for their help, support, and providing an stimulating environment for research. I would like to specially thank Dr. Diomidis Michalopoulos for his guidance and technical feedback during the last part of my research. Special thanks to my friends, Asef Javan, Yalda Mahmoudi, Pirooz Darabi, Mohammad Mohammadnia, Alireza Kenarsari, Ali Abdulhussein, Ali Nezampour, Vahid  Acknowledgments  xxi  Shah Mansouri, and many others for helping me, one way or another. Above all, I am deeply indebted to my lovely family, my parents, Hossein and Marzieh, and my sisters, Elnaz and Mehrnaz, for their endless and unconditional love, believing in me, and their constant encouragement and support in all stages of my life. This thesis is dedicated to them as a token of my gratitude. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Vancouver, BC, Canada September 2011  1  1  Introduction  With the widespread and fast growing application of wireless devices, the demand for high–rate and reliable wireless communication technologies is higher than ever before. While radio frequency (RF) communication has been the focus of most technological advancements during the past few decades, free–space optical (FSO) communication has recently gained considerable attention for last–mile terrestrial applications and wireless backhaul in mobile communication systems. FSO provides optical fiber data rates with the advantage of cheap and simple deployment. As a result, this technology is a prime candidate for the extension of already existing optical fiber networks. The possibility of fast deployment of FSO links is a huge asset in establishing emergency communication links in disaster recovery. Furthermore, the complementary nature of radio and optical communication channels also motivates hybrid RF/FSO communication systems that are suitable for a wider range of atmospheric conditions. In this chapter, we first review the favorable characteristics of FSO communications. We then study the historical developments that have led to the state of the art in optical wireless communications. Next, design challenges in FSO systems are discussed and a review of the contributions that have been already made in the FSO literature is provided. The final part of this chapter summarizes our contributions and includes an overview of the following chapters of this dissertation.  1. Introduction  1.1  2  FSO Communication Systems  FSO is a line–of–sight (LOS) technology that uses the propagation of light in free space for data transmission. This technology is of special interest for last–mile communications [1, 2], where the so-called “last mile” from the fiber backbone to the point of interest is missing. Since FSO is capable of providing Giga bit per second (Gbps) data rates, it is an appropriate candidate for last-mile communications. There are several other advantages to FSO technology, including: 1. FSO is a license-free technology and therefore its application results in huge savings in licensing fees. Besides, the time consuming and bureaucratic process of obtaining permits is not required for FSO. 2. It is easy, fast, and fairly cheap to deploy FSO devices and the established links will be operational within hours. 3. The very directional and narrow beam optical link makes interception and jamming nearly impossible and therefore FSO is also a favorable option for applications where high security is required. 4. The directional nature of the FSO link enables full duplex operation. Note that RF communications does not provide this luxury. 5. Unlike RF communications, FSO signals are immune to electromagnetic radiation emitted from external sources known as electromagnetic interference (EMI).  1. Introduction  1.2  3  History of FSO Communications  Optical communication has been in use for thousands of years in various forms. From signal fires on hilltops to lanterns on ships [3], light has been employed for data transmission since ancient times. However, the real invention in optical transmission technology did not take place until 1880, when Alexander Graham Bell and his assistant built the “Photophone” [3]. The photophone employed intensity modulated sunlight to transmit voice signals between two buildings more than 200 meters apart. In May 1960, the first laser was successfully built by Theodore Maiman [4]. The invention of laser was a revolutionary step in optical communications. Laser had the potential to realize a single frequency carrier signal combined with a sufficiently large bandwidth. Soon after, engineers at Bell Labs set up an experimental framework to employ the newly invented laser for long range optical wireless communications [4]. The results of the experiments were discouraging and revealed that the optical beam is adversely affected by atmospheric conditions such as rain, snow, and fog. The advent of optical fiber in the 1970s further resulted in a decline in free space optics research. In the meantime, however, military applications were still of interest due to the highly secure nature of optical beams. In the past few years, the rapid increase in the demand for high rate wireless applications has resulted in a renewed interest in FSO communications. Radio communications requires licensing and is limited in terms of the available spectrum. The data rates are also not comparable to what is offered by optical fiber. On the other hand, fiber is expensive, time consuming, and inconvenient to deploy. With a good portion of the buildings in densely populated urban areas not connected to the fiber network, FSO is more promising than ever to fill a crucial gap in today’s communication networks.  1. Introduction  1.3  4  Design Challenges in FSO Systems  FSO is a wireless technology and therefore terrestrial optical wireless communication systems are susceptible to the atmospheric conditions. The effect of the wireless channel is twofold. On the one hand, the signal power can be substantially reduced as a result of power attenuation over the FSO channel. On the other hand, turbulence– induced fading results in irradiance fluctuations in the received signal. Other design challenges stem from the LOS nature of FSO, safety issues associated with the application of laser, and the negative effects of background radiation. In summary, the following are the main issues to be considered for the design of FSO communication systems [5, 6]: 1. Scattering: Adverse weather conditions such as rain, snow, fog, and haze can result in detrimental effects when it comes to light propagation. The rain and snow droplets are much larger compared to the laser wavelengths employed for FSO communications. As a result, while some attenuation results, the effects are not as harmful as in case of fog and haze. The particles present in fog and haze have diameters that are comparable to the optical wavelengths and therefore cause major deflection of the optical beam from its initial direction. This phenomenon is known as scattering and the transmitted signal is attenuated and distorted in the presence of scatterers. 2. Absorption: The molecules present in the atmosphere absorb light energy and therefore result in power attenuation. This phenomenon is dependent on the wavelength of the laser employed and the type of molecules present in the atmosphere. Changes in atmospheric conditions that also translate to changes in the molecular contents of the propagation environment are equivalent to rapid changes in the level of attenuation and might adversely affect the availability of  1. Introduction  5  the optical link. 3. Scintillation: Atmospheric turbulence is caused by variations in the refractive index due to inhomogeneities in temperature, pressure fluctuations, humidity variations, and motion of the air along the propagation path of the laser beam. This phenomenon introduces irradiance fluctuations in the received signal and the resulting signal fading thus causes severe performance degradations. 4. Building sway: Since FSO equipment is normally mounted on tall buildings in densely populated urban areas, the LOS path might be disturbed due to the loss of alignment between the transmitter and receiver due to building sway. Seasonal changes in temperature affect the building dimensions. Besides, wind and weak earthquakes are other sources of misalignment and are also unpredictable. Spatial diversity is an effective solution to this problem. 5. Physical obstructions: Temporary obstructions by sources such as construction equipment, flying birds, etc have the potential to disturb the optical beam. Employing divergent optical beams or multiple lasers (spatial diversity) improves system performance under these circumstances. 6. Eye safety: While light emitting diodes (LEDs) are mostly used for indoor wireless communication systems, lasers are the preferable choice in terrestrial applications. The possibility of the exposure of human eye to the optical beam employed in FSO systems raises concerns about the safety of such systems. A standard has been therefore established for eye safety, where optical sources are classified with respect to their total transmit power. Table 1 in [7] summarizes the safety standard for a point–source laser emitter.  1. Introduction  6  In this thesis, we aim to address major FSO design challenges that engineers and researchers face. The next two sections provide a summary of the work accomplished in the FSO literature and the contributions made in this thesis towards providing solutions to the design obstacles in FSO communication systems.  1.4  FSO Literature Review  In this dissertation, we will address several relevant topics on FSO communications including uncoded single–input single–output (SISO) and multiple–input multiple– output (MIMO) FSO systems performance analysis, space–time code (STC) design for direct and coherent/differential detection FSO systems, and all–optical relaying in FSO systems. The details are provided in Section 1.5. Below, we review some of the work in the literature pertinent to the topics we will cover in this thesis. As was mentioned before, FSO systems are susceptible to atmospheric turbulence [6]. Recently, it has been shown that similar to RF communications, the effect of fading in FSO can be substantially reduced by creating a MIMO FSO system with multiple lasers at the transmitter and multiple photodetectors at the receiver [8]– [12]. In order to evaluate the impact of atmospheric turbulence and the effectiveness of corresponding countermeasures, accurate models for the fading distribution are important. A few different models have been used in the literature to model the effects of turbulence–induced fading including lognormal fading [13, 14], Gamma–Gamma fading [15, 16], K–fading [17], and negative exponential fading [18]. While lognormal distribution is often used to model weak turbulence conditions, k–fading and negative exponential fading are more relevant for scenarios where strong turbulence is present. Among different fading models, the Gamma–Gamma distribution has recently received considerable attention because of its excellent fit with measurement data for  1. Introduction  7  a wide range of turbulence conditions (weak to strong) [19, 20]. However, despite the popularity of the Gamma–Gamma distribution in the FSO literature [15]–[25], a basic understanding of the effects of Gamma–Gamma fading on the performance of (MIMO) FSO systems is not available. For FSO systems with intensity modulation and direct detection (IM/DD), heuristic STC designs such as repetition codes (RCs) [11] and orthogonal space–time block codes (OSTBCs) [26] have been proposed. Furthermore, the concatenation of FSO STCs with forward error correcting codes has been considered in [27]. In [28], it was shown that, seemingly surprisingly, simple RCs always outperform OSTBCs. Furthermore, FSO STC design was discussed in [29]. However, unlike in the RF case [30], performance bounds and systematic design guidelines for general FSO STCs are not available. In particular, from [28, 29] it is not clear whether FSO STCs that outperform RCs may exist. Although IM/DD is a simple approach to low–complexity FSO, a higher performance can potentially be achieved with FSO systems employing coherent or differential modulation at the expense of a higher receiver complexity. Nevertheless, the performance and design of STCs for coherent and differential FSO systems is not well understood. The performance of coherent and differential Alamouti–based space–time block codes (STBCs) was investigated via simulations for Rayleigh fading and lognormal fading FSO channels in [31] and [32], respectively. However, the simulations in [31, 32] do not offer insight into STC design. In [33], assuming lognormal fading, an STC design criterion for coherent FSO systems was derived based on the central limit theorem implying a large number of transmit and/or receive apertures. However, in practice, the number of transmit and receive apertures may be small. Also, while Rayleigh and lognormal fading are suitable models for strong and weak turbulence conditions, respectively, Gamma–Gamma fading is a more versatile model which  1. Introduction  8  can represent a wide range of turbulence conditions [19]. A comprehensive model that reflects the effects of all relevant signal and noise terms in coherent/differential space–time receivers has not been provided in the FSO literature. Also, a framework enabling a fair comparison of space–time coded IM/DD FSO systems and their coherent/differential counterparts is not available. The models provided in [31]–[33] for coherent detection FSO receivers are not well justified and the underlying assumptions are not provided. Besides, the quantitative effects of relevant system parameters on the provided models are not studied. In order to combat both fading and propagation loss in FSO systems, relay– assisted communication has been recently considered in the context of FSO, where both amplify and forward (AF) and decode and forward (DF) relaying schemes have been studied [23, 34]. Unlike in the RF channel, the severity of the small–scale fading in the optical wireless channel is distance–dependent. Consequently, in FSO systems, relays do not only reduce the path loss but improve also the small–scale fading statistics of the channel. Furthermore, since FSO is an LOS technology, relaying is attractive in situations where source and destination do not have a line of sight. Most of the existing work on FSO relaying considers relays employing electrical amplification (referred to as electrical relays in this thesis), where the received optical signal is converted to an electrical signal via a photodetector, the electrical signal is amplified via an electrical amplifier, and the amplified electrical signal is converted back to an optical signal employing a laser [23]–[28]. In [28, 34], relay–assisted FSO was investigated and the outage probability of AF and DF relaying was analyzed in lognormal fading. Thereby, the noises at the input of the relay and the destination node were assumed to be dominated by the shot noise caused by the background radiation leading to a conventional additive white Gaussian noise (AWGN) channel model. The simple AWGN model was also used for a diversity gain analysis in [36],  1. Introduction  9  a multi–hop performance analysis in [23], an outage probability analysis in [35], and a bit error rate analysis in [37]. Thanks to their widespread use in optical fiber communications as preamplifiers and in–line amplifiers in the 1550 nm range, erbium–doped fiber amplifiers (EDFAs) are a mature technology [3, 38]. While the use of EDFAs in optical fiber systems has been extensively studied [39], their application in FSO systems has not been well investigated. FSO relays with optical amplification have been recently considered for dual–hop transmission in [40, 41]. However, in [40, 41], simplified signal and noise models were adopted which may not reveal the true performance of optical relays. In particular, [40] only includes the effects of amplified spontaneous emission (ASE) noise and shot noise caused by background radiation, while [41] also takes the thermal receiver noise at the destination into account. However, signal-dependent noises and beat noises are neglected in [40, 41] but may have a significant impact on system performance. The references above should not be considered as a complete set of publications relevant to FSO. However, the already mentioned list along with further references which will be accordingly given in the following chapters, should serve as a good starting point for the interested reader to further investigate the literature in this area.  1.5  Thesis Contributions and Organization  In this thesis, we consider four main areas of interest pertinent to FSO communication systems: 1. Performance analysis of uncoded transmission over SISO and MIMO FSO channels suffering from Gamma–Gamma fading.  1. Introduction  10  2. Derivation of the asymptotic pairwise error probability (PEP) and design criteria for space–time coded IM/DD FSO systems in Gamma–Gamma fading. 3. Derivation of comprehensive received signal and noise models for coherent (homodyne and heterodyne) detection, asymptotic PEP analysis, and design criteria for space–time coded coherent FSO systems in Gamma–Gamma fading. 4. Derivation of a comprehensive received signal and noise model for dual-hop and multi-hop IM/DD FSO communication systems employing electrical and all-optical (EDFA-based) relays in lognormal fading. Each of the above topics will be addressed in detail in the following chapters. In Chapter 2, we will provide background information that will be useful in the context of the following chapters. This includes details on FSO channel fading distributions and comprehensive models for different detection schemes. Chapter 3 deals with the performance analysis of FSO systems in Gamma–Gamma fading. More specifically, we consider an FSO system with an arbitrary number of transmit and receive apertures and express the PEPs of SISO and MIMO IM/DD FSO systems as generalized infinite power series with respect to the signal–to–noise ratio (SNR). For numerical evaluation these power series are truncated to a finite number of terms and an upper bound for the associated approximation error is provided. The resulting finite power series enables fast and accurate numerical evaluation of the bit error rate (BER) of IM/DD FSO with on–off keying (OOK) and pulse position modulation (PPM) in SISO and MIMO Gamma–Gamma fading channels. Furthermore, we extend the well–known RF concepts of diversity and combining gain to FSO and Gamma–Gamma fading. In particular, we provide simple closed–form expressions for the diversity gain and the combining gain of MIMO FSO with RC across lasers at the  1. Introduction  11  transmitter and equal gain combining (EGC) or maximal ratio combining (MRC) at the receiver. In Chapter 4, we first investigate STC design for IM/DD FSO systems. For this purpose, we derive a closed–form expression for the asymptotic PEP of general FSO STCs for two lasers and an arbitrary number of photodetectors for channels suffering from Gamma–Gamma fading. Furthermore, we provide a simple design criterion for FSO STCs which is used to establish the quasi–optimality of previously proposed FSO repetition codes. We also show that STCs optimized for RF systems achieve full diversity in FSO systems but are suboptimal as far as the coding gain is concerned. Simulation results will be provided to validate the analytical findings of this chapter. Following the STC design for IM/DD systems, we derive a comprehensive received signal and noise model for coherent detection FSO systems. We then present an asymptotic performance analysis for MIMO FSO systems employing coherent and differential space-time codes over Gamma–Gamma fading channels. We consider the practically important case of two transmit and an arbitrary number of receive apertures and provide a simple STC design criterion. Our analysis reveals that, in contrast to IM/DD FSO systems, for coherent and differential FSO systems Alamouti’s STC is preferable over repetition STCs, which do not yield full diversity. Chapter 5 is concerned with dual-hop and multi-hop FSO communication systems in the presence of lognormal fading. More specifically, in this chapter, we advocate the use of all-optical relays equipped with EDFAs, which, in contrast to conventional FSO relays with electrical amplification, avoid optical-to-electrical and electrical-to-optical conversions. We develop accurate signal and noise models for fixed and variable gain all-optical and electrical relaying which include the effects of all relevant system parameters and types of noise. For performance evaluation, we analyze the outage probability of all-optical relaying in lognormal fading for dual-hop  1. Introduction  12  and multi-hop transmission. Our results show that all-optical relays, while simpler from an implementation point of view, outperform electrical relays unless the number of relays is very large. Moreover, for a fixed source-destination distance, performance improves as the number of hops (relays) increases up to a certain point beyond which adding more hops deteriorates performance. Finally, in Chapter 6, we summarize and provide additional perspective on the results obtained in previous chapters. We also provide a number of topics for possible future work on the grounds of the research presented in this dissertation.  13  2  FSO Fading Distributions and  Detection Schemes In this chapter, we study the most commonly used models for turbulence–induced fading in the FSO literature. These models will be later used in the following chapters for performance analysis and design purposes. Moreover, comprehensive models for direct and coherent detection are derived taking all system parameters and relevant signal and noise terms into account for a simple SISO link. Later in this dissertation, these models will be extended to more general cases wherever applicable and will be used for performance analysis and code design in MIMO and multi–hop FSO communication systems.  2.1  Fading Distributions  One of the main challenges in the design and operation of FSO systems is the adverse effect of turbulence–induced fading which is caused by variations in temperature, pressure, and humidity, and the motion of the air along the propagation path. The mathematical fading model is one of the building blocks for performance evaluation and design in any communication system. The parameters of such fading model should be determined so as to either fit measurement data or be represented  2. FSO Fading Distributions and Detection Schemes  14  in terms of atmospheric parameters. Among different fading distributions used in the FSO literature, lognormal and Gamma–Gamma distributions are the most common. While the lognormal distribution is a better fit for scenarios where weak turbulence is present, the Gamma–Gamma distribution is a more comprehensive model that represents a wider range of turbulence conditions [20]. In this section, we provide details on lognormal and Gamma–Gamma distributions, respectively. Note that regardless of the fading distribution, the severity of fading is measured by a quantity known as scintillation index (S.I.), which is defined as  S.I.  E{I 2 } , (E{I})2  (2.1)  where I represents the intensity gain factor of the FSO channel.  2.1.1  Lognormal Distribution  The lognormal fading distribution is more relevant for weak turbulence conditions or short distances [13]. The probability density function (pdf) of lognormal fading is given by fI (I, µI , σI ) = √  (ln(I) − µI )2 1 exp − 2σI2 2πσI I  ,  I > 0,  (2.2)  where µI and σI2 are the mean and variance of ln(I). The wireless channel, on average, is not supposed to either attenuate or amplify the intensity of the transmit signal and therefore the normalization E{I} = eµI +  2 σI 2  = 1 is necessary. As a result of this  normalization, the following relationship holds between the distribution parameters of the lognormal distribution, µI = −0.5σI2 .  (2.3)  2. FSO Fading Distributions and Detection Schemes  15  Therefore, based on the characteristics of the lognormal distribution and from (2.1), the scintillation index is given by S.I. = exp(σI2 ) − 1 [13]. Also, the distribution parameters of the lognormal distribution can be expressed in terms of the scintillation index as follows:  µI = −  ln(S.I. + 1) , 2  σI2 = ln(1 + S.I.).  (2.4)  In practice, it is more beneficial to derive the parameters of the lognormal distribution in terms of the physical characteristics of the channel. Assuming spherical wave propagation, the variance of the small-scale fading gain is given by [42] 7  11  σI2 = 0.496 (2π/λ) 6 Cn2 L 6 ,  (2.5)  where Cn2 , λ, and L are the refractive index structure constant, the wavelength, and the link distance, respectively. Fig. 2.1 shows the lognormal pdf for three different link distances in haze (Cn2 = 1.7 × 10−14 [43]). In Fig. 2.1, as the link distance increases, the scintillation index and equivalently the strength of the turbulence–induced fading also increases. As one expects, the corresponding curve further shifts to the left with the increasing link distance, which is equivalent to a higher distribution weight for smaller intensity gain factors.  2.1.2  Gamma–Gamma Distribution  For a wide range of turbulence conditions, the intensity gain factor I in FSO systems can be modeled by a Gamma–Gamma distribution [15]–[19], [21, 22]. The pdf of the  2. FSO Fading Distributions and Detection Schemes  16  1.4  L = 1 km (S.I. = 0.1462)  1.2  L = 2 km (S.I. = 0.6264) L = 3 km (S.I. = 1.7811)  fI(I,µI,σI) −−−>  1  0.8  0.6  0.4  0.2  0  0  0.5  1  1.5  2  2.5 I −−−>  3  3.5  4  4.5  5  Figure 2.1: Lognormal fading distribution for different link distances in haze (Cn2 = 1.7 × 10−14 , λ = 1550 nm [43]). Gamma–Gamma distribution is given by  fI (I, α, β) =  2(αβ)(α+β)/2 (α+β)/2−1 I Kα−β 2 αβI , Γ(α)Γ(β)  (2.6)  where Kν (x) is the modified Bessel function of the second kind and parameters α > 0 and β > 0 are linked to the scintillation index via S.I. = 1/α + 1/β + 1/(αβ). The parameters α and β can be adjusted to achieve a good agreement between fI (I, α, β) and measurement data [19, 20]. Alternatively, assuming spherical wave propagation, α and β can be directly linked to the physical parameters of the atmospheric channel  2. FSO Fading Distributions and Detection Schemes  17  16 α β  14  Value of α or β −−−>  12 10 8 6 4 2 0 1000  2000  3000  4000 5000 6000 7000 Link Distance (m) −−−>  8000  9000  10000  Figure 2.2: Values of α and β vs. link distance in Gamma–Gamma fading and haze (Cn2 = 1.7 × 10−14 , λ = 1550 nm, and D/L → 0 [22]). via [19, 22]  where χ2  α =  exp  0.49χ2 (1 + 0.18d2 + 0.56χ12/5 )7/6  β =  exp  0.51χ2 (1 + 0.69χ12/5 )−5/6 (1 + 0.9d2 + 0.62d2 χ12/5 )5/6  0.5Cn2 κ7/6 L11/6 , d  (κD2 /4L)1/2 , κ  −1 −1  −1  and  −1  ,  (2.7) (2.8)  2π/λ, and D is the diameter of  the receiver’s aperture. We note that (2.6) contains the K–distribution (α > 0 and β = 1) and the negative exponential distribution (α → ∞ and β = 1) as special cases. The K–distribution is typically used to model strong turbulence conditions  2. FSO Fading Distributions and Detection Schemes  18  1.4  L = 1 km (S.I. = 0.1391)  1.2  L = 2 km (S.I. = 0.4756) L = 3 km (S.I. = 0.8803)  fI(I,α,β) −−−>  1  0.8  0.6  0.4  0.2  0  0  0.5  1  1.5  2  2.5 I −−−>  3  3.5  4  4.5  5  Figure 2.3: Gamma–Gamma fading distribution for different link distances in haze (Cn2 = 1.7 × 10−14 , λ = 1550 nm [43]). [15, 20, 21, 25], while the negative exponential distribution can be seen as a limit distribution for extremely strong turbulence [19] and has been used in the literature mostly because of its mathematical tractability [12, 26]. Fig. 2.2 depicts the values of α and β for different source-destination distances. As the link distance increases from 1 km to 3 km, the values of α and β rapidly decrease. This is equivalent to a rapid increase in the value of the scintillation index of the corresponding channel. The pdf of the Gamma–Gamma distribution for 3 different link distances is shown in Fig. 2.3. As the link distance increases, similar to lognormal fading and as one expects, the scintillation index increases and the  2. FSO Fading Distributions and Detection Schemes  19  turbulence–induced fading becomes more severe. A comparison between lognormal fading pdfs in Fig. 2.1 and their Gamma–Gamma fading counterparts in Fig. 2.3 is interesting. While both models predict the same behavior for shorter link distances, it is evident that lognormal fading has a different estimation of the channel behavior compared to Gamma–Gamma fading for larger link distances. In the legends of Figs. 2.1 and 2.3, the corresponding scintillation indices are shown and it can be seen that the Gamma–Gamma distribution predicts smaller scintillation indices compared to lognormal fading for similar link distances. The difference is negligible for smaller link distances. However, as the link distance increases, the gap widens. Since, Gamma–Gamma fading has generally proved to have a better fit with experimental data, it is concluded that the lognormal distribution does not provide an accurate estimate of system performance for larger link distances. In summary, as was also mentioned before, the lognormal distribution is a better model for scenarios where weak turbulence or shorter distances are considered.  2.2  FSO Detection Schemes  In this section, assuming a simple SISO link, we derive comprehensive models for direct and coherent detection schemes. The resulting models are comprehensive and take the effect of all system parameters and relevant signal and noise terms into account. Besides, the provided models in this section are extendible to the more general cases of space–time coded and multi–hop FSO communication systems in the following chapters.  2. FSO Fading Distributions and Detection Schemes  2.2.1  20  Direct Detection  In a direct detection FSO receiver, the photodetector responds to the intensity of the incident optical field. The incident field is converted to a photocurrent that is later processed electronically [3]. A direct detection receiver is shown in Fig. 2.4. The received optical field is the sum of the signal field, Es,dd (t), and background radiation optical field, Eb,dd (t), and is given by [3, 38]  Er,dd (t) = Es,dd (t) + Eb,dd (t) Mo  =  2xt Pt IZ0 cos(ω0 t) +  2Nb δνZ0 cos(ωl t + φl ),  (2.9)  l=−Mo  where xt ∈ {0, 1}, Pt , I, Z0 , and ω0 are the data–carrying transmit pulse, the transmit power, the channel intensity gain factor, the free–space impedance, and the carrier frequency, respectively. Note that the transmit pulse xt can be either an OOK transmit symbol or one of the individual data–carrying transmit pulses of a PPM modulation scheme. Nb  Pb Bo  is the background radiation power spectral density, where Pb and  Bo are the background radiation power at the receiver and the optical bandwidth, respectively. The received background radiation power is directly proportional to the optical bandwidth of the receiver. Assuming that the power spectrum of the background radiation is flat over the optical bandwidth of the receiver [3], the background radiation electric field is represented as a sum of 2Mo + 1 cosine terms at frequencies ωl = ω0 + 2πδνl with random phases φl uniformly distributed over [−π, π), where Mo  Bo 2δν  and δν is the spacing of the considered frequencies [38]. The photodetector  2. FSO Fading Distributions and Detection Schemes  21  Eb,dd(t)  ir,dd(t)  Vr,dd(t)  A  PD Es,dd(t)  RL  Er,dd(t)  Figure 2.4: Block diagram of a direct detection receiver (PD: photodetector, A: transimpedance amplifier). at the destination node converts the received electric field to the photocurrent [3] R 2 E (t) Z0 r,dd  ir,dd (t) =  (2.10) Mo  Mo  2  cos(ωl1 t + φl1 ) cos(ωl2 t + φl2 )  = 2Rxt Pt I cos (ω0 t) + 2RNb δν l1 =−Mo l2 =−Mo Mo  +4R  cos(ω0 t) cos(ωl t + φl ).  xt Pt INb δν l=−Mo  Here, R  ηq hν0  is the responsivity of the photodetector, where q = 1.6 × 10−19 C is the  charge of an electron, h = 6.6 × 10−34 Js is Planck’s constant, η is the photodetector efficiency, and ν0  ω0 . 2π  Considering the fact that double-frequency terms (the terms  at frequency 2ω0 ) are filtered out due to the low pass nature of the photodetector, the resulting photocurrent is given by Mo  Mo  ir,dd (t) = Rxt Pt I + RNb δν l1 =−Mo l2 =−Mo  cos(2π(l1 − l2 )δνt + φl1 − φl2 ) (2.11)  Mo  +2R  xt Pt INb δν  cos(2πlδνt + φl ). l=−Mo  The different terms of the generated photocurrent in (2.11) will be discussed in detail in the following.  2. FSO Fading Distributions and Detection Schemes  22  Signal Current: The first term in (2.11) represents the useful signal term that is given by Is,dd = Rxt Pt I.  (2.12)  This term also contributes to the total generated DC current and in turn the total shot noise variance. Note that for any DC current IDC , the corresponding shot noise variance is given by 2qIDC Be , where Be is the electrical bandwidth. Background–Background Radiation DC Current and Beat noise: The second term in (2.11), which includes a double sum, is the result of the background radiation electric field mixing with itself. Those components of the double sum in (2.11) for which l1 = l2 , generate a DC current. The value of this DC current is given by  Ib×b = 2RNb δνMo = RPb .  (2.13)  The remaining components of the double sum in (2.11) for which l1 = l2 , are zeromean and result in the background-background beat noise, denoted here as iac,b×b , whose total power is given by  E{i2ac,b×b } = R2 Nb2 δν 2 E        2 2 2 = 2R Nb δν E   Mo  l1 =−Mo ,l1 =l2 Mo  l1 =−Mo ,l1 >l2  2  Mo  l2 =−Mo  cos(2π(l1 − l2 )δνt + φl1 − φl2 )  Mo  l2 =−Mo  cos(2π(l1 − l2 )δνt + φl1 − φl2 )       2    .  (2.14)  The square of the double sum in the above equation includes the sum of individual cosines squared and also the cross product of independent cosines. The expected value of the cross terms is zero (note that the phases of the cosines are random  2. FSO Fading Distributions and Detection Schemes  23  Noise PSD  2R2Nb2Bo 2R2Nb2(Bo − Be)  Be  Bo Frequency (Hz)  Figure 2.5: Power spectrum of the background-background beat noise (PSD: power spectral density). and independent), while the expected value of each squared cosine is 1/2 (double frequency terms are ignored). There are a total of 1 + 2 + ... + 2Mo = 2Mo (2Mo + 1)/2 cosine terms at frequencies 2Mo δν, (2Mo −1)δν, ..., δν, and therefore, the background– background beat noise has a triangular spectrum extending from 0 to Bo . The total noise power of the background–background beat noise simplifies to E{i2ac,b×b } = 4R2 Nb2 δν 2 ×  2Mo (2Mo + 1) 1 × = R2 Pb2 , 2 2  (2.15)  where 2Mo + 1 ≈ 2Mo has been used. The power density of the background– background beat noise near DC is given by Nb×b = 2R2 Nb2 Bo (cf. Fig. 2.5). Assuming Be < Bo , which is valid in practice, by integrating the power spectrum from 0 to Be , the beat noise variance results as 2 σb×b = R2 Nb2 (2Be Bo − Be2 ).  (2.16)  Signal–Background Radiation Beat Noise: Finally, the last term in (2.11) represents the signal–background radiation beat noise, referred to as is×b,dd in this thesis, and  2. FSO Fading Distributions and Detection Schemes  24  Noise PSD  4R2xtPtINb  Be  Bo/2  Frequency (Hz)  Figure 2.6: Power spectrum of the signal-background beat noise (PSD: power spectral density). does not include any DC terms. The total power of is×b,dd is given by  E{i2s×b,dd } = 4R2 xt Pt INb δνE      2  Mo  cos(2πlδνt + φl ) l=−Mo  = 4R2 xt Pt INb δν × 2M0 ×  1 2  = 2R2 xt Pt INb Bo .     (2.17)  Based on its frequency content, the signal–background noise has a rectangular spectrum extending from 0 to Bo /2 (cf. Fig. 2.6). Similar to what was shown above for the background–background beat noise, assuming Be < Bo /2, by integrating the power spectrum from 0 to Be , the signal–background beat noise variance results as 2 σs×b,dd = 4R2 xt Pt INb Be .  (2.18)  Summary: The total shot noise variance is given by 2qIDC,dd Be [3], where the total DC current generated at the receiver is given by  IDC,dd = Is,dd + Ib×b .  (2.19)  2. FSO Fading Distributions and Detection Schemes  25  2 Factoring in the receiver thermal noise variance σth , the total noise variance at the  receiver for a SISO link with direct detection at the destination is given by 2 2 2 2 σr,dd = σth + σb×b + σs×b + 2qBe (Rxt Pt I + RPb ),  (2.20)  where the last term represents the shot noise resulting from signal and background 2 radiation DC currents. The thermal noise variance is given by σth =  4kT Be , RL  where  k = 1.38 × 10−23 J/K is Boltzmann constant, T is the temperature in Kelvin, and RL is the load resistance of the photodetector. The load resistance that is used in a receiver is often chosen to be small to improve the frequency response of the receiver circuit. As a result, the thermal noise generated at the receiver is pretty large and the direct detection noise variance is often dominated by the thermal noise variance. Note that if the link distance is fairly large, the effect of signal–dependent noise terms is almost negligible. Adding the background radiation beat noise variance, 2 σb×b , and background radiation shot noise variance, 2qRPb Be , to the thermal noise  results in a very precise approximation of the total receiver noise variance under these circumstances. Since multiple noise terms add up and the thermal noise and background radiation power are fairly large, the Gaussian approximation is valid. Under these assumptions, the sufficient statistic for the received signal over a SISO link with direct detection at the destination node is given by  r=  √  γdd xt I + z,  (2.21)  where z and γdd are the real–valued AWGN with variance σz2 and the SNR at the direct detection receiver, respectively. Note that in this disseration, the term SNR represents the electrical SNR unless otherwise is specified.  2. FSO Fading Distributions and Detection Schemes  2.2.2  26  Coherent Detection  In a coherent detection FSO receiver, a local oscillator laser is employed and is spatially mixed with the incoming field [3]. A coherent detection receiver is shown in Fig. 2.7. As will be later seen, the incoming signal phase information may or may not be exploited for detection purposes and in this sense, coherent detection in FSO systems has a different implication compared to RF communications, where coherent detection is equivalent to the use of a phase synchronous local oscillator [3]. In this section, we assume a single receiver with a local oscillator with power PL and angular frequency ωL = ω0 + ωIF , where ωIF is the intermediate frequency. The photocurrent generated at the receiver in response to the received optical field is given by [3] R 2 E (t) Z0 r,cd R = (Es,cd (t) + Eb,cd (t) + EL (t))2 , Z0  ir,cd (t) =  where Es,cd (t) = Mo  Eb,cd (t) =  √  2Pt Z0 |xt H| cos(ω0 t + φh + φxt ), EL (t) =  √  (2.22)  2PL Z0 cos(ωL t), and  2Nb δνZ0 cos(ωl t+φl ) represent the optical fields of the transmitter, l=−Mo  the local oscillator, and the background radiation, respectively. Also xt  |xt |ejφxt  represents the complex transmit symbol taken from a finite complex alphabet S of an M -ary modulation scheme such as M -ary phase shift keying (MPSK), M -ary pulse amplitude modulation (MPAM), etc. Here, H  hejφh is the complex channel  fading factor, where h > 0 and φh are, respectively, the positive real fading gain and the channel phase, which is uniformly distributed in [−π, π). Note that the channel intensity gain factor denoted here as I relates to h via I = h2 . Consequently, the pdf  2. FSO Fading Distributions and Detection Schemes  27  Eb,cd(t)  ir,cd(t)  Vr,cd(t)  A  PD Es,cd(t)  RL  Er,cd(t)  EL(t)  LO  Figure 2.7: Block diagram of a coherent detection receiver (PD: photodetector, A: transimpedance amplifier, LO: local oscillator). of the fading gain h =  √  I is given by fh (h) = 2hfI (h2 ). Expanding (2.22) results in Mo  Mo  2  ir,cd (t) = RPt |xt H| + RPL + RNb δν +2R  l1 =−Mo l2 =−Mo  cos(2π(l1 − l2 )δνt + φl1 − φl2 )  Pt PL |xt H| cos(ωIF t − φh − φxt ) Mo  +2R  Pt Nb δν|xt H| l=−Mo  cos(2πlδνt + φl − φh − φxt )  Mo  +2R  PL Nb δν l=−Mo  cos((ωL − ωl )t − φl ).  (2.23)  The different terms in (2.23) will be discussed in detail in the following. Signal DC Current: The DC current generated by the signal electric field is given by IDC,s,cd = RPt |xt H|2 .  (2.24)  Note that the above term only contributes to the total shot noise variance. Local Oscillator DC Current: The local oscillator DC current, which also contributes  2. FSO Fading Distributions and Detection Schemes  28  to the total shot noise variance is given by  IDC,L = RPL .  (2.25)  Background–Background Beat Noise and DC Current: The third term in (2.23) represents the mixing of the background radiation electric field with itself. The same term is also generated in a direct detection receiver as can be seen from (2.11). Therefore, the assumptions and derivations in Section 2.2.1 for the background–background 2 radiation DC current, Ib×b , and beat noise variance, σb×b , are equally valid for the  case of coherent detection and therefore the following hold,  Ib×b = RPb  (2.26)  2 σb×b = R2 Nb2 (2Be Bo − Be2 ).  (2.27)  Useful Signal Term: The useful signal is (t) is given by  is (t) = 2R  Pt PL |xt H| cos(ωIF t − φh − φxt ).  (2.28)  Unlike for direct detection, the signal term in coherent detection contains information about the frequency and phase of the received signal. The intermediate frequency ωIF is assumed to be nonzero for “heterodyne” detection, while in case of ideal “homodyne” detection with perfect channel phase tracking, the intermediate frequency is assumed to be zero. The respective signal power for heterodyne detection is therefore given by Ps,het = 2R2 Pt PL |xt H|2 .  (2.29)  In case of homodyne detection, ωIF = 0 and perfect phase tracking is also assumed.  2. FSO Fading Distributions and Detection Schemes  29  √ As a result, is (t) = 2R Pt PL |xt H| holds and consequently the signal power is given by Ps,hom = 2Ps,het = 4R2 Pt PL |xt H|2 .  (2.30)  Signal–Background Radiation Beat Noise: In (2.23), the fifth term represents the mixing of signal and background radiation electric fields, which results in a signal– background radiation beat noise, referred to as is×b,cd (t) in this thesis. is×b,cd (t) is similar to the signal–background beat noise term in direct detection (last term in (2.11)). The signal–background beat noise has a rectangular spectrum extending from 0 to Bo /2 and the arguments in Section 2.2.1 leading to (2.17) are equally valid for the case of coherent detection. Comparing is×b,cd (t) with the last term in (2.11) and from the total noise power calculation in (2.17), the total coherent detection signal–background noise power is E{i2s×b,cd } = 4R2 Pt Nb δν|xt H|2 × 2M0 ×  1 2  = 2R2 Pt Nb |xt H|2 Bo .  (2.31)  Assuming Be < Bo /2, the signal–background beat noise variance is given by 2 σs×b,cd = 4R2 Pt Nb |xt H|2 Be .  (2.32)  Local Oscillator–Background Radiation Beat Noise: The last term in (2.23) denotes a noise term that is generated as a result of the mixing of the local oscillator and background radiation electric fields. Assuming that ωL falls within the optical bandwidth of the system, the resulting beat noise also has a rectangular spectrum extending from 0 to Bo /2. This noise term has the exact same form as the signal–background beat noise is×b,cd (t). Therefore, the local oscillator–background beat noise variance is  2. FSO Fading Distributions and Detection Schemes  30  given by 2 σL×b = 4R2 PL Nb Be .  (2.33)  Summary: The total DC current generated at the receiver is given by  IDC,cd = IDC,s,cd + IDC,L + Ib×b .  (2.34)  2 Factoring in the receiver thermal noise variance σth , the total noise variance at the  receiver for a SISO link with coherent detection at the destination is given by 2 2 2 2 2 + σL×b + 2qIDC,cd Be , + σs×b,cd σr,cd = σth + σb×b  (2.35)  where the last term represents the total shot noise. Since in a coherent detection receiver, the local oscillator power is large [44, 45], IDC,L = RPL dominates other DC current terms1 and therefore, the effective shot noise variance is given by 2 σshot,cd = 2qRPL Be .  (2.36)  2 Due to the dominance of the local oscillator power, the shot noise with variance σshot,cd 2 and local oscillator–background radiation beat noise with variance σL×b dominate all  other noise elements2 . Note that thermal noise is also added. The thermal noise variance for typical values of T = 300 K, RL = 50 Ω, and Be = 2 GHz [47, 48] results 2 as σth = 6.62 × 10−13 A2 . The sum of local oscillator shot noise and local-oscillator-  background radiation beat noise variances results as 8.6×10−12 A2 , assuming a typical 1  Typical values for the background radiation power Pb are in the order of 10−8 W, while the local oscillator power PL is typically in the order of 10−3 W [45, 46]. 2  As will be later seen in the following chapters, the local oscillator power is orders of magnitude larger than the laser transmit power.  2. FSO Fading Distributions and Detection Schemes  31  local oscillator power of 5 mW [45]. As a result, the noise terms generated by the local oscillator also dominate the thermal noise at the photodetector. The larger the local oscillator power, the less significant the effect of thermal noise on coherent detection. This is in contrast to direct detection receivers, where the effect of thermal noise is significant. In summary, the total noise variance for a coherent detection receiver is well approximated by 2 σr,cd = 2qRPL Be + 4R2 PL Nb Be .  (2.37)  Since the local oscillator power is large, the Gaussian approximation is valid for oscillator–relevant noise terms and therefore, the sufficient statistic for the received signal over a SISO link with coherent detection at the destination node is given by  r=  √  γcd xt H + z,  (2.38)  where z and γcd are the complex AWGN with variance σz2 and the SNR of the coherent detection receiver, respectively. The signal models discussed in this chapter for direct and coherent detection in a SISO link schemes set the stage for extensions to MIMO and multi–hop cases, as will be seen in the following chapters of this dissertation.  32  3  Performance Analysis of MIMO  FSO Systems in Gamma–Gamma Fading In this chapter, we analyze the performance of uncoded transmission over SISO and MIMO FSO channels suffering from Gamma–Gamma fading. As was also mentioned in Chapter 2, the Gamma–Gamma distribution is a comprehensive model that fits experimental data for a wide range of turbulence conditions and is therefore considered in this chapter for our performance analysis. We assume an IM/DD MIMO FSO system with repetition coding across lasers at the transmitter [11, 12], and EGC and MRC at the receiver. Note that more sophisticated STCs will be considered in Chapter 3, where the quasi–optimality of RCs will be established for direct detection receivers. The difficulties that arise in the analysis of Gamma–Gamma fading channels have their origin in the fact that the Gamma–Gamma pdf contains a modified Bessel function of the second kind. This Bessel function precludes simple closed–form expressions for the BER. Therefore, existing performance analysis methods for IM/DD FSO systems resort to numerical integration techniques. For example, the BER for uncoded SISO FSO transmission is obtained via numerical integration in [18, 20]. Analysis of coded and MIMO FSO transmission is even more difficult since the BER  3. Performance Analysis of MIMO FSO Systems in ...  33  has to be averaged over the effective (combined) Gamma–Gamma fading channel which typically results in two–dimensional numerical integration [22, 24]. However, numerical integration obscures the impact of the basic system and channel parameters on performance and may become numerically unstable at high SNRs [18]. The BER of a SISO FSO link impaired by K–fading (a special case of Gamma–Gamma fading) and pointing errors was expressed in terms of the Meijer’s G–function in [25]. More recently, this approach has also been extended to MIMO FSO links [17]. However, these closed–form results are limited to K–fading and are expressed in terms of the complicated Meijer’s G–function. The presented novel approach to performance analysis is based on a generalized infinite power series representation of the modified Bessel function of the second kind. With this representation, we can express the PEPs of SISO and MIMO IM/DD FSO systems as power series with respect to the SNR. The coefficients of these series expansions only include elementary and Gamma functions and are easy to compute. We prove that these series converge for any finite SNR and we provide an upper bound for the approximation error that is caused by truncating the infinite series to a finite number of terms, which is necessary for numerical evaluation. Interestingly, unlike numerical integration techniques, the accuracy of the truncated series PEP approximation improves with increasing SNR. Furthermore, we extend the concepts of diversity and combining gain, which are well known from the RF communication literature [49, 50], to FSO and Gamma–Gamma fading1 . Diversity and coding gain allow us to provide simple, insightful, and accurate closed–form approximations for the PEP of SISO and MIMO FSO at high SNR. The remainder of this chapter is organized as follows. In Section 3.1, the system 1  We note that the diversity gain of Gamma–Gamma fading was derived independently also in [51, 52].  3. Performance Analysis of MIMO FSO Systems in ...  Laser 1  PD 1  Laser 2  PD 2  34  Combiner STC  and  Encoder  STC  Input Data  Output Data  Decoder Laser M  PD N  Figure 3.1: Block diagram of a MIMO FSO system with M transmit lasers and N photodetectors. The footprint of each laser illuminates the whole receiver array (PD: photodetector). model and the generalized power series representation of the Gamma–Gamma pdf are introduced. The performance of IM/DD systems in SISO and MIMO FSO channels is analyzed in Sections 3.2 and 3.3, respectively. Performance results are presented in Section 3.4 followed by the concluding remarks in Section 3.5.  3.1  Preliminaries  In this section, we present the equivalent discrete–time model for the received signal in a MIMO FSO system with IM/DD and introduce a novel series representation for the Gamma–Gamma distribution. Fig. 3.1 represents a MIMO FSO system with a general space–time coding scheme at the transmitter and an arbitrary combining scheme at the receiver. Furthermore, we also discuss optimum detection with and without channel state information (CSI) and error rate calculation for various popular modulation schemes.  3.1.1  Signal Model  In Section 2.2.1, we derived the AWGN signal model for a SISO IM/DD FSO communication system. In this section, we also assume an FSO system whose performance  3. Performance Analysis of MIMO FSO Systems in ...  35  is limited by background radiation and thermal noise such that the Gaussian approximation is applicable. The model in (2.21) represents direct detection for an individual transmit pulse over a SISO link. Here, we extend the model in (2.21) to MIMO FSO with transmit symbols drawn from general modulation schemes. We consider a long–haul FSO system with M lasers, which are simultaneously intensity modulated with identical signals (repetition coding across lasers, cf. e.g. [11, 12]), and N photodetectors [20]. We assume that the M lasers simultaneously illuminate the N photodetectors, see e.g. [53, Fig. 6] for a corresponding block diagram. We also [s[1] . . . s[S]]T drawn from a general PPM scheme,  assume a transmit symbol s  where S ≥ 1 is the number of pulses per symbol and s[k] ∈ {0, 1}, 1 ≤ k ≤ S, represents the data–carrying transmit pulse. Under these conditions, the sufficient statistic in the kth pulse interval can be modeled as [8, 22] √  M  γ rn [k] = s[k] Imn + zn [k], M m=1  1 ≤ n ≤ N, 1 ≤ k ≤ S,  (3.1)  where Imn > 0 with E{Imn } = 1 is the real–valued fading gain (irradiance) between laser m and photodetector n, zn [k] is real–valued AWGN with variance σz2  E{zn2 [k]},  and γ denotes the SNR, cf. also (2.21). We note that due to the high data rates (hundreds to thousands of Mega bits per second (Mbps)) typical for FSO systems, the fading gains Imn can be assumed to be constant for thousands of symbol durations. Furthermore, we assume that the Imn ’s are independent and identically distributed (i.i.d.) random variables (RVs), which is justified for link distances of the order of kilometers and for aperture separation distances of the order of centimeters2 , cf. [9, 12]. 2  For example, experiments recently reported in [53] show that for a link distance of L = 1.5 km, a wavelength of λ = 1550 nm, and an aperture diameter of 1 mm photodetectors separated by as little as 35 mm are practically uncorrelated.  3. Performance Analysis of MIMO FSO Systems in ...  36  The symbol s belongs to an M –ary alphabet S. For example, for OOK, S = 1 and S = {0, 1} and for binary PPM, S = 2 and S = {[1 0]T , [0 1]T }. For multipulse PPM (MPPM) and overlapping PPM (OPPM) with S ≥ 2, respectively, all symbols s ∈ S contain w “ON” pulses (s[k] = 1) and S − w “OFF” pulses (s[k] = 0) [12, 54]. While for OPPM all “ON” pulses are confined to consecutive pulse intervals, this restriction does not exist for MPPM, cf. [54, Figs. 1, 2]. We note that the normalization σz2 = cS , S k=1  where cS  E{s[k]}2 /S, ensures that γ in (3.1) is indeed the SNR of the FSO  link. For convenience, we rewrite (3.1) in vector form as  rn = M m=1 Imn ,  where In  3.1.2  √  γ In s + z n , M  1 ≤ n ≤ N,  [rn [1] . . . rn [S]]T , and z n  rn  (3.2)  [zn [1] . . . zn [S]]T .  Series Representation of Gamma–Gamma Distribution  Performance analysis of MIMO FSO systems in Gamma–Gamma fading is difficult because of the modified Bessel function of the second kind in (2.6). To gain more insight and to avoid having to deal with the modified Bessel function, we base our analysis on the generalized power series representation [55], [56, Eqs. (8.445), (8.485)]  Kν (x) =  π 2 sin(πν)  ∞ j=0  1 x Γ(j − ν + 1)j! 2  2j−ν  −  ∞ j=0  x 1 Γ(j + ν + 1)j! 2  2j+ν  , (3.3)  which is valid for ν ∈ ZZ and |x| < ∞. We note that for the case of integer ν a simple power series representation of Kν (x) does not seem to exist. Combining (2.6) and (3.3) leads to fI (Imn ) =  ∞ j=0  j+β−1 j+α−1 aj (α, β)Imn + aj (β, α)Imn ,  (3.4)  3. Performance Analysis of MIMO FSO Systems in ...  37  (α − β) ∈ ZZ, where aj (α, β)  π(αβ)j+β . sin[π(α − β)]Γ(α)Γ(β)Γ(j − α + β + 1)j!  (3.5)  We point out that the condition (α−β) ∈ ZZ is not a severe restriction since (α−β) ∈ ZZ holds for typical parameters λ, D, Cn2 , and L, cf. (2.7), (2.8), and if necessary, we can choose α and β such that (α − β) + ǫ ∈ ZZ with some small constant ǫ to approximate the case (α − β) ∈ ZZ.  3.1.3  Receiver Structure and Error Rate  We assume that after appropriate scaling and combining of the signals received at the N receive apertures (cf. Sections 3.2, 3.3 for details), the combined signal can be represented as r= where r  √  γIs + z,  (3.6)  [r[1] . . . r[S]]T , and I and z denote the effective (combined) fading gain  and the effective AWGN vector, respectively. Both the pdf f (I) of I and the variance σ 2 of the elements of z depend on the adopted combining scheme. In the following, we discuss detection based on (3.6) with and without CSI. Detection with CSI: If the effective fading gain I is known to the receiver, the optimum detector computes the estimate  ˆ = argmin ||r − s  √  γIs||2 .  (3.7)  s∈S  Detection without CSI: For MPPM and OPPM, ||s|| = w holds for all s ∈ S. ˆ = argmax r T s . Therefore, in this case, (3.7) is equivalent to the decision rule s s∈S  3. Performance Analysis of MIMO FSO Systems in ...  38  As long as CSI is not required for calculation of r, which is the case if only one receive aperture is used or EGC is employed, the latter decision rule does not require CSI. We note that detection without CSI for modulation schemes with non–constant ||s|| is considerably more challenging and has not been studied in this thesis, cf. [46]. ˆ ∈ S is detected Bit Error Rate: Using (3.7) we can evaluate the probability that s ˆ, was transmitted, which is also referred to as the PEP. The PEP if s ∈ S, s = s ˆ|| between s and s ˆ and can be ||s − s  depends only on the Euclidean distance d expressed as  Pe (d, I) = Pr{||r − where Q(x)  √1 2π  √  γIs||2 > ||r −  ∞ −t2 /2 e dt x  √  γIˆ s||2 } = Q( γd2 I 2 /(4σ 2 )),  (3.8)  is the Gaussian Q-function. Consequently, the PEP  averaged over the fading gain is ∞  Pe (d) =  Q  γ d2 2 I 4σ 2  f (I) dI.  (3.9)  0  The PEP is directly linked to the BER Pb . For binary modulation schemes Pb = √ Pe (d) holds, where e.g. d = 1 and d = 2 for OOK and binary PPM, respectively. For general M –ary modulation schemes and medium–to–high SNR, the BER can be accurately approximated by [49]  Pb ≈ m1 m2 Pe (dmin ),  (3.10)  where dmin denotes the minimum distance of constellation S, m1 is the average number ˆ ∈ S that have distance dmin from s ∈ S, s = s ˆ, and m2 is the average of symbols s number of bit errors corresponding to minimum–distance error events.  3. Performance Analysis of MIMO FSO Systems in ...  39  Considering (3.10), in the next two sections, we will concentrate on the evaluation of the average PEP (3.9) in SISO and MIMO Gamma–Gamma fading channels.  3.2  Performance Analysis for SISO Channels  In this section, we derive PEP formulas for SISO channels (M = N = 1) and introduce the concepts of diversity and coding gain for Gamma–Gamma fading.  3.2.1  PEP in SISO Channels  For SISO transmission r = r 1 , I = I11 , and z = z 1 are valid in (3.6). Therefore, the PEP formula in (3.9) is applicable with σ 2 = σS2  cS and f (I) = fI (I) for  detection with CSI for arbitrary modulation schemes and for detection without CSI for MPPM and OPPM, respectively. Applying the series representation (3.4) of fI (I) in (3.9) and exploiting (A.2) in Appendix A, we obtain the average PEP in SISO Gamma–Gamma fading as  Pe (d) = =  ∞ j=0 ∞  γd2 /(4σS2 ), j + β  aj (α, β)X ξj (α, β)γ −  j+β 2  + ξj (β, α)γ −  + aj (β, α)X  γd2 /(4σS2 ), j + α  j+α 2  (3.11)  j=0  α − β ∈ ZZ, where X(·, ·) is defined in (A.2) and ξj (α, β)  √ π(2 2αβσS )j+β Γ j+β+1 2 2dj+β sin[π(α − β)]Γ(α)Γ(β)Γ(j − α + β + 1)(j + β)j! √  (3.12)  3. Performance Analysis of MIMO FSO Systems in ...  40  Since (3.11) is an infinite series, the question of convergence arises. For this purpose, we calculate the convergence radius R1 of the first sub–series in (3.11) as  R1 = lim  j→∞  ξj (α, β) ξj+1 (α, β)  d(j + 1)(j + β + 1)(j − α + β + 1)Γ √ = lim j→∞ 2 2αβσS (j + β)Γ j+β+2 2 → ∞.  j+β+1 2  (3.13)  Similarly, we obtain for the convergence radius R2 of the second sub–series in (3.11) R2 → ∞. Thus, (3.11) converges for all γ < ∞. In practice, some finite value J has to be used for the upper limit of the sum in (3.11) and we denote the resulting PEP approximation by Pe (d, J). As shown in Appendix B, the approximation error ε(J)  |Pe (d) − Pe (d, J)| is bounded by  √  ε(J) <  πsmax xJ+1 ex0 0 , 2| sin[π(α − β)]| Γ(α)Γ(β)(J + 1)!  (3.14)  smax  max  (3.15)  j>J  xα0 Γ j+α+1 xβ0 Γ j+β+1 2 2 − Γ(j − α + β + 1)(j + β) Γ(j − β + α + 1)(j + α)  √ √ with x0 = 2 2αβσS /(d γ). This bound illustrates that the approximation error can be made arbitrarily small by increasing J and/or γ. Thus, since the coefficients ξj (α, β) in (3.12) only involve a sin(·) and Γ(·) functions, the truncated version of (3.11) can be used for fast performance evaluation of SISO FSO systems in Gamma– Gamma fading.  3.2.2  Diversity and Coding Gain  In the RF communication literature, it is customary to characterize fading channels in terms of their diversity gain Gd and coding gain Gc , cf. e.g. [49, 50]. In particular,  3. Performance Analysis of MIMO FSO Systems in ...  41  for high SNR the PEP can be approximated as Pe (d) ≈ (Gc γ)−Gd , i.e., on a log–log scale Gc and Gd specify, respectively, a relative horizontal shift and the slope of the PEP curves in the asymptotic regime of γ → ∞. From (3.11), we obtain the diversity and coding gains of SISO Gamma–Gamma fading as  Gd = min{α/2, β/2}, Gc =  d √ 2 2αβσS  2  (3.16) √ 2 πΓ(max{α, β})Γ(2Gd + 1) Γ(|α − β|)Γ Gd + 21  1 Gd  ,  (3.17)  where we have used the identity π/ sin(πx) = Γ(x)Γ(1 − x) [57, Eq. (6.1.17)]. We note that typically α > β is considered in the literature since it provides the best fit with measurements for most scenarios. However, α < β provides a better match to measurements in some cases, cf. e.g. [19, Figs. 9–11, 13]. The results in this section are significant since they show that, despite the complicated nature of the Gamma– Gamma pdf, for high SNR the performance of FSO in Gamma–Gamma fading can be characterized by two parameters only. Interestingly, we observe from (3.16) that the diversity gain of SISO FSO only depends on the minimum of α and β. In particular, in contrast to SISO RF systems under Rayleigh fading [58], the diversity gain of SISO FSO systems under Gamma–Gamma fading is in general different from one. Furthermore, (3.17) shows that at high SNR for a given Gd performance improves with increasing max{α, β}.  3.2.3  Special Case  For negative exponential fading (α → ∞, β = 1), (3.11) can be simplified considerably. In particular, it is easy to verify that the second term in the sum in (3.11) is  3. Performance Analysis of MIMO FSO Systems in ...  42  zero in this case. The coefficients ξj (α, β) of the first term can be simplified to √ √ π(2 2ασS )j+1 Γ(j/2 + 1) 2(−1)j ( 2σS )j+1 lim ξj (α, 1) = lim = , α→∞ α→∞ 2dj+1 sin[π(α − 1)]Γ(α)Γ(j − α + 2)(j + 1)j! Γ(j/2)dj+1 (3.18) √  where we used π/ sin[π(α − 1)] = (−1)j+1 π/ sin[π(α − j − 2)] = (−1)j (α − j − 2)Γ(α − j − 2)Γ(j + 2 − α) [57, Eq. (6.1.17)], Γ(α) = (α − 1)(α − 2) · · · (α − j − 2)Γ(α − j − 2), and the Duplication Formula [57, Eq. (6.1.18)] for the Gamma function. Therefore, for negative exponential fading (3.11) simplifies to  Pe (d) = 2  ∞ j=0  3.3  (−1)j Γ(j/2)  √  2σS √ d γ  j+1  .  (3.19)  Performance Analysis for MIMO Channels  In this section, we derive closed–form PEP expressions for MIMO FSO with EGC and MRC, respectively, and provide simple, closed–form expressions for the related diversity and coding gains.  3.3.1  EGC Receiver  FSO EGC receivers estimate the transmitted signal from r = M (3.6) is valid with I  M m=1  N n=1 Imn  and z  M  2 PEP formula in (3.9) is applicable with σ 2 = σEGC  N n=1  N n=1  r n [12]3 , i.e.,  z n . Therefore, the average M 2 N cS for detection with  CSI for arbitrary modulation schemes and for detection without CSI for MPPM and OPPM, respectively. For application of (3.9), we have to determine the pdf f (I) = fEGC (I) of the 3  We note that the multiplication of the combined signal with M is not necessary in practice but facilitates our exposition here.  3. Performance Analysis of MIMO FSO Systems in ...  43  combined channel. For this purpose, we introduce the moment generating function E{e−sImn } of a single link, which can be calculated as4  (MGF) ΦI (s)  ∞  ΦI (s) =  bj (α, β)s−(j+β) + bj (β, α)s−(j+α) ,  (3.20)  j=0  cf. (3.4), where bj (α, β) I=  M m=1  N n=1 Imn  Γ(j + β)aj (α, β). Since the Imn are i.i.d. RVs, the MGF of  is given by ΦEGC (s) = (ΦI (s))M N and can be expressed as  MN  ∞  MN k  ΦEGC (s) = k=0 MN  MN k  = k=0  bj (α, β)s−(j+β)  M N −k  j=0 ∞ j=0  ∞  k  bj (β, α)s−(j+α)  j=0  cj (M N − k, k)s−j−(M N −k)β−kα .  (3.21)  The coefficients cj (M N −k, k) in (3.21) can be efficiently calculated as [56, Eq. (0.316)] (µ)  (ν)  cj (µ, ν) = bj (α, β) ∗ bj (β, α),  (3.22)  (x)  (2)  where the superscript yj means that yj is convolved x−1 times with itself, e.g. yj = (1)  yj ∗ yj , yj  (0)  = yj , and yj  = 1. From (3.21) we can express the fading pdf for EGC  as MN  fEGC (I) = k=0  MN k  ∞ j=0  cj (M N − k, k) I j+(M N −k)β+kα−1 . Γ(j + (M N − k)β + kα)  (3.23)  Applying f (I) = fEGC (I) in (3.9) yields MN  Pe (d) = k=0  4  MN k  ∞ j=0  ξjEGC (M N − k, k) γ −  j+(M N −k)β+kα 2  ,  (3.24)  We note that the MGF of Gamma–Gamma fading was derived independently in [59] in terms of the hypergeometric function.  3. Performance Analysis of MIMO FSO Systems in ...  44  α − β ∈ ZZ, where we have exploited (A.2), and the SNR–independent coefficients ξjEGC (M N − k, k) are given by √ (2 + 12 cj (µ, ν) 2σEGC )j+µβ+να Γ j+µβ+να 2 √ j+µβ+να ξjEGC (µ, ν) = . 2 πd Γ(j + µβ + να + 1)  (3.25)  To arrive at (3.24) we basically added and multiplied convergent series having an infinite convergence radius. Thus, the series in (3.24) also converges for all γ < ∞ [56]. In practice, the upper limit of the inner (infinite) series in (3.24) has to be truncated to some finite value J, and the resulting PEP approximation is again referred to as Pe (d, J). While it is difficult to obtain a simple upper bound for the approximation error in this case, we expect that the approximation error for EGC behaves qualitatively similar to the approximation error in the SISO case given in (3.14). We point out that even for large J the coefficients ξjEGC (µ, ν) required for evaluation of (3.24) can be easily computed. The main complexity for computation of ξjEGC (µ, ν) stems from the convolution in (3.22), which can be performed very efficiently with standard software such as MATLABTM . Similar to the SISO case, more light can be shed onto the performance of MIMO with EGC by considering the high SNR regime, where the PEP is fully characterized by the coding gain and the diversity gain. In particular, for high SNR the term corresponding to the largest exponent of γ in the series in (3.24) dominates the PEP. Using this observation along with the definition of combining and diversity gain and  3. Performance Analysis of MIMO FSO Systems in ...  45  (3.22) and (3.25), we obtain for these quantities GEGC = M N Gd , d = GEGC c  (3.26)  d √ 2 2αβσEGC  2  Γ(max{α, β}) Γ(|α − β|)  1 Gd  √ 2 πΓ (2M N Gd + 1) Γ M N Gd + 21  1 M N Gd  (3.27)  where Gd denotes the diversity gain in the SISO case as specified in (3.16). Eq. (3.26) shows that repetition coding across lasers at the transmitter and EGC at the receiver can exploit the full diversity gain offered by the FSO MIMO channel. A similar observation is made in [11, 12] for the Poisson model for photodetection which shows that the diversity gain depends on the fading but not on the noise model.  3.3.2  MRC Receiver N 5 n=1 In r n /I ,  An FSO MRC receiver forms the combined signal r = M N 2 n=1 In ,  applicable with I  In =  M m=1 Imn  and z  M  i.e., (3.6) is  N n=1 In z n /I.  Note that  for MRC CSI is required for calculation of r even in case of MPPM and OPPM. The 2 PEP of MRC can be obtained from (3.9) with σ 2 = σMRC  M 2 cS .  Before we can calculate the average PEP, the pdf f (I) = fMRC (I) of the MRC combined channel has to be determined. The pdf fIn (In ) of In is given by M  fIn (In ) = k=0  M k  ∞ j=0  cj (M − k, k) I j+(M −k)β+kα−1 , Γ(j + (M − k)β + kα)  (3.28)  where the coefficients cj (µ, ν) are defined in (3.22). The pdf of the RV Xn = In2 is 5  N  Note that in a practical implementation we may equivalently use the combined signal n=1 In r n . The multiplication by M and division by I does not affect the performance of the optimum detector but simplifies our exposition.  3. Performance Analysis of MIMO FSO Systems in ...  46  √ Xn ). Hence, we can express the MGF of Xn as  √1 f ( 2 Xn In  given by fX (Xn ) =  M  ∞  ΦX (s) =  pj (k)s−  j+(M −k)β+kα 2  ,  (3.29)  k=0 j=0  where j+(M −k)β+kα cj (M − k, k) 2 M Γ . k 2Γ(j + (M − k)β + kα)  pj (k) The MGF of Y = I 2 =  N n=1  (3.30)  Xn is given by ΦY (s) = (ΦX (s))N and can be conve-  niently expressed as  ΦY (s) = i0 +...+iM =N  ∞  N i0 , . . . , iM  qj (i0 , . . . , iM )s  −  j+  PM k=0 ik ((M −k)β+kα) 2  (3.31)  j=0  with (i )  (i )  (i  )  pj 0 (0) ∗ pj 1 (1) ∗ · · · ∗ pj M (M ).  qj (i0 , . . . , iM )  (3.32)  Using (3.31) the pdf fY (Y ) of Y can be calculated via the inverse Laplace transform. This pdf is related to the pdf fMRC (I) of I via fMRC (I) = 2IfY (I 2 ). Therefore, we obtain for fMRC (I) the expression  fMRC (I) = i0 +...+iM =N  ∞  N i0 , . . . , iM  j=0  2qj (i0 , . . . , iM ) Γ  j+  PM  k=0 ik ((M −k)β+kα)  I j+  PM  k=0 ik ((M −k)β+kα)−1  .  2  (3.33) Applying f (I) = fMRC (I) in (3.9) and exploiting (3.33) yields  Pe (d) = i0 +...+iM =N  N i0 , . . . , iM  ∞ j=0  ξjMRC (i0 , . . . , iM )γ −  j+  PM k=0 ik ((M −k)β+kα) 2  ,  α − β ∈ ZZ, (3.34)  3. Performance Analysis of MIMO FSO Systems in ...  47  where the coefficients ξjMRC (i0 , . . . , iM ) are given by ξjMRC (i0 , . . . , iM ) = P PM √ j+ M k=0 ik ((M −k)β+kα) (2 2σMRC )j+ k=0 ik ((M −k)β+kα) Γ + 12 qj (i0 , . . . , iM ) 2 .(3.35) P PM √ j+ M k=0 ik ((M −k)β+kα) 2 πdj+ k=0 ik ((M −k)β+kα) Γ + 1 2 Using similar arguments as for EGC, it can be shown that the infinite sum in (3.34) converges for all γ < ∞. In practice, we have to truncate the upper limit of the infinite sum in (3.34) to some finite value J yielding the approximate PEP Pe (d, J). The associated approximation error is expected to have a behavior similar to that in the SISO case. We note that even for large J (i.e., J > 20) evaluation of (3.34) is very fast since computation of the coefficients ξjMRC (i0 , . . . , iM ) requires only convolution operations, cf. (3.22), (3.32). While (3.34) lends itself to fast numerical evaluation, significant insight can be gained by considering the high SNR regime and specifying the diversity and coding gains. In particular, from (3.34) we obtain for these parameters the expressions GMRC = M N Gd , d  (3.36)  GMRC = c d √ 2 2αβσMRC  2  Γ(max{α, β}) Γ(|α − β|)  1 Gd  2Γ(2M Gd ) Γ (M Gd )  1 M Gd  1 √ M N Gd 2 πΓ (M N Gd + 1) (3.37) Γ M N Gd + 21  where Gd is the diversity gain for SISO transmission, cf. (3.16). We will use the diversity and coding gains to compare EGC and MRC in Gamma–Gamma fading in the next section.  3. Performance Analysis of MIMO FSO Systems in ...  3.3.3  48  Comparison of EGC and MRC  Assuming detection with CSI, cf. (3.7), MRC is more complex to implement than EGC. In particular, while for EGC only the gain of the combined channel I = M m=1  N n=1 Imn  has to be estimated, N channel gains In , 1 ≤ n ≤ N , have to be  estimated to perform MRC. Furthermore, for MPPM and OPPM EGC even enables detection without CSI, whereas CSI is always required for MRC. Therefore, from a complexity point of view, EGC is preferable. In terms of performance, both schemes achieve the same diversity gain, cf. (3.26), (3.36). The coding gain advantage of MRC compared to EGC is  G(M, N )  GMRC c =N GEGC c  2Γ(2M Gd ) Γ(M Gd )  1 M Gd  Γ(M N Gd + 1) Γ(2M N Gd + 1)  1 M N Gd  (3.38)  i.e., for high SNR on a log–log scale the SNR required to achieve a certain error rate performance is 10 log(G(M, N )) dB lower for MRC than for EGC. As expected, (3.38) yields G(M, 1) = 1 since EGC and MRC are equivalent for single–photodetector √ reception. In contrast, using Stirling’s formula Γ(x) → e−x xx 2πx for x → ∞ [57, Eq. (6.137)], we can show that for a large number of receive apertures the coding gain approaches  G(M, ∞)  e lim G(M, N ) = N →∞ 4M Gd  2Γ(2M Gd ) Γ(M Gd )  1 M Gd  .  (3.39)  For example, with M = 1 we obtain gains of 2.38 dB, 1.33 dB, and 0.71 dB for Gd = 0.5, 1, and 2, respectively. On the other hand, using again Stirling’s formula, it can be shown that lim  M Gd →∞  G(M, N ) = 1,  ∀N ≥ 1,  (3.40)  3. Performance Analysis of MIMO FSO Systems in ...  49  0  10  −1  10  −2  10  BER  α = 2.1 −3  10  Simulation Analysis, J = 0 Analysis, J = 5  −4  Analysis, J = 10  10  Analysis, J = 20 Asymptotic BER 0  5  10  15  α = 4.1 20  25  30  35  40  45  50  SNR [dB] Figure 3.2: BER of OOK vs. SNR γ. SISO (M = N = 1), Gamma–Gamma fading, β = 2. i.e., MRC does not give any performance gain if the Gamma–Gamma fading channel inherently has a large diversity order (large Gd ) or a large number of lasers is used at the transmitter (large M ). We note that in lognormal fading the gain of MRC over EGC is negligible even for small M as was shown in [60].  3.4  Performance Results  In this section, we verify the analytical results from Sections 3.2 and 3.3 by computer simulations and use them to study the performance of SISO and MIMO FSO systems  3. Performance Analysis of MIMO FSO Systems in ...  50  in Gamma–Gamma fading. In the following, BER curves labeled with “analysis” have been obtained based on truncated versions of (3.11), (3.24), and (3.34), whereas curves labeled with “asymptotic BER” are based on Pe (d) ≈ (Gc γ)−Gd and (3.16), (3.17), (3.26), (3.27), (3.36), and (3.37). In Fig. 3.2, we consider the BER of OOK over a SISO FSO link with β = 2. Analytical results Pb (J) = Pe (1, J), asymptotic results, and simulation results are compared. From Fig. 3.2 we observe that for a given J the theoretical BER Pb (J) converges faster to the true BER for higher SNR and for smaller α. For J = 10 and J = 20 the theoretical BER is practically identical to the simulated BER even for SNR = 0 dB for α = 2.1 and α = 4.1, respectively. As expected from (3.16), (3.17) both α values yield the same diversity gain but α = 4.1 has a larger coding gain. The asymptotic BER approaches the true BER faster for α = 4.1. For α = 2.1 the √ √ convergence of the asymptotic BER is quite slow since in this case γ −β and γ −α are similar for small to medium SNRs. Consequently, the first term in the series in (3.11), which is considered for the asymptotic BER, becomes dominant only at very high SNRs. In Fig. 3.3, the approximation error ε(J) = |Pe (1)−Pe (1, J)| is compared with the corresponding upper bound in (B.2). Both the error and the bound are normalized by Pe (1) which was obtained by simulation. The same link parameters as for Fig. 3.2 are valid. Fig. 3.3 shows that the upper bound is tight, and thus, is a useful tool for predicting the accuracy of the approximate PEP Pe (d, J). As expected, the approximation error rapidly decreases with increasing J, increasing SNR, and decreasing α. The impact of EGC and MRC on the accuracy of the proposed BER approximation is studied in Fig. 3.4 for OOK and α = 3.1 and β = 2. As one would expect, the number of terms J required in the truncated series to get an accurate BER estimate  3. Performance Analysis of MIMO FSO Systems in ...  51  0  10  α = 2.1 α = 4.1 Approx. Error, SNR = 10 dB Bound, SNR = 10 dB Approx. Error, SNR = 20 dB Bound, SNR = 20 dB  −2  10  −4  10  −6  ε(J) Pe (1)  10  −8  10  −10  10  −12  10  0  2  4  6  8  10  12  14  16  18  20  J Figure 3.3: Normalized approximation error ε(J)/Pe (1) = |Pe (1) − Pe (1, J)|/Pe (1) and corresponding upper bound (B.2) vs. truncation constant J. SISO (M = N = 1), OOK, Gamma–Gamma fading, β = 2. increases with the diversity gain. While in the considered SNR range J = 20 is sufficient to approach the simulated BER for the SISO case (M = N = 1), J = 40 is required for N = 3 photodetectors. EGC and MRC behave very similar as far as convergence is concerned. The asymptotic BER also converges faster for the SISO channel than for N = 3. However, the asymptotic BER accurately predicts the performance difference between EGC and MRC even for small SNR. In Table 3.1, we show the required J to achieve a relative approximation error |Pe (1) − Pe (1, J)|/Pe (1) of less than 10−9 for M = 1, N = 1 and EGC with M = 1,  3. Performance Analysis of MIMO FSO Systems in ...  52  0  10  α = 2.1 α = 4.1 Approx. Error, SNR = 10 dB Bound, SNR = 10 dB Approx. Error, SNR = 20 dB Bound, SNR = 20 dB  −2  10  −4  10  −6  BER  10  Simulation, M = 1, N = 1 Simulation, M = 1, N = 3, EGC  −8  10  Simulation, M = 1, N = 3, MRC Analysis, J = 5 Analysis, J = 20  −10  10  Analysis, J = 40 Asymptotic BER −12  10  0  2 5  10 4  15 6  20 8  10 25  12 30  14 35  16 40  18 45  20 50  SNR [dB] Figure 3.4: BER of OOK vs. SNR γ. SISO (M = N = 1) and MIMO (M = 1, N = 3) with EGC and MRC, Gamma–Gamma fading, α = 3.1, β = 2. N = 3 for different fading parameters α and β and different SNRs. Pe (1) is calculated with J = 200 and confirmed by simulations. As can be observed, the required J rapidly decreases with increasing SNR. However, even for SNR = 10 dB, the required J is comparatively small and the evaluation of the analytical expressions for the PEP is very fast. In Figs. 3.5 and 3.6, we investigate the asymptotic performance gain G(M, N ) (3.38) of MRC over EGC. In Fig. 3.5, we show G(M, N ) as a function of the number of receive apertures N for different Gd M . As expected from the discussion in Section 3.3.3, the performance advantage of MRC increases with increasing N but decreases  3. Performance Analysis of MIMO FSO Systems in ...  53  Table 3.1: Required J to achieve a relative truncation error ε(J)/Pe (1) = |Pe (1) − Pe (1, J)|/Pe (1) of less than 10−9 for M = 1, N = 1 and EGC with M = 1, N = 3, respectively.  PP P  PP SNR PP α, β PP 2.1, 2 2.5, 2 3.1, 2 4.1, 2 7.1, 2 12.1, 2 4.1, 4 4.5, 4 5.1, 4 6.1, 4 9.1, 4 14.1, 4  M = 1, N = 1  M = 1, N = 3  10  30  50  10  30  50  11 11 13 15 20 27 19 20 22 25 31 40  6 6 7 8 9 8 9 9 10 11 13 16  3 4 4 5 5 4 5 5 6 6 7 5  22 24 28 32 45 65 44 48 54 61 80 106  9 10 12 14 14 12 16 17 20 23 28 25  5 5 7 7 5 5 7 8 10 11 8 6  with increasing M Gd . To get a better understanding of the performance gains that can be expected in practice, we show in Fig. 3.6 G(M, N ) as a function of the distance L between transmitter and receiver assuming spherical wave propagation. α and β were calculated from (2.7), (2.8) and following [22] we adopted λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0. Fig. 3.6 shows that the performance advantage of MRC grows with increasing link distance. For example, for M = 1 and N = 8 MRC yields performance gains of 0.9 dB and 1.7 dB for L = 3 km and L = 7 km, respectively. Nevertheless, in a practical system, EGC may be preferable over MRC since the relatively small performance gains may not justify the additional implementation complexity. This is especially true for PPM formats where CSI is not required for detection if EGC is applied. For the analytical BER results shown in the remaining figures, we choose J suffi-  3. Performance Analysis of MIMO FSO Systems in ...  54  0  102.5 α = 2.1 α = 4.1 Approx. Error, SNR = 10 dB Bound, SNR = 10 dB Approx. Error, SNR = 20 dB Bound, SNR = 20 dB  −2  10 2  −4  G(M, N ) [dB]  101.5  −6  10 1  −8  100.5  −10  10 0  M Gd ∈ {0.5, 1, 2, 3, 4, 5, 8, 40}  −12  10 −0.5  0 1  22  43  6 4  8 5  10  6 12  7 14  816  918  20 10  N Figure 3.5: Performance gain G(M, N ) of MRC over EGC vs. N . ciently large to guarantee that the approximation error is negligible. Fig. 3.7 shows the BER of OOK as a function of the link distance L. The link SNR is assumed to be equal to 20 dB independent of L in order to separate the effect of attenuation from the distance dependence of α and β, cf. (2.7), (2.8). Fig. 3.7 shows that even in the absence of attenuation the BER of SISO and MIMO FSO systems degrades with increasing distance since α and β decrease. Transmit and receive diversity are effective means to improve performance but the gains achievable with MRC are again relatively small. For all considered cases the BER obtained from the presented analysis accurately predicts the simulated performance. In Fig. 3.8, we study the influence of β on the BER of a 2–PPM FSO system  3. Performance Analysis of MIMO FSO Systems in ...  55  2.5 1.8 M=1  1.6  M=2  2 1.4  M=4 M=8  1.2 1.5  N=1 N=2  G(M, N ) [dB]  1  N=4 N=8  0.8 1 0.6 0.5 0.4 0.2 0 0 −0.5 −0.2  1 3  23.5  34  4 4.5  5  5  6  5.5 7  68  6.59  10 7  L [km] Figure 3.6: Performance gain G(M, N ) of MRC over EGC vs. link distance L. α and β from (4.30) and (4.36), respectively. λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. with N = 1 and N = 2 (EGC) photodetectors. Since α = 4.1 > β holds for all three values of β considered in Fig. 3.8, the diversity gain is determined only by N and β, cf. (3.16), (3.26), and (3.36). This is the reason, why the BER curve for N = 1 and β = 2 has the same asymptotic slope as the BER curve for N = 2 and β = 1. In Fig. 3.9, the BERs of MPPM with, respectively, S = 8, w = 1 and S = 8, w = 4 are shown assuming a link distance of L = 5 km and EGC at the receiver. α and β have been calculated from (2.7) and (2.8), respectively. For w = 1 and w = 4, 3 and 6 bits per symbol s are transmitted. For w = 4 we have selected 64 out of  3. Performance Analysis of MIMO FSO Systems in ...  56  −1  10 1.8 M=1  1.6 −2 10  M=2 M=4  1.4  M=8  −3  10 1.2  N=1 N=2  1 −4  BER  10  N=4 N=8  0.8 −5  10 0.6  Simulation, M = 1, N = 1 Simulation, EGC  −6 0.4  Simulation, MRC  10  Analysis, M = 1, N = 1 0.2  Analysis, M = 2, N = 1  −7  10  Analysis, M = 2, N = 2  0  Analysis, M = 2, N = 3 −8  −0.2 10  3  3.2 3.5  3.4 4 3.6  4.5 3.8  5 4  4.2 5.5  4.4 6 4.6  6.5 4.8  7 5  L [km] Figure 3.7: BER of OOK vs. link distance L. α and β from (4.30) and (4.36), respectively. SNR = 20 dB, λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. a possible 70 MPPM symbols. As expected, the higher transmission rate achievable with w = 4 entails a degradation in performance since the minimum distance of the symbol alphabet S is decreased. Since M N = 4 is valid for all transmitter/receiver configurations considered in Fig. 3.9 all curves have the same asymptotic slope. Note that for the considered non–binary modulation formats the simulation results agree with the analytical results obtained from (3.10) for high SNR where the minimum distance error events dominate the BER. For low SNR, the analytical results deviate from the simulation results because of the approximation involved in (3.10). Finally in Fig. 3.10, we compare the BERs of 2–PPM and OPPM with S = 5  3. Performance Analysis of MIMO FSO Systems in ...  57  0  10  −2  10  −4  BER  10  −6  10  Simulation, α = 4.1, β = 1 Simulation, α = 4.1, β = 2 Simulation, α = 4.1, β = 3  −8  10  M = 1, N = 1, Analysis M = 1, N = 2, EGC, Analysis Asymptotic BER  −10  10  0  5  10  15  20  25  30  35  40  45  50  SNR [dB] Figure 3.8: BER of 2–PPM vs. SNR γ. and w = 2 for a link distance of L = 5 km and different system configurations. As expected from Fig. 3.6, for the considered link distance MRC yields a performance gain of approximately 1.2 dB compared to EGC for N = 4. OPPM outperforms 2– PPM at the expense of a decrease in data rate. In particular, in the considered OPPM and PPM formats 2/5 = 0.4 and 1/2 = 0.5 bits per pulse interval are transmitted.  3.5  Conclusions  In this chapter, we presented a novel approach to performance analysis of SISO and MIMO FSO systems in Gamma–Gamma fading. The proposed technique is based on  3. Performance Analysis of MIMO FSO Systems in ...  58  0  10  −2  10  −2  10  −4  10  S = 8, w = 4  −4  BER  10−6 10  −8  10−6 10  Simulation, α = 4.1, β = 1 Simulation, N β= =4 2 Simulation, M α= = 1, 4.1, Simulation, N β= =2 3 Simulation, M α= = 2, 4.1,  −10  10  −8  Simulation, M Analysis = 4, N = 1 M = 1, N = 1, S Approx. M = 1, NAnalysis = 2, EGC, Analysis  10  −12  10  = 8, w = 1  Asymptotic Asymptotic BER BER −10  10−14  0  5  10  15  20  25  30  35  40  45  50  SNR [dB] Figure 3.9: BER of two different MPPM formats vs. SNR γ. EGC, spherical wave propagation, L = 5 km, λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. a generalized power series representation of the modified Bessel function of the second kind and enables fast and accurate performance evaluation. This is especially true for high SNR where the first few terms of the series suffice to get accurate results. This is in contrast to existing techniques which are mostly based on numerical integration and may suffer from stability problems at high SNR. Furthermore, the proposed approach has allowed us to extend the concepts of coding gain and diversity gain to FSO and Gamma–Gamma fading. Closed–form expressions for both gains have been provided for SISO FSO and MIMO FSO with EGC and MRC at the receiver. These expressions provide valuable insight into the impact of various system and channel parameters on  3. Performance Analysis of MIMO FSO Systems in ...  59  0  10  N =1 −2  10−2 10 −4  10 −4 10 −6  10  BER  −6  10  N =4  −8  10  Simulation, M = 1, N = 1 Simulation, M = 1, N = 4 Simulation, M = 1, N = 4, EGC Simulation, M = 2, N = 2 Simulation, M = 1, N = 4, MRC Simulation, M = 4, N = 1 Analysis, 2PPM Approx. Analysis Approx. Analysis, OPPM Asymptotic BER Asymptotic BER  −8  10 −10 10 −10  10−12 10  −14  −12 10  0  5  10  15  20  25  30  35  40  45  50  SNR [dB] Figure 3.10: BER of 2–PPM and OPPM (S = 5, w = 2) vs. SNR γ. Spherical wave propagation, L = 5 km, λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 [22]. performance. In particular, they show that in Gamma–Gamma fading MRC achieves only small to moderate performance gains compared to EGC. Therefore, EGC is more attractive in practice because of its considerably lower implementation complexity. The simple parameterization of Gamma–Gamma fading via combining and diversity gain will be used in the following chapters for the design of STCs for FSO systems.  60  4  Performance and Design of  Space–Time Coded FSO Systems In Chapter 3, we studied the performance of IM/DD MIMO FSO systems subject to Gamma–Gamma fading. Our analysis in Chapter 3 was based on the use of simple RCs at the transmitter. While repetition coding is a simple form of STCs in MIMO FSO systems, more complicated STC designs should be also considered. In the FSO literature, performance bounds and STC design guidelines are not available for neither IM/DD nor coherent detection FSO systems. As an example, it is not clear from the current body of literature on IM/DD FSO, whether STCs that outperform RCs exist. In FSO communications, IM/DD is a simple and cheap approach to low–complexity FSO. Also in practice, IM/DD is a more attractive solution in FSO communication systems. As a result, the main focus in this dissertation is devoted to IM/DD FSO systems. However, it should be noted that a higher performance can potentially be achieved with FSO systems employing coherent or differential modulation at the expense of a higher receiver complexity. In this chapter we therefore consider space–time code design for MIMO FSO systems with direct and coherent/differential detection schemes at the receiver, respectively. In both cases, following the SISO signal models presented in Chapter 2, MIMO signal models are derived for general space-time coded FSO systems. The derived models also provide a framework enabling a fair  4. Performance and Design of Space–Time Coded FSO Systems  61  comparison of space–time coded IM/DD FSO systems and their coherent/differential counterparts. Assuming two lasers at the transmitter and an arbitrary number of photodetectors at the receiver we then derive asymptotically tight approximations for PEP of general FSO STCs in spatially independent Gamma–Gamma fading for direct and coherent/differential detection, respectively. Our analyses can be used to obtain asymptotic upper bounds on the BER of both STBCs and space–time trellis codes (STTCs). Exploiting the obtained PEP expressions, we derive design guidelines for FSO STCs for two lasers for both IM/DD and coherent/differential detection receivers. The remainder of this chapter is organized as follows. In Section 4.1, the general system models are introduced for FSO systems with direct and coherent/differential detection. The expressions for the asymptotic PEP and code design rules for direct and coherent/differential detection schemes are presented in Sections 4.2 and 4.3, respectively. Corresponding performance results are presented in Section 4.4 and some conclusions are drawn in Section 4.5.  4.1  System Model  We consider an FSO system with M lasers at the transmitter and N apertures at the receiver. The M lasers transmit an space–time codeword C S×M = [cmn ]S×M in S consecutive symbol intervals. We assume that the M lasers simultaneously illuminate the N photodetectors and the photocurrent generated by each photodetector is integrated over one pulse interval to derive the decision variable. Note that each element of the space–time code matrix C corresponds to an intensity modulated transmit symbol for direct detection and a complex transmit symbol in case of coherent/differential detection.  4. Performance and Design of Space–Time Coded FSO Systems  4.1.1  62  MIMO FSO with Direct Detection  Following the derivations in Section 2.2.1, during the k-th symbol interval, the photocurrent generated at the n-th receiver in response to the square of the aggregate optical field is given by [3]  ik,n (t) =    R  Z0  M  2Pt Z0  Mo  ckm Imn cos(ω0 t) + m=1  l=−Mo  2  2Nb δνZo cos(ωl t + φl ) ,  (4.1)  where Imn is the intensity gain of the channel between the m-th laser and the nth photodetector and follows a Gamma–Gamma distribution. For convenience we M  define ck I n  ckm Imn , where ck is the k-th row of the space–time matrix C and m=1  In  [I1n ... IM n ]T represents the n-th column of the M × N channel intensity gain  matrix I. Note that the expression for the photocurrent in (4.1) is the same as that of the SISO case in (2.11) if xt I is replaced with ck I n . As a result the following DC currents and noise terms are generated at the destination node for the space–time coded IM/DD FSO system. DC Currents: The DC currents generated include the signal current Is,dd and the background-background DC current Ib×b , given by the following  Is,dd = RPt ck I n  (4.2)  Ib×b = RPb .  (4.3)  2 Noise Variances: The noise variance includes the total shot noise variance σshot,dd , the 2 , the signal-background beat noise background-background beat noise variance σb×b  4. Performance and Design of Space–Time Coded FSO Systems  63  2 2 variance σs×b,dd , and the thermal noise variance σth , which are given by  2 σshot,dd = 2q(Is,dd + Ib×b )Be 2 σb×b = R2 Nb2 (2Be Bo − Be2 ) 2 σs×b,dd = 4R2 Pt ck I n Nb Be 2 σth =  4kT Be . RL  (4.4) (4.5) (4.6) (4.7)  Summary: As was discussed in detail in Chapter 2, the sum of background radiation beat noise, background radiation shot noise, and thermal noise will dominate all the other noise terms for typical system parameters [23] and an AWGN model is valid. Therefore, the decision variable for the k-th symbol interval and n-th photodetector is obtained via rn [k] =  √  γdd ck I n + zn [k], M  (4.8)  where zn [k] represents the n-th receiver noise element during the k-th symbol interval. Here, zn [k] is a zero-mean unit variance real Gaussian random variable1 . The effective direct detection SNR2 γdd at the n-th direct detection receiver during the k-th symbol interval is therefore given by  γdd =  M 2 R2 Pt2 . 2 R2 Nb2 (2Be Bo − Be2 ) + 2qRPb Be + σth  (4.9)  The signal samples, rn [k], received at photodetector n in S consecutive pulse intervals 1  Note that while the noise variance is not generally equal to unity, here, for the ease of notation, we normalize the noise variance to unity and define the SNR accordingly. 2  In this chapter, the direct detection SNR is quantified in terms of relevant system parameters so that a fair comparison to coherent/differential detection will be possible. The analysis in Chapter 3 was for direct detection only and therefore no such parametrization was necessary. Also, similar to Chapter 2, the subscripts “dd” and “cd” are used in this chapter to distinguish between the direct and coherent detection schemes, respectively.  4. Performance and Design of Space–Time Coded FSO Systems  64  [rn [1], rn [2] . . . , rn [S]]T ,  (corresponding to one STC codeword) are collected in vector r n which can be modeled as  rn =  √  γdd C I n + Z n, M  1 ≤ n ≤ N,  (4.10)  where Z n denotes the noise vector at the n-th photodetector. We adopt EGC to combine the received signals of the N photodetectors [8] since for FSO systems, the performance loss of EGC compared to optimum combining is negligible (cf. Chapter 3). The received signal after EGC can be expressed as N  R=  rn = n=1  √  γdd C I + Z, M  (4.11)  N  where R is the S × N received signal matrix. Also, I  In  [I1 ... IM ]T and  n=1  N  Z n denote, respectively, the vector of effective channel gains and the effective  Z n=1  noise vector, whose elements are zero–mean, real–valued Gaussian random variables with variance σn2 = N . For future reference, we note that for Im  N n=1 Imn  →0  we can express the pdf f (Im ) of the equivalent channel gains Im , m ∈ {1, 2}, as (cf. (3.23)) N µ−1 N µ−1 ), f (Im ) = aI Im + o(Im  where aI  (π(µν)µ /[sin(π(ν − µ))Γ(ν)Γ(µ − ν + 1)])N /Γ(N µ), µ  (4.12) min{α, β}, ν  max{α, β}, and (α − β) = ZZ. Note that the condition (α − β) = ZZ is valid in practice (cf. (2.7) and (2.8)). Eq. (4.12) will be exploited for derivation of the asymptotic PEP in the next section. Finally, we note that the optimum detector will choose that C ∈ C which minimizes the metric ||R − possible FSO STC matrices C.  √  γdd C M  I||2 , where the code C contains all  4. Performance and Design of Space–Time Coded FSO Systems  4.1.2  65  MIMO FSO with Coherent and Differential Detection  An array of N coherent detection receivers is assumed. The array is equipped with a local oscillator with power PL and angular frequency ωL = ω0 + ωIF . During the k-th symbol interval, 1 ≤ k ≤ S, the photocurrent generated at the n-th receiver in response to the square of the aggregate optical field is given by [3]  ik,n (t) =  R (Ek,n (t) + EL (t) + Eb (t))2 . Z0  M  Ek,n (t) m=1 Mo  2Pt Z0 |ckm Hmn | cos(ω0 t+φckm +φhmn ), EL (t)  (4.13)  √  2PL Z0 cos(ωL t), and  2Nb δνZ0 cos(ωl t + φl ) represent the received optical field during the  Eb (t) = l=−Mo  k-th symbol interval at the n-th receiver, the optical field of the local oscillator, and the effective background radiation optical field, respectively. Also Hmn  hmn ejφmn ,  the element in row m and column n of the channel gain matrix H, represents the complex channel gain from the mth laser to the nth photodetector, which is assumed to be constant. Here hmn > 0 and φmn are, respectively, the positive real fading gain and the channel phase, which is uniformly distributed in [−π, π). We also assume that the irradiance follows a Gamma–Gamma distribution. Combining the Gamma– Gamma fading series representation (cf. Chapter 3) and (2.6), for hmn → 0, the pdf of hmn can be expressed as 2µ−1 fhmn (hmn ) = ah h2µ−1 mn + o(hmn ),  where ah  2π(µν)µ /[Γ(µ)Γ(ν) sin(π(ν − µ))Γ(µ − ν + 1)], µ  (4.14)  min{α, β}, and ν  max{α, β}. M  For convenience, we define ck hn  ckm hmn ejφmn , where ck is the  jφkn  |ck hn |e  m=1  4. Performance and Design of Space–Time Coded FSO Systems  66  [H1n ... HM n ]T represents the n-th  k-th row of the space–time matrix C and hn  column of the M × N channel gain matrix H. Consequently the received optical field Ek,n (t) is given by M  Ek,n (t) = m=1  2Pt Z0 ℜ ckm hmn ejφmn ejω0 t M  ckm hmn ejφmn }  =  2Pt Z0 ℜ{ejω0 t  =  jω0 t  2Pt Z0 ℜ{e  =  2Pt Z0 |ck hn | cos(ω0 t + φkn )  m=1  ck hn } (4.15)  Following the derivations in Section 2.2.2, the expression for the coherent detection photocurrent ik,n (t) will be the same as that of the SISO case in (2.23) if xt H is replaced by ck hn . The generated DC currents and noise terms are discussed in the following. DC Currents: The signal–generated DC current IDC,s,cd , the local oscillator DC current IDC,L , and the background radiation DC current Ib×b are given by  Ib×b = RPb IDC,s,cd = RPt |ck hn |2 IDC,L = RPL .  (4.16) (4.17) (4.18)  Noise Variances: The total noise variance includes the total shot noise variance 2 2 σshot,cd , the background–background beat noise variance σb×b , the signal–background 2 radiation beat noise variance σs×b,cd , the local oscillator–background noise variance  4. Performance and Design of Space–Time Coded FSO Systems  67  2 2 σL×b , and the thermal noise variance σth , which are given by  2 σshot,cd = 2q(Ib×b + IDC,s,cd + IDC,L )Be 2 σb×b = R2 Nb2 (2Be Bo − Be2 ) 2 σs×b,cd = 4R2 Pt |ck hn |2 Nb Be 2 σL×b = 4R2 PL Nb Be 2 σth =  4kT Be . RL  (4.19) (4.20) (4.21) (4.22) (4.23)  Useful Signal: Following the arguments in Chapter 2, the heterodyne detection signal power is given by Ps,het = 2R2 Pt PL |ck hn |2 .  (4.24)  Consequently, Ps,hom = 4R2 Pt PL |ck hn |2 is valid for homodyne detection. Summary: As was also discussed in Chapter 2, due to the dominance of the local oscillator power PL , the effective shot noise variance results as 2 σshot,cd = 2qRPL Be .  (4.25)  2 As was mentioned in Section 2.2.2, the shot noise variance σshot,cd and local oscillator– 2 background radiation beat noise variance σL×b dominate all other noise elements.  Consequently, upon detection, the decision variable for the k-th symbol interval at the n-th receiver, rn [k] is obtained via  rn [k] =  γcd ck hn + zn [k], M  (4.26)  where zn [k] represents the n-th receiver noise element during the k-th symbol interval. Here, zn [k] is a zero-mean unit variance complex Gaussian random variable. Also,  4. Performance and Design of Space–Time Coded FSO Systems  68  γcd represents either the heterodyne or homodyne detection SNR depending on the detection methodology used. The n-th heterodyne detection receiver SNR during the k-th symbol interval γhet is given by  γhet =  M Ps,het M RPt = . 2 + σL×b (q + 2RNb )Be  2 σshot,cd  (4.27)  The homodyne detection SNR is also given by γhom = 2γhet . Extending the received signal model to the matrix form where all symbol intervals and receive apertures are considered, the following model is obtained for the space–time coded coherent detection FSO system. R=  γcd CH + Z, M  (4.28)  where R is the S × N received signal matrix and Z represents the S × N noise matrix with elements that are identically and independently distributed. For differential reception, we add a subscript τ to R, C, and Z, which denotes the τ th matrix– symbol interval and consider STBCs with S = M . The transmitted STBC matrix is obtained recursively as C τ = V C τ −1 , where the information carrying matrix V is taken from a set of S × S unitary matrices [61].  4.2  Asymptotic Performance Analysis for General STCs  In this section we assume the practically important case of M = 2 lasers at the transmitter and an arbitrary number of receive apertures and provide simple closed– form expressions for the asymptotic PEP of general STCs in both direct and coherent/differential detection FSO systems.  4. Performance and Design of Space–Time Coded FSO Systems  4.2.1  69  Direct Detection STCs  ˆ Let C and C  ˆ2 ] = C denote the transmitted and the detected FSO STC ma[ˆ c1 c  trices, respectively. Assuming the optimum decision rule is applied, the corresponding PEP can be expressed as ∞ ∞  Pe =  Q(x) =  1 π  (4.29)  0  0  where E  γdd ||EI||2 fI (I1 )fI (I2 ) dI1 dI2 , NM2  Q  ˆ Using the alternative representation of the Gaussian Q–function, C − C.  π/2 0  2  x exp − 2 sin 2θ  dθ [58, Eq. (4.2)], (4.29) can be rewritten as  π/2 ∞ π/2  exp −  Pe = 0  ×f  0  0  ρ sin ϕ √ γdd d22  ρ2 (1 + ξ sin(2ϕ)) 2M 2 N sin2 Θ  f  ρ √ dϕdρdΘ, πγdd d11 d22  ρ cos ϕ √ γdd d11 (4.30)  ˆm ||2 , m ∈ {1, 2}, and d12 (c1 − c ˆ1 )T (c2 − c ˆ2 ) are the elements ||cm − c √ of 2 × 2 matrix D E T E, and ξ d12 / d11 d22 . Note that d11 (d22 ) is the Hamming  where dmm  ˆ1 (ˆ distances between c1 (c2 ) and c c2 ) and −1 ≤ ξ ≤ 1 holds. To arrive at (4.30), we have introduced the new variables 0 ≤ ρ < ∞ and 0 ≤ ϕ ≤ π/2 with ρ cos ϕ = √ √ γdd d11 I1 and ρ sin ϕ = γdd d22 I2 . Unfortunately, a closed–form expression for (4.30) does not seem to exist. However, for sufficiently high SNR, errors only occur if Im → 0, m ∈ {1, 2}, [50], and we may apply (4.12) in (4.30). With this simplification, we  4. Performance and Design of Space–Time Coded FSO Systems  70  obtain for high SNRs, γdd ≫ 1, and ξ > −1 the asymptotic PEP . a2 Γ(N µ + 1/2)(M 2 N )N µ D(N µ) −N µ Pe = γdd √ √ Nµ 2 πN µ d11 d22 π/2 √ (sin(ϕ))x−1 πΓ(x) dϕ = x D(x) x (1 + ξ sin(ϕ)) 2 Γ(x + 1/2)  (4.31)  0  1 1−ξ x, x; x + ; 2 2  ×F where we have used the identities  ∞ 0  ,  (4.32)  z u−1 exp(−sz 2 ) dz = s−u/2 Γ(u/2)/2, u > 0, s >  π/2 0  sinu−1 z dz = Γ(1/2)Γ(u/2)/[2Γ((u+1)/2)] [56, Eq. (3.6211)], √ Γ(z + 1) = zΓ(z), and Γ(1/2) = π. To find the solution for the integral in (4.32)  0 [56, Eq. (3.4782)],  we used the substitutions z = 1/ sin ϕ and s = (z − 1)/2 and [56, Eq. (3.197.9)]. For special cases, (4.32) can be further simplified. For example, for D(1) and D(2) we obtain  D(1) = D(2) =  2 1− 1 1 − ξ2  ξ2  1−ξ 1+ξ  arctan 1−  2ξ 1 − ξ2  arctan  (4.33) 1−ξ . 1+ξ  (4.34)  Exploiting the asymptotic PEP in (4.31) and the union bound, an asymptotic upper bound on the BER of general FSO STCs can be obtained. A simple asymptotic approximation is obtained by considering only the dominant term of the asymptotic upper bound. The corresponding asymptotic BER approximation can be expressed as . m1 m2 Pe,max , Pb = log2 Mc  (4.35)  where Pe,max is the maximum asymptotic PEP (4.31) for any pair of FSO STC matrices in C, m1 is the average number of error events causing the maximum asymptotic  4. Performance and Design of Space–Time Coded FSO Systems  71  PEP, m2 denotes the average number of bit errors caused by these error events, and Mc is the number of bits per STC matrix.  4.2.2  Coherent STCs  Similar to the direct detection case, the coherent STC PEP can be expressed in terms of the elements of D =  d11 d12 d∗12 d22  E H E with error matrix E  ˆ C − C.  Comparing the equivalent coherent FSO signal model in (4.28) and the asymptotic fading pdf in (4.14) with [62, Eq. (1)] and [62, Eq. (2)], respectively, we conclude that the results from [62] are directly applicable to the problem at hand. Thus, exploiting [62, Eq. (10)], we obtain for the asymptotic PEP of a space–time coded FSO system with coherent homodyne or heterodyne detection the closed–form expression −2µN . (πa2 Γ2 (2µ)F (µ, µ; 1; ξ 2 ))NR Γ(2µNR + 0.5)γcd R √ Pe = , 2 π(d11 d22 )µNR Γ(2µNR + 1)Γ2NR (µ + 0.5)  where ξ  (4.36)  √ |d12 |/ d11 d22 and 0 ≤ ξ < 1. As usual, the asymptotic PEP in (4.36) can  be combined with truncated union bounds to obtain asymptotic approximations for the frame, symbol, and bit error probabilities of coherent FSO systems in Gamma– Gamma fading.  4.2.3  Differential STCs  The differential ML receiver is based on the metric ||Rτ − V Rτ −1 ||2 [61]. Thus, assuming V 1 has been transmitted, the PEP is the probability of deciding in favor of V 2 and is given by Pe = Pr ||Rτ − V 1 Rτ −1 ||2 > ||Rτ − V 2 Rτ −1 ||2 .  (4.37)  4. Performance and Design of Space–Time Coded FSO Systems We use the fact that for any matrix A, ||A||2 = tr AH A  72  holds. We have also  H assumed C τ = V 1 C τ −1 and unitary matrices i.e V H 1 V 1 = V 2 V 2 = I S , where I S is  the S × S identity matrix. After some straightforward manipulations, the PEP can be rewritten as Pe = Pr{zeff > γcd ||E D H||2 }, where E D is defined as E D  zeff  (4.38)  (V 1 − V 2 )C τ −1 and the equivalent noise is given by  √ 2ℜ{tr{ γcd (E D H)H (Z τ − V 2 Z τ −1 )}} + 2ℜ{tr{Z H τ (V 1 − V 2 )Z τ −1 }}. (4.39)  Although zeff is non–Gaussian, for high SNRs, it can be approximated by its Gaussian √ component zeff ≈ 2ℜ{tr{ γcd (E D H)H (Z τ − V 2 Z τ −1 )}}. The variance of zeff is therefore well approximated by σz2eff ≈ 2γcd ||E D H||2 .  (4.40)  Thus, at high SNR, the differential system can be approximated by an equivalent coherent system with error matrix E D and SNR γD  γcd /2. The PEP of a space–time  coded FSO system with differential homodyne or heterodyne detection is therefore given by −2µN . (πa2 Γ2 (2µ)F (µ, µ; 1; ξ ′2 ))NR Γ(2µNR + 0.5)γD R √ ′ ′ µN Pe = , 2 π(d11 d22 ) R Γ(2µNR + 1)Γ2NR (µ + 0.5)  where ξ ′  4.3  |d′12 |/ d′11 d′22 , 0 ≤ ξ ′ < 1, and D ′ =  d′11 d′12 ′ d′∗ 12 d22  (4.41)  EH D ED.  FSO Space–Time Code Design  In this section, we provide design criteria for FSO STCs and compare these criteria with that of RF STCs.  4. Performance and Design of Space–Time Coded FSO Systems  4.3.1  73  Direct Detection STC Design  In this section, we show that simple RCs are quasi–optimal FSO STCs for direct detection FSO systems. We also briefly discuss the case M > 2. 1) FSO STC Design Criterion: Minimizing the worst–case PEP Pe,max is a popular design criterion for RF STCs [30]. We adopt the same criterion here for FSO STCs. . Furthermore, it is convenient to express the asymptotic PEP as Pe = (Gc γdd )−Gd [30, 50], where Gc and Gd denote the coding gain and the diversity gain, respectively. From (4.31) we observe that the minimization of Pe,max is equivalent to the maximization of ˆ Jdd,F SO (C, C)  (d11 d22 )N µ/2 , D(N µ)  (4.42)  ˆ > which is also equivalent to the maximization of the coding gain Gc . If Jdd,F SO (C, C) 0 holds for any pair of FSO STC matrices, the STC achieves the maximum diversity ˆ > 0 is gain of Gd = N µ. A necessary and sufficient condition for Jdd,F SO (C, C) d11 > 0, d22 > 0, and ξ > −1, since the integral in (4.32) diverges for ξ = −1 corresponding to a loss in diversity. Note that D(N µ) is minimized for ξ = 1. These design guidelines are not equivalent to the guidelines proposed in [29] which are sufficient but not necessary. For example, unlike the design guidelines in [29], our design rules show that the matrices C T =  1 1 0 0 1 1  ˆT = and C  1 0 1 1 1 0  from [29,  Eq. (10)] do achieve full diversity since d11 = d22 = 2 > 0, and ξ = −1/2 > −1. Note that our design guidelines can also be used to show that the generalized PPM codes proposed in [29] cannot achieve full diversity. In particular, the additional STC matrices of the generalized PPM code have distance d11 = 0 or d22 = 0 to at least one codeword of the original PPM code. 2) Quasi–Optimality of RCs: We can maximize the minimum d11 for a given rate by selecting for c1 that scalar code C1 which has maximum minimum Hamming  4. Performance and Design of Space–Time Coded FSO Systems  74  distance dmin . Thereby, the scalar code may be a convolutional code, a block code, or some other type of code. Once C1 is fixed, we can achieve the optimum ξ = 1 with an RC, i.e., by setting c2 = c1 , ∀C ∈ C. Obviously, for RCs d22 = d11 holds, i.e., the minimum distance error events for c1 and c2 occur concurrently, which may ˆ2 is a not be optimum. Instead, we could try to design new codes where c2 → c ˆ1 is a minimum distance non–minimum distance error event with d22 > dmin if c1 → c error event with d11 = dmin . However, these new codes would invariably result in a suboptimal ξ < 1. Using (4.42) we have performed an extensive search for ST block and trellis codes. However, we could not find an example for a code that outperformed the best RC (with maximum minimum distance d11 = dmin ) of the same rate. Since we cannot prove the strict optimality of RCs but both our extensive code search and the above theoretical observations suggest that RCs are optimal, we refer to RCs as “quasi–optimal” FSO STCs. 3) Comparison with RF STCs: RF STCs for two transmit antennas are optimized for maximization of the minimum [30] ˆ JRF (C, C)  d11 d22 (1 − ξ 2 ).  (4.43)  ˆ > 0 for any pair of RF STC Full diversity is achieved if and only if JRF (C, C) matrices, which is equivalent to d11 > 0, d22 > 0, and |ξ| < 1. Thus, full diversity RF STCs also achieve full diversity in FSO systems. Conversely, full diversity FSO STCs do not necessarily achieve full diversity in RF systems. In particular, RCs (for which ξ = 1 is valid) do not achieve full diversity in RF systems. Furthermore, STCs that perform well in RF systems are generally suboptimum in FSO systems. For example, the OSTBC for M = 2 (Alamouti’s code) C =  s1 s2 s¯2 s1  | s1 ∈ {0, 1}, s2 ∈ {0, 1} [26,  ˆ = 28], which performs well in RF systems, yields ξ = 0 and minimum Jdd,F SO (C, C)  4. Performance and Design of Space–Time Coded FSO Systems  75  √ 2Γ((N µ + 1)/2)/[ πΓ(N µ/2)]. In contrast, the corresponding RC of equal rate C = ˆ = 2N µ | s1 ∈ {0, 1}, s2 ∈ {0, 1} achieves ξ = 1 and minimum Jdd,F SO (C, C) √ Γ(N µ + 1/2)/[ πΓ(N µ)].3 This translates into a coding gain advantage of s1 s1 s2 s2  A(N µ) [dB]  10 log Gc,RC [dB] − Gc,OSTBC [dB] = Nµ  √  πΓ(N µ + 1/2) (Γ(N µ/2 + 1/2))2  , (4.44)  which is always positive. Thus, our analysis confirms the results from [28] in that OSTBCs are indeed not needed in FSO systems. However, in contrast to [28], we are also able to quantify the asymptotic performance difference between the OSTBCs and the RCs in Gamma–Gamma fading, and our results are not limited to OSTBCs but are applicable to any STC with M = 2. For example, we obtain coding gain advantages of A(1) = 10 log10 (π/2) = 2.0 dB, A(2) = 5 log10 (3) = 2.4 dB, and A(∞) = 3.0 dB, where the latter result was obtained with Stirling’s approximation of the Gamma function. 4) M > 2: It is desirable to extend the results of this dissertation to M > 2. However, in this case, the average PEP in (4.29) involves an M –dimensional integral, which does not seem to be analytically tractable even in the asymptotic regime of γdd ≫ 1. Therefore, the derivation of simple design criteria similar to (4.42) for M > 2 is difficult and remains an open topic for future research. Nevertheless, our own simulations and prior work [11, 28] show that RCs achieve an excellent performance also for M > 2 and better FSO STC designs are not known for this case either. ˆ For calculation of Jdd,F SO √ (C, C) for for RCs and OSTBCs we exploited F (x, x; x + 1/2; 0) = 1 and F (x, x; x + 1/2; 1/2) = πΓ(x + 1/2)/[Γ((x + 1)/2)]2 in (4.32), respectively [63]. 3  4. Performance and Design of Space–Time Coded FSO Systems  4.3.2  76  Coherent and Differential Detection STC Design  We concentrate on the coherent case but all results in this section are also applicable to differential STCs by replacing the elements of D = E H E with the elements of D′ = E H D E D as outlined in the previous section. Expressing the asymptotic PEP as . Pe = (Gc γcd )−Gd , where Gc and Gd are the coding gain and diversity gain, respectively, we observe from (4.36) that in Gamma–Gamma fading the maximum diversity gain of a coherent space–time FSO system having two transmit apertures is given by Gd = 2N min{α, β}. Similar to RF communication over Rayleigh fading channels, this diversity gain is achieved if 0 ≤ ξ < 1 for all possible error matrices E, i.e., if all possible E have full rank. Note that this result is in stark contrast to FSO systems with IM/DD, where it was shown in the previous section that non–full rank matrices may achieve full diversity. Considering the class of full rank STCs, the diversity gain of the worst–case error event can be maximized by maximizing  Jcd,F SO  (d11 d22 )µ = , F (µ, µ; 1; ξ 2 )  (4.45)  which is positive if d11 > 0, d22 > 0, and 0 ≤ ξ < 1. For given d11 and d22 , (4.45) is a monotonically decreasing function in 0 ≤ ξ < 1 which is maximized for ξ = 0. This behavior of Jcd,F SO is very similar to that of JRF = d11 d22 (1 − ξ 2 ), which is typically used for optimization of STCs in RF systems. Thus, while (4.45) can be used for the search of optimal STCs for Gamma–Gamma fading, we expect STCs optimized for Rayleigh faded RF systems to also perform well in coherent FSO systems. This allows us to draw from the large existing body of literature of coherent (and differential) STCs for RF communication [30, 61, 64]. In particular, we expect Alamouti’s STBC to perform well in coherent FSO systems as it achieves ξ = 0. This is in contrast to  4. Performance and Design of Space–Time Coded FSO Systems  77  0  10  RC (simulation) OSTBC (simulation) NFDC I (simulation) NFDC II (simulation) RC (asymptotic analysis) OSTBC (asymptotic analysis)  −1  10  −2  10  −3  BER  10  N =1  −4  10  N =2  −5  10  −6  10  −7  10  0  5  10  15  20  25  30  35  40  45  50  SNR [dB] Figure 4.1: BER vs. SNR γdd for M = 2 lasers and Gamma–Gamma fading parameters α = 2.5 and β = 1. FSO systems with IM/DD where repetition codes with ξ = 1 outperform Alamouti’s STBC.  4.4  Performance Results  In this section, we verify some of the results in this chapter by simulation and numerical evaluation. The results are presented in two separate sections, where direct detection and coherent/differential detection schemes are studied, respectively.  4. Performance and Design of Space–Time Coded FSO Systems  4.4.1  78  Direct Detection Results  For Fig. 4.1, we adopted M = 2 and Gamma–Gamma fading with α = 2.5 and β = 1, which are typical values for channels with strong turbulence, cf. [19, Figs. 3 and 4]. We show in Fig. 4.1 BER simulation results for an RC (S = 2), an OSTBC (S = 2), non–full diversity code I (NFDC I) with C =  s1 s2 s2 s1  | s1 ∈ {0, 1}, s2 ∈ {0, 1} , and  NFDC II from [29, Eq. (10)]. Since the NFDC in [29, Eq. (10)] has six codewords, to simplify the bit–to–codeword mapping, we selected only the first four codewords for the results shown in Fig. 4.14 . The two NFDCs do not achieve full diversity since there are codeword pairs with ξ = −1 and d11 = 0 or d22 = 0 for NFDC I and NFDC II, respectively. Thus, as expected from Section 4.3.1 1), for N = 1 the diversity gain of the RC and OSTBC is Gd = β = 1, while that of NFDC I and NFDC II is only Gd = β/2 = 1/2. Furthermore, as expected from Section 4.3.1 3), the coding gain advantage of the RC over the OSTBC is A(1) = 2.0 dB and A(2) = 2.4 dB for N = 1 and N = 2, respectively. The bold dashed and dash–dotted lines in Fig. 4.1 represent asymptotic analytical results for the RC and the OSTBC, which have been obtained by evaluating (4.35). For both the RC and the OSTBC, the accuracy of the analytical results at high SNR is nicely confirmed by the simulation results. In Fig. 4.2, we show the coding gain advantage A(N µ) of the RC over the OSTBC vs. the link distance L for M = 2 and S = 2. Here, we assume spherical wave propagation, i.e., α and β are given by (2.7) and (2.8), respectively. Furthermore, we adopt λ = 1550 nm, Cn2 = 1.7 · 10−14 , and D/L → 0 from [22]. We observe from Fig. 4.2 that the gain achievable with RCs over OSTBCs increases with the number of photodetectors N but decreases with link distance L since β = µ decreases with increasing L. As predicted in Section 4.3.1 3), for large arguments N µ (corresponding 4  We note that any set of four codewords of the STC in [29, Eq. (10)] includes at least one pair of codewords with d11 = 0 or d22 = 0.  4. Performance and Design of Space–Time Coded FSO Systems  79  3 N N N N  2.9  = = = =  1 2 3 4  2.8  2.7  A(N µ) [dB]  2.6  2.5  2.4  2.3  2.2  2.1  2 1000  1500  2000  2500  3000  3500  4000  4500  5000  L [m] Figure 4.2: Coding gain advantage A(N µ) of RC over OSTBC vs. link distance L. Spherical wave propagation, wavelength λ = 1550 nm, index of refraction structure parameter Cn2 = 1.7 · 10−14 , and D/L → 0 (D: receiver aperture, L: link distance), M = 2 lasers, and S = 2 pulse intervals. to small L), A(N µ) approaches 3 dB.  4.4.2  Coherent/Differential Detection Results  In this section, we illustrate the validity of the obtained analytical results for coherent detection in Sections 4.2.2 and 4.2.3. Moreover, with the fair framework that has been established in this chapter for direct and coherent/differential detection schemes, we compare the performance of coherent and IM/DD FSO systems employing STCs. Throughout this section we assume heterodyne detection for coherent/differential FSO systems. Relevant system parameters are shown in Table 4.1.  4. Performance and Design of Space–Time Coded FSO Systems  80  In Fig. 4.3, we show the BERs of FSO systems employing coherent detection, differential detection, and IM/DD vs. Pt Tb for M = 1, 2, and 4, N = 1, and Gamma– Gamma fading parameters α = 2.5 and β = 1, which are typical values for strong turbulence channels [19]. Repetition code with OOK is employed for IM/DD with any number of transmit apertures. We adopt Alamouti’s STBC with BPSK for coherent and differential detection and M = 2. While our analysis is limited to the case of M = 2 transmit lasers, we show simulation results for 4 transmit lasers in Fig. 4.3, where the 4 × 4 orthogonal STBC from [65] is used in the coherent detection FSO system. As can be observed from Fig. 4.3, for high enough SNR, the analytical results (dashed lines) obtained from (4.36) and the union bound, are in excellent agreement with the simulation results (solid lines with markers). All considered detection schemes achieve the same diversity gain5 . Nevertheless, Alamouti’s code and the 4 × 4 orthogonal STBC from [65] with coherent detection achieve a considerable performance gain compared to IM/DD FSO with repetition coding. A huge gain is also achieved if differential detection is employed. The analytical results in Section 4.3.1 suggest that repetition coding is quasi–optimal for two transmit apertures and the code search for larger number of transmit lasers suggests the quasi–optimality for those cases as well. Also, our analysis in Section 4.3.2 suggests that the same RF design criteria are also applicable to FSO systems with coherent/differential detection. The results in Fig. 4.3 prove the superior performance of coherent/differential detection compared to IM/DD, also for larger numbers of transmit lasers. Fig. 4.4 depicts the bit error probability of IM/DD and coherent/differential FSO 5  In Section 4.3.1, it was shown that the diversity gain for IM/DD FSO systems is M N min{α, β}/2, whereas Fig. 4.3 suggests that the diversity gain is M N min{α, β} for IM/DD. This discrepancy is due to the fact that here, we plot BER as a function of Pt Tb rather than the SNR. The coherent detection SNR γcd is proportional to Pt , whereas the IM/DD SNR γdd is proportional to Pt2 .  4. Performance and Design of Space–Time Coded FSO Systems  81  Table 4.1: System Parameters [38, 47, 48]. Optical wavelength (λ) Electrical bandwidth (Be ) Optical bandwidth (Bo ) Data rate (Rb ) Receiver noise temperature (T ) Receiver quantum efficiency (η) Photodetector load resistance (RL ) Background radiation energy (Pb Tb )  1550 nm 2 GHz 125 GHz 2 Gbps 300 K 0.75 50 Ω -170 dBJ  0  10  −1  10  −2  Average BER −−−>  10  −3  10  −4  10  M=1 −5  10  M=2 M=4  −6  10  IM/DD (OOK) Coherent (BPSK)  −7  10  Differential (BPSK) Asymptotic Analysis  −8  10 −180  −175  −170  −165 −160 −155 PtTb [dBJ] −−−>  −150  −145  −140  Figure 4.3: BER of FSO systems with coherent heterodyne, differential heterodyne, and direct detection schemes in Gamma-Gamma fading (α = 2.5, β = 1). The 4 × 4 space-time block code from [65] is employed for coherent/differential detection and M = 4. Repetition coding is used for direct detection.  4. Performance and Design of Space–Time Coded FSO Systems  82  0  10  −1  10  −2  Avaerage BER −−−>  10  −3  10  −4  10  IM/DD (8−PPM), M = 1  −5  10  IM/DD (8−PPM), M = 2 Coherent (8−PSK), M = 1  −6  10  Coherent (8−PSK), M = 2 Differential (8−PSK), M = 2  −7  10  Asymptotic Analysis  −8  10 −180  −175  −170  −165 −160 −155 PtTb [dBJ] −−−>  −150  −145  −140  Figure 4.4: BER of FSO systems with coherent heterodyne, differential heterodyne, and direct detection schemes in Gamma-Gamma fading (α = 2.5, β = 1). The more bandwidth efficient 8-PPM and 8-PSK modulation schemes are applied to direct and coherent/differential detection FSO systems, respectively. systems in Gamma–Gamma fading with the same fading parameters as in Fig. 4.3. 8-PPM and 8-PSK have been employed for IM/DD and coherent/differential detection schemes, respectively. Repetition coding is used for direct detection, while Alamouti’s STBC is applied to coherent/differential detection systems. As for direct detection, increasing the modulation order results in a better performance due to the inherent power efficiency of MPPM schemes. For the coherent and differential detection schemes, on the other hand, a higher order modulation scheme, while more bandwidth efficient, results in performance degradation compared to BPSK. The  4. Performance and Design of Space–Time Coded FSO Systems  83  0  10  −2  10  Average BER −−−>  N=1  −4  10  N=2  −6  10  Simulation Asymptotic Analysis TSC, 4 km BBH, 4 km  −8  10  TSC, 8 km BBH, 8 km  −180  −175  −170 −165 PtTb [dBJ] −−−>  −160  −155  Figure 4.5: BER of TSC [30] and BBH [64] STTCs in Gamma-Gamma fading and haze (Cn2 = 1.7 × 10−14 ) for a heterodyne detection receiver. asymptotic analysis presented in Section 4.2.2 predicts the performance of 8-PSK coherent/differential detection FSO systems well. In Fig. 4.5, we compare the BER of the rate–2 4–state STTCs from [30] (TSC) and [64] (BBH) for a Gamma–Gamma faded coherent FSO system. Simulation results (solid lines with markers) and an upper bound (dashed lines with markers) obtained by combining (4.36) with a truncated union bound which includes all error events up to length 3 are shown for transmitter–receiver distances of L = 4 km and L = 8 km. The receiver distances were used along with the refraction structure parameter Cn2 = 1.7 × 10−14 , wavelength λ = 1550 nm, and D/L → 0 to obtain the fading  4. Performance and Design of Space–Time Coded FSO Systems  84  parameters α and β from (2.7) and (2.8). While both codes have a very similar performance for N = 1, the BBH code outperforms the TSC code for N = 2, which is consistent with the relative performance of these codes in Rayleigh faded RF systems [64]. The asymptotic BERs converge to the simulated BER faster for 8 km than for 4 km, since the inherent diversity order of the channel decreases with increasing distance (i.e., the fading becomes more severe).  4.5  Conclusions  In this chapter, we have presented comprehensive models for space–time coded FSO communication systems with coherent, differential, and direct detection. We have also provided closed–form expressions for the asymptotic PEPs of space–time coded FSO systems with coherent and differential detection. The general Gamma–Gamma fading model and FSO systems with two transmit and an arbitrary number of receive apertures were considered. For IM/DD systems, while RCs have been proposed before, we have shown that these simple FSO STCs are quasi–optimal and finding better codes for M = 2 may not be possible. Furthermore, we have shown that full–diversity RF STCs also achieve full diversity in IM/DD FSO systems but suffer from a loss in coding gain compared to RCs. The asymptotic loss of OSTBCs compared to RCs has been quantified analytically. The derived STC design guidelines revealed that there are fundamental differences between STC design for FSO systems with IM/DD detection and coherent/differential detection. Furthermore, it was shown that STCs optimized for Rayleigh faded RF systems (e.g. Alamouti’s code and STTCs) also perform well in FSO systems with coherent and differential detection.  85  5  Multi–Hop FSO  Communication Systems Performance analysis of MIMO FSO systems and STC design were discussed in Chapters 3 and 4. While spatial diversity is one of the effective measures that is taken against the detrimental effects of turbulence–induced fading in FSO systems, the distance–dependence of attenuation and fading motivates the application of relays as another effective measure for performance enhancement. The current body of literature on FSO relaying mainly focuses on electrical amplification. The use of EDFA–based all–optical relays does not require the optical to electrical and electrical to optical conversions that are necessary in relays with electrical amplification. In this chapter, we therefore advocate the use of all–optical relays. Based on the general approach introduced in Chapter 2, we develop accurate models for the signals and noises at the destination receiver for IM/DD FSO systems employing fixed and variable gain optical and electrical amplification, respectively. The proposed models include the effects of thermal noise, background radiation, ASE, various beat noises, and signal–dependent noise. For dual–hop and multi–hop all–optical relaying, we present a comprehensive outage probability analysis, which includes the effects of lognormal fading and all relevant noises. As was also discussed in Chapter 4, IM/DD FSO systems are simpler and therefore cheaper to implement compared to coherent  5. Multi–Hop FSO Communication Systems  86  detection systems. IM/DD systems are thus more attractive from an implementation point of view and are more popular in practice. We therefore focus on direct detection in this chapter, while our results can be also extended to coherent detection. Our results show that all–optical relays outperform electrical relays because of the thermal noise that impairs the latter unless the number of relays is very large, in which case, the signal–dependent noise dominates the performance of all–optical relaying. Furthermore, for a given source–destination distance for both types of relays, performance first improves as more relays are added before it deteriorates again for very large numbers of relays because of noise accumulation. Also, a comparison of the proposed model for all–optical relaying with existing models reveals that existing models may lead to overly optimistic results for the outage probability since not all relevant noises are taken into account. We advocate the use of EDFAs for all–optical relaying since in the 1550 nm region, the EDFA is a relatively mature technology [3]. EDFAs are one example of a more general class of optical amplifiers known as doped fiber amplifiers. Praseodymium doped fiber amplifier (PDFA) can be employed for optical amplification in the 1300 nm range. However, since the 1300 nm wavelength range is not commercially favorable, PDFAs are not that common. Unlike EDFAs, semiconductor laser amplifiers (SLAs) can be manufactured for multiple wavelengths including the 830 nm, 1310 nm, and 1550 nm. They are less expensive compared to EDFAs and can be integrated with semicoductor lasers, modulators, etc. SLAs however have a higher noise, lower gain, and exhibit more nonlinearity compared to EDFAs and are therefore not as favorable. The remainder of this chapter is organized as follows. In Section 5.1, the channel model for multi–hop FSO relaying systems is introduced. The proposed signal and noise models for dual–hop FSO relaying with electrical and optical amplification are developed in Section 5.2, and the corresponding outage probability is derived in  5. Multi–Hop FSO Communication Systems Eb,r (t)  Input Data  Laser  EASE (t)  αsr Es(t)  Ith,d(t) αrd  EDFA  87  Decision  PD  Output Data  Er (t) (a)  Eb,r (t) Ith,d(t)  Ith,r (t) Input Data  Laser  αsr Es(t)  PD  Er (t)  Ge  Laser  αrd  PD  Decision  Output Data  (b)  Figure 5.1: Block diagram of FSO AF relay systems with (a) optical amplification using an EDFA and (b) electrical amplification (PD: photodetector). Section 5.3. In Section 5.4, the results for dual–hop relaying are extended to the multi–hop case. Numerical results are presented in Section 5.5, and some conclusions are drawn in Section 5.6.  5.1  Channel Model  The considered FSO system operates at a wavelength of 1550 nm, employs intensity modulation with OOK and direct detection, and consists of one source terminal, m relays R1 , ..., Rm , and one destination terminal. The block diagrams of dual–hop FSO systems employing all–optical and electrical relays are depicted in Figs. 5.1(a) and 5.1(b), respectively. The power gains of the intermediate channels are denoted by αi = ai h2i , i = 0, ..., m, where ai and hi are the path loss and the small-scale fading gain in the ith hop, respectively. The ith hop represents the channel between the ith and the (i + 1)th relay, where i = 0 and i = m + 1 represent the source and destination nodes, respectively. For convenience, for the dual hop case, we use also the notation αsr = α0 and αrd = α1 to denote the source-relay and relay-destination power gains,  5. Multi–Hop FSO Communication Systems  88  respectively. The path loss is modeled as ai [dB] = −aattn Li , where aattn and Li are the weather-dependent attenuation coefficient (in dB/km) and the link distance of the ith hop, respectively. Furthermore, we assume that hi follows a lognormal distribution, i.e., hi ∼ lnN (µhi , σh2i ). Since on average, the intensity gain of the channel is supposed to neither amplify nor attenuate the transmit signal, we use the normalization E{h2i } = 1, which leads to µhi = −σh2i . Compared to lognormal fading, the Gamma–Gamma fading model is a more versatile distribution that applies well to a wide range of turbulence conditions. However, since we will be studying multi–hop systems in this chapter, the presence of short link distances are more probable and thus the lognormal fading model, which is more mathematically tractable compared to Gamma–Gamma distribution applies well to the considered scenario. Assuming spherical wave propagation, the variance of the small-scale fading gain is given by [42] 7  11  σh2i = 0.124 (2π/λ) 6 Cn2 Li6 .  (5.1)  Because of the characteristics of the lognormal distribution, it is straightforward to show that αi ∼ lnN (µi , σi2 ), where µi = ln(ai ) + 2µhi , σi2 = 4σh2i .  5.2  Dual–Hop FSO Model  In this section, we develop models for the signals and noises at the destination terminal for dual–hop FSO communication systems employing relays with optical and electrical amplification, respectively. These models are obtained based on the approach introduced in Chapter 2.  5. Multi–Hop FSO Communication Systems  5.2.1  89  Optical Amplification  The total electric field at the input of the relay is the sum of the signal electric field and the background radiation electric field M  Er (t) =  2xt Pt αsr Z0 cos(ω0 t) +  2Nb δνZ0 cos(ωl t + φl ),  (5.2)  l=−M  where xt ∈ {0, 2} denotes the transmitted OOK symbol. The relay employs an EDFA to amplify the received electric field, which gives also rise to ASE noise, cf. Fig. 5.1. For amplification, a variable (fading dependent) gain or a fixed gain may be employed. The variable amplification gain keeps the relay output power limited at Pr at all times and is given by Go,var =  Pr + nsp hν0 Bo , Pt αsr + Pb + nsp hν0 Bo  (5.3)  where h denotes Planck’s constant and nsp is the amplifier spontaneous emission factor. In contrast, the fixed amplification gain keeps the average relay output power constant at Pr and is given by  Go,f ixed =  Pr + nsp hν0 Bo , Pt E{αsr } + Pb + nsp hν0 Bo  (5.4)  2  where E{αsr } = eµsr +σsr /2 . The gain of an EDFA depends on parameters such as the pump laser power, the doped fiber length, and the wavelength of the excitation light provided by the pump laser [66, 67]. A variable gain can be achieved by changing the pump laser power in response to the instantaneous variable input power. In practice, due to the possible saturation of the EDFA gain, a limited range of variable gain is achievable [66]. However, we neglect this saturation effect here for tractability. For a fixed gain, the pump laser power is fixed to a desired value. In the remainder of this  5. Multi–Hop FSO Communication Systems  90  chapter, for the sake of brevity and wherever applicable, Go represents both the fixed and variable gain of an all–optical relay. The total electric field at the destination, Ed (t), includes contributions from the signal electric field, Es (t), the relay background radiation electric field, Eb,r (t), the destination background radiation electric field, Eb,d (t), and the ASE electric field, EASE (t). In view of the above, Ed (t) is expressed as M  Ed (t) = Es cos(ω0 t) +  Eb,r cos(ωl t + φr,l )  (5.5)  l=−M M  +  M  EASE cos(ωl t + φASE,l ) +  l=−M  Eb,d cos(ωl t + φd,l ), l=−M  √ where the amplitudes of the electric fields are given by Es = 2xt Pt αsr Go αrd Z0 , √ √ √ Eb,r = 2Nb δνGo αrd Z0 , EASE = 2N0 δναrd Z0 , and Eb,d = 2Nb δνZ0 . Here, N0 = nsp (Go − 1)hν0 is the spectral density of the ASE noise. Furthermore, φr,l , φASE,l , and φd,l are the mutually independent random phases of the components of the respective electric fields. The destination employs a photodetector resulting in the photocurrent  iphoto (t) =  R 2 E (t). Z0 d  (5.6)  Inserting (5.5) into (5.6) results in 10 terms which include the useful signal and different noise terms. These terms will be discussed in detail in the sequel. Signal Current: The direct current resulting from the transmit signal is given by  Is = Rxt Pt αsr Go αrd .  (5.7)  The signal terms at frequency 2ω0 have no effect due to the lowpass nature of the photodetector.  5. Multi–Hop FSO Communication Systems  91  Background-Background Radiation DC Current and Beat Noise: The relay background radiation field mixes with itself and results in a DC current, Ib×b,r , given by Ib×b,r  2 REb,r = 2M = RNb Bo Go αrd . 2Z0  (5.8)  Removing the bias of the background radiation term, the power of the remaining 4 zero mean beat noise simplifies to E i2ac,b×b,r (t) = R2 Bo2 Eb,r /(4δν 2 Z02 ). The power  spectrum of the background-background radiation beat noise has a triangular shape 2 and extends from 0 to Bo [38] with a power density of N0,b×b,r = 2R2 Nb2 G2o αrd Bo near  DC. Assuming that the electrical bandwidth Be of the photodetector is smaller than the optical bandwidth Bo , which is the case in practice, the variance of the relay background-background radiation beat noise is given by 2 2 (2Be B0 − Be2 ). = R2 Nb2 G2o αrd σb×b,r  (5.9)  Similarly, the destination background-background radiation DC current and noise 2 variance are given by Ib×b,d = RNb Bo and σb×b,d = R2 Nb2 (2Be B0 − Be2 ), respectively.  Hence, the total background-background radiation beat noise variance is given by 2 2 2 2 σb×b = σb×b,r + σb×b,d = R2 Nb2 (2Bo Be − Be2 )(1 + G2o αrd ).  (5.10)  Relay-Destination Background Radiation Beat Noise: This noise term originates from the mixing of the relay and destination background radiation fields. Since the two electric fields have mutually independent random phases, a DC term is not generated. The beat noise has a triangular power spectrum extending from 0 to Bo [38] and its variance is given by 2 σb,r×d = 4R2 Nb2 Go αrd (2Bo Be − Be2 ).  (5.11)  5. Multi–Hop FSO Communication Systems  92  ASE-ASE DC Current and Beat Noise: As a result of photodetection, the ASE field mixes with itself and results in an ASE-ASE DC current given by  IASE×ASE = 2M  2 REASE = RN0 Bo αrd . 2Z0  (5.12)  The ASE-ASE beat noise also has a triangular power spectrum [38] and variance 2 2 σASE×ASE = R2 N02 αrd (2Bo Be − Be2 ).  (5.13)  Signal-Background and Signal-ASE Beat Noises: The bias of the signal-background 2 beat noise is zero and its total energy is E i2s×b,r (t) = R2 Es2 Eb,r M/Z02 . Its power  spectrum has a rectangular shape extending from 0 to Bo /2 [38] and therefore, assuming Be < Bo /2, its variance is obtained as 2 2 σs×b,r = 4R2 xt Pt αsr αrd G2o Nb Be .  (5.14)  Similarly, the variance of the signal-destination background radiation beat noise is given by 2 σs×b,d = 4R2 xt Pt αsr αrd Go Nb Be .  (5.15)  Consequently, the total signal-background radiation beat noise is 2 2 2 σs×b = σs×b,d + σs×b,r = 4R2 xt Pt αsr αrd Go Nb Be (1 + Go αrd ).  (5.16)  Applying the same approach to the signal-ASE cross term results in 2 2 σs×ASE = 4R2 xt Pt αsr αrd Go N0 Be .  (5.17)  5. Multi–Hop FSO Communication Systems  93  ASE-Background Radiation Beat Noise: Finally, the background radiation fields at the relay and destination will also mix with the ASE field. Since the phases of the mixing fields are statistically independent, a DC term is not generated. The power spectra of the relay background-ASE and destination background-ASE beat noises are both triangular and extend from 0 to Bo . As a result, the corresponding noise variance is given by 2 σb×ASE = 4R2 Nb N0 αrd (2Bo Be − Be2 )(1 + Go αrd ).  (5.18)  Based on the above considerations, for transmit symbol xt = 0, the total noise variance at the destination due to thermal noise Ith,d (t), shot noise caused by DC noise currents, and beat noise is given by 2 2 2 2 2 2 2 σof f = σth + σshot,of f + σASE×ASE + σb×ASE + σb×b + σb,r×d ,  (5.19)  2 2 with thermal noise variance σth = 4KT Be /RL and shot noise variance σshot,of f =  2qIof f Be , where the total DC current is given by Iof f = Ib×b,r + Ib×b,d + IASE×ASE . The noise variance corresponding to xt = 2 is given by 2 2 2 2 σon = σof f + 2qIs Be + σs×b + σs×ASE ,  (5.20)  2 where σof f is given in (5.19). The signal current, Is , in (5.7) and the noise variances  in (5.19) and (5.20) constitute the proposed model for dual–hop transmission with an EDFA-based all–optical relay.  5. Multi–Hop FSO Communication Systems  5.2.2  94  Electrical Amplification  Similar to the all–optical relay, the total electric field at the input of the electrical relay is also given by (5.2). However, the electrical relay employs a photodetector generating photocurrent ˜ir (t) = REr2 (t)/Z0 which results in the relay signal current I˜s,r = Rxt Pt αsr , the background-background radiation bias I˜b×b,r = RPb , the 2 background-background radiation beat noise variance σ˜b×b,r = R2 Nb2 (2Bo Be − Be2 ), 2 and the signal-background radiation beat noise variance σ˜s×b,r = 4R2 xt Pt αsr Nb Be .  The total noise variance also includes the effect of the thermal noise at the relay, 2 Ith,r (t), with variance σth . The bias of the background radiation is removed, the  photocurrent is amplified by employing an electrical amplifier with gain Ge , and the resulting amplified signal is modulated via an OOK optical carrier [28]. Similar to the optical case, variable and fixed gain electrical amplification may be considered. The variable gain keeps the output power of the electrical amplifier limited at all times and is given by  Ge,var =  Pt αsr + RL  Pr , 2 2 2 2 2 σth +σ ˜b×b,r + 0.5(˜ σs×b,r +σ ˜shot,on,r +σ ˜shot,of f,r )  (5.21)  2 2 ˜ where σ ˜shot,on,r = 2qBe (I˜b×b,r + I˜s,r ) and σ ˜shot,of f,r = 2qBe Ib×b,r . The fixed gain is  given by  Ge,f ixed =  (5.22)  Pr 2 2 + 2qRPb Be + 2qRL Pt Be E{αsr } + 4R2 Pt Nb Be E{αsr }) + RL (˜ σb×b,r Pt E{αsr } + σth and keeps the average relay output power fixed at Pr . In the following, for simplicity of notation, wherever applicable we will use Ge to denote both the fixed and variable electrical relay gains.  5. Multi–Hop FSO Communication Systems  95  The received electric field at the destination is given by M  Ed (t) =  2Nb δνZ0 cos(ωl t + φ˜d,l ),  2Pω0 Z0 cos(ω0 t) +  (5.23)  l=−M  where the received power at the optical center frequency ω0 is given by Pω0 = 2 2 2 2 2 2 (σth +σ ˜b×b,r +σ ˜shot,of ˜s×b,r +σ ˜b×b,r + f,r )RL Ge αrd and Pω0 = 2Pt αsr Ge αrd + (σth + σ 2 σ ˜shot,on,r )RL Ge αrd for the off and on cases, respectively, and φ˜d,l is a random phase. In  the expression for Pω0 in the on case, the first term represents the signal-dependent received power and the second term represents the noise power introduced by the relay. The photodetection at the destination results in the received signal current I˜s = Rxt Pt αsr Ge αrd and the background radiation bias I˜b×b,d = RPb . Assuming that the destination photodetector has the same load resistance RL as the relay, the equivalent receiver noise variance caused by the relay for the off and on cases, respectively, is given by 2 2 2 2 σ ˜eq,of ˜b×b,r +σ ˜shot,of f,r = Ge αrd (σth + σ f,r )  (5.24)  2 2 2 2 2 σ ˜eq,on,r = Ge αrd (σth +σ ˜b×b,r +σ ˜s×b,r +σ ˜shot,on,r ).  (5.25)  Also, the destination shot noise, the signal-background radiation beat noise, and 2 the background-background radiation beat noise variances are given by σ ˜shot,d = 2 2 2qBe (I˜s + I˜b×b,d ), σ ˜s×b,d = 4R2 xt Pt αsr Ge αrd Nb Be , and σ ˜b×b,d = R2 Nb2 (2Be Bo − Be2 ),  respectively. Consequently, the total noise variances at the destination for transmit symbols xt = 0 and xt = 2 are given by, respectively, 2 2 2 2 ˜ σ ˜of ˜eq,of ˜b×b,d f = σth + 2qBe Ib×b + σ f,r + σ  and  2 2 2 2 2 2 σ ˜on = σth +σ ˜shot,d +σ ˜eq,on,r +σ ˜s×b,d +σ ˜b×b,d .  (5.26) (5.27)  5. Multi–Hop FSO Communication Systems  96  Hence, the proposed model for dual–hop transmission with an electrical relay is given 2 2 by the signal model I˜s and the noise variances σ ˜of ˜on . f and σ  5.3  Outage Analysis of Dual–Hop All–Optical Relaying  In this section, we analyze the outage probability of all–optical relaying. Since the total noise at the destination consists of several contributions including background radiation noise and thermal noise, the total noise can be modeled as Gaussian distributed. Consequently, the photocurrent follows a Gaussian distribution where the mean and variance are different for the on and off cases, i.e., the noise is signaldependent. Let Ion and Iof f denote the means for the on and off cases and σon and σof f denote the corresponding standard deviations. The optimum decision threshold for deciding on the transmitted OOK symbol is given by τ =  Ion σof f +Iof f σon . σon +σof f  Thus,  for dual–hop all–optical relaying, the bit error rate conditioned on the channel gains Is ). In the light of this conditional error rate exis given by Pe (αsr , αrd ) = Q( σon +σ of f  pression, we define the equivalent conditional (instantaneous) electrical SNR relevant for the performance of dual–hop all–optical relaying as  γ=  Is2 (σon + σof f )2  (5.28)  and the instantaneous outage probability as  Pout (αsr , αrd ) = Pr(γ < γth ),  (5.29)  5. Multi–Hop FSO Communication Systems  97  where γth is the threshold SNR. In the following, the average outage probabilities for fixed and variable amplification gains are derived.  5.3.1  Fixed Gain  Adopting the fixed gain, Go = Go,f ixed , from (5.4) and employing it in (5.19) and 2 2 (5.20), σof f and σon can be expressed in a unified form as  2 2 σf2 (xt ) = C1f + C2f αrd + C3f αrd + C4f (xt )αsr αrd + C5f (xt )αsr αrd ,  (5.30)  f 2 2 2 2 where σof f = σf (0), σon = σf (2), and the coefficients, Ci , are given in Table 5.1.  The SNR can be equivalently written as γ =  Is2 , 2 +σ 2 +2σ σ σon on of f of f  where the third term  in the denominator of the SNR creates square root terms which are not easy to deal with for evaluation of the average outage probability. However, there exist tractable upper and lower bounds on the SNR. In particular, γl ≤ γ ≤ γu , where γl = Is2 2 +σ 2 3σon of f  and γu =  Is2 . 2 +3σ 2 σon of f  These bounds prove to be tight for all practical ranges  of the various system parameters. Based on the proposed lower and upper bounds on the SNR, upper and lower bounds on the outage probability can be obtained, i.e., Pout,l (αsr , αrd ) ≤ Pout (αsr , αrd ) ≤ Pout,u (αsr , αrd ). The bounds on the outage probability are given by  Pout,a (αsr , αrd ) = Pr(γa < γth ) = Pr  1 1 > γa γth  , a ∈ u, l,  (5.31)  where f f f f f C1,a C4,a C2,a C3,a C5,a 1 = f + f + f + f + 2 α2 2 α 2 γa Cs αsr Cs αsr Cs αsr Cs αsr αrd Csf αsr rd rd  (5.32)  f f with Csf = 4R2 Pt2 G2o , Ci,a = 4Cif , i = 1, ..., 3, Ci,a = 3Cif (2), i = 4, 5, a = u, and f Ci,a = Cif (2), i = 4, 5, a = l. The calculation of the exact distribution of the inverse  5. Multi–Hop FSO Communication Systems  98  SNR does not seem tractable since the inverse SNR in (5.32) is a sum of correlated lognormal distributed random variables. However, employing the same approach as in [42], the sum of lognormal distributed random variables can be conveniently 1 γa  approximated by a single lognormal distributed random variable as 5  i=1  = exp(z) ≈  2 exp(ui ), where z ∼ N (µz,a , σz,a ) and ui ∼ N (µi , σi2 ), i = 1, ..., 5. The mean and  variance of z are functions of the means, variances, and covariances of the individual terms in the sum and are given by 2 2 µz,a = ln(A) − σz,a /2 and σz,a = ln 1 + B 2 /A2 ,  (5.33)  respectively, with 5  σi2 A= exp µi + 2 i=1  5  5  σi2 + σj2 , and B = exp µi + µj + 2 i=1 j=1 2  (exp(σij ) − 1), (5.34)  where σij = σji = cov(ui , uj ) denotes the covariance of ui and uj . Hence, for the purpose of outage analysis, it suffices to determine the distribution parameters of the ui and the covariance of each possible pair (ui , uj ), i = j. Using the properties of the lognormal distribution, the relevant means, variances, and covariances can be obtained. For convenience, they are summarized in Table 5.1. With the mean and variance of z in hand, the outage probability is given by  Pout,a = Pr exp(z) >  1 γth  =Q  ln (1/γth ) − µz,a σz,a  .  (5.35)  5. Multi–Hop FSO Communication Systems  99  Table 5.1: Coefficients of the noise variance expression in (5.30) and means, variances, 2 and covariances in (5.34) for fixed gain amplification. Beq = (2Be Bo − Be2 ). C1f C2f C3f C4f (xt ) C5f (xt )  2 2 σth + 2RPb qBe + R2 Nb2 Beq 2 2 2qBe RPb Go + 2qBe RN0 Bo + 4R2 Nb N0 Beq + 4R2 Nb2 Go Beq 2 2 2 R2 N02 Beq + 4R2 Nb N0 Go Beq + R2 Nb2 Beq G2o 2qBe Rxt Pt Go + 4R2 xt Pt Go Nb Be 4R2 xt Pt Go N0 Be + 4R2 xt Pt G2o Nb Be  µ1 , σ12  ln(  µ2 , σ22  f C1,a  ln(  Csf  µ3 , σ32  2 2 ) − 2µsr − µrd , 4σsr + σrd  ln(  µ4 , σ42  ln(  µ5 , σ52 σ45 σ15 , σ25 , σ34 , σ35 σ13 , σ23 σ24 σ14 σ12  5.3.2  2 2 ) − 2µsr − 2µrd , 4σsr + 4σrd  Csf f C2,a  f C4,a  Csf  f C3,a  Csf  2 ) − 2µsr , 4σsr  2 2 ) − µsr − µrd , σsr + σrd  ln(  f C5,a  2 ) − µsr , σsr 2 σsr 2 2σsr 2 4σsr 2 2 2σsr + σrd 2 2 2σsr + 2σrd 2 2 4σsr + 2σrd Csf  Variable Gain  Adopting the variable gain, Go = Go,var , from (5.3) and inserting it into (5.19) and 2 2 (5.20), σof f and σon can be expressed as  2 2 σv2 (xt ) = C1v + C2v αrd + C3v αrd + C4v αsr + C5v αsr 2 2 +C6v (xt )αsr αrd + C7v (xt )αsr αrd + C8v (xt )αsr αrd ,  (5.36)  2 2 2 2 v where σof f = σv (0) and σon = σv (2). The coefficients, Ci , in (5.36) are provided in  Table 5.2. Similar to the fixed gain scenario, upper and lower bounds on the SNR can be  5. Multi–Hop FSO Communication Systems  100  Table 5.2: Coefficients of the noise variance expression in (5.36) for variable gain amplification. Cb,ase = Pb + nsp hνBo and Cr,ase = Pr + nsp hνBo . C1v C2v C3v C4v C5v C6v (xt ) C7v (xt ) C8v (xt )  2 2 2 2 2 σth Cb,ase + 2qBe RPb Cb,ase + R2 Nb2 Beq Cb,ase 2qBe RPb Cr,ase Cb,ase + 2qBe Rnsp hνBo Cr,ase Cb,ase 2 2 +4R2 Nb Beq nsp hνCr,ase Cb,ase + 4R2 Nb2 Beq Cr,ase Cb,ase 2 2 2 2 2 2 2 2 2 2 2 R Beq nsp h ν Cr,ase + 4R Nb Beq nsp hνCr,ase + R2 Nb2 Beq Cr,ase 2 2 2σth Pt Cb,ase + 4qBe RPb Pt Cb,ase + 2R2 Nb2 Beq Pt Cb,ase 2 2 2 2 2 2 2 σth Pt + 2qBe RPb Pt + R Nb Beq Pt 2qBe Rxt Pt Cr,ase Cb,ase + 2qBe RPb Cr,ase Pt + 2qBe Rnsp hνBo Cr,ase Pt 2 2 +4R2 Nb Beq nsp hνCr,ase Pt + 4R2 xt Pt Nb Be Cr,ase Cb,ase + 4R2 Nb2 Beq Cr,ase Pt 2 2 2 2qBe Rxt Pt Cr,ase Pt + 4R xt Pt Nb Be Cr,ase Pt 2 2 4R2 xt Pt Nb Be Cr,ase + 4R2 xt Pt nsp hνBe Cr,ase  obtained and the resulting expression for the inverse of the SNR is given by v v v v v v v v C4,a C1,a C2,a C3,a C5,a C6,a C7,a C8,a 1 = v 2 2 + v 2 + v 2 + v + + + + , 2 2 γa Cs αsr αrd Cs αsr αrd Cs αsr Cs αsr αrd Csv αrd Csv αsr αrd Csv αrd Csv αsr (5.37) v v where Csv = 4R2 Pt2 (Pr + nsp hνB0 )2 , Ci,a = 4Civ , i = 1, ..., 5, Ci,a = 3Civ (2) + Civ (0), v i = 6, ..., 8, a = u, and Ci,a = Civ (2) + 3Civ (0), i = 6, ..., 8, a = l. Following the  same approach as for the fixed gain, the outage probability can be approximated as 2 in (5.35), where µz,a and σz,a are still given by (5.33) but with  8  σ2 A= exp µi + i 2 i=1  8  8  σi2 + σj2 and B = exp µi + µj + 2 i=1 j=1 2  (exp(σij ) − 1). (5.38)  The corresponding parameters µi , σi2 , and σij are given in Table 5.3.  5. Multi–Hop FSO Communication Systems  101  Table 5.3: Means, variances, and covariances in (5.38) for variable gain amplification. µ1 , σ12 µ2 , σ22 µ3 , σ32 µ4 , σ42 µ5 , σ52 µ6 , σ62 µ7 , σ72 µ8 , σ82 σ35 , σ37 , σ58 , σ78 σ48 , σ68 σ27 , σ67 σ18 , σ28 , σ34 , σ36 , σ38 σ17 , σ25 , σ47 , σ56 , σ57 σ26 σ46 σ16 , σ24 σ13 , σ23 σ15 , σ45 σ12 σ14  5.4  v C1,a ) − 2µsr − 2µrd , Csv v C2,a ln( C v ) − 2µsr − µrd , s Cv ln( C3,a v ) − 2µsr , s v C4,a ln( C v ) − µsr − 2µrd , s Cv ln( C5,a v ) − 2µrd , s v C6,a ln( C v ) − µsr − µrd , s Cv ln( C7,a v ) − µrd , s v C8,a ln( C v ) − µsr , s  ln(  2 2 4σsr + 4σrd 2 2 4σsr + σrd 2 4σsr 2 2 σsr + 4σrd 2 4σrd 2 2 σsr + σrd 2 σrd 2 σsr  0 2 σsr 2 σrd 2 2σsr 2 2σrd 2 2 2σsr + σrd 2 2 σsr + 2σrd 2 2 2σsr + 2σrd 2 4σsr 2 4σrd 2 2 4σsr + 2σrd 2 2 2σsr + 4σrd  Multi–Hop FSO Relay Model  In this section, we extend the proposed dual–hop FSO signal model to the multi– hop case, i.e., m > 1 relays are used in cascade. Since a variable gain results in a better performance compared to a fixed gain and fixed relay output powers may be desirable in practice because of eye-safety regulations, we only consider variable gain relays for multi–hop transmission. However, following the development in Section 5.2, a similar analysis as presented in this section is also possible for the fixed gain case. For derivation of the signal and noise models presented in Sections 5.4.1 and 5.4.2,  5. Multi–Hop FSO Communication Systems  102  we used the same approach as in Sections 5.2.1 and 5.2.2, respectively. Because of space limitation, we omit the details of this derivation here and concentrate on the final results instead.  5.4.1  Optical Amplification  Because of the background radiation and ASE noise that affects each relay, the received signal current at the destination is given by m  Is,m = Rxt Pt  Go,i αi ,  (5.39)  i=0  where Go,i , i = 0, ..., m, represents the gain of the ith relay and Go,0 = 1 is assumed for ease of notation. The relay gains are given by  Go,i =  Pr,i + nsp hνBo , Pr,i−1 αi−1 + Pb + nsp hνBo  (5.40)  where Pr,i denotes the output power of the ith relay. The background-background radiation and ASE-ASE DC currents are obtained as m+1 m  Ib×b,m = RPb  Go,i αi  (5.41)  j=1 i=j m  IASE×ASE,m = RBo αm  m−1  N0,j j=1  αi Go,i+1 ,  (5.42)  i=j  where N0,j is the ASE noise power spectral density at the jth relay. The ASE electric fields of each relay mix and result in the beat noise variance m−1  m 2 σASE×ASE,m  2  = R (2Bo Be −  αi2 G2o,i+1 .  2 N0,j  2 Be2 )αm j=1  i=j  (5.43)  5. Multi–Hop FSO Communication Systems  103  Furthermore, for m ≥ 2 relays, the ASE electric fields of different relays also mix and result in a new noise with variance 2  2 Be2 )αm  = 4R (2Be Bo −  j−1  m−1  m 2 σASE,r a ×rb ,m  i=j  αl Go,l+1 .  N0,k  αi Go,i+1  N0,j j=2  m−1  l=k  k=1  (5.44) Similar to the ASE electric fields, the background electric fields mix and result in the beat noise variances m+1 m 2 σb×b,m = R2 Nb2 (2Be Bo − Be2 )  2 σb,r a ×rb ,m  = 4R  2  Nb2 (2Be Bo  −  G2o,i αi2  (5.45)  j=1 i=j  m+1 m 2 Be ) αi Go,i j=2 i=j  j−1 m  αl Go,l  .  (5.46)  k=1 l=k  Finally, the background-ASE, signal-ASE, and signal-background beat noise variances are given by m  m+1 m 2 σb×ASE,m = 4R2 Nb (2Be Bo − Be2 )αm 2 σs×ASE,m = 4R2 xt Pt αm Be  2  = 4R xt Pt Nb Be  i=j  j=1  αi Go,i+1  (5.48)  i=j  m+1 m  m 2 σs×b,m  αi Go,i+1(5.47)  m−1  N0,j  Go,i αi i=0  j=1  j=1 i=j m  m  m−1  N0,j  Go,i αi  Go,i αi i=0  Go,i αi  .  (5.49)  j=1 i=j  Consequently, the total noise variances corresponding to the transmit symbols xt = 0 and xt = 2 are given by, respectively, 2 2 2 σof f,m = σth + 2qBe (Ib×b,m + IASE×ASE,m ) + σASE×ASE,m 2 2 2 2 . + σb×ASE,m + σb,r + σb×b,m +σASE,r a ×rb ,m a ×rb ,m  (5.50)  5. Multi–Hop FSO Communication Systems  104  and 2 2 2 2 σon,m = σof f,m + 2qBe Is,m + σs×ASE,m + σs×b,m .  (5.51)  Eqs. (5.39) and (5.50), (5.51) constitute the proposed signal and noise models for EDFA-based variable gain all–optical relaying, respectively.  5.4.2  Electrical Amplification  As opposed to all–optical relaying, electrical relaying requires a photodetector at each relay. Photodetection at the destination node results in signal and background radiation DC currents and generates background-background and signal-background radiation cross noise terms. Thermal noise is present at each relay and at the destination. The noise generated in the photodetection process at a given relay is propagated to the next relay and finally appears at the destination node. The variable electrical gain of each relay Ge,i , i = 1, ..., m, is given by  Ge,i =  Pr,i−1 αi−1 +  2 RL σth  +  2 σ ˜b×b  Pr,i , (5.52) 2 2 2 + 0.5(˜ σs×b,r,i +σ ˜shot,on,r,i +σ ˜shot,of f,r,i )  2 2 2 where σ ˜b×b = R2 Nb2 (2Be Bo −Be2 ), σ ˜s×b,r,i = 4R2 xt Nb Be Pt ge,j , σ ˜shot,on,r,i = 2qBe (RPb + i−1  Rxt Pt ge,j ),  2 σ ˜shot,of f,r,i  = 2qBe RPb , ge,i =  αj Ge,j , and Ge,0 = 1. Thus, the DC signal j=0  current and the noise variance at the destination node are given by I˜s,m = Rxt Pt ge,m+1 and m+1 m 2 σ ˜eq (xt )  =  2 (σth  +  2 σ ˜b×b  + 2qBe RPb )  αi Ge,i + 2(m + 1)RBe (2RNb + q)Pt ge,m+1 xt , j=1 i=j  (5.53)  5. Multi–Hop FSO Communication Systems  105  Table 5.4: Coefficients of the noise variance in (5.54) for multi–hop variable gain all–optical relaying. C1m C2m (xt ) C3m C4m (xt ) C5m C6m  2 2 σth + 2RPb qBe + R2 Nb2 Beq 2qBe Rxt ρ0 P P1 + 4R2 xt ρ0 P P1 Nb Be 2 2 2qBe RPb + 2qBe RBo nsp hν + 4R2 Nb nsp hνBeq + 4R2 Nb2 Beq 4R2 xt ρ0 P P1 Be nsp hν + 4R2 xt ρ0 P P1 Nb Be 2 2 2 2 2 2 R2 Beq nsp h ν + 4R2 Nb Beq nsp hν + R2 Nb2 Beq 2 2 2 4R2 n2sp h2 ν 2 Beq + 4R2 Nb2 Beq + 8R2 Nb nsp hνBeq  respectively. The noise variances at the destination for the on and off cases are given 2 2 2 2 by σ ˜on,m =σ ˜eq (2) and σ ˜of ˜eq (0), respectively. f,m = σ  5.4.3  Outage Analysis of Multi–Hop All–Optical Relaying  In this section, we analyze the outage performance of multi–hop all–optical relaying. Due to the complexities entailed by an exact analysis, we derive a lower bound on the outage probability by approximating the relay gains by Go,i = Pr,i /(Pr,i−1 αi−1 ). The results in Section 5.5 reveal that the obtained bound is tight especially for high transmit powers. After some manipulations, (5.50) and (5.51) can be unified to 2 2 2 2 σof f,m = σm (0) and σon,m = σm (2) with m  2 σm (xt )  =  C1m  +  C2m (xt )αm  m  pj pj 2 + C4m (xt )αm α α j=1 j−1 j=1 j−1 j−1  m p2j pj pk m 2 + C α , 6 m 2 α α α j−1 k−1 j−1 j=2 k=1 j=1 m  2 + C5m αm  +  C3m αm  (5.54)  m  Pr,i /Pr,i−1 and the coefficients, Cim , are given in Table 5.4.  where pj = i=j  The inverse SNR required in (5.31) for calculation of the outage probability can  5. Multi–Hop FSO Communication Systems  106  Table 5.5: Multi–hop Relaying System Parameters [38, 47, 48]. Optical wavelength (λ) Electrical bandwidth (Be ) Optical bandwidth (Bo ) Data rate (Rb ) Receiver noise temperature (T ) Receiver quantum efficiency (η) Amplifier spontaneous emission factor (nsp ) Photodetector load resistance (RL ) Background radiation energy (Pb Tb ) Threshold signal-to-noise ratio (γth ) Source-destination link distance (dsd )  1550 nm 2 GHz 125 GHz 2 Gbps 300 K 0.75 1.4 50 Ω -170 dBJ 0 dB 5 km  be expressed as m m C2,a C1,a 1 = 2 + + γa αm αm  m C3,a m + C4,a αm  j−1  m m p2j pj pk pj m m + C + C5,a , 6,a 2 α α α α j−1 j−1 k−1 j−1 j=2 k=1 j=1 j=1 m  (5.55) m m m m where Ci,a = Ci,a /(4R2 Pt2 p21 ), i = 1, ..., 6, Ci,a = 4Cim , i = 1, 3, 5, 6, Ci,a = 3Cim (2) + m Cim (0), i = 2, 4, a = u, and Ci,a = Cim (2) + 3Cim (0), i = 2, 4, a = l. Finally, using the  same approach as in the dual–hop case, the inverse of the SNR can be approximated by a single lognormally distributed random variable and the outage probability can be thus obtained.  5.5  Numercial Results  In this section, we present analytical and simulation results for the outage probability of dual–hop (Figs. 5.2–5.5) and multi–hop (Figs. 5.6, 5.7) FSO systems employing optical or electrical amplification in lognormal fading. The parameters of the considered FSO system are provided in Table 5.5 and the caption of the figures.  5. Multi–Hop FSO Communication Systems  107  0  10  −1  10  Direct transmission Opt. Amp., Proposed Model  −2  10  Opt. Amp., Model in [16] Opt. Amp., Model in [17]  Poutage −−−>  −3  10  Elec. Amp., Proposed Model Elec. Amp., Model in [7, 9]  −4  10  −5  10  −6  10  −7  10  −8  10 −140  −138  −136  −134  −132 −130 −128 PTb [dBJ] −−−>  −126  −124  −122  −120  Figure 5.2: Outage probability of electrical and optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). dsd = 5 km. A single variable gain relay is placed dsr = 3.5 km from the source node. Source and relay transmit with equal power Pt = Pr = P/2. The total transmit power is denoted by P and, unless specified otherwise, all nodes transmit with identical powers. The relays are located on the straight line connecting the source and the destination. We consider a hazy environment with Cn2 = 1.7 × 10−14 and aattn = 4.2 dB/km in all figures, except for Fig. 5.6 where we consider a clear air environment with Cn2 = 5 × 10−14 and aattn = 0.43 dB/km [43]. The outage performance of a dual–hop FSO system with a single variable gain relay is shown in Fig. 5.2. Simulation results are shown for optical and electrical amplification for the comprehensive signal models proposed in Section 5.2 and the  5. Multi–Hop FSO Communication Systems  108  simplified signal models in [23, 35, 40, 41]. The results for the proposed signal model reveal that all–optical relaying outperforms electrical relaying which also entails the additional complexities caused by optical-to-electrical and electrical-to-optical conversions. Furthermore, for the considered system parameters, the simplified model for electrical relaying proposed in [23, 35], which assumes that the receiver noise is dominated by the shot noise caused by background radiation and thermal noise, yields an accurate estimate of the system performance. However, the simplified model for all–optical relaying in [40], which assumes that the noise at the destination is dominated by background radiation and ASE shot noise, severely overestimates the performance of all–optical relaying, since it does not take into account the effects of thermal noise and mixed noise terms such as background-background radiation and signal-dependent noises. The model advocated in [41] takes into account thermal noise and yields a better approximation of the true performance. However, for high transmit powers, signal-dependent noise terms gain in importance and thus the model in [41] overestimates the achievable performance. Fig. 5.3 shows the outage probability of the proposed EDFA-based all–optical relaying scheme for different source-relay distances, dsr . Both fixed gain and variable gain relaying are considered. Variable gain all–optical relaying outperforms fixed gain relaying unless the relay is placed very close to the source node which is not practical. As can be observed, an optimal location for the relay minimizing the outage probability exists. For all considered scenarios, all–optical relaying yields significant gains compared to direct transmission. The analytical results shown in Fig. 5.3 are based on the upper bound Pout,u in (5.35) and are in perfect agreement with the simulation results. The lower bound (not shown in the figure) yields practically the same results. In Fig. 5.4, we show the outage probability of fixed gain all–optical relaying as a  5. Multi–Hop FSO Communication Systems  109  0  10  Variable Gain −2  10  −4  Poutage −−−>  10  Variable Gain  −6  10  −8  10  Direct Transmission dsr = 1 km  Fixed Gain  dsr = 2.5 km dsr = 3 km  −10  10  dsr = 4 km Analysis  −12  10 −145  −140  −135  −130  −125 −120 PTb [dBJ] −−−>  −115  −110  −105  Figure 5.3: Outage probability of EDFA-based all–optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). Fixed gain and variable gain relaying are compared with direct transmission for different relay locations. Source and relay transmit with equal power Pt = Pr = P/2. dsd = 5 km. Markers indicate simulation results. function of the relay location for different total transmit powers P and different gains. Obviously, as the total transmit power increases, the outage probability performance improves. More interestingly, the optimal relay location minimizing the outage probability is different for each combination of total transmit power and constant gain value. Another interesting observation is that a larger amplification gain does not necessarily result in a better performance. For example, for P Tb = −115 dBJ and dsr = 4.5 km, Go = 20 dB yields a considerably better performance than Go = 30  5. Multi–Hop FSO Communication Systems  110  0  10  PTb = −135 dBJ −2  10  Poutage −−−>  PTb = −125 dBJ  −4  10  −6  10  Go = 10 dB  PTb = −115 dBJ  Go = 20 dB Go = 30 dB  −8  10  0  500  1000  1500  2000 2500 3000 dsr (m) −−−>  3500  4000  4500  5000  Figure 5.4: Outage probability of EDFA-based fixed gain all–optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). Source and relay transmit with equal power Pt = Pr = P/2. dsd = 5 km. Results were obtained from analysis. dB. In this case, the larger gain results in a total noise variance that is roughly 10 times larger while the received signal powers are almost comparable. Fig. 5.5 depicts the outage probability of variable gain all–optical relaying as a function of the relay location for different source-relay power allocations. In Fig. 5.5, ρ = Pt /P denotes the fraction of power allocated to the source. The optimal location of the relay minimizing the outage probability shifts closer to the destination as the power allocated to the source increases. Furthermore, performance can be optimized by jointly optimizing the power allocation and the relay location. For the considered case, ρ = 0.6 and dsr = 3450 m minimize the outage probability.  5. Multi–Hop FSO Communication Systems  111  0  10  −2  10  −4  Poutage −−−>  10  ρ = 0.1, 0.2, ..., 0.9  −6  10  −8  10  −10  10  −12  10  −14  10  0  500  1000  1500  2000 2500 3000 dsr (m) −−−>  3500  4000  4500  5000  Figure 5.5: Outage probability of EDFA-based variable gain all–optical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). 100 × ρ % of the total available power are allocated to the source. P Tb = −125 dBJ. dsd = 5 km. Results were obtained from analysis. Next, we compare simulation and analytical results for the outage probability of multi–hop variable gain all–optical relaying in clear air, cf. Fig. 5.6. The distances between any two neighboring nodes are identical and all transmitting nodes transmit with equal powers. Fig. 5.6 confirms the excellent agreement between simulations and analysis also in the multi–hop case. For the considered scenario, increasing the number of relays to five yields significant performance gains since additional relays do not only achieve path-loss gains but also have a positive effect on the severity of the fading, cf. (5.1).  5. Multi–Hop FSO Communication Systems  112  0  10  −1  10  −2  Poutage −−−>  10  −3  10  Direct Transmission −4  10  m = 1 relay m = 2 relays m = 3 relays  −5  10  m = 4 relays m = 5 relays  −6  10  Analysis  −7  10 −145  −140  −135  −130  PTb [dBJ] −−−>  Figure 5.6: Outage probability of EDFA-based multi–hop variable gain all–optical relaying in lognormal fading and clear air (Cn2 = 5×10−14 , aattn = 0.43 dB/km). Power is equally divided between the transmitting nodes (source and relays). All nodes have identical distances from their nearest neighbors. Markers indicate simulation results. This effect is investigated more in detail in Fig. 5.7 for all–optical and electrical relaying in haze. Again, all relays are equally spaced and all transmitting nodes employ identical powers. The performance of both all–optical and electrical relaying improves with increasing number of relays until m = 8 is reached. Increasing the number of relays further to m = 10 results in a performance degradation in both cases. However, the severity of this degradation is quite different for all–optical and electrical relaying. In particular, for electrical relaying, thermal noise is the dominant effect and for a given transmit power adding more relays deteriorates the performance once the  5. Multi–Hop FSO Communication Systems  113  0  10  −1  10  −2  10  Poutage −−−>  −3  10  −4  10  −5  10  −6  10  −7  10  Direct Transmission Optical Amp. Electrical Amp. m = 2 relays m = 4 relays m = 6 relays m = 8 relays m = 10 relays  −8  10 −145  −140  −135  −130  PTb [dBJ] −−−>  Figure 5.7: Outage probability of multi–hop variable gain all–optical and electrical relaying in lognormal fading and haze (Cn2 = 1.7 × 10−14 , aattn = 4.2 dB/km). Power is equally divided between the transmitting nodes (source and relays). All nodes have identical distances from their nearest neighbors. Results were obtained from analysis. negative impact of the accumulated thermal noise overshadows the positive impact on fading and path loss. However, this degradation is gradual and can be compensated by increasing the transmit power. In contrast, for all–optical relaying, the signaldependent noise terms become dominant if the number of relays exceeds a certain value and the resulting degradation cannot be compensated by increasing the transmit power. In fact, for the case considered in Fig. 5.7 electrical relaying outperforms all– optical relaying for P Tb > −134 dBJ and m = 10 relays. Nevertheless, in all other considered cases, all–optical relaying yields significant gains compared to electrical  5. Multi–Hop FSO Communication Systems  114  relaying. Fig. 5.7 nicely illustrates the importance of an accurate signal and noise model taking into account signal-dependent noises for all–optical relaying.  5.6  Conclusions  In this chapter, we proposed comprehensive models for dual–hop and multi–hop EDFA-based all–optical relaying and presented a comprehensive and accurate outage probability analysis. For the sake of comparison, a similar model was developed for AF relays with electrical amplification. We showed that for electrical amplification the existing simplified models in the literature accurately estimate the actual system performance. In contrast, for all–optical relaying, the existing models overestimate the true performance. Furthermore, the presented results showed that while EDFA-based all–optical relaying is faster and simpler to implement, it achieves a better performance than electrical relaying unless the number of relays is very large in which case signal-dependent noises have a negative impact on the performance of all–optical relaying. Their simplicity and high performance make FSO systems with all–optical EDFA relays attractive for application in last-mile terrestrial communications and wireless backhaul in mobile communication systems.  115  6  Summary of Thesis and Future  Work In this chapter, we summarize the main results obtained in this dissertation and suggest a number of topics for future work based on the research work in this thesis.  6.1  Summary of Results  In this thesis, we addressed some of the main challenges that researchers and engineers face in performance evaluation and design of FSO communication systems. Due to the popularity, simple receiver structure, and favorable performance of IM/DD FSO systems in many applications, the main focus of this dissertation was on direct detection. Coherent detection FSO receivers offer superior performance at the price of more complex receiver structure and are favorable in scenarios, where performance is of higher priority compared to cost. We therefore also established a fundamental signal model for coherent/differential detection schemes, extended it to space–time coded FSO systems, and employed the resulting model to derive STC design criteria. In the following, we briefly review the results of each of the chapters of this thesis. In Chapter 2, we introduced the most commonly used fading models in FSO communication systems which include the Gamma–Gamma and lognormal distributions.  6. Summary of Thesis and Future Work  116  The comparison between the two models revealed that while the Gamma–Gamma distribution is a more versatile model for a wide range of turbulence conditions, both distributions show the same behavior for shorter link distances. We also derived comprehensive models for IM/DD and coherent detection schemes assuming a simple SISO link. The derived models were quantified in terms of all relevant system parameters and consequently, the validity of the AWGN model for long–haul FSO systems with both direct and coherent detection was justified. In Chapter 3, we extended the IM/DD SISO model derived in Chapter 2 to the case of IM/DD MIMO. We assumed the use of simple RCs at the transmitter side and studied the performance of MIMO FSO systems suffering from Gamma–Gamma fading. We employed a novel series representation of the modified Bessel function present in the Gamma–Gamma distribution and derived closed–form expressions for the PEP of SISO and MIMO FSO systems in the form of an infinite series. We truncated the respective series and quantified the approximation error. We also introduced the concepts of diversity and combining gain in the context of FSO systems based on the results in Chapter 3. In Chapter 4, we extended the signal and noise models derived in Chapter 2 for SISO FSO systems with direct and coherent detection to the case of space–time coded FSO systems. This generalization allowed us to analyze the performance of such systems in Gamma–Gamma fading and led to the derivation of closed–form expressions for PEP in these systems. We later used these PEP expressions to establish code design criteria for both detection schemes. Our results showed that the criteria used for STC design in FSO direct and coherent/differential detection systems are fundamentally different. We showed the quasi–optimality of simple RCs for the case of direct detection while our results showed that the same RF STC design criteria can be also applied to FSO systems with coherent/differential detection.  6. Summary of Thesis and Future Work  117  Finally, in Chapter 5, we moved away from spatial diversity as a measure for performance improvement and instead resorted to the use of relays as another effective means against the harmful effects of turbulence–induced fading and path loss. Following the approach that was established in Chapter 2, we obtained comprehensive models for dual–hop and multi–hop FSO systems that include the effect of all relevant system parameters, and signal and noise terms. Our results showed that variable gain relaying has a superior performance compared to fixed gain relaying. Besides, the use of all–optical relaying was advocated instead of the more conventional electrical relaying. It was shown that all–optical relays generally have a better performance compared to electrical relays despite of the fact that they do not require the optical– to–electrical and electrical–to–optical conversion of electrical amplification. In summary, we proposed several effective approaches to overcome the multiple challenges in the design and operation of FSO systems. Our proposed performance evaluation and design frameworks also set the stage for future extensions to other scenarios including the use of other fading distributions, detection methods, modulation schemes, and relaying structures.  6.2  Future Work  In this section, we suggest a number of research proposals based on the contents of this dissertation.  6.2.1  MIMO FSO STC Design with M > 2 in Gamma–Gamma Fading  In Chapter 4 of this thesis, we provided a general model for space–time coded FSO systems with either direct or coherent/differential detection. While our simulation  6. Summary of Thesis and Future Work  118  framework is general and can be employed to evaluate the performance of MIMO systems with any number of transmit and receive apertures, our analysis is limited to the practically important case of two transmit lasers. As was discussed in Chapter 4, our code search for direct detection and M > 2 suggests that RCs are also optimal for larger numbers of transmit lasers in direct detection FSO systems. It is however interesting to derive systematic STC design criteria to analytically derive optimum STCs for any number of transmit lasers. As for coherent detection, the case of two lasers and Gamma–Gamma fading suggests the same design criteria as for Rayleigh fading RF communication systems can be employed. While the same behavior is expected for more transmit apertures, availability of general design criteria for any number of transmit lasers would be interesting for coherent/differential FSO systems.  6.2.2  Performance Analysis of MIMO FSO Systems with Pointing Errors  FSO links are highly directional point–to–point links and thus misalignment between the transmit and receive apertures due to the building sway can result in FSO link performance degradation known as pointing error (jitter). While statistical models for pointing errors are available in the literature [68], a mathematically tractable analysis has not been provided, yet. The combined effect of atmospheric turbulence and jitter has been already considered in the literature [25]. It should be noted, however, that in many cases the detector size has been assumed to be negligible compared to the beam width. While [25, 68] propose a statistical model for FSO links which takes both atmospheric turbulence and pointing errors into account, they ultimately resort to numerical integration techniques for performance evaluation. It is therefore interesting to provide closed–form expressions for the performance of MIMO  6. Summary of Thesis and Future Work  119  FSO systems with pointing errors for different fading distributions.  6.2.3  Parallel All–Optical Relaying in FSO Systems  In this thesis, we studied the performance of multi–hop FSO systems with serial all–optical relaying. The use of multiple electrical relays in both serial and parallel structures has been already studied in the literature [28]. The approach that was taken in Chapter 5 of this dissertation can be applied to the case of multiple parallel relays and comprehensive models can be derived for all–optical and electrical amplification schemes. The extension to the parallel relaying scenario is straightforward based on the results presented in this dissertation. The same analytical approach used for serial relaying in Chapter 5 can be also applied to the case of parallel relays with slight modifications. It will be interesting to compare the performance of parallel and serial relays with the evaluation framework that was established in this thesis.  6.2.4  Cooperative Diversity in FSO Systems  In Chapter 5, we only considered the scenario, where each serial relay receives the signal from the previous hop, either optically or electrically amplifies it, and forwards it to the next relay/destination. Another possible scenario is to employ relays for cooperative diversity. This scenario combined with parallel relaying will be of special interest, where the destination node uses the signals received from the transmitter and multiple relays to decide on the transmit symbol, possibly over multiple time slots. It will be interesting to obtain the diversity gain of such a cooperative diversity scheme as well. Cooperative diversity schemes in FSO have been recently proposed in [69, 70]. The effect of the comprehensive models that were proposed in Chapter 5 on such cooperative schemes is a research topic worth investigating.  6. Summary of Thesis and Future Work  6.2.5  120  Multi–Hop FSO Systems Subject to Gamma–Gamma Fading  The comprehensive models presented in Chapter 5 of this thesis for multi–hop FSO systems are comprehensive and can be used for performance evaluation under any fading assumption. However, the provided analytical framework is limited to the case of lognormal fading. The link distances in multi–hop FSO systems are often short and therefore the lognormal distribution is a valid fading model under those circumstances. It is however favorable to obtain analytical results also for the case of Gamma–Gamma fading, which is a more versatile fading model for FSO systems.  121  Bibliography [1] T. Rokkas, T. Kamalakis, D. Katsianis, D. Varoutas, and T. Sphicopoulos, “Free Space Optical Technology as an Alternative Last-Mile Solution: A TechnoEconomic Analysis,” in Proceedings of 6th Conference on Telecommunications Techno-Economics (CTTE), pp. 1–5, Jun. 2007. [2] E. Leitgeb, M. Loschnigg, U. Birnbacher, G. Schwarz, and A. Merdonig, “High Reliable Optical Wireless Links for the Last Mile Access,” in Proceedings of 10th Anniversary International Conference on Transparent Optical Networks (ICTON), vol. 4, pp. 178–183, Aug. 2008. [3] S. Alexander, Optical Communication Receiver Design, 2nd ed.  Bellingham,  WA: SPIE Optical Engineering Press, 1997. [4] R. Dettmer, “A Ray of Light,” IEE Review, vol. 47, pp. 32–33, Mar. 2001. [5] “Free Space Optics,” http://www.free-space-optics.org, 2007. [6] D. Kedar and S. Arnon, “Urban Optical Wireless Communication Networks: The Main Challenges and Possible Solutions,” IEEE Communications Magazine, vol. 42, pp. S2–S7, May 2004. [7] D. Heatley, D. Wisely, I. Neild, and P. Cochrane, “Optical Wireless: The Story So Far,” IEEE Communications Magazine, pp. 72–74, 79–82, 1998.  Bibliography  122  [8] X. Zhu and J. M. Kahn, “Free-Space Optical Communication Through Atmospheric Turbulence Channels,” IEEE Transactions on Communications, vol. 50, pp. 1293–1300, Aug. 2002. [9] E. Lee and V. Chan, “Part 1: Optical Communication Over the Clear Turbulent Atmospheric Channel Using Diversity,” IEEE Journal on Selected Areas in Communications, vol. 22, pp. 1896–1906, Nov. 2004. [10] M. Razavi and J. Shapiro, “Wireless Optical Communications via Diversity Reception and Optical Preamplification,” IEEE Transactions on Wireless Communications, vol. 4, pp. 975–983, May 2005. [11] S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free-Space Optical MIMO Transmission With Q–ary PPM,” IEEE Transactions on Communications, vol. 53, pp. 1402–1412, Aug. 2005. [12] S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical Repetition MIMO Transmission With Multipulse PPM,” IEEE Journal on Selected Areas in Communications, vol. 23, pp. 1901–1910, 2005. [13] K. Kiasaleh, “Performance of APD-Based, PPM Free-Space Optical Communication Systems in Atmospheric Turbulence,” IEEE Transactions on Communications, vol. 53, no. 9, pp. 1455–1461, 2005. [14] A. Jurado-Navas, A. Garcia-Zambrana, and A. Puerta-Notario, “Efficient Lognormal Channel Model for Turbulent FSO Communications,” Electronics Letters, vol. 43, no. 3, pp. 178–179, Feb. 2007. [15] M. Uysal, M. Navidpour, and J. Li, “Error Rate Performance of Coded Free-  Bibliography  123  Space Optical Links Over Strong Turbulence Channels,” IEEE Communication Letters, vol. 8, pp. 635–637, Oct. 2004. [16] J. A. Anguita, I. V. Djordjevic, M. A. Neifeld, and B. V. Vasic, “Shannon Capacities and Error-Correction Codes for Optical Atmospheric Turbulent Channels,” Journal of Optical Networking, vol. 4, pp. 586–601, 2005. [17] T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “FSO Links with Spatial Diversity Over Strong Atmospheric Turbulence Channels,” in Proceedings of IEEE International Conference on Communications (ICC), pp. 5379–5384, May 2008. [18] W. Gappmair and S. S. Muhammad, “Error Performance of PPM/Poisson Channels in Turbulent Athmosphere with Gamma-Gamma Distribution,” Electronics Letters, vol. 43, pp. 880–882, Aug. 2007. [19] M.A.Al-Habash, L. Andrews, and R. Phillips, “Mathematical Model for the Irradiance Probability Density Function of a Laser Beam Propagating Through Turbulent Media,” Optical Engineering, vol. 40, pp. 1554–1562, Aug. 2001. [20] L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications. Bellingham, WA: SPIE Press, 2001. [21] K. Kiasaleh, “Performance of Coherent DPSK Free-Space Optical Communication Systems in K-Distributed Turbulence,” IEEE Transactions on Communications, vol. 54, pp. 604–607, Apr. 2006. [22] M. Uysal, J. Li, and M. Yu, “Error Rate Performance Analysis of Coded FreeSpace Optical Links over Gamma-Gamma Atmospheric Turbulence Channels,”  Bibliography  124  IEEE Transactions on Wireless Communications, vol. 5, pp. 1229–1233, Jun. 2006. [23] T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and N. Sagias, “Multihop Free-Space Optical Communications Over Strong Turbulence Channels,” in Proceedings of IEEE International Conference on Communications (ICC), vol. 6, pp. 2755–2759, 2006. [24] Z. Ghassemlooy, W. O. Popoola, and E. Leitgeb, “Free-Space Optical Communication Using Subcarrier Modulation in Gamma-Gamma Atmospheric Turbulance,” in Proceedings of 9th International Conference on Transparent Optical Networks (ICTON), Jul. 2007. [25] H. Sandalidis, T. Tsiftsis, G. Karagiannidis, and M. Uysal, “BER Performance of FSO Links over Strong Atmospheric Turbulence Channels with Pointing Errors,” IEEE Communications Letters, vol. 12, pp. 44–46, Jan. 2008. [26] M.K.Simon and V. Vilnrotter, “Alamouti-Type Space-Time Coding for FreeSpace Optical Communication With Direct Detection,” IEEE Transactions on Wireless Communications, vol. 4, pp. 35–39, Jan. 2005. [27] I. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC–Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” IEEE Journal of Lightwave Technology, vol. 26, pp. 478–487, Mar. 2008. [28] M. Safari and M. Uysal, “Relay-Assisted Free Space Optical Communications,” IEEE Transactions on Wireless Communications, vol. 7, pp. 5441–5449, Dec. 2008. [29] M. Brandt-Pearce, S. Wilson, Q. Cao, and M. Baedke, “Code Design for Optical  Bibliography  125  MIMO Systems Over Fading Channels,” in Proceedings of Asilomar Conference on Signals, Systems and Computers, vol. 1, pp. 871–875, Nov. 2004. [30] V. Tarokh, N. Seshadri, and A. Calderbank, “Space–Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction,” IEEE Transactions on Information Theory, vol. 44, pp. 744–765, Mar. 1998. [31] A. Evans, B. Yoo, and Z. Ding, “Space-Time Coded Modulation and Detection in Coherent Freespace Optical Communications,” in Proceedings of IEEE Global Communications Conference (GLOBECOM), pp. 1–5, Nov. 2006. [32] H. Yamamoto and T. Ohtsuki, “Atmospheric Optical Subcarrier Modulation Systems using Space-Time Block Code,” in Proceedings of IEEE Global Communications Conference (GLOBECOM), Dec. 2003. [33] S. M. Haas, J. H. Shapiro, and V. Tarokh, “Space-Time Codes for Wireless Optical Communications,” Eurasip Journal on Applied Signal Processing, vol. 3, pp. 211–220, Jan. 2002. [34] M. Safari and M. Uysal, “Relay-Assisted Free-Space Optical Communications,” in Proceedings of the Forty-First Asilomar Conference on Signals, Systems, and Computers, pp. 1891–1895, 2007. [35] G. Karagiannidis, T. Tsiftsis, and H. Sandalidis, “Outage Probability of Relayed Free Space Optical Communication Systems,” Electronics Letters, vol. 42, pp. 994–995, Aug. 2006. [36] M. Safari and M. Uysal, “Diversity Gain Analysis of Free-Space Optical Communication Systems,” in Proceedings of Canadian Conference on Electrical and Computer Engineering (CCECE), pp. 1239–1244, 2008.  Bibliography  126  [37] C. Datsikas, K. Peppas, N. Sagias, and G. Tombras, “ Serial Free-Space Optical Relaying Communications Over Gamma-Gamma Atmospheric Turbulence Channels,” IEEE/OSA Journal of Optical Communications and Networking, vol. 2, pp. 576–586, Aug. 2006. [38] N. Olsson, “Lightwave Systems With Optical Amplifiers,” Journal of Lightwave Technology, vol. 7, pp. 1071–1082, Jul. 1989. [39] C. R. Gilles and E. Desurvire, “Modeling Erbium-Doped Fiber Amplifiers,” Journal of Lightwave Technology, vol. 9, pp. 271–283, Feb. 1991. [40] M. Karimi and M. Nasiri-Kenari, “Outage Analysis of Relay-Assisted Free-Space Optical Communications,” IET Communications, vol. 4, pp. 1423–1432, Aug. 2010. [41] ——, “Free Space Optical Communications via Optical Amplify-and-Forward Relaying,” Journal of Lightwave Technology, vol. 29, pp. 242–248, Jan. 2011. [42] S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER Performance of Free-Space Optical Transmission with Spatial Diversity,” IEEE Transactions on Wireless Communications, vol. 6, pp. 2813–2819, Aug. 2007. [43] B. He and R. Schober, “Bit–Interleaved Coded Modulation for Hybrid RF/FSO Systems,” IEEE Transactions on Communications, vol. 57, pp. 3753–3763, Dec. 2009. [44] E. Lee and V. Chan, “Diversity Coherent Receivers for Optical Communication over the Clear Turbulent Atmosphere,” IEEE International Conference on Communications (ICC), Jun. 2007.  Bibliography  127  [45] J. M. Hunt, F. Holmes, and F. Amzajerdian, “Optimum Local Oscillator Levels for Coherent Detection Using Photoconductors,” Applied Optics, vol. 27, no. 15, pp. 3135–3141, Aug. 1988. [46] M. Riediger, R. Schober, and L. Lampe, “Reduced-Complexity Multiple-Symbol Detection for Free-Space Optical Communications,” in in Proceedings of IEEE Global Communications Conference (GLOBECOM), Washington, DC, Nov. 2007, pp. 4548–4553. [47] Q. Cao, M. Brandt-Pearce, and S. G. Wilson, “Free Space Optical MIMO System Using an Optical Pre-Amplifier,” in Proceedings of IEEE Global Communications Conference (GLOBECOM), pp. 1–5, 2006. [48] N. Cvijetic and S. G. Wilson, “Performance Bounds for Free-Space Optical MIMO Systems with APD Receivers in Atmospheric Turbulence,” IEEE Journal on Selected Areas in Communications, vol. 26, pp. 3–12, Apr. 2008. [49] J. Proakis, Digital Communications, 4th ed. New York: McGraw–Hill, 2000. [50] Z. Wang and G. Giannakis, “A Simple and General Parameterization Quantifying Performance in Fading Channels,” IEEE Transactions on Communications, Aug. 2003. [51] N. Letzepis and A. G. Fabregas, “Outage Probability of the MIMO Gaussian Free-Space Optical Channel with PPM,” in Proceedings of IEEE International Symposium on Information Theory (ISIT), pp. 2649–2653, Jul. 2008. [52] ——, “Outage Probability of the Gaussian MIMO Free-Space Optical Channel with PPM,” IEEE Transactions on Communications, vol. 57, pp. 3682–3690, Dec. 2009.  Bibliography  128  [53] N. Letzepis, I. Holland, and W. Cowley, “The Gaussian Free Space Optical MIMO Channel with Q–ary Pulse Position Modulation,” IEEE Transactions on Wireless Communications, vol. 7, pp. 1744–1753, May 2008. [54] C. N. Georghiades, “Modulation and Coding for Throughput-Efficient Optical Systems,” IEEE Transactions on Information Theory, 1994. [55] “Wolfram  Mathworld,”  2008.  [Online].  Available:  http://functions.wolfram.com/BesselAiryStruveFunctions/BesselK/06/01/01/01/ [56] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic Press, 2000. [57] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, 1972. [58] M. Simon and M.-S. Alouini, Digital Communication over Fading Channels. Wiley, 2005. [59] N. Letzepis and A. G. Fabregas, Outage Analysis in MIMO Free-Space Optical Channels with Pulse Position Modulation.  Tech. Rep. CUED/FINFENG/TR  597 (ISSN: 0951-9211), University of Cambridge, Feb. 2008. [60] E. Shin and V. Chan, “Optical Communication Over the Turbulent Atmospheric Channel Using Spatial Diversity,” in in Proceedings of IEEE Global Communications Conference (GLOBECOM), Nov. 2002, pp. 2055–2060. [61] B. M. Hochwald and W. Sweldens, “Differential Unitary Space-Time Modulation,” IEEE Transactions on Communications, vol. 48, no. 12, pp. 2041–2052, Dec. 2000.  Bibliography  129  [62] A. Nezampour and R. Schober, “Asymptotic Analysis of Space-Time Codes in Generalized Fading Channels,” IEEE Communication Letters, vol. 13, no. 8, pp. 561–563, Aug. 2009. [63] Wolfram Mathworld, 2008, [Online]: http://functions.wolfram.com/ HypergeometricFunctions/Hypergeometric2F1/. [64] S. B¨aro, G. Bauch, and A. Hansmann, “Improved Codes for Space-Time TrellisCoded Modulation,” IEEE Communication Letters, vol. 4, pp. 20–22, Jan. 2000. [65] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space-Time Block Codes from Orthogonal Designs,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [66] R. Mears, L. Reekie, I. Jauncey, and D. Payne, “Low-Noise Erbium-Doped Fiber Amplifier at 1.54um,” IEEE Electronic Letters, vol. 23, pp. 1026–1028, 1987. [67] E. Desurvire, J. Simpson, and P. Becker, “High-Gain Erbium-Doped TravelingWave Fiber Amplifier,” Optics Letters, vol. 12, pp. 888–890, Nov. 1987. [68] A. A. Farid and S. Hranilovic, “Outage Capacity Optimization for Free-Space Optical Links With Pointing Errors,” IEEE Journal of Lightwave Technology, Jul. 2007. [69] C. Abou-Rjeily and S. Haddad, “Cooperative FSO Systems: Performance Analysis and Optimal Power Allocation,” Journal of Lightwave Technology, vol. 29, no. 7, pp. 1058–1065, Apr. 2011. [70] C. Abou-Rjeily and A. Slim, “Cooperative Diversity for Free-Space Optical Communications: Transceiver Design and Performance Analysis,” IEEE Transactions on Communications, vol. 59, no. 3, pp. 658–663, Mar. 2011.  130  A  A Useful Result  In this appendix, we establish a key result which is repeatedly used in the performance analysis in Sections 3.2 and 3.3. In particular, in the analysis the integral ∞  Q  X(c, y) =  √  c2 I 2 I y−1 dI  (A.1)  0  with c2 > 0, y > 0 has to be evaluated. Using the alternative representation of the Gaussian Q–function Q(x) =  π/2 0  1 π  2  x exp − 2 sin 2θ  dθ [58, Eq. (4.2)], we obtain for  (A.1) the closed–form expression  1 X(c, y) = π  π/2 ∞  exp − 0  0  2y/2−1 Γ(y/2) = πcy  c2 I 2 2 sin2 θ  I y−1 dI dθ  π/2  siny θ dθ 0  y/2−1  =  2  where we have used the identities π/2 0  Γ((y + 1)/2) √ y , πc y ∞ 0  (A.2)  xν−1 exp(−µx2 ) dx = µ−ν/2 Γ(ν/2)/2, ν >  sinµ−1 x dx = Γ(1/2)Γ(µ/2)/[2Γ((µ + 1)/2)] [56, √ Eq. (3.6211)], Γ(x + 1) = xΓ(x), and Γ(1/2) = π.  0, µ > 0 [56, Eq. (3.4782)],  131  B  Bound on Approximation Error  In this appendix, we develop an upper bound for the approximation error ε(J) |Pe (d) − Pe (d, J)| which can be expressed as √  π ε(J) = 2| sin[π(α − β)]|Γ(α)Γ(β) where sj and x0  ∞  xj0 sj , j! j=J+1  (B.1)  µj (α, β)−µj (β, α), µj (α, β) xβ0 Γ((j +β +1)/2)/[Γ(j −α +β +1)(j +β)], √ √ 2 2αβσS /(d γ). From (B.1), we observe that ε(J) can be bounded as √  ∞  xj0 πsmax ε(J) < 2| sin[π(α − β)]|Γ(α)Γ(β) j=J+1 j! √ πsmax xJ+1 ex0 0 < , 2| sin[π(α − β)]|Γ(α)Γ(β)(J + 1)! where we have used the definition smax  (B.2)  maxj>J |sj |. The quantity smax can be easily  computed since it can be shown that there is a j0 ≥ J such that sj monotonically decreases if j > j0 . For sufficiently large J (J ≥ 10 is sufficient for typical values of α and β), j0 = J holds.  

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