Contribution on the study of underwater wireless optical links: channel prediction and energy efficiency

Instituto Universitario para el Desarrollo Tecnológico y la Innovación en Comunicaciones Programa de doctorado Doctorado en Cibernética y Telecomunicación Contribution on the study of Underwater Wireless Optical links: Channel prediction and energy efficiency Víctor Guerra Yánez Dirigida por el Dr. Rafael Pérez Jiménez Codirigida por el Dr. José Alberto Rabadán Borges El Director El Codirector El Doctorando Las Palmas de Gran Canaria Abril de 2016 Si tuviese que agradecer literalmente a todos aquellos que han hecho posible que haya terminado por fin mi tesis doctoral, deber´ıa empezar diciendo algo como – Agradezco a Sir Isaac Newton su esfuerzo y dedicaci´on para que yo haya podido acabar mi doctorado – Sin embargo, ya que si nos pusi´esemos en ese plan tendr´ıa tambi´en que agradecer a Planck, Maxwell, Lord Rayleigh y todos los dem´as se˜nores con barba que me dejo por el camino, he preferido acotar estos agradecimientos a un c´ırculo, digamos, que implique una circunferencia menor. Antes de empezar, quiero que conste que el orden de aparici´on no implica necesariamente una mayor o menor importancia, pero por si acaso me gustar´ıa comenzar agradeciendo a Lara, mi mujer, su infinita calma y actitud cari˜nosa. Envidio su capacidad para hacerme sentir tranquilo, a´un en momentos de estr´es. AGiulia, mi hija. Esa peque˜na central nuclear de un metro de altura. Le agradezco simplemente su existencia, ya que es la motivaci´on intr´ınseca de mi mundo. Por supuesto a mis padres, Mari Carmen yV´ıctor, que siempre han apoyado mis decisiones. A mis hermanos Carlos yH´ector, por permitirme que de vez en cuando los hostigase con mis fotones, los cuales a veces eran coherentes, y otras no tanto. Y a mis abuelos, Andr´es yLuisa, a los que tanto quiero. A mis suegros, Loli y´ Angel. Sin su ayuda no podr´ıa haber finalizado esta carrera de fondo. No podr´ıa olvidar a Omar. El que soport´o estoicamente el yugo de mi tutela en su Proyecto Fin de Carrera y mi inmisericorde l´atigo a la hora de obtener parte de los resultados de esta tesis. Estoy inmensamente agradecido por su tan valioso apoyo. A mis directores Rafa yJose. No podr´ıa imaginar unos mejores conductores para mi tesis. Por un lado la fuente inagotable de ideas que es Jose, as´ı como su eterna predisposici´on a mancharse las manos con todo tipo de cacharros electr´onicos. Y por otro lado, la experiencia, los amplios conocimientos y la motivaci´on que ha sabido siempre generar Rafa en mi. Sin ellos, esta tesis no existir´ıa. Y no me refiero a los aspectos legales de la afirmaci´on. No me perdonar´ıa a mi mismo no nombrar a mi amigo Crisanto en estos agradecimientos. Gran parte de mi amor por la investigaci´on se lo debo a ´el. Su pasi´on es contagiosa, y espero llegar tener al menos una fracci´on de su ´ımpetu. Adem´as, es el ´unico que me ganaba al Squash. Por ´ultimo, a mi otro compa˜nero de galimat´ıas matem´aticos de tinta, Cristo. Con el que consegu´ı acaparar m´as del 90 % del gasto en rotuladores del instituto. Gracias por no coartar mi natural efervescencia creativa escuchando atentamente los a menudo sinsentidos que produce mi cerebro. A todos, gracias. Contents Acronyms 1 Introduction 1 1.1 Underwater Radiofrequency Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Underwater Acoustic Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Underwater Wireless Optical Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Motivation, Hypotheses and Contributions 9 2.1 Motivation .................................................. 9 2.2 Hypotheses .................................................. 9 2.3 Contributions................................................. 10 2.4 Organizationofthedocument........................................ 11 3 Underwater Wireless Optical Communications: a thorough analysis 13 3.1 Channelmodeling .............................................. 14 3.1.1 Physicaleffects............................................ 14 3.1.2 Channelresponse........................................... 19 3.1.3 Stochasticmodeling ......................................... 22 3.2 ModulationsandEncodings......................................... 22 3.3 Energyefficiency ............................................... 24 3.4 Networklayer................................................. 25 3.5 Applications.................................................. 25 3.5.1 Hardwaredesign ........................................... 25 3.5.2 Underwater Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5.3 Applications for mobile systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.5.4 Otherapplications .......................................... 33 3.5.5 Summary ............................................... 34 3.6 Existingsurveys ............................................... 34 3.7 Remainingchallenges............................................. 36 4 Impulse Response of the Underwater Wireless Optical Channel 37 4.1 RadiativeTransferTheory.......................................... 37 4.2 Channelcharacteristics............................................ 38 4.2.1 Absorption .............................................. 38 4.2.2 Scattering............................................... 39 4.2.3 RefractiveIndex ........................................... 42 4.2.4 Turbulences.............................................. 42 4.2.5 Surfacereflections .......................................... 44 4.2.6 Seabeddiffusion ........................................... 45 4.2.7 Opticalfouling ............................................ 45 4.2.8 Fauna ................................................. 46 4.3 Simulation of the UWOC impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.1 Simulatedscenario.......................................... 46 4.3.2 Monte Carlo integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.3 Collisionwithparticles........................................ 48 4.3.4 Description of the simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.5 Parallelization of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Simulationresults .............................................. 54 4.4.1 Effectofthelink’srange....................................... 55 4.4.2 Effectofthelink’sdepth....................................... 56 4.4.3 Effect of the surface agitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.4 Effect of the distance to seabed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4.5 Effect of the seabed’s reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.6 Effect of the emitter’s directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.7 Effectofthewavelength....................................... 59 4.4.8 Effectoftheparticlesize ...................................... 60 4.4.9 Effect of the concentration of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.10 Parallelization speedup and efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.5 Comparisonwiththeliterature ....................................... 65 5 Considerations in Underwater-to-Air links 67 5.1 Seawavespropagation............................................ 67 5.2 Underwater-to-air channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.1 Seasurfacemodel .......................................... 68 5.2.2 Receivedpower............................................ 69 5.2.3 Projection of the receiver’s area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3 Simulationprocedure............................................. 71 5.4 Simulationresults .............................................. 72 5.4.1 Effectoftheemitter’sdepth..................................... 72 5.4.2 Effect of the receiver’s height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4.3 Effectoftheseawaveheight .................................... 73 5.4.4 Effect of the sea wave wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.4.5 Effectofthewindspeed....................................... 74 5.4.6 Channelavailability ......................................... 74 6 Statistical modeling of the Underwater Wireless Optical Channel 79 6.1 Briefanalysisoftheproblem ........................................ 79 6.1.1 Analysis of the statistical nature of the channel gain . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Statisticalprocedure............................................. 82 6.3 Results obtained through simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.1 ChannelgainandBandwidth.................................... 83 6.3.2 Comments on the relationship between the distribution and the channel’s parameters . . . . 83 6.4 Fresnel zones and Beam Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.4.1 BeamSpreadFunction........................................ 87 6.4.2 Analysis of the resulting Fresnel zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5 Statistical model for big opaque particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.5.1 Definition of big opaque particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.5.2 Mathematicalformulation...................................... 90 6.5.3 Influence of the link’s parameters on the SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7 Measurements on a short-range Underwater Wireless Optical Channel 97 7.1 WSSUSprocesses............................................... 98 7.1.1 CoherenceTime ........................................... 98 7.2 Description of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2.1 Emitterimplementation....................................... 99 7.2.2 Receiverimplementation.......................................100 7.2.3 Experimentalmethodology .....................................101 7.3 Results obtained through measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.3.1 Effect of the concentration and the movement of particles . . . . . . . . . . . . . . . . . . . . 103 7.3.2 Effectofthewindspeed.......................................105 7.3.3 Validity of the WSSUS assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8 Strategies for energy-efficient transceiver design 107 8.1 SignaltoNoiseratioinUWOC.......................................107 8.2 Optical transmitters and receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.2.1 Currentdrivers............................................108 8.2.2 Opticalemitters ...........................................109 8.3 PowerControlAlgorithms..........................................109 8.3.1 Fixed-stepalgorithm.........................................111 8.3.2 Variable-stepalgorithm .......................................112 8.3.3 Adaptive-stepalgorithm.......................................112 8.3.4 Adaptive-damping-and-step algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3.5 Wavelengthswitching ........................................113 8.3.6 Simulationresults ..........................................113 8.4 Pulse Width Modulated Optical OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.4.1 OpticalOFDMschemes .......................................117 8.4.2 Proposedscheme...........................................118 8.4.3 Experimentalcurves .........................................120 8.4.4 Comments on the BER performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9 Conclusions and Future Research 125 9.1 FutureResearch ...............................................128 Appendices 131 A Demonstrations of Chapter 5 133 B Received light intensity in partially obstructed links 137 C Waveforms, Correlations and Probability Density Functions of Chapter 7 139 C.1 Movementofparticles ............................................139 C.1.1 Blueemission.............................................139 C.1.2 Redemission .............................................141 C.2 Near-surfacelinkmeasurements.......................................142 D Summary in Spanish 151 D.1 Introducci´on..................................................151 D.1.1 Radiofrecuencia............................................152 D.1.2 Comunicaciones ac´usticas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 D.1.3 Comunicaciones´opticas .......................................154 D.2 Motivaci´on, Hip´otesis y Objetivos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 D.2.1 Hip´otesis ...............................................157 D.2.2 Aportaciones .............................................158 D.3 Respuestaimpulsivadelcanal........................................161 D.3.1 Transferenciaradiativa........................................161 D.3.2 Caracter´ısticasdelcanal.......................................162 D.3.3 Simulaci´on de la respuesta al impulso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 D.3.4 Resultadosdesimulaci´on ......................................166 D.4 Canalesagua-aire...............................................167 D.4.1 Propagaci´on de ondas marinas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 D.4.2 Modelodepropagaci´on .......................................168 D.4.3 Simulaci´on ..............................................169 D.4.4 Resultadosdesimulaci´on ......................................169 D.5 An´alisisestad´ıstico..............................................171 D.5.1 An´alisisdelproblema ........................................171 D.5.2 Procedimientoestad´ıstico ......................................173 D.5.3 Resultadosdesimulaci´on ......................................173 D.5.4 Zonas de Fresnel y Beam Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 D.5.5 Modelo de part´ıculas opacas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 D.6 Medidasexperimentles............................................179 D.6.1 Descripci´on del experimento . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 D.6.2 Resultadosexperimentales......................................181 D.7 Estrategias para la mejora de la eficiencia energ´etica . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 D.7.1 SNRencanalesUWOC .......................................185 D.7.2 Emisores y receptores ´opticos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 D.7.3 Algoritmos de control de potencia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 D.7.4 Cambiodelongituddeonda.....................................189 D.7.5 OFDMmoduladaenPWM .....................................190 D.8 Conclusiones .................................................193 Acronyms ACV After Convergence Variability. ADC Analog to Digital Conversion. AOA Angle Of Arrival. APD Avalanche PhotoDiode. API Application Programming Interface. AUV Autonomous Underwater Vehicle. BER Bit Error Rate. BJT Bipolar Junction Transistor. BSF Beam Spream Function. CDF Cumulative Density Function. CDMA Code Division Multiple Access. CI Convergence Iteration. CLT Central Limit Theorem. CPU Central Processing Unit. CSK Color Shift Keying. DC Direct Current. DPIM Digital Pulse Interval Modulation. DPSK Differential Phase Shift Keying. DSP Digital Signal Processing. ELF Extreme Low Frequency. EM ElectroMagnetic. FEC Forward Error Correction. FEM Finite Elements Method. FET Field Effect Transistor. FPGA Field Programmable Gate Array. FSK Frequency Shift Keying. FSO Free Space Optics. GEV Generalized Extreme Value. GPU Graphics Processing Unit. HF High Frequency. HOV Human Operated Vehicle. IMDD Intensity Modulated Direct Detection. IMU Inertial Measurement Unit. IOWCC Integrated OWC Circuit. IR InfraRed. LED Light Emitting Diode. LMS Least Mean Squares. LOS Line Of Sight. LTI Linear Time Invariant. LTV Linear Time Variant. MAC Medium Access Control. MCRT Monte Carlo Ray Tracing. MIMO Multiple Input Multiple Output. MMCRT Modified Monte Carlo Ray Tracing. MOS Metal Oxide Semiconductor. MPI Message Passing Interface. NLOS Non Line Of Sight. OBTS Optical Base Transceiver Station. OFDM Orthogonal Frequency Division Multiplexing. ONC Optical Network Controller. OOC Optical Orthogonal Code. OOK On Off Keying. OWC Optical Wireless Communication. PAPR Peak to Average Power Ratio. PCA Power Control Algorithm. PDF Probability Density Function. PIN Positive Intrinsic Negative. PMMA PolyMethyl MethAcrylate. PMT PhotoMultiplier Tube. PPM Pulse Position Modulation. PSK Phase Shift Keying. PWM Pulse Width Modulation. RF RadioFrequency. RGB Red Green Blue. ROV Remotely Operated Vehicle. RTE Radiative Transfer Equation. RTV Random Variable Transformation. SCPI Standard Commands for Programmable Instruments. SDMA Spatial Division Multiple Access. SER Symbol Error Rate. SIMD Single Instruction Multiple Data. SINR Signal Interference Noise Ratio. SNR Signal Noise Ratio. SSIM Structural Similarity Index Method. TDMA Time Division Multiple Access. TDS Total Dissolved Solids. TTP Threshold Transmit Power. UAC Underwater Acoustic Communication. UMTS Universal Mobile Telecommunications System. US Uncorrelated Scattering. UV UltraViolet. UWOC Underwater Wireless Optical Communication. UWSN Underwater Wireless Sensor Network. VLC Visible Light Communication. VRTT Vector Radiative Transfer Theory. VSF Volume Scattering Function. WSS Wide Sense Stationary. YB-WLED Yellow Blue White Light Emitting Diode. Chapter 1. Introduction Figure 1.7: (A(d, f)N(f))−1product for different distances. Extracted from [6]. parameters of the link, such as the sea surface’s agitation. Generally, coherence times up to hundreds of milliseconds may be considered. Additionally to the physical constraints inherent to acoustic propagation, acoustic modems add several limitation to the design of acoustic networks. The power required by the transmitter to carry out the communication depends on the distance, but is normally in the range of tens of watts, whilst the power requirements on the receiver side are much more relaxed (about a few milliwatts). Power consumption is a critical issue in battery-powered isolated nodes, and energy efficiency is of capital importance. There are different ways to reduce the power consumption of the node, but the main ones are power control algorithms, retransmission-reduction techniques in random access networks, and bandwidth-adaptive systems. Finally, marine mammals, such as dolphins and whales, have an audible spectrum that can cover up to almost 200 KHz. These animals use sound with both social and echolocation purposes, and the use of UAC under 200 KHz may be extremely harmful due to the high acoustic powers generally used. Furthermore, even though a priori non-harmful frequencies were used, due to Doppler effect, the actual spectral density that moving mammals may experience in an ensonified ocean could be harmful as well, as Siderius and Porter commented in [7]. 1.3 Underwater Wireless Optical Communications Underwater Wireless Optical Communications is a subgroup of Optical Wireless Communications. Unlike in FSO, the propagation medium presents absorption in UWOC. This absorption depends on the inherent properties of the seawater and the relative concentrations of algal and non-algal matter. Furthermore, whilst aerosols and gases produce scattering in FSO, UWOC suffers this spatial dispersion mainly from phytoplankton. Regarding turbulences, the refractive index gradients induced by temperatures changes in FSO are not enough to model this phenomenon in UWOC, since the refractive index of seawater depends on temperature, salinity and pressure. In addition, the wavelength dependency of the extinction coefficient (absorption plus scattering) generally defines a best wavelength that ranges within the blue-green window. As it was commented above, underwater RF communications are power and bandwidth limited. The power consumption of ELF stations is absurdly high and the effective data rate is very low, but offers very good communication range. Regarding UAC, the available bandwidth can support a wide range of services, but is normally limited to a few thousands of bits per seconds. In the past few years, UWOC has demonstrated the best energy efficiency in terms of bits per Joule and a maximum range above 100 meters. Furthermore, the cost associated to the optical transceivers is much lower than the associated to UAC’s hydrophones and drivers. However, due to the range limitation of UWOC, the current trend is to use hybrid opto-acoustical solutions, where long range telemetry is acoustically performed whilst low range, low latency and high bandwidth communication is carried out optically. The recent proposals of UWOC as data interface for Underwater Wireless Sensor Networks is highly supported by the availability to download huge amount of long-term collected data in a fraction of time respect to acoustic transceiver. Furthermore, the use of optical interfaces lengthens the life of the deployed nodes. However, although the maximum allowed misalignment error has been demonstrated to be lower than expected due to the beam spreading of light, it stills lower than the case of UAC. Moreover, the low latency of the UWOC links allows a 6 Chapter 1. Introduction natural remote operation of underwater vehicles. The latency of an UWOC link can be even better than the latency of a fiber-tethered link, since the refractive index is lower in seawater than in plastic or glass, but this enhancement is usually weighed by the necessity of retransmission in the untethered scenario. There are several current challenges in UWOC regarding energy efficiency and channel modeling. Several contributions where the impulse response is modeled as in a LTI scenario can be found in the literature, but the UWOC channel is clearly LTV and only a few authors address this issue in laser-based links. Moreover, there is a lack of contributions regarding energy efficiency in UWOC, which is of capital importance in scenarios where battery replacement is normally prohibited by the huge cost of the operation. In this dissertation, a statistical approach of the channel’s main parameters (bandwidth, gain and coherence time) is performed. The analysis is based using theoretical approximations and simulation results. Two different scenarios are analyzed due to their applicability in UWSN: underwater-to-underwater and underwater-to-air links. Furthermore, a statistical model for big opaque particles is proposed. The proposal is focused on shallow water scenarios where the coastal currents generate a cloud of particles in sandy seabeds. Regarding energy efficiency, the use of different modulations and encodings, as well as power control algorithms is explored. 7 Chapter 1. Introduction 8 Chapter 2 Motivation, Hypotheses and Contributions This chapter comprises the statements to be proven through experimentation in this thesis, the main objectives that have motivated its actual development, and the contributions that have been made during the development of this work. 2.1 Motivation The topic of this thesis emerged as the natural evolution of the research lines of the Photonic Technology and Communications Division of the Institute for Technological Development and Innovation in Communications, of the University of Las Palmas de Gran Canaria. The group has been focused during the last few years in IR and VLC, concretely in simulation engines [8] and the development of proof-of-concept prototypes for different applications using the aforementioned technology, such as video streaming using VLC [9] or in-flight optical communications (VLC downlink with IR uplink) [10][11]. In this work, since it is the first in a novel area for the group, several fronts have been treated. In first place, a thorough analysis of the current state-of-the-art research was mandatory in order to detect the main weaknesses and hence, opportunities to work in. As it will be commented in Chapter 3, energy-efficient strategies have not been studied in depth, and different approaches are proposed during the following chapters. Furthermore, stochastic models to carry out performance predictions according to link’s parameters are not available yet, and a slight contribution in this regard is made. Summarizing, the main objectives of this thesis are: •Carry out a thorough analysis of the current state-of-the-art research. •Propose strategies to enhance the energy efficiency of UWOC systems. •Study the feasibility of performing stochastic modeling in UWOC links. 2.2 Hypotheses This thesis departs from two fundamental hypotheses, which are enumerated below. The first one regards channel modeling, whilst the second addresses energy efficiency in UWOC. Hypothesis 1 On the channel. The Underwater Wireless Optical Communications channel is linear and time variant but Wide Sense Stationary with Uncorrelated Scattering (WSSUS). •The variability of the UWOC channel, enclosed within the time-variant impulse response h(t, τ), is due to the different scattering and reflective phenomena that occur during propagation. This variability, if emitter and receiver are at fixed positions, possesses a invariant mean value. Furthermore, the power contributions incoming from scattering events are mutually uncorrelated due to the independence of the locations where these scatterings are produced. •In a scenario with suspended matter, the movement of the particles amid the link generates a variability on the received power that reduces the SNR. In order to verify this hypothesis, both simulation and experimental approaches have been used. 9 Chapter 2. Motivation, Hypotheses and Contributions •Underwater-to-air links are also variable due to the changing shape of the seawater surface. Furthermore, the optical power that arrives the receiver can be estimated by the energy of the illuminated seawater surface area which impacts on the receiver. This illuminated area changes with time, following the shape of the sea waves spectrum. Hypothesis 2 On the energy efficiency. The energy efficiency of an Underwater Wireless Optical link can be improved by means of power control algorithms, the use of the best transmission wavelength, and energy-efficient encodings and modulations. •Generally, UWOC links are performed in the blue-green region of the visible spectrum, since the minimum absorption is usually located between these wavelengths. However, taking into account the better response of long-wavelength emitters and receivers, the worse propagation of redder wavelengths is compensated by these better efficiencies, defining a critical distance at which it is better to perform a red transmission than a blue one. •Power control algorithms are a well-known strategy to optimize SNR in highly interfered environments. Furthermore, underwater remote sensor nodes , which are battery-powered, have prohibitively expensive replacement costs. Hence, strategies to reduce the power consumption are mandatory in this kind of device, and power control algorithms can be energy-optimized. •Modulations and encodings are the lowest level of the communications stack. Taking into account the electrical characteristics of the optical emitters, nonlinear current drivers are a better option than linear ones. Therefore, modulations which need linear transmission can be quantized to allow the use of energy-efficient nonlinear drivers. All these hypotheses will be discussed along this document. Section 2.3 presents a summary of the contributions presented in this thesis. 2.3 Contributions To serve as a guide to those who read this work, a summary of the contributions made by this work is presented. •Chapter 3. A thorough analysis of the current state-of-the-art research is presented. This in-depth analysis has been structured attending to an intuitive taxonomy and tries to serve as the starting point of this work. •Chapter 4. A Monte Carlo Ray Tracing algorithm for UWOC is presented. Unlike other authors who employed Henyey-Greenstein scattering phase functions to model scattering due to particles, in this work, Mie scattering has been used since it models the phenomenon more accurately. The influence of each channel parameter on the impulse response is also analyzed. Furthermore, the algorithm was parallelized using both multiprocessor and GPU implementations •Chapter 5. A model of underwater-to-air communications is presented, focusing on the channel availability. •Chapter 6 –Using the aforementioned Monte Carlo Ray Tracing algorithm, a statistical approach of both channel gain and bandwidth is made. –An empirical formula to predict the BSF is obtained. –After a rectangular approximation of the impulse response, a definition of Fresnel zone is performed in terms of received energy, allowing the reduction of the volume of interest in channel simulation. This simplification also allows the prediction of the channel bandwidth. –A statistical study to model the influence of opaque particles such as sand grains is presented in this work. The approach is based on geometrical relationships and some approximations, but may serve as baseline for further work. •Chapter 7 –In this chapter, the influence of moving particles on the SNR is demonstrated. The movement of particles produces a random variation on the received signal that can be modeled as a normal distribution of variance related to the density of particles. –The coherence time of a near-surface link and its relationship with the wind stress are obtained. Besides the wind speed, other parameters are taken into account, such as depth and wavelength. 10 Chapter 2. Motivation, Hypotheses and Contributions –The validity of the WSSUS approximation of a near-surface link is demonstrated through experimentation. •Chapter 8 –The use of red wavelength instead of blue under certain channel restrictions is justified in terms of energy efficiency. As red emitters are more energy-efficient than blue ones, and Si-based receivers are more sensitive to longer wavelengths, below a critical distance is better to perform the transmission in red, despite the higher attenuation of the medium at this frequency. –A power control algorithm with wavelength switching (red-blue) is presented and evaluated. The use of PCA is mandatory to reduce the power consumption of isolated underwater nodes. In this case, several enhancements are proposed to the classic Newton-Raphson gradient-descent algorithm (equivalent to a LMS algorithm). –The use of PWM combined to nonlinear drivers is explored as an efficiency-enhancement strategy for Optical OFDM. It will be discussed that due to the higher efficiency of nonlinear drivers, PWM modulation of OFDM samples could be an alternative to reduce the power consumption and hence, longer the lifespan of nodes which transmit OFDM signals. 2.4 Organization of the document After commenting the motivation, the hypotheses and contributions that are the baseline of this work, the next chapters are organized as follows. In Chapter 3, a profound analysis of the evolution of the research in UWOC is presented. The analysis comments most of the available contributions in UWOC regarding different aspects: channel modeling, modulations and encodings, energy efficiency, network layer, applications and surveys. In Chapter 4, the UWOC channel is studied in detail. The different phenomena that affect underwater optical propagation are discussed in this chapter. Furthermore, a Monte Carlo Ray Tracing algorithm using Mie’s scattering model is presented. Furthermore, a parallelization scheme is also presented to reduce the computation time. Chapter 5 is dedicated to underwater-to-air links, especially in the discussion of the channel availability related to the seawater-air interface motion. The studied scenario has important implications in shallow-water sensor reading and UUV-to-air communications. In Chapter 6, a statistical approach of the channel gain and bandwidth is developed. Massive data obtained from the implemented simulator of Chapter 3 is introduced in a decision algorithm to infer the best-fit option within a battery of possible probability distribution functions. Furthermore, a qualitative relationship between the distribution parameters and the channel’s geometrical and physical parameters is commented. In Chapter 7, the results of a near-surface underwater transmission are presented. Using these results, the WSSUS consideration of the varying channel is demonstrated and the coherence time of the channel is also calculated. Chapter 8 comments different energy-efficient strategies. The use of red wavelengths instead of blue ones is justified for short-range links, and a power control algorithm with wavelength-switching capabilities is also presented and analyzed. Finally, in Chapter 9, several conclusions are extracted and future research lines are exposed and commented. 11 Chapter 2. Motivation, Hypotheses and Contributions 12 Chapter 3 Underwater Wireless Optical Communications: a thorough analysis In the past few years, there has been a growing interest in submarine applications, such as surveillance, telecommand of robots, ocean monitoring and military communications. The development and enhancement of visible range emitters and receivers has led to an increment on research works related to the use of LED and laser devices to establish communication links in the underwater medium. Nowadays, Underwater Wireless Optical Communications (UWOC), may be considered an independent topic apart from Free Space Optics (FSO) and Visible Light Communications (VLC). This independence has been encouraged by the particularities of the underlying communication channel, and the novelty of the field has attracted the attention of research groups worldwide. This growing interest can be observed at Figure 3.1 where the number of contributions in the field over time is analyzed. Year (+2K) 56789101112131415 0 5 10 15 20 25 Figure 3.1: Evolution of the number of papers available in the IEEE database related to UWOC To have a better understanding of the actual research interests within UWOC, the contributions have been classified in six different categories: •Channel modeling •Modulations and Encodings •Energy efficiency •Network layer •Applications •Surveys 13 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.2 depicts the distribution of the research attending the the aforementioned classification. Note that the distribution is qualitative as each paper may belong to several categories. Channel Energy Modulations Network Surveys Applications Figure 3.2: Distribution of publications per category in UWOC As it can be observed, Applications and Channel modeling are the main interests. Due to the novelty of the topic and the complexity of the channel, there were a need of contributions supporting the feasibility of links and a mathematical apparatus to predict the behavior of the channel. Nonetheless, the lack of publications regarding energy efficiency seems contradictory taking into account the necessity of energy-saving techniques in scenarios where the actors are normally battery-powered nodes. Finally, there are several surveys in this topic, but are normally incremental and their scope is reduced in time. In this work, a deeper analysis is made increasing the time range as the topic is affordable in size yet. 3.1 Channel modeling As it was commented above, channel modeling has been one of the main interests in UWOC since its origin. In order to provide a more accurate view of this issue, this category has been subdivided in three subtypes: physical effects, stochastic modeling and channel response. This issue has suffered an evolution that is depicted in Figure 3.3. It can be observed that the last two years (2013-2015) comprise more than three quarters of the contributions in this aspect. It is common to perform experimental evaluations before establishing the mathematical background of a novel subject, as the scientific method underscores. In this aspect, UWOC has suffered the same treatment, centering the efforts in proving the feasibility of the technology. This will be further discussed in Section 3.5. The following subsections comment the contributions in the three abovementioned subcategories. 3.1.1 Physical effects Within this category are the publications that try to model or empirically evaluate the effects of different underwater phenomena. The main effects that have a significant weight in UWOC are: •Turbulence •Scattering and Absorption •Fauna and optical fouling •Misalignment The following subsections analyze the most relevant contributions up to the date at each enumerated topic. Fauna has an unpredictable behavior from the communications’ point of view. The pass amid the link of fishes and mammals has the potential to produce long-duration deep fading events, but normally, it is not considered and there are no works in this regard. Turbulence Turbulence is of capital importance in laser-based systems, as the energy is concentrated in a small solid angle. Islam et al. performed an experimental evaluation in a laboratory-controlled scenario [12], finding out that turbulent regimes affect the received SNR depending on its salt concentration. Furthermore, other conclusion of 14 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Year (+2K) 6789101112131415 0 1 2 3 4 5 6 7 8 9 10 Stochastic modeling Channel Response Physical effects Figure 3.3: Evolution of the contributions regarding Channel Modeling the experiment was that the lower the bandwidth, the lower the influence of the turbulence, as it was expected theoretically. Hou and Matt [13] studied the propagation of images in a turbulent scenario. Although the scope of the article is not related to communications, the obtained results can be extrapolated to the UWOC domain. After using OpenFOAM to model turbulence conditions in a water tank, a degradation statistic was obtained. The structure similarity index metric (SSIM) is used to measure the statistical differences between two images, commonly an undistorted one and a distorted version [14]. In this case, the results showed that extreme turbulence regimes degraded the images up to 50 %. As each group of pixels can be considered as a traditional photodiode, this results may be easily extrapolated to a UWOC scenario. This SSIM degradation can be directly associated to a SNR decrement. Figure 3.4 reproduces the results obtained by Hou and Matt regarding the SSIM. The SSIM index is calculated between two windows xand yas shows Equation 3.1. Figure 3.4: Time series of image degradation during Hou and Matt’s lab experiment, under strong and extreme turbulence case 15 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis the double gamma impulse response is as easy as applying the Fourier Transform to 3.4. Equation 3.5 shows the result. H(jω) = a (jω +b)2+c (jω +d)2ejωt0(3.5) The double gamma approximation was proven to be very accurate to the simulations, but the relationship between its parameters and the physical and geometrical characteristics of the link has not been studied yet. Finally, Dong et al. examined the possibility of a closed-form impulse response modeling in a MIMO-based UWOC link [43]. The approximation was based on a weighed superposition of double gamma functions, as it is the straightforward solution. In this version of the double gamma approximation, the author slightly modified Equation 3.4 to fit a 2x2 MIMO impulse response, adding two new parameters yielding Equation 3.6. h(t)≈a(t−t0)αe−b(t−t0)+c(t−t0)βe−d(t−t0)(3.6) Other contributions Cochenour et al. defined an experimental method to measure the temporal dispersion in laser-based UWOC links [44]. The authors demonstrate that Beer-Lambert’s law is not accurate enough to establish a baseline in NLOS link design. Other effects such as multiple scattering , which generates temporal dispersion and hence, a reduction on the available bandwidth, should be considered. An interesting conclusion is that widening the FOV of the receiver enhances the input power but lowers the bandwidth in scattered environments. Furthermore, this sensitivity is dramatically reduced for off-axis links. These last conclusions have a significant impact on the link designer, which must know the geometrical parameters of the link. Dai et al. studied in [45] the behavior of IR and UV light in seawater. The authors proposed a scattering model for this wavelength bands and contrasted it empirically. The results have direct application in seawater parameter monitoring and the sensing of chemical oxygen in water treatment facilities. The direct impact in communications in reduced, as these bands present distance-limiting attenuations. 3.1.3 Stochastic modeling An accurate stochastic model is the main objective that channel modeling must try to achieve. The scientific method states that after observing a phenomenon a sufficient amount of times, a generalization may be performed with an accuracy directly related to the times the phenomenon was observed. It is logical that the efforts in UWOC channel modeling have been focused on simulation and experimental evaluations, but a critical mass has been reached and the production of probabilistic estimations and stochastic models should be the next milestone to fulfill. Tang et al. studied the temporal statistics of a laser-based link in a turbulent scenario [46]. After considering Kolmogorov’s turbulence approximation and Taylor’s frozen turbulence hypothesis, which are widely valid for regions below 100 meters in oceanic environments, the authors studied the temporal correlation of the irradiance between two points. Figure 3.13 shows this correlation for link between 30 and 50 meters with two different turbulent regimes defined by the average vertical water speed through the link’s axis. It can be observed that the coherence time, which is the time at which this correlation function decays 3 dB, is almost independent of the distance and is highly affected by the turbulence’s regime. Furthermore, the authors conclude that the correlation is more affected by salinity fluctuations than by temperature variations within the weak turbulence region. This is due to the higher sensitivity of the refractive index to salinity. In [47], Zhang et al. developed a numerical method to calculate the spatial and temporal probability function of a LOS UWOC link, considering up to one single scattering event per photon. The final mathematical expression is the result of directly applying Random Variable Transformation (RVT) to the link’s equations. The same authors continued this research line and presented in [48] a more generalized version of the aforementioned work, considering an integer number of scatterings. The authors do not depict any figure of the resulting probability density functions (pdf) and use the method as an alternative to the Monte Carlo simulation path loss estimation. 3.2 Modulations and Encodings Modulations and encodings are a critical aspect in any communications link. The way the information is sent in a changing environment defines the BER, the effective throughput and has also influence on the channel availability. Cochenour et al. studied in [49] the use of phase-coherent laser-based systems in turbid environments. The paper explores the use of different M-PSK schemes within a water tank. As it is usual in their contributions, the authors used different concentrations of Maalox to generate synthetic turbidities. The conclusions suggest that for relatively short distances (below 100 meters), coherent schemes are a feasible alternative in UWOC. However, 22 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.13: Correlation function of a laser-based link in a turbulent scenario the authors also manifest the necessity of further work in long-range environments in order to establish a clearer relationship between SNR, multipath and BER. Sui et al. studied in [50] different modulation approaches for turbid environments. Unlike Cochenour’s group, the authors performed a simulated study considering only extinction in their framework. This work neglects the stochastic nature of the underwater environment, and only considers attenuation. The compared schemes are OOK, FSK, DPSK (coherent), 4-PPM and 8-PPM. In[51], the same authors proposed a variation of a PPM schemes named SPPM (Shorten PPM). Nevertheless, the results does not suggest any real improvement respect to PPM, as it presents a lower spectral efficiency and BER performance, and it has higher power requirements. In [52], Yu et al. explored the use of FEC (Forward Error Correction) codes in UWOC. As the latter work, the random behavior of the channel response is not considered and its PDF (Probability Density Function) is defined as a Dirac’s delta. The results show the reduction of the BER after using this codes. The use of Optical OFDM in UWOC was experimentally explored by Minev et al. in [53]. The authors thoroughly describe the experimental setup and the OFDM scheme. The tested distances ranged from 0.5 meters to 3 meters, which is a very short distance to present a significant multipath effect so as to justify the use of OFDM. Furthermore, the authors obtained this conclusions observing that the ISI was negligible, as the delay spread is much lower than the symbol time. Figure 3.14 depicts the BER vs Es/N0curve of the experiment. Figure 3.14: BER curve of the experiment performed from 0.5 to 3 meters 23 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Gabriel et al. carried out an comparative work [54], similar to the one presented by Sui. In this case, the authors considered OOK, PPM, PWM and DPIM. The main conclusion of this paper is that although DPIM presents a higher demodulation complexity, its spectral efficiency is the highest and must be considered when designing an UWOC link. The authors remark the low PAPR of DPIM, but considering the associated transmission electronics, this aspect presents no actual advantage, as ON/OFF schemes may be driven by MOSFET-based circuits. The use of Spread Spectrum techniques has been also considered for UWOC. Akhoundi et al. presented an analytical model and a experimental evaluation of a CDMA system using OOC (Optical Orthogonal Codes) in [55]. One of the key aspects of the paper is the definition of the Optical Base Transceiver Station (OBTS), which comprises a series of photodiodes and LED, and the Optical Network Controller (ONC), that manages part of the network in a cellular way. The proposed system can be observed in Figure 3.15 Both downlink and uplink are defined to use OCDMA. Figure 3.16 shows a diagram of the experimental setup used to validate the system. Figure 3.15: Cellular OCDMA underwater network Figure 3.16: Experimental setup used to validate the proposed OCDMA underwater network. Virtex 4 FPGA’s were used to implement the system. This contribution does not deeply explore the use of different Spread Spectrum techniques, but is more focused on the network description and the MAC layer behavior. 3.3 Energy efficiency Energy efficiency is one of the more important aspects when considering UWOC, and much more when considering UWSN. The use of energy-efficient strategies to longer the life of the batteries in autonomous nodes is of capital importance as the battery replacement is often much expensive than simply deploying new nodes. Srinivasan et al. proposed a joint source-channel coding technique to reduce the average energy per symbol in [56]. Even though the authors propose the scheme for UWOC, the scheme is suitable for any sensor network. BaniHassan et al. presented in [57] a power control algorithm based on the previous work of their group in Optical CDMA networks [55]. The authors propose two different approaches to perform power control. The first strategy consists on sectoring to reduce the emitted power, so as to enhance the energy efficiency by limiting the emission to a determined solid angle. The other strategy is to define rings. This discretization allows a better power control assigning a transmission power to each ring. The authors also consider a joint ring-sector strategy. The ring-system is defined as open-loop, as the nodes estimate their position in a received power-basis. The main issue of this proposal is that each node must know the extinction coefficient of the medium. Furthermore, this coefficient changes in time and the method does not takes this into account. The authors estimate that in a combined scheme, up to 15.5 dB of power can be saved in cells with 50 m of radius. 24 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis 3.4 Network layer Network layer is important when considering UWSN. Fading events may occur in UWOC, and may produce harmful effects in the connectivity of the network. The following lines comment different contributions that have been made during the last few years. Liu presented in [58] a paper where a topology-recovery strategy was presented. The author proposed the technique for an ultrasound-based network, but the proposal can be easily extrapolated to the optical domain. The recovery procedure is based on two main steps. When the connectivity is destroyed, the reference node increments its transmission power. If this action does not solve the problem, then one of the mobile elements of the network (an AUV i.e.), is moved toward the location of the lost node. Hu and Fei proposed in [59] a multilayer routing protocol for hybrid opto-acoustical networks. In this paper, the use of reinforcement learning, specifically Q-learning, is explored as an alternative in routing decision for multihop UWSN. The proposed network topology consists in an acoustic backbone and optical clusters conformed by remote optical nodes. The upper-layer (acoustic) nodes manage the optical clusters allowing fast intra-cluster routing. The results suggests that this layering strategy allows a lower latency network as well as a more energy-efficient alternative. Figure 3.17 shows the packet delivery rate, the delay and the inter-layer overhead respect to the packet rate. It can be observed that the delivery rate is almost independent of the packet rate in a layered system. Furthermore, the delay remains below the unlayered case at any packet rate. Figure 3.17: Delivery rate (left), delay (center) and inter-layer overhead (right) vs packet rate in an interlayer Q-learning routing scenario. Mora et al. implemented in [60] an ad-hoc multihop optical network using a TDMA-based MAC layer. The network is dynamically conformed using a tree-based management strategy. The main disadvantage of the system is that every node has the full information of the network, which may incur in large delays and less energy efficiency. A connectivity analysis was performed by Vavoulas et al. in [61]. The authors studied the node density requirements (number of nodes per linear meter) in order to establish a k-connectivity network. This requirements depend on the emitted optical power, the channel path loss and the wavelength. Furthermore, the authors considered an isotropic radiation patter, which is mostly improbable in an optical wireless transceiver. Figure 3.18 depicts some of the obtained results. 3.5 Applications As it was mentioned at the beginning of this chapter, the development of a topic is normally bound to the theoretical analysis and the push of the experimental evaluations. In this section, different applications of the UWOC technology are presented. The following taxonomy has been applied in order to sort the papers. •Hardware design •Underwater Wireless Sensor Networks •Applications for mobile systems •Other applications 3.5.1 Hardware design Anguita et al. implemented an IEEE 802.15.4 compatible PHY and MAC layer. In [62] presented the physical layer, which was implemented in a Spartan 3 FPGA, using LED devices and PPM modulation. Figure 3.19 shows the experimental setup used to validate the system. 25 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.18: Required node density to achieve node isolation probability equal to 10−4versus (a) operating wavelength, (b) transmitted power, (c) datarate, and (d) chlorophyll concentration for BER = 10−3(dashed–dotted line), BER = 10−6(dotted line), and BER = 10−9(continuous line). Figure 3.19: Experimental setup used by Anguita et al. to validate their FPGA implementations The corresponding MAC layer implementation was presented in [63], transforming the former transmitter from directional to omnidirectional, as shows Figure 3.20. The proposed MAC protocol was CSMA/CA. Finally, the whole system was evaluated in [64]. Figure 3.21 depicts the block diagram of the system. The main contribution of this works is the application of an existing standard to the underwater medium. Doniec and Rus presented AquaOptical II in [65]. This bidirectional system is based on an array of LED and an avalanche photodiode to perform transmission and reception respectively. The system achieved a 2.28 Mbps using DPIM encoding at a distance of 50 meters with a estimated SNR of 5.1 dB. Figure 3.22 shows the design of this transceivers. In [66], this device was used to perform a robust underwater video-streaming link. Doniec et al. performed 26 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.20: Used planar emitter Figure 3.21: Block diagram of the IEEE 802.15.4 compatible VHDL implementation tests up to 40 meters, using different video frame encodings, qualities and speeds. Figure 3.23 shows the obtained results regarding the interval between packets. Destrez et al. implemented an hemispheric LED emitter using 12 LED [67]. The main objective of the work was to emulate an hemispheric radiation pattern, but also electronic design considerations regarding both emitter and receiver are presented. Even though the authors theorize speeds up to 50 Mbps, the experiments were performed at 1 Mbps using a traditional OOK encoding, showing poor BER performance at distances above 2.5 meters. Swathi and Prince presented in [68] an study regarding considerations on UWOC transceiver design. The authors enumerate the influence of each design parameter into the system performance, but the main conclusion is that the authors propose the use of adaptive transceivers to take advantage of the channel’s characteristics. Although the authors do not expose it literally, this means the use of wavelength-selectable emitters. Depending on the suspended-matter concentration and link’s distance, an optimum wavelength may be estimated. In [69], Tang et al. presented different considerations on the use of APD receivers in UWOC. The main tradeoff of using APD is their inner gain, and a more compact size and higher quantum efficiency compared to PMT. An 27 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.22: Aquaoptical II transceivers Figure 3.23: Interval between packets for links ranging from 10 to 25 meters (left), and for links ranging from 25 to 40 meters (right). interesting gain control scheme is presented in order to maximize the SER. As the gain of an APD affects the SNR, for each scenario there is an optimal gain that maximizes this parameter. The authors presented a closed-form expression of the optimal gain, which depends on several parameters. Figure 3.24 depicts the optimal gain for a given example scenario. It can be observed that the optimal gain depends almost only on the link range. Tian et al. implemented an UWOC link as the one shown in Figure 3.25 [70]. The LED driver is based on a non-inverting topology followed by a BJT-based switch. The receiver side implements a transimpedance amplifier followed by an inverting second stage. The tests, performed in a swimming pool, demonstrated communication distances between 20 and 30 meters. Cossu et al. demonstrated a 2.5 meters link using a 40 dBm optical source at 470 nm in clear water [71]. Two receiver schemes were used, one based on APD and other in a PIN photodiode. This work does not provide any novel aspect regarding hardware design, but the authors performed an analysis of the impact of daylight into the BER performance. As the experiment were carried out in a pool, the near-surface condition of the link made it very sensitive to the Sun’s position, as Figure 3.26 shows. 3.5.2 Underwater Wireless Sensor Networks In 2006, Farr et al. proposed a laser-based underwater wireless optical modem [72]. In this work, the authors proposed the use of wireless optical technology using visible-light to establish a link in the underwater medium. The work covers the main aspects in the design of a link of this characteristics: transmitter devices, optics, 28 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.24: Optimal APD gain to maximize SER in a given scenario. The offset distance is the distance from the emitter’s axis. Figure 3.25: Block diagram of the system implemented by Tian et al. Figure 3.26: Impact of daylight in the BER performance for a near-surface link receiver design, phenomena, application scenarios, etcetera. The work proposes three different solutions to establish communications between a vehicle and a fixed node, using combinations of directional and omnidirectional emitters 29 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis and receivers. As it is usual in the literature, the proposed receiver is PMT-based. Simpson et al. implemented an underwater node that retrieves data from an ultrasonic receiver, encodes it using a return-to-zero Reed-Solomon code and transmits it using a high-power LED [73]. The authors demonstrated the feasibility of low-power and cost-effective UWSN nodes achieving datarates up to 5 Mbps. The works presented by Anguita et al. regarding hardware implementations where evaluated by simulation in terms of achievable data rate in [74]. Medium-related aspects such as turbidity were considered in the simulations. Figure 3.27 depicts the obtained maximum data rate for the aforementioned planar emitter. Figure 3.27: Maximum achievable data rate of Anguita’s planar emitter Farr et al. deployed an UWSN node in the northeast Pacific Ocean [75]. The main objective of the experiment was to gather geochemical data, but due to the high amount of data to collect, ultrasonic link were not feasible. The implemented system had an ultrasonic channel for telecommand, whilst the optical channel was used to download data up to 5 Mbps at a distance of 61 meters. Figure 3.28 depicts a concept artwork of the operation. Johnson et al. made other contribution on hybrid optical-acoustical network approaches [76]. The authors expose the advantages of using an acoustical downlink and highly-directive optical uplinks at the UWSN nodes. However, the authors claim that this kind of structure presents several disadvantages, such as power-limited range and sensitivity to refractive gradients (as it is a vertical link). As Cochenour demonstrated in his experimental works, due to the BSF, misalignment and refractive gradients decrease their effect with distance. Finally, the power limitation of laser emission due to fauna’s eye-safety is mentioned in this paper, being the first work in address this issue regarding UWSN. 3.5.3 Applications for mobile systems Fung et al. implemented an UWOC system for robotic swarms in [77]. The robots carried out a multi-channel algorithm in order to propagate the information within the swarm. The communications was performed transmitting a serial port at 115 Kbps through a green laser. The authors complaint about the necessity of pointing in the system. Nonetheless, at the tested distances, an LED-based system would have been a trade-off regarding reliability. The use of laser devices was justified imposing the necessity of long range communications. However, only at actual long distances with significant particle concentrations, laser-based system relax their maximum misalignment error as Cochenour suggested in [16]. Doniec et al. evaluated the use of UWOC for robot operation [78]. The aim of the experiment was to perform a cable replacement to control the robot. Normally, this kind of robots are tethered to a base station, limiting the maximum range of operation and its maneuverability. The obtained delays were not distinguishable from a tethered version of the experiment. Furthermore, the implemented optical devices were capable of establishing a 200-meters link in air at night, a 30-meters link in a pool, and a 7-meters link in a high-turbidity harbor environment. The merit figure used by the authors to measure the quality of the link was the packet delay since the last update of the IMU. Figure 3.29 depicts the obtained results for two different environments: low SNR and high SNR. Gao and Guo presented in [79] an implementation of a remotely operated microrobot. The robot implementation were intended to work under the command of a mother submarine. Nonetheless, the authors used infrared light to perform communication, limiting the maximum allowed distance of operation to a few dozens of centimeters. 30 Chapter 3. Underwater Wireless Optical Communications: a thorough analysis Figure 3.28: Concept of the system implemented by Farr et al. Figure 3.29: Packet delay for two different environmental conditions Rust and Asada proposed a dual use of visible light to provide positioning plus communication to a robot for nuclear reactor inspection [80]. The positioning system is based on the signal strength and an estimation of the angle based on a photodiode array at the gateway station. Furthermore, incoming data from the ROV’s IMU is used to filter the estimations. The system has not been tested in an real scenario, but has only been treated under simulations. The use of UWOC in a heavy water environment has not been reported yet, and possible the extinction coefficient would be different from the distilled water estimations present in the literature. In addition, effects such as Cherenkov radiation must be taken into account as background noise terms if a minimum link quality is intended to be provided. Bowen et al. implemented an untethered ROV based on a hybrid optoacoustical approach [81]. The need of high bandwidth and low latency for ROV manipulation makes optical communication the best alternative. In this work, the communications system of a Nereus ROV was modified to include an optical subsystem. Although the uplink was implement with the ultrasonic modem, a TDMA half-duplex access protocol was proposed to eliminate the necessity of the acoustic part, which adds a significant latency to the system. In [82], Han et al. evaluated through simulation the performance, in terms of energy efficiency and throughput, a hybrid optical-acoustical system for AUV communications and positions, as it is usual in the literature. The proposed scheme does not provide a significant enhancement of the throughput respect to the lower energy efficiency due to the use of the acoustic part. Regarding positioning, optical-based systems are not addressed, and the 31 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel submerged particles within the medium, and only numerical solutions are feasible. The Schuster-Schwarzschild approximation [113] proposes the division of Equation 4.1 into two main streams (inward and outward directions), but this algebraic artifact can be generalized into 2ndirections as follows: cos θi κ(ω)ρ dI(ω, z, θi, t) dz=I(ω, z, θi, t)−1 2X j ajp(cos θi,cos θj)I(ω, z, cos θj, t)/i =±1, ..., ±n(4.3) ajis the j-th Gauss-Legendre quadrature weight. This last simplification is valid for any C(1) phase function which can be expressed as a (2n-1)-order polynomial. This system of linear differential equations can be easily solved numerically. Nonetheless, the accuracy of the solution depends on the number of directions involved in the calculation. Furthermore, the division using the Gauss-Legendre quadrature can be reformulated using other type of numerical scheme, for instance, Monte Carlo. This integration scheme will be discussed during Section 4.3. 4.2 Channel characteristics The RTE defines the propagation of EM waves through spatially-dispersive media. Seawater is conformed by molecules of water and a mixture of biological matter (phytoplankton in the vast majority) and dissolved salts. Depending on the concentration of each type of additive, the effects on the propagation differ. The following subsections comment each effect separately. 4.2.1 Absorption Absorption is a wavelength-dependent process where electromagnetic energy is converted into other types of energy, typically heat or chemical. It is of capital importance because it defines the decay of the propagating energy through seawater, and hence, has a direct impact on the amount of photons that arrive the receiver. Since seawater is a mixture of different elements apart from pure seawater, the overall absorption can be expressed as a sum of partial absorptive contributions (Equation 4.4). a(λ) = N X i=1 Ciai(λ) (4.4) Generally, the considered absorptive contributions are: pure seawater (αw(λ)), phytoplankton (αφ(λ)), gelbstoff (αg(λ), decaying organic matter) and non-algal matter (αn(λ)). Figure 4.1 depicts the spectral response of these components. Wavelength (nm) 400 500 600 700 αw(λ) (m−1) 0 0.2 0.4 0.6 0.8 Wavelength (nm) 400 500 600 700 αφ(λ) (m2·mg−1) 0 0.02 0.04 0.06 Wavelength (nm) 400 500 600 700 αw(λ) (m−1) 0 0.5 1 Wavelength (nm) 400 500 600 700 αw(λ) (m−1) 0 0.5 1 Figure 4.1: Absorption spectra of each element that conform seawater. Seawater (NW - [114]), Phytoplankton (NE - [115]), Gelbstoff (SW - [116]) and Non-algal matter (SE - [117]) The concentration of each of the components conform the overall absorption spectrum. Figure 4.2 depicts an example of absorption spectra for open ocean and coastal waters. It must be taken into account that the relative concentrations are depth dependent, as Johson et al. studied in [29]. This fact has a direct impact on the design of vertical links, where the light rays cross different concentration 38 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Wavelength (nm) 400 500 600 700 α(λ) (m−1) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Coastal water Wavelength (nm) 400 500 600 700 α(λ) (m−1) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Open ocean water Figure 4.2: Absorption spectra of open ocean (left) and coastal water (right). layers. Finally, Haltrin [118] derived a model describing the absorption in terms of a single parameters, chlorophyll concentration, Cφ. Equation 4.5 illustrates the expression. α(λ) = αw(λ) + αφ(λ) Cφ C∗ φ!0.6 +αf(λ)Cfe−kfλ+ah(λ)Che−khλ(4.5) C∗ φ= 1 mg·m−3.αφ(λ), αf(λ) and αh(λ) are the phytoplankton, fulvic acid and humic acid absorption spectra respectively. Cfand Chare the fulvic and humic acid concentrations, and kfand khtheir spectral decaying constants. 4.2.2 Scattering Scattering is a physical process in which energy is spatially dispersed due to light-matter interaction. Figure 4.3 illustrates this phenomenon. Figure 4.3: Representation of the scattering phenomenon Depending on the relative size of the scattering centers, different approaches may be used to model the phenomenon. The relative size of a particle, a, is the relationship between its diameter and the wavelength of interest (Equation 4.6). Dis the diameter of the particle, λthe wavelength in vacuum and nwthe refractive index of the surrounding medium. a=πDnw λ(4.6) For very small relative sizes (a << 1), Rayleigh scattering is used to model this phenomenon. For near unitary relative sizes (a≈1), Mie’s approximation of Maxwell’s equations is used to describe scattering. Finally, very high relative sizes (a >> 1) produce very small scattering effects and their behavior can be explained by geometric optics. In this work, since the scattering centers are usually conformed by phytoplankton, and the wavelengths of interest are next to the size of these particles, only Mie scattering is going to be considered. 39 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Mie scattering Mie’s theory is the most general solution to the elastic scattering problem. Its applicability ranges from Rayleigh’s approximation for small particles to geometric optics on the limit, but is limited to spherical particles. However, the spherical assumption is not a problem regarding the error bias it may produce in a communications estimation problem. Lets consider a spherical particle of diameter Dand refractive index m=nr−jκ. The imaginary part of the refractive index is related to the absorption coefficient of the material (amat(λ)) through Equation 4.7. αmat(λ) = 4πκ λ(4.7) Although Mie’s theory defines the scattering process taking into account the polarization state, and since this work is oriented to the use of unpolarized radiation (LED emissions), this introductory formulation is presented in its horizontal-vertical averaged version. In a spherical coordinate system, the relationship between the scattered intensity and the incident intensity after scattering, is related to a function σ0 scat(θ) as shows Equation 4.8. Iscat(θ) = I0 σ0 scat(θ) r2(4.8) Note that there is no azimuthal (φ) dependency as the surrounding medium is isotropic and the incident light is unpolarized. Taking into account the energy transformations involved in the process, the extinction cross section σext, the scattering cross section σscat and the absorption cross section σabs are related by Equation 4.9. σext =σabs +σscat (4.9) Furthermore, the differential cross section σ0 scat(θ) and the cross section σscat are related by a integration over 4π steradians. In Mie’s equations, the differential cross section can be expressed as a combination of terms. Equation 4.10 starts the mathematical description. σ0 scat(θ) = λ2 8π2(i1+i2) (4.10) In this formulation, the intensity functions are calculated from an infinite series given by: i1= ∞ X n=1 2n+ 1 n(n+ 1) [anπn(cos θ) + bnτn(cos θ)] 2 i2= ∞ X n=1 2n+ 1 n(n+ 1) [anτn(cos θ) + bnπn(cos θ)] 2 (4.11) The angular dependent functions πnand τnof Equation set 4.11 are expressed in terms of the Legendre polynomials by: πn(cos θ) = P(1) n(cos θ) sin θ τn(cos θ) = dP(1) n(cos θ) dθ(4.12) Finally, the parameters anand bnare defined in terms of the Ricatti-Bessel functions Ψ and ξas shows Equation set 4.13. an=Ψn(a)Ψ0 n(ma)−mΨn(ma)Ψ0 n(a) ξn(a)Ψ0 n(ma)−mΨn(ma)ξ0 n(a) bn=mΨn(a)Ψ0 n(ma)−Ψn(ma)Ψ0 n(a) mξn(a)Ψ0 n(ma)−Ψn(ma)ξ0 n(a)(4.13) Finally, after integrating the differential scattering cross section over the sphere, the resulting scattering cross section may be expressed as: σscat =λ2 2π ∞ X n=0 (2n+ 1) |an|2+|bn|2(4.14) 40 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel The above equations express the angular dependence of scattering and hence, the volume scattering function. As it was commented, this function only depends on elevation and has revolution symmetry for spherical particles immersed in an isotropic medium such as seawater. Figure 4.4 depicts the phase function of a particle for a given wavelength and refractive index. 0.2 0.4 0.6 0.8 1 30 210 60 240 90 270 120 300 150 330 180 0 Phase function Figure 4.4: Phase function of a Mie scattering for a particle with unitary normalized size Volume Scattering Function The volume scattering function is the differential scattering cross section per unit volume. Using the above nomenclature, the VSF β(θ, λ) can be expressed as the volume derivative of the scattered intensity, respect to the incident intensity per unit area. β(θ, λ) = dIscat(θ, λ) Ei(λ)dV(4.15) Where Ei(λ) is the incident irradiance. This function is important because it takes into consideration the particle concentration of the medium and integrating it over all direction yields the scattering coefficient b(λ). This coefficient and the absorption coefficient presented above conform the overall extinction coefficient as: c(λ) = a(λ) + b(λ) b(λ) = 2πZπ/2 0 β(θ, λ) sin θdθ(4.16) The VSF can be rewritten as the product of the scattering coefficient β(λ) and a phase function ˜ β(θ, λ) which defines the angular distribution of the scattered radiation. This phase function can be characterized by the asymmetry parameter gor mean cosine, which is the average of the cosine of the scattering angle over all scattering directions (Equation 4.17). g= 2πZπ 0 ˜ β(θ, λ) cos θsin θdθ(4.17) This parameter models the “shape” of the phase function and has been widely used in simulation schemes through the Henyey-Greenstein approximation as it was commented in Chapter 3. This approximation of phase 41 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel functions is not able to model the “peaky” nature of Mie’s scattering for relative sizes near to one. In addition, Zhao and Sun demonstrated in [38] the poor accuracy of this approximation. Nonetheless, several authors use Henyey-Greenstein function (Equation 3.3) because its integral has inverse and can be used in a closed form to generate random directions in a Monte Carlo integration scheme. 4.2.3 Refractive Index The refractive index of a material is the relationship between the speed of light in it, respect to the speed of light in vacuum. This parameter defines the delay of communication, and also has an important role in the behavior of reflections (Snell’s law) and turbulences. Generally, the refractive index of seawater, noted by nw, depends on temperature, salinity and wavelength. The wavelength-dependence implies chromatic dispersion, but due to the use of monochromatic sources and the reduced range of UWOC links, this effect can be neglected. Actually, this effect can be neglected even if wideband emitters, such as YB-WLED, were used, due to the effect of absorption. Several authors have empirically fitted this dependence after performing different measurements. McNeil [119], in Equation 4.18, and Matth¨aus [120], in Equation 4.19, performed two different attempts to obtain a mathematical expression of the refractive index of seawater within the visible spectrum. nw(λ, S, T)=1.3247 −2.5·10−6T2+S2·10−4−8·10−7T+3300 λ2−3.2·107 λ4(4.18) nw(λ, S, T) =1.447824 + 3.011 ·10−4S−1.8029 ·10−5T−1.6916 ·10−6T2−0.489λ+ 0.728λ2−0.384λ3− S7.9362 ·10−7T−8.06 ·10−9T2+ 4.249 ·10−4λ−5.847 ·10−4λ2+ 2.812 ·10−4λ3(4.19) Temperature Tis in Celsius degrees, salinity Sis in h, and the wavelength in micrometers. Figure 4.5 depicts the refractive index of seawater at 15◦and 30 hof salinity for each wavelength. From the above equations , it can be observed that the refractive index of seawater is more sensitive to salinity than to temperature fluctuations. Wavelength (nm) 400 450 500 550 600 650 700 nw(S, T, λ) 1.335 1.34 1.345 1.35 Figure 4.5: Refractive index vs wavelength for seawater at 15◦and 30 hof salinity Johnson et al. studied the relationship between depth and refractive index in order to estimate the expected bending of the light rays [121], and its effect regarding the pointing between emitter and receiver. However, this estimation departs from a unrealistic hypothesis, since it considers a stratified scenario regarding refractive index. This assumption directly implies an also stratified distribution of temperature and salinity, whilst an actual scenario cannot be considered as a laminar perfectly-separable scenario. Regardless this physical unrealism, due to the lack of literature analyzing this issue, their approximation may be considered as a valid starting point. 4.2.4 Turbulences Optical turbulence is a rapid change of the seawater’s refractive index. These sharp changes may occur at any depth, and are normally attributed to temperature variations. Unlike turbulences in FSO, the power spectrum of the refractive index depends on both temperature and salinity. Therefore, the traditional Kolmogorov power 42 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel spectrum may present errors modeling turbulences in oceanic environments, as Tang et al. stated in [46]. After introducing the effect of salinity, it yields the power spectrum proposed by Nikishov et al. [122]: Φ(κ) = C0−1/3κ−11/3χT ω2h1+2.35(κη)2/3iφ(κ) (4.20) C0= 3.88 ·10−9,is the dissipation rate of turbulent kinetic energy per unit mass and ranges from 10−8to 10−2m2s−3, and χTis the rate of mean-square temperature dissipation ranging from 10−10 to 10−4K2s−1. These constants and ranges are based on the measurements provided by Korotkova et al. [123]. κis the scalar spatial frequency, η= 10−3is Kolmogorov’s micro-scale and ωis a parameter that defines the dominance of the turbulence respect to salinity (ωnear to 0) or temperature (ωnear to -5). Finally, φ(κ) is a function of the form presented in Equation 4.21. φ(κ) = ω2e−ATδ+e−ASδ−2ωe−AT S δ(4.21) AS= 1.9·10−4,AT= 1.863 ·10−2and AT S = 9.41 ·10−3.δ= 8.284(κη)4/3+ 12.978(κη)2. After numerically solving this equation for a general case (ω=−2), the resulting power spectrum presents two peaks which are related to temperature and salinity fluctuations (Figure 4.6). Figure 4.6: Turbulence power spectrum obtained by Tang et al. [46] for ω=−2 As it can be observed, the oceanic turbulence power spectrum is very different from Kolmogorov’s −11/3 power law used in FSO. There are two main aspects of interest regarding turbulence analysis in communications. The first one is the turbulence variance related to the link’s range (Rytov’s variance σ2 I), and the second one is the temporal correlation. Rytov’s variance, also known as scintillation index, was studied by Farwell in his PhD dissertation [124]. Farwell concluded that for gaussian-like laser emissions, the scintillation index takes the form presented in Equation set 4.22. σ2 I(r, L) =σ2 I,l(0, L) + σ2 I,l(r, L) σ2 I,l(0, L) =8π2k2LZ1 0Z∞ 0 κΦ(κ)exp −ΛLκ2ξ2 k1−cos Lκ2 kξ(1 −ξ(1 −Θ))dκdξ σ2 I,l(r, L) =8π2k2LZ1 0Z∞ 0 κΦ(κ)exp −ΛLκ2ξ2 k[I0(2Λrξκ)−1]dκdξ(4.22) ξ= 1 −z Lis a normalized distance variable and kthe wavenumber. I0(x) is the zeroth order modified Bessel function whilst Λ and Θ are the output plane parameters. For an incident plane wave, these last parameters are 0 and 1 respectively. Introducing this condition into Equation 4.22, it yields: σ2 I(L)=8π2k2LZ∞ 0 κΦ(κ)dκ−Z1 0Z∞ 0 κΦ(κ) cos Lκ2 kξdκdξ(4.23) 43 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Farwell left the analysis at this point, but taking into account the shape of the sinc in the second term, the integral may be approximated to the first root of the sinc, hence: σ2 I(L)≈8π2k2LZ∞ √πk L κΦ(κ)dκ(4.24) To obtain an approximation of the scintillation index, the integrals resulting of the product κΦ(κ) are of the form shown at Equation 4.26. Depending on the condition of Equation 4.25, δcan be approximated to the squared term or to the 4/3 power. These two approximations could be applied depending on the shape of the κΦ(κ) product. In the end, the approximations define an upper and a lower bound to the power dependence of the scintillation index. δ≈   12.978(κη)2/ κη > 9.12 8.284(κη)4/3/ κη < 1.3·10−5 8.284(κη)4/3+ 12.978(κη)2/otherwise (4.25) Z∞ √πk L κ−ae−bκcdκ=ba−1 c cΓ1−a c, b (πk)c/2L−c/2(4.26) Where Γ(s, x) is the upper incomplete Gamma function, and Γ(s, x)=(s−1)Γ(s−1, x) + xs−1e−x. Taking into account the decaying nature of the exponentials involved in the latter expressions, the Rytov’s variance of the ocean turbulent channel would directly depend on the higher power of the link’s range L. Taking into account the approximations made in Equation 4.25, the dependence of the scintillation index respect to the link’s range can be approximated to: σ2 I(L)∝Lα/α ∈(3/2,11/6) (4.27) After imposing Taylor’s frozen turbulence hypothesis, which implies that the temporal statistics are related to the spatial statistics, Tang et al. analyzed the temporal correlation of a laser-based emission under weak turbulence regime. They obtained that the correlation is almost independent on the link’s distance, whilst is highly dependent on the speed of seawater. These formulae were obtained considering plane wave propagation, coherent radiation and neglecting the effect of scattering. If scattering were considered, the correlation between points would change due to the BSF. Furthermore, if the radiation were non-coherent, such as LED, Rytov’s variance would be dramatically reduced since interference effects would not occur. Furthermore, collimated and coherent emissions would suffer from beam wander with a higher probability, whilst LED emissions not. 4.2.5 Surface reflections The ocean surface can be modeled as a superpositions of traveling sea waves, usually modeled by a sea wave spectrum, plus a noisy term which depends on the wind stress. This stress produces a random variation on the seawater’s surface, whose PDF follows a Gram-Charlier series as Cox and Munk demonstrated in [21]. The variation of the surface is usually modeled azimuthally uniform, and the variance of the slope depends on the wind speed. The relationship that can be obtained from Cox and Munk’s work regarding wind speed is presented in Equation 4.28. Actually, the variance for the crosswind direction and the upwind direction slightly differ, but for the purposes of this work the omnidirectional approximation is valid. σ2 surf = 0.003 + 0.00512vwind /1< vwind <14m ·s−1(4.28) In an underwater-to-underwater scenario, random surface reflections may introduce delayed additional power contributions in a horizontal link. Figure 4.7 depicts the scenario under consideration. After a reflection, the direction of the output ray ˆvref can be expressed as a linear combination of both surface normal vector ˆnsurf and incident direction ˆvi. ˆvref = ˆvi−2<ˆvi,ˆnsurf >ˆnsurf (4.29) After the impact, part of the incident power is reflected and part is transmitted. The amount of energy which is reflected, RF resnel(θi), follows Fresnel’s equation (Equation 4.30). Additionally, for incident angles above certain limit (critical angle), there is no transmission of energy and total internal reflection occur. This happens because the light rays travel to a medium with a lower refractive index, in the opposite case, there is no critical angle and always part of the energy is transmitted. 44 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Figure 4.7: Reflection of light rays on the seawater surface RF resnel(θi) = 1 2"sin(θt−θi) sin(θt+θi)2 +tan(θt−θi) tan(θt+θi)2#(4.30) θtis the refracted angle, which follows Snell’s law nair sin θt=nwsin θi. 4.2.6 Seabed diffusion Diffusive effects are a common phenomenon which is produced due to the scattering of light on a surface. There are many scattering functions which model the diffusion depending on the presence or not of specular components, such as Phong or Torrance. However, as the seabed is normally compound by sand, rock or coral extensions, a Lambertian approximation is a compromise solution. Equation 4.31 shows the reflectivity of a Lambertian scattering, whilst Figure 4.8 depicts the scenario. Figure 4.8: Light scattering due to seabed diffusion Rseabed(θ, θi, ρseabed) = ρseabed ·cos θi·cos θ(4.31) 4.2.7 Optical fouling Optical fouling is the deposition of matter, normally algae and phytoplankton, over the transparent shielding of optical emitters in UWOC applications. This deposition has a direct effect on the effective power radiated to the medium. Depending on the matter spectral response, the deposition density and its thickness, the power loss can be modeled as an exponential decay (Equation 4.32). The fouling resistance, β, is the rate at which the thickness of the fouling, τ(t) increases. αdepends on the fouling density and the type of particle, whilst βdepends on the type of particle, the water speed, the temperature of the surface and the bulk water temperature [125]. Ptx,eff =Ptxe−α(ρfouling,λ)τ(t)(4.32) There are different types of fouling resistances depending on the nature of the deposited matter. Generally, linear, falling and asymptotic curves are used to model this phenomenon. The behavior of the fouling rate is related to the deposition rate and the removal rate, being the difference of both. Normally, biological matter follows an asymptotic curve, and Equation 4.33 shows the mathematical description of the thickness in terms of the fouling resistance. τ(t) = τmax 1−e−βt(4.33) τmax is the asymptotic maximum of the thickness. Including Equation 4.33 into Equation 4.32, it yields: Ptx,eff =Ptxe−α(ρfouling,λ)τmax(1−e−βt)(4.34) Figure 4.9 depicts, for illustrative purposes, the effect of fouling in the effective emitted power. αfouling can be expressed in terms of the mass density of chlorophyll and its absorption spectrum, using Equation 4.5. 45 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (h) 0 10 20 30 40 50 60 70 80 90 100 Power Loss (dB) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 Figure 4.9: Effect of optical fouling on the effective emitted power. Ptx = 1 W, β= 0.1 h−1,τmax = 10−2 m and α(ρfouling, λ) = 75.71 m−1with Cφ= 1 g ·m−3 4.2.8 Fauna The effects of fauna on the performance of an UWOC link have not been studied in depth yet. Nonetheless, it would presumably depend on the concentration of fauna, the size of the individuals, its mobility and its awareness respect to the used wavelength. 4.3 Simulation of the UWOC impulse response The RTT equation is difficult to solve, and the task becomes harder when the random nature of the scattering by particles is introduced. However, Monte Carlo integration offers a simple and computationally efficient alternative to obtain a solution to the time-dependent RTT equation in any scenario. Analytical solutions can be obtained without taking into account neither the randomness of the medium nor the effect of the seawater surface or seabed, as Jaruwatanadilok carried out in [33]. The main objective of this simulation process is to obtain a representation of the UWOC impulse response and perform an analysis relating physical parameters to communication-performance parameters, such as channel gain and bandwidth. 4.3.1 Simulated scenario The simulated scenario consists in a three-dimensional volume bounded in the Y and Z axes. The Y-limits are the seabed and the seawater surface, whilst the Z-axis boundaries are the emission and reception planes. The phenomena included in the simulation process are absorption, scattering by particles, diffusive effects on the seabed and surface reflections. Turbulences and fauna have not been considered because of their stochastic nature. In a stochastic model, these effects can be included as multiplicative factors, since they present statistical independence. Furthermore, LED emission is considered and hence polarization and coherence are neglected, easing the calculation. Figure 4.10 depicts the effects considered in the simulation as well as the geometry of the problem. It can be observed that the emitter point is the coordinate origin. Therefore, the depth of the link is parametrized as a positive value, whilst the seabed’s depth is a negative value. 4.3.2 Monte Carlo integration scheme Monte Carlo integration is a numerical integration technique which randomly generates points within the integration domain. Let’s consider the general multidimensional definite integral in the domain Ω ∈Rnshown at Equation 4.35. I=ZΩ f(~x)d~x (4.35) The integration domain has a hyper-volume Vof the form: 46 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Figure 4.10: Simulated scenario V=ZΩ d~x (4.36) Finally, Monte Carlo integration states that the integral of Equation 4.35 can be approximated to a summation of the form: I≈V N N X i=1 f(~xn) (4.37) Where ~xnare random sample points within the domain Ω. This type of integration is particularly useful when dealing with complex problems such as the one under consideration in this work. The UWOC impulse response can be approximated using a Monte Carlo approach over the RTT equation (Equation 4.1). This leads to an expression of the form: h(t)≈ M X i=1 Piδt−nw(λ) c0 di(4.38) Where Piis the weighed contribution of the i-th arriving ray, diis the traveled distance, and Mis the number of effective contributions to the impulse response. Equation 4.39 shows how Piis obtained, and Equation 4.40 defines the delay suffered by an arriving ray. Pi=Ptx Ne−α(λ)di Nh Y j=1 Lj(4.39) di= Nh X j=1 dj(4.40) Nis the number of rays generated at the emitter. Intuitively, the radiation pattern is divided in N differential solid angles, resulting in a random generation of directions over the emitter’s hemisphere. Each ray is then scattered in many directions depending on the multiple random collisions due to particles and the seabed. Note that M6=N because after each collision with particles, a new bundle of rays is generated. Nhis the number of hops (collision events) that a ray has made before arriving the receiver, djis the traveled distance after the j-th collision, and Lj is the j-th power weighing applied to a ray. These last weights are classified as follows. 47 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel •GPU parallel code. Parallel computed code must be embedded within a CUDA kernel. This code executes its inner instructions in the GPU using N blocks of threads and M threads per block. Calls to kernels must be carried out as kernel <<<Blocks,Threads>>>(args). •Device functions. GPU code cannot use host-defined functions. In order to introduce auxiliar functions into the GPU, these must be defined as device . •Avoid conditional branches. As the used GPU executes the code using SIMD, when a conditional branch occurs (actually a dependence on data), some threads are deactivated and queued, blocking the execution. •Kernels are not blocking. When the host calls a kernel, the GPU starts the computation and the host is not blocked, allowing the possibility to perform some calculations until the GPU ends. The host only is blocked when a device-to-host memory operation is requested. The actual parallelization was performed defining the initial random ray generation (for loop) as a kernel, whilst function growAndHarvestTree() as a device function. Furthermore, CUDA 2.0 did not offer a random number generator in GPU. Because of that, a simple random number generator of period 220 [130] was implemented as a device function. Each thread is initialized with a host-generated random seed. Finally, all the host-version functions which used memcpy() operations were transformed to a literal version (for loop with write sentence) since the GPU ignored host-defined operations. 4.4 Simulation results In this section, several results are obtained from the implemented algorithm. As communication-performance characteristics, channel gain and bandwidth are calculated from the obtained h(t). Nonetheless, unlike indoor OWC channels, where the impulse response may be considered fixed; in the case of UWOC, the scenario is continuously varying due to movement of particles and the seawater’s surface. Furthermore, the pointing between underwater emitters and receivers is not perfect and presents a variation which depends on the marine currents. This effect has been neglected for simplicity, but as well as turbulence and fauna, it can be included as a multiplicative factor in an stochastic description of the UWOC channel. Channel gain is defined as: H(0) = Z∞ 0 h(t) dt(4.50) In the case of the Monte Carlo integration, using Equation 4.38 and including it into Equation 4.50,it yields Equation 4.51. From an implementation-related viewpoint, this magnitude can be obtained as the sum of all the contributions saved in the 1-femtosecond-sampled h(t) array. H(0) ≈ NT−1 X i=0 Pi(4.51) Regarding bandwidth, it is related to delay spread by Equation 4.52. B≈1 5τrms (4.52) Where τrms is the delay spread of the impulse response, defined by Equation set 4.53 τrms =s1 H(0) Z∞ 0 (τ−¯τ)2h(τ)dτ ¯τ=1 H(0) Z∞ 0 τh(τ)dτ(4.53) The actual implementation uses the same concept as Equation 4.51, converting integrals to sums as shows Equation set 4.54 τrms ≈10−15v u u t 1 H(0) NT−1 X i=1 i2Pi!−¯ i2 ¯ i≈H(0) NT−1 X i=1 iPi(4.54) 54 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel In order to analyze the effect of the geometrical and physical link’s parameters on the impulse response, a baseline scenario was defined. Table 4.1 shows the parameters. Parameter Value Receiver’s position (0,0,5 m) Photodiode’s area Apd = 20 mm2 Directivity m= 1 Transmitted power 1 W Wavelength 470 nm Particle radius 500 nm Particle refractive index 1.25 (lossless) Concentration of particles 3.3·1011m−3 Distance to surface 5 m Agitation of the surface σ2 surf = 0.0081 (vwind = 1 m ·s−1) Distance to seabed 5 m Seabed’s reflectivity ρseabed = 0.5 Number of rays 106 MAXHOPS 12 Table 4.1: Parameters of the baseline scenario After modifying the parameter of interest, several runs of the simulation are performed in order to average the results. Concretely, to obtain a trend of the influence of each parameter, 100 runs were carried out per specific scenario. The results regarding the impact of each aforementioned parameter on the impulse response are discussed in the following subsections. The swept interval of each parameter will be presented, sample images of the impulse response at the two boundaries will be depicted, and finally, a curve showing the influence on both channel gain and bandwidth will be also presented and commented. Furthermore, the obtained channel gain H(0) has been calculated taking into account the emitter’s radiation pattern but normalizing respect to the photodetector’s area. Therefore, the units of the calculated H(0) are W/m2. 4.4.1 Effect of the link’s range In this case, the link’s range has been swept from 1 meter to 10 meters. The impulse responses associated to the boundaries of the interval can be observed in Figure 4.14. The broadening of the response is clearly observed, whilst the reduction of the overall power is evident. Time (ns) 4.6 4.8 5 5.2 5.4 H(0) dB -140 -120 -100 -80 -60 -40 -20 0D= 1 m Time (ns) 44.6 44.8 45 45.2 45.4 45.6 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 D= 10 m Figure 4.14: Impulse responses at different ranges The exponential decay of the light intensity can be observed in Figure 4.15. This decay also affects the bandwidth, which also is diminished due to the greater amount of multipath components due to multiple scattering. It can be stated that there is an important exponential dependency of both parameters with the link’s range. 55 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Distance (m) 1 2 3 4 5 6 7 8 9 10 H(0) dB -50 -40 -30 -20 -10 0 Bandwidth (GHz) 1 2 3 4 5 6 H(0) Bandwidth Figure 4.15: Dependency of H(0) and Bwith dlink 4.4.2 Effect of the link’s depth A priori, the link’s depth should have a significant importance since it defines the strength of the reflective components of the impulse response. Nonetheless, the mean value of the channel gain does not reflect a clear relationship with the link’s depth, nor the bandwidth does. Figure 4.16 illustrates the boundaries of the swept range whilst Figure 4.17 shows the dependency of H(0) and Bwith the link’s depth. Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -120 -100 -80 -60 -40 -20 dsurf = 0.25 m Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 dsurf = 2 m Figure 4.16: Impulse responses at different depths The random nature of the impulse response will be discussed in detail in Chapter 6, but a Wilcoxon test to compare medians has been performed in order to analyze the influence of the depth on the studied channel parameters. The analysis yields a p-value of 0.0508 between the obtained values at dsurf = 0.25 meters and dsurf = 2 meters, which assures with almost a 5 % of confidence that the link’s depth has influence on the channel gain. In the case of the bandwidth, the tests returns a p-value of 3.5·10−6, which implies almost certainty about the dependency. 4.4.3 Effect of the surface agitation The wind speed produces an azimuthally-symmetric random slope on the seawater surface. This random variation has several effects. On the one hand, a variable surface has more probability to produce reflections directly to the receiver, whilst a quiet surface only has a small illuminated area whose reflections produce power contributions. 56 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Depth (m) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 H(0) dB -30.5 -30 -29.5 -29 Bandwidth (GHz) 2.75 2.8 2.85 2.9 H(0) Bandwidth Figure 4.17: Dependency of H(0) and Bwith dsurf On the other hand, the delay associated to the main geometrical reflective component (defined by Snell’s law) are “spread” as the agitation increases, since the Beam Spread Function of the medium (due to multiple scattering) widens the effective surface area with significant contributions. Figure 4.18 depicts the impulse responses for the boundaries of the swept wind speed. Note that the circled areas involve the main reflective component delay. In this case, due to the geometry of the scenario, is located close to 22.4 nanoseconds. Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 vwind = 0 m/s Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -120 -100 -80 -60 -40 -20 vwind = 10 m/s Figure 4.18: Impulse responses at different wind speeds for a depth of 0.25 meters Figure 4.19 shows the detail of the circled area. For a still seawater surface, the geometrical reflective components are located surrounding the theoretical delay. Nonetheless, for a agitated surface, this components are spread. Regarding the influence of the wind speed on the channel parameters, as the wind speed increases, the bandwidth and the channel gain increase too. This trend is absolutely compliant with the above discussion on the spreading of the reflective components. 4.4.4 Effect of the distance to seabed The seabed was modeled as a Lambertian scatter, whose reflectivity ranges in the interval [0,1]. Figure 4.21 illustrates the effect of the distance to seabed on the impulse response. A secondary spike can be observed next to 22.6 nanoseconds at the close-to-seabed scenario. This delay implies a traveled distance of 5.067 meters. Taking into account the geometry of the problem, there are two impact points on the projection of the pointing vector 57 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (ns) 22.32 22.34 22.36 22.38 22.4 H(0) dB -120 -110 -100 -90 -80 -70 -60 -50 -40 vwind = 0 m/s Time (ns) 22.32 22.34 22.36 22.38 22.4 H(0) dB -120 -110 -100 -90 -80 -70 -60 -50 -40 -30 vwind = 10 m/s Figure 4.19: Detail of the circled area of Figure 4.18 σ2 wind 0 0.01 0.02 0.03 0.04 0.05 0.06 H(0) dB -29.8 -29.75 -29.7 -29.65 -29.6 -29.55 Bandwidth (GHz) 2.75 2.76 2.77 2.78 2.79 2.8 H(0) Bandwidth Figure 4.20: Dependency of H(0) and Bwith vwind over the seabed that generate contributions with this delay: 4.5 meters and 0.5 meters (There is actually an ellipse on the seabed whose associated delay is 22.6 nanoseconds). Since the radiation pattern and the diffusion pattern are the same, the weighing of both main diffusive contributions are also the same by trigonometry. It can be observed that for 2 meters, there is no apparent contribution of the seabed. The effect of the distance to seabed is shown in Figure 4.22. As it is obvious, the bandwidth is increased as the distance to seabed is incremented. However, there is not an appreciable impact on the channel gain. As it was done above, a Wilcoxon test was carried out to prove the influence of the distance to seabed on the channel gain. In this case, the test returned a p-value of 0.65, being impossible to assure that the distance to seabed has influence on the channel gain. 4.4.5 Effect of the seabed’s reflectivity In the previous subsection, it was demonstrated that the distance to seabed does not have a significant impact on the channel gain. In this case, for dseabed = 0.25 meters, the expected impact of the reflectivity on the channel gain would be presumably the same. Figure 4.23 depicts the impulse response of the aforementioned scenario for a totally absorbing seabed and a very reflective one. There is no difference between the right graph of Figure 4.23 and the left graph of Figure 4.21. Figure 4.24 depicts the obtained mean channel parameters respect to the seabed’s reflectivity. There is no significant impact on any of the parameters, and the Wilcoxon test returns a p-value above 0.9 for both cases. 58 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 dseabed = 0.25 m Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 dseabed = 2 m Figure 4.21: Impulse responses at different distances to the seabed Distance to seabed 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 H(0) dB -30 -29.8 -29.6 Bandwidth (GHz) 1 2 3 H(0) Bandwidth Figure 4.22: Dependency of H(0) and Bwith dseabed However, only lambertian-scattering seabed has been tested and in further research, the use of other diffusion patterns should be tested. 4.4.6 Effect of the emitter’s directivity The emitter directivity is related to the concentration of the emitted energy in a narrower solid angle. It is logical that a more directive emitter would generate a better response on the receiver, whilst the multipath would be considerably reduced due to the decrement of the illuminated volume of water. Figure 4.25 depicts the impulse response for a pure lambertian emitter (left graph) and for an emitter with θ1/2= 12◦. It is easily noticed that the directive emitter presents a narrower and more energetic impulse response. The relationship between directivity and the channel parameters is presented in Figure 4.26. A log-like dependency can be easily extracted from the available data. In the limit, channel gain and bandwidth tend to the parameters of a laser-like link, which is defined by the BSF. 4.4.7 Effect of the wavelength The wavelength directly affects attenuation, scattering and propagation delay. Since the redder wavelengths are more attenuated than the ones in the blue-green region, the channel gain is directly affected by the behavior of the 59 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -120 -100 -80 -60 -40 -20 ρseabed = 0 Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 ρseabed = 1 Figure 4.23: Impulse responses at different seabed reflectivity coefficients for distance to seabed of 0.25 meters Reflectivity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 H(0) dB -29.8 -29.6 -29.4 Bandwidth (GHz) 2.75 2.8 2.85 H(0) Bandwidth Figure 4.24: Dependency of H(0) and Bwith ρseabed attenuation coefficient. Furthermore, the refractive index of seawater is higher for longer wavelengths. Therefore, the excess delays are higher in a red-emission scenario, reducing the bandwidth. Figure 4.27 depicts the impulse response for a 470 nm and a 660 nm emission, whilst Figure 4.28 illustrates the aforementioned dependency of the parameters. 4.4.8 Effect of the particle size As it was commented in Section 4.2.2, the normalized size of a particle directly affects the shape of the scattering phase function. The lower the normalized size, the more similar the phase function to a Rayleigh scattering, which presents a very broad dispersive profile. However, the higher the normalized size, the narrower the phase function. In the limit, the phase function tends to a Dirac’s delta, only affecting the propagation adding wavelength-dependent scalar losses. The left graph of Figure 4.29 illustrates the impulse response with a low normalized size, whilst the right one shows the channel response of a particle with normalized size close to 1. Regarding the effects of the particle radius on the impulse response, small radii have broad phase functions, which have associated also broad BSFs. This broadening that has been commented several times in this document is the main reason of the relaxation of the pointing between emitter and receiver. In the presented scenario, a wide BSF produces a low channel gain but a high bandwidth, as shows Figure 4.30. Therefore, it may be concluded that the channel gain is inversely proportional to the particle radius whilst the bandwidth presents direct proportionality. 60 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -120 -100 -80 -60 -40 -20 θ1/2= 60◦ Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -120 -100 -80 -60 -40 -20 θ1/2= 12◦ Figure 4.25: Impulse responses at different directivity values Directivity 0 5 10 15 20 25 30 35 H(0) dB -30 -25 -20 Bandwidth (GHz) 2 4 6 H(0) Bandwidth Figure 4.26: Dependency of H(0) and Bwith the emitter’s directivity A local maximum can be observed in the channel gain curve. Its location depends on the particle concentration, which implicitly defines the scattering coefficient b(λ). This effect has a special treatment in Chapter 6 4.4.9 Effect of the concentration of particles As it has been commented, the concentration of particles joint to the scattering phase function defines the scattering coefficient. The total extinction is, hence, defined by the last two terms combined to the inherent optical absorption of the medium. As in the case of the particle radius, the concentration broadens the BSF since it defines the amount of scatterings per unit volume. Figure 4.31 illustrates the impulse responses associated to the boundaries of the swept range. Figure 4.32 depicts the dependency of the channel parameters with the concentration of particles. For a particle radius of 100 nanometers, it can be observed that the channel gain increases in the swept range, but its concave curvature intuitively foresees a trend change. Regarding the bandwidth, it varies in a very narrow interval, but this parameter presents inverse proportionality. 61 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 λ= 470 nm Time (ns) 22.2 22.4 22.6 22.8 23 23.2 H(0) dB -130 -120 -110 -100 -90 -80 -70 -60 -50 λ= 660 nm Figure 4.27: Impulse responses at different wavelengths Wavelength (nm) 460 480 500 520 540 560 580 600 620 640 660 H(0) dB -40 -30 -20 Bandwidth (GHz) 1 2 3 H(0) Bandwidth Figure 4.28: Dependency of H(0) and Bwith the emitted wavelength 4.4.10 Parallelization speedup and efficiency In this section, the speedups and efficiencies (speedup respect to number of physical processors) relative to the parallelization of the algorithm are presented. As a first result, the iterative version of the code respect to the initial recursive version offered an speedup of between 18 and 22. The simulation scenario presented as baseline scenario, was executed in an average time of 68381.74 ms. This result will also serve as the reference time to calculate the speedups. OpenMP Table 4.2 shows the speedups and efficiencies for different OpenMP schedulings and and number of threads. Using OpenMP without any type of scheduling is very inefficient. The tree-like structures generated during the calculation process have random widths and depths. Therefore, the execution times of each thread could highly differ, fixing the computation time to the slower one. Because of that, static and dynamic scheduling types were used, looking for an efficiency enhancement. It can be observed that the chunk size does not affect static scheduling, as all the threads must be synchronized between the calculation of consecutive chunks. There is an efficiency enhancement respect to the no-scheduling case, but the synchronization problem still exists, even though at a lower scale. Finally, dynamic scheduling offers the best solution due to the randomness of the execution time. Each time 62 Chapter 4. Impulse Response of the Underwater Wireless Optical Channel Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 R= 100 nm Time (ns) 22.4 22.6 22.8 23 23.2 H(0) dB -150 -140 -130 -120 -110 -100 -90 -80 -70 R= 1µm Figure 4.29: Impulse responses at different particles sizes Radius (nm) 100 200 300 400 500 600 700 800 900 1000 H(0) dB -70 -60 -50 -40 -30 -20 Bandwidth (GHz) 0 2 4 6 8 10 H(0) Bandwidth Figure 4.30: Dependency of H(0) and Bwith the particle radius a thread finishes the computation of a chunk of iterations, it retrieves a new chunk of iterations from the pool. Nonetheless, the maximum speedup for each configuration is very similar. The difference between the best configuration and the worst is approximately an 8 %, which depending on the problem, it may represent a significant amount of time. Furthermore, since the used host was based on a Xeon processor (8 physical threads and 16 virtual threads), the efficiency at 32 threads is also very similar. CUDA The used GPU has 448 cores to perform calculations. However, speedups next to the number of cores can not be achieved because, as it was commented above, GPU’s operate in a thread-wise manner, and the code was programmed using conditional branches. Therefore, the majority of the multiprocessors will run only a single thread at a time. Regardless this limitation, the speedups obtained with this technology will surpass any multiprocessor architecture just by hardware brute force. The results have been obtained using 2Nthread blocks and 2Mthreads per block (N∈[0,10] and M∈[0,8]), in an effort to visualize how the GPU deals with different memory configurations. The total memory-transaction overhead was measured in 3 ms approximately. Hence, the CUDA kernel execution time and the whole algorithm execution time will be practically the same. The efficiency of a parallel algorithm is measured respect to the number of active processors during the execution. 63 Chapter 5. Considerations in Underwater-to-Air links Figure 5.2: Lensing effect of the seawater surface ray ˆrmay be defined as a linear combination of these vectors, as shows Equation 5.11. γris the refracted angle. ˆr=nwˆv+ (cos γr−nwcos γi) ˆn(5.11) The coefficients resulting from Snell’s law can be expressed in terms of ˆvand ˆn, yielding: ˆr=nwˆv+p1−n2 w(1−<ˆv, ˆn >2)−nw<ˆv, ˆn >ˆn(5.12) This last expression is computationally more efficient than Equation 5.11, since the scalar product <ˆv, ˆn >= cos γiis easily calculated and does not imply the use of transcendent functions. As it was commented above, a change of basis has been used to refer the integration limits to a virtual photodiode located over the surface. It is supposed that do not exist neither spray nor particles between seawater and photodiode. The proposed change of basis is presented in the next expression. x0=x+ ∆x(x, y) y0=y+ ∆y(x, y) (5.13) Figure 5.3 depicts the meaning of this nonlinear change of basis. Note that the terms ∆x(x, y) and ∆y(x, y) refer to the distance a ray travels in each direction before impacting the receiver’s XY plane. This deviation depends on the receiver’s height Hand the refraction, as Equation set 5.14 shows. Figure 5.3: Graphical interpretation of the nonlinear change of basis 70 Chapter 5. Considerations in Underwater-to-Air links ∆x(x, y)=[H−S(x, t)] rx rz ∆y(x, y)=[H−S(x, t)] ry rz (5.14) Finally, the projected area of the photodiode can be obtained in terms of the inverse transformation (Demonstration of the existence of inverse in Appendix A). Actually, due to the characteristics of the seawater surface, the rectangular area of the photodiode would not be projected as a rectangular area but as a curved one. However, due to the small-slope restriction imposed in this work, the error of assuming a rectangular projection will be negligible. Therefore, the projected area can be calculated using equation 5.15. A0 pd ≈x(2) L−x(1) L·y(2) L−y(1) L(5.15) Where x( Li) and y(i) Lcorrespond to the projected corners of the photodiode after solving the nonlinear systems of equations: +xL=x(1) L+ ∆x(x(1) L, y(1) L) +yL=y(1) L+ ∆x(x(1) L, y(1) L)) −xL=x(2) L+ ∆x(x(2) L, y(2) L) −yL=y(2) L+ ∆x(x(2) L, y(2) L))(5.16) Figure 5.4 shows the evolution of the average received power in the scenario defined in Table 5.1. The systems of equations were solved using the fixed point method. Parameter Value Wavelength 3 m Period 2 s Wave height 25 cm Depth 3 m Receiver’s height 2 m Photodiode’s area 9 cm2 Emitter power 1 W Directivity m= 1 Extinction coefficient c(λ)=0.305 Table 5.1: Underwater-to-air baseline scenario It can be observed that the energy loss due to the stretching of the projected area can imply a reduction of more than 3 dB for the example scenario. If wind were considered, each point of the sea water surface would behave following a normal distribution on its slope. The shear effect due to wind would produce a reduction of the received power and in this case, a pseudo-analytical solution as the presented above should not be used. The next section introduces the followed simulation procedure used to obtain the received power in a windy scenario. 5.3 Simulation procedure Unlike the simulation of underwater-to-underwater links, underwater-to-air links are limited by the seawater surface. As it was commented in Chapter 4, when a light ray impacts on the surface, part of the energy is reflected, and the other part refracted. The direction of the refracted ray depends on the surface’s normal vector and the incident ray’s direction. Since there is no diffusive effects, the approach used to accelerate the convergence in underwaterto-underwater scenarios can not be used in this case. Therefore, a traditional MCRT algorithm was implemented to calculate the received power. Algorithm 3 shows the pseudo-code of the implemented simulator. Routine calculateImpact() solves the next equation in ρ, which is the distance that a ray travels before impacting the surface in a direction defined by (θ, φ). ρ cos θ=D+η0cos (ωt −kρ sin θcos φ) (5.17) On the other hand, calculateRefraction() calculates the random incidence angle and the Fresnel loss. Finally, the output direction is projected to the receiver’s plane using the transformation of Equation set 5.13. 71 Chapter 5. Considerations in Underwater-to-Air links Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Prx (dBm) -58 -56 -54 -52 -50 -48 -46 -44 Figure 5.4: Evolution of the received power vs time for the parameters of Table 5.1 . Algorithm 3 Underwater-to-air simulator 1: for N Random rays do 2: ray ←newRandomRay: 3: calculateImpact(): Calculates the impact on the surface 4: calculateRefraction(): Calculates the output direction and Fresnel loss 5: if Impacts on the receiver then 6: Sum contribution 7: end if 8: end for 5.4 Simulation results In this section, several simulation results are presented in a similar way as in Chapter 4. In this case, the swept parameters are depth, receiver’s height, sea wave height and wind speed (respect to the baseline scenario of Table 5.1). Furthermore, after commenting the influence of each parameter on the received power, the channel availability will be briefly discussed. The channel availability is defined as the amount of time the received signal is over a threshold level S. This parameter is critical since it defines the probability to loss the connection. Expression 5.18 shows the mathematical description in terms of the PDF at each instant fX(X, τ), and the sea wave period T. Tch =1 TZT 0 τZ∞ S fX(X, τ) dXdτ(5.18) The following curves represent only one half of the sea wave period. Due to the space-time symmetry of the seawater surface, one half of the period is enough to visualize the behavior of the received power. 5.4.1 Effect of the emitter’s depth The portion of the link which is performed underwater is subject to extinction. In this case, an extinction coefficient of 0.305 (Coastal water) has been selected. If this coefficient were varied, only the average received power would be diminished or increased, not the shape of the received power respect time. Hence, the main effect of the emitter’s depth is the higher attenuation distance c(λ)D. In Figure 5.5, it can be observed that the received waveform is almost unchanged, but the offset level is defined by the attenuation distance c(λ)D. 5.4.2 Effect of the receiver’s height The height has direct influence on the lensing effect of the seawater. The higher the receiver is located, the wider the illuminated area during the concave periods of the sea surface that allow a bigger collection of light. Nonetheless, 72 Chapter 5. Considerations in Underwater-to-Air links Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Prx (dBm) -90 -80 -70 -60 -50 -40 -30 D=5 m D=7 m D=9 m D=3 m D=1 m Figure 5.5: Effect of the emitter’s depth on the received power when the height increases, the maximum-minimum distance is also increased, since the lensing effect acts in an opposite way during convex periods. This effect can be observed in Figure 5.6. Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Prx (dBm) -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 H=9 m H=7 m H=5 m H=3 m H=1 m Figure 5.6: Effect of the receiver’s height on the received power 5.4.3 Effect of the sea wave height The sea wave height, joint to the wavelength of the sea wave, defines the slope of a monochromatic wave. This slope has a direct influence on the deviation of the illuminated area that contributes to the received power. As the sea wave height increases, the maximum slope is incremented proportionally and also the maximum deviation of the photodetector’s projected area. Furthermore, this deviation has a Fresnel loss associated to the incidence angle. If this angle is increased, the minimum of the received waveform is reduced, as shows Figure 5.7. Figure 5.8 depicts the trajectory that follow the corners of the photodiode over the surface, according to Equation 5.13. The effect commented above can be observed for two different values of η0. Note that the higher η0, the larger the perimeter of the curve. Furthermore, Figure 5.9 illustrates the behavior of the projected area. It can be observed that high values of η0introduce fadings between the convex and concave situations, whilst lower values produce very small variations as expected. 73 Chapter 5. Considerations in Underwater-to-Air links Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Prx (dBm) -62 -60 -58 -56 -54 -52 -50 -48 η0=0.10 m η0=0.21 m η0=0.05 m η0=0.01 m η0=0.02 m Figure 5.7: Effect of the sea wave height on the received power ∆x -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ∆y ×10-3 -5 -4 -3 -2 -1 0 1 2 3 4 5 η0= 25 cm η0= 1 cm Lower corners Upper corners Figure 5.8: Trajectory followed by the corners of the photodiode over the seawater surface according to Equation 5.13 5.4.4 Effect of the sea wave wavelength As it was commented above, wavelength and period are related through the dispersion relation. In order to correctly analyze the influence of the wavelength on the received power, this relation must be satisfied for each pair (T, λ). The expected effect due to wavelength is similar to the effect because of η0. Figure 5.10 shows the influence of λ in normalized time units. As λincreases, the slope of the seawater surface is decreased, limiting the maximum deviation of the projected area. 5.4.5 Effect of the wind speed The wind speed introduces a random noise on the sea surface slope due to stress. The noise variance is linearly related to the speed and is normally distributed on elevation and uniformly distributed on azimuth. The a priori effect of the wind speed would be a reduction of the peak-to-peak value of the received power, since each illuminated spot of the sea surface would have a probability to generate energy on the receiver. Figure 5.11 depicts the effect of wind speed on the received envelope. 5.4.6 Channel availability The following figures depict the channel availabilities of the simulated scenarios respect to the receiver’s sensitivity. This sensitivity is the minimum detectable optical power. In Figures 5.12 and 5.13, it can be observed that the 74 Chapter 5. Considerations in Underwater-to-Air links Time (s) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Projected Area ×10-5 0 1 2 3 4 5 6 7 8 9 η0= 25 cm η0= 1 cm Figure 5.9: Projected photodiode’s area vs. time for different values of η0 Normalized time 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Prx (dBm) -62 -60 -58 -56 -54 -52 -50 -48 λ=7.00 m λ=5.00 m λ=3.00 m λ=9.00 m λ=11.00 m λ=13.00 m λ=15.00 m Figure 5.10: Effect of the sea wave wavelength on the received power distance of emitter and receiver to the seawater interface increment the sensitivity requirements of the receiver. Furthermore, the relaxed slope of the curves denote large peak-to-peak variability on the received power. Figure 5.14 illustrates the effect of the sea wave height on the channel availability. Large sea waves imply higher variations of the seawater’s slope, and hence, a greater peak-to-peak variation on the received power. This effect is similar to the observed in Figure 5.15, where the increment of the sea wave wavelength relaxes the surface’s slope and the sensitivity requirements. Finally, in Figure 5.16, the effect of wind speed is depicted. As it was observed in the previous subsection, the shear effect of the wind on the surface produces a reduction on the mean received power, but also a decrement of the maximum power deviation. Therefore, wind speed helps to mitigate the refractive losses due to the changes on the surface’s slope. During this chapter, the problem of an underwater-to-air link has been addressed. It has been observed that the sea surface parameters have critical importance on the performance of the communications link, incrementing the sensitivity requirements on the receiver. Furthermore, it has been observed that the wind speed acts as a smoothing parameter, reducing the harmful effect of highly variable surfaces. 75 Chapter 5. Considerations in Underwater-to-Air links Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Prx (dBm) -62 -60 -58 -56 -54 -52 -50 -48 -46 vwind=18 m/s vwind=0 m/s vwind=2 m/s Figure 5.11: Effect of the wind speed on the received power Receiver’s sensitivity (dBm) -100 -90 -80 -70 -60 -50 -40 -30 Tch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D=1 D=9 D=7 D=5 D=3 Figure 5.12: Channel availability for different depths Receiver’s sensitivity (dBm) -100 -90 -80 -70 -60 -50 -40 -30 Tch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 H=1 H=3H=5H=7H=9 Figure 5.13: Channel availability for different receiver heights 76 Chapter 5. Considerations in Underwater-to-Air links Receiver’s sensitivity (dBm) -70 -65 -60 -55 -50 -45 -40 Tch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η0=0.01 η0=0.21 η0=0.1 η0=0.02 η0=0.05 Figure 5.14: Channel availability for different sea wave heights Receiver’s sensitivity (dBm) -70 -65 -60 -55 -50 -45 -40 Tch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ=3 λ=5 λ=7 λ=9 λ=11 λ=13 λ=15 Figure 5.15: Channel availability for different sea wave wavelengths Receiver’s sensitivity (dBm) -70 -65 -60 -55 -50 -45 -40 Tch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1vwind=18 m/s vwind=0 m/s vwind=2 m/s Figure 5.16: Channel availability for different wind speeds 77 Chapter 5. Considerations in Underwater-to-Air links 78 Chapter 6 Statistical modeling of the Underwater Wireless Optical Channel An UWOC link is fully characterized by its time-dependent impulse response. Generally, the values of the impulse response are samples of a random process defined in two variables: time and multipath delay. Therefore, an actual impulse response would present the form h(t, τ), where tis time and τis delay. From this information, if the channel is WSSUS, the scattering function S(f, τ) of the channel can be extracted as: S(f, τ) = F∆t{E[h(t, τ)h∗(t+ ∆t, τ)]}(6.1) The scattering function is the Fourier transform of the autocorrelation function on the variable ∆t.fis the Doppler frequency and τconserves its original meaning. From this function, two important statistical functions can be derived, the Power Delay profile P(τ) and the Doppler Spectrum S(f). Equations 6.2 and 6.3 show the mathematical description of each function. P(τ) = Z∞ −∞ S(f, τ) df=Eh|h(t, τ)|2i(6.2) S(f) = Z∞ −∞ S(f, τ) dτ(6.3) From P(τ), the delay spread can be obtained as it was performed in Chapter 4, and its inverse is related to the coherence bandwidth. Regarding S(f), it allows the calculation of the coherence time by means of its inverse Fourier transform. In this part of the work, the WSSUS approximation is going to be assumed, but in Chapter 7 it will be demonstrated through experimentation. Furthermore, in this work, a statistical approach of both channel gain and bandwidth will be presented, since the implemented simulator only offers independent samples of h(t, τ) (no dependence with t). In the described situation, the mean delay spread could be approximated from: P(τ)≈1 NX ih(i)(τ) 2(6.4) Where Nis the number of random samples of h(τ). Note that τhas inverse meaning respect to the original formulation of Chapter 4. However, a statistical description of the delay spread and the channel gain is the main objective of this work, and the parameters obtained in Chapter 4 will be introduced into a statistical inference engine to perform a best fit analysis of their PDFs. As it was mentioned above, the variable twill be introduced into the analysis after obtaining actual measured data. 6.1 Brief analysis of the problem The simulator presented in Chapter4 calculated the impulse response using a MMCRT algorithm. The resulting impulse response was conformed by the sum of a finite number of Dirac’s deltas weighed by a factor that depended on the phenomena the ray suffered during its trajectory. The trajectory followed by each ray is randomly determined by the particle distribution. This particle distribution defines the scenario and hence, the power and delay of each contribution. Nevertheless, at each run of the algorithm, the positions of the particles randomly vary and are uncorrelated between iterations. This fact makes a delay-based simulation impossible, which is a requirement to obtain the scattering function S(τ, f). However, this simulator allows the calculation of the distributions of both H(0) and τrms. 79 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel Directivity 0 20 40 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ξ Directivity 0 20 40 0 20 40 60 80 100 120 140 µ Directivity 0 20 40 2 4 6 8 10 12 14 σ Figure 6.8: Evolution of the GEV parameters with the emitter’s directivity Radius (nm) 0 500 1000 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 ξ Radius (nm) 0 500 1000 0 20 40 60 80 100 120 140 µ Radius (nm) 0 500 1000 0 1 2 3 4 5 6 7 8σ Figure 6.9: Evolution of the GEV parameters with the particle radius Note that the BSF has generally units of watts per square meter. However, in this case, this BSF has been normalized and is expressed uniquely as m−2. The radiant intensity Ptx(θ) has units of watts per steradian. Since dΩ≈Aeff /d2, the equation is consistent in units with the squared Aeff term. Note that Prx(θ0) = Prx when θ0=π/2, and the BSF has been considered azimuthally symmetric, which is a common assumption in homogeneous media. This nomenclature eases the following description regarding the calculation of ∆τ1, which can be obtained in terms of a maximum elevation angle θmax. This θmax is the angle at which the integrated power is the 95 % of the total received power. Mathematically: θmax = arg θ{Prx(θ)=0.95Prx}(6.26) Under the geometrical restrictions of the problem, this θmax implies a hypothetical maximum traveled distance equal to dlink (cos θ+ sin θ). Introducing this distance into Equation 6.7 it yields the following approximation of the Fresnel zone’s radius. rellip ≈dlink 2tan θmax (6.27) 86 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel Concentration (·109) 0 10 20 30 0.45 0.5 0.55 0.6 0.65 0.7 0.75 ξ Concentration (·109) 0 10 20 30 0 5 10 15 20 25 30 µ Concentration (·109) 0 10 20 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2σ Figure 6.10: Evolution of the GEV parameters with the concentration of particles Figure 6.11: Integration of the received power using the Beam Spread Function 6.4.1 Beam Spread Function The BSF describes the spatial dispersion of an infinitesimal solid angle respect to the traveled distance, which is related to the scattering particles of the medium. The BSF can be understood as the light intensity on a plane perpendicular to the propagation distance at each traveled distance d. If the medium is homogeneous, this BSF would present revolution symmetry respect to the propagation axis, and each point of the plane would be defined by a radius rin polar coordinates. From Cochenour’s work in [16], the BSF can be expressed in terms of the spatial frequency κof the light intensity and the spatial Fourier transforms of both intensity and scattering. To simplify the calculation, the BSF is expressed as the following zeroth order Hankel transform. ξ(d, r) = 1 2πZ∞ 0 I(κ, d)S(κ, d)J0(κr)κdκ(6.28) For the approximation presented in Equation 6.25, the emitter’s radiation pattern is divided in an infinite number or rays. At each direction, the spatial distribution of the incident light can be considered as a Dirac’s Delta. Hence, I(κ, d) is an unitary constant. Cochenour et al. divided the resulting BSF into a non-scattered term ξNS(d, r) and a scattered term ξS(d, r), but in this case this approximation is not used. The scattering term S(κ, d) is defined in Equation 6.29 S(κ, d) = e−c(λ)deb(λ)Rd 0P(κ(d−z)) dz=e−c(λ)deb(λ)dR1 0P(dκν) dν(6.29) 87 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel P(x) is the Hankel transform of the scattering phase function and ν= 1 −z/d. Introducing Equation 6.29 into Equation 6.28, it yields: ξ(d, r) = e−α(λ)d 2πe−b(λ)dZ∞ 0 eb(λ)dR1 0P(dκν) dνJ0(κr)κdκ(6.30) This equation is numerically integrable. Figure 6.12 depicts ξ(d, r) at different distances and scattering coefficients for isotropic scattering and a Henyey-Greenstein scattering with g= 0.9 respectively. Radial distance (m) 0246 dB -14 -12 -10 -8 -6 -4 -2 0g = 0.9 d= 1 , b= 0.5 d= 1 , b= 2 d= 5 , b= 0.5 d= 5 , b= 2 Radial distance (m) 0246 dB -25 -20 -15 -10 -5 0Isotropic d= 1 , b= 0.5 d= 1 , b= 2 d= 5 , b= 0.5 d= 5 , b= 2 Figure 6.12: Normalized Beam Spread Function for isotropic scattering (right) and Henyey-Greenstein with parameter g = 0.9 (left) It can be observed that the scattering coefficient and the phase function shape affect the width of the BSF. Furthermore, the obtained BSF may be approximated by the sum of two Gaussians in a wide range of distances and scattering coefficients. Under this assumption, a fitting analysis can be performed to obtain a relationship between b(λ), g,dand the peak and width of ξ(d, r). Sweeping b∈(0,2), g∈(0,1) and d∈(1,10) the following relationships were obtained. ξ(d, 0) ≈8.231e−b(λ)(1−0.13g)d(6.31) η≈(1 −6.6·10−3b(λ))e1−e5·10−4g b(λ)d2(6.32) σ2 1≈4.6·10−3+ 2.22 ·10−4b(λ)+1.74 ·10−4e0.169g b(λ)d(6.33) σ2 2≈(3.87g+ 0.976b(λ)2)e(0.39b(λ)−0.663g)d(6.34) With: ξ(d, r)≈ξ(d, 0) ηe−r2 2σ2 1+ (1 −η)e−r2 2σ2 2/σ1< σ2(6.35) ηis a weighing factor between the two gaussians. Generally, the BSF is formed by a very narrow component due to the privileged forward direction and a wide component due to the multiple scattering phenomena. Note also that the expression of ξ(d, 0) is completely compliant with the experiments of Cochenour [16]. This terms describes that the higher the average cosine of the particle’s phase function, the lower the actual power loss in the forward direction due to beam spreading. Regarding σ2 1, the obtained approximation can be enhanced since for lower values of b(λ) (out of the swept ranges), the BSF must present a more abrupt behavior. The following R2 goodness-of-fit values were obtained for the approximations presented in this section. 6.4.2 Analysis of the resulting Fresnel zones Using the approximation obtained in the previous subsection, the analysis of the Fresnel zones can be reduced to the solution of Equation 6.25. Figures 6.13 and 6.14 illustrate θmax for the cases presented in Figure 6.12, but for 88 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel Parameter R2 Two-Gaussian approximation of the BSF R2 max = 1.0, R2 min = 0.85 ξ(d, 0) 0.99 η0.989 σ2 10.945 σ2 20.955 Table 6.2: Goodness-of-fit R2values for the approximations presented in this section. The support was conformed by 400 data points. a pure lambertian emitter (θ1/2=π/3) and a lambertian emitter with m= 20. Distance (m) 1 2 3 4 5 6 7 8 9 10 θmax 0 0.2 0.4 0.6 0.8 1 1.2 1.4 g= 0; b= 0.5 g= 0; b= 2 g= 0.95; b= 0.5 g= 0.95; b= 2 Figure 6.13: θmax for both isotropic and Henyey-Greenstein scatterings at different distances for a pure lambertian emitter Distance (m) 1 2 3 4 5 6 7 8 9 10 θmax 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 g= 0; b= 0.5 g= 0; b= 2 g= 0.95; b= 0.5 g= 0.95; b= 2 Figure 6.14: θmax for both isotropic and Henyey-Greenstein scatterings at different distances for a lambertian emitter with m= 20 It can be observed that the directivity of the emitter is inversely proportional to θmax, since the energy is concentrated in a smaller solid angle as the directivity increases. Furthermore, for small particle concentrations (low b(λ)),θmax decreases with the distance regardless the directivity. This occurs because the scattered energy is very small and the majority of the energy remains on the forward direction. Finally, high concentrations invert 89 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel the relationship of θmax with the link’s range, increasing with the distance. Performing an analysis similar to the one carried out in the previous subsection, an approximate formula to predict the θmax can be obtained. Equation 6.36 shows the result of the approximation, which was obtained with an R2value of 0.9604. θmax ≈0.2971b(λ)g m dlink 1−6.8·10−3b(λ)g2e−0.157m dlink + 9 ·10−4m dlink b(λ)2(6.36) Using Equation 6.36 combined with Equation 6.27, rellip can be easily calculated. This approximation has three main applications. Knowing the Fresnel zone to a 95% of the received power and considering steep impulse responses, the maximum bandwidth can be directly calculated using Equations 6.36 and 6.7. On the other hand, this approximation allows the link designer to consider whether an obstacle affects the transmission or not. Finally, the simulation of UWOC links is reduced to a very small volume if only the significant ellipsoid is considered. As it was commented, the available bandwidth can be estimated using Equation 6.6 and introducing the definition of the Fresnel zone’s radius. ˜ B=c0√12 5nwdlink cos θmax (6.37) The empirical formulae presented in this section have been obtained through simulation. Therefore, they may not accurately model reality and further research acquiring actual data from real scenarios should be carried out. Nonetheless, the obtained predictions seem to indicate a logical trend on the estimated bandwidths. 6.5 Statistical model for big opaque particles Generally, the literature only treats the problem of scattering. Nonetheless, in certain scenarios, big opaque particles such as sand grains are part of the heterogeneous underwater medium. In coastal scenarios, under shallow water regimes, the forces of the seawater mass transportation on the seabed generate a random distribution of big opaque particles amid the link. In this section, a purely geometrical model to predict the influence of this particles is presented. 6.5.1 Definition of big opaque particle A big opaque particle is a piece of matter whose spectral absorption is high enough to consider that there is no transmission and whose dimension assures that the diffraction is negligible. If a plane wave passes through a circular slit of radius R, the diffraction pattern I(θ) is determined by the following Airy disk: I(θ) = 4I0 J12πR λsin θ 2πR λsin θ!2 /∀φ(6.38) The angle θis defined from the center of the slit. However, since the interest of this work is focused on particles and not slits, from Babinet’s principle it can be stated that the diffraction pattern of the particle is the same as the slit’s except on the forward direction. For very high R/λ ratios, the overall diffracted intensity tends to zero, since: lim x→∞ J1(x) x= 0 (6.39) For near field, which occurs when the Fresnel number (Equation 6.40) satisfies F >> 1, the diffracted intensity is governed by the Kirchoff-Fresnel equation. The resulting near field patterns are also determined by the R/λ ratio and the result is similar to the far field case. F=R2 dλ (6.40) dis the distance from the particle to the plane of interest. Hence, a big opaque particle will only block the incident light producing a shadow in a far field situation. This shadow could be projected over the photodiode’s area producing a power loss. The following subsection introduces the mathematical formulation of the power loss. 6.5.2 Mathematical formulation Lets consider a LOS situation in an UWOC link, with an uniform distribution with density ρof big opaque particles or radius R. If the emitter has a radius Rtx and the receiver a radius Rrx, the volume enclosed by the LOS emission would be: 90 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel Vw=π 3dlink  R3 tx −R3 rx Rtx −Rrx  (6.41) Figure 6.15 depicts the situation. The total volume is the difference of two conic volumes. It is intuitive to assume that the number of particles within Vwwould follow a Binomial distribution. This distribution has a mean value of ρVw. Furthermore, the hypothetical maximum number of particles of radius Rin the volume Vwis Nmax =b3Vw/(4πR3)c. From Nmax and the mean of the distribution, it is straightforward to state that: n∼Bρ4π 3R3, Nmax(6.42) Figure 6.15: LOS component subject to the presence of big opaque particles Observing Figure 6.15 it is logical to notice that if the emitter were a point source, the projected shadow of a close-to-the-emitter particle would tend to infinity. Hence, the theoretical analysis of the problem must be performed considering the emitter as an extended source. Generally, the power loss due to an opaque object over an area Arx is determined by the linear projection of the object’s cross section according to the slope of the cone determined by the two areas (Figure 6.16). Appendix B, demonstrates this assumption. The section of the cone defined by emitter and receiver at a distance diis defined by: Figure 6.16: Power loss due to the effect of big opaque particles A(di) = πRtx +di dlink (Rrx −Rtx)2 (6.43) The power loss would be related to the amount of receiver’s area that is shadowed by the particle. This amount can be expressed as a ratio between the particle’s cross section and the area of Equation 6.43. The received power can be expressed as: Prx =Ptx m+ 1 2π Apd d2 link e−c(λ)dlink ζ(6.44) Where ζis a random variable that expresses the reduction of the photoreceiver’s area as: 91 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel ζ= 1 −πR2 n X i=1 γi A(di)(6.45) Where γiis a variable that indicates the overlapping degree of each particle with the rest. This analysis neglects the effect of the beam spreading and should be included in further research. Probably, the analysis would be analogous to the current proposal but after applying an integration of the form of Equation 6.25. In that case, Vwshould be increased to the volume of a truncated ellipsoid defined by the above-studied Fresnel zone and the emitter and receiver’s areas. Furthermore, A(di) must be transformed to the area of the intersected ellipsoid at di. If the effect of γiwere neglected, an unrealistic worse case scenario would be considered. In that situation, all the particles within Vwwould produce shadowing and hence, the power loss would decay beyond zero. However, for a number of particles relatively small, this situation would present a very small probability. difollows a truncated exponential distribution from di= 0 to di=dlink. Finally, the resulting distribution of A(di)−1can be expressed as follows according to the random variable algebra (Equation 6.46). Figure 6.17 depicts the experimental CDF of the inverse area vs the approximation. fx(x) = K·(πx)−3/2e−µdlink Rrx−Rtx ((πx)−1/2−Rtx)(6.46) K=π µ dlink 2|Rrx −Rtx|(6.47) 0 5 10 15 20 25 30 35 A(di)−1 0 0.2 0.4 0.6 0.8 1 Normalized probability Empirical Theoretical Figure 6.17: Comparison between the histogram and the PDF of Equation 6.46. ρ= 10−4, dlink = 5 m, Rtx = 5 cm, Rrx = 1 cm, R= 1 mm. Assuming a big number of particles and using Wald’s equation, the PDF of ζcan be approximated to a normal distribution N(µ, σ2) with the following parameters. µ≈1−πR2E[n]·E[x] σ2≈πR22E[n]σ2 x+E[x]2σ2 n(6.48) E[x] and σ2 xare the expected value and variance of Equation 6.46 respectively. This approximation would present an error that increases as the number of particles decrease. Furthermore, since γihas been neglected, the estimated loss may become negative. Therefore, this approximation is valid for a middle-range average number of particles. Figure 6.18 depicts three example situations where an experimental CDF following equation 6.45 is compared to the normal approximation. The first scenario is an scenario with a reduced average number of particles, the second corresponds to a very polluted environment and finally the third presents a moderate number of particles. As it was commented, the loss surpasses the threshold and becomes negative for the most polluted scenario. In order to fix this issue, the simulated PDF could be truncated at zero, introducing the left-side excess probability as a delta at the origin. 6.5.3 Influence of the link’s parameters on the SNR In a situation in which Johnson and shot noises could be neglected respect to the noise associated to this kind of particles, the SNR of the link would be defined by the quotient E2[ζ]/σ2 ζ. The baseline scenario of Table 6.3 is 92 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel 0.9 0.95 1 ζ 0 0.2 0.4 0.6 0.8 1 Cumulative probability Empirical Normal approximation 0.6 0.7 0.8 0.9 ζ 0 0.2 0.4 0.6 0.8 1 Cumulative probability Empirical Normal approximation -1 -0.5 0 ζ 0 0.2 0.4 0.6 0.8 1 Cumulative probability Empirical Normal approximation Figure 6.18: Comparison between the experimental CDF and the normal approximation in a link shadowed by big opaque particles. The left figure corresponds to a link in which the average number of particles is approximately 16, the center figure to 100 particles, and the right figure is associated to a very polluted environment with an average of 1000 particles within the volume. Note the precision of the normal approximation as the number of particles increase. defined to analyze the influence of different parameters in a scenario with big opaque particles. Note that this SNR must be understood as a lower bound since the overlapping of shadows has not been considered. Parameter Value Rtx 10 cm Rrx 1 cm R1 mm ρ103m−1 dlink 5 m Table 6.3: Baseline scenario to analyze the influence of the link parameters on the SNR First of all, the influence of the link’s range has been obtained through simulation. From the mathematical development presented above, the volume of water enclosed by emitter and receiver increases linearly with dlink. This increment is linked to a greater average number of particles, and hence, the SNR decreases. Figure 6.20 depicts the influence of the emitter radius. It can be observed that large emitters are subject to the influence of a higher amount of particles, shadowing the output power of the lamp. Furthermore, very small emitters are also highly influenced by this type of particles, since their emission surfaces are comparable to the particle cross section. Note that there is a local maximum at which the influence of the particles is minimized. This optimal emission area is of high importance in very turbid environments, where the maximization of the SNR is mandatory. In further research, the mathematical expression of the optimal emitter radius depending on the concentration, and the receiver and particle radii may be found. On the contrary, in Figure 6.21 it can be observed that bigger light collection areas diminish the influence of these particles, improving the SNR performance but saturating at certain level. The initial enhancement occurs because although there is a bigger amount of particles within the emission cone, the relationship between the receiver’s area and the particle area increases rapidly. However, beyond a threshold radius the marginal improvement of the SNR is close to zero, since the increment on the area is compensated by the greater amount of particles. Figure 6.22 shows the influence of the particle radius. This parameter has the bigger influence on the SNR since the particle cross section to receiver area ratio rapidly increases. Finally, the influence of the concentration of particles can be observed in Figure 6.23. The concentration of particle has two main effects according to the theoretical analysis above. The first one is to increment the average number of particles, and the second one is to produce more collisions with particles at closer distances, reducing the effective output power of the lamp. 93 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel 1 2 3 4 5 6 7 8 9 10 Distance (m) 32 34 36 38 40 42 44 SNR (dB) Figure 6.19: SNR of a link defined by the big opaque particle distribution vs the link’s range 0 5 10 15 20 25 30 Transmitter radius (cm) 30 32 34 36 38 40 42 SNR (dB) Figure 6.20: SNR of a link defined by the big opaque particle distribution vs the emission area 0 5 10 15 20 25 30 Receiver radius (cm) 25 30 35 40 45 50 55 60 65 SNR (dB) Figure 6.21: SNR of a link defined by the big opaque particle distribution vs the reception area 94 Chapter 6. Statistical modeling of the Underwater Wireless Optical Channel 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Particle radius -20 -10 0 10 20 30 40 SNR (dB) Figure 6.22: SNR of a link defined by the big opaque particle distribution vs the particle radius 101102103104 Concentration of particles 10 20 30 40 50 60 SNR (dB) Figure 6.23: SNR of a link defined by the big opaque particle distribution vs the concentration of particles 95 Chapter 7. Measurements on a short-range Underwater Wireless Optical Channel inside the water tank at the positions depicted in Figure 7.7. The particle mobility statistic has been matched to the mean absolute value of the water flow through the LOS line, which was previously simulated using a FEM software [145] since an actual measurement was not feasible. The resulting water speed distribution can be observed in Figure 7.8. The average speed of the particles is 1 cm ·s−1for pumps with a water flow of 800 liters per hour. The output water speed is calculated assuming a small diameter, which allows a constant output flux approximation yielding an output speed of 0.31m ·s−1. Figure 7.7: Positions of the water pumps used to agitate the scenario Figure 7.8: Water speed distribution of the experiment The influence of the two variables on the received power has been analyzed using a two-way ANOVA test over an estimation of the electrical SNR, which can be observed at Equation 7.10. The received signal follows a Normal distribution for all the cases, as the result of the Kolmogorov-Smirnov test suggests. SNRelec ≈E2[Vout] Var(Vout)(7.10) Vout is the voltage measured at the oscilloscope. Introducing the receiver chain in Equation 7.10, it yields Equation 7.11. Ptx is the emitted power, ¯ H(0) is the average channel gain, R(λ) is the photodetector’s responsivity, Bis the receiver’s bandwidth, qis the electron charge, Idthe dark noise, and σ2 ns represents all the non-shot noise contributions (Johnson noise and variability of the channel due to particles). 102 Chapter 7. Measurements on a short-range Underwater Wireless Optical Channel SNRelec ≈PtxH(0)R(λ)2 σ2 ns + 2qPtxH(0)R(λ) + IdB (7.11) Note that the shot noise term is negligible due to the reduced bandwidth of operation and the small received power. The particle movement experiment may be affected by the distance of the emitter to the surface and the bottom of the tank. To minimize the effect of the reflections, the link’s axis was fixed at the geometrical center of the tank, and the still water level was gradually incremented until the received power’s variation was negligible. At this point, the link can be considered independent of the surface reflections. Regarding the bottom diffusive effect, since it was covered by an absorbing material, it is considered to not affect the link. In order to analyze the actual influence of the suspended matter, the darkness noise power of the receiver was calibrated as baseline. The wind speed experiment was performed using clear tap water, since the presence of particles was not part of the designed experiment. The wind speed was measured using a digital anemometer at the middle of the water surface. The used wind generator presented only two different speeds: 8.1 m ·s−1and 13 m ·s−1. Both wind speed and link’s depth were swept to obtain long-duration waveforms on the receiver. Using the oscilloscope’s retrieved data, the coherence time and the demonstration of the WSSUS approximation were obtained. 7.3 Results obtained through measurements Figure 7.9 shows the actual implementation of the experimental setup. The results have been subdivided regarding the first and the second experiment commented above. Figure 7.9: Instrumentation configuration of the experimental setup As an example, Figure 7.10 depicts two captured frames. The left one is an only LOS scenario, whilst the right one is a near-surface link. The following subsections present the demonstration of the WSSUS validity of a near-surface link, and the results obtained for the two aforementioned experiments, as well as comments on the effects of the wavelength and the wind speed. 7.3.1 Effect of the concentration and the movement of particles Suspended matter contributes to the channel’s power extinction in two different manners. If the particles are steady because of a null flow scenario, each emitter’s outgoing ray suffers the same scattering events before arriving the receiver. On the other hand, if the particles present certain mobility, that almost-deterministic behavior is replaced by a random behavior whose mean value and variance depend on the evolution of the volumetric density of particles. In order to test this effects, two different particle mobilities and three different concentrations were used. Table 7.3 shows the parameters of the experiment. Mobility {0,1}cm ·s−1 Concentration of particles {0,6.12,12.24}mg ·l−1 Depth 30 cm Table 7.3: Parameters of the particle mobility experiment 103 Chapter 7. Measurements on a short-range Underwater Wireless Optical Channel Figure 7.10: Captured frames for a LOS scenario with a concentration of 6.12 mg l−1with agitated particles (left), and a near-surface link with an air flow of 13 m s−1at 5 cm depth (right). Both are obtained with a 470 nm emission and are expressed in optical power units (W) Figure 7.11 depicts the random nature of the SNR for red and blue wavelengths varying both particle concentration and mobility. As it can be observed there are 6 different scenarios, which are the combinations of mobility and particle concentrations. To statistically verify the influence of the particle mobility on the received power, an ANOVA test may be performed. To carry out the ANOVA test, a k-fold cross validation with 16 frames of 4096 samples has been used. The SNR of each frame is then estimated according to Equation 7.10. Afterwards it is tagged with its associated mobility and concentration, and finally used as input to the ANOVA test. As it is obvious, the test concludes that both mobility and concentration affect the electrical SNR. Figure 7.11: Boxplots of the SNR vs concentration and mobility for red (left) and blue (right) wavelengths Taking into account the normality of the received power, which was demonstrated through a KolmogorovSmirnov test (Figures C.4 and C.10) a simple linear model is proposed to relate the mobility and concentration with the observed SNR decrement. Equation 7.12 expresses it mathematically for the channel gain, H(0). H(0) (C,M) = α(C,M)H(0) (C,0) + V(C,M) (7.12) Cand Mare the concentration and mobility respectively. αexpresses the reduction of the mean received power and Vis the channel variability term, which follows a normal distribution of the type N(0, σ2 ch(C,M)). The variance due to the channel variability (extra noise due to particles moving) is related to the variance of H(0) (C,M) as shows Equation 7.13. 104 Chapter 7. Measurements on a short-range Underwater Wireless Optical Channel σ2 ch (C,M) = Var (H(0) (C,M)) −α2(C,M) Var (H(0) (C,0)) (7.13) Finally, to obtain the dimensionless channel gain variability through the observed voltage signals, a simple conversion may be made as shows Equation 7.14. Note than αremains the same regardless the domain, as the voltage and the optical power are linearly related. (GT Z R(λ)Ptx)2σ2 ch (C,M) = Var (V(C,M)) − α2(C,M) Var (V(C,0)) (7.14) GT Z is the transimpedance gain and R(λ) the responsivity of the photodiode at the used wavelength. Table 7.4 summarizes the values of αand σ2 ch for each (C,M) pair. Note that the variability of the red channel differs an order of magnitude respect to the blue channel measurements. This is because the dissolved alga presents an elevated concentration of chlorophyll, which implies a higher absorption at the red wavelength, reducing the effect of mobility. Wavelength (nm) Concentration (mg l−1)α σ2 ch 660 0 0.99 6.67 ·10−18 660 6.12 0.92 1.08 ·10−15 660 12.24 0.96 3.6·10−15 470 0 1 4.25 ·10−12 470 6.12 0.89 116.3·10−12 470 12.24 0.91 138.5·10−12 Table 7.4: Values of αand σ2 ch for each concentration and wavelength for a mobility of 1 cm s−1 This reduction of the SNR in channels with moving particles may be important on the calculation of link budgets in UWOC. In order to avoid this effect, an extra power margin according the noise term presented above should be considered. These results are the sum of the turbulence effects (no concentration) and the influence of the particle density. Since this experiment was performed in laboratory, the Kolmogorov microscale would be higher than the expected in an open ocean environment. However, the concentration increment generates a reduction of the SNR much higher than the expected due to turbulences in short range links. 7.3.2 Effect of the wind speed Near surface links are characterized by presenting a very important reflective component. Equation 7.15 shows the mathematical expression of the received power for this kind of links. H(0) = H(0)LOS +ZxZy R(ˆv(x, y),ˆn(x, y))L(x, y) dxdy(7.15) The net reflective component is the sum of the contributions of all the points of the water surface. The surface plane can be divided forming a disjoint union of uncorrelated scatterers, whose fading behaviors are governed by their random normal vectors. The size of each uncorrelated region would depend on the wind speed and the propagating waves. R(ˆvi,ˆni) is the Fresnel loss due to the water-air interface, ˆv(x, y) is the emission vector, which is bounded by the upper hemisphere of the radiation pattern; ˆn(x, y) is the random normal vector of the seawater interface, and L(x, y) depends on the product of the extinction loss and the approximation of the solid angle at the receiver (dΩ ≈Apd/d2). This last term would depend on the geometry of the link and goes to zero if the output ray does not impact on the receiver. As it was commented in Chapter 5, ˆn(x, y) depends on the wind speed and is commonly approximated by a normal distribution. The NLOS component of Equation 7.15 follows a Normal distribution since it can be understood as the sum of a high amount of independent Bernoulli processes (each scatterer has a probability pxy to contribute to the overall received power), but its time-dependent evolution is unknown. According to Equation 7.2, the reflective term R(t) could be isolated since the LOS component does not vary significantly as it was shown in Subsection 7.3.1. Four different depths, three different wind speeds and two wavelengths were tested in this experiment. The resulting R(t) signal was calculated and its probability distribution fits a normal distribution. Furthermore, its mean, variance and coherence time are shown at Table 7.5. Regarding the coherence time, it must be taken into account that a very high bandwidth has been considered. This way, the time-frequency analysis is reduced to an only-time analysis, yielding a very simple calculation of the coherence time using Equation 7.9. The results of Table 7.5 are referred to signals expressed in watts. It 105 Chapter 7. Measurements on a short-range Underwater Wireless Optical Channel Wavelength (nm) Depth (cm) Windspeed (m s−1)µ σ2Tc(ms) 660 2 8.1 0.231 ·10−40.116 ·10−11 140 660 2 13 0.21 ·10−40.064 ·10−11 85.2 660 5 8.1 0.256 ·10−40.164 ·10−11 178.4 660 5 13 0.222 ·10−40.075 ·10−11 73.3 660 10 8.1 0.247 ·10−40.176 ·10−11 98 660 10 13 0.233 ·10−40.1·10−11 70.7 660 15 8.1 0.25 ·10−40.132 ·10−11 58.4 660 15 13 0.237 ·10−40.078 ·10−11 50.9 470 2 8.1 0.294 ·10−40.25 ·10−11 267 470 2 13 0.24 ·10−40.1·10−11 79.5 470 5 8.1 0.355 ·10−40.475 ·10−11 255.4 470 5 13 0.321 ·10−40.215 ·10−11 93.2 470 10 8.1 0.417 ·10−40.331 ·10−11 90.6 470 10 13 0.375 ·10−40.226 ·10−11 74.1 470 15 8.1 0.424 ·10−40.326 ·10−11 209.3 470 15 13 0.398 ·10−40.165 ·10−11 49.2 Table 7.5: Mean, variance and coherence time of R(t) for each measured scenario can be observed that the received reflective power contribution tends to increase in the considered depth interval. However, this trend changes abruptly above this depth, reducing its influence dramatically and being negligible above 22 cm. The reduction of the coherence time with the increasing wind speed occurs because the number of uncorrelated regions on the surface is incremented. In other words, the spatial correlation between the points of the surface is reduced as the wind speed increases, incrementing the number of uncorrelated contributions on the receiver. Considering the near-surface UWOC link as WSSUS is obvious, since the scatterers (points on the surface) vary in a bounded region, and the emitter and receiver are at fixed positions, but the experimental demonstration can be found in the following subsection. Furthermore, all the obtained waveforms, correlations and probability density functions can be found in Appendix C. 7.3.3 Validity of the WSSUS assumption In a wide sense stationary process, the expected value is constant. In order to demonstrate this property in a near-surface link, an ANOVA test was performed. The source signals were different frames built from the original 50 seconds time series as a concatenation of samples separated a multiple value of the coherence time (to create an uncorrelated vector). The result can be seen at Figure 7.12 for the red, 10 cm and 13 m/s scenario. As it was expected, the mean value of the process can be considered constant (p-value higher than 0.9 for all cases). The uncorrelation of the scatterers can be interpreted as a wide sense stationarity in the frequency domain. In this case, as the bandwidth is considered very high for all the scenarios, this property is directly satisfied. Figure 7.12: Result of the ANOVA test to demonstrate the WSS property of a near-surface link 106 Chapter 8 Strategies for energy-efficient transceiver design Energy efficiency is one of the most important aspects in UWSN. The power consumption of battery-powered isolated nodes defines their lifespan, and taking into account the high replacement costs, it is a primary minimization objective. Generally, the power consumption of a node is a sum of partial contributions. The main consumptions are related to processing (CPU and control), communication interfaces and payload. This last aspect can widely vary depending on the purpose of the deployed node. For monitoring applications, the payload comprises sensors and acquisition circuitry, which can be normally neglected respect to communications. However, for applications where actuation is needed, the payload power consumption may be the primary source of energy usage. Regarding communications, for short-range links, UWOC has demonstrated to present the best bits per Joule efficiency, but for long-range links, UAC is still the best technology. Energy efficiency can be improved in different manners. Using a layered approach, the best options to enhance the efficiency are the physical and the medium access layer. Regarding the physical layer, pulsed modulations are normally better alternatives than continuous waveforms such as OFDM or CSK. However, these types of signals can be Pulse Width Modulated in order to take advantage of high efficiency nonlinear drivers. Moreover, optical emitters and receivers present a better response to long wavelengths than to short ones. Although the best transmission windows are generally located at the blue-green region, the commented response of the devices makes red transmissions more energy-efficient below a critical distance dcrit. On the other hand, between PHY and MAC layers, power control algorithms are an interesting option to adapt the emitted power of the isolated nodes and hence, reduce the power consumption. 8.1 Signal to Noise ratio in UWOC The SNR of an UWOC link is defined by Equation 8.1. SNR(λ) = (PtxH(0, λ)R(λ))2 2q(PtxH(0, λ)R(λ) + Id+Ib)B+4KT B RLFn (8.1) Ptx is the emitted optical power, H(0, λ) is the channel gain, R(λ) is the responsivity of the photoreceiver at the wavelength λ,Idis the darkness noise current, Ibis the background noise, qthe electron charge, Kis the Boltzmann’s constant, Tis the temperature in Kelvins, Bis the bandwidth, RLis the amplifier’s gain and Fnits noise figure. In UWOC, the background noise is the sum of the effects of sunlight and the seawater blackbody radiation. Generally, the latter is neglected due to the low temperatures, whilst the first one losses importance as the depth increases. In this work, these noise sources are being neglected for simplicity in the analysis without loss of generality. If the efficiency η(λ) of the emitter were defined, the last Equation could be rewritten in terms of the power consumed by the transmitter. This efficiency is the product of the optical efficiency of the LED source ηopt(λ) and the efficiency of the LED driver ηdriver. SNR(λ) = (η(λ)Ptx|elecH(0, λ)R(λ))2 2q(η(λ)Ptx|elecH(0, λ)R(λ) + Id+Ib)B+4KT B RLFn (8.2) If the ratio of the SNR induced by two different emitters with the same power consumption at two different wavelengths were introduced, the following merit figure could be defined. 107 Chapter 8. Strategies for energy-efficient transceiver design SNR(λ1) SNR(λ0)=η(λ1)H(0, λ1)R(λ1) η(λ0)H(0, λ0)R(λ0)22qη(λ0)Ptx|elecH(0, λ0)R(λ0) + 4KT RLFn 2qη(λ1)Ptx|elecH(0, λ1)R(λ1) + 4KT RLFn (8.3) Note that for shot-noise dominant scenarios (short distance) this merit figure is linear, but for Johnson dominant situations (long distance), the ratio becomes quadratic. However, in any of the two cases, the critical distance dcrit is defined as the distance at which the merit figure becomes one. Taking into account the conclusions of Chapters 4 and 6, the ratio of channel gains would depend on the difference of effective extinction coefficients, as shows Equation 8.4. H(0, λ1) H(0, λ0)≈e−(c(λ1)−c(λ0)dlink (8.4) Introducing Equation 8.4 into Equation 8.3 for the critical distance, it yields: 1 = η(λ1)R(λ1) η(λ0)R(λ0)e−(c(λ1)−c(λ0))dcrit (8.5) Solving the equation, the critical distance is of the form: dcrit =−1 c(λ1)−c(λ0)ln η(λ0)R(λ0) η(λ1)R(λ1)(8.6) This critical distance describes the range from which it is better to transmit at wavelength λ1in terms of power consumption. 8.2 Optical transmitters and receivers The transmission topology, the used encoding and the quantum efficiency of the light source are critical aspects regarding optical transmitters. On the other hand, optical receivers are determined by the opto-electrical device’s NEP, the responsivity of the substrate and the amplifier’s noise figure. 8.2.1 Current drivers Optical transmitters generally comprise an LED lamp and a current driver, whose efficiency highly depends on the linearity requirements of the transmitted signal. Theoretically, a current driver performs a transconductance amplification since it converts an input voltage into a driving current. Linear drivers are normally implemented using bipolar junction transistors, whilst nonlinear drivers usually comprise MOSFET devices. The first ones need polarization in the active components, and that bias produces an energy leakage that can drop down the efficiency of the system below 50% easily. On the other hand, MOSFET transistors only consume energy during state transitions, allowing high speed drivers with efficiencies up to 95 % [146]. Nonetheless, these high impedance devices linearly increase their power consumption according to the storage of energy on the parasitic capacitors of the gate-source junction. Equation 8.7 shows the power consumption of a MOSFET. PMOSF ET =1 2CV 2f(8.7) This term must be taken into account to calculate the efficiency of a MOSFET-based nonlinear driver. In Figure 8.1, a typical linear and a nonlinear driver can be observed. Equations 8.8 and 8.9 show the efficiency of both drivers. ¯ Iis the average excitation current of the LED. ηlinear =VD Vcc +POP A ¯ I (8.8) ηnonlinear =VD Vcc +1 2CV2 ¯ If(8.9) For frequencies above 2POP AC−1V−2, the linear driver turns more efficient than the nonlinear one, but this limit is normally far enough to consider the nonlinear driver more efficient under any circumstance. In the case of nonlinear drivers, higher dynamic ranges are achievable since the transmitted signal is immune to nonlinear distortion. Nevertheless, linear drivers suffer from two types of dynamic range limitation. The first one is directly derived from the driving circuitry, and depends on the polarization. The other one is the inherent nonlinear behavior of LED devices. 108 Chapter 8. Strategies for energy-efficient transceiver design + - Figure 8.1: Linear (left) and nonlinear (right) current drivers Figure 8.2: Quantum efficiencies for different types of substrate [147] 8.2.2 Optical emitters The energy efficiency of optical emitters is measured in terms of their wall-plug efficiency. This efficiency is the ratio between the total radiated optical power and the electrical power consumed. Mathematically: ηopt(λ) = Ptx VDID =ηext(λ)hc qλVD (8.10) ηext(λ) is the external quantum efficiency of the device., which is the ratio of the output photon flux and the injected electron flux. This ratio depends on the substrate and the manufacturing. However, the internal quantum efficiency is normally higher in AlGaInP (orange, red) devices than in GaN (blue) ones, as Figure 8.2 illustrates. It must be taken into account that ηopt(λ) is not a constant, since VDdepends on the driving current IDand ηext(λ) also presents nonlinearities for high values of ID. 8.3 Power Control Algorithms Power control is the selection of the transmission output power in a communications system, attending to an optimization criterion. This criterion is normally a combination of an energy minimization and the satisfaction of a minimum performance at the receiver. Traditionally, PCAs have been used to maximize the SNR whilst keeping the overall interference below a threshold in wireless communication channels, such as UMTS [148]. In this case, since all the signals are transmitted at the same time, each one spread by its corresponding orthogonal code, if there is no power control, the interference level may increase up to harmful levels, dramatically reducing the BER. In the case of UWSN, PCAs are not proposed as SNR-improving techniques, but as energy-saving algorithms. BER is important since it partially defines the number of packet retransmissions, which is very power consuming. However, the possibility to adapt the transmission power to an optimum value in terms of energy, has more weight 109 Chapter 8. Strategies for energy-efficient transceiver design Figure 8.3: Scenario under consideration to study power control algorithms in the design of optical UWSN nodes. Almost any energy-saving protocol or technique is justified in UWSN, because the extension of the nodes’ lifespan dramatically reduces the replacement costs. There are different taxonomies of power control algorithms, depending on the classification criterion. For instance, if each node takes its own decisions the algorithm is distributed whilst on the other case is centralized. If there is channel status exchange between nodes, ergo, there is a feedback of information, the algorithm is closedloop. On the contrary, it is open-loop. Finally, depending on how is calculated the step size of the iterative power control process, the algorithms can be fixed-step, variable-step or adaptive-step. In this work, a star-like network topology has been considered. This kind of network is a very general approach, but it is enough to study the impact of PCAs. Furthermore, a TDMA MAC protocol has been considered in order to simplify the optimization of the SNR and to isolate the study of the feasible Nchannels of a general scheme. Figure 8.3 represents the scenario under consideration, which may comprise a main energy-unlimited node and a bundle of remote nodes. This topology fits a buoy-nodes or a UAV-nodes scenario, depending on the mobility of the main node. In addition, the studied algorithms are centralized and closed-loop. The centralization of the algorithms has been proposed to reduce the remote node’s complexity. Furthermore, the estimation of the link’s performance takes place at the main node’s side whilst the uplink (main node to remote node) is always carried out at maximum power. The feedback of channel information is necessary to estimate with a lower error the needed transmit output power, but it also adds an error source that should be taken into account. Generally, a power control algorithm is derived from an optimization problem of the type: min X i Pi subject to: (Pigii)2 σ2 N+ 2qB PjPjgij +Pj6=i(Pjgij)2≥Ki(8.11) σ2 Nis the sum of the receiver’s inherent noise powers and Kiis the SINR threshold for the i-th channel. gij is the channel gain, including the responsivity, between the i-th receiver and the j-th emitter. Note that the quadratic term of the denominator is the optical interference term, whilst the linear term is the sum of all the shot noises due to both interference and wanted signals. These last two terms are nullified in the proposed scenario, due to the orthogonality of each channel after the use of a TDMA scheme, yielding the following simplified version of the convex minimization problem. min Pi subject to: (P·g)2 σ2 N+ 2qBP ·g≥K(8.12) Generally, the power is minimized iteratively, as shows Equation 8.13. 110 Chapter 8. Strategies for energy-efficient transceiver design Figure 8.4: Sequence of the proposed UWOC power control algorithms P(i+1) =P(i)+ ∆P(i)(8.13) P(i+1) is the next transmission power and ∆P(i)is the calculated step. Depending on how is this step calculated, the algorithm would be fixed-step, variable-step or adaptive-step. Other important aspect of the algorithms is how the SNR condition is estimated. The most common strategies are the following. •BER. BER is directly related to the SNR through the complementary error function.As the modulation or encoding spectral efficiency increases, the BER becomes more sensitive to the SNR. The main disadvantage of this indirect estimation is the requirement of long integration periods to retrieve a sufficient amount of data to perform the calculation. •Direct SNR estimation. The SNR is the ratio between the squared expected value and the variance of a signal. This estimation needs a high amount of samples to present a reliable confidence interval. However, if a soft signal detection were performed using a DSP or a FPGA, this estimation could be real-time performed. •Received power estimation. For situations in which the shot noise could be neglected, the SNR condition can be directly calculated using Equation 8.14. Furthermore, this technique is suitable for low-power devices, since only a few amount of samples per frame are needed. Low pass filtering may be used to reduce noise before sampling, or a few samples from a long-duration synchronization header could be acquired to estimate the received power. (P·g)2≥K·σ2 N P·g≥K1/2·σN(8.14) Some of the proposed centralized algorithms need knowledge of the remote node’s emission parameters: energy efficiency, allowed power codes, transimpedance conversion factor, etcetera. Therefore, an initialization stage to retrieve the necessary information is needed before the execution of the PCA. Figure 8.4 depicts the stages of the proposed PCAs. 8.3.1 Fixed-step algorithm Fixed-step algorithms are the easiest approach to energy minimization. Depending on the result of Equation 8.14, the output power is incremented Nδ or decremented Mδ. Mathematically: P(i+1) =P(i)+ ∆P(i) ∆P(i)=   Nδ P ·g < K1/2·σN −Mδ P ·g≥K1/2·σN (8.15) This algorithm is characterized by its low convergence speed and its low stability after reaching the optimum value. However, is very memory and computationally-efficient. 111 Chapter 8. Strategies for energy-efficient transceiver design Figure 8.10: Block diagram of different optical OFDM schemes ADO-OFDM scheme We have pointed out succinctly DCO-OFDM is inefficient in terms of optical power and so does ACO-OFDM regarding bandwidth. In Flip-OFDM, even though the complexity at the receiver is augmented, the overall performance takes advantage of the strengths of each technique. On the even subcarriers, DCO-OFDM is transmitted whilst on the odd ones ACOOFDM is used; consequently, the optical power efficiency is better than DCO and all the subcarriers are employed enhancing bandwidth efficiency with respect to ACO. At the receiver, the ACO symbols are extracted in the same way as conventional ACO, at the same time, DCO symbols require an interference cancellation method due to both clipping noises, odd and even, fall in the even subcarriers. Hence, the constellation sizes in the DCO subcarriers are smaller than usual. Flip-OFDM scheme As an alternative approach, this scheme splits the signal into positive and negative parts which then are serialized in two consecutive OFDM subframes. Regardless of the fact that it is a patent [157], which has not succeeded in the literature, several works analyze its goodness facing ACO-OFDM. The authors introduce a modification for a fair comparison. On the whole, Flip-OFDM performs almost identically when compared to ACO-OFDM saving 50% in receiver hardware complexity now that all subcarriers carry data. The penalty of transmitting two subframes per N samples brings the possibility of demodulating N symbols per IFFT operation, while ACO-OFDM needs two N-IFFT operations to achieve the same N samples. 8.4.2 Proposed scheme Optical OFDM is limited by the transmission of only positive signals. As it has been shown, this restriction has been solved using several methods, ranging from bias addition to intelligent mapping on the FFT block taking advantage of the predictable harmonics after a clipping operation. Another important issue in optical OFDM is the implicit nonlinearity of the emitters versus the necessity of linear drivers to handle the emission. If the LED devices were operated in a linear region, the dynamic range would be significantly diminished to avoid nonlinear distortion on the OFDM frame. A previous distortion to compensate the inherent LED behavior may be used, adding complexity to the design. In this thesis, a PWM encoding of the OFDM frame is proposed. By adding this operation, the linearity requirement is avoided and the use of nonlinear and efficient power drivers is allowed. Furthermore, this kind of codification is less sensitive to temperature variations on the emitter, because the information is encoded in the duty cycle of each PWM symbol. In the following subsections the advantages and disadvantages of the introduction of this block are discussed. Figure 8.11 depicts the block diagram of the proposed scheme. A PWM modulator has been added at the output stage of the emitter. This modulator is one of the main parts of a class D amplifier. However, the low pass filter has been removed from the emitter and placed at the receiver’s front end. From LTI systems theory, this simple change generates the same waveform at the receiver whilst dramatically increases the energy efficiency at the transmitter. However, the SNR is decremented adding an extra noise source derived from the N-bits quantization of the output OFDM waveform. Furthermore, the output waveform needs a sampling frequency 2Ntimes higher than the linear version if it is digitally generated. In addition, as the real-valued optical OFDM samples are normally distributed, a predistortion block such as a µ-law may be used to increase the distance between symbols before the ADC block. It may be also considered that all the subcarriers are used to carry information, as in DCO-OFDM, but with 118 Chapter 8. Strategies for energy-efficient transceiver design Figure 8.11: Block diagram of the proposed scheme Advantages Disadvantages Allows the use of nonlinear drivers Needs sampling frequency 2Ntimes higher Eases synchronization Decreases SNR Reduces the cost Presents a higher dynamic range Table 8.5: Summary of the characteristics of the proposed scheme the advantage of reducing the PAPR of the modulation to 3 dB because the peak power is fixed to a known and controlled value. The pulsed nature of a PWM signal allow an easier synchronization respect to traditional OFDM schemes. Furthermore, the advantages of OFDM against multipath dispersion and fading are conserved. Table 8.5 summarizes the main advantages and disadvantages of the proposed scheme. Mathematical description Using the signal yOF DM [n] as starting point, the driving current signal at the LED is shown at Equation 8.31. ILED(t) = Imax N−1 X i=0 Π (τ(yOF DM [i], t −i·Tsym) (8.31) Where τ(·) is a linear mapping function between the desired duty cycle and the OFDM samples, Tsym is the PWM symbol duration and Π(·) is the pulse function. If a predistorted version were carried out, τ(·) would represent the nonlinear mapping commented above. The mean electrical power of a PWM-OFDM frame may be expressed as follows. Pelec =ImaxV(Imax)E[τ] (8.32) Introducing the definition of PAPR, which is the ratio between the maximum and the average power, it can be easily shown that this scheme would present a PAPR of 3 dB for every frame. Optical OFDM techniques present a lower spectral efficiency regarding radio systems, because both real and imaginary parts cannot be transmitted simultaneously. ACO-OFDM uses half the subcarriers in order to produce a fully-recoverable clipped frame. Furthermore, as optical OFDM scheme only transmit the real part of the FFT, Hermitian symmetry must be applied resulting in a halving of the efficiency. This scheme is the less spectral-efficient of the common traditional optical OFDM techniques, but is also the most energy efficient. The spectral efficiency of ACO-OFDM is determined by Equation 8.33. ξACO ≈log2(M)Nsc 4(Nsc +Ncp)(8.33) Where Mis the number of symbols of the constellation, Nsc is the number of subcarriers and finally Ncp is the number cyclic prefix samples. In the case of PWMO-OFDM, the spectral efficiency is directly affected by the bandwidth of the PWM signal that encodes each IFFT sample, but it uses all the available subcarriers to transmit information. Both information density and bandwidth increase result in a spectral efficiency which is reduced by a factor of 2N−1. Therefore, this scheme dramatically increases the energy saving but reduces the spectral efficiency, being only suitable for medium-speed communication scenarios such as UWSN. 119 Chapter 8. Strategies for energy-efficient transceiver design Current (mA) 50 60 70 80 90 100 110 120 130 140 150 Optical Power (mW) 4 6 8 10 12 14 16 18 20 22 Red Blue Figure 8.12: Optical emitted power vs driving current The main advantage of ACO-OFDM against DCO-OFDM is the power reduction due to the clipping, which transforms it into one of the current most energy-efficient optical OFDM schemes. Taking into account that the probability density function of each sample within a sufficiently large OFDM frame is defined by a normal process, the clipped frame reduces the frame optical power to the half. Regarding DCO-OFDM, the frame energy must be increased in order to transmit it. The frame optical power, in this case, tends to the DC-bias optical power. When considering PWMO-OFDM, the mean transmitted optical power is defined by the PWM signal amplitude and the mean duty cycle as it has been already shown. Lets assume a same-electrical power scenario between ACO-OFDM and PWMO-OFDM. E[τ]ImaxV(Imax) = E[IACOV(IACO)] = Pelec (8.34) ACO-OFDM needs a linear driver, which is sensitive to temperature variations and to the operation region. On the other hand, PWMO-OFDM uses a nonlinear power driver which is immune to the operation point and temperature, regarding nonlinear distortions on the signal. Generally, the output optical power of an LED has a linear behavior respect to the driving current. Defining the luminous efficiency as: η(λ) = φ(ILED)·ILED ILED ·V(ILED)=φ(ILED) V(ILED)(8.35) As it was commented at the beginning of the chapter, linear LED drivers may present an electrical-to-electrical efficiency up to 80%, whilst class D amplifiers (which are the ones used for PWM signals) have efficiencies up to 95% [158]. In addition, using the proposed efficiency formula, it must be taken into account that the driver power will be determined by the commutation of the MOSFET driver transistor, which implies a small fixed power payback at a given Pmax. 8.4.3 Experimental curves During the measurements of Chapter 7, the used optical emitters were characterized using an integrating sphere. The obtained power-current curves are depicted in Figure 8.12. Furthermore, the voltage-intensity curves of both emitters are shown in Figure 8.13. The current at which a nonlinear driver is more energy-efficient than a linear one, taking into account the consideration stated in Equation 8.34, is defined by: ηlinear φ(IACO) V(IACO)≤ηnonlinear φ(Imax) V(Imax)(8.36) Rearranging the terms of Equation8.36 and defining relative measurements between fluxes and voltages, it yields: ηlinear ηnonlinear ≤∆φ ∆V(8.37) 120 Chapter 8. Strategies for energy-efficient transceiver design Current (mA) 1.5 2 2.5 3 3.5 4 4.5 Voltage -50 0 50 100 150 200 250 300 Blue Red Figure 8.13: V-I curve of the used emitters After including the measured efficiencies in the above Equations, the ratio ∆φ ∆Vfor a range of electrical powers has been obtained. The same emitted power condition of Equation 8.34 has been included. Figure 8.14 depicts this curve and compares it to the ratio of driver efficiencies. In order to obtain the mean current of an ACO-OFDM scheme, the probability density function of the OFDM samples should be considered. In ACO-OFDM, each sample follows a Half Gaussian distribution where the variance depends on the used mapping. For a QAM mapping, this variance is unitary and the expected value of each sample is 1/√2πas is commented in [153]. It can be observed that for almost every electrical power, nonlinear drivers are more energy efficient than linear drivers. Notice that these curves correspond to a single LED emitter. In the case of a LED array where the currents are divided, the use of PWM is recommended in terms of energy. From the data, only very efficient linear drivers (more that 80%) justify the use of continuous waveforms. 8.4.4 Comments on the BER performance As it was commented above, the pulse width modulation of the OFDM waveform introduces an extra noise term due to quantization, which affects the SNR in the following way: SNRP W MO−OF DM ≈S2 σ2+σ2 Q (8.38) Where σ2 Qis the quantization noise power, which depends on the quantization step ∆V= (Vmax −Vmin)/2N. It implies that there is a maximum achievable SNR bounded by this always-present noise source. The BER curve of the proposed scheme would be right-shifted respect to a non-quantized scheme, as shows Figure 8.15. It must be taken into account that incrementing the number of bits to enhance the BER performance has several harmful implications: •The spectral efficiency is reduced. The product ∆SNR ·∆ξis conserved in this type of system. Hence, an increment on the SNR is directly translated to a decrement of the same amount in the efficiency. Nevertheless, the value of Ncan be optimized. •The power consumption of the switching components is increased. As it was commented in the drivers Section, the power consumption of the MOS-based components increases linearly with the frequency. In this case, the ratio ∆SNR/∆PMOSF ET is constant. •The nonlinear driver efficiency is reduced because of the previous effect. Depending on the parasitic capacitance of the used device, this effect may be neglected. As it was commented at the beginning of the chapter, there is a switching frequency fsat which the linear driver surpasses the nonlinear driver efficiency, which is defined by: fs=2POP A(fs/2N) CV 2(8.39) 121 Chapter 8. Strategies for energy-efficient transceiver design Electrical Power (mW) 0 50 100 150 200 250 300 350 400 450 500 Merit Figure 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Red Blue Figure 8.14: Merit figure of Equation 8.37 for the LED devices used in Chapter 7. The horizontal dotted lines correspond to the ratio of driver efficiencies. The upper one is for a linear driver efficiency of 85%, whilst the lower corresponds to an efficiency of 50%. In both cases, the nonlinear efficiency is 95% 5 6 7 8 9 10 11 12 13 SNR (dB) 10-8 10-6 10-4 10-2 100 BER ACO-OFDM PWMO-OFDM Figure 8.15: BER vs SNR curves for an ACO-OFDM (512 subcarriers with QAM) scheme and a PWMO-OFDM (1024 subcarriers with QAM) scheme. Note the right 3 dB shift of the PWM-based scheme. 122 Chapter 8. Strategies for energy-efficient transceiver design Generally, the parasitic capacitance is very small (a few pF) and the switching voltage Vis determined by the used LED array. Moreover, the power consumption of an OPA-based linear driver also increments with the signal’s bandwidth, turning even higher the critical frequency. In conclusion, a compromise between power consumption and BER performance should be defined in order to optimize the number of bits of the transmitter’s DAC. 123 Chapter 8. Strategies for energy-efficient transceiver design 124 Chapter 9 Conclusions and Future Research During this thesis, several contributions have been made regarding different aspects of channel modeling and energy efficiency in UWOC. After analyzing the state-of-the-art research lines in Chapter 3, two clear conclusions were extracted. On the one hand, channel modeling in UWOC has been primarily directed by contributions at physical level, regarding different phenomena such as turbulences and multiple scattering. Traditionally, Beer-Lambert’s law has been used to describe extinction in UWOC, but Cochenour et al. demonstrated in [16] that scattering produces beam spreading and in consequence, the effective extinction is smaller than the sum of α(λ) and b(λ). Furthermore, there are different simulation engines using Monte Carlo schemes in the literature. Nonetheless, these algorithms use the Henyey-Greenstein scattering phase function to model the photon deviation due to particles, which is an oversimplification of a more realistic approach such as Mie scattering. The main advantages of Henyey-Greenstein are the easy parametrization of the scattering broadening with a single parameter and the fast calculation of random angles using this distribution, but the accuracy is reduced as it is commented in [38]. On the other hand, the contributions addressing applications range from very specific scenarios such as providing feedback to swimmers in a pool [109], to long-distance UWOC links in actual scenarios during ocean monitoring campaigns [86]. There are different works addressing hybrid opto-acoustic transceivers for UWSN [82]. These kind of devices take advantage of the high inherent energy efficiency and bandwidth of UWOC, whilst maintaining the possibility of long range communication using UAC. However, statistical approximations of the UWOC channel have been focused on providing a faster method to obtain channel gain estimations [47], but there are no statistical channel models for UWOC links in terms of channel gain or bandwidth. Finally, there is an evident lack of contributions regarding energy efficiency in UWOC even though it is of capital importance taking into account the replacement cost of underwater nodes. The UWOC channel is a linear and time variant channel, but its time variability is subject to different phenomena such as turbulences, misalignment and scattering. In FSO, turbulences are normally subject to Kolmogorov’s spectrum under weak turbulence regimes. Nonetheless, the optical properties of seawater present variability not only due to temperature gradients, but also because of salinity variations. In the literature, Kolmogorov’s spectrum has proven to be inaccurate, and Nikishov’s spectrum is widely used in the literature [122]. This approximation includes the two dependencies in the spatial frequency spectrum. It was proven during Chapter 4, that the UWOC scintillation index depends on the distance with an exponent ranging from 3/2 to 11/6, and not strictly on FSO’s traditional 11/6 exponent. The exponent is defined by the spectrum’s parameters. In Chapter 4, the impulse response of different UWOC scenarios was obtained using a Modified Monte Carlo Ray Tracing algorithm. Unlike other authors, Mie scattering was used as phase function in this work. Consequently, the complexity of the ray generation is increased in exchange for the enhanced accuracy. However, using the modified approach, a direct ray to the receiver is calculated after each scattering, reducing the volume of rays and the convergence time of the algorithm respect to the impact-testing conditions used in the literature. Furthermore, the algorithm was parallelized using both multiprocessor and GPU schemes. Speedups up to 42 were obtained using a NVidia Tesla M2050 GPU, allowing calculations of full impulse responses with 105rays in less than 2 seconds. The economical efficiency of both parallelization technologies was analyzed, showing that GPU implementations are more cost-effective than multiprocessor schemes. Moreover, the impact of the channel parameters into the received impulse response was analyzed. The following effects were observed: •The link range reduces exponentially the channel gain and the bandwidth. As the distance increases, the optical paths followed by the arriving rays is greater. Therefore, the absorption losses and the number of impacts with particles also increase. •Directive emitters concentrate energy in narrow solid angles, whilst other emitters disperse light. Taking into account the effect of multiple scattering, although the energy were collimated, a significant amount of energy may arrive the receiver. For perfectly aligned links, the higher the directivity, the higher the bandwidth and the channel gain. 125 Chapter 9. Conclusions and Future Research •The emission wavelength has two main effects. Firstly, the seawater absorption and the loss due to particles are wavelength dependent. Secondly, the normalized dimension of the suspended matter depends on the wavelength, and therefore, for a given particle size, the Mie scattering phase function may widely vary from a wavelength to another. •As it was commented above, the particle radius defines the shape of the scattering, but for a given concentration of particles per cubic meter, the number of scatterings increases with the particle radius. Therefore, the particle radius effect increments the number of scatterings but turns the phase function more forwarddominant. The observed simulated results suggests that the channel gain is reduced with the particle radius, due to the increasing number of impacts, whilst the bandwidth is increased due to the reduction of the average cosine of the phase function. •The concentration of particles has two effects. The first one is the same as the particle radius. As the concentration increases, the collision probability also increases and the channel gain is reduced. The other effect is to contribute to the broadening of the BSF, allowing greater misalignment errors between emitter and receiver. Depending on the distance and the particle radius, the concentration of particles may contribute to a greater light collection since a bigger solid angle has influence on the receiver. •The distance to seabed was demonstrated to have a very reduced effect on the channel gain. Nonetheless, the bandwidth is slightly reduced in horizontal deep links. Since the distance to seabed may be neglected in UWOC channel estimation, the seabed albedo has also negligible influence. •For near-surface links, the contributions due to surface reflections cannot be obviated since they present a significant importance. However, this effect is dispersed as the link range increases due to the effect of extinction. Depending on the depth and for a still water scenario, the illuminated surface area with greater influence on the received power is modified. Therefore, as the depth increases, the channel gain is reduced and the bandwidth increased. •Wind speed produces a random slope variation on the seawater surface. This random variation turns each point of the surface in a potential scatterer, where the overall average contribution of the illuminated surface is incremented. Regarding the bandwidth, the multipath component is strong in still water scenarios, whilst the random agitation of seawater disperses this component along the impulse response, increasing the average bandwidth. In Chapter 5, underwater-to-air links were studied. This type of vertical links are suitable for shallow water node deployment where the data acquisition is performed without submerging the transceiver, for instance, an operator in a ship or a drone. The sealing process of a transceiver to make it submersible usually increments fabrication costs. This possibility is a cost-effective solution for shallow water UWSN applications. The underwater-to-air problem was addressed form a geometrical point of view, associating the received power to the optical power that impacts on the surface and is forwarded to the photodetector. This way, the analysis is based on calculating how much energy arrives a certain area of the seawater surface. To simplify the analysis, only surfaces with shapes that ensure the bijectivity of a spatial transformation were considered. This transformation related the XY positions of a ray on the surface and the position at the receiver’s plane. After imposing several conditions regarding the sea wave spectrum, which was considered monochromatic, a numerical integration scheme was used to obtain simulations of the received power. It was observed that the received light intensity followed the shape of the sea waves, due to the lensing effect of a propagating plane wave. In addition, an analysis similar to the one made during Chapter 4 was performed to relate each channel parameter to the received intensity. Furthermore, the channel availability was studied in each scenario respect to the receiver sensitivity. The main obtained conclusions were: •The emitter depth produces an exponential decay on the received power, following the extinction curve. •The receiver height has a similar effect to the previous parameter, since the solid angle formed by the receiver and the emitter is reduced with the squared distance. •The sea wave height has a direct impact on the seawater surface slope, which is critical on the calculation of the projected photodetector area. Steeper slopes generate deeper fadings, whilst low-height sea waves produce slight variations on the received power. The peak-to-peak variations of the received power can be higher than 10 dB, producing a notable impact on the channel availability and increasing the requirements of the receiver. •The sea wave wavelength produces the same effect as the sea wave height. In this case, longer wavelengths imply more relaxed slopes and hence, smaller variations. 126 Chapter 9. Conclusions and Future Research •Wind shear effect has a very interesting effect on the variation of the received power. As the wind speed increases, the peak-to-peak variation of the received signal is reduced (and also the maximum value). Therefore, windy scenarios may increase the channel availability when the receiver’s sensitivity is within the variation range of the received optical power. The statistical modeling of the UWOC channel response was addressed in Chapter 6. The main contributions made during this chapter were: •Analyzing the most suitable probability distribution function for both channel gain and bandwidth. •Defining a Fresnel zone attending to the 95 % of the impulse response energy, after performing a rectangular approximation. •Proposing a model of the losses induced by big opaque particles. A brief analysis of the impulse response formula showed the possibility of defining an ad hoc probability density function for the channel gain. Nonetheless, after performing a maximum likelihood estimation and several hypothesis tests on the simulated data, the Generalized Extreme Value distribution obtained the best results in the benchmark for both channel gain and bandwidth. This benchmark consisted on a set of scenarios covering a wide range of parameter combinations. Nonetheless, the proposed distribution only covered a few cases less than GEV. The obtained distribution showed how varies the received power at uncorrelated time instants. From the obtained data, it was observed that generally: •The SNR increases with the directivity, since the variability of the received power is reduced. It is straightforward to notice that narrow emission cones are subject to a lower amount of possible scatterings. •The link range reduce the effect of scattering-induced variability. However, the increment of the SNR due to distance respect to the impact with particles is countered by the effect of turbulences, which increase with a power law with distance. •From the obtained data it can be inferred that for a given link range and concentration, there is a particle size that minimizes the SNR. As it has been commented several times during this document, large particle radii imply a high dominance of the forward direction of the scattering. This leads to an increasing SNR for decreasing particle radii. •The particle concentration reduces the SNR. This occurs because for high concentrations there are more possible scatterers within the link’s volume of influence. The impulse response can be divided in the sum of two rectangles due to its abrupt shape. The first rectangle may be defined to contain the 95 % of the energy. Its width, or delay spread, is associated to those delays at which there is significant energy contributions. Therefore, the maximum delay defines through the space-time relation the maximum traveled distance of a single-scattered ray. Taking into account the geometry of the scenario, this maximum distance generates an ellipsoid that can be referred to as the volume of interest or Fresnel zone. This Fresnel zone has several implications: •The definition of a volume of interest opens the possibility of time-dependent simulators where each particle is tracked within the ellipsoid. This kind of simulator would be able to model the time-frequency response of the channel, from which coherence time can be extracted. •The Fresnel zone describes the volume at which any intersecting object produces an effect on the received power. Furthermore, the impact of big opaque particles was also analyzed in Chapter 6. A big opaque particle is a particle that does not produce neither scattering nor diffraction, for instance, sand grains. Actually, sand grains would generate contributions due to their reflectivity, but this was obviated to isolate the shadowing effect of these particles. After statistically analyze the influence of these particles, it was found that big opaque particles produce an impact on the SNR that depends on their size, the link’s range, the concentration of particles and the radii of emitter and receiver. Following the tendency of this thesis, the influence of each of the aforementioned parameters was analyzed, showing that: •The SNR diminishes with the link range, since the average number of particles linearly depends on the distance. The higher the number of particles, the higher the probability to suffer fading. 127 Appendix A. Demonstrations of Chapter 5 This equation has two solutions: the trivial solution rx= 0 and ln rx rzapproximately constant in y. Performing the Taylor series of the latter around (0,0) up to the first derivative, and imposing that this derivative must be much lower than the constant term, it yields the following relation. |y|<<  rxrz rx,yrz−rz,yrx ln rx rz(0,0) (A.8) Solving Equation A.8 it yields that there is no limitation on the y-axis, since rx,y and rz,y tend to zero. For ∆y,x a similar condition can be obtained but in terms of x. |x|<<  ry rz(H−S) ln ((H−S)rz/ry) (H−S)(rz,xry−ry,xrz)−S,xrzry(0,0) (A.9) It is straightforward to demonstrate that ry,x and rz,x linearly depend on ∂(ˆn·ˆv)/∂x. In order to vanish this last term (eliminate the dependency on X), the propagating sea wave must satisfy that: λ >> πp2Dη0(A.10) This last condition arises from the following approximation: ∂(ˆn·ˆv) ∂x =∂ˆn ∂x ˆv+∂ˆv ∂x ˆn ˆv=(x, y, S) (x2+y2+S2)1/2→ˆv(0,0) ≈(0,0,1) ˆn=(−S,x,0,1) (1 + S2 ,x)1/2 ∂ˆv ∂x =(1,0, S,x) (x2+y2+S2)1/2−(x, y, S)x+SS,x (x2+y2+S2)3/2→∂ˆv ∂x(0,0) ≈(S−1,0,0) ∂ˆn ∂x =(−S,xx,0,0) (1 + S2 ,x)1/2−(−S,x,0,1) S,xS,xx (1 + S2 ,x)3/2(A.11) The term S−1≈D−1since D >> η0. Taking this into consideration and considering the worst case, the previous equations can be reduced to: S,xxD << 1 (A.12) Finally, introducing the approximation on Equation A.9, the maximum xdistance that assures that ∆y,x →0 is the same in the case of y, because ry= 0 for ˆv= (0,0,1). Hence, xand ydo not limit the vanishing of the cross partial derivatives if condition A.10 is satisfied. Finally, in order to ensure the presence of inverse, ∆x,x and ∆y,y must be bounded to the interval (−1,∞). These conditions are formulated in Equation set A.13. ∆x,x >−1 ∆y,y >−1 (A.13) Introducing the corresponding dependencies and assuming the same as before, it yields the following system of inequations: (rz)2+ (H−D)rx,xrz−(H−D)rz,xrx>0 (rz)2+ (H−D)ry,yrz−(H−D)rz,yry>0 (A.14) Since the sea wave propagates along the x-axis, the limiting range is imposed by the maximum deviation of x. This maximum deviation δx is related to the x-axis itself (y= 0). Introducing the assumptions made during the previous steps into Equation set A.13 yields: D+ (H−D)nw1 D−η0ksin(ωt −kδx)>0 D+ (H−D)nw D>0 (A.15) 134 Appendix A. Demonstrations of Chapter 5 Depth (m) 2345678910 δx(m) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 H-D = 1 H-D = 5 Figure A.1: Maximum deviation in the x-axis to assure bijectivity (η0= 0.25 m). The second inequation is satisfied for all Dand Hsince H > D. Regarding the first inequation, the following condition can be obtained: sin(ωt −kδx)<D2+ (H−D)nw nwDη0k(H−D)(A.16) This condition may be reduced to the worst case, which occurs when the sine tends to one. Making a second order approximation of the sine near to one, the maximum deviation of δx which assures the existence of inverse is: |δx|<2D2+ (H−D)nw(1 −Dη0k) nwDη0k3(H−D)(A.17) Finally, there is a minimum height over the still water level (H−D)min at which the last condition is satisfied. (H−D)min =D2 nw(Dη0k−1) (A.18) From the denominator of the last expression, an upper bound of the sea wave wavelength arises. Joint to condition A.10, to consider the approximations as valid, λmust be in the interval: πp2Dη0<< λ < 2πDη0(A.19) This last expression describes an absolute minimum of λ, which is defined by the intersection of the two boundary functions. The wavelength only complies with condition A.19 if Dη0>1/2, which is easily satisfied by a traveling sea wave under shallow water propagation. Figures A.1 and A.2 depict the last two limits in terms of (H−D) and Dfor a sea wave satisfying condition A.10. It can be observed that the area which assures bijectivity is wide enough compared to the photoreceiver’s area. 135 Appendix A. Demonstrations of Chapter 5 Depth (m) 2345678910 H-D (m) 8 9 10 11 12 13 14 15 16 17 18 Figure A.2: Maximum height at which there is inverse (η0= 0.25 m). 136 Appendix B Received light intensity in partially obstructed links In Chapter 6, a statistical approach of the received light intensity in a link partially obstructed by big opaque particles was presented. In this Appendix, the mathematical development from which the equations of Chapter 6 were derived is presented. In order to ease the analysis, the propagation medium will be considered lossless, the emitter will be considered as isotropic, and a single spherical particle will considered. Figure B.1 depicts the scenario under consideration. Figure B.1: Particle obstructing a link comprising an extended source and a photodiode Since the particle’s position is important, the source will be considered as an extended source of light, where each point of the surface has the same emission pattern and radiance. Furthermore, to take advantage of cylindrical symmetry, both emitter and receiver are considered circular. Under the depicted situation, for a particle-free scenario, the received power can be expressed as four cascaded integrals of the form: Prx =Zrtx Zϕtx ZθZϕrx Ptx 4πAtx rtx sin θdrtxdθdϕrxdϕtx (B.1) The received power depends on the solid angle formed by the receiver and each point of the emitting surface. If there is no obstructing particle, the last integral presents symmetry on the emitter, and the solid angle subtended can be approximated by dΩ ≈Arx/d2, yielding: Prx =Ptx 2Atx Arx ZRtx 0 rtxdlink (d2 link +r2 tx)3/2drtx (B.2) Solving the integral: Prx =Ptx 4πR2 tx Arx 1−dlink pd2 link +R2 tx !(B.3) For a small area emitter, which is a common consideration in OWC in general, the total received power depends on the following limit: lim Rtx→0 1 R2 tx 1−dlink pd2 link +R2 tx !=1 d2 link (B.4) After analyzing the particle-free scenario, lets introduce a single particle of radius Ramid the link. Now, in this situation, there would be certain angles at which the solid angle between emission point and receiver is reduced, as Figure B.1 showed. Now, the total received power can be integrated under two different intervals. An interval at which the particle does not produce any effect, and a particle-influenced interval. Figure B.2 shows this division of the integral. Figure B.2: Division of the integration domain due to the presence of a particle The integral of Equation B.2 can be rewritten as: 137 Appendix B. Received light intensity in partially obstructed links Prx =Ptx 2Atx (Arx −Apart)ZR0 0 rtxdlink (d2 link +r2 tx)3/2drtx+ Ptx 2Atx Arx ZRtx R0 rtxdlink (d2 link +r2 tx)3/2drtx (B.5) Naming Ithe integral of Equation B.3 and rearranging the terms of Equation B.5, the following expression is obtained: Prx =I−Ptx 2Atx Apart ZR0 0 rtxdlink (d2 link +r2 tx)3/2drtx (B.6) The second part of the equation is similar to the one solved in Equation B.3, but in this case, the upper integration limit is defined by Equation B.7. R0=dlinkR+diRrx dlink −di (B.7) Where diis the z-axis position of the particle. Introducing Equation B.7 into Equation B.3, it yields. Prx =I−Ptx 2Atx Apart 1−dlink pd2 link +R2 0!(B.8) Finally, the expanded version of Equation B.8 is: Prx =Ptx 4πR2 tx "Arx 1−dlink pd2 link +R2 tx !−Apart 1−dlink pd2 link +R2 0!# (B.9) The loss term depending on Apart can be associated to the angle Ψ subtended by R0and dlink. Mathematically: Prx =I−Ptx 4 Apart πR2 tx (1 −cos Ψ(di))−1(B.10) It can be shown that for di= 0, the power loss is a fraction of the emission surface, whilst for di=dlink the power loss is defined by a reduction on the photodiode’s illuminated area. Finally, for a given impact distance, the power loss is the quotient between the particle area and the area of the truncated cone formed by emitter and receiver at that distance. 138 Appendix C Waveforms, Correlations and Probability Density Functions of Chapter 7 In this chapter, all the measurements obtained during the experiments commented in Chapter 7 are presented. In order to organize this appendix, the figures have been classified depending on the associated experiment. Section C.1 comprises the acquired waveforms and the obtained probability density functions, whilst Section C.2 presents the obtained correlation functions and the waveforms. C.1 Movement of particles This section presents all the captured waveforms. All the measurements were performed in a very low ambient noise environment, so as to reduce possible background extra noise. The amplified photodiode was configured with a gain of 60 dB, which implies a bandwidth of 11 KHz according to [138]. Furthermore, in this configuration, the RMS noise voltage is 800µV . Figure C.1 depicts the noise of the receiver. Its standard deviation is 1.2 mV. Noise Voltage (mV) -8 -6 -4 -2 0 2 4 6 8 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.1: Receiver noise under absolute darkness conditions. Note the small offset due to the effect of the transimpedance amplifier. C.1.1 Blue emission Blue suffers less extinction in pure seawater than red, but as the concentration of scatterers increases, the optimum transmission wavelength shifts to a redder value. However, due to the reduced length of the used water tank (91 cm), the red wavelength presents a better response for all cases. The following subsections present the obtained waveforms and PDFs for the used blue wavelength (470 nm) in the particle movement experiment. 139 Appendix C. Waveforms, Correlations and Probability Density Functions of Chapter 7 Still water In still water, the particle positions are practically invariant with time and the resulting variability is only due to Johnson and shot noises. However, comparing the variances of Figures C.4 and C.10, it can be observed that shot noise has a negligible influence. Furthermore, these variances are the same as the one shown in Figure C.1. In this case, the standard deviation of the still water capture was 1.15 mV. Note that the mean value of the blue emission is dramatically reduced as the concentration of chlorophyll increases. Time (s) 0 10 20 30 40 50 60 Voltage 0.745 0.746 0.747 0.748 0.749 0.75 0.751 0.752 0.753 0.754 0.755 Voltage 0.746 0.748 0.75 0.752 0.754 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.2: Waveform and PDF for a blue emission in still tap water Time (s) 0 10 20 30 40 50 60 Voltage 0.072 0.074 0.076 0.078 0.08 0.082 0.084 0.086 Voltage 0.0740.0760.078 0.08 0.082 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.3: Waveform and PDF for a blue emission in still water with 6.12 mg/l of particles Moving water For a moving environment, the changing positions of the suspended matter produce a variability on the received signal, since the incoming multiple-scattered energy varies with time. Comparing the two no-concentration figures (Figure C.4 and Figure C.7, it can be observed that the mean value of the received signal is decremented whilst the variance is slightly modified. This occurs because tap water has a certain amount of total dissolved solids, which is suspended matter that affects transmission as well. The mean value is decremented for all cases, and the variance increases with the concentration as it was commented in Chapter 7. 140 Appendix C. Waveforms, Correlations and Probability Density Functions of Chapter 7 Time (s) 0 10 20 30 40 50 60 Voltage 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Voltage 0.0120.0140.0160.018 0.02 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.4: Waveform and PDF for a blue emission in still water with 12.24 mg/l of particles Time (s) 0 10 20 30 40 50 60 Voltage 0.62 0.622 0.624 0.626 0.628 0.63 0.632 Voltage 0.624 0.626 0.628 0.63 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.5: Waveform and PDF for a blue emission in agitated tap water C.1.2 Red emission In this subsection, the captures of the red wavelength are presented. In this case, the induced voltage is higher than the generated using the blue wavelength. Taking into account the length of the tank, the effective optical output power at each wavelength (Table 7.2), and the responsivity of silicon, it is straightforward to notice this fact. Still water Chlorophyll’s absorption spectrum has two fundamental peaks, one at blue and one at red. The blue peak may be higher depending on the relative concentrations of chlorophyll-a and chlorophyll-b, producing a red shift of the minimum absorption of water as the concentration is incremented. It can be observed that the difference of the mean values increases with the concentration level. 141 Appendix C. Waveforms, Correlations and Probability Density Functions of Chapter 7 Time (s) 0 10 20 30 40 50 60 Voltage 0.064 0.066 0.068 0.07 0.072 0.074 0.076 Voltage 0.066 0.068 0.07 0.072 0.074 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.6: Waveform and PDF for a blue emission in agitated water with 6.12 mg/l of particles Time (s) 0 10 20 30 40 50 60 Voltage 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Voltage 0.01 0.012 0.014 0.016 0.018 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.7: Waveform and PDF for a blue emission in agitated water with 12.24 mg/l of particles Moving water In the case of red light, due to the better propagation of this wavelength, the particle-induced variability is greater than the associated to blue emissions at this distance. This effect is easily noticeable comparing the figures of moving water for a red emission and the same for a blue emission. Furthermore, the scattering phase function also differs in both cases, and probably it is much wider for a red emission. C.2 Near-surface link measurements In the case of the second experiment, the acquired waveforms and the obtained correlations are presented. The autocorrelation function of the acquired waveforms defined the coherence time as it was discussed during Chapter7. The results show variations with wavelength, depth and wind speed. At each wind speed, only the boundaries of the swept depths are shown (2 cm and 15 cm). The general trend of the results is that the deeper the higher the coherence time due to the decrement of the surface’s influence. However, in the obtained results, the realizations at 15 cm have lower coherence times due to the increasing effect of total internal reflection. Beyond this limit for the used link range, the influence of the water surface decays abruptly. Regarding agitation, the more agitated the 142 Appendix C. Waveforms, Correlations and Probability Density Functions of Chapter 7 Time (s) 0 10 20 30 40 50 60 Voltage 0.794 0.796 0.798 0.8 0.802 0.804 0.806 0.808 Voltage 0.798 0.8 0.8020.8040.806 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.8: Waveform and PDF for a red emission in still tap water Time (s) 0 10 20 30 40 50 60 Voltage 0.124 0.126 0.128 0.13 0.132 0.134 0.136 Voltage 0.1260.128 0.13 0.1320.134 Normalized probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure C.9: Waveform and PDF for a red emission in still water with 6.12 mg/l of particles surface, the lower the coherence time. Finally, the red emission presents a worse coherence time due to the better response of the system at very short distances. 143