aalborg universitykom.aau.dk/group/10gr890/docs/main_gr890_mob.pdfaalborg university department of...

AALBORG UNIVERSITYDepartment of Electronic Systems

Near Field Macro Array Reception for

Small Radar Targets in Sea-clutter

8th. Semester / Group 890 - MOB / 2009-2010

Near Field Macro Array Reception forSmall Radar Targets in Sea-clutter


Report Info

TITLE:Near Field Macro Array Reception forSmall Radar Targets in Sea-clutter

PROJECT PERIOD:February 2010 – June 2010

PROJECT GROUP:Mobile Communication – Group 890

GROUP MEMBERS:Gerard CasellasLoan Sylvie DoSandra HermosoJose Alberto MendezIgnacio RodriguezAlexandru Tatomirescu

SUPERVISOR:Patrick Eggers

Number of pages in report: xx

Number of pages in appendix: xx

Total number of pages: xx

Number of copies printed: x

ABSTRACT

This report is dealing withan initial overview of anWB (Wide Band) SIMO(Single Input - Multi-ple Outputs) RADAR(Radio Detection AndRanging) System.

The main feature of thesystem is a macro arrayin reception to try to takeadvantage of the NearField propagationcharacteristics for thelocalization of smalltargets in Sea-Clutter.

The different componentsof the system are analyzedand compared with thetypical RADAR.

Channel and signalshas been completelycharacterized taking intoaccount the most realisticsea as possible.

A first version of anestimation algorithmhas been performed tocombine the informationcoming out of the differentreception antennas.

Group 890 - MOB 1



Preface

This report has been written by Group 890 in 2010 at Aalborg University. It is the 8th

semester project and it helped us to improve our skills working in group along this semester.

It has been written in LATEX and consists of the following sections: Introduction, ProblemDefinition, Radar Basics, System description, Simulation and Conclusions. All technicaldetails of the report are supported using appendixes.

MATLAB has been used to give support to the different calculations performed.

Literature references follow IEEE recommendations. Texts, figures, formulas and tablesare referenced using number in brackets which indicates the position on the reference list:

Text [Reference Number]Figure (number): Figure Description [Reference Number]Table (number): Table Description [Reference Number][Reference Number]: Formula [units]

The authors would like to acknowledge Mr.Patrick C.F. Eggers for his continuous assis-tance effort on the whole project work time.

Mobile Communication - Group 890Aalborg University, 1st June 2010

Gerard Casellas

Loan Sylvie Do

Sandra Hermoso

Jose Alberto Mendez

Ignacio Rodriguez

Alexandru Tatomirescu

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Contents

REPORT INFO 1

PREFACE 2

LIST OF FIGURES 6

LIST OF TABLES 8

ABBREVIATION LIST 9

SYMBOL LIST 11

1 INTRODUCTION 141.1 APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 MAIN PARTS OF A RADAR . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 TYPES OF RADAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 PROBLEM DEFINITION 182.1 SCENARIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Sea description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Boats description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.3 Coordinates system . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 PROBLEM DELIMITATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.3 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 RADAR BASICS 243.1 OPERATING PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 RADAR RANGE DETECTION . . . . . . . . . . . . . . . . . . . . . . . . 253.3 CLASSICAL RADAR EQUATION AND RADAR CROSS-SECTION . . . 273.4 ANTENNA FIELD REGIONS . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 SIGNAL MODEL DUE TO SEA CLUTTER 364.1 SEA WAVES MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 RADIO SIGNAL FROM SEA CLUTTER . . . . . . . . . . . . . . . . . . . 37

4.2.1 Distribution due to sea clutter . . . . . . . . . . . . . . . . . . . . . 374.2.2 Backscattering coefficient . . . . . . . . . . . . . . . . . . . . . . . 39

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4.2.3 Phase Characteristics of Sea Clutter . . . . . . . . . . . . . . . . . . 454.2.4 Setting of a Detection Threshold . . . . . . . . . . . . . . . . . . . . 45

5 SYSTEM DESCRIPTION 485.1 GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 CHANNEL MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 SIGNAL QUALITY DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . 57

6 SIMULATION 606.1 SCENARIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2 TRANSMISSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3 ILLUMINATED AREA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 DOWNLINK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.5 TARGET AND SEA CLUTTER . . . . . . . . . . . . . . . . . . . . . . . . 646.6 UPLINK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.7 WB (WIDE BAND) SIGNALS RECEIVED . . . . . . . . . . . . . . . . . . 66

6.7.1 NB (Narrow Band) Signals . . . . . . . . . . . . . . . . . . . . . . . 676.8 TARGET ESTIMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.9 COMPARISON WITH TYPICAL RADAR . . . . . . . . . . . . . . . . . . 696.10 SIMULATION RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A RADAR SIGNAL GENERATION 73A.1 SIGNAL GENERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.2 LINEAR-FREQUENCY MODULATION (LFM) . . . . . . . . . . . . . . . 76A.3 PHASE-CODING MODULATION . . . . . . . . . . . . . . . . . . . . . . . 77

B RADAR FREQUENCIES 80

C SEA ELECTROMAGNETICS 82C.1 POLARIZATION AND SCATTER ECHOES . . . . . . . . . . . . . . . . . 84

D SEA MODEL IMPLEMENTED 88D.1 PIERSON-MOSKOVITZ MODEL . . . . . . . . . . . . . . . . . . . . . . . 88

E TRIANGLES 91E.1 LAW OF SINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91E.2 LAW OF COSINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91E.3 LAW OF TANGENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

F MATLAB CODES 92F.1 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92F.2 Sea Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97F.3 Sea Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97F.4 Illuminated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101F.5 Illuminated Pixel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103F.6 Look for Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103F.7 Line of Sight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104F.8 Visible Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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F.9 Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107F.10 Downlink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107F.11 Sigma Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108F.12 Backscattered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109F.13 Uplink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110F.14 Normal Uplink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111F.15 Signal Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112F.16 Normal Signal Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114F.17 PFA Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115F.18 Peak Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116F.19 Normal Peak Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117F.20 Normal Position Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118F.21 Minimum Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119F.22 Find Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119F.23 Array Position Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

G INTERNAL ORGANIZATION 124G.1 WEB PAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124G.2 TIME PLAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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List of Figures

1.1 Main blocks of a radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Indian Ocean location [?] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Graphic of the vessel dimensions selected for the analysis . . . . . . . . . . 202.3 Graphic of the dinghy dimensions selected for the analysis . . . . . . . . . . 212.4 Spherical Coordinates System . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Reference System for the analysis. . . . . . . . . . . . . . . . . . . . . . . . 222.6 Radar system under study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Radar Pulse Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Elevation beamwidth in a vertical plane . . . . . . . . . . . . . . . . . . . . 253.3 Azimuth beamwidth in a horizontal plane (Top view) . . . . . . . . . . . . . 263.4 RCS of Spherical Equivalent Target . . . . . . . . . . . . . . . . . . . . . . 283.5 Field regions of an antenna [?]. . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Matlab simulation of the near field (Fresnel zone) for the vessel array when

is seen from different angles (flat Earth). . . . . . . . . . . . . . . . . . . . . 313.7 Altitude vs. range(horizon distance). . . . . . . . . . . . . . . . . . . . . . . 323.8 Visual size of the array. (r=24 km). . . . . . . . . . . . . . . . . . . . . . . 333.9 Typical changes of antenna amplitude pattern shape from reactive near field

toward the far field [?]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 The phenomegical model of sea surface and interaction with electromagneticwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Pie slices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Beamwidth Transmitter Antenna. . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Typical relation between clutter cross section and grazing angle. . . . . . . 414.5 error in the threshold approximation as a function of ν for various Pfa. . . . 47

5.1 Symmetric views and region of analysis . . . . . . . . . . . . . . . . . . . . 495.2 Top view of ranges and angles in different elements of the reception array . 495.3 Relation between range and angles translating top view to real distance . . 505.4 Transmitter and Receiver signals . . . . . . . . . . . . . . . . . . . . . . . . 515.5 Arriving reflections from the sea . . . . . . . . . . . . . . . . . . . . . . . . 525.6 Illuminating area for a classical radar . . . . . . . . . . . . . . . . . . . . . . 535.7 Illuminating area for a specific case (array of antennas) . . . . . . . . . . . 545.8 Highlighted is the illuminating area for the array radar for one tap where

Rres = cτtap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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5.9 Geometry for calculating the ellipse equation. . . . . . . . . . . . . . . . . . 565.10 Clutter geometry [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1 Functional block for the sea surface generation. . . . . . . . . . . . . . . . . 606.2 Sea surface generated (sea state = 2). . . . . . . . . . . . . . . . . . . . . . 616.3 Sea surface generated (sea state = 6). . . . . . . . . . . . . . . . . . . . . . 616.4 Functional block to represent the illuminated area. . . . . . . . . . . . . . . 626.5 Illuminated area from the transmitter antenna for a transmission angle of

45 and a beamwidth of 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.6 Functional block for downlink. . . . . . . . . . . . . . . . . . . . . . . . . . 646.7 Functional block for the backscattering from the sea. . . . . . . . . . . . . . 656.8 Functional block for uplink. . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.9 Example of received signals in 2 different antennas of the array. . . . . . . . 676.10 Example of thresholding in one signal from an antenna of the array. . . . . 686.11 Example of signal received in typical radar. . . . . . . . . . . . . . . . . . . 696.12 Example of thresholding in one signal received in typical radar. . . . . . . . 706.13 Information screen during simulation. . . . . . . . . . . . . . . . . . . . . . 716.14 Plotted results after one simulation. . . . . . . . . . . . . . . . . . . . . . . 72

A.1 Sinusoidal Pulse generation from I/Q components . . . . . . . . . . . . . . . 74A.2 Linear Frequency Signal generation from I/Q components . . . . . . . . . . 77A.3 Phase-Coding Signal generation from I/Q components . . . . . . . . . . . . 79

C.1 Permittivity and conductivity of the sea (A) [Rec. ITU-R P.527-3]. . . . . . 83C.2 Typical variation with grazing angle and polarization of sea clutter reflec-

tivity.[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.3 (a) Radar illumination of a sea wave. (b) Illumination geometry.[] . . . . . . 85

D.1 Wave spectra of a fully developed sea for different wind speeds accordingto Pierson-Moskowitz.[Oceanography dep.] . . . . . . . . . . . . . . . . . . . 89

D.2 Significant wave-height and period at the peak of the spectrum of a fully de-veloped sea calculated from the Pierson-Moskowitz spectrum.[Oceanographydep.] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

E.1 A triangle with sides of length a,b,c and angles A,B,C. . . . . . . . . . . . . 91

G.1 Group 890 Web-page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125G.2 Initial Timeplan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126G.3 Final Timeplan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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List of Tables

2.1 Vessels dimensions from different commercial companies [?] . . . . . . . . . 202.2 Dinghies dimensions from different commercial companies [?] . . . . . . . . 202.3 Boat dimensions for the analysis of detection . . . . . . . . . . . . . . . . . 21

3.1 Radar Cross Section of Ideal Geometric Shapes . . . . . . . . . . . . . . . . 293.2 Radar frequency bands vs. near-field outer boundary. . . . . . . . . . . . . 35

4.1 Radar cross section model capabilities. . . . . . . . . . . . . . . . . . . . . . 414.2 Various relationships between sea state, wind speed and wave height ??. . . 424.3 Paramaters for K-distribution threshold approximation. ??. . . . . . . . . . 47

B.1 Radar frequency bands [Radar handbook]. . . . . . . . . . . . . . . . . . . . 80B.2 Range and resolution versus frequency . . . . . . . . . . . . . . . . . . . . . 81

C.1 Sea parameters versus frequency (Sea water at 20C) . . . . . . . . . . . . . 82C.2 Typical parameters of marine radar antennas . . . . . . . . . . . . . . . . . 84

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Abbreviation List

CFAR Constant False Alarm RateCNR Clutter to Noise Ratiocm CentimeterCW Continuous wave radardB DecibeldBi Decibel referred to the isotropic antennadBm Decibel milliwattsDL Down LinkEIRP Effective Isotropically Radiated PowerERP Effective Radiated PowerFFT Fas Fourier TransformFS Free SpaceFSK Frequency Shift KeyingGIT Georgia Institute of TechnologyGE Ground EffectGHz Giga HertzHF High FrequencyHp Horizontal electrical field polarizationHH Horizontal polarization (transmitter and receiver)I In-phase componentITU International Telecommunication UnionIGF Illumination Gain FactorKHz KilohertzKW Slope of the gravity sea waveL Lengthm MeterMHz Megahertzmm MillimeterPFA Probability of False AlarmPRT Pulse Repetition TimePSK Phase Shift KeyingQ Quadrature componentRF Radio FrequencyRMS Root Mean SquareRx ReceiverSCR Signal to Clutter Ratio

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SNR Signal to Noise RatioSNIR Signal to Interference + Noise RatioRCS Radar Cross-SectionSS Douglas Sea StateTx TransmitterUHF Ultra High FrequencyV VoltVp Vertical electrical field polarizationVHF Very High FrequencyVS VersusVV Vertical polarization (transmitter and receiver)W Watt

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Symbol List

Constants

α = 0.0081 Phillips constantβ = 0.74 Sea Model (Pierson-Moskowitz) constantco = 3 · 108 Speed of light [m/s]εo = 8.85418 · 10−12 Vacuum permittivity [F/m]g = 9.81 Gravitational constant [m/s2]∞ InfinityK = 1.38065 · 10−23 Boltzmann Constant [J/K]µo = 4 ·π · 10−7 Magnetic permeability [N/A2]π = 3.14159 Pi number

Variables

A Signal amplitude [V]Aant Sector area of the antenna patternAe Effective antenna aperture [m2]Afield Area of the Field [m2]Ag Geometric antenna area [m2]Ap Area of the resolution pixel [m2]Asph Area of isotropic sphere [m2]Au Sea direction factorBW Bandwidth [Hz]BWθ Elevation Beamwidth [rad]BWϕ3dB Half-power antenna beamwidth [rad]BWϕ Azimuth Beamwidth []c Scale parameterd Distance [m]D DirectivityDb Draft of the boat [m]Da Longitudinal dimension of the antenna [m]∆ Across-ranges resolution [m]∆f Frequency band interval [Hz]∆τ Time resolution [s]

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∆R Range resolution [m]ε Complex permittivity [F/m]εr Relative permittivityη Intrinsic impedance of the mediumE Electromagnetic field [V/m]fo Carrier frequency [Hz]G Gain [dB]GA Angle factorGRx Receiver antenna gain [dBi]GTx Transmitter antenna gain [dBi]GU Aspect factorGw Wind speed factorH Height [m]H1/3 Significant wave height from Pierson-Moskowitz [m]

ζ2 Significant wave height [m]hav Average wave height [m]hHn Field maximums for horizontal polarizationhV n Field maximums for vertical polarizationhRx Receiver antenna height [m]hTag Tag antenna height [m]hTx Transmitter antenna height [m]Hh Hull height [m]Hb Bridge height [m]Ifr Power flux at the receiver [W/m2]Ift Power flux at the antenna [W/m2]K Antenna efficiencyKν Modified second-kind Bessel function of order νLb Length of the boat [m]Lchannel Losses calculated with the Free Space or the Ground Effect modelL(d) Radio link losses [W]

L(d0) Mean loss in the reference distance [dB]LRx Receiver losses [W]LShadowing Shadow fading margin (XdB)LTx Transmitter losses [W]m Modulation depth or modulation indexNTX Number of transmitters through the fieldν Shape parameterP Power [W]Pthreshold Threshold power [W]PR Mean power received by the radar [W]Pre Pre-radiated power by target [W]Preference Reference power [W]PRx Available power at receiver antenna [W]PTag Received power in the tag [W]

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∆φ Half-power antenna beamwidth []PTxTag Transmitted power by Tag [W]Pn Noise power [W]φ Deviation angle []φu Relative direction to wind direction angle [rad]q Departure of p(x) from Gaussian distributionR Resistance [Ω]RH Reflection coefficient (perfect conductor, horizontal polarization)Rmax Maximum unambiguous range [m]RV Reflection coefficient (perfect conductor, vertical polarization)SN Signal to noise ratioSA Angular resolution []SRx Receiver sensitivity [W]STag Tag sensitivity [W]σ Radar Cross-Section (RCS) [m2]σs Radar Cross-Section (RCS) of the sea [m2]σt Radar Cross-Section (RCS) of the target [m2]σz Surface height standard deviationσφ Multipath propagation contribution for a Gaussian distributionσα Angle standard deviation factorσc Conductivity [S/m]τ Transmit pulse time [s]τj Signal delay [s]τw Relaxation time of water vapor transitions [s]θ Elevation angle []θB Brewster angle []θBW Antenna pattern beamwidth [rad]θi Local grazing angle (elevation angle) []T Pulse width [s]Ti Subinterval pulse width [s]TB Backscattering transmission coefficient [dB]U10 Wind speed at the height of 10 m [m/s]Vw Wind speed [m/s]ϑ Direction relative to the wind direction angle [rad]ϕ Azimuth angle [rad]Ω Radar frequency [rad/s]Wb Width of the boat [m]ξ Signal phase []x Amplitude of the returning signal [V]Xσ Gaussian random variable with zero mean and standard deviation σXT Threshold valueβ Parameter related to the variance of xγ Probability of getting a concrete received powerλ Wavelength [m]ΩA Beam solid angle DegreeZq Normalization constant

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Chapter 1

INTRODUCTION

Radar is an electromagnetic sensor to detect and locate objects remotely. The term radaris an acronym for Radio Detection and Ranging, and it was first used by the US Navy in1940. [?]

In addition to the purposes of detection and location, the shape, size or velocity of atarget can also be determined by radar.

The principle of operation is that electromagnetic energy, typically a series of narrowrectangular pulses, is radiated by the radar to propagate in space, where it will be scat-tered off anything that it encounters. Some of this energy will reach a reflecting object,which is usually called target. Once target and radio waves come into contact, the in-tercepted energy is scattered in many directions and a small amount of this reradiatedenergy is returned to the radar antenna. After amplification by the receiver, the signalsare processed and a decision is made in the receiver output in order to sort out the requiredechoes from the unwanted ones. [?] [?]

Depending on the applications, radar targets might be from missiles, aircrafts or ships, tobuildings, icebergs or animals.

1.1 APPLICATIONS

There are many applications for radar, on scale which vary from a few centimeters tolong-range scales.

The application which has most contributed to radar development has been military use:defense has been a worry along history. Radar is used for control, missile guidance, mili-tary reconnaissance, and surveillance of helicopters, aircrafts and missiles.

Another important application is its use in air traffic control, which helped to improvesafety in air travel. Radars allow controlling air traffic on ground and on route from oneterminal to another one. Furthermore, radars can provide information about weather.Radars, concretely marine radars, are used to prevent collisions between ships and with

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other fixed references. They make control and manage sea traffic easier.

These examples give some idea of variety of applications, but there are many other ones,such as control monitor vehicles speeds on the road, provide information about weather,explore surface of planets, manipulate spacecraft or to map the composition of the Earth.

1.2 MAIN PARTS OF A RADAR

Along this subchapter we are going to focus on the main blocks a radar is consisted of andto expose briefly how we get target information (h(ϕ,r)). A general sketch of a radar canbe seen in the image below.

Figure 1.1: Main blocks of a radar

Transmitter. The transmitter generates a proper and stable waveform, which will havethe concrete peak and average powers depending on the application the radar has beendesigned for. One of the most common waveforms is the series of short pulses and typicalvalues for average transmitted power go from milliwatts to megawatts. [?]

Antenna. The waveform generated is propagated into space by the antenna, and in theconcrete case of short pulse waveform, the same antenna is used for transmitting andreceiving. Furthermore, the antenna acts also like a spatial filter. In our case, we workwith directive antennas (ϕ0) which the power is concentrated into a narrow beam. Whenthe same antenna is used by transmitter and receiver, a switch is needed. Its function isto isolate both parts, protecting the sensitivity receiver. In addition, echoes are directedby the duplexer to the receiver. [?]

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Receiver. The receiver must amplify the tiny amount of received energy in order toguarantee that a correct decision will be made and to extract the target information, inother words, its purpose is to achieve h(ϕ,r) response from the target.The first stage of the receiver is a low-noise amplifier, which represents the noise thataffects more seriously the radar in microwave frequencies. The noise and the unwantedechoes are the main limitations for radar. Because of this, receiver must produce verylittle noise and have a wide dynamic range. The dynamic range is defined as the ratio ofthe maximum to the minimum signal input power levels over which radar guarantees aspecified performance.The signal processor, which is often in the IF portion of the receiver, is the part of thesystem which separates the desired signal from the clutter. Here, measured delay andphase h(τ , ϕ) are employed to calculate target location. [?]

1.3 TYPES OF RADAR

There are many ways to classify a radar, depending on what feature you focus on.According to transmitter and receiver relative positions, radars can be divided into multi-static and monostatic radars. Both different situations can be seen in the below graphic. [?]

• Monostatic radars. Transmitter and receiver are at the same location. They aremost used for radio detection and ranging.

• Multistatic radars use antennas at separate sides at transmission and reception.When only one receiver is used, radar is called bistatic. Generally, locations arechosen to minimize transmission line losses.These kind of systems are more complex but larger ranges are reached using sametransmitters and receivers. Another advantage is that a better quality can beachieved if autocorrelation function for different receivers information is used. [?]Pictures of both types can be seen below.

According to signal types:

• Continuous wave radar (CW). The waveform is a continuous sine, and it usuallyemploys the doppler frequency shift to detect moving targets and to measure itsrelative velocity. These systems need less power to get the same range and theyhave not minimum distance. [?]

• Pulse radar. In this case a series of almost-rectangular pulses is transmitted.Inside this group, there are non-coherent and coherent radars, depending on if onlyamplitude or phase and amplitude of echoes are processed, respectively. [?]

Regarding to the applications, a possible classification would be:

• Surveillance radar. The function of this radar is detection and location of multipletargets inside range, determining its angle too. Furthermore, target can be observed

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(a) Monostatic radar [?] (b) Multistatic radar [?]

during the period of time in which radar is getting its track. [?]

• Tracking radar. These radars are used for detecting and measuring the positionof only one target constantly to maintain it inside antenna beam. [?]

• Multifunction. Multifunction radars carry out functions of the two previous types.To make this possible an electronic sweep antennas array is needed. Controllingtransmitted signal phase and received phase by each element of the array, the beamwill be position in a wanted way. [?] [?]

With regard to the resolution, there are low and high resolution radars. It is important toknow that resolution is the system capability to separate or distinguish different targetsby its position. In other words, range resolution tells us how far two targets must beseparated before we can see there are two targets instead of only larger one. [?]

• Conventional Radar. Resolution cell size is smaller than target size. Due to thisfact, only one echoe is produced by each target. [?]

• High-resolution Radar. In this case, resolution cell size is bigger than target’sone, by this one target will produce several echoes. Generally, high resolution implieshigh range resolution, although it can be referred to angle or velocity too. [?] [?]

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Chapter 2

PROBLEM DEFINITION

Pirate assault to vessels is an increasing threat in the seas of the world. While somesystems exist to alert actual boarding, actual pre-boarding/approach alert systems aretypically not found [?]. The pirates use small dinghies which are difficult to detect bytypical radar systems (they are small and made of non-reflective materials).

This is a big problem for Marine Radar: small radar targets are difficult to bedetected by normal navigation sensor systems in large distances (from severalhundred of meters to a few kilometers. This is due to two principal reasons:

• The short range detectability is often limited by shadowing and blind zones close tothe detection system.

• The signal strength from small targets is often less than the signal from the sur-rounding waves and the signal processing in standard navigation systems can notdifferentiate the wanted signal from the non-wanted ones.

All the problem can be summarized in the difficulty of the current radar systems to geta high resolution over long distances. This project will try to take a look at thereception part of the radar system changing the typical system with another involvingseveral antennas exploiting the huge dimensions of large vessels for a large array opera-tion (near field situation where also the phase front curvature and the amplitude of thereceived signal change to give more detail on both direction and range). The transmittingpart will not be changed respect to the typical systems.

The purpose of this report is to investigate the benefits of this large array respectto the typical Marine Radar system in:

• Resolution.

• Detection.

• Discrimination of small targets respect to sea-clutter.

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2.1 SCENARIO

Typically, these piracy actions take place in high open sea. To define the scenario, it isneeded to characterize the boats (dimensions of big vessels and small dinghies), the sea(location, environmental conditions, electromagnetic properties), the coordinates systemand the radar antenna system of the vessel (location of the antennas on the boat).

2.1.1 Sea description

The sea considered in the analysis will be the Indian Ocean. It is situated between 3continents (Africa, Asia and Australia - see Figure 2.1).

Figure 2.1: Indian Ocean location [?]

This Ocean is affected by a Monsoon climate with strong winds in the northern hemi-sphere and milder winds in the southern one. The Indian Ocean is the warmest ocean inthe world. [?]

Deep water circulation is controlled primarily by inflows from the Atlantic Ocean, the RedSea, and Antarctic currents. North of 20 south latitude the minimum surface tempera-ture is 22 C, exceeding 28 C to the east. Southward of 40 south latitude, temperaturesdrop quickly. [?]

Surface water salinity ranges from 32 to 37 parts per 1000, the highest occurring in theArabian Sea and in a belt between southern Africa and south-western Australia. [?]

Respect to the electromagnetic description of this sea, details can be seen in Appendix??. These are the most important parameters:

INCLUDE TABLE WITH PARAMETERS

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2.1.2 Boats description

Vessels are very big boats made of steel. They are normally used in freight traffic andthey are manned by groups of 10-15 people. Typical dimensions 1 are shown in Table 2.1.

VESSEL Lb [m] Wb [m] Bh [m] Hh [m] Db [m]

PANAMA 291.1 32.3 57.9 12

PANAMAX I 320.4 33.5 62.48 12.50

GAIA 319.5 54 6.2 18.1

EMMA MÆRSK 397 56 14.5 15.5

MSC 365.5 51.2 31.2 15.2 16

Table 2.1: Vessels dimensions from different commercial companies [?]

In this project, it is considered a typical radar transmitter antenna located on thetop of the bridge and an array of receiver antennas located on the sides of the boat atthe hull height.

Figure 2.2: Graphic of the vessel dimensions selected for the analysis

Dinghies are very small boats normally made in wood and steel and sometimes in plas-tic. They are very fast and they are manned by 5-10 pirates carrying weapons. Typicaldimensions of this kind of boats are shown in Table 2.2.

DINGHY Lb [m] Wb [m] Bh [m]

INVADER 6 1.8 1.3

MOTEREY 8-10 2.5 2.1

SPEEDBOAT 5 1.8 1.2

Table 2.2: Dinghies dimensions from different commercial companies [?]

1Dimensions: Lenght (Lb), Width (Wb), Bridge height (Bh), Hull height (Hh), Draft (Db). Bridge andhull height measured from waterline.

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Figure 2.3: Graphic of the dinghy dimensions selected for the analysis

The dimensions selected to be used in the analysis are shown in Table 2.3 and can be seengraphically in Figures 2.3 and 2.2.

BOAT Lb [m] Wb [m] Bh [m] Hh [m]

VESSEL 300 40 30 15

DINGHY 6 2 1.5

Table 2.3: Boat dimensions for the analysis of detection

2.1.3 Coordinates system

It is necessary to define a coordinate system as reference for the study. In this case, it isselected a spherical coordinates system.

Figure 2.4: Spherical Coordinates System

Where, XY is the horizontal (azimuth) plane, YZ is the vertical (elevation) plane, ϕ theazimuth angle, θ the elevation angle and r is the range.

The reference point of the system (origin of the coordinates systems) will be located inthe middle point of the receiving antenna array as shown in Figure 2.5

2.2 PROBLEM DELIMITATIONS

According to the description of the problem done at the beginning of this chapter, anoverview of the system can be made:

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Figure 2.5: Reference System for the analysis.

Figure 2.6: Radar system under study

2.2.1 Transmitter

Typical Marine Radar systems work with Pulse Radar. Depending on the purpose of thesystem, different modulations and techniques can be used. There is more informationabout the signal generation in Appendix A.

It is assumed no SNR (signal to noise ratio) calculations due to close range, but it is nec-essary to look at the SNIR (signal-to-interference + noise ratio) or SCR (signal to clutterratio).

It is not the main topic of this project to deal with the transmitting part of the system.

2.2.2 Channel

In this case, the signal sent by the transmitter travels through the air until it hits a target.In this moment, it is reflected (backscattered signal) and travels back through the air tothe receiver. Each receiver element is related to a channel function h(τ, ϕ).

In the open sea conditions described in Section 2.1, it is necessary to take into accountreflections of the signal on the sea waves, polarization behaviors, sea absorptions. This

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parameters are explained and described in tables in the Appendix ??.

To describe the dynamic channel, this report will deal with multi-path and shadowing.At the beginning it will be just a deterministic channel and it will become progressively astochastic channel.

2.2.3 Receiver

This is the main part of this analysis and it is focused only in estimating location fromthe phase and amplitude of the different samples of the signal received in the receptionarray.

Location will be performed in 2D (azimuth plane). Elevation angle will be considered onlywhere necessary with respect to link budget or channel modeling.

Only one target relative to sea-clutter is considered in this analysis.

This report is not dealing with tracking.

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Chapter 3

RADAR BASICS

In this chapter, the main features of radar systems such as operating principle or classicaldefinitions will be presented. Radar systems radiate each pulse at the carrier frequencyduring transmit time, and, range detection, Classical Radar Equation and Radar Cross-Section (RCS) become important parameters when analyzing the whole system understudy in this project. Finally, in the last section, the antenna field regions (definition ofNear/Far field) will be both explained and related to the scenario.

3.1 OPERATING PRINCIPLE

The main way to measure the distance from the radar to an object is to send an electro-magnetic pulse and, then, measure the time that the echo takes to return.

R =coτ

2[m] (3.1)

Where, R is the distance between target and antenna, c0 the velocity of electromagneticpropagation and τ the transmit pulse time.

The Figure 3.1 shows the characteristics of the transmitted signal in the time domain:

Figure 3.1: Radar Pulse Train

Radar systems radiate each pulse at the carrier frequency during transmit time (PulseWidth), wait for returning echoes during listening and then radiate the next pulse. The

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time between the beginning of one pulse and the start of the next pulse is called PulseRepetition Time (PRT) [?].

In order to obtain an unambiguous measurement of the target range, echoes from tar-gets must be detected and processed before the next transmitter pulse. The maximumunambiguous range is then given by Equation 3.2 [?].

Rmax =coPRT

2[m] (3.2)

3.2 RADAR RANGE DETECTION

The objective of this project is to work in an alternative Reception Antenna to everysignal received from the target. We will mention some of the features of the transmissionantenna which will limit the reception measurements.

The antenna beam scans a portion of space where targets are expected. When differenttargets are illuminated by the beam, it reflects a portion of energy. Then, the illuminatedarea by the antenna (radar) is the region of detection. [?]. Figure 3.2 shows a typical op-eration radar in elevation where R is the slant - range between target and antenna, BWθ

is the elevation angle and ∆θi is the local grazing angle. Figure 3.3 shows the azimuthbeamwidth in a top view, where R is the slant - range [?].

Figure 3.2: Elevation beamwidth in a vertical plane

Range detection is important due to its dependency on the whole system performance:dimensions of antenna and frequency to achieve the antenna beamwidth. Depending onthe transmitting antenna height, elevation and azimuth (or azimuth only) must be takeninto account. Therefore, a change in the azimuth and elevation leads to a variation in therange detection.

An important angle related with radar resolution is necessary to be considered: angularresolution is the minimum angular separation at which two equal targets can be separated

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Figure 3.3: Azimuth beamwidth in a horizontal plane (Top view)

within the same range. The angular resolution characteristics of a radar are determinedby the antenna beamwidth represented by the -3 dB (over BWϕ) which is defined by thehalf-power (-3 dB) points. The half-power points of the antenna radiation pattern (i.e.the -3 dB beam width) are normally specified as the limits of the antenna beamwidthfor the purpose of defining angular resolution. Therefore, two identical targets at thesame distance are resolved in angle if they are separated by more than the antenna -3 dBbeamwidth. [?,?]

The angular resolution as a distance between two targets can be found as:

SA = 2R sinBWϕ

2[m] (3.3)

Where, ∆φ is the half-power antenna beamwidth, SA the angular resolution as a distancebetween two targets and R the distance between target and antenna measured in meters.

For a given angular resolution (as a rule of thumb), the longitudinal dimensions of anantenna are related to its required angular resolution by the following equation [?]:

BWϕ = 70λ/Da[] (3.4)

Where, λ is the wavelength and D the antenna longitudinal dimension.

An important remark has been made looking at Figure 3.3: the smaller the beamwidth ϕ,the higher the directivity of the radar antenna.

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3.3 CLASSICAL RADAR EQUATION AND RADAR CROSS-SECTION

All considerations, when calculating the radar equation, have been made assuming thatthe electromagnetic waves propagate under ideal conditions without disturbing influences.Although, a number of losses should be considered since the effectiveness of the radar isconsiderably reduced when are taken into account.

The radar equation accomplishes the relationship between the signal power received andthe radar and target parameters. The equation describes the received power from inter-fering sources, thermal noise, clutter and jamming and it is used to predict echoes andinterfering power.

Several parameters affect the radar system: the operating parameters, target parame-ters and the propagation medium parameters. The first ones include power transmitted,antenna gain and effective aperture, receiver noise performance and radar system losses.Target parameters include radar cross-section (RCS) and radar range. And the last onesinclude RF energy absorption by gases and scattering.

In typical radar systems the antenna is directional, which concentrates the power towardsthe targets [?]. The Power Flux at the antenna is:

Ift =PtGTx4πR2

[W/m2] (3.5)

Where, PTx is the power transmitted by the antenna, GTx is the transmitter antenna gainand R is the range.

Those targets intercept a portion of the incident power, and re-radiate it. The measure ofthe incident power intercepted by the target and radiated back towards the radar is calledthe Radar Cross Section, σ. RCS indicates how large the target appears to be viewed bythe radar and depends on the angle of incidence at which it is viewed, the radar frequencyand the polarization used. Then, the re-radiated power by the target is now as follows:

Pre =PtGtσ

4πR2[W ] (3.6)

Where, σ is the radar cross- section. On the return path, this power (again) spreads outover the sphere of area 4πR2. The power density at the radar becomes the Power FluxReceiver given by:

Ifr =PtGtσ

(4πR2)2[W/m2] (3.7)

The amount of this returning power intercepted by the antenna, is determined by its ef-fective antenna aperture, Ae. The effective antenna aperture arises from the fact that anantenna suffer from losses. Therefore, the received power at the antenna is not equal to theinput power. As a rule, the efficiency of the antenna is around 0.6 to 0.7 (Efficiency K) [?].

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Ae = AgK[m2] (3.8)

Where, Ag is the geometric antenna area [m2] and K is the efficiency. The mean powerreceived by the radar, Pr, is the radar equation in its fundamental form [?]:

Pr =PtGtσAe(4πR2)2

[W ] (3.9)

The target is considered as a spherical reflector of radius, a. In the Figure 3.4 it is shownthat the RCS of the target is the area on the surface of the sphere that isotropically re-radiates all of its incident power at the same radiation intensity as the target re-radiatestoward the radar receiver. Then the RCS is equal to its projected area πa2 [?].

σ =4πIfr[W ]

If t[W/m2]= πa2[m2] (3.10)

Figure 3.4: RCS of Spherical Equivalent Target[?]

Then, rewriting the radar equation and solving for σ:

σ =Pr4πR

2

PtGt[m2] (3.11)

In Figure 3.4 is shown that the target is equivalent to a sphere, but in the following (Table3.1) are described for different geometries:

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Geometric Shape Dimension Cross-Sectional Area σmax

Sphere radius a πa2 πa2

Cylinder l x radius a 2la 2πal2

λ

Flat Plate a x a a2 4πa4

λ2

Ellipsoid semi-axes a ≥ b ≥ c - πa2b2

c2

Table 3.1: Radar Cross Section of Ideal Geometric Shapes[?]

3.4 ANTENNA FIELD REGIONS

The space around an antenna can be usually subdivided into three different regions: reac-tive near-field, radiating near-field (Fresnel zone) and far-field (Fraunhofer zone) regions asis shown in Figure 3.5. All of these regions are so designated to identify the field structurein each one. Since the Reactive near-field zone is stated as an electromagnetic criterionfor a single element of D size, it cannot be applied in our scenario (array of antennas, of Dsize, along the vessel length). Although no abrupt changes in the field configurations arenoted as the boundaries are crossed, there are distinct differences among them. As theboundaries separating these regions are not unique, the criteria stated in Antenna Theorybook will be presented in this section [?].

Figure 3.5: Field regions of an antenna [?].

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Now, according to the previous figure (Figure 3.5), the following definitions for the differentregions are given as below:

• Reactive near-field region: it can be defined as the portion of the near-field re-gion immediately surrounding the antenna wherein the reactive field predominates.For most antennas, the outer boundary of this region is commonly taken to exist atthe following distance:

R < 0.62√D3a/λ[m] (3.12)

Where λ is the wavelength and Da is the largest dimension of the antenna used tocalculate the boundary (R).

• Radiating near-field (Fresnel) region: is defined as the region of the field ofan antenna between the reactive near-field region and the far-field region whereinradiation fields predominate and wherein the angular field distribution is dependentupon the distance from the antenna. If the antenna has a maximum dimensionthat is not large compared to the wavelength, this region may not exist. For anantenna focused at infinity, the radiating near-field region is sometimes referred toas the Fresnel region on the basis of analogy to optical terminology. Here, the innerboundary is taken to be the distance below:

R ≥ 0.62√D3a/λ[m] (3.13)

Finally, the outer boundary is given by:

R < 2D2a/λ[m] (3.14)

Where Da is the largest dimension of the antenna in transmission, but it can bealso for reception (as is stated below) which is the case studied in thisreport: an array of antennas along the length of the vessel, which receive the signalfrom the target. The field which is reflected from the target has a spherical form,because of that, the angle referring the position of the target along the sea clutterbecomes important.

In the next figure, (Figure 3.6), is shown the symmetrical distribution of the field inthe two sides of the vessel. In this first simulation, the flat Earth model hasbeen supposed while, for the scenario calculations, the horizon distance has beentaken into account and the boundary achieved is shown in Figure 3.7. Note thatthis criterion is based on a maximum phase error of π/8 (it means 22.5). In the

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following equation is considered the angle of the target along the vessel plane:

Da = Lbsin(φ)[m] (3.15)

R = 2D2a/λ[m] (3.16)

Where Lb is the length of the boat and φ is the angle referring to the target position(seen from the middle of the vessel).

Figure 3.6: Matlab simulation of the near field (Fresnel zone) for the vessel array when isseen from different angles (flat Earth).

Now, using the geometrical calculations, the distance to horizon in this case is givenby:

Dt = D1 +D2[km] (3.17)

Dt =√

2Reh1 + h12 +√

2Reh2 + h22 (3.18)

Dt = 3.57(√h1 +

√h2) (3.19)

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Figure 3.7: Altitude vs. range(horizon distance).

Where, Re is the Earth radius and h1, h2 the height of both vessel and dinghy.Solving for the dimensions given in the next chapter (2):

Dt = 3.57(√

30 +√

1.5)[km] (3.20)

Dt = 24km (3.21)

The radar horizon is not the same as the geometrical horizon of the scanner’s height.Atmospheric density gradients bend radar rays as they travel to and from the tar-get. This bending is called refraction. The maximum radar range is determined bythe height of the scanner and the power output of the transceiver. Objects beyondthe radar horizon will not be detected unless a reflecting surface extends above thehorizon. Nothing is depicted on the display beyond the first object detected unlessit is taller.

Thereby, in the next simulation (Figure 3.8) is shown the antenna size in functionof the angle for the distance in analysis considered from the origin equal to 24 km(radius):

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Figure 3.8: Visual size of the array. (r=24 km).

• Far-field (Fraunhofer) region: it can be found as the region of the field of anantenna where the angular field distribution is essentially independent of the dis-tance from the antenna. If the antenna has a maximum 1 overall dimension D, thefar-field region is commonly taken to exist at distances greater than 2D2

a/λ from theantenna, λ being the wavelength. Whereas the outer boundary is taken to be atinfinity, the inner one is considered as the radial distance given by: [?].

R = 2D2a/λ[m] (3.22)

In this region, the field components are essentially transverse and the angular dis-tribution is independent of the radial distance where the measurements are made.Finally, the outer boundary is the one at infinity.

The amplitude pattern of an antenna, as the observation distance is varied from the re-active near field to the far field, changes in shape because of variations of the fields, bothmagnitude and phase. A typical progression of the shape of an antenna, with the largestdimension Da, is shown in Figure 3.9.

It is apparent that in the reactive near field region the pattern is more spread out andnearly uniform, with slight variations. As the observation is moved to the radiating near-field region (Fresnel), the pattern begins to smooth and forms lobes. In the far-field region(Fraunhofer), the pattern is well formed, usually consisting of few minor lobes and one,

1To be valid, D must also be large compared to the wavelength (Da > λ)

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Figure 3.9: Typical changes of antenna amplitude pattern shape from reactive near fieldtoward the far field [?].

or more, major lobes [Balanis].

Now, focusing into this project and as is stated in the next chapter (2) is necessary totake into account the scenario in study. Then, working in reception and assuming a vessellength of 300 meters, is known that the reception antenna array will be:

• For the outer boundary:

Da = 300m (3.23)

R2 = 2D2a/λ (3.24)

R2 = 2 · 3002/λ (3.25)

Where, using the operating frequency at 1 GHz:

λ = c/f (3.26)

R2 = 2 · 3002/3 · 108

109(3.27)

R2 = 600km (3.28)

Finally, taking into account the horizon distance (3.19) the boundary in the scenario whichis being studied will be around 24 km. All of these calculations can be proved using the

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simulation (assuming flat Earth) shown in Figure 3.6. Note that is not chosen each ele-ment of the array separately, otherwise it will be in the far-field (Fraunhofer) zone. Then,the situation in reception will be clearly a radiating near-field (Fresnel zone) region from,approximately, few meters (due to each element of the array will be Da 1m, and it meansto deal with coupling near-field instead of conduction near-field) as is shown in Figure 3.5and Figure 3.9.

In the next table (Table 3.2), is shown the relationship between the near-field outer bound-ary and the frequency. As can be seen, this table states how the outer boundary (equation3.14) changes when increasing the radar frequency band:

Band Specific ranges Theoretical boundary (Da=300m) Boundary(Da=1m)

HF 10 MHz R=6 km R=0.07 m

VHF 150 MHz R=90 km R=1 m

UHF 500 MHz - R=3 m

L 1 GHz - R=7 m

S 2 GHz - R=12 m

C 4 GHz - R=25 m

X 8 GHz - R=50 m

Ku 16 GHz - R=100 m

K 25 GHz - R=167 m

Ka 32 GHz - R=200 m

V 64 GHz - R=400 m

W 100 GHz - R=667 m

Table 3.2: Radar frequency bands vs. near-field outer boundary.

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Chapter 4

SIGNAL MODEL DUE TO SEACLUTTER

In maritime surveillance radar, modeling the sea is very important because sea clutterproperties have a big impact on detection process. Indeed, echoes from waves are mixedwith target echoes and make the detection more difficult. Along this chapter, the seasurface features and the returned signal to the receiver characterization will be described.

4.1 SEA WAVES MODELING

In order to study radar backscattering from sea, we need to characterize the sea surface.In general, modeling the sea surface is a very complex problem which is even more com-plicated if we consider wind interaction, currents or breaking waves. An ocean wave isthe undulation, the rising and falling movement, of the sea surface and is usually causedby winds. Waves are generated (born) in the fetch area (where wind and water are inter-acting) and travel across the sea until their collapse (death) as breakers on some distantshore. This is usually called: the life cycle of a wave.

The highest part of the wave is called the crest, whilst the lowest part of the wave is calledthe trough. Waves can be described by their: height, wave-length, and wave period. Thewave-height is the vertical distance from the crest to the trough. The wavelength is thehorizontal distance between the crest of one wave and the crest of the next (successive)wave. Simplifying the problem, we are going to consider two types of waves: gravity andcapillary waves. [?]

• Gravity waves. Waves generated by the accumulation of gravitational forces andwhose velocity of propagation is controlled mainly by gravity. This kind of waveshas wavelengths of meters. [?]

• Capillary waves. In this case, the surface tension of the water controls propagationvelocity of these waves. Capillary waves have smaller amplitudes and wavelengthsaround 2 cm. As it can be seen in the below graphic, this kind of waves are at the

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surface, superposed onto the gravity waves. [?] [?]

Thus, ocean waves will be approximated by a small scale random roughness which is su-perimposed on a large scale surface.

Figure 4.1: The phenomegical model of sea surface and interaction with electromagneticwaves.

[?]

4.2 RADIO SIGNAL FROM SEA CLUTTER

Once all the information related to sea waves has been given, we are going to focus on signalprocessing. So henceforth, our objective is to achieve power of sea clutter returned signal.After selecting a distribution to model sea clutter amplitude, all the needed parametersare explained, just as the models used to calculate it.

4.2.1 Distribution due to sea clutter

There are several distributions used to model sea clutter depending on the wanted res-olution, the polarization and the grazing angle, which is the angle between the radarillumination wave vector and the mean sea surface plane, measured in the plane of inci-dence.In the case of working with low resolution, and for grazing angles greater than 10 degrees,Rayleigh or Weibull distributions can be used to model sea clutter amplitude.But for smaller grazing angles, spiky results and long fails are observed when the afore-mentioned distributions are used. In this case, to get better resolution the K distributionfits better. [?] [?]

K - Distribution

This distribution is a good fit to the real clutter signal amplitude, which is mod-eled as the product of a Rayleigh-distributed term and a root-gamma distributed

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term. Gamma distribution shows the underlying mean level of the clutter, rep-resents the slowly varying component, and it is the part in which shadowing is taken intoaccount. While the Rayleigh distribution represents the speckle component. [?] [?]

The overall amplitude distribution is given by:

p(x) =2c

Γ(ν)

(cx2

)νKν−1(cx) (4.1)

Where, x is the amplitude of the returning signal and (defined for the range 0< x < ∞),ν the shape parameter (ν > 0), Kν is the modified second-kind Bessel function of order νand c is the scale parameter (c > 0) and it is given by: c = 2b

√(π/4).

Furthermore, K distribution allows explaining the temporal and spatial correlation char-acteristics. This distribution will be used and in order to realize the simulations the saidparameters will be calculated as follows.

• An empirical model for νThe shape parameter ν provides information about the amplitude statistics and rep-resents the ”spikiness” of the clutter. Moreover, it gives some of the correlationproperties. [?] [?]

This parameter depends on the grazing angle, the polarization, radarparameters, the cross-range resolution and the sea swell direction; andaccording to the empirical model developed by Ward, ν can be calculated by usingthe next formula.

logν =2

3logθi +

5

8log∆ + ζ − k (4.2)

Where, θi is the grazing angle in degrees, ∆ the across-ranges resolution [m].ζ : = −1/3 for up or down swell directions.= 0 for intermediate directions or no swell exists.= +1/3 for across swell directions .k : = 1 for vertical polarization.= 1.7 for horizontal polarization.

As it can be seen, the smaller is the grazing angle, the smaller is the shapeparameter. Furthermore, for horizontal polarization the clutter is spikier, due tothe fact that in this case ν is smaller than for vertical polarization.In practice, the value of the shape parameter varies within the range:

0 <1

ν< 10 (4.3)

Smaller values correspond to very spiky data, and the bigger ones correspond toapproximately Gaussian distributed data. [?] [?]

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• Scale parameter cThe parameter c describes only the mean power return, which are related by thenext expression. [?] [?]

Pc =4ν

c2(4.4)

According to above formula, if c decreases, the reflected signals from the sea surfacewill be more powerful. Working up the value of c, it can be calculated as follows.

c =

√4ν

PtG2tλ2f4

(4πR)3σ0θBW

clτ2

(4.5)

Where, Pt is the transmitted power [W], Gt the transmitter antenna gain, λ theradar wavelength [m], f4 is the two-ray antenna pattern value at the surface, R isthe slant range to the clutter cell, θBW is the antenna pattern beamwidth, τ is thepulse width and σ0 is the mean clutter cross section.

4.2.2 Backscattering coefficient

The backscattering coefficient, also called sea clutter cross section, σ0, is the normalizedclutter radar cross section and it is given by the following expression:

σ0 = σ/Ap (4.6)

Where, σ is the equivalent radar cross-section of the sea clutter return for that pixel andbeing Ap the area of the resolution pixel.

5.2.2.1 Illuminated Area

In order to calculate σ, we have to define the illuminated area, which will be given bya simplification of the radiation antenna pattern. We will work with a radiationfootprint as pie slice, as it can be seen in the image below.During our study, we found out that azimuth angle is very small (1), getting a longnarrow beam. Instead, we have worked with big elevation angles (22.5) in comparison toazimuth one, and we have assumed that antenna’s gain along elevation does notvary. As a simplification, in the simulation we have divided the illuminated area intosquare pixels, and basing on the position of the pixel center, the pixel will be illuminatedor not.

5.2.2.2 Sea Clutter Cross Section σ0

The clutter cross section increases with frequency, sea state and grazing angle. Dependingon the grazing angle, two different models will be used to characterize sea clutter in orderto be more accuracy for each situation. The first one describes the situation where theangle is low, between 0and 3, in which the dependence of the clutter cross section on

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Figure 4.2: Pie slices.

Figure 4.3: Beamwidth Transmitter Antenna.

the grazing angle is strong. The second model is used for greater angles, from 3to 90.The next figure shows a general overview variation of backscattering coefficient.

As it is shown in the following table, the GIT model can be used to describe lowgrazing angle situation and the TSC model for the case where angles aregreater than 3.

Next, both used models are going to be detailed.

GIT sea clutter model

The Georgia Institute of Technology (GIT) [?] sea clutter model is one of the available

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Figure 4.4: Typical relation between clutter cross section and grazing angle.[?]

Parameter GIT model TSC model

Indicated carrier frequency, GHz 1-100 0,5-35

Average wave height (m ) 0 to 4 sea state 4.2

Wind (kts) 3-30 0-5

Grazing angle (degrees) 0,1 - 10 0,1 - 90

Polarisation HH,VV HH,VV

Table 4.1: Radar cross section model capabilities.[?]

models to estimate the sea clutter coefficient, σ0 . This model provides prediction ofthe mean sea clutter reflectivity (σ0) from 1 GHz to 100 GHz. The value returnedwill be a function of grazing angle, wind speed, average wave height, polarization, radarwavelength and the look direction with respect to the wind direction. The dependence ongrazing angles is stronger at low grazing angles, and GIT model gives us more accuracyvalues for grazing angles not greater than 3. For the rest cases, other models will be used.To calculate σ0 three different contributions are taken into account and each one is goingto be detailed below.

• Multipath factor

This factor represents multipath propagation contribution for a Gaussian distribution ofwave heights, and it can be obtained using equations below.

σφ = (14.4λ+ 5.5)θihav/λ (4.7)

Al = σ4φ/(1 + σ4

φ) (4.8)

Where, hav is the average wave height [m].

• Sea direction factor

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This factor is obtained empirically and it is given by:

Au = exp[0.2cosϑ(1− 2.8θi)(λ+ 0.02)−0.4] (4.9)

Where, ϑ is the looking direction relative to wind direction angle [rad].

• Wind speed factor

Now, referring to the wind speed factor, the equations stated below are used:

qw =1.1

(λ+ 0.02)0.4(4.10)

Aw =1.94Vw

(1 + Vw/15.4)qw(4.11)

Where, Vw is the wind speed which must be input separately for the conditions of changingwind and is given by:

Vw = 8.67(hav)0.4 (4.12)

For a ”fully arisen” sea under conditions of stationary equilibrium. Once this relationshipsare proved, in the next table (Table 4.2) are shown the speed of wind in function of bothsea state and wave height.

Wind speed [m/s] Wave height [m] Sea state

<0.51 0 0

0.51 - 1.54 0 0

2.06 - 3.09 0 - 0.3 1

3.6 - 5.1 0.3 - 0.61 2

5.7 - 8.2 0.61 - 1.2 3

8.7 - 10.8 1.2 - 2.4 4

11.3 - 13.9 2.4 - 4 5

14.4 - 17 4 - 6.1 6

17.5 - 20.6 4 - 6.1 6

21.1 - 24.2 4 - 6.1 6

24.7 - 28.3 6.1 - 9.1 7

28.8 - 32.4 9.1 - 13.7 8

>32.9 >13.7 9

Table 4.2: Various relationships between sea state, wind speed and wave height ??.

Where, for our scenario (Indian Ocean), in the National Data Buoy Center (Center of ex-cellence in marine technology) are found both recent and historical data about the oceansand its properties ??. Then, taking into account the last year data stored from the stationnumber 41025 and calculating the mean value from it, the average wind speed for theIndian Ocean is: V w = 6.3m/s.

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Once all the previous parameters have been calculated, the following expressions are usedto achieve the reflectivity depending on the polarization.

• Horizontal polarization

σ0(HH) = 10 ∗ log(3.9 ∗ 10−6λθ0.4i AlAuAW ) (4.13)

• Vertical polarization

When 3GHz < f <10GHz:

σ0(V V ) = σ0(HH)− 1.05ln(hav + 0.015) + 1.09ln(λ) + 1.27ln(θi + 0.0001) + 9.70(4.14)

When f < 3GHz:

σ0(V V ) = σ0(HH)− 1.73ln(hav + 0.015) + 3.76ln(λ) + 2.46ln(θi + 0.0001) + 22.2(4.15)

In both cases, results are given in dB m2/m2. As it may be concluded from this equa-tions, the mean clutter reflectivity increases with frequency, grazing angle or sea state andit decreases with the look direction moves from the upwind direction. Furthermore, thegot value for vertical polarization is greater than for horizontal one.

TSC sea clutter model The Technology Services Corporation (TSC) is a sea clut-ter model that works from 1 to 35 GHz and is available for angles from 3to 90. It is amodel using Douglas sea state number, grazing angle, wind aspect angle, radar wavelengthand polarization. Moreover, the TSC model have a similar functional form that the GITmodel. So, like the GIT model, different contributions are needed and each one is goingto be described now.

• Angle factor

This factor GA can be obtained using equations below :

σz = 0.115SS1.95 (4.16)

σα = 14.9θi(σz + 0.25)/λ (4.17)

GA = σ1.5α /(1 + σ1.5

α ) (4.18)

(4.19)

Where, σz is the surface height standard deviation and SS is the sea state defined byDouglas.

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• Wind speed factor

Now, to calculate the wind speed factor Gw, the equations stated below are used:

Vw = 6.2SS2.8 (4.20)

Q = θ0.6i (4.21)

A1 = (1 + (λ

0.03)3)0.1 (4.22)

A2 = (1 + (λ

0.1)3)0.1 (4.23)

A3 = (1 + (λ

0.3)3)Q/3 (4.24)

A4 = 1 + 0.35Q (4.25)

A = 2.63A1/(A2A3A4) (4.26)

Gw = [(Vw + 4.0)/15]A (4.27)

(4.28)

Where, Vw is the wind speed [m/s].

• Aspect factor

This factor GU verifies this equation below:

When θi=π/2:GU = 1 (4.29)

Otherwise:GU = e0.3cos(φu)∗e−θi/0.17/(λ2+0.005)0.2 (4.30)

Where, φu is the looking direction relative to wind direction angle [rad].

Once all the previous parameters have been calculated,the reflectivity depending on thepolarization can be calculated following these expressions:

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• Horizontal polarization

σ0(HH) = 1.7 ∗ 10−5θ0.5i GuGwGA/(λ+ 0.05)1.8 (4.31)

• Vertical polarization

When f <2GHz:

10log(σ0(V V )) = 10log(σ0(HH))− 1.73ln(2.507σz + 0.05) + 3.76ln(λ)

+2.46ln(sin(θi) + 0.0001) + 19.8 (4.32)

When f >= 2GHz:

10log(σ0(V V )) = 10log(σ0(HH))− 1.05ln(2.507σz + 0.05) + 1.09ln(λ)

+1.27ln(sin(θi) + 0.0001) + 9.65 (4.33)

In both cases, results are given in dB m2/m2.

4.2.3 Phase Characteristics of Sea Clutter

Complex received signal is measured by coherent radars, in other words, a measurementof both amplitude and phase is made in coherent detection. Due to this fact, the rate ofchange of sea clutter phase allows to evaluate motions, making easier target detection bycomparing velocities.Sea clutter return from the illuminated area is the result of a lot of individual reflectors,which returns are interacting. As a conclusion, it is generally accepted that the phasecomponent of sea clutter is uniformly distributed on 2π. [?] [?]

4.2.4 Setting of a Detection Threshold

The detection problem can be formulated with two hypotheses which are the hypothesis ofno target, H0, and the hypothesis of detection (true or false) where the signal is identifiedas target plus clutter, H1.

H0 : x = cH1: x = t + c

Under theses hypotheses, different probabilites can be defined. First, there is the proba-bility of false alarm, Pfa, the probability to choose H1 whereas H0 is true. It is givenby:

Pfa =

∫ ∞XT

P (x|H0)dx (4.34)

Where:XT is the threshold.

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P (x|H0) is the pdf of the signal under the hypothesis H0 that is to say from the clutter only.

There is also the probability of detection, Pd, describing the situation where H1 ischosen correctly. Pd is defined by:

Pd =

∫ ∞XT

P (x|H1)dx (4.35)

Where:P (x|H1)is the pdf of the signal in the presence of target.

The threshold is set to be the limit to detect target.The value of threshold is cal-culated to guarantee a specific probability of false alarm, α. Therefore, withthis constraint, the threshold for single pulse detection can be determinated by using theexpression below:

α =

∫ ∞XT

P (x|H0)dx =2cν

Γ(ν)XνTKν(2cXT ) (4.36)

Where, XT is threshold value, x is a random sample of the K distribution.

As enviromental conditions and radar parameters change, the threshold should vary tomaintain a constant false alarm rate (CFAR). [?]

Thus, for a given Pfa, 10−2 in this study case, the threshold is calculated by solvingnumerically the equation 4.36. From the equation, an approximation of the solution forthe threshold in a K-distributed clutter has been developed and evaluated in IEEE paperwritten by D.A.Abraham in 2010 [?]. The approximation threshold, depending onthe Pfa and the shape parameter from the K-distributed clutter, is calculated with thefollowing equations:

X0 = −log(Pfa) (4.37)

XT ' X0 +a1X0a2

νa3X0.20

(4.38)

Where, X0 is the threshold when the clutter is considered as gaussian, ν is the shapeparamater characterising the K-distributed clutter, and a1, a2, a3 are empirical parame-ters chosen according to Pfa.

The next table (Table 4.3) presents the set of values of parameters a1, a2, a3 according tothe given Pfa.

As it can be seen in the following graph, the figure 4.5 shows the accuracy of the approx-imation of the threshold which presents errors less than 0.2 dB compared to real valuesover a wide range of Pfa and ν.The detection process using likelihood ratio test is based on Neymann-Pearson criterionwhich maximises the probability of detection, Pd, while maintaining the false alarm rateat a constant value α. The likelihood ratio test, ∆, is given by

∆(x) =P (x|H1)

P (x|H0)〈XT (4.39)

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Pfa a1 a2 a3

1−2 0.112 2.256 0.516

1−3 0.098 2.376 0.525

1−4 0.128 2.245 0.519

1−5 0.149 2.179 0.509

1−6 0.145 2.189 0.500

1−7 0.116 2.267 0.491

1−8 0.234 2.018 0.481

1−9 0.236 2.018 0.473

1−10 0.274 1.972 0.466

Table 4.3: Paramaters for K-distribution threshold approximation. ??.

Figure 4.5: error in the threshold approximation as a function of ν for various Pfa.[?]

Where, XT is the threshold, and P (x|Hi) is the pdf of the observed signal x conditionedin the hypothesis Hi

The likelihood ratio test is built in order to decide whether the pdf of the signal is betterdescribing by P (x|H0) or P (x|H1). Thus, if ∆(x) < XT , that means the presence ofthe target is detected. Otherwise, the signal is identified as clutter only. The thresholddepending on the probability of false alarm and the shape parameter, ν, of K-distributedsea clutter is calculated with the approximation given in the equation 4.38. XT has to below enough to detect target especially small targets yet high enough to reject clutter andavoid false alarm.

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Chapter 5

SYSTEM DESCRIPTION

In this chapter, all the main blocks of the whole system and assumptions made will bedescribed. First of all, the geometry has to be settled down: region of analysis, symmetry,ranges and angles for each element of the reception array. Then, the channel modeling(up-link/down-link) will be presented for both normal and antenna array radar. For thereceived signal, the clutter behavior will be taken into account. Furthermore, definitionsabout signal quality such as Signal-to-Noise Ratio (SNR), Signal-to-Clutter Ratio (SCR),Signal-to-Interference plus Noise Ratio (SNIR) need to be presented. Finally, the lastsubsection will be focused in the system implementation and all the steps followed.

5.1 GEOMETRY

The target can be located in any position surrounding the vessel with the macro array fordetection. As it can be seen in Figure 5.1, due to symmetry, it is only necessary to analyzea quarter of the space (in the other three possibilities, the relations will be the same buttaking into account the reflection with respect to the appropriated axis).

To describe the geometry of the reception part, range to the target (Ri) and angle respectto the vessel side (φi) are considered for all the elements of the array 1. The centralelement used as reference is described with R0 and φ0. This can be seen in Figure 5.2.It is necessary to relate all the ranges and angles from the different single elements of thearray to the central element which will be the reference in order to estimate R0 and φ0

from every single sample of the array. According to the basic triangle trigonometry, therelation from the elements at the two sides of the reference will be different. They can becalculated using the relations from Appendix E.

1The array must content 2N+1 elements and the element N+1 will be the reference situated in themiddle of the array. Each element will be referred with the index i = 1, 2, ..., 2N + 1

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Figure 5.1: Symmetric views and region of analysis

Figure 5.2: Top view of ranges and angles in different elements of the reception array

For the elements at the side of the center farther from the target (elements with R1 andφ1 in Figure 5.2):

R20 = R2

1 + d21 − 2R1d1cos(φ1) (5.1)

φ0 = acos

(R2

0 +R21 − d2

1

2R0R1

)+ φ1 (5.2)

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For the elements at the side of the center nearer from the target (elements with R2 andφ2 in Figure 5.2):

R20 = R2

2 + d22 − 2R2d2cos(π − φ2) (5.3)

φ0 = φ2 − acos(R2

0 +R22 − d2

2

2R0R2

)(5.4)

Once obtained these relations, it is necessary to introduce the elevation angle from thetarget respect to the vessel in order to understand how to translate the measurementsobtained from the samples in each antenna (distance d0 and angle φ0) to the real rangeR0. This situation has been represented in Figure 5.2.

Figure 5.3: Relation between range and angles translating top view to real distance

With the help of basic trigonometry, the next relations are obtained:

R0 =hhull

tan(θ0)(5.5)

R0 = d0cos(θ0) (5.6)

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5.2 CHANNEL MODELING

Figure 5.4: Transmitter and Receiver signals

In a typical wireless communication channels, when a pulse is transmitted through theradio channel, multiple pulses are received that spread out in time. The amount of time-spreading of the transmitted signal depends on the amount of multipath present in theenvironment. In highly cluttered areas such as urban regions the time-varying channel issaid to be time-dispersive [?].

In this project, a narrow pulse is transmitted in a environment were two distinctions mustbe done. The whole system taking into account the arriving reflections from the sea, isthe first case. In this case, the reflections from the sea could be treated as multi-pathenvironment, so the complete system is time-dispersive. The second case, in a local area(local area are the pixel of the illuminating area) the local multipath is neglected withinthe illuminating area because is received only one reflection per pixel. These reflectionsare shown in both Figures 5.4 and 5.5.

In radar systems, is necessary to model the channel considering the downlink and uplink.In the case of a typical wireless environment, the convolution between the transmittedsignal with the downlink channel is needed to obtain the signal which is incident to bothsea and target. After modeling the interaction effect of the incident wave and the contactsurface, a convolution of the reflected signal and the uplink channel is done.

The assumption that the interactions between the incident wave and the surface happenswithout angular dispersions 2 allows us to consider that there is no interactions between

2Angular dispersion is when local multi-path occurs throughout the area where the incident wave hitsproducing many echoes from this area

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the reflections from different points. Due to that and the fact that the system simulatedin this project has a channel that is not considered local time-dispersive, intermediaryconvolutions are not necessary. Instead, the complete channel, downlink, interaction withthe surface and uplink, can be expressed by the impulse response of the channel, as shownin equation 5.20 .

The channel reception is treated independently for each antenna because the signal re-ceiver hits every element of the array in different instants of time. Hence, it is made theconvolution of the signal with the channel (DL+UL) depending of the delay in order tomodel one channel for each antenna.

Figure 5.5: Arriving reflections from the sea

Considering the transmitted signal as a train of modulated pulses (with pulse duration w,and repetition time T):

p(t, w) = rect

(t

w

)ej2πfct (5.7)

pt(τ, w, T ) = p(τ, w) ∗∑n

δ(τ − nT ) =

=∑n

rect

(τ − nTw

)ej2πfc(τ−nT )

(5.8)

The variable w and T are understood, so it is not necessary to carry them in every equa-tion, in particular because the baseband frequency is the one taken into account. Finally,the transmitted signal is given by:

sTx(τ, ϕ, ϕ0, θi) = pt(τ)ATx(ϕ− ϕ0, θi) (5.9)

Where ATx(ϕ − ϕ0, θi) is the antenna directivity pattern with the main orientation ϕ0

and the elevation angle θi. However, ATx(ϕ − ϕ0, θi) is considered as this following step

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function in azimuth where the value for the gain is an empirical data:

ATx(ϕ− ϕ0, θi) =

35[dB] if ϕ0-BW2 < ϕ < ϕ0+BW

20 otherwise

(5.10)

In addition, it is assumed that the gain does not change with the elevation angle in thearea considered illuminated in one period of the transmitted signal (T) because typicalvalues for fan beam antenna have an elevation directivity in radar systems between 20 and30 degrees[ [?]] with fairly constant gain. Thus, the antenna directivity pattern is only afunction of ϕ. Therefore, the signal transmitter is rewritten as follow:

sTx(τ, ϕ0) = pt(τ)ATx(ϕ0) (5.11)

After, the signal transmitter is convoluted with the channel.

In Figure 5.6 and Figure 5.7 are shown the different cases for classical radar and thespecific one of this project with an array of antennas in reception.

Figure 5.6: Illuminating area for a classical radar

In order to model the channel, the downlink and uplink processes are distinguished. Then,to model the channel assuming that the whole system is time-dispersive channel, all thecontributions of the illuminating area are calculated in the equation 5.20. Firstly, as it isshown in the next figure 5.8, we will focus on modeling the impulse response for a specificpoint defined by (R,φ) in each tap (this will be detailed in the following) delimited by twoellipses. In the same delay, only signal from this particular tap will come.

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Figure 5.7: Illuminating area for a specific case (array of antennas)

hDL(t, ϕ,R) =λ

4π

1

Rej2π

Rλ δ(t− R

c) (5.12)

hUL(t, ϕ,R, ϕRx, RRx) =λ

4π

1

cτUL(ϕ,R)ej2π

cτUL(φ,R)

λ δ(t− τUL(φ,R)) (5.13)

Where, the delay taken into account in uplink is given by the following relation:

τUL(ϕ,R) =

√R2 +R2

Rx − 2RRRxcos(ϕ− ϕRx)

c(5.14)

To calculate the signal backscattered from the clutter it should be taken into account twodifferent possibilities due to the possibility that the incoming signal is illuminating bothtarget and sea surface. The reflected signals will be determined by the coefficients αt(amplitude change depending on the area of the target illuminated,with the simplificationthat the target is made out of a perfect conductor) or by αs and ξs (amplitude and phasechanges due to sea clutter effects).

α(ϕ,R)ejξ(ϕ,R)dis(ϕdis) =

(αte

jπdis(ϕdis)αse

jξsdis(ϕdis)

)(5.15)

Moreover, theoretically in the point of contact,there is dispersion. However, a simple casewhere there is locally no dispersion will be considered. Therefore, the angular dispersion

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function dis(ϕdis) will be neglected. So, at the point of contact, there is no dispersion butjust a change of phase, direction of propagation and amplitude of the signal.

hscattered = α(ϕ,R)ejξ(ϕ,R) (5.16)

Here, when the downlink and the uplink are taken into account, the channel as the multi-plication of the 3 components is modeled. When moving from continuous time to discretetime, the impulse response of the channel must be calculated for each sample. In thispaper, the taped delay model is used, so the impulse response of the channel is calculatedfor each tap.

The time distance between two taps is τtap. For a classical radar, the area illuminatedin the time τtap for tap number n is the subtraction of two sectors that are delimited bythe beamwidth of the transmitter antenna with the ranges nτtapc and (n − 1)τtapc . Forthe array described, the area illuminated in the duration of one tap is an area delimitedby two ellipses and the beamwidth of the transmitter antenna, as shown in Figure 5.8,because the transmitter antenna and Rx antenna are not in the same position. So thesignals that arrive in the same time come from points that are positioned on a ellipse noton a circle as for traditional radars.

The signal in one tap is the integration of the impulse response of each point in thearea illuminated in one tap. The limits of this area for ϕ are: ϕ1 = ϕ0 − BW/2 andϕ2 = ϕ0 +BW/2. The range limits for tap number n are: Rn(ϕ) and Rn−1(ϕ), calculatedusing equations 5.17 and 5.18. The first one is the equation of the ellipse in polar coor-dinates when the axes origin is in one of the foci, in this case in the transmitter antennaposition as represented in Figure 5.9. With 5.17, 5.18 and the point P that has the coordi-nates RP = nτtapc and ϕP = ϕ0, the parameters a and ε can be obtained thus describingthe ellipse Rn(ϕ).

R(ϕ) =a(1− ε2)

1 + εcos(ϕ)(5.17)

ε =c

a=RRx2a

(5.18)

Where, ε is the eccentricity of the ellipse, dcenter is the distance between the center ofthe ellipse and the foci, a is half of the major axis and RRx is the distance between thetransmitter and receiver antennas. [?]

After these considerations concerning integration limits the impulse response for one tapn is calculated with equation 5.20.

hT (t, ϕ0, ϕRx, RRx, n) =(5.19)

=

∫ ϕ0+BW2

ϕ0−BW2

∫ Rn(ϕ,ϕRx,RRx)

Rn−1(ϕ,ϕRx,RRx)hDL(t, ϕ,R, n)hscattered(ϕ,R)hUL(t, φ,R, ϕRx, RRx, n)dRdϕ

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Figure 5.8: Highlighted is the illuminating area for the array radar for one tap whereRres = cτtap.

Figure 5.9: Geometry for calculating the ellipse equation.

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In order to have the total impulse response of the channel, all the impulse responses previ-ously calculated have to be summed for each tap. Thus, the model of the channel is givenby:

hT (t, ϕ0, ϕRx, RRx) =∑

1〈n〈N

hTn(t, ϕ0, ϕRx, RRx) (5.20)

Finally, the received signal is the convolution of the transmitted signal with the channel.Here below is the expression of the received signal coming out from each receiver antennadescribed by its position (RRx, φRx) in polar coordinates where the origin is in the trans-mitter antenna:

SRx(t, ϕ, ϕ0, ϕRx, RRx) = ARx(ϕ,ϕRx, RRx, θ)(STx(t, ϕ0) ∗ hT (t, ϕ0, ϕRx, RRx)(5.21)

But, we assumed that every receiver antenna is omni-directional with a constant gain equalto 1, so the expression of the signal at a certain receiver antenna output is finally given by:

SRx(t, ϕ0, ϕRx, RRx) =

∫ ∞−∞

STx(τ, ϕ0)hT (t− τ, ϕ0, ϕRx, RRx)dτ (5.22)

5.3 SIGNAL QUALITY DEFINITIONS

In radar, the signal-to-noise ratio (SNR) is known as the ratio of the power correspondingto a specified target measured at some point in the receiver to the noise power at thesame point in the absence of the received signal [?]. This project is based in so closeranges and there is no need to consider the thermal noise as a dominant source to localizeimperfections. The signal-to-noise ratio is defined as follows:

SNR =PRtPn

(5.23)

On the other hand, the signal-to-interference + noise ratio (SNIR) can be defined as thereceived signal quality at a receiver which is the ratio of the power of the wanted signal tothe total residue power of the unwanted signals and, where ”interference” means clutterpower plus noise power.

The lower the SNIR, the bigger the measurement error will be. The radar scenario isalways a low-SNIR scenario. In some applications the radar performance is noise lim-ited, while in other applications the performance is interference limited. The reflection ofa target is almost always accompanied by reflections from the surrounding environment(ground, sea), referred to as clutter, or by reflections from neighboring targets or targets

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farther away.

In order to correctly interpret the wanted signal, the SNIR must be above a given thresh-old. This parameter can be found as follow:

SNIR =PRt

PRn + Pn(5.24)

SNIR =SNR

1 + CNR(5.25)

SNIR =SNR

1 + SNRSCR

(5.26)

Where, PRt and PRc are target and clutter returns and Pn is the noise power. The situa-tion presented in this report, only deals with the own system, it means, no other signalswill contribute to the ones that are received. Finally, the clutter-to-noise ratio (CNR) isdefined as follows:

CNR =PRcPn

(5.27)

Figure 5.10: Clutter geometry [4].

The signal-to-clutter ratio (SCR) is defined, by the IEEE, as the ratio of target echo powerto the power received from clutter sources lying within the same resolution element. In

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other words, is the ratio of the return from a point target in the sea at the same rangeas the center of the clutter patch and also at the peak of the antenna beam [?]. In thefollowing equation the SCR is defined as:

• For low grazing angle:

SCR = (PRtPRc

)smallθi =σcos(θi)

σo∆R∆φ(cT/2)(5.28)

• For large grazing angle:

SCR = (PRtPRc

)largeθi =σsin(θi)

σo∆R2(∆φ∆θ)(5.29)

Where, σ is the radar cross section, ∆R is the range, c the speed of propagation, T is thepulse width, σo is the surface clutter a cross section per unit area, being ∆φ and ∆θ thebeamwidths. The only controllable variables in 5.28 and 5.29 are the pulse width (T ) andthe beamwidths ∆φ and ∆θ, so these relationships present possible trade off problems indeciding on those parameters in design.

As previously stated in 3.3, the radar cross-section (RCS) of a target is known as the(fictional) area intercepting that amount of power which, when scattered equally in all di-rections, produces an echo at the radar equal to that from the target; or in other terms [?]:

σ =power − reflected− toward− source/unit− solid− angle

incident− power − density/4π(5.30)

σ = limR→∞4πR2|ErEi|2 (5.31)

Where, R is the distance between radar and target, Er and Ei the reflected field strengthat radar and strength of incident field at target respectively.

The radar cross section does not necessarily bear a simple relationship to the physicalarea, except that the larger the target size, the larger the cross section is likely to be.Finally, the surface clutter a cross section σc per unit area Ac is defined as below:

σo =σcAc

(5.32)

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Chapter 6

SIMULATION

In this section, all the steps followed in order to simulate the whole system will be ex-plained. All the different parts in which it has been split up, will be presented in the nextsubsections: scenario modeled, detection, downlink, uplink and signal construction.

6.1 SCENARIO

First of all, the scenario has to be defined. In order to create the sea representation,a matrix with the sea waves heights is created as is shown in the next functional block(Figure 6.1). Taking into account the sea state (2, 4 or 6 have been simulated), the windspeed and the average wave height are calculated. Finally, these parameters are modelingthe sea surface (sea grid) as it can be seen in Figure 6.2 and Figure 6.3.

Figure 6.1: Functional block for the sea surface generation.

From this function comes out a matrix with all the height values (positives and negatives).Since is better to work with positive values (for future calculations), the zero (x axis) willbe scrolled down (only positive height values).

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Figure 6.2: Sea surface generated (sea state = 2).

Figure 6.3: Sea surface generated (sea state = 6).

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As it is plotted in the previous figures; it is obvious that the higher sea state, the roughersea surface. It can be seen by comparing both figures. In future simulations, these values2, 4 and 6 of sea state will be analyzed.

6.2 TRANSMISSION

All the simulations will be performed in base band, and the energy of a pulse will becompressed into a delta with constant complex amplitude.

6.3 ILLUMINATED AREA

The illuminated area can be determinated through the beamwidth azimuth. It is assumedthat if the center of a pixel is illuminated within the beam range, then the whole pixel(5m x 5m cell) becomes illuminated as well. The functional block used in this subsectionis detailed in the figure (Figure 6.4) below:

Figure 6.4: Functional block to represent the illuminated area.

The previous block is composed by structural data such as the sea grid coordinates (statedabove) and transmission/reception distances. In the following figure (Figure 6.5) an ex-ample for a random target within the sea grid created is shown:

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Figure 6.5: Illuminated area from the transmitter antenna for a transmission angle of 45and a beamwidth of 1.

The x axis of Figure 6.5 starts at -150 m due to the center of the vessel is set in zero inorder to represent the total area illuminated using beamwidth and range only. Finally,changing the beamwidth azimuth and the orientation of the transmitter antenna, the il-luminated area will vary.

Then, when the sea surface is modeled (see Appendix ??), the shadowing in this surfacehas to be taken into account: depending on the different angles and heights of the surfacegenerated, the pixels which are in shadowing, or not, will be studied. The percentageof the target located in a shadowing position will be figured out through calculating theangular difference between a shadowing point and the position when this point stops beingin within the shadowing zone.

6.4 DOWNLINK

This functional block will generate the incident signal hitting the illuminated sea surface.Now, taking into account the distances from each illuminated pixel, the amplitude andphase of the transmitted signal will change according to free space conditions (Friis model).Using the following expressions, another data matrix is generated with the complex valuesof the incident signal and the delay:

A = Ainiλ

4πcτj(6.1)

Where, A is the amplitude of the signal, Aini is the initial amplitude, τj = Rj/c is thesignal delay hitting each pixel and Rj is the distance from the transmitter to each pixel.

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In this case, the starting signal comes from the transmitter so it is neccesary to apply theantenna gain (GT ) to the amplitude:

Atx = A√GT (6.2)

ξ = ξini +2πRjλ

(6.3)

Where, ξ is the phase of the signal, and ξini is the initial phase.

The functional block implemented in this subsection is shown in the figure below (Figure6.6):

Figure 6.6: Functional block for downlink.

6.5 TARGET AND SEA CLUTTER

In this subsection, the changes on the signal incoming on the sea surface are explained.The backscattered signal will depend on each point illuminated over the sea surface andthe target (only if it is in the illuminated area):

• Sea:

– Amplitude (A): The amplitude is given following a K-distribution (see Chapter4).

– Phase (ξ): The phase is given following a random uniform distribution from 0to 2π.

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The amplitude in this case, will follow the next formulation:

Aα(cellsize)2σs (6.4)

Where, σs is different for each pixel and depends on the grazing angle (calculatedusing the GIT Model 4.2.2).

• Target:

– Amplitude (A): The amplitude is given by the Radar Cross-Section (RCS) ofthe target.

– Phase (ξ): The phase will suffer a 180 change (reflection from target).

The amplitude now, will change as follow:

AαRCSσt (6.5)

Where, σt is the RCS of the target and is given by the next equation:

σt = SCR+ σs (6.6)

As expected, the lower SCR, the harder detection.

The functional block describing this effect, could be something like Figure 6.7:

Figure 6.7: Functional block for the backscattering from the sea.

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6.6 UPLINK

As previously stated in the downlink subsection, now the signal reflected from the surfacewill get both amplitude and phase modified. Assuming free space conditions and usingthe same expressions above, a new data matrix is generated with the amplitude and phaseof each element (incoming reflection from each pixel) of the reception array as shown inFigure 6.8. Furthermore, the delay suffered by this signal in the way back to the receptionarray is added in this data matrix.

Figure 6.8: Functional block for uplink.

6.7 WB (WIDE BAND) SIGNALS RECEIVED

Finally, in this subsection, the signals received are sorted by delay (from lower to higher).In order to generate the Ultra Wide Band (UWB) estimation, it has been taken intoaccount the delay of each echo (reflection) produced when transmitting the pulse. Thissignal is reflected in each pixel of the illuminated area (now the grazing angle becomesimportant), the backscattered signal generated is normal to the direction of propagationbut in opposite way (180), which means that there is only one reflection per pixel (nolocal time-dispersive).

The reflections from the closest pixels, with the same time delay (τj = Rj/c), are summed(averaged) to obtain only one sample of the signal with the same delay. Then, the samemethod is applied with all the other delays received as is shown in Figure 6.9, where all theamplitudes can be related with its corresponding time delay. Last, with this time delaybetween samples, the time resolution can be calculated (through the time resolution, thepeaks of the signal may be distinguished in order to decide if the signal hit either the seaor target).

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Figure 6.9: Example of received signals in 2 different antennas of the array.

6.7.1 NB (Narrow Band) Signals

For the Narrow Band (NB) case, the delay becomes unnecessary. Now, the samples withhigher amplitude are filtered (the ones with the same delay) and its magnitude (ampli-tude) and phase are extracted in order to achieve in a straightforward manner the NBsignal. During the construction of the system, it was realized that the narrow band in-formation based on amplitude and phase differences was not credible due to the differentcontributions of the clutter over the different antennas.

6.8 TARGET ESTIMATION

From the Wide Band signals, information can be extracted through thresholding. In thiscase, it is used a combination of PFA (Probability of False Alarm) and spatial properties.

To apply this threshold it is necessary to estimate the mean of the received signal. InFigure 6.10 it can be seen an example:

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Figure 6.10: Example of thresholding in one signal from an antenna of the array.

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All the peaks over the threshold are detected and treated with the WB estimator explainedin Section ??.

6.9 COMPARISON WITH TYPICAL RADAR

In Chapter ??, it was explained the behavior of the typical radar. In the simulation it hasbeen also implemented to try to appreciate the possible differences with the receiver array.

In Figures 6.11 and 6.12 it can be seen the signals received by the normal radar and howthey are used to detect the peaks.

Figure 6.11: Example of signal received in typical radar.

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Figure 6.12: Example of thresholding in one signal received in typical radar.

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6.10 SIMULATION RESULTS

During the simulation all the steps can be tracked in the main window as it can be seenin Figure 6.13.

Figure 6.13: Information screen during simulation.

After the simulation, all the results (both from normal radar or the reception array) areplotted to make comparisons (Figure 6.14). Several measures of error are performed aswell, as explained in the next chapter.

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Figure 6.14: Plotted results after one simulation.

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Appendix A

RADAR SIGNAL GENERATION

A radar system uses a radio frequency electromagnetic signal reflected from a target todetermine information about that target (i.e. position). In any radar system, the sig-nal generated will exhibit many of the characteristics described in this appendix. Thein-phase and quadrature components, range resolution and pulse compression definitionswill be given in the next sections.

A.1 SIGNAL GENERATION

Marine radars are usually pulse radars which send out signals in short (few millionthsof a second) but powerful bursts or pulses.

In the simplest case, a pulse radar can transmit is a sinusoidal signal of amplitude Aand carrier frequency f0, truncated by pulse of width T .1 We can write the signal usingcomplex notation [?]:

s(t) = Resc(t)ej2πf0t 0 ≤ t ≤ T (A.1)

sc(t) = Aeφ(t) (A.2)

φ(t) = 0 (A.3)

Where, s(t) is the signal transmitted and sc(t) is the baseband signal and its phase isφ(t) = 0.

If the previous expression is related to I (in-phase) and Q (quadrature) components:

1The pulse is transmitted periodically, but this is not the main topic of this appendix.

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sc(t) = I(t) + jQ(t) (A.4)

sc(t) = Ae0 = A (A.5)

I(t) = A (A.6)

Q(t) = 0 (A.7)

s(t) = ReAej2πf0t = Acos(j2πf0t) 0 ≤ t ≤ T (A.8)

In Figure A.1, it is represented a sinusoidal pulse of amplitude A = 2V. with carrierfrequency f0 = 3GHz. The pulse length has been set to be 10 periods of the signal(T = 3.33ns).

Figure A.1: Sinusoidal Pulse generation from I/Q components

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The range resolution with this sinusoidal pulse is v T2 . This means, that to increase theresolution, the pulse length must be reduced.

The energy of the signal (E) is directly proportional to the length of the pulse:

E =

∞∫−∞

|s(t)|2dt =

T∫0

|Aej2πf0t|2dt = A2T [V 2s] (A.9)

The received signal depends on the transmitted signal, so the energy of the received signalwill also be directly proportional to the pulse length. This means that the SNR is higherfor longer pulses (detailed information about signal quality (SNR, SNIR, etc. is given inthe System Description chapter 5).

The effect between better resolution and higher SNR must be balanced. To achieve this,pulse compression is used and it can be done with two different modulations: Linear-Frequency Modulation (’Chirping’) and Phase-Coding Modulation.

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The basic principle of pulse compression is the following [?]:

• A code is modulated on the pulsed signal.

• This signal is transmitted with a long enough length so that the energy budget iscorrect.

• In the receiver, the code is used to combine the signal information to achieve a highrange resolution.

A.2 LINEAR-FREQUENCY MODULATION (LFM)

In radar applications, linear chirps are the most typically used signals to achieve pulsecompression. They are rectangular pulsed signals in amplitude with finite length andsweep linearly the frequency band ∆f centered on the carrier f0. These signals can bewritten [?]:

s(t) = Resc(t)ej2πf0t 0 ≤ t ≤ T (A.10)

sc(t) = Aeφ(t) (A.11)

φ(t) = 2π

(∆f

Tt− ∆f

2

)t (A.12)

f(t) =1

2π

(dφ

dt

)= f0 +

∆f

Tt− ∆f

2(A.13)

Rewriting the signal expression into I/Q format:

sc(t) = I(t) + jQ(t) (A.14)

sc(t) = Acos(φ(t)) + jAsin(φ(t)) (A.15)

s(t) = I(t)cos(2πf0t)−Q(t)sin(2πf0t) 0 ≤ t ≤ T (A.16)

Where, f(t) is the instant frequency. In Figure A.2, it can be seen an example of this kindof modulation using a pulse with a length T = 3.33ns, sweeping frequencies in the range(bandwidth) ∆f = f2 − f1 = 5GHz − 1GHz = 4GHz.

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Figure A.2: Linear Frequency Signal generation from I/Q components

The distance resolution reachable with a linear frequency modulation of a pulse on abandwidth ∆f is: c

∆f . The pulse compression ratio can be expressed as T∆f and it isgenerally greater than 1. After pulse compression, the power of the received signal can beconsidered as being amplified by T∆f .

A.3 PHASE-CODING MODULATION

In this case, the pulse is divided in N time slots of duration TN for which the phase at the

origin is chosen according to a pre-established convention (code). To implement this kindof pulses, the most widely used type of phase coding is binary coding. [?]

The binary code consists of a sequence of either +1 and -1. The phase of the transmittedsignal alternates between 0 and 180 in accordance with the sequence of elements. Sincethe transmitted frequency is usually not a multiple of the reciprocal of the sub pulse width,the coded signal is generally discontinuous at the phase-reversal points. [?]

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The general expression for this signals is:

s(t) =

N∑i=1

Resci(t− T

Ni

)ej2πf0(t−

TNi) 0 ≤ t ≤ T (A.17)

sci(t) = Aeφi(t) (A.18)

φi(t) =

0 if 1π if -1

(A.19)

Rewriting the signal expression into I/Q format:

sci(t) = Ii(t) + jQi(t) (A.20)

Ii(t) =

Acos(0) + jsin(0) = A if 1Acos(π) + jsin(π) = −A if -1

(A.21)

Qi(t) = 0 (A.22)

s(t) =N∑i=1

Ii

(t− T

Ni

)cos

(2πf0

(t− T

Ni

))(A.23)

Figure A.3, shows an example of this Phase-Coding modulation using a pulse with alength T = 5 ns. This pulse has been coded following the next sequence: ’-1,1,1,-1,1’.Each subinterval has a length of Ti = 1 ns. The modulation works on a carrier frequencyof f0 = 3GHz.

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Figure A.3: Phase-Coding Signal generation from I/Q components

The range resolution with this modulation is v T2N and the pulse compression ratio is lower

than in the linear frequency modulation because the bandwidth is smaller.

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Appendix B

RADAR FREQUENCIES

In radar applications there are a lot of frequency operating bands (as is shown in Ta-ble B.1), but there are two basic marine radar frequencies commonly known as ”X” and”S” band. The ”X” band, because of its higher frequency, at 10 GHz provides a higherresolution and a crisper image while ”S” band, at 3 GHz is less affected by rain and fog. Inmost situations larger vessels are fitted with both ”X” and ”S” band radars while smallervessels will only have an ”X” band.

Mostly, radars at X band are generally of a convenient size and are, therefore, of interestfor applications where mobility and light weight are important and very long range is nota major requirement. The relatively wide range of frequencies available at X band and theability to obtain narrow beamwidths with relatively small antennas in this band requiresimportant considerations for high resolution applications. Because of the high frequencyof X band, rain can sometimes be a serious factor in reducing the performance of X-bandsystems.[ref] In Table B.1, are shown the specific frequency ranges for radar applicationsbased on ITU (International Telecommunication Union) frequency assignments.

BAND FREQUENCY RANGE SPECIFIC RANGES(RADAR)

HF 3-30 MHz

VHF 30-300 MHz 138-144 MHz and 216-225 MHz

UHF 300-1000 MHz 420-450 MHz and 890-942 MHz

L 1 - 2 GHz 1215-1400 MHz

S 2 - 4 GHz 2.3-2.5 GHz and 2.7-3.7 GHz

C 4 - 8 GHz 4.2-4.4 GHz and 5.25-5.925 GHz

X 8 - 12 GHz 8.5-10.68 GHz

Ku 12 - 18 GHz 13.4-14 GHz and 15.7-17.7 GHz

K 18 - 27 GHz 24.05-24.25 GHz and 24.65-24.75 GHz

Ka 27 - 40 GHz 34.4-36 GHz

V 40 - 75 GHz 59-64 GHz

W 75 - 110 GHz 76-81 GHz 92-100 GHz

Table B.1: Radar frequency bands [Radar handbook].

Bandwidth is a fundamental parameter of any imaging system and determines the ulti-

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mate resolution available (from the Table B.1), we can find the bandwidths for all thefrequency-bands as is shown in Table B.2. The radar resolution is its ability to displaymultiple targets clearly and separately. Range resolution refers to targets oriented alongthe beam axis as viewed from the position of the antenna. Longer pulses have poorer rangeresolution and targets too close together lose definition and become blurred. It must bemore than one-half pulse length apart or will occupy the pulse simultaneously and appearas a single target. Finally, as is shown in the following table, the bandwidth (BW) growsas higher is the frequency-band used. This provides more resolution (∆τ) and range (∆R)for the target detection.[ref] Then, the resolution and range can be calculated as below,

∆τ = 1/BW (B.1)

∆R = co∆τ (B.2)

Where co = 3 · 108m/s is the speed of light, Bandwidth (BW), Range (∆R) and Resolu-tion (∆τ).

BAND CENTER FREQUENCY BANDWIDTH RESOLUTION RANGE

VHF 141 MHz 6 MHz 167 ns 50 m

UHF 435 MHz 30 MHz 33 ns 10 m

L 1.3 GHz 185 MHz 5.4 ns 1.622 m

S 2.4 GHz 200 MHz 5 ns 1.5 m

X 9.55 GHz 2.18 GHz 0.46 ns 13.7 cm

Ka 34.7 GHz 2.6 GHz 0.38 ns 11.5 cm

V 61.5 GHz 5 GHz 0.2 ns 6 cm

Table B.2: Range and resolution versus frequency

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Appendix C

SEA ELECTROMAGNETICS

In marine radar, parameters depending on the frequency such as conductivity, permittiv-ity, resolution and range have to be analyzed. Now, referring to the sea parameters, in thefigure below (Figure C.1) are shown the permittivity and conductivity versus frequencyin a sea surface.

In the table below (Table C.1) is found the relationship between frequency and sea pa-rameters such as conductivity and permittivity,

BAND FREQUENCY CONDUCTIVITY (σ) PERMITTIVITY (εr)

VHF 141 MHz 5 S/m 70

UHF 435 MHz 5 S/m 70

L 1.3 GHz 5 S/m 70

S 2.4 GHz 6 S/m 70

X 9.55 GHz 17 S/m 50

Ka 34.7 GHz 60 S/m 19

V 61.5 GHz 70 S/m 11

Table C.1: Sea parameters versus frequency (Sea water at 20C)

ε = εrεo (C.1)

Where ε is the medium permittivity, εr the relative permittivity and εo the vacuum per-mittivity.

As is known, reflectivity is the fraction of incident radiation reflected by a surface (seaclutter in our study case). Then, the reflection phenomenon, occurs when light movesfrom a medium with one index of refraction into a second medium with a different indexof refraction.In Table C.2 below are some examples of radar antenna systems and its fea-tures depending on the operation frequency from some European manufacturers [].

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Figure C.1: Permittivity and conductivity of the sea (A) [Rec. ITU-R P.527-3].

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BAND BANDWIDTH GAIN BEAMWIDTH -3dB LENGHT

S 100 MHz more than 31 dB 1.1 6.3 m

X 400 MHz more than 35 dB 0.29 - 0.43 5 m - 9.5 m

Ka 400 MHz more than 49 dB 0.59 1.1 m

Table C.2: Typical parameters of marine radar antennas

C.1 POLARIZATION AND SCATTER ECHOES

There are many examples of strong polarization differences in the low grazing radar seascatter results in the literature. Here below are shown some general definitions in thischapter:

• Polarization of an electromagnetical wave is specifying the orientation of the wave’selectric field at a point in space over one period of the oscillation. In this paper onlyV polarization (vertical, the electrical field vector is perpendicular to the surfacein the direction of propagation) and H polarization (horizontal,the electrical fieldvector is parallel with the ground in the direction of propagation) are used.

• Grazing angle. Known as the angle between a ray incident on a surface and thesurface at the point of incidence.

• Brewster angle.(Also known as the polarization angle) is an angle of incidence atwhich light with a particular polarization is perfectly transmitted through a surface,with no reflection (C.2).

θB = arctan(

√ε2

ε1) (C.2)

Where θB is the Brewster angle and ε is the respective permittivity in each medium (ε1

for the air and ε2 for the sea surface).

In [ref] it is argued that Brewster angle damping and local multipath effects are consideredas sources of polarization differences. This is done by using a model that describes themarine waves as a sinusoidal with an cylinder like crest in the moment of braking. Inthe generating of the model the assumption that the illuminating radar pulse has a smallduration of just a couple of waves length was made. In the following section the connectionbetween polarization and sea scatter is shown.

As is shown in Figure C.2, the dependence of the reflectivity on radar polarization isimportant. As has been seen in this section, many other clutter characteristics are depen-dent on radar polarization and often quite different scattering mechanisms are present fordifferent polarizations [3].

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Figure C.2: Typical variation with grazing angle and polarization of sea clutter reflectiv-ity.[3]

During a field experiment using horizontal (HH) and vertical (VV) polarizations for lightwind and wave conditions using a marine imaging radar in coastal regions, strikingly dif-ferent polarization characteristics appeared. As the phenomena of braking of the sea wavesbegan to appear, HH echoes became larger than the VV polarized ones. The separationbetween illumination effects and scattering phenomena is done in [2] by using very shortpulses of just a few radar wavelengths so the multipath length differences are consideredto be longer then the pulse length thus eliminating the multipath interference. Only themultiple echoes in time are observed due to local multipath.

Figure C.3: (a) Radar illumination of a sea wave. (b) Illumination geometry.[]

As is shown in the previous Figure, C.3(a), radar illumination of a sea wave is depictedwhich is confined to a narrow height range of the order of a few radar wavelengths (thetop of the wave that is braking). Where, h is the crest front height, this results in ascattering lobe with the dominant fraction of energy scattered away from the radar un-der the geometrical optics approximation, but also allowing multipath illumination of thebore which rises sharply above the local mean surface. Referring to the second one (b),the illumination geometry is shown for illumination of a perfectly conducting dihedral,

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by a direct path and one scattered via the surface forward vertical face, resulting in fieldmaximal indicated by filled and shaded circles for V and H, respectively.

The area is illuminated by a plane wave, with one path to the bore propagating directlyfrom the radar, and a second via the scattered path from the water surface a distance,A, forward of if as shown by the lines drawn. The bidirectional arrows indicate that bothlines are possible return paths to the radar as well. In Figures C.3 is represented thephenomena of local multipath. In C.3 a) a multipath bore model is represented withthe crest front a height, h, above the local tilted mean surface. Figure C.3 b) showsthe multipath illumination of a dihedral, with the x-axis taken along the direction of theincoming radiation. The incident fields are propagating at a local grazing angle (elevationangle), θi relative to its horizontal face. The total field at a height, h, on its perpendicularface is the sum of the fields via the direct path from the radar, plus the once-reflected pathfrom the surface at a distance, A, forward of the vertex. For the H case, a perpendicularline to the propagation direction intersects the horizontal surface at a distance from thevertex, AH0. The path length from this point is L’ for the direct path and L for theforward scattered path. The total electrical field at a height h is calculated using equationC.3 :

EI = EIejkx+jkL′ +RjEje

jkx+jkL (C.3)

Where EI is the field with polarization H either V. The path distance between the directand reflected ray to the point h is L-L’. Considering the reflection coefficients (perfect con-ductor) , RV = 1 and RH = −1, the distance between two field maximums or minimumson h is k(L−L′) = 2nπ for V and k(L−L′) = 2(n+ 1)π for H. Using simple geometry itcan be derived that field maximums are distributed along the vertical axes are given foran integer n, for H and V polarization by the following equations:

hHn =2n+ 1

2λ/2cosθi(C.4)

and

hV n =fr2nλ

2cosθi(C.5)

The resulting illumination gain factor (IGF), the ratio of the total power incident to thatreflected off the wave, for a perfect conductor has a distribution along the height h asrepresented in figure ??. For sea water, which is an imperfect conductor, Brewster angleeffects will modify the reflection coefficient for polarization V , but not for polarization Has shown in Figure ??. The Fresnel equations for the reflection coefficients of the sea forthe two polarization are [Rec. ITU-R P.680-3]:

RH =sinθi −

√η − cos2θi

sinθi +√η − cos2θi

(C.6)

RV =sinθi −

√η−cos2θi

η2

sinθi +√

η−cos2θiη2

(C.7)

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Where, the reflection coefficient for circular polarization, RC , is given by:

RC =RH +RV

2(C.8)

With, the intrinsic impedance of the medium for this case as below:

η = εr(f)− j60σ(f) (C.9)

Where εr is the relative permittivity of sea water (in function of frequency), σ is the con-ductivity of sea water (in function of frequency) and θi is the local grazing angle.

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Appendix D

SEA MODEL IMPLEMENTED

In this appendix, some of the background knowledge and concepts required for understand-ing the wind-generated ocean surface model implemented with Matlab are introduced. Astatistical description of wind-generated surface waves becomes very important when ex-amining the effects of the propagation through a rough surface such as the ocean one.Much work has been done to the study of the statistics of the ocean surface. Oceanogra-phers Pierson and Moskowitz developed spectral density functions and analytical solutionsfor the height variance of the ocean surface versus wind speed. Their model was later im-proved by the North Sea Wave Project (JONSWAP) spectrum.

D.1 PIERSON-MOSKOVITZ MODEL

In 1964, Pierson and Moskowitz developed a model for the spectrum of fully developedwind seas. A fully developed wind sea is a stage in the growth of a wind-driven oceanwhere, given a constant wind velocity and, an adequate fetch and duration, the spectrumwill no longer grow. This model was based on several ship-recorded wave records collectedover five years. To obtain the spectrum of the sea, they used measurements of waves madeby accelerometers on British weather ships. First, they selected wave data for times whenthe wind had blown steadily for long times over large areas. Then, they calculated thewave spectra for various wind speeds. The modeled spectrum (Figure D.1) may be statedmathematically as below [Ocean Scenes]:

S(ω) =αg2

ω5exp[−β(ωo/ω)2] (D.1)

Where, ω = 2πf , f is the wave frequency in Hertz, α = 0.0081 (Philips constant), β = 0.74,the peak frequency ωo is directly related to the wind speed at a height of 10 m above thesea surface, U10 (height of the anemometers on the weather ships used). Finally, the peakfrequency is given by:

ωo =g

1.026U10(D.2)

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And, directly, the period:

To =2π

ωo(D.3)

Figure D.1: Wave spectra of a fully developed sea for different wind speeds according toPierson-Moskowitz.[Oceanography dep.]

To describe the ocean surface in spatial coordinates, the Pierson-Moskowitz spectrum hasto be rewritten in terms of the wave number k. Using the deep water dispersion relation-ship:

ω2 = gk (D.4)

The wavenumber spectrum in one dimension will be as follow:

S(k) = S(ω)dk

dω(D.5)

Then, the significant wave height is calculated from the integral of S(ω) and it gives:

< ζ2 > =

∫ ∞0S(ω) dω (D.6)

< ζ2 > =2.81 · 10−3(U10)4

g2(D.7)

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Remembering that H1/3 = 4 < ζ2 >1/2, the significant wave-height calculated from thePierson-Moskowitz spectrum is:

H1/3 =0.22(U10)2

g(D.8)

Finally, the next figure (Figure D.2) gives significant wave heights and periods calculatedfrom the Pierson-Moskowitz spectrum.

Figure D.2: Significant wave-height and period at the peak of the spectrum of a fullydeveloped sea calculated from the Pierson-Moskowitz spectrum.[Oceanography dep.]

For the implementation, synthetic ocean images may be generated from white noise im-ages in a very straightforward manner using the forward FFT and the inverse FFT. Thewhite-noise image is generated by adding uniformly distributed noise, having intensitiesbetween -127 and 127, to a constant intensity image of gray level 128. This results inrandom gray shades between 0 and 255, inclusive. A two-dimensional forward FFT isperformed on the image to generate a magnitude and a phase image. The result of theFFT is a complex number file [Ocean Scenes].

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Appendix E

TRIANGLES

Figure E.1: A triangle with sides of length a,b,c and angles A,B,C.

E.1 LAW OF SINES

a

sin(A)=

b

sin(B)=

c

sin(C)(E.1)

E.2 LAW OF COSINES

a2 = b2 + c2 − 2bccos(A) (E.2)

b2 = a2 + c2 − 2accos(B) (E.3)

c2 = a2 + b2 − 2abcos(C) (E.4)

E.3 LAW OF TANGENTS

a− ba+ b

=tan(1

2(A−B))

tan(12(A+B))

(E.5)

[?]

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Appendix F

MATLAB CODES

F.1 Main

Main program to set the parameters and run the complete simulation.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/main.m

1 % main.m2 % RADAR project − Group 890 MOB − Aalborg University 20103

4 clear all5 close all6 clc7

8 %% Data9

10 % 0.Global11

12 c=3e8; % speed of light [m/s]13 x min=−150; % minimum distance over y axis [m]14 y min=0; % minimum distance over y axis [m]15 x max=2500; % maximum distance over x axis [m]16 y max=2500; % maximum distance over y axis [m]17 SCR=40; % sea to clutter ratio [dB]18

19 % 1.Vessel20

21 length boat=300; % lenght of the vessel [m]22 width boat=40; % width of the vessel [m]23 height boat=30; % height of the vessel [m]24 hull boat=15; % height of the hull of the vessel [m]25

26 % 2.Dinghie27

28 length din=6; % lenght of the dinghie [m]29 width din=2; % width of the dinghie [m]30 height din=0.2; % height of the dinghie [m]31

32 % 3.Sea33 sea state=2; % sea state 2[4 m/s, 0.46 m] 4[9.7 m/s, 1.8 m] 6[15.8 m/s 5 m]

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34 wind direction=pi/4; % direction wrt x axis [rad]35

36 % 4.Transmission37

38 tx pos=[−130;−20]; % [x,y] position [m,m]39 tx angle=pi/4;% angle the transmitter is pointing to [rad]40 %tx bw elevation=22.5; % elevation beamwidth [ ]41 tx bw elevation=22.5∗pi/180; % elevation beamwidth [rad]42 %tx bw azimuth=1; % azimuth beamwidth [ ]43 tx bw azimuth=1∗pi/180; % azimuth beamwidth [rad]44 tx height=height boat; % transmitter height [m]45 fc=1e9; % carrier frequency [Hz] (L−band)46 lambda=c/fc; % wavelenght of the carrier [m]47 ph ini=0; % initial phase [rad]48 tau ini=0; % initial delay [s]49 amp ini=100∗sqrt(30); % initial amplitude [V]50 power ini=amp iniˆ2; % initial power [W]51 G tx=35; % transmitter antenna gain [dB]52 tx gain=10ˆ(G tx/10); % transmitter antenna gain53 BW=60e6; % bandwidth of transmitted signal [Hz]54 T res=1/BW; % time resolution [s]55 R res=c∗T res; % spatial resolution [m]56 fs=2∗BW; % Nyquist frequency [Hz]57 oversampling=10; % oversampling rate58 samples=fs∗oversampling; % number of samples/pulse59 w=T res; % pulse duration [s]60 T=x max/c; % repetition time [s]61

62 % 5.Target63

64 target pos=[890;980]; % [x,y] position [m,m]65 target height=height din; % target height [m]66 cross section=mean([length din∗width din height din∗width din ...67 height din∗length din]); % average rectangular cross section [m2]68

69 % 6.Reception70

71 M=15; % number of elements in reception (samples) − odd number72 N=(M−1)/2; % number of elements at each side of the center73 %d ant=length boat/(M−1); % distance between array elements [m]74 d ant=20; % distance between array elements [m]75 d local=lambda/2; % distance between antennas for angle resolution [m]76 rx pos=[−N∗d ant:d ant:N∗d ant;zeros(1,M)]; % [x,y] position [m,m]77 rx height=hull boat; % transmitter height [m]78 grx=1; % receiver antenna gain79 PFA=10ˆ−3; % expected Probability of False Alarm80

81 % 7.Scenario82

83 cell size=R res; % cell size [m] (according to resolution)84 x axis=x min:cell size:x max; % index vector for x axis [m]85 y axis=y min:cell size:y max; % index vector for y axis [m]86 blind zone=2∗length boat; % blind distance [m]87 blind zone t=blind zone/c; % blind delay [s]88

89 %% Create Scenario90 tini=clock;

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91 t1=tini;92

93 disp(0.Creating scenario...)94

95 % sea conditions96 [wind speed,avg wave]=sea conditions(sea state);97 t2=clock;98 time=etime(t2,t1);99 disp([ − Sea conditions: OK ... ,num2str(time),s])

100

101 % matrix with sea heights [m]102 [sea grid,Tp,fm,Sk,kx,ky]=sea surface(x axis,y axis,wind speed,...103 wind direction,none);104

105 % convert all elements to positive heights [m]106 sea grid=sea grid−min(min(sea grid));107 h sea max=max(max(sea grid)); % maximum height of the sea [m]108 t1=clock;109 time=etime(t1,t2);110 disp([ − Sea surface: OK ... ,num2str(time),s])111

112 % calculation of the illuminated area and link distances113 [data,data s,ones grid,target percent,percent tx]=illuminated(x axis,...114 y axis,sea grid,cell size,tx pos,tx angle,tx height,tx bw azimuth,...115 tx bw elevation,rx pos,rx height,target pos,target height,lambda);116 t2=clock;117 time=etime(t2,t1);118 disp([ − Illuminated area and link distances: OK ... ,num2str(time),s])119

120 %% Transmission121 disp(1.Transmission...)122

123 % generate signal124 s tx=signal generator(amp ini,ph ini,tau ini);125 t1=clock;126 time=etime(t1,t2);127 disp([ − Signal generation: OK ... ,num2str(time),s])128

129 %% Channel130 disp(2.Channel...)131

132 % downlink133 data d=downlink(data,abs(s tx(1)),phase(s tx(1)),s tx(2),c,lambda,tx gain);134 t2=clock;135 time=etime(t2,t1);136 disp([ − Downlink: OK ... ,num2str(time),s])137

138 % target and sea139 [nu,sigma s]=sigma calculator(data(3,:),cell size,tx height,lambda,...140 avg wave,wind direction);141 target pos index=look for target(data(1:2,:),target pos);142 if target pos index 6=0143 disp( ∗ Target in illuminated area)144 else145 disp( ∗ NO Target in illuminated area!!)146 end147

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148 [data t,data t n]=backscattered(data d,target pos index,sigma s,...149 target percent,percent tx,nu,SCR);150 t1=clock;151 time=etime(t1,t2);152 disp([ − Backscattered signals from sea and target: OK ... ,...153 num2str(time),s])154

155 % uplink156 [data u,data u s]=uplink(data,data s,data t,c,lambda,grx);157 t2=clock;158 time=etime(t2,t1);159 disp([ − Uplink ARRAY: OK ... ,num2str(time),s])160

161 [data u n]=uplink normal(data,data t n,c,lambda,tx gain);162 t1=clock;163 time=etime(t1,t2);164 disp([ − Uplink NORMAL: OK ... ,num2str(time),s])165

166 %% Reception167 disp(3.Recepection...)168

169 [tapped,amplitudes,tapped s,amplitudes s]=signal reception(data u,...170 data u s,T res,c,x max,y max);171 t2=clock;172 time=etime(t2,t1);173 disp([ − Received signals ARRAY: OK ... ,num2str(time),s])174 disp([ ∗ Shadow coefficients ARRAY: ,num2str(target percent)])175

176 [tapped n,amplitudes n]=normal signal reception(data u n,T res,c,...177 x max,y max);178 t1=clock;179 time=etime(t1,t2);180 disp([ − Received signal NORMAL: OK ... ,num2str(time),s])181 disp([ ∗ Shadow coefficient NORMAL: ,num2str(percent tx)])182

183 % figure184 % plot(tapped n/(1e−6),20∗log10(abs(amplitudes n)))185 % title(NORMAL signal received)186 % xlabel(delay [us])187 % ylabel(signal level [dBW])188 %189 %190 % figure191 % subplot(2,1,1)192 % plot(tapped(1,:)/(1e−6),20∗log10(abs(amplitudes(1,:))))193 % title(ARRAY signals received − Antenna 1)194 % xlabel(delay [us])195 % ylabel(signal level [dBW])196 % subplot(2,1,2)197 % plot(tapped(13,:)/(1e−6),20∗log10(abs(amplitudes(13,:))))198 % title(ARRAY signals received − Antenna 13)199 % xlabel(delay [us])200 % ylabel(signal level [dBW])201

202 %% Detection203 disp(4.Recepection...)204

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205 % threshold206 threshold=PFA tr(x max,y max,tx height,PFA)−8;207 t2=clock;208 time=etime(t2,t1);209 disp([ − Threshold calculation: OK ... ,num2str(time),s])210 disp([ ∗ PFA: ,num2str(PFA), − Threshold: ,num2str(threshold), dB.])211

212 % peak detection213 [delays peak,amp peak,amp s peak]=peak detection(tapped,amplitudes,...214 amplitudes s,threshold);215 t1=clock;216 time=etime(t1,t2);217 disp([ − Peak detection ARRAY: OK ... ,num2str(time),s])218

219 [delays n peak,amp n peak]=normal peak detection(tapped n,amplitudes n,...220 threshold);221 t2=clock;222 time=etime(t2,t1);223 disp([ − Peak detection NORMAL: OK ... ,num2str(time),s])224

225 %% Estimation226 disp(5.Estimating position...)227

228 % WB array229 [est]=WBposition(delays peak,d ant,rx pos,tx pos,tx angle,tx bw azimuth,M);230 t1=clock;231 time=etime(t1,t2);232 disp([ − Detection ARRAY: OK ... ,num2str(time),s])233 disp([ ∗ Targets: ,num2str(size(est,2)),])234

235 % normal236 [est n]=normal position estimator(tx pos,tx angle,c,delays n peak,...237 amp n peak);238 t2=clock;239 time=etime(t2,t1);240 disp([ − Detection NORMAL: OK ... ,num2str(time),s])241 disp([ ∗ Targets: ,num2str(size(est n,2))])242

243 % figure244 % hold on245 % scatter(target pos(1),target pos(2),x,LineWidth,2)246 % scatter(est(1,:),est(2,:),o)247 % scatter(est n(1,:),est n(2,:),d)248 % scatter(tx pos(1),tx pos(2),+b,LineWidth,2)249 % plot (rx pos(1,:),rx pos(2,:),xk,LineWidth,2)250 % plot ([−150 ,150 ,150 ,−150 ,−150], [0 , 0, −40 , −40 , 0],k,LineWidth,2)251 % axis([−150 2000 −40 2000])252 % legend(Real target,ARRAY estimations,NORMAL estimations)253

254 %% Simulation time255

256 tfin=t2;257 time=etime(tfin,tini);258 disp([ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ TOTAL TIME: ,num2str(time),s ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗])

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F.2 Sea Conditions

Selection of wind speed and average height of the waves in certain conditions.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/seaconditions.m

1 function [wind speed,avg wave]=sea conditions(sea state)2

3 % sea conditions.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of wind speed and average wave height for different sea7 % conditions.8 %9 % [wind speed,avg wave]=sea conditions(sea state)

10 %11 % Outputs:12 % wind speed: normal wind speed for the selected condition.13 % avg wave: average height of the sea waves for the selected condition.14 % Inputs:15 % sea state: number associated to different sea states.16 % sea state 2[4 m/s, 0.46 m] 4[9.7 m/s, 1.8 m] 6[15.8 m/s 5 m]17

18 if sea state==219 wind speed=6.8;20 avg wave=0.46;21 elseif sea state==422 wind speed=8;23 avg wave=1.8;24 else % sea state=6 by default25 wind speed=9.25;26 avg wave=5;27 end

F.3 Sea Surface

Sea surface calculation: matrix of heights. [?]

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/seasurface.m

1 function [s,Tp,fm,SK,kx,ky] = sea surface(x,y,U,thetaU,spreading)2

3 % SEA SURFACE: generates sea surface realizations for a given intensity4 % and direction of wind velocity (uses the Pierson−Moskowitz spectrum).5 %6 % Usage: [s,Tp,fm,Sk,kx,ky] = sea surface(x,v,vtheta,spreading)7 %8 % where x,y are vectors defining the surface grid, v is the intensity of9 % the wind speed, vtheta is the wind direction (in radians) and spreading

10 % is a string defining the angular spreading of the sea surface spectrum11 % (none for no spreading, cos2 for cosine−squared spreading, mits12 % for Mitsuyasu spreading and hass for Hasselmann spreading). The

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http://kom.aau.dk/group/10gr890/docs/code/seaconditions.m

http://kom.aau.dk/group/10gr890/docs/code/seasurface.m



13 % output is a matrix s (size(s) = [length(y) length(x)]), the peak period14 % Tp, the peak frequency fm, the spectrum Sk and the wavenumber vector15 % arguments kx and ky, which define the grid of Sk. The Pierson−Moskowitz16 % spectrum is valid for v ≥ 10.17 %18 % Examples:19 % nx = 101; xmin = 0; xmax = 100; x = linspace(xmin,xmax,nx);20 % ny = 51; ymin = 0; ymax = 50; y = linspace(ymin,ymax,ny);21 % [s,Tp,fm,Sk,kx,ky] = sea surface(x,y,10,pi/4,none);22 % figure(1)23 % subplot(211),mesh(x,y,s),ylabel(y(m)),xlabel(x(m))24 % subplot(212),mesh(kx,ky,Sk),ylabel(ky (1/m)),xlabel(kx (1/m))25 % [s,Tp,fm,Sk,kx,ky] = sea surface(x,y,10,pi/4,cos2);26 % figure(1)27 % subplot(211),mesh(x,y,s),ylabel(y(m)),xlabel(x(m))28 % subplot(212),mesh(kx,ky,Sk),ylabel(ky (1/m)),xlabel(kx (1/m))29 % [s,Tp,fm,Sk,kx,ky] = sea surface(x,y,10,pi/4,mits);30 % figure(1)31 % subplot(211),mesh(x,y,s),ylabel(y(m)),xlabel(x(m))32 % subplot(212),mesh(kx,ky,Sk),ylabel(ky (1/m)),xlabel(kx (1/m))33 % [s,Tp,fm,Sk,kx,ky] = sea surface(x,y,10,pi/4,hass);34 % figure(1)35 % subplot(211),mesh(x,y,s),ylabel(y(m)),xlabel(x(m))36 % subplot(212),mesh(kx,ky,Sk),ylabel(ky (1/m)),xlabel(kx (1/m))37 %38 % References:39 % 1) Directional wave spectra observed during JONSWAP 197340 % D. E. Hasselmann et al. 198041 % 2) Directional wave spectra using cosine−squared and cosine 2s42 % spreading functions43 % Coastal Engineering Technical Note 198544 % 3) Fourier Synthesis of Ocean Scenes45 % Gary A. Mastin et al. 198746 % 4) The generation of a time correlated 2d random process for ocean47 % wave motion48 % L. M. Linnet et al. 199749 % 5) Acoustic wave scattering from rough sea surface and sea bed50 % Chen−Fen Huang, Master Thesis. 199851 % 6) Tutorial 2: Ocean Waves (1)52 % http://www.naturewizard.com53 % 7) matlabwaves.zip54 % http://neumeier.perso.ch/matlab/waves.html55

56 %∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗57 % First version: 30/07/200858 %59 % Contact: [email protected] %61 % Any suggestions to improve the performance of this62 % code will be greatly appreciated.63 %64 %∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗65 s = [];66 Sk = [];67 Tp = [];68 fm = [];69 kx = [];

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70 ky = [];71

72 imunit = sqrt( −1 );73

74 g = 9.80665;75 gg = g∗g;76 alpha = 8.1e−3;77

78 UU = U∗U;79

80 nx = length( x );81 ny = length( y );82

83 %======================================================================84 % Surface generation:85 % since most expressions for the spectrum are given in the frequency86 % domain we need to convert wavenumbers to frequencies, apply the formulas87 % and go back to the wavenumber domain:88

89 fm = 0.13∗g/U;90 Tp = 1/fm;91 dx = x(2) − x(1);92 kxmax = 1/( 2∗dx );93 kx = linspace(−kxmax,kxmax,nx);94 dy = y(2) − y(1);95 kymax = 1/( 2∗dy );96 ky = linspace(−kymax,kymax,ny);97 [Kx,Ky] = meshgrid(kx,ky);98 K = sqrt( Kx.ˆ2 + Ky.ˆ2 );99 F = sqrt( g∗K )/( 2∗pi ); % Valid for surface waves over deep oceans

100 F( F == 0 ) = Inf ;101 K( K == 0 ) = Inf ;102 dFdK = sqrt( g./K )/( 4∗pi );103

104 %======================================================================105 % Calculate the spectrum in the frequency domain:106

107 SF = alpha∗gg/( (2∗pi)ˆ4 )∗( F.ˆ(−5) ).∗exp( −5/4∗( fm./F ).ˆ4 );108

109 %======================================================================110 % Convert spectrum from frequency domain to wavenumber domain:111

112 SK = SF.∗dFdK;113

114 %======================================================================115 % A real sea surface requires a symmetric spectrum in the wavenumber116 % domain; thus, wherever required, additional calculations will ensure117 % that the spreading matrix is indeed symmetrical:118

119 THETA = angle( Kx+imunit∗Ky ) − thetaU;120

121 switch spreading122

123 case none % no spreading124

125 D = ones(size(K));126

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127 case cos2 % cosinus−squared spreading128

129 D = cos( THETA ).ˆ2;130

131 case mits % Mitsuyasu spreading132

133 ss = 9.77∗( F/fm ).ˆ(−2.5);134 indexes1 = ( F < fm );135 ss( indexes1 ) = 6.97∗( F(indexes1)/fm ).ˆ5;136 Nss = gamma( ss + 1 )./( 2∗sqrt(pi)∗gamma( ss + 0.5 ) );137 D = ( ( cos( THETA/2 ).ˆ2 ).ˆ(ss) ).∗Nss;138 D = D + fliplr( flipud( D ) );139

140 case hass % Hasselmann spreading141

142 Mu = 4.06∗ones( size(K) );143 indexes1 = ( F > fm );144 Mu( indexes1 ) = −2.34;145 pp = 9.77∗( F/fm ).ˆMu;146 Npp = pi∗2.ˆ( 1 − 2∗pp ).∗gamma( 2∗pp + 1 )./( gamma( pp + 1 ) ).ˆ2;147 D = cos( THETA/2 ).ˆ(2∗pp)./Npp;148 D = ( ( cos( THETA/2 ).ˆ2 ).ˆpp )./Npp;149 D = D + fliplr( flipud( D ) );150

151 otherwise152

153 disp(Unknown sea surface spreading.)154

155 end156

157 %======================================================================158 % Get the power spectrum:159

160 D = D/max( D(:) )∗2/pi; % spreading normalization161 SK = SK.∗D; % power spectrum(k,theta) = spectrum(k)∗spreading(theta)162

163 %======================================================================164 % Get the surface realization from the spectrum:165

166 white noise = unifrnd(−127,127,ny,nx)/127;167 WHITE NOISE = fft2( white noise );168 NOISE amplitude = abs( WHITE NOISE );169

170 NOISE energy = sum( WHITE NOISE(:).ˆ2 );171 WHITE NOISE = WHITE NOISE/NOISE energy;172

173 centered WHITE NOISE = fftshift( WHITE NOISE );174 NOISE phase = angle( centered WHITE NOISE );175

176 % Modulate noise amplitude with the power spectrum:177 NOISE amplitude = NOISE amplitude .∗ SK;178

179 % Randomize modulated noise in the wavenumber space combining180 % modulated amplitudes with original phases:181 NOISE ipart = NOISE amplitude .∗ sin( NOISE phase );182 NOISE rpart = NOISE amplitude .∗ cos( NOISE phase );183

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184 filtered NOISE = NOISE rpart + imunit∗NOISE ipart;185 filtered NOISE = fftshift( filtered NOISE );186

187 % Get the 2D surface through an inverse fft:188 s = ifft2( filtered NOISE );189 s = real( s );

F.4 Illuminated

Calculation of a matrix with all the possible information relative to the cells illuminatedin the sea.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/illuminated.m

1 function [data,data s,ones grid,target percent,sh tx]=...2 illuminated(x axis,y axis,sea grid,cell size,tx pos,tx angle,...3 tx height,beamwidth,elevation,rx pos,rx height,target pos,...4 target height,lambda)5

6 % illuminated.m7 % RADAR project − Group 890 MOB − Aalborg University 20108 %9 % This function calculates if a point is illuminated by the transmitter,

10 % giving out a matrix with all the information related to each pixel.11 %12 % [data,data s,ones grid,target percent,sh tx]=...13 % illuminated(x axis,y axis,sea grid,cell size,tx pos,tx angle,...14 % tx height,beamwidth,elevation,rx pos,rx height,target pos,...15 % target height,lambda)16 %17 % Outputs:18 % data: matrix with information about position in the two first rows (x,y),19 % distances from each illuminated pixel to the Tx in the 3rd row,20 % distances from each illuminated pixel to the Rx in the succesives21 % rows from the 4th (one row per receiver element).22 % data s: the same as data, but with respect to the small elements at23 % distance lambda/2 in each antenna element.24 % ones grid: sea grid with ones only for pixels inside of the Tx beamwidth.25 % target percent: vector with shadow coefficients for each antenna.26 % sh tx: shadow coefficient for the transmitter antenna.27 % Inputs:28 % x axis: x data for the sea grid.29 % y axis: y data for the sea grid.30 % sea grid: matrix with the values of the sea height.31 % cell size: size of the cells of the sea grid.32 % tx pos: transmitter position.33 % tx angle: angle of transmission.34 % tx heigth: height of the transmitter over the boat.35 % beamwidth: azimuth beamwidth.36 % elevation: elevation beamwidth to correct position for shadowing.37 % rx pos: position of the receiver antennas.38 % rx height: height of the receiver antennas.39 % target pos: position of the target.40 % target height: height of the target.

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41 % lambda: wavelength.42

43 illum cont=0;44 data=zeros(3+length(rx pos(1,:)),round((length(sea grid(:,1))ˆ2)/2));45 data s=zeros(3+length(rx pos(1,:)),round((length(sea grid(:,1))ˆ2)/2));46 ones grid=zeros(size(sea grid));47

48 for i=1:length(sea grid(1,:))49 for j=1:length(sea grid(:,1))50 x0=(i−1)∗cell size;51 y0=(j−1)∗cell size;52 % check illumination53 flag=ilum pixel(tx pos(1),tx pos(2),x0,y0,tx angle,beamwidth);54 if flag==1 % if illuminated55 ones grid(j,i)=1;56 illum cont=illum cont+1;57 data(1,illum cont)=x0; % x position58 data s(1,illum cont)=x0;59 data(2,illum cont)=y0; % y position60 data s(2,illum cont)=y0;61 data(3,illum cont)=sqrt(abs(x0−tx pos(1))ˆ2+...62 abs(y0−tx pos(2))ˆ2); % distance to tx63 data s(3,illum cont)=sqrt(abs(x0−tx pos(1))ˆ2+...64 abs(y0−tx pos(2))ˆ2);65 for k=1:length(rx pos)66 data(3+k,illum cont)=sqrt(abs(x0−rx pos(1,k))ˆ2+...67 abs(y0−rx pos(2,k))ˆ2); % distance to M rx68 % distance to M small rx69 data s(3+k,illum cont)=sqrt(abs(x0+(lambda/2)−...70 rx pos(1,k))ˆ2+abs(y0−rx pos(2,k))ˆ2);71 end72 else73 ones grid(j,i)=0;74 end75 end76 end77

78 data=data(:,1:illum cont); % eliminate zero colums79 data s=data s(:,1:illum cont);80

81

82 % correct initial data of the transmitter over the grid83 tx pos ini=[tx pos(1)+abs(tx pos(2)/tan(tx angle));0];84 r tx ini=sqrt((tx pos ini(1)−tx pos(1))ˆ2+(tx pos ini(2)−tx pos(2))ˆ2);85 tx height ini=tx height−(r tx ini/tan(pi/2−elevation/2));86

87 % target shadow calculation88

89 target percent=zeros(1,length(rx pos(1,:)));90

91 % shadowing from the transmitter92 [r to tx,h to tx]=line of sight(x axis,y axis,cell size,sea grid,...93 tx pos ini(1),tx pos ini(2),target pos(1),target pos(2));94 sh tx=visible area(r to tx,h to tx,tx height ini,target height);95

96 % total shadowing from transmitter to receivers97 for i=1:length(target percent)

Group 890 - MOB 102



98 [r,h]=line of sight(x axis,y axis,cell size,sea grid,rx pos(1,i),...99 rx pos(2,i),target pos(1),target pos(2));

100 sh rx=visible area(r,h,rx height,target height);101 if sh tx≥sh rx102 target percent(i)=sh rx;103 else104 target percent(i)=sh tx;105 end106 end

F.5 Illuminated Pixel

Decision about if a pixel is illuminated or not.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/ilumpixel.m

1 function flag=ilum pixel(xt,yt,x0,y0,tx angle,beamwidth)2

3 % ilum pixel.m4 %5 % RADAR project − Group 890 MOB − Aalborg University 20106 % This function calculates if a point is illuminated by the transmitter.7 %8 % flag=ilum pixel(xt,yt,x0,y0,tx angle,beamwidth)9 %

10 % Outputs:11 % flag: 1 (if illuminated) or 0 (if not illumated)12 % Inputs:13 % [xt,yt]: position of the transmitter.14 % [x0,y0]: position under analysis.15 % tx angle: angle the transmitter is pointing to.16 % beamwidth: transmission azimuth beamwidth.17

18 if (y0−yt+tan(tx angle)∗xt)/tan(tx angle)< x0 && ...19 (y0−yt+tan(tx angle−beamwidth)∗xt)/tan(tx angle−beamwidth)> x020 flag=1;21 else22 flag=0;23 end

F.6 Look for Target

Function that determines if a target is situated in one of the illuminated cells of the sea.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/lookfortarget.m

1 function target pos index=look for target(data,target pos)2

3 % look for target.m4 % RADAR project − Group 890 MOB − Aalborg University 2010

Group 890 - MOB 103

http://kom.aau.dk/group/10gr890/docs/code/ilumpixel.m

http://kom.aau.dk/group/10gr890/docs/code/lookfortarget.m



5 %6 % Function which calculates the index of the cell in which the target is7 % located into the illuminated area matrix.8 %9 % target pos index=look for target(data,target pos)

10 %11 % Outputs:12 % target pos index: cell index.13 % Inputs:14 % data: matrix with information about position in the two first rows (x,y),15 % distances from each illuminated pixel to the Tx in the 3rd row,16 % distances from each illuminated pixel to the Rx in the succesives17 % rows from the 4th (one row per receiver element).18 % target pos: target position.19

20 flag=0;21 i=2;22

23 while (flag==0) && (i≤length(data(1,:))−1)24 if (target pos(1)==data(1,i−1)) && (target pos(2)==data(2,i−1))25 target pos index=i−1;26 flag=1;27 elseif (target pos(1)>data(1,i−1)) && (target pos(1)<data(1,i+1))&&...28 (target pos(2)>data(2,i−1)) && (target pos(2)<data(2,i+1))29 target pos index=i;30 flag=1;31 else32 target pos index=0;33 end34 i=i+1;35 end

F.7 Line of Sight

Calculation of the line of sight between two cells of a matrix.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/lineofsight.m

1 function [r,h]=line of sight(x axis,y axis,cell size,a,x0,y0,xt,yt)2

3 % line of sight.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of the cells between one antenna and a point of the grid.7 %8 % [r,h]=line of sight(x axis,y axis,cell size,a,x0,y0,xt,yt)9 %

10 % Outputs:11 % r: vector with the distance from the selected point to the antenna.12 % h: vector with the heights of the different cells selected.13 % Inputs:14 % x axis: x data for the sea grid.15 % y axis: y data for the sea grid.16 % cell size: size of the cells of the sea grid.

Group 890 - MOB 104

http://kom.aau.dk/group/10gr890/docs/code/lineofsight.m



17 % a: matrix with heights.18 % x0: x coordinate of the antenna.19 % y0: y coordinate of the antenna.20 % xt: x coordinate of the point under study.21 % yt: y coordinate of the point under study.22

23 index0=[round((x0−x axis(1))/cell size)+1,...24 round((y0−y axis(1))/cell size)+1];25 indext=[round((xt−x axis(1))/cell size)+1,...26 round((yt−y axis(1))/cell size)+1];27 start=index0;28 ending=indext;29

30 if indext(1)>index0(1)31 st=1;32 else33 st=−1;34 end35

36 angle=atan(abs(ending(2)−start(2))/abs(ending(1)−start(1)));37

38 int cont=0;39

40 r=zeros(round(sqrt(length(x axis)ˆ2+length(y axis)ˆ2)));41 h=zeros(round(sqrt(length(x axis)ˆ2+length(y axis)ˆ2)));42

43 if(yt<10)% for very small angles to target (aprox 0 )44 for i=start(1):ending(1)45 for j=1:246 int cont=int cont+1;47 r(int cont)=sqrt(x axis(i)ˆ2+y axis(j)ˆ2);48 h(int cont)=a(j,i);49 end50 end51 else52 for i=start(1):st:ending(1)53 for j=start(2):ending(2)54 % look into cells next to the one under study.55 if (atan(abs(j−start(2))/abs(i−start(1)))==angle) | | ...56 (atan(abs(j−start(2))/abs(i−1−start(1)))==angle) | | ...57 (atan(abs(j−start(2))/abs(i+1−start(1)))==angle) | | ...58 (atan(abs(j−start(2))/abs(i−2−start(1)))==angle) | | ...59 (atan(abs(j−start(2))/abs(i+2−start(1)))==angle) | | ...60 (atan(abs(j−start(2))/abs(i−3−start(1)))==angle) | | ...61 (atan(abs(j−start(2))/abs(i+3−start(1)))==angle)62 int cont=int cont+1;63 r(int cont)=sqrt(x axis(i)ˆ2+y axis(j)ˆ2);64 h(int cont)=a(j,i);65 end66 end67 end68 end69

70 r=r(1:int cont); % eliminate zero elements71 h=h(1:int cont); % eliminate zero elements

Group 890 - MOB 105



F.8 Visible Area

Calculation of the shadow coefficient from an antenna to an specific target in a specificpoint from the LOS.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/visiblearea.m

1 function percentage=visible area(x,z,h0,ht)2

3 % visible area.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of the percentage of a certain element situated at the end7 % of the LOS from a selected location.8 %9 % percentage=visible area(x,z,h0,ht)

10 %11 % Outputs:12 % percentage: percentage seen from the object.13 % Inputs:14 % x: vector with the distances in the LOS.15 % z: vector of elevations of the LOS.16 % h0: height of the antenna.17 % ht: height of the object at the end of the selected LOS.18

19 if ¬isempty(x) && length(x)>120 angle point=atan(x./abs(h0−z));21 angle shadow=angle point(2);22 shadow vec=zeros(length(x));23

24 for i=2:length(z)25 if angle point(i)<angle shadow;26 shadow vec(i)=angle shadow−angle point(i);27 else28 angle shadow=angle point(i);29 shadow vec(i)=0;30 end31 end32

33 if shadow vec(length(x))==034 percentage=1;35 else36 h s p=h0−z(length(x))−x(length(x))/tan(angle point(length(x))+...37 shadow vec(length(x)));38 if h s p>ht39 percentage=0;40 else41 percentage=1−(h s p/ht);42 end43 end44 else45 percentage=0;46 end

Group 890 - MOB 106

http://kom.aau.dk/group/10gr890/docs/code/visiblearea.m



F.9 Signal Generation

Initial signal sent from the transmitter to the sea surface.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/signalgenerator.m

1 function [s]=signal generator(A ini,ph ini,tau ini)2

3 % signal generator.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % This function generates the baseband signal for the simulation.7 %8 % [s]=signal generator(A ini,ph ini,tau ini)9 %

10 % Inputs:11 % A ini: initial amplitude [V]12 % ph ini: initial phase [rad]13 % tau ini: initial delay [s]14 % Outputs:15 % s: signal vector (complex magnitude, delay) [V,s]16

17 s=[A ini∗exp(1i∗ph ini);tau ini];

F.10 Downlink

Calculation of the signal in the downlink from the transmitter to the illuminated area overthe sea.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/downlink.m

1 function data d=downlink(data,A ini,ph ini,tau ini,c,lambda,g)2

3 % downlink.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of the downlink signal, incoming to the sea and the target.7 %8 % data d=downlink(data,A ini,ph ini,tau ini,c,lambda,g)9 %

10 % Outputs:11 % data d: vector with the data of the downlink signal.12 % Inputs:13 % data: matrix with information about position in the two first rows (x,y),14 % distances from each illuminated pixel to the Tx in the 3rd row,15 % distances from each illuminated pixel to the Rx in the succesives16 % rows from the 4th (one row per receiver element).17 % A ini: initial amplitude of the signal.18 % ph ini: initial phase of the signal.19 % tau ini: initial delay of the signal.20 % c: light speed.21 % lambda: wavelength.

Group 890 - MOB 107

http://kom.aau.dk/group/10gr890/docs/code/signalgenerator.m

http://kom.aau.dk/group/10gr890/docs/code/downlink.m



22 % g: transmitter gain;23

24 data d=zeros(2,length(data(1,:)));25

26 A=A ini∗sqrt(g); % equal gain for all directions27

28 amp=A.∗(lambda./(4∗pi.∗data(3,:))); % amplitude in each point [V]29 pha=ph ini+(2∗pi.∗(data(3,:)./lambda)); % phase in each point [rad]30 delay=tau ini+(data(3,:)./c); % delay to each point [s]31

32 data d(1,:)=amp.∗exp(1j.∗pha);33 data d(2,:)=delay;

F.11 Sigma Calculation

Calculation of sigma and nu for the different illuminated points of the sea surface.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/sigmacalculator.m

1 function [nu,sigma s]=sigma calculator(range,cell size,h tx,lambda,...2 h wave avg,wave angle)3

4 % sigma calculator.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of sigma and nu for the different point over the sea.8 %9 % [nu,sigma s]=sigma calculator(range,cell size,h tx,lambda,...

10 % h wave avg,wave angle)11 %12 % Outputs:13 % nu: parameter for sea shape calculated in the worst case.14 % sigma s: sigma parameter for each point.15 % Inputs:16 % range: distance to each point under study in the illuminated area.17 % cell size: cell size of the sea matrix.18 % h tx: height of the transmitter.19 % lambda: wavelenght.20 % h wave avg: average height of the sea.21 % wave angle: wind propagation angle.22

23 % for the sea24 gr angle=atan(h tx./range); % grazing angle in each point [rad]25 sigma phi=(14.4∗lambda+5.5).∗gr angle.∗h wave avg/lambda;26 Al=(sigma phi.ˆ4)./(1+(sigma phi.ˆ4));27 Au=exp(0.2.∗cos(wave angle).∗(1−2.8.∗gr angle).∗(lambda+0.02).ˆ(−0.4));28 qw=1.1/(lambda+0.02)ˆ(−0.4);29 Vw=8.67∗h wave avgˆ0.4;30 Aw=(1.94∗Vw/(1+Vw/15.4))ˆqw;31 Area=cell sizeˆ2;32 sigma s=10∗log10(3.9∗10ˆ(−6)∗lambda.∗(gr angle.ˆ(0.4)).∗Al.∗Au∗Aw);%[dB/m2]33

34 sigma s=Area∗10.ˆ(sigma s/10);35

Group 890 - MOB 108

http://kom.aau.dk/group/10gr890/docs/code/sigmacalculator.m



36 % parameter nu for sea shape37 nu=10.ˆ(((2/3).∗log10(gr angle.∗180/pi))+((5/8)∗log10(800))+(−1.7));

F.12 Backscattered

Calculation of the signal coming back from the sea and the target.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/backscattered.m

1 function [data t,data t n]=backscattered(data d,target pos index,...2 sigma s,percent,percent tx,nu,SCR)3

4 % backscattered.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of the signal backscattered from the sea in function of the8 % incoming power ang grazing angles.9 %

10 % function [data t,data t n]=backscattered(data d,target pos index,...11 % sigma s,percent,percent tx,nu,SCR)12 %13 % Outputs:14 % data t: vector with the data prepared for the uplink to the array.15 % data t n: vector with the backscattered data for the uplink to the...16 % normal radar.17 % Inputs:18 % data d: incoming signal into the sea (after the downlink).19 % target pos index: target position index into the illuminated area matrix.20 % sigma s: value to characterize the sea in each point.21 % percent: vector with shadow coefficients for each array element.22 % percent tx: shadow coefficient for normal radar.23 % nu: sea shape parameter.24 % SCR: sea to clutter ratio to characterize the target.25

26 data t=zeros(2∗length(percent),length(data d));27 data t n=zeros(2,length(data d));28

29 % parameter mu for sea signal30 mu=abs(data d(1,:)).∗sqrt(sigma s);31

32 % signal backscattered from the sea33 coeff=−gamrnd(nu,1).∗log(rand(1,1));34 amp s=mu.∗coeff; % amplitude changes according to K−distribution [V]35 % random phase rotation (0,2pi) [rad]36 pha s=phase(data d(1,:))+rand(1,length(data d(1,:)))∗2∗pi;37 signal s=amp s.∗exp(1j.∗pha s);38

39 % signal backscatteresd from the target40 ph t=pi; % reflection on the target [rad]41

42 % signal level correction factors (shadowing)43 correction=percent;44 correction n=percent tx;45

Group 890 - MOB 109

http://kom.aau.dk/group/10gr890/docs/code/backscattered.m



46 % combine signals in target pixel (+SCR)47 for i=1:2:2∗length(percent)−148 data t(i,:)=signal s;49 if target pos index>0 % if the target is in the illuminated area50 data t(i,target pos index)=((10ˆ((20∗log10(abs(data t(i,...51 target pos index)))+SCR)/20))∗correction((i+1)/2))∗...52 exp(1j∗(phase(data t(i,target pos index))+ph t));53 end54 data t(i+1,:)=data d(2,:); % no delay!55 end56

57 % for normal radar58 data t n(1,:)=signal s;59 if target pos index>0 % if the target is in the illuminated area60 data t n(1,target pos index)=((10ˆ((20∗log10(abs(data t n(1,...61 target pos index)))+SCR)/20))∗correction n)∗exp(1j∗...62 (phase(data t n(1,target pos index))+ph t));63 end64 data t n(2,:)=data d(2,:);

F.13 Uplink

Calculation of the signal in the uplink from the sea and the target to the receiver antennas.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/uplink.m

1 function [data u,data u s]=uplink(data,data s,data t,c,lambda,grx)2

3 % uplink.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of the uplink signal, reaching the array and coming form the7 % illuminated area over the sea.8 %9 % [data u,data u s]=uplink(data,data s,data t,c,lambda,grx)

10 %11 % Outputs:12 % data u: vector with the data of the uplink signal.13 % data u s: the same as data u but for the antennas at lambda/2 distance.14 % Inputs:15 % data: matrix with information about position in the two first rows (x,y),16 % distances from each illuminated pixel to the Tx in the 3rd row,17 % distances from each illuminated pixel to the Rx in the succesives18 % rows from the 4th (one row per receiver element).19 % data s: as data, but for the small antennas.20 % data t: data with the distances from the illuminated cells to the21 % receiver antennas.22 % c: light speed.23 % lambda: wavelength.24 % grx: receiver antennas gain;25

26 data u=zeros(size(data t));27 data u s=zeros(size(data t));28

Group 890 - MOB 110

http://kom.aau.dk/group/10gr890/docs/code/uplink.m



29 for i=1:2:size(data t,1)−130 amp=abs(data t(i,:)).∗(lambda./(4∗pi.∗data(3+...31 ((i+1)/2),:))).∗sqrt(grx); % amplitude in each point [V]32 pha=phase(data t(i,:))+(2∗pi.∗(data(3+...33 ((i+1)/2),:)./lambda)); % phase in each point [rad]34 data u(i,:)=amp.∗exp(1j∗pha);35 delay=data t(i+1,:)+(data(3+((i+1)/2),:)./c); % delay to each point [s]36 data u(i+1,:)=delay;37 amp s=abs(data t(i,:)).∗(lambda./(4∗pi.∗data s(3+...38 ((i+1)/2),:))).∗sqrt(grx); % amplitude in each point [V]39 pha s=phase(data t(i,:))+(2∗pi.∗(data s(3+...40 ((i+1)/2),:)./lambda)); % phase in each point [rad]41 data u s(i,:)=amp s.∗exp(1j∗pha s);42 delay s=data t(i+1,:)+(data s(3+...43 ((i+1)/2),:)./c); % delay to each point [s]44 data u s(i+1,:)=delay s;45 end

F.14 Normal Uplink

Calculation of the signal in the uplink from the sea and the target to normal radar antenna.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/uplinknormal.m

1 function [data u]=uplink normal(data,data t n,c,lambda,grx)2

3 % uplink normal.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of the uplink signal for the normal radar, coming back from7 % the illuminated area over the sea to the transmitter antenna.8 %9 % [data u]=uplink normal(data,data t n,c,lambda,grx)

10 %11 % Outputs:12 % data u: vector with the data of the uplink signal.13 % Inputs:14 % data: matrix with information about position in the two first rows (x,y),15 % distances from each illuminated pixel to the Tx in the 3rd row,16 % distances from each illuminated pixel to the Rx in the succesives17 % rows from the 4th (one row per receiver element).18 % data t n: data with the distances from the illuminated cells to the19 % receiver antenna.20 % c: light speed.21 % lambda: wavelength.22 % grx: receiver antenna gain;23

24 data u=zeros(2,length(data(1,:)));25

26 amp=abs(data t n(1,:)).∗(lambda./...27 (4∗pi.∗data(3,:))).∗sqrt(grx); % amplitude from each point [V]28 pha=phase(data t n(1,:))+...29 (2∗pi.∗(data(3,:)./lambda)); % phase from each point [rad]30 delay=data t n(2,:)+(data(3,:)./c); % delay from each point [s]

Group 890 - MOB 111

http://kom.aau.dk/group/10gr890/docs/code/uplinknormal.m



31

32 data u(1,:)=amp.∗exp(1j.∗pha);33 data u(2,:)=delay;

F.15 Signal Reception

Construction of the WB signals with from the cells with similar in the different antennasof the receiver array.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/signalreception.m

1 function [tapped,amplitudes,tapped s,amplitudes s]=...2 signal reception(data t,data t s,T res,c,xmax,ymax)3

4 % signal reception.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of the signals received in array.8 %9 % function [tapped,amplitudes,tapped s,amplitudes s]=...

10 % signal reception(data t,data t s,T res,c,xmax,ymax)11 %12 % Outputs:13 % tapped: matrix with delays from the different signal contributions14 % in each antenna (one in each row).15 % amplitudes: matrix with signal values associated to the same delays.16 % tapped s: the same as tapped but for the lambda/2 elements.17 % amplitudes s: the same as amplitudes but for the lambda/2 elements.18 % Inputs:19 % data t: data with delays and signals from the different illuminated20 % cells after the uplink for each element of the array.21 % data t s: the same as data t but for the lambda/2 elements.22 % T res: spatial resolution to separate the tapped delays.23 % c: speed of the light.24 % xmax: maximum x coordinate to calculate the maximum delay allowed.25 % ymax: maximum y coordinate to calculate the maximum delay allowed.26

27 signal=zeros(size(data t,1)/2,size(data t,2));28 delays=zeros(size(data t,1)/2,size(data t,2));29 index=zeros(size(data t,1)/2,size(data t,2));30 tapped=zeros(size(data t,1)/2,size(data t,2));31 signal s=zeros(size(data t,1)/2,size(data t,2));32 delays s=zeros(size(data t,1)/2,size(data t,2));33 index s=zeros(size(data t,1)/2,size(data t,2));34 tapped s=zeros(size(data t,1)/2,size(data t,2));35

36 max delay=2∗sqrt(xmaxˆ2+ymaxˆ2)/c;37

38 for i=2:2:size(data t,1) % order delays,signals and indexes39 [delays(i/2,:),index(i/2,:)]=sort(data t(i,:),ascend);40 [delays s(i/2,:),index s(i/2,:)]=sort(data t s(i,:),ascend);41 for j=1:length(delays)42 signal(i/2,j)=data t(i−1,index(i/2,j));43 signal s(i/2,j)=data t s(i−1,index s(i/2,j));

Group 890 - MOB 112

http://kom.aau.dk/group/10gr890/docs/code/signalreception.m



44 end45 end46

47 % create vectors of WB tapped delays48 tapped(:,1)=delays(:,1);49 tapped s(:,1)=delays s(:,1);50 for i=1:size(index,1)51 j=2;52 while (tapped(i,1)+(j−1)∗2∗T res≤max delay)53 tapped(i,j)=tapped(i,1)+(j−1)∗2∗T res;54 j=j+1;55 end56 end57 for i=1:size(index,1)58 j=2;59 while (tapped s(i,1)+(j−1)∗2∗T res≤max delay)60 tapped s(i,j)=tapped s(i,1)+(j−1)∗2∗T res;61 j=j+1;62 end63 end64

65 % cut to maximum delays allowed66 cont=0;67 for i=1:size(tapped,2)68 if(tapped(1,i)6=0)&&(tapped(1,i)<max delay)69 cont=i;70 end71 end72 tapped=tapped(:,1:cont);73 tapped s=tapped s(:,1:cont);74

75 amplitudes=zeros(size(tapped));76 amplitudes s=zeros(size(tapped s));77

78 % create vector of WB amplitudes79 amplitudes(:,1)=signal(:,1);80 amplitudes s(:,1)=signal s(:,1);81

82 for j=1:size(delays,1)83 in=2;84 sum=0;85 i=2;86 while in<size(tapped,2)87 if(delays(j,i)<tapped(j,in))88 sum=sum+signal(j,i);89 i=i+1;90 else91 amplitudes(j,in)=sum;92 sum=signal(j,i);93 i=i+1;94 in=in+1;95 end96 end97 amplitudes(j,in)=sum;98 in=2;99 sum s=0;

100 i=2;

Group 890 - MOB 113



101 while in<size(tapped s,2)102 if(delays s(j,i)<tapped s(j,in))103 sum s=sum s+signal s(j,i);104 i=i+1;105 else106 amplitudes s(j,in)=sum s;107 sum s=signal s(j,i);108 i=i+1;109 in=in+1;110 end111 end112 amplitudes s(j,in)=sum s;113 end

F.16 Normal Signal Reception

Construction of the WB signal with from the cells with similar delay in normal radar.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/normalsignalreception.m

1 function [tapped,amplitudes]=normal signal reception(data u n,T res,c,...2 xmax,ymax)3

4 % normal signal reception.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of the signal received in the normal radar.8 %9 % [tapped,amplitudes]=normal signal reception(data u n,T res,c,xmax,ymax)

10 %11 % Outputs:12 % tapped: vector with delays from the different signal contributions.13 % amplitudes: vector with signal values associated to the same delays.14 % Inputs:15 % data u n: data with delays and signals from the different illuminated16 % cells after the uplink.17 % T res: spatial resolution to separate the tapped delays.18 % c: speed of the light.19 % xmax: maximum x coordinate to calculate the maximum delay allowed.20 % ymax: maximum y coordinate to calculate the maximum delay allowed.21

22

23 signal=zeros(1,size(data u n,2));24 max delay=2∗sqrt(xmaxˆ2+ymaxˆ2)/c;25

26 % Order delays and signals27 [delays,index]=sort(data u n(2,:),ascend);28 for i=1:length(delays)29 signal(i)=data u n(1,index(i));30 end31

32 % create vectors of WB tapped delays33 tapped=zeros(1,length(delays));34 tapped(1)=delays(1);

Group 890 - MOB 114

http://kom.aau.dk/group/10gr890/docs/code/normalsignalreception.m



35 j=2;36 while (tapped(1)+(j−1)∗2∗T res≤max delay)37 tapped(j)=tapped(1)+(j−1)∗2∗T res;38 j=j+1;39 end40 tapped=tapped(1:j−1);41

42 % create vector of WB amplitudes43 amplitudes=zeros(1,length(tapped));44 amplitudes(1)=signal(1);45 in=2;46 sum=0;47 i=2;48 while in<length(tapped) % accumulate amplitudes of the same delay49 if(delays(i)<tapped(in))50 sum=sum+signal(i);51 i=i+1;52 else53 amplitudes(in)=sum;54 sum=signal(i);55 i=i+1;56 in=in+1;57 end58 end59 amplitudes(in)=sum;

F.17 PFA Threshold

Threshold calculation from the PFA (Probability of False Alarm).

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/PFAtr.m

1 function t=PFA tr(xmax,ymax,tx height,PFA)2

3 % PFA tr.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % Calculation of the detection threshold in function of the probability of7 % false alarm.8 %9 % t=PFA tr(xmax,ymax,tx height,PFA)

10 %11 % Outputs:12 % t: detection threshold adapted to the PFA requirement.13 % Inputs:14 % xmax: maximum x coordinate to calculate the maximum grazing angle.15 % ymax: maximum y coordinate to calculate the maximum grazing angle.16 % tx height: transmitter height;17 % PFA: probability of false alarm.18

19 gr angle=atan(tx height/sqrt((xmax)ˆ2+(ymax)ˆ2));20 h0=−log(PFA);21 a1=0.112;22 a2=2.256;

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23 a3=0.516;24 nu=10.ˆ(((2/3).∗log10(gr angle.∗180/pi))+((5/8)∗log10(800))+(−1.7));25 t=h0+a1∗h0ˆ(a2)/(nuˆ(a3∗h0ˆ0.2));

F.18 Peak Detection

Detection of the peaks from the WB signals from the different elements of the array.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/peakdetection.m

1 function [delays peak,amp peak,amp s peak]=peak detection(tapped,...2 amplitudes,amplitudes s,treshold)3

4 % peak detection.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of the delays and the correspondent signals in peak detection8 % for the array radar.9 %

10 % [delays peak,amp peak,amp s peak]=peak detection(tapped,amplitudes,...11 % amplitudes s,treshold)12

13 % Outputs:14 % delays peak: vector with the delays where the peaks over a certain15 % threshold are stored.16 % amp peak: signals associated to the different delays detected.17 % amp s peak: signals in the antennas placed at lambda/2 and associated18 % to the different delays detected.19 % Inputs:20 % tapped: vector with the delays of the WB signal received in the21 % different antennas of the array (stored in different rows).22 % amplitudes: vector with the signals associated to the WB delays in23 % the different elements of the array.24 % amplitudes s: the same as amplitudes but for the lambda/2 antennas.25 % threshold: threshold to perfom detection.26

27 delays peak=zeros(size(tapped));28 amp peak=zeros(size(tapped));29 amp s peak=zeros(size(tapped));30

31 for i=1:size(amplitudes,1)32 % Threshold creation33 val=20∗log10(abs(amplitudes(i,:)));34 vale=val(1:2:length(amplitudes(i,:)));35 pol=polyfit(tapped(i,1:2:length(amplitudes(i,:)))∗1e6,vale,4);36 zero=polyval(pol,tapped(i,1:2:length(amplitudes(i,:)))/1e−6);37 tr=zero+treshold;38 % if i==139 % figure40 % plot(tapped(i,1:2:length(amplitudes(i,:)))/(1e−6),vale)41 % hold on42 % plot(tapped(i,1:2:length(amplitudes(i,:)))/(1e−6),zero,−,LineWidth,2)43 % plot(tapped(i,1:2:length(amplitudes(i,:)))/(1e−6),tr,−−)44 % title(ARRAY signal thresholding − Antenna 1)

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45 % xlabel(delay [us])46 % ylabel(signal level [dBW])47 % legend(Received signal,Estimated mean level,Calculated Threshold)48 % end49

50 % Comparation of the signal levels with respect to the threshold.51 cont=1;52 for k=1:length(tr)53 if(tr(k)<val(2∗k−1))54 delays peak(i,cont)=tapped(i,2∗k−1);55 amp peak(i,cont)=amplitudes(i,2∗k−1);56 amp s peak(i,cont)=amplitudes s(i,2∗k−1);57 cont=cont+1;58 elseif(2∗k<length(val))&&(tr(k)<val(2∗k))59 delays peak(i,cont)=tapped(i,2∗k);60 amp peak(i,cont)=amplitudes(i,2∗k);61 amp s peak(i,cont)=amplitudes s(i,2∗k);62 cont=cont+1;63 end64 end65 end

F.19 Normal Peak Detection

Detection of the peaks from a WB signal in typical radar.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/normalpeakdetection.m

1 function [delays n peak,amp n peak]=normal peak detection(tapped n,...2 amplitudes n,treshold)3

4 % normal peak detection.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of the delays and the correspondent signals in peak detection8 % for normal radar.9 %

10 % [delays n peak,amp n peak]=normal peak detection(tapped n,...11 % amplitudes n,treshold)12 %13 % Outputs:14 % delays n peak: vector with the delays where the peaks over a certain15 % threshold are stored.16 % amp n peak: signals associated to the different delays detected.17 % Inputs:18 % tapped n: vector with the delays of the WB signal received in the19 % normal radar.20 % amplitudes n: vector with the signals associated to the WB delays in21 % normal radar.22 % threshold: threshold to perfom detection.23

24 delays n peak=zeros(size(tapped n));25 amp n peak=zeros(size(tapped n));26

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27 % Threshold creation28 val=20∗log10(abs(amplitudes n));29 vale=val(1:2:length(amplitudes n));30 pol=polyfit(tapped n(1:2:length(amplitudes n))∗1e6,vale,4);31 zero=polyval(pol,tapped n(1:2:length(amplitudes n))/1e−6);32 tr=zero+treshold;33

34 % figure35 % plot(tapped n(1:2:length(amplitudes n))/(1e−6),vale)36 % hold on37 % plot(tapped n(1:2:length(amplitudes n))/(1e−6),zero,−,LineWidth,2)38 % plot(tapped n(1:2:length(amplitudes n))/(1e−6),tr,−−)39 % title(NORMAL signal thresholding)40 % xlabel(delay [us])41 % ylabel(signal level [dBW])42 % legend(Received signal,Estimated mean level,Calculated Threshold)43

44 % Comparation of the signal levels with respect to the threshold.45 cont=1;46 for k=1:length(tr)47 if(tr(k)<val(2∗k−1))48 delays n peak(cont)=tapped n(2∗k−1);49 amp n peak(cont)=amplitudes n(2∗k−1);50 cont=cont+1;51 elseif(2∗k<length(val))&&(tr(k)<val(2∗k))52 delays n peak(cont)=tapped n(2∗k);53 amp n peak(cont)=amplitudes n(2∗k);54 cont=cont+1;55 end56 end

F.20 Normal Position Estimator

Estimation of the points in the plane from the peaks detected in normal radar.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/normalpositionestimator.m

1 function [est n]=normal position estimator(tx pos,tx angle,c,...2 delays n peak,amp n peak)3

4 % normal position estimator.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Calculation of the position of the targets detected from the different8 % peaks analyzed in the WB signal.9 %

10 % [est n]=normal position estimator(tx pos,tx angle,c,delays n peak,...11 % amp n peak)12 %13 % Outputs:14 % est n: matrix with x coordinates of the targets detected in the first row15 % and the y coordinates in the second row..16 % Inputs:17 % tx pos: position of the transmitter antenna.

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18 % tx angle: angle to which is pointing the transmitter.19 % c: speed of the light.20 % delays n peak: vector with the delay of the different peaks.21 % amp n peak: vector of signals amplitude from the WB signal peaks.22

23 i=1;24 while delays n peak(i)6=025 i=i+1;26 end27 r=delays n peak(1:i−1)./2;28

29 est n(1,:)=tx pos(1)+c.∗r.∗cos(tx angle);30 est n(2,:)=tx pos(2)+c.∗r.∗sin(tx angle);

F.21 Minimum Counter

Calculation of the row of a matrix with the minimum number of elements different fromzero.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/mincounter.m

1 function num=min counter(x)2

3 % min counter.m4 % RADAR project − Group 890 MOB − Aalborg University 20105 %6 % This function looks for the minimum number of elements differents from7 % zero in the different rows of a matrix.8 %9 % num=min counter(x)

10 %11 % Outputs:12 % num: vector of non−zero elements in each of the rows of a matrix.13 % Inputs:14 % x: matrix under analysis.15

16 cont=1;17 for i=1:size(x,1)18 while (x(i,cont)6=0)19 num(i)=cont;20 cont=cont+1;21 end22 end

F.22 Find Values

Function that finds numbers between two values in a vector.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/findvalues.m

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http://kom.aau.dk/group/10gr890/docs/code/mincounter.m

http://kom.aau.dk/group/10gr890/docs/code/findvalues.m



1 % find values.m2 % RADAR project − Group 890 MOB − Aalborg University 20103 %4 % This function finds the numbers between two values in a vector.5 % Only for a maximum of 100 positions.6 %7 % [row col]=find values(x,up,down)8 %9 % Outputs:

10 % row: indexes of rows.11 % col: indexes of columns.12 % Inputs:13 % x: vector under analysis.14 % up: upper value.15 % down: lower value.16

17 function [row col]=find values(x,up,down)18

19 row=1;20 col=1;21

22 for i=1:size(x,1)23 for j=1:10024 if x(i,j)<up && x(i,j)> down25 row=[row i];26 col=[col j];27 else28 end29 end30 end

F.23 Array Position Estimator

Estimation of the points in the plane from the peaks detected in the array antennas.

This code can be downloaded from: http://kom.aau.dk/group/10gr890/docs/code/WBposition.m

1 function poz=WBposition(delays peak,d ant,rx pos,tx pos,tx angle,...2 tx bw azimuth,M)3

4 % WBposition.m5 % RADAR project − Group 890 MOB − Aalborg University 20106 %7 % Estimation of the position of the targets evaluating the different values8 % over the detection threshold in the different elements of the receiver9 % array.

10 %11 % poz=WBposition(delays peak,d ant,rx pos,tx pos,tx angle,tx bw azimuth,M)12 %13 % Outputs:14 % poz: matrix with x coordinates of the targets detected in the first row15 % and the y coordinates in the second row.16 % Inputs:17 % delays peak: matrix with the delays detected in the different antennas

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18 % (each antenna is represented in a row).19 % d ant: distance between antennas.20 % rx pos: vector with the position of the receiver antennas.21 % tx pos: position of the transmitter antenna.22 % tx angle: angle to which is pointing the transmitter.23 % tx bw azimuth: azimuth beamwidth.24 % M: number of receiver elements.25

26

27 c=3E8;28 poz=0;29

30 num=min counter(delays peak);31 [y,i] = min(num);32 [row,col] = find(delays peak);33 delayNr=col(length(col));34 antena=11;35

36 cont=0;37 for k=1:delayNr38 currT=delays peak(antena,k);39 S=currT∗c;40 dloc=sqrt(tx pos(2)ˆ2+(tx pos(1)−rx pos(1,antena))ˆ2);41 if rx pos(1,antena)>tx pos(1)42 if tx angle−tx bw azimuth/2−atan(tx pos(2))/...43 abs(tx pos(1)−rx pos(1,antena))>044 trid=tx angle−tx bw azimuth/2−atan(abs(tx pos(2))/...45 abs(tx pos(1)−rx pos(1,antena)));46 RtRx=S−(−dlocˆ2+Sˆ2)/(−2∗dloc∗cos(trid)+2∗S);47

48 else49 trid=atan(abs(tx pos(2))/abs(tx pos(1)−rx pos(1,antena)))−...50 tx angle−tx bw azimuth/2;51 RtRx=S−(dlocˆ2−Sˆ2)/(+2∗dloc∗cos(trid)−2∗S);52

53 end54 else55 trid=atan(abs(tx pos(1)−rx pos(1,antena))/abs(tx pos(2)))+...56 pi/2−tx angle−tx bw azimuth/2;57 RtRx=S−(dlocˆ2−Sˆ2)/(+2∗dloc∗cos(trid)−2∗S);58 end59

60 P(1)=(S−RtRx)∗cos(tx angle)−abs(tx pos(1));61 P(2)=(S−RtRx)∗sin(tx angle)−abs(tx pos(2));62

63 target(1)=P(1);64 target(2)=P(2);65 for i=1:size(rx pos,2)66 R(i)=sqrt((target(1)−rx pos(1,i))ˆ2+...67 (target(2)−rx pos(2,i))ˆ2);68 end69 [row,col]=find values(delays peak, currT+(−R(antena)+...70 R(1))∗1/c+d ant/c, currT−(R(15)−R(antena))∗1/c−d ant/c ) ;71 if delayNr==172

73 [row,col]=find(delays peak< currT+(antena−1)∗...74 d ant∗cos(tx angle)/c & delays peak >currT−...

Group 890 - MOB 121



75 (M−antena)∗d ant∗cos(tx angle)/c )76 else77 end78 if length(row)> M−579 cont=cont+1;80 poz(1,cont)=P(1);81 poz(2,cont)=P(2);82

83 for k=1:size(delays peak,1)84 if k==185 indices=find(delays peak(k,:)<currT−...86 (R(antena)−R(k))/c +d ant/c & delays peak(k,:)>...87 currT−(R(antena)−R(k))/c −(R(k)−R(k+1))∗1.25/c )88 elseif k==size(delays peak,1);89 indices=find(delays peak(k,:)<currT−(R(antena)−...90 R(k))/c +(R(k−1)−R(k))∗1.25/c &...91 delays peak(k,:)>currT−(R(antena)−R(k))/c−d ant/c)92 else93 indices=find(delays peak(k,:)<currT−(R(antena)−R(k))/c+...94 (R(k−1)−R(k))∗1.25/c & delays peak(k,:)>currT−...95 (R(antena)−R(k))/c −(R(k)−R(k+1))∗1.25/c)96 end97 if length(indices)== 198 t(k,cont)=delays peak(k,indices);99 elseif length(indices)> 1

100 t(k,cont)=mean(delays peak(k,indices));101 else102 end103 end104 else105 end106 end107

108

109 for i=1:size(t,1)110 for j=1:size(t,2)111 if t(i,j)6=0112 S(i,j)=t(i,j)∗c;113 dloc(j)=sqrt(tx pos(2)ˆ2+(tx pos(1)−rx pos(1,i))ˆ2);114 if rx pos(1,i)>tx pos(1)115 if tx angle−tx bw azimuth/2−atan(tx pos(2))/abs(tx pos(1)−rx pos(1,i))>0116 trid(i,j)=tx angle−tx bw azimuth/2−atan(abs(tx pos(2))/abs(tx pos(1)−...117 rx pos(1,i)));118 RtRx(i,j)=S(i,j)−(−dloc(j)ˆ2+S(i,j)ˆ2)/(−2∗dloc(j)∗cos(trid(i,j))+...119 2∗S(i,j));120 else121 trid(i,j)=atan(abs(tx pos(2))/abs(tx pos(1)−rx pos(1,i)))−...122 tx angle−tx bw azimuth/2;123 RtRx(i,j)=S(i,j)−(dloc(j)ˆ2−S(i,j)ˆ2)/(+2∗dloc(j)∗cos(trid(i,j))−2∗S(i,j));124 end125 else126 trid(i,j)=atan(abs(tx pos(1)−rx pos(1,i))/abs(tx pos(2)))+pi/2−...127 tx angle−tx bw azimuth/2;128 RtRx(i,j)=S(i,j)−(dloc(j)ˆ2−S(i,j)ˆ2)/(+2∗dloc(j)∗cos(trid(i,j))−2∗S(i,j));129 end130 else131 end

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132 end133 end134 for j=1:size(RtRx,2)135 for l=1:size(RtRx,1)−1136 alpha2(l)=acos((RtRx(l,j)ˆ2+d antˆ2−RtRx(l+1,j)ˆ2)/(2∗d ant∗RtRx(l,j)));137 poz(l,j)=RtRx(l,j)∗cos(alpha2(l))+rx pos(1,l)+...138 1i∗(RtRx(l,j)∗sin(alpha2(l)));139 end140 end

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Appendix G

INTERNAL ORGANIZATION

G.1 WEB PAGE

In order to organize ourselves, we have created a web page in which we have centralizedthe access to the working stuffs, to access to some documents and to our forum.

G.2 TIME PLAN

Figure G.3 shows the timeplan that we have followed during the development of ourproject. It was necessary to make some changes in order to adapt it to our work pace.

Group 890 - MOB 124



Figure G.1: Group 890 Web-pageGroup 890 - MOB 125



Fig

ure

G.2

:In

itia

lT

imep

lan

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Fig

ure

G.3

:F

inal

Tim

epla

n

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References

[1] International Hydrographic Organization, Accessed on 22nd of March 2010.http://www.iho.int/.

[2] Constantine A.Balanis. Antenna Theory: Analysis and Design (second edition).

[3]

[4] Mondial Broker (Boats for sale), Accessed on 12th of March 2010.http://www.mondialbroker.com/.

[5] Shaun Quegan Simon Kingsley. Understanding Radar systems. McGraw Hill, firstedition, 1992.

[6] Merril I. Skolnik. Radar Handbook. McGraw Hill, third edition, 2008.

[7] Alberto Asensio. Radar Systems Notes, 2008.

[8] Modern High Seas Piracy, Accessed on 12th of March 2010.http://www.cargolaw.com/presentationspirates.html.

[9] Rasul B. Rais. The Indian Ocean and the Superpowers. Croom Helm, 1986.

[10] Keith D. Ward, Robert J. A. Tough, and Simon Watts. Sea Clutter: Scattering, the KDistribution and Radar Performance. The Institution of Engineering and Technology,first edition, 2006.

[11] Felix Totir, Emanuel Radoi, Lucian Anton, and Cornel Ioana. Advanced Sea ClutterModels and their Usefulness for Target Detection, 2008.

[12] A. Parthiban, J. Madhavan, P. Radhakrishna, D. Savitha, and L. Sathish Kumar.Modeling and Simulation of Radar Sea Clutter using K-distribution, 2004.

[13] H.C. Chan. Radar Sea-Clutter at Low Grazing Angles, 1990.

[14] Simon Haykin, Rembrandt Bakker, and Brian W. Currie. Uncovering NonlinearDynamics - The Case Study of Sea Clutter.

[15] Irina Antipov. Simulation of Sea Clutter Returns, 1998.

[16] Brian L. Reid. Detection of Small Targets in Sea Clutter Limited Environments UsingPhase Information, 1995.

[17] C.J. Baker. K-Distributed Coherent Sea Clutter, 1991.

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http://www.iho.int/

http://www.mondialbroker.com/

http://www.cargolaw.com/presentationspirates.html



[18] D.A. Abraham. Detection Threshold Approximation for Non Gaussian Backgrounds,2010.

[19] Fan-Beam Antenna, Accessed on 23st of May 2010.http://www.radartutorial.eu/06.antennas/an11.en.html.

[20] J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim. The Theory andDesign of Chirp Radars. Bell System Technical Journal 39,745, 1960.

[21] Antonio De Maio and Alfonso Farina. Code Selection for Radar Performance Opti-mization. IEEE.

[22] M. H. Ackroyd. Amplitude and Phase Modulated Pulse Trains for Radar. The Radioand Electronic Engineer, Vol. 41. No. 12.

[23] David E. Joyce. The Laws of Cosines and Sines, 2002.

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http://www.radartutorial.eu/06.antennas/an11.en.html

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