Passive bistatic radar imaging of aircraft using FM broadcast signals

... main goal of our work is to build radar images of aircraft using passive bistatic radar signals The use of the FM band for that purpose is motivated by the geographic prevalence of radio broadcasting... 4.7 Simulation Of PBR Images Using Low Frequency Signals 4.7.1 Simulation Of Passive Bistatic Radar Image 4.7.2 Simulation Of Passive bistatic Radar Image Using Actual Airplanes... 1.4 Contribution Of The Thesis The main contributions of the thesis are : • The feasibility study of a passive bistatic radar system using the radio broadcasting FM signals for imaging airplanes

PASSIVE BISTATIC RADAR IMAGING OF

AIRPLANES BY USING FM RADIO

departe-I also wish to thank Doctor Jonathan Pisane, research scientist at the Temasek oratory of Nanyiang Technological University, for the technical discussions that we hadduring the last 10 months of my thesis and for his comments that helped me to writethis thesis I also wish to thank Doctor Danny Tan Kai Pin for his invaluable help forthe experiments that could not have been without him.

Lab-I also wish to thank Doctor Fr´ed´eric Brigui and Doctor Mehdi Air ighil for the informaldiscussion on their respective field of expertise which provides me a better understanding

on passive radar technologies

I wish to thank all my family and friends that supported me during the thesis cially, I would like to thank my friends who convinced me to join the NUS running teamand allowed me to meet a large number of amazing persons

Espe-Finally, I wish to thank all the persons I forgot to mention here

A conventional Airport Surveillance Radar (ASR) system must transmit a signal in order

to detect approaching aircraft, and thus a frequency band has to be allocated to the ASRsystem Passive bistatic radar (PBR) systems, on the other hand, reuse electromagneticsignals already present in order to detect, localize and identify an object within a givenarea PBR is a well-known topic but prior research has mainly focused on detection andestimation, and relatively little work has been done on PBR imaging The use of the FMband for that purpose is motivated by the geographic prevalence of radio broadcastingand the large size of an FM cell, and is the subject of this thesis

Firstly, the tomography principles applied to radar imaging are presented for themonostatic configuration and then generalized to the bistatic configuration Secondly, thefeasibility of using FM radio broadcasting signals is studied for imaging airborne aircraftusing a realistic configuration and a validation of the theoretical work is done using abright point model The feasibility of the PBR imaging is considered for the Singaporeanconfiguration which has two transmitters, one in Johor Bahru (Malaysia) and the other

in Bukit Timah(Singapore) Thirdly, a second validation of the theory is undertaken bydeveloping a new tool for simulating the electromagnetic field reflected off an object based

on the NEC2 program It is based on the transformation of a CAD model given by freelicense software to an interpretable model for NEC2 that composes of wire coordinatesonly PBR images are built from the simulated RCS obtained Finally an experimentaldata collection campaign has been executed to compare the theory and the simulationwith the reality in the Singapore vicinity by using the FM radio broadcasting signal

transmitted from Bukit Batok (Singapore) and airplanes approaching Changi Airport.

A PBR system has been built and is able to track and detect targets but the too lowpower of the reflected signals prevents us from extracting the RCS of the targets andfrom generating their PBR images

The main contribution is the successful construction of interpretable PBR imagesgiven a realistic configuration and limited trajectories for the airplanes based on simulateddata It also provides a new tool for obtaining an estimation of the RCS of an airplane

at a low frequency for a limited cost Moreover, a PBR system has been built and wasable to detect and to track real targets landing and taking off at Changi airport Finally,all the work was derived for a special configuration but the methodology can be easilyapplied in other configurations

1.1 Targets Considered 1

1.2 Radar 1

1.2.1 Monostatic Radar Description 2

1.2.2 Bistatic Radar Description 3

1.2.3 Passive Radar 4

1.2.4 Radar Imaging 5

1.3 Notation 6

1.4 Contribution Of The Thesis 6

1.5 Organisation Of The Thesis 7

2 Principles Of Passive Bistatic Radar Imaging 8 2.1 Introduction 8

2.2 Radar Cross-Section And Radar Image Definition 9

2.2.1 Electrical Field Modeling 9

2.2.2 Polarization 10

2.2.3 Radar Cross Section 11

2.2.4 Radar Image 13

2.3 A Tomography Approach To Bistatic Radar Imaging 14

2.3.1 Radon Transform And Projection Slice Theorem 14

2.3.2 Tomography Principles Applied To Monostatic Radar imaging 16

2.3.3 Tomography Principles Applied To Bistatic Radar Imaging 22

2.4 Conclusion 25

3 Characteristic Of The Singaporean Configuration 26 3.1 Introduction 26

3.2 Practical PBR imaging 27

3.2.1 Block Diagram Of The PBR System Considered 27

3.2.2 PBR imaging implementation 28

3.2.3 Resolution 29

3.3 Actual Configuration Constraints 30

3.3.1 General Presentation Of The Bistatic Configuration 30

3.3.2 Transmitters Constraints 31

3.3.3 Airplanes Trajectory Constraints 34

3.3.4 Consequence On The Fourier Space Coverage 35

3.4 Example Of Radar Images Generated By Using Bright Point Model And Actual Airplane Trajectory 37

3.4.1 Bright-Point Model 37

3.4.2 Passive Bistatic Radar Images 38

3.5 Conclusion 40

4 Simulated Electromagnetic Airplane Model 41 4.1 Introduction 41

4.2 General Description 42

4.3 Electromagnetic Model And Simulation 43

4.3.1 The Importance Of A Conformal Mesh 43

4.3.2 Maxwell’s Equation 44

4.3.3 E-Field Reflected Off A Sphere And The Electromagnetic Regions 46 4.3.4 Electromagnetic Simulation 47

4.3.5 The Methods Of Moments (MoM) 48

4.3.6 The NEC2 Software 48

4.3.7 Constraints Of NEC2 49

4.4 From CAD Model To NEC2 Interpretable Model 51

4.5 Validation Of The Approach And Limitations 57

4.5.1 Verification On A Sphere 57

4.5.2 Estimation Of The Computation Time And Memory Required 58

4.5.3 Remaining Limitations 60

4.6 Simulation Of Radar Cross Section Of Airplanes 61

4.6.1 RCS At Different Frequencies 61

4.6.2 Variation Of RCS In Function Of An Orientation Error Of The Airplanes 62

4.7 Simulation Of PBR Images Using Low Frequency Signals 64

4.7.1 Simulation Of Passive Bistatic Radar Image 64

4.7.2 Simulation Of Passive bistatic Radar Image Using Actual Airplanes Trajectory 65

4.8 Conclusion 67

5 Passive Bistatic Radar Using Real Data 69 5.1 Introduction 69

5.2 Design Of The Experiment 70

5.2.1 General Description Of The Real Data Collection And Processing 70 5.2.2 Acquisition Of Measured And Reference Signals 71

5.2.3 Generation Of RD Map And Extraction Of RCS 72

5.2.4 Synchronization And Filtering Of The ADSB Data 74

5.3 Experimental Results 76

5.3.1 Range Doppler Map Obtained 76

5.3.2 RCS Extraction Of Commercial Airliner By Using Real Data 77

5.3.3 Discussion On The Passive Bistatic Radar System Built 81

5.4 Conclusion 81

6 Conclusion 836.1 Summary Of The Thesis 836.2 Future Work 856.2.1 Overcoming NEC2 limitations 856.2.2 Improving the extraction of the CRCS from real data measurements 866.2.3 Use of multiple FM station transmitters and receivers 866.2.4 Considering another signal 86

List of Tables

3.1 Parameters used for the bistatic radar 32

List of Figures

1.1 A general description of a radar system 2

1.2 The bistatic configuration 4

1.3 Block diagram of radar imaging processing developed in this thesis 5

2.1 Block diagram of radar imaging processing and focus on this Chapter 9

2.2 Description of the polarization 12

2.3 A monostatic configuration 17

2.4 A bistatic configuration 23

3.1 Block diagram of the simulated PBR imaging system 27

3.2 Geometric configuration used for the ISAR imaging using a passive bistatic radar 28

3.3 Difference between the Mean Bandwidth (MBW) and the Full Width at Half Maximum (FWHM) Reproduced from [1] 29

3.4 Singaporean configuration considered for the ISAR imaging using a passive bistatic radar 30

3.5 Presentation of the parameters used in the power link budget equation [2] 32 3.6 Representation of the frequency, expressed in MHz, repartition available from the two transmitters considered 33

3.7 Trajectories of commercial airliners in the Cartesian representation 34

3.8 Trajectory represented in the (α, β)-representation 35

3.9 Covered Fourier domain with those transmitters by five targets 36

3.10 Covered Fourier domain with those transmitters by only one target 37

3.11 Bright point model descritpion 38

3.12 PBR images of 5 bright points using a real trajectory 39

3.13 PBR image of 9 bright points using a real trajectory 39

4.1 Block diagram of the radar imaging processing 43

4.2 CAD model and conformal wire model of a F16 fighter airplane 44

4.3 Radar cross section of metal sphere from Mie’s theory 47

4.4 Block diagram of the processing for obtaining a NEC2c interpretable model from a CAD model 52

4.5 Example of a face divided in four sub-faces and the wires model derivated from it 54

4.6 Example of a face crossed by a wire and the modification of the wires structure induced 56

4.7 RCS of a 10-m sphere with 5376 faces 57

4.8 RCS of a 10-m sphere with 5376 faces when ka = 5.3, i.e at 50 MHz 58

4.9 CPU time needed in function of the number of segments, with a fixed number of incident and scattered angle 59

4.10 CPU time needed in function of the number of incident and reflected angles 60 4.11 Airplane CAD model used for PBR imaging 61

4.12 Influence of the frequency on the bistatic RCS 62

4.13 Influence of an azimuth error on the bistatic RCS 63

4.14 Influence of an elevation mistake on the bistatic RCS 64

4.15 Monostatic radar images two different airplanes with or without their shape using a 360 degrees range of angles 66

4.16 PBR images of two different airplanes with a 20 degree constant bistatic angle and an aspect angle ranging from 230 to 350 degrees 67

4.17 Fourier space considered for the PBR imaging 67

4.18 Influence of an elevation mistake on the bistatic RCS 68

5.1 Block diagram of the processing chain for experimental data 70

5.2 Block diagram of the extraction of RCS from real data The improvement provided by our work is emphasized by the yellow background 71

5.3 Example of ADSB data smoothed by a polynomial function of degree 6 and its influence on the aspect angle and Doppler shift at the different instant of a trajectory 75

5.4 RD map obtained by using FM signals and the station at 93.8 MHz 76

5.5 Example of target tracking on RDMap3 77

5.6 Example of Doppler shift and bistatic range estimated from ADSB data and tracked from RDMap3 78

5.7 Representation of the ratio between the peak value corresponding to the reflected path and the noise level on RDMap3 79

5.8 Example of power received and RCS computed 80

5.9 Polar representation of the RCS of a target in function of the aspect angle α and its bistatic angle 80

ADC Analog to digital converter

ADSB Automatic dependent surveillance-broadcast.ASR Airport surveillance radar

ATR Automatic target recognition

B-Field Magnetic field

CAD Computer-aided design

CIT Computation integration time

COLLADA Collaborative design activity

CPU central processing unit

CRCS Complex radar cross-section

E-Field ElectricalField

FM Frequency modulation

FWHM Full width at half maximum

HFA High frequency asymptotic

LNA Low noise amplifier.

MBW Mean bandwidth

NEC Numerical electromagnetic code

NTU Nanyang Technological University

OS Operating system

PBR Passive bistatic radar

PEC Perfect electric conductor

RCS Radar cross-section

RD Range-Doppler

RF Radio frequency

SNR Signal-to-noise ratio

SR-CRCS Square root of the complex radar cross-section

TL@NTU Temasek Laboratory at Nanyang Technological University.TL@NTU Worldwide Interoperability for Microwave Access

VAC Visual approach chart

XML Extensible markup language

Chapter 1

Introduction

The main goal of our work is to build radar images of aircraft using passive bistaticradar signals The use of the FM band for that purpose is motivated by the geographicprevalence of radio broadcasting and the large size of an FM cell The thesis covers thefeasibility study of such a system, its design, its simulation, and its implementation

In the first part of the thesis, the targets considered are air target models obtained fromCAD models The targets ranged from fighter jets to private jets In the second part, thetargets considered are actual commercial airplanes thanks to the proximity with ChangiAirport and the possibility to obtain exact flight information of the airplanes in real time.Moreover, the trajectories considered for the imaging are actual trajectories of airplanes

in and around Singapore

The term radar comes from RAdio Detection And Ranging and originally described asystem that uses the electromagnetic wave for detecting the presence of an object and

Chapter 1 1.2 Radar

determining its position A full radar system is composed of a transmitter, a receiver and

a signal processing chain, as represented in Figure 1.1

Figure 1.1: A general description of a radar system

A radar system is called monostatic if the transmitting antenna and the receiving antennaare co-located In a monostatic configuration, the delay τ is related to the range R fromthe antenna to the receiver, the Doppler shift fD to the range-rate VR (also called radialvelocity), and the power to the reflectivity coefficient of the object such that

Chapter 1 1.2 Radar

for detection

A bistatic radar refers to a radar system where the receiver and the transmitter are notco-located The bistatic radar configuration is described in Figure 1.2 New parametersare introduced for describing the bistatic configuration such as the baseline distance L,and the bistatic range RB such that

where RRS is the distance between the scattered point and the receiver and RT S thedistance between the scattered point and the transmitter In that configuration, thedelay can be related to the bistatic range RB, the Doppler shift to the bistatic range-rate, and the ratio between the transmitted and received power to the bistatic reflectivityfunction The relation between the signal parameters and the geographic parameters aregiven by

RSR2

T SLLoss

(1.7)where σB the bistatic reflection coefficient Moreover, a new parameter appears in abistatic configuration called the bistatic angle β and is defined as the angle between thetransmitter, the scattered object, and the receiver

The monostatic configuration corresponds to a bistatic configuration where the bistaticangle and the baseline distance are null The bistatic detection range is an oval of Cassinidefined by a constant product of the distance transmitter-object and receiver-object, and

Chapter 1 1.2 Radar

Figure 1.2: The bistatic configuration

it is derived from Equation (1.7)

A passive radar is a radar system which takes advantage of an illuminator of opportunity

to replace the transmitter in a radar system Such systems make use of existing radiatedsignals already available and so have discretion properties for the radar itself, and do notrequire a new frequency spectrum slot [3] Recently, a renewed interest in bistatic radarhas been seen, due to the widespread deployment of cellular communication equipment,and thus the possibility of using the cellular infrastructure for detection, estimation, andtracking [4, 5]

In [6], Griffiths investigated the use of various wireless digital signals for detectionand estimation of targets, which presents good ambiguity functions, but their low geo-graphic spread or the small coverage of their cells decreases the interest for those signals

or increases the complexity of those systems For instance, the use of WiMAX as an minator of opportunity was discussed in [7] for ground imaging but its limited availabilityimpairs its practical value

illu-In [6], Griffiths also analyzed various analog signals as illuminators of opportunity andproved that the resolution achievable by using FM band signals is too low for practicaldetection and estimation applications

However, despite the low image resolution, radio broadcasting FM signals appears to

be a good candidate for imaging airplanes thanks to the widespread use of the technology

Chapter 1 1.2 Radar

and the typical large size of the FM cells In [8], Daoult et al investigated the tion of passive radar images of synthetic air targets flying synthetic trajectories from thesignal of a single FM station In this thesis, the construction of passive radar imagesusing CAD-modeled and real airplanes flying actual trajectories

The radar image of a target is the distribution of its elementary scattering coefficients

A radar image is generally presented as a two dimension map where each pixel sity/color represents the amplitude of the elementary scattering coefficient at this posi-tion, usually represented in a dB scale A radar image is obtained by doing a multitude

inten-of processing In this thesis, the processing, as represented in Figure 1.3, is divided intotwo main parts:

• The extraction of the radar cross section (RCS) of airplanes

• The construction of a passive bistatic radar (PBR) image

The first block calculates the RCS of the target given simulated or real observations,

a process that will be described in the next chapter The extracted RCS is then passed

to the second block to obtain a radar image This second process will also be discussed

in the next chapter

Environmentparameters

RCSFigure 1.3: Block diagram of radar imaging processing developed in this thesis

Chapter 1 1.3 Notation

In the thesis, unless otherwise specified, the following notation will be used for a variable

“x”:

• a real number in R is denoted by x,

• a complex number in C is denoted by x,

• a vector in Rn

or Cn is denoted by ~x,

• a matrix (in R or C) is denoted by X

The main contributions of the thesis are :

• The feasibility study of a passive bistatic radar system using the radio broadcasting

FM signals for imaging airplanes applied to the Singaporean configuration

• The use of low frequency signals to build radar image in a constrained environmentbased on the principles of tomography from simulated data

• The construction of a conformal model interpretable by NEC2 based on a basicCAD model available online

• The simulation of a scattered field of small airplanes using NEC2

• The realization of a tracking system based on both FM data and ADSB data

• The processing of the FM data to find the complex RCS of airplanes

Chapter 1 1.5 Organisation Of The Thesis

The thesis is organized as follows

Chapter 2 presents the electromagnetic and radar background needed for the standing of the radar image generation It presents the principles of tomography used forthe construction of radar images

under-Chapter 3 presents the feasibility study of the passive bistatic radar system designed

in this thesis The chapter discusses the constraints introduced by a true bistatic figuration and their influence on the quality of the radar images obtained by our passivebistatic radar

con-Chapter 4 presents the electromagnetic tools used for the electromagnetic tion and the simulation results It also describes the method used for building a NEC2interpretable model from a CAD model

simula-Chapter 5 presents the design of the passive bistatic radar system built to extract theradar cross section of airplanes The limitations of the system built are underlined andsolutions for overcoming them are presented

Chapter 6 summarizes the work performed in this thesis and provides ideas for futuredevelopments

In Section 2.2, we define the concepts of CRCS and radar image In Section 2.3, afterintroducing the Radon Transform and the projection theorem, the relation between theRCS and the radar images is derived for the monostatic configuration and then for thebistatic configuration.

The development performed in this Chapter is then used in Chapter 3, 4 and 5 for theconstruction of radar images The diagram in Figure 2.1 represents the passive bistaticradar considered and it underlines the focus put on the radar imaging part in this Chapter

Chapter 2 2.2 Radar Cross-Section And Radar Image Definition

Environmentparameters

Chapter 2 focusRCS

Figure 2.1: Block diagram of radar imaging processing and focus on this Chapter

Defini-tion

A real-valued sinusoidal signal x(t) = A cos(ωt + θ) can be expressed as Re[Aejωtejθ] and

we refer to Aejθas the fixed phasor, and Aejωtejθ as the time-varying phasor The electricfield (E-Field) transmitted by a point source can be modeled as in [10]

~E(~r, t) = ~E0(~r)ej(ωt−~k·~r), (2.1)where the vector ~E(~r, t) is the E-Field at a position ~r of polar coordinates centered onthe point source at a time t, ~E0(~r) its amplitude, ω its angular frequency, and ~k is thewave vector The E-Field is propagating in the direction of the wave vector ~k, thereforethe E-Field is described by two components orthogonal to the wave vector ~k Moreoverthe wave vector ~k generally depends on the position

From Equation (2.1), it is obvious that the E-Field can be separated into time andspace functions, as follows

Chapter 2 2.2 Radar Cross-Section And Radar Image Definition

~E(~r, t) = ~E0(~r)e−j~k·~rejωt

to point out that both phasors are complex-valued and that the E-Field is the real part

of the time-varying phasor

The amplitude of the E-Field ~E0(~r) depends on the model considered In a nearfield condition, the usual assumption is that the amplitude of the E-Field is inverselyproportional to the range from the transmitter, i.e

where ~E0 is a constant that can be determined by boundary conditions If the amplitude

of the E-Field respects the condition in Equation (2.3), the wave is called a sphericalwave In a far-field condition, a usual hypothesis is that the amplitude of the E-Field isnot range dependent and the wave is then called a plane wave

As said previously, the E-Field can be described by two orthogonal components to thedirection of propagation given by the wave vector ~k For the purpose of the demonstrationand without any lost of generality, we assume a propagation over the z-axis and so theE-field initial can be described as:

Chapter 2 2.2 Radar Cross-Section And Radar Image Definition

com-• Circular polarization : If the two orthogonal components |E0x| and |E0y| are equal

but E0x and E0y have a 90-degrees phase difference

• Linear polarization : If the two orthogonal components E0x or E0y have the same

phase

• Elliptical polarization : If the two orthogonal components E0x and E0y have

90-degrees phase difference and amplitude, none of them are null

The linear polarization is the most used for radar system and two polarizations aremainly used called the Horizontal (H) polarization and the Vertical (V) polarization Byconsidering cylindrical coordinates centered at the antenna, we can defined in each point

of the space a local basis represented in Figure 2.2 The V polarization is defined as apolarization such that the E-Field is only in the ~Uθ direction whereas in a H polarizationthe E-Field is pointing in the ~Uφ

Monostatic Radar Cross-Section Definition

Previously, we spoke about the power received by the receiver and it was linked to thereflectivity coefficient σ, which is also called the radar cross-section (RCS) It is defined

as in [11] by

σ = lim

|~ r|→∞4π|~r|2| ~ER(~r)|2

Chapter 2 2.2 Radar Cross-Section And Radar Image Definition

Figure 2.2: Description of the polarization

where ~ER(~r) denotes the reflected E-Field observed by a receiver located at a position ~rsufficiently far away from the object and ~ET(~0) the incident electric field at the target

in a far field condition The RCS of an object depends mainly on its shape, its size, itsmaterials, the frequency, the polarization, and the aspect angle

The complex radar cross-section (CRCS) σ is introduced from the definition of theRCS given in [12]

Chapter 2 2.2 Radar Cross-Section And Radar Image Definition

dependency is pointed out by denoting the RCS σ(α, k) where α and k are the aspectangle and the wavenumber

Bistatic Radar Cross Section Definition

In a similar way we can also define the bistatic radar-cross section σB by Equation (2.8),

As mentioned in Chapter 1, the radar image is the distribution of its elementary scatteringcoefficients In this thesis, we only consider a two dimensional radar image, i.e thescattering coefficients are projected onto a two dimensional plane If we consider a scene

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

Ω with only discrete scattering points, the radar image is denoted as S(x, y) and definedas

Imag-ing

The tomography principle used for the radar image generation is based on the Radontransform and the projection slice theorem It consists in building a 2D image of ascene from a multitude of 1D information of the scene with different points of view [9].Therefore, Radon transform and projection slice theorem are presented before derivingthe tomography principle applied to radar image generation The relation between theSR-CRCS √

σB and the radar image S(x, y) is used in Chapters 3, 4 and 5

Radon Transform

The Radon transform is the integral transform of a 2D-function over a straight line For

a given function f defined on R2, the Radon transform is defined as [9]

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

Projection Slice Theorem

The Radon transform is closely related to the Fourier Transform and this relation iscaptured in the projection slice theorem or Fourier slice theorem [9] The one-dimensionalFourier transform of a function f : R2 → R is given by

f (u cos(α) + v sin(α), u sin(α) − v cos(α))e−j2πuωudvdu (2.16)

A rotation matrix Rα of rotation angle α is introduced ~u is a vector from R2 withcoordinates (u, v), and Px is the operator for the projection onto the x-axis We canrewrite the previous equation as

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

The change of variable ~u = Rα~x where ~x is a vector from R2 with coordinates (x, y)gives us

imag-ing

The monostatic configuration is described in this section in order to explain the mographic principles applied to radar imaging It is described in [9] for a monostaticconfiguration and a chirp pulse In this section, assuming a monostatic configurationand an ISAR configuration, the imaging principle is described for a more general signal.Figure 2.3 shows the monostatic configuration used for the mathematical description ofthe tomography principles applied to the monostatic radar imaging where ω denotes anelementary element of the object being tracked, O is the center of the object, r(x, y) therange between the radar and the elementary points considered and rO the range betweenthe center of the object and the radar First, we consider an aspect angle α null, i.e theline of sight and the x-axis are collinear

to-The incident E-Field is assumed to be far field and a bright point model is used forthe object, i.e the object can be described as a sum of discrete scatterers, so that the

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

Figure 2.3: A monostatic configuration

reflected total E-Field is a discrete sum of reflected, attenuated, and delayed incidentE-Field reflected by each scatter points Thus, the reflected field is modeled by thefixed-phasor given by

~

ER(~rR) =X

ω∈Ω

AωSωE~I(~rT), (2.19)

where ~rR and ~rT are the receiver and transmitter positions in the local basis centered at

O of the object Ω, respectively, ~EI(~rT) is the incident E-Field fixed-phasor, ~ER(~rR) is thereflected E-Field fixed-phasor, Aω the attenuation and delay coefficient associated withthe scattering point ω , and Sω the reflection coefficient associated with the scatteringpoint ω Since a monostatic configuration is considered, thus we have ~r = ~rR = ~rT Thereflection coefficient depends on the polarization of the incident E-Field and the reflectedE-Field of interest The reflection coefficients between the different polarization HH, HV,

VH and VV define the scattering matrix

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

By using a first order approximation of the denominator, and using the Cartesiancoordinates, we have

2√

πr0 · ~ET(~r)dxdy, (2.24)

Then we consider a second order approximation for the exponent of the numeratorand we find that

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

By using Equation (2.6), the CRCS is

= lim

|~ r|→∞

4π|~r|24πr2 0

ˆ

RS,0(2k) =√

where √

σ(0, k) denotes the SR-CRCS presented previously in Section 2.2.3

Second, we assumed that the radar is still over the x-axis as plotted in Figure 2.3.The radar is still considered at a distance r0 but now the angle between the line of sight

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

and the x-axis is −α, it is as we considered a rotation of the object of an angle α Thenthe Equation (2.25) changes such that

|r(x, y)| =

q(r0+ x cos(α) + y sin(α))2+ (−x sin(α) + y cos(α))2

And then we find that for any aspect angle α the SR-CRCS is the Fourier transform

of the Radon transform of the object’s reflectivity function By using the projection slicetheorem stated in Equation (2.18), we can rewrite the Fourier transform of the reflectivityfunction S such that

ˆS(~α2k) =√

where ~α is the vector of coordinates (cos(α), sin(α)) corresponding to an aspect angle α

We introduce a change of variable such that

σ(α(ωx, ωy), k(ωx, ωy)) (2.38)

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

Using the inverse Fourier transform, we have

S(x, y) = 1

(2π)2Z

(ω x ,ω y )∈R 2

√σ(α(ωx, ωy), k(ωx, ωy))ej(ωx x+ω y y)dωxdωy, (2.39)

By using the aspect angle α and the wavenumber k we have

S(x, y) = 1

(2π)2Z

k∈RZ

α∈[0,2π]

√σ(α, k)ej2k(cos(α)x+sin(α)y)|4k|dαdk, (2.40)

Equation (2.40) then shows the relation between the SR-CRCS and the radar image

Imag-ing

Second, a bistatic configuration is considered in order to apply the tomography principles

to passive bistatic radar imaging Figure 2.4 shows the bistatic configuration used forthe mathematical description of the tomography principles applied to the bistatic radarimaging, where ω denotes an elementary element of the object, O the center of the ob-ject A Cartesian and a polar basis are centered in O and the x-axis is in the direction ofthe bisector of the angle between the transmitter, the center O and the Receiver More-over, rT(x, y) is the range between the transmitter and the elementary points considered,

rR(x, y) the range between the receiver and the elementary points considered, rTO therange between the center of the object and the transmitter, rR O the range between thecenter of the object and the receiver and β the bistatic angle between the line of sightbetween the receiver and the object center and the line of sight between the transmitterand the center of the object

In a same way that in Section 2.3.2, the incident E-Field is assumed in a far fieldconditions and it is then considered as a plane wave Therefore, the received E-Field can

Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

Figure 2.4: A bistatic configuration

wavenum-Chapter 2 2.3 A Tomography Approach To Bistatic Radar Imaging

rR(x, y) + rT(x, y) ≈ RT0 + RR0+ 2 cos(β/2) (x cos(α) + y sin(α)) (2.46)

By going through the derivation in Section 2.3.2 again by using the relation given in

Sin-is presented.

In Section 3.2, the practical implementation of the theory developed in Chapter 2 ispresented and the resolution of an image is introduced as a performance measure InSection 3.3, the performance of the system is studied by analyzing the link budget equa-tion, the available frequencies in Singapore and the different bistatic parameters Finally,the Fourier space domain fulfilled is estimated and the theoretical achievable resolution

is computed In Section 3.4, the quality of the radar image is discussed according to thelimited FM Band and the limited variation of the aspect angle in a Singaporean con-

Chapter 3 3.2 Practical PBR imaging

text The discussion is done on image generated from a bright-point model and the radarimaging presented in Chapter 2

The influence of actual airplanes trajectories on PBR images is studied in this chapter

In that purpose,PBR simulated system is built such that the inputs are the frequencyused, the bright points considered and the ADSB data measured and the output of thePBR simulated system is a radar image The block diagram of the PBR system built is

in Figure 3.1

ComputeE-Field

RCSExtraction

BuildPBR image

Extractaspect andbistaticangles

Angles

Angles

Figure 3.1: Block diagram of the simulated PBR imaging system

The bistatic and aspect angles are extracted from actual trajectories extracted fromthe information given by the ADSB system available on all commercial airliner TheE-Field is computed according to those angles and to those frequencies, the bright-pointspositions and their scattering coefficients The CRCS is then extracted from the E-Field.Finally, radar images are constructed from the CRCS obtained and the different aspectand bistatic angles using the Equation (2.49)