forward imaging using synthetic aperture...

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FORWARD IMAGING USING SYNTHETIC

APERTURE RADARS

Thesis submitted in partial fulfilment of the requirements for the degree of

MS by Research

in

Electronics and Communication Engineering

by

Palash Jain

200831009 [email protected]

International Institute of Information Technology

Hyderabad - 500032, INDIA

MAY, 2015

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Copyright Palash Jain, 2015

All Rights Reserved

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International Institute of Information Technology

Hyderabad, India

CERTIFICATE

It is certified that the work contained in this thesis, titled Forward imaging using

Synthetic Aperture Radars by Palash Jain, 200831009, has been carried out under our

supervision and is not submitted elsewhere for a degree.

Date Adviser: Dr. K.R. Sarma

Adviser: Dr. P.R.K. Rao

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Acknowledgments

I would like to take this opportunity to acknowledge and appreciate the efforts of the people

who have helped me during my research and documenting this thesis. I am grateful to Dr. K.R

Sarma for providing his immense support and guidance and helping me build my foundations of

the subject. I am also grateful to Dr. P.R.K Rao for his invaluable advice, that have helped me

shape this thesis, and furthermore, which will be helpful to me in all of my life, and also for

teaching me the proper line of investigation to study a subject.

I would like to acknowledge the Institution of IIIT-Hyderabad, for providing support and

accommodation for me to carry out my work. I would also like to thank my friends Kritika Jain,

Neeraj Pradhan, Roopak Dubey, Ankit Gandhi and Varun Ramchandani for their support and

encouragement. I also wish to thank everyone at the Communication Research Centre for

building a congenial environment for discussion and learning at the lab. I take this opportunity to

thank all the anonymous reviewers of my work at SIP-2013 and APSAR-2013 for evaluating my

work and providing valuable feedback that helped me improve my work. Also for recognizing the

novelty and effort put into the work.

On a personal front, I am grateful to my parents for supporting me to pursue all my endeavors.

This acknowledgement would not be complete without recognizing the great atmosphere, ample

resources provided by IIIT-Hyderabad and all the faculty members. Lastly, I wish to thank the

Almighty for all his grace and generosity.

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Abstract

A typical microwave Synthetic Aperture Radar (SAR) is an aerial based system for high

resolution imaging of terrain. In contrast to an optical imaging system, its operation is not limited

to daytime and its performance is not adversely affected by severe weather conditions like fog,

snow or clouds. A conventional Side-looking Synthetic Aperture Radar provides high azimuth

resolution but in the side looking mode. In situations like aircraft landing or ground vehicle

piloting, the radar has to image a forward swath. But the SAR in the forward-looking mode not

only suffers from target ambiguities, but more importantly, it also cannot yield an azimuth

resolution comparable to that of the side looking mode.

In this thesis, we explore the possibility of an imaging system, which draws on the principle of

functioning of an SAR and is able to image the terrain ahead of a vehicle. Three approaches are

proposed and analyzed. The first approach, termed as Azimuth Triangulation Method,

abbreviated as ATM, deploys a multiple antenna SAR to solve the problem of low azimuth

resolution and high azimuth ambiguity in a forward-looking SAR (FSAR). As is to be expected,

the performance improves with a larger number of antennas. In view of the complexity of the

system based on the ATM approach, as the second approach, a SAR in a squint-mode is

considered. For purposes of comparison, a SAR in the forward mode is also simulated. The

simulations performed for the three approaches indicate while the range resolution is same in all

the cases (approximately 0.3 m, under the stipulated conditions of simulations) the azimuth

resolution varies considerably. The ATM approach achieved an azimuth resolution of

approximately 12m, while the forward-mode SAR and the squint-mode SAR yielded an azimuth

resolution of 18m and 1.5m respectively.

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Contents

Chapter Page

1. Introduction ...10 1.1 Problem statement...11 1.2 Radar fundamentals.11 1.3 Synthetic Aperture Radar....12

1.3.1 Background of SAR.......13 1.4 Modes of SAR.....14 1.5 Chirped Radars15 1.6 Summary.....17 1.7 Contribution.18 1.8 Thesis outline and organization...19

2. Synthetic Aperture Radar : Basics.........20 2.1 SAR Equation: Incident field.....20 2.2 SAR Equation: Scattered field........23 2.3 SAR Equation: Image formation.........24

3. Signal Model......27 3.1 Signal Model for MIMO-SAR....27 3.2 Coherent summation of signals...29

4. Azimuth Triangulation Method for forward imaging........31 4.1 Proposed architecture......32 4.2 Target model and signal model.......33 4.3 Algorithm........34 4.4 Simulation parameters.........35 4.5 Results.........36 4.6 Resolution............37

5. Azimuth Resolution Enhancement using FSAR....39 5.1 Problems with FSAR...........39 5.2 Target model and signal model.......40 5.3 Resolution........41

5.3.1 Azimuthal resolution......42 5.3.2 Range resolution.........43

5.4 Simulation results........44

6. Squint beam Synthetic Aperture Radar..................50 6.1 Target Model and signal model...51 6.2 Azimuthal resolution...51 6.3 Simulation results....54

7. Simulation Methodology.......60 7.1 Received SAR Signal......60

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7.2 Terrain Map.........63

8. Conclusion and Future Work........65

Related Publications....66

Bibliography....67

APPENDIX......70

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List of Figures

Figure Page

1.1 Multiple Radar Beams..12

1.2 Modes of SAR..14

1.3 Analysis of chirp signal resolution capabilities....17

1.4 Analysis of matched filtering in AWGN channel.17

2.1 Geometry of a SAR system...21

4.1 Geometric configuration of the proposed FSAR system..32

4.2 Pictorial representation of ATM algorithm...35

4.3 Simulation result with 5 antenna elements...36

4.4 Simulation result with 9 antenna elements...36

4.5 Resolution study........37

5.1 Proposed radar configuration39

5.2 Result: Case-1 : Discrete point target simulation.44

5.3 Result: Case-2 : Discrete point target simulation in presence of noise.45

5.4 Result: Case-2 : Discrete point target simulation in presence of noise.46

5.5 Result: Case-3 : Resolution capability analysis....47



5.8 Result: Case-4 : Real world target scene simulation...48

6.1 SAR in squint mode......50

6.2 SAR imaging schematic for the squinted beam strip-map mode......51

6.3 Result: Case-1 : Discrete point target simulation.....54

6.4 Result: Case-2 : Discrete point target simulation in presence of noise....55

6.5 Result: Case-2 : Discrete point target simulation in presence of noise56

6.5 Result: Case-3 : Resolution capability analysis........57

6.6 Result: Case-3 : Resolution capability analysis........57




6.10 Result: Case-4 : Real world target scene simulation.59

7.1 Result: Case-3 : Range reference signal...........60

7.2 Result: Case-4 : Correlation with range reference signal........61

7.3 Result: Case-4 : Image of single target after azimuth correlation, after compression.....62

7.4 Result: Case-4 : Detection of a single target....62

7.5 Result: Case-4 : Terrain Map in MATLAB.....63

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List of Tables

Table Page

1. Table 4.1 : Simulation parameters....35 2. Table 4.2 : Resolution Result.......37 3. Table 5.1 : Accuracy analysis of FSAR in absence of noise....45 4. Table 5.2 : Accuracy analysis of FSAR in presence of noise..45 5. Table 6.1 : Accuracy analysis of Squint SAR in absence of noise..54 6. Table 6.2 : Accuracy analysis of Squint SAR in presence of noise.....55 7. Table 7.1 : Comparison with Zaharris implementation......64

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Chapter 1

Introduction

All transport systems encounter the problem of poor visibility due to adverse weather conditions

such as fog, rain, snow, storms, and other natural phenomenon. As microwaves propagate

seamlessly, even in such adverse conditions, radar based imaging systems can be built using

microwaves for visibility-aid in transportation systems. But the radar system used in such

application must produce high resolution imagery to enable the pilot of the vehicle to distinguish

objects in front of the vehicle. As the imaging system is to be part of the air or ground based

vehicle its size and weight are limited by physical constraints. Classical radar systems used large

antennas to obtain high angular resolution equal to its beam width. However, Synthetic Aperture

Radar (SAR), which makes use of the motion of the vehicle provides high angular resolution with

small antennas and generates high resolution images of the terrain. Our primary goal is to develop

an imaging system using the capabilities of SAR to image the area ahead of the vehicle to help

the pilot to guide the vehicle smoothly, without accidents.

SARs have been in use since 70s for civilian applications after it was declassified by the U.S.

military. It is still being used extensively by military for aerial reconnaissance. It has made

inroads into satellite imaging for remote sensing. In recent times it is also being used to provide

landing support for aircrafts in bad weather. However, SAR has not been explored for use in

ground based vehicles due to some fundamental limitations which will be discussed later.

Conventional SAR works in side looking mode and images the underlying terrain to the right

and left of the flight path. But if the area of interest is in front of the flight, there exits an inherent

visualization gap. Many applications require forward imaging radars, such as airplane landing or

ground vehicle guidance systems like railways in adverse weather condition and several other

automated vehicular applications. Here, our intent is to explore the use of SAR for imaging the

area that lies in front of the radar platform, by finding solutions to the problems associated with

forward looking imaging mode of SAR.

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1.1 Problem Statement

The problem addressed in this thesis is of finding an imaging sensor for ground based

locomotive which will operate in adverse conditions like fog etc.. Railway system experiences

inordinate delays during the winter season due to severe fog affecting the visibility, quite often

resulting in accidents. We explore the use of Forward-looking SAR (FSAR) operating in Ku-

Band to provide the driver with an unambiguous image of the terrain in front of the locomotive

with high resolution, to enable him to pilot the vehicle safely and without delays.

1.2 Radar Fundamentals

Radar (Range Detection and Ranging) works by transmitting pulses of electromagnetic energy,

in a narrow beam which are propagated at speed of light and bounced off the target surfaces

producing return echoes that are received by the receiving antenna. The time delay thus incurred

by the signal and the beam position provides information about the location of the target with

respect to the antenna. The strength of the received echo conveys information regarding the

reflectivity and size of the target. For mono-static radar (radar system in which, the transmitting

and receiving antenna are same), the target range R can be determined from the measured time

delay , incurred by the echo due to its two-way propagation, by the equation

=.

2 (1.1)

where c is the speed of light. The angular coordinates of target are obtained from beams angular

position. Radar resolution is defined as the minimal distance at which two close spaced

discrete targets can be unambiguously separated. For conventional short-pulsed radar, high range

resolution can be achieved by utilizing a shorter pulse. But, the shorter the pulse duration, the

higher is the peak power requirement, since radar detection performance requires high pulse

energy to be transmitted. Generating high peak power signals becomes impractical, especially for

a system that needs to be mounted on a moving platform. The solution to this problem is use of

radar signals with high time-bandwidth product. Pulse compression waveforms is a class of

signals with large time bandwidth product, an example of which, is the chirp signal where pulses

of large time duration are modulated by linear frequency modulated carrier resulting in large

spectral bandwidth. These pulse compressed waveforms are decompressed at the receiver using

matched filtering, resulting in narrow pulses of duration which is inversely proportional to the

signal bandwidth B and the resolution obtained is given by

=

2~

2 (1.2)

High-resolution capabilities of chirp signals are utilized to achieve fine range resolutions in

almost all radars applications. A chirp signal with compressed pulse duration = 0.1ms and an up-

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chirp rate of 10MHz/s results in a bandwidth of 1GHz, which gives the resolution of 15cms,

which is an acceptable value for most practical applications including the problem pertaining to

railways. Details of chirp pulse-compressed radar are discussed in section 1.5.

1.3 Synthetic Aperture Radar

In conventional radar the angular coordinates of the target are obtained by the look angle of the

radar beam. All targets falling in the patch illuminated by the radar beam on the terrain will be

interpreted as having same angular position and all targets falling on a constant range locus will

add up to a single target located at that range. The only way to increase the angular resolution is

to make the beam width narrower. However, to obtain narrow beam width, large aperture

antennas are required. For example at 10 GHz, a 10 meter parabolic dish is required to obtain 0.2-

degree beam width (Compare this with 1 minute of arc resolution of the human eye!). Carrying

such large antennas on moving vehicles is practically infeasible.

Figure 1.1: Multiple Radar Beams

In Synthetic Aperture Radar (SAR), the vehicle carries an antenna, which has a known radiation

pattern and the vehicle moves with a constant velocity along a straight line. The antenna transmits

a known waveform, commonly the chirp signal and a part of the radiated power reaches the target

in the illuminated area. The target receives the radiation and reradiates a part of the incident

radiation in the direction of the receiving antenna depending on its scattering cross-section,

which, in general is time varying, depends on polarization, physical size of the target, reflectivity

of surface and its aspect angle relative to the antenna. The target also imposes its own radiation

pattern on the reradiated power. The receiving antenna then receives the reradiated power

depending on its effective aperture, also known as effective area of the antenna. The effective

area is similar to radar cross-section of target and it depends on the same variables associated

with it. Although it may appear from the discussion of SAR till now that it is same as the

conventional radar, the main difference comes from the movement of the antenna, which results

in synthesis of a spatial chirp over and above the temporal chirp in the waveform. The

mathematical details are postponed to later chapters and it suffices to say that this additional chirp

is what is responsible for the enhanced resolution obtained in the azimuth direction. Moreover the

resulting azimuth resolution increases with the decrease in the physical size of the antenna where

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as conventional radar wisdom tells us that a larger antenna would have to be used to generate a

narrow beam pattern to distinguish closely located targets. Although some simplistic reasoning is

normally given to explain the resolution improvement, we prefer to go through a rigorous

analysis to prove it in a later chapter.

1.3.1 Background: History of SAR

In 1950 Carl Wiley, a mathematician at Goodyear Aerospace, now Lockheed Martin, was the

pioneer who invented SAR. He showed that post processing of the Doppler shift information

provides the ability to obtain finer resolution in the direction of the travel of the beam. However,

no computers (even digital computers were in early stages at that time) were available to process

the amount of reflected waveform data. Fortunately, just at that time, Fourier optics was

rediscovered by Stroke and Marechal and with the availability of laser, providing coherent

radiation, optical signal processing techniques using Fourier optics were proposed. With the

inherent two-dimensional nature of the optical systems and their parallel processing abilities, vast

amount of data could be processed using such methodology. The discovery of holography by

Dennis Gabor also gave a great deal of fillip. Leith and Upatnieks used optical signal processing

on SAR signals under a highly classified defense project at University of Michigan. SAR was

declassified in early sixties. The radar data, after being collected and stored on high-resolution

photographic film via intensity modulation of a cathode tube by the radar signal from the receive

antenna, was sent for further processing. Using a coherent laser and a carefully designed and

positioned system of lenses, Fourier transform of the data and filtering in the spatial frequency

domain was performed before performing another Fourier transform resulting the final image in

spatial domain. This processing was not done in real time, but was done after all the data had

been gathered. Harger, in his pioneering book on SAR makes a statement that; digital computers,

which work at speeds greater than the speed of light, will be required to process SAR data. His

predictions were proved wrong in just a few years of time with the advancement of

semiconductor technology and signal processing methodologies. Today, SAR signal acquisition

and processing is being done in real time on inexpensive digital computers. Concurrently, new

variants of SAR like interferometric SAR and spotlight SAR have also been developed.

McDonald Dettwilder and others at Jet Propulsion Lab developed the first digital processing

algorithm in 1978, for use on SEASAT. The technique of processing the SAR data separately in

the range and azimuth domains was implemented digitally with the use of Fast Fourier transforms

(FFT). In 1978, this algorithm could develop a 40 square kilometer image with a resolution of 25

square meters in about 40 hours. Today, an average modern computer can process the same

image in less than a quarter of a second. SEASAT was operational for 105 days after its launch

on October 10, 1978 before a critical short circuit failure ended the mission. However, the

implementation of the digital SAR processor on SEASAT proved operational and effective and

paved the way for many signal-processing algorithms that are widely used in various applications

even today.

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1.4 Modes of SAR

The SAR was originally used in side looking mode where the antenna is mounted on the side of

the aircraft and the beam illuminates a swath, which moves parallely along with the trajectory of

the plane covering a strip. This mode of operation is called the Strip-map SAR. In this mode,

target is illuminated only for the time duration the radar beam takes to traverse it. A modification

of this is the Spotlight SAR where antenna beam is continuously reoriented to illuminate the same

target as the plane moves. In Strip-map SAR, a large area is imaged with relatively lower

resolution when compared to Spotlight SAR, in which, a smaller area is imaged but with higher

resolution. Figure 1.2 depicts these two modes of SAR.

It may be noted that in the other modes of SAR only stationary targets can be imaged. Moving

targets will produce ambiguous outputs. A more sophisticated SAR forms an interference signal

pattern using all the signals received in forward and backward travel of the plane and then the

interference pattern is analyzed which gives information on moving targets. However, moving

target imaging is not covered within the scope of work done under this thesis.

Figure 1.2 Modes of SAR (a) Strip-map imaging (b) Spotlight imaging

The spot-light imaging mode may contribute towards an alternative solution of forward

imaging, but is disregarded in this work due to direct correlation between strip-map imaging and

the problem of forward imaging. Phased array radars can also be utilized to enhance resolution

problem for the railways imaging system but they rely on the physical aperture of the antenna for

their resolution capabilities and hence are disregarded since large antennas are not feasible to

install on a moving train. However, the radar configuration proposed in Chapter-4 implements an

approach closely associated with phased array radars and analyzes the results obtained.

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1.5 Chirped (Pulse-compressed) Radars

As discussed in section 1.2, in conventional radar, the range resolution is restricted by the width

of the pulse transmitted. To achieve ideal resolution, the pulse width needs to be as narrow as

possible, ideally resembling a thumb-tack, (t) function. But an ideal thumb-tack function is

impossible to synthesize in practice, because even for a small signal energy to be transmitted, the

signal will have to carry an infinite amount of power. If the pulse is made wider, the resolution

suffers. The following analysis explains it.

The simplest of signals a pulse radar can transmit is a sinusoidal pulse of amplitude A, and

carrier frequency f0, truncated by a rectangular function of width T. Such pulses are transmitted

periodically, but as the system is linear; we will only consider a single pulse in our calculations. If

we assume the pulse to start at time t=0, the transmitted signal can be written as

() = 20 if 0< t () = ()( )

0

(1.4)

If the transmitted signal is delayed by , which is the two-way travel time, and is attenuated by a

factor k, the received signal can be written as

() = 20() + () < < + (1.5)

= () Since we know the transmitted signal, we obtain:

http://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Rectangular_functionhttp://en.wikipedia.org/wiki/Doppler_effecthttp://en.wikipedia.org/wiki/White_noisehttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Matched_filterhttp://en.wikipedia.org/wiki/Complex_conjugate

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< , > () = 2(

)20() + () (1.6)

where () is the result of the correlation between the noise and the impulse response of the

matched filter. The signal component is the autocorrelation of the transmitted signal, which is a

triangle function (t), which increases linearly on [-1/2, 0] and reaches its maximum value 1 at t

= 0, and it decreases linearly on [0, 1/2] until it reaches 0 again, and is 0 otherwise. Figures 1.3

shows the shape of the correlation for a sample signal (in red), which is assumed to be a real

truncated sine, of duration T =1 seconds, of unit amplitude, and frequency f0 =10 hertz. Two

echoes (in blue) come back with a delay of 3 and 5 seconds respectively, and have amplitudes

equal to 0.5 and 0.3.

If two targets are close by, the pulses from the two targets return with nearly the same delay.

The output of the matched filter is then equal to the sum of the two identical autocorrelations of

the transmitted signals. To distinguish the triangular envelope of one pulse from that of the other

pulse, it is obvious that the times of arrival of the two pulses must be separated at least by the

width of the autocorrelation function T, so that the maxima of both pulses can distinguished. If

this condition is not met, both triangles will be superimposed, making it impossible to distinguish

them. On the other hand if T is decreased to increase resolution, the average energy decreases

proportionally, resulting in erroneous detection.

In a chirped radar, the pulse is allowed to be of much longer duration. A longer pulse carries

more energy, and hence the detection probability increases. The wide pulse is linearly frequency

modulated mimicking the birds chirp (hence the name). When the chirped signal is passed

through a matched filter at the receiver, the pulse compresses to a narrow pulse whose width is

inversely proportional to the bandwidth of the chirp signal. The rate of chirp and the pulse width

determine the bandwidth. In most analog systems, a dispersive delay line like a SAW device, acts

as the matched filter as it has an impulse corresponding to the matched filter required for the

chirp signal. The chirp signal p(t) is given by:

() = 2[(0

2)+

2

2] if 0

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Figure 1.3: Transmitted signal in red (carrier 10 hertz, amplitude 1, duration 1 second) and two echoes (in blue). From

left, first image depicts the return signal, second image depicts the signal after simple processing without chirp, third

image depicts the result after chirp processing. Top set of images show that the echoes are separable if they are far apart

using either kind of processing. Bottom set shows that if the return echoes close they cannot be separated with

rectangular pulse, but they can be, with chirp processing.

Figure 1.4: Transmitted signal in red (carrier 10 hertz, amplitude 1, duration 1 second) and two echoes (in

blue); noise constitutes most of the return signal. This result shows the power of matched filtering.

1.6 Summary

The main objective of this project is to develop a system, which images the terrain in front of

the vehicle to help the driver in piloting the vehicle safely. We started our work with one-

dimensional implementation of radar range finding using chirp waveforms and matched filtering

[2][3][4][5][8]. After simulating chirp signal processing, which was speculated to be useful for

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future simulations also, we moved on to implementation of conventional side-looking SAR

[2][5][8][20][21][22], with strip-map imaging mode to analyze and explore its capabilities

[6][23][24][25][26]. This was analyzed as a prelude to forward-looking SAR (FSAR)

[1][14][15][27]. The results produced by side-looking SAR set standards for our future

approaches [28][29][31] because in case of cross-range (azimuth) resolution, forward imaging is

inferior to side-looking imaging because of lack of Doppler gradient in the return signals [5][31].

We analyzed the limitations of FSAR [10][11][12] by means of simulation in Chapter-5. To

overcome ambiguities of FSAR, a multiple antenna based SAR configuration was explored in

Chapter-4.

A binary detection and estimation algorithm for discrete targets [16][17] in front of the vehicle

was developed and termed as Azimuth Triangulation Method (ATM) in Chapter-4. Movement of

platform, though utilized to achieve finer resolution and final image formation, it is not analogous

to SAR, as the spatial chirp that results due the movement of the antenna is not utilized [31][34].

The ATM approach was communicated to SIP-2013, A conference for signal and image

processing and was accepted for publication in the proceedings.

The well-known Range-Doppler algorithm (RDA) [7][13][14][22] in conjunction with Doppler

beam sharpening (DBS) [6][9][22] for side-looking SAR was simulated for the purpose of

comparison with our radar models. Following system configurations were also studied and

analyzed to improve low azimuth resolution in case of forward imaging; 1) Radar platform

movement in azimuth direction [15][31]; 2) Chirp aperture illumination function based spatial

processing to achieve higher azimuth resolution [6][11][35]; 3) Forward looking Linear phased

array radar [31][36][37].

A FSAR system with signal processing similar to SAR [5][8][10][15] was simulated in

Chapter-5 and the results were compared with the results from RDA algorithm. A paper based on

this study was communicated to APSAR-2013, 4th conference on Synthetic Aperture Radar and

its use in disaster management. The paper was accepted for publication in the Proceedings.

In Chapter-6, we configure FSAR in squint-mode [34][38][39] to study the trade-off between

side-looking SAR and FSAR, as the former is unable to provide imaging in advance of time

making it unsuitable for forward imaging and the latter lacks high-resolution capabilities

[34][39]. The squint-mode FSAR was found to deliver higher resolution compared to other

approaches that were studied. Results are discussed in later chapters.

1.7 Contribution

Azimuth Triangulation Method (ATM) algorithm has been developed and implemented.

Its analysis with discrete point targets and resolution capabilities have been analyzed and

presented.

An FSAR approach to ground based forward imaging problem has been proposed and

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simulated. Simulation of RDA algorithm has been effected. RDA is suited for side-

looking SAR and has been widely utilized to perform the imaging in this mode. This

gives us a point of reference to measure the performance of our approaches in

comparison with the conventional side-looking SAR.

Squint FSAR (SFSAR) configuration is studied and implemented by the means of

computer simulation. Its resolution capabilities are theoretically analyzed and also

compared with FSAR and other previously implemented approaches.

All simulation work (ATM, RDA-SAR, FSAR and SFSAR) has been developed using

MATLAB.

1.8 Thesis Outline and Organization

Chapter-2 formulates the mathematical foundation of wave propagation in space and wave

scattering by targets. Based on equations obtained for propagation, scattering and beam formation

in Chapter-2, a signal model is derived for multiple antenna based SAR in Chapter-3. Signal

processing methodology is also formulated. Chapter-4 describes the ATM approach developed as

a part of this work, for detection and estimation of discrete point targets. Chapter-5 presents a

FSAR approach to the forward imaging problem. Chapter-6 explores a side-looking SAR

operating in squint mode, termed as SFSAR, to achieve higher resolution in azimuth direction.

Chapter-7 presents simulation methodology and optimization techniques. Conclusions are drawn

in Chapter-8 and scope for future studies on the topic has been discussed.

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Chapter 2

Synthetic Aperture Radar: Basics

Synthetic-Aperture Radar (SAR) is a form of radar whose defining characteristic is its use of

relative motion between the antenna and the target, to provide a spatial-chirp in the received

signal, which is exploited to obtain finer spatial resolution than is possible with conventional

beam-scanning methods. In a typical SAR, a radar antenna is attached to an aircraft or spacecraft

so as to radiate a beam whose wave-propagation direction has a substantial component

perpendicular to the flight-path direction. The wave component perpendicular to the flight

direction is responsible for the differential Doppler in the echoes that leads to the enhanced

azimuth resolution. The antenna beam illuminates a target scene with modulated pulses at carrier

frequencies in GHz region. The reflected waveforms received by the antenna as it moves with the

platform are coherently detected, stored and then post-processed to resolve spatial elements to

form a high-resolution image of the target region.

Current airborne systems provide resolutions to about 10 cm, ultra-wideband systems provide

resolutions of a few millimeters, and experimental terahertz SAR has provided sub-millimeter

resolution in the laboratory [40][41][42].

2.1 SAR Equation: Incident field on the target

In the conventional strip-mode Synthetic Aperture Radar (SAR) imaging shown in Fig 2.1, an

airplane or satellite flies along a straight track. The antenna emits pulses of electromagnetic

radiation in the beam, directed perpendicular to the flight track. These waves scatter off the

terrain, and the scattered waves are received by the same antenna. The received signals are then

used to produce an image of the terrain.

http://en.wikipedia.org/wiki/Radar

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Figure 2.1: Geometry of a SAR system

In this section, closely following [5], we present the equations describing the far-field from the

transmitting antenna to the target, scattered field from the target and the propagated field from

the target back to the antenna. We also derive the signal after performing matched filtering on the

received signal and analyze the resolutions in range and azimuth achievable in SAR.

The coordinate system is shown in Fig 2. A point x (Note - All vectors are represented using

bold notation.) in space is defined by ( x1 , x2 , x3 ). The vehicle is moving along x2 with a uniform

velocity v. The ground is situated at x3 = 0. All the components of electric and magnetic fields

satisfy the wave equation. Representing any of the components of field as U(t , x) at time t and at

a point x, U satisfies:

2 =1

2

2

2 (2.1)

The Greens function G0(t , x y) appropriate to equation 2.1 is given by::

0( , ) =(||

0)

4|| (2.2)

which satisfies the equation

20( , ) 1

2

20(,)

2 = ( )( ) (2.3)

Clearly 0( , ) is the field at point x due to an impulse at time located at y.

We assume that the signal transmitted by the antenna is

() = ()0 (2.4)

where the frequency 0/2 is the carrier frequency and A(t) is a slow varying amplitude that is

allowed to be complex. As the medium is linear and time invariant, by superposition the field

(, ) at time t and at some point z is given by

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(, ) = (

||

)

(4||)() =

(||

)

4||

= (

| |

)

(4| |)

0(||

) (2.5)

The antenna, however, is not a point source. Most conventional SAR antennas are either slotted

waveguides or micro-strip antennas, and in either case, a good mathematical model is a

rectangular distribution of point sources. We denote the length and width of the antenna by L and

D, respectively. We denote the centre of the antenna by x; thus a point on the antenna can be

written y = x + q, where q is a vector from the centre of the antenna to any point y on the antenna.

We also introduce coordinates on the antenna: q = 1 + 2 , where and are unit

vectors along the width and length of the antenna respectively. The vector points along

direction of flight; for the straight flight track shown in Figure 2.1, this would be the x2 axis. For

side-looking systems as shown in Figure 2.1, is tilted with respect to the x1 axis.

We consider points z that are far from the antenna; for such points, for which |q| q and that A is assumed to be slowly varying to

write

(, ) = (

| | )

4| |. (2.8)

Note that this is the field at points z far away from the antenna due to a point source at y = x + q.

To get the field at a far point due to the whole antenna, we will need to integrate this field over all

possible point sources within the antenna dimensions, which is assumed to be rectangular with

length, L and width, D.

Thus, far from the antenna, the field from the whole antenna is

23

(, ) = +1+ 2 (, )12

/2

/2

/2

/2

~ (

| |0

)

4| | 1 . 1

/2

/2

2 . 2

/2

/2

~ (

| |0

)

4| |() (2.9)

where

w() = 2Dsinc(ke1D/2).2Lsinc(e2L/2) (2.10)

is the antenna beam pattern and where sinc(x) = sin(x)/x. The sinc function has its main peak at x

= 0 and its first zero at x = ; this value of x gives half the width of the main peak. Thus the main

beam of the antenna is directed perpendicular to the antenna. The first zero of sinc(ke1D/2)

occurs when ke1D/2 = . Using the fact that 2/k is precisely the wavelength , we can write

this as . = /D. To understand this condition, we write . = cos(/2) = sin , an

approximation that is valid for small angles . Thus when

24

polarized and the scattering is linear and isotropic. Thus the received signal from a point scatterer

at z, while the antenna is at location , is

(, ) = (

| |

||

)

4| |

()

4|| () () (2.11)

where V(z) denotes the scattering coefficient of the point scatterer at z. We can now write the

complete received signal at time t by integrating over all possible target points, as

() = (

| |

||

)

4| |

()

4|| () () (2.12)

In case of SAR, the antenna emits a series of fields of the form (2.11) as it moves along its

trajectory. In particular, we assume that the antenna is located at position at time = nT,

when it emits a field of the form (2.11) and is at locations at time of reception. Then the

received signal for nth transmission can be written as

() = (

| |

| |

)

4||

()

4| | ( ) () (2.13)

We note that | |

+

| |

is the two-way travel time from the center of the antenna to the

point z and then from z to the center of the antenna. The antenna beam pattern illuminates a patch

on the ground. The integral is over the patch illuminated by the antenna beam. The terms

| |, | | in the denominator can be approximated by 0, in the summation, where 0 is

the nominal distance between the position of the antenna and centre of the scattering patch.

2.3 SAR Equation: Image Formation

To form an image, first, a matched filter is applied to the received signal Sn(t). Approximating

the denominator terms, each by 0 , as stated above, and performing the matched filtering

operation on each of the received signals (), the output of the matched filter () is given by:

() = (

| |

0

| |

0) () (2.14)

where * denotes complex conjugate. This is called a matched filter because one matches the

received signal against a signal proportional to that due to a point scatterer at position y i.e.,

V(z) = (zy). This is equivalent to matching the return signal with a sample signal, which

assumes that the scatterer at y is a point scatterer with unit reflectivity. A matched filter is used

because it is the optimal linear filter in the sense of providing the best signal-to-noise ratio.

Substituting () in (2.14) and changing order of integration:

25

() = (, )()

(40)2

(2.15)

where

(, ) = ( ) () (

| |

| |

) (

| |

| |

) (2.16)

represents the point spread function of this single-look imaging system: if V(z)=(zz0), then In(y)

= Wn(y, z0) would be proportional to the resulting image of V. Our goal is to make this point

spread function as close to a delta function as possible. The key idea of SAR is that this point

spread function can be improved by summing over n, i.e., by combining information from

multiple looks. Thus, we consider

() = () = (, )()

(40)2

(2.17)

with the point spread function

(, ) = ( , ) (, ) (2.18)

The point-spread function (, ) is called the generalized ambiguity function of the SAR

system. The weighting function (, ) determines if the target is in the beam emitted at . It

has a value equal to 1 if target is in the beam emitted at otherwise it has a value of 0.

Implicitly it determines the limit of n.

In the analysis we assumed that the pulse is emitted at position at time and received at

position as the vehicle moved during the two way travel time of the pulse. If the target

distance and or the vehicle speed are small, the two positions and nearly coincide and this

approximation is called the start-stop approximation. Under this approximation, the point spread

function (, ), is given by

(, ) = ( ) ( ) (

| |

) ( 2

| |

) (2.19)

Assuming that the complex transmitted signal () = ()0, with A(t) the slowly varying

complex envelope and using t for t-nT

(, ) = ( , )

20

( | || | ( ) (( )

( | |

) ( 2

| |

) (2.20)

The function (, ) can be written as

(, ) = (, ) (, )

26

with

( , ) = (

| |

) ( 2

| |

) (2.21)

and

( , ) = (, )

20

( | || | ( ) (( ) (2.22)

The two functions in (2.21) and (2.22) decide the resolutions in range and azimuth respectively,

the former depending on A(t) and the later on geometry of SAR and the antenna radiation

pattern. (, ) is the autocorrelation function of A(t). In this work A(t) is simulated to be a

chirp waveform, most common in SARs, its autocorrelation has already been discussed in section

1.5. (, ) will be analyzed to obtain the resolution in azimuth later in Chapters 5 and 6 for

specific geometries of SAR.

27

Chapter 3

MIMO Synthetic Aperture Radar

With the advent of MIMO communication, the MIMO concept has been extended to Radar

and to SAR. In MIMO SAR, the vehicle carries many antennas which are used for transmission

and reception. In the set of antennas available, some or all can be used as transmitting antennas

putting out different signal on each antenna. Similarly, some or all antennas can be used for

reception of some or all the transmitted signals. The physical placement of antennas along with

the signals they transmit can provide beam forming capability in transmission and in receiving

mode, which can, along with a combiner, provide direction of arrival (DOA) capability to the

radar. MIMO SAR is a very versatile and flexible system, which, in principle, can be structured to

gain performance advantages over conventional phased array radars and other radar

configurations. In this Chapter we develop a generic model for a MIMO radar based SAR, which

is used in approaches discussed in Chapters 4 through 6.

3.1 Signal Model for MIMO-SAR

In MIMO SAR, the vehicle carries a linear array of 2N+1 antennas symmetric to the flight axis

N on either side with the central one on the axis spaced d metres apart with the physical length of

2N.d metres. The array is perpendicular to the direction of flight. However, in principle, it can be

a two dimensional array, unequal spacing and non identical array elements.

All the signals {Pi(t)} transmitted by the antenna elements i = -N to +N may be assumed to

have the same carrier frequency . If there is a difference in the carrier frequency it can be

absorbed in the complex envelope Ai(t).

Let the target area on the ground be divided into resolution cells denoted by their azimuth ,

and elevation , angles subtended to the reference antenna which is the antenna element at the

centre of the antenna array. The signal Pi(t) from the ith antenna element incident upon the (,)

patch on the ground is given by:

28

(. 0(, ) (, )) = ( 0(, ) (, ))0(0(,)(,)) (3.1)

where 0(,) = One-way delay from the reference antenna i = 0 to (,) patch on the ground.

i(,) = One-way delay difference between the reference antenna to (,) resolution cell

and ith antenna element to the same resolution cell =

The signal reflected from the patch can be written as

(, ). . . ( 0(, ) (, ))0(0(,)(,)) (3.2)

where V(,) is the Reflectivity of the (,) resolution cell or patch

The received signal at jth receiver from (,) patch becomes

,(, , ) = (, ). . . ( 2. 0(, ) (, ) (, )) (3.3)

This composite signal received at jth element from (,) patch due to all antennas is given by

(, , ) = (, ). . . ( 2. 0(, ) (, ) (, )) (3.4)

=

After demodulating the signal to baseband and taking into account the slow variation of Ai(t) as

compared to the fast varying 0, we can write the above equation as

(, , ) = (, ). . . 200(,) ( 20(, ))

=

0((,)(,)) (3.5)

Let (,) = [0(,), 00(,). . .0(,) ]T be a (2N+1)x1 column vector and

A(t-20 (,)) be a (2N+1)x1 column vector with entries Ai(t-20 (,))

Then

(, , ) = (, ). . . 200(,)0(,)(, )( 20(, )) (3.6)

The composition of received signal for all 2N+1 receive antennas is a (2N+1)x1 vector given by

(, , ) = {(, ). . . 200(,)(, ). (, )( 2. 0(, ))} (3.7)

The term 00(,) can be absorbed into V(,) and the received signal from the whole

29

illuminated area can be written as

() = (, ). (, , )

(3.8)

Where S(t) = [S-N(t), .. S0(t), ... SN(t)]T and F(,) determines the range of the summation. F(,)

is assumed to be:

(, ) = 1 if (,) patch is in the beam illumination area

= 0 Otherwise.

The antenna emits and receives a series of pulses as it moves along the rail track. Let the

subscript n denotes the occurrence of an even at time t = nT. We determine the range of

summation F(,) mentioned in (3.8). Let L be the size of the antenna, which emits a narrow

beam with the angular width of approximately 2/L; Then, one can see that the target z =(z1, z2,

z3) is in the beam as long as the target point falls within the elliptical beam region on the ground

as beam transverses it, hence

(, ) = 1; if 1 0

1

1 +0

and

(, ) = 0; if 1 < 1

0

1

> 1 +0

(3.9)

where 1 is the azimuth coordinate of centre of the beam at nth pulse.

3.2 Coherent summation of signals

Applying matched filtering to the received signal after each pulse results in an intermediate

image during real-time processing. These intermediate images are aligned coherently in spatial

domain, according to the beam illumination region and beams position with respect to the

antenna platform, and then fused together to form final target image. Note that in real time this

process is done after each pulse to obtain the final image of the strip that the beam has surpassed.

A major overhead of this approach is computational load. Matched filtering is a computation

intensive process. As matched filtering is a linear process, instead of matched filtering after every

pulse we can coherently add the returns for all n pulses and apply matched filtering only once,

without affecting the final result. This approach gives us clear processing advantage over

Zaharris approach [45]. Processing time comparison is done under Chapter-7.

The target points that leave the nth beam illumination can be given by the (, ) patches for

which Fn(,) Fn+1(,) = 1. This means, these target points were present in the nth beam but

will not be present in the (n+1)th beam due to antenna movement. The received signal at jth

antenna at time t = nT is given by

30

,() = (, ). (, , )

And the image of the scatterer at (,) is given by matched filtering as

(, ) = (

2,0(, )

) ,() (3.10)

The MIMO SAR described here is a very general form. The SAR formulations discussed in this

Chapter will be used in later chapters for specific SAR geometries.

31

Chapter 4

Azimuth Triangulation Method for Forward Imaging

The work done under this thesis is focused on developing an imaging system to guide a vehicle

on its path, at times when human vision proves to be inadequate and unreliable. Our paper

Azimuth Triangulation Method for forward looking Synthetic Aperture Radars based on

forward looking mode of SARs has been accepted in IASTED-SIP-2013. In this Chapter, we

describe the work done under that publication, in coherence with the signal model developed in

Chapter-3.

Recently, the German Aerospace Center (DLR) has developed an innovative radar system,

SIREV (sector imaging radar for enhanced imaging) [4], having the potential to supply high

quality radar images of a sector in front of the aircraft. Same as side-looking SAR, the forward

looking imaging radar also utilizes the concept of azimuth sub-aperture, which, in this case is

spanned by a physically existent linear array of single elements, and not due to the motion of

antenna along the azimuth direction. This gives N-fold resolution advantage over a single antenna

configuration, similar to phased array radars. We intend to use the configuration discussed in

SIREV and simulate a forward-looking radar system and image a sector in front of the vehicle.

Note that this approach is not based on synthetic aperture formation in azimuth direction like

conventional side-looking SARs.

Our paper, Azimuth Triangulation Method for forward looking Synthetic Aperture Radars,

presents a novel approach to mitigate the problem of low azimuth resolution in forward imaging

systems and also solves the ambiguity problem [10][11] by the use of image superposition

principle in case of multiple antenna SAR. Here, we study a linear array radar configuration with

2M+1 antenna elements for signal transmission and reception to triangulate targets azimuth

location after its range has been determined by range compression of the return signal. This

method is named Azimuth Triangulation Method (ATM). ATM also solves the left-right

ambiguity problem, which is a major issue in conventional forward-looking imaging systems, by

utilizing the geometry relationship between multiple receiver antennas and targets, and amongst

32

antennas themselves. Following sections in this Chapter describe the proposed architecture,

processing algorithm and analyses the resolution capabilities of this approach in detail.

4.1 Proposed Architecture

The configuration of radar system studied here is shown in Fig.4.1. Top left part of the figure

shows beam footprint. The geometric shape of the beam is simulated to be elliptical in case of a

conical beam, which is obtained by using a 2-D antenna array with phase shifted signals on

different elements [34]. Range is the direction along the flight path (y-axis) and azimuth is the

direction perpendicular to the flight path (x-axis). We have 2M+1 antenna elements where E0 acts

as the reference antenna, analogous to previously formed signal model. The spacing between

consecutive antenna elements is considered to be uniform, however, the model and simulations

can be easily extended to include variable spacing between antennas.

Total length of the radar platform is L = |xM-x-M|. The velocity of radar platform is v, and y-

coordinate of the platform is defined by y = v.t at any time t; assuming simulation start time as

t=0. Transmitter position at any time t is, (xT =0, yT =v.t, zT =h). Height and velocity of the

platform is assumed to be constant, however, it is straightforward to accommodate their variations

into analysis.

Figure 4.1: Geometric configuration of the proposed radar system (Top-view). (Inset depicts system alignment with

reference to coordinate system.)

In fig.4.1, point P is any stationary point-target located at distance d from transmitter footprint

(footprint of E0) and is the angle subtended by the line joining the target and the transmitter

footprint with the positive y-axis at time t. The coordinates of P are x = d.sin, y = yT +d.cos and

z = 0. Note that the problem addressed here is of ground imaging, hence the z-coordinate of any

target point is 0 but d and are functions of slow-time, n. In SAR terminology, fast-time refers to

the instantaneous time and slow-time refers to an instance when a pulse is transmitted [4][11].

The transmitter illuminates the region ahead of it with an up-chirped pulse signal and targets

present in the illuminated region scatter the signal and it is received by 2M+1 receiver after

different time delays. The radar is assumed to be stationary during this process, which is a

reasonable assumption [1][33] and is called start-stop approximation in SAR terminology.

33

4.2. Target and Signal Model

Target: This approach simulates a simplistic version of the forward imaging problem using

linear array of radars antennas and assuming the target area to be constituted of discrete point

targets in the presence of ground clutter and the channel to be AWGN. At this point we refrain

from modeling multi-path, terrain blocking and interference, which are concluded to be less

important in railways problem. Hence, only additive white gaussian noise and background clutter

is compensated.

Signal: Transmitted signal model remains the same as developed in Chapter-3. Return signal is

processed using a matched filtering in intermediate frequency domain. Note that differential

Doppler information in return signal from the target is not utilized to enhance the azimuth

resolution, this will be discussed in detail in Chapter 5 and 6. Return signals after each pulse are

imaged independently and are mapped onto a fine-dimension grid.

Following the generic signal model developed in Chapter-3 for a linear array radar, we derive

the signal model and processing for this approach. From equation 3.4 we see that the signal

received by jth antenna element after being reflected by (, ) patch is

(, , ) = (, ). . . ( 2. 0(, ) (, ) (, ))

=0

which can be written in vectored form as

(, , ) = (, ). . . 200(,)0(,)(, )( 2. (, ))

We ignore the Doppler delays from the high frequency domain (carrier) and demodulate the

signal down to baseband

(, , ) = (, ). . . ( 2. (, )) (4.1)

The complete signal received at jth antenna from the target field is given by summing over all (,

) patches that fall in the beam illumination region as

() = (, ). (, , )

(4.2)

where the range of summation is governed by (, ), which is derived in equation 3.9. This

signal is then dechirped to construct a range imaging of the target field, mentioned in equation

3.10, as

34

,(, ) = ( 2(,)

0)

() (4.3)

where (,) denotes the distance of (, ) patch from the jth receiver. Using transmitter position

described in section 4.1, we can write

(,) = 2 + (. . )2 + (. )

2 (4.4)

Substituting (,) in equation 4.3, we have

,(, ) = ( 2 + (. . )2 + (. )

2

0)

() (4.5)

The complete image for jth antenna at slow-time n is constructed by conjoining all the images

obtained by varying (, ) in the summation range.

, = ,(, )

(4.6)

Images from all 2N+1 antennas are superimposed to obtain aggregate images for each n, which

are again summed over the slow-time n, as:

= ,

=

=0 (4.7)

4.3 Algorithm

This approach uses a novel method developed by us to triangulate the azimuth location of a

target point after having identified its range using range domain chirp processing [4]. This

method is termed as Azimuth Triangulation Method (ATM). In ATM, azimuth resolution is

improved using three main steps: 1) Triangulating the target by taking advantage of geometry of

multiple antenna elements and their relative distance from the target. 2) Populating the range bins

with target indication for each receiver antenna. 3) Superposition of images from several antennas

for one instance of slow-time and finally summation of all the images over slow-time to obtain

the final image.

This approach takes advantage of high range resolution obtainable using chirp signals. We

exploit the fact that even a slight difference (few cms) in distance between target and different

antenna elements will result in target falling into different range bins for different antenna

elements. Arguably, more number of antenna elements will yield more geometrical information

which will aid in further improving the azimuth resolution. This conjecture is tested and verified

by the means of simulation, presented at the end of this Chapter.

35

For visualization purposes, we assume that the beam illumination region on the ground is a

circular patch (in practice, the beam shape on ground tends to be elliptical), which, when

differentiated into range bins for different antennas, looks like the image below.

(a) (b) (c)

Figure 4.2: Range bin representation for a 3-antenna configuration. (a), (b) and (c) represent range bins for R1, R0 and

R-1 respectively [Refer Figure 4.1]. Figure at the bottom represents the crossover of the range bins when images from

different antennas are overlapped, thus forming finer resolution cells. This is a pictorial representation of the signal

processing involved.

These bins are populated with target reflection coefficient, obtained from equation 4.5, for each

antenna separately. Thereafter, images from individual antennas are superimposed to get an

intermediate image, which is then summed up over slow-time and aligned spatially to account for

the movement of the platform along y-axis, to produce the final image as described by the

summation in equation 4.7. One significant advantage of ATM algorithm is that the resolution is

marginally better at the centre of the beam than the area near the beam edges, which is crucial to

ascertain smooth guidance of the vehicle on its trajectory and may prove to be advantageous in

applications like railways, where the main intent is to image the rail-track with high resolution,

paying little heed to surroundings areas.

4.4 Simulation Parameters

The basic parameters given here are used in all the computer simulations that have been

performed as a part of various configurations studies in this thesis. However, some case-specific

parameters have been given along with the specific example, to which they refer.

System Parameters Value used in the simulation

Antenna length and width 1 m

Near Range 500 m

Velocity of the platform 30 m/s

Flight duration 3 s

Signal Parameters Value used in the simulation

Carrier Frequency 5.33 GHz

Data Sampling Rate 19.27 MHz

36

Pulse repetition frequency (PRF) 500 per second

Chirp Bandwidth 1 GHz

Chirp Rate 10 MHz/s

Chirp Duration (Tp) 100 s

Table 4.1: Simulation parameters

4.5 Results

We study the case with 4 spread stationary point targets located arbitrarily in x-y plane, each

having unit reflectivity. This experiment is performed to study the behavior of resolution

capabilities of ATM approach as number of antennas are varied. Two simulations are presented

here, with N = 2 and N = 4. The details of other parameters is listed in Table 4.1.

Figure 4.3: Result of simulation with five antennas (N=2), performed on the input image given on the left.

The output obtained is shown on the right side

As evident, this approach is prone to detect false positives (several dark patches in the output

image, which are not present in the input) on the places where multiple antennas give detection,

but not all. However this is the case only when the number of antennas is very less. In figure 4.3,

the value of N is 2 (2N+1 = 5 antennas), however, with N = 4 (2N+1 = 9 antennas), we get the

following result.

Figure 4.4: Results of Simulation performed with N=4. Input image is on the left, ouput is on the right.

37

4.6 Resolution

To study the resolution of this system, we simulate a single target located at any angular

coordinate (, ) with respect to the reference antenna footprint and measure the dimensions of

target in the resulting image. From (4.1) we have

(, , ) = (, ). . . ( 2. (, ))

For 2N+1 antennas, the summation is done as

(, , ) = (, ). . . ( 2. (, ))= (4.8)

To get the final image, we correlate this with the transmitted signal and then, to measure the

resolution, we measure the span of non-zero values in both x and y direction.

(, ) = ( 2(,)

0)

(, , ) (4.9)

Figure 4.5: Resolution study image. Figure shows a point target on the left simulated and resulted on the

right. N=2, top image. N=4, bottom image.

The dimensions of the input image - Azimuth: 1m, Range: 2m.

Number of Antenna

Elements

Range dimensions

observed (m)

Azimuth dimensions observed

(m)

Range

Resolution(m)

Azimuth

Resolution (m)

5 2.05 19.82 0.05 19.82

9 2.03 11.87 0.03 11.87

13 2.03 9.95 0.03 9.95

Table 4.2: Resolution result of ATM

38

As evident from Table 4.2, an increase in the number of antennas increases the azimuth

resolution, as is expected from a phased array radar, but to be physically feasible the number of

antennas in the radar system cant be extended beyond a certain limit, especially if the radar

system has to be mounted on a moving platform. Hence, in Chapters 5 and 6 we explore synthetic

aperture radar to tackle the problem of forward imaging since, SARs can accomplish high

resolution imaging with much smaller antennas.

39

Chapter 5

Azimuth Resolution Enhancement using FSAR

Forward Looking SAR (FSAR) imaging radar utilizes the motion of the radar platform to

generate the required Doppler information to resolve targets in azimuth, however as we shall see

in section 5.2, the gradient of Doppler is not sufficient to obtain the desirable azimuth resolution

unlike the side-looking SAR.

Figure 5.1: Proposed radar system with 2N+1 antenna elements; N=2.

5.1 Problems with FSAR

There are two major problems with the imaging mode FSAR. The first one is that the gradient

of the Doppler frequency is very small which makes it impossible to achieve high azimuth

resolution. The second one is that the terrain targets situated symmetrically about the flight path

40

have same Doppler history, which causes ambiguity in distinction of the targets in azimuth

direction. The configuration of FSAR with multiple receiving antennas as in Fig 5.1 solves the

left-right ambiguity problem by fusing the results from multiple antennas, but high azimuth

resolution is a little more difficult to achieve. Traditionally proposed methods to solve the

azimuth resolution problem are Doppler beam sharpening (DBS) [6][9][22] and phased array

antennas [5]. In DBS mode, a sophisticated antenna illumination function is designed, which in

turn provides a sharper/narrower beam and hence the resolution. To obtain high azimuth

resolution, a phased array antenna requires a large number of sensor elements. However, in both

cases, resolution obtained is not satisfactory for the problem of forward imaging.

In case of side looking SAR, the most common approach is to correlate the azimuth signal from

a range gate with corresponding azimuth reference signals to achieve pulse compression in

Doppler-time (f-t) domain. The Doppler history signals in FSAR, though similar to those of SAR,

lack in the gradient of Doppler frequency, which is much smaller as compared to that of side-

looking SAR, leading to lower resolution. Here, we simulate a FSAR based on processing similar

to side looking SAR and study its performance, for the purpose of comparison with ATM

approach discussed in Chapter-4 and squint beam mode SAR, which is discussed in Chapter-6.

5.2 Target and Signal Model

Targets are considered to be stationary, in presence of clutter. The problem is viewed as an

imaging problem rather than a detection and estimation problem, hence, everything, including

ground is considered to be a possible target point. Signal model remains the same as formulated

in Chapter-3 and processing remains the same as described in Chapter-2. With these assumptions,

an analysis of the proposed FSAR system is presented below.

In figure 5.1 the distance between any antenna Ek, to the reference antenna E0 is |lk|, v is the

velocity of radar platform, and y = vt is the radars y-coordinate at a time t. The point (x0, r0) is

any target at azimuth position x0 in the illumination area of the beam. In this approach, as single

transmitter and multiple receivers are assumed. The transmitter is located at (x, y) = (0, 0) at t =

0. The two-way distance travelled by the signal after being transmitted from E0, hitting the target

and then being received at Ek is expressed as,

(, 0, 0) = (0 )2 + 0

2 + (0 )2 + (0 + )

2 (5.1)

Using Taylor series, we get,

(, 0, 0) = (0 + ) (00

+0

) . +1

2(

1

0+

1

02

03

02

3) . ()

2

+1

2(

1

03 +

1

3

03

05

03

5) . ()

3 + (5.2)

41

Where 0 = 02 + 0

2 and = 02 + (0 + )

2 ,

The Doppler frequency fDK(t,x0,r0) of the target to Ek is

(, 0, 0) = (, 0, 0)/

(, 0, 0) =1

[ (

00

+0

) . + (1

0+

1

02

03

02

3) .

2

+3

2(

1

03 +

1

3

03

05

03

5) .

32 + (5.3)

In the above equation, the slant range (Doppler or phase) histories of targets are dependent on y-

coordinate of the antenna platform, the reference range r0, the coordinate x0 of the target, and lk. It

shows that the Doppler frequency of a target is very large (the linear term of v) while the gradient

of the Doppler frequency (the 2nd and higher terms of v) is relatively small. Therefore, it is

impossible to get high azimuth resolution for FSAR by performing a similar azimuth compression

method like in side-looking SAR. Nonetheless, we compute the resolution capabilities of such

radar and simulate it for the purpose of analysis and comparison.

5.3 Resolution

The resolution of FSAR system can be determined by an investigation of the generalized

ambiguity function (2.20). Under the assumption (3.8) we write W as

(, ) = (, )

(, ) = ( )

( 2| |

0)

(

2| |

0)

(, ) = ( )

( 2| |

0)

0(2||

0)

( 2| |

0)

0(2||

0)

(5.4)

In (5.4), we make the change of variables tnT t, and then use the fact that A is slowly

varying. In this approximation, the As no longer depend on n and can be pulled out of the sum.

The time dependence in the exponentials cancels out, and thus the ambiguity function factors as

(, ) = (, )(, ) (5.5)

42

where

(, ) = ( 2| |

0)

(

2| |

0) (5.6)

And

(, ) = ( )0(

2||0

)

0(2||

0)

(, ) = ( )2(||||)

(5.7)

5.3.1 Azimuthal resolution

To study the azimuthal resolution, we consider two points y and z at the same range, i.e., at the

same distance 0 from the flight track. In particular, we write y = (r, 0, 0) and z = (r, z2, 0). We

assume that the range 0 is much greater than the distances along the flight path, in the course of

nth beam, so that | 2|

43

The range of n values in this sum is effectively determined by the width of the antenna beam

pattern ( ). To determine the limits, we have to discuss the beam dimensions and its

impact on the ground, which is at a distance R0. We assume, for the purpose of resolution analysis

that the 3dB beam shape is important and within the shape, the beam intensity is assumed to be

constant. The above expression controls the azimuth resolution of the system, which is majorly

restricted to the linear phased array capabilities.

5.3.2 Range resolution

The transmitted waveform P(t) must be chosen to get the best range resolution by proper choice

of A(t). Consider the problem of resolving of two point scatterers whose positions differ by a

range r. To get perfect resolution, we would want to use a Dirac delta function pulse; then the

time separation of the return pulses would be t = 2r/c0. A system in which the time sampling is

finer than this would be able to determine that the two scatterers were located at different ranges.

Unfortunately, it is not practical to transmit an infinite peak power pulse such as the Dirac delta

function. Instead, if one uses a function that is zero outside a time window of length , then the

return waves dont overlap if < 2r/c0. Thus a waveform of length can resolve two scatterers

if their ranges differ by r > c0/2. Thus, to get good resolution, we would like to use a very short

wavetrain.

Unfortunately, because any transmitted field is necessarily limited in amplitude, short

wavetrains also have very low energy. This means that little energy is reflected from the target,

and noise will drown these low-energy scattered waves. Thus short pulses cannot be used. To

circumvent this difficulty, most radar systems, including SAR systems, use pulse compression, in

which one transmits a complex waveform and then compresses the received signal, generally by

matched filter processing, to synthesize the response from a short pulse. The most commonly

used modulated pulse is a chirp, which involves linear frequency modulation described

elaborately in Section 1.5 upon following [32].

Here, in this approach, we make () = /2()(()) in which the instantaneous frequency

given by ()

changes linearly with time. Here /2() denotes the function having unit value in

the time interval [/2, /2], and zero outside. In an up-chirp, the instantaneous frequency

increases linearly with time as d(t)/dt = 0 +2Bt/ , where 0/2 is the carrier frequency and B

is called the deviation. To determine (t), we simply integrate to obtain (t) = 0t + Bt2/ . Thus

() is a wavetrain of the form

() = /2()20 (5.13)

where = B/ is called the up-chirp rate. We note that such a pulse is of the form (2.4), where

() = 2

()2

44

We use (5.13) in (5.6) and use the shorthand notation Ry = |xy|, Rz = |xz|:

(, ) = /2 ( 2

0)

(20

)2

/2 ( 20

) (

20

)2

(5.14)

We expand the squares in the exponentials, and find that the terms associated with t2 cancel out

and we are left with

(, ) = 4(

22)/0

2 4()/0

/2+2max(Ry,Rz)/2

/2+2max(Ry,Rz)/2

(5.15)

The t integral here gives rise to a sinc function of the form

(4( )

20) = (

2( )

0) (5.16)

The main lobe of this sinc function has a half-width determined by setting the argument equal

to , which means 2( )/0 should be equal to . This shows that two scatterers can be

resolved if their range difference is = 0/2. In other words, better resolution is

obtained by using a chirp with large frequency deviation B, as expected.

5.4 Simulation Results

This section presents the simulation results obtained via the means of computer simulation

using MATLAB. The computer simulation parameters remain the same as described in table 4.1.

Case 1: We test the system against a set of discrete, discernible targets in the absence of noise

and observe their location and intensity in the result. Table 5.1 compares input and output

parameters.

Figure 5.2: Discrete point target simulation in the absence of noise

1

2

3

4

8

5

6

7

1

2

3 5

4

7

6

8

45

Input Target parameters

(z1, z2, reflectivity)

Observed Target parameters

(y1, y2, reflectivity)

P1(-28, 55, 2.0) P1(-26.5, 55.7, 1.68)

P2(-23, 12, 2.0) P2(-21.9, 12.1, 1.67)

P3(-17, 40, 1.2) P3(-17.6, 40.2, 1.14)

P4 (-5, 15, 1.8) P4 (-4.2, 15.4, 1.53)

P5 (2, 40, 0.6) P5 (2.9, 40.2, 0.42)

P6 (29, 18, 1.1) P6 (28.2, 18.1, 1.0)

P7 (30, 52, 1.9) P7 (29.7, 51.8, 1.74)

P8 (33, 8, 2.0) P8 (33, 8.1, 1.78)

Table 5.1: Accuracy analysis of FSAR in the absence of noise

We now move on to test the system against the same input test case but in presence of noise.

Case 2: We test the system against a set of discrete, discernible targets in the presence of noise

and observe their location and intensity in the result. Noise is assumed to be White Gaussian.

White noise is shown in spatial domain. Table 5.2 compares input and output parameters.

Figure 5.3: Discrete point target simulation in the presence of noise.

Input Target parameters

(z1, z2, reflectivity)

Observed Target parameters

(y1, y2, reflectivity )

P1(-28, 55, 2.0) P1(-26.5, 55.7, 1.75)

P2(-23, 12, 2.0) P2(-21.9, 12.1, 1.68)

P3(-17, 40, 1.2) P3(-17.6, 40.2, 1.13)

P4 (-5, 15, 1.8) P4 (-4.2, 15.4, 1.51)

P5 (2, 40, 0.6) P5 (2.9, 40.2, 0.41)

P6 (29, 18, 1.1) P6 (28.2, 18.1, 0.89)

P7 (30, 52, 1.9) P7 (29.7, 51.8, 1.44)

P8 (33, 8, 2.0) P8 (33, 8.1, 1.68)

Table 5.2: Accuracy analysis of FSAR in the presence of noise.

46

SNR Analysis:

We analyze the SNR advantage of matched filtering processing for the above case. SNR is

calculated by the means of measurement of extreme intensity values of signal and noise at the

input and output.

Max. Input Signal intensity: 2.0

Max. Input Noise intensity: 1.8

Max. Output Signal intensity: 1.89

Max. Output Noise intensity: 0.23

/ = (1.89/0.23)

(2.0/1.8)

Calculated SNR gain is / 7.8

Shown below are two more cases pertaining to target detection in the presence of noise. We

increase the noise power and observe the output.

Figure 5.4: Discrete point target simulation in the presence of noise. N0=0.7 (top) and N0=0.8 (bottom).

47

Case 3: We test the resolution capabilities, both in range and azimuth with the help of following

example. We place three targets, namely 1, 2 and 3. Target 2 and 3 are collocated in range

dimension, whilst 1 is distinguished from them in range.

Figure 5.5: Closely located discrete point target simulation to verify resolution capabilities.

Target 1 is separated from 2 and 3 in range by a gap of 3 pixels. Each pixel is calibrated to

account for 50cm x 50cm area on the ground. This means that range separation is 1.5m. Similarly,

azimuth separation is also kept to be 3 pixels = 1.5m. We observe that targets in range are

discriminated in the output, but same cannot be said for targets 2 and 3. We now increase the

distance between 2 and 3 and get the results.

Figure 5.6: Closely located discrete point target simulation to verify resolution capabilities. (Azimuth distance = 14m)

We keep the range separation same, but increase the azimuth separation to 28 pixels = 14m. We

can see that the target begin to separate but the separation is still not discernible. We increase the

distance to 36 pixels = 18m and simulate again.

48

Figure 5.7: Closely located discrete point target simulation to verify resolution capabilities. (Azimuth distance = 18m)

As clearly visible, the targets appear to be distinguished in azimuth at a resolution = 18m. This

is justifiable as we are using phased array radars with /2 dipoles. The number of antennas used is

10. So, we get a length of 10/2 = 5. Thereby making beam-width = 2/L = 2/5 = 2/5 radians.

At a distance R0 = 500m, this gives a length of around 180m. Now, we further use n separate

antennas to make an independent image of the target scene. This gives rise to a phased array

resolution factor approximately equal number of antenna elements, which is 10. Thereby making

the resolution 180/10 = 18m.

One important point to note here is that the azimuth resolution is not improved over the ATM

approach mentioned in Chapter-4. The reason being the lack of Doppler variation in return

signals in the azimuth domain, as discussed in section 5.2. In Chapter-6, we introduce changes to

the radars physical configuration to generate Doppler variation in azimuth direction.

Case 4: We present a real-world test cases to speculate the performance of squint SAR model

developed here.

Figure 5.8: Left shows a typical rail track target field that lies at a distance of 500m from the radar and is

100mx100m in dimensions as input image and the SAR reproduction of the image is shown on the right.

Here, we perform a simulation test on a rail-track image. As is clearly visible, the resolution

achievable by FSAR approach is not good enough to enable the pilot/controller to navigate the

49

vehicle. Therefore, in the next Chapter we discuss a new approach using squint beams and side

looking SAR to achieve better resolution in azimuth direction.

50

Chapter 6

Squint beam Synthetic Aperture Radar Model

SAR in any mode, whether forward looking or side looking, is basically an imaging system. In

the context of railway applications, there is a need for imaging as well as an early warning

regarding presence of an obstacle on the track. The conventional SAR cannot meet the dual

requirements. To circumvent this problem, we propose to divide the target field into two halves

along the azimuth direction, each half to be scanned by a separate SAR. We further propose to

squint the physical beams in a forward direction for each SAR, carry out the typical SAR

processing and fuse the results from both radar systems to obtain the final target image. The

proposed system is shown in figure 6.1.

Figure 6.1: SAR in squint mode.

51

6.1 Target Model and Signal Model

The target points are considered to be stationary in the presence of clutter. Noise signal is

modelled as White Gaussian noise signal with variance N0/2. Ground clutter constitutes a major

part of the signal and hence significant SNR advantage is required out of the radar (SNR analysis

is also done through the means of simulation in Section 6.4). Signal model remains the same as

formulated in Chapter-3 and processing remains the same as formulated in Chapter-2. Here, we

analyze of the proposed Squint FSAR (SFSAR) system.

The main difference between the proposed SFSAR system and conventional side-looking SAR

is the squinting of the beam and hence, appropriate changes in processing and resolution analysis

have been made and discussed later. The area on ground spanned by the beam now depends on

the squint angle. Figure 6.2 shows the detailed configuration.

Figure 6.2: SAR imaging schematic for the squinted beam strip-map mode with the antenna

pointing forward with beam squint angle being c

6.2. Azimuthal resolution

Analogous to section 5.3, we will derive the expressions for azimuth resolution for squint mode

of SAR. Hereafter, we will denote individual Cartesian coordinates by subscripts: x = (x1, x2, x3),

y = (y1, y2, y3), z = (z1, z2, z3). To analyze the azimuthal resolution we will assume that both the

target z and the reference point y are located on the Earths surface at the same perpendicular

range bin at a distance R0 from the orbit; see Figure 6.2. Let c denote the angle between the

velocity v of the satellite (positive x1 direction) and the direction of the antenna beam (i.e., its

z

R0

y

v

2

1

3

c

q

52

central line). If c < /2, then the antenna is pointing forward; if c > /2, then the antenna is

pointing backward; and the value = /2 corresponds to broadside imaging (Conventional SAR).

Let us denote the center of the antenna beam footprint on the Earths surface by q = (q1, q2, q3).

Clearly, we have q1 = x1 + R0cot(c); see Figure 6.2. Hereafter, we will assume with no loss of

generality that both the target z and the reference point y are close to the center of the footprint q

so that |q1 z1|

53

Let x1 be the distance along the orbit between the successive emissions of pulses so that 1 =

1. Let us also recall that 1 = 1

forward imaging using synthetic aperture...

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