stripmap sar mocomp2005

7/27/2019 Stripmap SAR Mocomp2005

1/105

Stripmap Mode Synthetic Aperture

Radar Imaging

with Motion Compensation

R O B E R T E R I K S S O N

Master's Degree Project

Stockholm, Sweden 2005

IR-SIP-EX-0507


2/105

Saab Bofors Dynamics AB 1 (104)Utfrdare, tjnstestlle, telefon Issued by, department, telephone Datum Date Utgva

Issue IDDokumentnummerDocument ID

Robert Eriksson 2005-06-28 1.0 PM366953Godknd avApproved by Antal bilagor

No. of appendicesInformationsklass Classification

PPENMikael Hmlinen 0

p16/allmnblankett.dot/utg5

MottagareAddressee(s)

Abstract

This Master thesis investigates two different SAR focusing methods. One methodapplies low-pass filtration and the other is based on FFT and is called the step-transform. The evaluated techniques are used for focusing of the raw-data collected

by airborne radars. The radar data is collected in stripmap mode and both thebroadside and the squint case is regarded. The two implemented methods areevaluated with different types of motion deviation, some deviations taken from realmeasurements. Variations in the crab angle have an effect in the step-transformwhereas the low-pass algorithm handles the variations well.Large variations in thecrab angle results in mainlobe broadening and increased sidelobe levels for the step-transform. The low-pass filtering algorithm could be adjusted to handle SAR data inhigh squint mode. The adjustment was a squint compensation to the line

perpendicular to the antenna pointing direction. The step-transform was not able tohandle SAR data in squint mode for aircraft application. This was concluded by acareful analysis of the algorithm and by applying simulated radar data.


3/105







Acknowledgement

This Master of Science Thesis was carried out between December 2004 and June2005 at SAAB Bofors Dynamics AB in Jrflla, Sweden.

I would like to thank my supervisors Mikael Hmlinien at Saab Bofors Dynamics,Jonas Lindblom at the Signals, Sensors and Systems (S3) department of KTH. I alsowould like to thank my examiner Bastiaan Kleijn at the Signals, Sensors andSystems (S3) department of KTH. Their guidance and support have been reallyhelpful, and without their advice this thesis would not have been the same.

I would also like to thank everybody from Saab Bofors Dynamics in Jrflla for

their kindness and hospitality.


4/105







Table of contents

1 Background .................................................................................... 51.1 This project ....................................................................................... 61.2 Previous work................................................................................... 7

2 SAR fundamentals...................................................................... 82.1 SAR modes ...................................................................................... 122.2 SAR antenna ................................................................................... 15

3 The simulation model............................................................. 173.1 Pulse compression........................................................................ 193.1.1 A mathematic description.......................................................... 20

4 Sidelobe suppression ............................................................. 23

5 Motion error compensation ................................................ 25

6 SAR signal processing ........................................................... 266.1 Broadside mode ............................................................................ 266.2 High squint mode ......................................................................... 276.3 Range-cell migration ................................................................... 276.3.1 A mathematical description of broadside mode ............... 28

6.3.2 A mathematical description of high squint mode ............ 30

7 Azimuth focusing ...................................................................... 367.1 Preprocessing of data.................................................................. 367.2 Theoretical resolution in azimuth direction ........................ 387.2.1 Real aperture radar...................................................................... 397.2.2 Synthetic aperture radar ........................................................... 397.3 The low-pass filtering algorithm ............................................. 417.4 The step-transform ...................................................................... 447.4.1 The step-transform in high squint mode............................. 57

8 Implementation......................................................................... 628.1 Implementing the filter algorithm.......................................... 628.2 Implementing the step-transform ......................................... 65

9 Simulation results .................................................................... 669.1 Simulations of straight flight path in broadside mode... 679.1.1 Low-pass filter algorithm........................................................... 679.1.2 Step-transform .............................................................................. 69


5/105







9.1.3 Resolution comparison ............................................................... 719.2 Simulations of motion deviations in broadside mode .... 729.3 Example of target scene in broadside mode ..................... 909.4 Simulations of motion deviations in squint mode............ 94

10 Computational load analysis ............................................. 9710.1 The computational load of the step-transform ................. 9710.2 The computational load of the low-pass filter algorithm9810.3 Computational load comparison ............................................. 99

11 Conclusions ................................................................................ 101

12 Bibliography............................................................................... 102

13 List of acronyms ...................................................................... 104


6/105







1 Background

Radar is a shortening for RAdio Detection And Ranging and was developed in the1930s to detect and track aircrafts and ships. Radars can be classified as ground

based, airborne, spaceborne or ship based radar system. They can also be classifiedinto several different categories for example operating frequency band, type ofantenna and transmitting waveforms. They can also be divided into the utilization ofthe radar. The radars are divided into two main categories continuous wave (CW) or

pulsed radars (PR), which differ in the way of transmitting. CW radars transmitelectromagnetic energy continuously and uses separate transmit and receiveantennas. PR transmit pulse trains of electromagnetic pulses, and are often using the

same antenna for transmitting and receiving of the backscattered signal. In thisthesis pulsed radar will be regarded.

Imaging radars sends out electromagnetic waves, with a wavelength longer thanvisible light. Objects in the illuminated scene reflects portions of this energy back tothe radar. Imaging radars are generating an electromagnetic map of the illuminatedsurface to separate different type of objects on the surface. Radar can operate dayand night trough clouds, fog and rain, as well as at very long ranges.

High range resolution was obtained by using pulses with high bandwidths. For realaperture radars the azimuth resolution was limited by the antenna beamwidth, twotargets could only be resolved if they were separated with at least one beamwidth.To improve the resolution the beamwidth had to be reduced, which was enabled byincreasing the transmitter frequency or by utilizing physically larger antennas. Forairborne radars these requirements can however not be satisfied, whereby theresolution was limited.

In 1951 Carl Wiley of the Goodyear Aircraft Corporation noted that the reflectionsfrom two fixed targets in the antenna beam, but at different angular positions relativeto the velocity vector of the platform, could be resolved by frequency analysis. Eachtarget had different Doppler characteristics because of its relative position to theradar platform. The use of this knowledge was taken into account in a side lookingaperture radar (SLAR), whereby the azimuth resolution of the real aperture radar

could be improved by proper Doppler processing.

The technique is called synthetic aperture radar because a synthetic aperture antennais used. This is done by taking advantage of the forward motion of the radar

platform and coherently combining the received signals of the real antenna along theflight track to produce a very long array antenna. Coherence means that the phase ofthe received signals should be preserved.


7/105







The utilization of synthetic aperture radar (SAR) techniques is becomingincreasingly widespread in many applications ranging from satellite remote sensingof land and sea, through to target imaging from airborne radar for military purposes.

1.1 This project

To be able to produce a high resolution SAR image accurate signal and motionanalysis need to be performed, furthermore it uses high computational load. Thecomputational load must be kept low to be able to produce a real time application,since the received quantity of radar data per time is high. Therefore in many

applications a real time stripmap SAR image with lower computational load isdesirable to effectively get an image of the illuminated area and to be able toevaluate the collected data. There are several things that must be taken in account,for example motion deviations. All motion deviations will have a large affect on thecollected data, especially when working at high frequencies.

The input data used in this report is a simulation of collected data from a radarplaced onboard an airplane. The simulated target scene consists of stationary pointtargets i.e. the radar clutter is neglected.

The received energy from a point target in the raw SAR data is spread in range andazimuth. The purpose of the SAR focusing is to collect this dispersed energy into asingle point in the output image. The coherent combining of the received pulsesduring the flight path is producing a synthetic aperture radar.

There are many different SAR-focusing methods, each has its own advantages anddisadvantages. Within the topic stripmap mode SAR focusing there are for instancethe low-pass filtering and the matched filtering methods. In this project one method

based on low-pass filtering and one method based on matched-filtering will beanalyzed. The methods fast convolution, basic spectral analysis and step-transformare all methods within the category of matched filtering. The step-transform ischosen to be analyzed since it does not have the limitations of loss of computationalefficiency at or near spatial resolution and no variation of the output data rate which

the other two algorithms have.

The step-transform is developed for the broadside application. For the high squintapplication the step-transform is adjusted to a method called the time-varying step-transform developed by Xiaobing Sun et al [1], [2] and [3]. The challenge is toadjust the step-transform algorithm for this airborne SAR application and to do anevaluation. The time-varying step-transform is developed for satellite applicationswhich mean that the range is much larger than for the application used in this report.


8/105







The range for a satellite is about 1000 km and for airplane radar application therange is only about 5 km. The methods will be evaluated with real and simulatedflight paths to create raw-data from point targets. The methods will be compared inazimuth resolution, computational load and motion error robustness.

By using motion sensors the actual position can be measured, whereby the collecteddata can be motion compensated. How large and what types of deviations can becompensated and what affect will it give in the final results? High squint (see 6.2) ofthe antenna will introduce some other aspects that need to be taken into account,which can be neglected in the broadside mode (see 6.1). Range cell migration (see6.3) will arise and affect the phase of the received signal, and must also becompensated.

1.2 Previous work

The two SAR focusing techniques analyzed in this project are both well-knownapproaches for the broadside situation [4],[5]. For the high squint situation the time-varying step-transform has only been evaluated for satellite SAR applications[1],[2],[3],[6]. The performance of the method for low-altitude applications (e.g.airplanes, missiles etc.) will be investigated in this project.


9/105







2 SAR fundamentals

The range direction is the antenna pointing direction, and the azimuth direction isperpendicular to the range direction.

Before considering the properties of a synthetic aperture radar system, the propertiesof real aperture radar (RAR) are mentioned [7]. For RAR systems only theamplitude of the backscattered signal is of concern to form an image. A fine rangeresolution is obtained by transmitting broadband signals, where the bandwidth isimproved by shortening the pulse width or by modulating the signal waveform,which is explained in section 3.1 Pulse compression. The azimuth resolution is given

by the width of the antenna beamwidth, which is dependent upon the physicalantenna length and the transmitter frequency. The width of the beam is alsoincreasing with the distance from the antenna to the surface. Since the antenna is

placed above the surface at an altitude and the illuminated scene is far away, thebeamwidth need to be really narrow. Two targets on the ground can be separatedonly if they are not both in the radar beam at the same time, i.e. separated by at leastone beamwidth.

Instead of using a long real antenna a synthetic antenna can be created, by takingadvantage of the forward motion of the radar platform. As the radar is moving alongthe flight path a pulse is transmitted and received at equally distance. By coherentlycombining the received signals along the flight track, a very long array antenna is

produced. A synthetic antenna is thereby created which is equivalent to a long realantenna, see Figure 2-1.

Figure 2-1. An overview of the SAR geometry.


10/105







The SAR processing is a two-dimensional problem. In the received SAR data, thesignal energy from a point target is spread in both range and azimuth, and the

purpose of SAR focusing is to compress this energy into a point in the output image.The range direction spreading arises by the width of the transmitted pulse. Inazimuth direction the signal is spread by the duration it is illuminated by the antenna

beamwidth.

Pulsed radars are using a train of pulses often with some modulation, where thepulse repetition frequency (PRF) is the rate for the emitted pulses. The maximumPRFis set to avoid range ambiguity see equation (4.) and the minimumPRFis setby the Doppler content see equation (3.), which is given according to:

)cos(2 sDoppler

vf = (1.)

wherev is the platform velocity is the wavelengths is the angle to the target, see Figure 2-2.

Figure 2-2. SAR target Doppler return from stationary targets A, B, C.

The maximum Doppler frequency arises at the greatest angle, which is given by realantenna beamwidth.


11/105







ThePRFinterval is given by:

maxmin PRFPRFPRF (2.)

maxmin

2 RPRF

= or maxmin 2 DopplerfPRF = (3.)

c

RPRF maxmax

2= (4.)

whereRmax is the maximum range of the illuminated scene

2max R is the maximum range difference between two adjacent sample points,

i.e. the phase shift between two adjacent sample points must be less than .fDoppler max is the maximum Doppler frequency of a target.

The low-pass filter algorithm is based on the knowledge that the Doppler frequencyis zero for a target, when the antenna is exactly perpendicular to the target relativethe flight path. Hence can a low-pass filter be applied on the received data to get thetargets azimuth positions.

The step-transform uses the concept of deramping the Doppler frequency of the

targets followed by spectral analysis. Targets distributed in azimuth will beseparated by its constant frequency after the FFTs. Step-transform is a two-stepalgorithm which divides the received signal (in azimuth) into subapertures. The datain each subaperture is correlated with the matched reference function in order toobtain the low resolution images. By adding all the subapertures coherently, a highresolution image is obtained.

In general the radar platform is traveling in a straight line and a single antenna isused for both transmission and reception of the backscattered signal. In practice it isimpossible to fly in a perfectly straight line at a constant velocity. There are alwayssome perturbations which cause deviation from the ideal path of flight, whereby amotion compensation must be performed on the received radar signals.

A SAR system consists of three basic blocks, SAR sensor, motion sensor and imageformation processor, shown in Figure 2-3. The outputs from the two sensor blocksare the input to the image formation processor which creates the final SAR image.The SAR sensor generates, transmits and receives the electromagnetic pulses.The received signal is sampled and the term "range cell" refers to samples in therange or "fast time" direction.


12/105







Figure 2-3. An overview of the SAR system

The motion sensor measures the movement of the radar platform with accurateprecision. The attitude of the airplane is defined by three axis, roll, pitch and yaw.

The yaw axis is defined to be perpendicular to the vertical axis and perpendicular tothe wings of the plane, its origin is placed at the center of gravity and directedtowards the bottom of the aircraft. A yaw motion is a movement of the nose of theaircraft from side to side, see Figure 2-4.

Figure 2-4. The figure shows the yaw rotion of the airplane.

The pitch axis is perpendicular to the yaw axis and is parallel to the plane of thewings with its origin at the center of gravity and directed towards the right wing tip.

A pitch motion up or down is a movement of the nose of the aircraft, see Figure 2-5.


13/105







Figure 2-5 The figure shows the pitch rotation of the airplane.

The roll axis is perpendicular to the yaw and the pitch axis the other two axes withits origin at the center of gravity, and is directed towards the nose of the aircraft.Rolling motion is an up and down movement of the wing tips of the aircraft, seeFigure 2-6.

Figure 2-6. The figure shows the roll rotation of the airplane

The sensor also measures the heading which is the direction of the real motionvector in the XY-plane. The crab angle is defined by the heading angle minus theyaw angle. In addition, the altitude, latitude and longitude position is estimated, inthe basis of measuring the acceleration. With these measurements the transportationof the antenna position is given, and used for the necessary compensations.

The image formation processor comprises components for the data processing toachieve the final image. It uses the data from the two sensor blocks to process the

final image.

2.1 SAR modes

There are four different modes of SAR, spotlight, stripmap, scan and inverse SAR.The modes differ is the way they illuminates the surface.


14/105







In stripmap mode, shown in Figure 2-7, the antenna is pointed at a fixed angleduring the data collection along the flight path. This means that the antenna scansover a particular area which is parallel to the flight path. The antenna can besquinted backward, forward or just perpendicular relative the flight path. In thismode a target is only illuminated for a limited amount of time.

When using spotlight mode, shown in Figure 2-8, the antenna is steered(mechanically or electrically) to illuminate a particular area during the entire flight

path. This mode creates a very long synthetic antenna. The most important featuresthat differ spotlight mode and stripmap are:

1. Spotlight mode illuminates the same area during the entire flight path which

is not the case for stripmap. Hence becomes the length of the syntheticantenna much longer for the spotlight mode than for the stripmap mode.

2. The spotlight mode gives a better resolution then stripmap to the price ofsmaller illuminated target scene.

3. The angle of illumination for the stripmap mode is fixed and given by thesquint angle. In spotlight mode the target are illuminate from a much widerangular variation.

In scan mode, shown in Figure 2-9, the antenna is steered to illuminate a strip ofterrain at any angle relative the flight path. The difference between scan mode andstripmap is the removal of the antenna pointing direction.

The mode called inverse SAR, shown in Figure 2-10, works in a similar way asspotlight mode. Instead of movement of the antenna it uses a stationary antenna anda moving target.

Figure 2-7. SAR in stripmap mode.


15/105







Figure 2-8. SAR in spotlight mode.

Figure 2-9. SAR in scan mode.

Figure 2-10. SAR in inverse mode.


16/105







2.2 SAR antenna

Normally the same antenna is used for both transmitting and receiving of the signals.In SAR is the real antenna used to produce a synthetic antenna, by combing thereceived signal to a synthetic antenna. Real versus synthetic arrays, the radiation ofthe two types of antenna is compared. The mainlobe of the synthetic array is twiceas wide as the mainlobe for the real array. This means that to get the same azimuthresolution as with a real array of length L, a synthetic array of the twice length i.e.2L is required [8]. In the Figure 2-11 is the array pattern for a synthetic and a realarray of the same length plotted.

Figure 2-11. The array pattern for real versus synthetic antenna.

In the simulation model the antenna with the antenna pattern shown in the Figure2-12 is used. The beamwidth is given by the -3 dB antenna radiation amplificationwidth. The antenna radiation beamwidth used in the report is 5.6 degrees, see Figure2-12.


17/105







Figure 2-12. Shows the amplification of the antenna.

Figure 2-13. A magnified view of the antenna amplification.


18/105







3 The simulation model

A simulation model was created to simulate raw-data, i.e. produce data forstationary point targets to be processed. In reality there are endless amounts ofreflecting objects, which all contribute to the final result. The reflections fromvegetation, terrain, sea etc. is usually referred to as clutter, and is not included in theapplied simulation model.

The measurements of the antenna position used in the simulation model are assumedto have no error. In reality the position of the antenna is measured with very high

precision, thus can the measurement error be neglected.

Simply the received signal is a copy of the transmitted signal but with a time delayand reduced amplitude. The time delay is the durations for the signal to go to and tocome back from the target, which is twice the range divided by the propagationvelocity. The amplitude is proportional to target radar cross section (RCS), antennagain and range attenuation. The influence by the antenna is described in the section2.2 SAR antenna. In this simulation model the data from the SAR sensor in Figure2-3 is simulated. The simulated data is created with the measurements from themotion sensor in Figure 2-3.

The transmitted chirp signal is:

]22

(2[)()(

ttcfie

T

nTtrectAts

p

t

+

=

(5.)

where

pT

B= is the chirp rate

B is the bandwidth of the transmitted signalTp is the length of the transmitted signal

fcis the chirp center frequencyAtis the transmitted amplitudetis the time in the range direction, often called fast time

rectis the rectangular function defined by:

>

=

1

1

0

1)(

t

ttrect (6.)

A chirp is a swept frequency where the instantaneous frequency varies with time.


19/105







The transmitted signal is plotted in Figure 3-1.

The received echo from the targets is:

)]2)(2

)((2[)()( d

ttd

ttcfiet

T

nTtrectAtr d

p

r

+

=

(7.)

whereAris the received amplitude from a target

c

Rtd

2= is the time delay

R is the distance to the target, which is dependent on antenna phase center and targetposition.

2arg

2arg

2arg )()()( ettAPCettAPCettAPC ZZYYXXR ++=

),,( APCAPCAPCAPC ZYXR = is the position of the antenna

),,( argargargarg ettettettett ZYXR = is the position of the target

In this simulation model the measurement errors is assumed to be zero.

Figure 3-1. The frequency variation of the transmitted signal.


20/105







The final single pulse raw-data is created by the sum of the all backscattered(delayed) signals from one transmitted pulse. Repeating this for all transmitted

pulses along the azimuth direction and the final raw-data is produced.

The model is designed to be flexible i.e. the included parameters should easily bechanged. It should take the flight path as input and include the amplification of theantenna. The input signal consists of the motion of the antenna, it can come fromreal measurements or from simulations. For this simulation an ideal flight path isassumed.

3.1 Pulse compression

For older radar systems where no pulse compression was possible, the rangeresolution was improved by reducing the pulse length. The disadvantage is that thetransmitted power is also decreased which is directly linked to the signal to noiseratio (SNR). By instead transmitting frequency modulated pulses and using pulsecompression techniques, the pulse length can be increased and still achieve finerange resolution. This also allows the pulses to be transmitted with a lower peak

power.

The linear FM pulse has the property that, when correlating it with a matched filter,it results in a narrow pulse in which all the pulse energy has been collected.

Pulse compression techniques can separate two targets in range, with overlappedradar returns. The filter has the property of introducing a time lag that decreaseslinearly with frequency. It decreases at exactly the same rate as the frequencyincreases at the transmitted pulse. Range compression is performed on each received

pulse. In practice this is performed in the frequency domain using the FFT. Thefrequency response of the matched filter is generated only once, by taking thecomplex conjugate of the Fourier transform of the transmitted frequency modulatedsignal, see Figure 3-2.

Figure 3-2. Pulse compression.


21/105







Weighting functions can be used on the received pulses in order to reduce thesidelobe levels. The disadvantage is mainlobe broadening, for more details seesection 4 Sidelobe suppression.

The range resolution is dependent on the transmitted signal bandwidth B, accordingto the equation:

B

cr

2=

(8.)

To avoid aliasing the sampling rate in range must be higher than the bandwidthB,

since the transmitted signal varies in frequency between B/2. If a weightingwindow is applied an additional constantKwin must be added, whereKwin is aconstant and corresponds to the mainlobe broadening. Hence the final rangeresolution is given by:

B

cKwinr

2=

(9.)

3.1.1 A mathematic description

Filtering the received signal r(t) by a filter with a pulse response h(t), it can bedescribed mathematically by convolution. If the signals are sampled properly thiscan be described in the frequency domain according to:

)()()()( fHfRthtr (10.)

If a weightings function is used it is applied to the signal in the frequency domain.

The time delay for different frequencies is given by:

f=

(11.)

wherefis the frequency is the chirp rate


22/105







The reference function in Figure 3-2 is in the time domain given by:

)2

2(2

)(t

tcfieth

+

= (12.)

The pulse compression factor is the ratio of the pulse duration before compression toits length after compression [9], i.e. Tp divided by 1/B. It is expressed as following:

2pp TBTPCR == (13.)

It represents the improvement in range resolution of the system.

Figure 3-3 and Figure 3-4 show the result before and after range compression forone target return.

Figure 3-3. One received pulse from a target placed at a range of 5000 m.


23/105







Figure 3-4. The compressed signal after range compression with a Taylor window.


24/105







4 Sidelobe suppression

The primary source which arises to sidelobes are finite data processing apertures andthe presence of phase errors in the SAR data. The phase errors in the SAR data arereduced by the motion compensation. By applying weight functions in azimuth andrange directions the sidelobes levels can by reduced, but this will increase themainlobe width. There are many different types of weightings functions, where themost commonly used window in SAR applications is the Taylor window. TheTaylor window provides strong, selectable sidelobe suppression with a minimum

broadening of the mainlobe.

The discrete Taylor window is specified by three parametersN, n and SLL where,Nis the window size, n is the number of nearly constant level sidelobes adjacent to themainlobe and SLL is the sidelobe suppression in dB relative the mainlobe. A popularchoice of the parameters are n=5 and SLL=-35 dB [9], which gives a mainlobe

broadening of 1.34 times the mainlobe width for a rectangular window. The Taylorcoefficients are given by

=

++=

1

1

)2/12/(2cos21)(

n

m

mN

NnmFnTay

(14.)

forn=0,1,,N-1, whereNis the number of coefficients.

=

=

+

+

=1

1

22

1

122

221

)/1(2

)2/1(

/1)1(

n

mjj

n

i

m

m

jm

iA

m

F

(15.)

where

)1ln( 2 +=

BBA

(16.)

2010SLL

B

= (17.)

22

22

)2/1(

)(

+=

nA

n

(18.)


25/105







In the Table 4-1 some data for common used weighting functions [10] is presented.

Table 4-1. Apperture wieghtening windows, Ref[10].

Window Sidelobe level, dB -3 dB bandwidth,3dBRectangular -13 0.89 2/N

Taylor, with SLL -35 dB and =5 -35 1.19 2/NBartlett -27 1.28 2/NHanning -32 1.44 2/NHamming -43 1.30 2/N

Blackman -58 1.68 2/N

Figure 4-1. Impluse responses of different windows, the length is 64 and is zeropadded to get a good resolution.The Taylor window has =5 and SLL=-35 dB.

The Figure 4-1 compares different types of windows. The Taylor window has thenarrowest mainlobe and the first sidelobe is of intensity -35 dB. Since a sidelobesuppression of more than -35 dB is of no concern in the most SAR applications theTaylor window with SLL of -35 dB is often used.


26/105







5 Motion error compensation

Optimally the flight path is a straight line, but in reality this is not the case sincesome portion of deviation from the ideal path is impossible to avoid. A smalldisturbance from the ideal line will have a major affect on the signal phase,especially for high frequency systems, and must be compensated for. To enable thisthe antenna position must be measured with high accuracy, which is done by themotion sensors.

In stripmap mode, compensations for the distance between the actual position andthe planned position is done for every received pulse along the flight path. This isdone in two steps, first by a phase and time compensation in range and secondly in

azimuth by an interpolation.

The first step compensates the data to the straight planned flight path. The distancealong the antenna direction from the actual position to the planned flight path iscalculated and compensated. The distance AB in the Figure 5-1 shows the distancewhich needs to be phase and time compensated. Secondly the interpolationcompensates the data to the planned position along the planned flight path, given bythe distance AC in the Figure 5-1.

There are still some deviations left which can not be compensated for. For instance,if the platform has involuntary variations in yaw, the antenna is illuminating anotherarea than computed which is not able to compensate for. This is crucial because itgives rise to amplitude modulation, which can be hard to deal with.

Figure 5-1. The straight line shows the planned line of flight and the curve is the actual path. The position C is theplanned one and the postion B is the virtual one.


27/105







6 SAR signal processing

There are many signal processing algorithms developed for SAR in stripmap mode,for instance basic SPECtral ANalysis (SPECAN) and fast convolution, described in[6]. The used algorithm must be chosen under the valid conditions.

6.1 Broadside mode

Broadside mode means the antenna is pointing perpendicular to the flight path. Inthis application the distance to a target is decreasing until the angle is 90 degrees and

then starts increase again, see Figure 6-1. This gives a Doppler frequency that in anear region is varying linearly. It starts with positive frequency and passes throughzero when the target is at an angle of 90 degrees and ends up with negativefrequency. This distance difference is equivalent to a phase shift of the received

pulses, hence gives this phase shift a Doppler frequency because it is the derivativeof the pulse-to-pulse phase shift.

Figure 6-1 An example of the distance to a target along the fligth path, the target is placed at a distance of 5000 m

at an angle of 90o relative the pulse number 2000.


28/105







6.2 High squint mode

For conventional strip-mapping the antenna is directed perpendicular to the flightpath. For some applications however, the antenna has to be squinted (e.g. missilesetc.). In squint mode the antenna is directed forward or backward, which willintroduce certain phenomena that need to be taken into consideration. In this reportthe antenna is squinted in a forward direction. Additional squint can occur due to

platform rotation, discussed in section 2 SAR fundamentals.

The radar cross section of a target can be highly dependent of the illumination angleor some objects can be in radar shadow in the broadside mode. Strong side wind canalso cause drift in the yaw angle which means that the antenna is squinted.

The squint mode will affect the Doppler shift of the received signals. The distancebetween a target and the antenna is changing more rapidly hence the phase is alsochanging more rapidly. The Doppler frequency which is dependent of the phaseshifting is then also shifting more rapidly. In the broadside application the Dopplerfrequency was centered around zero, but in the squint mode is it shifted up infrequency, which is shown in Figure 6-2.

Figure 6-2. An example of f requency shift broadeside mode versus squint mode.

6.3 Range-cell migration

The varying distance between the antenna and the target along the flight path iscalled range-cell migration (RCM). Hence will the signal from a target not lie alonga straight line in the azimuth signal memory. The RCM for broadside respectivelysquint mode is analyzed in section 6.3.1 and 6.3.2.


29/105







6.3.1 A mathematical description of broadside mode

During the flight path targets will be illuminated only for a limited amount of time.The size of the angle interval which the targets will be illuminated is given by the

beamwidth of the antenna. In this report the beamwidth of the antenna is rathernarrow (3) which gives approximately linear Doppler frequency in broadsidemode. In applications with a much wider beamwidth a more complicated modelmust be applied. The distance to a target will vary as following:

{ } 222222 0)sin()sin(2),(ssssss RtvvtRRtvRtR ++= (19.)

wherev is the velocity of the radar platform

sR is the slant range to the target, i.e. the closest approach

t is the time along the flight path when the target is illuminated, i.e. duration of atarget in antenna footprint t0. If a target is illuminated during a time t0 the timevector becomes:

2200 tt

t . (20.)

The equation of distance (19.) can be expanded in its Taylor series as following:

{ }

2 32 3

2 3

2 2 3 22 3

2

2 22

2

1 1( , )

2! 3!

cos ( ) cos ( )sin( )sin( )

2 2

cos ( )sin( ) 0

2

4

s s

s s ss s

s s

ss s

s

s r

dr d r d r R t R R t t t

dt dt dt

v vR v t t t

R R

vR t

R

R f t

+ + +

= + +

+

=

(21.)

where

s

sr

R

v

dt

Rdf

)(cos22 22

2

2

== is the Doppler frequency chirp rate (22.)

is the wavelength

The assumption 1)cos( s is also used, because s is varying between 3 degrees.


30/105







In broadside mode the RCM is caused by the quadratic term, terms of higher ordercan be neglected.

An example of the varying distance in a broadside mode application is shown inFigure 6-1. The Figure 6-1 shows clearly the quadratic behavior of the RCM in the

broadside mode.

If the RCM is smaller than a range cell the signal will be lying in a straight line afterthe range compression and the RCM is only shown as a phase variation.

The pulse-to-pulse phase variation, due to target distance variation is givenaccording to:

02

22),(

4)(

+= t

R

vRtRt

s

s (23.)

where

0 is stationary phase for the target

An example of the phase for a target is plotted in the Figure 6-3.

Figure 6-3. An example of phase variation from pulse-to-pulse, a target placed perpendicular the the position ofpulse number 2000.

Since the frequency is the time derivate of the phase, the frequency can be expressedaccording to:


31/105







tR

vtf

s

24)( (24.)

An example of the Doppler frequency for a target is plotted in the Figure 6-4.

Figure 6-4. An example of the Doppler frequency along the azimuth direction, one target is placed perpendicular tothe position of the pulse number 2000.

This is approximately a linear frequency term which only is valid for a certain angleinterval. The phase and frequency equation are equal to their true values when t isequal to zero and the approximation becomes worse for larger values of|t|.

6.3.2 A mathematical description of high squint mode

When the antenna is squinted there will arise additional range terms. Firstly, thesquint gives arise to a linear term denoted range-walk, which must becompensated for. In addition, the size of the cubic range term becomes of such orderthat its phase contribution must be taken into consideration. The envelope movementof the cubic term can however be neglected in most situations (compared to the sizeof a range cell).

A mathematical description [6],[2],[3] of the influence of squint mode is describedbelow.


32/105







The distance to a target is given by:

)sin(2),( 222 ssss vtRRtvRtR += (25.)

wherev is the velocity of the platformtis the azimuth time along the flight path

Rs is the slant range at the center of the apertures is the squint angle between rs and flight path

The equation of distance (25.) can be expanded in its Taylor series [3] as following

2 32 3

2 3

2 2 3 22 3

2

2 3

1 1( , )2! 3!

cos ( ) cos ( )sin( )sin( )

2 2

2 4 12

s s

s s ss s

s s

s dc r r

dr d r d r R t R R t t tdt dt dt

v vR v t t t

R R

R f t f t f t

+ + +

= + +

=

(26.)

where

cf

c= = wavelength

dcf = Doppler-frequency centroid, also denoted range-walk

rf = Doppler-frequency chirp rate, also denoted range-curvature

rf = changing rate of rf , also denoted range-migration

The range-walk, -curvature and -migration can be expressed as:

)sin(2

2sdc

v

dt

dRf

==

(27.)

s

sr

R

v

dt

Rdf

)(cos22 22

2

2

==(28.)

2

23

3

3 )sin()(cos62

s

ssr

R

v

dt

Rdf

==

(29.)

In broadside mode with a fairly narrow beamwidth was the distance to a targetapproximately constant term plus the quadratic term, because s is approximatelyequal to zero ( )1(cos2 s ). This gives an approximately linear Doppler frequency.


33/105







In low squint mode must only the linear and the quadratic term be considered and allthe terms of higher order can be neglected in the Taylor expansion. But in highsquint SAR also the cubic term must be taken into account. The range-walk causesrange-cell migration of several range cells and must be removed. The removal of therange-walk can be viewed as a phase and time compensation of the received signalto a fictive flight path perpendicular to the antenna angle. The data is taken along thelineRs - v sin(s)t and is shifted in phase and time. The Figure 6-5 shows thegeometry of the phase compensation. This phase removal gives the Dopplerfrequency along the azimuth direction lying around zero. The RCM is now muchsmaller than before and is caused by the quadratic and the cubic terms. The Figure6-6 and Figure 6-7 show the signal before and after range-walk removal.

The first termRs in equation (26.) is a constant and gives rise to a phase 0 whichdoes not change in time, hence it will not affect the Doppler frequency.

The second term in equation (26.) is the linear phase term which gives rise to theDoppler-frequency centroid. This term causes the largest RCM and must beremoved.

The third term in equation (26.) is the same as in the broadside application, but thedifference is the scaling of thefrfactor. It is the one which gives rise to the linearterm in the Doppler frequency. Thefrfactor includes the velocity to the power oftwo times )(cos2 s , which gives the velocity along the fictive flight path. Hence is

the velocity along this fictive flight path slower than in the broadside application.Due to the fact of slower velocity in the high squint application, the phase variationbecomes slower.

The forth term in equation (26.) rf is the changing rate of rf . It does not influence

the phase or the frequency in the middle of the antenna footprint and has maximuminfluence in the beginning and at the end.

In this application it can be shown that cubic term can be neglected in the envelopevariation because it is much smaller then a range cell. This envelope variation [3] isexpressed according to:

)tan(88

)sin()cos(2

33

3 bbs

s

sb R

R

tvr

=

(30.)

where

3r is the RCM caused by the cubic term

b is the beamwidth, which is the angle where the antenna amplification is -3 dB


34/105







tis the duration of a target in the antenna footprint

ExampleRs = 5000mb=3gives:

mr 7.13

which is smaller then a range used in this report.

On the other hand the phase influence of this range movement will affect and causenon-neglectable phase errors that must be taken into account. This phase term can be

expressed as following:

33

4r=

(31.)

Using the same parameters as the example above and a wavelength of 0.02 m whichgives a maximal phase term of 1068 rad.

Figure 6-5. The geometry of squint mode.

After range-walk removal the signal content in a single range cell consists of targetsfrom different distances in the original simulation geometry, see Figure 6-6 and


35/105







Figure 6-7. Since the target Doppler rate is inversely proportional to the target slantrange the Doppler rate content will also vary within each range cell, according to:

s

sr

R

vf

)(cos2 22 ==

(32.)

Figure 6-6. Signal RCM before range-walk removal.

Figure 6-7. Signal RCM after range-walk removal.


36/105







To sum it up it is important to distinguish between the phase shift caused by rangecell migration and the fact that the signal does not lie along a straight line in thesignal memory. If the curve the signal lie along is crossing one range cell the imagequality may be significantly affected if it is not taken into account.


37/105







7 Azimuth focusing

The azimuth focusing can be performed in several different ways. In this masterthesis project two different methods are evaluated. The first method is a low-passfiltering method and the second is the step-transform. A method called the time-varying step-transform will also be investigated for the high squint mode. The time-varying step-transform is a further development of the step-transform performed bySack et al and Sun et al [6], [1], [2], [3]. Before applying any algorithm the raw-datamust be preprocessed. Both motion compensation and squint compensation must be

performed before entering the both of the algorithms. The azimuth focusingalgorithms process the data independently for each range-cell and in the end put the

range cells together.

The azimuth resolution of the two algorithms can be compared to the possibletheoretical resolution. The azimuth resolution of a target is given by the -3 dB targetresponse width. The sidelobe levels from a target must be suppressed more than -35dB to be considered enough.

7.1 Preprocessing of data

The received raw-data must be preprocessed before entering the algorithms. The

first thing is to compensate for motion errors. From the motion sensors the real pathof flight can be calculated. The influence of the antenna being out of the ideal flight

path is given by the longitude and latitude.

The distance from the ideal position and the real position must be compensated. InFigure 5-1 is the actual point B and the planned is the point C. The compensationthat needs to be performed is a phase and time compensation, see Figure 7-1, whichis dependent of the distance difference between the real position and the planned

position. The distance is calculated by taking the distance from the real position tothe planned flight path in the squint direction, i.e. the distance AB in Figure 5-1 .The importance of accurate measurements is obvious because of the short wavelength. The compensation is performed on a pulse-to-pulse basis, by first taking FFTof each received pulse, phase shift the signal and thereafter transform the signal backto the time-plane.

Thereafter the interpolation step is performed along the azimuth direction for eachrow of the received data, see Figure 7-1. The interpolation is performed to createequidistant sampled pulses, which was not the case in Figure 5-1 because of thedistance AC.


38/105







The interpolation is performed with the Matlab function interp1 with cubicinterpolation. The actual sample is the point A in Figure 5-1 and the interpolated and

planned point is the point C.

Figure 7-1. Motion compensation in stripmap SAR.

Figure 7-2. A block diagram over the motion error compensation.

At this stage the disturbances caused by motion error are partly compensated. It iscompensated as good as it can be, but it will still remain some errors compared tothe ideal application. The roll, pitch and yaw will influence the angle which thetargets are illuminated by the antenna. The influence of the angle to a target gives an


39/105







amplitude variation of the signal from the antenna which hits the target, see section2.2 SAR antenna. The influence of this can not be compensated for. In section 9Simulation results, the influence of this is investigated.

If the antenna is squinted the range-walk removal must be performed before theinterpolation. The range-walk removal is performed at the same time as the phaseand time compensation. The range-walk term is given by the following equation:

)sin(2

sdcwalkrange vttfR

==(33.)

wheretis the time vector along the flight path

The range-walk walkrangeR distance starts at zero and grows along the flight path. As

mentioned before it gives rise to range-cell migration, and in addition it gives rise toa constant Doppler offset frequency. Since the sampling frequency is not highenough it will cause aliasing. By the phase and time compensation of the range-walkthis problem got solved and the range-walk is completely removed.The phase and time compensation is performed as following:

c

compR

cf

sf

dfi

eYY receivedomppha

)(4

sec

+

=

(34.)

where

receivedY is the FFT of the received signal

ompphaY sec is the FFT of the phase compensated signal

df is the normalize frequency vector

sf is the sampling frequency

c is the speed of lightRcomp is the distance to be phase compensated for

7.2 Theoretical resolution in azimuth direction

In the following section the azimuth resolution for real aperture radar and syntheticaperture radar is reviewed.


40/105







7.2.1 Real aperture radar

In real aperture radar the width of the illuminated scene is dependent on the antennabeamwidthb [9], which is given by:

DKwindowb

(35.)

where

D is the length of the physical antenna.Kwindow is the mainlobe broadening caused by the aperture weighting. is the wavelength

The azimuth resolution is dependent on the target distance [11] and is given by:

D

RKwindowreal

= (36.)

Hence becomes the azimuth resolution very poor for targets far away.

Example:

When using an antenna of length D=1m, a wavelength of 0.015m and a target placedat a distance of R=5000m the resolution becomes 75m (Kwin=1).

7.2.2 Synthetic aperture radar

Instead of using real aperture a synthetic aperture is created. The azimuth resolutionproperties are explained below. The theoretical resolution in azimuth direction isdependent on the synthetic aperture length [9].The synthetic aperture length is thelength of the flight path where the target is in the antenna footprint see Figure 7-3,and is given by:

)sin()sin( s

bwindow

s

windowsynthetic

RK

D

RKL

= (37.)

where


41/105







s is the squint angle relative the flight path

b is the angular interval where the target is illuminated, shown in Figure 7-3.

Figure 7-3. The geometry of SAR. b is the antenna beamwidth

The synthetic beamwidth is approximately given by:

)2

cos()sin(2 bssynthetic

windowsynthetic

L

K

= (38.)

Where the beamwidth is a factor two smaller for the synthetic aperture then it is forthe real aperture. The cosinus factor indicates a broadening of the synthetic aperture

beamwidth for wide angle synthetic apertures.

The beamwidth times the range to the target gives the azimuth resolution asfollowing:

)2

cos()sin(2 bssynthetic

windowsynthetic

L

RK

= (39.)


42/105







R/Lsyntheticis often approximated with the following expression:

)2

tan(2

)sin(

b

s

syntheticL

R

(40.)

and this approximation gives the resolution:

)2

cos()2

tan(4 bbwindowsynthetic K

(41.)

which often is simplified to:

)2

sin(4 bwindowsynthetic K

(42.)

For an antenna with a narrow beamwidth this correlation can be simplified to:

b

windowsynthetic K

2 (43.)

In the broadside application the theoretic azimuth resolution becomes D/2independent of range if the targets are illuminated during the entire antennafootprint. This can be realized by using the equations (35.) and (43.) and theknowledge that b=b which is valid during the cited conditions.

To achieve the SAR resolution, the raw-data has to be focused. In the next sectionstwo different approaches are discussed.

7.3 The low-pass filtering algorithm

The algorithm using low-pass filtering is an unfocused SAR method. The idea ofthis method is to low-pass filter in the azimuth direction, this takes directly


43/105







advantage of Doppler frequency. Since the Doppler frequency is zero when theantenna position is exactly perpendicular to the target, the position can be resolved.

The phase from a target in broadside mode varies according to:

02

22),(

4)(

+= t

R

vRtRt

s

s (44.)

wheretis the time of the antenna foot print

The Figure 7-4 shows an example of the phase variation along the flight path for atarget. This is placed perpendicular to the radar platform at pulse number 2000.

Figure 7-4. Angle variation to a target placed perpendicular to radar platform at pulse 2000.

The Doppler frequency for the target is:

tR

vtf

s

24)(

(45.)

which is illustrated in Figure 7-5.


44/105







Figure 7-5. Doppler frequency for a target.

In a narrow neighborhood of t equal to zero the phase can be considered to beconstant. Since the frequency is the derivate of the phase and the phase is a constantthis will produce a frequency that is approximately zero in this neighborhood. Atzero Doppler frequency is the target positioned perpendicular to the travel direction

and the frequency does not change very quickly from pulse to pulse. A low-passfilter will obviously have an azimuth focusing effect.

The limitation in the filtering method is to create a sharp low-pass filter with a lowcut of frequency. By using the equation (45.) a theoretical azimuth resolution can becalculated. The passband spectrum ofw corresponds then to the time:

24

2

v

Rwt sres

= (46.)

wherew is the cut-off frequency, i.e. w/PRFis the normalized cut-off frequency.

The azimuth resolution is then given by the velocity times tres.

Example:Reasonable value of the passband of the low-pass filter is frequencies around wnorm=0.025 normalized frequency, i.e. 5 % of the spectrum. With=0.015m,Rs=5000mand v=50m/s gives an azimuth resolution of 6.0m.


45/105







Another aspect to be considered in the filter design is that it is of no benefit to createa filter with a stopband suppression of more then -40 dB.

The computational load for this method is dependent on the length of the filterand/or the filtering method.

7.4 The step-transform

The step-transform is a variation of an algorithm called basic SPECtral ANalysis

(SPECAN) [6]. The step-transform applies some additional steps. The basic idea isto separate azimuth distributed targets in frequency. A Fourier analysis is then usedto analyze the frequency content.

The basic SPECAN de-chirps the signal in the time-domain by a reference functionof different slope. Thereafter it applies a single FFT to separate the targets, seeFigure 7-6. For a long flight path the FFT must be overlapped, which reduces theefficiency of the SPECAN significantly since the most data from each FFT isdiscarded [9]. Variations in the azimuth FM rate causes also problems, to be able toachieve constant output rate an interpolation is needed [1].

Figure 7-6. Computational stages in basic SPECAN.

The step-transform comprises a number of sub-stages, which are shown in the blockdiagram in Figure 7-7.


46/105







Figure 7-7. The figure shows the block diagram of the step-transform.

The input data to the step-transform is the motion compensated and rangecompressed signal, which is divided into subapertures along the azimuth directionfor every range-cell. The de-ramping step comprises an operation by a referencefunction. The reference function has the opposite slope but the same rate as thetarget returns. There will be for each target a sinusoid left with a constant frequency,which is dependent of the placement of the target. Since the Doppler slope varies for

different ranges the reference function will also be different. The rate is given by:

sR

vK

22= [Hz/s] (47.)

The target return for a target placed in a time position ttar=m, see Figure 7-8, isgiven by:

2)()( tarttKietS = (48.)

where

2200 ttt

tt tartar +

t0 is the time the target is in the antenna footprint.m is the target position measured in sample points.

PRF

1= is the time between adjacent samples


47/105







Figure 7-8. The target scene, where one target is placed at m.

Multiplying the target return with the reference function placed at tref=n accordingto:

22 )()()*( refref

ntKittKieetS

==

(49.)

gives the following result:

))(2(

)()()()(

2222

2222

)*()(nmtmnKi

ntKimtKittKittKi

e

eeeetStS reftar

+

+ ===

(50.)

Since the two terms m22 and n22 in equation (50.) are independent of time theywill be constant. The only time dependent term is 2t(m-n) which is a linearrelationship dependent of the displacement of target returns. The relationship

between target placement and the frequency is given by:

= )( nmKf (51.)

Converting the equation (50.) to discrete time by setting t=k+n-T1/2

where

T1is the length of a subaperture


48/105







kis the sample number in a subaperture

)2()()())(2( 2122

1222

)*()( += mnKimTmKinTnKimnkKi eeeetStS (52.)

The frequency relationship between targets depends only on the first term. Theremaining phase terms are:

))2()()(( 2122

122 +++= mnmTmnTnK (53.)

When the de-ramping is performed a FFT is applied, which is called Coarse-resolution FFT (CRFFT). The CRFFTs can be viewed as a set of band-pass filters,

which are dividing the frequency spectrum into several pass-bands. Each output binin CRFFT corresponds to one band-pass filter, see Figure 7-9. To avoid sidelobeleakage a Taylor window with sidelobe suppression of -35 dB is used.

Figure 7-9. The bandpass filtering effect of the CRFFTs, the normalized frequency spans from -PRF/2 to PRF/2.

One disadvantage of the step-transform is that the subapertures need to beoverlapped. The requirement of overlapping the subapertures arise since the FFTand the de-ramping consist of different types of regions in time-frequency space, seeFigure 7-10. The FFT occupies a rectangular region in time-frequency space whilethe de-ramping procedure produces a parallelogram shaped processing region. Thus,one FFT can not produce data for all targets return in a particular processing region.


49/105







This cited problem is solved by overlapping the subapertures, unfortunately itreduces the efficiency of the algorithm.

Figure 7-10. The upper figure shows the targets return before de-chirping. The lower figure shows the de-rampedtarget returns.

The length of the CRFFT corresponds to the number of samples in a subaperture.The signal energy from a target will be lying in different frequency bin in severalCRFFT, see Figure 7-11. The difference in number of frequency bins between twoadjacent CRFFT is given by:

221 = KNNH (54.)

whereN1 is the number of samples in a subapertureN2 is the number of samples between the centers of two adjacent subapertures.


50/105







Figure 7-11. The de-ramping proces in the step-transform.

Using the x-axis as the frequency axis and the y-axis as the subaperture number thedata is placed along the diagonals, according toH. Figure 7-12 shows how the datais placed along the diagonals.


51/105







Figure 7-12. The energy from three targets is along the diagonals, where white is high amplitude and black is lowamplitude.

The value ofN1 andN2 must be carefully chosen, there are several things that needto be fulfilled [2]. The requirements are listed below:

1. The range curvature creates range cell migration and the data from one targetmust be lying in the same range resolution cell. I.e. the range cell migration cause bythe range curvature must for that reason be less then a half resolution cell. Therebythe subaperture length is limited by the following equation:

v

rT

b

rcr

1 (55.)

where

s

rcrf

cr

2= is the size of a range cell

2. The length of the subapertures,N1 must be a power of two, to be able to apply theefficient FFT.

3. The subapertures must be sufficiently overlapped to avoid frequency aliasing.


52/105







To obtain full resolution the data has to be chosen for a target from the CRFFToutput bins and perform second FFT, which is called fine-resolution FFT (FRFFT).As mentioned before the data is lying along diagonals with a slopeH. The dataalong these diagonals will be sent to the FRFFT. It is explained in [2] that theHmust be an integer to avoid amplitude modulation between successive subapertures.

To be able to keep theHvalue to an integer for the different ranges some of theparameters in the equation for theHmust be changed. This problem arises due to thefact of the variation of the chirp rate along different ranges and thereby variation ofthe K. The length of a subaperture must be a power of two, to be able to use theefficient FFT, so theN1 value can not be changed. The only parameter that can bechanged is theN2 value i.e. the overlap between subapertures. But one disadvantage

of changing the overlap comes up when choosing the data from the final FRFFT.

To come around this problem the data is delayed so that the data will be lying alongvertical lines. The displacement is performed by multiplying each row with a delayincreasing linearly with subaperture number. By delaying to rows theHdoes notneed to be an integer and the overlap between subapertures can be kept constant.The delay can be expressed as the following function:

1

)1(2

Nd

Hfli

edelay

=

(56.)

wherelis the subaperture number

df is the normalized frequency vector

The delay is just a complex operation and is performed before the CRFFT. The dataafter the delay is shown in Figure 7-13.


53/105







Figure 7-13. The signal after the CRFFT from three targets with the delay included.

Before the FRFFT is performed on each column, the data has to be phase corrected.The phase term does not depend on time (or sample) and must be canceled. The

phase term is dependent on the subaperture center position and target position. It isconstant in every subaperture and does not contain information to separate targets inthe FRFFT. The phase term is applied on each row and is updated for eachsubaperture. The phase term is:

)(* 122 TnnK = (57.)

where

K is the chirp rate

n is the center position of the subaperture= 11 NT is the length of a subaperture

The phase compensation is performed by multiplying each subaperture with theconjugate of the phase term, i.e.

)1

22(*

TnnKi ee

=

(58.)


54/105







The FFT is applied in the columns which contain valid data. The number of columnswhich are containing valid data is dependent on the length of the scene in azimuthdirection and the slope of the Doppler frequency. Because of the subapertures areoverlapped each FRFFT also contains data which is discarded. The quantity ofdiscarded data is determined by the overlapping factor, which is given by:

1

21N

N= (59.)

An overlapping factor of zero is no overlap and an overlapping factor close to one isclose to full overlap. The overlapping factor is usually larger than 0.6 [6], to be able

to suppress the sidelobe levels. However every target will appear in multiple in thefinal image, this is cause by the sidelobe leakage, which leads to ghosts see Figure7-14. For that reason the sidelobe suppression must be kept high and thesubapertures must be overlapped to get low amplitude of the ghost targets.

Figure 7-14. Multiple target reurn after FRFFT from a single target return.

The final image is created by picking appropriate data from the FRFFTs. Thefrequency spectrum of the valid part of the FRFFTs contains a frequency spectrumofPRFHz. The frequency spectrum of each FRFFT is containing one divided by T2Hz. Also in the FRFFT the signal is windowed by a Taylor window with sidelobesuppression of -35 dB.


55/105







But which data is the appropriate and which data must be discarded?

There is a relation between frequency and azimuth position of the targets after theFRFFT. Hence by ensuring there are no gaps in the frequency domain the wholeazimuth axis is covered. The output bins of one CRFFT coversPRFHz, and oneFRFFT coversPRF/N2 Hz. The new frequency substance in every FRFFT isPRF/N1Hz. A target shifted one second is shiftedKHz i.e. the Doppler-frequency rate frmultiplied with the time. The data must be selected from frequency-frequency spacethe along, where the x-axis is the frequency axis from the CRFFT and the y-axis isthe frequency axis from the FRFFT. The appropriate data will be lying alongdiagonals with slopeN1/N2 , the slope means the number of FRFFTs for onefrequency spectrum ofPRF/N2 Hz, see Figure 7-15. ExampleN1/N2 is equal to three,

from one FRFFT that coversPRF/N2 Hz onlyPRF/N1 Hz is taken and the remainingdata will be discarded [4]. The length in azimuth direction of every diagonal is given

by:

KN

vXazimuth

2= ,[m] (60.)

Since the distance of every diagonal is dependent on the slopeKwhich changesalong the range the diagonals will cover different distance in the azimuth direction.Hence alter the number of samples between different range-gaps and thereby alsothe azimuth resolution. For a short flight path the azimuth resolution is by default

different due to the fact of different Doppler frequency width, i.e. different angularinterval. The fact of fewer samples per meter for high ranges increasers thedifference in resolution between near and far ranges.


56/105







Figure 7-15. The signal after the second FFT, with three target.

A target located in the middle of the scene in azimuth direction becomes located inthe middle in the frequency-frequency space. The data can be chosen along thediagonals, but since the frequency-frequency matrix is discrete it is impossible toavoid amplitude difference between targets, of the same recei

stripmap sar mocomp2005

Documents