CW_chap04

Chapter 4

Synthetic Aperture Concepts

4.1 Introduction SynApt intro.tex Jan 12, 2004

The purposes of this chapter are to explain the term “synthetic aperture”in the radar context, and to derive associated parameters such as azimuthbandwidth and resolution. The chapter begins with an explanation of theSAR geometry in Section 4.2 and the special terminology that is used in theimaging radar. Then, Section 4.3 outlines the “range equation,” detailing howthe distance from the sensor to the target changes with time.

The next three sections describe how the SAR signal is acquired. First,the form of the transmitted radar pulse is given in Section 4.4, pointing out therelationship between transmitted bandwidth and achievable processed rangeresolution. The received echo from the pulse is a convolution of the pulse andthe ground reflectivity.

Second, the form of the SAR signal in the azimuth direction is discussed inSection 4.5. The notion of pulse coherency and the timing of the transmittedpulses are presented. The timing, as characterized by the pulse repetitionfrequency or PRF, is affected by many of the SAR system design parameters,and its choice is quite restricted in a satellite sensor. After discussing thefactors affecting the received signal strength, the important signal parametersof exposure time, Doppler frequency and bandwidth are discussed.

Third, Section 4.6 explains how the received signal can be considered asa two-dimensional signal, and how it is written into the range and azimuthdimensions of the signal processor memory. This structure is needed so thesesignals can be processed into a two-dimensional image of the Earth’s surface.The concept of the impulse response of the SAR sensor is introduced, andtypical aircraft and satellite SAR parameters are given.

The central idea of SAR processing is based upon matched filtering of

107

108 Chapter 4. Synthetic Aperture Concepts

the received SAR signal in both the range and azimuth directions. Matchedfiltering is possible because the acquired SAR data are modulated in thesedirections with an appropriate phase function. The modulation in range isprovided by the phase encoding of a transmitted pulse, while the modulationin azimuth is created by the motion of the radar platform.1 The phase containsthe most important information in the signal, so the phase characteristics ofthese modulations are examined throughout this chapter. More details of theSAR signal properties are given in Chapter 5.

So far, the ground work for matched filtering and range resolution hasbeen established. By now, the concept that a high azimuth resolution canalso be obtained by matched filtering should be apparent. The classical limitof azimuth resolution, which is half the antenna aperture, is derived in Section4.7.1 from the viewpoint of processed bandwidth and SAR system velocities.

Finally, the foregoing discussion leads to the concept of synthetic aperture,which is presented in Section 4.7.2. The signal processor operates on a group ofsignals obtained during the time that the sensor illuminates a selected target,and in doing so, creates the effect that would be obtained by a single antennawith a very long aperture. This concept of synthetic aperture also leads to analternate derivation of azimuth resolution.

The chapter ends with three appendices. The first appendix derives asimple form of the range equation for a satellite orbit that is locally circular,and an Earth that is locally spherical. It justifies the approximate radarvelocity used in Section 4.3.1. The second appendix describes quadraturedemodulation in detail, including how to correct for calibration errors betweenthe two quadrature channels. The third appendix takes another look at themeaning of “synthetic aperture,” this time from an antenna viewpoint.

4.2 SAR Geometry SynApt sarGeom.tex

The purposes of this section are to describe SAR data acquisition geometryand to define the geometry-related terms used in the text.

4.2.1 Definition of Terms

Figure 4.1 shows a simple geometric model of the radar location and the beamfootprint on the Earth’s surface. Radar systems can be monostatic, bistaticor multistatic, depending upon the location of the receiver in relation to the

1 The term “platform” or “sensor” is used for aircraft and/or satellite in this book forsimplicity. The terms “aircraft” and “satellite” are used in the respective specific situationswhen the distinction is important.

4.2 SAR Geometry 109

transmitter. A monostatic radar, in which the same radar antenna is usedfor transmission and reception, is assumed throughout this book. Monostaticradars are typically used in remote sensing, although an interesting multistaticconfiguration known as the interferometric cartwheel has been proposed byMassonnet [1, 2].

Target

Sensor path

Radar

Beam footprint

Nadir

Squint

angle

Ground range

(after processingto zero Doppler)

P2

R0

Slant range

(after processing

to zero Doppler)

Slant range

(before processing)

radar track (azimuth)

R

R0

X

Plane of zero

Doppler

P1

Figure 4.1: Radar data acquisition geometry.

The terms used to describe the SAR geometry are defined as follows:

Target: This is a hypothetical point on the Earth’s surface that the SARsystem is imaging. The SAR system actually images an area on theground, but to develop the SAR equations, a single representative pointon the ground is considered. This point is called a “point target” or“point scatterer,” or simply “target” or “scatterer.”

Beam footprint: As the platform advances, pulses of electromagnetic en-ergy are transmitted at regular intervals towards the ground. During thetransmission of a particular pulse, the radar antenna projects a beamonto an area of the ground referred to as the beam “footprint.” Theposition and shape of the footprint is dictated by the antenna beampattern and the sensor/Earth geometry. This footprint is said to be“illuminated” by the radar beam.


Nadir: The nadir is the point on the Earth’s surface directly below thesensor, so that the “normal” to the Earth’s surface at the nadir passesthrough the sensor. For a spherical Earth model, the vector from thesensor to the Earth’s center intersects the Earth’s surface at the nadir,but not for an ellipsoidal model.

Radar track: As the nadir point moves along the Earth’s surface, it tracesout the radar track.

Velocities: There are two system velocities to consider:2

• Platform velocity: This is the velocity, denoted by Vs, of the plat-form along the flight path.

• Beam velocity: This is the velocity, denoted by Vg, with which thezero Doppler line sweeps along the ground.

For the satellite case, Vs is the orbital velocity, which can be expressedin either Earth centered inertial (ECI) coordinates or Earth centeredrotating (ECR) coordinates. The set of ECI axes does not move withEarth rotation, while the set of ECR axes does, as discussed in Chap-ter 12. For a circular orbit with a constant angular velocity, Vs is aconstant in ECI coordinates, but varies in ECR coordinates due to thedifference in Earth tangential speeds at different latitudes.3 From nowon, Vs is assumed to be expressed in ECR coordinates, as this simplifiessome formulations.

The velocity Vg is the speed of the zero Doppler line along the Earth’ssurface. Assuming the satellite attitude is controlled so that the beamcenter is approximately steered to zero Doppler (or other suitable ref-erence), Vg can be considered to be the velocity of the beam sweepingalong the surface. For a satellite with an altitude of 800 km, Vg is about12% less than Vs, because the orbit “circumference” is greater than thetrack circumference (see Figure 4.5). In addition, Vg varies around theorbit as the Earth’s radius and tangential speed change.

For the aircraft case, Vs is the nominal aircraft speed relative to theEarth. It can be assumed that Vg = Vs for the aircraft case. The trueaircraft speed varies, but is compensated by changing pulse repetitionfrequency (PRF) to make the “pulses” evenly spaced on the ground.

2 Strictly speaking, velocity is a vector, but in this book, “velocity” is loosely used tomean “magnitude of the velocity vector.” The term “velocity vector” is used when the truevector velocity is meant.

3 The difference of Vs between ECR and ECI coordinates is only a few percent. Theconversion of this velocity between these two sets of coordinates is treated in Chapter 12.The orbit model can express Vs accurately in both ECI and ECR coordinates.


Azimuth: In the context of SAR processing, this is a direction aligned withthe relative platform velocity vector (or sensor velocity vector in ECRcoordinates). It can be considered as a vector parallel to the net sensormotion, as in Figure 4.1, or as a vector in the slant range plane, as inFigure 4.2.

Zero Doppler plane: This is the plane containing the sensor that is per-pendicular to the platform velocity vector (in ECR coordinates). It isapproximately perpendicular to the azimuth axis, where the approxi-mation comes from the fact that the platform may be climbing or de-scending. The intersection of this plane with the ground is called thezero-Doppler line. When this line crosses the target, the relative radialvelocity of the sensor, with respect to the target, is zero.

Range of closest approach: The distance from the radar to the targetvaries with time as the platform moves. When the range is a minimum(when the zero Doppler line crosses the target), it is called the range ofclosest approach, denoted by R0 in Figure 4.1.

Position of closest approach: The position of closest approach is the po-sition of the radar when it is closest to the target, as shown by pointP in Figure 4.1. Note that the target may not be illuminated when thesensor is at this point, because of beam squint.

Zero Doppler time: This is the time of closest approach, measured rel-ative to an arbitrary time origin.4 Most SAR processing algorithms,including the ones discussed in this book, register targets to positionscorresponding to their zero Doppler times, referred to as “compressionto zero Doppler.”

Beamwidth: The radar beam can be viewed as a cone and the footprintviewed as the intersection of the cone with the ground. The beam hastwo significant dimensions: its angular width in the azimuth and el-evation planes respectively. In each plane, the half power beamwidthor simply beamwidth is defined by the angle subtended by the beam“edges,” where the beam edge is defined where the radiation strength is3-dB below the maximum.5

In azimuth, with a uniformly illuminated aperture, the beamwidth isapproximately the wavelength divided by the antenna length in this di-rection. In elevation, the beamwidth governs the width of the imaged

4 For a single point target, this origin is often chosen to be the time of closest approach,for convenience of analysis.

5 In a monostatic SAR, the same antenna is used for receiving as for transmitting. Uponreception, the signal has been influenced by the beam pattern twice, so the received signalstrength at the beam edges is 6-dB below the maximum strength.


“range swath.” Its formula is more complicated, as the elevation radia-tion pattern is usually shaped with a non-uniform illuminated aperture.

The radar beamwidth is not affected by Earth curvature or rotation, butit is shown later that the exposure time, the azimuth bandwidth and theresolution are affected (see Section 4.5.5).

Target trajectory: The range from the radar to a target changes duringthe time that the target is illuminated by the radar beam. When drawnon a two-dimensional plot vs. range and azimuth, the locus of receivedtarget energy is curved, and is referred to as the target trajectory insignal space (see Panel (b) of Figure 4.2).

Beam center crossing time: This is the difference between the time whenthe zero Doppler line crosses the target and the time when the beam cen-terline crosses the target. It is positive when the beam points backwardsrelative to the zero Doppler line, in other words, when the beam centercrosses the target after the zero Doppler line crosses the target. It issometimes referred to as the beam center offset time.

Signal space and image space: There are two two-dimensional spacesused for the SAR data in the signal processor. The signal space containsthe received SAR data and the image space contains the processed data.If the data in the signal are displayed, features imaged by the radarare not recognizable. The features will emerge only after extensive pro-cessing is performed on the input data. The processed data are definedin the image space since the data now form a meaningful image. SeeFigure 4.2.

Radar

(c) SAR image space

Range Range

Azim

uth

Beamcentre A A

B

D

(a) Data acquisition (b) SAR signal space

Azim

uth

Azim

uth

A

B

C DCC

B

D

R ' R'0

R

Figure 4.2: Definitions of range at different points in the SAR system andprocessor.

Range: First, the generic term “range” can mean slant range or groundrange, as shown in Figures 4.1 or 4.3. The former is measured along


the radar line-of-sight, while the latter is measured along the ground.Because all SAR processing operations use the slant range definition,the usual convention is that “range” defaults to “slant range” when notspecified.

Second, there are two cases to consider in the definition of range: signalspace and image space. In signal space, range is a distance measuredfrom the radar antenna to the target on the ground — it is not orthogonalto the azimuth axis unless the squint angle, as defined in Figure 4.1, iszero. This range direction is called the radar “line-of-sight” — it isapproximately along the beam centerline or boresight, but the directionvaries with the location of the target within the beam. After the SARprocessing, the image is registered to the azimuth position of closestapproach, and to the range of closest approach. At this point, the rangeaxis is perpendicular to the azimuth axis.6

Figure 4.2 shows the difference between range in the input signal spaceand in the final image after compression to zero Doppler. Panel (a)shows the physical coordinates, with four targets on the Earth’s surface.The antenna is assumed to look ahead (i.e., squinted forward), as inFigure 4.1, except that the antenna is looking left in this figure. Theradar is moving down the page, and the beam center crosses Targets Aand B at the same time. Later it crosses Target D and finally, TargetC. The range R is measured along the radar beam, as in Figure 4.1.

In Panel (b), the target trajectories are shown in the signal memory atthe input to the SAR processor. They are located according to theirrange (horizontally) and their beam center crossing time (vertically). Inthis memory, the range R′ is relative to the first sample, as controlledby the range gate delay, RGD

R = R′ + RGD c/2 (4.1)

The RGD is the difference in time between the transmission of the pulseand the recording of the first sample of the associated echo, and c =2.997925×108 m/s is the speed of light.

In Panel (c), the targets are focused in image space to positions accordingto their zero Doppler times. Range now lies in the zero Doppler direction,like R0 in Figure 4.1. The zero Doppler time is independent of theantenna squint angle, so the target positions in the final image do notdepend upon the squint angle. Similar to the signal space, the rangevariable R′

0 is relative to the first processed sample, and for a giventarget, its slant range of closest approach is R0 = R′

0 + RGD c/2.

6 Between the signal and image stages of the SAR processing, range can either be line-of-sight or orthogonal to azimuth.


Slant range plane: This is the plane containing the relative (ECR) sensorvelocity vector and the slant range vector for a given target. The orien-tation of this plane, relative to the local vertical, changes with targetsat different ranges R0, so the plane is only used for conceptual purposes.

Ground range: This is the projection of slant range onto the ground. If theimage is to be presented in a map-like format, the slant range variable isconverted to ground range. Assuming that the data is registered to zeroDoppler, ground range is the direction orthogonal to the azimuth axisand parallel to the Earth’s surface with its origin at the nadir point, asshown in Figure 4.1.

Squint angle: This is the angle θsq that the slant range vector makes withthe zero-Doppler plane,7 and is an important component in the descrip-tion of the beam pointing direction. It is measured in the slant rangeplane. If viewed from above (i.e., projected to the ground plane), itcoincides with the beam yaw angle. The squint angle depends upon thetarget range R0 for a given beam pointing direction.

Note that the zero Doppler time of a target is independent of the squintangle, but the beam center crossing time does depend upon the squintangle. Since the zero Doppler plane in ECR coordinates (from whichθsq is measured) accounts for Earth curvature and rotation, θsq is notthe squint angle in inertial space. The computation of θsq from beampointing and Earth/platform geometry is discussed in Chapter 12.

Cross range: This is a direction orthogonal to the radar’s line-of-sight. Un-less the squint angle is zero, the cross range and azimuth axes are notparallel. Theoretically, “azimuth” resolution is developed along the crossrange axis instead of the azimuth axis. But in stripmap SAR, the crossrange resolution does not differ significantly from the azimuth resolution,because the squint angle is usually small. Since this book concentrates onprocessing stripmap data, the generic definition of “azimuth resolution”is used throughout the book, and the cross-range/azimuth distinction ispointed out as necessary.

4.2.2 Satellite Slant Range vs. Ground Range Geometry

The SAR focusing steps produce an image in slant range vs. azimuth coor-dinates, as in Figure 4.2(c). It is often desirable to resample the image tocoordinates corresponding to those of a map or an optical sensor, where therange and azimuth axes have equal scales. In this post-processing step, the

7 Sometimes in the literature, the squint angle is measured from the sensor velocity vectorinstead of the plane of zero Doppler.


concept of ground range arises, which is a distance measured along the sur-face of the Earth, approximately perpendicular to the azimuth direction. Theconversion from slant range to ground range provides a geometrically-realisticimage, approximately aligned with the radar track.8

Target

Radar

E

Re Re

h

Center of earth

R0

G θi

βe

θn

TargetGround

Ver

tical

Radar beam

∆R

∆Gθi

θi

Blowup of target area

Figure 4.3: Satellite cross-track geometry, illustrating slant range Rvs. ground range G, and the associated sample spacings ∆R and ∆G.

For the case of zero squint and a locally circular Earth, the relationshipbetween slant range and ground range coordinates is illustrated in Figure 4.3.Let the line joining the radar and the Earth’s center intersect the Earth atPoint E. Ground range is the arc length along the Earth’s surface from E tothe target. It is marked by G in the figure, and βe is the angle subtendedby G at the Earth’s center. Re is the local radius of the Earth, taken atthe scene center. Also, let h be the altitude of the platform with respect toE, θn the off-nadir angle and θi the incidence angle. For the locally circularEarth approximation, the geometric variables in Figure 4.3 are related by thelaw of sines and cosines

Re

sin θn=

Re + h

sin θi=

R0

sinβe(4.2)

8 If target altitudes are available, they can be included in the ground range conversion,making the converted image more geometrically correct.


cosβe =R2

e + (Re + h)2 −R20

2Re (Re + h)(4.3)

and byG = Re βe (4.4)

The incidence angle θi is larger than the off-nadir angle θn by the angle βe.The difference is negligible in the airborne case, but is a few degrees in thesatellite case.

An expanded view of the target area is shown on the right side of Fig-ure 4.3. The length ∆R represents the distance between two slant rangesamples and the dotted line is a small part of a “constant range” circle. Theground is assumed to be locally flat and the dashed line is the local vertical.Then ∆G is the distance along the ground represented by the range sample.

260 280 300 320 340 360 380 400 420840

860

880

900

920

Sla

nt r

ange

(km

) R = 6368 kme

h = 800 km

260 280 300 320 340 360 380 400 420

20

22

24

26

28

30

32

Inci

denc

e an

gle

(de

g)

260 280 300 320 340 360 380 400 42025

30

35

40

Ground range (km) −−−>

Gro

und

rang

e re

sol.

(m

)

Slant range resolution = 13.6 m

Figure 4.4: The variation of ground range resolution for the RADARSAT-1W1 beam.

For a given radar mode, the slant range sample spacing ∆R is a constant,and the ground sample spacing ∆G varies with the local incidence angle

∆G =∆R

sin θi(4.5)


The quantities ∆R and ∆G can be thought of as one range resolution cell,and sin θi gives the ground range resolution as a factor of the slant rangeresolution. Equation (4.5) shows how the ground range resolution degradeswith a decreasing off-nadir angle, with the worst case occurring when θi iszero. This is unlike optical sensors, which enjoy the best resolution whenlooking straight down.

The variation in ground range resolution with range is largest when theincidence angle is small. It is interesting to consider the RADARSAT-1 beams,as RADARSAT exhibits a large variety of beams with different incidence an-gles. There are seven regular beams, called S1 to S7, and three wide swathbeams, called W1 to W3 [3]. The largest variation of ground range resolutionoccurs in Beam W1, which is the widest non-ScanSAR beam with the smallestincidence angle. Figure 4.4 shows the slant range, incidence angle and groundrange resolution for the W1 beam. As the slant range resolution is 13.6 m forthis RADARSAT-1 beam, the ground range resolution varies from 27 m at farrange to 40 m at near range.

4.2.3 Satellite Orbit Geometry

A generic satellite geometry is illustrated in Figure 4.5. The satellite orbit isapproximately a low-eccentricity ellipse, defined by the length and hour angleof the semi-major and semi-minor axes, and by the inclination of the orbitplane with respect to the equator.9

The choice of the orbit parameters for remote-sensing SARs involves anumber of complicated tradeoffs, and a few considerations are mentioned here.The orbit altitude above the Earth’s surface is often around 800 km, beinga compromise between power requirements and atmospheric drag. The orbiteccentricity is close to zero so that the altitude is nearly constant around theorbit. If the orbit is circular, the square of the orbit period P is related tothe cube of the orbit radius Rs by

P 2 =4π2 R 3

s

µe(4.6)

where µe = 3.9860× 1014 is the gravitational constant of the Earth, and theperiod is expressed in seconds. This corresponds to a satellite angular velocityof

ωs =2π

P=

√µe /R 3

s (4.7)

9 The inclination is the angle that the normal of the orbit plane makes with the vectorfrom the center of the Earth to the north pole. In Figure 4.5, by applying the right hand rule,the normal points westward and slightly to the south of the equator; hence the inclinationangle is greater than 90◦.


Lat. and long. lines are drawn 10 deg. apart (1100 km at the equator)

RADARSAT−1 altitude ≈ 800 kmInclin = 98.6 deg. period ≈ 100 min

The Earth rotates 25 deg. to the eastevery orbit (2830 km at the equator)

Equator

Satellite orbit

Radar beam S Radar beam

Imagedswath

50−150 km

Satelliteorbit

Figure 4.5: Earth/Satellite Geometry with RADARSAT-1 parameters.

radians/s and to a satellite inertial speed of

Vs = Rs ωs =√

µe /Rs (4.8)

For example, a satellite with a nominal altitude of 800 km (i.e., an orbitradius of 7168 km) has a period of 100.66 minutes, an angular velocity of1.0403 milliradians/s, and a tangential speed of 7457 m/s, assuming a circularorbit.

The orbit inclination is usually set at around 98◦, so that the orbit issun-synchronous. With this inclination, the Earth’s oblateness causes theorbit plane to precess once per year, so that it has a fixed angle with respectto the sun. This simplifies the power collection strategy of the solar panels,an important issue as the radar coverage and signal-to-noise ratio (SNR) isdirectly proportional to the available power. An inclination greater than 90◦

indicates that the satellite is orbiting the Earth in a westerly direction. As apoint on the Earth’s surface is moving eastward, the average relative targetvelocity and the average beam velocity is higher than if the satellite weremoving eastward. This gives a slightly higher coverage rate over the Earth’ssurface.

4.3. THE RANGE EQUATION 119

4.3 The Range Equation SynApt RangeEqn.tex

The single most important parameter in SAR processing is the slant rangefrom the sensor to the target. This range varies with azimuth time, and isdefined by the so-called range equation.10 As the sensor approaches the targetthrough the motion of the radar platform, the range decreases with everypulse. After the sensor passes the target, the range increases with every pulse.

This change in range has two important implications. It causes a phasemodulation from pulse to pulse, which is necessary to obtain fine azimuthresolution in the SAR processor. However, it also causes the received data tobe skewed in computer memory, an effect called range cell migration (RCM).As shown later, this range/azimuth coupling must be taken into account inthe SAR processing.

To get the exact range equation, one must be able to model the sensormotion, plus the motion of the target or surface, if any. This can get quitecomplicated but, in most cases, the simple geometry of Figure 4.1 can be used,with an appropriate choice of sensor velocity. This results in a hyperbolic formof the range equation, which allows the signal properties in various domains tobe represented conveniently and the processing equations to be derived easily(see Chapter 5). For these reasons, this section focuses its discussion on ahyperbolic model.

4.3.1 Hyperbolic Form of the Range Equation

To develop the hyperbolic form of the range equation, a simplified form of thegeometry of Figure 4.1 is considered. In this case, the flight path is assumedto be locally straight, and the Earth is assumed to be locally flat and notrotating. This is a good model for the aircraft case, where the distancesare much shorter than the satellite case, and the aircraft follows the Earth’satmosphere as it rotates.

Assuming a velocity Vr pertaining to the simplified case, the distance Xin Figure 4.1 equals Vr η, where η is the azimuth time referenced to the timeof closest approach. Then, using the Pythagoras theorem, the range to thetarget R(η) is given by the hyperbolic equation:

R 2(η) = R 20 + V 2

r η2 (4.9)

where R0 is the slant range when the radar is closest to the target, that is,R0 is the range of closest approach.

10 The range equation should not be confused with the radar equation (1.1), which pertainsto transmitted power and received SNR.


For the aircraft case, the beam is assumed to be stationary with respectto the flight direction, so that the geometry of Figure 4.1 remains stable fora given radar scene. The variable Vr is the nominal aircraft speed, andalso equals the speed of the beam footprint along the surface. However, inthe satellite case, the geometry is more complicated, as the orbit is curved,the Earth’s surface is curved, and the Earth is rotating independently of thesatellite orbit [4, 5].

When an accurate, curved-geometry model is used, it turns out that therange equation is still very close to a hyperbola, within the limits of the targetexposure time. Thus, the parameter Vr can still be used as a type of “velocity”in the satellite case, provided it is interpreted in a special way. Specifically,when R 2(η) is expressed as a power series in η, the cubic and higher termsare very small for typical remote sensing SARs, particularly the C-band andhigher frequency SARs. Then, V 2

r is the quadratic coefficient of (4.9), and canbe calculated from a geometry model as half the second derivative of R 2(η).

With this assumption, the hyperbolic range equation (4.9) holds for asatellite, except that Vr is not a physical velocity, but a pseudo-velocity,selected so that the hyperbola (4.9) models the actual range equation. Theparameter Vr has been called the “radar velocity,” “effective velocity” or“speed parameter” by several authors [4]. A better term may be “effectiveradar velocity,” although some authors prefer not call it a velocity at all [6].

Important differences from the aircraft case include the fact that Vr varieswith range, and varies slowly with azimuth as the satellite orbit and the Earthrotation component changes. Its numerical value lies between the satelliteplatform velocity, Vs, and the lower speed, Vg, with which the beam movesalong the ground. The hyperbolic model is adequate over the duration of thetarget exposure time, which is typically in the order of a second.

To see how the radar velocity Vr relates to the physical velocities of thesatellite and the beam, it is useful to consider the two geometry models inFigure 4.6. Panel (a) shows the radar/beam geometry in the slant rangeplane, assuming a curved orbit and a curved Earth. The satellite moves withtangential velocity Vs and the beam footprint moves along the surface with avelocity Vg. Assume that the beam centerline CB makes a squint angle θsq

with respect to the zero Doppler line CA, and illuminates a point, B, on theground.

Now, can this curved geometry be related to the rectilinear geometry ofPanel (b)? A rectilinear geometry can be formed out of the curved geometryif tangential lines are drawn through points C and A, and the angle θsq isincreased by swinging the vector CB outwards, until it meets the tangentialline through A (keeping its length R the same). Using the primed notationfor the rectilinear geometry in Panel (b), the vectors C ′A′ and C ′B′ are the

4.3 The Range Equation 121

D

qsq

Satellite

Orbit

Earth's

surface

Xg

AB

C

qr

A' B'

C' D'

Vr

Vr

Vg

Vs

(a) Curved earth geometry (b) Rectilinear geometry

R0

R0

R0

R0

RR

Xr

qg

Xs

Figure 4.6: Approximation of curved Earth geometry by rectilinear geometry.

same length as their associated vectors in Panel (a), but the distance Xr

is larger than Xg because the angle ACB has increased. The new angleA′C ′B′ is used in SAR processing, and it is called θr. Distances and angleshave been distorted in Panel (b), but time is unaltered, i.e., the time takenfor the satellite to move from C to D is the same as that from C ′ to D′,and is the same as it takes the beam centerline to go from A to B and fromA′ to B′.

By comparing the two geometries in Figure 4.6, it can be seen that A′B′ >AB, and that C ′D′ < CD, so that the effective radar velocity Vr > Vg

and Vr < Vs. Note that Vg < Vs models the property that the satelliteattitude changes by 2π radians every orbit. As shown in Appendix 4A, anapproximation for Vr is

Vr ≈√

Vs Vg (4.10)

Note that Vs and Vg vary with orbit position and range, because the magni-tude and relative direction of the Earth rotation changes and Vr is expressedin ECR coordinates. In this way, Vr changes with time and range, and must beupdated around the orbit. The main approximation in (4.10) comes from thefact that the orbit is not circular. An example in Section 13.3 indicates thatthe approximation is accurate to approximately 0.6% for typical RADARSATparameters.

While the approximation (4.10) is not accurate enough for calculating theazimuth matched filter coefficients in precision SAR processing, it is adequate


for analysis purposes such as finding the exposure time and Doppler band-width in Section 4.5.5. For precision SAR processing, the velocity Vr mustbe calculated from a refined geometric model, as shown in Chapter 12.

Note that Vr and Vg vary with range. They are calculated at the zeroDoppler point of the target, so they are a function of R0, and do not vary alongthe target exposure. That means Vr is a constant for a particular target, animportant fact to note in a point target simulator. This property is also usedwhen the target’s spectrum is derived in Chapter 5.

Which velocity is to be used depends upon the application. As shownlater, Vr is the velocity used to obtain the RCM and the azimuth FM rate inthe azimuth SAR processing, and Vs is used to obtain the Doppler bandwidth.Finally, when ground resolution and distance are concerned, Vg is used.

4.3.2 Relationships between Velocities and Angles

To help understand the physical meaning of the various velocities, distancesand angles in Figure 4.6, it is useful to examine the relationship between:

• Vs, Xs and θsq in curved Earth geometry along the orbit,• Vg, Xg and θg in curved Earth geometry on the ground, and• Vr, Xr and θr in rectilinear geometry.

Using small angle approximations in the curved Earth geometry of Fig-ure 4.6(a), the squint angle θsq is defined as

sin θsq =Xg

R(η)= − Vg η

R(η)(4.11)

and a new angle θg is defined as

sin θg =Xs

R(η)= − Vs η

R(η)(4.12)

The negative signs are due to the fact that for a positive θsq and θg, theradar looks ahead, and hence the position of closest approach, where η isdefined to be zero, has not been reached yet. In other words, η is negative forpositive squint angles. Note that θg is not the radar incidence angle, becauseFigure 4.6 depicts the geometry in the slant range plane.

In the rectilinear geometry of Figure 4.6(b), a new squint angle θr canbe defined that is useful for analysis and, sometimes, for data processing.Rearranging the range equation (4.9) as

R0 =√

R2(η)− V 2r η 2 = R(η)

√1−

[Vr η

R(η)

]2(4.13)

4.4. SAR SIGNAL IN THE RANGE DIRECTION 123

the new squint angle is defined in the equivalent rectilinear geometry using

sin θr =Xr

R(η)= − Vr η

R(η)(4.14)

Similar to the variable Vr, the angle θr is not a physical angle, but servesa useful purpose in SAR signal analysis. Since this is the angle often usedin SAR system analysis instead of the physical squint angle, it is called the“squint angle” for brevity.

Combining (4.11) and (4.14), and using small angle approximations, thefollowing ratios are equal

θsq : θr : θg = Vg : Vr : Vs = Xg : Xr : Xs (4.15)

From the above equation and (4.10), it is found that the squint angle θr is ascaled version of the physical squint angle θsq

θr =Vr

Vgθsq =

Vs

Vrθsq (4.16)

For a typical satellite case, θr is about 6% larger than θsq, but the cosines ofthese two angles differ by less than 0.08% for a squint angle as large as 6◦.However, it is important to distinguish between these angles when their sinesor tangents are invoked.

The following formulae are also useful. Using (4.14), cos θr can be written

cos θr =

√1−

[Vr η

R(η)

]2

(4.17)

and from Figure 4.6, it is seen that

R0 = R(η) cos θr (4.18)

Since the hyperbolic equation assuming rectilinear geometry is used inthe processing, the angle θr is relevant, but the angles θs and θg are rarelyused. For example, the cross track direction is at an angle θr with respect toazimuth (Section 5.5 addresses this), hence the effective velocity and groundvelocity components in the cross range direction are Vr cos θr and Vg cos θr

respectively, and the latter velocity component is used to derive the (cross-range) resolution.

4.4 SAR Signal in the Range Direction. SynApt RangeDir.tex

It is convenient to first consider the SAR signal in the range or beam direction,then in the azimuth direction. Later, the signal coupling between the range


and azimuth coordinates will become apparent. In the range direction, theradar sends out an FM pulse given by

spul(τ) = wr(τ) cos

(2π

N∑

n=0

Pn τn

)(4.19)

where τ is the range time and Pn are the phase coefficients, when the signalphase is expressed as a power series. The pulse envelope is usually approxi-mated by a rectangular function

wr(τ) = rect(

τ

Tr

)(4.20)

where Tr is the pulse duration. Even if the pulse envelope is not quite rect-angular, it is usually safe to assume a rectangular envelope when defining thematched filter for the processing. In early radar systems, the pulse was gen-erated by an analog Surface Acoustic Wave (SAW) device [7], but now it isgenerated by a digital synthesizer.

The most commonly used pulse is a linear FM one

spul(τ) = wr(τ) cos{

2π f0 τ + π Kr τ2}

(4.21)

where Kr is the FM rate of the range pulse. Here τ is referenced to thecenter of the pulse for convenience.11 In this form, the phase coefficients areP0 = 0, P1 = f0, P2 = ±Kr/2, and Pn = 0 for n > 2. For ease of analysis,this simple linear FM form is assumed from now on, unless otherwise stated.

The instantaneous frequency of the signal spul(τ) varies with fast timeτ . For a linear FM pulse, given by (4.21), the instantaneous frequency isfi = f0 +Krτ . As the radar wavelength is c/fi, it also varies within the pulse.But it is assumed to be the wavelength corresponding to the center frequency,λ = c/f0, which unless otherwise stated is the definition of λ used in thisbook.

The radar designer has a choice of the sign of the FM rate, i.e., the signpreceding Kr. When the sign is positive, the pulse is an “up chirp” becausethe pulse frequency increases with time. Similarly, when the sign is negative,the pulse is said to be a “down chirp.” The direction of the chirp neitheraffects the structure of the SAR processing nor the quality of the processedimage.

The signal bandwidth is a very important parameter as it governs the rangeresolution and the sampling requirements. The signal bandwidth is given by

11 Sometimes the leading edge of the pulse is taken as the range time reference. Care mustbe taken to define the range time reference, to avoid azimuth focusing and range registrationerrors.

4.4 SAR Signal in the Range Direction 125

KrTr and, when the demodulated received signal is sampled, the complexsampling rate Fr must be higher than the bandwidth to prevent aliasing.The range oversampling ratio αr is the sampling rate divided by the signalbandwidth and, in practice, is usually between 1.1 and 1.4.

Thus the range complex sampling rate is

Fr = αr |Kr|Tr (4.22)

The range resolution is approximately 1/(|Kr|Tr) in seconds, or c/(2KrTr)in meters. The pulse compression gain is the time-bandwidth product, KrT

2r ,

as derived in Chapter 3.

Figure 4.7 shows how the data are acquired across a range swath. Theradar beam has a certain 3-dB width in the elevation plane, called the “el-evation beamwidth.” The beam illuminates a section of the ground, lyingbetween “near range” and “far range” in the figure. At a given instant intime, the pulse has a finite extent, between the two dashed arcs in the figure.

ORadar

Nearrange Far

range

NadirEarth’s surface

Lead

ing

edge

of p

ulse

Trai

ling

edge

of p

ulse

ElevationBeamwidth

Figure 4.7: Illustrating the radar beam’s 3-dB elevation beamwidth and theradar pulse spreading outward in concentric spheres.

The pulse expands outwards in concentric spheres, expanding at the speedof light. The lower dashed arc in Figure 4.7 shows the pulse at the instant itreaches the ground, at a time t1 after it leaves the transmitting antenna. Attime t2, a fraction of a millisecond later, the trailing edge of the pulse passes


the “far range” point. In this way, each point on the ground, between nearrange and far range, is illuminated by the beam for a duration of Tr. Notethat, at any instant, only a portion of the beam footprint is illuminated by thepulse, and this portion sweeps outwards at the speed of light divided by sin θi,which θi is the local beam incidence angle shown in Figure 4.3. The reflectedenergy at any illumination instant is a convolution of the pulse waveform andthe ground reflectivity gr within the illuminated patch:

sr(τ) = gr(τ)⊗ spul(τ) (4.23)

This energy arrives back at the receiving antenna between times 2 t1 and2 t2. The receiver starts sampling a few microseconds before 2 t1 and finishes afew microseconds after 2 t2, thereby recording the ground reflections betweennear range and far range. If the elevation beam is too wide in relation to theinterpulse period, range ambiguities may occur, which result from the mixingof reflected energy from consecutive pulses at the receiver.

Consider a point target at a distance Ra away from the radar, with amagnitude A′0, which models the backscatter coefficient, σ0. This means thatgr(τ) = A′0 δ(τ−2Ra/c) in (4.23), where 2Ra/c is the delay time for thatreflector. The signal received from the point target, from (4.21) and (4.23), isthen:

sr(τ) = A′0 spul(τ−2Ra/c)

= A′0 wr(τ−2Ra/c)

cos{

2π f0 (τ−2Ra/c) + πKr (τ−2Ra/c)2 + ψ}

(4.24)

The scattering process may cause a phase change in the radar signal uponreflection from the surface, which is accounted for by the variable ψ in theequation. The present analysis is unaffected, as long as the phase change isconstant for a given reflector within the radar illumination time. Note thatall variables in (4.24) are real.

The echo sr(τ) contains a high frequency component cos(2π f0 τ) whichis the radar carrier frequency, and a low frequency component consisting ofthe rest of the terms in (4.24). Appendix 4B shows how the high frequencycomponent is removed by a quadrature demodulation process, so that themaximum signal frequency is in the order of the transmitted signal bandwidth.

The data often has a radiometric variation in the range direction causedby several factors. First, the echo power is inversely proportional to the fourthpower of the slant range. Second, the elevation beam pattern of Figure 4.7is not uniformly weighted. The antenna gain at the upper elevation angle issometimes designed to be higher than that at lower angles, to compensate

4.5. SAR SIGNAL IN THE AZIMUTH DIRECTION 127

for the 1/R4 law. Third, the reflectivity of the ground is a function of thebeam incidence angle θi. Finally, there is a geometrical 1/ sin θi term as theground area is converted to an equivalent area, perpendicular to the radarbeam. These effects, if uncorrected, will cause a variation of intensities acrossthe range swath in the processed image. The correction can be performed inthe processor, assuming a knowledge of the above factors.

4.5 SAR Signal in the Azimuth Direction. SynApt AzimDir.tex

In the previous section, the signal received from a single pulse was discussed.As the sensor advances along its path, subsequent pulses are transmitted andreceived by the radar. The pulses are transmitted every 1/PRF of a sec-ond, where PRF is the pulse repetition frequency. But before getting too farinto a discussion of azimuth parameters, an intuitive explanation of Dopplerfrequency is needed.

4.5.1 What is Doppler Frequency in the SAR Context?

Consider a radar that transmits a pure tone, which is generated by a localoscillator. The signal is transmitted through the antenna, and the resultingelectromagnetic (EM) wave travels to the ground where it hits an object andis reflected (scattered). The reflected EM wave travels back to the antenna,where it is converted into a voltage. The received signal has the same wave-form as the transmitted signal, but is much weaker and has a frequency shiftgoverned by the relative speed of the sensor (antenna) and the scatterer. Ifthe distance from the antenna to the scatterer is decreasing, the frequency ofthe received signal increases. On the contrary, if the distance to the scattereris increasing, the frequency of the received signal decreases. The situation isanalogous to a high (low) frequency heard from the siren on an approaching(receding) ambulance.

It is this frequency, governed by the relative speed of the sensor andthe target, which is called the SAR Doppler frequency, in analogy with thewell-known effect in physics. This discussion has taken a few shortcuts, butdoes give an intuitive description of the Doppler effect in coherent radars.The radar is ”coherent” if the consistent timing of the local oscillator allowsone to observe the change in phase and frequency of the received signal veryaccurately.

Omitted from the above discussion are the following:

1. The radar system generates and transmits a finite-duration pulse, rather


than a pure tone.

2. The radar electronics upconverts the pulse to a very high frequency (theradar carrier frequency) and subsequently downconverts the receivedsignal to the original frequency (or even lower, to baseband).

3. The pulse has a linear FM waveform rather than a tone and, after re-ception and downconversion, it is converted (compressed) to a sharpimpulse that has the approximate shape of a sinc function.

4. The Doppler frequency is a function of the carrier frequency rather thanthe original baseband pulse frequency.

5. The pulses are repeated at a precisely controlled time interval, calledthe pulse repetition interval or PRI. The inverse of this interval is thepulse repetition frequency or PRF.

Despite these simplifications in the analogy, the concept of Doppler fre-quency is still valid in the modulated, pulsed radar. The effect of the pulsesis to sample the waveform representing the Doppler-shifted, received signal,with the sampling frequency being the PRF.

The sampling of the continuous signal creates an aliasing effect when theDoppler frequency exceeds the sampling frequency (the samples are complex,so the aliasing rules, including the folding or Nyquist frequency, follow therules for complex signals). It is the sampling that profoundly affects how theDoppler frequency is observed and how it is estimated.

4.5.2 Coherent Pulses

The transmitted pulses are evenly-spaced, as shown in Figure 4.8, with eachpulse represented by (4.19) or (4.21). “Coherent” means that the start timeand phase of each pulse is carefully controlled. The receiver and demodulatormust also maintain high timing accuracy. This coherency is an importantproperty, necessary to obtain high azimuth resolution in the SAR system.

Pulse Repetition Interval = 1/PRF

τl

Transmit Receive

Figure 4.8: Timing of transmitted radar pulses (not to scale).

4.5 SAR Signal in the Azimuth Direction 129

When the radar is not transmitting, it can receive echoes reflected backfrom objects and surfaces on the ground. A time line of transmitted pulsesand received echoes is shown in Figure 4.9. In an airborne case, each echois received directly after the transmitted pulse, before another pulse is trans-mitted. In a satellite-borne case, the echo from a specific pulse is receivedafter 6 to 10 intervening pulses have been transmitted, because of the muchlonger ranges involved. For a satellite SAR with a PRF of 1700 Hz and a pulseduration of 34µs, the time available to receive the echo is 554µs, althougha few µs are needed at either end of the receive window to switch the signalpath. This time allows a slant range swath width of up to 80 km, althoughother constraints usually keep it smaller, such as a varying satellite altitudethat requires the receive window to be moved.

Pulse Echo

Time

Mag

nitu

de

Figure 4.9: Illustrating the transmit and receive cycles of a pulsed radar.

Between successive pulses, the radar platform advances in the azimuthdirection by a small amount. The separation between the footprints of eachpulse, also known as azimuth sample spacing in the input data, is the footprintvelocity divided by the PRF. For an aircraft, the footprint velocity equals theplatform velocity, but for a satellite, the footprint velocity is about 12% lessthan the satellite velocity, as discussed in Section 4.2. The separation offootprints is generally about 40% of the SAR antenna length, although it canbe smaller than 40% in aircraft cases (because aircraft SARs do not come closeto hitting the range and azimuth ambiguity limitations [8]). The separationis about 4 m for the ERS/ENVISAT satellites and 5 m for RADARSAT.

4.5.3 Choice of PRF

The azimuth sampling rate or PRF is selected by considering the followingparameters and criteria:

Nyquist sampling rate: The PRF should be larger than the significant az-imuth signal bandwidth, as it corresponds to a complex sampler. Theazimuth oversampling factor Os is usually about 1.1 to 1.4. If the PRFis too low, azimuth ambiguities caused by aliasing will be troublesome.The azimuth oversampling ratio is usually higher than the range over-

sampling ratio because the azimuth spectrum rolls off more slowly thanthe range spectrum.


Range swath width: The sampling window can be up to 1/PRF− Tr

seconds long, corresponding to a slant range interval of (1/PRF−Tr) c/2meters. The PRF should be low enough so that most or all of the nearrange to far range interval illuminated by the beam (the swath width)falls within the receive window, as shown in Figure 4.9. If the PRF is toolarge in relation to the echo duration, range ambiguities occur becauseof echoes from different pulses overlapping in the receive window. If therange ambiguities are too large and the PRF cannot be lowered, theantenna’s elevation beamwidth can be reduced by making the antennawider or by adjusting the antenna weighting.

Receive window timing: The significant energy from the ground must ar-rive at the receiving antenna between the pulse times. Unlike the pre-vious criterion, which concerns the length of the receive window, thisconcerns the start time of the window. The start time is particularlyaffected by the PRF in the satellite case when a given transmitted pulseis not received until several pulse intervals have elapsed.

Nadir return: Sometimes a significant amount of energy arises from groundreflections at the nadir point, and causes a bright streak in the image.This nadir return is bright because, when the incident angle is small,each range cell covers a large area and specular reflections occur. Thisenergy is unwanted in satellite SARs as it is range ambiguous, and it isusually possible to choose a PRF for which the nadir return does notfall within the receive window (or at least not within the main part ofthe imaged swath).

Each of these criteria is in conflict with some or all of the other criteria, soa compromise is needed, especially in the satellite case. The tradeoff involvesmany of the SAR system parameters, notably the platform height and velocity,operating range, radar wavelength, antenna length and swath width. Thetradeoff is mainly between range ambiguity levels, azimuth ambiguity levelsand swath width, and results in a lower limit for the antenna area [8, 9].However, there have been systems built which accept more compromises, anduse an antenna area smaller than this lower limit [10].

In the aircraft case, these restrictions are usually not a limiting factor,because the platform velocity is lower, and the beam geometry restricts theswath width to well below the ambiguity limit. This means that the PRF canbe made higher than that needed to support the azimuth bandwidth. A higherPRF allows the transmission of a higher average power, without raising thepeak power or the pulse length, thereby improving the SNR. When this occurs,the azimuth signal can be filtered and the sampling rate reduced, to increasethe efficiency of subsequent processing steps. This filtering and sample rate


reduction is called “presumming,” which lowers the PRF, resulting in moreefficient SAR processing.

4.5.4 Azimuth Signal Strength and Doppler History

As the platform advances, a target on the ground is illuminated by manyhundreds of pulses. For each pulse the strength of the signal varies, primarilybecause of the azimuth beam pattern.12 The azimuth beam pattern for a zerosquint case is shown in the top part of Figure 4.10 for three positions of thesensor, as seen in the slant range plane. At sensor position A, the target isjust entering the main lobe of the beam. The received signal strength is shownin the middle part of the figure. The signal strength increases until the targetlies in the center of the beam, as shown at position B.

Azimuth −−−−>

Beam pattern

Target

Sensor position A B C

targetslantrange

R0

Azimuth time −−−−>

Received signal strength

0

Azimuth time −−−−>

Frequency

0

Figure 4.10: Azimuth beam pattern and its effect upon signal strength andDoppler frequency.

After the beam center crossing time, the signal strength decreases untilthe targets lies in the first null of the beam pattern, when the sensor is at

12 Any small dependence of the complex backscattering constant on the viewing angle isignored in this book.


position C. From then on, a small amount of energy will be received fromside lobes of the beam pattern. The energy in the outer edges of the main lobe,as well as the energy in the side lobes, contribute to the azimuth ambiguities inthe processed image, as discussed in Chapter 5. Doppler centroid estimationerrors aggravate these ambiguities (see Chapter 12).

The bottom part of the figure shows the Doppler frequency history of thetarget. The Doppler frequency is proportional to the target’s radial velocitywith respect to the sensor. When the target is approaching the radar theDoppler frequency is positive, and is negative when the target is receding.Thus the frequency vs. time curve has a negative slope.

Revisiting the middle part of Figure 4.10, note that the received signalstrength is governed by the azimuth beam pattern. As most SAR antennas areunweighted in the azimuth plane, the one-way beam pattern is approximatelya sinc function [9]

pa(θ) ≈ sinc(

0.886 θ

βbw

)(4.25)

where θ is the angle measured from the beam center in the slant range plane,βbw is the azimuth beamwidth, 0.886λ/La, and La is the antenna lengthalong the azimuth direction. The received signal strength is given by thesquare of pa(θ) because of the two-way propagation of the radar energy, andis usually expressed as a function of azimuth time η

wa(η) = p 2a { θ(η) } (4.26)

Equation (4.11) shows how θsq is related to the azimuth time η.13

The beam center crosses the target when the radar is at point B inFigure 4.10. The received signal strength is greatest at that time, and in thecase drawn the beam center has zero squint. However, this is usually not thecase, as a certain amount of beam squint is inevitable, due to such causesas platform or antenna attitude, beam alignment, Earth curvature, Earthrotation and, in the aircraft case, side winds.

In the general nonzero squint case, the target is illuminated by the beamcenter at a “beam center crossing time,” ηc, referenced to the time of zeroDoppler. When the beam squints forward, ηc is negative, following the pre-vious convention. Conversely, when the antenna squints backward, ηc is pos-

13 At this point it is useful to review the different squint angles used. The angle θ ismeasured from the beam center and it varies with time η. The angles θsq and θr definedin Figure 4.6 are measured from the plane of zero Doppler and they also vary with η.

In contrast, the angles θsq,c and θr,c, used in (4.27) and (4.28), are constants over theexposure time of a target, but are a function of the pointing angle of the antenna, whichmay vary slowly with time.


itive. The time ηc is given by

ηc = − R0 tan θsq,c

Vg= − R(ηc) sin θsq,c

Vg(4.27)

where R(ηc) is the slant range to the target at the time it is illuminated bythe beam center, and θsq,c is the value of θsq at this time. In the rectilineargeometry discussed in Section 4.3, ηc can be expressed as

ηc = − R0 tan θr,c

Vr= − R(ηc) sin θr,c

Vr(4.28)

where θr,c is the value of θr at the beam center crossing time.

In this squinted case, the angle θ measured from the beam center in(4.25) equals θsq− θsq,c. Also, using small angle approximations, the two-waybeam pattern becomes

sinc2{

0.886 (θsq − θsq,c)βbw

}≈ p 2

a

{arctan

(Vg (η − ηc)

R0

) }(4.29)

which is the expression for wa(η − ηc) in (4.26).

Following this discussion, it is seen that the term Ra in (4.24) changeswith slow time η and is denoted by R(η). The signal received from the targetis then

sr(τ, η) = A0 wr(τ−2R(η)/c) wa(η − ηc)

cos{ 2πf0 (τ−2R(η)/c) + πKr (τ−2R(η)/c)2 + ψ } (4.30)

This is the real-valued signal received from a point target, having R0 as itsrange of closest approach and a range R(η) defined by the range equation(4.9).

Very often, the SAR data are manipulated in the azimuth frequency do-main, where the beam center crossing time is transformed into an equivalentDoppler centroid frequency.

4.5.5 Azimuth Parameters

Other azimuth parameters derived in this section include the exposure time,FM rate and Doppler bandwidth. These parameters depend on the antennasquint and are evaluated at η = ηc, hence the squint angle θr,c is used insteadof the variable θr .


Doppler centroid

The Doppler centroid frequency at η = ηc is proportional to the rate of changeof R(η) which is given by (4.9)

fηc = − 2λ

dR(η)dη

∣∣∣∣η = ηc

= − 2 V 2r ηc

λ R(ηc)= +

2 Vr sin θr,c

λ(4.31)

in units of Hz. Equation (4.14) has been used to obtain the final equality. Itis possible to express the Doppler frequency in terms of the physical satellitevelocity Vs and the physical squint angle θsq,c. Using (4.16), fηc can bewritten as

fηc =2Vs sin θsq,c

λ(4.32)

This expression can be visualized from the fact that Vs sin θsq,c is the radialvelocity of the radar along the line-of-sight to the target, and Vs is in ECRcoordinates.

Doppler bandwdith

From (4.31), the azimuth bandwidth of the target can be derived as

∆fdop =∣∣∣∣2Vr cos θr,c

λ

Vs

Vrθbw

∣∣∣∣ =2Vs cos θr,c

λθbw (4.33)

in which the scaling factor Vs/Vr is due to the rectilinear geometry assump-tion. This equation makes use of the fact that the bandwidth is the frequencyexcursion experienced by the target during the time in which the target is illu-minated by the 3-dB width of the radar beam, θbw = 0.886λ/La. Therefore,the Doppler bandwidth is

∆fdop = 0.8862Vs cos θr,c

La(4.34)

This bandwidth governs the sampling requirements, that is, it defines thelower limit of the PRF. However, the signal strength is only down by 6 dB atthe beam edges defined by θbw, and the azimuth spectrum rolls off slowly. Anoversampling factor of 1.1 to 1.4 is usually used to reduce azimuth ambiguitypower, that is, the PRF is set to the oversampling factor times ∆fdop (seeSection 5.4).

Target exposure time

Two other important parameters are target exposure time Ta and the azimuthFM rate Ka. The exposure time, as defined by how long the target stays in

4.6. THE TWO-DIMENSIONAL SIGNAL 135

the 3-dB beam limits, is given by

Ta = 0.886λ R(ηc)

La Vg cos θr,c(4.35)

In the above equation, 0.886λ/La is the azimuth beamwidth, therefore 0.886R(ηc) λ/La is the projection of this beamwidth on the ground. This projectionis lengthened in azimuth by the factor 1/cos θr,c for a nonzero squint angle.Again, note that the use of the velocity, Vg, the smallest of the three velocitiesdefined in Figure 4.6, takes into account the satellite attitude drift throughwhich the sensor nadir remains pointed at the local vertical as it proceedsaround its orbit. This attitude drift has the effect of lengthening the exposuretime.

Azimuth FM rate

The azimuth FM rate is the rate of change of azimuth or Doppler frequency

Ka =2λ

d 2R(η)dη2

∣∣∣∣∣η = ηc

=2V 2

r cos2 θr,c

λ R(ηc)=

2V 2r cos3 θr,c

λ R0(4.36)

where the azimuth frequency is 2/λ times the first derivative of range. Also, as-suming the velocity approximation (4.10) and the relation R0 = R(ηc) cos θr,c ,an alternate derivation of the Doppler bandwidth of (4.34) is obtained by mul-tiplying (4.35) and (4.36).

4.6 The Two-Dimensional Signal. SynApt 2Dsignal.tex

This section first shows how the received radar signal is configured as a two-dimensional signal, within the signal processor. Then the important SARsensor impulse response is presented. Finally, typical values of the parametersof the two-dimensional signal are given.

4.6.1 Data Arrangement in Signal Memory

In a simple sense, the received radar signal is one-dimensional — a voltage as afunction of time. In accordance with the transmit/receive cycles of Figure 4.9,the waveform of the received signal could take the form shown in Figure 4.11,where each segment of the signal represents the ground echo received duringone pulse cycle. The gaps between each segment represent the time whenthe receiver is turned off, which includes the pulse transmission time, plus


an allowance for switching of the signal paths. The format of the signal inFigure 4.11 is how it might appear in a one-dimensional storage media, suchas a magnetic tape.

Time −−−>Am

plitu

de −

−>

Figure 4.11: Voltage or amplitude of the received radar signal.

To see how this signal can be considered as a two-dimensional signal,it is useful to re-examine the SAR data collection scenario, as portrayed inFigure 4.12. For simplicity, assume that the radar beamwidth is finite inazimuth. When the sensor is at Point A, a target is just entering the radarbeam. The received signal from the target, which is part of the echo from agiven transmitted pulse, is written into one row of SAR signal memory. Whilethis memory may actually be on tape or in downlink memory for the timebeing, it can be considered as being in the memory at the input to the SARsignal processor.

Then, as the sensor advances, more pulses are transmitted, and the as-sociated echoes are written into successive rows in the signal memory. Whenthe sensor is at Point B, the target leaves the beam, and the last receivedenergy of that target is written into SAR signal memory. Naturally, the signalmemory contains data from many targets, not just the one shown in the fig-ure. Also, the azimuth beamwidth is not finite in practice, which means thatenergy received from the azimuth side lobes from each target is also recordedbefore A and after B.

Going back to the “one-dimensional” signal, shown in Figure 4.11, onecan also think of this format as two-dimensional, where the received signalis sampled and written into a computer memory, as in Figure 4.13. Thedata from each segment or pulse are written into a new row in memory. Thebeginning of each row occurs at a fixed time delay with respect to the pulsetransmission time, referred to as the range gate delay in (4.1). In this way, thehorizontal axis in Figure 4.13 represents the travel time τ or “range” fromthe sensor to the ground. From another viewpoint, one can consider a singlecolumn in Figure 4.13, in which each sample corresponds to the same rangefrom the sensor. A column is sometimes called a range gate and a row calleda range line.

Now consider the vertical axis of the two-dimensional memory of Fig-ure 4.13. While each sample in a given column is at the same range, thesensor has moved a small amount in the azimuth direction from one sample

4.6 The Two-Dimensional Signal 137

Flight path

A

B

Nadir

AzimuthRange

Target

SAR

R(hA)

R(hB)

Beam motion along the surface

SAR Signal Memory

Figure 4.12: How the received SAR data fits into a two-dimensional signalmemory.

to the next, so this vertical axis can be labelled “azimuth” or azimuth timeη. Data have now been recorded, corresponding to two near-orthogonal direc-tions on the Earth’s surface, which is appropriate as the objective is to makea two-dimensional image of the Earth’s surface.

In this way, one can see how a two-dimensional signal is obtained fromthe radar system, with the coordinates being range time and azimuth time.14

The range time is also called “fast time” and the azimuth time is called “slowtime,” because the range distance is related to range time by the speed oflight and the azimuth distance is related to azimuth time by the much slowerforward motion of the beam footprint.

A more realistic view of the two-dimensional memory is given in Fig-

14 Where the geographer thinks of distance along the Earth’s surface, the radar engineerthinks of “time” in the radar system or computer memory. Thus, time and distance are usedinterchangeably in this context.


Range time −−−>

<−

−−

− A

zim

uth

time

Figure 4.13: How the voltage of the received radar signal of Figure 4.11 iswritten into the two-dimensional signal processor memory.

ure 4.14, where the locus of energy from a single point target is outlined.Items to note are the extent of the echo in the range dimension (the trans-mitted pulse duration), the extent of the echo in the azimuth direction (theexposure time or the synthetic aperture length), and the range migration.The sketch is not to scale, as the range and azimuth extents of the energy areusually many hundreds of samples. However, the range migration may onlybe a few cells.

To summarize, the radar samples originate from the sampling of a contin-uous-time analog signal, i.e., the echo received from a transmitted pulse, andthese samples are written along the horizontal range axis. In the azimuthdirection, the signal is inherently in discrete-time from the outset, owing tothe discrete nature of the transmitted pulse events. The azimuth samples arewritten along the vertical azimuth axis in the two-dimensional signal memory.

4.6.2 Demodulated Baseband Signal

The received signal sr contains the radar carrier, cos(2πf0τ), which is re-moved before the sampling by a quadrature demodulation process, as dis-cussed in Appendix 4B. The demodulated baseband signal from a single pointtarget can be represented by the complex signal

s0(τ, η) = A0 wr

(τ−2R(η)/c

)wa(η − ηc)

exp {−j 4π f0 R(η)/c} exp{

jπ Kr

(τ−2R(η)/c

)2}

(4.37)

where the coefficient A0 is now a complex constant

A0 = A′0 exp(i ψ) (4.38)


0 5 10 15 20 25 30

45

40

35

30

25

20

15

10

5

0

Slant range (samples or cells) −−−−>

<−

−−

− A

zim

uth

(sa

mpl

es o

r pu

lse

num

ber)

η0

ηc

End of target exposure

Start of exposure

Syn

thet

ic a

pert

ure

leng

th

Radar pulse length

Figure 4.14: The locus of energy of a single point target in the two-dimensionalsignal processor memory, within the exposure time (the target energy extendsbeyond these limits with a smaller magnitude).

where A′0 is the real coefficient in (4.24).

The signal can now be sampled in range. Since it has a range bandwidthof KrTr and is complex, the complex sampling rate should be

Fr > |Kr|Tr (4.39)

to satisfy the Nyquist sampling criterion.

Equation (4.37) represents the demodulated baseband SAR signal receivedfrom a point target, with coefficient A0. It is the signal that is usually recordedor downlinked in a SAR system and is referred to as the “raw data,” “SARsignal data” or “SAR phase history.” The signal is at baseband in the rangedirection only, as the azimuth signal often has a nonzero center frequency (seeChapter 5).


4.6.3 The SAR Impulse Response

If A0 is ignored, (4.37) is the impulse response of a point target having unityamplitude. Thus the important SAR sensor impulse response is given by

himp(τ, η) = wr

(τ−2R(η)/c

)wa(η − ηc)

exp {−j 4π f0 R(η)/c} exp{

j π Kr

(τ−2R(η)/c

)2}

(4.40)

To model the signal received from a general ground surface, the groundreflectivity is convolved with this impulse response in two dimensions to givethe baseband SAR signal data

sbb(τ, η) = g(τ, η) ⊗ himp(τ, η) + n(τ, η) (4.41)

where n(τ, η) is an additional noise component that is present in all practicalsystems. The noise originates mainly from the front end receiver electronics,and can be modelled as Gaussian white noise. The SAR system model cor-responding to (4.41) is shown in Figure 4.15. In the following developmentof matched filtering, the noise can be ignored. In simulation experiments,one may wish to include noise in order to see the power of matched filteringat work — the wanted signal emerges from the noise floor as illustrated inFigure 3.7.

Figure 4.15: SAR system model with additive noise.

SAR processing is assumed to start with this demodulated baseband sig-nal. SAR processing algorithms attempt to solve for g(τ, η), which is a de-convolution process. The difficulty, and also the challenge, lies in the factthat the impulse response is both range and azimuth dependent and containsrange-varying RCM.

4.6.4 Typical Radar Parameter Values

Table 4.1 gives a representative set of SAR parameter values for both asatellite-borne case and an airborne case. For the satellite case, a set of param-eters broadly representative of the SEASAT, J-ERS, ERS, RADARSAT and


ENVISAT remote sensing satellites is used, assuming a C-band wavelengthand a 10 m antenna.

For the aircraft case, a generic X-band system with a 1 m antenna isconsidered. The table is separated into those parameters which primarilyaffect range processing, and those which mainly affect azimuth processing.The squint in the satellite case is caused by Earth rotation, assuming no yawsteering. In the airborne case, the squint is caused by side winds and physicalantenna motion.

Table 4.1: Representative Airborne and Spaceborne SAR Parameters

Parameter Name Symbol Aircraft Satellite Units

Range parameters

Slant range of scene center R(ηc) 30 850 km

Transmitted pulse duration Tr 10 40 µsec

Range FM rate Kr 10 0.5 MHz/µsec

Signal bandwidth 100 20 MHz

Range sampling rate Fr 120 24 MHz

Range swath width 30 100 km

Azimuth parameters

Effective radar velocity Vr 250 7100 m/s

Radar center frequency f0 9.4 5.3 GHz

Radar wavelength λ 0.032 0.057 m

Azimuth FM rate Ka 130 2070 Hz/sec

Synthetic aperture length 0.85 4.3 km

Target exposure time Ta 3.4 0.65 sec

Antenna length La 1 10 m

Doppler bandwidth ∆fdop 440 1340 Hz

Azimuth samp. rate (PRF) Fa 600 1700 Hz

Beam squint angle θsq,c < 8 < 4 deg


4.7 SAR Resolution and Synthetic Aperture. SynApt SARres.tex

The purpose of this section is to derive the azimuth resolution obtainablefrom the SAR system. The azimuth resolution is derived from the conceptof azimuth bandwidth here, and is also derived from antenna concepts inAppendix 4C.15

4.7.1 Resolution Derived from Bandwidth

In the range direction, the received signal has FM characteristics, inheritedfrom the transmitted pulse. A high resolution can be obtained by matchedfiltering. The resolution is governed by the pulse bandwidth as shown in (3.28)of Section 3.3. In time units (seconds), it is 0.886 times the reciprocal of therange bandwidth in Hz. Multiplying further by c/2 gives the resolution inslant range units (meters).

In the azimuth direction, the beamwidth is given by 0.886λ/La. WithoutSAR processing, the azimuth resolution is the projection of the beamwidthonto the ground

ρ′a = R(ηc) θbw =0.886 R(ηc) λ

La(4.42)

This is called the resolution of a real aperture radar. It is in the order of severalhundreds of meters in an airborne case and several kilometers in a satellitecase.

In Section 4.5, it is shown that the signal in the azimuth direction is alsofrequency modulated by virtue of motion of the platform. Hence, as in therange direction, one would expect to obtain a high resolution by matchedfiltering. The azimuth resolution that can be obtained (in time units) is 0.886times the reciprocal of the bandwidth (4.33). In distance units, the obtainableresolution is

ρa =0.886 Vg cos θr,c

∆fdop=

La

2Vg

Vs(4.43)

The azimuth resolution normally quoted is ρa = La/2, which ignores theVg/Vs ≈ 0.88 factor in the satellite case. This means that the inherent azimuthresolution is approximately one half the antenna length and is independent ofrange, velocity or wavelength. This is the most distinguishing result in SARsystems and is widely quoted in the literature. Equation (4.43) represents

15 As pointed out in Section 4.2, the azimuth resolution derived here is actually the crossrange resolution. Since the squint angle being considered is small, the azimuth resolutiondoes not differ significantly from the cross range resolution. A special mention is made whenthe azimuth resolution is not the cross range resolution.

4.7 SAR Resolution and Synthetic Aperture 143

the cross range resolution, as cos θr,c is used for projecting the resolution inthat direction. But if the squint angle is small, (4.43) effectively gives theachievable azimuth resolution.

The above result is an approximation because it has been assumed thatthe antenna pattern in the azimuth frequency domain is a rectangular window,spanning the width of the 6-dB (two-way) bandwidth. In practice, the beampattern is not flat (it is usually a sinc squared function, as in (4.29)), and ithas significant energy beyond the 6-dB points.16 The actual resolution is afunction of how much of the bandwidth is processed, and the combined shapeof the beam pattern and the weighting function. Therefore, processing the en-tire spectrum would give a slightly better resolution than just processing the6-dB bandwidth. However, because of the roll-off, processing extra bandwidthdoes not have a large effect on the resolution, so the effective obtainable az-imuth resolution is quite close to (4.43). The actual resolution can be derivedexperimentally, for a given PRF, antenna pattern, processed bandwidth andweighting function.

Ls

qsyn

qbw

qbw

AzimuthSatellite

Orbit

Earth's

surface

R0

Target

qbw

AB

C

D EF

Figure 4.16: Antenna azimuth beamwidth and synthetic angle. For clarity, thebeamwidth and synthetic angle are exaggerated, and a zero squint angle is used.

Sometimes the resolution is expressed in terms of the synthetic angle,θsyn. This angle is the change in the viewing angle of the target betweenthe times when the target enters and exits from the beam. This angle isshown in Figure 4.16. For an airborne case, the synthetic angle is equal to

16 The PRF is usually higher than the 6-dB bandwidth, so that some spectral energybeyond the 6-dB points is available for processing. However, the energy outside the 6-dBlimits has a higher level of azimuth ambiguities, and is usually not processed.


the beamwidth.17 For a satellite case, the synthetic angle is a little largerthan the beamwidth, because the antenna rotates slowly as the nadir remainspointed towards the Earth’s center. In the satellite case, examining θsq andθg in Figure 4.6(a) and using (4.15), the synthetic angle is

θsyn =Vs

Vgθbw (4.44)

which can be determined by comparing the approximate triangles AEC andDBF in Figure 4.16 (the ratio of the path lengths ABC to DEF is Vs/Vg

using small angle approximations). Substituting this angle into (4.33), theDoppler bandwidth ∆fdop can be expressed in terms of θsyn as

∆fdop =2Vg cos θr,c

λ| θsyn | (4.45)

By combining the first part of (4.43) and (4.45), the desired result isobtained, showing that the resolution is

ρa =0.886 λ

2 θsyn(4.46)

The resolution is now independent of the squint angle. This form of the reso-lution equation is used more frequently in spotlight SAR [11, 12] and inverseSAR (ISAR) [7, 13] than in stripmap SAR (see Section 1.3).

4.7.2 Synthetic Aperture

The purpose of this section is to explain the term “synthetic aperture” in theSAR context. This gives another derivation of the azimuth resolution.

The azimuth resolution of a conventional radar, or a SAR before process-ing, is given by the azimuth beamwidth. The beamwidth is determined bythe radar wavelength λ and the antenna length or aperture, La. Both theseparameters are fixed for a given radar system. To improve the resolution, itwould be desirable to reduce the effective beamwidth.

Similar to creating (synthesizing) a narrow pulse in range, the trick is touse signal processing to synthesize a narrow beamwidth in azimuth. Since thebeamwidth is inversely proportional to the antenna aperture, synthesizing anarrow beamwidth is equivalent to synthesizing a large aperture. In practice,the synthesized aperture can be several hundred meters long in the airborne

17 This definition assumes that the target is not rotating. If the target is rotating, therotation angles, caused by the changing beam viewing angle and by the target rotation angle,must be added to get the synthetic angle.

4.8. SUMMARY 145

case and several thousand meters long in the satellite-borne case, whereas theantenna’s real aperture, La, is only in the order of 1 to 15 meters in length.

The synthetic aperture is shown by Ls in Figure 4.16. It is the lengthof the sensor path during the time that a target stays within the radar beam.This length governs the amount of data that is available for processing froma given target. The synthetic aperture Ls is given by

Ls =R0 θbw

cos θr,c

Vs

Vg=

0.886R0 λ

La cos θr,c

Vs

Vg(4.47)

where θbw = 0.886λ/La and the ratio Vs/Vg is due to the difference betweenthe beamwidth θbw and the synthetic angle θsyn.

Appendix 4C shows that this definition of synthetic aperture gives a null-to-null beamwidth of λ/Ls. From the sinc function presented in Section 2.3.4,the synthesized half-power beamwidth is

φs =0.886λ

2 Ls(4.48)

where the factor of 2 accounts for the two-way path, as shown in Appendix 4C.Assuming an antenna having a beamwidth φs , the azimuth resolution is thenR(ηc) φs = R0 φs / cos θr,c. Using (4.47) and (4.48), the azimuth resolutionbecomes

ρa =0.886R0 λ

2Ls cos θr,c=

La

2Vg

Vs(4.49)

As an example, let La = 10 m, λ = 0.057 m, R0 = 900 km, Vg/Vs = 0.88and assume a small squint angle. Then the obtainable resolution from (4.43)is 4.4 m, the real aperture required to yield this resolution from (4.42) isabout 10.3 km, and the synthetic aperture from (4.47) is about 5.15 km. Thisexample shows that the equivalent real aperture is twice the length of the SARsynthetic aperture.

4.8 Summary SynApt summary.tex

In this chapter, the geometry of the SAR data collection is discussed. Ofparticular importance is the origin of the frequency modulations in range andazimuth. The range modulation is achieved by the design of the transmittedpulse. The azimuth modulation is introduced by the motion of the platform.The SAR signal is the convolution of the ground reflectivity with the SARsystem impulse response, which is range varying and possibly azimuth varyingalso. SAR processors solve for the ground reflectivity from this convolutionequation.


An important result, derived in this chapter, is that the available azimuthresolution is about half the antenna length. Two different methods have beenshown to arrive at the same result. One is by utilizing the bandwidth ofa signal captured within the exposure time. Another method starts withthe definition of synthetic aperture and then derives the resolution from thesynthesized, much narrower, beam width.

It appears that SAR violates two “common sense” principles:

Resolution vs. range: In general, the closer a sensor is to a target, themore the target details are revealed. But in SAR the azimuth band-width, and hence the resolution, is independent of range and this canbe explained as follows. The exposure time is proportional to range,but the azimuth FM rate is inversely proportional to range; hence thesignal bandwidth, being the product of the exposure time and FM rate,is independent of range. Note however, that the SNR decreases withR3, and under power-limiting conditions, this SNR loss obscures targetdetails at longer ranges.

Sensor size: Also, in general, a larger sensor can “see” more details than asmaller sensor (as in a telescope or microscope). This is true for a realaperture radar, but the opposite is true if the received data are processedin SAR mode. The beamwidth increases with a smaller antenna and,in turn, the exposure time and signal bandwidth increase, leading to afiner resolution. However, ambiguities and SNR place a lower limit onthe antenna size.

The important equations derived in this chapter are summarized in Ta-ble 4.2. In this table, use Vs = Vr = Vg for an airborne case, and cos θr,c = 1for a negligible squint angle.

4.8 Summary 147

Table 4.2: Summary of Key Synthetic Aperture Equations

Parameter Name Symbol Expression Units

Range bandwidth |Kr| Tr Hz

Slant range R(η)√

R20 + V 2

r η2 m

wr(τ − 2R(η)/c) wa(η − ηc)Impulse response himp(τ, η) exp {−j 4π f0 R(η)/c}

exp{j π Kr (τ − 2R(η)/c)2

}

Azimuth beamwidth θbw 0.886λ/La rad

Azimuth beamfootprint ρ′a 0.886R(ηc) λ /La m

Synthetic aperture Ls [0.886R(ηc)λ/(La cos θr,c) ] (Vg/Vs) m

Synthetic angle θsyn (Vg/Vs) θbw rad

Doppler frequencyat η = ηc

fηc 2 Vr sin θr,c/λ = 2 Vs sin θsq,c/λ Hz

Beam centercrossing time ηc −R0 tan θr,c/Vr = −R0 tan θsq,c/Vg s

Doppler bandwidth ∆fdop 0.886 (2Vs cos θr,c /La) Hz

Exposure time Ta 0.886 R(ηc) λ/(La Vg cos θr,c) s

Azimuth FM rate Ka 2 V 2r cos2 θr,c / [ λR(ηc) ] Hz/s

Azimuth resolution ρa Vg cos θr,c / ∆fdop m

Azimuth resolution ρa (La/2) (Vg/Vs) ≈ La/2 m

Azimuth resolution ρa 0.886 λ / (2 θsyn) m


4.8.1 Example of a SAR Image

An example of a SAR image taken by the RADARSAT-1 sensor is shown inFigure 4.17. The image was acquired on March 22, 1996, and its size is 102by 102 km. It was processed by Radarsat International Inc. using the rangeDoppler algorithm to a resolution of 25 m, 4 looks.18

The image shows the Dead Sea at 31.5◦N and 35.4◦ E. Israel and theWest Bank lie to the east, and Jordan lies to the west of the sea and the RiftValley. The city of Jerusalem is near the top left and the city of Amman nearthe top right of the image.

Figure 4.17: RADARSAT-1 S4 ascending image of the Dead Sea Rift Valleyin the Middle East. Copyright, Canadian Space Agency.

18 Note that because of file size and printing limitations, the image quality portrayed inthe book is not representative of the full quality available from the sensor.

References 149

References

[1] D. Massonnet. Capabilities and limitations of the interferomet-ric cartwheel. IEEE Trans. on Geoscience and Remote Sensing,39(3): pp. 506–520, March 2001.

[2] T. Amiot, F. Douchin, E. Thouvenot, J.-C. Souyris, and B. Cugny. Theinterferometric cartwheel: a multi-purpose formation of passive radarmicrosatellites. In Proc. Int. Geoscience and Remote Sensing Symp.,IGARSS’02, volume 1, pages 435–437, Toronto, June 2002.

[3] R. K. Raney, A. P. Luscombe, E. J. Langham, and S. Ahmed.RADARSAT. Proc. IEEE, 79(6): pp. 839–849, 1991.

[4] J. Curlander and R. McDonough. Synthetic Aperture Radar: Systemsand Signal Processing. Wiley, New York, 1991.

[5] R. K. Raney. A Comment on Doppler FM Rate. International Journalof Remote Sensing, 8(7): pp. 1091–1092, January 1987.

[6] R. K. Raney. Radar Fundamentals: Technical Perspective, chapter 2,page 59. Manual of Remote Sensing, Volume 2: Principles and Appli-cations of Imaging Radar. John Wiley & Sons, New York, 3rd edition,1998.

[7] D. R. Wehner. High Resolution Radar. Artech House, Norwood, MA,2nd edition, 1995.

[8] R. W. Bayma and P. A. McInnes. Aperture size and ambiguity constraintsfor a synthetic aperture radar. In J. J. Kovaly, editor, Synthetic ApertureRadar. Artech House, Norwood, MA, 1978.

[9] S. W. McCandless. SAR in Space — The Theory, Design, Engineeringand Application of a Space-Based SAR System, chapter 4. Space-BasedRadar Handbook, L. J. Cantafio, editor. Artech House, 1989.

[10] A. Freeman, W. T. K. Johnson, B. Honeycutt, R. Jordan, S. Hensley,P. Siqueira, and J. Curlander. The “Myth” of the Minimum SAR An-tenna Area Constraint. IEEE Trans. Geoscience and Remote Sensing,38(1):320–324, January 2000.

[11] W. G. Carrara, R. S. Goodman, and R. M. Majewski. Spotlight SyntheticAperture Radar: Signal Processing Algorithms. Artech House, Norwood,MA, 1995.

150 REFERENCES

[12] C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A.Thompson. Spotlight-Mode Synthetic Aperture Radar: A Signal Process-ing Approach. Kluwer Academic Publishers, New York, 1996.

[13] R. J. Sullivan. Microwave Radar Imaging and Advanced Concepts. ArtechHouse, Norwood, MA, 2000.

Appendix 4A. Derivation of the Effective Radar Velocity 151

Appendix 4A: Derivation of Approximate Form ofthe Effective Radar Velocity SynApt Derive Vr approx.tex

Equation (4.10) states that the effective radar velocity Vr is approximatelythe geometric mean of the satellite velocity, Vs, and the beam velocity overthe ground, Vg. The relationship is accurate for zero Doppler pointing and asatellite orbit that can be assumed to be circular within the target exposuretime. The purpose of this appendix is to prove this relationship. It should beemphasized that the approximation is adequate for simple geometric analyses,but is not sufficiently accurate for precision focusing of the SAR data.

AB

CEarth's surface

Satellite velocity Vs

Beam footprint velocity Vg

Re

R0

R(h)

Center of Earth

be

wsh

x

y

z

O

H

Figure 4.18: The SAR geometry, showing the satellite velocity and the beamfootprint velocity.

Figure 4.18 can be used to illustrate the satellite/Earth geometry of theradar system. Let C be a target on the Earth’s surface under consideration,and A be the satellite position when the target at C is at zero Doppler.The relative orbit time at A is set to zero. Let the satellite, travelling withangular velocity ωs, advance from position A to B in a time η. All velocities,including Vs, are in ECR coordinates in which Earth rotation is not a factor.


Assume a right-handed, Earth-centered coordinate system in which thez-axis points from the center of the Earth to Point A, the y-axis points inthe direction of the satellite ECR velocity vector at Point A, and the x-axispoints to the right to complete the orthogonal system. At η = 0, the satelliteposition is [0 0 H]T . Then, the position of the satellite at time η is at B,given by

PB =

0H sinωsηH cosωsη

(4A.1)

and the target position C is

PC =

Re sinβe

0Re cosβe

(4A.2)

where H is the local orbit radius, Re is the local Earth radius at the target,and βe is the angle between OC and the orbit plane.

The range to the target at η = 0 is R0 and the range at time η is

R(η) = |PC − PB|

=√

(Re sinβe)2 + (H sinωsη)2 + (H cosωsη − Re cosβe)2 (4A.3)

Using the small angle approximations, sinωsη ≈ ωsη and cosωsη ≈ 1 −ω2

sη2/2, and ignoring terms in η4, the range can be expressed as

R(η) =√

H2 + R2e − 2H Re cosβe + (H ωs) (Re ωs cosβe) η2

=√

R20 + (H ωs) (Re ωs cosβe) η2 (4A.4)

where the last equation uses the triangle relationship R20 = H2 + R2

e −2HRe cosβe. From the figure, Hωs is the satellite velocity Vs arising fromthe locally circular orbit assumption, and Re ωs cosβe is the beam footprintvelocity, Vg. This value of Vg assumes that the Earth is locally spherical inthe vicinity of C, so that Vg is measured parallel to Vs.

Hence the final result is

R2(η) = R20 + Vs Vg η2 (4A.5)

which shows that the effective radar velocity, Vr in (4.9), is equal to√

Vs Vg

under the conditions of a locally circular orbit and an antenna centered onzero Doppler.

Appendix 4B. Quadrature Demodulation 153

Appendix 4B: Quadrature Demodulation. SynApt hetero.tex

The pulses transmitted and received by the radar system are real signals.This appendix explains how the received signal can be bandshifted, to obtaina complex baseband signal by a quadrature demodulation process. The de-modulation removes the high frequency carrier but may create some signalerrors, so the compensation of the errors is also discussed.

4B.1 Theory of Quadrature Demodulation

Let a general real-valued signal, having a high-frequency carrier and a low-frequency modulation, be represented by

x(τ) = cos{ 2π f0 τ + φ(τ) } (4B.1)

where the frequency of the carrier, f0, is several orders of magnitude higherthan the bandwidth of the modulation φ(τ) (GHz versus MHz).

Figure 4.19 shows the quadrature demodulation process produces twochannels that represent a complex-valued output [11]. First, consider the up-per channel, where the signal is multiplied by cos(2πf0τ). Using the trigono-metric identity

cos θ1 cos θ2 =12

cos(θ1− θ2) +12

cos(θ1+ θ2) (4B.2)

the result of the multiplication is

xc1(τ) =12

cos{φ(τ) } +12

cos{ 4π f0 τ + φ(τ) } (4B.3)

The first cosine term of (4B.3) has an upper frequency governed by thebandwidth of φ(τ), while the second cosine term has a much higher frequency,centered around 2f0. Therefore, the second term can be removed by a lowpass filter, giving the result

xc2(τ) =12

cos{φ(τ) } (4B.4)

Similarly, the lower channel of Figure 4.19 is multiplied by − sin(2πf0τ)and the following trigonometric identity is used

sin θ1 cos θ2 =12

sin(θ1 − θ2) +12

sin(θ1 + θ2) (4B.5)


ADCLow Pass

Filter

Low Pass

Filter ADC

cos( 2p f0t )

-sin( 2p f0t )

x(t)

Real

channel

Imaginary

channel

I

Q

xs1

(t)

xc1

(t) xc2

(t)

xs2

(t)

Figure 4.19: Quadrature demodulation of a signal to remove the carrier.

to express the signal as the sum of high and low frequency components. Afterlow pass filtering, the signal xs2(τ) is

xs2(τ) =12

sin{φ(τ) } (4B.6)

The signals xc2(τ) and xs2(τ) are then sampled by the analog-to-digitalconverters (ADCs) at a rate at least equal to the bandwidth of φ(τ). Becauseof the cosine and sine multiplication, the two signals are in phase quadrature,and represent a complex signal

x3(τ) = xc2(τ) + j xs2(τ) =12

exp{j φ(τ)} (4B.7)

The two individual signals are called the quadrature components of the complexsignal, or the I and Q channels for in-phase and quadrature. The signal x3(τ)is the required baseband signal, which is used in the processing of the SARsignals.

When demodulation is applied to the real SAR echo data from a pointtarget (4.30) to obtain the complex baseband signal, the phase term φ(τ) is

φ(τ) = − 4π f0 R(η)c

+ π Kr

[τ − 2R(η)

c

]2

+ ψ (4B.8)

The demodulated baseband signal is then given by (4.37).

4B.2 Errors and Corrections

The following errors can occur in the demodulation process:

Appendix 4B. Quadrature Demodulation 155

Frequency mixing: The input signal in analog form goes through two chan-nels. One channel is multiplied by cos(2π f0 τ), and the other channelby sin(2π f0 τ). A constant phase error can be introduced in the mul-tiplier in each channel, caused by the electronics. The phase differencebetween these two paths, rather than the absolute phase angles, is ofimportance for the SAR processing. Let this phase difference be ∆θ.

The frequency beating can then be viewed as follows. The upper channelis multiplied by cos(2π f0 τ), and the lower channel by sin[2π f0 (τ +∆θ)]. The additional phase ∆θ is an error. In other words, the signalsin the two channels are no longer orthogonal.

Low pass filter: Ideally, the gains of the low pass filters in the two channelsare equal. Because of imbalance in the electronics, this may not alwaysbe the case. When this imbalance occurs, the signal powers after filteringare not equal. The ratio of the two gains, rather than the absolute valuesof the gains, are of importance. Similarly, each channel has a DC bias,and the two biases can be different.

Analog to digital conversion: In the ADCs, the gains of the two channelsmay not be balanced, and there may be a timing error between the twochannels.

The corrections for gain, DC bias and phase should be performed in thefollowing order: bias removal in both channels, power balancing between thechannels, and phase correction in one channel. The first two corrections canbe performed quite easily:

1. Determine the DC bias in each channel from the data.

2. Correct the DC bias for each channel.

3. Determine the relative gain of the channels, after the bias removal.

4. Select a channel and correct for the gain.

Correcting the phase

The phase correction is not as straightforward. One method is describedbelow, assuming that bias removal and power balancing have been performed.Let I and Q denote the pixel intensity in the I and Q channels, A themagnitude and θ the phase, where the symbol ˆ denotes a random variable.Then the channel intensities can be written

I = A cos θ (4B.9)

Q = A sin(θ + ∆θ) (4B.10)


where the phase error ∆θ has been included. The probability distributionof A is immaterial in the subsequent analysis. The angle θ has a uniformdistribution between −π and π. The two random variables A and θ arestatistically independent. The random variables I and Q each have zeromean, because of the cosine and sine factors.

For perfect orthogonality, the cross-channel covariance must be zero. Hence,any non-orthogonality can be detected from the off-diagonal terms of the co-variance matrix, and the angle ∆θ can be determined. Let the covariance beC, which can be expressed by

C = E{ I Q } (4B.11)

where E is the expectation over the received data set. Substituting (4B.9) and(4B.10) into (4B.11), and recognizing the fact that A and θ are statisticallyindependent, the covariance is

C = E{(A)2 cos θ sin(θ + ∆θ)}= E{(A)2} E{cos θ sin(θ + ∆θ)}

=12

E{(A)2}E{sin(2θ + ∆θ) + sin ∆θ} (4B.12)

Recognizing that E{sin(2 θ + ∆θ)} = 0 and E{sin∆θ} = sin ∆θ, the covari-ance is

C =12

E{(A)2} sin∆θ (4B.13)

The value of E{(A)} can be obtained from (4B.9) or (4B.10)

E{(A)2} = 2E{I2} = 2 E{Q2} (4B.14)

Hence, the required phase correction ∆θ can be determined by combining(4B.11), (4B.13) and (4B.14)

sin∆θ =E{I Q}E{I2} =

E{I Q}E{Q2} (4B.15)

The phase correction need only be applied to one channel, say the Q chan-nel. Using ∆θ of (4B.10), the phase error for this channel can be expressedusing

Q = A sin(θ + ∆θ) (4B.16)

where the ˆ has been dropped because the equation now refers to each partic-ular pixel. The desired result is

Q′ = A sin θ (4B.17)

Appendix 4C. Concept of Synthetic Aperture 157

where ∆θ has been compensated. Now Q′ has to be expressed in terms of theknown variables I, Q, and ∆θ. Equation (4B.10) can be re-written as

Q = A sin θ cos∆θ + A cos θ sin∆θ (4B.18)

Combining (4B.9), (4B.17) and (4B.18), the phase-corrected channel is

Q′ =Q − I sin∆θ

cos ∆θ(4B.19)

The data I and Q′ now form an orthogonal set. It is seen that if ∆θ = 0 tostart with, then Q′ = Q and no correction is needed. From here on, in thisbook, it is assumed that the input data have been properly corrected.

Appendix 4C: Concept of Synthetic Aperture. SynApt Antenna.tex

In Section 4.7, the SAR azimuth resolution is derived from the processed band-width, using the resolution formula from the pulse compression developmentof Section 3.3. In this appendix, the azimuth resolution is derived from adifferent viewpoint, using antenna beamwidth concepts. In doing so, an intu-itive explanation of the term “synthetic aperture” is given. For simplicity, thedevelopment assumes a zero squint case, but it can be extended to a nonzerosquint easily.

4C.1 Antenna Beamwidth

The presentation starts by taking a brief look at antenna theory, and discussesthe meaning of “aperture” and the resolving power of an antenna. Then, theoperations with which the SAR processor creates or synthesizes an antennaare examined, which gives an alternate way of deriving azimuth resolution.

Consider a radar antenna consisting of a linear array of identical radiatingelements, as shown in Figure 4.20. The length of the antenna, La, is calledthe real aperture (or simply the aperture) of the antenna. This is in analogywith the aperture or diameter of a lens and is the “opening” through whichthe sensor views the imaged terrain.

Consider the far-field radiation pattern of the antenna beam as it strikesthe Earth’s surface. The main lobe of the pattern illuminates a patch of theground at any instant and, in a simple sense, this patch is what is “observed”at that time. Consequently, the azimuth extent of this patch defines theresolving power of the antenna in the azimuth direction. More specifically,


Azimuth

radiating elements

La

q

q

perpendicular to

viewing direction q

Voltmeter

x

R0

R

Earth's surface

Figure 4.20: Far-field radiation pattern of a planar antenna array.

the 3-dB width of the radiation pattern is usually taken as the resolution ofthe unprocessed received signal.

Suppose it is desired to measure the far-field strength of the radiatedenergy at a point on the ground, using a field strength meter, placed as shownin Figure 4.20. Consider a line from the field strength meter to the centerof the antenna that makes an angle θ with the normal to the surface of theantenna. The far-field assumption states that the rays from each elementto the field strength meter can be considered to be parallel. The distancefrom each element to the field strength meter is R0+x θ, assuming that theangle θ is small. Assuming that the radiation from each element is of equalamplitude at the field strength meter, and neglecting the constant phase dueto the range R0, the net voltage is given by the sum of the radiation from allof the radiating elements

pa(θ) =∑n

exp{−j 2π

x(n) θ

λ

}(4C.1)

where n is the element number.

This sum can be recognized as the DFT of a rectangular function. Asthe number of radiating elements increases and the elements become closertogether, this sum converges to the familiar Fourier integral that gives theone-way beam pattern

pa(θ) =∫ +La/2

−La/2exp

{−j 2π

x θ

λ

}dx = sinc

(La θ

λ

)(4C.2)


The beam pattern is a sinc function, as illustrated in Figure 2.3.

This field strength on the ground has a maximum at the boresight direc-tion, θ = 0, and the limits of the main lobe are defined by the width of thetwo zero crossings adjacent to the peak. Referring to Figure 2.3, the angu-lar width between the zero crossings can be deduced to be 2λ/La. Also, bycomparing (4.25) and (4C.2), the half power width is 0.886λ/La.

The width between the zero crossings can be interpreted as the resolv-ability of the radar beam, although the 3-dB width, which is about one halfthe zero crossing width, is normally considered to be the “resolution” in radarterminology. At range R0, the 3-dB resolution is 0.886λR0/La.

An intuitive way of finding the width between the two zero crossings is tofind the smallest value of θ that makes the sum in (4C.1) go to zero. Notethat the contribution of each radiating element to the field strength is a phasorexp{−j2π x θ/λ} with constant amplitude and a phase angle proportional toθ. The sum in (4C.1) is zero when the n phasors form a circle, as shown inFigure 4.21.

Complex

plane

Real

Imaginary

1

2 3

n -1

n

Figure 4.21: Summation of electric vectors from each radiating element, atthe first null of the beam pattern.

From Figure 4.21, it can be seen that the vector sum is zero if the phasorsfrom the first and last radiating elements are aligned in almost the samedirection. Beginning from the antenna boresight and moving outwards, thephasors are next aligned when the path difference, La θ for small θ, betweenthe two ends of the aperture to the field strength meter, is one wavelength.This occurs at the beam angle of θ = λ/La so, from symmetry, the beamwidthbetween the nulls adjacent to the main lobe is 2λ/La. This agrees with theprevious result.


4C.2 Synthetic Aperture

Now, how does the SAR signal processor contribute to the resolution? Inthe SAR system, there are two main differences with respect to the antennamodel described in the preceding section. First, the location of the radiatingelements in the SAR case is given by the location of the sensor when thepulses are transmitted and received. Thus, the antenna phase center locationat each pulse epoch is analogous to each element in Figure 4.20, as each pulseis acting as one contribution to the received signal in the SAR system.19

Second, the signal strength is being observed at the receiver, rather thanon the ground. This means that the ranges in the analysis of the precedingsection must be doubled. Then, in order to complete the analogy with theprevious section, the field strength meter in Figure 4.20 is replaced by anideal reflector (corner reflector), and the field strength meter is replaced by avoltmeter in the SAR receiver.

X

R0

R0

R0

R1

R2

Synthetic aperture length Ls

= 0.886 l R0

/ La

corner

reflector

SAR sensor

location at pulse i

azimuth

O

Figure 4.22: Sensor locations where the data are collected, illustrating theconcept of synthetic aperture.

19 While the SAR antenna actually moves a few meters between the transmission andreception of a given pulse, this distance is very small compared with the ranges involved, soit is permissible to assume that the antenna is stationary between the transmit and receiveevents. This is sometimes called the “start-stop” assumption in SAR signal analysis.


This analogy is expressed in Figure 4.22, where the synthetic array lengthis given by the distance that the sensor travels while the corner reflector isilluminated by the radar beam. The length 0.886λR0/La is used, whichcorresponds to the length where the received signal strength is within 6-dB ofits maximum.

The corner reflector is located a distance X away from the central axisof the synthetic array. The perpendicular distance from the corner reflectorto the array is R0. The distances from the corner reflector to either end ofthe synthetic array are R1 and R2. Then, for large range distances, the totalpath length difference is

2 (R2 −R1) ≈ 2Ls X

R0(4C.3)

and, using the same argument as that associated with Figure 4.21, the firstnull occurs when the corner reflector is at

Xnull =R0 λ

2 Ls(4C.4)

and the null-to-null separation of two corner reflectors located symmetricallyabout the central axis is 2Xnull. Thus, taking the resolution as 0.886 timeshalf the null-to-null distance, the azimuth resolution of the processed SARdata is

ρa = 0.886Xnull = 0.886R0 λ

2Ls= 0.886

R0 λ

2La

0.886R0 λ=

La

2(4C.5)

Section 4.7 shows that the achievable resolution is better than this by a ratioof footprint velocity to satellite velocity.

An alternate way to arrive at the resolution is to follow the developmentleading to (4C.2), in which La is replaced by Ls, and the phase inside exp{.}is increased by a factor of two due to the two-way path. Then the synthesizedradiation pattern is

ps(θ) = p2a(θ)

∫ Ls/2

−Ls/2exp

{−j4π

x θ

λ

}dx ≈ sinc

(2Ls

λθ

)(4C.6)

The term p2a(θ) outside the integral is the antenna beam pattern at each

element position, and the integral is the Fourier transform of a rectangularfunction — a sinc function. The width of p2

a(θ) is much wider than thatof the sinc function because the former is the beamwidth of the original realaperture, while the latter is of the synthesized beamwidth. The factor p2

a(θ)can then be ignored; hence an approximation sign is used in the last step.

Drawing a further analogy between the synthetic aperture Ls and a realaperture antenna, it would take a conventional antenna of length 2Ls to


obtain a resolution of La/2 without SAR processing. The synthetic array isonly half this long because the SAR system benefits from the two-way radarpath length.

Index

ambiguitiesazimuth, 129, 132, 134range, 126, 130tradeoff, 130

antennabeamwidth, 132, 157–159size, 146

azimuthambiguities, 129bandwidth, 134beam pattern, 131direction, 111FM rate, 135resolution, 142–145signal strength, 132

backscatter coefficient, 126, 138baseband signals, 153beam

footprint, 109velocity, 110

beam center crossing time, 112

compressiongain, 125to zero Doppler, 113

convolutionwith ground reflectivity, 126

cross range, 114

demodulationerrors, 154

Dopplerbandwidth, 135frequency, 127, 132, 134

down chirp, 124

ENVISAT, 141ERS, 129, 141exposure time, 135

far range, 126fast time, 137

geometrycurved Earth, 120orbit, 117rectilinear, 119

ground range, 114

image space, 112

linear FM, 124

Massonnet, 109

nadirpoint, 110return, 130

near range, 126Nyquist sampling rate, 129

orbitaltitude, 117eccentricity, 117inclination, 118period, 118radius, 118

orbital velocity, 110oversampling ratio

azimuth, 129, 134range, 125

614

INDEX 615

platform velocity, 110point target, 109PRF

choice of, 129minimum, 134

pulse envelope, 124pulse timing, 128

quadrature demodulation, 153–157I and Q components, 154phase correction, 155

radarbeamwidth, 111coherence, 128real aperture, 142track, 110transmitted pulses, 124, 128

RADARSAT, 117, 129, 141radiation pattern, 158radiometric

variation with range, 126range, 112–114

ambiguities, 130bandwidth, 125FM rate, 124of closest approach, 111oversampling ratio, 125resolution, 125sampling rate, 125slant vs. ground, 114–117swath, 125swath width, 130

range equation, 119–122hyperbolic, 119

receive window, 130

SAR parametersantenna beamwidth, 157–159aperture length, 160–162

SAR signalat baseband, 138azimuth, 127–135

convolution, 126data acquisition, 136deconvolution, 140demodulated baseband, 139demodulation, 153–157range, 123–127raw data, 139signal memory, 136two-dimensional, 135–140

SAR systemgeometry, 108–118impulse response, 140parameters, 140

satellite speed, 118SEASAT, 141signal memory, 135–138signal space, 112slow time, 137squint angle, 114, 122–123start-stop assumption, 160synthesized beamwidth, 145synthetic angle, 143synthetic aperture, 144–145

beamwidth concept, 160–162

target trajectory, 112time bandwidth product, 125transmitted bandwidth, 125transmitted pulse, 124

up chirp, 124

velocitybeam, 110effective radar, 120–123orbital, 110platform, 110

zero Dopplercompression to, 113line, 111plane of, 111time of, 111

CW_chap04

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