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2.0 SYNTHETIC APERTURE RADAR

2.1 INTRODUCTION

The term synthetic aperture radar (SAR) derives from the fact that the

motion of an aircraft (airplane, satellite, UAV, etc.) is used to artificially create,or synthesizea very long, linear array. The reason for creating a long array is toprovide the ability to resolve targets that are closely spaced in angle, or crossrange (usually azimuth). This, in turn, is driven by one of the main uses ofSARs: to image the ground or targets. In both cases, the radar needs to be ableto resolve very closely spaced scatterers. Specifically, resolutions in the order ofa few feet are needed. To realize such resolutions in the range coordinate theradar uses wide bandwidth waveforms. To realize such resolutions in crossrange very long antennas are required.

To get an idea of what we mean by long antenna, lets consider anexample. Suppose we are trying to image a ground patch at a range of 20 Km.We decide that to do so we want a cross range resolution of 1 m. We can

approximately relate cross range distance, y , to antenna beamwidth,B

, and

range, R , by

By R (1)

as shown in Figure 1. For 1 my and 20 KmR we get 55 10 radB or

about 0.003!

The beamwidth of a linear array with uniform illumination can beapproximately related to antenna length by

B L

. (2)If we assume that the radar of the above example operates at X-band and that

0.03 m we get

600 mBL . (3)

Clearly, it would not be practical to use a real antenna that is as long as sixfootball fields. Instead, a SAR synthesizes such antenna by using aircraftmotion and signal processing.

Figure 1Relation of Cross-Range distance to Beamwidth

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2.2 BACKGROUND

2.2.1 Linear Array Theory

Before we discuss SAR processing we want to discuss some properties of

SAR. We will start with a review linear arrays since a SAR synthesizes a lineararray. Suppose we have a 2 1M N 1 element linear array as shown in Figure

2. We have a target located at some ,n nx y that emits an E-field2 cj f t

oE e

that

eventually reaches each antenna element. We can write the E-field at the kthelement as

2 2 2c k k cj f t r c j r j f t o okk k

E EE t e e e

r r

. (4)

The resulting voltage at the output of the kth element is

2 2k cj r j f tokk

Vv t e e

r

. (5)

1 We choose this form ofMto simplify some of the notation to follow.

Figure 22N+1 Element Linear Array

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We can writek

r as

22 2 2 2 2

2 2 2

2

2 sin

k n n n n n

o o

r x y kd x y k d kdy

r k d kdr

(6)

where we used sinn oy r .

Since or kd we can approximate kr as

2 22 sin 1 sin sink o o o o

o

kdr r kdr r r kd

r . (7)

We next substitute Equation (7) into the exponent of kv t (Equation (5)) and

assume that k or r in the o kV r term. This gives,

2sin

2 2o c

kdj

j r j f tok

o

V

v t e e er

. (8)

To form the total output of the array we sum the kv t to get

2

sin2 2o c

kdN N jj r j f to

k

k N k N o

Vv t v t e e e

r

. (9)

We next use Equation (9) to form the antenna radiation pattern as

22

2

2

sin sin1

sin sin

M d

o

do

rR v t

V M

(10)

where we included the2 2

0 or V term to normalize it out of R .

When we formulated the antenna radiation pattern as above we were

interested in how R varied with target angle, . We found that the peak of

R occurred at a target angle of 0 as shown in Figure 3.

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As an extension to the above we can devise a means ofsteering the beamby including a linear phase shift across the array elements as shown in Figure

4. This phase shift is included in the equation for v t in the form

2 2

sin sin2 2 So c

kd kd N N j jj r j f to

k k

k N k N o

Vv t a v t e e e e

r

. (11)

This leads to a more general R of

2

sin sin sin1

sin sin sin

M dS

dS

RM

. (12)

Again, for standard array theory, we were interested in how R varies with

for afixedS

. In this case the peak of R would occur atS

, as shown in

the example ofFigure 5.

Figure 3Normalized Radiation Pattern vs. Target Angle

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Figure 4Linear Array with Phase Shifters

Figure 5Normalized Radiation Pattern vs. Target AngleBeam Steered to -0.01 deg

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2.2.2 Transition to SAR Theory

In SAR theory we need or reorient ourselves by thinking of the target

angle, , as being fixed and examining how R varies withS

. In other

words, we consider a fixed, , and plot SR . An example plot of SR for

0.01 (i.e. the target location is fixed at 0.01 deg) is shown in Figure 6. Inthis plot, SR peaks when the beam is steered to an angle of 0.01.

Figure 7contains a plot of SR for the case where there is a target at0.01 and a second target at -0.02. Further, the second target has twice the

RCS (radar cross-section) of the first target. Here we note that the plot of SR tells us the location of the two targets and their relative amplitudes. This is thetype of information we want when we form SAR images.

Figure 6Normalized Radiation Pattern vs. Beam Steering AngleTargetlocated at 0.01 deg

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It will be noted that the plot of SR is analogous to a plot of the power

out of a stretch processor, P r . P r is the other information we use to we

form SAR images. Specifically, we will compute SR and P r for various

values ofS

and r and then plot SR P r as intensities on a rectangular

grid. The discrete values ofS

and r will be separated by the angle resolution

of the SAR array and the range resolution of the stretch processor. The

resulting image will be a SAR image.

2.3 DEVELOPMENT OF SAR SPECIFIC EQUATIONS

With the above background we now want to start addressing the issues

associated with forming SR in practical SAR situations. We begin bymodifying the above array theory so that it more directly applies to the SARproblem.

In standard array theory we generate a one-wayantenna patternbecause we consider an antenna radiating toward a target (the transmitantenna case) or a target radiating toward an array (the receive antenna case).In SAR theory we need to consider a two-wayproblem since we transmit andreceive from each element of the synthetic array. If we refer to Figure 2, we canthink of each element as the position of the SAR aircraft as it transmits and

receives successive pulses. When the aircraft is located at y kd the

normalized transmit voltage is

Figure 7Normalized Radiation Pattern vs. Beam Steering AngleTwo

Targets located at -0.02 and 0.01 deg

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2 cj f tv t e .2 (13)

The resultant received signal (voltage) from a scatterer at ,n nx y is

, ,2 2 4 2

, ,2 2 2

, , ,

2 c n k n k cj f t r c j r j f tSn Sn Sn

n k n k

n k n k n k

P P Pv t v t r c e e e

r r r

(14)

whereSn

P is the return signal power and is determined from the radar range

equation.,n k

r is the range to the nth scatterer when the aircraft is at y kd .

We note that the difference between Equations (5) and (14) is that the latter

includes an2

r term in the denominator and has twice the phase shift as theformer.

If we modify Equation (7) as

2

, ,0 ,0 ,0 ,0

,0

22 sin 1 sin sin

n k n n n n

n

kdr r kdr r r kd

r

(15)

and repeat the math of Section 2.2.1, we get the following equation for theradiation pattern of a SAR antenna

22

2 2,0

sin sin sin1

sin sin sin

M dSSo

S dn S

PR

r M

. (16)

Figure 8contains plots of SR for the standard linear array (Equation(12)) and the SAR array (Equation (16)). In both cases the peak is normalized tounity. The notable difference between the two plots is that the width of the

main beam of the SAR array is half that of the standard linear array. This leadsto one of the standard statements in SAR books that a SAR has twice theresolution of a standard linear array. In fact, this is not quite true. If we wereto consider the two-wayantenna pattern of a standard linear array we wouldfind that its beamwidth lies between the one-way beamwidth of a standardlinear array and the beamwidth of a SAR array. The reason that the two-waybeamwidth of a standard linear array is not equal to the beamwidth of a SARarray has to do with the interaction between elements in the two arrays. In astandard linear array each receive element receives returns from all of theelements of the transmit array: However, in the SAR array, each receiveelement only receives returns from itself.

2 In our initial discussions we will be concerned only with cross-range imaging and can thus usea CW signal. We will consider a pulsed (LFM) signal when we add the second dimension.

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If we adapt the Equation (2) we have, for the SAR array, that

2B L . (17)

If we combine this with the equation for cross-range distance we get, again forthe SAR array,

2y R L (18)

which is also termed the cross-range resolution of the SAR. This equationindicates that the cross-range resolution of a SAR can be made arbitrarily small(fine) by increasing the length of the SAR array. In theory, this is true for a

spotlight SAR. In the case of strip map SAR, the size of the actual antenna onthe SAR aircraft (the element of the SAR array) is the theoretical limitingfactor on resolution. In either case, there are several other factors related tophase coherency that place further limits on the cross-range resolution.

Figure 8Normalized Radiation Patterns for a Standard Linear Array (topplot) and a SAR Array (bottom plot)

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2.4 TYPES OF SAR

At this point we want to discuss the basic differences between spotlightand strip map SAR and show how the actual antenna limits resolution for thestrip map SAR.

Figures 9and 10contain illustrations of the geometry associated with

strip map and spotlight SAR, respectively. With strip map SAR, the actualantenna remains pointed at the same angle while the aircraft flies past the areabeing imaged. This angle is shown as 90 in Figure 9but can be any angle. Forspotlight SAR, the actual antenna is steered to constantly point towards thearea being imaged. The term strip map derives from the fact that this type ofSAR can continually map strips of the ground as the aircraft flies by. The termspotlight derives from the fact that the actual antenna constantly illuminates,or spotlights, the region being imaged. A spotlight SAR must map a strip ofground in segments.

As might be deduced from Figure 9a limitation of the strip map SARgeometry is that the length of the synthetic array is limited by the fact that theregion imaged must remain in the actual antenna beam as the aircraft flies byit. For the case of spotlight SAR the antenna is always pointed at the regionbeing imaged so that the length of the synthetic array can, in theory, be as large

as desired. In practice, the length of the synthetic array for the spotlight SAR islimited by other factors such as range coherency and signal processinglimitations. Since the cross range resolution of a SAR is related to the length ofthe synthetic array, spotlight SARs can usually attain finer cross rangeresolution than strip map SARs

Figure 9Strip Map SAR Geometry

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2.4.1 Theoretical Limits for Strip Map SAR

The theoretical limit on cross range resolution for a strip map SAR canbe deduced with the help ofFigure 11. As illustrated in this figure, anddiscussed above, the point to be imaged must be in the actual antenna beamfrom the beginning to the end of the aircraft motion. The cross range span ofthe main beam of the actual antenna is

n ANT L r (19)

wheren

r is the perpendicular range from the aircraft flight path to the point

being imaged. From the geometry ofFigure 11 it clear that the point beingimaged will remain in the main beam of the actual antenna as the aircrafttraverses a distance of L . Thus the length of the synthetic array is L .

Using the Equation (2), we can write the beamwidth of the actualantenna as

ANT ANTL (20)

where ANTL is the horizontal width of the actual antenna. If we substitute

Equation (20) into Equation (19) we get

n ANT L r L (21)

which we can combine with Equation (18) to get

22

nANT

n ANT

ry L

r L

. (22)

Thus, the finest cross range resolution one can expect from a strip map SAR ishalf of the horizontal width of the actual antenna. This cross range resolution

Figure 10Spotlight SAR Geometry

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only applies to the case where a point is being imaged. The resolution for afinite sized area will be slightly worse.

2.4.2 Effects of Imaged Area Width on Strip Map SAR Resolution

Figure 12illustrates a case where the width of the region to be imaged is

w . It can be observed from this figure that 'L L w . From this we concludethat the modified cross range resolution is

'2 ' 1

nr yy yL w L

. (23)

In practice the term w L will be small so that 'y y . As an example of this,lets us consider the earlier example where we had a synthetic antenna length of

600 mL . From Equation (18), the resulting resolution for a point target is, intheory, 0.5 my . Suppose we wanted to image an area with a width of 50 m.For this case we would need to shorten the synthetic array to

' 550 mL L w . As a result, from Equation (23), the resolution would be0.546 m instead of 0.5 m.

Figure 11Resolution Limit for Strip Map SAR

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Figure 12Effect of Finite Area Width on Strip Map SAR resolution