2.0 SYNTHETIC APERTURE RADAR

2.1 INTRODUCTIONThe 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 synthesize a very long, linear array. The reason for creating a long array is to provide the ability to resolve targets that are closely spaced in angle, or cross range (usually azimuth). This, in turn, is driven by one of the main uses of SARs: to image the ground or targets. In both cases, the radar needs to be able to resolve very closely spaced scatterers. Specifically, resolutions in the order of a few feet are needed. To realize such resolutions in the range coordinate the radar uses wide bandwidth waveforms. To realize such resolutions in cross range very long antennas are required.

To get an idea of what we mean by “long” antenna, let’s consider an example. 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, , to antenna beamwidth,

, and range, , by

(1)

as shown in Figure 1. For and we get or about 0.003°!

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

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

. (3)Clearly, it would not be practical to use a real antenna that is as long as six football fields. Instead, a SAR synthesizes such antenna by using aircraft motion and signal processing.

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Figure 1 – Relation of Cross-Range distance to Beamwidth

2.2 BACKGROUND

2.2.1 Linear Array TheoryBefore 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 linear array. Suppose we have a 1 element linear array as shown in Figure 2. We have a target located at some that emits an E-field

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

. (4)

The resulting voltage at the output of the kth element is

. (5)

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

Figure 2 – 2N+1 Element Linear Array

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We can write as

(6)

where we used .

Since we can approximate as

. (7)

We next substitute Equation (7) into the exponent of (Equation (5)) and assume that in the term. This gives,

. (8)

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

. (9)

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

(10)

where we included the term to normalize it out of .When we formulated the antenna radiation pattern as above we were

interested in how varied with target angle, . We found that the peak of occurred at a target angle of as shown in Figure 3.

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As an extension to the above we can devise a means of steering the beam by including a linear phase shift across the array elements as shown in Figure 4. This phase shift is included in the equation for in the form

. (11)

This leads to a more general of

. (12)

Again, for standard array theory, we were interested in how varies with for a fixed . In this case the peak of would occur at , as shown

in the example of Figure 5.

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Figure 3 – Normalized Radiation Pattern vs. Target Angle

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Figure 4 – Linear Array with Phase Shifters

Figure 5 – Normalized Radiation Pattern vs. Target Angle – Beam Steered to - 0.01 deg

2.2.2 Transition to SAR TheoryIn SAR theory we need or reorient ourselves by thinking of the target

angle, , as being fixed and examining how varies with . In other words, we consider a fixed, , and plot . An example plot of for

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

Figure 7 contains a plot of for the case where there is a target at 0.01° 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

tells us the location of the two targets and their relative amplitudes. This is the type of information we want when we form SAR images.

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Figure 6 – Normalized Radiation Pattern vs. Beam Steering Angle – Target located at 0.01 deg

It will be noted that the plot of is analogous to a plot of the power out of a stretch processor, . is the other information we use to we form SAR images. Specifically, we will compute and for various values of and and then plot as intensities on a rectangular grid. The discrete values of and 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 EQUATIONSWith the above background we now want to start addressing the issues

associated with forming in practical SAR situations. We begin by modifying the above array theory so that it more directly applies to the SAR problem.

In standard array theory we generate a one-way antenna pattern because we consider an antenna radiating toward a target (the transmit antenna case) or a target radiating toward an array (the receive antenna case). In SAR theory we need to consider a two-way problem since we transmit and receive from each element of the synthetic array. If we refer to Figure 2, we can think of each element as the position of the SAR aircraft as it transmits and receives successive pulses. When the aircraft is located at

the normalized transmit “voltage” is

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Figure 7 – Normalized Radiation Pattern vs. Beam Steering Angle – Two Targets located at -0.02 and 0.01 deg

.2 (13)

The resultant received signal (voltage) from a scatterer at is

(14)

where is the return signal power and is determined from the radar range equation. is the range to the nth scatterer when the aircraft is at .We note that the difference between Equations (5) and (14) is that the latter includes an term in the denominator and has twice the phase shift as the former.

If we modify Equation (7) as

(15)

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

. (16)

Figure 8 contains plots of for the standard linear array (Equation (12)) and the SAR array (Equation (16)). In both cases the peak is normalized to unity. 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 leads to one of the standard statements in SAR books that a SAR has twice the resolution of a standard linear array. In fact, this is not quite true. If we were to consider the two-way antenna pattern of a standard linear array we would find that its beamwidth lies between the one-way beamwidth of a standard linear array and the beamwidth of a SAR array. The reason that the two-way beamwidth of a standard linear array is not equal to the beamwidth of a SAR array has to do with the interaction between “elements” in the two arrays. In a standard linear array each receive element receives returns from all of the elements of the transmit array: However, in the SAR array, each receive “element” only receives returns from itself.

2 In our initial discussions we will be concerned only with cross-range imaging and can thus use a 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. (17)

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

(18)which is also termed the cross-range resolution of the SAR. This equation indicates 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 on the SAR aircraft (the “element” of the SAR array) is the theoretical limiting factor on resolution. In either case, there are several other factors related to phase coherency that place further limits on the cross-range resolution.

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

2.4 TYPES OF SARAt this point we want to discuss the basic differences between

spotlight and strip map SAR and show how the actual antenna limits resolution for the strip map SAR.

Figures 9 and 10 contain illustrations of the geometry associated with strip map and spotlight SAR, respectively. With strip map SAR, the actual antenna remains pointed at the same angle while the aircraft flies past the area being imaged. This angle is shown as 90° in Figure 9 but can be any angle. For spotlight SAR, the actual antenna is steered to constantly point towards the area being imaged. The term strip map derives from the fact that this type of SAR can continually map strips of the ground as the aircraft flies by. The term spotlight derives from the fact that the actual antenna constantly illuminates, or spotlights, the region being imaged. A spotlight SAR must map a strip of ground in segments.

As might be deduced from Figure 9 a limitation of the strip map SAR geometry is that the length of the synthetic array is limited by the fact that the region imaged must remain in the actual antenna beam as the aircraft flies by it. For the case of spotlight SAR the antenna is always pointed at the region being 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 is limited by other factors such as range coherency and signal processing limitations. Since the cross range resolution of a SAR is related to the length of the synthetic array, spotlight SARs can usually attain finer cross range resolution than strip map SARs

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Figure 9 – Strip Map SAR Geometry

2.4.1 Theoretical Limits for Strip Map SARThe theoretical limit on cross range resolution for a strip map SAR can

be deduced with the help of Figure 11. As illustrated in this figure, and discussed above, the point to be imaged must be in the actual antenna beam from the beginning to the end of the aircraft motion. The cross range span of the main beam of the actual antenna is

(19)

where is the perpendicular range from the aircraft flight path to the point being imaged. From the geometry of Figure 11 it clear that the point being imaged will remain in the main beam of the actual antenna as the aircraft traverses a distance of . Thus the length of the synthetic array is .

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

(20)

where is the horizontal width of the actual antenna. If we substitute Equation (20) into Equation (19) we get

(21)which we can combine with Equation (18) to get

. (22)

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

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Figure 10 – Spotlight SAR Geometry

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

2.4.2 Effects of Imaged Area Width on Strip Map SAR ResolutionFigure 12 illustrates a case where the width of the region to be imaged

is . It can be observed from this figure that . From this we conclude that the modified cross range resolution is

. (23)

In practice the term will be small so that . As an example of this, lets us consider the earlier example where we had a synthetic antenna length of . From Equation (18), the resulting resolution for a point target is, in theory, . 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

. As a result, from Equation (23), the resolution would be 0.546 m instead of 0.5 m.

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Figure 11 – Resolution Limit for Strip Map SAR

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

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