George W. Stimson introduction to Airborne Radar (Se)

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CHAPTER 31 Principles of Synthetic Array Aperture Radar

407

Signal Processing Required. In the foregoing discussion,

the inputs to the SAR processor were referred to only as the

digitized radar returns. In an actual radar, the inputs would

be the digitized I and Q components (x

, y

) of the returns

(translated to video frequencies). The sums that build up in

the range bins, then, would be the vector sums of the accu-

mulated values of x

and y

. And the quantities transferred

from the range bins to the display memory would be the

magnitudes of the vector sums. The signal processing

required to synthesize a simple array of this sort is summa-

rized in the panel at the top of the page. Note that the

equations shown there are identical to those which must be

solved to form a doppler filter tuned to zero frequency (or

to the PRF).

Limitation of Unfocused Array. The rudimentary array

just described is called an “unfocused” array. For it must be

short enough in relation to the range to the swath being

mapped that the lines of sight from any one point at the

swath’s range to the individual array elements are essentially

parallel. In this respect, the array is similar to a pinhole

camera: both are focused at infinity.

Now, if the array length is an appreciable fraction of the

swath’s range, the lines of sight from a point at that range to

the individual elements will diverge slightly. Then, even if

the point is on the boresight line, the distances to the ele-

SIGNAL PROCESSING FOR UNFOCUSED ARRAY

The signal processing required to synthesize an unfocused

array antenna can be summarized mathematically as follows:

Inputs: For each resolvable range (R

), N successive pairs of

numbers are supplied.

Each pair represents the I and Q components of the return,

received from range R

by a single array element.

Integration: To form the beam (azimuth processing), the I and

Q components are summed.

Magnitude Detection: The magnitude or the vector sum of

l and Q is computed and output.

S is the amplitude of the total return from a single-resolution

ceII on the boresight line at range R

(An intermediate step not shown in the above diagram is

the scaling of the detected magnitudes to the values of

intensity—gray levels—that are to be displayed.)

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PART VII High Resolution Ground Mapping and Imaging

408

ments will not all be the same (Fig. 8, above). Since the

wavelength is generally fairly short, very small differences

in these distances can result in considerable differences in

the phases of the returns which the individual array ele-

ments receive from the point.

Because these phase errors are not compensated in the

unfocused array, its capabilities are quite limited. While the

azimuth resolution distance the array provides at any given

range can initially be reduced by increasing the length of

the array, a point is soon reached beyond which any further

increase in length only degrades performance.

Degradation begins with a gradual increase in gain of the

sidelobes relative to the mainlobe and a merging of the

lower order sidelobes with the mainlobe (Fig. 9).

This effect continues increasingly and is accompanied by

a progressive fall-off in the rate at which the mainlobe gain

increases with array length.

The reason for the fall-off in gain can be seen if we exam-

ine Fig. 8 again. It shows the distance from point P on the

boresight line to each element of an array of length L. For

elements near the array center, there is very little difference

in this distance. But for elements farther and farther

removed from the center, the difference grows increasingly.

As the array is lengthened, therefore, the phase of the

returns received by the end-most elements falls increasingly

far behind the phase of the sum of the returns received by

the other elements.

This progressive phase rotation and its effect on the

mainlobe gain of the synthetic array is illustrated in Fig. 10

(top of facing page). The phasors shown there represent the

returns received by the individual elements of a 27-element

array from a distant point (P) on the boresight line. The

phase of the return received by the middle element (14) is

taken as the reference. The gain in the boresight direction

corresponds to the sum of the phasors.

The returns received by the central elements (9 through

19) are so close to being in phase that their sum is virtually

undegraded by the lack of focus. But the phases of the

returns received by elements farther and farther out are

8. Distance from a point (P) on the boresight line of an array antenna to the individual array elements. If array length (L) is an appreciable frac-

tion of the range (R), the distances to the end elements will be appreciably greater than the distances in the central element.

9. Radiation pattern of an unfocused synthetic array showing

increase in relative gain of sidelobes and merging of side-

lobes with mainlobe as array length (L) is increased.

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rotated increasingly. The returns received by elements 4 and

24 are nearly 90˚ out of phase with the sum of the returns

received by the elements closer in, hence, contribute only

negligibly to that sum. The returns received by elements 1,

2, and 3 and 25, 26, and 27 are actually subtractive.

Obviously, under the conditions for which Fig. 10 was

drawn, the gain would have its maximum value if the array

were only 21 elements long (elements 4 through 24).

The degradation of beamwidth closely parallels that of

mainlobe gain. Initially, the gain at angles slightly off bore-

sight is degraded by the lack of focus to nearly the same

extent as the gain in the boresight direction. So at first,

there is little reduction in the rate at which the beam nar-

rows as the array is lengthened. But when the length reach-

es the point where the gain in the boresight direction stops

increasing, the beamwidth stops decreasing. If we lengthen

the array beyond this point, the beam starts spreading.

From the standpoint of both gain and beamwidth, the max-

imum effective length has been reached.

In Fig. 11, the gain and beamwidth for an unfocused

array are plotted versus array length.

11. Effect of increased array length on gain and beamwidth of an

unfocused synthetic array. Gain is maximum and beamwidth

is minimum when length, L = 1.2

公

僓

λR.

CHAPTER 31 Principles of Synthetic Array Aperture Radar

409

10. Degradation in the gain of an unfocused array. Phasors repre-

sent returns received from a distant point P by individual array

elements. Gain is the sum of the phasors. In this case, the gain

could be increased by decreasing the length of the array.

It can be shown from the geometry of the situation that

the array length for which given values of gain and

beamwidth are obtained varies in proportion to 公

僓

λR ,

where

λ is the wavelength and R is the range. To make the

graph of Fig. 11 applicable to any combination of

λ and R,

the array length is plotted there in terms of 公

僓

λR. As you

can see, the maximum effective array length is

eff

= 1.2 λR.

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PART VII High Resolution Ground Mapping and Imaging

410

Another way of looking at the effects of defocusing is

this. Imagine that you are approaching an unfocused array

of a given length from a long enough distance that the lines

of sight to each “elements” of the array are essentially paral-

lel. The beamwidth of the array at your range, therefore,

does not change as you advance.

However, as you approach the range for which the array

length is optimum and defocusing comes into play, the

beamwidth starts increasing.

The azimuth resolution distance, hence the finest achiev-

able resolution at that range, turns out to be roughly 40

percent of the array length.

min

0.4 L

eff

Moreover, beyond that range, we cannot make the

azimuth resolution of the radar independent of range, as we

would like. For when we lengthen the array further, the res-

olution distance increases as the square root of the range

(Fig. 12).

12. Maximum effective length of unfocused array increases only as

square root of range. Resolution distance at range for which

length is optimized is roughly 40 percent of array length.

Focused Array

The limitation on array length may largely be removed

by focusing the array. Then, by suitably increasing the

length of the array in proportion to the range, virtually the

same resolution may be obtained at any desired range.

How Focusing Is Done. In principle, to focus an array all

you need to do is apply an appropriate phase correction

(rotation) to the returns received by each array element. As

illustrated in Fig. 13 (top of facing page), the phase error

for any one element, hence the phase rotation needed to

cancel the error, is proportional to the square of the dis-

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∴ PHASE ERROR,

____

2π

(2∆R

) ≈

____

2π

(

____

)

Radians

λλR

Accounts for Round-Trip Travel

CHAPTER 31 Principles of Synthetic Array Aperture Radar

411

tance of the element from the center of the array.

Phase correction = –

2π

λR

where

= distance of element n from array center

λ = wavelength (same units as d

)

R = range to area being mapped (same units as λ)

In some cases, it may be possible to presum the returns

received by blocks of adjacent elements without impairing

performance. By rotating only the phases of the sums, com-

puting and storage requirements may be eased.

To simplify the description, however, we will omit presum-

ming here. Also, we will assume that the combination of array

length, PRF, and range is such that the resolution distance,

, is roughly equal to the spacing between array elements.

To focus an array when no presumming is done, we must

provide as many rows of storage positions in the bank of

13. Phase error for return received by any one array element (n) is proportional to the square of the distance (d

) from the element to the array

center. Factor of two by which ∆R

is multiplied accounts for the phase error being proportional to the difference in

round-trip

distance from

the element to point P (see page 416).

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PART VII High Resolution Ground Mapping and Imaging

412

14. How array is focused in a line-by-line processor. To simplify the description, conditions are assumed to be such that the resolution distance,

, equals the spacing between array elements. Hence, every time a pulse is transmitted a new array must by synthesized.

range bins as there are array elements (Fig. 14, above). As

the returns from any one transmitted pulse (array element)

come in, they are stored in the top row. When the return

from the most distant range increment has been received,

the contents of every row are shifted down to the row

below it to make room for the incoming returns from the

next transmitted pulse. The contents of the bottom row are

discarded.

Between these shifts, the column of numbers in each bin

is read serially and the numbers are appropriately phase

shifted and summed—a process called azimuth compression.

The magnitude of the sum for each range bin is entered in

the appropriate range positions in the top row of the dis-

play memory. Thus (for the conditions assumed in this

example), every time the returns from another transmitted

pulse have been received—i.e., every time the radar has

advanced a distance equal to the spacing between array ele-

ments—another array is synthesized.

3. If the resolution distance, d

were greater than the spacing

between array elements, an

array would be synthesized

only after the radar had

advanced a distance d

CHAPTER 31 Principles of Synthetic Array Aperture Radar

413

Signal Processing Required. The computations (per

range bin) required to perform the phase rotation and sum-

ming for a focused array are shown in the panel, above.

They are exactly the same, you will notice, as the computa-

tions required to form a doppler filter with the DFT.

Azimuth Resolution. With focusing, the length of the

array can be greatly increased. But as with all things, a point

is ultimately reached where a further increase in length does

not improve resolution. In the case of an array whose

azimuth angle is fixed—that is, one which looks out at a

constant angle relative to the flight path—this limit is estab-

lished by the physical size of the elemental radiator, the real

antenna. Surprisingly, the smaller this antenna is, the longer

the array can be made.

The reason is simple enough. For return to be received

from any one ground patch by all elements of the array, the

beam of the real antenna must be wide enough for the

patch to fall within the beam for every position of the

antenna in the entire length of the array (Fig. 15). For that

condition to be satisfied, the width of the beam at the range

of the patch must at least equal the length of the array. The

smaller the real antenna is, the wider its beam; hence, the

longer the array can be made and the finer the resolution

that can be achieved.

For a given real-antenna size, how fine can that be?

Before answering this question, we must consider an impor-

15. Each line that is mapped must be within the mainlobe of the

real antenna while the radar traverses the entire length of the

array, L.

SIGNAL PROCESSING FOR A FOCUSED ARRAY

To focus an array, for every range bin the signal processor

must mathematically perform the equivalent of rotating the

phasor representation (A) of the return received by each

successive array element (n) through the phase angle,

␾

(the

value of which was derived in Figure 13).

The I and Q components of the phasor before rotation are the

inputs, x

and y

After rotation, we’ll represent the components by

x⬘

and y⬘

To perform the rotation, the following algorithms must

be computed.

The values of x⬘

and y⬘

for the total number of array elements (N)

must then be summed separately

and the magnitude of the vector sum of x and y must be calculated.

These, you’ll recognize, are the same algorithms that are

computed when forming a digital filter with the DFT.

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PART VII High Resolution Ground Mapping and Imaging

414

tant difference between the beam of a synthetic array and

the beam of a real array.

A real array has both a one-way and a two-way radiation

pattern. The one-way pattern is formed upon transmission

as a result of the progressive difference in the distances

from successive array elements to any point off the bore-

sight line (Fig. 16). (The phases of the radiation from the

individual elements arriving at that point differ in proportion

to the differences in distance.) This pattern has a sin x/x

shape. The two-way pattern is formed upon reception,

through the same mechanism. Since the phase shifts are the

same for both transmission and reception, the two-way

pattern is essentially a compounding of the one-way pat-

tern, and so has a (sin x/x)

shape.

A synthetic array, on the other hand, has only a two-way

pattern. For the array is synthesized out of the returns

received by the real antenna, which sequentially assumes

the role of successive array elements. Because each element

receives only the returns from its own transmissions, how-

ever, the element-to-element phase shifts in the returns

received from a given point off the boresight line correspond

to the differences in the round-trip distances from the indi-

vidual elements to the point and back (see panel, page 416).

This is equivalent to saying that the two-way pattern of the

synthetic array has the same shape as the one-way pattern of

a real array of twice the length, sin 2x / 2x (Fig. 17).

For a uniformly illuminated real array, the one-way 3-dB

beamwidth is 0.88 times the ratio of the wavelength to the

array length. Consequently, for a uniformly illuminated syn-

thetic array, the two-way 3-dB beamwidth is

3 dB

= 0.44

radians

The point on the radiation pattern where the beamwidth is

measured, of course, is fairly arbitrary. It turns out that by

measuring the beamwidth at a point 1 dB lower down, the

factor 0.44 can be increased to one half. To simplify the

beamwidth equation, therefore, the minus 4-dB point is

commonly used.

4 dB

___

radians

The azimuth resolution distance, then is

___

Armed with this expression, we can now go back and

answer the question raised earlier: if the length of the syn-

thetic array is limited to the width of the beam of the real

16. One-way radiation pattern of a real array is formed during

transmission as a result of progressive difference in distance

from successive array elements to observation point.

17. Comparison of mainlobes of real and synthetic arrays of same

length. Synthetic array has no one-way radiation pattern since

the array is synthesized from the radar return.

Click for high-quality image

antenna at the range being mapped, for a given sized anten-

na how fine can the resolution of the synthetic array be?

If the real antenna is a linear array and its length is

then its one-way 4-dB beamwidth is λ /

l, where l is the

array length. Multiplying this expression by the range, R, of

the swath being mapped gives the maximum length of the

real array.

max

Substituting L

max

for L in the expression for azimuth

resolution distance (Fig. 18), we find that the minimum

resolution distance d

min

is half the length of the real

antenna.

min

Length of real antenna

This, then, is the ultimate resolution of a synthetic array

whose beam is positioned at a fixed angle relative to the

flight path, as in strip map radars. As we will see in the next

chapter, this limitation is removed in the “spotlight” mode,

by keeping the beam of the real antenna continuously

trained on the area being mapped.

Reducing the Computing Load: Doppler Processing

As is clear from Fig. 14 (page 412), if the array is very

long, an immense amount of computing is required for

line-by-line processing of a focused array. In the simple

example illustrated there, every time the radar transmits a

pulse, it must perform the phase correction all over again

for every pulse that has been received over an entire array

length and sum the results.

Put another way, if there are N elements in the array,

every time the radar advances a distance equal to the array

length, it must phase correct and sum (N x N) returns for

every range bin. This load may be reduced somewhat by

presumming (if conditions permit presumming). But it is

still formidable.

The computing load can, however, be dramatically

reduced by processing the data in parallel for many lines of

the map at one time, rather than serially, a line at a time.

For parallel processing, the returns from different

azimuth angles are isolated with doppler filters. But before

getting into the details of that, we must see how doppler

frequency is related to azimuth angle.

CHAPTER 31 Principles of Synthetic Array Aperture Radar

415

18. If array length, L, is made equal to beamwidth of real anten-

na at range, R, azimuth resolution of synthetic array will

equal 1/2 length of real antenna.

PART VII High Resolution Ground Mapping and Imaging

416

WHY THE ELEMENT-TO-ELEMENT PHASE SHIFT

IS DOUBLE IN A SYNTHETIC ARRAY

A linear-array radar antenna achieves its directivity by virtue of

the progressive shift in the phases of the returns received by

successive array elements from points off the antenna boresight

line. For any one angle off boresight, this shift is twice as great

for a synthetic array as for a real array having the same inter-

element spacing. The reason can be seen by considering the

returns received from a distant point, P, displaced from the

boresight line by a small angle,

␪

Real Array. In the case of a real array, transmission from all

array elements is simultaneous. Every time a pulse is

transmitted, the radiation from all elements arrives at P

simultaneously—albeit staggered in phase as a result of the

progressive differences in the distances from successive

elements to P. The phase differences naturally reduce the

amplitude of the sum of the radiation received at P from the

individual elements. (This reduction gives the one-way radiation

pattern its sin x/x shape.) But for each pulse, the phase shift of

the sum is determined by the distance from the central element

(3) to P. Therefore, if the position of the antenna is not changed,

all of the pulses reflected by P have the same phase.

The portions of each reflected pulse that are received by the

individual array elements similarly differ in phase as a result of

the progressive difference in the distances from P to the

elements. As with transmission, the phase differences reduce

amplitude of the sum of the outputs the received pulse produces

from the individual elements. This reduction, compounded with

the reduction in the amplitude of the pulses reflected ffom P,

gives the two-way radiation pattern its (sin x/x) shape. But the

phase differences are again due only to the differences in the

one-way

distances from P to the elements.