George W. Stimson introduction to Airborne Radar (Se)

Подождите немного. Документ загружается.

PART VI Air-to-Air Operation

384

vers, to eliminate as much noise as possible without intro-

ducing excessive lag.

Control is the generation of a command calculated on the

basis of the filter’s outputs to reduce the tracking error as

nearly as possible to zero.

Response is the response of the hardware and/or software

to which the command is given. The difference between the

response and the current actual value of the parameter

feeds back to the input, closing the loop, and the entire

process repeats. Through successive iterations, the parame-

ter may be tracked with extreme precision.

Special Terminology. Before proceeding further, it will be

well to introduce two important technical terms used by

tracking-loop designers: discriminant and estimate.

Discriminant is the term for the calibration of the mea-

surement function. It is commonly represented by a plot of

the output of the hardware and/or software that performs

the measurement versus the true value of the tracking error

(Fig. 3). The slope of the linear portion of the plot deter-

mines the sensitivity of the measurement. Typically, the

slope increases as signal-to-noise ratio increases.

An important feature of discriminants is that they are

dimensionless (normalized). Consequently, precise mea-

surement of voltage or power levels isn’t required. More-

over, except for the influence of signal-to-noise ratio, the

measured values of the tracking error don’t vary with signal

strength. They are independent of the target’s size, its range,

its maneuvers, and fluctuations of its RCS. If desired,

though, a discriminant can be given a dimension simply by

multiplying it by a precomputed constant.

Estimate is the term applied to the value of any parame-

ter that is

(a) measurable only in combination with corrupting

interference—e.g., thermal noise (Fig. 4), or

(b) not directly measurable, e.g., range rate based on a

sequence of range measurements.

According to this definition, virtually every parameter mea-

sured or computed by a radar, no matter how precisely, is

an estimate.

With these definitions in mind, let us take a quick look

at the angle-tracking loops commonly incorporated in sin-

gle-target tracking modes.

Range-Tracking Loop. This loop has two primary goals:

to continuously and accurately determine the target’s cur-

rent range, and to keep a range-gate—actually two adjacent

sampling times—centered on the target’s echoes to isolate

them for doppler and angle tracking.

True Value

(+)

(–)

Measured Value

SNR

(dB)

(+)

(–)

3. A tracking discriminant may be represented by a normalized

plot of the measured value of the tracking error versus the true

value. The steeper the linear portion of the discriminant, the

more sensitive the measurement.

Estimate

Measurements

4. The value of any parameter that can only be determined on

the basis of successive measurements which are corrupted by

noise or other interference is termed an estimate.

To facilitate forming the range discriminant, the video

output of the receiver is passed through a low-pass filter,

stretching the target’s echoes to roughly twice the radar’s

pulse width and giving them a more “rounded” shape.

Assuming that the video is sampled at intervals equal to the

pulse width, this results in two samples being taken of each

target echo and in the amplitudes of the samples differing

in proportion to the displacement of the range gate from

the center of the echo (Fig. 5). Because successive samples

are stored in separate range bins, the first sample is called

the early range bin; the second, the late range bin.

The goal being to keep the range gate centered on the

target echoes, the range discriminant is formed by measur-

ing the difference between the amplitudes of the two sam-

ples: R

– R

. The measurement is normalized by dividing it

by the sum of the amplitudes (Fig. 6). The result is sup-

plied to the range filter.

CHAPTER 29 Automatic Tracking

385

1. The filter removes the pulse’s

higher frequency compo-

nents, which contribute to the

sharpness of its leading and

trailing edges.

On the basis of the range discriminant and the previous

range-gate command, the range filter produces best esti-

mates of the target’s range and range rate, a measure of the

range acceleration, and a new range-gate command (Fig. 7).

The range-gate command is essentially a prediction of

what the target’s range will be when the next target echo is

sampled. Typically, the command is formed by taking the

filter’s latest estimates of the target’s range and range rate

and linearly extrapolating the range.

To carry out the range-gate command, the predicted tar-

get range is first corrected for radar peculiarities (such as

sampling-time granularity) and distortion of the pulse-

shape in going through the receiver and pulse-stretching

low-pass filter. The prediction is then converted into units

of time measured from the trailing edge of the immediately

preceding transmitted pulse, hence to the estimated arrival

time of the next echo (Fig. 8).

Time

Stretched Envelope

of Target Echo

Sampling

Times

= Magnitude of sample in late range bin

= Magnitude of sample in early range bin

e = Tracking error

Range Gate

e ∝ R

– R

5. The range gate is centered between two adjacent sampling

times. To track a target in range, the sampling times must be

shifted to center the range gate on the target’s echoes. The

tracking error, e, is proportional to the difference between

the early and late samples, R

and R

∆R =

– R

+ R

(M +

) – (M –

)

(M +

) + (M –

)

RANGE DISCRIMINANT, ∆R

e = Tracking error = 2

M = Mean value of samples

6. Range-tracking error is proportional to the difference between

the magnitudes of the samples stored in the early and late

range bins. Dividing by their sum yields a nondimensional

ratio of the error to twice the mean of the samples.

∆R

Range*

Range Rate*

Range Acceleration*

*Best estimate

Range

Filter

Range Gate

Command

New Range Gate

Command

7. Inputs and outputs of the range filter. ∆R is the range discrimi-

nant.

Time

Range

Gate

Time Until Arrival

of Next Echo

Transmitted

Pulse

8. Positioning of the range gate in response to range-gate com-

mand. For this, the predicted range is converted to time.

PART VI Air-to-Air Operation

386

Doppler (Range-Rate) Tracking Loop. The purpose of

this loop is two-fold: (a) to provide a directly measured,

more accurate value of the target’s range rate than is avail-

able from the range-tracking loop, and (b) to isolate the tar-

get’s returns for angle tracking by keeping a so-called

“velocity gate” centered on the target’s doppler frequency.

The simplest velocity gate is the crossover point of two

adjacent doppler filters,

called the low- and high-frequen-

cy filters. Any error in the alignment of the velocity gate, of

course, shows up as a difference between the outputs of

these filters. The discriminant is formed by taking the dif-

ference between the magnitudes of the outputs, V

– V

and normalizing it by dividing by their sum (Fig. 9). The

result is supplied to the velocity filter.

The functions of this filter almost exactly parallel those

of the range filter. The velocity filter’s outputs are simply

more accurate estimates of the target’s range rate and range

acceleration.

Based on the velocity filter’s most current range-rate and

range acceleration estimates, a velocity-gate command is

produced. It is essentially a prediction of what the target’s

doppler frequency will be when the next set of doppler fil-

ters is formed.

The command is applied to a variable-frequency RF

oscillator. Its output is mixed with the received signal,

thereby shifting its frequency so that the target’s predicted

doppler frequency will be centered in the velocity gate.

The sum of the oscillator’s frequency and the velocity gate’s

fixed frequency then is the target’s predicted doppler fre-

quency (Fig. 10).

Angle-Tracking Loop. The role of this loop is to (a) accu-

rately determine the target’s direction (angle) relative to a

chosen coordinate system, (b) determine the target’s angle

rate, and (c) keep the antenna boresight precisely trained

on the target. Commonly used coordinate systems are

defined in the panel on the facing page.

What the angle tracking loop measures is the angle

between the antenna boresight and the line of sight to the

target. This angle, ε, is called the angle off boresight, AOB

(Fig. 11), and is generally resolved into azimuth and eleva-

tion coordinates.

Previous chapters introduced three techniques for sens-

ing the AOB: sequential lobing, amplitude-comparison

monopulse, and phase-comparison monopulse. Since

they’re basically quite similar, we’ll consider only one here:

amplitude-comparison monopulse. For it, you’ll recall, dur-

ing reception, the antenna’s radiation pattern is split into

two lobes which cross at their half power points.

3. If the PRF is less than the tar-

get’s doppler frequency, some

multiple, n, of the PRF must

be added to this sum. See

Chap. 21, page 286.

2. Two separate banks of filters

are formed by integrating the

samples collected in the early

and late range bins. The

velocity gate may be formed

in either or both of them.

High

Frequency

Filter

Low

Frequency

Filter

Target Return

Doppler Frequency

Velocity Gate

∆V

Tracking Error

Velocity Discriminant =

– V

+ V

Voltage

9. The simplest velocity gate is the intersection of two adjacent

doppler filters. The velocity discriminant is the difference

between the output voltage the target return produces from the

two filters divided by the sum of the two voltages.

AOB

Line of Sight to Target

Antenna Boresight

11. What the angle tracking loop measures is the angle, AOB,

between the line of sight to the target and the antenna bore-

sight line.

Oscillator

Frequency

Target’s Predicted Doppler Frequency

Velocity Gate

Gate’s

Frequency

Target

10. When the oscillator has moved the target into the gate, the

sum of the oscillator’s frequency and the velocity gate’s fixed

frequency is the target’s predicted doppler frequency.

As can be seen from Fig. 12, the difference between the

amplitude of the target’s echoes as received through the left

and right lobes, V

– V

, is roughly proportional to the

AOB. Dividing this difference by the sum of the two ampli-

tudes yields a dimensionless discriminant for the azimuth

component of the AOB. A discriminant for the elevation

component is similarly formed.

The measured components of the AOB are supplied to

the angle-tracking filter along with the following environ-

mental information:

• Signal-to-noise ratio

• Radar-bearing aircraft’s velocity

• Target range and range rate

• Antenna’s current angle rate

From these inputs, the filter produces best estimates of

the azimuth and elevation components of the AOB, the

angle rate of the line of sight to the the target, and the tar-

get’s acceleration (Fig. 13).

To reduce the AOB and keep the antenna boresight

trained on the target, azimuth and elevation rate commands

are generated. Each of these is the algebraic sum of (a) the

filter’s best estimate of the respective line-of-sight rate and

(b) a rate proportional to the filter’s best estimate of the

respective component of the AOB.

The rate commands are fed to the antenna stabilization

system (Fig. 14). There they control the rate of precession

of gyros that inertially establish azimuth and elevation axes

in space to which the antenna is tightly slaved.

In the case of an electronically steered antenna, steering

commands for both angle tracking and space stabilization

must be provided. To continuously correct for changes in

aircraft attitude, no matter how small, new commands must

be computed and fed to the antenna at a very high rate.

CHAPTER 29 Automatic Tracking

387

COMMON

COORDINATE

SYSTEMS

Measurements of

distances and angles

make sense only if

referenced to a

coordinate system.

Several common

systems are shown here.

Antenna

Stabilized

Aircraft

Fixed, Nonrotating

(May be used for short time periods.)

(Tail to nose)

(Top to bottom)

(Right wing,

root to tip)

Aircraft

N = north

E = east

D = down

r = range

e = east

= down

AOB

Left Lobe

Right Lobe

AOB =

– V

+ V

ANGLE DISCRIMINANT

12. Angle-tracking discriminant for amplitude comparison

monopulse. The antenna lobes cross on the boresight line; so

the angle AOB is roughly proportional to the difference

between the voltage of returns received through the two lobes.

Acceleration*

Angular Rate*

AOB*

Az. & El.

AOB

Estimates

Aircraft

Velocity

* Az & El Components,

best estimate.

Antenna

Motion

TARGET

Angle-Tracking

Filter

Range

Rate

SNRRange

Antenna

Stabilization

System

Aircraft

Motion

Achieved

Boresight

Angular Rate

Azimuth &

Elevation Rate

Commands

13. Inputs and outputs of the angle-tracking filter.

14. The antenna is stabilized against changes in aircraft attitude

by slaving it to azimuth and elevation axes established by

rate-integrating gyros mounted on it. The rate commands pre-

cess the gyros.

PART VI Air-to-Air Operation

388

Track-While-Scan

Track-while-scan (TWS) is an elegant combination of

searching and tracking. To search for targets, the radar

repeatedly scans a raster of one or more bars (Fig. 15). Each

scan is independent of all the others. Whenever a target is

detected, the radar typically provides both the operator and

the TWS function with estimates of the target’s range, range

rate (doppler), azimuth angle, and elevation angle. For any

one detection the estimates are referred to collectively as an

observation.

In pure search, the operator must decide whether targets

detected on the current scan are the same as those detected

on a previous scan or scans. With TWS, however, this deci-

sion must be made automatically. The algorithm used to

make it is one of the most complex algorithms in the radar.

In the course of successive scans, TWS maintains an

accurate track of the relative flight path of each valid target.

This process is iteratively carried out in five basic steps:

preprocessing, correlation, track initiation and deletion, fil-

tering, and gate formation (Fig. 16).

Preprocessing. In this step, two important operations

may be performed on each new observation. First, if a tar-

get having the same range, range rate, and angular position

has been detected on a preceding, overlapping bar of the

scan, the observations are combined. Second, if not already

so referenced, each observation is translated to a fixed coor-

dinate system, such as the NED. The angle estimates are

conveniently formulated as direction cosines—cosines of

the angles between the direction of the target and the N, E,

and D axes. Range and range rate may be projected onto

the N, E, and D axes simply by multiplying them by the

respective direction cosines.

Correlation. The purpose of this step is to determine

whether a new observation should be assigned to an exist-

ing track. On the basis of the observations assigned to the

track thus far, tracking filters accurately extend the values

of the N, E, and D components of each parameter of the

track to the time of the current observation. The filters then

predict what the values of these components will be at the

time of the next observation.

On the basis of accuracy statistics derived by the filters, a

gate scaled to the maximum error in measurement and pre-

diction is placed around each component of the prediction

for the track, as illustrated in Fig. 17. If the next observa-

tion falls within all of the gates for the track, the observa-

tion is assigned to the track.

Naturally, when closely spaced observations are received,

conflicts in assignments are likely to occur. To facilitate

3 dB beamwidth

< 3 dB beamwidth

Start

Finish

15. A representative four-bar raster scan. So that targets won’t be

missed, spacing of bars is less than the 3-dB beamwidth.

Consequently, the same target may often be detected on more

than one bar—one of several conflicts TWS resolves.

New

Obser-

vations

Updated Gates

Correlate

with

Tracks

Initiate or

Delete

Tracks

Filter

Form

Gates

Pre-

process

16. The five basic steps in track-while-scan processing.

Distance on Coordinate

Gate

Predicted

Position,

Observation

Time since initial observation

Target Observation

Computed Track

Actual Track

17. Representative track of one component (N, E, or D) of one of

a target’s parameters, illustrating its predicted value at the

time of the next observation and the gate for correlating the

observation with the track.

their resolution, a statistical distance of each observation

from the track or tracks involved is computed by normaliz-

ing and combining the differences between measurement

and prediction for all components of the observation. Each

track is centered in a gate, the radius of which corresponds

to the maximum possible statistical distance between mea-

surement and prediction.

CHAPTER 29 Automatic Tracking

389

4. A restriction applied in this

case is that a tentative track

cannot be initiated for an

observation that falls within

the gate of an existing track.

Accordingly, because a com-

peting observation is assigned

to the track O

falls in, O

discarded.

i, j

18. Gate for correlating an observation, O

, with a track, T

. Size

of the gate corresponds to the maximum possible statistical

distance, d, a valid observation may be from the track.

A representative conflict is illustrated in Fig. 19.

Observation O

falls within the gates of two different tracks:

and T

. Observations O

and O

both fall within the gate

of track T

. Conflicts such as this are typically resolved as

follows.

• Observation O

is assigned to track T

because it is

the only observation within the gate of T

, while T

has other observations, O

and O

, within its gate.

• Observation O

is assigned to track T

because its dis-

tance, d

2,2

, from the center of the gate is less than that

of O

Track Creation or Deletion. When a new observation,

such as O

in Fig. 19 does not fit in the gate of an existing

track, a tentative new track is established. If, on the next

scan (or possibly the next scan after that) a second observa-

tion correlates with this track, the track is confirmed. If not,

the observation is assumed to have been a false alarm and is

dropped. Similarly, if for a given number of scans no new

observation correlates with an existing track, the track is

deleted.

Filtering. This is similar to the filtering performed in sin-

gle-target tracking. On the basis of the differences between

the predictions and new measurements for each track, the

track is updated, new predictions are made, and accuracy

statistics for both observations and predictions are derived.

2,3

2,2

19. Typical conflicts arising when targets are closely spaced.

Here, gates for tracks T

and T

overlap. Observation O

falls in both gates, and observations O

and O

both fall in

the gate for track T

PART VI Air-to-Air Operation

390

Gate Formation. From the prediction and accuracy sta-

tistics derived by the filter, new gates are formed and sup-

plied to the correlation function.

As a result of the filtering, the longer a target is observed,

the more accurately the new gates are positioned, and the

closer the computed track comes to the actual track.

Summary

For single-target tracking, semi-independent tracking

loops are generally provided for range, doppler frequency,

azimuth, and elevation. Each loop includes four basic func-

tions: measurement, filtering, control, and system response.

The range-tracking error is measured by taking the dif-

ference between early and late samples of the target echoes;

the doppler-tracking error, by taking the difference between

the outputs of two adjacent doppler filters; the angle-track-

ing errors, by taking the difference between the returns

received through two antenna lobes.

The “scale factor” of each measurement, commonly rep-

resented by a plot of the measured value of the tracking

error versus the true value is called a discriminant. So that

the measurement will be largely independent of signal

strength and precise measurement of voltages or powers

won’t be required, the discriminant is normalized.

Successive measurements are, in effect, passed through a

low-pass filter whose gain and cut-off frequency are con-

stantly adjusted in light of the SNR, potential target maneu-

vers, and the aircraft’s own maneuvers to eliminate as much

noise as possible without introducing excessive lag.

From the filter outputs, a command calculated to reduce

the tracking error to zero is produced. For range tracking,

the command adjusts the radar’s sampling times; for

doppler tracking, it shifts the frequency of the received

echoes; for angle tracking it precesses the rate gyros of the

antenna stabilization system.

In track-while-scan, targets detected in successive search

scans are accurately tracked by filtering their parameters,

much as in single-target tracking. For each track, gates

based on the filtered parameters are used to determine

whether new detections should be assigned to existing

tracks or tentative tracks should be established for them,

and whether any existing tracks should be dropped.

Measure Filter Control

Respond

Actual

Value of

Parameter

Radar’s Knowlege of

Parameter

Tracking

Error

Time

Stretched Envelope

of Target Echo

Sampling

Times

= Magnitude of sample in late range bin

= Magnitude of sample in early range bin

e = Tracking error

Range Gate

e ∝ R

– R

New

Obser-

vations

Updated Gates

Correlate

with

Tracks

Initiate or

Delete

Tracks

Filter

Form

Gates

Pre-

process

Time

Range

Gate

Time Until Arrival

of Next Echo

Transmitted

Pulse

393

Meeting High

Resolution Ground

Mapping Requirements

1. Resolution distance is the minimum distance two points on

the ground can be separated and still be discerned sepa-

rately. A resolution cell is a rectangle whose sides are the

range and azimuth resolution distances.

n increasingly important airborne radar applica-

tion is making radar maps of sufficiently

fine

resolution that topographic features and objects

on the ground can be recognized.

In this chapter, we will learn how ground map resolution

is defined and see what the optimum resolution is for vari-

ous uses; then, review the approaches taken to providing it.

How Resolution Is Defined

The quality of the ground maps produced by a radar is

gauged primarily by the ability of the radar to resolve closely

spaced features of the terrain. This ability is generally

defined in terms of resolution distance and cell size.

Resolution distance is the minimum distance by which

two points on the ground may be separated and still be dis-

cerned individually by the radar. The separation is usually

expressed in terms of a range component, d

, and an

azimuth or cross range component, d

—the component at

right angles to the line of sight from the radar.

A resolution cell, or “pixel” (for picture element),

is a

rectangle whose sides are d

and d

(Fig. 1). Because fea-

tures of the terrain may be oriented in any direction, ideally

and d

are equal, making the cell a square.

As a rule, however, one does not deliberately restrict the

resolution in one direction to make the cells square. In real-

beam mapping for instance, where fine azimuth resolution

is difficult to obtain, d

is typically a small fraction of d

(see

Fig. 7). Nor is the resolution cell a sharply delineated rec-

tangle, as shown in Fig. 1. Rather, it is usually a rounded

rectangular “blob” whose brightness falls off at the edges.

1. Pixel and resolution cell are

not exactly synonymous.

Their dimensions may differ

considerably depending

upon how the radar’s signal

processor and display are

mechanized.

Click for high-quality image

RESOLUTION REQUIRED FOR

VARIOUS MAPPING APPLICATIONS

Features to be Resolved Cell size

Coast lines, large cities, and the

500 ft

outlines of mountains

Major highways, variations in fields 60–100 ft

“Road map” details: city streets,

30–50 ft

large buildings, small airfields

Vehicles, houses, small buildings 5–10 ft

PART VII High Resolution Ground Mapping and Imaging

394

Factors Influencing Choice of Cell Size

Among the more important considerations influencing

the choice of cell size are the sizes of the objects that must

be resolved, the amount of signal processing required to

produce the maps, cost, and finally the task of interpreting

the maps once they have ben made.

Size of Objects to Be Resolved. How large the resolution

cells can be and still provide useful ground maps depends

upon what the maps are used for. For discerning gross fea-

tures of the terrain such as coastlines and the outlines of

cities and mountains, a resolution of 500 feet or so will do.

For recognizing major highways, variations in the texture of

fields, and the like, a resolution of around 100 feet is need-

ed. To recognize city streets, large buildings, and small air-

fields—the sort of details commonly included in a road

map—resolution on the order of 30 to 50 feet is required.

To recognize the shapes of objects on the ground—such

as vehicles, houses, and small buildings—the resolution

must be considerably finer. Exactly how fine varies with

both the sizes and the shapes of the objects. As a rule, the

required resolution distance is somewhere between 1/5th

and 1/20th of the major dimension of the smallest object to

be recognized.

This is illustrated in Fig. 2 (facing page). It shows two

silhouettes of the same airplane. Over one is superimposed

a grid of resolution cells whose sides are 1/5th of the

wingspan. Over the other is superimposed a grid of cells

whose sides are 1/20th of the wingspan.

Alongside each silhouette is a simplified representation

of the ground map corresponding to the indicated cell size.

In these maps, cells that are filled completely by the silhou-

ette are shown as yellow; cells that are partly filled are

shown in shades of green corresponding to the percentage

filled; cells that do not include the airplane at all are shown

as dark green. For this particular shape, a resolution of

1/5th of the major dimension enables some shape recogni-

tion, while a resolution of 1/20th of this dimension enables

good recognition.

It should be pointed out, though, that in preparing Fig.

2 all elements of the airplane were assumed to reflect radio

waves in the radar’s direction equally. Actually, for any one

combination of look angle, radio frequency, and polariza-

tion, only a few bright scattering centers might be map-

pable. So even though the cell size was only 1/20th of the

major dimension, the airplane’s shape might still be difficult

to recognize. However, as we shall see, by repeatedly map-

ping the same area from different directions and with differ-

ent radio frequencies and polarizations, we can substantial-

ly increase the fraction of an object’s surface from which

mappable reflections are received. Through such tech-

niques, we can come quite close to realizing the kind of

shape recognition illustrated in Fig. 2 (above).

Amount of Signal Processing Required. A major con-

straint on the fineness of resolution that one would like to

provide is the amount of signal processing it requires.

CHAPTER 30 Meeting High Resolution Ground Mapping Requirements

395

2. Cell size required for shape resolution. Silhouettes (left) are identical. Radar maps, right, are simplified representations. Assuming that

all elements of the plane reflect equally in the radar’s direction, a cell size of 1/20 of the silhouette’s major dimension enables good

shape recognition.