Diss / 10

4.4  SUMMARY AND DISCUSSION

Target range tracking is accomplished by continuously measuring the time delay between the transmission of an RF pulse and the echo signal returned from the target, and converting the roundtrip delay into units of distance. The target range measurement may also be performed with continuous wave (CW) radar systems using the frequency-modulated continuous wave (FM-CW) that is typically a linear-ramp FM. The target range is determined by the range-related frequency difference between the echo-frequency ramp and the frequency of the ramp being transmitted. The first function of the target range tracker is acquisition of a desired target. Although this is not a tracking operation, it is a necessary first step before the target range tracking or target angle tracking (the azimuth tracking) may take place in a typical radar system. The range-tracking portion of the radar has the advantage of seeing all targets within the beam from close range out to the maximum range of the radar. It typically breaks this range into small increments, each of which may be simultaneously examined for the presence of a target. When beam scanning is necessary, the target range tracker examines the increments simultaneously for short periods, such as 0.1 s, makes its decision about the presence of a target, and allows the beam to move to a new location if a “no” target is present.

Once a target is acquired in range, it is desirable to follow it in the range coordinate to provide distance information or slant range to the target. Appropriate timing pulses provide a target range gating so the angle-tracking circuits and automatic gain control (AGC) circuits look at only the short target range interval, or time interval, when the desired echo pulse is expected. The target range tracking operation is performed by closed-loop tracking similar to the azimuth tracker. Error in centering the range gate on the target echo pulse is sensed, error voltages are generated, and circuitry is provided to respond to the error voltage by causing the gate to move in a direction to recenter on the target echo pulse. Automatic target range tracking can generally be divided into five steps: (a) radar detection acceptance, accepting or rejecting detections for insertion into the tracking process—the purpose of this step is to control false track rates; (b) association of accepted detections with existing tracks; (c) updating the existing tracks with associated detections; (d) new track formation using unassociated detections; and (e) radar scheduling and control.

In the course of target range track detection the following operations are carried out: (a) gating and selection of target pip within the gate, (b) checking the detection criterion, and (c) estimation and extrapolation of target range track parameters. Target range tracking is sequential from measuring to measuring procedure of newly obtained target pip binding to target range track and accurate target range track determination. Under target range autotracking, the following operations are carried out: (a) accurate definition of target range track parameters in the course of newly obtained target pip binding, (b) extrapolation of target range track parameters for the next radar antenna scanning, (c) range gating of possible newly obtained target pip positions, and (d) selection of the definite target pip within the gate if there are several target pips within the gate area. When there are several target pips within the tracking gate, we expand the target range track binding for each target pip. When the target pip is absent within the tracking gate, the target range track is prolonged by the corresponding extrapolated point, but the next gate is increased in area to take into consideration the errors increasing in the course of extrapolation. If there are no target pips during k radar antennas scanning following one after another, the target range tracking is canceled. Thus, at various stages of target range track detection and target range tracking, we carry out the same operations in reality:

(a) radar antenna scanning area gating, (b) selection and identification of target pips within the gate, and (c) filtering and extrapolation of target range track parameters.

While the CRS computer subsystem processes a high number of targets in real time, computation of dimensions and orientation of parallelepiped gate sides is very cumbersome. In this case, there is a need to use a simple version of gating, which is described as follows. The simplest gate shape is chosen for definition in that coordinate system, wherein radar information is processed. In the case of spherical coordinate system, the simplest gate is given by a linear dimension in target range rgate and by two angular dimensions, namely, by the azimuth Δβgate and elevation Δεgate

(see Figure 4.4). These dimensions can be set in advance, based on the maximal magnitudes of random and dynamic errors while processing all target range tracks. In short, in this case, the gate dimensions are chosen so that the ellipsoid of all possible (under all directions of target moving and flight) total deviations of true target pips from corresponding extrapolated points will be easily fit and rotated in any direction inside the gate. This is the crudest technique of gating. We note that all procedures to choose dimensions of 3D gate can be used in full under gating in plane for binding newly obtained target pips in 2D radar systems.

The quality of target pip indication process can be evaluated by the probability of correct indication; in other words, the probability of event that the true target pip will be indicated to prolong the target range track in the next radar antennas scanning. The problem of definition of the probability of correct indication can be solved analytically if we assume that a presence of false target pips within the gate is only caused by a stimulus of the noise and interference and these target pips are uniformly distributed within the limits of radar antenna scanning area. Under indication within 2D rectangle gate circumscribed around the ellipse with the parameter equal λmax ≥ 3, the probability of correct indication is determined by

where

ρS is the density of false target pips per unit square of gate

ση and σξ are the mean square deviations of true target pips from the gate center along the axes η and ξ

In the case of target pip indication in 3D gate that can be presented in the form of parallelogram circumscribed around the total error ellipsoid, the probability of correct indication is determined by

Pind ≈ 1 − 21.33πσησξσζρV ,

2π

where ρV is a density of false target pips per unit volume of gate.

We can note that in addition to target pip deviations from the gate center, the weight features of target pips formed in the course of signal preprocessing, as some analog of SNR, can be used to indicate the target pips. In the simplest case of binary quantized target return pulse train, we can use the number of train pulses or train bandwidth to generate the target pip weight features. Weight features of target pips can be used under target range track indication jointly with features of target pip deviations from the gate center or independently. One of the possible ways to use the target pip weight features jointly with the target pip deviations from the gate center is as follows. All target pips within the gate are divided with weight ν1 and with weight ν0 depending on whether the target return pulse train bandwidth exceeds the earlier-given threshold value or not, which in turn depends on target range magnitude. If there are target pips with weight ν1 we indicate the target pip that is nearest to the gate center as the true target pip from the observed target pip group. If there are no target pips with weight ν1, we indicate the target pip with weight ν0 that is nearest to the gate center as the true target pip. If we can characterize the target pip weight by the number of pulses in the target return pulse train, then we are able to indicate the target range track by the maximum number of pulses in the target return pulse train. In this case, features of target pip deviations from the gate center are only used under equality of weights of several target pips.

Under a hard noise environment, several gates formed around extrapolated points of prolonged target range tracks will be overlapped. Moreover, false and newly obtained target pips will appear within the gates. In this case, the problem of newly obtained target pip distribution and binding to

target range tracking becomes more complex. In addition, the problem of fixing the beginning of a new target range track arises. First of all, this complication is associated with the necessity to use a joint target pip binding to target range track after getting newly obtained target pips liable to binding or, in other words, if these newly obtained target pips are within overlapping gates. There is a need to generate all possible ways or hypotheses for target pip binding and choose the most plausible version. After the determination of probabilities of events that the target pips belong to the existing target range tracks, new target range tracks, and false target range tracks, we are able to determine the probabilities of all versions of the target pip binding and to choose the version with the highest probability as a true version. Solution of this problem is very tedious even for the case that is considered the simplest. This problem can be simplified if the following practical rules are followed: (a) each target range track must be attached by the target pip, and (b) if the target pip is not associated to the given-before target range track, then a new target range track must be started independently of the probability of event that this target pip belongs to the new target range track or false target range track; in this case, we should compare only versions when there are target pip bindings for each target range track.

The process of target range track detection is divided into two stages: (a) the first stage is the target range track detection by criterion “2/m” and (b) the second stage is the confirmation of beginning the target range track, that is, the final target range track detection by criterion “l/n.” In particular cases of the second stage, the target range track detection may be absent. Algorithm of the target range track detection by criterion “2/m” jointly with the confirmation algorithm or final target range track detection by criterion “l/n” forms a united target range track detection algorithm by criterion “2/m + l/n.” The main computer subsystem operations carried out under target range track detection are as follows: (a) evaluation of target velocity under moving, (b) extrapolation of coordinates, and (c) gating the target pips. Target range track detection quality can be evaluated by the following characteristics: (a) the probability of detection of true target range track, (b) average number of false target range tracking per unit time, and (c) speed of computer systems used to realize the target range track detection algorithm.

Results of determination of the average number Nftr of target range tracks that are needed to be tracked as a function of the average number of false target pips within the radar antenna scanning that are subjected to signal reprocessing by computer subsystems are presented in Figure 4.9, for the purpose of comparing the filtering ability. Analyzing these results, we can conclude as follows:

(a) filtering ability of target range track detection algorithms, realizing that the criterion “2/m + l/n” increases with a decrease in m and n and with an increase in l, and (b) incrementing l on unit leads to a great increase in the filtering ability, compared to a corresponding decrease in m and n. These features must be taken into consideration while choosing the target range track detection algorithm employed in practice. If the CRS computer subsystem has limitations in the number of false target range tracks that are needed to be tracked, then a combination of the average number Nftr and criteria of target range track detection algorithms during production tests allows us to raise a demand to the noise and interference level at the reprocessing computer subsystem input, that is, SNR. While designing the reprocessing computer subsystem of a CRS, information about the average number of false target range tracks that are under detection in the steady working state of a CRS is needed. Evidently, all false target range tracks, with respect to which the final decision about tracking or cancellation has not been made, are under detection process.

Analysis and comparison of performance shown in Figure 4.11 allow us to conclude that it is worthwhile to employ the criterion “2/m + 1/n” based on the viewpoint of decreasing the number of steps under detection of true target range track. In doing so, small deviations in m and n do not lead to great changes in the number of steps to ensure the probability of true target range track detection close to the unit. The probability of true target range track detection equal to 0.95 using the discussed criteria is obtained for target range corresponding to 0.75 of the maximum radar system range. Implementation of confirmation criteria “l/n” (l > 1) leads to the prolongation of time to detect the true target range tracks.

In the course of each target range tracking, two main problems are solved: gating and selection of newly obtained target pips for the prolongation of target range track, estimation of target range track parameters, and definition of variations in these parameters as a function of time. In principle, solutions to both problems can be obtained by employing the same detection algorithm. In this case, the required quality of a solved target range track estimation parameter problem must be in agreement with the customer requirements. However, there is a version of target range autotracking system in which the autotracking algorithm hunts only for target range track. To achieve a high-quality estimate of the target range track parameters, the individual target range track detection algorithm must be designed.

In general, in addition to the presence of the target pips within the gate to prolong the target range track, we should take into consideration a set of other factors such as the target importance; target maneuverability, that is, to change a target range track during the flight; current target coordinates; direction of target moving; and length of target visibility within radar antenna scanning area and so on to make a decision about the target range tracking cancellation. However, a recordkeeping of these factors is very difficult and is not accessible forever owing to limitations in the computer system speed. For this reason, the main criterion for decision about the target range tracking cancellation is the appearance of some threshold series kth of target pip misses within tracking gates. This criterion of cancellation of the target range tracking does not take into consideration individual peculiarities of each target range track and does not use information about accumulated accuracy level at the instant of appearance of the target pip miss series. A single advantage of this criterion is its simplicity.

In practice, in the majority of cases, it is very convenient to employ a structure of signal reprocessing subsystems in which the target range track detection and tracking algorithms are united and presented in the form of a single algorithm of target range track detection and tracking, and realization of this united algorithm is carried out by the individual computer subsystem. If the criterion of beginning the target range track “2/m,” the criterion of confirmation of the target range track “l/n,” and the criterion of cancellation of target range tracking, for example, the criterion using kth misses one after another are given, then the united criterion of detection and tracking of the target range track can be written in the following symbolic notation “2/m + l/n − kth.” The graph of the united algorithm under detection and tracking of target range track by criterion “2/m + l/n − kth” is shown in Figure 4.14. It allows us to analyze the processes of detection and tracking of target range tracks as a whole instead of analysis by parts, as discussed previously. On the basis of this graph, we can present an accurate formula that defines the initial points of false target range tracks forming in the course of the steady working state (see (4.48)).

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stronger point-source signal. Because most targets, such as aircraft, are complex in shape, the total echo signal is composed of the vector sum of a group of superimposed echo signals from the individual parts of the target, such as the engines, propellers, fuselage, and wing edges. The motions of a target with respect to the CRS cause the total echo signal to change with time, resulting in random fluctuations in the radar measurements of the parameters of the target. These fluctuations caused by the target only, excluding atmospheric effects and radar receiver noise contributions, are called the target noise. This discussion of target noise is based largely on aircraft, but it is generally applicable to any target, including land targets of complex shape that are large with respect to a wavelength. The major difference is in the target motion, but the discussions are sufficiently general to apply to any target situation.

The echo return from a complex target differs from that of a point source by the modulations that are produced by the change in amplitude and relative phase of the returns from the individual elements. The word modulations is used in plural form because five types of modulation of the echo signal that are caused by a complex target affect the CRS. These are amplitude modulation, phase front modulation (glint), polarization modulation, Doppler modulation, and pulse time modulation (range glint). The basic mechanism by which the modulations are produced is the motion of the target, including yaw, pitch, and roll, which causes the change in relative range of the various individual elements with respect to the radar. Although the target motions may appear small, a change in relative range of the parts of a target of only one-half wavelength (because of the two-way radar signal path) causes a full 360° change in relative phase.

5.1  INITIAL CONDITIONS

The problem to define the estimation θˆ (ti) of target track parameters θ(t) by the observed coordinate sample Y(ti), i = 0, 1, …, n, … is solved in all stages of signal reprocessing in a CRS. In the course of signal reprocessing, we can distinguish the following kinds of target track parameter estimations [5,6]:

•Estimation obtained at the instant tj = tn: the last measurement, that is, target track parameter filtering

•Estimation obtained at predicted coordinates at the point tj > tn—the extrapolation of target track parameters

•Estimation obtained at the points lying within the limits of the interval of observation 0 ≤ tj < tn—the target track parameter smoothing

In this chapter, we discuss only the first and second problems of target track parameter estimation definition. The class of considered objects is mainly limited by air targets. There is a need to take into consideration that in this chapter we discuss only the main results of filtering theory and extrapolation methods with reference to signal reprocessing problems in CRSs. We use the representation form that is understandable to designers and engineers of signal processing algorithms, and computer architecture systems and other experts exploiting CRSs for various areas of application.

5.2  PROCESS REPRESENTATION IN FILTERING SUBSYSTEMS

5.2.1  Target Track Model

In solving the filtering problems, a presentation procedure of variation of the filtered target parameters as a function of time takes a principal meaning. In our case, this corresponds to the choice of target track model. In the case of signal reprocessing problems in CRSs, taking into consideration sampling and discretization of target coordinate measuring and radio frequency interference, the

target track model can be described by a system of linear difference equations that can be presented in the following vector form:

where

θn is the s-dimensional target track parameter vector at the nth step Φn is the known s × s transfer matrix

ηn is the h-dimensional vector of target track parameter disturbance Γn is the known s × h matrix

q′ + is the deterministic (undisturbed) component of the target track parameter vector at the

n 1

(n + 1)th step

Under polynomial representation of independent target coordinates, a prediction of parameters of an undisturbed target track, for example, by target range coordinate r(t), is carried out in the following manner:

............................

where

τn = tn+1 − tn; tn and tn+1 are the readings of the function r(t)

rn ,rn ,rn ,… are the target track parameters representing the target track coordinate, velocity of

Formulas for undisturbed target track parameters by other independent coordinates take an analogous form.

The second term in the target track model given by (5.1) must take into consideration, first of all, the target track parameter disturbance caused by the nonuniform environment in which the target moves, atmospheric conditions, and, also, inaccuracy and inertness of control and target parameter