Diss / 10

Optimization of the target pip indication process by deviation from the gate center is carried out using a criterion of maximal likelihood, on the basis of which we believe that the likelihood function is maximal only for the true target pip. Under target pip indication within 3D gate with facets that are parallel to the main axles of the total error ellipsoid (see Figure 4.3), a condition of maximal likelihood takes the following form:

where itrue is the number of target pip considered as true, i = 1, 2,…, m, m being the number of target pips within a gate. Condition given by (4.15) is equivalent to the condition

Consequently, we should take into consideration only that target pip to prolong the target range track, the elliptical deviation of which from the gate center is minimal. A natural simplification of optimal operation discussed is indicated by the minimum of squared sums of linear deviations of the target pip from the gate center that corresponds to an assumption about the equality of the variances σ2ηi, σ2ξi, and σζ2i. If we assume that the target pip indication is carried out in a spherical coordinate system within the gate represented in Figure 4.4, then a criterion of target pip indication takes the following form:

A quality of the 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 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 [9,10]

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 (see Figure 4.3), the probability of correct indication is determined by

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

FIGURE 4.5  Algorithm of target pip indication within 2D gate.

The algorithm of target pip indication within 2D gate using a minimal linear deviation from the gate center and the polar coordinate system is represented in Figure 4.5. Blocks that are not related to the algorithm are shown by the dashed line. Sequence of operations in the considered algorithm is the following.

Step 1: Using signal processing results of the previous (n − 1)th radar antenna scanning, we are able to define and compute the gate dimensions for the next nth radar antenna scanning. For choosing the gate dimensions, we should take into consideration the information about target maneuver and target pip misses.

Step 2: We select the target pips within the gate (block 2, Figure 4.5) using the following formulae:

and determine their number. If there is a single target pip within the gate (m = 1), then this target pip is considered as true and comes in both at the filter input and at the input of the block of target range track parameters extrapolation (the block 5). If there are several target pips within the gate, then all of them come in at the input of computer block 3 that carries out computations of squared distances between the each target pip and the gate center by the following formula:

where i = 1, 2,…, m, m being the number of indicated target pips within the gate.

Step 3: We compare the squared distances (the block 4) and choose the only target pip, for which

Step 4: If there are no target pips within the gate, we check the criterion of cancellation of target range tracking (the block 6). If the criterion is satisfied, then the target range tracking is canceled. If the criterion of cancellation is not satisfied, then a command to continue to expand the target range tracking by extrapolation of target range track coordinates and target range track parameters is generated.

In conclusion, 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 a 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 the following. All target pips within the gate are divided on the target pips with the weight ν1 and target pips with the weight ν0 depending on either the target return pulse train bandwidth exceeds the earlier-given threshold value or not, which depends on target range magnitude. If there are target pips with weight ν1, we indicate the target pip nearest to the gate center as the true target pip from observed target pip group. If there are no target pips with weight ν1, we indicate the target pip with weight ν0 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 this case, features of target pip deviations from the gate center are only used under the equality of weights of several target pips.

4.1.3  Target Pip Distribution and Binding within Overlapping Gates

In the hard noise environment, several gates formed around extrapolated points of prolonged target range tracks will overlap. Moreover, false and newly obtained target pips will appear within 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 a beginning of 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. Henceforth, there is a need to generate all possible ways or hypotheses for target pip binding and choose the most plausible version.

Consider one approach to solve the problem of target pip distribution and binding in overlapping gates under the following initial premises:

•The gates formed by each target range track have such dimensions that the probability of hitting of true target pips belonging to the given target range track is very close to unit.

•Groups of overlapping gates are selected in such way that each group can be processed individually.

•The problem is solved within 2D gates.

2.The probability of event that a newly obtained target pip is false can be determined in the following form:

where Lgate is the number of resolved elements within the gate.

3. The probability of event that the target pip belongs to a newly detected target is given by

where

ρscan is the density of newly obtained target pips per unit area of scanning Sgate is the gate area

After 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 choose the version with the highest probability as a true version. Solution of this problem is very tedious even for the considered simplest case. This problem can be simplified if we will follow the evident practical rules:

1.Each target range track must be attached by the target pip.

2.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 the target pip bindings for each target range track. In the considered case, there are two such versions.

a.The first target pip corresponds to the first target range track, and the second target pip corresponds to the second target range track.

b.The second target pip corresponds to the first target range track, and the third target pip corresponds to the second target range track.

For this case, the third target pip at the first version and the first target pip at the second version are considered as newly obtained target pips and allow us to start a new target range track.

4.2  TARGET RANGE TRACK DETECTION USING SURVEILLANCE RADAR DATA

4.2.1  Main Operations under Target Range Track Detection

In accordance with general principles discussed in Section 4.1, the detection process of new target range tracking is started with a formation of an initial gate with primary lock-in around a single target pip. Dimensions of initial gate are selected based on the possible target moving over the radar antenna scanning period. If in the course of next radar antenna scanning period one or several target pips appear within the gate with primary lock-in, the new target range track is started using each target pip. If there are no target pips within the gate with primary lock-in, the initial target pip is canceled as false (the criterion of beginning a new target range track is “2/m”) or is reserved for confirmation during the next radar antenna scanning. In doing so, dimensions of the gate with primary lock-in will be increased (fractional criteria “2/m,” where m > 2). After target range track is begun, the target moving direction and velocity is defined. This allows us to extrapolate and gate the target position for the next radar antenna scanning. If the newly obtained target pips appear within these gates with primary lock-in, we can make a final decision about target range track detection.

Thus, the process of target range track detection is divided into two stages:

•The first stage: the target range track detection by criterion “2/m.”

•The second stage: 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:

•Evaluation of the target velocity in the course of moving

•Extrapolation of coordinates

•Gating the target pips

With respect to these operations, in future we assume the following prerequisites:

•Coordinate extrapolation is carried out in accordance with a hypothesis of uniform and straight-line target moving.

•Gates at all stages of target range track detection have a shape of spherical element (see

Figure 4.4). Gate dimensions rgate, Δβgate, Δεgate are defined based on total errors of measurement and coordinate extrapolation at the corresponding target range track detection stage.

•Radar system range resolution by corresponding coordinates is considered as the unit volume element of gate. In this case, the gate dimensions do not depend on target range, and moreover, a distribution of false target pips within radar antenna scanning area can be considered as uniform because the probability of appearance of false target pip within each elementary gate volume is the same.

Target range track detection quality can be evaluated by the following characteristics:

•Probability of detection of true target range track

•Average number of false target range tracking per unit time

•Speed of computer subsystems used by CRSs to realize the target range track detection algorithm

Henceforth, we discuss the main results of analysis of target range track detection algorithm applying to surveillance radar systems with omnidirectional or sector radar antenna scanning.

4.2.2  Statistical Analysis of “2/m + 1/n” Algorithms under False Target Range Track Detection

First, we consider the simplest algorithm realizing the criterion “2/m + 1/n.” Sequence of operations employing this algorithm under the false target range track detection can be presented using a graph with random transitions (see Figure 4.8). Beginning of algorithm functioning coincides with transition from state a0 to state a1 when a single target pip considered as the starting point of detected target range track appears. Henceforth, we check the appearance of newly obtained target pips within the gate Vi with primary lock-in, where i = 1, 2,…, m − 1. If even a single target pip appears within one from (m − 1) gates with primary lock-in, the graph passes into the state am corresponding to making decision about the detection of target range track beginning. Otherwise, the graph passes from the state am−1 to the state a0 (the initial point of assumed target range track is canceled as false point). After getting the second target pip by criterion “2/m,” the second stage of target range track detection is started. At this stage, we check the presence of newly obtained target pips within gates Vm+j, where

FIGURE 4.8  False target range track detection algorithm by criterion “2/m + 1/n.”

j = 1, 2,…, n − 1. Gate center position for the next radar antenna scanning is defined by coordinate extrapolation using two target pips. The dimension of gates is defined based on the measurement and coordinate extrapolation errors. If a newly obtained target pip appears within one of the n gates of the second stage, the graph passes into the state am+n that corresponds to making a final decision about the target range track detection. Otherwise, the graph passes into the state a0 from the state am+n−1 and a beginning of new target range track is canceled as false one. Volume of gates with primary lock-in defined by the number of elementary or resolved volumes is determined in the following form [11–13]:

ViI = V1i3,

where

The volume of gates at the second stage of target range track detection is determined by

where k ≈ 3 is the coefficient of gate dimension increasing in comparison with magnitudes of the total mean square deviations of target pips from the gate center. There is a need to emphasize that the gate with definite dimensions corresponds to each graph state from 1 to (m + n − 1).

To define the probability of false target range track detection, first of all, there is a need to define the conditional probability p1 of event that a random point starting its movement from state a1 can reach anywhere in state am+n and will stay at this state. This probability can be defined as solving a system of linear equations that can be found based on the graph presented in Figure 4.8:

where gfi , i = 1, 2,…, m − 1, is the probability of appearance of the false target pips within the gates with primary lock-in:

gf j , j = m, m + 1,…, m + n − 1, is the probability of appearance of the false target pips within the confirmation gates:

Equation 4.33 allows us to define the conditional probability of detection of the false target range track. This probability can be presented in the following form:

where

pbeg is the conditional probability of new target range track beginning

pconf is the conditional probability of confirmation of new target range track beginning

The unconditional probability of detection of the false target range track is given by

where P1 is the probability of a single false target pip appearance that can be considered as the new target range track beginning. If we consider a general case for the confirmation criterion “l/m,” it is impossible to obtain the probability pconf in a general form. For this reason, these criteria are analyzed individually.

Filtering ability of the target range track detection algorithm can be characterized by average number Nftr of false target range tracks that will be analyzed by radar target trackers within the radar antenna scanning period. The number Nftr is related to the probability p1, which can be presented by the following formula:

are considered as initial points of false target range tracks in steady working state of the CRS.

ber N1. This formula can be obtained in the case of the simplest algorithm realizing the criterion “2/m + 1/n.”

The number of individual target pips becoming initial points of newly obtained false target range tracks after the (r + 1)th radar antenna scanning can be determined by the following formula:

j=1

where

j = 1, 2,…, m + n − 1

Nf(r + 1) is the number of false target pips that is reprocessed by computer subsystem within the (r + 1)th radar antenna scanning

gfj is the probability of appearance of false target pips within the gate of volume Vj; the number of such gates is equal to m + n − 1

Nj(r) is the number of gates with the volume Vj formed within the rth radar antenna scanning ∑mj=+1n−1 gf j N j (r) is the number of false target pips of the current radar antenna scanning appearing

within the gates of all false target range tracks that are under detection process, but a condition of no overlapping gates must be satisfied

In turn,

where

N1(i) is the number of initial points of the false target range tracks formed within the ith radar antenna scanning

is the probability of system transition (see Figure 4.8) from the initial state a1 to the state aj for (r − j) steps

Using (4.37) we should take into consideration that

Introduce a new variable s = r − i in (4.39) that corresponds to the relocation of the origin at the instant of finishing the ith radar antenna scanning. The variable s represents the number of steps (the radar antenna scanning) necessary for the transition of target range track detection algorithm graph from the initial state a1 to the state aj. In the case of target range track detection algorithm with criterion “2/m + 1/n,” the maximum value of s corresponds to the maximum number of steps needed for transition from the initial state a1 to the state am+n−1 and is equal to

It is easy to verify this condition if we can define the longest branch in the algorithm graph presented in Figure 4.8. Taking into consideration all transformations discussed earlier, we obtain

s= 0