4.1.1 TARGET PIP GATING; SHAPE SELECTION AND DIMENSIONS OF GATES

In accordance with the main principles of target range autotracking, a newly obtained target pip can be used to suggest a target range tracking if a target pip deviation from the gate center does not exceed a given before fixed value defined by gate dimensions or, in other words, if the following condition is satisfied:

where

is a set of coordinates of the ith newly obtained target pip;

is a set of the gate center coordinates for the ith target range track;

are the gate dimensions by coordinates r, β, and ε for the ith target range track. One of the problems arising during the prolongation of the target range tracks by gating is a problem of selection of gate shape and dimensions based on known statistical characteristics of deviations of true (belonging to prolongated target range tracks) target pips from their corresponding extrapolated points. Deviation of the true target pip from the gate center is defined by total (random plus dynamic) errors of target coordinate extrapolation by previous smoothed parameters of target range track and coordinate measurement errors of newly obtained target pips. These errors are independent and identically distributed, subjected to the normal Gaussian pdf.

Let the target range track coordinate extrapolation for the next nth radar antenna scanning be carried out using the data of previous (n- 1)th radar antennas scanning. Position of the extrapolated point is denoted by O (see Figure 4.3). We think that the origin of Cartesian coordinate system is located at the point O. The axis Y is matched with “radar system-target” direction, the axis X is a perpendicular to the “radar system-target” direction away from radar antenna rotation, and the axis Z is directed in such way that we have a right-handed coordinate system. Then the random deviations Δx, Δy, Δz of the newly obtained under nth radar antenna scanning target pip from the gate center are defined in the following form:

where

• Δrnextr,Δβnextr, and Δεnextr are the random errors of coordinate extrapolation in the course of the nth radar antenna scanning

• Δr_n, Δβ_n, and Δε_n are the random errors of coordinate measuring in the course of the nth radar antenna scanning

Under the definition of gate dimensions we can believe that the components Δr_n, Δβ_n, and Δε_n are statistically independent, do not depend on the scanning number step n, and are subjected to the normal Gaussian pdf with zero mean and variances σx2,σy2, and σz2, respectively. Consequently, the joint pdf can be defined in the following form:

and the surface corresponding to equal pdf is defined by the following equation:

where λ is the arbitrary constant. Dividing the left- and right-hand side of (4.7) by λ², we obtain

Equation 4.8 is the ellipsoid equation with conjugate axles λσ_x, λσ_y, and λσ_z. At λ = 1, we obtain the unit ellipsoid—see ellipsoid 1 in Figure 4.3.

Henceforth, we believe that dynamical errors of extrapolation caused by contingency target maneuver are also normally distributed and have independent components on the F, G, and H axes. The F axis is matched with the target velocity vector; the G axis is directed in opposition to a tangential acceleration; and the H axis supplements the system to the right-handed coordinate one. Origin of the obtained coordinate system, as well as in the case of the previous coordinate system, coincides with the extrapolated point O. For better visualization the origin is replaced at the point O′ in Figure 4.3.

In 3D space the dynamic errors form an ellipsoid of equal probabilities, an equation of which takes the following form:

the ellipsoid 2 in Figure 4.3 at λ = 1. If we add ellipsoids 1 and 2, then ellipsoid 3 will be formed, and directions of conjugate axles (the directions of axes of the Cartesian coordinate system Oηξζ with respect to the axes of the Cartesian coordinate system OXYZ and variances ση2,σξ2, and σζ2 by these axles) are defined by summing rules in space of independent vectorial deviations, caused by random and dynamic errors. For better visualization, the origin of the coordinate system Oηξζ is replaced at the point Oηξζ in Figure 4.3.

The joint pdf of random components Δη, Δξ, and Δζ takes the following form:

where

Thus, a surface of equally probable deviation of true target pips from the gate center represents an ellipsoid, value and direction of conjugate axles of which relative to the direction “radar system-target” depend on coordinate measurement errors, target maneuver intensity, and vectorial direction of target moving. Under ellipsoid distribution of true target pip deviation from the gate center, it is evident that the gate must have an ellipsoid shape with the conjugate axles λσ_η, λσ_ξ, and λσ_ζ, where λ is the enlargement factor of the gate dimensions in comparison with the dimensions of unit ellipsoid to ensure the given before expectancy of hitting of true target pip within the gate.

The expectancy of random point hitting into the ellipsoid that is similar and has a similar disposition as ellipsoids of equal probabilities is defined in the following form:

where Φ₀(λ) is the standard normal Gaussian pdf given by (3.110). At λ ≥ 3, the probability P(λ) is very close to unit. There is a need to choose just the values of λ, forming an ellipsoid gate.

In practice, forming the ellipsoidal gates is impossible both under physical and mathematical gating. By this reason, the best thing that can be done is to form a gate in the shape of parallelepiped defined around the total error ellipsoid, as shown in Figure 4.3 (3). Parallelepiped side dimensions are equal to 2λσ_η, 2λσ_ξ, and 2λσ_ζ and its volume is defined as

Taking into consideration the fact that the volume of total error ellipsoid is defined as

the gate volume is increased twice in comparison with the optimal volume. This phenomenon leads to an increase in the expectancy of hitting of false target pips within the gate or an increase in the number of target pips belonging to another target range tracks and, consequently, to deterioration of selection and resolution during gating operation.

During the processing of high number of targets in real time by the computer subsystem of a complex radar system (CRS), 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, a sense of which is the following. The simplest gate shape is chosen for definition in that coordinate system, in which radar information is processed. In the case of spherical coordinate system, the simplest gate is given by linear dimension in target range Δr_gate and by two angular dimensions, namely, by the azimuth Δβ_gate and elevation Δε_gate (see Figure 4.4).

These dimensions can be preset based on 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 fight) 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. In conclusion, we note that all procedures, which are discussed in this subsection, to choose dimensions of 3D gate can be used in full during gating in plane for binding newly obtained target pips in 2D radar systems.

4.1.2 ALGORITHM OF TARGET PIP INDICATION BY MINIMAL DEVIATION FROM GATE CENTER

We consider a case of target pip indication under definition of target range track for a single target. At the same time, we assume that in addition to true target pips, the false target pips caused by noise and interferences passed over preprocessing filters can be present within the gate. Given the earlier-provided situation, the following varied decisions could be made:

• When there are several target pips, there is a need to continue to define the target range tracks using each target pip; in other words, we can suppose multiple target range tracks. Prolongation of target range tracks using false target pips will be canceled over several radar antennas scanning owing to absence of confirmations, but prolongation of the target range tracks using the true target pips will be continued. Such way of newly obtained target pip binding is very tedious. Moreover, when the density of false target pips is high, an avalanche-like increase in false target tracks leading to overload of memory devices in computer subsystems is possible.

• We must choose a single target pip within the gate, and we should be sure that the probability of the event that this target pip belongs to the target range tracking is the highest. Other target pips can be rejected as false. Such approach is appropriate with a view to decreasing the computation cost, but it requires a solution for the optimal target pip indication problem.

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 i^true 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 σηi2,σξi2 and σζi2. 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 λ_max ≥ 3, the probability of correct indication is determined by [9,10]

where

• ρ_S is the density of false target pips per unit square of gate

• σ_η and σ_ξ 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 ρ_v is the density of false target pips per unit volume of 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 v₁ and target pips with the weight v₀ 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 v₁, 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 v₁, we indicate the target pip with weight v₀ 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.

A logical flowchart of target pip distribution and binding algorithm within the isolated group of overlapping gates is presented in Figure 4.6. Block 1 selects the isolated groups with overlapping gates. Block 2 generates possible target pip binding versions taking into account information about the sources of newly obtained target pips forming a group. The following hypotheses are considered with respect to each target pip: The first hypothesis is that the target pip belongs to an earlier-detected target range track or to a new target range track, and the second hypothesis is that the target pip is false. Block 3 computes probabilities of all versions of the target pip binding. Block 4 defines an appropriate version of the target pip binding to begin a new target range track or to continue the earlier-obtained target range tracks.

Consider the simplest case of two overlapping gates shown in Figure 4.7a, as an example. We can see two extrapolated points of target range tracks 1 and 2 and three newly obtained target pips I, II, and III, one of which is within the overlapping gate region. Each newly obtained target pip is considered both as one belonging to the target range track 1 or 2 if this target pip is within the gate of corresponding target range track and as one belonging to a new target range track, namely, three for target pip I, four for target pip II, and five for target pip III. In the other case, the target pip is considered false (the target range track 0). Possible versions of target pip binding forms a tree of binding hypotheses presented in Figure 4.7b. Each branch of this tree is considered as one of possible versions of newly obtained target pip distribution. For example, the first (left) branch presents a version when all newly obtained target pips are false. The last (right) branch of this tree is considered as a version when all newly obtained target pips belong to detected target range tracks with numbers 3, 4, and 5. For the considered case, the total number of versions is equal to 30.

The probabilities of events that the target pips belong to the existing target range tracks, new target range tracks, and false target range tracks are defined as follows:

1. The probability of event that the ith newly obtained target pip belongs to the observed before jth target range track, if, of course, the target pip is within the gate for this target range track, is defined by the probability of detection Ptgpitrue of a newly obtained target pip and by a distance between this newly obtained target pip and the extrapolation point, in other words, between the newly obtained target pip and the gate center of the jth target range track. The probability of target pip detection depends on the target range, the effective reflective surface of the target, and power of radar system. In the case of target range tracking, all these characteristics are known very well. For this reason, a determination of the probability of detection Ptgpitrue of a newly obtained target pip does not pose any difficulties. As before, we assume that a distance between the target pip belonging to the given target range track and the gate center along each coordinate of the Cartesian coordinate system is subjected to the normal Gaussian pdf with zero mean and the variance that can be determined in the following form:

   σΣij2=σextrj2+σmeasi2,⁢(4.23)

   where

   • σextrj2 is the variance of errors under coordinate extrapolation of the jth target range track

   • σmeasi2 is the variance of errors of coordinate measurement of the ith target pip

   Hence, the probability of deviation of the ith target pip from the extrapolated point of the jth target range track, for example, by the coordinate x, is given by

   Pxij=12πσΣij2exp[−(xi−x^extri)22πσΣij2],⁢(4.24)

   where

   • x_i is the measured target pip coordinate

   • x^extri is the extrapolated point coordinate

   Thus, if the following condition

   σxi2=σyi2and(σxiextr)2=(σyiextr)2⁢(4.25)

   is satisfied, then the probability of event that the ith target pip belongs to the jth target range track is defined in the following form:

   Pijtrue=PtgpitruePxijPyij.⁢(4.26)

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

   Pifalse=PtgpifalseLgate,⁢(4.27)

   where L_gate 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

   Pnewi=PtgpitrueρscanSgate,⁢(4.28)

   where

   • ρ_scan is the density of newly obtained target pips per unit area of scanning

   • S_gate 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.

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

   2. 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.

4.2.1 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 Δr_te, Δβ_gate, Δε_gate 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 V_i 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 a_m corresponding to making decision about the detection of target range track beginning. Otherwise, the graph passes from the state a_m−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 V_m+j, where 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 a_m+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 a_m+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]:

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

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

Solution of the linear equation system for arbitrary m and n is given by

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

• p_beg is the conditional probability of new target range track beginning

• p_conf 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 p_conf 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 N¯ftr of false target range tracks that will be analyzed by radar target trackers within the radar antenna scanning period. The number N¯ftr is related to the probability p₁, which can be presented by the following formula:

where N¯1 is the average number of individual target pips within the radar antenna scanning that are considered as initial points of false target range tracks in steady working state of the CRS. Thus, to define N¯ftr, first of all, there is a need to obtain the formula to determine the average number N¯1. 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:

where

• j = 1,2,…, m + n − 1

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

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

• N_j(r) is the number of gates with the volume V_j formed within the rth radar antenna scanning ∑j=1m+n−1gfjNj(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

• P1j(r−i) is the probability of system transition (see Figure 4.8) from the initial state a₁ to the state a_j 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 a_j. 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 a_m+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

Analyzing (4.41), we see that the terms for which the following condition P1j(s)≠0 is satisfied are not equal to zero. Furthermore, we are going to determine the probabilities of transitions P1j(s). System of recurrent equations to determine the probabilities P1j(s) takes the following form:

Taking into consideration (4.41), we can present (4.37) in the following form:

In steady working state (r → ∞) we can assume

Then we obtain

and, finally we obtain

Formula (4.47) combined with linear equation system (4.42) allows us to determine the average number of starting points of false target range tracks per radar antenna scanning in the steady working state of radar system. Knowing N¯1 and the conditional probability of lock-in p₁ and using (4.36), we can define a function between the average number of false target range tracking N¯1 and the average number of false target pips N_f Equation 4.47 is satisfied for target range track detection algorithms when the confirmation criterion is differed from “1/n.” It is necessary only to make the information with respect to upper limits of summing by j and s more exact. The upper limit of summing by s is defined by the total number of gates formed in the course of realization of target range track detection algorithms. In the case of target range track detection algorithm with the confirmation criterion “l/n” (l > 1), this number is greater than m + n − 1. For this reason, in general the upper limit of summing by s will be defined by the maximum number of steps that can be made in the course of transition from state a₁ to the state that is previously relative to the accepted state. It is easy to show that this number does not depend on l if we use the criterion “l/n” and always equals m + n − 2. Thus, in a general form, the formula defining the number of individual target pips considered as the false target range tracks beginning under the use of criterion “2/m + l/n” is as follows:

where v is the number defined from target range track detection algorithm graph.

Results of determination of the average number N¯ftr 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, with the purpose of comparing the filtering ability. Analyzing results presented in Figure 4.9, we can conclude as follows:

• Filtering ability of target range track detection algorithms realizing the criterion “2/m + l/n” increases with a decrease in m and n and with an increase in l.

• Incrementing l on unit leads to a great increase in the filtering ability, in comparison with a corresponding decrease in m and n.

These features must be taken into consideration while choosing the target range track detection algorithm implemented in practice. If the computer subsystem of a CRS has limitations in the number of false target range tracks that are to be tracked, then joint choice of the average number N¯ftr 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 of the CRS computer subsystem, there is a need to have information about the average number of false target range tracks that are under detection in the steady working state of a CRS. Denote this number by Nftrd. It is evident that all false target range tracks, with respect to which the final decision about tracking or cancellation has been not made, are under the detection process. The number of such target range tracks in the steady working state of the CRS is determined by

where in accordance with (4.41) in the steady working state of the CRS we have

Substituting (4.49) in (4.50), we obtain

or taking into consideration (4.48), we obtain finally

4.2.3 STATISTICAL ANALYSIS OF “2/m + 1/n” ALGORITHMS UNDER TRUE TARGET RANGE TRACK DETECTION

First, we consider the simplest algorithm realizing the criterion “2/m + 1/n.” Functioning of the true target range track detection algorithm by the criterion “2/m + 1/n” is explained using Figure 4.10. A great feature of the true target range track detection algorithm graph in comparison with the false target range track detection algorithm graph shown in Figure 4.8 is the presence of nonzero probabilities of transitions to the initial state a₀ from any intermediate state a_j, where j = 1, 2,…, m + n − 1. The probability of replacement from the intermediate state a_i, where i = 1, 2,…, m − 1, to the initial state a₀ is equal to the probability of two simultaneous events. The first event is that the true target pip does not appear in the corresponding gate with primary lock-in; the second event is that at least one false target pip appears within the gate with primary lock-in, that is,

where

• p_tr(r) is the probability of true target pip detection in the course of the rth radar antenna scanning, which is independent of the gate volume V_i

• g_fi is the probability of detection at least one false target pip within the gate volume V_i

The probability of replacement to the initial state a₀ from any intermediate state a_j, where j = m, m + 1,…, m + n − 1, is equal to the sum of the probabilities of two inconsistent events A_r and B_r:

where j = m, m + 1,…, m + n − 1;

P_ind(V_j) is the probability of indication of true target pip among false target pips within the gate volume V_i.

Thus, based on the general principles of target range track detection algorithm functioning and taking into consideration (4.53) through (4.56), we can define the probabilities of transitions p_ij(r) for the algorithm graph shown in Figure 4.10 or elements of corresponding transient probability matrix. These probabilities are determined as follows:

We would like to stress again that the probability of the target return signal detection for the formulas of probabilities of transition given by (4.57) is determined by the following [14–17]:

where

• P(r − 1) is the row vector of probabilities of states at the previous (r − 1)th radar antenna scanning step

• π(r) is the matrix of transient probabilities at the rth radar antenna scanning step

In accordance with (4.58) and taking into consideration (4.57), these equations define the elements of the transient probability matrix π(r), and the system of recurrent equations to define components of the vector P(r) can be presented in the following form:

The last line in (4.59) defines the total probability of true target range track detection at the rth radar antenna scanning increasing from scanning to scanning.

If we consider a general case of the criterion “2/m+ 1/n,” it is impossible to obtain the formula for increasing the probability of true target range track detection at an arbitrary value of l if l > 1. There is a need to carry out analysis for each individual target range track detection algorithm [18-20]. To compare various criteria of the type “2/m + l/n” used under the true target range track detection, the probability of detection as a function of normalized target range d_r/d_max determined based on discussed procedure is presented in Figure 4.11. In accordance with adopted notations, d_max is the maximal horizontal radar range; d_r is the current horizontal radar range:

where d(T_sc) is the variation of radar range coordinate within the limits of the radar antenna scanning period T_sc. The probability of target detection at the rth radar antenna scanning is determined by the formula

The probability of true indication of target pips within confirmation gates is considered constant and equal to P_ind = 0.95. Density of false target pips per unit volume of radar antenna scanning area is equal to 10⁻⁴.

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 maximal radar system range. Implementation of confirmation criteria “l/n” (l > 1) leads to the prolongation of time to detect the true target range tracks.

4.3.1 TARGET RANGE AUTOTRACKING ALGORITHM

In the course of each target range tracking, two main problems are solved: gating and selection of newly obtained target pips for prolongation of target range track, estimation of target range track parameters, and definition of variation of these parameters as a function of time. In principle, solution of both problems can be realized employing the same detection algorithm. In this case, a required quality of the solved target range track estimation parameter problem must be in agreement with the customer. However, there is such a 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. Henceforth, we term this detection algorithm as target range track detection algorithm computations. Expediency to design the target range track detection algorithm computations arises from the following principles:

• Operations of estimation and extrapolation of target range track parameters must be carried out in the radar coordinate system to ensure the continuity of target range autotracking as soon as the input information is changed. There are no hard restrictions over the accuracy of these operations that allow us to use simple formulas for computations based on a hypothesis about straight-line target moving.

• Computations of target range track parameters must be carried out using very precise formulas, taking into consideration all available information about the target movement character (air or cosmic target, maneuvering or nonmaneuvering target, etc.) in the interests of customers. In doing so, the parameter output can be presented in other coordinate systems that are different from a radar coordinate system, for example, the Cartesian coordinate system with the center at the point gathering information. Moreover, to estimate the necessary target range track parameters, for example, a course and velocity vector modulus under aircraft autotracking, the parameters that are not associated with the target range autotracking can be selected based on the needs of the customer or with the purpose of matching them with other detection algorithms of a CRS.

• The customer is interested, first of all, in target information that is very important for a CRS, for example, the information about the type and number of aircraft following for landing stored into the automatic control system of airport. The exact target range track parameters must be determined using just these targets. Naturally, not all detected targets within the radar antenna scanning area are important and some of them are not interested in a CRS, for example, recessive targets, transiting targets, and so on. Consequently, an estimation of target range track parameters with high accuracy is necessary only for a very small part of autotracked targets. In the considered case, definition of individual target range autotracking algorithm allows us to reduce requirements to computer system speed.

Logical block diagram of the target range tracking algorithm is presented in Figure 4.12 based on the statements discussed earlier. Block 1 solves the problem of selection and indication of target pip to continue the target range tracking. The gating algorithm and indication of target pips within the gate is designed in accordance with theoretical assumptions discussed in Section 4.1. The indicated target pip is given the number of tracked target range trajectory and it is processed by the block of target range track computations (the block 6). Simultaneously, the newly obtained target pip is used to estimate the parameters of target range track and to extrapolate the target coordinate for the next radar antenna scanning process, that is, to prepare a new gating cycle and indication. For this purpose the following operations are carried out:

• Estimation of target range track parameters under simplest conditions of target moving and coordinate measurement errors (the block 2).

• Determination of the extrapolated coordinates for the next radar antenna scanning (the block 3). Extrapolation is carried out by linear law.

• Determination of gate dimensions (the block 4). At the same time, the accurate characteristics of measured and extrapolated coordinates and information about a target pip miss within the gate are used.

• When the newly obtained target pip is absent, we check the criterion of cancellation of the target range tracking with the purpose to prolong the target range track (the block 5). If the cancellation criterion is satisfied, the target range tracking is stopped and the previous information about this target range track is removed. When the cancellation criterion is not satisfied, the coordinates of extrapolated point are used as the coordinates of newly obtained target pip and the computation process is repeated.

In general, in addition to the presence of the target pips within the gate that prolongs 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 fight; 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 speed of a computer system. For this reason, the main criterion while making a decision about the target range tracking cancellation is the appearance of some threshold series k_th 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.

While choosing k_th there is a need to proceed from the following assumptions. The greater is the value of k_th, the smaller is the probability to make a false decision about cancellation of the true target range track. On the other hand, with an increase in the value of k_th, the number of false target range tracking and its average length is increased. Because of this, while choosing k_th there is a need to take into consideration the statistical characteristics of true target pip misses (no detections). The final choice of the value of k_th is usually carried out in the course of testing the signal processing subsystem.

Taking into consideration the criterion of target range track cancellation by k_th misses one after another, the target range tracking process is described by the graph with random transitions (see Figure 4.13). The character of states and transitions of this graph allows us to select the following modes of the target range tracking:

• The mode of stable target range tracking characterizing that the graph is at the initial state a_m+n. For the first time, this state is obtained when the criterion of target range track detection is satisfied.

• The mode of unstable target range tracking corresponding to one of the intermediate states of the graph, namely, a_j, where j = m + n − 1,…, m + n + k_th − 1.

• The mode of target range tracking cancellation indicating the fact that the number of target pip misses one after another could reach the threshold level k = k_th and the graph has been passed into the state a_m+n+kth.

In this case, the target range tracking algorithm graph is analogous to the graph of binary quantized target return pulse train latching algorithm [21–23]. For this reason, a procedure of analysis of these algorithms is the same.

Under statistical analysis of target range tracking algorithms, the main interest is the average time of false target range track observation and the average number of false target range tracking in the steady working state, which is associated with the average time of false target range track observation. Moreover, it is very interesting to define the probability of true target range tracking cancellation at the given probability of target pip detection within the gate. Let us define a function between the average number of false target range tracking for each radar antenna scanning period and the average number of false target range tracks in the steady working state. For this purpose, first of all, there is a need to define the probability to stop exactly the false target range tracking process at the μth step (the radar antenna scanning period) after starting the false target range tracking at the instant μ = 0. In the case of cancellation criterion by k_th misses one after another, the probability to stop exactly the false target range tracking process at the μth step (the radar antenna scanning period) is equal to the probability of graph transition from the state a_m+n to the state a_m+n+kth in the course of μ steps (see Figure 4.13):

To determine the probability P_m+n+kth (µ), we can use the following recurrent formulas:

The average length of the false target range tracks expressed as the number of radar antenna scanning periods is defined in the following form:

Furthermore, when we know the average number of the false target range tracks, the average number of the false target range tracks being under tracking is determined by

The average number of the false target range tracks being under tracking is taken into account under the definition of computer subsystem cost and speed.

4.3.2 UNITED ALGORITHM OF DETECTION AND TARGET RANGE TRACKING

Until now, we have assumed that the target range track detection and tracking algorithms are realized individually, that is, by the individual computer subsystems. In practice, for the majority of cases, it is very convenient to employ such structure of signal reprocessing subsystems where 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 individual computer subsystem. Henceforth, we consider this version of structure of signal reprocessing 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 k_th misses one after another, are given, then the united criterion of detection and tracking of the target range track can be written in the symbolic notation “2/m + l/n - k_th.” The graph of the united algorithm under detection and tracking of target range track by criterion “2/m + l/n - k_th” is shown in Figure 4.14. This graph allows us to analyze the processes of detection and tracking of target range tracks as a whole instead of analysis by parts discussed earlier. Based on this graph, we can present an accurate formula defining the initial points of false target range tracks forming in the course of the steady working state (see (4.48)).

In the united algorithm realizing the criterion “2/m + l/n − k_th,” the number of gates is equal to m + v + k_th − 1. Because of this, the upper limit of summation by j in (4.48) will be j_max = m + v + k_th − 1. The upper limit of summation by s will be defined as s_max → ∞ since the number of steps under transition from the state a₁ to the state a_m+n+kth − 1 will be arbitrary large.

Thus, in the united algorithm case, the number of false target pips obtained at the initial points of new target range tracks is determined by the following formula:

If the criterion of confirmation takes the form “1/n,” we obtain v = n. The average number of false target range tracks for tracking is determined as

which is less than that of individual realization since the number of initial points is decreased. The number of false target range tracks being under tracking will also decrease correspondingly.