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Национальный исследовательский университет «МИЭТ»

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Algorithms of Target Range Track Detection and Tracking

133

Analyzing (4.41), we see that the terms for which the following condition P1(js) ≠ 0 is satisfied are not equal to zero. Furthermore, we are going to determine the probabilities of transitions P1(js). System of recurrent equations to determine the probabilities P1(js) takes the following form:

			1		at		s = 0,
P(s)		=
	11				at		s > 0,
			0		at		s > 0,

P(s)		=	P(s−1)q			,
	12		11		f1
		= P12(s−1)qf2 ,
P13(s)		= P12(s−1)qf2 ,
………………,
									(4.42)
									(4.42)
									m−1
P1(ms)		= P11(s−1)gf1 + P12(s−1)gf2 + + P1,(ms−−1)1gfm−1 = ∑P1(is−1)qfi ,
									i=1
									i=1

P(s)		−1	= P(s−1)q					,
	1,m	−1		1,m		fm
………………,

P(s)				= P(s−1)				q	.
	1,m+ n−1				1,m+ n− 2 fm+n−2

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

m+ n−1

m+ n− 2

N1(r + 1) = Nf (r + 1) − ∑ gf j

∑ N1(r − s)P1(js).

(4.43)

j=1

s= 0

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

N1(r + 1) = N1(r) = N1(r − 1) = = N1[r − (m + n − 2)] = N

(4.44)

Then we obtain

Nf (r + 1) = Nf (r) = = Nf [r − (m + n − 2)] = N

f ;

(4.45)

m+ n−1

m+ n− 2

1 ∑ gf j

∑ P1(js)

(4.46)

= N

f − N

j=1

s= 0

and, finally we obtain

N1 =

(4.47)

1 + ∑m+ n−1 gf j ∑m+ n− 2 P1(js)

j=1

s= 0

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 N1 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 Nftr and the average number of false target pips Nf. Equation 4.47 is satisfied for target range track detection

134	Signal Processing in Radar Systems

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:


		N	f
N1 =		N	f	,	(4.48)
N1 =	1 + ∑m+ ν−1 gf j ∑m+ n− 2 P1(js)			,	(4.48)
	j=1		s= 0

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

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, 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 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 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

			2		+		1
					+
		4				4
Nftr		4				4
Nftr
50
40
				2		+			1
30				4		+			3
30				4					3
20
				2		+			1
10				3		+			3
				3					3
					2	+				2
				4		+			4
				4					4

100	300	500
100	300	500						Nf

FIGURE 4.9 Comparison of filtering ability.

Algorithms of Target Range Track Detection and Tracking

135

working state of a CRS. Denote this number by Nfdtr. 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

			m+ ν−1
	Nfdtr =			∑ N	j ,	(4.49)
				j=1
where in accordance with (4.41) in the steady working state of the CRS we have
				m+ n− 2
				∑ P1(js).		(4.50)
N		j = N1		∑ P1(js).		(4.50)

s= 0

Substituting (4.49) in (4.50), we obtain

m+ ν−1 m+ n− 2

Nfdtr = N1 ∑ ∑ P1(js), j=1 s= 0

or taking into consideration (4.48), we obtain finally

		f ∑m+ ν−1	∑m+ n− 2 P1(js)
	N	f ∑m+ ν−1	∑m+ n− 2 P1(js)
Nfdtr =		j=1	s= 0		.
Nfdtr =	1 + ∑m+ ν−1 gf j ∑m+ n− 2				.
	1 + ∑m+ ν−1 gf j ∑m+ n− 2			P1(js)
		j=1	s= 0

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

(4.51)

(4.52)

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 aj, where j = 1, 2,…, m + n − 1. The probability of replacement from the intermediate state ai, 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,

pi0 (r) = [1 − ptr (r)]gfi

(4.53)

p1, m

p2, m

pm+1, m+n

m + n

0 p01 1 p12

m – 1

m + 1

m + n – 1

. . .

p10 p20

pm–1,0

pm–1,m pm, m+1

pm,0

pm+1,0

p00

FIGURE 4.10 True target track detection algorithm graph by criterion “2/m + 1/n.”

136	Signal Processing in Radar Systems

where

ptr(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 Vi

gfi is the probability of detection at least one false target pip within the gate volume Vi

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

pi0 (r) = Pj ( Ar ) + Pj (Br ),							(4.54)
where j = m, m + 1,…, m + n − 1;
Pj ( Ar ) = [1 − ptr (r)]gfi ;							(4.55)
P (B ) = p (r)g		1 − P		(V	)	;	(4.56)
j r	tr	fi	ind	j

Pind(Vj) is the probability of indication of true target pip among false target pips within the gate

volume Vi.

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 pij(r) for the algorithm graph shown in Figure 4.10 or elements of corresponding transient probability matrix. These probabilities are determined as follows:

p00 (r) = 1 − ptr (r);
p01(r) = ptr (r);
p10 (r) = gfi [1 − ptr (r)];

	= (1 − gfi )[1 − ptr (r)];
p12 (r)	= (1 − gfi )[1 − ptr (r)];
	= ptr (r);
p1m (r)	= ptr (r);
…………………………


pm−1,0 (r) = 1 − ptr (r);
		= ptr (r);
pm−1,m (r)		= ptr (r);			(4.57)
					(4.57)
…………………………


pm0 (r) = [1 − ptr (r)]gfm + ptr (r)gfm [1 − Pind (Vm )];
pm,m+1(r)		= [1 − ptr (r)](1 − gfi );

pm,m+ n (r)		= ptr (r)(1 − gfi ) + ptr (r)gfm Pind (Vm );


…………………………
pm+ n−1,0 (r) = [1 − ptr (r)]gf			+ ptr (r)gf	m+n−1	[1 − Pind (Vm+ n−1 )];
		m+n−1		m+n−1

pm+ n−1,m+ n(r) = ptr (r)(1 − gfm+n−1 ) + ptr (r)gfm+n−1 Pind (Vm+ n−1 );

p (r) = 1.

m+ n,m+ n

Algorithms of Target Range Track Detection and Tracking

137

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]:

P(r) =				P0(r)P1(r)…Pm+n (r)				= P(r − 1)P(r),	(4.58)

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:

	m+ n−1
P0 (r) = P0 (r − 1) p00 (r) + P1(r − 1) p10 (r) + + Pm+ n−1(r) pm+ n−1,0 (r) =		∑ Pj (r − 1) pj0 (r);
		j=1
		j=1

P1(r) = P0 (r − 1) p01(r);

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

		m+ n−1
	(r − 1) p1,m (r) + P2 (r − 1) p2,m (r) + + Pm−1(r − 1) pm−1,m (r) = ∑ Pj (r − 1) pj,m (r);
Pm (r) = P1
		j=1

Pm+1(r) = Pm (r − 1) pm,m+1(r);

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

	m+ n−1
	∑ Pj (r − 1) pj,m− n (r) + Pm+ n (r − 1) = Ptrue (r).	(4.59)
Pm+ n (r) =	∑ Pj (r − 1) pj,m− n (r) + Pm+ n (r − 1) = Ptrue (r).	(4.59)
	j= m

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 dr /dmax determined based on discussed procedure is presented in Figure 4.11. In accordance with adopted notations, dmax is the maximal horizontal radar range; dr is the current horizontal radar range:

dr = dmax − r d(Tsc )

(4.60)

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

	−	0.68dr4		(4.61)
Ptg (r) = exp	−	4	.	(4.61)
		dmax

138												Signal Processing in Radar Systems
PD
1.0


0.9
0.9
0.8
0.8
0.7					2			+	2			2		1
0.7
0.6													+
						3				4
						3				4	4
0.5											4		4
0.5
0.4
0.4
0.3							2		+		2
0.3
0.2
			2	+	3		3				2
			2		3		3				2
0.1			3		3
0.1			3		3
0.0																dr
0.0																dr
0.4	0.5	0.6	0.7		0.8						0.9		1.0
0.4	0.5	0.6	0.7		0.8						0.9		1.0		dmax
															dmax

FIGURE 4.11 Probabilities of true target range track detection for various criteria “2/m + l/n.”

The probability of true indication of target pips within confirmation gates is considered constant and equal to Pind = 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 TARGET RANGE TRACKING USING SURVEILLANCE RADAR DATA

If the earlier-given criterion of target detection is satisfied, the signal processing of the target return signal is finished at the stage of target range track detection. After that initial parameters of detected target range track are determined, and this target range track will be under autotracking. Henceforth, the target autotracking is considered as automatic target range track prolongation of target moving and accurate definition of prolongated target range track parameters. Thus, the terms “target autotracking” and “autotracking” (or “tracking”) must be understood in the same sense. Furthermore, we prefer to use the term “target range autotracking” or “target autotracking.”

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

Algorithms of Target Range Track Detection and Tracking

139

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

		6
			Target track	To user
			computation
Target	1	2			3
pips		Target pip	Target track		Coordinate
			Target track
			parameter
		selection	parameter		extrapolation
		selection	estimation		extrapolation
			estimation
		m = 0
	5				4
		Cancellation		Features	Gate
		criterion			dimensions
		verification			computation
		Target tracking
		cancellation

FIGURE 4.12 Block diagram of the target range tracking algorithm.

140	Signal Processing in Radar Systems

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 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 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 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.

While choosing kth there is a need to proceed from the following assumptions. The greater is the value of kth, 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 kth, the number of false target range tracking and its average length is increased. Because of this, while choosing kth 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 kth is usually carried out in the course of testing the signal processing subsystem.

Taking into consideration the criterion of target range track cancellation by kth 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

am+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, aj, where j = m + n − 1,…, m + n + kth − 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 = kth and the graph has been passed into the state am+n+kth .

Algorithms of Target Range Track Detection and Tracking							141
m + n	m + n + 1 qm+n+1	m + n + 2			m + n + kth – 1		m + n + kth
qm+n	m + n + 1 qm+n+1	m + n + 2		. . .	m + n + kth – 1
				. . .		qm+n+kth–1		1
gm+n+1	gm+n+2		gm+n+kth–1			qm+n+kth–1		1
gm+n

FIGURE 4.13 Target range tracking graph with random transitions.

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 kth 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 am+n to the state am+n+kth in the course of μ steps (see Figure 4.13):

		Pcan ( ) = Pm+n+kth ( ).					(4.62)
To determine the probability Pm+n+kth ( ), we can use the following recurrent formulas:
Pm+ n (0) = 1,

			m+ n+ ktr −1
Pm+ n ( ) =			∑	Pj ( −1)gfi ,
			j= m+ n
			j= m+ n
	( )		= Pm+ n ( −1)gfm+n ,				(4.63)
Pm+ n+1	( )		= Pm+ n ( −1)gfm+n ,

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

P		( ) = P		+ n+ kth −1	( −1)g	fm+n+kth −1	.
m+ n+ kth			m	+ n+ kth −1		fm+n+kth −1

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

∞
= ∑ Pm+n+kth ( ).	(4.64)

=kth

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

			.	(4.65)
N	tracking = Nf	track
	ftrack

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.

142	Signal Processing in Radar Systems

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 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 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. 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 − kth,” the number of gates is equal to

m + ν + kth − 1. Because of this, the upper limit of summation by j in (4.48) will be jmax = m + ν + kth − 1. The upper limit of summation by s will be defined as smax → ∞ since the number of steps

under transition from the state a1 to the state am+ 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:


		=		N	f	.	(4.66)
N	1			N	f
N	1
	′
		1 + ∑m+ ν+ kth −1 gf j ∑∞ P1(js)
			j=1		s= 0

If the criterion of confirmation takes the form “1/n,” we obtain target range tracks for tracking is determined as

′tr			′
Nf	= N		1 p1,

ν = n. The average number of false

(4.67)

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.

gm+n+kth–1

gm+1

0 g1

m – 1

gm+n+1

m + n – 1

. . .

+ 1 . . .

gm–1

m m + 1

m + n

m + n + kth – 1

qm–1	gm+n
	qm+n–1
	qm+n+kth–1

FIGURE 4.14 Graph of the united algorithm under detection and tracking of target range track by criterion “2/m + l/n − kth.”

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