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Filtering and Extrapolation of Target Track Parameters Based on Radar Measure

163

period Teq and measure errors are uncorrelated. In this case, the coordinate is varied according to the following law:

xi = x(ti ) = xN − (N − i) 1xN − (N − i)2 2 xN , i = 1, 2,…, N,

(5.52)

where

	(5.53)
1xn = Teq xN

is the first increment of the coordinate x;

2	(5.54)
2 xN = 0.5Teq xN

is the second increment of the coordinate x; x.N is the velocity of changes of the coordinate x; x¨N is the acceleration by the coordinate x.

An order of obtaining the corresponding formulas to estimate the parameters of a second-order polynomial target track is the same as in the case of a linear target track. Omitting intermediate mathematical transformations, we can write the final formulas in the following form:

αN ∑wi xi + γ N ∑wi (N − i)xi + δN ∑wi (N − i) xi ;

(5.55)

i=1

1xN =

γ N ∑wi xi − ξN ∑wi (N − i)xi + ηN ∑wi (N − i)

;

(5.56)

i=1

2 xN =

δ N ∑wi xi − ηN ∑wi (N − i)xi + µ N ∑wi (N − i)

;

(5.57)

i=1

where

α N = hN eN − dN2 ;	(5.58)
γ N = gN eN − hN dN ;	(5.59)
δ N = gN dN − hN2 ;	(5.60)
ξN = fN eN − hN2 ;	(5.61)
ηN = fN dN − gN hN ;	(5.62)
µ N = fN hN − gN2 ;	(5.63)

164	Signal Processing in Radar Systems
	N
	dN = ∑wi (N − i)3;	(5.64)
	i=1
	N
	eN = ∑wi (N − i)4;	(5.65)
	i=1
	IN = eN ( fN hN − gN2 )+ dN (gN hN − fN dN ) + hN (gN dN − hN2 ) .	(5.66)

The error correlation matrix of second-order polynomial target track parameter estimation takes the following form:

YN = B−N1 =	1	α N	− γ N	δ N

		−γ N	ξN	− ηN	.	(5.67)
	IN
		δ N	− ηN	−µ N

In the case of uniformly precise measurements, at wi = w from (5.58) through (5.65) it follows that

dN = wi ∑(N − i)3 = N2 (N − 1)2 w; 4

i=1

eN = w∑(N − i)4 = N(N − 1)(2N − 1)(3N2 − 3N − 1) w. 30

i=1

The required target track parameter estimations can be determined in the following form:

xˆN = ∑ηxˆ (i)xi;

i=1

1xˆN = ∑η 1xˆ (i)xi;

(5.68)

(5.69)

(5.70)

(5.71)

i=1
N
ˆ	(i)xi ,	(5.72)
2 xN = ∑η 2 xˆ	(i)xi ,	(5.72)

i=1

where ηxˆ (i), η 1xˆ (i), and η 2 xˆ (i) are the discrete weight coefficients of measure records under the determination of estimations of the coordinate, the first coordinate increment, and the second coordinate increment, respectively:

	3	(N + 1)(N + 2) − 2i(4N + 3) + 10i2
ηxˆ (i) =			;	(5.73)
ηxˆ (i) =		N(N + 1)(N + 2)	;	(5.73)
		N(N + 1)(N + 2)

Filtering and Extrapolation of Target Track Parameters Based on Radar Measure

165

(N + 1)(N + 2)(6N − 7) − 2i(16N 2 − 19)

+ 30i2 (N − 1)

xˆ (i) =

;

(5.74)

N(N 2 − 1)(N 2 − 4)

(N +

1)(N + 2) − 6i(N + 1)

+ 6i2

xˆ (i) =

(5.75)

N(N 2 − 1)(N 2 − 4)

Formulas (5.70) through (5.75) show us that in the case of equally discrete and uniformly precise measures, the optimal estimation of the second-order polynomial target track parameters is determined by the weighted summing of measured coordinate values. The weight coefficients are the functions of the sample size N and the sequence number of sample i in the processing series. When the sample size is minimal, that is, N = 3, the target track parameters are determined by the following formulas:

			ˆ	;	(5.76)
			x3 = x3
ˆ				− 2x2 + 1.5x1;	(5.77)
1x3		= x3Teq = 0.5x3
ˆ			2		(5.78)
2 x3	= 0.5x3Teq = 0.5(x3 − 2x2 + x1).

In the considered case, the error correlation matrix of target track parameter estimation is produced from the matrix given by (5.67) by substituting for IN, αN, γN, δN, ξN, ηN, and μN of the corresponding values fN, gN, hN, dN, eN given by (5.42), (5.68), and (5.69). As a result, the elements of the error correlation matrix of target track parameter estimation ΨN can be presented in the following form:

3(3N

− 3N + 2)

ψ11

σ2x;

N(N + 1)(N + 2)

18(2N − 1)

ψ12

= ψ 21

= −

σ2x;

N(N + 1)(N + 2)

ψ13

= ψ 31

σ2x;

N(N +

1)(N + 2)

(5.79)

12(2N − 1)(8N − 11)

ψ 22

σ x

;

N(N

− 4)(N

− 1)

180

ψ 23

= ψ32

= −

σ2x;

N(N

2 −

4)(N + 1)

180

ψ33

σ x

;

N(N

− 4)(N

− 1)

166					Signal Processing in Radar Systems

			2
GN		1	σˆx
GN		1	σ2y
2.0			σ2y
2.0			σ2
			σ2
				1ˆ
		2		x
1.6		2	σ2y
			σ2y
			σ2
			σ2
			2ˆ
		3		x
1.2		3	σ2y
1.2			σ2y
0.8					1
0.8

0.4					2
0.4

0			3			N

2	4	6	8		10

FIGURE 5.4 Coefficient of accuracy in determination of the normalized elements of error correlation matrix of second-order polynomial target track parameter estimation versus the number of measurements.

For example, at N = 3 we obtain

			1	−	3	1
			1	−	2	2
					2	2
Y3	=	−	3		13	−3	σ2x.	(5.80)
Y3	=	−	2		2	−3	σ2x.	(5.80)
			2		2
			1	−3		3
			2
						2
						2

Coefficient of accuracy under determination of the normalized elements of error correlation matrix of the second-order polynomial target track parameter estimation versus the number of measurements is shown in Figure 5.4. Comparison of the diagonal elements of the matrix (5.80) that are characteristics of accuracy under estimation of the second-order polynomial target track coordinate and its first increment with analogous elements of the error correlation matrix of linear target track parameter estimation (see Figure 5.4) shows that at low values of N, the accuracy of linear target track parameter estimation is much higher than the accuracy of the second-order polynomial target track parameter estimation. Consequently, within the limits of small observation intervals, it is worth presenting the target track as using the first-order polynomial. In this case, the high quality of cancellation of random errors of target track parameter estimation by filtering is guaranteed. Dynamic errors arising owing to mismatching between the target moving hypotheses can be neglected as a consequence of the narrow approximated part of the target track.

5.4.4 Algorithm of Extrapolation of Target Track Parameters

The extrapolation problem of target track parameters is to define the target track parameter estimations at the point that is outside the observation interval using the magnitudes of target track parameters determined during the last observation or using a set of observed coordinate values. Under polynomial representation of independent coordinates, the target track parameters extrapolated using the time τex are defined by the following formulas:

		ˆ	ˆ	τex2		(s) τ(exs)
ˆ	ˆ				ˆ			,	(5.81)
xex = xN + xN τex + x				2	+ + xN s		!	,	(5.81)

Filtering and Extrapolation of Target Track Parameters Based on Radar Measure									167
ˆ	ˆ	ˆ	ˆ		τex2		(s)	τ(exs−1)
						ˆ			(5.82)
xex = xN + xN τex + x					2 + + xN			(s − 1)! ,	(5.82)
			ˆ(s)		ˆ(s)	,			(5.83)
			xex	= xN		,			(5.83)

where τex = tex − tN is the interval of extrapolation time. Equations 5.81 through 5.83 allow us to define the extrapolated coordinates for each specific case of target track representation. For example, in the case of linear target track at equally discrete coordinate measurement we obtain the following:

		ˆ	τex					τex
ˆ	ˆ					ˆ	ˆ
xN+ p	= x + xN		T0	Teq		= xN +	1xN T0		,	(5.84)
			ˆ		=	ˆ				(5.85)
		1xN+ p			=	1xN .				(5.85)

Substituting in (5.84) and (5.85) the corresponding formulas for smoothed parameters, we obtain

		1		τex		N	τex		N
xN+p	=		hN +		gN	∑wi xi − gN +		fN	∑wi
xN+p	=		hN +	T0	gN	∑wi xi − gN +	T0	fN	∑wi
		GN		T0		i=1	T0		i=1
						i=1			i=1

If, additionally, these measurements are uniformly precise, we obtain

τex

∑

ηˆ (i) +

(i)

x .

N+ p

Teq

i=1

At τex = Teq, we have

xN+1 = ∑ηxˆN +1 (i)xi ,

i=1

where

ηxˆ

(i) =

2(3 − N − 2)

N +1

(N − 1)

(N − i)xi .

(5.86)

(5.87)

(5.88)

(5.89)

is the weight function of measured coordinate magnitudes under the target track parameter extrapolation per one measurement period.

The error correlation matrix of linear target track parameter extrapolation takes the following form at equally discrete coordinate measurement:

τex

τex 2

τex

hN +

gN +

YN+p =

(5.90)

gN +

τex

168	Signal Processing in Radar Systems

If, additionally, these measurements are uniformly precise, we obtain the following elements of the error correlation matrix of linear target track parameter extrapolation:

(N − 1)(2N − 1)

+ 6(τex /Teq )(N − 1) + 6 (τex /Teq )

ψ11 =

σ2x

;

(5.91)

N(N2 − 1)

(N − 1) + (τex /Teq )

ψ12 = ψ 21 =

σ2x;

(5.92)

N(N2 − 1)

ψ 22

σ2x.

(5.93)

N(N2

− 1)

If the independent target track parameter coordinate is represented by the polynomial of the second order, the target track parameter extrapolation formulas and the error correlation matrix of linear target track parameter extrapolation are obtained in an analogous way.

5.4.5 Dynamic Errors of Target Track Parameter Estimation Using Polar Coordinate System

Sometimes we use the polynomial deterministic model of target track parameter estimation to represent changes in polar coordinate rtg and βtg of the target track. The polynomial representation of polar coordinates does not reflect the true law of target moving and allows us only to approximate with the previously given accuracy the target movement law within the limits of finite observation interval. In the case of a uniform and straight-line target moving with arbitrary course at the fixed altitude with respect to the stationary radar system, the law of changing the polar coordinates (see Figure 5.5) is determined by the following form:

β0

βtg (t)

Target rtg(t)

FIGURE 5.5 Target moving in polar coordinates.

Filtering and Extrapolation of Target Track Parameters Based on Radar Measure

169

where

r0 and β0 are the range and azimuth of the nearest to origin point on the target track (the point A on Figure 5.5)

t0 is the time of target flight to the point A

Vtg is the target velocity. In the case of a target moving on circular arc within the limits of maneuver segment or mutual target moving and replacement of the CRS, we obtain the formulas that are analogous to (5.94) and (5.95) by structure but complex by essence

As follows from (5.94) and (5.95), even in the simplest cases of linear target movement and a stationary CRS, the polar coordinates are changed according to nonlinear laws. This nonlinearity becomes intensified as it is passed to complex target moving models, especially under the moving CRS. Inconsistency between the polynomial model and nonlinear character in changing the vector of estimated target track parameters leads to dynamical errors in smoothing, which can be presented in the form of differences between the true value of the vector of estimated target track parameters and the mathematical expectation of estimate of this vector, namely,

qg = [q− E(q
ˆ)].	(5.96)

To describe the dynamical errors of estimations of the target track parameters by algorithms that are synthesized by the maximum likelihood criterion and using the target track polynomial model, we employ the well-known theory of errors in the approximation of arbitrary continuously differentiable function f(t) within the limits of the interval tN − t0 by the polynomial function of the first or second order using the technique of least squares. This function takes the following form:

(t)

2 (t)

= a0	+ a1	(t − t0 ),	(5.97)
			(5.97)

= a0 + a1(t − t0 ) + 0.5a2 (t − t0 )2 .

In the case of linear target track, the coefficients a0 and a1 are defined from the following system of equations:

tN
	∫[ f (t) − a0 − a1(t − t0 )]dt = 0,
t0		(5.98)
		(5.98)

tN
	∫[ f (t) − a0 − a1(t − t0 )]t dt = 0.

t0

Solution of the equation system (5.98) is found under expansion of the function f(t) by Taylor series at the point 0.5(tN + t0), that is, in the middle of the approximation interval, namely,

	∞
f (t) = f [0.5(t0 + tN )] + ∑			bk	[t − 0.5(t0 + tN )]k ;	(5.99)
f (t) = f [0.5(t0 + tN )] + ∑			k!	[t − 0.5(t0 + tN )]k ;	(5.99)
	k =1		k!
	k =1
bk =	dk [ f (t)]			.	(5.100)
	dk [ f (t)]
	dtk
	dtk	t =0.5(t0 +tN )
		t =0.5(t0 +tN )

170	Signal Processing in Radar Systems

The error of approximation at the point t = tN can be determined in the following form:

f (tN ) = f (tN ) − 1(tN ).

(5.101)

Investigations and computations made by this procedure give us the following results. Under linear approximation of polar coordinates, the maximum dynamic errors take the following form:

		(N −1)2 T 2V 2
rdmax	≈	eq	tg			,
rdmax	≈	12rmin				,
		12rmin

		3(N −1)2 T			2V 2
βdmax ≈				eq			tg	;
βdmax ≈		2						;
		32rmin
max		(N −1)T V	2
max	≈	eq tg			,
rd
rd		2rmin
		2rmin

max		3 3(N −1)T V 2
max				eq			tg
βd	≈							;
βd	≈	2						;
		16rmin

(5.102)

(5.103)

where N is the number of coordinate readings within the limits of the observation interval tN − t0. In the case of quadratic approximation of the polar coordinates, the formulas for the maximum dynamic errors are more complex and we omit them.

Comparison of the dynamic errors of approximation of the polar coordinates by the polynomials of the first and second power at the same values of the target track parameters Vtg, rmin, and Teq and the sample size (N − 1)Teq shows that the dynamic errors under linear approximation are approximately one order higher than those under quadratic approximation. Moreover, under approximation of rtg(t) and βtg(t) by the polynomials of the second power, the dynamical errors are small in comparison with random errors and we can neglect them. However, as shown by simulation, if the target is in an in-flight maneuver and the CRS is moving, the dynamical errors increase essentially, because nonlinearity in changing the polar coordinates increases sharply at the same time.

5.5 RECURRENT FILTERING ALGORITHMS OF UNDISTORTED POLYNOMIAL TARGET TRACK PARAMETERS

5.5.1 Optimal Filtering Algorithm Formula Flowchart

Methods of estimation of the target track parameters based on the fixed sample of measured coordinates, which are discussed in the previous sections, are used, as a rule, at the beginning of the detected target trajectory. Implementation of these methods in the course of tracking is not worthwhile owing to the complexity and insufficient accuracy defined by the small magnitude of employed measures. Accordingly, there is a need to employ the recurrent algorithms ensuring a sequential, that is, after each new coordinating measurement, adjustment of target track parameters and their filtering purposes. At the recurrent filter output, we obtain the target track parameter estimations caused by the last observation. For this reason, a process of recurrent evaluation is called further the sequential filtering, and corresponding algorithms are called the algorithms of sequential filtering of target track parameters.

Filtering and Extrapolation of Target Track Parameters Based on Radar Measure

171

In a general case, the problem of synthesis of the sequential filtering algorithm for a set or vector of target track parameters is defined in the following way. Let the model of the undistorted target track be given by the following difference equation:

qn = Fnqn−1‘ ,	(5.104)
and the observed random sequence is given by
Yn = Hnqn + DYn,	(5.105)

where

θn is the (s + 1)-dimensional vector of filtered target track parameters Yn is the l-dimensional vector of observed coordinates

Yn is the l-dimensional vector of measurement errors

The sequence of these vectors is the uncorrelated random sequence with zero mean and known correlation matrix Rn; Φn and Hn are the known matrices defined in Section 4.2. Further, we consider that θˆ n−1 is the vector of estimations of target track parameters determined by (n − 1) coordinating measurements; Ψn−1 is the corresponding correlation matrix of estimation errors. There is a need to obtain the formulas for θˆ n, using for this purpose the vector θn−1 of previous estimations and results of new measurement Yn, and also the formula for the error correlation matrix Ψn based on the known matrices Ψn−1 and Rn.

In accordance with a general estimation theory, the optimal solution of the sequential filtering problem is reduced, first of all, to a definition of the a posteriori pdf of the filtered target track parameters, because this pdf possesses all information obtained from a priori sources and observation results. Differentiating the a posteriori pdf, we can obtain the optimal estimation of target track parameters, which are interesting for us, by the maximum a posteriori probability criterion. Henceforth, we consider the optimal sequential filtering, namely, in this sense.

Thus, let the estimation vector θˆ n−1 of the target track parameter vector θn obtained by obser-

vations of the previous (n − 1) coordinating measurement be given. We assume that pdf of
ˆ		and correlation
the vector θn−1 is the normal Gaussian with the mathematical expectation θn−1
ˆ	is extrapolated during the next nth measurement in accordance
matrix Ψn−1. The vector θn−1
with the equation
	ˆ	(5.106)
	ϑn\|n−1 = qexn = Fnqn−1.

The specific form of the extrapolation matrix Φn is defined by the target track model. For example, in the case of the only coordinate xn that can be presented by the polynomial of the second order

θn−1 =	ˆ	ˆ	ˆ	T
		ˆ	ˆ	T
				,	(5.107)
	xn−1	xn−1	xn−1	,	(5.107)

we have

	1	τex	0.5τex2
Fn =	0	1	τex	(5.108)
	0	0	1

172	Signal Processing in Radar Systems

and (5.107) can be presented in the following form:

		ˆ					xˆn−1
		xexn		1	τex	2
		xexn		1	τex	0.5τex
ˆ	=	ˆ	=			τex	ˆ		(5.109)
ˆ				0	1			,	(5.109)
qexn		xexn		0	1		xn−1	,
		ˆ		0	0	1	ˆ
		xexn					xn−1

where τex = tn − tn−1. The correlation matrix Ψn−1 is also extrapolated at the instant tn by the following formula:

Y n|n−1 = Yexn = FnY n−1FnT .

(5.110)

Taking into consideration a linearity property of the extrapolation operator Φn, the pdf of the vector of extrapolated target track parameters will also be normal Gaussian:

ˆ	ˆ	T	−1	ˆ	− qn)],	(5.111)
p( qexn ) = C1 exp[ −0.5	(qexn	− qn)	Yexn	(qexn	− qn)],	(5.111)

where

θn is the vector of true values of target track parameters at the instant tn C1 is the normalizing factor

The pdf given by (5.111) is the a priori pdf of the vector of estimated target track parameters before the next nth measurement. At the instant tn, the regular measurement of target coordinates is carried out. In a general case of a 3-D CRS we obtain

Yn =				rn βn εn				T .	(5.112)

It is assumed that the errors under coordinating measurements are subjected to the normal Gaussian distribution and uncorrelated between each other for neighbor radar antenna scanning. Consequently,

p(Yn | qn ) = C2 exp[−0.5(Yn − Hnqn )T Rn−1(Yn − Hnqn )],

(5.113)

where Rn−1 is the inverse correlation matrix of measure errors.

Under the assumption that there is no correlation between the measurement errors in the case of the neighbor radar antenna scanning, the a posteriori pdf for the target track parameter θn after n measurements is defined using the Bayes formula

ˆ	ˆ	(5.114)
p(qn \| Yn ) = C3 p( qexn ) p(Yn \| qn ),

and owing to the fact that pdfs of components are the normal Gaussian, the a posteriori pdf (5.115) will also be the normal Gaussian:

p(qn \| Yn ) = C4 exp[−0.5( qn − qn )			Y n	(qn − qn )],
ˆ	ˆ	T	−1	ˆ	(5.115)

where

θˆ n is the vector of estimated target track parameters by n measurements Ψn is the error correlation matrix of estimated target track parameters

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