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Usually, the autocorrelation function of the target maneuver intensity can be presented in exponential form as follows:

where

αis the value that is inversely proportional to the average duration of target maneuver Tm, that is, α = Tm−1

σ2m is the variance of the target maneuver intensity

In the case of equivalent discrete time, the autocorrelation function of the target maneuver intensity can be presented in the following form:

Tnew is the period of getting new information about the target. Further values of the target maneuver intensity can be presented using the previous one, for example,

where ξn is the white noise with zero mean and the variance equal to unit.

In practice, in designing the CRSs, we can conditionally think that a set of targets can be divided into maneuvering and nonmaneuvering targets. The target is not considered as a maneuvering target if it moves along a straight line with the constant velocity accurate within the action of the control noise intensity. In other cases, the target is maneuvering. For example, in the case of aerodynamic objects, the nonmaneuvering target model is considered as the main model and each coordinate of this model is defined by the polynomial of the first order. However, this classification has a sense only in the case when the filtering parameters of target track are represented in the Cartesian coordinate system under signal processing in CRSs. If the filtering parameters of the target track are represented in the spherical coordinate system, then they are changed nonlinearly even when the target is moving along the straight line and uniformly. In this case, the polynomial of the second order must be applied to represent the independent target coordinates.

5.2.2  Measuring Process Model

In the solution of the filtering problems, in addition to the target track model there is a need to specify a function between the m-dimensional vector of measured coordinates Yn and s-dimensional vector of estimated parameters θn at the instant of nth measuring. This function, as a rule, is given by a linear equation:

where

Hn is the known m × s matrix defining a function between the observed coordinates and estimated parameters of target track

Yn is the error of coordinate measuring

154 Signal Processing in Radar Systems

In the considered case, the observed coordinates are the current target coordinates in a spherical coordinate system—the target range rn, the azimuth βn, the elevation εn, or some specific coordinates for CRSs—and radar coordinates, for example, the radar range, cosine between the radar antenna array axis and direction to the target. In some CRSs, the radial velocity r.n can serve as the measured coordinate. The matrix Hn = H takes the simplest form and consists of zeros and units when the target track parameters are estimated by the observed coordinates in the spheri-

When the target track parameters are filtered by measured spherical coordinates in the Cartesian coordinate system, then computation of elements of the matrix Hn is carried out by differentiation of transformation formulas from the spherical coordinate system to the Cartesian one:

Analogously, we can define the elements of matrix Hn for other compositions of measured coordinates and filtered parameters.

Errors in coordinate measuring given by the vector DYn = DY(tn ) in (5.11) can be considered, in a general case, as the random sequence subjected to the normal Gaussian pdf. The following initial conditions can be accepted with respect to this random sequence:

•Errors in measuring the independent observed coordinate are independent. This condition allows us to solve the filtering problems by each observed coordinate individually.

•In a general case, a set of errors under measurements of each coordinate at the instants

t1, t2, …, tn is the n-dimensional system of correlated normally distributed random variables with the n × n correlation matrix Rn:

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

Rn1 Rn2 Rn3 σ2n

Yn

Φnθn

FIGURE 5.2  Flowchart of the united dynamic model.

The elements of the correlation matrix Rn are symmetric relative to the diagonal and equal between each other, that is, Rij = Rji. It means that RTn = Rn. When measurement errors are uncorrelated, all elements of the correlation matrix Rn excepting diagonal elements are equal to zero. That matrix is called the diagonal matrix.

In conclusion, we note that the target track model combined with the model of measuring process forms the model of united dynamic system representing a process subjected to filtering. The flowchart of the united dynamic model is shown in Figure 5.2, where the double arrows denote multidimensional or vectors relations.

5.3  STATISTICAL APPROACH TO SOLUTION OF FILTERING PROBLEMS OF STOCHASTIC (UNKNOWN) PARAMETERS

The filtering problem of stochastic (unknown) parameters is posed in the following way. Let the sequence of measured coordinate vectors {Y}n = {Y1,Y2 ,…,Yn}, which is statistically related to the sequence of dynamic system state vectors {q}n = {θ1,θ2 ,…,θn} in accordance with (5.1) and (5.11), be observed. There is a need to define the current estimation θˆ n of the state vector θn. The general approach to solve the assigned problem is given in the theory of statistical decisions. In particular, the optimal estimation of unknown vector parameter by minimal average risk criterion at the quadratic function of losses is determined from the following equation:

Q

where

p(qn|{Y}n ) is the a posteriori pdf of the current value of parameter vector θn by sequence of measured data {Y}n

Θ is the space of possible values of the estimated vector parameter θn

If the a posteriori pdf is the unimodal function and symmetric with respect to the mode, the optimal parameter estimation is defined based on the solution of the following equation:

In this case, the optimal parameter estimation is called the optimal estimation by criterion of the maximum a posteriori pdf. Thus, in the considered case and in the cases of any other criteria to define an estimation quality, a computation of the a posteriori pdf is a sufficient procedure to define the optimal estimations.

In accordance with developed procedures and methods to carry out statistical tests in mathematical statistics, the following approaches are possible to calculate the a posteriori pdf and, consequently, to estimate the parameters: the batch method when the fixed samples are used and the recurrent algorithm consisting in sequent accurate definition of the a posteriori pdf after each new measuring.

Using the first approach, the a priori pdf of estimated parameter must be given. Under the use of the second approach, the predicted pdf based on data obtained at the previous step is used as the a priori pdf on the next step. Recurrent computation of the a posteriori pdf of estimated parameter, when a correlation between the model noise and measuring errors is absent, is carried out by the following formula [6]:

where

p(qn | {Y}n−1) is the pdf of predicted value of the estimated parameter θn at the instant of the nth measuring by sequence data of previous (n − 1) coordinate measurements

p(Yn | qn ) is the likelihood function of the last nth coordinate measuring

In a general case of nonlinear target track models and measuring process models, computations by the formula (5.17) are impossible, as a rule, in closed forms. Because of this, in solving filtering problems in practice, various approximations of models and statistical characteristics of CRS noise and measuring processes are used. Methods of linear filtering are widely used in practice. Models of system state and measuring of these methods are supposedly linear, and the noise is considered as the Gaussian noise. Further, we mainly consider algorithms of linear filtering.

5.4  ALGORITHMS OF LINEAR FILTERING AND EXTRAPOLATION UNDER FIXED SAMPLE SIZE OF MEASUREMENTS

Algorithms of linear filtering and target track parameter extrapolation are obtained in this section using the following initial premises:

1.The model of undisturbed target track by each of the independent coordinates is given in the form of polynomial function:

s

X(q, t) = ∑θl tl!l , (5.18)

l = 0

where the power s of this polynomial function is defined by the accepted hypothesis of target moving. In (5.18) the polynomial coefficients take a sense of coordinates, coordinate change velocity, acceleration, etc., which are the parameters of target track. The set of parameters θl presented in the column form forms (s + 1)-dimensional vector of the target track parameters

We assume that this vector is not variable within the limits of measurement time.

2.Measurement results of the coordinate Yi at discrete time instants t1, t2, …, tn are linearly related to the parameter vector by the following equation:

where Yi is the measure error.

3. The conditional pdf of measure error under individual measurement takes the following form:

where σY2i is the measure error variance.

4. In a general case, a totality of coordinate measurement errors Y1, Y2 ,…, YN represents the N-dimensional system of correlated and normal distributed random variables and is characterized by N × N-dimensional correlation matrix RN (5.14). In solving the filtering problems, this matrix must be known. The conditional pdf of N-dimensional sample of correlated normally distributed random variables can be presented in the following form:

where

R−N1 is the inverse correlation matrix of measurer error RN is the determinant of correlation matrix

5.A priori information about the filtered parameters is absent. This corresponds to the case of parameter evaluation at the initial part of target track, that is, at the start of the target track by a set of target pips selected in a special way. The estimations obtained in this way are used at a later time as a priori data for the next stages of filtering. When a priori information is absent, the optimal filtering problems are solved using the maximum likelihood criterion. Thus, in the present section we consider the filtering algorithms and extrapolation of polynomial target track parameters using the fixed sample of measurements, which are optimal by the maximal likelihood criterion.

5.4.1  Optimal Parameter Estimation Algorithm by Maximal Likelihood Criterion for Polynomial Target Track: A General Case

The likelihood function of the vector parameter θN estimated by sequent measurements {YN} takes the following form in vector–matrix representation:

This is analogous to the conditional pdf of the N-dimensional sample of the correlated normally distributed random variables. It is more convenient to use the natural logarithm of the likelihood function, that is,

5.4.2  Algorithms of Optimal Estimation of Linear Target Track Parameters

Let us obtain the optimal algorithms of target track parameter estimations of the coordinate x(t), which is varied linearly using discrete readings xi, i = 1, 2, …, N, characterized by errors σ2xi . We consider the coordinate xN and its increment 1xN as the estimated parameters at the last observation point tN. For simplicity, we believe that measurements are carried out with the period Teq and the measure errors are uncorrelated from measure to measure. The coordinate is varied according to the following equation:

where

is the coordinate increment. Thus, in the considered case, the vector of estimated parameters takes the following form:

and the transposed matrix of differential operators given by (5.28) takes a form

In the considered case, the error correlation matrix will be diagonal. By this reason, the inverse error correlation matrix can be presented in the following form:

where wi = σ−xi2; δij = 1 if i = j and δij = 0 if i ≠ j. Substituting (5.37) and (5.38) in (5.27), we obtain the two-equation system to estimate the parameters of target linear track:

FIGURE 5.3  Coefficient of accuracy in determination of the normalized elements of correlation error matrix of target linear track parameters as a function of the number of measurements.

For example, at N = 3 the error correlation matrix of parameter estimation of linear target track takes the following form:

Consequently, the variance of error of the smoothed coordinate estimation by three uniformly precise measurement is equal to 5/6 of the variance of individual measurement error. The variance of error of the coordinate increment estimation is only one half of the variance of error of individual measurement error owing to inaccurate target velocity estimation, and the correlation moment of relation between the errors of coordinate estimation and its increment is equal to one half of the variance of error of individual coordinate measurement. Figure 5.3 represents a coefficient of accuracy under determination of the normalized elements of error correlation matrix of linear target track parameter estimation versus the number of measurements. As follows from Figure 5.3, to obtain the required accuracy of estimations there is a need to carry out a minimum of 5–6 measurements.

Turning again to the algorithms of linear target track parameter estimations (5.43) and (5.44), we can easily see that these algorithms are the algorithms of nonrecursive filters and the weight coefficients ηxˆ(i) and η 1xˆ (i) form a sequence of impulse response values of these filters. To apply filtering processing using these filters, there is a need to carry out N multiplications between the measured coordinate values at each step, that is, after each coordinate measurement, and the corresponding weight coefficients and, additionally, N summations of the obtained partial products. To store in a memory device (N − 1) records of previous measurements, there is a need to use high-capacity memory. As a result, to realize such a filter, taking into account that N > 5, there is a need to use a high-capacity memory, and this realization is complex.

5.4.3  Algorithm of Optimal Estimation of Second-Order Polynomial Target Track Parameters

parameters. As in the previous section, we believe that measurements are equally sampled with the