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Extrapolation of target track parameters is carried out in accordance with the hypothesis about the target straight-line movement, and the correlation matrix is extrapolated by the following rule:

where

Extrapolated polar coordinates are determined from the Cartesian coordinates by the following formulas:

The vector of measured target track parameter coordinates and the error correlation matrix take the following form:

To establish a relationship between the measured coordinates and estimated target track parameters, we use the linearized operator:

186 Signal Processing in Radar Systems

5.6.3  Adaptive Filtering Algorithm Version Based on Bayesian Approach in Maneuvering Target

Under adaptive filtering, the linear dynamic system described by the state equation given by (5.144) is considered as the target track model. Target track distortions caused by a deliberate target maneuver are represented as a random process, the mean E(gmn) of which is changed step-wise taking a set of

fixed magnitudes (states) within the limits of range [−gmmax , + gmmax ]. Transitions of a step-wise process from the state i to the state j are carried out with the probability Pij ≥ 0 defined by a priori data about the target maneuver. The time when the process is in the state i before transition into the state j is the random variable with arbitrary pdf p(ti). The mathematical model of this process is the semi-Markov random process. Distortions of the target track caused by a deliberate target maneuver and errors of intensity estimations of deliberate target maneuver are characterized by the random component ηn in an adaptive filtering algorithm. The matrices Φn, Γn, and Kn are considered as the known matrices.

Initially, we consider the Bayesian approach to design the adaptive filtering algorithm for the case of continuous distortion action gmn. As is well known, an optimal estimation of the parametric

(Θ )

where

(Θ) is the space of possible values of estimated target track parameter

p(qn | {Y}n ) is the a posteriori pdf of the vector θn by data of n-dimensional sequence of measurements {Y}n

Under the presence of the distortion parameter gm, the a posteriori pdf of the vector θn can be written in the following form:

Thus, the problem of estimating the vector θˆ n is reduced to weight averaging the estimations qˆn (gmn ), which are the solution of the filtering problem at the fixed magnitudes of gmn. The estima-

tions qˆn (gmn) can be obtained in any way that minimizes the MMSE criterion including the recurrent linear filter or Kalman filter. The problem of optimal adaptive filtering will be solved when the a posteriori pdf p(gmn | {Y}n ) is determined at each step. Determination of this pdf by sample of measurements {Y}n and its employment with the purpose of obtaining the weight estimations is the main peculiarity of the adaptive filtering method considered.

In the case considered here, when the parametric disturbance takes only the fixed magnitudes gm j, j = −0.5m,…, −1, 0, 1,…, 0.5m, m is even, we obtain the following formula instead of (5.171):

j= −0.5m

{Y}n. To determine the a posteriori probability P(gm jn |{Y}n ), we use the Bayes rule, in accordance with which (5.17) we can write

is the conditional probability of the transition of disturbance process from the state i at the (n − 1)th step to the state j at the nth step; p(Yn | gm j,n−1 ) is the conditional pdf of observed magnitude of the coordinate Yn when the parametric disturbance at the previous (n − 1)th step takes the magnitude gm j. This pdf can be approximated by the normal Gaussian distribution with the mean determined by

and the variance given by

Taking into consideration (5.177) and (5.178), we obtain the a posteriori probability in the following form:

The magnitudes Pnj for each j are the weight coefficients at averaging of estimations of filtered target track parameters.

Henceforth, we assume that the target track parameters are filtered individually by each Cartesian coordinate of the target. Measured values of spherical coordinates are transformed into Cartesian coordinates outside the filter. Correlation of measure errors of the Cartesian coordinates is not taken into consideration. The Cartesian coordinates fixing the intensity gmx, gmy , and gmz of deliberate target maneuver are also considered as independent between each other and gmx = xm , gmy = ym , and gmz = zm. In the following, the equations of adaptive filtering algorithm applying to the Cartesian coordinate x are written in detailed form.

FIGURE 5.10  Adaptive filter flowchart: general case.

this weight, to compute the filtered target track parameters by the usual (undivided) filter instead of weighting the output estimations of filtered target track parameters. The equation system of the simplified adaptive filter is different from the previous one by the fact that the extrapolated values of filtered target track parameters determined by (5.180) are averaged with the following weights:

After that, we can define more exactly the estimations of filtered target track parameters taking into consideration the nth coordinating measure using the well-known formulas for the Kalman filter. A flowchart of the simplified adaptive filter is shown in Figure 5.11. It consists of blocks for computation of the correlation matrix of errors Ψn, the filter gain Gn, and the probabilities P(xm jn |{Y}n )

that are shared by the filter as a whole; the block to compute the estimations x and ˆ of target

ˆn xn

track parameters; (m + 1)th blocks of target track parameter extrapolation for each fixed magnitude of acceleration x¨mjn; the weight device to compute averaged extrapolated coordinates. From Figure 5.11, it is easy to understand the interaction of all blocks. Employment of adaptive filters allows us to decrease essentially the dynamic error of filtering the target track parameters within the limits of the target maneuver range. In doing so, when there is no target maneuver, the root-mean-square magnitude of random filtering error is slightly increased, on average 10%–15%.

The relative dynamic errors of filtering on the coordinate x by the adaptive filter (solid lines) and nonadaptive filter (dashed lines) with σ2x = 0.5g for the target track shown in the right top are presented in Figure 5.12. The target makes a maneuver with accelerations d1 = 4g and d1 = 6g, where g is the gravitational acceleration, moving with the constant velocity Vtg = 300 m/s. Maneuver is observed within the limits of six scanning periods of the radar antenna. In the case of the adaptive filter, x1 = −8g, x2 = 0, and x3 = 8g are considered as the discrete magnitudes of acceleration. As follows from Figure 5.12, the adaptive filter allows us to decrease the dynamic error of filtering

FIGURE 5.11  Flowchart of simplified adaptive filter.

1.8

6g

1.6

4g

1.4

4g

1.2

x

1.0

0.86g

4 6 8 10 12 14 16 18 20 22 24 26 28 30

FIGURE 5.12  Dynamic errors of filtering: the adaptive filter (solid line) and nonadaptive filter (dashed line).

twice in comparison with the nonadaptive filter even in the case of low-accuracy fragmentation of the possible acceleration range. In the case considered in Figure 5.12, these errors do not exceed the variance of errors under the coordinating measurement. There is a need to take into consideration that the labor intensiveness of the considered adaptive filter realization by the number of arithmetic operations is twice higher in comparison with the labor intensiveness of the nonadaptive filter realization. With an increase in the number m of discrete values of maneuver acceleration (fragmentation with high accuracy) within the limits of the range (−xmax xmax ), the labor intensiveness of the considered adaptive filter realization is essentially increased.