# Algorithm for calculating the covariance matrix of the noise jammer estimated position

### Abstract

#### For citation:

Devyatyarov I.O., Dobrozhanskiy V.A. Algorithm for calculating the covariance matrix of the noise jammer estimated position. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(4):43-49.
https://doi.org/10.38013/2542-0542-2019-4-43-49

## Introduction

A promising anti-aircraft missile system (AAMS) includes multifunction radars (MFR), launchers (L), and a combat control post (CCP).

An important task of the CCP is generation of a common airspace environment based on the data coming in from subordinate sources of radar information. This task is solved by the algorithm of uniform tracks array (UTA) formation.

Under present-day conditions, operation of a promising anti-aircraft missile system will be conducted in a complex multi-target and jamming environment. One of the possible jamming options is active noise jamming (ANJ), for which the MFR determines and transmits to the CCP angular coordinates and jamming bit only. If AAMS structure features at least two spatially separated MFR, it becomes possible to determine the coordinates of ANJ jammer presumed position. However, in the presence of several simultaneously acting ANJ jammers, it becomes a vital task to correctly identify track information received from several MFR.

This paper provides an algorithm of identifying tracks of ANJ jammers, part of which is an algorithm for estimation of presumed target position coordinates. An algorithm for calculating a covariance matrix of such estimation is proposed.

## The algorithm of identifying tracks of ANJ jammers

To solve the task of estimating the coordinates of ANJ jammers, it is necessary to decide first that the information on an ANJ jammer received from different sources is referred to the same ANJ jammer. For aerodynamic targets, the identification task is already solved at the CCP by calculating the value of the highest likelihood ratio and subsequently comparing it with the threshold [1]. The likelihood ratio is built using target coordinates and information on the measurement accuracy of those coordinates (expressed in the form of a covariance matrix) as received from the MFR (in the Cartesian local Earth-based coordinate system (LECS) MFR coordinates - x, y, z).

For unification purposes it was proposed to build an algorithm of identifying ANJ jammers in a similar way, taking into account that the information coming in from MFR in the LECS is generated based on the measurement of two independent angular coordinates. To that end, on each of the straight lines containing bearings to target from each of the MFR, points with the smallest distance between them are selected. This distance will be the common perpendicular to the two lines [1].

An AAMS diagram and bearings are given in Fig. 1. We shall use the following designations:

Q - coordinates of point on the bearing received from MFR No. 2;

L - coordinates of point on the bearing received from MFR No. 1;

H - coordinates of MFR No. 1;

E - coordinates of MFR No. 2;

N - coordinates of presumed position of ANJ jammer on the bearing received from MFR No. 2;

P - coordinates of presumed position of ANJ jammer on the bearing received from MFR No. 1.

**Fig. 1.** Schematic view of AAMS facilities and bearings to ANJ jammer’s presumed position

For the vectors of Cartesian rectangular coordinates EN and HP the following expressions are fair [2]:

EN = EQ_{t}; (1)

HP = HL_{s}. (2)

Here, coordinates t and s are found from the condition of coplanarity of vectors HN, HL, [ ] HL × EQ and EP, EQ, [ ] HL × EQ , namely, equality to zero of their mixed product:

Similarly, for coefficient s, the following is fair:

The covariance matrices in points N and P are calculated by the formulas:

where K_{Q} – covariance matrix of point Q coordinates errors;

K_{P} – covariance matrix of point P coordinates errors.

Further, identification of ANJ jammers’ tracks is performed through comparison with generalised distance threshold [3].

(λ_{N} - λ_{Р} )^{T} (К_{N} + K_{Р} )^{-1} (λ_{N} -λ_{Р}) < С. (9)

Here, λ_{N}, λ_{p} - vectors of the coordinates of the ANJ jammer presumed position points, with the CCP coordinates given in the Cartesian (rectangular) local Earth-based coordinate system;

C – threshold value.

The generalised distance is a random value with distribution χ^{2} . A degree of freedom of the distribution is determined by the number of independent coordinates [4][5]. In case of identification of ANJ jammers, the number of independent coordinates is equal to 2. The threshold is selected proceeding from a required distribution percentage point χ^{2} with account of the degree of freedom. If generalised distance has not exceeded the threshold, it is inferred that the two given tracks belong to one target, i. e., ANJ jammer.

## Algorithm for estimating triangulation point coordinates

After successful identification of ANJ jammers using the above algorithm it is necessary to estimate ANJ jammer coordinates as target coordinates at the CCP. As an estimate of these coordinates, it is proposed to use a linear unbiased estimate with the minimal dispersion based on two measurements of the nearest points on the bearings

where ξ_{1} =(ξ_{1x}, ξ_{1y}, ξ_{1z} ), ξ_{2} =(ξ_{2x}, ξ_{2y}, ξ_{2z}) - vectors of the measurement noise.

The covariance matrices of the noise vectors are calculated by the formulas:

Measurements of the nearest points on the bearings are dependent quantities, with covariance:

A linear estimate will look as follows

– weight matrices.

From the condition of unbiasedness of estimate, considering that M ξ_{1} = M ξ_{2} = 0, we have:

Here, I – unity matrix;

M – mathematical expectation.

To determine weight matrices A_{1} and A_{2} , we shall use the criterion of the minimum rms deviation of the estimate

The expression for dispersion can be rewritten as

Each of the three summands in expression (18) is non-negative. Let us consider more closely the first one of them, and expressions for the rest will be similar:

Since then, from the condition of the minimum rms deviation of the estimate, after removal of parentheses and differentiating each of the components, we have:

or, in the matrix form,

where K = K^{1}+ K^{2} - K^{12} - K^{21}

ПHaving obtained similar expressions for other components A^{1} , we shall write down

Thus, the expression for A^{1} will look as follows

As a result, for linear estimate of the triangulation point coordinates vector

## Algorithm for estimating triangulation point covariance matrix

The covariance matrix for estimating the nearest points on the bearings contains two components, the first of which is conditioned by the identification method being used. The second component is associated with inaccuracy of angular coordinates measurement. Considering this, the following expressions for the covariance matrices are fair [6]:

where K _{м}__{сск} - covariance matrix of the errors of spherical coordinates conditioned by the identification method being used;

K _{и}__{сск} - covariance matrix of the errors of spherical coordinates conditioned by the characteristics of the current target coordinates estimation by the radar

The covariance matrix of the measurements of spherical coordinates of the nearest point is found from the covariance matrix of the Cartesian coordinates measurement by the formula

Here, S – the matrix of derivative spherical coordinates of the nearest point as per rectangular Cartesian coordinates in LECS, calculated by the formula

R – azimuth in the spherical coordinate system;

β – range in the spherical coordinate system;

ε – elevation angle in the spherical coordinate system;

K _{и}__{пск} - covariance matrix of the errors of rectangular Cartesian coordinates, received from MFR.

The covariance matrix of errors caused by the identification method being used looks as follows:

An estimate of the dispersion of the nearest point range is drawn up as follows:

where grad - vector of partial derivatives by the Cartesian coordinates of respective tracks.

The covariances of the range of triangulation point with angles is determined as follows:

Calculation of the elements of the covariance matrix of coordinates errors for the second measurement is done similarly.

The coordinate values of points on the bearings used in estimation of the ANJ jammer coordinates are dependent. The covariance matrix

of two measurements has the following view in spherical coordinates:

The covariances are calculated as follows:

After calculation of covariance matrices in the spherical coordinates it is necessary to perform transition to the Cartesian coordinates by the formulas:

Here, V – matrix of the derivative Cartesian coordinates in LECS, as per LECS of respective tracks, namely:

The coordinates of triangulation point, in accordance with the previous section, are determined as follows:

Hence, covariance matrix of triangulation point is determined as follows:

## Research results

For insight into the operation of the algorithms, a model for shaping and processing of track information was developed. The air environment consisted of a single ANJ jammer. The AAMS

included two spatially separated MFR. In the biconical coordinate system, root-mean-square deviations (RMSD) of the angular coordinates amounted to 3' for one and 9' for the other

MFR.

As a simple estimate of the triangulation point coordinates, with account of the offered identification algorithm, a half-sum of the coordinates of the nearest points on the bearings can

be taken. Fig. 2 shows a histogram of triangulation point coordinate y obtained using this simple method and the method described in this paper.

The RMSD of coordinate y in the modelled case was smaller by 9.4 %.

The covariance matrix of triangulation point in the Cartesian rectangular coordinates can be transformed, using a transfer matrix, to obtain a covariance matrix in the spherical coordinates. Its diagonal elements are dispersions by range, elevation angle, and azimuth. The RMSD of triangulation point range can be determined as the root of range dispersion. The RMSD of range is also estimated by means of statistical modelling. Fig. 3 shows the results of triangulation point range RMSD calculation, determined by two methods.

The vertical lines in Fig. 3 show possible values of the triangulation point range RMSD obtained from covariance matrix. The points designate mathematical expectations of the obtained RMSD. The RMSD mathematical expectations coincided with RMSD determined by the mathematical model to an accuracy of 1 %.

## Conclusion

This paper considers the algorithms of ANJ jammers identification and estimation of the coordinates of ANJ jammer’s presumed location ensuring minimum dispersion of errors. Besides, an algorithm for calculating a covariance matrix of the estimate obtained is proposed. A comparison between RMSD of the triangulation point range calculated by the covariance matrix and the modelling results is drawn.

The obtained results can be used for development of algorithms ensuring operation against an ANJ jammer..

## References

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### About the Authors

**I. O. Devyatyarov**Russian Federation

Devyatyarov Igor Olegovich – engineer. Science research interests: algorithms for tertiary processing of radar data, computer methods for processing and analyzing data.

Moscow

**V. A. Dobrozhanskiy**Russian Federation

Dobrozhanskiy Vladimir Alekseevich – Head of Department. Science research interests: algorithms for secondary and tertiary processing of radar data, mathematical and simulation modeling of radar systems and control centers.

Moscow

### Review

#### For citation:

Devyatyarov I.O., Dobrozhanskiy V.A. Algorithm for calculating the covariance matrix of the noise jammer estimated position. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(4):43-49.
https://doi.org/10.38013/2542-0542-2019-4-43-49