RU2392640C1 - Method for identification of parametres of trajectory instabilities of small-sized flying object in form of radial acceleration of motion for accompaniment mode with help of signals with per pulse carrier frequency tuning - Google Patents

Abstract

FIELD: measurement equipment.

SUBSTANCE: measurement method fit for small-sized fling objects with minor azimuths of motion relative to the radiolocation station. The method may find application in prospective double-purpose antijamming radiolocation stations with signals tunable in terms of frequency including those for aerodrome landing radiolocators, aircraft landing radiolocation stations, accompaniment radiolocation stations for military purpose, onboard radiolocation stations of aircraft carrying ships etc. to provide for measurement of radial acceleration (as one of the forms of an object flight trajectory instability) a package of signals with per pulse random tuning of frequency is radiated into the space. Randomness of frequency tuning is formed algorithmically from a linear graded law of frequency change in a package of pulse signals. Centimetre wave band is used while frequency tuning range is within 150 MHz. Number of signals with frequency tuning in the package is 2^k, where k-6…9. The method is based on rephrasing the object's frequency characteristic by way of compensation of phase components connected with its radial motion with account for the frequency tuning randomness.

EFFECT: elaboration of a method whose advantages consist in simplicity of digital implementation and high precision of measurement of radial acceleration of an object being accompanied.

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Description

The invention relates to the field of radar and is intended to measure the parameters of trajectory instabilities in the form of radial acceleration of a small airborne object (AT) with pulse-wise tuning of the carrier frequency according to a random law.

As is known, the use of pulsed signals with pulse-wise tuning of the carrier frequency (with a stepwise change in frequency from pulse to pulse) according to a random law can increase the noise immunity of a radar system [1]. The use of pulse-frequency tuning of the carrier frequency together with a change in the repetition period of the probing signals reduces the effectiveness of traditional and promising types of interference [2, 3]. In addition, the use of frequency tunable signals (HF) allows you to expand the information capabilities of radar stations in the following areas: increasing resolution in range [1], providing recognition of airborne objects [4, 5] and selection of false objects [6], measuring the radial extent of aircraft [7], as well as determining the composition of group HE [8].

Radial acceleration is one of the types of trajectory instabilities of the VO flight. Its measurement and accounting are important for reliable identification of signs of classification, ensuring air traffic control in the area of aerodromes, when landing an aircraft on board an aircraft carrier, etc. In this regard, the urgent task of measuring radial acceleration in advanced radar prospective in real time.

The use of signals with pulse-wise tuning of the carrier frequency according to a random law is one of the promising directions for improving existing radars. However, for radar systems focused on the use of these impulse signals, a method for measuring the radial acceleration of an air object even in its tracking mode has not yet been proposed.

Currently, the development of aircraft shipbuilding is on the path to creating and improving unmanned aircraft. According to forecasts of American experts [9], by 2025, the share of unmanned aerial vehicles will amount to 90% of the total number of military aircraft. The vast majority of modern unmanned aerial vehicles are small (up to 5 m). Consequently, promising radar systems, which include radars with pulse-frequency tuning of the carrier frequency, should be equipped with algorithms for detecting and measuring the motion parameters of small-sized objects, including parameters of this type of flight path instabilities (VT) such as radial acceleration.

A known method for identifying the parameters of VT flight VO in the form of radial acceleration, used in single-frequency coherent-pulse radar [10], which consists in the fact that they emit pulsed sounding signals into space, receive signals reflected from the VO, reduce the frequency of the received signals to an intermediate, amplify the received power signals, perform their amplitude limitation and phase detection, translate the amplitude and phase of each reflected signal into digital form, record the values of the amplitudes and phases of the reflected signals memory, convert the received signals into video pulses whose amplitude from period to period modulated with the Doppler frequency F _d, is measured Doppler frequency of the reflected signal F _d1 and F _d2 in two successive points in time with intervals of the order of Δt = l to determine the radial velocity of the object V _rl and V _r2 at these times according to the formula V _{r1 (2) =} F _{dl (2)} λ / 2, where the index “r” means belonging to the radial component of the velocity vector, λ is the wavelength, roughly determine the value of the radial acceleration VO a _r according to the formula a _r = (V _r2 -V _rl ) / Δt, select a private array of N elements is taken from the general set of digital parameters of the reflected signals, Doppler portraits of the object are formed from this digital array using fast Fourier transform, and before the formation of the next Doppler portrait, the phase value calculated from the phase of each digitized signal entering the private array is calculated by the formula Δφ = 2π (nT _i ) ² a _r * / λ, where n is the number of the stored pulse from the private array, T _i is the pulse repetition period, and _r * is the estimated value of the radial acceleration In order to achieve a range of angles, selected from the range [a _r ± 5 m / s ² ] with a step of 0.1 m / s ² , one obtains a set of Doppler portraits of the object for various assumed values of acceleration a _r *, in each generated Doppler portrait they find the sum of the amplitudes of all components (elements), the resulting amounts are compared, taken for the measured value of the radial acceleration IN the value at which the calculated amount is minimal.

The disadvantage of this method [10] is the impossibility of its implementation in the mode of pulse frequency tuning according to a random law, since an unknown phase shift due to a change in the carrier frequency is added to the regular Doppler frequency shift. In addition, the single-frequency sounding mode used by the method has low noise immunity, and under the influence of high intensity interference-frequency interference, the extraction of useful information from the set of reflected signals becomes problematic. It should also be noted the inoperability of this method in the case of the location of small airborne objects, since the Doppler portrait in this case appears to be one narrow glider component, and it is difficult to detect a change in the width of the portrait.

The objective of the invention is the provision of identifying the parameters of the flight parameters of a small airborne object in the form of radial acceleration in the mode of pulse-wise tuning of the carrier frequency according to a random law.

To solve the problem of the invention, it is proposed to use the advantages of the method of measuring the radial speed of an air object in the frequency tuning mode from pulse to pulse [11]. This method of measuring radial velocity consists in emitting a packet of signals with frequency tuning according to a random law, receiving signals reflected at the n-th frequencies from the airborne object, lowering the frequencies of the received signals to an intermediate one, and using quadrature phase detectors, quadrature components of the received signals are extracted, convert the quadrature components into digital form using analog-to-digital converters and get the frequency response (FX) of the air object, representing the second vector G from the set of responses of the matched receiver to the reflected FH. The method is based on the rephasing of the frequency response by multiplying its elements by a complex phase factor, depending on the radial velocity of the HE. Due to the fact that the radial velocity of the object needs to be determined, in the interests of maximum compensation for phase incursions associated with the radial movement of the HE (correct rephasing of the frequency response), the possible values of the radial velocity of the object are enumerated, as a result of which a two-dimensional matrix D is formed using the vector G, which lines are a set of rephased frequency response. When enumerating all the possible (assumed) radial velocities of the HE in one of the cases (in one of the rows of the matrix D), the best compensation for the negative phase incursions associated with the radial movement of the HE will occur. As a tool for determining the maximum coincidence of the true radial velocity of an object with its estimated value, a range VO portrait [12] and the entropy value of the data constituting its vector are used. When the true and assumed radial velocities of the VO coincide, the most informative range portrait (DP) is formed, the data entropy of which is minimal. To do this, form the matrix D1 by performing the inverse Fourier transform with complex data vectors of each row of the matrix D, find the maximum value of the modulus of the complex signal in the matrix D1 and divide the complex values of all elements of the matrix D1 by this value (normalize it), calculate the data entropy for each row of the matrix D1, find the value of the estimated radial velocity VO at which the entropy of the data constituting the range portrait is minimal, and take the found value as ve assessment of the radial velocity VO.

The advantages of the method are high noise immunity and indifference to the complexity and size of the geometric structure of VO. These advantages, together with the essence of the method [11] create the basis for its use in the interests of identifying the parameters of the flight parameters of a small airborne object in the form of radial acceleration.

To solve the problem of identifying the flight parameters of the flight of a small airborne object in the form of radial acceleration, it is proposed to radiate a burst of pulsed signals in the direction of the VO with the tuning of the carrier frequency according to a random law with a wobble of the repetition period. The number of signals in the packet N is chosen equal to 2 ^k where k is an integer in the range from 6 to 9. The choice of the specified value of the number N is determined by the desire to use fast Fourier transform (FFT) algorithms in processing, which can significantly reduce the amount of calculations [13]. Wobble of the signal repetition period inside the packet is necessary to eliminate the ambiguity of the radial velocity measurement by the method [11].

The duration of the FH bundle Δt is chosen equal to the time interval during which the change in the radial velocity of the HE would be guaranteed to exceed the value of the potential accuracy of estimating the radial velocity by the method [11]. The indicated accuracy depends on the number of pulses in the packet, the working signal-to-noise ratio, and amounts to no more than 0.5 m / s. From this it can be seen that for measuring accelerations of units m / s ² , the duration of the HF pack is of the order of 1 s. Here, the limited nature of the proposed method for estimating radial acceleration a _r is manifested explicitly. If the VO moves with a large heading angle, then the radial component of the acceleration changes slowly, and its measurement is difficult. Therefore, the method is suitable only in the case of radial movement of the aircraft (in the direction of the radar), which is typical for landing radars of aerodromes, aircraft carriers and aircraft carriers.

The next limitation of the proposed method is the ability to measure the radial acceleration of only small BO. The presence of trajectory instabilities in the flight of large-sized objects in the form of yaw leads to an unpredictable change in the distance to the scattering centers of the HE and a significant change in the phase characteristic of the received packet of reflected signals. The change in the distance to the scattering centers of the small-sized HE due to its yaw is negligible and has virtually no effect on the phase characteristic of the packet of received signals. A set of packs of reflected HFs for 1 s should not exclude radiation of other sounding signals in the same interval.

, определяется по формулеTo form an FH bundle in the radar RAM, a sequence of numbers and values of the frequencies used is used from f ₀ to f ₀ + F _per , where f ₀ is the main carrier frequency of the probe signal of the centimeter range, F _per = 150 MHz is the range in which tuning is performed frequency from pulse to pulse in increments of Δf = F _per / (Nl). Then, the radiation frequency numbers are distributed according to a random law, in which the radiation time t _n pulse at the n-th frequency f _n = f ₀ + nΔf, where n is the number of the used frequency

determined by the formula

where T _i - the main period of the sequence (repetition) of pulses inside the packet;

- порядковый номер излучения импульса на n-й частоте f_n, принимающий значение от 0 до N-1, единожды повторяющееся в пределах пачки СПЧ. Например, если в 26-м периоде излучен импульс на частоте f₂=f₀+2Δf, то

=26 при n=2. Порядок использования частот в пачке и величины t_n запоминаются в интересах последующей обработки.ΔT _n is the temporary addition necessary to ensure the wobble of the pulse repetition period, randomly selected from the range [-0,05T _i ; 0.05T _i ];

- the serial number of the pulse radiation at the n-th frequency f _n , taking on a value from 0 to N-1, repeating itself once within the FH bundle. For example, if in the 26th period a pulse is emitted at a frequency f ₂ = f ₀ + 2Δf, then

= 26 for n = 2. The order in which the frequencies are used in the packet and the values of t _{n are} stored in the interests of subsequent processing.

It is further proposed to receive signals reflected from the VO at different frequencies f ₀ + nΔf + F _dn , where F _dn is the Doppler frequency addition of the reflected signal at the nth frequency, due to the radial velocity of the VO. Then it is proposed to lower the frequency of the received signals to an intermediate f _pr + nΔf + F _dn , where f _np is the value of the intermediate frequency, and amplify the signals using broadband amplifiers of the intermediate frequency.

Subsequently, it is necessary to filter signals using band-pass filters, as a result of which signals with carrier frequencies f _n enter the corresponding nth frequency channels (the number of frequency channels is equal to the number of frequencies used), quadrature phase detectors [14, 15] should be selected components of the received signals and convert the quadrature components of the received signals into digital form using analog-to-digital converters.

отклика согласованного приемника на сигнал n-й частоты.Then it is proposed to carry out digital matched filtering of each received pulse signal separately, to arrange the received signals in the order of a linear increase in frequency and to form a vector G consisting of N elements. At the same time, the nth elements of the specified vector write the complex value

response matched receiver to the signal of the n-th frequency.

After this, it is necessary to form a three-dimensional data matrix W (Fig. 1), consisting of V × Z longitudinal rows, N × V transverse rows and N × Z columns, where V = (2V _{r max} / dV + l); V _{r max} - the maximum possible radial velocity VO; dV is the sampling interval (search step) of the estimated radial velocity of the object; Z = (2a _{r max} / da + 1); a _rmax is the maximum possible radial acceleration of VO; da is the sampling interval (enumeration step) of the estimated radial acceleration a _{r of the} object. The number 2 determines the ability to measure positive and negative radial velocities and accelerations.

трехмерной матрицы, у которых изменяется только индекс n, а индексы v и z остаются неизменными, т.е. изменение величин откликов согласованного приемника определяется только изменением частоты излучаемых сигналов (фиг.1). У элементов

, составляющих поперечную строку, изменяется только индекс z при неизменности индексов n и v. У элементов трехмерной матрицы, входящих в состав столбцов, изменяется только индекс v при неизменности индексов n и z (фиг.1).A longitudinal line refers to a collection of elements

three-dimensional matrix, in which only the index n changes, and the indices v and z remain unchanged, i.e. the change in the response values of the matched receiver is determined only by the change in the frequency of the emitted signals (figure 1). The elements

constituting the transverse row, only the z index changes with the indices n and v unchanged. For the elements of the three-dimensional matrix included in the columns, only the index v changes with the indices n and z unchanged (Fig. 1).

рассчитанную по формулеIn the elements of the three-dimensional matrix W, the complex quantity should be written

calculated by the formula

- комплексная величина элементов вектора G, с - скорость распространения электромагнитных волн. В результате проведения указанных операций совокупность продольных строк матрицы W будет представлять собой набор перефазированных частотных характеристик ВО.Where

is the complex magnitude of the elements of the vector G, and c is the propagation velocity of electromagnetic waves. As a result of these operations, the set of longitudinal rows of the matrix W will be a set of rephased frequency characteristics of VO.

матрицы W1 будут аналогичны номерам столбцов и строк (индексов элементов

исходной матрицы W. Другими словами, необходимо нормировать элементы матрицы W1.It is further proposed by conducting an inverse FFT with complex data vectors of each longitudinal row of the matrix W to form a three-dimensional matrix W1 (Fig. 1), and then find in it the maximum value Wl _{max of the} modulus of its constituent complex quantities. After dividing the values of the matrix elements by the found maximum value, it is proposed to write the division results in the elements (cells) previously corresponding to the elements used as dividends in the division. Numbers of columns, longitudinal and transverse rows (element indices

matrices W1 will be similar to the numbers of columns and rows (element indices

of the original matrix W. In other words, it is necessary to normalize the elements of the matrix W1.

Thus, the elements of the vectors that are the result of the inverse FFT over the vectors of the longitudinal rows of the original matrix W will be placed in the elements of the longitudinal rows of the matrix W1, and then the results of dividing the recorded complex quantities by the maximum value of the module of complex quantities W1 _max .

Further, it is proposed to calculate the entropy of the data H _{v, z} [16] for each longitudinal row of the matrix W1 according to the formula [11]

^- элементы матрицы W1, полученной после проведения обратного БПФ с комплексными векторами данных каждой продольной строки матрицы W и нормировки ее элементов.Where

^- elements of the matrix W1 obtained after performing the inverse FFT with complex data vectors of each longitudinal row of the matrix W and normalizing its elements.

It is further proposed to form a two-dimensional matrix H consisting of V rows and Z columns (FIG. 1).

By the vth row we mean the set of elements of the two-dimensional matrix H, in which only the index z changes, and the index v remains unchanged. For elements H _{v, z} that are part of the z-th column, only the index v changes with the index z being unchanged. The numbers vx of the rows and zx columns of the matrix H are similar to the numbers vx of the transverse rows and zx columns of the original matrix W1. Thus, the rows of matrix H will contain the values of entropy that change in accordance with the change in the estimated radial acceleration of VO, and the columns will contain the values of entropy that change in accordance with the change in the estimated radial velocity.

по формулеAt the final stage, it is proposed to find in the generated matrix H the column number Z _minH containing the lowest entropy H _v , _{z min} and determine the radial acceleration estimate by the number of the found column

according to the formula

The essence of the method for measuring the radial acceleration of small-sized VOs when using signals with pulse-frequency tuning of the carrier frequency is as follows.

When a packet of signals is emitted with randomization of the carrier frequency and reception of reflected signals, the n-th term of the received frequency response of an airborne object moving with a speed V _r and radial acceleration a _r is determined by the formula

where K (n) is the coefficient determined by the properties of the radar receiver when processing the signal of the nth frequency; m - serial number of the scattering center (RC); M is the number of RCs in the airframe; σ _m is the effective scattering area (EPR) of the mth RC [14]; R _m∥ is the distance from the reference range point to the mth RC along the radial coordinate at the time of the emission of the first pulse from the composition of the HF pack; ψ _m is the phase value due to the reflection of the pulse signal from the m-th RC.

It is proposed to carry out the frequency response processing similarly to the method for estimating the radial velocity given in [11]. Carrying out the inverse FFT with the elements of the frequency response of a stationary air object allows one to obtain its DP [12]. In the interests of ensuring that the parameters of the processed HF bundle match the immobility of the object, it is necessary to rephase the frequency response by multiplying its elements by a complex phase factor

Due to the fact that the values of radial velocity and radial acceleration are unknown, it is necessary, by analogy with [11], to use the method of their selection. When enumerating all possible (assumed) values of the radial velocity in the range ± V _{r max} and radial acceleration in the range ± a _{r max} in one of the cases (when the true radial velocity and acceleration of the object are equal to their expected values), the best compensation for phase distortions associated with radial movement of VO. As a result of the inverse Fourier transform with the rephased frequency response, an informative DP of the object will be formed (Fig. 2), in which the mth diffuser on the BO surface corresponds to a well-defined impulse response [4, 5]. Expressions for coherent DP are known and presented in [12].

The geometric design of the object (the number of scatterers on its surface and the distance between them), as well as the type of the corresponding DP, are also unknown. As a criterion for determining the maximum coincidence of the true radial velocity and the true radial acceleration of the HE with their expected values, it is advisable to use the minimum entropy of the system [16]. The entropy of the data constituting the DP is minimal when the true radial acceleration coincides with its expected value, which should be chosen as an estimate of the radial acceleration of VO by analogy with the method [11].

определяется по номеру столбца z_minH, содержащего наименьшее значение энтропии Н_{v,z min}.Thus, to ensure the measurement of radial acceleration, it is necessary to conduct a double enumeration according to the parameters: first, according to the radial velocity, and then according to the radial acceleration of the HE. As a result, V × Z long-range portraits will be formed. With the coincidence of the true and selected values of the radial velocity and the radial acceleration of the HE, the most informative coherent DP of the object will be formed, the data entropy of which is minimal. Radial Acceleration Estimation

determined by the column number z _minH containing the lowest entropy value H _{v, z min} .

The introduced restriction on the suitability of the method only with respect to small-sized objects is due to the following. Any VO in flight experience random angular displacements due to the influence of turbulent flows of air masses. At large and medium sizes of HE during the Δt interval for receiving a packet of reflected HF, phase signal incursions will include components due to random turns of the aircraft. In the presence of such phase additives in the reflected signals, the radial velocity estimation method [11] ceases to function. In order for this method to work efficiently, a limitation of the duration of the FH bundle was introduced, which in method [11] should be less than the interval of angular correlation VO of 5 ms. For the proposed method for measuring radial acceleration, such a duration of an FH bundle is unacceptable, and yawing of large-sized HEs destroys the phase characteristic of the received packet of reflected signals. Therefore, the method is suitable only for measuring the radial acceleration of a small BO.

To test the operability of the proposed method for measuring the radial acceleration of HE, the method of mathematical modeling was used. In studies, VO models of the F-15, F-16, F / A-18, A-10A, Q-5 type were used, constructed by the method of approximating their design by bodies of simple geometric shape [17]. The effective scattering area of the RC was calculated taking into account the angle of the glider VO. The change in the noise level was simulated by the additive addition of additional components distributed according to the Gauss law to the quadrature components of the reflected signal [18].

The values of the estimated velocities of VO during modeling were varied in the range of ± 700 m / s, and the values of the expected accelerations were varied in the range of ± 10 m / s ² . The search step in the radial speed of flight of the aircraft was 0.1 m / s, and the search step in radial accelerations was 0.1 m / s ² . The result of processing the FH bundle was displayed in the form of a three-dimensional high-speed scan, a variant of which is shown in FIG. 3, showing the dependence of the entropy of the data vector constituting the DP on the assumed values of the radial velocity and radial acceleration of the HE. To plot the graph, the F-15 type of motion was simulated with a radial velocity V _r = -340 m / s and with a radial acceleration a _r = -5 m / s ² . For clarity, the graph is inverted along the axis H.

Figure 3 confirms the possibility of measuring radial acceleration IN in the mode of pulse tuning of the carrier frequency. The position of the global minimum of the velocity sweep along the axis of the assumed velocities of the HE corresponds to its true radial velocity, and along the axis of the assumed accelerations to the true radial acceleration of the HE. Analysis of 1000 high-speed scans built for each of the 5 types of HE models confirmed the efficiency of the proposed method for estimating the radial acceleration of small-sized aircraft.

в_r=-5,1 м/с². При проведении 1000 расчетов заявляемым способом ошибка измерения радиального ускорения не превысила 0,2 м/с² при отношении сигнал-шум 13 дБ на входе системы обработки, что говорит о работоспособности способа и высокой точности измерения.The estimation of the radial acceleration of VO in the case shown in figure 3 was

in _r = -5.1 m / s ² . When conducting 1000 calculations by the claimed method, the error in measuring radial acceleration did not exceed 0.2 m / s ² with a signal-to-noise ratio of 13 dB at the input of the processing system, which indicates the operability of the method and high measurement accuracy.

The proposed method is easy to implement and has the following advantages: high noise immunity due to the pulse-wise tuning of the carrier frequency according to a random law, which eliminates the negative impact of interference aiming in frequency; high accuracy of radial acceleration estimation; simplicity of digital implementation; the possibility of obtaining DP air objects in the interests of their recognition. The proposed method can find application in promising dual-purpose radars with pulse-frequency tuning of the carrier frequency.

Information sources

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Claims (1)

A method for detecting the parameters of trajectory instabilities of a small airborne object in the form of radial acceleration of motion for the tracking mode using signals with pulse-frequency tuning of the carrier frequency, which consists in using a radar station to radiate a packet of signals with tuning of the carrier frequency according to a random law, receiving reflected from the airborne object signals at n-th frequencies, lower the frequencies of the received signals to an intermediate, amplify the signals at an intermediate frequency, with the help of The filtering of signals is carried out by means of the bandpass filters, as a result of which the signals with the n-th carrier frequencies fn arrive at the corresponding n-th frequency channels, the quadrature components of the received signals are extracted using quadrature phase detectors, and the quadrature components of the received signals are converted to digital form using analog digital converters, characterized in that the duration of the emitted packet of signals with frequency tuning Δt is chosen equal to the time interval of about 1 s, in t whose importance change object radial velocity air exceeds 0.5 m / s, the radiation bundle of N signals with the rearrangement of the carrier frequency at random in the random access memory of the radar generates a sequence of numbers and quantities used carrier frequencies from f ₀ to f ₀ + F _lane increments Δf = F _trans / (N-1) where f ₀ - primary carrier frequency of the probing signal centimeter range, F = 150 MHz _lane - The range in which the rearrangement of the carrier frequency from pulse to pulse, partitioned prefecture pa carrier frequency radiation at random, whereby the time t _n the pulse radiation on the n-th frequency f _n = f ₀ + nΔf, where n - the number of the signal frequency determined by the formula

where T _i is the main pulse repetition period (repetition) inside the packet, ΔT _n is the time supplement necessary to ensure the wobble of the pulse repetition period, randomly selected from the range [-0.05T _i ; 0.05T _i ];

- the serial number of the pulse radiation at the n-th frequency f _n , taking on a value from 0 to N-1, repeating itself once within a packet of signals with tuning of the carrier frequency, remember the order of using carrier frequencies when emitting a packet of signals with tuning of the carrier frequency, conduct digital coordinated filtering each received pulse signal separately, the received signals are arranged in the order of a linear increase in frequency, a vector G consisting of N elements is formed from the parameters of these signals, recorded in the nth element said vector complex value

the response of the matched receiver to the signal of the n-th frequency, form a three-dimensional data matrix W, consisting of V × Z longitudinal rows containing elements with constant indices v and z, N × V transverse rows containing elements with constant indices n and v, and N × Z columns containing elements with constant indices n and z, where V = (2V _{r max} / dV + 1); V _{r max} - the maximum possible radial speed of an air object, dV - sampling interval (search step) of the estimated radial speed of the object, Z = (2a _{r max} / da + 1); a _{r max} is the maximum possible radial acceleration of an air object, da is the sampling interval (search step) of the estimated radial acceleration of a _r , object, complex values are written into the elements of the three-dimensional matrix W

calculated by the formula

Where

- the complex value of the nth element of the vector G, by performing the inverse fast Fourier transform with complex data vectors of each longitudinal row of the matrix W, we obtain a three-dimensional matrix W1, in which the numbers of columns, longitudinal and transverse rows are similar to the numbers of columns and rows of the original matrix W, but in the elements of the longitudinal rows are the values of the elements of the vectors resulting from the inverse fast Fourier transform over the vectors of the longitudinal rows of the original matrix W, find the maximum value in the matrix W1 W1 _max modulus of its constituent complex values, find the quotients of the division of the complex values of all elements of the matrix W1 by the value W1 _max and write the results of division into elements that previously corresponded to the elements used as dividends, calculate the entropy of the data H _{v, z} for each longitudinal rows of matrix W1 by the formula

Where

- the elements of the matrix W1, form a two-dimensional matrix H, consisting of V rows and Z columns corresponding to the transverse rows and columns of the original matrix W1, the entropy values H _{v, z} calculated according to the data of the longitudinal rows of the matrix W1 are recorded in the elements of the matrix H while maintaining the identity of these longitudinal lines z-m columns and v-th transverse rows of the original matrix, are formed in matrix H z _minH column number having the smallest value of the entropy, based on the found number column determining an estimate of the radial acceleration of air ekta

according to the formula

if the obtained estimate differs from zero, they decide that the airborne object moves with trajectory instabilities due to the presence of radial acceleration equal to

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