RU2716004C1 - Method for spatial localization of radio transmitters - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: invention relates to radio engineering and can be used in single-position systems for concealed control of ground, sea and air space, performing trajectory tracking of mobile objects on direct radio signals of their on-board radio transmitters and copies of these radio signals, reflected by foreign reflectors in form of natural irregularities of terrain or stationary and mobile objects of artificial origin. Said result is achieved by enlarging the range of measured parameters of reflected signals (time delays and Doppler frequency shifts instead of time delays) and performing hypothesis testing on values of Doppler shifts of reflected signals instead of checking hypotheses on values of spatial coordinates of reflectors, which are difficult to implement in stationary and virtually impossible in mobile concealed monitoring systems.

EFFECT: technical result is increase in information value (definition of speed vector in addition to spatial coordinates) and efficiency of spatial localization of wide range of radio transmitters by single-position control system under conditions of a priori uncertainty of shape, dimensions, reflecting properties and spatial coordinates of foreign reflectors of radio signals.

1 cl, 2 dwg

Description

The invention relates to radio engineering and can be used in one-position systems of covert control of land, sea and air space, trajectory tracking of moving objects by direct radio signals of their airborne transmitters and copies of these radio signals reflected by extraneous reflectors in the form of natural inhomogeneities of the terrain or stationary and moving objects artificial origin.

The technology of trajectory tracking of targets by the radiations of their radio transmitters, using the reflections naturally occurring during the emission of radio signals from extraneous reflectors randomly distributed in space, has not yet received sufficient distribution, despite the fact that it is feasible with a single-position monitoring system and can significantly increase the operational efficiency of a wide class of systems and complexes of secretive remote detection, tracking and control.

The known method of spatial localization of radio transmitters [1], which consists in the fact that the antenna array receives the radio signals of the transmitters, converts the received radio signals into two-dimensional complex angular spectra of the received radio signals, the azimuth and elevation bearings of the transmitters are determined from the angular spectra, and after comparing the angular bearings β with the separation threshold transmitters to ground and air and determine the slant range R to the radio transmitters of air targets by the formula R = N / sinβ, where N is the known height from summer goals.

This method provides spatial localization (determination of spatial coordinates and trajectory tracking) of airborne targets radio transmitters with a one-position control system. However, this method requires a priori information about the height of the transmitter and in its absence loses its effectiveness.

A known method of spatial localization of radio transmitters [2], which consists in the fact that

receive, at a given frequency of reception by the array of antennas, a direct radio signal of the transmitter and copies of this radio signal reflected from extraneous reflectors,

synchronously transform the ensemble of radio signals received by the antennas into digital signals,

digital signals are converted into phased signals for selected azimuthal elevation directions of reception,

phased signals are divided into direct and reflected signals, remember the direction of their reception,

direct and reflected signals determine and remember the time delay of each reflected signal relative to the direct signal,

the delays calculate the path length differences of the direct and each reflected signal,

according to the directions of receiving the reflected signals and the calculated differences of the path lengths, the values of the hypothetical coordinates of the reflectors are calculated depending on the values of the hypothetical range to the radio transmitter,

comparing the values of hypothetical and previously measured coordinates of the reflectors,

when the hypothetical and previously measured coordinates of the reflectors coincide with the given accuracy, the most probable range value to the radio transmitter is recorded,

by the value of the direction of reception of the direct signal and the most probable value of the range, the spatial coordinates of the radio transmitter are found.

The prototype method does not require a priori information about the height of the radio transmitter and ensures its spatial localization by a one-position control system.

However, the prototype method has the following disadvantages that reduce its effectiveness:

1) performs hypothesis testing operations on the values of the spatial coordinates of the reflectors, requiring a priori information on the spatial coordinates of the possible reflectors of radio signals in the area where the one-position monitoring system is located and, as a result:

a) it is difficult to implement in stationary complexes of covert monitoring, since it requires the operation of the complex to take long (several years) and laborious (using flight and lifting means) operations to select possible reflectors of radio signals and measure their spatial coordinates.

In this case, the preliminary selection of reflectors involves measuring the bistatic effective scattering area of each of the possible reflectors at each of the many possible operating frequencies of the monitoring system and at each of the possible angular reception directions in azimuth and elevation. For example, under the most typical conditions, when measuring at each of 1000 frequencies and the number of discrete values in azimuth of 360, and in elevation angle 90, we can find that the total number of measurements can reach very large values of 1000 × 360 × 90 = 32400000. If we assume that 100 measurements are performed during the day, then the total duration of the measurement of the spatial coordinates of possible reflectors of radio signals in the area where the one-position monitoring system is located can reach 324,000 days;

b) we will not carry out covert control in mobile complexes, as a rule, designed for multiple operational relocation to new positions and, as a result, requiring a minimum deployment time;

2) does not contain operations for determining the velocity vector of a radio transmitter, which indicates the limitations of its information content.

The technical result of the invention is to increase the information content (determination of the velocity vector in addition to spatial coordinates) and the speed of spatial localization of a wide class of radio transmitters with a one-position monitoring system under the conditions of a priori uncertainty in the shape, size, reflective properties and spatial coordinates of extraneous reflectors of radio signals.

The increase in information content and efficiency is achieved by expanding the range of measured parameters of the reflected signals (time delays and Doppler frequency shifts instead of time delays) and performing hypothesis testing operations on the values of the Doppler shifts of the reflected signals instead of testing the hypothesis on the values of the spatial coordinates of reflectors, which are difficult to implement in stationary and practically not feasible in mobile systems of covert control

- номер отражателя, по значениям задержек и направлений приема вычисляют гипотетические значения дальности до каждого отражателя

в зависимости от гипотетических значений дальности до передатчика

где h - текущий номер гипотетического значения дальности, с - скорость света, e_t и e_i - соответствующие запомненным направлениям приема прямого сигнала передатчика и сигнала i-го отражателя единичные векторы-пеленги, по гипотетическим значениям дальностей и единичным вектор-пеленгам находят и фиксируют гипотетические координаты отражателей и передатчика, формируют и запоминают матрицу Q^h, элементы которой

где m=1, 2, 3, e_m - единичные векторы осей декартовой системы координат, с точностью до множителя, равного обратной длине волны λ на частоте приема, являются проекциями на оси декартовой системы координат суммы направлений из точки приема на гипотетическое положение передатчика и из гипотетического положения передатчика на каждую из точек гипотетических положений отражателей, из запомненных значений доплеровских сдвигов частоты ω_i формируют и запоминают вектор-столбец ω измеренных доплеровских сдвигов частоты отраженных сигналов, находят вектор-столбец гипотетической скорости передатчика

где (Q^h)^H - матрица, эрмитово сопряженная с Q^h, который запоминают и преобразуют в вектор-столбец гипотетических доплеровских сдвигов частоты отраженных сигналов ω^h=Q^hv^h, для каждого значения гипотетической дальности до передатчика

вычисляют невязку между вектор-столбцами гипотетических ω^h и измеренных ω доплеровских сдвигов отраженных сигналов по формуле

где ω^H - вектор-столбец, эрмитово сопряженный с ω, по глобальному минимуму невязки определяют дальность

до радиопередатчика, по которой находят вектор скорости

и пространственные координаты

где r₀ - радиус-вектор положения однопозиционной системы контроля в декартовой системе координат, радиопередатчика.The technical result is achieved by the fact that in the method of spatial localization of radio transmitters, which consists in the fact that a direct radio signal of the transmitter and copies of this radio signal reflected from extraneous reflectors are received at a given antenna receiving frequency, the ensemble of radio signals received by the antennas is simultaneously converted into digital signals, digital signals are converted into phased signals for the selected azimuthal elevation directions of reception, phased signals are divided into direct and reflected signals, to which, together with the selected reception directions, store, according to the invention, the direct and reflected signals are converted into signals of complex frequency-time images for each direction of reception of the reflected signals, the number M of reflectors, the Doppler frequency shift ω _i and the delay according to time-frequency images are determined and stored time τ _{i of the} signal of each reflector relative to the direct signal, where

- reflector number, according to the values of delays and directions of reception, hypothetical values of the distance to each reflector are calculated

depending on hypothetical values of the distance to the transmitter

where h is the current number of the hypothetical value of the range, c is the speed of light, e _t and e _i are the unit bearing vectors corresponding to the remembered directions of receiving the direct signal of the transmitter and the signal of the i-th reflector, find and fix by the hypothetical range values and unit bearing vectors the hypothetical coordinates of the reflectors and the transmitter form and store the matrix Q ^h , the elements of which

where m = 1, 2, 3, e _m are the unit vectors of the axes of the Cartesian coordinate system, up to a factor equal to the inverse wavelength λ at the reception frequency, are the projections on the axis of the Cartesian coordinate system of the sum of directions from the reception point to the hypothetical transmitter position and from the hypothetical position of the transmitter to each of the points of the hypothetical positions of the reflectors, from the stored values of the Doppler frequency shifts ω _i form and store the column vector ω of the measured Doppler frequency shifts of the reflected signals, find column vector of hypothetical transmitter speed

where (Q ^h ) ^H is a Hermitian conjugate matrix with Q ^h , which is stored and converted into a column vector of hypothetical Doppler shifts of the frequency of the reflected signals ω ^h = Q ^h v ^h , for each value of the hypothetical range to the transmitter

calculate the residual between the column vectors of the hypothetical ω ^h and the measured ω Doppler shifts of the reflected signals by the formula

where ω ^H is the column vector Hermitian conjugate to ω, the distance is determined from the global minimum of the residual

to the radio transmitter by which the velocity vector is found

and spatial coordinates

where r ₀ is the radius vector of the position of the one-position control system in the Cartesian coordinate system, the radio transmitter.

The operation of the method is illustrated by drawings:

FIG. 1. The structural diagram of a device that implements the proposed calibration method.

FIG. 2. The mutual arrangement of the one-position control system, radio transmitter and reflector in the Cartesian coordinate system.

The device (Fig. 1), which implements the proposed method, contains series-connected reception and preprocessing systems 1 and computing system 2.

System 1 includes an antenna array 1-1, a radio signal receiving path including a frequency converter 1-2, an ADC 1-3 and a spatial filtering device 1-4.

System 2 has an output intended for connection to external systems. In addition, the system 2 has a control output for tuning to a given frequency of reception of the frequency converter 1-2 (to simplify the control output in Fig. 1 is not shown).

System 1 is an analog-to-digital device and is designed to receive at a given frequency the operating frequency range and spatial filtering of the direct radio signal of the transmitter and copies of this radio signal reflected from extraneous reflectors.

Каждая антенна обеспечивает прием прямых и отраженных радиосигналов.Antenna array 1-1 consists of N antennas with numbers

Each antenna provides direct and reflected radio signals.

The spatial configuration of the antenna array can be arbitrary: planar rectangular, planar annular or surround, in particular conformal.

Frequency converter 1-2 is an N-channel, made with a common local oscillator and with the bandwidth of each channel, which is changed in accordance with the width of the spectrum of the received radio signal. A common local oscillator provides multi-channel coherent signal reception.

ADC 1-3 is also N-channel and is synchronized by the signal of one reference oscillator (for simplicity, the reference oscillator is not shown in Fig. 1). If the resolution and speed of the ADC are sufficient for direct analog-to-digital conversion of the input signals, then frequency selective bandpass filters and amplifiers can be used instead of the frequency converter 1-2. In addition, the frequency converter 1-2 provides the connection of one of the antennas instead of all the antennas of the array for periodic calibration of the receiving channels using an external signal source. Calibration is possible using an internal generator, the output of which is also connected instead of all antennas for periodic channel calibration (to simplify, the internal generator is not shown in Fig. 1).

The device 1-4 is a computing device that provides, for example, the formation of a classical beam-forming method or adaptive spatial filtering method [3], phased signals for selected azimuth-elevation directions of reception and their subsequent separation into direct and reflected radio signals.

Computing system 2 is intended for generating signals of complex frequency-time images of radio signals scattered by extraneous reflectors in the analyzed area of Doppler frequencies and time delays, determining the number of reflected signals, their Doppler frequency shifts and time delays relative to the direct signal, as well as calculating coordinates and velocity vector controlled radio transmitter.

The device operates as follows.

In system 2, based on data from external systems, the parameters of radio transmitters subject to spatial localization are periodically updated.

The parameters of the radio transmitters (carrier frequency and the width of the spectrum of the radio signal) are stored in system 2, and are also used to configure the converter 1-2. To simplify the control circuit of the converter are not shown.

The frequency converter 1-2 according to the signals of system 2 is tuned to a given frequency of reception.

The multi-beam electromagnetic field of the direct and scattered radio signals received by each antenna with array number 1 of the array 1-1 in the form of time-dependent radio signals s _n (t) is fed to the inputs of the frequency converter 1-2.

In the frequency converter 1-2, each received radio signal s _n (t) is filtered by frequency and transferred to a lower frequency.

где

- номер временного отсчета сигнала, которые поступают в устройство 1-4, где запоминаются.The ensemble of radio signals formed in the converter 1-2 is synchronously converted using digital-to-analog converters 1-3 into digital signals

Where

- the number of time reference signals that enter the device 1-4, where they are remembered.

In device 1-4, digital signals are converted into direct and scattered signals for selected azimuthal elevation directions of reception.

The conversion of digital signals into a direct signal and scattered signals for the selected azimuth-elevation directions of reception is carried out by a known method of classical beam formation or adaptive spatial filtering [3].

размером N×Z. Из матричного цифрового сигнала S формируется сигнал пространственной корреляционной матрицы R размером N×N. Сигнал корреляционной матрицы R преобразуется в сигнал оптимального весового вектора

размером N×1 для формирования фазированного сигнала в выбранном азимутально-угломестном направлении приема, где η - вектор наведения размером N×1, определяемый выбранным азимутально-угломестным направлением приема радиосигнала, длиной волны (частотой) и геометрией решетки.For example, when using the adaptive spatial filtering method, the digital signals of individual antennas s _n are combined into a matrix digital signal

size N × Z. From the matrix digital signal S, a signal of a spatial correlation matrix R of size N × N is formed. The signal of the correlation matrix R is converted into a signal of the optimal weight vector

size N × 1 for the formation of a phased signal in the selected azimuthal elevation direction of reception, where η is the guidance vector of size N × 1 determined by the selected azimuthal elevation direction of reception of the radio signal, wavelength (frequency) and lattice geometry.

где

- символ эрмитова сопряжения. Фазированный сигнал является векторным сигналом

включающим Z временных отсчетов.After that, the matrix digital signal S is converted into a phased signal for the selected azimuthal elevation direction of reception

Where

- a symbol of Hermitian conjugation. A phased signal is a vector signal

including Z time samples.

Similarly, phased signals are generated for all selected azimuthal elevation directions of reception.

Physically described adaptive spatial filtering operations provide simultaneous directional reception of useful signals in selected reception directions. At the same time, spatial filtering of signals from each selected direction is provided with simultaneous suppression of a wide class of interference coming from other directions.

The obtained phased signals together with the values of the azimuthal elevation directions of reception are received in the computing system 2, where they are stored.

In computing system 2, the following actions are performed:

- phased signals are divided into direct and reflected signals, which are stored together with the selected directions of reception.

When splitting the signals, the following actions are performed:

the values of the modules of the phased signals s are compared with the threshold, and when the threshold is exceeded, a decision is made to detect the direct signal, or otherwise, the reflected signal, in the selected azimuth-elevation direction.

The threshold is selected based on minimizing the likelihood of confusion between the direct and reflected signals;

- direct and reflected signals are converted into signals of complex time-frequency images for each direction of reception of reflected signals.

The conversion of direct and reflected signals into signals of complex time-frequency images is possible in various ways: the classical method of cross-correlation [4] or modern iterative methods [5-7].

For example, when using the method [4] for each direction of receiving reflected signals, a complex two-dimensional cross-correlation function between direct and reflected signals is formed depending on the time and frequency shifts.

The module of the complex two-dimensional cross-correlation function describes the time-frequency image of the energy distribution of the reflected signals in the analyzed region of delays and Doppler frequencies and allows you to determine the number of reflectors, Doppler frequency shift and time delay of the signal of each reflector.

Iterative methods [5-7] due to additional operations of non-linear signal processing provide the formation of frequency-time images of the energy distribution of the reflected signals in the analyzed region of delays and Doppler frequencies with a high dynamic range and resolution;

- номер отражателя.- from the time-frequency images, the number M of reflectors, the Doppler frequency shift ω _i and the time delay τ _{i of the} signal of each i-th reflector relative to the direct signal are determined and stored, where

- reflector number.

The described operations of generating signals of two-dimensional complex time-frequency images in given reception directions are key for increasing information content and efficiency, since they allow us to describe the distribution of reflected radio signals not only in the field of time delays, but also in the field of Doppler shifts.

As a result of these operations, it is possible to increase the number of measured and modeled parameters in the form of relative time delays and Doppler frequency shifts of the reflector signals, instead of relative delays.

At the subsequent stages of signal processing, this opens up the possibility of determining not only the coordinates of the transmitter, but also its velocity vector.

Due to this, the informativeness of the spatial localization of a wide class of radio transmitters with a one-position control system is increased. Moreover, an a priori knowledge of the coordinates of the reflectors is not required, that is, the second main drawback of the prototype method is eliminated - the complexity of feasibility in stationary and the practical impracticability in mobile complexes of covert control.

In addition, in computing system 2, the following actions are performed:

в зависимости от гипотетических значений дальности до передатчика

где h - текущий номер гипотетического значения дальности, с - скорость света, e_t и e_i - соответствующие запомненным направлениям приема прямого сигнала передатчика и сигнала i-го отражателя единичные векторы-пеленги;- based on the values of the delays τ _i and the directions of reception, hypothetical values of the distance to each i-th reflector are calculated

depending on hypothetical values of the distance to the transmitter

where h is the current number of the hypothetical range value, c is the speed of light, e _t and e _i are the unit bearing vectors corresponding to the remembered directions of receiving the direct signal of the transmitter and the signal of the i-th reflector;

и единичным вектор-пеленгам e_i и e_t находятся и фиксируются гипотетические координаты отражателей и передатчика.- by hypothetical range values

and unit vector bearings e _i and e _t are found and fixed hypothetical coordinates of the reflectors and the transmitter.

и

где r₀ - радиус-вектор положения однопозиционной системы контроля в декартовой системе координат;The hypothetical coordinates of the reflectors and the transmitter are calculated using the following formulas:

and

where r ₀ is the radius vector of the position of the one-position control system in the Cartesian coordinate system;

где m=1, 2, 3, e_m - единичные векторы осей декартовой системы координат, с точностью до множителя, равного обратной длине волны λ на частоте приема, являются проекциями на оси декартовой системы координат суммы направлений из точки приема с радиус-вектором r₀ на гипотетическое положение передатчика с радиус-вектором r^h и из гипотетического положения передатчика на каждую из точек с радиус-вектором

гипотетических положений отражателей (фиг. 2);- the matrix Q ^h is formed and stored, the elements of which

where m = 1, 2, 3, e _m are the unit vectors of the axes of the Cartesian coordinate system, up to a factor equal to the inverse wavelength λ at the receiving frequency, are the projections on the axis of the Cartesian coordinate system of the sum of directions from the receiving point with the radius vector r ₀ to the hypothetical position of the transmitter with the radius vector r ^h and from the hypothetical position of the transmitter to each of the points with the radius vector

hypothetical positions of the reflectors (Fig. 2);

- from the stored values of the Doppler frequency shifts ω _i , a column vector ω of the measured Doppler frequency shifts of the reflected signals is generated and stored;

где (Q^h)^H - матрица, эрмитово сопряженная с матрицей Q^h.- there is a column vector of the hypothetical transmitter speed

where (Q ^h ) ^H is the Hermitian conjugate matrix of Q ^h .

может быть получена из переопределенной системы Q^hv^h=ω, включающей М уравнений относительно трех компонент вектора-столбца гипотетической скорости передатчика v^h;Note that the formula

can be obtained from the overdetermined system Q ^h v ^h = ω, including M equations for the three components of the column vector of the hypothetical transmitter speed v ^h ;

запоминается и преобразуется в вектор-столбец гипотетических доплеровских сдвигов частоты отраженных сигналов ω^h=Q^hv^h;- column vector of hypothetical transmitter speed

is stored and converted into a column vector of hypothetical Doppler shifts of the frequency of the reflected signals ω ^h = Q ^h v ^h ;

вычисляется невязка между вектор-столбцами гипотетических ω^h и измеренных ω доплеровских сдвигов отраженных сигналов по формуле

где ω^H - вектор-столбец, эрмитово сопряженный с вектор-столбцом ω;- for each value of the hypothetical range to the transmitter

the discrepancy between the vector columns of the hypothetical ω ^h and the measured ω Doppler shifts of the reflected signals is calculated by the formula

where ω ^H is the column vector, Hermitian conjugate to the column vector ω;

до радиопередатчика, по которой находят вектор скорости

и пространственные координаты

где r₀ - радиус-вектор положения однопозиционной системы контроля в декартовой системе координат, радиопередатчика.- the global minimum of residuals determines the range

to the radio transmitter by which the velocity vector is found

and spatial coordinates

where r ₀ is the radius vector of the position of the one-position control system in the Cartesian coordinate system, the radio transmitter.

находятся элементы

матрицы Q, которая подставляется в формулу для вычисления вектора скорости

где (Q)^H - матрица, эрмитово сопряженная с матрицей Q.When determining the velocity vector v of the radio transmitter from the resulting range estimate

there are elements

matrix Q, which is substituted into the formula for calculating the velocity vector

where (Q) ^H is a Hermitian conjugate matrix with Q.

From the above description it follows that the device that implements the proposed method provides an increase in information content (determination of the velocity vector in addition to spatial coordinates) and the efficiency of spatial localization of a wide class of radio transmitters by a one-position control system under the conditions of a priori uncertainty in the shape, size, reflective properties and spatial coordinates of extraneous reflectors radio signals.

Thus, by expanding the range of measured parameters of the reflected signals (time delays and Doppler frequency shifts instead of time delays) and performing hypothesis testing operations on the values of the Doppler shifts of the reflected signals instead of testing the hypothesis on the spatial coordinates of the reflectors, which are difficult to implement in stationary and practical not feasible in mobile complexes of covert control, it is possible to solve the problem with the achievement of the specified nical result.

Sources of information:

1. RU, patent, 2158002, cl. G01S 13/14, 2000

2. RU, patent, 2457505 C2, class. G01S 5/04 (2006.01), 2012

3. Ratynsky M.V. Adaptation and superresolution in antenna arrays. M .: Radio and communication. 2003 year

4. Shirman Y.D., Manzhos V.N. The theory and technique of processing radar information against the background of interference. - M.: Radio and Communications, 1981.

5. RU, patent, 2521608 C1, cl. G01S 13/02 (2006.01), 2014

6. RU, patent, 2524401 C1, cl. G01S 13/02 (2006.01), 2014

7. RU, patent, 2557250 C1, cl. G01S 13/02 (2006.01), 2015

Claims (1)

The method of spatial localization of radio transmitters, which consists in receiving a direct transmitter radio signal and copies of this radio signal reflected from extraneous reflectors at a given antenna receiving frequency, synchronously converts the ensemble of radio signals received by the antennas into digital signals, digital signals are converted into phased signals for the selected azimuth-angle signals directions of reception, phased signals are divided into direct and reflected signals, which together with the selected directions n The reception is remembered, characterized in that the direct and reflected signals are converted into signals of complex time-frequency images for each direction of reception of the reflected signals, the number M of reflectors, the Doppler frequency shift ω _i and the time delay τ _{i of the} signal are determined and stored on the time-frequency images each reflector relative to the direct signal, where

- reflector number, according to the values of delays and directions of reception, hypothetical values of the distance to each reflector are calculated

depending on hypothetical values of the distance to the transmitter

where h is the current number of the hypothetical value of the range, c is the speed of light, e _t and e _i are the unit bearing vectors corresponding to the remembered directions of receiving the direct signal of the transmitter and the signal of the i-th reflector, find and fix by the hypothetical range values and unit bearing vectors the hypothetical coordinates of the reflectors and the transmitter form and store the matrix Q ^h , the elements of which

where m = 1, 2, 3, e _m are the unit vectors of the axes of the Cartesian coordinate system, up to a factor equal to the inverse wavelength λ at the reception frequency, are the projections on the axis of the Cartesian coordinate system of the sum of directions from the reception point to the hypothetical transmitter position and from the hypothetical position of the transmitter to each of the points of the hypothetical positions of the reflectors, from the stored values of the Doppler frequency shifts ω _i form and store the column vector ω of the measured Doppler frequency shifts of the reflected signals, find column vector of hypothetical transmitter speed

Where

- a Hermitian conjugate matrix with Q ^h , which is stored and converted into a column vector of hypothetical Doppler frequency shifts of the reflected signals

for each hypothetical transmitter distance value

calculate the residual between the column vectors of the hypothetical ω ^h and the measured ω Doppler shifts of the reflected signals by the formula

where ω ^H is the column vector Hermitian conjugate to ω, the distance is determined from the global minimum of the residual

to the radio transmitter by which the velocity vector is found

and spatial coordinates

where r ₀ is the radius vector of the position of the one-position control system in the Cartesian coordinate system, the radio transmitter.

Images