RU2380719C2 - Method for location finding of radiation sources at one frequency - Google Patents

Abstract

FIELD: radio engineering.

SUBSTANCE: improved efficiency is achieved by application of efficient method for detection of radio signal parametres, which are represented in the form of exponential functions, which comprise bearing functions. Besides proposed method excludes mathematical operations, which require significant computer costs, such as Fourier transform. Additionally, for reliable location finding, interval assessments of radiation source bearings and signal amplitudes are defined on the basis of according dispersions.

EFFECT: increased efficiency of signals processing during location finding of radio signals from several sources of radiation that operate at one frequency, with application of antenna systems, comprising weakly directional elements, and provision of interval assessments of bearings.

2 cl

Description

Technical field

The invention relates to radio engineering, in particular to direction finding, and can be used in systems for determining the direction of radio sources operating at the same frequency.

State of the art

Direction finding of several sources of radio emission (IRI) takes place in the process of monitoring the radio-electronic situation in the case of multipath propagation of radio waves, exposure to deliberate and unintentional interference, signal reflections from various objects and layers of the atmosphere.

The direction finding problem in the general case is incorrect in mathematical terms. Most known methods of multi-signal direction finding at a single frequency rely on statistical methods for testing hypotheses, on the maximum likelihood method, on superresolving methods (for example, MUSIC - multiple signal classification), etc. However, the IR direction finding problem, as an incorrect task, cannot be solved reliably either statistical methods, the reliability of the result of which is determined by the accuracy of the obtained estimates of signal parameters; nor by the least squares method (OLS) due to nonlinearity and poor conditionality of the system of equations being solved; nor by superresolving methods that give acceptable results only at high signal-to-noise ratios and do not provide the resolution of IRIs with similar bearings.

A known method of direction finding [1], adopted as a prototype, which is carried out as follows.

1. The radio signals of the sources are received through the antenna system (AC), consisting of M vibrators.

2. Get the complex amplitudes of the signals at the outputs of the antennas (amplitude-phase distribution vector (AFR)). The mth element of the AFR vector has the form

where K is the number of radio signal sources (IRI), u _k is the amplitude of the signal of the kth IRI, φ _m (θ _k , β _k ) is the phase of the signal of the kth IRI on the mth vibrator, depending on the azimuthal and elevation bearings k- Iran θ _k and β _k, respectively, n _m is the noise occurring on the m-th vibrator, including the noise of the world background and equipment.

3. Get the average antenna power of the received radio signals and the angular spectrum of the first order, respectively:

,

,

where * is the designation of the complex conjugation operation. The power value P and each value of the angular spectrum of the first order (for each azimuthal elevation bearing with a resolution of 1 °) are obtained in one reading cycle, including M clocks. A complete set of values of the angular spectrum of the first order is formed after 360 · 90 cycles.

4. Get the square modulus of the angular spectrum of the first order, the totality of the values of which determine the coordinates and the value of its maximum, which is normalized to the power value.

и

, соответствуют азимутальному и угломестному пеленгам ИРИ с максимальной амплитудой сигнала. При отсутствии радиосигналов это положение случайно. На данной операции завершается первый этап обработки сигнала.5. If there are energy and spatial conditions for resolving the maximum coordinate

and

correspond to azimuthal and elevation bearings of the IRI with maximum signal amplitude. In the absence of radio signals, this position is accidental. In this operation, the first stage of signal processing is completed.

и

максимума углового спектра первого порядка6. The second stage with angular ranges of 0 ... 360 ° and 0 ... 90 ° and a grid resolution of 1 ° is carried out for 360 · 90 cycles and consists in the following. The values of the normalized radiation pattern of the speakers in the direction of coordinates are determined

and

maximum angular spectrum of the first order

7. Determine the angular spectrum of the second order by the formula

8. Next, determine

и

соответствуют азимутальному и угломестному пеленгам ИРИ с меньшей амплитудой радиосигнала. В отсутствие радиосигналов это положение случайно.9. The coordinates of the maximum points

and

correspond to azimuthal and elevation bearings of the IRI with a smaller amplitude of the radio signal. In the absence of radio signals, this situation is accidental.

10. The final decision on the availability of radio signals is made according to the values of the maximum squares of the angular spectrum modules. For this, the maximum values L ₁ and L ₂ are summarized and compared with the first threshold C ₁ . If the threshold is not exceeded, a decision is made about the absence of radio signals, otherwise a relation of the form is determined and compared with the second threshold C ₂

и

. Значения порогов устанавливают по критерию Неймана-Пирсона исходя из заданной вероятности ложной тревоги, взаимного расположения антенн, длины волны излучения и заданной погрешности пеленгования.If the second threshold value is exceeded, a decision is made about the presence of a radio signal of one source with azimuthal and elevation bearings

and

. The thresholds are set according to the Neumann-Pearson criterion based on the given probability of false alarm, the relative position of the antennas, the wavelength of the radiation and the given bearing error.

The specified method has the following disadvantages.

1. The main disadvantage arises from the fundamental principles of the method, which is based on probabilistic concepts and relationships. It is known that probabilistic laws are satisfied only when processing mass events, and the more accurately, the greater the number of events. Direction finding deals with single events (bearing determination operations). Thus, the results obtained in this way are quite unreliable and, therefore, practically inapplicable.

2. To obtain the result, it is necessary to perform a large number of complex mathematical operations, such as calculating the average power of radio signals, angular spectra of the first and second orders and their maxima, thresholds, etc.

3. Despite the probabilistic basis of the construction, the method does not give interval estimates of the results, which is incorrect in principle, since it is known that it is impossible to unambiguously characterize a random variable with a point estimate.

These shortcomings, apparently, will not allow the use of this method in real conditions.

Disclosure of invention

The proposed method is free from these disadvantages and is a parametric method of multi-signal direction finding at a single frequency.

The technical result achieved is a significant increase in the processing speed (speed) of signal processing during direction finding of radio signals from several sources of radio emission operating at the same frequency using antenna systems (AS) consisting of weakly directed elements (vibrators), as well as obtaining interval bearings estimates. The increase in performance is achieved through the use of an effective method for determining the parameters of radio signals, presented in the form of sums of exponential functions. Moreover, in the proposed method there are no mathematical operations that require large computational costs, such as, for example, the Fourier transform.

A method of multi-signal direction finding of radio-frequency sources at a single frequency includes receiving a multi-beam signal by means of a multi-element antenna system, synchronously converting an ensemble of received signals depending on the time and number of an element of the antenna system into digital signals, converting digital signals into an amplitude-phase distribution signal describing the distribution amplitudes and phases on the elements of the antenna system, and is characterized in that the bearings are obtained by determining the parameter in exponential functions containing bearings function.

Additionally, for reliability of direction finding, interval estimates of bearings of radio emission sources and signal amplitudes are determined based on the corresponding variances obtained as variances of the scalar function of a random argument (for bearings [3. P.388]) and variances of the elements of the solution vector system of linear algebraic equations obtained by the least squares (for amplitudes (powers) [3. P.374]).

The implementation of the invention

As a multi-element speaker system, we consider a linear system consisting of several weakly directed elements (vibrators). As the phase center (the point with respect to which the phases of the signals arriving at the elements of the antenna system are measured), we select one of the vibrators.

It is necessary to determine the following parameters of IRI present on the air:

- amplitudes (power) of the emitted signals;

- azimuthal and elevation bearings.

As a practically justified assumption for the proposed method, the signals are considered as deterministic, subject to additive interference, the parameter estimates of which are to be determined.

Since interference is inevitably imposed on the measurement results, and there are also measurement errors due to the equipment used, it is necessary to obtain not only point estimates of the desired parameters, but also their covariance matrices or, at least, the variances.

угломестными пеленгами

и амплитудами (мощностями) излучаемых сигналов

- сигнал комплексного амплитудно-фазового распределения, описывающий амплитуды и фазы сигналов, принятых элементами АС, где М - количество элементов АС. Используемый вид модуляции (амплитудная, частотная, фазовая и др.) не имеет принципиального значения.We believe that on the air there are K pieces of IRI with azimuth bearings

elevation bearings

and amplitudes (powers) of the emitted signals

- a signal of a complex amplitude-phase distribution that describes the amplitudes and phases of the signals received by the elements of the AC, where M is the number of elements of the AC. The type of modulation used (amplitude, frequency, phase, etc.) does not matter.

In the general case, the problem has the form of a system of nonlinear equations with respect to the unknown θ, β, and u:

where the matrix A (θ, β, t) is formed taking into account the type of direction-finding signals of the IRI and the spatial configuration AC, n (t) is the additive interference vector with zero mathematical expectation and the covariance matrix of the form σ ² I, I is the identity matrix, σ is the mean square deviation (RMS).

For a linear AC with a phase center located on the extreme vibrator, the elements of the matrix A (θ, β, t) have the form

m = 1, 2, ... M; k = 1, 2, ..., K,

Where:

j - imaginary unit

f ₀ - the frequency of the signals emitted by direction finding IRI,

φ _k is the initial phase of the k-th signal,

R is the radius of the circle along which the elements of the antenna system are located,

λ is the wavelength of the IRI signals,

d is the distance between adjacent elements of the antenna system,

γ _i , i = 1; 2; ...;

M is the angle between the reference line of bearings and the line drawn through the center of the circle and the i-th element of the antenna system (for a circular speaker),

M is the number of elements in the antenna system.

In nonlinear system (1), it is required to determine the amplitude (power) u _k , azimuth bearing θ _k and elevation bearing β _k for each of the signals simultaneously arriving at the AS.

To significantly simplify the solution of the problem, we reduce the definition of bearings to the determination of the exponents of the sums of exponential functions. This method leads to the estimation of bearings by solving algebraic equations (square, cubic, fourth degree, etc.) by introducing new variables ξ _i , i = 1; 2; ...; K. If the step of setting these variables is uniform (which is preferable), then we obtain a complete algebraic equation (linear AC), and if the step for ξ _i is uneven, we obtain an incomplete algebraic equation or an equation with fractional exponents (non-linear AC). The roots of the obtained algebraic equation are the values of ξ _i .

Consider the case when on the air there are K pieces of IRI. The output of the m-th element of the linear AC has the form

where u _i is the amplitude of the i-th signal on the m-th vibrator.

i=1; 2; …; K. Тогда можно записать следующую систему уравненийWe introduce the notation of an exponential function

i = 1; 2; ...; K. Then we can write the following system of equations

We introduce the polynomial of exponential functions

To determine the coefficients C _i , i = 0; one; ...; K-1, we construct a system of linear algebraic equations (SLAE) with K unknowns. Multiply the first K + 1 equations of system (4) by C ₀ , C ₁ , ..., C _K-1 and 1, respectively, and add them

At the next shift, we multiply K + 1 equations by C ₀ , C ₁ , ..., C _K-1 and 1, respectively, starting from the 2nd equation. As a result, we get:

Continuing this process, we obtain SLAE with respect to the coefficients of the polynomial C ₀ , C ₁ , C ₂ , ..., C _K-1 .

We pay attention to the construction of system (7). The matrix elements of its left side are obtained by shifting the values of the measured amplitudes (Toeplitz matrix). This kind of system matrix allows you to quickly get a solution (effective methods have been developed for inverting Toeplitz matrices, for example, in [2]). Moreover, the quantities y _i (u, θ, β), i = 1; 2; ...; M are known (this is the AFR vector obtained on the basis of voltage measurements at the outputs of the AC elements).

In the general case, system (7) is poorly conditioned. The value of the condition number N depends on the difference in the values of ξ _i , i = 1; 2; ...; K and interference intensity.

This system can be redefined if the values y _i , i> M are recorded, preferably up to i = kM, k = 2; 3; ... For the direction finding problem, each equation of system (4) is a record of a signal from one vibrator.

You can increase the number of equations in the system by taking into account the values of the AFR vector obtained at different instants of time.

Having determined from system (7) С ₀ , С ₁ , ..., С _K-1 , we obtain the polynomial of exponential functions P (ξ) in explicit form. Next, we find the roots of the polynomial (parameters) of the exponential functions ξ _i , i = 1; 2; ...; K.

Elements of the SLAE matrix (7) were obtained as a result of measurements, i.e. include some error (are random variables). Exact formulas for calculating the covariance matrix of decision estimates are cumbersome. Here is a simplified formula. Let the standard deviation (RMS) of the elements of the matrix of system (7) A _C be equal to σ _y . We use the properties of the dispersion operator to transfer it from the left side of the SLAE (7) to the right. Thus, the formula for calculating the covariance matrix of the solution, taking into account the error of the elements of the matrix of the system, has the form

- СКО элементов вектора комплексной огибающей выходов элементов линейной АС,Where

- standard deviation of the elements of the vector of the complex envelope of the outputs of the elements of the linear AC,

- матрица СЛАУ (7),

- matrix SLAE (7),

C = [C ₀ ... C _K-1 ] ^T.

This method of calculating the covariance matrix of solution estimates is valid only if the solution to SLAE (7) is carried out by the least squares method (LSM). If SLAE is solved (7), for example, using one of the regularization methods, then the covariance matrix of the solution can be calculated as the inverse matrix of the second derivatives of the likelihood function, which in this case is constructed for a given regularizing functional.

The roots ξ _{i of the} resulting polynomial of exponential functions can be written as functions of random variables C _i , i = 0; 1; ...; K-1, and calculate the variances of root values as for functions of random arguments. Knowing ξ _i , we find the value of the product cosθ _i cosβ _i , i = 1; 2; ...; K by the formula

If β _i = 0, then the value of θ _i can be calculated by the formula

Where

In the case where β ≠ 0, simple trigonometric formulas can be used to calculate azimuthal and elevation bearings.

We find the amplitudes for each signal by substituting the found values of ξ _i in system (4) and solving it. Regarding unknown amplitudes, system (4) is also a SLAE. Knowing the analytical expressions for bearings and amplitudes, it is not difficult to determine their dispersion.

Based on the formula (9), it can be argued that θ _i = θ _i (C ₀ , C ₁ , C ₂ ), i = 1; 2; 3. To find the variance of bearings θ _{i we} use the following formula [3, p. 388]:

- элемент Where

- element

matrices D _c with index (i, j).

If we solve system (4) with respect to the amplitudes by the least squares method, then the covariance matrix of the solution D _u can be found by a formula similar to (8)

- СКО элементов матрицы А_ξ,Where

- the standard deviation of the elements of the matrix A _ξ ,

u = [u ₁ ... u _K ] ^T ,

- матрица СЛАУ (4).

- matrix SLAE (4).

Based on the obtained dispersions, the corresponding confidence intervals are built.

. В идеальном случае (помеха отсутствует) количество сигналов равно количеству ненулевых собственных чисел. Под «большими» понимаются числа, превышающие по величине некоторую заранее заданную на этапе калибровки пеленгатора величину (порог), значение которой зависит от соотношения сигнал/шум.To determine the number of signals received by the speakers, we formulate the following criterion. The number of signals received by the speaker is equal to the number of "large" eigenvalues of the matrix

. In the ideal case (no interference), the number of signals is equal to the number of nonzero eigenvalues. By "large" we mean numbers that exceed a certain value (threshold) predetermined at the stage of direction finder calibration, the value of which depends on the signal-to-noise ratio.

The increase in performance is achieved due to the lack of mathematical operations in the proposed method that require large computational costs, such as, for example, the Fourier transform.

The solution of SLAE (7) is carried out by the least squares method. Therefore, the solution vector is determined by the formula

Where A _with - matrix of system (7),

y _c is the vector of the right-hand side of system (7).

In modern computing systems, the operations of matrix multiplication and inversion are performed in fractions of a second. In addition, as mentioned above, the matrix A _c is the Toeplitz matrix, and effective methods have been developed [2] to reduce the computational cost by several times to solve the least-squares problem for SLAEs with Toeplitz matrices.

In practical terms, the method is as follows.

1. The radio signals of the sources are received through the antenna system (AC), consisting of M vibrators.

2. Get the complex amplitudes of the signals at the outputs of the antennas (amplitude-phase distribution vector (AFR)). The mth element of the AFR vector has the form

- фаза сигнала k-го ИРИ на m-м вибраторе, зависящая от азимутального и угломестного пеленгов k-го ИРИ θ_k и β_k соответственно, n_m - шум, имеющий место на m-м вибраторе, включающий в себя шум мирового фона и аппаратуры.where K is the number of radio signal sources (IRI), u _k is the signal amplitude of the k-th IRI,

is the phase of the signal of the kth IRI on the mth vibrator, depending on the azimuthal and elevation bearings of the kth IRI θ _k and β _k, respectively, n _m is the noise occurring on the mth vibrator, including the noise of the world background and equipment.

3. Form and solve the system (7), remember the resulting coefficient vector of the polynomial C.

запоминают их.4. Find the roots (parameters) of exponential functions ξ _i , i = 1; 2; ...; K polynomial of exponential functions

remember them.

5. Using the formula (9b), find the azimuthal bearings of the sources of radio emission (if the elevation bearings are equal to zero) or (if the elevation bearings are not equal to zero) using the formula (9a) find the values of the cosines of the azimuthal and elevation bearings, and then find the values using simple trigonometric formulas azimuthal and elevation bearings, remember the values of bearings.

6. Solve system (4) with respect to the amplitudes of the signals and find their values.

7. Find the variance of the coefficients of the polynomial C by the formula (8).

8. Find the dispersion of bearings by the formula (10).

9. Find the variance of the amplitudes according to the formula (11).

10. Based on the variances obtained in clauses 8 and 9, the corresponding confidence intervals are constructed [3, p. 388].

11. Visualize the results.

Here is a model example of the implementation of the method. Consider the direction finding of two IRI operating at a frequency of 20 MHz. Bearings

, углы места

и амплитуды u=[10 мВ 1 мВ]^T. Помеха имеет математическое ожидание, равное нулю, и СКО σ=0,1 мВ. Пеленгацию будем осуществлять в диапазоне пеленгов [0°; 180°], посредством линейной АС, состоящей из 16 вибраторов, отстоящих друг от друга на расстояние, равное половине длины волны сигнала (7,5 м).

elevation angles

and amplitudes u = [10 mV 1 mV] ^T. The interference has a mathematical expectation of zero, and the standard deviation σ = 0.1 mV. Direction finding will be carried out in the range of bearings [0 °; 180 °], by means of a linear speaker system consisting of 16 vibrators spaced apart by a distance equal to half the wavelength of the signal (7.5 m).

Point estimates of IRI parameters:

RMSE of the estimates obtained by formulas (10), (11):

(

(

RMSE estimates obtained by statistical tests in 1000 experiments:

(

(

On a computer with a processor with a clock frequency of 2 GHz, the time taken to calculate point estimates is 0.015 s. The time spent on calculating the standard deviation of estimates by formulas (10), (11) is 0.31 s. For comparison, in practical applications, it is necessary to ensure the time for obtaining bearing estimates within 1 s. The calculations were carried out in the environment of Matlab 7.0, which is not optimized for speed.

Thus, from the above example, the advantages of the proposed method in terms of improving performance and obtaining reliable interval estimates of the results are clearly visible.

Information sources

1. RF patent No. 2289146, IPC G01S 3/00, G01S 5/04, 2006

2. G. Rodriguez, D. Theis Least squares solution of large Toeplitz linear systems. VI Congresso SIMAI - Societa Italiana di Matematica Applicata e Industriale, Chia Laguna (Cagliari), Italy, May 27-31,2002.

3. A.A. Greshilov. Mathematical decision-making methods: Textbook. manual for universities. - M.: Publishing House of MSTU. N.E. Bauman, 2006 .-- 584 p.

Claims (2)

1. A method of multi-signal direction finding of radio-frequency sources at a single frequency, which includes receiving a multi-beam signal through a multi-element antenna system, synchronously converting an ensemble of received signals depending on the time and number of the antenna system element into digital signals, converting digital signals into an amplitude-phase distribution signal describing the distribution of amplitudes and phases on the elements of the antenna system, characterized in that of the amplitudes of the complex signals of the amplitude-phase distribution Lenia forming a system of linear algebraic equations y ₁ ... y _m, describing the amplitude of the signals received elements of the antenna system, where M - the number of elements of the antenna system, each line of the system of algebraic equations is a sum of products C _i exponential functions polynomial coefficients, where i = 1, 2, ..., K, K - number of emitters, and the amplitudes of the signals y _i, where i = 1, 2, ..., M, with each subsequent equation system formed of linear algebraic equations is shifted right n one position in relation to the previous equation, and the exponential function represents ξ _i = exp (j (2π / λ) dcosθ i sosβ _i), where i = 1, 2, ..., K, j - imaginary unit, λ - wavelength signals of radio emission sources, d is the distance between adjacent elements of the antenna array, θ _i is the azimuth bearing of the radio source, β _i is the elevation bearing of the radio source, then the roots of the polynomial of exponential functions, which are the parameters of the exponential functions, are determined using the parameters of the exponential functions using logarithm and arccosine operations determine bearings of radio emission sources. 2. The method according to claim 1, characterized in that for known values of bearings and amplitudes, interval estimates of these parameters of signal sources are determined based on variances obtained as variances of the values of scalar functions of signals ξ _i , i = 1, 2, ..., K, for bearings and as the variance of the elements of the signal vector y obtained by minimizing the square of the difference of the product of the matrix composed of the signals ξ _i by the amplitude u of the desired signal and the signal y recorded from the outputs of the elements of the antenna system for amplitudes.