RU2651640C1 - Method of the signals spline-filtering - Google Patents

Abstract

FIELD: radio engineering; electronics.

SUBSTANCE: invention relates to the field of radio electronics and hydro acoustics, namely to methods of signal detecting and processing in the presence of interferences. Invention consists in implementing of an optimal method development for the hydro acoustic signal detecting under conditions of interferences described by nonlinear stochastic equations. Method of signals spline filtering is based on the conditional Markov filtering method, which includes: solving the filtering equations for the hypothesis of signal presence / absence; likelihood equations; equation for gain ratios calculating. Distinctive feature of the spline filtering method is that, in order to obtain interference estimates for a nonlinear equation of state, the nonlinear function spline interpolation is additionally used, and the nonlinear function changes dynamic range area is divided into intervals, in each of which an equation of state linear representation is implemented, which allows to implement a Kalman-Bucy filter on each of the sub-bands, including: two equations of state estimation with hypotheses of the signal presence / absence; dispersion estimation equation on the various intervals and the likelihood equation including these estimates, based on the calculation results a decision is made on the signal detection or non-detection.

EFFECT: technical result is the reduction of computational costs, enabling high efficiency of the signal detection with various interferences, obtaining a more efficient filtering algorithm and the ability of operation in real time.

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Description

The present invention relates to the field of electronics, and in particular to methods for processing and detecting a signal against a background of interference. The problem to which the claimed invention is directed, is to implement the development of an optimal method for detecting a hydroacoustic signal under interference conditions described by nonlinear stochastic equations. The problem is solved due to the fact that in the claimed method, the nonlinear function is represented through splines, which makes it possible to carry out linear processing on each interval and present the nonlinear algorithm as a composition of a linear Kalman-Bucy filter.

Achievable technical result consists in reducing computational costs, ensuring high detection efficiency and in the versatility of processing with various interference. The ability to work in real time, to obtain a more efficient filtering algorithm that requires less computational cost, leads to a high economic benefit that can be used when implementing sonar signals in a processing system.

To date, approximate solutions of the filtration equations are based on approximating the solution — an a posteriori probability density by some function from a parameterized class, using the normal probability density [1]. However, in some cases, the posterior probability density differs significantly from normal (for example, the near-reverb process), and when large filtering errors occur (small signal-to-noise, noise-to-noise), more accurate approximations are required. From this point of view, approximations based on spline representations are especially attractive, since in their application no assumptions are made about the laws of distributions.

Of the previous methods, the conditional Markov filtering method (prototype) is known (2, p. 213-222], in which the signal-to-noise relationship is expressed by some deterministic function and the white noise is additive. The main task of the method is to have one of components of the multidimensional Markov process, calculate the probability distribution for the values of the observed component, where the signal is an arbitrary determinate function in time S (x, t) = S (t), and the noise V (η, t) = η (t) is represented by a diffusion Gaussian process m. The essence of the method of conditional Markov filtration can be expressed in the system of equations:

- оценка реверберационной помехи при гипотезе наличия сигнала;

- оценка реверберационной помехи при гипотезе отсутствия сигнала; σ^*2 - дисперсия оценки помехи; N₀ - величина спектральной плотности белого шума; γ - коэффициент диффузии.where: s (t) is the deterministic signal; u (t) is the incoming input implementation; z (t) is the logarithm of the likelihood ratio;

- evaluation of reverberation noise under the hypothesis of the presence of a signal;

- evaluation of reverberation noise in the absence of a signal hypothesis; σ ^{* 2} is the variance of the interference estimate; N ₀ is the spectral density of white noise; γ is the diffusion coefficient.

A scheme that implements the specified method of conditional Markov filtering (prototype) [2, p. 213-222], shown in FIG. 1. where:

;block 1 - gain block by coefficient

;

;block 2 - gain block by coefficient

;

blocks 3, 7, 18 - integrators:

;block 5 - gain block by coefficient

;

blocks 4, 8, 15 - quadrators;

blocks 6, 9, 16 — gain blocks by a factor of [-1];

blocks 10, 13 - gain blocks by a factor [2];

blocks 11, 14 are multipliers;

block 12 - block forming a reference signal;

;block 17 - gain block by coefficient

;

block 19 is a two-threshold device;

block 20 is a single-threshold device;

blocks 21, 24 - decision blocks under the hypothesis of the presence of a signal;

block 22 is a decision block in the absence of a signal hypothesis;

block 23 is a block that implements the continuation of the observation.

The principle of the method is as follows: under the hypotheses of the presence and absence of a signal, the estimated values of the noise filtered out from white noise are calculated, based on them form the logarithm of the likelihood ratio and then make a decision in one of two threshold schemes. The conditional Markov filtering method allows to obtain structural diagrams of optimal receivers for various interference situations, but does not evaluate the noise immunity of such receivers.

Also known is the method of piecewise decomposition of estimates (patent No. 2257610), which is based on dividing the initial discrete implementation into concealed intervals of the same length, followed by the evaluation of the useful signal (analog) on each of them [3, p. 4-10]. Based on this method, it is assumed that the useful signal is described by a piecewise-continuous smooth function that satisfies the conditions of the Weierstrass approximation theorem on local segments. This approach allows you to get many estimates of the useful component in each section of the process with their subsequent averaging. The use of the orthogonal polynomial system in solving the approximation problem allows one to obtain only a general solution to the signal estimation problem and process one-dimensional discrete implementations of signals of a limited volume under conditions of nonparametric a priori uncertainty, which makes the spline filtering method more universal in comparison with the piecewise decomposition of estimates.

In the transition from approximations by polynomials to approximation by splines, there are two goals. The first is an improvement in the quality of approximation: for identical computational costs, the absolute errors of approximation by splines are smaller than the absolute errors of approximation by polynomials, and with the same errors, the amount of computation decreases. Splines allow one to avoid oscillations [4-6]. For the approximation to converge to the approximated function, weaker requirements are imposed than in the case of polynomials. For example, interpolation by splines of low degrees converges even for continuous functions. The second goal is a sharp reduction in computational difficulties both in constructing algorithms for solving problems and in further work with approximants, which at each link are polynomials of low degrees or other elementary functions [4,5].

The asymptotically best uniform approximation by a spline of the first degree S ₁ (x) of the function ƒ (x) is determined by the expression [7]:

where: x _i , x _{i + 1} - nodal points on the interval; Δ _i is the interval with the number of section i. A spline representation of the test signal is shown in FIG. 2. where:

x ₁ , x ₂ , x ₃ - nodal points on the interval;

Δ ₁ , Δ ₂ , Δ ₃ - intervals;

r (x) is a test signal with a drift coefficient of r (x) = - 0.5x ² -x;

s (x) is the spline representation of the signal r (x).

In the framework of the present invention, in order to satisfy the requirement of probability density continuity, the operation of smoothing high-frequency components of the characteristic function by replacing high-frequency samples with zeros was used. The spectral probability densities of the test process ps (x) and its approximation by splines are shown in FIG. 3. where:

x - values of the test process;

ps (x) is the probability density of the test process;

phx (x) is the probability density of the approach of the test process by splines.

Both curves (Fig. 3) have a similar normal distribution shape with positive asymmetry coefficients: for the probability density of the test signal, it is 0.77, and for the probability density of the approach of the test process with splines, 0.75; also, both distributions have only one possible value, corresponding to the highest probability of occurrence, and are single-mode.

The initial function can be approximated in various ways, but to solve the linear filtering problem, expression (2) on the intervals [x _i , x _{i + 1} ) is considered as a linear stochastic equation. As a result, the filtering algorithm on the interval can be represented as a Kalman filter with parameters that vary depending on the number of the section to which the current process assessment belongs. It is known that in linear filtering problems there is no need to calculate the probability density, it is enough to find the estimate of the mathematical expectation of the process at each point and the variance of this estimate [5-7]. Thus, we can write the filtering equation of the random process x (t) ∈Δ _i :

where: x ^* - current assessment of the process; h is the function of the rectangular window; a _i , b _i - coefficients whose values are determined through the nodal points x _i and the values of the function ƒ (x _i ) through the system of equations:

,

, …σ_n), при котором имеет место соответствие:

, тогда система уравнений фильтрации представляется уравнениями:If we consider the entire interval of variation x∈X, then the variance of the filtering of the random process x (t) will be represented by the vector (

,

, ... σ _n ), for which there is a correspondence:

, then the system of filtration equations is represented by the equations:

. Важным фактором с вычислительной точки зрения является то, что стационарные коэффициенты усиления

могут быть рассчитаны априори и могут представляться в виде набора чисел, поэтому реализация метода фильтрации со стационарными коэффициентами требует значительно меньших объемов памяти и вычислительных затрат [5].It follows from expression (5) that in the stationary mode, the dispersion of the filtration is constant at the intervals Δ _i and changes in a stepwise manner when switching to another interval. It should be noted that at each interval, a linear Kalman-Bucy filter is implemented. Moreover, the filtering variance is a function of only the estimate

. An important factor from a computational point of view is that the stationary gains

can be calculated a priori and can be represented as a set of numbers, therefore, the implementation of the filtering method with stationary coefficients requires significantly less memory and computational costs [5].

The spline filtering method allows you to evaluate random processes defined not only by a scalar, but also by a matrix-vector equation. The advantages of the proposed approach in computational terms, in comparison with the known methods of nonlinear filtering, are more significant. In the case of solving the problem of detecting a signal against a background of noise and white noise, the detection and filtering equations have the form [4-7]:

The proposed method of spline filtering based on the prototype and the system of equations (6) is presented in FIG. 4, where:

;block 14 - gain block by coefficient

;

;block 7 - gain block by coefficient

;

blocks 8, 12, 26 - integrators;

;block 10 - gain block by coefficient

;

blocks 9, 13, 21 - quadrators;

blocks 11, 15, 22 - gain blocks by a factor of [-1];

blocks 16, 19 - gain blocks by a factor [2];

blocks 17, 20 - multipliers;

block 18 - block forming a reference signal;

;block 25 - gain block by coefficient

;

block 27 is a two-threshold device;

block 28 is a single threshold device;

blocks 29, 32 - decision blocks under the hypothesis of the presence of a signal;

block 30 - decision block in the absence of a signal hypothesis;

block 31 is a block that implements the continuation of the observation;

additionally included blocks:

block 1 - block determining the nodal points on the interval;

block 2 - block forming a system of intervals;

block 3 is a block comparing the assessment of interference with the system of intervals;

block 4 - block determining the value of the interference estimate | x ^* (1)];

block 5 - block determining the magnitude of the coefficient [a _i ];

block 6 - block determining the magnitude of the coefficient [b _i ];

block 23 - block calculating the values of the following interference estimates [x ^* (2)];

block 24 is a block for calculating the value of the next interference estimate [x ^* (3)].

, которые используются для определения следующей оценки x^*(2). С приходом отсчета реализации u₁ на основании системы уравнений [3] вычисляется следующая оценка помехи х^*(2). Полученное значение x^*(2) соотносится с множеством интервалов {Δ_i} и снова определяется тройка {a _i, b_i,

}. С учетом полученных значений коэффициентов a _i(2), b_i(2), дисперсии

и системы [3] находится следующее значение оценки помехи х^*(3). Процедуры определения коэффициентов для следующих оценок на основании системы уравнений повторяется. Получение оценок помехи при нелинейном уравнении состояния происходит при использовании сплайн-интерполяции нелинейной функции, где область динамического диапазона изменений нелинейной функции разбивается на интервалы, в каждом из которых реализуется линейное представление уравнения состояния, что позволяет на каждом из поддиапазонов реализовать фильтр Калмана-Бьюси. Фильтр включает в себя: два уравнения оценки состояния при гипотезах наличия/отсутствия сигнала; уравнения оценки дисперсии на различных интервалах и уравнения правдоподобия, включающего эти оценки, и по результатам вычисления которого выносится решение об обнаружении или необнаружении сигнала.The invention works as follows: the value of the interference estimate x ^* (1) is found. The value of the estimate x ^* (1) is related to the system {Δ _i }. As a result, the values of the coefficients a _i (1), b _i (1) and the variance are determined

which are used to determine the next estimate x ^* (2). With the arrival of the implementation count u ₁ based on the system of equations [3], the following interference estimate x ^* (2) is calculated. The obtained value x ^* (2) correlates with the set of intervals {Δ _i } and the triple { a _i , b _i , is determined again

}. Given the obtained values of the coefficients a _i (2), b _i (2), variance

and system [3], the following value of the interference estimate x ^* (3) is found. The procedure for determining the coefficients for the following estimates based on a system of equations is repeated. Obtaining noise estimates for the nonlinear equation of state occurs when spline interpolation of the nonlinear function is used, where the region of the dynamic range of changes of the nonlinear function is divided into intervals, in each of which a linear representation of the equation of state is realized, which allows implementing a Kalman-Bucy filter on each of the subbands. The filter includes: two equations for assessing the state under the hypotheses of the presence / absence of a signal; equations of variance estimation at various intervals; and likelihood equations that include these estimates, and the calculation of which makes a decision about detecting or not detecting a signal.

The spline filtering method is based on approximating the dependence ƒ (x) by the spline function S ₁ (x), which allows the use of a linear Kalman filter with coefficients whose values are calculated in advance. The values of a _i and b _i can be obtained a priori based on the form of the dependence ƒ (x) and the necessary approximation accuracy, since the Riccati equation for the gain of the Kalman filter does not contain the measured data.

Simulation of the spline filtering method is performed in the environment of the MathCAD package based on a system of equations (6). At the first stage, noise is generated from the formation and observation of the implementation of white Gaussian noise, after which, based on the initial nonlinear dynamic equation, the noise structurally related to the signal is modeled, which is shown in FIG. 5, where:

i is the number of discrete samples;

j is the number of spectral readings;

n _i is the interference;

Fz _j is the spectral density of the input implementation.

The test signal is presented in the manner described above and the implementation of the input random process is generated, which are shown in FIG. 6, where:

i is the number of discrete samples;

z _i is the input implementation;

St _i - test signal.

Given the specifics of the reverb, the spectral densities of the noise and signal are chosen the same, as shown in FIG. 7, where:

i, j is the number of spectral samples;

Fs _j is the spectral density of the test signal;

Fr _i is the spectral density of the interference.

Direct modeling of equations (5) is performed under conditions where the drift coefficient in the dynamic interference equation is determined by the function r (x) = - 0.05x ² -0.3x, which is approximated by a piecewise-linear function, which is two straight line segments in which the input the signal-to-noise ratio is 0.5, and the number of discrete samples is 64. In FIG. Figure 8 shows the interference value curves and its estimates, where:

i is the number of spectral readings;

n _i is the interference;

n ^* _i is the interference estimate.

Considering the interference values and its estimation (Fig. 8), the relative error of the spline filtering method was determined in the MathCAD environment, which amounted to 0.05% and on the basis of which it can be stated that the spline filtering method can quite effectively solve the problems of nonlinear filtering of interference and detection signals by representing the original nonlinear algorithm as a composition of a linear Kalman-Bucy filter.

A comparative analysis of the spline filtering method with the prototype and analogue showed that the proposed approach is not inferior in such an indicator as the output signal-to-noise ratio, and the difference is minimized if the number of approximating segments increases. The spline filtering method allows you to evaluate random processes specified not only by a scalar, but also by a matrix-vector equation. It is not difficult to extrapolate the obtained results to this case, and the advantages of the proposed approach in computational terms, in comparison with the known nonlinear filtering methods, become even more significant.

List of sources used

1. Tikhonov V.I., Kharisov V.N. Statistical analysis and synthesis of radio engineering devices and systems. M .: Radio and communications, 1991, 608 p.

2. Kazakov V.A. Introduction to the theory of Markov processes and some radio engineering problems. M .: Soviet Radio, 1973.232 p. (prototype, pp. 213-222).

3. Marchuk V.I., Sherstobitov A.I., Gavrilenko D.S. Features of filtering digital signals in conditions of limited sampling of its initial implementation. Successes of modern radio electronics. Foreign electronics. 2011. No9. S. 4-10 (analogue, p. 4-10).

4. Burova I.G., Demyanovich Yu.K. Theory of minimal splines. - St. Petersburg: Publishing House St. Petersburg State University, 2001. - 315 p.

5. Zavyalov Yu.S., Kvasov BN, Miroshnichenko V.L. Spline function method. M .: Nauka, 1980.

6. Rozov A.K. Nonlinear filtering of signals. - St. Petersburg: Polytechnic, 1994. - 381 p.

7. Butyrsky E.Yu. Signal detection against the background of Markov reverberation noise // Scientific instrument engineering. - 2012. - T. 22. - No. 1. - S. 87-95.

Claims (1)

A spline filtering method for signals based on the conditional Markov filtering method, comprising: operations for solving filtering equations for the hypothesis of signal presence / absence; operations to solve the likelihood equation and equations for calculating the gain, characterized in that to obtain estimates of the interference in the non-linear equation of state and reduce computational costs in non-linear filtering, the spline interpolation of the non-linear function is additionally used by dividing the dynamic range of changes of the non-linear function into intervals, in each of which implements the operation of linear representation of the equation of state, which allows for each of the subranges ealizovat Kalman-Bucy filter, comprising: two operation state estimation equations for hypothesis presence / absence signal; the operations of solving the dispersion estimation equation at various intervals; and the operations of solving the likelihood equation including these estimates, and the calculation of which makes a decision about detecting or not detecting the signal.

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