RU2782160C1 - Method for signal processing - Google Patents

Abstract

FIELD: radio engineering.

SUBSTANCE: area of application: invention relates to the field of radio electronics and hydroacoustics, namely, to methods for signal processing in case of interference. Invention consists in developing an optimal method for detecting a hydroacoustic signal under conditions of interference conditions described by nonlinear stochastic equations. Substance: method for signal processing is based on the method for conditional Markov filtering, including: solving filtering equations for the hypothesis of the presence/absence of a signal; a likelihood equation; and an equation for calculating the gain coefficients. The method is characterised by the area of the dynamic range of change in the nonlinear function being divided into intervals, wherein the nonlinear function included in the equation of state describing the dynamics of change in the interference is independently approximated in each of said intervals by a linear function, and the nonlinear function in general is approximated by a chain function with discontinuities of the first kind at the boundaries of the intervals, thereby allowing for the implementation of a Kalman-Bucy filter at each of the intervals, including: two state estimation equations based on the hypotheses of the presence/absence of a signal; dispersion estimation equations at various intervals, and a likelihood equation including said estimates, based on the results of calculation whereof a decision is made on having detected or not detected a signal; and the integral square error criterion is applied to obtain the value of spline coefficients.

EFFECT: lower computational costs with respect to the existing and promising methods for signal processing; high effectiveness of signal detection under influence of various interferences; a more effective filtering algorithm; and possibility of working at a real time scale.

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Description

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

The achieved technical result consists in increasing the accuracy of the interference assessment and the efficiency of signal detection against its background and in the versatility of processing with various interferences. The ability to work in real time, obtaining a more efficient filtering algorithm that requires less computational costs, determines the high economic benefits that can be used when introducing hydroacoustic signal processing into a system.

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

There is a known method of piecewise decomposition of estimates (patent No. 2257610), which is based on splitting the original discrete implementation into covering intervals of the same length, followed by evaluation of the useful signal (analogue) on each of them [2, p. 4-10]. Based on this method, it is assumed that the useful signal is described by some piecewise continuous smooth function that satisfies the conditions of the Weierstraße theorem on approximation on local segments. This approach makes it possible to obtain many estimates of the useful component in each section of the process with their subsequent averaging. The use of a system of orthogonal polynomials in solving the approximation problem makes it possible to obtain a general solution to the signal estimation problem and process one-dimensional discrete realizations of signals under conditions of nonparametric a priori uncertainty, which makes the spline filtering method more universal than the method of piecewise decomposition of estimates.

There is a method for spline filtering of signals (patent No. 2651640), based on the method of conditional Markov filtering, which includes: solving filtering equations for the hypothesis of the presence/absence of a signal; likelihood equations; equations for calculating the gain (prototype) [3]. A distinctive feature of the spline filtering method is that in order to obtain estimates of interference that is not Gaussian and is described by a stochastic differential equation of state, spline interpolation of a nonlinear function is additionally used, and the region of the dynamic range of changes in a nonlinear function is divided into intervals, in each of which linear representation of the equation of state, which allows to implement the Kalman-Bucy filter on each of the subbands, which includes: two equations for estimating the state under hypotheses of the presence/absence of a signal; the equation for estimating the variance at different intervals and the likelihood equation, which includes these estimates, and based on the calculation results of which a decision is made on the detection or not detection of a signal [4, 5]. Approximation is carried out by nodal points, i.e. at points where the value of the non-linear function and its approximant of the piecewise linear function coincide. The spline filtering method is shown in Fig. 1 where:

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

block 2 - block of formation of the system of intervals;

block 3 - block comparison of interference assessment with a system of intervals;

block 4 - block for determining the value of the evaluation of interference [x*(1)];

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

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

block 7 - block amplification factor

blocks 8, 12, 26 - integrators;

blocks 9, 13, 21 - quadrators;

block 10 - block amplification factor

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

block 14 - block amplification factor

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

blocks 17, 20 - multipliers;

block 18 - block generating the reference signal;

block 25 - block amplification factor

block 27 - two-threshold device;

block 28 - one-threshold device;

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

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

block 31 - a block that implements the continuation of the observation.

block 23 - block for calculating the value of the next estimate of interference [x*(2)];

block 24 - block for calculating the value of the next estimate of the interference [x*(3)];

An increase in the accuracy of approximation by spline functions is achieved by reducing the estimation error of the observed process at the stage of its representation by a linear spline at each linearization interval. When the estimated process is described by a stochastic dynamic system of the form:

and restrictions of the form:

where: ƒ(.), h(.) are non-linear functions ∈ L ² .

Then the SSS representation (1) can be approximated by a system of differential equations of the form:

The corresponding detection filtering equation will be:

Then, based on the fact that in each interval the filter is a linear Kalman-Bucy filter and satisfies the stability conditions [11], and the errors are generally and locally less than the threshold ones, it follows that the spline approximation is stable.

A distinctive feature of the proposed method compared with the prototype is the choice of coefficients a _i , b _i . In the prototype, the coefficients a _i , b _i , are determined through the nodal points, which are determined using the Newton or Lagrange interpolation polynomials. The disadvantage of this approach is that the approximation error of the nonlinear function is determined not through the optimal procedure, but only by the fact of the coincidence of the nonlinear function and the linear spline at the nodal points, which is quite acceptable if the rearrangement of the coefficients of the Kalman-Bucy filter was affected by discontinuities of the 1st kind, approximating function. But since the gaps do not affect, since the coefficients always change abruptly. Therefore, there is no need to impose such a strict condition on the approximating function as its continuity. The latter allows finding the coefficients a _i , b _i independently in each interval (section) using the error minimization criterion.

Let's determine the optimal values of the coefficients a _i , b _i , which are selected in each section (interval [x _i , x _i+1 ]), based on the criterion of minimum mean square error. When calculating the coefficients for solving expression (3), we will perform the following calculations:

Thus, for [x _i , x _i+i ] we have

Denote:

Then:

Let's use Cramer's method and then:

The error value in each section will look like this:

In the prototype, the non-linear function was approximated by a linear function over points, and the error value was calculated by the formula (figure 3) [10]:

In the proposed method, even if calculated by the formula of medium rectangles, the error is calculated by the formula (figure 4) [10]:

From the analysis of 6 and 7 it can be seen that in the proposed method the error is at least 2 times less. However, figure 4 clearly shows that when approximating using the minimum mean squared error criterion, discontinuities of the first kind occur at the boundaries of the intervals, however, this fact can be neglected during implementation due to the fact that the operation of the implemented filter is carried out by rearranging the filter from one coefficient to another (figure 2).

The proposed spline filtering method based on the prototype and the system of equations (23) is shown in Fig.5, where:

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

block 2 - block of formation of the system of intervals;

block 3 - block comparison of interference assessment with a system of intervals;

block 4 - block for determining the value of the evaluation of interference [x*(1)];

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

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

block 7 - block amplification factor

blocks 8, 12, 26 - integrators;

blocks 9, 13, 21 - quadrators;

block 10 - block amplification factor

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

block 14 - block amplification factor

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

blocks 17, 20 - multipliers;

block 18 - block generating the reference signal;

block 25 - block amplification factor

block 27 - two-threshold device;

block 28 - one-threshold device;

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

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

block 31 - a block that implements the continuation of the observation.

block 23 - block for calculating the value of the next estimate of interference [x*(2)];

block 24 - block for calculating the value of the next estimate of interference [x*(3)];

In the prototype, the calculation of the values of a _i , b _i can be obtained a priori, based on the form of dependence ƒ(x) and the required approximation accuracy, since the Riccati equation for the Kalman filter gain does not contain measured data (figure 5):

The total noise n _∑ (t) is the sum of the true value n ₁ (t) of the noise plus the approximation error:

n _Σ (t)=n ₁ (t)+n _cko (t), n _cko (t) - proposed method,

n _Σ (t)=n ₁ (t)+n _clf (t), n _clf (t) - prototype.

In the proposed method, taking into account the error in accordance with expression (3) (figure 6), the calculation of the values a _i , b _i can also be obtained a priori, but the errors will be less than in the prototype, and, as a result, the signal-to-noise ratio the output of the detector will be higher (because _ncko (t) of the proposed method is less than that of the prototype, and hence n _Σ (t)).

Geometric interpretation of the comparison of the proposed method and prototype (figures 7-10).

Figure 7 shows a method for approximating a non-linear function on an interval by a piecewise linear function. Figure 8 shows the same approximation method on one interval Δx _i. The error on this interval is determined by the area E _i → (S _clf (i). Accordingly, the total error is equal to

так как для каждого интервала имеет место неравенство S_ско(i)<S_клф(i) ⇒ (S_ско(i)<S_клф(i). Как следствие получаем, что дисперсия процесса n_cko(t) меньше, чем дисперсия процесса n_клф(t), а ОСП_ско>ОСП_клф и соответственно задача обнаружения решается более эффективно.Figure 9 shows a method for approximating a non-linear function over an interval by a linear function based on the mean squared error. In figure 10, the same approximation method is shown in one interval Δх _i . The error on this interval is determined by the area E _skoi → (S _sko (i). The total error is equal to

since for each interval there is an inequality S _sko (i)<S _clf (i) ⇒ (S _sko (i)<S _clf (i). As a consequence, we obtain that the variance of the process n _cko (t) is less than the variance of the process n _klf (t), and OSP _sco > OSP _klf and, accordingly, the detection problem is solved more efficiently.

The proposed signal processing method makes it possible to evaluate random processes specified not only by scalar, but also by matrix-vector equations. Extrapolation of the obtained results to this case is not difficult, and the advantages of the proposed method in solving the detection problem in comparison with the known methods of nonlinear filtering and the prototype 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 communication, 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. (Analogue, pp. 213-222).

3. Butyrsky E.Yu., Vasiliev V.V., Shklyaruk O.N., Obukhov E.V. Method for spline filtering of signals, patent for invention 2651640, 2018 (Prototype).

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

5. Zavyalov Yu.S., Kvasov B.N., Miroshnichenko V.L. spline function method. Moscow: Nauka, 1980.

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

7. Butyrsky E.Yu. Detection of signals against the background of Markov reverberation noise. Nauchnoe priborostroenie. - 2012. - V. 22. - No. 1. - S. 87-95.

8. Butyrsky E.Yu. Fundamentals of spline filtering of signals // Information and space-2010. No. 1. - S. 34-39.

9. Butyrsky E.Yu. Spline signal models and spline filtering // National Security and Strategic Planning. - 2014. - No. 2 (6). - S. 43-56.

10. Samarsky A.A., Gulin A.V. Numerical methods. M.: Nauka, 1989. - 430 p.

11. Ashinyants R.A., Morozova T.Yu. Regularization of the Kalman-Bucy filtering algorithm under poor conditionality of the correlation noise matrix // Digital Signal Processing, No. 4, 2007. P. 29-32.

Claims (1)

A method for processing signals based on the operation of spline filtering of signals, comprising: the operation of solving filtering equations for the hypothesis of the presence/absence of a signal; the operation of solving the likelihood equation; the operation of solving the equation for calculating the gain coefficients, in order to obtain estimates of interference with a nonlinear equation of state and reduce computational costs in nonlinear filtering using spline interpolation of a nonlinear function, the region of the dynamic range of changes in the nonlinear function is divided into intervals, in each of which the operation of linear approximation of the nonlinear function is implemented. a function included in the equation of state that describes the noise dynamics, which makes it possible to implement the Kalman-Bucy filter on each of the subbands, which includes: the operation of solving two equations for estimating interference under the hypotheses of the presence/absence of a signal; the operation of solving the equation for estimating the variance at different intervals and the operation of solving the likelihood equation, which includes these estimates, and based on the results of the calculation of which a decision is made whether the signal is detected or not, characterized in that at the ends of the intervals, the operation of approximating a nonlinear function by a piecewise linear function is carried out with discontinuities of the first kind, which approximates a non-linear function to a linear one independently in each of the intervals in accordance with the minimum mean square error criterion, additionally, operations are performed to determine the values of the coefficients of the piecewise broken curve of the approximation of the non-linear function in accordance with the selected error value.

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