Improving the precision of the selection of piecewise linear useful signal component under conditions of a priori uncertainty

The paper is devoted to the analytical substantiation of a new approach to the processing of signals representing a collection of some piecewise linear signals under conditions of a priori uncertainty about its parameters. The results of computer simulation fully confirm the main theoretical results and allow us to conclude that the new method is highly efficient in processing piecewise linear signals under conditions of a priori uncertainty about the useful signal function and the statistical characteristics of the additive noise component. The developed method makes it possible to reduce the dispersion of the additive noise component up to 10 times, however, as the dispersion of the noise increases, the efficiency decreases.

In addition, there is no doubt that for each 1 ≤ ≤ as the values , in (2) coefficients should be taken * , * OLS-line equations, built from the points ( , ), −1 + 1 ≤ ≤ . In this regard, condition (1) is strengthened by supplementing it with the requirement ≥ −1 + , = 1, 2, … , , where − is a predetermined integer from the segment [2; ]. Note that in fact the value of l should be greater than 2. This is explained by the fact that the equality = −1 + 2 implies the equalities −1 = −1 , = , which reduce the quality of the approximation. Let us give a recommendation for choosing the value of . We will consider the moments equidistant on the time axis (which is typical for discretization of a continuous signal): = ( − 1)ℎ, = 1, 2, … , , where ℎ(> 0) − is a given constant. Note that the approximation of a pure signal by piecewise linear functions makes sense only if the step ℎ is sufficiently small in comparison with the lengths f linearity intervals of the function ( ). Let Δ = min . Let us show that if Δ exceeds the value ℎ, each of the intervals contains at least, points even in the case when the ends of the interval do not belong to it. First, we note that at least one of these points will appear in (otherwise it would entail an impossible inequality: ℎ ≥ ≥ Δ > ℎ ⇒ < 1). Let ( ) − be the smallest (largest) integer for which ℎ ∈ (respectively, ℎ ∈ ). Then ℎ < Δ ≤ ≤ ( + 1)ℎ − ( − 1)ℎ = ( − + 2)ℎ, whence < − + 2 ⇔ ≤ − + 1, as required to prove ( − + 1 is the number of points from the interval ).
Let us now assume that the value of Δ 0 , is known, practically reliably less than Δ. Then the choice of the step ℎ ≤ Δ 0 / (< Δ/ ) with the same reliability will allow us to assume that constraint (3), in fact, is not such. This follows from the just proved statement: since each linearity interval of a pure signal contains at least points , it is natural to demand the same from its approximation. The above-mentioned requirement > 2 ⇔ ≥ 3 is updated in the case of choosing the step ℎ ≤ Δ 0 /3. After setting the step ℎ, the best allowable value will be 0 = [Δ 0 /ℎ] (the largest integer , satisfying the inequality ℎ ≤ Δ 0 / . Here [ ] − is the integer part of the number ). If it is impossible to set Δ 0 due to the lack of a priori information about the lengths of the intervals 0 = 3 should be taken. Due with the above, filtering the signal ( ) is reduced to minimizing the sum ∑ ∑ ( * ℎ( − 1) + * − ) 2 over the set of all sequences ( ) =0 , satisfying conditions (1) and inequalities ≥ −1 + 0 , = 1, 2, … , . In turn, this problem can be solved by plotting the shortest route on a weighted digraph with vertices 0, 1, 2, … , and arcs ( , ), where 0 ≤ < ≤ .  (for > the sum of the form ∑ = is considered equal to zero). We divide both sides of Eq. (5) by ℎ and introduce the notation = ∑ =1 ,̂= ∑ ( − 1), =1 = 0, 1, 2, … , . Then the right side of equation (4) (equation (5)) can be given the form − (respectively, the form ̂−̂) . Now, according to Cramer's rule, The obtained processing results were checked using machine modeling on a piecewise linear function model in the presence of additive normal noise with zero mathematical expectation, which are shown in Figure 1. The simulation was carried out using the model of a piecewise linear function presented in Figure 1 and an additive noise component with variance taking values from 1 to 5 with a step equal to 1. Figure 2 shows the results of processing the measured process ( ) + ( ) when the noise variance is equal to 3. When analyzing the research results, it should be taken into account that there was no a priori information about the initial and final coordinates of each of the steps. There was also no information about the slope of the piecewise linear function on each of the intervals. In the figure presented, the solid line shows the model of the useful component of the initial implementation of the measurement results, in the dotted line the estimate of the useful signal obtained as a result of using the proposed method of processing the measurement results. Comparative analysis of the simulation results allows us to conclude that the evaluation of the useful signal with almost a single probability determined the initial and final coordinates of each of the steps. The error lies in determining the slope of the piecewise linear function, which is explained by the influence of the noise structure.
In conclusion, let us consider the dependence of the error on the variance of the additive noise component, which is shown in Figure 3. The results presented in Figure 3 allow us to conclude that the efficiency of the proposed method decreases with increasing variance. For example, when 2 =1, there is a decrease in variance by more 5 than 9 times, with 2 =3 only 3 times, and with 2 =5 only by 15 percent, which is explained by the fact that the probability of correct estimation of the slope decreases piecewise linear function.

Discussion
A new method of filtering a useful signal presented as a piecewise linear function is proposed. Optimization of this method using the least squares method is analytically proved.

Conclusions
1. The results of computer modeling confirmed the theoretical results of processing piecewise linear functions under conditions of a priori uncertainty. 2. The developed method makes it possible to reduce the dispersion of the additive noise component up to 10 times, however, as the dispersion of the noise increases, the efficiency decreases.