DEVELOPMENT OF A DEVICE FOR THE OPTIMAL RECEPTION OF SIGNALS AGAINST THE BACKGROUND OF AN ADDITIVE THREE-COMPONENT INTERFERENCE

particularly unacceptable where a communication system is integrated into the overall system of a technological process safety system. The described situation occurs specifically during railroad transportation in the technological communication systems and subsystems that transfer signals from railroad automatics. As the saturation of industries, transport, and even households, with such integrated systems grows, solving the tasks on noise-resistant receival of information signals becomes increasingly relevant.


Introduction
Current communication systems operate under increasingly complex electromagnetic environment. This environment is formed due to the working and side emissions from third-party radio-electronic devices and technological installations. The number of such devices and installations has been steadily increasing over time. A series of natural processes is also accompanied by the formation of disturbing electromagnetic radiation. The undesired electromagnetic processes can also occur within communication systems. Typical causes are the uncontrolled fluctuations in the parameters of communication channels. The impact of listed factors is a prerequisite for a relative increase in the number of errors that a recipient makes while recognizing the received information signals. Such an increase may prove
The complexity of current electromagnetic environment implies that the types and parameters of noise can vary over short time intervals. The duration of these intervals is practically commensurate to the duration of an information signal. In communication systems that ensure the safety of railroad traffic every such signal has a considerable informational value. Therefore, an optimal receiver must generate a solution for each successive signal. That means that the algorithm for optimized processing must be built based on such computational and logical structures that would enable the rapid formation of the above-specified solution. It was proposed earlier in work [10] to use, for the optimal detection and recognition of signals from automated locomotive signaling system, a method that implied a joint evaluation of signal and noise parameters. In addition, this work substantiated and constructed a mathematical model of the additive mixture of a signal and a three-component interference, describing a typical electromagnetic environment. Paper [11] showed that the use of this model when solving a task on the joint estimation of signal and noise parameters makes it possible to derive a rather compact expression for the mean square of approximation error. Here, an approximation error refers to the difference between the magnitude of sum of the signal and structurally deterministic interference components and the magnitude of total voltage at a receiver's input. The author of article [11] left the above expression in a very general form and confined himself to the proposal to search for the required estimates based on the criterion of a minimum of the mean square of an error. However, there are neglected mathematical and technical means that could be used to compute parameters' estimates and then recognize the information signals. The issue of minimizing the related computational and hardware costs has remained unresolved. Finally, there are no methods that would ensure addressing the situation of an arbitrary number of interferences.
In the target field of subsystems that transmit signals from railroad automatics along rail lines, the research and development of the specified systems were originally conducted, and are still performed, mainly by specialists in circuitry, as well as specialists in operation, who apply specific concepts and categories that have no direct equivalents to the concepts and categories in communication theory.
The research into the field of railroad automation signal reception along rail lines, undertaken in recent years [1,9,10], made it possible to estimate the probabilities of correct and incorrect solutions on the type of a received signal for each act of reception. However, it is still difficult to unambiguously compare the results with the characteristics of the systems that operate in railroad transportation.
Thus, the methods and means for suppressing multi-component interference, constructed up to now, contributed to addressing many tasks related to communication and automation. However, there are no identified results that would be applicable beyond the situation when disturbances are similar or when the number of disturbances does not exceed two. In this regard, it is of interest to consider a possibility to solve the problem on the optimal signal reception against the background of at least three different types of interference, which would ensure high noise immunity for the system of automatic locomotive signaling over a wide range of interference parameters.

The aim and objectives of the study
The aim of the research that we conducted was to synthesize the structure of a device for the optimal reception of signals, observed against the background of an additive three-component interference. In this case, the two components of an interference were the structurally deterministic oscillations of different type, while the third one was a wideband Gaussian noise.
To achieve the set aim, the following tasks have been solved: -based on the predefined optimality criterion, to substantiate mathematically the objective function that corresponds to it; in this case, the objective function must be represented in the form that would provide for the lowest possible computational cost of its minimization; -to determine the types and sequence of mathematical operations over the input signal that would match the above-specified criteria and objective function, as well as the composition of technical tools that implement these operations; -to quantify the noise immunity of the designed device.

1. Substantiation of the objective function
To further develop the above-described results reported in [10,11], the present paper considers a method of the joint estimation of signal and interference parameters based on the criterion of a minimum mean square of approximation error. The objective function, constructed in study [11], takes a rather general form and needs further transformation to make its form computationally efficient.
Given the form of expressions that describe in work [10] the information signals and structurally deterministic interference, it can be assumed that the objective function is multi-extremal. The desired values of assessments in this case correspond to the position of the point of a global minimum in coordinate space. Finding this point is the task of global minimization. The number of unknown parameters in the problem under consideration is seven. It is known that solving a problem with such a dimensionality puts forward an unacceptably large requirements to random-access me mory and computational power of a signal processing device [12]. However, resolving a multidimensional minimization problem can be technically implemented if the objective function is subjected to a procedure of the multi-step dimensionality reduction [12]. It is possible if for vector X  of unknown parameters the objective function ( ) F X  can be represented in the following form: where F i (…) are the two-dimensional functions; x(i) are the components of vector X  . Note that in this case one imposes a requirement that the numeric values for the components of vector X  should belong to the singular N-dimensional cube [ ] 0,1 . N Such a normalization of parameters is easily feasible in our case. Thus, first of all, we shall focus our efforts on transforming the expression, derived in article [11], which describes the average squared error, to form (1), that is to the sum of two-dimensional functions. By maintaining the terminology and notation adopted in article [11], we note that functions and G B (…) violate the requirements imposed on the form of the objective function by expression (1). Can we make any assumptions acceptable from a practical point of view, and in so doing make it possible to comply with the specified requirements?
To answer this question, we shall clarify the meaning of terms in a given expression within the theory of measurement of signals parameters. According to classic work [13], functions represent the likelihood sampling relationship logarithms of input voltage { } … 1 2 ,� ,� ,� , k u u u calculated for the relevant groups of parameters and taken with a negative sign. When estimating the parameters of signals, one searches for the maxima of these logarithms. Accordingly, following the sign inversion, the minima should be derived. Therefore, the resulting expression from article [10] can be regarded as the sum of logarithms, taken with a minus sign, of the ratio of the likelihood of sample { } These additional terms introduce corrections to the isolated likelihood relations. These corrections are predetermined by the cross-correlational connections between a signal and the interference. Suppose one has found the magnitudes  mp U and  , p τ which minimize the function W(U mP , τ P ) only. These magnitudes can be considered to be approximate values for the maximum likelihood estimates. How much will affect ignoring the terms H A (…) and H В (…) the degree of this approximation when calculating the estimates obtained? If the pulse disturbance has a small energy, the true magnitude of U mP is small. In this case, the pulse of disturbance itself is similar in shape to a low and gentle «bell». Under these circumstances, the correctional mutually-correlated terms H A (…) and H В (…) are small and thus do not make a significant contribution to the overall error. However, the substitution of magnitudes  j Similar reasoning could be made to the situation of an arbitrary number of disturbances. That would make it possible to represent a multidimensional objective function as a sum of two-dimensional component, which is rational in terms of subsequent minimization [12].
Consequently, the resulting expression from paper [11] can be rewritten as follows: where R А (…) equals the sum of all the components that are included in expression (5) with index A, and R В equals the sum of all the components that are included in expression (5) with index B. Expression (6) describes the objective function as a sum of two-dimensional functions. Such a form of the function, according to paper [12], ensures a substantial reduction in computational costs during its minimization.

2. Procedure and technical implementation of signal processing
Since In this case, the first two terms of a given expression yield the estimates  , j necessary to minimize its last term. To find these estimates, we shall apply, according to paper [12], Following this rule, applied to W(U mP , τ P ), we obtain x = τ P , y = U mP . Then, taking into account the model for a pulse interference component, introduced in work [ This is a quadratic (and thus a single-extremal) function of the magnitude U mP . Therefore, the estimate of  �� mP U is to be found from the condition that the first derivative from the contents of braces equals zero. It is easy to see that this estimate, as a function of parameter τ P , takes the following form: According to (8), the estimate of  �� P τ is to be found as a point at which This requires a one-dimensional global minimization of the contents of a brace in the right-hand side of expression (10) for parameter τ P . Such an operation can be performed, for example, based on a simple tabulation within the observation interval [T 1 , T 2 ] or by other known methods [12].
The function Y(U mE , φ E ) has a structure similar to the structure of function W(U mP , τ P ), which is why the course of its minimization is similar. It is easy to show that ω + j j = ω + j ∑ ∑ (11) and that the estimate of  �� E j is derived as a point where the following ratio holds in this case, global minimization should be performed within an interval [ ) 0,2 �. π . Next, to fully minimize the objective function, we have to find the last term from expressions (7). Given that functions R A (A, τ s ) and R B (B, τ s ) «overlap» for argument τ s , dimensionality reduction can be performed according to the following algorithm [12] ( ) ( ) In this case, the components x(i) of vector X  must belong to a single cube [0, 1] N . To satisfy this condition, one must perform the appropriate normalization of parameters of an actual problem prior to starting the construction of a minimization procedure. The A and B parameters will be normalized relying on the definition given in paper [10]. We shall consider that in practice of operation the U ms amplitude ranges from 0 to a certain a priori known boundary magnitude U M , while sine and cosine lie within [−1, 1]. It is easy to verify that the normalized parameters that match the A and B magnitudes are the following variables: The magnitude of signal delay τ s ranges from 0 until the moment T 2 of observation end, therefore, the corresponding normalized parameter: The relations between the actual and normalized parameters that follow from ratios (16) to (18) are: Next, one can build a function that is included in the right-hand side of equality (14) in the following form: This is a quadratic function relative to variable x(1). Because a coefficient at the square of this variable is positive, then the only extremum of this function is a minimum. We shall equate a derivative from expression (22) Given this result, expression (14) can be written in the form: Following algorithm (13) to (15), we then search for function: In line with (1) and (6), we obtain In accordance with work [12], in order to compute each value for the function ( ) Note that specific algorithms for the rapid one-dimensional global minimization can be built based on known methods [12].
x y As a result, we shall find among the above-mentioned totalities a set at which variable x(3) takes such a numeric value 3 , χ so ,� � ,� � χ χ χ will, in accordance with expressions (19) to (21), make it possible to obtain estimates for  , A  B and  . s τ An automated locomotive signaling system employs three information code signals: Green (G), Yellow (Y), Red-yellow (R). To form a decision on the form of a received signal, it is necessary, in accordance with the final expression from article [11], to calculate the magnitudes of mean squared errors 2 0 ε along three processing channels. Each of these channels has its own code envelope f(t). It should be assumed that such a signal is received for which the magnitude of 2 0 ε is the smallest. A structural diagram of the corresponding device for the optimal reception of code signals is shown in Fig. 1.   Fig. 1. Structural diagram of a device for the optimal reception of information signals from an automated locomotive signaling system: 1 -shaper U mP , τ P ; 2 -shaper U mE , φ E ; 3 -shaper А, В, τ s ; 4 -computational unit 2 0 ; ε 5 -channel G; 6 -channel Y; 7 -channel R; 8 -a minimum selection unit; 9 -decision on G, Y, or R The auxiliary units that perform the conversion of an analog signal into digital and store constant magnitudes are not shown in the diagram.

3. Quantification of noise immunity of the designed device
Computer simulation was conducted in order to numerically evaluate the characteristics of the designed optimal receiver. Fig. 2 shows a chart of the code signal Green (Y), distorted by a pulse interference, by an interference from a power line (PL) and by a Gaussian noise. Numerical magni-tudes of the signal model parameters are as follows: signal amplitude U ms = 0.2 V, signal carrier frequency f s = 25 Hz, initial signal phase φ s = 75°, time offset of the signal relative to the onset of observation τ s = 0.4 s. The model of a pulse interference corresponds to the experimental curve reported in article [14]. A given curve was obtained when a locomotive moved over a track switch at speed 75 km/h. The amplitude of a pulse interference U mP = 1.5 V, the pulse interference attenuation α = 10 3 , the angular frequency of a pulse interference β = 60 rad/s, the position of a maximum of a pulse interference over time relative to the onset of observation τ P = 1.17 s, the coefficient of the initial phase of a pulse interference γ = 0.5, the amplitude of the PL-induced interference U mE = 0.05 V, the frequency of the PL-induced interference f E = 50 Hz, the initial phase of the PL-induced interference φ E = 248°, the standard Gaussian noise voltage is 0.005 V, the noise bandwidth is 250 Hz. At such an arrangement of the interference pulse a typical receiver in the system of automated locomotive signaling makes an erroneous decision on the reception of Yellow signal (Y) [15]. To more broadly assess the noise immunity of the designed device, computer simulation was performed. The result is the magnitudes for the mean squares of errors 2 0 ε along each of the three processing channels. A Green (G) signal was chosen as the signal to be recognized. The following source data were accepted for simulation: 1) amplitude U mS of the code signal (identical for signals R, G, Y); 2) frequency f s and initial phase φ s of the code signal (identical for all signals); 3) time-delay τ s of the coded signal relative to the onset of observation (same for all signals); 4) the pulse interference parameters, the PL-induced interference parameters, and the Gaussian noise parameters (their definitions are given in explanations to Fig. 2).
All oscillations were sampled over time with an interval of ∆ = 0.004 s. t Numerical magnitudes of the code signal parameters are the same as in annotations to Fig. 2. When computing Table 1, the numerical magnitudes of all other model parameters coincided with those given in annotations to Fig. 2, except for a pulse interference amplitude U mP . The magnitude of this amplitude varied in the range 0...1.5 V, typical for practical situations. When computing Table 2, the magnitude of pulse interference amplitude U mP was taken to be equal to 1 V. The numerical magnitudes of all other model parameters coincided with those given in annotations to Fig. 2, except for the amplitude of the PL-induced interference U mE . Its magnitude varied in a range of 0…0.5 V, which is also typical for practical situations. Based on the results of simulation, we have built the curves for dependences of the mean-squared error 2 0 ε along each of the three processing channels on magnitude U mP (Fig. 3) and magnitude U mE (Fig. 4), respectively. Based on the results of computer simulation, we can conclude that the synthesized device makes it possible to reliably distinguish the transmitted code signal (in this case, G) from other code signals that can arise in an automated system of locomotive signaling. Such a high ability to discriminate is maintained over a wide range of amplitudes of a pulse interference and a PL-induced interference, characteristic of actual operating conditions. The magnitude of RMS voltage of the Gaussian noise, accepted during simulation, is relatively small, which is also characteristic of such conditions. The processing of input oscillations in the synthesized device for optimal signal reception is based on the previously known maximization of the input sample likelihood ratio in first approximation only. The improvement of noise immunity is achieved in it due to accounting for the corrections to the isolated likelihood ratios (this feature was not considered in previous works). The specified corrections take into account the cross-correlational connections of a signal and each interference component. The research underlying the present work were aimed primarily at easing the impact of structurally deterministic interference. As a result, the designed device is capable to respond quickly to changes in the parameters of such an interference. A noise interference was assumed to be stationary. This limits the applicability of the device. Accounting for the non-stationarity of random noise is regarded as one of the avenues for further research.
In the cases, characteristic of actual operation, of a negligible magnitude of the intra-interference correlation, the formula that describes the magnitude of a squared error is represented as the sum of computational segments. Each of these segments describes the contribution of only one type of noise. For example, in expression (6), the computational segment [W(…)+H A (…)+H B (…)] describes the individual contribution of a pulse interference to the total magnitude of error 2 0 . ε This segment becomes zero and hence can be excluded from the procedure if there are any a priori reasons to believe that there are no pulse interferences. A similar conclusion can be easily extended to the case of an arbitrary number of mutually uncorrelated interferences. The proposed method of segmentation provides for the fundamental possibility to build an optimal receiver in a modular fashion. In this case, the modules can be connected and disconnected in accordance with the a priori predicted composition of the interference set, while the «library» of modules can be replenished when the new types of interference emerge. The main difficulties are expected when looking for ways to represent an arbitrary set of interferences in the form that allows the application of a multi-step procedure of dimensionality reduction.
We consider the estimation of noise immunity, based on the results of software simulation, to be preliminary. Strictly speaking, it is necessary to calculate the magnitudes of the first-and second-kind error when making decisions. The standard DSTU 4178-2003 «Complexes of technical means for systems of control and regulation of train motion. Functional safety and reliability. Requirements and test methods» (its European analog is CENELEC EN 50126: Railway Application -The Specification and Demonstration of Reliability, Availability, Maintainability and Safety) requires that the likelihood of failures that lead to dangerous malfunctions should be at the level 10 -10 per one hour of operation. However, we do not have enough computing power to identify such a small error. Therefore, the possibility is being investigated at present to analytically estimate the noise immunity of the designed device.
We did not consider the related energy costs because when designing and investigating communication devices and, in particular, their processing units, it is a commonly accepted practice to evaluate the number of basic mathematical operations required to achieve a set goal, and the corresponding number of arithmetic and logical units (adders, multipliers, comparators, etc.). Determining the magnitude of associated energy costs in the synthesis of receiving devices is assumed by default to be an additional task to be resolved within a framework of specialized research and development.

Conclusions
1. The optimality criterion for processing an additive mixture of signal and a three-component interference has been selected. Based on realistic assumptions about the statistical relationships between the signal and interference components, it has been demonstrated that the objective function is the sum, taken with an opposite sign, in which the number of summands is limited by the quantity of isolated logarithms of the likelihood and correction functions ratios. A special feature of the proposed approach is that it makes it possible to build an optimal receiver based on a flexible modular design.
2. We have synthesized a signal processing device that recognizes the received information signal through the multiparameter minimization of the objective function. A decrease in the computational cost has been achieved by a multi-step reduction of dimensionality. Namely, according to paper [12]: while minimizing via direct search, one needs a total number of computations at the order of N n (where n is the dimensionality of an objective function, N is the number of its computations for each parameter), while at a multi-step dimensionality reduction, the total number of calculations is a much less magnitude of the order Nn.
3. By employing computer simulation, it has been shown that in a channel corresponding to the formation of a correct solution the magnitude of an approximation error is about 6 times less than that in the other two channels. This ratio holds when the amplitude of a pulse interference exceeds the signal amplitude by up to 7.5 times. The above ratio is true in the case when pulse interferences exceed the useful signal by 5 times, and interference from power linesup to 2.5 times. These indicators characterize high noise immunity when the designed device recognizes code signals over a wide range of interference parameters.