Algorithm for Verifying the Stability of Signal Separation for Objects with Varying Characteristics Valery Zasov Samara State Transport University Samara, Russia vzasov@mail.ru Abstract—This paper proposes an algorithm for verifying constrains the application of the methods and the algorithm the stability of a solution to the inverse problem of separating in real-time systems. individual signals from an additive mixture of several signals. The algorithm is designed for objects whose characteristics Therefore, developing algorithms to verify the stability vary depending on a certain parameter vector. The paper also of solutions to the signal separation problem in objects with considers a version of the algorithm for objects whose changes varying characteristics is a relevant problem. in characteristics are described by deterministic functions. A feature of the proposed algorithm is preliminary learning, II. RESEARCH AREA which can help reduce by far its computational complexity and the stability verification time by building a singularity To state the problem formally, we will consider a boundary to separate the spaces of stable and unstable mathematical signal formation model presented as a linear solutions. This paper also presents the computer modeling multivariable system that has N inputs and M outputs. The results for the proposed algorithm. model’s input signals are s n  k  and n  1 , 2 , ..., N ; output Keywords—signals, mixture, separation, algorithm, signals, x m  k  and m  1 , 2 , ..., M . characteristics, determination, variation, parameter, stability, boundary, calculation, complexity, learning The mathematical model of signal formation is described by equations of discrete convolution type (1), where the m th observed signal is an additive mixture of channel- I. INTRODUCTION distorted source signals [1]: Signal separation involves solving the problem of extracting individual signals from an additive mixture of N G 1 several signals that come to measurement points from x m  k     hm n  g , I  s n  k  g  , (1) various sources inaccessible for direct measurement. n 1 g  0 The problem of signal separation relates to the class of inverse problems, which may be ill-posed, generally. From where h  g , I  is the element N  M of the mixing matrix mn that it follows that a solution to the problem may be unstable [1]. For a stable solution to exist, parameters of the object h  g ,I  for the impulse characteristics of channels; and described by the signal formation model (parameters of the g  0 , ...,G - 1 and k  0 , ..., K - 1 are the counts for the mixing matrix H [1]) must satisfy several prior restrictions impulse characteristics of channels and signals, respectively. [2,3]. Generally, the solution to the inverse problem of Under real operating conditions, the prior restrictions separating source signals is the solution to (1), and it can be assumed in developing signal separation algorithms may fail expressed as to be satisfied. This leads to solutions that are unstable and therefore unsuitable for practical applications. M G 1 At present, the stability of a solution to the inverse sn  k     w nm  g , I  xm  k  g  , (2) problem of signal separation is verifiable by using the m 1 g  0 condition numbers c o n d  H  [4] and the matrix norm where w nm  g , I  are the impulse characteristics of the ΔH 2 [5] of the mixing matrix H ; the singular-direction method [6]; and the algorithm for calculating singular separating filters that form the separating matrix w  g ,I  , intervals, as well as through comparison with given intervals which is equal or close, by a given criterion (in the case of of stable separation [7]. ill-posedness), to the matrix inverse to the matrix h  g , I  . These methods and the algorithm are effective for static objects, the parameters and characteristics of which virtually In the frequency domain, equation (2) can be written as do not change during operation or slowly change because of unstable environmental conditions, wear, and the like. S     W   ,I  X    , But for dynamic objects whose characteristics vary during operation, applying the methods and the algorithm where W   , I   H -1   , I  . [7] is inefficient because of their high computational complexity. Indeed, in this case, for each of the many We propose using the singular intervals for the varying states of objects, complicated and time-consuming parameters of the mixing matrix H   , I  , whose calculations are necessary to verify stability, and this Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) Data Science calculation algorithms are given in [7], as parameters that The purpose of this paper is to develop an algorithm to make the solution stable. verify the stability of solutions to the problem of signal separation through calculating singular intervals—an We assume that the current state of the mathematical algorithm differing from the known one [1,7] in that it offers model is unequivocally determined by the state vector l , extended functionality, allowing signal separation to be whose parameters are set by the current characteristics of verified in objects with varying characteristics. the object. For example, in mobile communication systems, the III. ALGORITHM TO VERIFY THE STABILITY OF SIGNAL parameters l specify the distance between mobile receivers SEPARATION FOR OBJECTS WHOSE CHANGES IN and base transmitters; in vibroacoustic diagnosis systems, CHARACTERISTICS ARE DESCRIBED BY DETERMINISTIC these parameters specify the relative positions of FUNCTIONS mechanisms, such as those determined by the rotation angle of a shaft. Thus, the impulse characteristics of channels The algorithm consists of two stages—learning and h m n  g ,l  in the signal formation model change depending verification—which include the following steps. on a certain vector, l [1,7]. The parameters of this vector Step 1. Identify the possible path of variation in the are determined by on-site sensors measuring displacements, object’s state corresponding to the values of the state vector rotation angles, distances, coordinates, and the like. l 0 , l 1 , K , l d —that is, describe the region of possible model Therefore, the states of the object (its characteristics) states with the discrete set H 0  ω g , l 0  , K , H d  ω g , l a  . vary during operation as shown in Fig. 1, and they take values corresponding to those of the state vector Step 2. Calculate the norms  H ωg ,l  Е of mixing l 0 , l1 , K , l d . matrices for various object states, and determine the parameters of vector l for which the matrix norms differ by the given value γ . Thus, a list of object states is compiled for which variation in characteristics is substantial, calling for stability to be verified. Step 3. For the object states determined in step 2 and the selected type of perturbation (absolute, relative, critical, or their combinations), calculate the following parameter matrices using the algorithm proposed in [1,7]: The singular matrices H    , which set a singularity g boundary for the region of stable solutions Matrices of singular intervals for model parameters,  H  g , which determine the intervals of model Fig. 1. Graphical representation of an object with varying characteristics. parameters from the initial ( H   g  ) to the singular The region of possible states for the mathematical model ( H   g  ) state is described by the discrete set H 0  ω g , l 0  , K , H d  ω g , l d  , The threshold matrices H th  ω g , l 0  , K , H th  ω g , l a  — which we assume is bounded and finite. We also assume that the matrix of the maximum allowable variation intervals mixing matrices for each of which the condition number for the parameters Δ H  ω , l  is known beforehand for each c o n d H   g , l  exceeds a given threshold max g state of the object set by the parameter vector l . In Fig. 1 the area highlighted in gray represents the maximum The matrices  H R  g  and % ΔH S  ω g , l  for the allowable variation interval for object parameters set by intervals of model parameters corresponding to stable and prior restrictions. unstable separation of signals Let us consider objects for which the elements of the set The parameters of these matrices and the parameters of  H 0  g , l0 ,   , H d  g , la  , which defines the possible the associated state vectors l are written to a database. states of the mathematical model for the object, and the Step 4. For each object state determined in step 2, verify parameters of the vector l are linked by a functional the condition relationship. For purposes of further discussion, we will divide objects into two groups. In group 1 objects, variation in characteristics is described by deterministic functions, as  Δ H m ax ω g , l   Δ H% R  ω g , l  . (3) in radio communication systems in which mobile receivers follow routes such as roads or railways. In group 2 objects, variation in characteristics is described by random functions. This verifies whether the model with the preset matrix Δ H m a x  ω g , l  for maximum allowable parameter variation VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 256 Data Science intervals falls into the stability region determined by the matrix Δ H%  ω , l  . If condition (3) is not fulfilled, a R g message is displayed that a stable separation of signals in the object is impossible and that the given mathematical model cannot be used. In step 4 the algorithm completes learning, resulting in the parameters for singularity and stability boundaries being calculated and stored in the database according to the parameters of the vector l . Thus, a function is calculated that determines the stability boundary for signal separation when the object’s characteristics vary. Learning is completed in free time and involves averaging the measured parameters of the mixing matrices Fig. 2. Graphical interpretation of the algorithm for verifying the stability H   , I  , allowing a diagnostic model to be obtained for the of signal separation. object. This model is then used at the second stage, described in step 5, to monitor in real time whether signal Thus, in the proposed algorithm, time constraints (real- separation is stable during object operation. time requirements) are only imposed in step 5. This step is simple and comes down to comparing the intervals of Step 5. For each object state identified in step 2, verify perturbations for the object parameters and the parameter the following condition for stable separation: intervals for stable signal separation—that is, to verifying conditions (4) or (5).  Δ H per ω g , l   Δ H% R  ω g , l  . (4) This expands the possibilities of applying the algorithm to objects with dynamically changing characteristics. Complicated calculations of matrix intervals for the Under this condition, the matrix  Δ H per ω g , l  for parameters of stable signal separation are removed from real-time constraints and are performed instead in free time parameter perturbation intervals is determined from at the learning stage as part of building an object model, which is updated rarely (when major changes are made to the object).    Δ H per ω g , l  H var ω g , l  H ω g , l   , IV. COMPUTER MODELING RESULTS where  H ωg ,l  and  H var ω g , l  correspond to the Let us consider monitoring a railroad infrastructure parameter matrices for the model and the object at a facility by using specialized mobile laboratory cars, with the frequency of ω g for the given state vector l . For the same facility including a track, a contact network, a train radio communication system, and the like. state vectors l , the matrices % ΔH  R ωg ,l  of parameter We assume that the signal generation model for a intervals for stable separation are retrieved from the communication system with two transmitters (mounted at database for verification under condition (4). stations) and two mobile receivers (in cars) is described by the mixing matrix M  N  2 with frequency-dependent If condition (4) is not satisfied, then stable separation of channels. Signals from the two transmitters as well as signals for the frequency  g is not guaranteed. reflected signals that form an additive mixture of signals can enter the mobile receivers. Therefore, to make messages The condition for the stable separation of signals can encoded in signals accurate, the system should provide also be expressed as stable separation of signals according to their source. The frequency response of the channels changes when  Δ H per ω g , l   Δ H m a x  ω g , l  < m in Δ H%m n  ω g , l  , (5) the receivers are moving on the rail-track in relation to the transmitters. An example of the measured frequency response of communication channels for a specific track where m in Δ H%m n ω g , l   is the module of the minimum coordinate (the state parameter l is the distance) is shown in Fig. 3(а). singular interval for the matrix H   g  . For certain track coordinates of the receivers, a change A graphical interpretation of the proposed algorithm is in the frequency response of the channels simulates a stable shown in Fig. 2. and unstable separation of signals, and the separation is confirmed by the condition number of the mixing matrix c o n d H  ω g , l  (Fig. 3(b-1) and 3(b-2), respectively). VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 257 Data Science Fig. 3. a) Frequency response of the channels of matrix H  ω g , l  ; b) the Fig. 4. Modeled stable signal separation [7]. relationship between the condition number c o n d H  ω g , l  of the mixing matrix and the frequency. At the learning stage, the parameters of singularity and stability boundaries for a 75 km track section were calculated and stored in the database according to the track coordinate changed in 1 km increments. Thus, a function was determined that set a stability boundary for signal separation. Next, at the verification stage, condition (5) for stable separation was verified for all values of track coordinates for the channels’ randomly perturbed frequency responses. The modeling showed that the time taken to monitor the stability of signal separation for each of the coordinates (object states) did not exceed 6 s. This time makes monitoring possible when the receivers are moving at a speed up to 100 km/h, as opposed to static monitoring with algorithm [7]. This enhances the algorithm’s functionality and therefore reduces monitoring times. The receivers’ speed was modeled on the speed of data transfer to a program that used the stability verification algorithm. The reliability of the verification results obtained from the proposed algorithm was confirmed by comparing them with those of the known algorithm [7], shown in Fig. 4 and Fig. 5. Modeled unstable signal separation [7]. Fig. 5. tability was verified for two track coordinates for which the conditions of stable and unstable signal separation The modeling also showed that the verification results were simulated. for the coordinates of stable and unstable signal separation in [7] (at rest) and the verification results for the same If condition (5) is fulfilled as shown in Fig. 4, then the conditions obtained in the proposed algorithm (in solution to the problem of signal separation is stable, and postlearning motion) are virtually identical. triangular test signals are separated from the additive mixture. Otherwise (Fig. 5), the solution is unstable, and no This provides a proof of continuity of the algorithm’s signal separation takes place. proposed generalized version with its earlier published version [7]. VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 258 Data Science The proposed algorithm is effective for verifying V. PRIMARY CONCLUSIONS whether solutions to the signal separation problem are stable We developed an algorithm to verify the stability of when object characteristics change anomalously. solutions to the problem of signal separation through The computational complexity involved and the time calculating singular intervals. The algorithm is characterized spent on learning are substantial and require special by extended functionality that allows the stability of signal individual operating modes. As a result, the diagnostic separation to be verified for objects whose characteristics model is only updated when major changes are made to the are described by deterministic functions. facility. With learning incorporated in the proposed algorithm, it Therefore, the learning process (building a diagnostic takes far less time to verify stability, making the algorithm model) should run when the facility is operating. One of the suitable for use in real-time systems. methods used to follow this approach is the adaptive Our computer modeling results confirmed the efficiency parametric identification method [8]. Fig. 6. shows a block of the solutions proposed. diagram for it. s(n) REFERENCES x(n) Monitored object [1] T.N. Bushtruk and V.A. Zasov, “Prospects for Modeling and Identifying Dynamic Systems: A Monograph,” Samara State Transport University Press, 2019. Learning object model y(n) [2] V.F. Kravchenko, “Digital Signal and Image Processing in Radiophysical Applications,” Moscow: Fizmatlit, 2007. [3] A. Cichocki and Sh. Amari, “Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications,” John Wiley & Sons Ltd, 2002. d(n) Model parameters (n) [4] J.W. Demmel, “Applied Numerical Linear Algebra,” SIAM, Optimization algorithm Philadelphia, PA, 1997. [5] Ye.Ye. Tyrtyshnikov, “Matrix Analysis and Linear Algebra,” Fig. 6. Block diagram of adaptive parametric identification. 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