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Analysis of autoregressive sequences with multiple roots of characteristic equations


K.K. Vasilev – Dr. Sc. (Eng.), Professor, Head of Department «Telecommunications», Ulyanovsk State Technical University
N.A. Andriyanov – Post-graduate Student, Department «Telecommunications», Ulyanovsk State Technical University

A lot of real processes that vary in time, such as for example the output signal from various measuring sensors or the trajectory of the vehicle, have a smooth appearance. The known autoregressive signal models either do not adequately describe such processes even with strong correlation (first order), or require rather complex operations to calculate the set of correlation parameters (with increasing order). Meanwhile, autoregressive models of random processes (RP), generated by characteristic equations, are known. Such models are called autoregressions with multiple roots. At the same time, the application of algorithms based on such models, when processing real signals and in various applied problems, can improve the efficiency of solving such problems.
Among them, the important task is to suppress the noise or filter signals, the solution of which is often found in the frequency domain. However, the transformations used in this approach can significantly increase the time spent on solving this problem. The solution in this situation is the use of recurrent filtration procedures.
Thus, the problem of investigating the correlation properties of autoregressions with multiple roots of characteristic equations, as well as the problem of optimal filtration generated by such models of the RP, are of particular interest.
The article gives a rather complete description of autoregressive models with multiple roots of characteristic equations. An explicit formulas also were obtained for the special cases of small multiplicities. In addition, the results of statistical properties of autoregressions with multiple roots research are presented, the dependencies of the correlation coefficients for models of different orders on the correlation interval equal for all models are found. It is shown that as the correlation interval increases, the multiple root value tends to unity so that the product of the correlation interval and the difference of unity and the correlation coefficient tends to a constant. This constant increases with the multiplicity of the model.
The optimum filter for RP generated by autoregressive models of various multiplicities is also considered. On the example of models of the first and second orders, a comparative analysis of the filtration efficiency is performed. The dependences of the established filtration error dispersion on the correlation interval of the model for different signal-to-noise ratios are found.
Thus, in the article autoregressive models with multiple roots of characteristic equations are considered. It is shown that an increase in the multiplicity of such models leads to the formation of smoother and closer to real physical processes even with the same correlation intervals. Dispersion of filtering errors for models of the first and second orders is found. It is shown that the gain of the second order model with a correlation interval of 20 samples is from 10% (for small correlation intervals) to 70% (for large correlation intervals).

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