This book deals with the autoregressive method for digital processing of random oscillations. The method is based on a one-to-one transformation of the numeric factors of the Yule series model to linear elastic system characteristics. This parametric approach allowed to develop a formal processing procedure from the experimental data to obtain estimates of logarithmic decrement and natural frequency of random oscillations. A straightforward mathematical description of the procedure makes it possible to optimize a discretization of oscillation realizations providing efficient estimates. The derived analytical expressions for confidence intervals of estimates enable a priori evaluation of their accuracy. Experimental validation of the method is also provided. Statistical applications for the analysis of mechanical systems arise from the fact that the loads experienced by machineries and various structures often cannot be described by deterministic vibration theory. Therefore, a sufficient description of real oscillatory processes (vibrations) calls for the use of random functions. In engineering practice, the linear vibration theory (modeling phenomena by common linear differential equations) is generally used. This theory’s fundamental concepts such as natural frequency, oscillation decrement, resonance, etc. are credited for its wide use in different technical tasks. In technical applications two types of research tasks exist: direct and inverse. The former allows to determine stochastic characteristics of the system output X(t) resulting from a random process E(t) when the object model is considered known. The direct task enables to evaluate the effect of an operational environment on the designed object and to predict its operation under various loads. The inverse task is aimed at evaluating the object model on known processes E(t) and X(t), i.e. finding model (equations) factors. This task is usually met at the tests of prototypes to identify (or verify) its model experimentally. To characterize random processes a notion of "shaping dynamic system" is commonly used. This concept allows to consider the observing process as the output of a hypothetical system with the input being stationary Gauss-distributed ("white") noise. Therefore, the process may be exhaustively described in terms of parameters of that system. In the case of random oscillations, the "shaping system" is an elastic system described by the common differential equation of the second order: X ̈(t)+2hX ̇(t)+ ω_0^2 X(t)=E(t), where ω0 = 2π/Т0 is the natural frequency, T0 is the oscillation period, and h is a damping factor. As a result, the process X(t) can be characterized in terms of the system parameters – natural frequency and logarithmic oscillations decrement δ = hT0 as well as the process variance. Evaluation of these parameters is subjected to experimental data processing based on frequency or time-domain representations of oscillations. It must be noted that a concept of these parameters evaluation did not change much during the last century. For instance, in case of the spectral density utilization, evaluation of the decrement values is linked with bandwidth measurements at the points of half-power of the observed oscillations. For a time-domain presentation, evaluation of the decrement requires measuring covariance values delayed by a time interval divisible by T0. Both estimation procedures are derived from a continuous description of research phenomena, so the accuracy of estimates is linked directly to the adequacy of discrete representation of random oscillations. This approach is similar a concept of transforming differential equations to difference ones with derivative approximation by corresponding finite differences. The resulting discrete model, being an approximation, features a methodical error which can be decreased but never eliminated. To render such a presentation more accurate it is imperative to decrease the discretization interval and to increase realization size growing requirements for computing power. The spectral density and covariance function estimates comprise a non-parametric (non-formal) approach. In principle, any non-formal approach is a kind of art i.e. the results depend on the performer’s skills. Due to interference of subjective factors in spectral or covariance estimates of random signals, accuracy of results cannot be properly determined or justified. To avoid the abovementioned difficulties, the application of linear time-series models with well-developed procedures for parameter estimates is more advantageous. A method for the analysis of random oscillations using a parametric model corresponding discretely (no approximation error) with a linear elastic system is developed and presented in this book. As a result, a one-to-one transformation of the model’s numerical factors to logarithmic decrement and natural frequency of random oscillations is established. It allowed to develop a formal processing procedure from experimental data to obtain the estimates of δ and ω0. The proposed approach allows researchers to replace traditional subjective techniques by a formal processing procedure providing efficient estimates with analytically defined statistical uncertainties.

This book exposes the theory under the phenomenon of economic cycles and provides mathematical tools for their quantitative description. The approach allows to provide the stochastic description of the investment function, to recover the income function from GDP estimates, and to propose the identification procedure for pseudo-stationary models of economic cycles. .

This book offers an in-depth discussion related to metrological aspects of automated tests. The accuracy of experimental estimates of test object performance is examined from the standpoint of their statistical variance and systematic biases. The proposed metrological model of automated tests allows to determine the metrological characteristics of measurement means using data from their static and dynamic calibrations. Knowledge of these characteristics provides an ability to examine their impact on the accuracy of test results for the purposes of estimating statistical uncertainties caused by instrumentation errors and eliminating biases that occur as a consequence of inertial properties of measurement means. Optimization of requirements for measurement errors to ensure a given accuracy of test results is discussed as well. Proposed approaches and described methods are illustrated by test examples of turbomachinery products.

This volume brings together a comprehensive selection of over fifty reprints on the theory and applications of chaotic oscillators. Included are fundamental mathematical papers describing methods for the investigation of chaotic behavior in oscillatory systems as well as the most important applications in physics and engineering. There is currently no book similar to this collection.

Additive and multiplicative noise in the information signal can significantly limit the potential of complex signal processing systems, especially when those systems use signals with complex phase structure. During the last few years this problem has been the focus of much research, and its solution could lead to profound improvements in applications of complex signals and coherent signal processing. Signal Processing Noise sets forth a generalized approach to signal processing in multiplicative and additive noise that represents a remarkable advance in signal processing and detection theory. This approach extends the boundaries of the noise immunity set by classical and modern signal processing theories, and systems constructed on this basis achieve better detection performance than that of systems currently in use. Featuring the results of the author's own research, the book is filled with examples and applications, and each chapter contains an analysis of recent observations obtained by computer modelling and experiments. Tables and illustrations clearly show the superiority of the generalized approach over both classical and modern approaches to signal processing noise. Addressing a fundamental problem in complex signal processing systems, this book offers not only theoretical development, but practical recommendations for raising noise immunity in a wide range of applications.