Linear Filtering and Spectral Analysis

Abstract

The study of vibrations of numerous real physical systems leads to nonlinear problems. Linearization often provides a satisfactory approximate solution. Even when it does not, one frequently begins the study of the given system by investigating the linearized problem so as to compare results between the linearized solution and the nonlinear solution, when the latter can be constructed. On the other hand, the stochastic problems of nonlinear vibrations are often very difficult to solve, and many of them still await a constructive solution. Hence the interest attached to the study of the wide class of linear vibration problems encountered in practice. The results concerning linear stochastic oscillators are of extreme practical importance. Even in the infinite dimensional case, constructive methods for finding solutions involve finite dimensional problems. The first three sections of this chapter form the first part of a study of deterministic aspects of linear transformation of signals that are considered here and in Chapter XI. In the first section we specify the mathematical framework of the linear transformations considered in this book and define certain notions, such as the impulse response, frequency response, physically realizable operators, transfer function, etc. The second section is devoted to the study of an important linear transformation arising in the physics of vibrations: convolution filtering. A particular case of this filtering, associated with the linear oscillator, is investigated in detail in Section 3.