Time‐variant filtering for nonstationary random processes

Abstract

This paper is concerned with the effects of time‐variant (including time‐invariant) linear systems (filters) upon the nonstationarity of an acquired random process. The basis approach is the frequency domain description and, therefore, the relationship between the modulation function (MF) and the power spectral density function is emphasized though the both are time‐dependent for nonstationary processes. In particular, this paper proposes a general series form of the input–output relationship expressed in terms of the relevant modulation functions of the processes. Based upon this form, the problems of filtering a nonstationary input can be conveniently handled. The system performance is also described by a time‐variant differential equation and the technique proposed in this paper is the so‐called Modulation Function Equation (MFE) method which is an alternative time‐variant differential equation, derived from an ordinary one, yet with the randomness excluded. The principle of deriving a class of MFE is presented and the technique is applied to some practical problems, e.g., trolley wires, in order to describe their responses to a random nonstationary excitation (noise and/or vibration). The solutions of the formed MFE are, however, essentially those of the equivalence of the frequency implementation which is now defined as being the frequency/time two‐dimensional function for the time‐variant systems (filters). By invoking the MFE technique the implementation can be conducted numerically or analytically. This paper discusses such possibilities and, in particular, develops two approximations: namely the System Iterative Method (SIM) and the Power Series Method (PSM) for a class of time‐variant MFEs; they both may be usefully applied to certain nonstationary problems.