Nonstationary White Noise Research Articles

Some properties and examples of random processes that are almost wide sense stationary

A wide sense stationary (WSS) process X(t), t \in I , has shift operators T_h: X(t) \rightarrow X(t + h), h \in I , which are unitary operators in the Hilbert space H(X) generated in the usual way by X(t) . We study the class of uniformly hounded linearly stationary (UBLS) processes; This is the class of processes having shift operators T_h that are linear and bounded with \parallel T_h \parallel ^ 2 \leq M , for some constant M . Examples are given of UBLS processes resulting from linear transformations on non-stationary white noise. The notion of an UBLS almost white noise process is defined, and some special cases are studied. Also, possible applications to time series modeling are indicated. The canonical structure of a finite-dimensional deterministic UBLS process is obtained. Theorems for superposition and multiplication of UBLS processes are presented. Finally, continuous-time white noise is given a rigorous treatment in terms of generalized processes, and conditions for UBLS are given.

Random Processes for Earthquake Simulation

Criteria presented for the evaluation of earthquake simulation processes relate to the autocovariance function, maximum acceleration response spectra, and nonstationarity of the process. Two previously used and two new processes are examined with respect to these criteria by means of theoretical and simulated results. The previous processes develop a nonstationary white noise for input to a linear filter and take the filter output as the simulated ground acceleration. They are identical except for filter properties. Neither of these processes complies with all the criteria. One fails to provide reasonable response spectra; the other produces an unrealistic ground-velocity variance function that does not disappear with time. The new processes incorporate weighting functions that produce correlated, nonwhite filter inputs. One of these produces response spectra with undesirable irregularities. The other, which employs an exponential decay-type weighting function, complies with all criteria and is recommended as a suitable model for earthquake simulation. Additional relationships are examined for rms and maximum oscillator response spectra.

The response of linear vibratory systems to random impulses

A theory for predicting the response of a linear vibratory system to impulses which occur at random times, and are of random strengths, is considered. It is shown that in a limiting case, the excitation may be regarded as a continuous, normally distributed process, called “nonstationary white noise”, and the practical significance of this process is discussed. The theory is used to find the mean square response of a single degree of freedom system to two simple types of random impulse excitation, and the concept of “mean square resonance” is introduced.