Stochastic processes power spectrum

For our purpose the important moments are ( x p), sometimes referred to as the average strengths of the Fourier components The power spectrum I (co) of the stochastic process is defined as the T -> oo limit of the average intensity at frequency m ... 

If X is a complex fimction, this theorem holds for C(t) = x (O xlt). The proof of this relationship is given in Appendix 7D. The power spectrum of a given stochastic process is thus identified as the Fourier transform of the corresponding time correlation fimction. 

The power spectrum was defined here as a property of a given stochastic process. In the physics literature it is customary to consider a closely related fimction that focuses on the properties of the thennal environment that couples to the system of interest and affects the stochastic nature of its evolution. This is the spectral density that was discussed in Section 6.5.2. (see also Section 8.2.6). To see the connection between these functions we recall that in applications of the theory of stochastic processes to physical phenomena, the stochastic process x(Z) represents a physical observable A, say a coordinate or a momentum of some observed particle. Suppose that this observable can be expanded in harmonic nonnal modes uj as in Eq. (6,79)... 

A Fourier transform of the autocorrelation function of a stochastic process gives the power spectrum function which shows the strength or energy of the process as a function of frequency [17]. Frequency analysis of a stochastic process is based on the assumption that it contains features changing at different frequencies, and thus it can be described using sine and cosine functions having the same frequencies [16]. The power spectrum is defined in terms of the covariance function of the process, Vk = Cov(e,. et k). as... 

The stochastic characteristics of the wind process are described with the mean wind velocity Up (z) and the power spectrum of the wind. The following equations (see (61 for further details) describe these quatities as a function of Up, pgf,z and Zg where z is the height above ground. 

As noted before, the Brownian force n t) may be modeled as a white noise stochastic process. White noise is a zero mean Gaussian random process with a constant power spectrum given in (72). Thus,... 

Time series, power spectra, and phase portraits are shown in Fig. 1 for the BZ reaction for three different flow rates [21]. The power spectra for the periodic states in Figs. 1(a) and (c) contain an instrumentally sharp fundamental frequency and its harmonics, while the spectrum in (b) consists of broadband noise that is well above the instrumental noise level. This spectral noise could arise from either stochastic or deterministic processes. However, at least in principle it should be possible to distinguish stochastic and deterministic processes from the behavior of the power spectrum in the high frequency limit [34] for stochastic differential equations of order n, P(a)) 0) ", while for nonperiodic behavior given by deterministic differential equations, P( o)) exp(-ra)). To our knowledge this... 

In the previous sections the most common procedures to determine stationary/quasi-stationary power-spectral density functions compatible with a given response spectrum have been presented. It should be noted at the outset that the ground motion time histories generated from quasi-stationary stochastic processes have energy content only variable with time. Although this approach is convenient and accurate for the seismic analysis of traditional linear behaving struc-mres, it does not lead to a comprehensive description of the seismic phenomenon. 

In postulating the stationarity of the stochastic process, very strong assumptions regarding the structure of the process are made. Once these assumptions are dropped, the process can become nonstationary in many different ways. In the framework of the spectral analysis of nonstationary processes, Priestley (see, e.g., Priestley 1999) introduced the evolutionary power spectral density (EPDS) function. The EPSD function has essentially the same type of physical interpretation of the PSD function of stationary processes. The main difference is that whereas the PSD function describes the power-frequency distribution for the whole stationary process, the EPSD function is time dependent and describes the local power-frequency distribution at each instant time. The theory of EPSD function is the only one which preserves this physical interpretation for the nonstationary processes. Moreover, since the spectrum may be estimated by fairly simple numerical techniques, which do not require any specific assumption of the structure of the process, this model, based on the EPSD function, is nowadays the most adopted model for the analysis of structures subjected to nonstationary processes as the seismic motion due to earthquakes. 

Clearly, the proper definition of the power-spectral density functirai is the crucial step to address the stochastic modeling of the seismic action. In the context of defining spectrum-compatible ground motion processes, the issue consists in determining which is the inverse relationship between the power-spectral density function and the target response spectrum. It has to be emphasized that the evaluation of the response-spectrum-compatible power-spectral... 

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