Time-dependent power spectral density

The time-dependent power spectral density is given by the Fourier transform of the correlation function [161] ... 

Time-Dependent Power Spectral Densities of Randomly Vibrating Kirchhoff-Plates... 

Noise is characterized by the time dependence of noise amplitude A. The measured value of A (the instantaneous value of potential or current) depends to some extent on the time resolution of the measuring device (its frequency bandwidth A/). Since noise always is a signal of alternating sign, its intensity is characterized in terms of the mean square of amplitude, A, over the frequency range A/, and is called (somewhat unfortunately) noise power. The Fourier transform of the experimental time dependence of noise intensity leads to the frequency dependence of noise intensity. In the literature these curves became known as PSD (power spectral density) plots. 

The frequency noise power spectral density of a SL typically exhibits a 1/f dependence below 100 kHz and is flat from 1 MHz to well above 100 MHz. Relaxation oscillations will induce a pronounced peak in the spectrum above 1 GHz. The "white" spectral component represents the phase fluctuations that are responsible for the Lorentzian linewidth and its intensity is equal to IT times the Lorentzian FWHM.20 xhe 1/f component represents a random walk of the center frequency of the field. This phase noise is responsible for a slight Gaussian rounding at the peak of the laser field spectrum and results in a power independent component in the linewidth. Figure 3 shows typical frequency noise spectra for a TJS laser at two power levels. 

A gravity dam-reservoir system is selected for studying the interaction of structure-fluid systems under nonstationary random excitation. For an idealized random excitation with a zero start and a white power spectrxjun, the nonstationary power spectral density for the structural displacement is obtained by a frequency domain approach due to Shinozuka. The spectral density functions are then integrated to check the transient variance solution obtained previously by a time domain approach. Using this power spectral solution the random interaction effect is examined for the entire frequency range in detail. This interaction problem of the structure-fluid system is important because it simulates random and time dependent structural response to earthquake ground accelerations. 

First, a distinction can be made between non-parametric and parametric identification. Non-parametric system identification involves the estimation of an impulse response function, frequency response function (FRF), correlation function, or power spectral density (PSD), not as a mathematical function depending on a few parameters, but as a set of tabulated values for each considered time lag or frequency. Although nonparametric models are sometimes directly used for modal analysis, they are most often used as preprocessed data for parametric identification since the estimation accuracy of parametric approaches is much higher than that of nonparametric approaches (Peeters and De Roeck 2001 Reynders 2012). 

Finally, in the particular case in which the modulating function presented in Eq. 2 figures as a t) = 1, the process is called stationary. Note that in this case the power-spectral density function depends only on the circular frequency and not on the time. 

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. 

Jeener and Broekaert introduced, in 1%7, a three-pulse B,(r) sequence to measure the relaxation time Tm of the dipolar order of / = 1 spin systems in the presence of a conventional high Zeeman field, Bq, which is based on the decay time of the so-called Jeener echo . It was later extended by Spiess and Kemp-Harper and Wimperis to study in a similar way the quadrupolar order in / a 1 systems. The appearance of a Jeener echo depends upon the existence of interactions that are not averaged out by molecular motions on the considered time scale. The method has become of great importance in recent relaxation studies, in particular of liquid crystals because, in standard spin relaxation theories, it provides a power l means to separate and analyse the spectral densities / v) and /2) j. i4,is,2025 ggg... 

If time dependencies of the measurement error shall be considered, the error can be described by the autocorrelation function of the error, the referred autocorrelation function of the error, and/or the spectral power density function of the error. 

Note For photographic detection of line spectra, it is actually better to really use the lower limit bmm for the width of the entrance slit, because the density of the developed photographic layer depends only on the time-integrated spectral irradiance [W/m ] rather than on the radiation power [W]. Increasing the slit width beyond the diffraction limit b , in fact, does not significantly increase the density contrast on the plate, but does deaease the spectral resolution. 

The stochastic analysis of structural vibrations deals with the description and characterization of structural loads and responses that are modeled as stochastic processes. The probabilistic characterization of the input process could be extremely complex in time domain where the probability density functions depend on the autocorrelation functions which experimentally have to be specified over given set points. Since this approach is difficult to be used in applications, stochastic vibration analysis of structural linear systems subjected to Gaussian input processes is quite often performed in the frequency domain by means of the spectral analysis. This analysis is a very powerful tool for the analytical and experimental treatment of a large class of physical as well as structural problems subjected to random excitations. The main reasons are... 

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