Spectral density discrete

In practice one can hardly ever evaluate all of the moments of a spectral density, but rather one evaluates only the first few, until the calculations become too difficult or lengthy. If only the first 2M moments are evaluated, then /(a) is never known uniquely, except in the special case that 1(a)) consists of discrete contributions at M or fewer frequencies. Nevertheless, one expects that the knowledge of even a few moments of a spectral density furnishes some useful constraints on the possible forms of /(a). The problem is to translate these moment constraints, and the other general properties of /(a), into useful forms. Several schemes for making use of this information are outlined in the next section. 

The interpretation of relaxation data is most often performed either with reduced spectral density or the Lipari-Szabo approach. The first is easy to implement as the values of spectral density at discrete frequencies are derived from a linear combinations of relaxation rates, but it does not provide any insight into a physical model of the motion. The second approach provides parameters that are related to the model of the internal motion, but the data analysis requires non-linear optimisation and a selection of a suitable model. A graphical way to relate the two approaches is described by Andrec et al Comparison of calculated parametric curves correlating 7h and Jn values for different Lipari-Szabo models of the internal motion with the experimental values provides a range of parameter values compatible with the data and allows to select a suitable model. The method is particularly useful at the initial stage of the data analysis. 

The initial state is discrete, but the final state is in the continuum and transitions are described by a spectral density, slowly varying function of the energy or frequency. 

The first source of nonequilibrium noise, described as early as 1918 (23) (in fact 10 years earlier than Johnson noise), was shot noise that stems from the discrete nature of charge transfer. The current spectral density, Sj(/), of this noise is white (independent of frequency /) up to frequencies of the order of the inverse time of elementary charge transfer and is given by... 

Here, we briefly describe MFDA. Considering the usual situations where the trajectory is sampled with discrete time steps, we focus on discrete time and continuous frequency spaces. For simplicity, we set the sampling time interval of the trajectory data. At, to unity. We define the spectral density matrix S(f) as the inverse Fourier transform of a lagged variance-covariance matrix C(r) = (v(f)v (f+x)). 

The discrete stochastic process y, being the measurement of the system governed by Equations (3.1) or (3.6), and Equation (3.19), is Gaussian with zero mean and so do the random variables dcok) and Furthermore, for a given value of T, the random variables c(k) are independent with equal variance asymptotically as At 0+ [277]. Therefore, it follows that the spectral density estimator Sy N a>k) has the following asymptotic behavior ... 

Consider a discrete stochastic vector process y and a finite number of discrete data points = y , n = 1,2,..., A. Based on D a discrete estimator of the spectral density matrix of the stochastic process y is introduced ... 

The Bayesian spectral density approach, which is a frequency-domain method, transforms the response time history to the spectral density estimator by discrete Fourier transform. 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

The Bayesian fast Fourier transform approach uses the statistical properties of discrete Fourier transforms, instead of the spectral density estimators, to construct the likelihood function and the updated PDF of the model parameters [292]. It does not rely on the approximation of the Wishart distributed spectrum. Expressions of the covariance matrix of the real and imaginary parts of the discrete Fourier transform were given. The only approximation was made on the independency of the discrete Fourier transforms at different frequencies. Therefore, the Bayesian fast Fourier transform approach is more accurate than the spectral density approach in the sense that one of the two approximations in the latter is released. However, since the fast Fourier transform approach considers the real and imaginary parts of the discrete Fourier transform, the corresponding covariance matrices are 2No x 2Nq, instead of No x No in the spectral density approach. Therefore, the spectral density approach is computationally more efficient than the fast Fourier transform approach. 

The electronic density of states (DOS) describes energy of electrons in solids. By definition, the density of states p(B) is a value, which being multiplied by a small interval of energies dE equals to a number of the electronic states with energies in an interval from Eto E dE. In solids, the density distributions are not discrete like a spectral density but continuous. A high density of states at a specific energy level means that there are many states available for occupation. A density of states of zero means that no states can be occupied at that energy level. Local variations, most often due to distortions of the system, are often called local density of states (LDOS). 

Spectral density 8 discrete frequency data point per decade... 

Abstract Photoinduced processes in extended molecular systems are often ultrafast and involve strong electron-vibration (vibronic) coupling effects which necessitate a non-perturbative treatment. In the approach presented here, high-dimensional vibrational subspaces are expressed in terms of effective modes, and hierarchical chains of such modes which sequentially resolve the dynamics as a function of time. This permits introducing systematic reduction procedures, both for discretized vibrational distributions and for continuous distributions characterized by spectral densities. In the latter case, a sequence of spectral densities is obtained from a Mori/Rubin-type continued fraction representation. The approach is suitable to describe nonadiabatic processes at conical intersections, excitation energy transfer in molecular aggregates, and related transport phenomena that can be described by generalized spin-boson models. 

The effective-mode construction can thus be employed both for a discrete set of vibrational modes (e.g., in a polyatomic molecule) and for typical system-bath type situations where the spectrum of bath modes is dense. In the latter case, the environment and its coupling to the electronic subsystem are entirely characterized by a spectral density. Approximate spectral densities can be constructed from few effective modes, representing a simplified realization of the true environmental spectral density that is designed to give a faithful representation of the dynamics on short time scales [32,33]. Thus, even a highly structured, multi-peaked spectral density can be reduced to an effective, simplified spectral density on ultrafast time scales. Importantly, the procedure converges, as has recently been shown in Ref. [34]. 

If the frequency distribution of the bath modes is dense, it is natural to characterize the influence of the bath on the subsystem in terms of a spectral density, or its discretized representation. In the case where the bath modes couple only to one of the subsystem operators, for instance... 

The FSMs are able to restitute both power spectral density (see Eq. 16) and the correlation function (see Eq. 27) in the whole domain with exclusion of cu = 0 or t = 0, respectively. This pathological behavior, due to the discretization in t] axis, may be overcome by assuming the value in zero as the value in CO = exp(—b). 

A time discretization step equal to At = 0.01 [s] is used to model the ground acceleration. Thus, the discrete representation of the white noise signal is (o tk) = yinSjAt = 1,. .., 1501, where S = 10 [m /s ] is the spectral density of the white noise and z, k = 1,..., 1501 are independent, identically distributed standard Gaussian variables. 

---