White noise process

Alternatively, the white-noise processes W(f) could be replaced by colored-noise processes. Since the latter have finite auto-correlation times, the resulting Lagrangian correlation functions for U and would be nonexponential. However, it would generally not be possible to describe the Lagrangian PDF by a Fokker-Planck equation. Thus, in order to simplify the comparison with Eulerian PDF methods, we will use white-noise processes throughout this section. 

The dWi are Gaussian white noise processes, and their strength a is related to the kinetic friction y through the fluctuation-dissipation relation.72 When deriving integrators for these methods, one has to be careful to take into account the special character of the random forces employed in these simulations.73 A variant of the velocity Verlet method, including a stochastic dynamics treatment of constraints, can be found in Ref. 74. The stochastic... 

It has the form (IV.4.8), as it should. The coefficients are the rm that characterized the A. Vice versa, it is now possible to construct a white noise process by taking any process with independent increments and differentiating it its rm are given by (6.4). 

A non-linear time series model transforms an observed signal x[t into a white noise process e[t, and may be written in discrete form [Priestley, 1988] as ... 

Thus, since the fractional-difference dynamics are linear, the system response is Gaussian, the same as the statistics for the white noise process on the right-hand side of Eq. (22). However, whereas the spectrum of fluctuations is flat, since it is white noise, the spectrum of the system response is inverse power law. From these analytic results we conclude that Xj is analogous to fractional Brownian motion. The analogy is complete if we set a = // 1/2 so that the... 

The term refers to a white noise process with zero mean and (constant) variance... 

Taking the first differences results in a stationary time series since the white noise process remains... 

Here, the stochastic process x represents the response of a single-degree-of-freedom (SDOF) system or the response of a particular degree of freedom of a multi-degree-of-lfeedom (MDOF) system. The prediction error is modeled as a zero-mean discrete (band-limited) white noise process e with variance and spectral intensity ... 

Discrete data with a time step Af are collected in the experiment. Let y be the measured surface fluctuation at time t = nAt. There is a difference between the measured response y and the actual response x nAt) due to measurement noise and this is modeled here by a discrete white noise process e with zero mean and variance... 

A further extension of these ideas, in which multiple states that evolve in time are possible, is obtained when one models the speech signal by a hidden Markov process (HMP) [8]. An HMP is a bivariate random process of states and observations sequences. The state process S t = 1,2,... is a finite-state homogeneous Markov chain that is not directly observed. The observation process yf,t = 1,2,...) is conditionally independent given the state process. Thus, each observation depends statistically only on the state of the Markov chain at the same time and not on any other states or observations. Consider, for example, an HMP observed in an additive white noise process W),t = 1,2,...). For each t, let Zt = Yt + Wt denote the noisy signal. Let Z = Zi,..., Z,. Let / denote the number of states of the Markov chain. The causal MMSE estimator of Y, given Z is given by [6]... 

The system mechanical parameters involved are coj = / kT/tm), cos = y/(ks/fns), It = (ct= cs/2y/msks), /u= and So is the Gaussian zero mean white noise process whose intensities, cof and are the base filter frequency and damping. 

A stochastic process is also characterized by its spectral density, the Fourier transform of its autocorrelation function. The autocorrelation function of a (stationary stochastic process) measures the correlation of the process at different time intervals while the spectral density measures the amplitudes of the component waves of different frequencies. A white noise process has a constant spectral density (i.e., the same amplitude for all frequencies) and the band-limited noise has a frequency band over which the spectral density is nearly constant. 

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