Autocorrelation function spectral density

Hence, is the spectral density of the autocorrelation function of density... 

Goldfisher, Autocorrelation function and power spectral density of laser-produced speckle pattern . J. Opt. Soc. Am., vol.55, p.247(1965). 

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

Autocorrelation function Power spectrum (Spectral power density)... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

We may recall and emphasize that the autocorrelation function obtained in the three representations I, II, and III must be equivalent, from the general properties of canonical transformation which must leave invariant the physical results. Thus, because of this equivalence, the spectral density obtained by Fourier transform of (43) and (45) will lead to the same Franck-Condon progression (51). 

On the other hand, the undamped autocorrelation function (17) we have obtained within the standard approach avoiding the adiabatic approximation must lead after Fourier transform to spectral densities involving very puzzling Dirac delta peaks given by... 

Note that this last expression is nothing but the closed form [90] of the autocorrelation function obtained (as an infinite sum) in quantum representation III by Boulil et al.[87] in their initial quantum approach of indirect damping. Although the small approximation involved in the quantum representation III and avoided in the quantum representation II, both autocorrelation functions are of the same form and lead to the same spectral densities (as discussed later). 

The spectral density is the Fourier transform of this autocorrelation function, that... 

It is assumed that the noise voltage n(t) is the result of a real stationary process (Davenport and Root, 1958) with zero mean. Because it can be shown that the spectral density function S(f) is the Fourier transform of the autocorrelation function of the noise, it follows that the rms noise is given by... 

As an example, we consider these error bounds for the cumulative distribution of the spectral density of the velocity autocorrelation function,... 

Fig. 2. Error bounds for the cumulative frequency distribution of the spectral density for the velocity autocorrelation function using jU.0, /n2, and ja evaluated for a classical model of liquid argon.29...

Fig. 4. Spectral density for the velocity autocorrelation function for vibrations of abody-centered cubic lattice, extrapolated from 7 even moments, by the method of Section IV.

Knowledge of the spectral density J co) is equivalent to knowledge of the dipole autocorrelation function C(t), and vice versa. 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

More precisely, die quantity displayed is the signal power estimated from 10ms frames. As die power spectral densities of die two types of noise exhibit a strong peak at the null frequency, the two noises were pre-whitened by use of an all-pole filter [Cappe, 1991]. This pre-processing guarantees that the noise autocorrelation functions decay sufficiently fast to obtain a robust power estimate even with short frame durations [Kay, 1993]. 

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